text:
Five Trends Shaping the Cloud Security Conversation
By Steve Wilson
This blog is based on a new Constellation Research report Preparing for the New Age of Cloud Security, Dec 7, 2021, by Liz Miller and me,
1. Zero Trust
One of the most popular slogans today is also one of the most confusing: Zero Trust. Of course we want to trust business partners and service providers to provide predictable outcomes. But "zero trust" is a technicality, a policy perspective that dispels the traditional class system which grants selected individuals or systems more privileged access than others. That's the road to security hell, for it is increasingly possible to fake or falsely assume a trusted position, and thence wreak limitless damage.
A zero-trust access control policy means that all agents within an system are dealt with equally, regardless of identity or history. Essentially it equates to the "need to know" principle: all actors must have a demonstrable need for a given level of access.
Sadly "zero trust" has become a catch phrase, much abused in security marketing. All it really means is do not trust people on their word. Equally, we should not accept "zero trust" as a technical description on its word alone. It must go with tangible security controls.
2. Exponentially Increasing Client-Device Capabilities
Moore's Law leads to periodic paradigm shifts in computing focus, from the server to the client and back again. In the 1980s and 90s, mainframe computing gave way to minicomputers, workstations, and microcomputers. Then the sheer volume of data processing required computing to swing back to huge back-end systems and shared resources, which became known as the cloud aka "someone else's computers".
The pendulum has swung yet again, driven by cryptography, with mounting preference (nay, mandates) for secure microprocessing units (MCUs) on the client side. Local encryption key generation and storage, and integrated transaction signing are now standard in mobile devices, and IoT device capability is heading the same way. Managing fleets of connected automobiles, smart electricity meters, and medical devices -- to cite some popular examples -- takes a dynamic mix of processing and storage at the edge and in the cloud.
High quality client-side cryptography is the key (pardon the pun) to genuine strong authentication; that is, using personally controlled tangible devices, individually accountable and resistant to phishing.
The FIDO Alliance has brought about the biggest improvements ever seen in end user authentication, by consumerising cryptography. Most users are blithely unaware that when they unlock a virtual credit card on their mobile phone via facial recognition, they are invoking hardware security and public key cryptography which until recently was confined to HSMs costing $25,000 each.
3. Quantum-Safe Cryptography
Quantum computing promises radical new architectures surpassing the classical limits of arithmetic and sequential data processing. In cybersecurity, there is a dark side top this power: many encryption algorithms, especially those used for digital signatures and authentication, are based on "one-way functions" that are easy to compute in one direction but practically impossible to reverse. There has always been an arms race between cryptologists and attackers who uncover secret keys through brute force attacks using powerful conventional computers (like graphics cards and custom gate arrays). The main weapon against these brute force attack is for vendors and sysadmins to regularly increase the lengths of the keys used to protect data.
Quantum computers will one day punch through the computational barriers that underpin most of today's digital signature algorithms. For now, the threat tends to be exaggerated; some of the world's best cryptologists advise the arms race will continue, with longer keys still offering protection for many years to come. Meanwhile the National Institute of Technology and Standards (NIST) is conducting a methodical search -- and sponsoring competitions -- the next generation of quantum-safe cryptographic techniques, and practical quantum engineering progresses in leaps and bounds.
This is an especially difficult area to predict with confidence. The maintenance of enterprise cryptographic services must be left to experts. The better cloud providers, offering managed services, original research and insights, will be watching and preparing for the next generation of encryption.
4. Cloud Hardware Security Modules
For decades, hardware security modules (HSMs) have been specialised core components of critical security operations such as payments gateways, ATMs, and high-volume website encryption. Cloud HSMs are critical new building blocks for the future of the virtualized cryptographic processing, offering:
compact, certifiable, tamper-resistant cryptographic primitives running in firmware
secure execution environments for running partitioned custom code
secure elements (dedicated microchips) holding private keys which rarely if ever are released outside the safe hardware
physically robust enclosures with tamper detection to scrub code and keys in the event of a mechanical or electronic intrusion
certification to high levels of assurance such as Common Criteria Evaluation Assurance Level 6 (EAL6) or FIPS 140-2 Level 4, and
options to rent either shared or dedicated modules.
5. Data Protection as a Service
Many security challenges have more to do with regulatory trends and competitive pressure than fraud or overt criminality. Increasingly stringent data protection rules are driving demand for encryption and confidentiality.
Managed cryptography is needed in response to the long-standing challenge of keeping encryption systems up to date. I see encryption, tokenization and anonymization (with qualifications) being delivered to customers as services via APIs, with the algorithmic complexity abstracted away. Ideally, higher order data protection services will come to be expressed in an algorithm-neutral manner.
True anonymisation is a promise that is difficult to keep, but a range of technologies are available for providing qualified degrees of data-hiding -- including homomorphic encryption, fully homomorphic encryption (FHE), and Differential Privacy. These are highly technical methods, needing careful fine tuning and awareness of the compromises they entail.
Traditionally encrypted data loses all structure and cannot be processed for routine tasks such as sorting, reporting and statistical analysis. Homomorphic encryption is a class of algorithms that preserve some structure and enable some processing. Fully homomorphic encryption promises to enable all regular processing to be performed on encrypted data. This is new technology, not yet fully proven or accepted by the academic and regulatory communities. With FHE and similar techniques still in flux, I see these being offered as a cloud service in innovative new ways.
"Infostructure"
In conclusion, with data itself becoming a critical new asset class, we need a broader, more integrated definition of "data protection" to transcend today's siloed approaches to privacy and cybersecurity. The technical complexity, regulatory risk, and intensity of ongoing R&D will all drive the virtualization of cloud-based data protection, and a new breed of infostructure services.
Digital Safety & Privacy
Chief Data Officer, Chief Digital Officer, Chief Executive Officer, Chief Information Officer, Chief Information Security Officer, Chief Privacy Officer
Data Protection, GDPR, anonymization, encryption, quantum safe, CPRA, cloud, FIDO, HSM, TPM


Download our app to get live step-by-step directions, real-time schedule estimation, and which nearby transit line will get you to Port Of Brisbane in the shortest time. Download Moovit to see why the Google Play Store and the Apple App Store voted us as one of the best transit apps available.... 10/02/2018�� Planning to drive to Brisbane from Adelaide and return, in April. Which, in people's opinion, is the better set of roads, via Hay(Sturt Hwy) or via Broken Hill(Barrier Hwy).
All times are in Australian Eastern Standard Time (Queensland) [*] These are request pick up and drop off locations only. Please call Warwick office to make a booking.
The first free settlement in Queensland after Brisbane, Warwick is a historic country town perched on the Condamine River. Just an hour's drive from Toowoomba and two hours from Brisbane, Warwick is a unique place to get to know country Queensland.
The plan is to get to Brisbane via New England. My flight arrives in Sydney from Hong Kong at 6.45am, giving me enough time to get to the station to catch the Xplorer to Armidale (I've never been on the Main North Line beyond Maitland).
Find cheap flights from Brisbane to Cyprus with Cheapflights.com.au. The quick and easy way to find the lowest prices on Brisbane to Cyprus flights. The quick and easy way to find the lowest prices on Brisbane to Cyprus flights.
Wanting to hear from anyone who has driven from Canberra to Brisbane, or vice versa, within the last 6 months. I always take the inland route, however haven�t driven the coastal route for a couple of years and am wondering if it is faster than going inland nowadays.


Recently, the government of Germany announced plans to invest $3 billion into Artificial Intelligence R&D through 2025 in order to increase the company's competitiveness on the global stage. Investments in AI will be critical to drive productivity and economic growth in coming decades, and Germany has lagged world leaders in development of AI. Over the next five years, France plans to invest 1.5 billion euro in AI, and China has plans to create a state-funded $150 billion industry by 2030.
Germany – and Europe's economies are still heavily concentrated in industrials. An Ernst and Young study found that 55 of Germany's top 100 companies by sales are in industrial sectors, while in the US 20 of the top 100 companies are in tech and media. IT and tech are also rewarded by investors with 5 of the top 6 US companies by market cap in the tech sector (as of September 30, 2018). Apple, Amazon, Alphabet, Microsoft and Facebook are all actively investing in AI while only Berkshire Hathaway is not a tech-focused company.
Germany plans to present its digital strategy in early December which embraces five areas: Digital competence, which involves educating citizens of all ages about digitalization; infrastructure, which will include high-speed Internet nationwide by 2025; innovation and digital transformation applied to business around the "Industrie 4.0." concept; digital impact on society, including ethics and jobs; and digital government, replacing paper based processes and bureaucracy.
A pair of recent studies from McKinsey Global Institute (MGI) and PwC highlight the potential value-add that artificial intelligence can create for the global economy. According to the McKinsey Global study, artificial intelligence has the potential to increase economic output by 16% or roughly $13 trillion by 2030, which equates to an annual average increase of 1.2% to productivity growth percent between now and 2030. This finding is consistent with a recent report by PwC, which found that AI technologies and applications could increase global GDP by up to 14% through 2030.
The MGI study found that AI has the potential to increase competitiveness gaps between countries. According to the study, the (mostly developed) countries that establish themselves as AI leaders could capture an additional 20-25% in economic benefits compared with today, while emerging economies may capture only half their upside.
China and the US lead the world in terms of AI-related R&D, at least among commercial companies. The U.S. is the global leader in terms of R&D, with over 15,000 patents filed between 2010 and 2014, nearly double second place China. Chinese companies Baidu, Tencent, JD.com and other firms continue to invest heavily in machine learning and AI technologies, aided by support from Chinese government (although there are concerns about the Chinese government's use of technology to control its citizens). In the U.S. private investment accounts for over 60% of AI R&D, led by companies including Amazon, Apple, Microsoft, IBM, Facebook and others. In addition to software, the US is home to more companies creating specialized hardware for AI processing, notably Nvidia. Other countries investing actively include Canada, the UK, France, South Korea, and Japan. Germany's announcement is a significant move by a leader across several industrial sectors of the need to invest in AI to advance digitalization, building on the efforts of leading firms including Siemens and SAP to expand its footprint for the future.


A Hawaiian Studies public charter school in Kalihi is on the brink of closure. Halau Lokahi Charter School opened 13 years ago. It has had financial problems in the past and now has $417,000 of debt.
Testimony in support of Halau Lokahi School was passionate trying to convince the State Public Charter School Commission not to close it.
"Shutting down Halau Lokahi is not a great idea to do. Keeping Halau Lokahi alive is the thing to do," said Hinano Paia, Halau Lokahi 9th grade student.
"As a teacher trust me I didn't get paid but I understand the bigger picture. I think we all need to understand the bigger picture," said Junior Coleman, Halau Lokahi Teacher.
"All I ask is that you guys find a way to make it work," said Kilinahe Nahoi, parent.
"I don't want them in any other school I mean what works for one is not going to work for another," said Kraig Kina, parent.
Emotions aside, finances matter. The K-12 school is $417,000 in debt. It has had trouble paying rent since February and didn't pay teachers last pay period.
Enrollment has dropped to 169 students and with it State funding.
Some on the Commission called school leaders financially irresponsible.
The school's director remained defiant.
"Until the day I die I will continue this cause. Why? Because evidently the Department of Education who has been running system here for so many years hasn't bothered," said Laara Allbrett, Halau Lokahi School Director.
"We should have just quit years ago. We should have quit when we saw the frickin government, excuse me, when the government was not going to play fair," said Allbrett, later in the meeting.
Allbrett had complained her school gets less funding from the state per student.
The school says it has a new financial plan. The Commission gave administrators until 5:00 Friday to submit it. It will meet again next Wednesday to decide the school's fate.
"We certainly want to give them every chance to succeed because we do support the children and what is going on, but any organization that doesn't have the resources simply can't continue. You have to be able to continue to pay your staff and pay your rent and that is not happening right now," said Catherine Payne, State Public Charter School Commission Chair.
Enrollment is crucial. They have 169 students now. They need at least 200 to make the financial plan work.


Let's face it. Sometimes negative events occur when we least expect them (just ask Crock-Pot). From customer complaints that go viral, to internal team issues that impact productivity and morale, to supplychain issues that frustrate customers... it's not always or only about the big crises, but the every day issues that can have an impact on your business's operations and reputation. When these events occur–and better yet, before they do–if yours is like many organizations, then your team could benefit from having a partner that will be committed to helping you spot risk and turn it into opportunity before it materializes.
Loved how you "woke up our organization" Melissa! For any organization .... Partner with Melissa Agnes and you WILL be Crisis ready!
Help in effectively responding to real-time risks as they occur.
Analysis of reputational and other risks involved with new business strategies, industry and/or regulatory changes, product launches, marketing campaigns, communications, etc., in order to mitigate any oversight of potential risk and its escalation.
Help in developing the organization's crisis ready program.
Advising on the evolution of the organization's high-risk scenarios and stakeholder expectations, e.g.: new real-time risks and challenges to be aware of and incorporate into the organization's crisis ready program.
If this type of partnership could benefit your team, giving management comfort in knowing that the team has access to an experienced crisis management professional acting in the best interest of the organization, then reach out to learn more about Melissa's advisory services and availability.
"Melissa Agnes proved to be highly effective in helping Domtar modernize its crisis preparedness program. As a result, we are better prepared to manage issues as they arise, while avoiding mistakes and minimizing risks. Melissa accomplished this with a critical review of our procedures and plans, a customized exercise that engaged our entire team, and continued assessments of areas for continued improvement."
"We brought Melissa in to advise on crisis communication with embassies with which we have close contact in Seoul. Crisis communication is of utmost importance in our environment, as tension between South and North Korea fluctuates on a weekly basis. Melissa led the discussion on various crisis communication strategies where she provided the benefits and disadvantages of each. She introduced many of the diplomates to various new communication tools and strategies that could be implemented when communicating with our citizens in South Korea. Her command of the subject, as well as her presentation skills were phenomenal, and each embassy representative commented on the high value she brought to the discussion. Her advisory was a catalyst which enabled discussion on the creation and application of common communication strategies in crisis situations, by embassies in Seoul."
No matter the size, type, or industry of your business, Crisis Ready was designed to provide your team with the insight into how to be perfectly prepared for anything life throws at you. Organizations that are crisis ready are more than just resilient. They're INVINCIBLE. Crisis Ready is the ultimate roadmap to business invincibility.


My Alternate Reality: Holy Macro!
Yes, I have quite a few obsessions in my life… and macro photography is one of them! And the funny part is, I don't even own a real macro lens!
So I have found a few work-arounds that work just fine!!
Ok, now that the gear is out of way, on to the fun part! The comparisons!
As you can see, very yummy close… but we can get better!
As you can see in this method… very close, very cool… a little vignette but it just makes it look artsy :) It does take some practice with this method though to get the focus just right, a lot of moving closer, farther from the object… but once you get it, WOW!
I also wanted to give credit to the site that got me started with this method, DPS… gotta love em'! If you love digital photography, they are a must visit!
Just as using the 70-200mm you can get scary close, however using a shorter lens allows for much more vignette. Hard to pass these off as an artsy effect… Unless you do some major cropping. I would definitely recommend using a telephoto for the reverse ring method… But this is still fun!
Well, that's about it for my macro fun for today! Unfortunately I couldn't ramble on longer due to camera issues, but maybe someday soon I'll have some more comparisons to share! Also, I'd love to hear from anyone if they have any macro tips/techniques or photos to share!!
Wow! I love the shots with the reverse ring! Very nice!
Nicely written up - glad you could use that image and thanks for the shout out!
I'm not sure how to actually attach the lens... how do you do it?
Awesome! Thanks for sharing these. I'm new to the Canon family and find this very useful.


text:
\section{\label{sec:introduction} Introduction}

In modern physics, the idea of extra dimensions is used for explanation of a variety
of phenomena. This idea is applied for elaboration of physics beyond the Standard
Model, cosmological scenarios including inflationary models and the origin of the dark
component of the Universe (dark matter and energy), for explaining its baryon asymmetry,
the number of fermion generations and so on. Gradually, it is becoming
the main element for a future theory.

With their whole diversity, the models involving extra dimensions must overcome
two common problems: the smallness of the cosmological constant they induce
\cite{Lambdasmall1, Lambdasmall2}, and the stability of an extra space metric.
Observations indicate that our Universe is nowadays expanding with acceleration
\cite{accel}, which can be explained in Einstein gravity by the existence of a
cosmological constant $\Lambda \sim 10^{-3}~{\rm eV}^4$ or something very close
to it, called dark energy, see a review in \cite{StarL}. We live in space-time
approximately described by the de Sitter metric, which is, in a sense, extremely
close to the Minkowski one. Any study should be in agreement with this
observational fact.

There are a lot of approaches to solving the problem: one introduces scalar fields
(see, e.g., \cite{Carugno}), form fields \cite{Hawk}, invariants other than
powers of $R$ \cite{Ru16}, or considers compact extra spaces with zero curvature,
e.g., Calabi-Yau manifolds \cite{Zhuk13}, one-dimensional extra spaces
\cite{Ponce} and flat tori \cite{Flatorus}. As shown in \cite{Navarro},
insertion of form fields leads to an extra space with the metric of a sphere
with a deficit angle and an observationally acceptable value of $\Lambda$. The study appeared recently is based on Horndeski scalar-tensor theories
of gravity involving Poincare invariance breaking \cite{FabFour} that comes
together with a self-tuning mechanism driven by scalar field dynamics. It was
pointed out by Babichev and Esposito-Farese that a local deviation from general
relativity would rule out a model even with good $\Lambda$ tuning. The accordance
of Hordeski-like theories (generalized Gallileon) with solar system gravity
tests were studied in \cite{localtest}.

On the other hand, inhomogeneous extra spaces are a promising tool that
could underlie such effects of low-energy physics as the formation of fermion
generations \cite{Gogber}, explanation of dark matter \cite{RDG}, and creation
of branes \cite{RuLocali}.
Evidently, this direction deserves a deep study. In particular, both problems
mentioned above, stability of compact extra dimensions and smallness of the
cosmological constant, should be solved at least at the classical level.
In this paper we  discuss these problems for an inhomogeneous (deformed) compact
extra space with a nonzero Ricci scalar, related to \cite{RDG, RuLocali}.

This study is performed on the basis of pure multidimensional $f(R)$ gravity.
No other tools listed above are used. The interest in $f(R)$ theories is motivated
by inflationary scenarios starting from Starobinsky's pioneering work
\cite{Starf}. A number of viable $f(R)$ models in 4D space that satisfy the
observational constraints have been proposed in
\cite{Amendola,Starobinsky,Cognola,Criscienzo}. At first glance,
it seems to be a hard task to obtain a zero or extremely small curvature of our
4D space if the Ricci scalar of extra space is not negligibly small.
A reason is that the Ricci tensor of the extra space contributes
to the 4D part of the multidmensional field equations, and therefore 4D flat
or weakly curved 4D space-times fail to be solutions of these equations; see
some details in Section 3 where we prove that if the extra space is a direct
product of maximally symmetric manifolds, the 4D Minkowski metric cannot appear
in the solutions, be it in the Einstein or Jordan conformal frame.

Nevertheless, we show that the problem can be solved if the extra space
and hence its curvature is inhomogeneous. We discuss the ways to construct
a set of ``almost Minkowski 4D spaces". In what follows, except for
Section 6, we omit the word "almost" and simply mention flat Minkowski space
as a representative of the set.

A separate problem is that of stability of the extra space geometry
\cite{Green,Carroll,Nasri}. A static solution can be obtained by
taking into account the Casimir effect \cite{Candelas} or, for example,
form fields \cite{Freund,Kubyshin}. Another possibility was discussed
in \cite{Yoshimura,Svadk}: it was shown that if the scale factor $a(t)$
of our 3D space is much larger than the scale factor $b(t)$ of the extra
dimensions, a contradiction with observations can be avoided. Section 6
of the present paper is devoted to the stability problem for our deformed
extra space models.

\section{Basic equations}

Consider, in $D$-dimensional space-time with the metric $g_{AB}$, a theory
of gravity with higher-order derivatives and the action in the form
\beq          \label{act1}
S=\frac{m_D ^{D-2}}{2}\int d^D x \sqrt{g_D}[f(R) ]
\eeq
with an arbitrary function $f$ of the the D-dimensional scalar curvature
$R = R_D$ and the D-dimensional Planck mass $m_D$; $g_D = |\det g_{AB}|$,
and the indexes $A,B, \ldots$ refer to all $D$ dimensions. In the following
consideration we choose the units in which $m_D = 1$.

The corresponding equations of motion read
\beq                    \label{eq-gen}
-\frac{1}{2}f(R)g_{AB}
+ (R_{AB} + \nabla_{A}\nabla_{B} - g_{AB}\square_D) f_R = 0,
\eeq
where $\square_D = \nabla^A \nabla_A$, $f_R \equiv df/dR$.
Assuming $D=n+4$, we will consider metrics of the form
\beq                                 \label{ds1}
ds^2 = g_{AB} dx^A dx^B = g\mn(x) dx^\mu dx^\nu + g_{ab}(x,y) dy^a dy^b,
\eeq
where the indexes $\mu, \nu, \ldots$ refer to the 4D part of the metric, and
$a,b,\ldots$ to its extra-dimensional part; $(x)$ and $(y)$ mark the
dependence on $x^\mu$ and $y^a$, respectively.

Notice that the $(\mu,\nu)$ components of the metric tensor are assumed
to be independent of the extra coordinates $y^a$. This leads to some
integro-differential equations for $g_{AB}$ instead of the equations
following directly from \eqref{eq-gen}, see a discussion after
\eq \eqref{ab-trace}.

A substantial simplification of the field equations is achieved if we
suppose that our metric varies slowly along the 4D coordinates of our
space-time as compared to the extra coordinates $y^a$. More specifically,
\beq          						\label{ep}
\partial_{\mu} g_{AB} \sim \epsilon \partial_a  g_{AB},
\eeq
that is, each $\partial_\mu$ contains a small parameter $\epsilon$.

The general expression for the scalar curvature of space-time with
the metric \rf{ds1} is
\bearr                \label{R_D}
R = R_4 (x) + R_n (x,y) + f_X (x,y),
\nnnv
f_X  \equiv -\nabla_{\mu} X^{\mu}  + \qua X_{\mu} X^{\mu}
- \qua (\partial^{\mu} g^{ab}) (\partial_{\mu} g_{ab}),
\cm
X^{\mu} \equiv g^{ab} \partial^{\mu} g_{ab},
\ear
where $R_4 = R_4[g\mn] = R^\mu_\mu$, $R_n = R_n[g_{ab}] = R^a_a$ are the
scalar curvatures of the corresponding subspaces, and $\nabla_{\mu}$ is
the covariant derivative in the metric $g\mn$. It follows from \eqref{ep}
that the 4D space is very weakly curved as compared to the extra dimensions,
that is, the Ricci tensor $R\mN = O(\ep^2)$, and, since $X^\mu = O(\ep)$,
\beq
R \equiv R^A_A = R_n + O(\ep^2).       \label{ep-R}
\eeq

Under the condition \eqref{ep}, the Taylor expansion of $f(R)$ in
Eq.\,(\ref{act1}) gives
\bearr   \label{act2}
S= \frac{1}{2}\int d^4 x d^n y \sqrt{g_4 g_n}~ f(R_4 + R_n + f_X)
\nnn \quad
\simeq \frac{1}{2}\int d^4 x d^n y\sqrt{g_4 g_n}
[  f(R_n) + f_R(R_n) (R_4 + f_X)  + o(\ep^2)],
\ear

In the next section we show that maximally symmetric extra spaces restrict our abilities in the description of the modern stage of our Universe.

\section{A maximally symmetric extra space is incompatible with 4D flat space}

In this section we prove a \textit{no go theorem: in the framework of pure $f(R)$ gravity, 4D Minkowski space-time is incompatible with compact, maximally symmetric extra spaces of nonzero curvature.}

The metric of our 4D space is extremely weakly curved, i.e., almost flat, as compared with possible extra-dimensional curvatures. This theorem
shows that to be in agreement with the modern state of the Universe,
a compact extra space must either have zero curvature or not be
maximally symmetric, if not accompanied by other fields. In this paper
we consider in detail a compact inhomogeneous extra space with
nonzero curvature. It is its inhomogeneity that leads to a small
or zero cosmological constant without violating the theorem.

To prove the theorem, let us analyze the expression \eqref{act2} with
the metric \eqref{ds1} where
\begin{equation}     \label{eq_8}
g_{ab}(x,y) = e^{2\beta(x)} \og_{ab}(y),
\end{equation}
and $\og_{ab}(y)$ describes a space of constant nonzero curvature.

A few words to justify this ansatz.
The characteristic size $l$ of a compact extra space is smaller than
$10^{-18}$ cm, or the energy scale is higher than 10 TeV. Excitations of a
compact extra space geometry are known to form a Kaluza-Klein tower with
energies $E > l^{-1}$. The energies we deal with are many orders of magnitude
smaller, therefore excitations are suppressed, and the extra space metric
$g_{ab}$ represents a stationary configuration described by the $(ab)$ part of
the classical equations of motion and a uniform distribution in our space.
Hence, if we decompose the metric $g_{ab}(x,y)$ as
\[
g_{ab}(x,y) = \sum_k f_n(x)h_{ab}^{(k)}(y)
\]
where $h_{ab}^{(k)}(y)$ is an orthonormal set of solutions to the corresponding
equations of motion, there remains only the first term with $k=1$, and we
arrive at Eq.\,\eqref{eq_8}. The latter leads to the following expressions
\bearr
\sqrt{g_n(x,y)} = e^{n\beta(x)}\sqrt{\og_n(y)},
\yyy                    \label{phi}
R_n(\beta(x),y) = \bar{R}_n \cdot e^{-2\beta(x)} \equiv \phi(x),
\ear
where $\bar{R}_n \ne 0$ is a constant.
The expression \eqref{act2} can be rewritten as
\begin{multline} \label{act3}
S \simeq \frac{1}{2}\int d^4 x d^n y\sqrt{g_4 g_n}
[  f(R_n) + f_R(R_n) (R_4 + f_X)  + o(\ep^2)]
\\
= \frac{m_4^2}{2} \int d^4 x\sqrt{g_4}e^{n\beta(x)}
[ f(\phi) + f_R(\phi) (R_4 + f_X) + o(\ep^2)],  \inch
\end{multline}
where
\begin{equation}
m_4^2 = \int d^n y\sqrt{\bar{g}_n}, \qquad
g_n(x,y) = e^{2n\beta(x)}\bar{g}_n(y).
\end{equation}

The expression \rf{act3} represents a 4D scalar-tensor theory of gravity with
the metric $g\mn$ and the scalar field $\beta$ (or $\phi$), specified in a
Jordan conformal frame. We want to obtain a constant equilibrium value of
$\beta$, which can be achieved after a transition to the Einstein frame in which
the scalar field dynamics is separated from that of the metric. This is
done using the conformal mapping \cite{BlackHoles}
\begin{equation}
\tilde{g}_{\mu\nu}(x) = |Y(\phi(x))|g_{\mu\nu}(x), \qquad
Y(\phi(x)) \equiv e^{n\beta(x)} f_R(\phi(x)).
\end{equation}
Now the action acquires the form \cite{BlackHoles}
\begin{equation} \label{ScalarAction}
S\simeq\frac{m_4^2}{2} \int d^4 x\sqrt{\tilde{g}_4}
[ (\sign\,F_R(\phi)) (\tilde{R}_4 + K(\phi) - 2W(\phi)],
\end{equation}
where $K(\phi)$ is a kinetic term of the field $\phi(x)$, and
\begin{equation}
-2 W(\phi) = e^{-n\beta}\frac{f(\phi)}{f_R(\phi)^2}
= \left(\frac{\phi}{\bar{R}_n}\right)^{n/2} \frac{f(\phi)}{f_R(\phi)^2},
\end{equation}
so that $W(\phi)$ is the potential energy density of $\phi$.

We intend to consider the physical situation in the modern epoch, such
that the scalar field $\phi(x)$ corresponds to a uniform distribution of
dark energy, which is almost constant. Therefore we assume that $\phi$ has
been settled at an extremum of its potential.\footnote
{Depending on the particular form of $f(R)$, the kinetic term
	$K(\phi)$ can have any sign. If it is positive, we are dealing with a
	usual, canonical scalar field, and its stable equilibrium corresponds to
	a minimum of $W$. If $K(\phi) <0$, the field is phantom and has a
	stable equilibrium at a maximum of $W$. Thus it is more precise
	to speak of an extremum.
}
Let there be an extremum of $W(\phi)$ at some $\phi=\phi_c$.
To get flat 4D space, we should require
\beq		\label{zeroLa}
W(\phi_c) = 0;\qquad \frac{dW}{d\phi} (\phi_c) = 0.
\eeq
This gives
\bearr \label{w0}
|\phi_c|^{n/2}\frac{f(\phi_c)}{f_R(\phi_c)^2} = 0,
\yyy       \label{w'0}
\left[\frac{n}{2\phi}\frac{f}{f_R^2}+ \frac 1{f_R}
- \frac{2f f_{RR}}{f_R^3} \right]_{\phi = \phi_c} = 0,
\ear
where $f_{RR}\equiv d^2 f(R)/dR^2$. Note that $\phi\neq 0$ due to \eqref{phi},
and since $\bar{R}_n\neq 0$, \eq \eqref{w0} gives $f(\phi_c)=0$,
then \rf{w'0} gives $f_R(\phi_c)=\infty$, which is unacceptable.
This completes the proof.

A question arises: can it happen that, unlike a precisely flat 4D space-time, a weakly curved, say, de Sitter space-time can still be compatible with a
homogeneous extra space? In other words, does the above theorem cover not   only precisely Minkowski but also weakly curved 4D space-times?

The answer depends on how small is the deviation from the flat metric.
The actually observed 4D curvature is very small: the cosmological almost
dS metric has a curvature radius around $10^{28}$ cm, enormously huge as
compared to the length scale ($\lesssim 10^{-18}$ cm) of the extra space,
making the curvature smaller by a factor of $\gtrsim 10^{92}$.
Even if one tries to obtain, by choosing the form of $f(R)$ and other
parameters, the cosmologically observed value of
$\Lambda \sim 10^{-123} m_{\rm Pl}^2$, where $m_{\rm Pl}$ is the Planck mass,
then, in any case, the calculations are performed with some finite accuracy
with errors manifestly larger than $10^{-123}$, therefore such a choice will
not be distinguishable from that leading to $\Lambda =0$. The lesson is that
it does not make sense to lose time for such attempts, and one should instead
try to involve new ingredients like matter fields or deformed extra spaces.

The above theorem also holds for a direct product of any number of
maximally symmetric compact extra spaces. In this case, in \rf{ds1}
we have
\beq
g_{ab}(x,y) dy^a dy^b = \sum_{i=1}^{N} e^{2\beta_i(x)} \og^{(i)},
\eeq
where $N$ is the number of $d_i$-dimensional extra factor spaces of
constant curvatures
$\bar{R}_i \ne 0$, with $x$-independent metrics $\og^{(i)}$. Then,
\bearr
R = R_4 (x) + \sum_i R_i (x,y) + f_X (x,y)
= R_4 (x) + \sum_i e^{-2\beta_i(x)} \bar{R}_i (y) + f_X (x,y),
\yyy
R_i (\beta(x),y) = \bar{R}_i \cdot e^{-2\beta_i(x)} \equiv \phi_i(x),
\qquad  \phi(x)\equiv \sum_i \phi_i(x),
\yyy
f(R) = f(\phi) + f_R(\phi) ( R_4 + f_X ) + o(\ep^2),
\yyy
m_4^2 = m_D^{D-2}
\int d^{d_1} y d^{d_2} y...\sqrt{\bar{g}_1 \bar{g}_2 ...},
\yyy
-2W(\phi) = \exp {\Big(-\sum_i d_i\beta_i}\Big)
\frac{f(\phi)}{f_R(\phi)^2}
= \left(\frac{\phi_1}{\bar{R}_{n_1}}\right)^{d_1/2}
\left(\frac{\phi_2}{\bar{R}_{n_2}}\right)^{d_2/2} ... \cdot
\frac{f(\phi)}{f_R(\phi)^2}.
\ear
The condition \eqref{zeroLa} now turns into
\begin{equation}
W(\phi_{1c},\phi_{2c},...) = 0, \qquad
\d W/\d \phi_i (\phi_{1c},\phi_{2c},...) = 0, \qquad i=1,2,...N,
\end{equation}
where $\phi_{i}=\phi_{ic}$ is the assumed extremum of $W$. This gives:
\bearr   \label{wi0}
\left(\frac{\phi_{1c}}{\bar{R}_1}\right)^{d_1/2}
\left(\frac{\phi_{2c}}{\bar{R}_2}\right)^{d_2/2} ... \cdot
\frac{f(\phi_c)}{f_R(\phi_c)^2} = 0,
\yyy       \label{wi'0}
\left[\frac{d_i}{2\phi_i}\frac{f}{f_R^2}+ \frac 1{f_R}
- \frac{2f f_{RR}}{f_R^3} \right]_{\phi_i = \phi_{ic}} = 0
\ear
with $i = 1,2,...N$. As above, it is easy to see that \eqref{wi0} leads
to $f(\phi_c)=0$ and $f_R(\phi_c) = \infty$, which is unacceptable.
Therefore flat 4D space cannot be obtained if the extra spaces are
maximally symmetric.

Another feature of interest follows from the condition \eqref{wi'0}
in the case with a nonzero extremum of $W$:
$W(\beta_{1c},\beta_{2c},...) = \const\ne 0$, so that $f(R)\ne 0$
at $\phi_i = \phi_{ic}$. One has
\beq         \label{cond}
\frac{\phi_{ic}}{d_i} = \frac{f f_R}{2[ 2 f f_{RR} - f_R^2]}
\,\biggr|_{\phi_k = \phi_{kc}}, \qquad i,k = 1,2,...N.
\eeq
This equation is correct for any sign of $\phi_i$. (Note that
$\sign \phi_i = \sign\bar{R}_i$, the curvature sign of a factor spaces. )

Immediate consequences of the expressions \eqref{cond} are:
\begin{enumerate}
	\item
	At an extremum of the potential, there are simple relations between the Ricci
	scalars of the extra factor spaces:
	\beq
	R_{n_i}/R_{n_k} = d_i/ d_k,  \quad i, k = 1,2,...N.
	\eeq
	\item
	The Ricci scalars of all extra spaces should have the same sign for $W$
	to have an extremum.
\end{enumerate}

All this was obtained in the Einstein frame in four dimensions. However,
at constant $\beta_i$ the conformal factor $Y$ between the two frames is constant,
therefore all conditions obtained for an equilibrium state of $\beta_i$ are
valid if we work in the Jordan frame as well.

One can also note that one cannot exclude that, in addition to the extra
factor spaces with nonzero curvature, there are some others, with
$\bar{R}_i =0$ (e.g., toroidal ones). Such factor spaces simply
do not take part in the above relations, which thus remain unaffected.

\section{Zero-order approximation. Analytical treatment}

Let us now return to configurations with a single but deformed
extra factor space in order to show that this concept can open new ways
in the description of our universe evolution.
In what follows we restrict ourselves to static extra space-times
and impose certain boundary conditions determining the structure
of extra dimensions. We also use the slow-change approximation
\rf{ep}.

If the extra-space metric $g_{ab}$ is $x^\mu$-independent,
(that is, $\beta (x)=0$ without loss of generality),
then in our 4D space-time we obtain general relativity with
a cosmological constant. Indeed, a comparison of the second
line in \eqref{act2} with the Einstein-Hilbert action
\beq \label{HilbEin}
S_{\rm EH}=\frac{m_4^2}{2} \int d^4x  \sqrt{|g(x)|}(R_4 -2\Lambda_4)
\eeq
gives the Planck mass and the cosmological constant:
\bearr                \label{Lam}
m_4^2 = m_D^{D-2} \int d^n y \sqrt{g_n(y)}f_R(R_n),\qquad
\Lambda_4 =  -\frac{1}{2m_4^2}\int d^n y \sqrt{g_n(y)}\,f(R_n).
\ear
both depending on the static geometry $g_{ab}(y)$.

\subsection{General consideration}

In the order of magnitude $O(1)$ in terms of the small parameter $\ep$,
the curvature of 4D space-time is negligible. Therefore, assuming for
simplicity that there are only two extra dimensions, we consider the
following metric with, in general, inhomogeneous curved compact extra space:
\beq          \label{ds_6}
ds^2 = g\mn dx^\mu dx^\nu + g_{ab} dy^a dy^b \equiv
\eta\mn dx^\mu dx^\nu  - e^{2\lambda}(dy^2  + dz^2),
\eeq
where $\eta\mn = \diag(1, -1, -1, -1)$ is the Minkowski metric, while $y$ and
$z$ are extra-space coordinates; the 2D metric $g_{ab}$ is chosen in a conformally
flat form without loss of generality, but we assume that $\lambda$ is a function of $y$ only.

In the previous section we have shown that with a maximally symmetric compact
extra space the Minlowski 4D geometry is impossible. This, however, does not
exclude other geometries, which can, in particular, have the form of a
deformed 2-sphere, but in any case the requirement $\Lambda_4 =0$ should
be satisfied. (Let us note for clarity that an ordinary 2-sphere
of radius $r_0$ is described in \rf{ds_6} by the function
$\e^\lambda = r_0/\cosh y$,
and the usual angular coordinates $\theta$ and $\varphi$ are connected with $y$ and $z$
by the transformation $1/\cosh y = \sin \theta$, $z=\varphi$.)

Let us try to find such solutions.
The $y$ dependence of the 2D metric to be found means that it is in general not
maximally symmetric but is deformed in some way. In the present approximation,
nonzero are only the diagonal Ricci tensor components $R^a_b$.
The 6-dimensional Ricci tensor and scalar are expressed in terms of $\lambda(y)$ as
(the prime means $d/dy$)
\beq          \label{R_2}
R^y_y  = R^z_z = - e^{-2\lambda}\lambda'', \qquad
R = R_2 = -2 e^{-2\lambda}\lambda''.
\eeq
The ${a \choose b}$ components of (\ref{eq-gen}) read
\beq           							\label{ab}
- \delta^a_b f(R)/2 + (R^a_b + \nabla^a \nabla_b - \delta^a_b \Box_6)f_R (R) = 0,
\eeq
where $\Box_6 f_R = \Box_2 f_R = \nabla^a \nabla_a f_R$.
The trace of (\ref{ab}) gives
\beq                                        		    \label{ab-trace}
\Box f_R = -f + R^a_a f_R.
\eeq

The mixed ${\mu \choose a}$ components of (\ref{eq-gen}) are trivial.
As to the ${\mu \choose \nu}$ components,  there is a subtle point: instead of directly
writing these components of \eqref{eq-gen},  it makes sense to return to the variation
procedure taking into account the independence of $g\mn$ of $y^a$. Then, instead of
the $({}\mN)$ components of \eqref{eq-gen}, variation of the action in $g\mn$ leads to
\beq  \label{mn-int0}
\int \sqrt{g_2[y]} d^2 y [f(R) + 2\Box f_R] =0,
\eeq
if we assume the Minkowski 4D metric. Taking into account (\ref{ab-trace}), this can be transformed to
\beq                                             \label{mn-int}
\int \sqrt{g_2[y]} d^2 y [2R f_R -f(R)] =0,
\eeq
where (recall) $R = R^a_a$. Thus in this approximation we must solve  (\ref{ab}) and
apply the integral condition (\ref{mn-int}) to the solutions obtained.

More than that, the condition $\Lambda_4 =0$ with \eqn{Lam} together with \rf{mn-int}
imply that, separately,
\beq                                             \label{int-0}
\int \sqrt{g_2[y]} f(R)\, d^2 y=0, \qquad
\int \sqrt{g_2[y]} R\,f_R\,d^2 y=0.
\eeq

There are two noncoinciding equations in (\ref{ab}), and their difference reads
\beq                                                          \label{vv1}
(\nabla^y\nabla_y - \nabla^z\nabla_z) f_R =0
\ \ \then \ \
\Box f_R =  2 \nabla^z\nabla_z f_R = -2e^{-2\lambda}\lambda' (f_R)'
\eeq
Writing this difference in its explicit form and integrating, we have
\beq                                                          \label{vv2}
(f_R)'' = 2\lambda' (f_R)' \ \ \then \ \
(f_R)' = A_0 e^{2\lambda}, \cm   A_0 = \const,
\eeq
and a further substitution into the trace equation \rf{ab-trace} gives a
second-order equation with respect to $\lambda(y)$:
\beq
f(R) - R f_R  = 2A_0 \lambda'.                 \label{eq-2o}
\eeq

We had two components of \eqs \rf{ab}, or, equivalently, their sum
\rf{ab-trace} and difference \rf{vv1}, which are of fourth order with respect
to $\lambda(y)$. Since there is only one unknown function
$\lambda(y)$, these equations must not be independent, otherwise the
system will be overdetermined. Let us show that both \rf{ab-trace} and
\rf{vv1} follow from the single second-order equation \rf{eq-2o}.

First, applying $d/dy$ to \rf{eq-2o} and expressing $\lambda''$
in terms of $R$ using \rf{R_2}, we obtain $A_0 R e^{2\lambda} = R (f_R)'$,
whence, for $R \ne 0$, it follows precisely the equality \rf{vv2}, from
which \rf{vv1} is trivially obtained.

Second, an explicit form of \rf{ab-trace} is
$e^{-2\lambda}(f_R)''= f - R f_R$.
The l.h.s. of this equality can be rewritten by substituting $(f_R)''$
from \rf{vv1} (already proven) and further $(f_R)'$ from \rf{vv2},
giving $2A_0\lambda'$. For the r.h.s. of \rf{ab-trace} the same
expression follows directly from \rf{eq-2o}. This completes the proof.
Thus a solution of \rf{eq-2o} is automatically a solution of the whole
set of vacuum field equations.

After solving \eqn{eq-2o}, it is necessary to substitute the solution to the integral
condition \rf{mn-int}, which, using \eqref{eq-2o}, can be rewritten as
\beq                                        \label {mn-int'}
\int \e^{2\lambda} dy [f(R) - 4A_0 \lambda'] =0.
\eeq
since $\sqrt{g_2[y]} = \e^{2\lambda}$, and there is no $z$ dependence.
Thus the extra space geometry to be discussed below must satisfy the condition
\eqref{mn-int} or \eqref{mn-int'}. Furthermore, since we are seeking a solution with flat
4D space-time, the condition $\Lambda_4 =0$ must be obtained for it automatically.

The order of \eqn{eq-2o} may be further lowered. Since its left-hand side is a function
of $R$, say, $Q(R)$, assuming that it is monotonic, we can consider its inverse,
$R = R(Q)$, and then \eqn{eq-2o} may be rewritten as (since $Q = 2A_0 \lambda'$)
\beq
R  = R(Q) = R (2A_0 \lambda') = -2\e^{-2\lambda} \lambda''.
\eeq
Next, let us use a standard trick and consider the new variable $v(\lambda) = \lambda'(y)$,
hence $\lambda'' = v\, dv/d\lambda$, and therefore,
\beq
-\e^{-2\lambda} v \frac {dv}{d\lambda} = R(2A_0 v)
\ \ \then \ \
\e^{2\lambda}d\lambda = - \frac{v\,dv}{R(2A_0 v)},
\eeq
an equation integrable by quadratures for an arbitrary (up to monotonicity
of $Q(R)$) function $f(R)$.

\subsection{Two exceptional cases}

There are two exceptions: $\lambda' = \const$ and $A_0 =0$ which must be considered separately.

First, $\lambda' = \const$ leads to $R^y_y  = R^z_z = R=0$, see \rf{R_2}, that is, flat 2-space
(being conformally flat, its Riemann tensor is completely determined by the Ricci tensor components),
and we return to the flat toroidal version of extra dimensions discussed above.

Second, $A_0=0$ leads to $f(R) = Rf_R$. The case $R = f_R =0$ is obviously unacceptable
(by \rf{Lam}, it gives $m_4=0$). Next, $R = R_0 = \const \ne 0$ means that the constants
$f(R_0)$ and $f_R(R_0)$ are either both zero or both nonzero. In any of these cases, according to \rf{Lam}, it is impossible to obtain simultaneously $m_4 > 0$ and $\Lambda_4 =0$.

It remains to suppose $R \ne \const$, then the equation $f(R) = Rf_R$ leads to
$f(R) = \const\cdot R$ (which is 6D general relativity). The field equations \rf{ab} in this case
read  $R^a_b - \half\delta^a_b R =0$ and are satisfied by an arbitrary function $\lambda (y)$
owing to the expressions \rf{R_2}. This total uncertainty in the shape of the extra space
cannot be regarded as a viable solution of our problem.

\subsection{Quadratic gravity}

Let us return to the general case $A_0\ne 0$, $\lambda'\ne\const$,
and choose the function $f(R)$ in the form
\beq                                 \label{f2}
f(R) = R^2 + bR + c
\eeq
then from \eq \eqref{eq-2o} one can get
\beq
R^2 = c - 2A_0\lambda',
\eeq
whence
\beq
\pm \sqrt{c - 2A_0 \lambda'} = -2\e^{-2\lambda}\lambda''
\eeq
Substituting, as was suggested, $\lambda'  = v(\lambda)$, we obtain the relation
\beq
\pm\frac{1}{2}\e^{2\lambda}d\lambda = \frac{v\,dv}{\sqrt{c - 2A_0 v }}.
\eeq
Its integration results in
\beq
\frac{16}{9} \frac{c^3}{A_0^4} \left( 1-\frac{2A_0}{c} \lambda' \right)
\left( \frac{A_0}{c} \lambda' + 1 \right)^2 = (\e^{2\lambda} + c_1)^2,
\eeq
where $c_1$ is an integration constant. With the notation
$\chi := 2A_0\lambda/c$, this equation can be rewritten as
\beq                \label{chi'}
(1-\chi') (\chi' + 2)^2 = K (\e^{c\chi/A_0} + c_1)^2
\eeq
where $K = 9A_0^4/(8 c^3)$. Equation \rf{chi'} can be presented in the form
\beq                 \label{chi}
\chi = \psi (\chi') = \frac{A_0}{c}\ln\Big (-c_1 \pm (2+\chi')
\sqrt{(1-\chi')/K}\Big).
\eeq

Introducing the parameter $p=\chi'$, we can write the solution in
a parametric form:
\bearr
\chi(p) = \psi(p),
\nnn
y(p) = \int \frac{-3 A_0 dp}
{2c\big[(2+p)(1-p) \pm c_1\sqrt{K(1-p)}\big]}.
\ear
This solution is rather cumbersome and inconvenient for analysis.
Therefore, in what follows we will find solutions for $\lambda(y)$ numerically.

\subsection{Numerical results}

In this section, we will develop a numerical approach to make the results more convincing and evident. The metric \eqref{ds_6} is convenient for the above analytic calculations, but now,  to facilitate numerical simulations, we transform
the extra-space metric to spherical-like coordinates:
\beq                                                 	\label{ds_7}
g_{ab} dy^a dy^b =  - e^{2\lambda}(dy^2 + dz^2)
=  - \e^{2\mu (\theta)} [d\theta^2 + \sin^2\theta\, d\phi^2].
\eeq
where $\theta$ and $\phi$ are related to $y$ an $z$ by
\beq
z = \phi, \qquad y = \ln \tan (\theta/2), \qquad
\e^\lambda = \e^\mu \sin \theta.
\eeq
The scalar curvature is expressed in terms of $\mu (\theta)$ as
\beq \label{Ricci}
R = 2\e^{-2\mu}(1 - \mu_{\theta\theta} - \mu_\theta \cot \theta).
\eeq
where $\mu_\theta = d\mu/d\theta$ and
$\mu_{\theta\theta} = d^2\mu/d\theta^2$.
The special case $\mu = \const$ corresponds to a 2D sphere with
$R = \const = 2\e^{-2\mu}$. It has been already discussed in Section 3
with a conclusion that it is incompatible with $m_4 >0$ and $\Lambda_4 =0$.

We therefore assume $R \ne \const$ and return to \eqn{vv2} which now reads
\beq
(f_R)' = A_0 e^{2\lambda} \equiv A_0 e^{2\mu}\sin^2\theta.
\eeq
Under the assumption \eqref{f2}, it becomes (since $d\theta/dy = \sin\theta$)
\beq \label{eq-3o}
R_\theta =\frac{A_0}{2} e^{2\mu}\sin(\theta),
\eeq
it is a third-order linear equation with respect to $\mu(\theta)$.
To solve it, we need three boundary conditions, which we impose
at $\theta=0$: $\mu(0)=0$, $\mu_\theta(0)$ and $R(0)$. The constant $R(0)$
is connected with the previously introduced integration constant $A_0$.
Indeed, from \eqref{eq-2o} it follows
\beq          \label{eq-2o2}
f(R) - R f_R  = 2A_0 (\mu_\theta\sin\theta +\cos\theta),
\eeq
which, taken at $\theta =0$, gives the relation
\beq            \label{A0}
f(R(0))-R(0)f_R (R(0)) = 2A_0 .
\eeq
Equation \rf{eq-2o2} is a nonlinear second-order differential equation
with respect to $\mu$, and \eqn{eq-3o}, being linear, is more
convenient for numerical integration.

The solution must satisfy the integral condition \eqref{mn-int}, which can be rewritten in the form
\beq                                        \label {mn-int1}
\int\sqrt{g_2[y]} d^2 y[f(R) + 4A_0(\mu_\theta \sin\theta +\cos\theta)] =0.
\eeq
This condition follows from \eqref{mn-int0}, where the second term
is total derivative, so that
\beq           \label{Lameq}
\int d\theta \Box f_R = \sin\theta \cdot (f_R)_{\theta}\Big|_0^{\pi}=0.
\eeq
The second equality holds if the Ricci scalar behavior is not too pathological.
It means that the conditions \eqref{Lam} and \eqref{mn-int0}
coincide provided $\Lambda_4 =0$. We have verified this statement numerically.

We have solved \eqn {eq-3o} numerically with \eqref{Ricci} under some
boundary conditions for $\mu(\theta)$ and $R(\theta)$ (see Fig.\,1).
By varying these initial parameters one can obtain a very small or
even zero $\Lambda_4$ and thus make it possible to have a
flat 4D space-time (see Fig. 2).
In this case, the integral condition \rf {mn-int1} is satisfied.

\begin{figure}[h]
	\begin{minipage}[h]{0.32\linewidth}
		\center{\includegraphics[width=1.\linewidth]{radius.eps}}
	\end{minipage}
	\hfill
	\begin{minipage}[h]{0.34\linewidth}
		\center{\includegraphics[width=1.4\linewidth]{egg.eps}}
	\end{minipage}
	\hfill
	\begin{minipage}[h]{0.32\linewidth}
		\center{\includegraphics[width=0.8\linewidth]{hole.eps}}
	\end{minipage}
	\\
	\hspace*{3cm} (a) \hspace{5.5cm} (b)  \hspace{4cm} (c)
	\caption{
		(a) Radius $r(\theta)=e^{2\mu}$ in extra space for the parameter
		values $b=-10^{-2}$, $c = 2.1 \cdot 10^{-5}$,  $\mu_\theta(0) =0,
		\mu(0) = \ln (100)$, $R(0) = 1.01 \cdot 10^{-3}$.
		(b) 3D plot of the solution. (c) A small part of the  3D plot near   $\theta=\pi$ (left ``end'').
	}
	\label{ris:image1}
\end{figure}
As a result, we have proved that a compact extra space with a
deformed metric is compatible with 4D Minkowski space. In the next
section we consider the next approximation in $\ep$.
.
\begin{figure}[h!]
	\center{\includegraphics[width=0.5\linewidth]{Zerolambda1.eps} }
	\caption{
		The cosmological constant $\Lambda_4$ as a function of the boundary condition $R(0)$.
		The other parameter values are $b=-10^{-2}$, $c = 2.1 \cdot 10^{-5}$, $\mu_\theta(0) =0$,
		$\mu(0) = \ln (100)$. It is seen that $\Lambda_4=0$ at $R(0)\simeq 1.025\cdot 10^{-3}$. }
	\label{ris:image1}
\end{figure}
\section{First-order approximation. Stability of the deformed extra space}

Let us study the stability of the solution obtained above under small
$x$-dependent perturbations. We suppose that the 4D metric is flat,
as before (since its perturbations should be of the order $O(\ep^2)$),
and consider the perturbed extra-space metric in the form
\beq     	\label{ds_8}
g_{ab} dy^a dy^b =
=- \e^{2\mu (\theta)+2\beta (x)} [d\theta^2 + \sin^2\theta\, d\phi^2].
\eeq
Then, after integrating out the extra dimensions in the action,
$\beta(x)$ behaves as a scalar field in 4D Minkowski space-time
\cite{BlackHoles}. In the previous sections $\beta = 0$.

The field $\beta$ can in general have complicated kinetic and potential
terms. If the potential has a minimum, and $\beta$ settles at this minimum,
then such a background configuration with its extra-space geometry
is stable.

Consider the case $\beta(x) \ll 1$ with the extra metric obtained in
\eqref{ds1} and \eqref{ds_7}. A back reaction of $\beta$ on the extra
metric is neglected. As before,
\beq
f(R) = R^2 +bR +c,~ \cm
S = \frac{m_D^4}{2}\int d^4x d^2y \sqrt{g_4g_2} f(R).
\eeq
In general, the action contains three parameters $b$, $c$ and $m_D$.
For the 2D extra metric me have the expressions
for the determinant and the Ricci scalar:
\beq 		\label{key}
\sqrt{g_2} = e^{2\beta (x)}r^2(\theta)\sin \theta,\qquad
R_2 = e^{-2\beta (x)}\bar{R}(\theta),
\eeq
The 6D Ricci scalar has the form
\beq                      \label{2+4}
R = R_2 + R_4 + 4 \Box \beta + 6 (\d\beta)^2 + o(\ep^2),
\eeq
where $\Box \beta$ and $(\d\beta)^2$ are the 4D d'Alembertian and
squared gradient of $\beta(x)$. Using the expansion \rf{act2} for the
action $S$ with $R$ given by \rf{2+4} and $R_2$ given by \rf{key},
integrating out the extra dimensions and excluding a total divergence
related to $\Box \beta$, we obtain
\beq \label{act4}
S\simeq \frac{m_6^4}{2} \int d^4 x \sqrt{g_4}
\left[ h(\beta) R_4 +  (6 - 4h_\beta)(\d \beta)^2 - 2W(\beta)\right],
\eeq
where $h_\beta \equiv dh/d\beta$, the function $h(\beta)$ and the potential
$W(\beta)$ having the form
\bearr                  \label{def-h}
h(\beta(x)) = 2I_1 + e^{2\beta(x)}bI_0,
\yyy                    \label{W}
W(\beta(x)) = -\frac{1}{2}\left(I_2 e^{-2\beta(x)}
+ cI_0 e^{2\beta(x)} + bI_1\right),
\ear
and $I_k$ are the integrals
\beq
I_k = 2\pi \int d\theta r^2(\theta)\sin \theta\, \bar{R}(\theta)^k,
\qquad k=0,1,2.
\eeq
We have restored the 6D Planck mass $m_6$, which was equal to unity
in the previous sections.
The parameters $I_k$ depend on the metric $g_{ab}(y)$. The latter is assumed
to be one of the solutions to the field equations discussed in the previous
section.

The action \rf{act4} is the vacuum action of a 4D scalar-tensor theory of
gravity  written in its Jordan frame.  The conformal mapping
\beq
\tilde{g}_{\mu\nu} = h(\beta)g_{\mu\nu}
\eeq
transforms it to the Einstein frame. The result is
\beq                   \label{SE}
S = S_E = \frac{m_6^4}{2} \int d^4 x \sqrt{\tilde g_4} (\sign h)
\left[ \tilde R_4 + K(\beta) ({\tilde \d}\beta)^2 - 2U(\beta)\right],
\eeq
where the tilde denotes quantities obtained from or with ${\tilde g}\mn$, and
\beq
K(\beta) = \frac{6 - 4 h_\beta}{h} + \frac 32 \frac{h_\beta^2}{h^2},
\cm
U(\beta) = (\sign h) \frac{W(\beta)}{h^2(\beta)}.
\eeq
More explicitly,
\bearr
K(\beta) = \frac{2}{h^2}\Big[6 I_1 + (3 - 8 I_1) bI_0 \e^{2\beta}
- 2b^2 I_0^2 \e^{4\beta}  \Big],
\nnn
U(\beta) = - \frac{(\sign h)}{2 h^2}
\Big(I_2 \e^{-2\beta} + b I_1 + c I_0 \e^{2\beta} \Big).
\ear

Evidently, $\beta =0$ should correspond to our ``background'' solution from
the previous section, with 4D Minkowski space and an inhomogeneous 2D
extra space geometry. For this solution to be in stable equilibrium under
the emergence of the 4D scalar $\beta(x)$, we should have
\beq                     \label{min-U}
U(0) = 0,\qquad  U_\beta(0) = 0,\qquad U_{\beta\beta}(0) > 0,
\eeq
provided that $K(0) > 0$. Let us make sure that this is really the case for the parameter
values used in Figs.\,1 and 2, that is, $b = -10^{-2}$ and $c = 2.1\ten{-5}$.

To this end, it is useful to recall that for solutions with flat 4D space
(that is, for $\beta = 0$), as follows from \eqn {int-0}, we have
\beq                       \label{III}
I_2 = cI_0, \qquad \ bI_1 = -2I_2, \ \ \then\ I_1 = -2c I_0/b.
\eeq
Then, for the potential at small $\beta$ we have
\beq                \label{UE}
U(\beta) =  \frac{\sign h}{2 h^2} c I_0\ [\cosh (2\beta) -1],
\eeq
which evidently satisfies the conditions \rf{min-U} if $c \sign h > 0$
since $I_0>0$. This is a general result in quadratic 6D $f(R)$ gravity \rf{f2}
for any model from the class under consideration.

Next, for the function $h(\beta)$ and the kinetic term at $\beta =0$ we have
due to \rf{III}
\beq
h(0) = \frac{I_0}{b}(b^2 - 4c), \qquad
K(0) = \frac{2}{(b^2-4c)^2}\Big(5b^4 -8 b^2 c + 48 b^4\Big).
\eeq
Substituting the values $b = -10^{-2}$ and $c = 2.1\ten{-5}$, we obtain
\beq
h(0) = -1.6\ten{-3}, \qquad  K(0) \approx 425.
\eeq
Thus we have $h(0)< 0$ (which is unimportant) and $K >0$, so that, according
to \rf{UE}, our model is stable under slow $x$-dependent changes in the extra-space metric.

The parameter values were selected so that the final expression for the action \eqref{SE} contains proper signs of the kinetic term $K$ and the potential $V$, as follows from Figure 3.
The common sign of the total action does not affect both the classical behavior and quantum transitions, see a discussion in \cite{BlackHoles}. Indeed, the transition amplitude is expressed in the path integral technique as $\int\exp{(iS[q])}Dq$ where $q(t)$ is some dynamic variable. The transition probability $\int\exp{ (iS[q_1]- iS[q_2])}Dq_1Dq_2$
is invariant under the substitution $i \to -i$ (with interchanging the integration
variables $q_1$ and $q_2$ ).

\begin{figure}[ht]
	\centering
	\includegraphics[width=0.5\linewidth]{PotBeta.eps}
	\caption{The potential $U(\beta)$ for the parameter values $b = -10^{-2}$,
		$c = 2.1\cdot10^{-5}$, $\mu(0) = \ln(100)$.}
	\bigskip
	\label{ris:image1}
\end{figure}

Specifying the boundary values $R(0)$ and $\mu(0)$ (equivalently, $r(0)$)
and calculating the parameter $A_0$ using \rf{A0}, we can solve \eqs
(58) and (59) with a given function $f(R)$, in particular, \rf{f2}
with its parameters $b$ and $c$. The potential $U(\beta)$ for a certain
range of $\beta$, the same parameter values and different boundary values
$R(0)$ is presented in Fig.\,\ref{ris:image1}.
One can see that the behavior of $U(\beta)$ confirms the existence of a
minimum with $U=0$ at $\beta =0$ for $R(0)\approx 1.0025$, which corresponds
to our model obtained in Section 4, with $\Lambda_4 = 0$ and flat 4D space.

Returning to the action \rf{act4} with $\beta =0$, we can identify the
constant coefficient of $R_4$ with $m_4^2/2$, where $m_4$ is the 4D Planck mass
related to the Newtonian gravitational constant $G_N$:
\beq
m_4^2 = (8\pi G_N)^{-1} = m_6^4 |h(0)| = m_6^4 I_0 |b-4c/b|\simeq 16m_6^2,
\eeq
where the integral $I_0 =8\cdot 10^{3}/m_6^2$ according to the numerical
simulations. As a result, the $D$-dimensional Planck mass is only a few
times smaller than the 4D one.

\section{Conclusion}

We have shown that an inhomogeneous (deformed) compact extra space may
be considered as a promising tool for physics beyond the Standard Model.
Such spaces satisfy the necessary conditions: (i) the deformed extra space
does not contradict the observable value of the cosmological constant,
(ii) the extra dimensions are stable at least relative to the ``radion mode''.
The latter represents a scalar field in 4D space. To be considered as the
inflaton, this mode should have a mass of order $10^{13}$ GeV, which needs
fine tuning since natural values of this mass are sub-Planckian.

We have also proved that maximally symmetric extra spaces prevent the
existence of solutions with 4D Minkowski space or de Sitter space with
a cosmological constant compatible with observations.

The analysis was performed on the basis of pure gravity without invoking any additional
ingredients. This lays a foundation for further research based on inhomogeneous extra
spaces.

\subsection*{Acknowledgments}

This work was supported by Russian Science Foundation and fulfilled in the framework of MEPhI Academic Excellence Project (contract № 02.a03.21.0005, 27.08.2013) and according to the Russian Government Program of Competitive Growth of Kazan Federal University. The
work of S.G.R. was also supported by the Ministry of Education and Science of the Russian Federation, Project № 3.4970.2017/BY. The research of K.B. was supported within the RUDN-University program 5-100 and by RFBR grant 16-02-00602.

\small

redpajama_set_name: RedPajamaArXiv


Each Christmas we like to give our children something with meaning. The Mr. and I like to hand make something that we hope they will treasure not only the item itself but the meaning behind why we chose to give them what we did.
This year our gift to them was a charcuterie board made from our cherry tree.
It wasn't until I personally started researching the value and blessing of trees that I could not have been more excited to give our children not only a piece of history but also some pretty fabulous information that I was quite sure they were not aware of.
Since the dawn of time humans have sought shelter in trees, trees provide oxygen, paper and building materials. Trees also provide physical security in the form of shelter, windbreaks and a sense of place. We clothed, built homes, provided food and even sailed ships and boats for thousands of years with trees. Olive trees provide olive oil, Fruit such as apples, cherries, pears, walnuts, pecans, dyes and jams all come from trees.
Trees "talk" to one another by exchanging chemicals. They communicate through underground fungi, and when they can recognize their relatives, they share nutrients. A great metaphor for the human life - that families need and should always support and help one another.
We talked about the love that both the Mr and I have for wood from a tree. For me it was the sound of the saws, the cutting of the wood, the smell of the polish, the pounding of the nails as I watched in admiration my grandfather as he repaired antique furniture to provide for his family. And the Mr's love for trees that comes from watching his father plant hundreds of tree's on their property over the years as well as a lineage of craftsman of wood and stone.
There is something special about the texture of wood, it's natural warmth and it's strength is miraculous. They are looming, quiet souls that watch over us as we develop, imperfectly, they are witnesses to our history, good or bad, without judgment. Like a companion who will always be a reminder of the past, trees are a constant.
The Mr. has fond memories of planting his first tree when he was four years old and watching the trees grow "while I grew into manhood, each one having a spiritual connection with my parallel growth."
Trees play a role in the context of play and recreation, as well. We use trees for crafting musical instruments and constructing boats and canoes. We have picnics under the trees and take walks through the woods.
At the spiritual level, trees help us become more aware of our connections with something larger than ourselves. We had the opportunity to visit the Redwood forests this past summer, it was the most spiritual place we have ever experienced. It took our breath away at their grandeur and beauty with the Sequoia roots deep into the earth and branches reaching 380 feet high to the sky.
We presented the beautiful cherry wood boards the Mr had created from the very tree that he had cut down in 1994 from our yard as the tree had given in to insects and disease. It is the tree that we watched our children climb in and build forts in when they were small. He had saved a piece of the trunk for it's beauty and strength in hopes of some day giving it purpose.
We pointed out the rings on the trunk and how each ring represents one year of growth. We made note of the size of the rings and their representation of drought or heavy rains. Cherry trees grow hard and slow but their density of the fiber and brilliant red color is characteristic of only Cherry. And like humans their midlife is strong and vibrant and much slower growth near the end of the life. At times we are strong and at times we are weak. Sometimes we blossom and at times we are more stagnant. We are naturally protectors.
All that we learned and shared really is connected to our lives here on this earth. It is our wish that our children and grandchildren learn to respect and love the simple cellulose fiber that we take for granted each day. We don't often think about the beauty that surrounds and protects us as well as how connected our human lives are to the earth.
It was a wonderful learning lesson for all of us and I am quite certain the Charcuterie boards will be treasured for years to come.


text:
January 19, 2022 | 2020 Unemployment Called. They Want Their Money Back.
January 18, 2022 | Why They Want To Keep the "Health Emergency" Going Forever
January 18, 2022 | It's Now Cheaper to Rent a Robot than Pay a Human; "It might eventually become normal to pay robots to do all sorts of jobs"
January 18, 2022 | Will 2022 Be "The Year of Sound Money" in the States?
January 18, 2022 | Visiting Myrtle Village Green Community Garden in Winter
5000 Anti-Mandate Protesters at New York Capitol
TOPICS:EmploymentMandatory vaccinesMichael KaneNew YorkProtest
By Michael Kane
5,000 peaceful protesters rallied at the New York Capitol in Albany on January 5th to say no to all covid and vaccine mandates. The 2022 legislative session in Albany is loaded with bills attempting to codify multiple covid and vaccine mandates into state law.
Driving to the protest, road conditions were treacherous where black ice, rain and sleet caused multiple accidents including one 6-car pile up. Hundreds of protesters who were headed to Albany turned back due to the horrible road conditions. Even many confirmed speakers turned back when trapped in horrible traffic and icing roads, including Kevin Jenkins.
Despite these major obstacles the turnout was still enormous! 5000 people NEVER show up in Albany over any issue, ever. The point was made loud and clear, New York is completely against codifying covid mandates into law.
Children's Health Defense President Mary Holland gave an outstanding speech where she called dozens of children to stand with her as she spoke against requiring covid vaccination for children.
Assemblyman David DiPietro addressed the crowd, clearly impressed by the size of the crowd he was addressing. DiPietro is the sponsor of a bill to bring back the Religious Exemption to Vaccination in New York for children to attend school.
Perhaps the highlight of the event was when New York City's Transit & Labor Leader Tramell Thompson announced that he was fully vaccinated yet anti-mandate. Tramell took out his vaccine card and tore it up, stating he is against vaccine passports. This was a very powerful message, making clear this is not about vaccinated vs. unvaccinated, it's about fundamental rights.
Watch Tramell tear up his vaccine card here:
Whoa. At the NY protest, @progressiveact, who is fully vaxxed rips up his vax card in front of thousandspic.twitter.com/A2yS88B1Ga
— Election Wizard (@ElectionWiz) January 5, 2022
What's left of my segregation card 😃 pic.twitter.com/qGVzKo85Ni
— Progressive Action (TV) (@progressiveact) January 6, 2022
There were many reports in the mainstream press. All of them seemed to avoid the "anti-vaxxer" misnomer but often focused on "rising covid cases." They left out that the new Omicron variant of covid is significantly less virulent than any previous variant, and that almost all cases of covid now are equivalent to flu or the common cold. We should be celebrating that we have gotten past the worst of covid, not clinging to mandates that infringe on civil liberties. We can be safe, with common sense, and not force medical mandates on anyone.
Governor Hochul is clearly quite afraid of the rising anti-mandate movement throughout the entire state of New York. She is so afraid that at 7pm the night before our protest she announced her State of the State speech would be delivered entirely remotely. This had NOTHING to do with public health. If it did, she would have made the announcement days or weeks earlier. This was a way to avoid having to deal with the thousands of us there railing against her support to codify more covid mandates, including forcing covid vaccination on children to attend school.
Here is some of the Mainstream Press coverage of our Jan 5 Protest in Albany, NY:
Newsweek https://www.newsweek.com/protesters-against-covid-mandates-descend-ny-capitol-ahead-hochuls-state-address-1665895?amp=1
Reuters — https://www.reuters.com/news/picture/americans-protest-covid-19-rules-as-infe-idUSRTS49K62
News 10 ABC — https://www.news10.com/news/ny-news/rally-against-mask-vaccine-mandates-for-healthcare-workers-held-at-capitol/
Times Union — https://www.timesunion.com/state/article/Hochul-pledges-to-build-a-brighter-future-for-16751410.php
NEWSDAY — https://www.newsday.com/opinion/mask-mandate-protest-1.50467573
DENVER GAZETTE picks up piece of REUTERS coverage – https://denvergazette.com/news/protestors-demonstrate-against-covid-19-vaccine-mandates-during-rally-outside-new-york-state-capitol-in/image_73f39d2d-1f5f-5cec-9dd2-74be6d214f15.html
FOX NEWS (social media) https://www.instagram.com/foxnews/p/CYX2z8NNmvP/?utm_medium=copy_link
Local FOX affiliate – https://www.news10.com/news/ny-news/rally-against-mask-vaccine-mandates-for-healthcare-workers-held-at-capitol/
NBC 10 Local Affiliate — https://www.whec.com/news/state-of-the-state-draws-protest-against-mandated-covid-vaccines/6350430/
YOUTUBE – https://www.youtube.com/watch?v=H8FYu1xesfM
Source: Teachers for Choice
Or support us at SubscribeStar
Donate cryptocurrency HERE
Subscribe to Activist Post for truth, peace, and freedom news. Follow us on Telegram, HIVE, Flote, Minds, MeWe, Twitter, Gab and What Really Happened and GETTR.
Provide, Protect and Profit from what's coming! Get a free issue of Counter Markets today.
Apple AirTags: "especially powerful for stalking in a way that goes beyond what we have seen in previously existing physical trackers"
FAA Eases 5G Restrictions; "Even with these new approvals, flights at some airports may still be affected," cautioned the agency."
2020 Unemployment Called. They Want Their Money Back.
Why They Want To Keep the "Health Emergency" Going Forever
Visiting Myrtle Village Green Community Garden in Winter
Students Describe Life on Campus Under "Health Surveillance"

redpajama_set_name: RedPajamaCommonCrawl


The beautiful manicure sets in the Bijoux collection are the pure essence of glamour and luxury, made especially for a woman who loves the good things in life and who loves to be pampered. These lovely sets are decorated with designs kissed with genuine Swarovski crystals, giving them a new definition of exquisite elegance.
Majestic and purely feminine, this lovely black suede case shimmering with the brilliance of Swarovski crystals, featuring 5 superb pieces made of highest quality TopInox stainless steel in a matte finish, including a pair of nail scissors, cuticle scissors, slanted tweezers, sapphire nail file, cuticle pusher / nail cleaner.
These remarkable fully handmade Niegeloh nippers are individually, precisely ground, sharpened, assembled, adjusted and tested by hand.


Selecting Grayson Loveseat By Wayfair Custom Upholstery?
Looking For Tomas Loveseat By Ophelia & Co.
Excellent Alvinholmes Loveseat By Langley Street is important have in almost any home. You want for the greatest items, so you want to ensure that you never ever overpay to them. Seems just a little challenging, correct? Nicely, this article is in this article to help you. Read more and find some terrific specialist tips for finding the offers on furnishings sections you are going to enjoy. Before buying any cupboards, open every one of the storage and look within. You're not merely being sure that each of the compartments are made to previous and open without the hitches, you also want to be sure that the inner from the compartments have some form of concluding also. When buying Alvinholmes Loveseat By Langley Street, evaluate your room before going shopping. There exists nothing at all even worse than selecting the ideal set of Alvinholmes Loveseat By Langley Street only to find out that it does not easily fit in your bedroom once you get it house. As an alternative, accurately measure your areas sizes before heading remembering where by doorway and home windows are located.
Bar furniture are not only specific furniture intended to be utilized in pubs as the name suggests. If youre looking to provide your kitchen, dining area of home a company-like look, you will want to buy the best bar furniture. The dinning tables are earning a constantly-growing existence in individuals homes as they search for that distinctive company-like existence at home. However, with a large number of club furniture to select, from, getting the best offer can be a daunting job for most owners. For all those looking to change the diner right into a vibrant modern club, we have you covered with some of the best bar furniture available on the market currently. This club desk is a breakfast established well suited for homes with little dining areas. It is a fairly small table that won't consider too much space in the dining area. It features caster wheels that allow for easy motion within the home particularly when cleaning. There is also some bar stools and decrease leaf giving you good bang for your buck. Made of the natural Beachwood, this table is a great buy and arrives in the round and sq . shape.
If youre looking for some severe relaxation, this loveseats strong pillow back again and rolled hands will deliverbut at a higher cost. Each seat cushioning consists of 30 individual pocketed coils for spring and assistance, and resilience chair foam for an additional coating of plushness and sturdiness. Plus, the froth cushions are topped with super soft, combined down feathers. For your extra comfort, you will find three regular jobs: sitting, a slight recline with an elevated footrest, and a full lie down with an elevated footrest. It does not hurt that this lying loveseat appears lux, possibly. Its all-leather-based furniture and ornamental toe nail mind trim provide a classic look that can function with lots of home decorations styles, from traditional to country to crafts and arts. Plus, leather holds up nicely against daily use, children and dog locks and nails. Just be sure you situation the leather-based each and every 6 to 12 months.
This established features a one left arm couch established, two armless couch sets and one corner sofa established. This provides enough space to accommodate your friends and relations. The material is 100% rayon for sufficient comfort and durability. The good thing about this sofa established may be the matching and mixing of seats in the available space in the room for a ideal form. And also allows for fitted even just in small areas. It takes just light putting together.It also features drive cushions for optimum comfort. You may want to try this established. It does the job well.
This established is padded inside a drive and comfy distinctive padded purple velvet. This gives it the amazing comfort that its cherished for. Again it facilitates chaise design seating for adequate comfort. Furthermore, the sinus spring foundation increases the durability and comfort. Furthermore, the sofa has hard wood frames that add to its toughness and sturdiness. The sofa has cushions on the seat and also the back again. Yet still, the sofa can move from a seated position to some reclined position effortlessly. Having said that, simply to repeat, the sofa is very gentle and gives outstanding comfort.
The Lawson could be known as quintessential United states couch style. It's comfy and straightforward: The boxy form is large and characteristically has three back again soft cushions as well as three chair cushions. The traditional Lawson also offers A higher back and container-shaped soft cushions that have welts in the edges, just like the rear cushions. Ideal for snuggling and sleeping, this couch design was designed for Jones Lawson, a united states copper tycoon in the change of the century. He desired a sofa which was quite different from the picky Victorian designs that were common in those days. Present versions of the Lawson can also include metal or wood contained in the arms. Most significantly, this sofa design is the one which will morph its feel with the fabric you select for the upholstery. The shape is so flexible that it can be glamorous, luxurious, industrial, casual or official depending on your material option.
Puffy, overstuffed loveseat cushions arent your thing? This lying loveseat from GDF has crisper, solution lines than your traditional reclining loveseat, making it an elegant accessory for a middle-century contemporary, modern, eclectic or contemporary type of house. And, at just 46.46 inches in width, its easy to squeeze into a smaller space, like flats, dens or offices. Obtainable in a number of colour and material options, such as grilling with charcoal material, navy blue material and slate microfiber, its simple to find the perfect match for your home decorations, and the durability your family needs for everyday use. The days are gone of good-searching, but unpleasant furniture you will no longer need to give up comfort and ease for style. This reclining loveseat functions an oversized, soft, plush-stuffed seat cushion with enough space for two or enough space for one to disseminate and relax.
This reclining loveseat requires the 2nd place because its an excellent inexpensive, extremely comfortable option that may participate in most house dcor styles. Its solid wood body is cushioned with coils (for many spring) and high-density froth (for many gentle), and padded with brownish fake leather which makes it an ultra-versatile furniture piece. Combine it with the man cave for any comfy spot to view the game (and dont worry about any spillsits vinyl fabric furniture makes clean-up simple) or perhaps your family room with a couple of decorative throw cushions for the ideal place to comfortable up after a long day. Real-existence clients love this reclining loveseat simply because its comfy, durable, easy to put together one reviewer even mentioned that they could assemble two units in under forty-five minutes! and shipping in the vendor is a breeze.
Fluffy, overstuffed loveseat soft cushions arent your lifestyle? This lying loveseat from GDF has crisper, cleaner lines than your traditional lying loveseat, which makes it a stylish addition to a middle-century contemporary, contemporary, eclectic or modern type of home. And, at only 46.46 in . in width, it's very easy to match an inferior room, like flats, dog dens or office spaces. Available in several color and material options, such as grilling with charcoal material, navy blue material and slate micro-fiber, its easy to find the perfect match for your home decorations, and the durability your loved ones needs for daily use. The days are gone of excellent-looking, but uncomfortable furniture you no longer need to give up comfort and ease for design. This reclining loveseat features a big, soft, luxurious-stuffed seat cushion with enough room for two or sufficient room for one to disseminate and unwind.
This couch is appreciated for its distinctive design and comfort. Its striking pazazz-equip couch with chairs offers you maximum room for stretching out and relaxing in your room. So generally the modern sofa chaise increases the great thing about this established. The established also includes pillow leading arms cushioned back and chairs for maximum comfort. The sofa includes a corner-blocked wood frame that provides the sofa enough toughness. The sofa steps 89 in . broad x 62 in . in depth by 4 inches tall which makes it ideal for moderate dimension rooms. You may want to do minor putting together eg of the thighs.
This set is padded inside a push and comfortable textured cushioned purple velvet. This gives it the amazing comfort that it is loved for. Once again it facilitates chaise style seats for sufficient comfort and ease. Moreover, the nose springtime foundation adds to the comfort and durability. Furthermore, the sofa has hardwood frames that add to its sturdiness and sturdiness. The couch has cushions on the seat and also the back again. But still, the sofa can move from a sitting placement to a reclined position easily. Having said that, just to replicate, the sofa is extremely soft and provides exceptional comfort.
Typically now used in houses having a great room or open up layout, they are often large configurations that chair lots of visitors. In smaller rooms, they works well for seats in an region which has an odd corner or any other room restriction. The ability to mix part units, finish units and reclining sections according to room and person choice makes this couch design extremely flexible. Sectional couches are also available in a multitude of styles, from extremely contemporary or super luxe, to more loved ones-pleasant contemporary designs.
Copyright © Alvinholmes Loveseat By Langley Street in Loveseats All right reserved.


The BTM Wallet is an innovative tool made to make it easier for companies to manage digitized assets, solving the issues presented from the administration between physical and digitized businesses. This promising project was developed by DuanXinxing and Chang Jia. His idea when designing the platform would have been to allow users to join up assets as fast and quickly as possible. This project is able to allow users, businesses and small merchants to make applications of financial and digital assets through Bytom Wallet Download.
Enter the current world, allowing your money to begin to come back a new turn, creating channels between physical and digital businesses, therefore we recommend employing this new application since its architecture are operating in three layers: the applying Layer, anything Layer, and the Shared Distributed Ledger Book Layer, allowing a fabulous organization and simplification inside the accounting process for anyone people who lack time.
The process to obtain the program is pretty simple; download the Bytom Wallet Github, then access the platform where you can register the assets and projects, this configuration works with for Mac, Windows, and Linux programs. Then a file is extracted, giving an individual click for the icon that will be displayed, opening the configuration window as well as in this way you need to choose the word "join the key Bytom network", creating the alias and password in the account, and From that moment on, your digitized portfolio is going to be created, start to store your Bytom cryptocurrencies inside.
With this striking medium, it is going to allow the user to express, build and consume information, from applications of financial and digitized assets, without interference from the centralized intermediary, where different assets from the companies (bonds, dividends, stocks, and the like) , can be stored and marketed with different companies worldwide from the blockchain protocol, in the role of a link to build the chains with registered users, keeping the input total privacy, and allowing users to get their cryptocurrencies secure in their folders .


The first step in any skincare regime is always to clean your face to remove dirt, oil and what not on your face. I used the natural herbal facial cleanser - Rosemary & Calendula Cleansing Milk 120ml RM85. It smells like herbal infusion to me. The texture is milky and I use it with or without a cotton pad to clean my face. It can also remove daily make up and is gentle for sensitive skin users. This is just like your cleansing milk at home only better because it does not contain harmful chemicals and is made organically. A natural herbal cleanser that can be used for up to a year after opening.
P.S: Paul Penders use cold process technology so their products can be used for longer period of time without using harmful preservatives such as parabens.
After cleansing, a toner is used to restore the skin's ph balance back. Having dry and sensitive skin, I am always on the look out for an alcohol-free toner and Paul Pender's Chamomile & Moringa Skin Toner 120ml RM85 is my absolute favorite at the moment. It's watery, non-drying and refreshing to use. The scent is amazing, it smells like cooling baby powder that I use when I was young. The best part about this toner is that it leaves my skin moisturized unlike other toners in the market.
To keep my face moisturized at night, I used the Avocado & Cranberry P.M. Moisturizer 40g RM118. It's a light creamy night time moisturizer for dry and sensitive skin. Smells like herbs as well and leaves my skin hydrated. It feels a bit greasy on my skin. I read that it covers the skin with emollients derived from selected natural moisturizing agents, including the finest avocado and jojoba plant oils. Contains natural vitamins, ceramides and liposomes to keep skin smooth, supple, and resilient. The first few days I was using it generously at night, and a pimple pop out! So I only use a very small amount to spread around my face. It makes my skin moisturized the next day, I don't experience any skin peeling during 3 weeks under the current heat condition.
If I am going out in the day, I would use the Herbal Sunscreen SPF22 70g RM95 before putting on make up. I am surprise the thick cream scared me off at first. During application, the thick creams quickly turns into a lotion spread and disappears on my skin. It's non-sticky but I feel it's a bit greasy, probably due to the emollients used. The sunscreen contains ethyl hexyl methoxycinnamat (occurring naturally in balsam and cinnamon leaves), titanium dioxide (a naturally occurring mineral), vitamin E and vitamin A (retinyl palmitate).
The herbal scent in this product isn't as strong as the others. I am not fond of cream products in our hot weather and would prefer non-cream based sunscreen. Spray mists, lotions, gel or water based sun protection would be nice. I guess thats one of the reasons Paul Penders is popular in countries like America, Australia and U.K. Cream based skincare is really important during the winder when gel and lotions won't offer the hydration and protection needed.
The last product that I would like to include is the Intensive Clarifying Therapy (ICT) 110g. I was given the travel size version to use and I could use for up to 4 times. In my Paul Penders Langkawi post, I explain what it is and a showed pictures during the demonstration. I have already finished this product and well the results were immediate - smoother, brighter looking skin. It does makes my skin dry and red. I don't think this suits sensitive skin, but it didn't irritate my skin other than being flushed. I am kinda impressed by this product and may consider purchasing it.
To conclude, my favorite Paul Penders products are the cleansing milk, toner and ICT from Paul Penders and I would consider purchasing these 3 after finishing. It's after all organic and made from herbal ingredients that makes its product smells so good, like in a spa!.
Hay supper ya i like your post y.


text:
Rawlings never visited Abossey Okai – Spare Parts Dealers Association Co-Chair
Almost all the presidential candidates for political parties since time immemorial made a stop at Abossey Okai, one of the country's trading hub to interact with them.
The visit and interaction was to bring to bear their challenges and concerns for both the traders and the sitting government to devise means of finding a lasting solution to their problems.
This was not so during former president Jerry John Rawlings' era as he reportedly failed to turn up there.
According to the Co-chairman of the Abossey Okai Spare Parts Dealers Association, Clement Boateng, J.J Rawlings never stepped foot in Abossey Okai when he was President.
"Since 1996 or ever since Ghana was ushered into democratic dispensation, apart from Rawlings who never stepped foot in Abossey Okai here, all the presidents whether aspiring or incumbent had come to Abossey Okai, everybody was welcomed," Mr Boateng revealed in an interview with GhanaWeb.
Aside from him, the likes of the late John Evans Atta-Mills, John Agyekum Kufuor, John Dramani Mahama passed through the trading community while in office.
The recent to follow the lead was President Nana Addo Dankwa Akufo-Addo.
On Tuesday, October 27, 2020, the NPP flagbearer visited the trading community as part of his Greater Accra regional tour.
Speaking on the President's visit, Mr Boateng said the Abossey Okai Spare Parts Dealers Association is apolitical and welcomes any presidential candidate from all political parties who show interest in visiting them.
He noted that their gesture is aimed at listening to all political parties on policies they'll put in place to make sure the business environment is friendly for the Association.
"We the executives here are non-political. Any presidential aspirant who wants to come to Abossey Okai here is welcomed. We don't discriminate. We are businessmen so whatever government that comes on board, comes out with policies that will be of beneficial to the business community and for that matter the spare parts industry is welcomed," he said in an interview with GhanaWeb.
Clement Boateng, however, noted that the President's visitation was successful as he was met with a cheerful crowd.
Election 2020: EC to commence printing of voters register on Wednesday
Withdraw Agyapa deal and sack Ofori-Atta – Mahama to Akufo-Addo
School reopening committee submits final report to govt
A committee chaired by former Minister of Education, Prof. Dominic Fobih, inaugurated to advise government on the reopening of pre-tertiary schools for the 2020/21 academic year, have presented their report to the Ministry of Education.In a release signed by the…
Komfo Anokye Teaching Hospital nurses resume work
Nurses at the Komfo Anokye Teaching Hospital (KATH) in Kumasi in the Ashanti Region have on Thursday, 24 September 2020 resumed work. This follows the suspension of a strike action by the Ghana Registered Nurses and Midwives Association (GRNMA) and its allied…


It's hardly a secret that Millennials are a force to be reckoned with in modern American business. They make up an increasingly significant portion of the 25-45 age group business covets and targets. This is as true of the beer industry as any other.
The problem for business is the Millennial demographic is not receptive to traditional marketing strategies and tactics. You cannot effectively reach them via print, radio or television advertising. One might argue that's because they are fixated on computer, tablet and phone screens. But never mind.
There are a lot of things businesses have to keep in mind with Millennials and plenty of places to find that intel if that's your thing. Generally speaking, these kids identify more personally and emotionally with brands than prior generations. They demand to interact and be part of the brands they respect.
The platform that makes this stuff happen is provided by the combined emergence of the smart phone and social media, which allow good and bad experiences to be widely spread electronically in the blink of an eye. This kind of instant publicity was unheard of 10 years ago. Today it drives the success of many businesses...and Millennials are heavily immersed in it.
When it comes to beer, Millennial tastes are vague and transient. They want to experience a wide range of flavors, which means they are receptive to inventive approaches and bizarre blends, and somewhat bored by traditional styles. In short, they are regularly looking for something new and different.
That reality is forcing suppliers and retailers to radically increase the number of available choices. A reliable industry source expects the number of SKUs on the market to double within 10 years. That's a significant increase given there are already something like 10,000 SKUs out there.
These changes mean chaos for big beer, whose leadership is dominated mostly by folks who have been around for 20 or more years. Age alone isn't the issue. The more serious problem is they are stuck in an antiquated mindset. Expensive ad campaigns and traditional media are yesterday's news...they fall flat with Millennials.
Craft beer is another story. Small breweries and pubs never had the luxury of using expensive, traditional media. Building a brand identity took years. Then came social media and the smart phone. Today, craft-centric businesses are leveraging the digital space and effectively engaging with the all-important Millennial demographic. Craft brands are being built quickly, almost overnight in some cases.
In other words, the brand building shoe is now on the other foot. Being small and unable to afford expensive advertising tactics has put the little guys ahead of the big guys in the digital, Millennial age. If nothing else, you have to appreciate the irony.


text:
Town Bylaws
Attorney General Rulings
Conflict of Interest/Ethics
Conflict of Interest Law Summary
Ethics Training
Monthly Archives February 2020
Presidential Primary EARLY VOTING Feb. 24-28, 2020
Published February 24, 2020 | Town Clerk | Automatically Archived on 2/28/2020
Early Voting for the Presidential Primary will be held at Lower Town Hall from Monday, February 24 – Friday, February 28, 2020 for all precincts. More →
Beth R. Klein
The Sudbury Town Clerk is appointed by the Town Manager. The Clerk holds the rank of department head, and has a staff which includes three full-time and one part-time employees.
State law provides that the Town Clerk is the official keeper of the Town Seal.
The Town Clerk serves as the custodian of Town records including Vital Records and other official documents filed in the Town Clerk's Office. She is responsible for the maintenance, preservation and disposition of Town records in her custody.
If you are a new resident of the Commonwealth,
Welcome to Massachusetts!
From the Town Clerk, Beth R. Klein, CMC, CMMC
Monday, Wednesday, Thursday: 8:30a.m. - 4:30 p.m.
Friday: 8:30 a.m. - 12:30 p.m.
Email: clerk@sudbury.ma.us
Building: Town Hall
Positions Available for the 2022 Annual Town Election January 3, 2022
2022 Dog Licensing December 31, 2021
Sudbury 2021 Precinct's 1,2,3,4,5,6 December 30, 2021


Created by Rand Holmes this humorous cartoon strip appeared in the Georgia Straight in the early 1970's.
Hedd's adventures ranged from selling oregano to tourists in Vancouver's Gastown to flying back [to Vancouver} a plane load of pot from Mexico. His adventures often paralleled events in Vancouver. Holmes mocked then Mayor Tom Campbell, who was anti-pot and anti-hippie and Vancouver's vice squad. In this way "Harold Hedd" tended to straddle the worlds of the underground or alternate and the editorial cartoon.
Harold Hedd, 1, Last Gasp Eco-Funnies, 1973.
Front: Illus., R. Holmes. Kitchen Sink Press, 1984.
Harold Hedd, 1, Car., Rand Holmes. Last Gasp Eco-Funnies, 1973.
Harold Hedd: Hilter's Cocaine. Car., R. Holmes. Kitchen Sink Press, 1984.
Part., 1 of 2. Part, 2 of 2.
"Harold Hedd in Wings over Tijuana." Car., R. Holmes.
Harold Hedd. 2, Last Gasp Eco-Funnies, 1973.
Harold Hedd. 2, Print run 10,000 copies, December 1973, no publisher named. Distributed in Canada only by Greater Vancouver News Ltd., Identical format to Eco-Funnies issue.
5-185, July 20-21 1971: 1 & 9.
5-193, Aug. 17-20 1971: 8.
5-195, Aug. 24-27 1971: 19.
5-199, Sept. 7-11 1971: 24.
5-201, Sept. 14-17, 1971: 24.
5-211, Oct. 19-22 1971: 1+.
, 5-217, Nov. 18-25 1971: 1+ 6-224, Jan. 13-20 1972: 1 & 24.
6-225, Jan. 20-27 1972: 1&4.
6-228, Feb.10 – 17 1972: 5.
6-231, Mar. 2-9 1972: 5.
Rand Holmes The Artist Himself: A Rand Holmes Retrospective. Writ., Patrick Rosenkranz. Fantagraphic Books Inc., 2010.
"'Harold Hedd' creator dies." Writ., Dana Larsen. Cannabis Culture Magazine, 18 Aug. 2002. Retrieved http://www.cannabisculture.com/v2/articles/2486.html 7/26/2010.
Illus., Rand Holmes. Harold Head, 2, 1973, Front cover.
Illus., Rand Holmes. Harold head: Hitler's Cocaine, part 1. Front cover.


We teach English GCSE in small group classes.
At Tuition Works we ensure our tutors are qualified and have correct experience to to really add value to the GCSE English Tuition they provide. We truly believe our bespoke GCSE English Tuition courses provide students with all the tips & tricks on how to get a top grade for their exams.


Can you guess where this unique angle can be found? It's kind of mysterious, and quite dramatic. The cool blues and purples add an ominous glow, and yet the mosaic tiled curves nestled within the hugging branches of this most unusual tree suggest something mechanically organic. It's almost as if a large UFO had touched down, announcing its presence, ready for exploration.
In a sense this is a space craft. But not the extraterrestrial kind... this a craft that articulates space--a house, in plain speak. And this house can be found within Innoventions, showcasing the technological "features of tomorrow." Of course, predictions for the future in Tomorrowland have a tendency of becoming outdated before they're even unveiled, but the whole presentation is certainly not unflattering to look at!


text:
Der Lichtenstättgraben ist ein Bach in den Ammergauer Alpen im Landkreis Garmisch-Partenkirchen in Bayern. Er ist etwa 1,3 Kilometer lang, hat ein Einzugsgebiet von etwa 0,8 Quadratkilometern und mündet von rechts in den Manndlbach.

Verlauf
Der Lichtenstättgraben entspringt nördlich der Ortschaft Ettal am Südwesthang von Schartenkopf und Latschenköpfl auf einer Höhe von etwa . Er verläuft im Wesentlichen nach Südwesten, unterquert die Bundesstraße 23 und biegt dann nach Nordwesten ab. Etwa 250 Meter hinter der Bundesstraße mündet er am Westrand des Weidmooses auf einer Höhe von  von rechts in den Manndlbach.

Weblinks
 Karte des Lichtenstättgrabens auf:  mit Hintergrundlayer Topographische Karte

Einzelnachweise

Fließgewässer im Landkreis Garmisch-Partenkirchen
Geographie (Ettal)
Gewässer in den Alpen
Ammergauer Alpen

redpajama_set_name: RedPajamaWikipedia


Who wants to come play with buckets and buckets of cut flowers? We are so excited to invite you to part 2 of our Cut Flower Growing Workshop Series. Come join Nina Foster and friends for a flower filled 4 hour workshop. We will go over harvesting and post harvest care of your cut flowers. Then, we will spend the bulk of our time making magical arrangements that will take your breath away and that you can take home with you. Nina will demonstrate pro techniques for making a loose, flowing and wild arrangement, and will support each participant in making their own lush bouquet. Come and meet like-minded flower friends! This workshop will sell out quickly, and space is limited. We cannot wait to see you! $120 workshop fee includes all materials, flowers, and a light lunch. Please register here by purchasing your workshop slot.
Newer PostWhat is to love about the 2017 garden season ?


text:
function wait(ms) {
  return new Promise(r => setTimeout(r, ms)).then(() => "Yay");
}

// do some work in background.
setInterval(() => 42, 200);

async function test() {
  debugger;
  const hello = "world";
  const response = await fetch('index.html');
  const tmp = await wait(1000);
  console.log(tmp);
  return hello;
}

async function runTest() {
  let result = await test();
  console.log(result);
}

redpajama_set_name: RedPajamaGithub