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

Quantum mechanics (QM) and general relativity (GR) are the two cornerstones of modern physics. Yet, merging them together in a quantum theory of gravity (QG) is still elusive despite nearly century long efforts of vast numbers of physicists and mathematicians. While the majority of the attempts were focused on trying to formulate the full theory of a quantised gravity, such as the string theory, loop quantum gravity, non-commutative geometry, and causal set theory, to name a few, a number of recent studies embraced a rather more modest approach by exploring possible consequences of basic features and principles of QM and GR, and their status, in a tentative theory of QG. Acknowledging that the superposition principle, as a defining characteristic of any quantum theory, must be featured in QG as well, led to a number of papers studying gravity-matter entanglement~\cite{kay:98,oni:wan:16,bru:16,bos:etal:17,mar:ved:17,mar:ved:18,pau:voj:18}, genuine indefinite causal orders~\cite{ore:cos:bru:12,ara:bra:cos:fei:gia:bru:15,vil:17,ore:19,pau:voj:20,vil:col:22,vil:ren:22,orm:van:bar:22} and quantum reference frames~\cite{van:hoh:gia:cas:20,kru:hoh:mul:21,ahm:etal:22,ham:kab:cas:bru:21}, to name a few major research directions. Exploring spatial superpositions of masses, and in general of gravitational fields, led to the analysis of the status of various versions of the equivalence principle, and their exact formulations in the context of QG. In particular, in a recent work~\cite{pip:pau:voj:19} it was shown that the weak equivalence principle (WEP) should generically be violated in the presence of a specific type of superpositions of gravitational fields, a result seemingly at odds with some previous studies~\cite{gia:bru:20,ham:kab:cas:bru:21,gia:bru:22,mar:ved:20,mar:ved:22} (for details, see the discussion from Section~\ref{sec:conclusions}).

Modern formulation of WEP is given in terms of a {\em test particle} and its {\em trajectory}: it is a {\em theorem} within the mathematical formulation of GR stating that the trajectory of a test particle satisfies the so-called geodesic equation~\cite{EinsteinInfeldHoffmann,Mathisson,Papapetrou,Tulczyjew,Taub,Dixon1,Dixon2,Dixon3,Dixon4,Dixon5,YasskinStoeger,ShirafujiNomuraHayashiOne,ShirafujiNomuraHayashiTwo,VasilicVojinovicJHEP,VasilicVojinovicDirakovaCestica,VasilicVojinovicJedan,VasilicVojinovicDva,VasilicVojinovicTri,VasilicVojinovicCetiri,VasilicVojinovicPet}, while it is {\em violated} within the context of QG, as shown in~\cite{pip:pau:voj:19}. In this paper, we present a brief overview of WEP in GR and a critical analysis of the notions of particle and trajectory in both classical and quantum mechanics, as well as in the corresponding field theories, with the aim of generalising and studying WEP in the context of QG.

The paper is organised as follows. In Section~\ref{sec:wep}, we give a brief historical overview of various flavours of the equivalence principle, with a focus on WEP. In Section~\ref{sec:trajectory_in_mechanics} we analyse the notion of a trajectory in classical and quantum mechanics, while in Section~\ref{sec:particle_in_gr} we discuss the notion of a particle in field theory and QG. Finally, in Conclusions, we briefly review and discuss our results, and present possible future lines of research.

\section{Equivalence principle in general relativity}
\label{sec:wep}

The equivalence principle (EP) is one of the most fundamental principles in modern physics. It is one of the two cornerstone building blocks for GR, the other being the principle of general relativity. While its importance is well understood in the context of gravity, it is often underappreciated in the context of other fundamental interactions. In addition, there have been numerous studies and everlasting debates whether EP holds also in quantum physics, if it should be generalised to include quantum phenomena or not, etc. Finally, EP has been historically formulated in a vast number of different ways, which are often not mutually equivalent, leading to a lot of confusion about the actual statement of the principle and its physical content \cite{AcciolyPaszko,Longhi,Chowdhury,Rosi}. Given the importance of EP, and the amount of confusion around it, it is important to try and help clarify these issues.

The equivalence principle is best introduced by stating its purpose --- in its traditional sense, the purpose of EP is to {\em prescribe the interaction between gravity and all other fields in nature, collectively called matter}. This is important to state explicitly, since EP is often mistakenly portrayed as a property of gravity alone, without any reference to matter. In a more general, less traditional, and often not appreciated sense, the purpose of EP is to prescribe the interaction between {\em any gauge field} and all other fields in nature, which we will reflect on briefly in the case of electrodynamics below.

Given such a purpose, let us for the moment concentrate on the gravitational version of EP, and provide its modern formulation, as it is known and understood today. The statement of the equivalence principle is the following:

\medskip

\begin{quote}
\medskip
  {\em The equations of motion for matter coupled to gravity remain locally identical to the equations of motion for matter in the absence of gravity.}
\medskip
\end{quote}
This kind of statement requires some unpacking and comments.
\begin{itemize}
\item When comparing the equations of motion in the presence and in the absence of gravity, the claim that they remain identical may naively suggest that gravity does not influence the motion of matter in any way whatsoever. But on closer inspection, the statement is that the two sets of equations remain {\em locally} identical, emphasizing that the notion of locality is a crucial feature of the EP. While equations of motion are already local in nature (since they are usually expressed as partial differential equations of finite order), the actual interaction between matter and gravity enters only when {\em integrating} those equations to find a solution. In other words, the statement of EP is that interaction between matter and gravity appears in a specifically nonlocal sense.

\item In order to compare the equations of motion for matter in the presence of gravity to those in its absence, the equations themselves need to be put in a suitable form (typically expressed in general curvilinear coordinates, as tensor equations). The statement of EP relies on a theorem that this can always be done, first noted by Erich Kretschmann~\cite{kretschmann}.

\item Despite being dominantly a statement about the interaction between matter and gravity, EP also implicitly suggests that the best way to describe the gravitational field is as a property of geometry of spacetime, such as its metric \cite{mis:tho:whe:73}. In that setup, EP can be reformulated as a statement of {\em minimal coupling} between gravity and matter, stating that equations of motion for matter may depend on the spacetime metric and its first derivatives, but not on its (antisymmetrised) second derivatives, i.e., the {\em spacetime curvature does not explicitly appear in the equations of motion for matter}.

\item The generalisation of EP to other gauge fields is completely straightforward, by replacing the role of gravity with some other gauge field, and suitably redefining what matter is. For example, in electrodynamics, the EP can be formulated as follows:
\vspace*{-0.3cm}
\begin{quote}
\medskip
  {\em The equations of motion for matter coupled to the electromagnetic field remain locally identical to the equations of motion for matter in the absence of the electromagnetic field.}
\medskip
\end{quote}
This statement can also be suitably reformulated as the minimal coupling between the EM field and matter, i.e., coupling matter to the electromagnetic potential $A_\mu$ but not to the corresponding field strength $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$, which is in fact the standard way EM field is coupled to matter. Even more generally, the gauge field sector of the whole Standard Model of elementary particles (SM) is built using the minimal coupling prescription, meaning that the suitably generalised version of the EP actually prescribes the interaction between matter and all fundamental interactions in nature, namely strong, weak, electromagnetic and gravitational. In this sense, EP is a cornerstone principle for the whole fundamental physics, as we understand it today.
\end{itemize}
Of course, much more can be said about the statement of EP, its consequences, and various other details. However, in this work our attention will focus on the so-called {\em weak equivalence principle} (WEP), which is a reformulation of EP applied to matter which consists of mechanical particles. To that end, it is important to understand various flavors and reformulations of EP that have appeared through history.

As any deep concept in physics, EP has been expressed historically through painstaking cycle of formulating it in a precise way, understanding the formulation, understanding the drawbacks of that formulation, coming up with a better formulation, and repeating. In this sense, EP, as quoted above, is a modern product of long and meticulous refinement over several generations of scientists. Needless to say, each step in that process made its way into contemporary physics textbooks, leading to a plethora of different formulations of EP that have accummulated in the literature over the years. This can bring about a lot of confusion about what EP actually states \cite{AcciolyPaszko,Longhi,Chowdhury,Rosi}, since various formulations from old and new literature may often be not merely phrased differently, but in fact substantively inequivalent. To that end, let us comment on several most common historical statements of EP (for a more detailed historical overview and classification, see \cite{oco:cal:11,cas:lib:son:15}), and their relationship with the modern version:
\begin{itemize}
\item {\em Equality of gravitational and inertial mass.} This is one of the oldest variants of EP, going back to Newton's law of universal gravitation. The statement claims that the ``gravitational charge'' of a body is the same as body's resistance to acceleration, in the sense that the mass appearing on the left-hand side of Newton's second law of motion exactly cancels the mass appearing in the Newton's gravitational force law on the right-hand side. This allows one to relate it to the modern version of EP, in the sense that a suitably accelerated observer could rewrite the Newton's law of motion as the equation for a free particle, exploiting the cancellation of the ``intertial force'' and the gravitational force on the right-hand side of the equation. Unfortunately, this version of EP is intrinsically nonrelativistic, and applicable only in the context of Newtonian gravity, since already in GR the source of gravity becomes the full stress-energy tensor of matter fields, rather than just the total mass. Finally, this principle obviously fails when applied to photons, as demonstrated by the gravitational bending of light.

\item {\em Universality of free fall.} Going back all the way to Galileo, this statement claims that the interaction between matter and gravity does not depend on any intrinsic property of matter itself, such as its mass, angular momentum, chemical composition, temperature, or any other property, leading to the idea that gravity couples universally (i.e., in the same way) to all matter. Formulated from experimental observations by Galileo, its validity is related to the quality of experiments used to verify it. As we shall see below, in a precise enough setting, one can experimentally observe direct coupling between the angular momentum of a body and spacetime curvature \cite{EinsteinInfeldHoffmann,Mathisson,Papapetrou,Tulczyjew,Taub,Dixon1,Dixon2,Dixon3,Dixon4,Dixon5,YasskinStoeger,ShirafujiNomuraHayashiOne,ShirafujiNomuraHayashiTwo,VasilicVojinovicJHEP,VasilicVojinovicDirakovaCestica,VasilicVojinovicJedan,VasilicVojinovicDva,VasilicVojinovicTri,VasilicVojinovicCetiri,VasilicVojinovicPet}, invalidating the statement.

\item {\em Local equality between gravity and inertia.} Often called Einstein's equivalence principle, the statement claims that a local and suitably isolated observer cannot distinguish between accelerating and being at rest in a uniform gravitational field. While this statement is much closer in spirit to the modern formulation of EP, it obscures the crucial aspect of the principle --- coupling of matter to gravity. Namely, in this formulation it is merely implicit that the only way an observer can {\em attempt to distinguish} gravity from inertia is by making local experiments using some form of {\em matter}, i.e., studying the equations of motion of matter in the two situations and trying to distinguish them by observing whether or not matter behaves differently. Moreover, the statement is often discussed in the context of mechanics, arguing that any given particle does not distinguish between gravity and inertia. This has two main pitfalls --- first, the reliance on particles is very misleading (as we will discuss below in much more detail), and second, it implicitly suggests that gravity and inertia are the same phenomenon, which is completely false. Namely, inertia can be understood as a specific form of gravity, but a general gravitational field cannot be simulated by inertia, since inertia cannot account for tidal effects of inhomogeneous configurations of gravity.

\item {\em Weak equivalence principle.} Stating that the equations of motion of particles do not depend on spacetime curvature, or equivalently, that the motion of a free particle is always a geodesic trajectory in spacetime, WEP is in fact an application of modern EP to mechanical point-like particles (i.e., test particles). One can argue that, as far as the notion of a point-like particle is a well defined concept in physics, WEP is a good principle. Nevertheless, as we will discuss below in detail, the notion of a point-like particle is an idealisation that does not actually have any counterpart in reality, in either classical or quantum physics. Regarding a realistic particle (with nonzero size), WEP {\em never holds}, due to the explicit effect of gravitational tidal forces across the particle's size. In this sense, WEP can be considered at best an {\em approximate} principle, which can be assumed to hold only in situations where particle size can be approximated to zero.

\item {\em Strong equivalence principle (SEP).} This version of the principle states that the equations of motion of all fundamental fields in nature do not depend on spacetime curvature. To the best of our knowledge so far, fields are indeed the most fundamental building blocks in modern physics (such as SM), while the strength of the gravitational field is indeed described by spacetime curvature (as in GR). In this sense, the statement of SEP is actually an instance of EP applied to field theory, and as such equivalent to the modern statement of EP. So far, all our knowledge of natural phenomena is consistent with the validity of SEP.
\end{itemize}
As can be seen from the above review, various formulations of EP are both mutually inequivalent and have different domains of applicability. In the remainder of the paper, we focus on the study of WEP, since recently it gained a lot of attention in the literature~\cite{gia:bru:20,ham:kab:cas:bru:21,gia:bru:22,mar:ved:20,mar:ved:22}, primarily in the context of its generalisation to a ``quantum WEP'', and in the context of a related question of particle motion in a quantum superposition of different gravitational configurations, the latter being a scenario that naturally arises in QG. Since WEP is stated in terms of a test particle and its trajectory, in order to try and generalise it to the scope of QG one should first analyse these two notions in classical and quantum mechanics and field theory in more detail.

\section{The notion of trajectory in classical and quantum mechanics}
\label{sec:trajectory_in_mechanics}

A trajectory of a physical system in three-dimensional space is a set of points that form a line, usually parametrised by time. More formally, a trajectory is a set $\{(x(t),y(t),z(t)) \in \mathbb{R}^3 | t \in [t_i, t_f] \subset \mathbb R \land t_i < t_f \}$, given by three smooth functions $x, y, z : \mathbb R \mapsto \mathbb R$. Depending on the nature of the system, the choice of points that form its trajectory may vary.

In classical mechanics, one often considers an ideal ``point-like particle'' localised in one spatial point $(x(t),y(t),z(t))$ at each moment of time $t$, in which case the choice of the points forming the system's trajectory is obvious. In case of systems continuously spread over certain volume (``rigid bodies'', or ``objects''), or composite systems consisting of several point-like particles or bodies, it is natural to consider their centres of mass as points that form the trajectory. While this definition is natural, widely accepted, and formally applicable to any classical mechanical system, there are cases in which the very notion of a trajectory looses its intuitive, as well as useful, meaning.

Consider for example an electrical dipole, i.e., a system of two point-like particles with equal masses and opposite electrical charges, separated by the distance $\ell(t)$. As long as this distance stays ``small'' and does not vary significantly with time, the notion of a trajectory of a dipole, defined as the set of centres of mass of the two particles, does meet our intuition, and can be useful. Informally, if the trajectories of each of the two particles are ``close'' to each other, they can be approximated, and consequently represented, by the trajectory of the system's centre of mass. But if the separate trajectories of the two particles diverge, one going to the ``left'', and another to the ``right'', one could hardy talk of a trajectory of such a composite system, although the set of locations of its centres of mass is still well defined.

Moving to the realm of quantum mechanics, due to the superposition principle, even the ideal point-like particles do not have a well defined position, which is further quantified by the famous Heisenberg uncertainty relations. Thus, the trajectory of point-like particles (and any system that in a given regime can be approximated to be point-like) is defined as a set of expectation values of the position operator. Like in the case of composite classical systems, here as well the definition of a trajectory of a point-like particle is mathematically always well defined, yet for the very similar reason is applicable only to certain cases. Namely, in order to give a useful meaning to the above definition of trajectory, the system considered must be {\em well localised}. Consider for example the double-slit experiment, in which the point-like particle is highly delocalised, so that we say that {\em its trajectory is not well defined}, even though the set of the expectation values of the position operator is.

We see that, while in mechanics both the notions of a particle and its trajectory are rather straightforward and always well defined, the latter make sense only if our system is well localised in space.

\section{The notion of a particle in field theory}
\label{sec:particle_in_gr}

While in classical mechanics a point-like particle is always well localised, we have seen that in the quantum case one must introduce an additional constraint in order for it to be considered localised --- the particle should be represented by a wave-packet. The source for this requirement lies in the fact that quantum particles, although mechanical, are represented by a {\em wavefunction}. Thus, it is only to be expected that when moving to the realm of the field ontology, the notion of a particle becomes even more involved and technical.

\bigskip

In field theory, the fundamental concept is the {\em field}, rather than a particle. The notion of a particle is considered a derived concept, and in fact in QFT one can distinguish two vastly different phenomena that are called ``particles''.

The first notion of a particle is an elementary excitation of a free field. For example, the state
\begin{equation*}
\ket{\Psi} = \hat{a}^\dag(\vec{k}) \ket{0} \,,
\end{equation*}
is called a {\em single particle state} of the field, or a {\em plane-wave particle}. It has the following properties:
\begin{itemize}
\item It is an eigenstate of the {\em particle number operator} $ \hat{N} = \int d^3 \vec{k} \; \hat{a}^\dag(\vec{k}) \hat{a}(\vec{k})$, for the eigenvalue $1$.
\item It has a sharp value of the momentum $\vec{k}$, and corresponds to a completely delocalised plane wave configuration of the field.
\item It has no centre of mass, and no concept of ``position'' in space, since the ``position operator'' is not a well defined concept for the field.
\item States of this kind are said to describe {\em elementary particles}. For a real scalar field, an example would be the {\em Higgs particle}. For fields of other types (Dirac fields, vector fields, etc.) examples would be an {\em electron}, a {\em photon}, a {\em neutrino}, an asymptotically free {\em quark}, and so on. Essentially, all particles tabulated in the Standard Model of elementary particles are of this type.
\end{itemize}
The second notion of the particle in field theory is a bound state of fields, also called a {\em kink solution}. This requires an interacting theory, since interactions are necessary to form bound states. This kind of configuration of fields has the following properties:
\begin{itemize}
\item It is not an eigenstate of the particle number operator, and the expectation value of this operator is typically different from $1$.
\item It is usually well localised in space, and does not have a sharp value of momentum.
\item As long as the kink maintains a stable configuration (i.e., as long as it does not decay), one can in principle assign to it the concept of {\em size}, and as a consequence also the concepts of {\em centre of mass}, {\em position in space}, and {\em trajectory}. In this sense, a kink can play the role of a test particle.
\item States of this kind are said to describe {\em composite particles}. Given an interacting theory such as the Standard Model, under certain circumstances quarks and gluons form bound states called a {\em proton} and a {\em neutron}. Moreover, protons and neutrons further form bound states called {\em atomic nuclei}, which together with electrons and photons form {\em atoms}, {\em molecules}, and so on.
\end{itemize}
For a kink, the notions of centre of mass, position in space and size are described only as classical concepts, i.e., as expectation values of certain field operators, such as the stress-energy tensor. Moreover, given the nonzero size of the kink, its centre of mass and position are not uniquely defined, even classically, since in relativity different observers would assign different points as the centre of mass.

Given the two notions of particles in QFT, one can describe two different corresponding notions of WEP. In principle, one first needs to apply SEP in order to couple the matter fields to gravity, at the fundamental level. Assuming this is done, the motions of both the plane-wave particles and kink particles can be derived from the combined set of Einstein's equations and matter field equations, without any appeal to any notion of WEP. In this sense, once the trajectory of the particle in the background gravitational field has been determined from the field equations, one can verify {\em as a theorem} whether the particle satisfies WEP or not.

Specifically, in the case of a matter field coupled to general relativity such that it  locally resembles a plane wave, one can apply the WKB approximation to demonstrate that the wave $4$-vector $k^{\mu}(x)$, orthogonal to the wavefront at its every point $x\in \ensuremath{\mathbb{R}}^4$, will satisfy a geodesic equation,
\begin{equation} \label{eq:geodesic_plane_wave}
k^\mu(x) \nabla_\mu k^\lambda(x) = 0\,.
\end{equation}
However, given that the plane-wave particle is completely delocalised in space, the fact that the wave $4$-vector satisfies the geodesic equation could hardly be interpreted as ``the particle following a geodesic trajectory'', and thus obeying WEP. Indeed, identifying the vector field orthogonal to the wavefront to the notion of ``particle's trajectory'' is at best an abuse of terminology.

Next, in the case of the kink particle coupled to general relativity, one assumes the configuration of the background gravitational field is such that the particle maintains its structure and that its size can be completely neglected. One can then apply the procedure given in \cite{EinsteinInfeldHoffmann,Mathisson,Papapetrou,Tulczyjew,Taub,Dixon1,Dixon2,Dixon3,Dixon4,Dixon5,YasskinStoeger,ShirafujiNomuraHayashiOne,ShirafujiNomuraHayashiTwo,VasilicVojinovicJHEP,VasilicVojinovicDirakovaCestica,VasilicVojinovicJedan,VasilicVojinovicDva,VasilicVojinovicTri,VasilicVojinovicCetiri,VasilicVojinovicPet,pip:pau:voj:19} to demonstrate that the $4$-vector $u^\mu(\tau)$, tangent to the kink's world line (i.e., its trajectory), will satisfy a geodesic equation ($\tau \in \ensuremath{\mathbb{R}} $ represents kink's proper time),
\begin{equation} \label{eq:geodesic_kink}
u^\mu(\tau) \nabla_\mu u^\lambda(\tau) = 0\,.
\end{equation}
Thus, one concludes that the kink obeys WEP as a {\em theorem} in field theory, without the necessity to actually postulate it.

Note the crucial difference between equations (\ref{eq:geodesic_plane_wave}) and (\ref{eq:geodesic_kink}) --- while the former features $4$-dimensional variable $x$, the latter is given in terms of only $1$-dimensional proper time $\tau$. This reflects the fact that the plane-wave particle is a highly delocalised object, with no well defined position and trajectory, while the kink is a highly localised object, with well defined position and trajectory. As a consequence, WEP can be formulated only for the kink, and not for the plane-wave particle.

In the case of the kink it is also important to emphasise that the zero size approximation of the kink is crucial. Namely, without this assumption, the particle will feel the tidal forces of gravity across its size, effectively coupling its angular momentum $J^{\mu\nu}(\tau)$ to the curvature of the background gravitational field \cite{EinsteinInfeldHoffmann,Mathisson,Papapetrou,Tulczyjew,Taub,Dixon1,Dixon2,Dixon3,Dixon4,Dixon5,YasskinStoeger,ShirafujiNomuraHayashiOne,ShirafujiNomuraHayashiTwo,VasilicVojinovicJHEP,VasilicVojinovicDirakovaCestica,VasilicVojinovicJedan,VasilicVojinovicDva,VasilicVojinovicTri,VasilicVojinovicCetiri,VasilicVojinovicPet}. This will give rise to an equation of motion for the kink of the form
$$
u^\mu(\tau) \nabla_\mu u^\lambda(\tau) = R^\mu{}_{\lambda\rho\sigma} u^\lambda(\tau) J^{\rho\sigma}(\tau) \,,
$$
which features explicit coupling to curvature (absent from (\ref{eq:geodesic_kink})) and thus fails to obey WEP. In this sense, for realistic kink solutions WEP is {\em always violated}, and can be considered to hold only as an approximation, when the size of the particle can be completely neglected compared to the radius of curvature of the background gravitational field.  If in addition the kink has negligible total energy, it can be used as a point-like test particle.

In the above discussion, while matter fields are described as quantum, using QFT, the background gravitational field is considered to be completely classical. It should therefore not be surprising that WEP may fail to hold if one allows the gravitatational field to be quantum, like matter fields, and one needs to revisit all steps of the above analysis from the perspective of QG. In fact, the case of the kink particle has been studied in precisely this scenario \cite{pip:pau:voj:19}, and it has been shown that if the background gravitational field is in a specific type of quantum superposition of different configurations, the kink will fail to obey WEP even in the zero size approximation. Simply put, the equation of motion for the kink will contain extra terms due to the interference effects between superposed configurations of gravity, giving rise to an effective force that pushes the kink off the geodesic trajectory. And of course, similar to the case of the classical gravity, the resulting conclusion is a {\em theorem}, which follows from the fundamental field equations of the theory.

For the case of the plane-wave particle travelling through the superposed background of two gravitational field configurations, the analysis has not been done so far (to the best of our knowledge). But in principle one can expect a similar interference term to appear in the WKB analysis, and give rise to a non-geodesic equation for the wave $4$-vector field $k^\mu(x)$ as well. In this sense, it is to be expected that generically even the wavefronts of such plane-wave particles would fail to obey WEP.

\section{Conclusions and discussion}
\label{sec:conclusions}

In this paper, we give an overview of the equivalence principle and its various flavors formulated historically, with a special emphasis on the weak equivalence principle. In order to generalise WEP to QG, we performed a critical analysis of the notions of particle and trajectory in various frameworks of physics, showing that the notion of point-like particle and its trajectory are not always well defined. This in turn suggests that WEP is not a good starting point for generalisation to QG, as we argue in more detail below.

As discussed in Section~\ref{sec:particle_in_gr}, in~\cite{pip:pau:voj:19} it was shown that if superpositions of states of gravity and matter are allowed, WEP can be violated. It is important to note that the cases considered in~\cite{pip:pau:voj:19} feature a specific type of superpositions of three groups of states: the first consists of a single so-called dominant state --- a classical state whose expectation values of the metric and the stress-energy tensors satisfy Einstein field equations; the second consists of states similar to the dominant one, with arbitrary coefficients; and the third consists of states quasi-orthogonal to the dominant one, but with negligible coefficients. Only then one can talk of a (well localised) trajectory of the test particle in the overall superposed state and consequently about the straightforward generalisation of the classical WEP to the realm of QG. Considering that for the dominant state, being classical, the trajectory of the test particle follows the corresponding geodesic, we see that in the superposed state its trajectory would deviate from the geodesic of the dominant state, thus violating WEP. Note that, as discussed in Section~\ref{sec:particle_in_gr}, this deviation, in addition to classically weighted trajectories of the individual branches, also features purely quantum (i.e., off-diagonal) interference terms.

On the other hand, in recent studies~\cite{gia:bru:20,ham:kab:cas:bru:21,gia:bru:22,mar:ved:20,mar:ved:22} it was proposed that WEP is {\em not violated} in QG, thus being at odds with our result. The authors of the mentioned studies considered superpositions of an arbitrary number of classical quasi-orthogonal states with arbitrary coefficients. Their argument goes as follows: since WEP is valid in each classical branch, it is valid in their superposition as well. If taken as a {\em definition} of what does it mean that a certain principle is satisfied in a superposition of different quantum states, then the above statement is manifestly true. But, as such, being a definition, it tells little about physics --- it merely rephrases one statement (``principle A is separately satisfied in all branches of a superposition'') with another, simpler (``principle A is satisfied in a superposition''). Moreover, it can be somewhat misleading, for the following three reasons:

\begin{itemize}
	\item[1.] When applied to the case studied in~\cite{pip:pau:voj:19} and in this paper, the reasoning from~\cite{gia:bru:20,ham:kab:cas:bru:21,gia:bru:22,mar:ved:20,mar:ved:22} would lead to the conclusion that even in our case WEP is not violated. Which, as long as understood in terms of the above mentioned definition, is a valid point of view. But it is misleading, as we have seen that in our case one can talk of particle's trajectory and its deflection from the dominant geodesic, precisely in the same way one talks about WEP in GR, which is a way that contains non-trivial information about physics. In other words, in our approach the violation of WEP is a theorem, rather than a definition, and its consequences can in principle be experimetally tested.

    \item[2.] Second, the authors of~\cite{gia:bru:20,ham:kab:cas:bru:21,gia:bru:22,mar:ved:20,mar:ved:22} often focus on approximately balanced superpositions of macroscopically distinguishable quasi-orthogonal configurations of gravity and matter. Such branches correspond to different ``worlds'' of the Many Worlds Interpretation, which manifestly cannot interfere in the future. Even within the standard quantum theory, it becomes increasingly difficult to discriminate such coherent superpositions from their classical mixtures, and it was suggested to be for all practical purposes impossible to do so~\cite{per:ros:64}. In light of this, one could argue that in such scenarios the mentioned statement regarding WEP says little about quantum theory.

    \item[3.] Moreover, the above definition is not in the spirit of quantum mechanics. Recall that, as discussed above and in \cite{pip:pau:voj:19}, the modern formulation of WEP is given precisely in terms of a particular property of a system --- its position, and consequently trajectory. To say that whenever the weak equivalence principle is satisfied in a superposed state if it is so in each of the branches, would then mean that the position and the trajectory of a test particle would be well defined not only in each separate branch, but in a superposed state as well. When applied to the case of the double-slit experiment, this would mean that the position of a particle and its trajectory are well defined even in superposition. Such a statement is very difficult to find in literature, to say the least, while its exact opposite can be found in almost any textbook on QM.

\end{itemize}

Finally, we would like to argue that the very notion of WEP is ill-defined in such ``extreme'' scenarios of superposed quasi-orthogonal classical states, and should not be even discussed. After all, we have seen that what we call a principle is nothing but a theorem in GR about the approximate trajectory of a point-like test particle, both of which are not always possible to define. Consequently, studying generalisations of WEP might not be very useful for inferring the features of QG. In our opinion, it would be better to instead study other principles and their possible generalisations to QG, such as SEP (see Section 4.2 in \cite{pip:pau:voj:19}), background independence, quantum nonlocality, and so on.

\bigskip

\centerline{\bf Acknowledgments}

\bigskip

The authors wish to thank \v{C}aslav Brukner, Flaminia Giacomini, Chiara Marletto and Vlatko Vedral for useful discussions.

NP's work was partially supported by SQIG -- Security and Quantum Information Group of Instituto de Telecomunica\c{c}\~oes, by Programme (COMPETE 2020) of the Portugal 2020 framework [Project Q.DOT with Nr.\ 039728 (POCI-01-0247-FEDER-039728)] and the Funda\c{c}\~ao para a Ci\^{e}ncia e a Tecnologia (FCT) through national funds, by FEDER, COMPETE 2020, and by Regional Operational Program of Lisbon, under UIDB/50008/2020 (actions QuRUNNER, QUESTS), Projects QuantumMining POCI-01-0145-FEDER-031826, PRE\-DICT PTDC/CCI-CIF/29877/2017, CERN/FIS-PAR/0023/2019, QuantumPrime PTDC/EEI-TEL/8017/2020, as well as the FCT Est\'{i}mulo ao Emprego Cient\'{i}fico grant no. CEECIND/04594/2017/CP1393/CT000.

MV was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, and by the Science Fund of the Republic of Serbia, grant 7745968, ``Quantum Gravity from Higher Gauge Theory 2021'' --- QGHG-2021. The contents of this publication are the sole responsibility of the authors and can in no way be taken to reflect the views of the Science Fund of the Republic of Serbia.

redpajama_set_name: RedPajamaArXiv


text:
Hepatic fibrosis is usually a scarring process that may progress to
immune Uncategorized AZD8055, Rabbit polyclonal to ABCG1.
Hepatic fibrosis is usually a scarring process that may progress to hepatic cirrhosis and even hepatic carcinoma if left untreated. p53 transforming growth factor (TGF-?1) and ?-easy muscle mass actin (?-SMA) which is a marker of activated HSCs was detected by immunohistochemical assays and RT-qPCR. (19). Artesunate has also been found to inhibit HSC proliferation in a time- and dose-dependent manner by increasing p53 expression (20). Evidence from and studies has exhibited that recombinant human adenovirus-p53 (Ad-p53) as a novel drug for gene therapy has therapeutic effects on various types of tumors including breast prostate head and neck and ovarian malignancy (21). However there have been AZD8055 no studies to date to the best of our knowledge examining the mechanism responsible for the effects of Ad-p53 in hepatic fibrosis. To further examine the effect of Ad-p53 around the development of hepatic fibrosis a rat model of hepatic fibrosis was established and HSC-T6 cells AZD8055 were cultured under different conditions. We found that Ad-p53 promotes apoptosis and inhibits HSC proliferation in a time- and dose-dependent manner by modulating the expression of p53 transforming growth factor (TGF)-?1 and ?-SMA. Materials and methods Reagents Ad-p53 (1×1012 computer virus particles/ml) were obtained from Shenzhen SiBiono GeneTech Co. Ltd. (Shenzhen China). Cell culture media high glucose Dulbecco's altered Eagle's medium (DMEM) and supplements were purchased from HyClone (Burlington ON Canada). Carbon tetrachloride (CCl4) was purchased from Xi'an Helin Biological Engineering Co. Ltd. (Xi'an China). 3 3 (DAB) mix was purchased from Beyotime Institute of Biotechnology (Beijing China). TRIzol reagent was purchased from Life Technologies (Gaithersburg MD USA). The primary antibodies anti-p53 (sc-13580) TGF-?1 (sc-66904) and ?-SMA (sc-324317) were obtained from Santa Cruz Biotechnology Inc. (Santa Cruz CA USA) and horseradish peroxidase (HRP)-conjugated secondary antibody was purchased from Amersham Pharmacia Biotech (Piscataway NJ USA). Cell culture HSC-T6 cells (Fuxiang Biological Co. Ltd Shanghai China) were managed in high glucose DMEM medium supplemented with 10% fetal bovine serum (FBS) and 1% penicillin-streptomycin in a 5% CO2 incubator at 37°C. The cells in the logarithmic growth phase were utilized for all experiments. Establishment of a model of hepatic fibrosis Forty Sprague Dawley (SD) rats (male weighing 240-260 g) were purchased from your Experimental Animal Center of the Medical School of Xi'an Jiaotong University or college (Xi'an China). The animals were housed and dealt with in accordance with the approved guidelines of the Animal Welfare Committee of Xi'an Jiaotong University or college. All rats AZD8055 were randomly divided into either the fibrosis model group or the AZD8055 normal control group. The fibrosis model group (5 out of 32 rats died during model preparation) was prepared by subcutaneously injecting the fibrosis-inducer 40 CCl4 diluted in salad oil (Jinlongyu Xi'an China) (an initial dose of 0.5 ml/100 Rabbit polyclonal to ABCG1. g body weight followed by 0.3 ml/100 g body weight thereafter) twice per week for 12 weeks. This group was further divided randomly into the following three subgroups: 8 rats in the experimental subgroup (Ad-p53 group) 8 rats in the control subgroup (normal saline group) and 11 rats in the model test subgroup (hepatic fibrosis model group). Eight rats in the normal control group received routine nursing. The animal research protocol was examined and approved by the Animal Care and Use Committee of Xi'an Jiaotong University or college (Xi'an China). Immunohistochemical assay The animals were euthanized and the liver tissues were dissected and fixed immediately with 4% paraformaldehyde for 2 days and subsequently embedded in paraffin and sectioned (4 value of control group ? value of experimental group)/value of control group] ×100%. Cell cycle analysis To analyze HSC-T6 cell cycle distribution DNA content AZD8055 was determined by circulation cytometry using propidium iodide (PI) staining. Briefly the cells were pre-treated with Ad-p53 (5×106 1 and 2×107 PFU/ml respectively) for 24 or 48 h harvested washed twice with phosphate-buffered saline (PBS) and fixed in 75% ethanol (in 0.01 mol/l PBS) at ?20°C overnight. Following centrifugation the cells were incubated in PBS made up of 100 (30) found that insulin-like.

redpajama_set_name: RedPajamaCommonCrawl


The Summer Institute on Innovation and Entrepreneurship is a week-long program on global competitiveness, emphasizing innovation and entrepreneurship. Hosted by the Ahlers Center for International Business at the University of San Diego (USD), graduate students around the world will participate in an international learning opportunity featuring USD professors and experts in their field.
This institute is a one-week program held on the University of San Diego campus from August 19-23, 2019. The tentative schedule is from 9:30 am to 5:00 pm with a lunch break each day. Activities will include classroom seminars, a company visit and receptions.
We will provide all instructional materials in a flash drive. Pre-reading material may be shared via Dropbox prior to the program start date. You will need a laptop or tablet with a USB drive to access the program materials on the flash drive. Internet access will be available at USD.
Participants will receive a certificate of completion on the last day of the program. USD does not provide academic credit for the Summer Institute.
The Summer Institute on Innovation and Entrepreneurship is designed for graduate students currently enrolled in an institution of higher education. We may ask for verification of enrollment (i.e., contact information of your graduate advisor, transcript) during the application process.
Participants will receive a certificate for completing this educational program.
Please note that participants will not be enrolled as students at USD nor will any academic credit from USD be granted for participating in the Summer Institute. USD does not issue student visas for this program.
The first step is to complete the online interest form. Upon review of your interest, we will invite you to register for the program by email. We have a rolling application and acceptance policy. You must complete the interest form to receive the registration link by email.
Should you be invited to participate in the program, full program payment of US$500* will be due by July 31, 2019. Registration link and payment information will be sent upon review of your interest form.
*Institutions that send five (5) or more students may qualify for a discount. Please send an email to ahlersevents@sandiego.edu for more details.
We have a rolling acceptance policy and will review interest forms until the registration deadline. The email with the registration and payment link will be sent as the interest forms are received and reviewed. In the email, you will also receive additional program and logistical information for the program.
Should you need to cancel, a 50% refund will be available if cancellation is received in writing by August 9, 2019. Cancellations made past this date will not qualify for a refund.


text:
The School of Engineering launches MyKompas
The School of Engineering at EPFL has launched a new communication tool for its faculty members that aims at providing them with access to different dashboards under a unique platform. The dashboards provide faculty with continuous and up-to-date statistical information about teaching, research, and infrastructure activities related to their laboratories, and allow a global view of the contributions of the School of Engineering in these domains.
MyKompas brings together institutional data, all visualized graphically. Each faculty member will now have access to their own data aggregated together into a single accessible interface. Faculty can also provide feedback to improve and correct the different figures and metrics.
© 2020 EPFL, Example of figures
The first version summarizes data originating from EPFL institutional databases with regards to teaching, human resources, and space. Other developments are ongoing, namely in relation to publications and impact. The tool aims to integrate continuous improvement, based on feedback received, in order to provide a valuable service to faculty members and the School.
The main goal of the interface is to provide easy, transparent and pertinent access to institutional data, while encouraging feedback to improve the quality of the data and processes.


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Monicca Thulisile Bhuda, a master's student in Indigenous Knowledge Systems (IKS), at the North-West University (NWU), is using ethno-mathematics as part of her research to trace symmetrical geometry in Ndebele mural art and beadwork.
Prof David Mello, an associate professor in the subject group Public Administration and Local Government at the North-West University's (NWU's) campus in Mahikeng, delivered his inaugural lecture titled "Fusing social and human capital for optimum performance of municipalities" on 21 June 2018.
The North-West University's (NWU's) Research Chair in Early Detection and Prevention of Cardiovascular Disease in Africa, held by Prof Alta Schutte of the Faculty of Health Sciences, has received a successful evaluation from the National Research Foundation.
Zuziwe Nkosi, a PhD student in crop science at the North-West University's (NWU's) campus in Mahikeng, uses a unique and creative technique called permaculture for food gardening to address food insecurity and nutrition among NWU students and surrounding communities.
Following the remarks made by Mr Theo Venter in an interview on RSG on 5 June 2018, the executive dean of the North-West University's (NWU's) Faculty of Economic and Management Sciences has engaged with Mr Venter on the matter.
Prof Masood Khalique from the North-West University's (NWU's) campus in Mahikeng participated in the 12th International Conference on Recent Advances in Applied Dynamical Systems which was held from 8 to 10 June at Chongqing Normal University in China.
A collaborative venture between the North-West University (NWU), the Midvaal Local Municipality and the New Hope Group (NHG) will not only benefit the local community, but also positively impact the future of South Africa's agricultural industry.
Over the past years the School of Agricultural Sciences at the North-West University's (NWU's) campus in Mahikeng has steadily proved its dedication to community development and education.
When it comes to mind sports – and more particularly the game of Morabaraba – the North-West University (NWU) is dominating the national rankings.
A bodybuilding fanatic and final-year student at the North-West University's (NWU's) campus in Mahikeng, David Tshoose is set to take the sport of bodybuilding by storm during this year's upcoming competitions.


As part of my recipe development process, I do a lot of research, whether related to ingredients or techniques. One of my favorite things to look into are cuisines that I am not closely familiar with. It's always exciting to think about the flavor combination treasures to be found in parts of the world, that I haven't been fortunate to know. Toffee banana is one of those – I made a note of it some time ago and recently got around to playing with the idea.
Toffee bananas are a classic Chinese dessert, which consists of deep-fried bananas, generously topped with toffee and garnished with sesame seeds. You'll see that the tart I came up with has very little to do with the original idea, but it's those kinds of things – recipe titles, ingredient pairings, or even color combinations, that often give me a push to come up with new dishes that I'm excited about.
The thought of cooking whole bananas has always intrigued me, but my fear was that the already sweet fruit would become unbearably saccharine. This tart turned everything around for me – the cooked bananas are sweet of course, but they also get a heavenly, creme brûlée-like caramel crust and become even more buttery and tender.
Under the layer of the transformed, cardamom-kissed bananas in this tart is a slather of date 'toffee.' You couldn't find a more perfect ingredient than dates for creating a healthy and whole-food based toffee. They have it all – the thick, rich texture, the sweetness, the caramel color.
The spelt crust is my go-to quick tart crust – it's unsweetened and with a touch of salt, making a perfect bed for the decadent filling. Together, everything combines into an impressive whole. I've made this tart a couple of times, and it's been hugely popular with my friends. I get requests to re-make it more than anything else I've been cooking lately.
And I want to say a few words about the elegant plum-tree spatula in the photos. It was sent to me by a reader, Violet, who has an online shop, where she sells wooden fruit tree wares, hand-carved by her father. I was incredibly touched to receive a few of their items in the mail all the way from Europe, pictured below. Each piece has character, warmth, beauty and utility. The pricing is very fair too, especially considering the unique quality of each ware. Head over to Whole Grain Homes – perhaps you'll find a beautiful piece for your own kitchen.
Place flour and salt into a food processor. Cut oil into small pieces and add to the flour. Pulse until mixture resembles sand. Add 4 tablespoons cold water and pulse to combine. Test the mixture by pressing it between your fingers - it should come together. If not, add more water, 1 tablespoon at a time, until it sticks between your fingers.
Oil a 9-inch tart pan thoroughly. Press the dough against the bottom of the pan into the even crust, starting in the middle and working up the sides. Prick with a fork several times and let chill in the refrigerator for 1 hour.
Cover the crust with a piece of parchment paper and place baking beans inside. Bake blind for 20 minutes. Let cool.
Spoon toffee inside the crust and even out with a spoon.
Pre-heat oven to 400° F (200° C).
In a spacious pan, warm oil over medium heat. Add bananas, cut side down, followed by cardamom. Cook for 2-3 minutes, until lightly golden, flip carefully and cook the other side for 1 more minute.
Arrange banana slices on top of the toffee inside the crust. Sprinkle with coconut sugar.
Bake for 30-40 minutes, until the crust is golden and bananas are caramelized. Let cool before slicing.
Your recipe is likely to increase my appetite more & more… its been delicious & fantastic, no doubt… keep writing more yummy dishes… Thanks for sharing & your kind attention to detail!!!
this looks delicious! Will have to try.
Omg, I've no words!! I've never tried cooked banana – but in a banana bread, but it doesn't count – and this tart seems just perfect, creamy and voluptuous *____* I really have to make it!
This tart is soooo inviting! Fabulous recipe!
And I'm so glad you found something at Whole Grain Homes for your mom, their products are beautiful.
It's a week of no granola/no pie/no cake, our very own kind of Survivor!
That date toffee is a game changer! This tart is gorgeous and sounds SO GOOD! I can't wait to try!
This looks so delicious! Difficult not to save this recipe for later! Thanks Anya!
I made it today and we almost finished it so lovely it was! Such an interesting combination between cinnamon and cardamon. I have to say that I didn't use amond butter(i didn't have it home) but a mix of peanut and tahini! Too good!
Hello Anya, Thanks for your research on this yummy delicious food. I love the addition of cardamom too. Dates are natures toffees aren't they what incredible fruit. The ingredients and process are so easy. Its a healthy dish also. All food lovers definitely love this.
MY GOODNESS! This looks absolutely incredible! Thanks for the recipe!
Hello, i noticed that your website is very slow, it took around 7 seconds to load this post.
now? If you speed up your page loading time you can rank higher and get more targeted traffic.
This really looks mouthwatering good!


text:
Effort To Grow Support For Same Sex Marriage Is Underway
Wyoming Public Radio | By Bob Beck
Published March 21, 2014 at 3:57 PM MDT
This week a new statewide public education campaign was launched called Wyoming unites for marriage. The idea is to get support for same sex couples. Earlier this year a lawsuit was filed to try and make same sex marriage legal in Wyoming. Jeran Artery of the group Wyoming Equality tells Bob Beck that the lawsuit seems like the quickest way to get equality.
Open Spaces same-sex marriagegay marriageJeran ArteryWyoming Equality
Bob Beck
Bob Beck retired from Wyoming Public Media after serving as News Director of Wyoming Public Radio for 34 years. During his time as News Director WPR has won over 100 national, regional and state news awards.
See stories by Bob Beck
Wyoming Senators are split on support for same-sex marriage protections
Wyoming senators undecided on same-sex marriage protections
Pandemic Exacerbates Disparities For LGBTQ Wyomingites Without Support Or Legal Protections
LCSD#1 Announces Homophobic And Racist Behavior Is Under Investigation


text:
Is Vaping Safe For Teens?
A vape or electronic cigarette is a better alternative to smoking. It is a battery-powered device that uses a special liquid to create an aerosol that contains nicotine and other chemicals. It is often advertised as a healthier alternative to smoking. However, vaping is still a risky habit and can lead to serious health problems. For those who have just about any questions regarding where by as well as how to use พอต, it is possible to contact us in our own site.
The nicotine found in vape pens or pods is harmful because it contains other chemicals. The chemicals could cause cancer or lung disease. They can also cause brain damage and other tissue damage. Many teens do not realize that they are using nicotine-based products and can become addicted to them.
E-cigarettes and vapes are commonly used by teenagers. They are convenient and can be purchased as an alternative. Many e-cigarettes, vapes, and other vapes may not be properly labeled and could contain harmful chemicals. Even though vaping devices are more popular among teens than ever, there has not been enough research to confirm their safety.
Teenagers are using vapes more frequently than ever. In fact, over 78% of high school seniors used vapes in their last year. Several studies show that e-cigarettes and vapes may be less harmful than cigarettes, but that doesn't mean that they are harmless. Teens are increasingly addicted to nicotine-based products and may even become depressed.
It is easy to disguise vaping devices so that they are accessible to children. They can look like USB flash drives, cell phones, and other gadgets. Many children are attracted to the appealing advertisements on vape devices.
The vaping industry has grown rapidly in recent years. Initially, find out here the earliest vape devices looked like cigarettes. The liquid was a glycerin-based liquid which produced vapor. The device appears like a small pod in newer models. These pods can be made from many metal alloys including nickel and chrome. They may also contain waxy concentrates, which are used to make the vapor.
There are many flavors available for vaping devices. However, some of these flavours can pose unknown health risks. The liquids might contain chemicals which can cause harm to the eyes, throats, lungs and skin. Vitamin E acetate may be added to some of these products to dilute the liquid.
The aerosol that is produced by vape devices contains chemicals that can irritate the eyes, throat, lungs, and other tissues. The aerosol can also cause fatigue and headaches. It can also cause breathing difficulties, such as EVALI or e-cigarette-associated pulmonary injury.
Teens should not use vaping devices. Research shows that teens who use vaping devices are twice as likely to smoke cigarettes or tobacco products than young adults. The brain's development could also be affected by vaping nicotine. It may also cause the user to develop other addictions. These risks should be made known to parents and counselors. You probably have any type of inquiries relating to where and the best ways to use pod, you could contact us at our site.


text:
Home/arithmetic progressions
deepaksoniGuru
Asked: July 1, 2021 In: CBSE
Q74. students decided to beautify the school by fixing flags on the straight passage of the school. They have 27 flags to be fixed at intervals of 2 m. The flags are stored at the position of middle most flag. Ruchi kept her books where the flags were stored. She carry only one flag at a time. How much distance did she cover in completing this job and returning to collect her books? What is the maximum distance she travelled carrying a flag?
One of the most important and exam oriented question from Chapter name- Arithmetic Progression Class 10th Chapter number- 9 Exercise ...Read more
arithmetic progressionsclass 10thrd sharma
Question 69. A sum of Rs 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs 20 less than its preceding prize, find the value of each prize.
arithmetic progressionsrd sharma class 10th
Question 63. A man saved Rs 16500 in ten years. In each year after the first year, he saved Rs 100 more than he did in the preceding year. How much did he save in the first year?
arithmetic progressionsclass 10thrd sharmac
Question 62. Ramkali would need Rs. 1800 for admission fee and books etc., for her daughter to start going to school from the next year. She saved Rs. 50 in the first month of this year and increased her monthly saving by Rs. 20. After a year, how much money will she save? Will she be able to fulfill her dream of sending her daughter to school?
This is the basic and conceptual question from Chapter name- Arithmetic Progression Chapter number- 9 Exercise 9.6 In this question we ...Read more
Question 61. In a school, students decided to plant trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be double of the class in which they are studying. If there are 1 to 12 classes in the school and each class has two sections, find how many trees were planted by the students.
arithmetic progressionsclass 10tjrd sharma
Question 60. Resham wanted to save at least Rs. 6500 for sending her daughter to school next year (after 12 months). She saved Rs. 450 in the first month and raised her saying by Rs.20 every next month. How much will she be able to save in next 12 months? Will she be able to send her daughter to the school next year?
This is the basic and conceptual question from Chapter name- Arithmetic Progression Chapter number- 9 Exercise 9.6 In this question its ...Read more
Question 57. If Sn denotes the sum of first n terms of an A.P., prove that S12 = 3(S8 – S4).
arithmetic progressionsrd sharma
Question 56. Let there be an A.P. with first term a and common difference d. If an denotes its nth term and Sn is the sum of first n terms, then find:(vii) k, if Sn = 3n2 + 5n and ak = 164.
This is an arithmetic progression based question from Chapter name- Arithmetic progressions In this question we have been given that there ...Read more
Question 56. Let there be an A.P. with first term a and common difference d. If an denotes its nth term and Sn is the sum of first n terms, then find:(ii) n and a, if an = 4, d = 2 and Sn = –14
This is an arithmetic progression based question from Chapter name- Arithmetic Progression Chapter number- 9 Exercise – 9.6 In this question ...Read more
arithmetic progressionsrd sharma class 10
Question 56. Let there be an A.P. with first term a and common difference d. If an denotes its nth term and Sn is the sum of first n terms, then find: (i) n and Sn, if a = 5, d = 3 and an = 50.


text:
U.S. Republicans join call to delay vote on Trump pick to head Latam bank
By Andrea Shalal Posted on September 5, 2020
Former US Secretaries of State Shultz and Albright testify at Senate Armed Services hearings in Washington
WASHINGTON (Reuters) – Three senior former U.S. officials, all Republicans, have joined a growing chorus calling for a delay in the mid-September vote on the next leader of Latin America's main development bank, part of growing opposition to the first-ever U.S. candidate.
Former U.S. Secretary of State George Shultz, former World Bank President Robert Zoellick and former U.S. Trade Representative Carla Hills urged a delay in a joint statement with Senator Patrick Leahy, the top Democrat on the powerful U.S. Senate Appropriations Committee.
Mauricio Claver-Carone, President Donald Trump's nominee to head the Inter-American Development Bank (IDB), would be the first person from outside the region to head the IDB in its 61-year history.
Claver-Carone, known for his hardline stance on Cuba and Venezuela, is backed by 17 of the IDB's 28 member countries, but several others – Argentina, Mexico, Costa Rica and Chile, as well as the European Union – are urging a delay in the vote.
It remains unclear if those countries, which control about 22% of the vote, will persuade enough members to block the needed 75% quorum to hold the vote, now slated for Sept. 12-13.
A group of former Latin American heads of state has also called for a delay, concerned that Trump could lose the Nov. 3 U.S. election, and his candidate could find it difficult to work with a Democratic administration.
"Electing Mr. Claver-Carone to a five-year term only weeks before the U.S. presidential election would trigger an immediate institutional crisis at the IDB and put in jeopardy U.S. support for the Bank, regardless of who wins the U.S. election," Leahy, Shultz, Hills and Zoellick wrote in a statement.
Leahy would assume control of the Senate appropriations committee if Democrats win control of the Senate, giving him the power to block U.S. contributions to the Bank.
Democratic presidential candidate Joe Biden, who is leading Trump in national polls, opposes Claver-Carone's candidacy.
Claver-Carone argues that a minority of countries should not be allowed to override the views of a majority of IDB members.
Top U.S. officials, including Secretary of State Mike Pompeo, say Claver-Carone is the right man for the job and the vote should not be delayed.
(Reporting by Andrea Shalal; Editing by Alistair Bell)
World Bank President Robert Zoellick attends a panel during a B20 meeting prior to the G20 summit in Los Cabos
Biden marks 50th anniversary of death of wife, daughter
Power outages could last days after shootings at North Carolina substations
Supreme Court weighs 'most important case' on democracy


text:
Why the Empire State Building has a 200-foot mast
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New York City's newly renovated Empire State Building opens to the public Saturday. Tony Dokoupil, who got an exclusive look, looks back at the iconic building's history.
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Each weekday morning, "CBS This Morning" co-hosts Gayle King, Anthony Mason and Tony Dokoupil deliver two hours of original reporting, breaking news and top-level newsmaker interviews in an engaging and informative format that challenges the norm in network morning news programs. The broadcast has earned a prestigious Peabody Award, a Polk Award, four News & Documentary Emmys, three Daytime Emmys and the 2017 Edward R. Murrow Award for Best Newscast. The broadcast was also honored with an Alfred I. duPont-Columbia Award as part of CBS News division-wide coverage of the shootings at Sandy Hook Elementary School in Newtown, Connecticut. Check local listings for "CBS This Morning" broadcast times.
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We're happy announce the release of Mautic 1.0.0-beta4, our strongest version yet. Your favorite new marketing automation software just got better, with tons of fixes, new features, and integrations we're sure you'll love.
As requested, you can now import your existing leads into Mautic via CSV, making it easier than ever to use Mautic for all your leads.
Thank you to all of our testers for helping us make these changes with your feedback and involvement.
If this is your first introduction to Mautic then we welcome you! You are about to experience the very best in marketing automation. You have two ways to take advantage of Mautic. You can download directly from the Mautic website. Use this method if you want to host your own marketing automation platform (that's right, no more SaaS for you!). The second option is to register for an account on Mautic.com where you can be up and running in less than 3 minutes. This account is free and never expires!
Beta4 is our first release that supports updates through Mautic itself.
You should then see a new notification with the update. Click Update Now, then follow the prompts.
If you have any problems with the process, post your issues on the community forum or under 'Issues' on our GitHub project page and we will do our best to get them quickly resolved. Please provide as much detail as possible. Note that Mautic logs its error to apps/logs/mautic_prod.php.
Thanks again to everyone involved in the beta process for allowing us to reach this point. An RC update will be coming soon!
If you like what Mautic is doing, share our project with your friends and colleagues. Help us achieve global equality in access to growth through open source software!


Welcome to Bronx Burger Bar on Vandkunsten in downtown Copenhagen. Bronx Burger Bar is located in this captivating area close to Strøget, which is the longest and one of the oldest shopping streets in Europe. Copenhagen city center is the ideal shopping destination for everyone! Bronx Burger Bar on Vandkunsten is only a few minutes walk from public transportation, so feel free to go by any time.
At Bronx Burger Bar on Vandkunsten you will get the best professional service. In addition, there are pay lots throughout Copenhagen city centre, which makes it easier and more convenient for you to visit Bronx Burger Bar on Vandkunsten. You are more than welcome to contact or come by Bronx Burger Bar in downtown Copenhagen.


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(1) Any person who willfully or knowingly violates paragraph (1), (3), (5), or (6) of subsection (a) of section 4909 of this title shall be punished by a fine of not more than $25,000 per day of violation, or by imprisonment for not more than one year, or by both. If the conviction is for a violation committed after a first conviction of such person under this subsection, punishment shall be by a fine of not more than $50,000 per day of violation, or by imprisonment for not more than two years, or by both.
(2) Any person who violates paragraph (1), (3), (5), or (6) of subsection (a) of section 4909 of this title shall be subject to a civil penalty not to exceed $10,000 per day of such violation.
For the purpose of this section, each day of violation of any paragraph of section 4909(a) of this title shall constitute a separate violation of that section.
The district courts of the United States shall have jurisdiction of actions brought by and in the name of the United States to restrain any violations of section 4909(a) of this title.
(1) Whenever any person is in violation of section 4909(a) of this title, the Administrator may issue an order specifying such relief as he determines is necessary to protect the public health and welfare.
(2) Any order under this subsection shall be issued only after notice and opportunity for a hearing in accordance with section 554 of title 5.
The term "person," as used in this section, does not include a department, agency, or instrumentality of the United States.
1978—Subsec. (a). Pub. L. 95–609 redesignated existing provisions as par. (1) and added par. (2).


text:
<!DOCTYPE HTML>

    <meta http-equiv="X-UA-Compatible" content="IE=edge">
    <meta http-equiv='Content-Type' content='text/html;charset=UTF-8'/>
    <title>Color Swatcher</title>
    <link rel="stylesheet" href="css/color-swatcher.css" type="text/css">

redpajama_set_name: RedPajamaGithub


A man in a wheelchair-type electric scooter lit his jacket on fire outside the White House fence on Friday and was hospitalized with what appeared to be non-life threatening injuries, the U.S. Secret Service said.
Secret Service officers responded to the incident on the north side of the executive mansion, extinguished the fire and provided first aid, the agency said on Twitter.
The Secret Service closed Pennsylvania Avenue in front of the White House to pedestrian traffic, and a nearby block of 17th Street was closed as well. At least portions of the White House complex were locked down.
News video showed emergency medical personnel arriving to tend to a person who police had surrounded on the ground. A CNN reporter on Twitter cited a Secret Service spokeswoman as saying there was no threat to U.S. President Donald Trump.
The incident occurred shortly after Trump spoke at the White House about the nation's 5G network.


Our favorite UK duo, Disclosure, bring their live show to the legendary Madison Square Garden on October 24th! Their recently released sophomore record 'Caracal' is packed with anthemic, UK garage and house influenced pop jams that we can't wait to experience live. After some amazing guests at their LA show, we're excited to see who they might bring in NYC. Joining them is the undeniable Dirtybird head-honcho Claude VonStroke and Canadian house-tinged pop producer Pomo.


New technologies and big data are causing disruptions in many industries. But how will these evolutions impact our lives and the cities of tomorrow? Two experts who know what they're talking about, are Bart Preneel (Cryptographer and professor at KU Leuven) and Mieke De Ketelaere (Director Customer Intelligence at SAS).
The technological innovations of the past few years clearly show that digitization is advancing rapidly. Mieke De Ketelaere predicts that 5 new technologies will be key in the coming years.
A treasure trove of data: what's the added value?
"Digitization indeed entails a treasure trove of data", believes Mieke. "And that big data is the fuel of almost all new tech that we can expect in the near future. But all depends on what is being done with that big data."
"Big data can only offer real added value if specific conditions are met", says Mieke. "The most important one probably being the transparency towards citizens and customers. Customers should have the right to decide what happens with their data. And we still have to find a good balance in the way man and machine work together to achieve optimal results. The approach as defined by Human Computation initiatives is certainly a right approach for this", says Mieke.
"Our research group at imec carefully evaluates the tension between public data and privacy protection", Bart explains. "We're also trying to develop solutions that reduce that tension."
"In that sense, we can state that a privacy debate is a necessity," says Mieke. "I'm thinking of a demystifying debate about what is or is not possible. People who are not active in our ICT world clearly need to be more aware. Apart from that, I also share the opinion that it's up to each individual to decide what can be shared. That freedom is really important. Compare it with building a house: some want a large fence to protect their privacy, others don't care. Privacy differences exist in the offline world too.
"But it's important to understand that the data you share as an individual can also be used to profile others," Bart says. "So, privacy – just like pollution – is also a collective problem." Enough food for thought!
10 free tickets to win!
Triggered to learn more about the impact on our society of new technologies and datastreams in the cities of tomorrow? The first 10 people who send a message to kim.indeherberg@sas.com will get a free ticket to join us in Leuven on 3 and 4 May during the and& summit & festival. Check out the full program and the exclusive SAS sofa sessions!


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Q: NIntegrate and Indefinite boundaries -- Why does it do this? Could someone help me understand why I'm seeing the following behavior?
I'm trying to evaluate the following function
NIntegrate[{Exp[- x]/(Exp[-x] + 1), 1/(Exp[-x] + 1)} PDF[LogNormalDistribution[0, sdM], Exp[x ]] , {x, -\[Infinity], \[Infinity]}]

Which I expect to converge to a value of 0.5 as sdM approaches 0.
For 'large' values of sdM, such as 0.35, Mathematica has no problem with the integration.  Specifically,
NIntegrate[{Exp[- x]/(Exp[-x] + 1), 1/(Exp[-x] + 1)} PDF[LogNormalDistribution[0, sdM], Exp[x ]] , {x, -\[Infinity], \[Infinity]}]/.{sdM -> 0.35}

returns,
0.563165

However, if I make sdM small, such as 0.001, I get a value of 0,
NIntegrate[{Exp[- x]/(Exp[-x] + 1), 1/(Exp[-x] + 1)} PDF[LogNormalDistribution[0, sdM], Exp[x ]] , {x, -\[Infinity], \[Infinity]}]/.{sdM -> 0.35}

returns,
0.

I'm not too surprised by this. As I understand, this results from the indefinite boundaries and the limited region of the density function that has a value greater than $MachineEpsilon.
However, if I split the integration into two parts to ensure that the function is evaluated at x = 0, I still get 0.  That is,
(NIntegrate[{Exp[- x]/(Exp[-x] + 1), 1/(Exp[-x] + 1)} PDF[LogNormalDistribution[0, sdM], Exp[x ]] , {x, -\[Infinity], 0}]+ NIntegrate[{Exp[- x]/(Exp[-x] + 1), 1/(Exp[-x] + 1)} PDF[LogNormalDistribution[0, sdM], Exp[x ]] , {x, 0, \[Infinity]}])/.{sdM -> 0.0001}

This is despite the fact that the density function is clearly non zero at x = 0.
Questions

*

*Why does it behave like this?

*Is there a better way to assign an evaluation point in NIntegrate besides splitting the evaluation at that point?

Much obliged.

A: Hello sir i think you have to work with WorkingPrecision
i1 = NIntegrate[{Exp[-x]/(Exp[-x] + 1), 1/(Exp[-x] + 1)} PDF[
     LogNormalDistribution[0, sdM], Exp[x]], {x, -\[Infinity], 0},
   WorkingPrecision -> 10] /. {sdM -> 0.0001}

i2 = NIntegrate[{Exp[-x]/(Exp[-x] + 1), 1/(Exp[-x] + 1)} PDF[
     LogNormalDistribution[0, sdM], Exp[x]], {x, 0, \[Infinity]},
   WorkingPrecision -> 10] /. {sdM -> 0.0001}

i1 + i2

A: Use both MinRecursion and WorkingPrecision
$Version

(* "12.3.0 for Mac OS X x86 (64-bit) (May 10, 2021)" *)

int[sdM_?NumericQ] := NIntegrate[
  {Exp[-x]/(Exp[-x] + 1), 1/(Exp[-x] + 1)} *
   PDF[LogNormalDistribution[0, sdM],
    Exp[x]], {x, -∞, ∞},
  MinRecursion -> 4,
  WorkingPrecision -> 15]

int /@ {0.35, 0.005, 0.001, 0.0005, 0.0001}

(* {{0.563164672134101, 0.500000000000000}, {0.500012500075248,
  0.499999999997122}, {0.500000500000259,
  0.500000000000134}, {0.500000125000008,
  0.500000000000000}, {0.500000005000000, 0.500000000000000}} *)

redpajama_set_name: RedPajamaStackExchange


If you become sick or injured while traveling, modern airline travel makes it quick and easy to be whisked from virtually anywhere in the world to the facilities of the best-qualified specialist.
But that same air travel makes it just as quick and easy, not only for you to be set down in a strange climate and environment filled with microorganisms against which your body has few if any defenses, but for those microorganisms to find their way to where you wait, unsuspectingly, at home.
In addition, there are problems caused by such factors as jet lag, altitude changes, and most definitely the special characteristics of the locale you're visiting. For example, some 50 million people suffer from allergic reactions, including 35 million people susceptible to seasonal allergies, which can flare up the minute you get off the plane in a high-pollen-count area.
But depending on where you travel, there are dangers far worse than allergies. Malaria is rampant in the Ethiopian lowlands and also in Kenya, but there's none in Nairobi or in the Ethiopian highlands.
who stay for 14 days or longer will develop some form of "traveler's stomach."
There are presently outbreaks of cholera in South America, of encephalitis in Japan, of necrotizing fasciitis in England and the U.S., and of diptheria in Russia and the Ukraine. St. Petersburg (formerly Leningrad) remains the world capital of giardiasis, a parasitic infection of the small intestine with very uncomfortable symptoms.
Your personal physician may be one of your best sources of information. You can call with specific questions before or during your trip.


text:
Jana Verdura i Brugarola (Arenys de Munt, Maresme, 14 de novembre de 1989) és una jugadora d'hoquei sobre patins i investigadora catalana.

Formada al Centre d'Esports Arenys de Munt, va debutar amb el primer equip l'any 2003. Va aconseguir un Campionat d'Espanya el 2004 i un subcampionat de la Copa d'Europa el 2006 i subcampionat de la Copa de la Reina el 2007. La temporada 2012-13 va jugar amb l'Sfèric de Terrassa, tornant l'any següent al club arenyenc. La temporada 2015-16 va fitxar pel Citylift Girona Hoquei Club, amb el qual va guanyar una Copa Generalitat el 2017. Internacional amb la selecció espanyola en categories inferiors, va proclamar-se campiona d'Europa sub-19 el 2007. Amb la selecció catalana d'hoquei sobre patins va aconseguir una medalla de bronze a la Copa Amèrica de 2006 i una Blanes Golden Cup el 2009.

En la seva vessant professional, és doctorand de ciències ambientals de la Universitat de Girona i del Centre d'Estudis Avançats de Blanes (CEAB), que investiga l'impacte del canvi climàtic en els boscos submarins així com de la seva conservació i restauració. Forma part del grup d'investigació MedRecover.

Palmarès
Clubs

 1 Copa Generalitat d'hoquei sobre patins femenina: 2016-17
 1 Campionat d'Espanya d'hoquei sobre patins femení: 2003-04

 Selecció catalana
  1 medalla de bronze a la Copa Amèrica d'hoquei patins femenina 2006
 1 Golden Cup: 2009

Referències

Enllaços externs
 Fitxa de Jana Verdura Brugarola - RinkHockey.net
 Fitxa de Jana Verdura Brugarola - MedRecover

Alumnes de la Universitat de Girona
Arenyencs d'Arenys de Munt
Jugadores d'hoquei sobre patins del Maresme
Jugadores d'hoquei sobre patins del CE Arenys de Munt

redpajama_set_name: RedPajamaWikipedia


Total Population 1881 to 1961: Population fell from 41 in 1881 to 21 in 1961.
F. Youngs, Local Administrative Units: Northern England (London: Royal Historical Society, 1991), p. 517.
YORKSHIRE Ancient County F. Youngs, Local Administrative Units: Northern England (London: Royal Historical Society, 1991), p. 517.
SKIPTON PLU/RegD Poor Law Union/Reg. District soon after 1837 1930 F. Youngs, Local Administrative Units: Northern England (London: Royal Historical Society, 1991), p. 517.
WEST RIDING Riding Division of Ancient County F. Youngs, Local Administrative Units: Northern England (London: Royal Historical Society, 1991), p. 517.
SKIPTON RD Local Government District F. Youngs, Local Administrative Units: Northern England (London: Royal Historical Society, 1991), p. 517.
SKIPTON RSD Sanitary District F. Youngs, Local Administrative Units: Northern England (London: Royal Historical Society, 1991), p. 517.
STAINCLIFFE AND EWCROSS Wap Ancient District F. Youngs, Local Administrative Units: Northern England (London: Royal Historical Society, 1991), p. 517.
BURNSALL CP/AP Parish-level Unit F. Youngs, Local Administrative Units: Northern England (London: Royal Historical Society, 1991), p. 517.
GB Historical GIS / University of Portsmouth, Bordley Tn/CP through time | Census tables with data for the Parish-level Unit, A Vision of Britain through Time.


The BMW E30 is one of the most sought-after BMWs of all time. With it's sporty looks, resolute performance, and capable braking and handling, this 3 series has made quite the name for itself over the years.
This 1988 325i Convertible has been meticulously maintained with extensive service records upon request. It features original paint and along with the current 15" wheels, it includes the original 14" BBS factory wheels as well. Original paint, recent reupholstering of the seats, new soft top, and H&R springs and shocks.
Whether it's winding mountain road trips or a family outing to the beach, this is the ultimate summer cruising car.


The DG400 is a high performance gearbox which is perfectly suited for use in front engine, rear wheel drive cars with medium capacity (turbo) engines. Its reliable design has been used in several motorsport categories and can handle up to 500 Nm of torque in "Heavy Duty" specification. A variety of standard applications are available such as BMW E30 / E36 / E46, Ford Type 9 and Honda S2000. Drenth Gearboxes can also design bellhousings for customer specific applications.
Please contact us for more information by clicking here.


In honor of Veterans and Military Family Appreciation Week at Drexel University, flags will be placed on the lawn beside Perelman Plaza and Gerri LeBow Hall to recognize veterans who are currently enrolled at Drexel University.
Drexel University is proud to be a Yellow Ribbon School and strives to be an inclusive campus that welcomes all diverse communities. A special thanks to Drexel University departments of Campus Services and University Facilities and Real Estate for their sponsorship and support.
We will start placing the flags at 6:30AM on Thursday, November 8th and then have a photograph and probably interviews with University Communications --- come on out! What a great way to start our day!


text:
3 new COVID-19 cases reported in Herkimer County: Sept. 30, 2020
Times Telegram
Three new COVID-19 cases were reported in Herkimer County as of noon Wednesday, according to the latest daily report from Herkimer County Public Health.
The county has recorded 346 total positive cases since mid-March. There were 16 active cases in the county at the time of the daily report with one person hospitalized. Three hundred-and-twenty people have recovered. Ten have died.
The number of negative tests was up to 37,182 as of Wednesday.
Thirty-eight people are under mandatory quarantine with public health orders; no one was released. Ninety-nine are under precautionary quarantine; 23 were released.
© 2021 www.timestelegram.com. All rights reserved.


"Killing My Old Man." I was both horrified and admittedly curious what Petra's (a Christian rock band in the '80s) new album had produced. It revealed a timeless truth of one's conversion: the old me dies and a new me is created. Not "a better version of myself" as the popular slogan goes, or a new Me 2.0, but a radically different me that Christ wants to live through.


Frank Ntilikina has been diagnosed with a left ankle strain and will not return to Friday's game.
Ntilikina lasted one minute before exiting. X-rays came back negative but it'll be Trey Burke owning the backup spot tonight.
That makes it three straight games of 17 points and two threes for Schroder. It'll be interesting to see if he can carve out a slightly bigger role as he returns to OKC next season, as the Thunder could look at making some changes.


ASB junior banking: What do they offer kids?
ASB has won Canstar's 2016 Award for Bank of the Year – Junior Banking. What is this award, who is ASB Bank, and what do they offer kids?
When kids develop good financial habits young, it can set them up well for the rest of their lives. If you've grown up saving your pocket money or birthday money for something you really want, then when you get your first "real job", you're more likely not to blow each pay check on shopping sprees.
Here at Canstar, we recognise that parents can't be everywhere at once when it comes to learning how to manage money. That's why our Bank of the Year – Junior Banking Award recognises the institution that is able to provide appropriate support for financial education, school banking, content or resources for both schools and parents, and a savings account to give kids a head start.
Canstar is an information provider and in giving you product information Canstar is not making any suggestion or recommendation about a particular product. If you decide to apply for a product, you will deal directly with a financial institution, and not with Canstar. Rates and product information should be confirmed with the relevant financial institution. For more information, read our detailed disclosure, important notes and additional information.
The Bank of the Year – Junior Banking award seeks to recognise the level of support provided to juniors (under 12s). The award seeks to focus on instructional support in financial education, parental support and availability of workshops and resources to schools.
Canstar also compares institutions on the availability of and coverage across banking services in the school banking programs. Additionally the savings account with the highest average overall interest and comprehensive array of features will achieve the highest product score.
The institution which has performed exceptionally across each component and has accumulated the highest overall score wins the award.
The first Auckland Savings Bank (ASB) was founded in 1847 with the goal "to serve the community and to help Kiwis grow". Today, they have more than 1.3 million customers across the country.
What do ASB offer kids?
ASB's Getting Kids Started website makes it easy to navigate ASB's products and educational resources designed for parents and kids beginning to learn about money. The website provides specific information to help parents with the financial education of the junior or youth savers in their household.
Clever Kash is an impressive innovation launching in 2016, too. He's an upgrade to the traditional ASB elephant moneybox, Kashin – but instead of putting in coins, you use the ASB mobile banking app to transfer money. The money shows up in the app and then kids flick the digital coins into Clever Kash's tummy, helping kids to understand the value of money even when it's not in physical form.
ASB has maintained its lead for Education this year due to a comprehensive offering of financial education workshops for both primary and secondary students. Last year they managed to conduct a total of 4,134 workshops in New Zealand schools.
ASB's workshops revolve around learning the basics, including money, saving, spending, banking, setting goals, and budgeting. Seminars and workshops for older students also include a focus on interest, tax, currency, and income and risk.
You can check out what ASB offer on their GetWise website. The ASB GetWise education program impressed our researchers because it covers not just primary schools across New Zealand but also intermediate and secondary schools as well.
ASB Bank started New Zealand's first school banking program back in 1926, and it's still running today.
Today, a whopping 722 schools participate in ASB's school banking program, the largest number of schools of any of the institutions we rated. This is vital because there is certainly no value for customers if they choose a school banking program that is only offered in a few schools.
The advantage of ASB's system for parents is that they gain access to further learning resources, and the ability to control how their child uses their account.
How does it work? There are 11 managers dedicated to running these school banking programs and collecting money weekly. Kids can open an account at school with assistance from staff, and they can make a deposit at school during each weekly session. Students who actively make an effort to save regularly will receive a certificate of achievement each school term.
Canstar congratulates ASB Bank on winning the 2016 Award for Bank of the Year – Junior Banking.
Compare Junior Banking savings and transaction accounts with the Canstar Award report, and see if you can't find outstanding value for your kids. Help them sign up for an account and it could be the beginning of a beautiful friendship between your kids and savvy saving habits.


Whether you are out at the market or snuggled up at home with a good book, the Market Shawl is a beautiful accessory that is soft and elegant. It features a pointed lace pattern throughout, with garter and picot edge finishing. The sheen and silky softness of Panda Pearl make this shawl a delight to knit and wear.


To the uninitiated, the phrase "yoga practitioner" brings to mind images of petite, limber young women bending and stretching into pretzel-like poses.
While yogis of that description certainly do exist, there are also male yogis. What's more, their numbers are increasing. In case you haven't noticed yet, more and more men are getting into yoga, and loving it.
These words aren't ones a non-practitioner would normally associate with yoga, but the ancient practice is all that and more. It's also intensely rewarding and uplifting, as any yogi will happily tell you.
Up until recently, however, yoga has been perceived as being too easy, too "spiritual," and too (gasp!) feminine. In the past, many men, especially those who prefer to improve their health via traditionally male sports and programs, viewed yoga as an activity that's about as appealing as watching paint dry.
Thankfully, traditional male attitudes on yoga are changing, as evidenced by the results of a recent study. According to the 2016 Yoga in America study, more men are getting on the mat. Currently, 28 percent of all yoga practitioners in the United States are male, up from 17.8 percent in 2012. That's 10 million male yogis!
So why are more men getting their yoga on?
Mostly, it's because of the physical and mental health benefits of yoga, like increased flexibility, improved mental focus, and pain relief. But the social benefits of yoga also play a part in its sudden spike in popularity among men.
Yoga and Dating: A Match Made in Heaven?
Joseph Alpern, a musician and yogi, told the New York Post, "[!yoga!] helps with the way that I carry myself and that's a huge thing that girls look [!at!] it shows confidence." He also said that he makes no effort to hide the fact that he does yoga from women he's dating. Furthermore, Alpern believes that being a male yogi also indicates that you're interested in a healthy lifestyle and in being fit, which helps if you're looking for a partner with similar interests and values.
The fact that more and more professional male athletes are getting on the mat might also be helping to dispel the notion that only women can benefit from yoga. Lebron James, Kevin Garnett, Kareem Abdul-Jabbar, the Philadelphia Eagles, and the Seattle Seahawks are just a few examples.
Yoga studios that value inclusiveness and diversity are also attracting a slew of men who want to get into yoga and be assured of a welcoming space where they can stretch, bend, and meditate without judgment.
A regular yoga practice provides amazing, lifelong benefits to all yogis, regardless of gender. If you're a male yogi (or want to be one), check out the Men's 30 Day Yoga Challenge and gain mantastic strength, flexibility, and peace of mind.