It does be one high efficiency and low cost crushing machine. . crushing machine in shahdol region - . stone crusher project report in bhopal - . Bhopal Region.
crushing machine in shahdol region. betul coal mine area. kangala is an open pit coal mine situated in the witbank region of mpumalanga 2012 · coal mines shahdol .
crushing machine in shahdol region. Area Of Aggregate Crushing Zones In Madhya Pradesh. crushing machine in shahdol region - gujaratgenomics. area of aggregate .
coal mines in shahdol - Newest Crusher, Grinding Mill .
crushing machine in shahdol region « Crusher South Africa . . Madhya Pradesh Coal Mining Projects - Mining Industry . pulverizer mining 200 mesh; Get A Free Quote.
Machine Striker Crushing: Crushing Machine In Shahdol Region [IMG]crusher is a crushing machine researched and More About; Get More Info.
crushing machine in shahdol region. betul coal mine area. kangala is an open pit coal mine situated in the witbank region of mpumalanga 2012 · coal mines shahdol mp .
crushing machine in shahdol region . Crushing Machine In Shahdol Region [IMG]crusher is a crushing machine researched and manufactured by .
crushing machine in shahdol region - villasovaggio.it. autocatalyst crushing plant - sevenstarcricket. crushing machine in shahdol region; fruit crushing machine; .
Crushing Machine In Shahdol Region; Rebar Milling Used Machine For Sale; Machine To Separate Manganese Ore And Iron Ore; How To Order Crushing Machine .
crushing machine in shahdol region . rajendra coal mines sohagpur area. shahdol census of india. there is a virateshwar temple of lord shiv in sohagpur vanganga .
coal mines shahdol mp - bdpublicschool. sohagpur coalfield is located in shahdol district in the indian state of madhya pradesh in the coal was discovered in this region, . crushing machine double .
Crushing Machine In Shahdol Region; 100 T/h Cone Crushing Plant Chiness Supplier; Crushing Production Line Siderite Jaw Crusher Processing Of Crushing .
crushing machine in shahdol region - gujaratgenomicsin area of aggregate crushing zones in madhya, sohagpur coalfield is located in shahdol district in the indian .
crushing machine in shahdol region - debpaints. ivrcl crusher machine venusvascular. crushing machine in shahdol region gosupsurf. ivrcl crusher site in bhopal crusher machine for sale.the xsm is professional ore crushing .
crusher plants in ajmer region stone crusher machine. crushing machine in shahdol region icrap. Work orders are placed for Crushing Plant . Ajmer has ..
crushing machine in shahdol region - gosupsurfcoza. rajendra coal mines sohagpur area shahdol census of india there is a virateshwar temple of lord shiv in sohagpur .
Crushing Machine In Shahdol Region - . hot sale gold ore crushing machine for sale gold ore hot sell mobile jaw crusher/ small .
home >> crushing machine in shahdol region. crushing machine in shahdol region. ball mill rubber lining parts - jkhealthcare.
26521 Crusher Machine Region New Condition And Sand Making Machine Stone Crushers Plant,lizenithne Mining Room Of Rock Crushing Equipment Crusher Crusher Machine For Sale. The Xsm Is New Fine A Crusher Machine Polluting Our Area 26521 Crusher Machine Region Advisorpubliions. Ltd.
crushing machine for market garbage for anaerobic .
crushing machine in shahdol region « Crusher South . crushing machine for market garbage for anaerobic digestion. crushing machine for market garbage for .
crushing crushing machine in shahdol region - ivrcl crusher site in bhopal mining shri premchandra mishra, m/s mishra stone crusher, budhar, .


What marketing tactics will make your business soar to the top in 2017?
We asked the Social Media Rockstar Event Speakers what they believe will be the "killer marketing tactic" for 2017.
Relevant and Credible Content, Instagram Stories, Customers and Relationships, Story Telling and Visuals, Providing the Experience, Live Video and much more!
Want to see these experts in action?
Join us at the Social Media Rockstar Event on Wednesday, September 14th at the Willmar Conference Center in Willmar, MN.
HINT: There's a coupon code in the eBook!


In a 2006–2008 study, seven aphid species (Hemiptera: Aphididae) were detected on apple trees in South Bulgaria: Rhopalosiphum insertum Walk. (apple-grass aphid), Dysaphis plantaginea Pass (rosy apple aphid), Dysaphis devecta Walker (rosy leafurling aphid), Macrosiphum euphorbiae Thomas (potato aphid), Aphis pomi De Geer (green apple aphid), Aphis spiraecola Patch. (spiraea aphid/green citrus aphid) and Eriosoma lanigerum Nausm. (wooly apple aphid). The dominant species were D. plantaginea, found in 97.8% of the surveyed orchards. Two green aphid species from the genus Aphis were both found in 96.4% of the orchards. The rosy apple aphid was a more important pest of bearing orchards, where infestation in spring always exceeded the economic injury level (1–2% of infested shoots). The green aphids were more important in young orchards and nurseries, where they developed during the entire vegetation period and infestation sometimes exceed 50–80% of shoots. A. pomi was still more widespread, but A. spiraecola was found throughout all of South Bulgaria. As pests of apple, the other species were less common, causing lower infestations and minor economic importance.
Andreev R., Rasheva D., Kutinkova H. 2007. Aphids in apple orchards in Central-South Bulgaria. J. Plant Prot. Res. 47 (1): 109–112.
Blackman R.L., Eastop V.F. 1989. Aphids on the World's Crops. An Identification and Information Guide. Chichester, Wiley, UK, 470 pp.
Grigorov S. 1980. Listni Vashki i Borbata s Tyakh. [Leaf Aphids and their Control]. Zemizdat, Sofia, Bulgaria, 285 pp.
Leclant F. 2000. Les Pucerons des Plantes Cultivees Clefs D'identificatin. III – Cultures Fruitieres. [Aphids of Cultivated Plants. III – Fruit Crops]. INRA, Paris, France, 98 pp.
Mikhailova P., Straka F., Apostolov I. 1982. Rastiteľno-Zashtitna Prognoza i Signalizatsiya. [Plant Protection Prognosis and Signalization]. Zemizdat, Sofia, Bulgaria, 344 pp.
Pelov V. 1977. Listni vashki po yabalkata i lyucernata i tekhnite estestveni vragove. [Aphids on apple and alfalfa and their natural enemies]. Spisanie Rastiteľna Zashtita 25 (2): 3–6.
Rasheva D., Andreev R. 2007. Aphis spiraecola Patch. (Hemiptera: Aphididae) nov nepriyatel po yabalkata v Balgariya. [Aphis spiraecola (Hemiptera: Aphididae) – a new pest on apple in Bulgaria]. Acta Entomologica Bulgarica 13 (1, 2): 91–97.
Shaposhnikov G. 1964. Podotryad Aphidodea. [Subfamily Aphidodea – Aphids]. p. 489–616. In: "Opredeliteľ Nasekomakh Evropeĭskoĭ Chasti SSSR" [Identification Guide of Insects in the European Part of USSR] (G.Y. Beĭ-Bienko, ed.). Izdateľstvo "Nauka", Moskva-Leningrad, Russia, 1128 pp.
Stancheva Y., Borovinova M., Andreev R., Kalinova St., Balevski N., Simova S., Velcheva N., Staneva E., Draganova S., Arnaudov V., Kolev K., Stoev A., Rankova Z., Georgieva M. 2008. Guide for Integrated Pest Management in Fruit Crops. National Service of Plant Protection, Sofia, Bulgaria, 152 pp.
Stekolshikov A.V. 2004. The species complexes of aphids closely related to Dysaphis devecta (Walker) (Homoptera, Aphididae): Possible ways of its formation. p. 183–188. In: "Aphids in a new millennium" (J.-Ch. Simon, ed.). Proc. Sixth International Symposium on Aphids, INRA/ENSAR, Renne, France, September 2001, 549 pp.


text:
MtLeg Viewpoint: Unleash Montana's Energy Potential
Sen. Duane Ankney
/ Legislative Viewpoints
by Sen. Duane Ankney
"We must unleash Montana's energy potential so that our kids and grandkids will have a good future."
Energy is the lifeblood of human civilization and progress. Transportation, manufacturing, the Internet, heated homes, clean drinking water, agricultural production, and almost every other vital component of our modern lives relies on having consistent and reliable access to energy. Energy is also essential to our national security and independence. Just look at what is going on in Europe.
Montana can help secure that energy future. Our state can be an energy producing powerhouse, for the nation as well as for ourselves, if we unleash our state's potential. We have coal, oil, gas, wind, sunlight, geothermal, hydropower, and the infrastructure to transport nuclear-produced energy.
Radical environmental ideology, politics, bureaucracy, and red tape are putting all of that at risk. What started as the War on Coal is quickly becoming a war against most all energy sources. Many of the same people who've been working to shut down Colstrip are now opposing nuclear and natural gas. Solar projects sometimes face lawsuits from the radical environmentalists. Dams have already been breached in Oregon and a strong movement, backed by politicians and environmental groups in Washington and Oregon, have proposed pulling dams on the Columbia river.
If the answer to every energy project is "no," the people of Montana will not have a good future. That's especially true when it comes to projects that would produce reliable, baseload power. I'm not opposed to wind, solar, and batteries, but those things alone aren't going to cut it. When the demand for power peaks, we need to be able to generate energy at will so we have enough supply. That's where coal and nuclear come in. The Legislature is currently studying the viability of future nuclear power production. In my opinion, nuclear will be the long-term answer to our baseload power needs.
Nuclear can provide the reliable energy we need. While not as affordable as coal, it is available 24/7/, 365 days a year and modern reactor technology makes it incredibly safe. Natural gas is an option, but the difficulties in permitting pipelines and the volatility of pricing make it more uncertain.
With Colstrip's massive transmission infrastructure already available, the long-term solution absolutely must include nuclear. But because of the long lead time in sighting, such as permitting, reactor design, and construction, nuclear can't be the short-term fix. The existing coal-fired power plant is the solution to our immediate baseload power needs. Colstrip power production, coal mining, and related indirect roles employ 3,000-4,000 Montanans in high paying jobs. Nuclear is very similar in number of jobs, with the average pay even a bit higher. These jobs add to the economic health of our state and keep our most precious resource—the people of Montana—working.
To have reliable baseload power, to have the energy we need to power our lives, we must reject radical ideology and get unnecessary red tape out of the way. We must unleash Montana's energy potential so that our kids and grandkids will have a good future.
Sen. Duane Ankney, R-Colstrip, is a former coal miner and longtime leader on Montana energy policy.
Frontier Institute's Mtleg Viewpoint series provides an opportunity once per month for Montana legislators to deliver an update about topics that matter to our followers. Offers to publish columns were made and remain open to both the Legislative Majority and Minority. Opinions expressed by guest authors do not necessarily represent the positions of the Frontier Institute.

redpajama_set_name: RedPajamaCommonCrawl


text:
1913 in Canada
Events from the year 1913 in Canada.
Years in Canada: 1910 1911 1912 1913 1914 1915 1916
Centuries: 19th century · 20th century · 21st century
Decades: 1880s 1890s 1900s 1910s 1920s 1930s 1940s
Years: 1910 1911 1912 1913 1914 1915 1916
IncumbentsEdit
CrownEdit
Head of state (monarch) – George V
Federal governmentEdit
Governor general – Prince Arthur, Duke of Connaught and Strathearn
Prime minister – Robert Borden
Chief Justice – Charles Fitzpatrick (Quebec)
Parliament – 12th
Provincial governmentsEdit
Lieutenant governorsEdit
Lieutenant Governor of Alberta – George H. V. Bulyea
Lieutenant Governor of British Columbia – Thomas Wilson Paterson
Lieutenant Governor of Manitoba – Douglas Cameron
Lieutenant Governor of New Brunswick – Josiah Wood
Lieutenant Governor of Nova Scotia – James Drummond McGregor
Lieutenant Governor of Ontario – John Morison Gibson
Lieutenant Governor of Prince Edward Island – Benjamin Rogers
Lieutenant Governor of Quebec – François Langelier
Lieutenant Governor of Saskatchewan – George William Brown
PremiersEdit
Premier of Alberta – Arthur Sifton
Premier of British Columbia – Richard McBride
Premier of Manitoba – Rodmond Roblin
Premier of New Brunswick – James Kidd Flemming
Premier of Nova Scotia – George Henry Murray
Premier of Ontario – James Whitney
Premier of Prince Edward Island – John Alexander Mathieson
Premier of Quebec – Lomer Gouin
Premier of Saskatchewan – Thomas Walter Scott
Territorial governmentsEdit
CommissionersEdit
Commissioner of Yukon – George Black
Gold Commissioner of Yukon – George P. MacKenzie
Commissioner of Northwest Territories – Frederick D. White
EventsEdit
March 27 – Le Droit first published in French
April 17 – 1913 Alberta general election: Arthur Sifton's Liberals win a third consecutive majority
November 7 – November 8 – A storm on the Great Lakes sinks some thirty-four ships
November 17 – The National Transcontinental Railway is completed
March 1 – The Quebec Bulldogs win their 2nd Stanley Cup
March 7 – The Victoria Senators win their First PCHA Championship
November 29 – The Hamilton Tigers win their 1st Grey Cup by defeating the Toronto Parkdale Canoe Club 44 to 2 in the 5th Grey Cup played at Hamilton, Ontario's A.A.A. Grounds
Unknown dateEdit
June – Start of the Canadian Arctic Expedition 1913–1916 a scientific expedition in the Arctic Circle organized and led by Vilhjalmur Stefansson.
Laura Secord Chocolates opens
Arts and literatureEdit
New BooksEdit
BirthsEdit
January to JuneEdit
January 13 – Philip Gaglardi, politician (d. 1995)
March 11 – John Weinzweig, composer (d. 2006)
March 24 – Émile Benoît, musician (d. 1992)
April 4 – Jules Léger, diplomat and 21st Governor General of Canada (d. 1980)
April 24 – Violet Archer, composer, teacher, pianist, organist and percussionist (d. 2000)
April 30 – Edith Fowke, folk song collector, author and radio presenter (d. 1996)
May 27 – James Page Mackey, chief of Toronto Police Service (d. 2009)
June 3 – Lloyd Percival, Canadian tennis player, boxer, cricketer, and coach (d. 1974)
June 12 – Jean Victor Allard, general and first French-Canadian to become Chief of the Defence Staff (d. 1996)
June 14 – Joe Morris, trade unionist and president of the Canadian Labour Congress (d. 1996)
June 18 – Wilfred Gordon Bigelow, heart surgeon (d. 2005)
July to DecemberEdit
July 6 – J. Carson Mark, mathematician who worked on development of nuclear weapons (d. 1997)
July 16 – Woodrow Stanley Lloyd, politician and 8th Premier of Saskatchewan (d. 1972)
August 28 – Robertson Davies, novelist, playwright, critic, journalist and professor (d. 1995)
September 20 – Robert Christie, actor and director (d. 1996)
October 5 – Horace Gwynne, boxer and Olympic gold medalist (d. 2001)
November 7 – Elizabeth Bradford Holbrook, portrait sculptor (d. 2009)
November 8 – June Havoc, actress, dancer, writer, and theater director (d. 2010)
November 16 – Dora de Pedery-Hunt, sculptor and coin and medal designer (d. 2008)
November 21 – Stewart McLean, politician (d. 1996)
December 7 – Donald C. MacDonald, politician (d. 2008)
December 12 – Clint Smith, ice hockey player and coach (d. 2009)
December 16 – George Ignatieff, diplomat (d. 1989)
December 27 – Elizabeth Smart, poet and novelist (d. 1986)
DeathsEdit
March 7 – Pauline Johnson, poet, writer and performer (b. 1861)
April 12 – Alexander Francis Macdonald, politician (b. 1818)
April 23 – Richard William Scott, politician and Minister (b. 1826)
July 15 – Hugh Richardson, jurist (b. 1826)
August 6 – John M. Bryant, first aviator killed in a crash in Canada
List of Canadian films
Retrieved from "https://en.wikipedia.org/w/index.php?title=1913_in_Canada&oldid=903750099"


This multinational Pharmaceutical Shared Services organisation provides healthcare service and support the medical and animal healthcare products across the EMEA region. Due to phenomenal success in delivering customer service, accountancy and supply chain support throughout the EMEA region, they are now seeking a HR Shared Services Manager. This person will work closely with the leadership team and will have the opportunity to drive the business forward.
Working with your team of 6 HR Analysts and Associates, you will be responsible for the delivery of HR Services to 72 countries. This team is responsible for global HR Data Management including all processes, interpretation, integration, and reporting of business needs into data systems for data intelligence, maintenance and accuracy.
You will provide resource planning, capacity of management of tasks; direction and coaching to ensure execution with quality and integrity; ensure collaboration, support, and consistent execution across HR Operations. Identify and solve a range of organizational and operational problems and technical challenges, as well as forecast future business planning needs in conjunction with Global HR Operations Director.
Supervisory responsibility of HR Analysts and Associates , including staffing, training, performance, management, compensation, administration, people development & coaching.
You will have 2 – 3 years people management experience in a HR function along with strong data analytics capability within a HR Shared Services Capacity (ideally SAP or Workday, though this is not essential).
You will be a natural leader who can motivate and coach their team. This organisation continuously displays world calls employee engagement, as they firmly belief that it is their people that bring success. You will have a similar mind-set.
Excellent salary (DOE), performance bonus, full family healthcare, pension (up to 8%), educational assistance.


text:
\section{Introduction}

The existence of resolution of singularities in arbitrary dimension over a field of characteristic zero was solved by Hironaka in his famous paper \cite{existence}. Later on, different constructive proofs have been given, among others, by Villamayor \cite{villa}, Bierstone-Milman \cite{bm}, Encinas-Villamayor \cite{cour}, Encinas-Hauser \cite{strong} and Wodarczyk \cite{Wo}.

This paper is devoted to study the complexity of Villamayor's algorithm of resolution of singularities. This algorithm appears originally in \cite{villa} and we will use the presentation given in \cite{cour}. In this paper, the authors introduce a class of objects called \emph{basic objects} $B=(W,(J,c),E)$ where $W$ is a regular ambient space over a field $k$ of characteristic zero, $J\subset \mathcal{O}_W$ is a sheaf of ideals, $c$ is an integer and $E$ is a set of smooth hypersurfaces in $W$ having only normal crossings. That is, they consider the ideal $J$ together with a positive integer $c$, or \emph{critical value} defining the \emph{singular locus} $Sing(J,c)=\{\xi \in W|\ ord_{\xi}(J)\geq c \}$, where $ord_{\xi}(J)$ is the \emph{order} of $J$ in a point $\xi$.

Let $W\stackrel{\pi}{\leftarrow} W'$ be the monoidal transformation with center $\mathcal{Z}\subset Sing(J,c)$, $\pi^{-1}(\mathcal{Z})=Y'$ is the exceptional divisor. Let $\xi$ be the generic point of $\mathcal{Z}$, $ord_{\xi}(J)=\theta$, the total transform of $J$ in $W'$ satisfies $J\mathcal{O}_{W'}=I(Y')^{\theta}\cdot J^{\curlyvee}$ where $J^{\curlyvee}$ is the \emph{weak} transform of $J$, (see \cite{cour} for details).

A \emph{transformation} of a basic object $(W,(J,c),E)\leftarrow (W',(J',c),E')$ is defined by a monoidal transformation $W\stackrel{\pi}{\leftarrow} W'$ and defining $J'=I(Y')^{\theta-c}\cdot J^{\curlyvee}$, the \emph{controlled transform} of $J$.

A sequence of transformations of basic objects \begin{equation} (W,(J,c),E)\leftarrow (W^{(1)},(J^{(1)},c),E^{(1)})\leftarrow \cdots \leftarrow (W^{(N)},(J^{(N)},c),E^{(N)}) \label{resolution} \end{equation} is a \emph{resolution} of $(W,(J,c),E)$ if $Sing(J^{(N)},c)=\emptyset$.

\begin{remark} Superscripts $^{(k)}$ in basic objects will denote the $k$-stage of the resolution process. Subscripts $_i$ will always denote the dimension of the ambient space $W_i^{(k)}$.
\end{remark}

Villamayor's algorithm provides a \emph{log-resolution} in characteristic zero. A log-resolution of $J$ is a sequence of monoidal transformations at regular centers as (\ref{resolution}) such that each center has normal crossings with the exceptional divisors $E^{(i)}$, and the total transform of $J$ in $W^{(N)}$ is of the form  $$J\mathcal{O}_{W^{(N)}}=I(H_1)^{b_1}\cdot \ldots \cdot I(H_N)^{b_N}$$ with $b_i\in \mathbb{N}$ for all $1\leq i \leq N$  and $E^{(N)}=\{H_1,\ldots,H_N\}$.

In \cite{cour} it is shown that algorithmic principalization of ideals reduces to algorithmic resolution of basic objects. That is, starting with $c\!=\!max\ ord(J)$, the maximal order of $J$, we obtain a resolution of $(W,(J,c),E)$ as (\ref{resolution}). At this step $max\ ord(J^{(N)})\!=\!c^{(N)}\!<c$. If $c^{(N)}\!>1$, we continue resolving $(W^{(N)},(J^{(N)},c^{(N)}),E^{(N)})$ and so on, until have $max\ ord(J^{(\mathcal{N})})\!=\!c^{(\mathcal{N})}\!=1$. Finally, a resolution of $(W^{(\mathcal{N})},(J^{(\mathcal{N})},1),E^{(\mathcal{N})})$ provides a log-resolution of $J^{(\mathcal{N})}$, and therefore a log-resolution of $J$.

In \cite{cour} it is also shown that algorithmic principalization of ideals leads to embe\-dded desingulari\-zation of varieties. That is, given a closed subscheme $X\subset W$, the algorithmic principalization of the ideal $I(X)$ provides an embedded desingularization of $X$. See also \cite{EncinasVillamayor2003} for more details.

A key point in the definition of the algorithm is to use induction on the dimension of the ambient space $W$ to define an upper-semi-continuous function $t$. The set of points where this function attains its maximal value, $\underline{Max}\ t$, is a regular closed set, and defines a regular center for the next monoidal transformation.

A resolution of the basic object $(W,(J,c),E)$ is achieved by a sequence of monoidal transformations as in (\ref{resolution}), with centers $\underline{Max}\ t^{(k)}$ for $0\leq k\leq N-1$. That is, the sequence of
monoidal transformations is defined by taking successively the center defined by the upper-semi-continuous function. The algorithm stops at some stage because the maximal value of the function $t$ drops after monoidal transformations, that is, $max\ t^{(0)}>max\ t^{(1)}>\ldots >max\ t^{(N-1)}.$

This function $t$ will be the resolution invariant. We shall work with the invariant defined in \cite{cour},
using the language of mobiles developed in \cite{strong}. We
remind briefly the main notions.

Let $J\subset \mathcal{O}_W$ be an ideal defining a singular algebraic set $X\subset W$. The ideal $J$ factors into $J=M\cdot I$, with $M$ the ideal defining a normal crossing divisors, and $I$ some ideal
still unresolved.

By induction on the dimension of $W$, we will
have this decomposition at every dimension from $n$ to $1$, that
is $J_i=M_i\cdot I_i$, for $n\geq i \geq 1$, are defined in local flags $W_n\supseteq W_{n-1}\supseteq \cdots \supseteq W_i\supseteq  \cdots \supseteq W_1$, where each $J_i,M_i,I_i \in \mathcal{O}_{W_i}$ are in dimension $i$. There is a
critical value $c_{i+1}$ at each dimension $i$, ($c_{n+1}=c$), see \cite{strong} for details. All the basic objects $(W_i,(J_i,c_{i+1}),E_i)$, for $n\geq i \geq 1$, will be resolved during the process of the algorithm.

Let $E$ be the exceptional divisor of previous monoidal transformations, and consider $E=\cup_{i=1}^nE_i$ where $E_i$ applies to dimension $i$. Obviously, we start with $E=\emptyset$.

For any point $\xi \in Sing(J,c)$, the function $t$ will
have $n$ coordinates, with lexicographical order, and it will be
one of the following three types: \begin{equation} \label{invt} \hspace*{-0.2cm} \begin{array}{ll} (a) &
t(\xi)=(t_n(\xi),t_{n-1}(\xi),\ldots,t_{n-r}(\xi),\ \infty,\
\infty,\ldots,\infty)
\\ (b) & t(\xi)=(t_n(\xi),t_{n-1}(\xi),\ldots,t_{n-r}(\xi),
\Gamma(\xi),\infty,\ldots,\infty)
\\ (c) & t(\xi)=(t_n(\xi),t_{n-1}(\xi),\ldots,t_{n-r}(\xi),\ldots
\ldots \ldots,t_1(\xi)) \end{array}\  \text{ with }
t_i=\left[\frac{\theta_i}{c_{i+1}},m_i\right] \end{equation} where
$\theta_i=ord_{\xi}(I_i)\ ,$ $m_i$ is the number of exceptional
divisors in $E_i$, and $\Gamma$ is the resolution function corresponding to the so-called \emph{monomial case}, following the notation of \cite{cour}, pages $165-166$. We will recall the definition of $\Gamma$ in equation (\ref{gamma}).

For simplicity, let assume that we start with a polynomial ring, $W=Spec(k[X_1,\ldots,X_n])$. In $\mathcal{O}_W=k[X_1,\ldots,X_n]$ the ideal $J$ is locally given by a monomial with respect to a regular system of parameters $$J=<X_1^{a_1}\cdot \ldots \cdot X_n^{a_n}>\subset \mathcal{O}_W \text{ with } a_i\in \mathbb{N}, \text{ for } i=1,\ldots,n.$$

Note that, in this situation, the center of the next monoidal transformation is combinatorial, it is a linear combination of $X_1,\ldots,X_n$. And this is also true after monoidal transformations, since Villamayor's algorithm applied to a monomial ideal provides always combinatorial centers, and after a monoidal transformation in a combinatorial center we obtain again a monomial ideal.

So, at any stage of the resolution process, $W=\cup_i U_i$, where $U_i\cong\mathbb{A}^n_k$. Thereafter, we shall work locally, so we will assume that $W$ is an affine space.

To resolve the toric hypersurface $\{f=0\}=\{Z^c-X_1^{a_1}\cdot \ldots \cdot X_n^{a_n}=0\}$ we note that its singular locus $Sing(<f>,c)$ is always included in $\{Z=0\}$, so we argue by induction on the dimension and reduce to the case
where the corresponding ideal $J$ is of the form
\begin{equation} \label{jota} J=<X_1^{a_1}\cdot \ldots \cdot
X_n^{a_n}>\subset \mathcal{O}_W \text{ with }\ 1\leq a_1\leq a_2\leq \ldots \leq a_n,\
\sum_{i=1}^n a_i=d,\ d\geq c,\end{equation} where $c$ is
the critical value. If $a_i=0$ for some $i$, then we may assume $dim(W)<n$.

After a monoidal transformation, we always consider the controlled transform of $J$ with respect to $c$, $J'=I(Y')^{-c}\cdot J^{*}$ where $J^{*}$ is the total transform of $J$ and $Y'$ denotes the new exceptional divisor. For the toric problem $J=<Z^c-X_1^{a_1}\cdot \ldots \cdot X_n^{a_n}>$, taking the origin as center of the next monoidal transformation, at the $i$-th chart: $$J^*=<Z^c\cdot X_i^{c} - X_1^{a_1}\cdots X_i^{d} \cdots X_n^{a_n}>=<X_i^{c}\cdot(Z^c-X_1^{a_1}\cdots X_i^{d-c} \cdots X_n^{a_n})>,$$ and we can only factorize $c$ times the exceptional divisor.

\begin{remark} We will denote as {\it $i$-th chart} the chart where we divide by $X_i$. When the center of the monoidal transformation is the origin, this monoidal transformation is expressed:
$$\begin{array}{ccc} k[Z,X_1,\ldots,X_n] & \rightarrow & k[Z,X_1,\ldots,X_n,\frac{Z}{X_i},\frac{X_1}{X_i},\ldots,\frac{X_{i-1}}{X_i},\frac{X_{i+1}}{X_i},\ldots,\frac{X_n}{X_i}] \\ Z & \rightarrow & \frac{Z}{X_i} \\ X_i & \rightarrow & X_i \\ X_j & \rightarrow & \hspace*{1.5cm} \frac{X_j}{X_i} \text{ for } j\neq i \\
\end{array} $$  where $k[Z,X_1,\ldots,X_n,\frac{Z}{X_i},\frac{X_1}{X_i},\ldots,\frac{X_{i-1}}{X_i},\frac{X_{i+1}}{X_i},\ldots,\frac{X_n}{X_i}]\cong \\ k[\frac{Z}{X_i},\frac{X_1}{X_i},\ldots,\frac{X_{i-1}}{X_i},X_i,\frac{X_{i+1}}{X_i},\ldots,\frac{X_n}{X_i}]$. For simplicity, we will denote each $\frac{X_j}{X_i}$ again as $X_j$, and $\frac{Z}{X_i}$ as $Z$.
\end{remark}

So we will apply the resolution algorithm to the basic object $(W,(J,c),\emptyset)$ for $J=<X_1^{a_1}\cdot \ldots \cdot X_n^{a_n}>$, which is already a monomial ideal, but it is not necessarily supported on the exceptional divisors.

\section{Monomial case (exceptional monomial)}

The \emph{monomial case} is a special case in which J is a ``monomial ideal" given locally by a monomial that can be expressed in terms of the exceptional divisors. This case arises after several monoidal transformations.

This means that we have a basic object $(W,(J,c),E)$ where $J$ is locally defined by one monomial supported on the hypersurfaces in $E$. In this case, the ideal $J$ factors into $J=M\cdot I$ with $J=M$ and $I=1$. We can also call it {\bf exceptional monomial}.

\begin{theorem} \label{mon} Let $J\subset \mathcal{O}_W$ be a monomial ideal as in equation (\ref{jota}). Let
$E=\{H_1,\ldots,H_n\}$, with $H_i=V(X_i)$, be a normal crossing divisor. \\ Then an upper bound for
the number of monoidal transformations to resolve $(W,(J,c),E)$ is given by $$\frac{d - c +
gcd (a_1,\ldots,a_n,c)}{gcd (a_1,\ldots,a_n,c)}.$$
\end{theorem}

\begin{proof} We may assume that the greatest common divisor of the
exponents $a_i$ and the critical value $c$ is equal to $1$, because
both the simplified problem and the original problem have the
same singular locus. That is, if $gcd (a_1,\ldots,a_n,c)=k$ then $d=k\cdot d_1$, $c=k\cdot c_1$, $a_i=k\cdot b_i$ for all $1\leq i\leq n$ and $gcd (b_1,\ldots,b_n,c_1)=1$. The ideal $J$ can be written as $J=(J_1)^k$ where $J_1=<X_1^{b_1}\cdot \ldots \cdot X_n^{b_n}>$ therefore $$Sing(J,c)=\{\xi \in X|\ ord_{\xi}((J_1)^k)\geq k\cdot c_1 \}=\{\xi \in X|\ ord_{\xi}(J_1)\geq c_1 \}=Sing(J_1,c_1),$$ where $X$ is the algebraic set defined by $J$.

For a point $\xi \in \mathbb{A}^n_k$, $\Gamma(\xi)=(-\Gamma_1(\xi),\Gamma_2(\xi),\Gamma_3(\xi))$ where \begin{equation} \label{gamma} \vspace*{0.15cm} \begin{array}{l} \Gamma_1(\xi)=\min\{p\ |\ \exists\ i_1,\ldots,i_p, a_{i_1}(\xi)+\cdots+a_{i_p}(\xi)\geq c,\ \xi\in H_{i_1}\cap\cdots \cap H_{i_p}\}, \vspace*{0.15cm} \\ \Gamma_2(\xi)=\max\left\{\frac{a_{i_1}(\xi)+\cdots+a_{i_p}(\xi)}{c}\ |\ {\scriptstyle p=\Gamma_1(\xi),\ a_{i_1}(\xi)+\cdots+a_{i_p}(\xi)\geq c,\ \xi\in H_{i_1}\cap\cdots \cap H_{i_p}}\right\},\vspace*{0.15cm} \\  \Gamma_3(\xi)=\max\{(i_1,\ldots,i_p,0,\ldots,0)\in \mathbb{Z}^n\ |\ {\scriptstyle \Gamma_2(\xi)=\frac{a_{i_1}(\xi)+\cdots+a_{i_p}(\xi)}{c},\ \xi\in H_{i_1}\cap\cdots \cap H_{i_p}}\} \vspace*{0.1cm} \end{array} \end{equation} with lexicographical order in $\mathbb{Z}^n$.

The center $\mathcal{Z}$ of the next monoidal transformation is given by the set of points where $\Gamma$ attains its maximal value. It is easy to see that $\mathcal{Z}=\cap_{i=n-(r-1)}^n H_i$.

So at the $j$-th chart, the exponent of $X_j$ after the monoidal transformation is $(\sum_{i=n-r+1}^n a_i)-c$ and   $$\left(\sum_{i=n-r+1}^n a_i\right)-c< \min_{n-r+1\leq i \leq n}a_i=a_{n-r+1}$$ because $\sum_{i=n-r+2}^{n} a_i<c$ by construction of the center $\mathcal{Z}$.

This shows that the order of the ideal drops after each monoidal transformation by at least
one, so in the worst case, we need $d-(c-1)$ monoidal transformations to obtain an
order lower than $c$.
\end{proof}

\begin{remark} Note that it is necessary to consider the monomial case. On one hand, this case may appear in dimension $n$, and also in lower dimensions, $n-1,\ldots,1$, when we resolve any basic object $(W,(J,c),E)$ (where $J$ is any ideal). So we need to resolve the monomial case in order to obtain a resolution of the original basic object $(W,(J,c),E)$.

On the other hand, the algorithm of resolution leads to the monomial case, since given any ideal $J$, the algorithm provides a log-resolution of $J$. And it is necessary to continue to a resolution within the monomial case.
\end{remark}

\begin{remark}
The bound in theorem \ref{mon} is reached only for the following values of $c$: $$1,\
a_n+ \ldots + a_j+1 \text{ for } n\geq j\geq 2,\ d.$$ For these values of $c$, the order of the ideal drops after each monoidal transformation exactly by one:
\begin{itemize}
	\item If $c=1$, the monoidal transformation is an isomorphism. The exponent of $X_n$ after the monoidal transformation is $a_n-1$.
	\item If $c=a_n+ \ldots + a_j+1$, for $n\geq j\geq 2$, the center of the monoidal transformation is $\mathcal{Z}=\cap_{i=j-1}^n H_i$. At the $l$-th chart, for $n\geq l\geq j-1$, the exponent of $X_l$ after the monoidal transformation is $(\sum_{i=j-1}^n a_i)-c=(\sum_{i=j-1}^n a_i)-(\sum_{i=j}^n a_i)-1=a_{j-1}-1$.

In particular, at the $(j-1)$-th chart, the exponent of $X_{j-1}$ after the monoidal transformation has droped exactly by one.
	\item If $c=d$, we finish after only one monoidal transformation.
\end{itemize}
\end{remark}

\begin{remark} If $gcd (a_1,\ldots,a_n,c)=k>1$, then the bound of the theorem \ref{mon}
is $(d-c+k)/ k$. \\ As $(d-c+k)/ k<d-c+1,$ we can use in practice the bound for the case $gcd
(a_1,\ldots,a_n,c)=1$.
\end{remark}

\section{Case of one monomial}

To construct an upper bound for the number of monoidal transformations needed to resolve the basic object
$(W,(J,c),E=\emptyset)$, where $J$ is locally defined by a unique monomial, we estimate the number of
monoidal transformations needed to obtain $(W',(J',c),E')$, a transformation of the original basic object, with $J'=M'$ (an exceptional monomial), and then apply theorem \ref{mon}. In order to use theorem \ref{mon}, we need an estimation of the
order of $M'$. This estimation will be valid at any stage of the resolution process.

\begin{lemma} \label{grad} Let $(W,(J,c),\emptyset)$ be a basic object where $J$ is a monomial ideal as in equation (\ref{jota}). Let $J=M\cdot I$ be the factorization of $J$, where $M=1$, because of $E=\emptyset$, and $J=I$. After $N$ monoidal transformations we have $(W^{(N)},(J^{(N)},c),E^{(N)})$. Let $\xi\in W^{(N)}$ be a point. Then $$\ ord_{\xi}(M^{(N)}) \leq (2^N-1)(d-c)$$
where $ord_{\xi}(M^{(N)})$ denotes the order at $\xi$ of $M^{(N)}$, the (exceptional)
monomial part of $J^{(N)}$.
\end{lemma}

\begin{proof} It follows by induction on $N$:
\begin{itemize}
\item that if $N=1, \  ord_{\xi}(M^{(1)})=d-c.$ \\ At the beginning, the first center defined by this algorithm is always the origin, so at the $i$-th chart: $$J^{(1)}=M^{(1)} \cdot I^{(1)}=<X_i^{d-c}>\cdot <X_1^{a_1}\cdot \stackrel{\widehat{i}}{\ldots} \cdot X_n^{a_n}>$$ with $E^{(1)}=\{H_i\}$, where $H_i=V(X_i)$.
\item We assume that the result holds for $N=m-1$. $$J^{(m-1)}= M^{(m-1)} \cdot I^{(m-1)}= <X_{i_1}^{b_1}\cdots
X_{i_s}^{b_s}>\cdot <X_{i_{s+1}}^{a_{i_{s+1}}}\cdots X_{i_n}^{a_{i_n}}>$$ with $\sum_{i=1}^s b_i=d'.$ By inductive hypothesis, after $m-1$ monoidal transformations, the maximal order $d'$ of the (exceptional) monomial part $M^{(m-1)}$ satisfies $$d'\leq (2^{m-1}-1)(d-c).$$
For $N=m$, there are two possibilities:
\begin{enumerate}
\item If $max\ ord(I^{(m-1)})=\sum_{j=s+1}^n a_{i_j}\geq c$ then the center of the next monoidal transformation contains only variables appearing in $I^{(m-1)}$.
\item If $max\ ord(I^{(m-1)})=\sum_{j=s+1}^n a_{i_j}<c$ then the center of the next monoidal transformation contains variables appearing in $I^{(m-1)}$ and also variables appearing in $M^{(m-1)}$.
\end{enumerate}
\begin{itemize}
\item[\underline{Case $1$}:] \ \\ If the center of the monoidal transformation is as small as possible, that is $\mathcal{Z}\!=\!\cap_{j=s+1}^n V(X_{i_j})$, at the $i_l$-th chart,
$$J^{(m)}\!=\!M^{(m)}\cdot I^{(m)}\!=<\!X_{i_1}^{b_1}\!\cdots\!X_{i_s}^{b_s}\cdot
X_{i_l}^{d-\sum_{j=1}^s a_{i_j}-c}\!>\!\cdot\!
<\!X_{i_{s+1}}^{a_{i_{s+1}}}\!\stackrel{\widehat{i_l}}{\cdots}\!X_{i_n}^{a_{i_n}}\!>\!.$$ The exponent of $X_{i_l}$, $\ d-\sum_{j=1}^s a_{i_j}-c=\sum_{j=s+1}^n a_{i_j}-c \ $ is as big as possible, so this is the worst case, because the increase in the order of the exceptional monomial part after the monoidal transformation will be greater than that for another centers.

The highest order of $M^{(m)}$ is $$\sum_{i=1}^s b_i + d-
\sum_{j=1}^s a_{i_j}-c =d'+d-c- \sum_{j=1}^s a_{i_j} \leq
d'+d-c,$$ so by inductive hypothesis $$d'+d-c \leq (2^{m-1}-1)(d-c)+d-c=2^{m-1}(d-c)\leq
(2^m-1)(d-c).$$
\item[\underline{Case $2$}:]
\begin{enumerate}
\item[-] At the $i_j$-th chart, for $1\leq j\leq s$
$$ J^{(m)}=M^{(m)}\cdot I^{(m)}=<X_{i_1}^{b_1}
\stackrel{\widehat{i_j}}{\cdots} X_{i_s}^{b_s}\cdot
X_{i_j}^{\square}>\cdot <X_{i_{s+1}}^{a_{i_{s+1}}}
\cdots X_{i_n}^{a_{i_n}}>.$$
\item[-] At the $i_l$-th chart, for $s+1\leq l\leq n$
$$J^{(m)}=M^{(m)}\cdot I^{(m)}=
<X_{i_1}^{b_1}\cdots X_{i_s}^{b_s}\cdot
X_{i_l}^{\vartriangle}>\cdot <X_{i_{s+1}}^{a_{i_{s+1}}}
\stackrel{\widehat{i_l}}{\cdots} X_{i_n}^{a_{i_n}}>.$$
\end{enumerate}
As above, if we are in the worst case, when the center of the monoidal transformation is as small as possible, that is, the center is a point,
$$\square=\vartriangle= d'+d-\sum_{j=1}^s a_{i_j}-c\ .$$ Therefore in both cases the highest order of $M^{(m)}$ satisfies $$\leq 2d'+d-c\ \leq \ 2(2^{m-1}-1)(d-c)+d-c=(2^m-1)(d-c)\ .$$
\end{itemize}
\end{itemize}
\end{proof}

\begin{remark} Due to its general character, this bound is
large and far from being optimal.
\end{remark}

\begin{remark} \label{constructM} The ideals $M_i$ are supported on a normal crossing divisors $D_i$. Recall that their transformations after monoidal transformations, in the neighbourhood of a point $\xi\in W_i$, are $$D_i'=\left\{ \begin{array}{ll} D_i^*+(\theta_i-c_{i+1})\cdot Y' &
\text{ \scriptsize if } {\scriptstyle
(t_n'(\xi'),\ldots,t_{i+1}'(\xi')=(t_n(\xi),\ldots,t_{i+1}(\xi))}  \\ \emptyset & \text{ \scriptsize in other case} \end{array} \right. ,\ n\geq i\geq 1,$$
\hspace*{2.5 cm} (${\scriptstyle D_n'=D_n^*+(\theta_n-c)\cdot Y'}$ {\scriptsize always}) \vspace*{2 mm} \\ where $D_i^*$ denotes the pull-back of $D_i$ by the monoidal transformation $\pi$, $Y'$ denotes the new exceptional divisor, the point $\xi'\in W_i'$ satisfies $\pi(\xi')=\xi$, $\theta_i=ord_{\xi}(I_i)$ and $c_{i+1}$ is the corresponding critical value.
\end{remark}

In what follows we will define the ideals $J_{i-1}$, $n\geq i> 1$. We need some auxiliary definitions: the companion ideals $P_i$ and the composition ideals $K_i$, see \cite{strong} for details. \\ We construct the companion ideals to ensure that $Sing(P_i,\theta_i)\subset Sing(J_i,c_{i+1})$,
\begin{equation} \label{pes}
P_i= \left\{\begin{array}{ll} I_i & \text{ if }\ \theta_i\geq
c_{i+1} \\ I_i+M_i^{\frac{\theta_i}{c_{i+1}-\theta_i}} & \text{
if }\ 0< \theta_i< c_{i+1} \end{array}\right.
\end{equation} where $\xi\in \mathbb{A}^n_k$ is a point, $\theta_i=ord_{\xi}(I_i)$ and $c_{i+1}$ is the corresponding critical value.

Let $J_i=M_i\cdot I_i$ be the factorization of an ideal $J_i$ in $W_i$, where $M_i,I_i$ are ideals in $W_i$ in the neighborhood of a point $\xi\in \mathbb{A}^n_k$. Let $E_i$ be a normal crossing divisor in $\mathbb{A}^n_k$.

The composition ideal $K_i$ in $W_i$ of the product $J_i=M_i\cdot I_i$, with respect to a control $c_{i+1}$, is
\begin{equation} \label{qus}
K_i= \left\{\begin{array}{ll} P_i\cdot I_{W_i}(E_i\cap W_i) & \text{ if }\ I_i\neq 1, \\ 1 & \text{
if }\ I_i=1. \end{array}\right.
\end{equation}

The critical value for the following step of induction on the dimension is $c_i=ord_{\xi}(K_i)$.

The construction of the composition ideal $K_i$ ensures normal crossing with the exceptional divisor $E_i$.

We say that an ideal $K$ is \emph{bold regular} if $K=<X^a>$, $K\in k[X]$, $a\in \mathbb{N}$.

Finally, construct the junior ideal $J_{i-1}$
\begin{equation} \label{jotas}
J_{i-1}= \left\{\begin{array}{ll} Coeff_V(K_i) & \text{ if }\ K_i \text{ is not bold regular or } 1 \\ 1 & \text{
otherwise }\ \end{array}\right.
\end{equation} where $V$ is a hypersurface of maximal contact in $W_i$ (see \cite{strong} page $830$) and $Coeff_V(K_i)$ is the coefficient ideal of $K_i$ in $V$ (see \cite{strong} page $829$). The junior ideal $J_{i-1}$ is an ideal in this suitable hypersurface $V$ of dimension $i-1$.

If $\frac{\theta_{n}}{c}\geq 1$ we are in the first case of equation
(\ref{pes}), $\frac{\theta_{n-1}}{c_{n}}=\frac{\theta_{n-2}}{c_{n-1}}=\ldots
=\frac{\theta_j}{c_{j+1}}=1$ and $t_{j-1}=\ldots=t_1=\infty$ for $n-1\geq j\geq 1$, because $D_{n-1}=\ldots=D_1=\emptyset$ and $P_i=I_i$, and hence $J_{i-1}$
is always given by a unique monomial.

\begin{remark}
For an ideal $J=<X_1^{a_1}\cdot \ldots \cdot X_n^{a_n}>$ as in equation (\ref{jota}), if we assume $a_n\geq a_{n-1}\geq \ldots \geq a_1\geq c$, then at every stage $\frac{\theta_{n}}{c}\geq 1$
, so we are always in the previous situation. The singular locus of $(J,c)$ is always a union of hypersurfaces $\cup_{i=1}^r\{X_i=0\}$, $1\leq r \leq n$, and the center of the next monoidal transformation will be the intersection of some of these hypersurfaces. So we will call this case the {\bf minimal codimensional
case}.
\end{remark}

\begin{remark} \label{remgrande}
If there exists some $a_{i_0}<c$, at a certain stage of
the resolution process it may occur $\frac{\theta_{n}}{c}< 1$. Then
we are in the second case of equation (\ref{pes}), the (exceptional) monomial part $M_n$
can appear in some $J_j$ for $n-1\geq j\geq 1$, and
$\frac{\theta_{j}}{c_{j+1}}$ can be much greater than $1$, what
increase the number of monoidal transformations. Now its singular locus is a union of intersections of hypersurfaces of the type $\cup_{l_j}(\{X_{l_1}=0\}\cap \ldots \cap \{X_{l_i}=0\})$. This is the {\bf higher codimensional case}.
\end{remark}

\section{Bound in the minimal codimensional case}

\begin{remark} From now on, we always look to the points where the function $t$, defined in (\ref{invt}), is maximal. So the following results concerning the behaviour of the function $t$ always affect the points where it reaches its maximal value.
\end{remark}

\begin{proposition} \label{bajatn} Let $(W,(J,c),E)$ be a basic object where $J$ is a monomial ideal as in equation (\ref{jota}), with $a_i\geq c$ for all $1\leq i \leq n$. We can factor $J=J_n=M_n\cdot I_n$, and after $r-1$ monoidal transformations, $J_n^{(r-1)}=M_n^{(r-1)}\cdot I_n^{(r-1)}$. Let $\xi\in W^{(r-1)}$ be a point where  $ord_{\xi}(I_n^{(r-1)})=\theta_n$. After each monoidal transformation $\pi$, the resolution function in a neighbourhood of $\xi'$, where $\pi(\xi')=\xi$ and $ord_{\xi'}(I_n^{(r)})=\theta_n'<\theta_n$, is of the form $$\left(
\left[\frac{d-\sum_{j=1}^sa_{i_j}}{c},s\right],[1,0],\ldots,[1,0]\right)
\text{ for some }\  1\leq s\leq n-1. $$
\end{proposition}

\begin{proof}
After monoidal transformations, $$J_n^{(r)}=M_n^{(r)}\cdot I_n^{(r)}=<X_{i_1}^{b_1}\cdots
X_{i_s}^{b_s}>\cdot <X_{i_{s+1}}^{a_{i_{s+1}}}\cdots
X_{i_n}^{a_{i_n}}>$$  with
$d-\sum_{j=1}^sa_{i_j}=\sum_{j=s+1}^n a_{i_j}\geq c $ then,
$P_n^{(r)}=I_n^{(r)}$ and the (exceptional) monomial part does not appear in $J_l^{(r)}$
for all $n\geq l\geq 1$.

We have $\theta_n'\neq \theta_n$, then $E_n^{(r)}=Y'+|E|^{\curlyvee}$ and $m_n=s$, we count all the exceptional divisors of the previous steps and the new one. There are no exceptional divisors in lower dimension because $E_{n-1}^{(r)}=(Y'+|E|^{\curlyvee})-E_n^{(r)}=\emptyset$ and,
in a similar way, we obtain $E_l^{(r)}=\emptyset$ for all $n-1 \geq l\geq 1$.

The normal crossing divisors $D_i^{(r)}=\emptyset$ for all $n-1\geq i \geq 1$, so the corresponding ideals $M_{n-1}^{(r)}=\ldots=M_1^{(r)}=1$. In particular, $M_{n-1}^{(r)}=1$, hence
$$c_n'=ord_{\xi'}(K_n^{(r)})=ord_{\xi'}(Coeff(K_n^{(r)}))=ord_{\xi'}(J_{n-1}^{(r)})=ord_{\xi'}(I_{n-1}^{(r)})=\theta_{n-1}'$$ with $\xi'\in W^{(r)}$ such that $\pi(\xi')=\xi$, because $ord(Coeff(K))=ord(K)$ when $K$ is a monomial ideal, therefore $\frac{\theta_{n-1}'}{c_n'}=1$. By the same argument we obtain $\frac{\theta_{n-2}'}{c_{n-1}'}=\ldots=\frac{\theta_1'}{c_{2}'}=1$.
\end{proof}

\begin{remark} After each monoidal transformation, the exceptional divisors at each dimension are:
$$E_j'=\left\{ \begin{array}{ll} {\scriptstyle E_j^{\curlyvee}} &
\text{ \scriptsize if } {\scriptstyle
(t_n'(\xi'),\ldots,t_{j+1}'(\xi'))=(t_n(\xi),\ldots,t_{j+1}(\xi))} \text{ \scriptsize
and } {\scriptstyle \theta_j'=\theta_j} \\ {\scriptstyle
(Y'+(E_1\cup \ldots \cup E_n)^{\curlyvee})-(E_n'+\cdots
+E_{j+1}')} & \text{ \scriptsize in other case}
\end{array} \right.$$  \hspace*{0.5 cm}
${\scriptstyle ( E_n'=E_n^{\curlyvee}}$ {\scriptsize if}
${\scriptstyle \theta_n'=\theta_n}$ {\scriptsize or}
${\scriptstyle E_n'=Y'+(E_1\cup \ldots \cup E_n)^{\curlyvee}}$
{\scriptsize otherwise)} \vspace*{2 mm} \\ for $n> j\geq 1$, where $E_j^{\curlyvee}$
denotes the strict transform of $E_j$ by the monoidal transformation $\pi$, $Y'$ denotes
the new exceptional divisor, the point $\xi'\in W_i'$ satisfies $\pi(\xi')=\xi$, $\theta_j'=ord_{\xi'}(I_j')$ and $\theta_j=ord_{\xi}(I_j)$. We denote $|E|=E_1\cup \ldots \cup E_n$.

Hence, after the first monoidal transformation,
since $\theta_n'<\theta_n$ we have $E_n'=Y'$ and
$E_{n-1}'=\cdots=E_1'=\emptyset$. After the second monoidal transformation, at the
chart where $\theta_n''=\theta_n'$ we obtain
$E_n''=(E_n')^{\curlyvee}=\emptyset$, $E_{n-1}''=Y''$, and
$E_{n-2}''=\cdots=E_1''=\emptyset$ and so on. We call this
phenomena propagation because every exceptional divisor appears in the resolution function $t$
firstly in dimension $n$, then in dimension $n-1$, $n-2$, and so on.
\end{remark}

\begin{definition} We will call {\bf propagation}, ${\bf p(i,j)}$, for $1\leq i\leq
j-1$, $1\leq j\leq n$, to the number of monoidal transformations needed to
eliminate $i$ exceptional divisors in dimension $j$, when we remain
constant $(t_n,t_{n-1},\ldots ,t_{j+1})$ and $\theta_j$, and there are
no exceptional divisors in lower dimensions $j-1,\ldots,1$. That is, passing from the stage  $$([\theta_n,m_n],\ldots,[\theta_{j+1},m_{j+1}],[\theta_j,i],[1,0],\ldots,
[1,0])$$ to the stage $$([\theta_n,m_n],\ldots,[\theta_{j+1},m_{j+1}],[\theta_j,0],[1,0],\ldots,
[1,0],\overbrace{\infty,\ldots,\infty}^{i}).$$
\end{definition}

\begin{lemma}{\bf Propagation Lemma} Let $(W,(J,c),E)$ be a basic object where $J$ is a monomial ideal as in equation (\ref{jota}) with $a_l\geq c$ for all $1\leq l \leq n$. Let $p(i,j)$ be the propagation of $i$ exceptional divisors in dimension $j$ in the resolution process of $(W,(J,c),E)$. \\ Then, for all $1\leq j\leq n$,
\begin{equation} \label{pro}
p\ (i,j)=\left\{ \begin{array}{ll} i+\sum_{k=1}^i p\ (k,j-1) &
\text{\ if \ } 0\leq i\leq j-1  \\ 0  & \text{\ if \ } i=j \\
\end{array} \right.\end{equation}
\end{lemma}

\begin{proof}
\begin{itemize}
\item If there are $i$ exceptional divisors in dimension $i$, $K_{i+1}$ is bold regular, $t_i=\infty$ then $p(i,i)=0$. We can not propagate these $i$ exceptional divisors at this stage of the resolution process.

If there are $s$ exceptional divisors at this step of the resolution process, then there are $n-s$ variables in $I_n$. On the other hand, from dimension $n$ to dimension $i+1$ there are $s-i$ exceptional divisors.

When we construct $J_{n-1},\ldots ,J_{i+1}$, add to the corresponding composition ideal $K_j$ the variables in $I_{W_j}(E_j\cap W_j)$, so in these dimensions there are $(n-s)+(s-i)=n-i$ variables.

When we make induction on the dimension, at each step lose one variable, so in $n-i-1$ steps obtain that $K_{i+1}$, that corresponds to the $n-(n-i-1)=i+1$ position, is bold regular. And the variables appearing in these $i$ exceptional divisors do not appear in the center of the next monoidal transformation.

\item By induction on the dimension:
\begin{itemize}
\item[-] If $j=1$, $p(1,1)=0$ by the previous argument.
\item[-] If $j=2$, $p(1,2)=1$ because when we propagate $1$ excepcional divisor from dimension
$2$ to dimension $1$, $K_2'$ is bold regular.

\begin{center} \vspace*{2 mm}
\begin{tabular}{c}
  $([\theta_n,m_n],\ldots,[\theta_2,1],[1,0])$  \\
  $\downarrow {\scriptstyle X_i}$ \\
  $([\theta_n,m_n],\ldots,[\theta_2,0],\infty)$ \\
\end{tabular} \vspace*{2 mm}
\end{center}
Then $p(1,2)=1=1+0=1+p(1,1)$.

\item[-]We assume that the result holds for $j\leq s-1$. For $j=s$:
$$\begin{array}{c} ([\theta_n,m_n],\ldots,[\theta_{s+1},m_{s+1}],[\theta_s,i],[1,0],\ldots,
[1,0])  \\ \downarrow  \\
([\theta_n,m_n],\ldots,[\theta_{s+1},m_{s+1}],[\theta_s,i-1],[1,1],[1,0],\ldots,[1,0])  \\ \vspace*{0.2cm}
\left. \hspace*{2 cm} \begin{array}{r} \downarrow  \\ \vdots
\\ \downarrow  \end{array} \right\} p(1,s-1) \\ \vspace*{0.2cm} ([\theta_n,m_n],\ldots,[\theta_{s+1},m_{s+1}],[\theta_s,i-1],[1,0],\ldots,
[1,0],\infty)  \\ \downarrow  \\
([\theta_n,m_n],\ldots,[\theta_{s+1},m_{s+1}],[\theta_s,i-2],[1,2],[1,0],\ldots,[1,0])
\\ \end{array}$$

We want $([\theta_n,m_n]\ldots [\theta_{s+1},m_{s+1}])$ and $\theta_s$ remain constant. So after the first monoidal transformation look to some suitable chart where $m_s=i$ drops. As $m_s$ drops then $m_{s-1}=i-(i-1)=1$ and propagate this exceptional divisor in dimension $s-1$, making $p(1,s-1)$ monoidal transformations. Otherwise, to keep $([\theta_n,m_n]\ldots [\theta_{s+1},m_{s+1}])$ and $\theta_s$ constant, the only possibility is to look to a suitable chart where $m_s$ drops from $i-1$ to $i-2$. But in this case this would provide the same resolution function that appears after the propagation. As we want to construct the largest possible sequence of monoidal transformations, we follow the propagation phenomenon as above.

After more monoidal transformations:
$$\begin{array}{c} \left. \hspace*{2 cm} \begin{array}{r} \downarrow  \\ \vdots
\\ \downarrow  \end{array} \right\} p(2,s-1) \vspace*{0.2cm} \\
 \downarrow  \\ \vdots
\\ \downarrow  \\ ([\theta_n,m_n],\ldots,[\theta_{s+1},m_{s+1}],[\theta_s,1],[1,0],\ldots,
[1,0],\overbrace{\infty,\ldots,\infty}^{i-1}) \\  \downarrow  \\
([\theta_n,m_n],\ldots,[\theta_{s+1},m_{s+1}],[\theta_s,0],[1,i],[1,0],\ldots,[1,0])
\\ \end{array}$$
Then, for $1\leq i \leq s-1$,
$$p(i,s)=1+p(1,s-1)+1+p(2,s-1)+ \cdots +1+p(i,s-1)$$ with
$p(l,s-1)$, $1\leq l \leq i$, defined by the induction hypothesis.
\end{itemize}
\end{itemize}
\end{proof}

\begin{remark} Computation of examples in Singular with \emph{desing} package has been useful to state this behaviour of the exceptional divisors after monoidal transformations. The implementation of this package is based on the results appearing in \cite{paperlib}.
\end{remark}

\begin{theorem} \label{inv} Let $(W,(J,c),\emptyset)$ be a basic object where $J$ is a monomial ideal as in equation (\ref{jota}) with $a_i\geq c$ for all $1\leq i \leq n$. Then, the resolution function corresponding to $(W,(J,c),\emptyset)$ drops after monoidal transformations in the following form:
\begin{center}
\begin{tabular}{cl}
 \multicolumn{1}{c}{} &
 \multicolumn{1}{c}{} \\
$([\frac{d}{c},0],[1,0],\ldots,[1,0])$ & \\ $\hspace*{0.4 cm}
\downarrow {\scriptstyle X_i}$ & $1^{st}$  monoidal transformation \\
$([\frac{d-a_i}{c},1],[1,0],\ldots,[1,0])$ & \\ $ \hspace*{0.6 cm}
 \begin{array}{cc} \downarrow {\scriptstyle X_i} & \\ \vdots
\\ \downarrow {\scriptstyle X_i} & \\ \end{array} $ & $\left.
\hspace*{-7 mm} \begin{array}{c} \\ \\ \\ \end{array} \right\} p\
(1,n)$ monoidal transformations \\
$([\frac{d-a_i}{c},0],[1,0],\ldots,[1,0],\infty)$ & \\
$\downarrow$ & center defined only by variables in $I$ \\
$([\frac{d-a_i-a_j}{c},2],[1,0],\ldots,[1,0])$ & \\
$ \hspace*{0.3cm} \begin{array}{cc} \downarrow & \\ \vdots & \\ \downarrow &
\\ \end{array}$ & $\left. \hspace*{-7 mm} \begin{array}{c} \\ \\ \\
\end{array} \right\} p\ (2,n)$ monoidal transformations \\
$([\frac{d-a_i-a_j}{c},0],[1,0],\ldots,[1,0],\infty,\infty)$ & \\
$\downarrow$ & center defined only by variables in $I$ \\ $\vdots$ &
\hspace*{1 cm} $\vdots$ \\ $\downarrow$ & \\
$([\frac{a_l}{c},n-1],[1,0],\ldots,[1,0])$ & \\ $ \hspace*{0.3 cm}
\begin{array}{cc} \downarrow  & \\ \vdots & \\ \downarrow & \\
\end{array}$ & $\left. \hspace*{-7 mm}
\begin{array}{c} \\ \\ \\ \end{array} \right\} p\ (n-1,n)$ monoidal transformations \\
$([\frac{a_l}{c},0],\infty,\ldots,\infty)$ \hspace*{0.6 cm} & \\
\end{tabular}
\end{center}

At this stage, $a_l\geq c$ by hypothesis, so the center of the next monoidal transformation is $\{X_l=0\}$, and then we obtain an exceptional monomial.
\end{theorem}

\begin{proof}
It follows by the propagation lemma and the fact that each time that $\theta_n$ drops
$E_n'=Y'+|E|^{\nu}\neq\emptyset$, and $E_l'=(Y'+|E|^{\nu})-(E_n'+\cdots +E_{l+1}')=\emptyset$ for all
$n-1\geq l\geq 1$.
\end{proof}

\begin{remark}
Following the propagation in the previous way provides the largest branch in the resolution tree, because in other case, for example after the first monoidal transformation
$$\begin{array}{c} ([\frac{d-a_i}{c},1],[1,0],\ldots,[1,0])  \\
{\scriptstyle X_i}\swarrow \qquad \searrow {\scriptstyle X_j} \\ ([\frac{d-a_i}{c},0],[1,1],[1,0]\ldots,[1,0]) \qquad ([\frac{d-a_i-a_j}{c},2],[1,0],\ldots,[1,0])  \end{array}$$ looking to some chart $j$ with $j\neq i$
we obtain an invariant which will appear later in the resolution process, after the propagation $p(1,n)$.
\end{remark}

\begin{corollary}
Let $(W,(J,c),\emptyset)$ be a basic object where $J$ is a monomial ideal as in equation (\ref{jota}) with $a_i\geq c$ for all $1\leq i \leq n$. Therefore the number of monoidal transformations needed to transform $J$ into an exceptional
monomial is at most \begin{equation} \label{ec} 1 + p(1,n) +1 +
p(2,n)+ \ldots + 1 + p(n-1,n) + 1 =\  n + \sum_{j=1}^{n-1}p(j,n).
\end{equation}
\end{corollary}

\begin{remark} In this case we always have $\theta_n\geq c$, so
$Sing(J,c)\neq \emptyset$ at every stage of the resolution process. Therefore, in the resolution tree, the branch of
theorem \ref{inv} effectively appears, and it is the largest, hence (\ref{ec}) is exactly the number of monoidal transformations to obtain $J'=M'$.
\end{remark}

\begin{proposition} \label{cata} Let $(W,(J,c),\emptyset)$ be a basic object where $J$ is a monomial ideal as in equation (\ref{jota}) with $a_i\geq c$ for all $1\leq i \leq n$. Then the previous sum of propagations is a partial sum of Catalan numbers.  $$n +
\sum_{{\scriptscriptstyle j=1}}^{{\scriptscriptstyle n-1}}p(j,n)=
\sum_{{\scriptscriptstyle j=1}}^{{\scriptscriptstyle n}}C_j\
\text{  where }\ C_j=\left\{\frac{1}{j+1}\left(\begin{array}{c}
2j\\ j \end{array} \right)\right\}\ \text{ are Catalan numbers.}$$
\end{proposition}
\begin{proof}
\begin{enumerate}
\item[(1)] Extend $p$ to arbitrary dimension: $$n + \sum_{j=1}^{n-1}p(j,n)=p(n,n+1).$$ Because of the form of the recurrence equation defining $p(i,j)$, and the fact that $p(n,n)=0$ by definition, it follows that \begin{equation} \label{rec} p(n,n+1)=n + \sum_{j=1}^{n}p(j,n)=n + \sum_{j=1}^{n-1}p(j,n). \end{equation}
\item[(2)] Solve the recurrence equation defining $p(i,j)$:
\begin{enumerate}
\item We transform the recurrence equation (\ref{pro}), defining $p(i,j)$ for $0\leq i\leq j$ and $1\leq j\leq n$, to another recurrence equation defined for every $i,j\geq 0$:

By sending the pair $(i,j)$ to the pair $(i,j-i)$ we extend the recurrence to $i,j\geq 0$, that is, we consider $$\tilde{p}(i,j)=p(i,i+j)$$ then $p(i,j)=\tilde{p}(i,j-i)$. As $p(i,j)$ is defined for $0\leq i \leq j$ then $\tilde{p}(i,j)$ is defined for $0\leq i \leq i+j$ for every $i,j\geq 0$.

\item Note that $$\tilde{p}(i,j)-\tilde{p}(i-1,j+1)=p(i,i+j)-p(i-1,i+j)$$ $$=i+\sum_{k=1}^i p(k,i+j-1)-(i-1)- \sum_{k=1}^{i-1}p(k,i+j-1)=p(i,i+j-1)+1=\tilde{p}(i,j-1)+1.$$ Therefore, we have the following recurrence equation involving $\tilde{p}(i,j)$
\begin{equation} \label{chis} \left\{\begin{array}{ll}\tilde{p}(i,j)=1+\tilde{p}(i-1,j+1)+\tilde{p}(i,j-1) & \text{ for } i,j \geq 1 \\ \tilde{p}(0,j)=\tilde{p}(i,0)=0 & \end{array}\right. \end{equation}
Take $r(i,j)=p(i,i+j)+1=\tilde{p}(i,j)+1$ and replace $\tilde{p}(i,j)$ with $r(i,j)$ in the equation (\ref{chis}). It follows the auxiliary recurrence equation:
\begin{equation} \label{erres} \left\{\begin{array}{ll} r(i,j)=r(i-1,j+1)+r(i,j-1)& \text{ for } i,j \geq 1\\ r(0,j)=\tilde{p}(0,j)+1=1,\   r(i,0)=\tilde{p}(i,0)+1=1 & \end{array}\right. \end{equation}

\item Resolving the auxiliary recurrence equation (\ref{erres}) by generating functions:

Define $r_{i,j}:=r(i,j)$ and the generating functions $$R(x,y)=\sum_{i,j\geq 0}r_{i,j}x^iy^j \in \mathbb{C}[[x,y]], \   R_s(x,y)=\sum_{i,j\geq 1}r_{i,j}x^{i-1}y^{j-1} \in \mathbb{C}[[x,y]].$$ Note that $R(x,y)$ is, by definition, the generating function of the sequence $r(i,j)$.

By the recurrence equation (\ref{erres}) involving $r(i,j)$, it follows
$$\hspace*{-3.5cm} R_s(x,y)=\sum_{i,j\geq 1}r_{i-1,j+1}x^{i-1}y^{j-1}+\sum_{i,j\geq 1}r_{i,j-1}x^{i-1}y^{j-1}$$
$$ \hspace*{1cm} \begin{array}{l}
=\sum\limits_{i\geq 0,j\geq 1}r_{i,j+1}x^{i}y^{j-1}+\frac{1}{x}\sum\limits_{i\geq 1,j\geq 0}r_{i,j}x^{i}y^{j} \vspace*{0.15cm} \\
=\frac{1}{y^2}\sum\limits_{i\geq 0,j\geq 1}r_{i,j+1}x^{i}y^{j+1}+\frac{1}{x}\left[\sum\limits_{i\geq 1}r_{i,0}x^{i}+ \sum\limits_{i\geq 1,j\geq 1}r_{i,j}x^{i}y^{j}\right]  \vspace*{0.15cm} \\
=\frac{1}{y^2}\sum\limits_{i\geq 0,j\geq 2}r_{i,j}x^{i}y^{j}+\frac{1}{x}\left[\sum\limits_{i\geq 1}x^{i}+ \sum\limits_{i\geq 1,j\geq 1}r_{i,j}x^{i}y^{j}\right]  \vspace*{0.15cm} \\
=\frac{1}{y^2}\left[\sum\limits_{i\geq 0,j\geq 1}r_{i,j}x^{i}y^{j}-\sum\limits_{i\geq 0}r_{i,1}x^{i}y\right] +\frac{1}{x}\left[\frac{1}{1-x}-1+ xyR_s(x,y)\right]  \vspace*{0.15cm} \\
=\frac{1}{y^2}\left[\sum\limits_{j\geq 1}r_{0,j}y^{j}+xyR_s(x,y)-y\sum\limits_{i\geq 0}r_{i,1}x^{i}\right] +\frac{1}{x}\left[\frac{x}{1-x}+ xyR_s(x,y)\right] \vspace*{0.15cm} \\
=\frac{1}{y^2}\left[\frac{y}{1-y}+xyR_s(x,y)-y\sum\limits_{i\geq 0}r_{i,1}x^{i}\right] +\frac{1}{1-x}+ yR_s(x,y) \vspace*{0.15cm} \\
=\frac{1}{y(1-y)}+\frac{x}{y}R_s(x,y)-\frac{1}{y}\sum\limits_{i\geq 0}r_{i,1}x^{i}+\frac{1}{1-x}+ yR_s(x,y).
\end{array}$$

Then $$\left(1-y-\frac{x}{y}\right)R_s(x,y)=\frac{1}{y(1-y)}+\frac{1}{1-x}-\frac{1}{y}\sum_{i\geq 0}r_{i,1}x^{i}$$ multiplying the equality by \emph{y} we have $$(y-y^2-x)R_s(x,y)=\frac{1}{1-y}+\frac{y}{1-x}-\sum_{i\geq 0}r_{i,1}x^{i}$$
$$=\frac{1}{1-y}+\frac{y}{1-x}-r_{0,1}-\sum_{i\geq 1}r_{i,1}x^{i}=\frac{y}{1-y}+\frac{y}{1-x}-\sum_{i\geq 1}r_{i,1}x^{i}.$$ Therefore $$(y-y^2-x)R_s(x,y)=\frac{y}{1-y}+\frac{y}{1-x}-\sum_{i\geq 1}r_{i,1}x^{i}$$ which defines an equation of the form $$Q(x,y)R_s(x,y)=K(x,y)-U(x).$$ Now apply the \emph{kernel method} used in \cite{pet}, algebraic case $4.3$:

If $Q(x,y)=0$ then $y=\frac{1\pm \sqrt{1-4x}}{2}$. We take the solution passing through the origin, $y=\frac{1- \sqrt{1-4x}}{2}$ and $y=xC(x)$ where $C(x)$ is the generating function of Catalan numbers.

On the other hand, $Q(x,y)=0$ gives $K(x,xC(x))=U(x)$, $$K(x,y)=\frac{y}{1-y}+\frac{y}{1-x}=\frac{-y^2+y-x+1}{(1-x)(1-y)}-1$$ so $K(x,xC(x))=\frac{1}{(1-x)(1-xC(x))}-1$ and using $\frac{1}{1-xC(x)}=C(x)$ we have $$U(x)=\frac{C(x)}{1-x}-1.$$ Making some calculations and using that $R(x,y)$ satisfies $$R(x,y)=xyR_s(x,y)+\sum_{j\geq 0}r_{0,j}y^j+\sum_{i\geq 0}r_{i,0}x^i-r_{0,0}$$ we obtain the generating function of $r(i,j)$ $$R(x,y)=\frac{xyC(x)+x-y}{(y^2-y+x)(1-x)}.$$
\end{enumerate}

\item[(3)] Compute the generating function of the sequence $p(n,n+1)$:

The coefficient of \emph{y} in $R(x,y)$ is just $\sum_{i\geq 0}r_{i,1}x^{i}$ then $$\sum_{i\geq 0}r_{i,1}x^{i}=\frac{\partial R(x,y)}{\partial y}\Big|_{y=0}= \frac{C(x)}{1-x}$$ is the generating function of the elements in the first column.

If $C(x)$ is the generating function of $C_n$ then the convolution product $C(x)\cdot \frac{1}{1-x}$ is the genera\-ting function of $\sum_{k=0}^nC_k=S_n$ therefore $$r_{n,1}=\sum_{k=0}^nC_k.$$ As $r(n,1)=p(n,n+1)+1$ then $p(n,n+1)=r(n,1)-1=\sum_{k=0}^nC_k-1$, as $C_0=1$ we have $$p(n,n+1)=\sum_{k=1}^nC_k$$ where $C_k$ are Catalan numbers.
\end{enumerate}
\end{proof}

See \cite{stan} for more details about Catalan numbers and the web
page \cite{slo} for further details about their partial sums.

\begin{theorem} Let $(W,(J,c),\emptyset)$ be a basic object where $J$ is a monomial ideal as in equation (\ref{jota}) with $a_i\geq c$ for all $1\leq i \leq n$. Then the number of monoidal transformations required to resolve $(W,(J,c),\emptyset)$ is at most $$\sum_{j=1}^{n}C_j+(2^{\sum_{j=1}^{n}C_j}-1)(d-c)-c+1$$ where $C_j$ are Catalan numbers.
\end{theorem}

\begin{proof} It follows by theorem \ref{mon}, lemma \ref{grad} and  proposition \ref{cata}.
\end{proof}

\begin{example}
The following table shows some values of the bound for any monomial ideal $J$ as in equation (\ref{jota}) with $a_i\geq c$ for all $1\leq i \leq n$.

\vspace*{0.6cm}

\begin{table}[ht] \vspace*{-0.6cm}
\caption{Values of the bound}
\begin{center}
\begin{tabular}{||c|c|c||} \hline
 {$n$}  & $\sum_{j=1}^{n}C_j$ & global bound \\ \hline
 $1$ & $1$ & $1+(d-c)-c+1$ \\ \hline
 $2$ & $3$ & $3+7(d-c)-c+1$ \\ \hline
 $3$ & $8$ & $8+255(d-c)-c+1$ \\ \hline
 $4$ & $22$ & $22+4194303(d-c)-c+1$ \\ \hline
\end{tabular}
\end{center}
\end{table}
\end{example}

\vspace*{-0.5cm}

\begin{remark} Note that, as a consecuence of proposition \ref{cata}, the number of monoidal transformations needed to transform $J$ into an exceptional monomial only depends on $n$, the dimension of the ambient space.
\end{remark}

\begin{corollary} Let $J=<Z^c-X_1^{a_1}\cdot \ldots \cdot X_n^{a_n}>\subset k[X_1,\ldots,X_n,Z]$ be a toric ideal with $a_i\geq c$ for all $1\leq i \leq n$. Then the number of monoidal transformations needed to resolve $(\mathbb{A}^{n+1}_k,(J,c),\emptyset)$ is at most $$\sum_{j=1}^{n}C_j+(2^{\sum_{j=1}^{n}C_j}-1)(d-c)-c+1 $$ where $C_j$ are Catalan numbers and $d=\sum_{i=1}^n a_i$.
\end{corollary}

\section{Higher codimensional case}

In the minimal codimensional case, the way in which the invariant drops essentially
depends on the number of accumulated exceptional divisors. Because the first components of the invariant, $\theta_{n},\ldots,\theta_{1}$, defined in equation (\ref{invt}), only depend on the order of the ideals $I_n,\ldots,I_1$. Recall that, for each $J_i$, we use the ideal $M_i$ (see remark \ref{constructM}), to define the ideal $I_i$.

But in this case the first components of the invariant play an important
role. They can also depend on the order of the (exceptional) monomial
part $M_n$, see remark \ref{remgrande}. So they may increase suddenly when some $\theta_{j}$ is given by the order of the ideal $M_n$. We will call this situation the {\it higher codimensional case in dimension $j$}.

Note that after some monoidal transformations, we can obtain a new higher codimensional case in another dimension.

So, we must compute the number of monoidal transformations while $\theta_n\geq c$, with a
suitable sum of propagations. Then, estimate the number of monoidal transformations needed to get the higher codimensional case in dimension $1$, and use the known estimation for the
order of $M_n$ to give an upper bound for the number of monoidal transformations needed to get the
following higher codimensional case inside this one (if it is possible). Afterward, estimate the number of monoidal transformations needed to get the higher codimensional case in dimension $2$,
and so on.

Hence, it has not been possible to obtain a bound for this case in
the same way as above, due to the complications of the
combinatorial problem, that perform that we can not know what
branch is the largest in the resolution tree (to obtain an
exceptional monomial).

Furthermore, if we could find such bound, the large number of potential cases we expect, suggests that this bound would be very huge, even to estimate only the number of monoidal transformations needed to obtain an exceptional
monomial.

\begin{example} If we consider the basic object $(W,(J,c),\emptyset)=(\mathbb{A}^3_k,(X_1^5X_2^4X_3,4),\emptyset)$, there exists a branch of height $15$ to obtain $J'=M'$ or
$Sing(J',c)=\emptyset$. So, in dimension $3$, we need a bound
greater than or equal to $15$ for a higher codimensional case, in front of the $8$ monoidal transformations needed for a minimal codimensional case.
\end{example}

\begin{remark}
In any case, both theorem \ref{mon} and lemma \ref{grad} are valid also in the higher codimensional case. So the open problem is to find a bound $C$ to obtain an
exceptional monomial, to construct a global bound of the form
$$C+(2^C-1)(d-c)-c+1 .$$
\end{remark}

\begin{remark}
For $n=2$ the higher codimensional case appears only in dimension $1$ and making
some calculations we obtain $C=3$, that gives the same bound as in
the minimal codimensional case. This bound can be improved by studying the
different branches.
\end{remark}

\section*{Acknowledgments}

I am very grateful to Professor O. Villamayor for his suggestions to improve the writing of this paper, and Professor S. Encinas for his continued advice and help. I also would like to thank the anonymous referee for useful comments to improve the presentation of the paper.

redpajama_set_name: RedPajamaArXiv


text:
Besides being scientifically illiterate, why does Abbot place so much importance on acquiring a foreign tongue?
"Mr Abbott's goals for Australia's linguistic future can be achieved only with the express support of state leaders, who must mandate the study of at least one language other than English for all students from Year 4/5 to Year 10." And "The Leader of the Opposition's proposal to have 40% of Year 12 students learning a foreign language within a decade is seriously intelligent policy." Fiona Mueller
More seriously unintelligent thinking from Tony Abbot?
Our children's study time is limited. What they learn before they move out of the education system and into the rest of their lives is crucial. The most successful people are those who are comfortable with their own existence. We have a responsibility to shape the minds of the young when it is most plastic in a way which will enhance their chances of being at ease with their own existence. In other words, we must ensure that they have the necessary life-philosophy.
In the matter of language, John McCrone, author of The Ape that Spoke,has something of far more substance to say than has Tony Abbot:
We arrive in the world with the naked brain of an animal and through the moulding power of speech, we become equipped with the thought habits which make us human.
What makes us human! Contemplations at that level are seriously intelligent thinking.
Tony Abbot has studied Latin and Ancient Greek. An interesting and probably enjoyable mental exercise, but it did nothing to warn him against making bizarre statements regarding climate change and the national optic fibre rollout.
Besides being scientifically illiterate, why does Abbot place so much importance on acquiring a foreign tongue? It's the global market place he sees where riches beckon. But the evolution of differing languages was never a barrier to the spread of ideas. In the 21st century, this has never been truer.
Setting the right priorities
In my final two years at school, the question as to what use was a BA majoring in a language going to be to anyone was occasionally asked. The stock answer was that it might lead to a career in the diplomatic service. The clear message was that learning a foreign language was not the means to the making of a living to aim for.
Today the focus is no longer on doing business with Britain. We are on the world stage and, if you can speak Chinese, a company which does business with China could use you.
However, in the bigger picture, I cannot see much urgency in learning Chinese or Japanese when public signage in China and Japan is becoming bilingual - with that second language being our native tongue. English is the first language of India's middle class. Indonesia is doing what the rest of South East Asia has been doing for years - which is to use English as the language to do business in. So, why not just sit back and wait for the rest of the word to learn to speak the way we already can?
Besides, English is the world language of science and technology - and that is what really matters.
How important is knowing why we are who we are?
Such knowledge has become increasingly important as society became less religious. Religion built a deep structure into our lives. It was the reference point. It was where we touched base. Scientific revelation can now fulfil this role. An awareness of the process of the acquisition of language is an example.
"Astonishing" hardly seems to be the appropriate word to describe the acquisition of language. At two and one half years my grand-daughter was putting sentences of three or four words together, but her words were difficult to understand. At age five she can now express simple ideas.
While others thought her way of speaking as being cute, as a reader of the new books of revelation, I saw the wonder in it. I was aware that I was observing the results of the neural networks in her speech centres being steadily hardwired - seemingly by the week. This was not cute. This was awesome.
The brain cells of my grand-daughter function very similarly to those of a fish and the structure of her brain is almost the same as an ape. It is social contact which is accelerating the growth and abilities of her frontal cortex in a feedback with the developing speech centres in her brain.
René Descartes famously said: "I think - therefore I am." Could Descartes think without using words? No, he could not. You might say that you are not thinking in words about what you are doing when you are driving (e.g. you don't have to say: "I now have to turn the steering wheel to turn around that corner.") However, when learning to drive, then talking to yourself is what you did. Now the network of brain cells that was constructed by words holds the program enabling you to drive.
If Descartes was alive today he would be saying: "I speak - therefore I am."
Abbot fails to see wealth where it really is
The humanities people fear that an increasing emphasis on science and mathematics will desensitise the young mind to the feeling side of human nature - the side which enjoys poetry and music. It would be a very rare scientist who does not have a keen (even passionate) interest in some aspect of life outside of science. And science tells the human side of history which the history of nations our children study does not.
Imagine a camp fire scene of 50,000 years ago. Those huddled together from the cold and staring into the light are making many sounds that those of 50,000 years earlier were not making. The sounds we now call words are connected into strings we now call sentences. The strings are very short. Linking the names of objects we now call nouns there are action words we now call verbs. The human ear is ready for many more differing words and the human tongue and larynx ready to utter many more differing words. Words we now call adjectives will be added. This will occur around camp fires in the millennia ahead when ideas of increasing complexity will be exchanged and entertaining stories told.
In the above scene stretched over millennia the most important series of events in the history of the planet is occurring. The software and the hardware are developing each other in the brains of our ancestors. One day that brain being shaped around campfires will be composing symphonies, designing 100-story buildings, replacing defunct human organs sending probes to the edge of the solar system.
Science tells other stories which remind us of the other life forms who we should be unselfishly sharing the planet with and not destroying in the pursuit of economic growth. To regain the spiritual that religion once provided, there is a lot to learn from the new books of revelation.
Our children do not have the time to be distracted by Tony Abbot's interpretation of enlightenment. We don't need 40% of our senior students ploughing through text books and sitting for exams in some foreign tongue because some politicians believe we need a sales advantage in the world market place more than we need the holistic and nourishing life-philosophy that scientific revelation is waiting there to give all who show an interest in receiving it.
Brian Holden has been retired since 1988. He advises that if you can keep physically and mentally active, retirement can be the best time of your life.
This article was first published online at Online Opinion


text:
Ksienija Nikołajewna Pałkina, też Ksienija Ułukan (ur. 13 grudnia 1989 w Chabarowsku) – kirgiska tenisistka.

Kariera tenisowa
Zadebiutowała w październiku 2004 roku, otrzymując od organizatorów dziką kartę, na udział w kwalifikacjach do turnieju WTA w Taszkencie. Debiut nie był udany i tenisistka odpadła już po pierwszym meczu. W maju 2005 roku wygrała kwalifikacje do turnieju ITF w Ahmadabadzie, w Indiach i w turnieju głównym dotarła do ćwierćfinału gry pojedynczej. W końcówce roku odniosła swoje pierwsze sukcesy w grze podwójnej, w rozgrywkach cyklu ITF, rozgrywanych w Indiach. Najpierw w Bombaju dotarła do finału, a potem w Pune wygrała turniej, w obu tych przypadkach partnerując Włoszce Nicole Clerico. W styczniu 2006 roku, po raz pierwszy w karierze, osiągnęła finał gry singlowej w Nowym Delhi a tydzień później wygrała turniej singlowy w Muzaffarnagar, pokonując w finale reprezentantkę gospodarzy Meghhę Vakarię. W sumie wygrała jedenaście turniejów singlowych i dwadzieścia siedem deblowych rangi ITF.

We wrześniu 2007 roku zagrała w kwalifikacjach do turnieju WTA w Taszkencie, które wygrała, pokonując w decydującym meczu Marię Korytcewą i po raz pierwszy zagrała w turnieju głównym. Osiągnęła w nim ćwierćfinał, wygrywając w dwóch pierwszych rundach z Kaciarynę Dziehalewicz z Białorusi i Vladę Yekshibarovą z Uzbekistanu. W ćwierćfinale przegrała z Wiktoryją Azaranką. W następnym roku zagrała w kwalifikacjach do turnieju wielkoszlemowego US Open, ale przegrała z Marią Eleną Camerin. Potem ponownie wygrała kwalifikacje w Taszkencie, ale w turnieju głównym wygrała tylko jeden mecz, z Olgą Goworcową, a w drugiej rundzie, po trzysetowym meczu przegrała z Rumunką Ioaną Raluca Olaru. W roku 2009 wzięła udział w kwalifikacjach do wszystkich czterech turniejów wielkoszlemowych, ale nie udało jej się przejść etapu kwalifikacji i zagrać w fazie głównej.

W marcu 2009 roku osiągnęła najwyższą w karierze pozycję w światowym rankingu, plasując się na 163 miejscu.

Tenisistka była także reprezentantką kraju w rozgrywkach Pucharu Federacji.

W 2022 roku została zdyskwalifikowana na szesnaście lat (w tym sześć lat w zawieszeniu), z rozpoczęciem upływu kary datowanym na 22 listopada 2019 roku, za ustawianie meczów i naruszenie zasad antykorupcyjnych.

Wygrane turnieje rangi ITF

Gra pojedyncza

Przypisy

Bibliografia

Kirgiskie tenisistki
Urodzeni w 1989
Ludzie urodzeni w Chabarowsku
Tenisiści ukarani dyskwalifikacją

redpajama_set_name: RedPajamaWikipedia


NASHVILLE (CelebrityAccess) — Nashville-based management duo of Doc McGhee and Scott McGhee are officially parting ways, at least as business partners.
The two brothers announced the windup of their 25-year business partnership in McGhee Entertainment with Scott McGhee launching his own company 1˚ Entertainment Music & Management Group.
1˚ Entertainment will provide brand consulting, digital and social media strategy consultation, as well as content creation for alternative distribution with an eye towards artist tie-ins.
Doc McGhee Entertainment will continue "business as usual" and Doc is currently working on an autobiography about his experiences working with artists such as Kiss, Bon Jovi, Motley Crue, and Hootie & The Blowfish.
According to Music Row, the two brothers will continue to share an office at 21 Music Square West in Nashville.


text:
L'antigene Ki-67 è una proteina nucleare strettamente associata con la proliferazione cellulare. Esso può essere riscontrato esclusivamente all'interno del nucleo durante l'interfase e, dal momento che è presente durante tutte le fasi del ciclo cellulare (G1, S, G2, mitosi) ma è assente nella fase G0, il Ki-67 rappresenta un utile marker della cosiddetta frazione di crescita di una data popolazione di cellule.
In ambito clinico, l'indice del Ki67 (la frazione di cellule tumorali positive all'antigene espressa in percentuale), è spesso correlato con il decorso clinico della malattia neoplastica. Esso viene determinato mediante l'utilizzo di anticorpi monoclonali.

Altri progetti

Proteine umane
Citologia
Oncologia
Anatomia patologica


text:
Leonhard Stock (Finkenberg, 14 maart 1958) is een Oostenrijks oud-alpineskiër.

Stock was tweemaal deelnemer op de Olympische Winterspelen (in 1980 en 1988), waarbij hij in 1980 de gouden medaille op de afdaling veroverde. In 1980 veroverde hij ook brons bij de Wereldkampioenschappen alpineskiën op de combinatie welke tegelijkertijd met de Winterspelen plaatsvond.

In de wereldbeker alpineskiën boekte hij drie keer een dagzege op de afdaling. Op 6 januari 1989 boekte hij zijn eerste zege in Laax (Zwitserland), zijn tweede zege was op 8 december 1990 in Val-d'Isère (Frankrijk) en de derde en laatste zege noteerde hij op 12 december 1992 in Val Gardena (Italië).

Hij stond tussen 1977 en 1992 in totaal 27 keer in de top-3 en 112 keer in de top-10 op de afdaling, Super G en combinatie op de OS, bij de WK en in de wereldbeker.

Erelijst

OS / WK
 N.B. de Olympische Spelen van 1980 waren tevens de wereldkampioenschappen alpineskiën (de combinatie was geen olympische discipline)
 1980 · OS Lake Placid
  op de afdaling (OS/WK)
  op de combinatie (WK)

Oostenrijks alpineskiër
Oostenrijks olympisch kampioen


The flaunt-worthy Samsung Galaxy A9 Pro offers glitch-free performance along with a long battery life.
The Samsung Galaxy A9 Pro, with its unique glass and metal design, and narrow bezel, ensures you stay in vogue.
The 15.19 cm FHD sAMOLED screen offers an immersive viewing experience so you can enjoy watching movies on Netflix and YouTube with a theatre-like clarity. The Gorilla Glass 4 adds to the durability of this smartphone.
If you are always on the move and your phone is your favorite companion, then you will love this Android phone. Do more, even when you are on the go, without worrying about running out of charge as the Samsung Galaxy A9 Pro is powered by a 5000 mAh battery and offers fast charging.
Do you hate it when your phone hangs while switching between apps? With this dual SIM smartphone, you won't face lag as it's powered by a 64-bit octa-core processor and coupled with 4 GB of RAM.
Never miss an important moment. With this Samsung phone's 16 MP rear camera and 8 MP front camera, you can capture beautiful memories without worrying about the lighting as it comes with an F1.9 brighter lens and OIS (optical image stabilization).
Movies, old and new, trending viral videos, selfies in all angles, music from the 80s to the top-trending list. Store all that you want without worrying about running out of space as this phone comes with 32 GB internal space which can be expanded by up to 256 GB via a microSD card.
Hey guys. Below are some pros and cons which i felt would come to your help while purchasing this phone.
Pictures taken at low light are bad.
Black piece is a master of finger print magnet.
Feels a bit heavy due to tge bigger battery.
Display is ultimate at mid brightness.
Basic specs are top notch but could have had a 64gb ROM.
Design goes well with the weight.
The delivery was on time. First let me say the mobile its very very good. I was not so happy with two things.
One - After using A7 i felt the screen resolution of A9 was not as sharp. It is because the PPI is lesser than A7.
Two - A9 a bit heavier.
But overall a good phone.
It is FAST Memory is Good.
but i am facing heating issues while gaming and charging.
i am satisfied with the overall performance of the Phone.
Good phone. Only difficulty is that no back cover and screen guards available for this model still date.
But features are Good. Slim phone.
Looks Stylish (I have a black).
Feels good in hand )I read people complaining about weight issue trust me it's not that heavy).
Q:Specs says that no hybrid sim slot. Does it mean separate slots for 2 SIMs & 1 SD card? Earlier answered questions says only 2 slots are there. Is it modified to 3 slots?
Q:can Gorill glass breakable ?
A:Yes, it dam breaks and costs you 12,000Rs....!
Q:This phone has fingerprint sensor????
A:yes.. it has.. I'm using it..
Q:Is anyone using samsung pay on A9 Pro? Is it Supports Samsung Pay?
A:yes .it supports samsung pay.
Q:why it is not available on SBI credit card?
Q:what is the cost of this phone in market????


text:
People, not passwords, are the key to security
People are the biggest security threat facing IT, a report says. That is not where the problem lies
By Leader | November 1, 2004 -- 12:44 GMT (04:44 PST) | Topic: Networking
Students of the human condition will find little to surprise them in Gartner's latest report. It says that social engineering – duping people – will be the biggest security risk for companies and individuals over the next decade. In other words, there's no point in setting up a biometric access, triple-DES encryption, policy managed and physically secure server if the users can be persuaded to misbehave.
Confidence tricksters, like the poor, will be with us always. Mankind's earliest myths talk of deception and lies, and we have yet to break the habit. As the siege of Troy showed, when the physical defences get good enough, humans become the weakest link – and while we can always re-engineer our machinery, we are stuck with people.
All of which teaches us a lesson that IT would much rather ignore: people should come first, programmers second. We see it in email systems that can embed live data objects in messages because that is cool and easy to program, but do not have proper message threading. We see it in open source, where usability is harder to come by than a copy of the GPL with Bill Gates' signature on the bottom.
We especially see it in online security, where the user is supposed to remember all manner of things – tiny yellow padlocks, checking URLs for https://, and a different password for every site – and to be responsible for filtering safe options from heavily disguised con jobs. People cannot manage security well in real life, so why do security designers assume otherwise in the virtual world where by definition nothing is quite what it seems?
Computer security is designed by engineers and sold by marketing departments. Neither group is known for its deep insights into human behaviour, although both have considerable self-confidence that their way is the right way if only the rest of the world would fall into line. Well, that ain't going to happen – something the open source community is discovering now that most of its users aren't also developers.
There are two groups of people who must get much more involved in IT design, security and otherwise, now that the days of the expert user are irretrievably past. Humanities experts are one group – anthropologists, sociologists, psychologists, graphics designers, even dramatists – while the other is the user base itself.
Look at the Bugtraq entries for any major open source effort, and more than 90 percent of the problems reported are feature-based. The problems ordinary people have with software are overwhelmingly usability related – but the reporting mechanisms that reach designers might as well be written in Sanskrit.
There are no forums for the feedback of ordinary users to design teams. There are no wide-scale usability studies by security companies, let alone ones that use the Internet to reach out to the very people most at threat. Saying that 'people are the problem' is getting the issue precisely wrong: people - - not data, not security, not network management -- are the very core of IT's purpose and reason to exist.
It is sad that after more than fifty years of commercial computing this lesson still has to be learned. It has never been more pressing.
Cloud Internet of Things Security Data Centers
More from Leader
Government : UK
Microsoft vs Google: Birth of a new world
Stop the data-roaming madness
Dell and HP: a last chance for relevance
Vote for the future we need
A decline in usage of voice minutes from mobiles joins the trend of disappearing fixed-line calls, as users switch to over-the-top services.
Huawei and 5G: Why the UK's decision is getting tougher every day
The prime minister's decision about Huawei will say much about the UK's place in the world after Brexit.
Mozilla lays off about 70 employees including senior staffers
Known best for its Firefox browser, Mozilla laid off approximately 6% of its staff in the face of diminishing revenue.
ACMA orders ISPs to block nine more gambling sites
It comes as the regulator continues to exercise its power under s313.
Senator unveils bill to stop the US from sharing intel with countries using Huawei 5G
The legislation comes just as the UK is set to decide whether to upgrade its telecom network with Huawei.
TPG complains of 'wireless attack' by 5G on fixed line broadband
If only a certain telco had not dropped plans to create its own mobile networks, nor try to merge with the number three mobile operator.


text:
//Webix CDN files are available online for HTTP or HTTPS including their fonts
//Place the following lines in your HTML file

//    <link rel="stylesheet" href="https://cdn.webix.com/edge/webix.css" type="text/css">
//

// webix supports a simple require, but only relative to the 'codebase'
// folder, so we can't go there much...

//console.log ( 'codebase' , webix.codebase );
//console.log ( 'url' , window.location , location );

//Here for reference, not needed for this script
// webix has no concept of query string params or pathnames in any
// reasonable form, so we just use a home grown query string hack
/*
function GetQueryStringParams(sParam) {
    var sPageURL = window.location.search.substring(1);
    var sURLVariables = sPageURL.split('&');
    for (var i = 0; i < sURLVariables.length; i++) {
        var sParameterName = sURLVariables[i].split('=');
        if (sParameterName[0] == sParam)
        {
            return sParameterName[1];
        }
    }
    return null;
}
var proxy = GetQueryStringParams('p');
*/

//Variable for the initial number of columns before we add EAs, Options, etc.
var NUM_COLUMNS=8;

//---------------------------------
// set up the REST location
//var myServer = proxy ? location.host + '/proxy/' + proxy : location.host ;
var myServer = location.host ;
var base_url = "https://" + myServer + "/wapi/v1.4/";
//console.log ( 'rest server' , base_url );

//---------------------------------
//Initialize the Webix Ready and Helper functions

//---------------------------------
// MAIN ENTRY POINT

webix.ready(function(){
    renderMain();
});

//---------------------------------
// HELPER Functions
webix.error = function(myText) {
    webix.message({type:"error", text: myText});
};

//---------------------------------
// Data controls

//---------------------------------
// Define our UI

var gridPanel = {
    rows: [
//---------------------------------
//First row is our form
//---------------------------------
    {
        view:"toolbar",
        id:"myToolbar",
        cols:[
            { view:"text",
                id: 'query',
                //label:"Hostname|Prefix",
                placeholder: 'hostname/ip (regex)',
                width: 200,
                //value:'linux',
                name: 'query' },
			{ view:"select",
				id: 'querytype',
				name: 'querytype',
				options: ['Name','Alias','IPv4','IPv6','MAC','Comment'],
				width: 100
			},
            { view:"button", id:"gridDelBut", value:"Search",
//When this search button is clicked, we will do an onGridSearch()
                click: 'onGridSearch',
                width:100, align:"left" },
            { view:"text",
            	id: 'max_results',
                //label:"Hostname|Prefix",
                label: 'Max Results',
                labelWidth: 100,
                labelAlign: 'right',
                placeholder: 'Max Results',
                width: 200,
                value:'100',
                name: 'max_results' },
            { view:"text",
            	id: 'total_records',
                //label:"Hostname|Prefix",
                label: 'Returned Records:',
                labelWidth: 125,
                labelAlign: 'right',
                placeholder: '0',
                width: 200,
//                value:'0',
                name: 'total_records'},
        ]
    },
//---------------------------------
//Second row is our table
//---------------------------------
    {
        id: "gridPanelGrid",
//Helper functions
        on:{
          onBeforeLoad:function(){
            this.showOverlay("Loading...");
          },
          onAfterLoad:function(){
            if (!this.count()){this.showOverlay("Sorry, there is no data");}
            else{this.hideOverlay();}
          }
        },
        view:"datatable",

//Set all of the table attributes
        select:"cell",
        multiselect:true,
        clipboard:"selection",
        resizeColumn: true,
//Uncomment the following if you want pagination
/*
//Link Pager to "pager" div ID from main HTML file
        pager:{
          template:"{common.first()} {common.prev()} {common.pages()} {common.next()} {common.last()}",
          container:"pager",
          autosize:true,
          group:10
    	},
*/
        leftSplit:3,
        minColumnWidth:150,
		//autoheight:true,
		//autowidth:true,
        //footer:true,
        //autoConfig:true
        // we /could/ use autoConfig, but it sets default widths (50px)
        // and the effort to walk the dataaset and resize everything
        // calling datatable.adjustColumn("title", "header");
        // is ineffecient since we are calling things like 'parse' that
        // will have done it N times already
        //
        // So we manually define the base columns, then add additional
        // columns later
//---------------------------------
//Define our Columns in the table here
//---------------------------------

        columns:[
            //autowidth:true,
            // sort == int or string
            // adjust:"data" : set col width to widest item
            // adjust:"header" : set col width to header string
            // fillspace:true : column widens to all avallable space
//            {id:"_ref", hidden: true, header:"ref", adjust:"data" },
            {id:"view", header:["View",{content:"selectFilter"}], adjust:"data", sort:"string"},
            {id:"name", header:["Name",{content:"textFilter"}], adjust:"data", sort:"string"},
            {id:"aliases", header:["Aliases",{content:"textFilter"}], adjust:"data", sort:"string"},
            {id:"ipv4addrs", header:["IPv4 Address",{content:"textFilter"}], adjust:"data", sort:"string"},
            {id:"ipv6addrs", header:["IPv6 Address",{content:"textFilter"}], adjust:"data", sort:"string"},
            {id:"zone", header:["Zone",{content:"selectFilter"}], adjust:"data", sort:"string"},
            {id:"comment", header:["Comment",{content:"textFilter"}], sort:"string"},
//Have a hard time sorting and filtering on boolean values, even if I convert them .toString(), so just make it sortable
            {id:"disable", header:"Disabled", rowspan:2, adjust:"header", sort:"string"}
//Can't do this because the select box uses the original values
//            {id:"disable", header:["Disabled",{content:"selectFilter"}], adjust:"header", sort:"string", format:function(value){
//				value=='true' ? value="Yes" : value="No";
//				return(value);
//            	}
//            }
        ]
    },
//---------------------------------
//Third & Forth row is our table to bound the datatable
//---------------------------------
    {
    view:"resizer",
    },
    {
    view:"template",
    template:"Written by Brad Andersen (brad@bradandersen.com)",
    type:"clean",
    height:40
    }
    ]

};

//---------------------------------
// UI Controllers

//Run this when the search button is pressed
function onGridSearch() {
    var myGrid = $$("gridPanelGrid");

//Remove any rows from the table from a previous run
    myGrid.clearAll();
    myGrid.showOverlay("Loading...");

// Setup the AJAX query
    var params={
            '_max_results' : $$("myToolbar").getValues().max_results,
            '_return_fields%2B' : 'comment,disable,view,zone,extattrs,aliases'
    };
    var query=$$("myToolbar").getValues().query;
    var querytype=$$("myToolbar").getValues().querytype;
    var queryop="~:"; //Case-insensitive regular expression search
    var queryobj="name";
    if(querytype=="Alias"){var queryobj="alias";}
    if(querytype=="Comment"){var queryobj="comment";}
    if(querytype=="MAC"){var queryobj="mac";}
    if(querytype=="IPv6"){var queryobj="ipv6addr";}
//IPv4 search do not have letters, so case-sensitivity does not work in the Infoblox API
    if(querytype=="IPv4"){var queryop="~"; var queryobj="ipv4addr";}
//Set a query operation if the query field has data
    if(query){
      var toquery=queryobj+queryop;
      toquery=toquery.toString();
      params[toquery]=query;
//console.log(toquery);
    }

//console.log(params);
    webix.ajax().get( base_url + 'record:host', params,
        // handlers
        {
            error:function(text, data, request){
                webix.error(request.status+':'+text);
            },
//Take the data.json from Infoblox API and add it to the table
            success:function(text, data, request){
                //console.log ( 'hosts' , data.json() );
                var myGrid = $$("gridPanelGrid");
                myGrid.hideOverlay();
                myGrid.showOverlay("Processing...");
                addResultsToGrid( data.json() );
            }
        });

}

//---------------------------------
//Add results to Webix datatable
//---------------------------------

function addResultsToGrid( json ) {
    var myGrid = $$("gridPanelGrid");

    // if you add rows to the grid, they don't autosize or support
    // autoConfig, so you should parse the whole dataset at once, clean
    // it up, then punt it to the renderer

    // walk each element of the grid and clean it up
    // and track all unique EAs
    var eaValues = {};
    var jsonarray=webix.toArray( json );
    jsonarray.each( function(rec) {

// flatten addresses, aliases, macs, Extensible Attributes
        var ipv4addrs = [];
        var ipv6addrs = [];
        var aliases = [];
//Does not work trying to turn a boolean into a string, webix.datatable still can't sort/filter
//        if (rec.disable){
//           rec.disable=rec.disable.toString();
//        }
        if ( rec.ipv4addrs ) {
            // use the webix toArray() + each()
            webix.toArray( rec.ipv4addrs ).each( function(ip) {
                ipv4addrs.push( ip.ipv4addr );
            });
            if ( ipv4addrs.length > 0 ) {
              rec.ipv4addrs = ipv4addrs.join(", ");
       		}

        }
        if ( rec.ipv6addrs ) {
            webix.toArray( rec.ipv6addrs ).each( function(ip) {
                ipv6addrs.push( ip.ipv6addr );
            });
            if ( ipv6addrs.length > 0 ) {
              rec.ipv6addrs = ipv6addrs.join(", ");
            }
        }
        if ( rec.aliases ) {
            rec.aliases=rec.aliases.join(", ");
//			console.log(rec.aliases);
        }

        // flatten EAs and add them as unique fields
        if ( rec.extattrs ) {
            Object.keys(rec.extattrs).forEach(function(key,index) {
//console.log(key+'='+JSON.stringify(index));
                var eaname = 'EA_'+key;
				//Replace spaces and dashes with underscores or webix.datatable will wrap the headers
                eaname=eaname.replace(/[\s\-]/ig,"_");
                rec[eaname] = rec.extattrs[key].value;
//console.log(rec.extattrs[key].value," -",eaname," - ",rec[eaname]);
                // and track this new EA in a hash
                eaValues[eaname] = 1 ;
            });

        }
//console.log ( 'rec' , rec );

    });

    // now add unique EA values as extra columns to the config at the end
    var columns = webix.toArray(myGrid.config.columns);

//If we do multiple searches our columns keep getting appended, so we must remove what was added previously.
    for (i = columns.length; i >= NUM_COLUMNS; i--) {
      columns.removeAt(i);
//      console.log(columns.length,'- removing ', i);
    }

    Object.keys(eaValues).sort(function(s1, s2){var l=s1.toLowerCase(), m=s2.toLowerCase();return l===m?0:l>m?1:-1;}).forEach(function(key,index) {
//console.log(key+'='+index);
        columns.insertAt({
            id: key,
            header: [key,{content:"textFilter"}],
            adjust:"header",
            sort:"string"
        },columns.length);
    });

//Various methods for refreshing columns, header, etc.
//    myGrid.refreshHeaderContent();
    myGrid.refreshColumns();
//    myGrid.refresh();
    myGrid.parse( json );
//    datatable.adjustColumn("title", "data");

//    var numrows=webix.toArray(json).length;

//  if you run this as text/javascript you have to do it this way
//    var elem = document.getElementById("total_records");
//    elem.value = numrows;
//    console.log(elem);
//  however, if you run this as application/javascript, this works
//    document.getElementById("total_records").value = numrows;
    document.getElementById("total_records").value = jsonarray.length;
//    webix.alert("Rows Returned: "+numrows);

}

//---------------------------------
// on Ready :

function renderMain() {
//    console.log ( "page rendered : ok" );

    // define the UI,
    // this is all done with objects, so we can use vars

    webix.ui( gridPanel );

}

redpajama_set_name: RedPajamaGithub


text:
Q: Why does android ImageSpan show my picture twice (when setBounds exceed certain magic width)? Here is my code to put an ImageSpan in an EditText.
public void onActivityCreated(Bundle savedInstanceState) {
    super.onActivityCreated(savedInstanceState);
    EditText et = (EditText) getActivity().findViewById(R.id.html_text);

    SpannableString ss = new SpannableString("ABC");
    Drawable d = getResources().getDrawable(R.drawable.ic_launcher);

    d.setBounds(0,0,2256,760);

    ImageSpan span = new ImageSpan(d, "haha", ImageSpan.ALIGN_BASELINE);
    ss.setSpan(span, 0, 3, Spannable.SPAN_EXCLUSIVE_EXCLUSIVE);
    et.setText(ss);
}

While playing around, I noticed something strange.  My AVD is 2560x1600.  When the setBounds call with width less or equal to 2256, the picture is showing correctly.  But for width larger than 2256, the picture is shown twice.!  The magic value is the same for different pictures.  I also tried AVD with different size.  For 720x1080, the magic width is 608.  What is this magic width?  How can I make sure my picture doesn't exceed this magic width?
BTW, here is the layout for the EditText:
<EditText
    android:id="@+id/html_text"
    android:layout_width="match_parent"
    android:layout_height="match_parent" />

A: this worked for me
int width;
int height;
@Override
protected void onCreate(Bundle savedInstanceState) {
    super.onCreate(savedInstanceState);
    setContentView(R.layout.activity_matn_activity3);
Display display = getWindowManager().getDefaultDisplay();
     width = display.getWidth();  // deprecated
     height = display.getHeight();  // deprecated
 SpannableString ss = new SpannableString("ABC");
Drawable d = getResources().getDrawable(R.drawable.ic_launcher);
double ratio= (double)((double)(d.getIntrinsicWidth())/(double)(d.getIntrinsicHeight()));
//this shows the original ratio of image

                if(d.getIntrinsicWidth()<width){
                d.setBounds(0, 0, d.getIntrinsicWidth(), d.getIntrinsicHeight());
                Log.v("small","small");
                }else{
                    Log.v("big","big");
                    d.setBounds(0, 0,(int)(width*.95),(int)(d.getIntrinsicHeight()/ratio*.95));
//i dont know why but if you put 1 instead of .95 again you will see 2 images
//this is exactly the magic width you have said
                }
}

it may have little problems because my project is  big and different i cant put the exact code
.but it generally works

redpajama_set_name: RedPajamaStackExchange