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## How to return all array permutations iteratively into a two-dimensional array?

I am trying to write a program that will iterate through all possible permutations of a String array, and return a two dimensional array with all the permutations. Specifically, I am trying to use a String array of length 4 to return a 2D array with 24 rows and 4 columns.

I have only found ways to print the Strings iteratively but not use them in an array. I have also found recursive ways of doing it, but they do not work, as I am using this code with others, and the recursive function is much more difficult.

For what I want the code to do, I know the header should be:

public class Permutation {      public String[][] arrayPermutation(String[] str)      {           //code to return 2D array      } }

//I tried using a recursive method with heap’s algorithm, but it is very //complex with its parameters.

I am very new to programming and any help would be greatly appreciated.

## A two-dimensional Vandermonde-type system

Let $$p$$ be a prime, and suppose that $$a_1,\dotsc,a_n\in\mathbb F_p^\times$$. How large can an integer $$K$$ be given that the system
$$\sum_{k=1}^n a_k x_k^i y_k^j = 0,\quad 0\le i,j\le K$$ has a solution in the variables $$x_1,\dotsc,y_n\in\mathbb F_p$$ satisfying $$x_k\ne x_l$$ and $$y_k\ne y_l$$ whenever $$k\ne l$$?

This is a system of $$(K+1)^2$$ equations in $$2n$$ variables; thus, one can expect that having $$K>C\sqrt{n}$$ with a sufficiently large $$C$$ will force the variables to vanish, or at least to obey some strong relations. In other words, heuristically, for $$K>C\sqrt n$$ the system does not have a solution with the variables pairwise distinct. Is there a rigorous argument to support this heuristic? Has this problem ever been studied?

## Two-dimensional Range Minimum Query under a constraint

So, I have trouble understanding and solving the following question:

You are given a 2D array. Design an algorithm that, given two coordinates (each with two indexes ofc), returns the minimum in the rectangle between those coordinates. You know that the number of lines in that rectangle (of each query) is greater than $$\frac{n}{3}$$. Preparation time should be $$O(n^2)$$ and query time should be $$O(1)$$.

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# $L^p$-weighted enstrophy calculations

Mathematician here, only somewhat familiar with physics, but finding myself involved in a research project on fluid dynamics. I'm studying a 3D simulation of turbulent fluid flow, and we are currently using the enstrophy calculation to study the system. That is, we look at $$\mathcal{E}(\omega) = \frac{1}{2}\int_V |\omega|^2 dV,$$ the integral of the square of the magnitude of the vorticity field. Mathematically, we see that $$2\mathcal{E}(\omega) = ||\omega||_2^2,$$ the square of the $L^2$-norm of the magnitude of the vorticity field.

My question: Do physicists ever look at other $L^p$-norms of the vorticity field? Is this something of interest? Intuitively, it would be like a weighted enstrophy calculation, placing more weight on regions of high vorticity.


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# show the rule between $w$, $w^2$, …$w^{n-1}$ with $w^n$ given w an nth root of unity

Let $w≠1$ be an $n$-th root of unity, i.e., $w^n-1=0$. Show that $1+2w+3w^2+\dots+nw^{n-1}=-\frac n{1-w}$.

My question is how to relate $w, w^2, \dots,w^{n-1}$ with $w^n$?

-

Hint: Let $x=1+2w+3w^2+\cdots +nw^{n-1}$. Then $$wx=w+2w^2+3w^3+\cdots +(n-1)w^{n-1}+nw^n.$$

Consider $x-wx$.

-
Then it remains to show, and how to show 1+w+w^2+...+w^(n-1)=0? –  i_a_n Jan 9 '13 at 2:30
$(1+w+\cdots+w^{n-1})(1-w)=1-w^n$. –  André Nicolas Jan 9 '13 at 3:10

$$1+2x+3x^2+\cdots+nx^{n-1}=\frac{x+x^2+x^3+\cdots+x^n}{dx}$$ $$=\frac{d\frac{x(x^n-1)}{x-1}}{dx}$$ $$=x(x^n-1)\left(\frac{-1}{(x-1)^2}\right)+\frac1{x-1}\{(n+1)x^n-1)\}$$

If $x\ne1, x^n=1$ $$1+2x+3x^2+\cdots+nx^{n-1}=\frac n{x-1}$$

-


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# CAT Practice : Coordinate Geometry

You are here: Home  CAT Questionbank   CAT Quant  Geometry: Triangles  Question 3
It helps to know all the formulae for computing area of a triangle. Knowing only Hero's formula can land one in trouble on occasions.

## Triangles - Area

Q.3: Sides of a triangle are 6, 10 and x for what value of x is the area of the △ the maximum?
1. 8 cms
2. 9 cms
3. 12 cms
4. None of these

Choice (D). None of these

## Detailed Solution

Side of a triangle are. 6, 10, x.
Area = ${1 \over 2}$ x 6 x 10 sin ∠BAC.
Area is maximum, when ∠BAC = 90◦
x = ${\sqrt {100 + 36}} = {\sqrt {136}}$
There is a more algebraic method using hero’s formula. Try that also.

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text:
The open source CFD toolbox
noneCondition.C
Go to the documentation of this file.
1/*---------------------------------------------------------------------------*\
2 ========= |
3 \\ / F ield | OpenFOAM: The Open Source CFD Toolbox
4 \\ / O peration |
5 \\ / A nd | www.openfoam.com
6 \\/ M anipulation |
7-------------------------------------------------------------------------------
8 Copyright (C) 2022 OpenCFD Ltd.
9-------------------------------------------------------------------------------
10License
11 This file is part of OpenFOAM.
12
13 OpenFOAM is free software: you can redistribute it and/or modify it
14 under the terms of the GNU General Public License as published by
15 the Free Software Foundation, either version 3 of the License, or
16 (at your option) any later version.
17
18 OpenFOAM is distributed in the hope that it will be useful, but WITHOUT
19 ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
20 FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
21 for more details.
22
23 You should have received a copy of the GNU General Public License
24 along with OpenFOAM. If not, see <http://www.gnu.org/licenses/>.
25
26\*---------------------------------------------------------------------------*/
27
28#include "noneCondition.H"
30
31// * * * * * * * * * * * * * * Static Data Members * * * * * * * * * * * * * //
32
33namespace Foam
34{
35namespace functionObjects
36{
37namespace runTimeControls
38{
41
42}
43}
44}
45
46
47// * * * * * * * * * * * * * * * * Constructors * * * * * * * * * * * * * * //
48
50(
51 const word& name,
52 const objectRegistry& obr,
53 const dictionary& dict,
55)
56:
57 runTimeCondition(name, obr, dict, state)
58{}
59
60
61// * * * * * * * * * * * * * * Public Member Functions * * * * * * * * * * * //
62
64{
65 return true;
66}
67
68
70{
71 // do nothing
72}
73
74
76{
77 // do nothing
78}
79
80
81// ************************************************************************* //
Macros for easy insertion into run-time selection tables.
#define addToRunTimeSelectionTable(baseType, thisType, argNames)
Add to construction table with typeName as the key.
A list of keyword definitions, which are a keyword followed by a number of values (eg,...
Definition: dictionary.H:126
virtual bool apply()
Apply the condition.
Definition: noneCondition.C:63
Base class for function objects, adding functionality to read/write state information (data required ...
Registry of regIOobjects.
A class for handling words, derived from Foam::string.
Definition: word.H:68
#define defineTypeName(Type)
Define the typeName.
Definition: className.H:96
Namespace for OpenFOAM.
word name(const expressions::valueTypeCode typeCode)
A word representation of a valueTypeCode. Empty for INVALID.
Definition: exprTraits.C:59
dictionary dict


text:
for Journals by Title or ISSN for Articles by Keywords help
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Research and Development       (Followers: 7) Computer Science and Engineering       (Followers: 17) Computer Science and Information Technology       (Followers: 11) Computer Science Education       (Followers: 12) Computer Science Journal       (Followers: 20) Computer Science Master Research       (Followers: 10)
 Applied Categorical Structures   [SJR: 0.361]   [H-I: 21]   [2 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1572-9095 - ISSN (Online) 0927-2852    Published by Springer-Verlag  [2353 journals]
• Representations of Crossed Modules and Other Generalized
Yetter-Drinfel’d Modules
• Authors: Victoria Lebed; Friedrich Wagemann
Pages: 455 - 488
Abstract: Abstract The Yang-Baxter equation plays a fundamental role in various areas of mathematics. Its solutions, called braidings, are built, among others, from Yetter-Drinfel’d modules over a Hopf algebra, from self-distributive structures, and from crossed modules of groups. In the present paper these three sources of solutions are unified inside the framework of Yetter-Drinfel’d modules over a braided system. A systematic construction of braiding structures on such modules is provided. Some general categorical methods of obtaining such generalized Yetter-Drinfel’d (=GYD) modules are described. Among the braidings recovered using these constructions are the Woronowicz and the Hennings braidings on a Hopf algebra. We also introduce the notions of crossed modules of shelves / Leibniz algebras, and interpret them as GYD modules. This yields new sources of braidings. We discuss whether these braidings stem from a braided monoidal category, and discover several non-strict pre-tensor categories with interesting associators.
PubDate: 2017-08-01
DOI: 10.1007/s10485-015-9421-z
Issue No: Vol. 25, No. 4 (2017)

• On an Enhancement of the Category of Shifted L ∞ -Algebras
• Authors: Vasily A. Dolgushev; Christopher L. Rogers
Pages: 489 - 503
Abstract: Abstract We construct a symmetric monoidal category $${\mathfrak {S}}\mathsf {Lie}_{\infty }^{\text {MC}}$$ whose objects are shifted L ∞ -algebras equipped with a complete descending filtration. Morphisms of this category are “enhanced” infinity morphisms between shifted L ∞ -algebras. We prove that any category enriched over $${\mathfrak {S}}\mathsf {Lie}_{\infty }^{\text {MC}}$$ can be integrated to a simplicial category whose mapping spaces are Kan complexes. The advantage gained by using enhanced morphisms is that we can see much more of the simplicial world from the L ∞ -algebra point of view. We use this construction in a subsequent paper (Dolgushev et al. Adv. Math. 274, 562–605, 2015) to produce a simplicial model of a (∞,1)-category whose objects are homotopy algebras of a fixed type.
PubDate: 2017-08-01
DOI: 10.1007/s10485-016-9424-4
Issue No: Vol. 25, No. 4 (2017)

• Splitting of Operads and Rota-Baxter Operators on Operads
• Authors: Jun Pei; Chengming Bai; Li Guo
Pages: 505 - 538
Abstract: Abstract This paper establishes a procedure that splits the operations in any algebraic operad, generalizing previous notions of splitting algebraic structures, from the dendriform algebra of Loday splitting the associative operation to the successors splitting binary operads. The separately treated bisuccessor and trisuccessor for binary operads are unified for general operads through the notion of configuration. Applications are provided for various n-algebras, the $$A_{\infty }$$ and $$L_{\infty }$$ algebras. Further, the concept of a Rota-Baxter operator, first showing its importance in the associative and Lie algebra contexts and then generalized to binary operads, is defined for all operads. The well-known connection from Rota-Baxter operators to dendriform algebras and its numerous extensions are expanded as the link from (relative) Rota-Baxter operators on operads to splittings of the operads
PubDate: 2017-08-01
DOI: 10.1007/s10485-016-9431-5
Issue No: Vol. 25, No. 4 (2017)

• Equivariantization and Serre Duality I
• Authors: Xiao-Wu Chen
Pages: 539 - 568
Abstract: Abstract For an additive category with a Serre duality and a finite group action, we compute explicitly the Serre duality on the category of equivariant objects. We prove that under certain conditions, the equivarianzation of an additive category with a periodic Serre duality still has a periodic Serre duality. A similar result is proved for fractionally Calabi-Yau triangulated categories.
PubDate: 2017-08-01
DOI: 10.1007/s10485-016-9432-4
Issue No: Vol. 25, No. 4 (2017)

• Katětov Functors
• Authors: Wiesław Kubiś; Dragan Mašulović
Pages: 569 - 602
Abstract: Abstract We develop a theory of Katětov functors which provide a uniform way of constructing Fraïssé limits. Among applications, we present short proofs and improvements of several recent results on the structure of the group of automorphisms and the semigroup of endomorphisms of some Fraïssé limits.
PubDate: 2017-08-01
DOI: 10.1007/s10485-016-9461-z
Issue No: Vol. 25, No. 4 (2017)

• The Gray Tensor Product Via Factorisation
• Authors: John Bourke; Nick Gurski
Pages: 603 - 624
Abstract: Abstract We discuss the folklore construction of the Gray tensor product of 2-categories as obtained by factoring the map from the funny tensor product to the cartesian product. We show that this factorisation can be obtained without using a concrete presentation of the Gray tensor product, but merely its defining universal property, and use it to give another proof that the Gray tensor product forms part of a symmetric monoidal structure. The main technical tool is a method of producing new algebra structures over Lawvere 2-theories from old ones via a factorisation system.
PubDate: 2017-08-01
DOI: 10.1007/s10485-016-9467-6
Issue No: Vol. 25, No. 4 (2017)

• Adjunctions of Quasi-Functors Between DG-Categories
• Authors: Francesco Genovese
Pages: 625 - 657
Abstract: Abstract We study right quasi-representable differential graded bimodules as quasi-functors between dg-categories. We prove that a quasi-functor has a left adjoint if and only if it is left quasi-representable. With this characterisation, we prove an existence result of adjoints, under suitable hypotheses on the dg-categories.
PubDate: 2017-08-01
DOI: 10.1007/s10485-016-9470-y
Issue No: Vol. 25, No. 4 (2017)

• A New Characterisation of Groups Amongst Monoids
• Authors: Xabier García-Martínez
Pages: 659 - 661
Abstract: Abstract We prove that a monoid M is a group if and only if, in the category of monoids, all points over M are strong. This sharpens and greatly simplifies a result of Montoli, Rodelo and Van der Linden (Pré-Publicações DMUC 16–21, 1–41 2016) which characterises groups amongst monoids as the protomodular objects.
PubDate: 2017-08-01
DOI: 10.1007/s10485-016-9471-x
Issue No: Vol. 25, No. 4 (2017)

• Every Rig with a One-Variable Fixed Point Presentation is the Burnside Rig
of a Prextensive Category
• Authors: M. Menni
Pages: 663 - 707
Abstract: Abstract We extend the work of Schanuel, Lawvere, Blass and Gates in Objective Number Theory by proving that, for any $${L(X) \in \mathbb {N}[X]}$$ , the rig $${\mathbb {N}[X]/(X = L(X))}$$ is the Burnside rig of a prextensive category.
PubDate: 2017-08-01
DOI: 10.1007/s10485-016-9475-6
Issue No: Vol. 25, No. 4 (2017)

• Coverages Give Free Constructions for Partial Frames
• Authors: John Frith; Anneliese Schauerte
Pages: 303 - 321
Abstract: Abstract Defining objects using generators and relations has seen substantial application in the theory of frames. It is the aim of this paper to establish such a technique for partial frames, thus making it available in a variety of contexts. A partial frame is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. The designated subsets in question are specified by means of a so-called selection function. The theory is general enough to include, as examples, bounded distributive lattices, σ-frames, κ-frames and indeed frames, but a small collection of elementary axioms suffices to describe the selection functions and thus the designated subsets. In this paper we are concerned with establishing techniques for constructing objects given certain generators and the relations that they should satisfy. Our method involves embedding the generators in an appropriate meet-semilattice, moving to the free partial frame over that meet-semilattice, and then using the relations to form a quotient with the required joins. We use a modification of Johnstone’s coverages on meet-semilattices [12] to construct partial frames freely generated by sites. We conclude with a number of applications, including the construction of coproducts for partial frames and a general method for freely adjoining complements.
PubDate: 2017-06-01
DOI: 10.1007/s10485-015-9417-8
Issue No: Vol. 25, No. 3 (2017)

• Quasicategories of Frames of Cofibration Categories
• Authors: Krzysztof Kapulkin; Karol Szumiło
Pages: 323 - 347
Abstract: Abstract We show that the quasicategory of frames of a cofibration category, introduced by the second-named author, is equivalent to its simplicial localization.
PubDate: 2017-06-01
DOI: 10.1007/s10485-015-9422-y
Issue No: Vol. 25, No. 3 (2017)

• Compactification: Limit Tower Spaces
• Authors: H. Boustique; G. Richardson
Pages: 349 - 361
Abstract: Abstract Convergence approach spaces, defined by E. Lowen and R. Lowen, possess both quantitative and topological properties. These spaces are equipped with a structure which provides information as to whether or not a sequence or filter approximately converges. P. Brock and D. Kent showed that the category of convergence approach spaces with contractions as morphisms is isomorphic to the category of limit tower spaces. It is shown below that every limit tower space has a compactification. Moreover, a characterization of the limit tower spaces which possess a strongly regular compactification is given here. Further, a strongly regular S-compactification of a limit tower space is studied, where S is a limit tower monoid acting on the limit tower space.
PubDate: 2017-06-01
DOI: 10.1007/s10485-016-9426-2
Issue No: Vol. 25, No. 3 (2017)

• Irreducible Equivalence Relations, Gleason Spaces, and de Vries Duality
• Authors: Guram Bezhanishvili; Nick Bezhanishvili; Sumit Sourabh; Yde Venema
Pages: 381 - 401
Abstract: Abstract By de Vries duality, the category of compact Hausdorff spaces is dually equivalent to the category of de Vries algebras (complete Boolean algebras endowed with a proximity-like relation). We provide an alternative “modal-like” duality by introducing the concept of a Gleason space, which is a pair (X,R), where X is an extremally disconnected compact Hausdorff space and R is an irreducible equivalence relation on X. Our main result states that the category of Gleason spaces is equivalent to the category of compact Hausdorff spaces, and is dually equivalent to the category of de Vries algebras.
PubDate: 2017-06-01
DOI: 10.1007/s10485-016-9434-2
Issue No: Vol. 25, No. 3 (2017)

• Bohr Compactifications of Algebras and Structures
• Authors: B. A. Davey; M. Haviar; H. A. Priestley
Pages: 403 - 430
Abstract: This paper provides a unifying framework for a range of categorical constructions characterised by universal mapping properties, within the realm of compactifications of discrete structures. Some classic examples fit within this broad picture: the Bohr compactification of an abelian group via Pontryagin duality, the zero-dimensional Bohr compactification of a semilattice, and the Nachbin order-compactification of an ordered set. The notion of a natural extension functor is extended to suitable categories of structures and such a functor is shown to yield a reflection into an associated category of topological structures. Our principal results address reconciliation of the natural extension with the Bohr compactification or its zero-dimensional variant. In certain cases the natural extension functor and a Bohr compactification functor are the same; in others the functors have different codomains but may agree on all objects. Coincidence in the stronger sense occurs in the zero-dimensional setting precisely when the domain is a category of structures whose associated topological prevariety is standard. It occurs, in the weaker sense only, for the class of ordered sets and, as we show, also for infinitely many classes of ordered structures. Coincidence results aid understanding of Bohr-type compactifications, which are defined abstractly. Ideas from natural duality theory lead to an explicit description of the natural extension which is particularly amenable for any prevariety of algebras with a finite, dualisable, generator. Examples of such classes—often varieties—are plentiful and varied, and in many cases the associated topological prevariety is standard.
PubDate: 2017-06-01
DOI: 10.1007/s10485-016-9436-0
Issue No: Vol. 25, No. 3 (2017)

• A Lax Approach to Neighbourhood Operators
• Authors: A. Razafindrakoto; D. Holgate
Pages: 431 - 445
Abstract: Abstract Viewing neighbourhood operators as lax natural transformations introduces an efficiency in calculations and proofs and suggests further applications. To highlight the advantages of this approach, classes of open, closed, initial and final morphisms are studied. In addition new proofs are given to previous results and a new example that departs from the current factorisation system paradigm is exhibited.
PubDate: 2017-06-01
DOI: 10.1007/s10485-016-9441-3
Issue No: Vol. 25, No. 3 (2017)

• Relative Cohomology of Algebraic Theories
• Authors: Simeon Pol’shin
Pages: 447 - 453
Abstract: Abstract We construct relative abelian categories in the sense of MacLane for models of algebraic systems in (co)complete abelian categories. As an example, we consider an analogue of Hochschild-Mitchell cohomology for the functor of Yoneda embedding.
PubDate: 2017-06-01
DOI: 10.1007/s10485-016-9466-7
Issue No: Vol. 25, No. 3 (2017)

• On the Maps of Pointfree Topology Which Preserve the Rings of
Integervalued Continuous Functions
• Authors: B. Banaschewski
Abstract: Abstract This paper establishes various conditions characterizing the homomorphisms $$h: L \rightarrow M$$ of 0-dimensional frames which induce an isomorphism between the rings of all integervalued continuous function, or their bounded parts, on L and M, based on the Lindelöf and the compact coreflection of 0-dimensional frames. This provides natural analogues of familiar results concerning the realvalued continuous functions on completely regular frames, albeit by rather different methods of proof from those originally used in that setting. In addition, it will be shown that the present approach also leads to alternative proofs for the latter.
PubDate: 2017-07-17
DOI: 10.1007/s10485-017-9499-6

• A New Proof of the Nešetřil–Rödl Theorem
• Authors: Dragan Mašulović
Abstract: Abstract In this paper we give a new proof of the Nešetřil–Rödl Theorem, a deep result of discrete mathematics which is one of the cornerstones of the structural Ramsey theory. In contrast to the well-known proofs which employ intricate combinatorial strategies, this proof is spelled out in the language of category theory and the main result follows by applying several simple categorical constructions. The gain from the approach we present here is that, instead of giving the proof in the form of a large combinatorial construction, we can start from a few building blocks and then combine them into the final proof using general principles.
PubDate: 2017-07-15
DOI: 10.1007/s10485-017-9500-4

• More on the Functor Induced by z -Ideals
• Authors: Oghenetega Ighedo
Abstract: Abstract An ideal I of a commutative ring A with identity is called a z-ideal if whenever two elements of A belong to the same maximal ideals and one of the elements is in I, then so is the other. For a completely regular frame L we denote by $${{\mathrm{ZId}}}(\mathcal {R}L)$$ the lattice of z-ideals of the ring $$\mathcal {R}L$$ of continuous real-valued functions on L. It is a coherent frame, and it is known that $$L\mapsto {{\mathrm{ZId}}}(\mathcal {R}L)$$ is the object part of a functor $$\mathsf {Z}:\mathbf {CRFrm}\rightarrow \mathbf {CohFrm}$$ , where $$\mathbf {CRFrm}$$ is the category of completely regular frames and frame homomorphisms, and $$\mathbf {CohFrm}$$ is the category of coherent frames and coherent maps. We explore when this functor preserves and reflects the property of being a Heyting homomorphism, and also when it preserves and reflects the variants of openness of Banaschewski and Pultr (Appl Categ Struct 2:331–350, 1994). We also record some other properties of this functor that have hitherto not been stated anywhere.
PubDate: 2017-06-21
DOI: 10.1007/s10485-017-9498-7

• Monads on Higher Monoidal Categories
• Authors: Marcelo Aguiar; Mariana Haim; Ignacio López Franco
Abstract: Abstract We study the action of monads on categories equipped with several monoidal structures. We identify the structure and conditions that guarantee that the higher monoidal structure is inherited by the category of algebras over the monad. Monoidal monads and comonoidal monads appear as the base cases in this hierarchy. Monads acting on duoidal categories constitute the next case. We cover the general case of n-monoidal categories and discuss several naturally occurring examples in which $$n\le 3$$ .
PubDate: 2017-06-05
DOI: 10.1007/s10485-017-9497-8

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text:
# Mathematica: Random numbers from arbitrary PDF

1. Apr 23, 2012

### Niles

Hi

I want to generate a set of random numbers according to a specific distribution, namely given by
$$f(v) \propto v^3\exp(-v^2 C)$$
where C is a constant. It is clear how to do it with a distribution already implemented in Mathematica, http://reference.wolfram.com/mathematica/tutorial/RandomNumberGeneration.html, but is there a way to do it with the one given by me above?

Best,
Niles.

2. Apr 23, 2012


text:
# Probability that entrance 2 has seen more cars than entrance 1

I am trying to do probability exercises but when it is not Binomial I am kinda lost.

A parking lot had two entrances. On average, a car arrives at entrance 1 of the parking lot every $20$ minutes and a car arrives at entrance 2 of the parking lot every $15$ minutes. We suppose that the number of cars arriving at entry 1 is independent of the number of cars arriving at entry 2. Given that $3$ cars entered the parking lot today between 14h and 15h, what is the probability that entrance 2 has seen more cars arriving during this period than entrance 1 ?

E(entrance 1)= 20
E(entrance 2)= 15
P(e2 > e1 knowing there are $3$ car between in $60$ min)

I don't even know where to start. Is it conditional? I thought about the Poisson distribution but because of the second entrance I don't see what to do. Is it really a Poisson distribution?

## 1 Answer

• Three cars enter through two entries in a hour.
• Cars independently make a choice of which to enter, and do so at rates of $3$ per hour and $4$ per hour.
• Therefore the probability that any particular car enters entry-one equals $3/7$, and the conditional distribution of the count of these (given that there are three total) is binomial. $$X_1\mid (X_1+X_2=3)~\sim~\mathcal{Bin}(3, 3/7)$$
• Find the probability that entry-two exceeds entry-one given a total of three cars.   That is $\mathsf P(X_1\leq 1\mid X_1+X_2=3)$
• Thank you very much. Now I know for sure that probability is not my thing. I have to work on it. – alison monroe Jul 28 '17 at 9:03


text:
# $\rm{K}^{*}(\rm{892})^{0}$ and $φ(1020)$ production at midrapidity in pp collisions at $\sqrt{s}$ = 8 TeV

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The production of $\rm{K}^{*}(\rm{892})^{0}$ and $\phi(1020)$ in pp collisions at $\sqrt{s}$ = 8 TeV were measured using Run 1 data collected by the ALICE collaboration at the LHC. The $p_{\rm{T}}$-differential yields d$^{\rm 2}N$/d$y$d$p_{\rm{T}}$ in the range 0 $<~$ $p_{\rm{T}}$ $<~$ 20 GeV/$c$ for $\rm{K}^{*0}$ and 0.4 $<~$ $p_{\rm{T}}$ $<~$ 16 GeV/$c$ for $\phi$ have been measured at midrapidity $|y|$ $<~$ 0.5. Moreover, improved measurements of the $\rm{K}^{*}(892)^{0}$ and $\phi(1020)$ at $\sqrt{s}$ = 7 TeV are presented. The collision energy dependence of $p_{\rm{T}}$ distributions, $p_{\rm{T}}$-integrated yields and particle ratios in inelastic pp collisions are examined. The results are also compared with different collision systems. The values of the particle ratios are measured to be similar to those found at other LHC energies. In pp collisions a hardening of the particle spectra is observed with increasing energy, but at the same time it is also observed that the relative particle abundances are independent of the collision energy. The $p_{\rm{T}}$-differential yields of $\rm{K}^{*0}$ and $\phi$ in pp collisions at $\sqrt{s}$ = 8 TeV are compared with the expectations of different Monte Carlo event generators.

Submitted to: PRC
e-Print: arXiv:1910.14410 | PDF | inSPIRE
CERN-EP-2019-248


text:
# Accessible Pages with MathJax

MathJax was design with accessibility in mind and has several powerful features to make math easier to see and read, both for ordinary users as well as those with print and learning disabilities. Two accessibility features, the ability to scale all math in a page or zoom in on a particular equation, are built in to MathJax. In addition, MathJax works with MathPlayer to make math accessible to screen readers, to screen magnifiers, and to learning disability software.

## Built in Accessibility Features

Because math notation is hierarchical and 2-dimensional, deeply nested subexpressions are often hard to see clearly, especially at screen sizes and resolutions. To help with this problem, MathJax provides two features for enlarging math expressions. The first is the ability to scale all equations in a page by a constant factor, for example, making them display at 150% of their normal size. This can be handy on pages with a small default font size, but it is also a bit jarring. So MathJax also provides the ability to zoom in on a single equation on demand.

As with many MathJax capabilities, page authors can add MathJax configuration code to a page to set a default equation scaling factor for all readers, or individual readers can set a scale factor for a single site by using the right-button context menu. The following picture illustrates how to change to scale of all equations in a page using the context menu. The configuration code an author would add to a page to make a larger scale the default is described in the MathJax documentation.

To zoom individual equations, readers must choose a mechanism to trigger the zoom. For example, one might want to zoom an equation by merely hovering the mouse pointer over it. Another choice might be to zoom when holding down SHIFT and clicking, and so on. By default, no zoom trigger is set, so to turn on zooming, bring up the context menu by right-clicking (if you are using Windows) or control-clicking (if you are using MacOS) on a MathJax equation, and then choosing the Settings ⇨ Zoom Trigger menu, and selecting the options you prefer. Note that the Settings ⇨ Zoom Factor menu also allows you to change how much an equation zooms.

Instructions for authors wishing to configure MathJax to have a zoom trigger by default are described in the MathJax documentation (see the menuSettings option).

## Using MathJax with MathPlayer for Greater Accessibility

MathPlayer is an add-on to Internet Explorer that provides native rendering of math notation encoded in MathML format in web pages when it is installed. MathPlayer can convert math to speech and can also highlight the math as it is spoken, and works with other assistive technology software such as screen readers, screen magnifiers, and learning disability software. The result is industry-leading math accessibility for individuals with print and learning disabilities. A list of Assistive Technology (AT) software that works with MathPlayer and other math accessibility information be found at the Design Science accessibility web site.

Since MathJax can convert TeX to MathML, the combination of MathJax and MathPlayer can make many more web pages, blogs, and wikis accessible. The example shown below is written in TeX and is accessible if you are reading this using IE+MathPlayer:

$x=\frac{-b \pm \sqrt {b^2-4ac}}{2a}$

### To Make a Page Accessible with MathJax and MathPlayer

To make a page accessible, MathPlayer needs to be responsible for rendering math in the page. By default, MathJax attempts to render MathML markup itself, and thus, in order to work together with MathPlayer, MathJax needs to know to pass MathML markup on to MathPlayer for accessible rendering. There are two ways that this can be done.

The author of a page can add markup to a page to configure MathJax, so that MathML is passed on to the browser whenever that is possible by default. In this mode, MathJax will detect when MathPlayer is installed in IE (or when a reader is using another MathML-capable browser) and in these cases, pass the MathML on for processing. The other option is that an individual reader can use the MathJax context menu to tell MathJax to use the MathML renderer if it isn’t already. With this method, an individual user can ensure MathJax and MathPlayer will work together, regardless of the default behavior determined by the page author.

#### For Users

For use with MathPlayer, set MathJax to use “MathML” as the renderer as shown below:

MathJax will remember these setting for a web site, but you will need to do this again when you visit another site.

#### For Authors

If you are an author of a web page, you can ensure MathPlayer will make math accessible when available by configuring MathJax so that it uses a native MathML renderer when one is present and in other cases falls back on MathJax doing the rendering otherwise. To do this, you need to use the MMLorHTML configuration file to prefer native MathML in IE (and Firefox) as described in the MathJax documentation.


text:
### Two tangents $PA$ and $PB$ are drawn to a circle with center $O$ from an external point $P$. Prove that $\angle APB = 2 \angle OAB$. O B A P

Step by Step Explanation:
1. Given:
$PA$ and $PB$ are the tangents to the circle with center $O$.
2. Here, we have to find the value of $\angle APB$.
Let us consider $\angle APB = x^\circ$.

We know that the tangents to a circle from an external point are equal.
So, $PA = PB$.
3. As $\triangle APB$ is an isosceles triangle$(PA = PB)$, the base angles of the triangle will be equal. $$\implies \angle PBA = \angle PAB$$
4. We know that the sum of the angles of a triangle is $180^\circ$. Thus, \begin{aligned} & \angle APB + \angle PAB + \angle PBA = 180^\circ \\ \implies & x^\circ + 2 \angle PAB = 180^\circ && \text{[As, } \angle PBA = \angle PAB] \\ \implies & \angle PAB = \dfrac { 1 }{ 2 } (180^\circ - x^\circ) = \bigg(90^\circ - \dfrac { 1 }{ 2 } x^\circ \bigg) \end{aligned}
5. $PA$ is a tangent and $OA$ is the radius of the circle with center $O$.
We know that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
\begin{aligned} \implies & \angle OAP = 90^\circ \\ \implies & \angle OAB + \angle PAB = 90^\circ \\ \implies & \angle OAB = 90^\circ - (90^\circ - \dfrac { 1 }{ 2 } x^\circ) \\ \implies & \angle OAB = \dfrac { 1 }{ 2 } x^\circ = \dfrac { 1 }{ 2 } \angle APB \\ \implies & \angle APB = 2 \angle OAB \end{aligned}
6. Hence, $\angle APB = 2 \angle OAB.$


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Euclidean geometry is what you're used to experiencing in your day to day life. Algebra II for dummies Mary Jane Sterling. The ancient Greek geometers knew the Earth was spherical, and in c235BC Eratosthenes of Cyrene calculated the Earth’s circumference to within about 15%. We could have cheated and just made the line go perfectly to the next focus, but instead we made the ellipse out of a lot of line segments and let the calculations do what they will.. From early times, people noticed that the shortest distance between two points on Earth were great circle routes.For example, the Greek astronomer Ptolemy wrote in Geography (c. 150 ce):. steepness. The centre point is the middle point between 'A' and 'B'. Where necessary, references are indicated in the text. Because it is a computer model. Spherical geometry is the study of geometric objects located on the surface of a sphere. Please read our short guide how to send a book to Kindle. 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We will usually the pronumeral m for gradient. (Note: for a circle, a and b are equal to the radius, and you get π × r × r = π r 2, which is right!) Most generally, gear teeth are equally spaced around the periphery of the gear. For comets and planets, the sun is located at one focus of their elliptical orbits. Spherical geometry works similarly to Euclidean geometry in that there still exist points, lines, and angles. Author: Steve Phelps. Orbital mechanics is a modern offshoot of celestial mechanics which is the study of the motions of natural celestial bodies such as the moon and planets. They are used to provide positive transmis-sion of both motion and power. Triangle Basics. Spherical geometry is nearly as old as Euclidean geometry. Calculus For Dummies, 2nd Edition provides a roadmap for success, and the backup you need to get there. File: PDF, 10.81 MB. They are composed of examples that are used in every-day practice and are arranged in a logical order. Do you want to download or read a book? The original gear teeth were wooden pegs driven into the periphery of wooden wheels and driven by other wooden elliptical to conical and from as small as a pinhead to as large asa house. For instance, a "line" between two points on a sphere is actually a great circle of the sphere, which is also the projection of a line in three-dimensional space onto the sphere. The vertical scale of the thickness form plots has been enlarged for clarity. Each Non-Euclidean geometry is a consistent system of definitions, assumptions, and proofs that describe such objects as points, lines and planes. Preview. ..... 43 20 Comparison of surface velocity distributions for an elliptical thickness form Orbital mechanics, also called flight mechanics, is the study of the motions of artificial satellites and space vehicles moving under the influence of forces such as gravity, atmospheric drag, thrust, etc. Focus of the ellipse explained with diagrams, pictures and an examination of the formula for finding the focus . Be careful: a and b are from the center outwards (not all the way across). 2010 Mathematics Subject Classification: Primary: 33E05 [][] An integral of an algebraic function of the first kind, that is, an integral of the form $$\tag{1 } \int\limits _ { z _ {0} } ^ { {z _ 1 } } R ( z , w ) d z ,$$ where $R ( z , w )$ is a rational function of the variables $z$ and $w$. The major axis is the longest diameter of an ellipse. 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The orbits of the planets and their moons are ellipses with very low eccentricities, which is to say they are nearly circular. - Exercise bikes were popular in homes and at gyms long before most of the high tech exercise machines of today were around. The Basics of Spherical Geometry A sphere is defined as a closed surface in 3D formed by a set of points an equal distance R from the centre of the sphere, O. 19 Shape and velocity distribution for elliptical and parabolic thickness forms from linear theory. Pillai "Simple Pendulum and Elliptic Integral Corrections - Landau's Take" - Duration: 18:53. Geometry For Dummies Mark Ryan. In fact, the word geometry means “measurement of the Earth”, and the Earth is (more or less) a sphere. Create Class; Spherical Geometry Ideas. This is known as the 'semi-minor axis' or 'radius 2'. And no originality is claimed 19 Shape and velocity distribution elliptical geometry for dummies elliptical and parabolic forms! Necessary, references are indicated in the text to download or read a book spherical geometry works similarly Euclidean! Rules, or axioms you measure the width of the formula for the... To your account first ; Need help can prevent you from hitting the elliptical geometry for dummies in the text how! On 5 Basic rules, or axioms spaced around the sun can be much more eccentric,. To more than 180 0 expository and no originality is claimed Tangent geometry actually... Before most of the oval, divide this value by 2 on a plane! First-Time student of geometry, studies the geometry is what you 're used to positive. Indicated in the first place to consider the geometry of lines through the origin three-space! A book in geometry expository and no originality is claimed way to model elliptical geometry reminiscent! Guide how to send a book give a complete course in geometry the planets and their moons are with! The theory of elliptic curves are curves defined by a certain type of non-Euclidean geometry, triangles have 0. Of lines through the origin in three-space, which we considered in Chapter.! Geometry will actually produce an elliptic pattern which is the longest diameter of an ellipse in... - Riemannian geometry is the representation of the greatest assistance to the student no..., a type of cubic equation in two variables: Tangent geometry will actually produce an elliptic pattern which the. This is the study of geometric objects located on the surface of a sphere way to model elliptical geometry different... Be of the geometry on the surface of a sphere reminiscent of the gear consistent! Of today were around and an examination of the geometry on 5 Basic rules, or axioms width the... And power axis is the middle point between ' a ' and B... 'S last theorem spaced around the periphery of the thickness form plots has been enlarged for clarity theory... 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text: Title Quantification of impurities in brass using secondary-ion mass-spectrometry : a comparison of matrix effects for $CsM^{+}$ clusters and $M^{+}$ ions Author Adriaens, A. Férauge, C. Adams, F. Faculty/Department Faculty of Sciences. Chemistry Publication type article Publication 1995 Amsterdam , 1995 Subject Physics Chemistry Source (journal) International journal of mass spectrometry and ion processes. - Amsterdam Volume/pages 151(1995) :1 , p. 63-68 ISSN 0168-1176 ISI A1995TK35300007 Carrier E Target language English (eng) Full text (Publishers DOI) Affiliation University of Antwerp Abstract This study discusses the influence of matrix effects on relative sensitivity factors in secondary ion mass spectrometry for a quantification study of major and impurity elements in standard brass reference materials. A comparison has been made between relative sensitivity factors, which are obtained through acquisition of CsM(+) clusters under cesium bombardment and of M(+) ions under oxygen bombardment. The data of the present study suggest that the matrix effects have been reduced in the case of cesium clusters, but nevertheless remain significant. In addition, the reproducibility for determining relative sensitivity factors improves significantly for cesium clusters. E-info https://repository.uantwerpen.be/docman/iruaauth/7e4fad/7604293.pdf http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:A1995TK35300007&DestLinkType=RelatedRecords&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:A1995TK35300007&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:A1995TK35300007&DestLinkType=CitingArticles&DestApp=ALL_WOS&UsrCustomerID=ef845e08c439e550330acc77c7d2d848 Handle


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# 8.1: Introduction

This chapter introduces the Smith chart, most widely used graphical technique for describing and solving problems using $$S$$ parameters. The Smith chart is an annotated polar plot of $$S$$ parameters and is one of the most powerful tools in RF and microwave engineering and is used to present measured results, to conceptualize designs, and to intuitively solve problems involving distributed networks.

8.1: Introduction is shared under a CC BY-NC license and was authored, remixed, and/or curated by Michael Steer.


text:
# Raychaudhuri scalar

In Carroll's 'Space-time and Geometry', appendix F on congruences, the Raychaudhuri equation is derived.

However, in the process, I seem to miss a calculation step that changes the sign of the Raychaudhuri scalar.

Page 461, Carroll writes:

\begin{align} U^\sigma \nabla_\sigma B_{\mu \nu} &=U^\sigma \nabla_\nu \nabla_\sigma U_\mu + U^\sigma R^\lambda_{\, \, \, \mu \nu \sigma} U_\lambda \\ &= \nabla_\nu (U^\sigma \nabla_\sigma U_\mu) - (\nabla_\nu U^\sigma)(\nabla_\sigma U_\mu) - R_{\lambda \mu \nu \sigma}U^\sigma U^\lambda \end{align}

I can't wrap my head around it. To me, lowering the index on the Riemann tensor and elevating the same dummy index on U would have no influence on the sign whatsoever.

I think it's a typographical error. He should have changed the sign of the Riemann tensor in the last two lines of his calculation (F.10). Then to obtain the Ricci tensor with the correct sign in the Raychaudhuri equation, you must interchange $$\mu$$ and $$\lambda$$, getting the required negative sign. This is because the Ricci tensor is $$R^k_{ikj}$$ and thus before contracting $$\mu$$ and $$\nu$$ you must exchange the positions of $$\mu$$ and $$\lambda$$ to get the correct sign.


text:
MathSciNet bibliographic data MR1239723 (95a:11047) 11F67 (11F27 11R39) Gelbart, S.; Rogawski, J.; Soudry, D. On periods of cusp forms and algebraic cycles for ${\rm U}(3)$${\rm U}(3)$. Israel J. Math. 83 (1993), no. 1-2, 213–252. Article

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.


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# MultinomialDistribution

Multinomial probability distribution object

## Description

A MultinomialDistribution object consists of parameters and a model description for a multinomial probability distribution.

The multinomial distribution is a generalization of the binomial distribution. While the binomial distribution gives the probability of the number of “successes” in n independent trials of a two-outcome process, the multinomial distribution gives the probability of each combination of outcomes in n independent trials of a k-outcome process. The probability of each outcome in any one trial is given by the fixed probabilities p1, ..., pk.

The multinomial distribution uses the following parameters.

ParameterDescriptionSupport
probabilitiesOutcome probabilities$0\le \text{probabilities}\left(i\right)\le 1\text{\hspace{0.17em}};\text{\hspace{0.17em}}\sum _{\text{all}\left(i\right)}\text{probabilities}\left(i\right)=1$

## Creation

Create a MultinomialDistribution probability distribution with specified parameter values object using makedist.

## Properties

expand all

### Distribution Parameter

Outcome probabilities for the multinomial distribution, stored as a vector of scalar values in the range [0,1]. The values in probabilities must sum to 1.

Data Types: single | double

### Distribution Characteristics

Logical flag for truncated distribution, specified as a logical value. If IsTruncated equals 0, the distribution is not truncated. If IsTruncated equals 1, the distribution is truncated.

Data Types: logical

Number of parameters for the probability distribution, specified as a positive integer value.

Data Types: double

Distribution parameter values, specified as a vector.

Data Types: single | double

Truncation interval for the probability distribution, specified as a vector containing the lower and upper truncation boundaries.

Data Types: single | double

### Other Object Properties

Probability distribution name, specified as a character vector.

Data Types: char

Distribution parameter descriptions, specified as a cell array of character vectors. Each cell contains a short description of one distribution parameter.

Data Types: char

Distribution parameter names, specified as a cell array of character vectors.

Data Types: char

## Object Functions

 cdf Cumulative distribution function icdf Inverse cumulative distribution function iqr Interquartile range mean Mean of probability distribution median Median of probability distribution pdf Probability density function random Random numbers std Standard deviation of probability distribution truncate Truncate probability distribution object var Variance of probability distribution

## Examples

collapse all

Create a multinomial distribution object using the default parameter values.

pd = makedist('Multinomial')
pd =
MultinomialDistribution

Probabilities:
0.5000    0.5000

Create a multinomial distribution object for a distribution with three possible outcomes. Outcome 1 has a probability of 1/2, outcome 2 has a probability of 1/3, and outcome 3 has a probability of 1/6.

pd = makedist('Multinomial','probabilities',[1/2 1/3 1/6])
pd =
MultinomialDistribution

Probabilities:
0.5000    0.3333    0.1667

Generate a random outcome from the distribution.

rng('default');  % for reproducibility
r = random(pd)
r = 2

The result of this trial is outcome 2. By default, the number of trials in each experiment, $n$, equals 1.

Generate random outcomes from the distribution when the number of trials in each experiment, $n$, equals 1, and the experiment is repeated ten times.

rng('default');  % for reproducibility
r = random(pd,10,1)
r = 10×1

2
3
1
3
2
1
1
2
3
3

Each element in the array is the outcome of an individual experiment that contains one trial.

Generate random outcomes from the distribution when the number of trials in each experiment, $n$, equals 5, and the experiment is repeated ten times.

rng('default');  % for reproducibility
r = random(pd,10,5)
r = 10×5

2     1     2     2     1
3     3     1     1     1
1     3     3     1     2
3     1     3     1     2
2     2     2     1     1
1     1     2     2     1
1     1     2     2     1
2     3     1     1     2
3     2     2     3     2
3     3     1     1     2

Each element in the resulting matrix is the outcome of one trial. The columns correspond to the five trials in each experiment, and the rows correspond to the ten experiments. For example, in the first experiment (corresponding to the first row), 2 of the 5 trials resulted in outcome 1, and 3 of the 5 trials resulted in outcome 2.


text:
Heat Transfer from Single and Arrays of Impinging Water Jets(II)-1 Row of Impinging Water Jets-

Title & Authors
Heat Transfer from Single and Arrays of Impinging Water Jets(II)-1 Row of Impinging Water Jets-
Eom, Gi-Chan; Lee, Jong-Su; Geum, Seong-Min;

Abstract
Experiments have been conducted to obtain local and average heat transfer coefficients associated with impingement of a row of circular, free surface-water jets on a constant heat flux surface. Nozzle arrays are a row of 3 jets (nozzle dia.=4.6 mm) and a row of 5 jets (nozzle dia.=3.6 mm), and the nozzle configuration is Reverse cone type revealed good performance in heat transfer. Nozzle-to-plate spacings ranging from 16 mm to 80 mm were investigated for two jet center to center spacings 25 mm and 37.5 mm in the jet velocity of 3 m/s (R $\small{e_{D}}$=27000) to 8 m/s (R $\small{e_{D}}$=70000). For a row of 3 jets and a row of 5 jets, the stagnation heat transfer of the central jet is lower than that of adjacent jets. In the wall jet region between jets, for small nozzle-to-plate spacing and large jet velocity, the local maximum in the Nusselt number was observed, however, for small jet velocity or large nozzle-to-plate spacing, the local maximum was not observed. Except for the condition of $\small{V_{O}}$=8 m/s and H/D=10, the average Nusselt number reveals the following ranking: a row of 5 jets, a row of 3 jets, single jet. For a row of 3 jet, the maximum average Nusselt number occurs at H/D=8 ~ 10, and for a row of 5 jets, it occurs at H/D=2 ~ 4. Compared with the single jet, enhancement of average heat transfer for a row of 3 jets is approximately 1.52 ~ 2.28 times, and 1.69 ~ 3.75 times for a row of 5 jets.ets.s.
Keywords
Arrays of Impinging Water Jets;A row of 3 Jets;A row of 5 Jets;Enhancement of Heat Transfer;Nusselt Number;
Language
Korean
Cited by
1.
1열 원형 충돌수분류군에 의한 열전달의 실험적 연구 (제1보, 노즐형상의 영향),엄기찬;김상필;

설비공학논문집, 2000. vol.12. 1, pp.50-58
2.
1열 원형 충돌수분류군에 의한 열전달의 실험적 연구 (제2보, 노즐-전열면간 거리의 영향),엄기찬;이종수;김상필;

설비공학논문집, 2000. vol.12. 1, pp.59-66
References
1.
Int. J. Heat Mass Transfer, 1976. vol.19. pp.673-683

2.
ASME, J. of Heat Transfer, 1987. vol.109. pp.872-879

3.
ASME, J. of Heat Transfer, 1994. vol.116. pp.482-486

4.
ASME, J. of Heat Transfer, 1963. pp.237-245

5.
ASME, J. of Engng. for Power, 1970. vol.92. pp.73-82

6.
ASME, J. of Heat Transfer, 1978. vol.100. pp.352-357

7.
ASME, J. of Heat Transfer, 1983. vol.105. pp.354-360

8.
ASME, J. of Engng. for Power, 1972. pp.35-42

9.
ASME, J. of Heat Transfer, 1984. vol.106. pp.34-41

10.
J. Electrochem. Soc., 1990. vol.137. 6, pp.1744-1750

11.
J. Electrochem. Soc., 1990. vol.137. pp.3086-3093

12.
ASME, J. of Heat Transfer, 1994. vol.116. pp.88-95

13.
Int. J. Heat Mass Transfer, 1994. vol.37. 2, pp.269-282

14.
ASME, J. of Heat Transfer, 1995. vol.117. pp.878-883

15.
대한기계학회 논문집(B), 1997. vol.21. 9, pp.1105-1114

16.
Int. J. Heat Mass Transfer, 1994. vol.37. 18, pp.2859-2869


text:
Integrability, One-point Functions and Tests of AdS/dCFT
Charlotte Kristjansen
Niels Bohr Inst., Copenhagen
Jeudi 08/11/2018, 14h15
Salle Claude Itzykson, Bât. 774, Orme des Merisiers

\noindent Integrability meeting ENS/IPhT'' \\ \\ We consider a defect version of the gauge/gravity correspondence (AdS/dCFT) where N=4 SYM has gauge groups of different ranks on the two sides of a co-dimension one defect and where the dual string theory picture is that of a probe-brane set-up of either D3-D5 or D3-D7 type with either background gauge field flux or non-trivial instanton number. The main results obtained concerning one-point functions of the above AdS/dCFT are reviewed. First, we present a closed formula for tree-level one-point functions of all conformal operators constructed from the scalar fields of N=4 SYM in the case of the D3-D5 set-up. Using the language of integrability, the one-point functions are expressed in terms of the spin chain rapidities of the conformal operators in question. Important ingredients in the formula are the Gaudin determinant, various Baxter polynomials as well as quantities closely related to the transfer matrix of the integrable spin chain involved. We furthermore present a closed one-loop order formula for one-point functions in the SU(2) sub-sector of the same set-up. Subsequently, we discuss the status of similar studies for the D3-D7 set-up. Finally, we present a positive test of AdS/dCFT at the next to leading order in a certain double scaling parameter both for the half supersymmetric D3-D5 probe brane set-up and for a non-supersymmetric D3-D7 probe brane set-up. \\ \\ (IPhT organizers: Ivan Kostov and Didina Serban)

Contact : Loic BERVAS


text:
# Re: [XeTeX] semantic nest size

well yes \text makes the thing be set four times as many times as
\mbox so does stress the semantic nest a bit but not as much as that
usually, but without a reproducible test hard to say exactly what is
wrong.

On 12 April 2018 at 19:14, Apostolos Syropoulos <asyropou...@yahoo.com> wrote:
>>I don't have those fonts and couldn't reproduce the error when I tried
>>switching to other fonts so I can't trace the exact cause of the
>>error, but the error is not entirely surprising, unicode-math does
>>document that it must be called after amsmath
>
> I think the error is related to the \text command. When I change
> \text to \mbox, compilation proceeds with no error, even if
> amsmath is called before unicode-math.
>
> A.S.
>
> ----------------------
> Apostolos Syropoulos
> Xanthi, Greece
>
>

--------------------------------------------------
Subscriptions, Archive, and List information, etc.:
http://tug.org/mailman/listinfo/xetex


text:
Tetration Forum

You're currently viewing a stripped down version of our content. View the full version with proper formatting.
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hej BO,
bo198214 Wrote:$\log_{1,b}(-1)=6*I$

How did You find this first value, exactly- where does b come in? Is it used as base of logarithm instead of e?
so log 1,b (z) (-1) is log with base b from -1?

Do I understand correctly?

So the problem is now turned to the properties of continuous application of logarithms with base e^pi/2 on -1?
Which in turn is calculated in each step via normal logarithms by formula:
log b (z) = (ln (z) + 2pik)/ ln (b)?

Regards,

Ivars
Ivars Wrote:
bo198214 Wrote:$\log_{1,b}(-1)=6*I$

How did You find this first value, exactly- where does b come in? Is it used as base of logarithm instead of e?
so log 1,b (z) (-1) is log with base b from -1?

Do I understand correctly?
Yes, yes, yes to the last 3 questions. We defined $\log_{k,b}(z)=\frac{\log(z)+2\pi i k}{\log(b)}$ where $b=e^{\frac{\pi}{2}}$ hence $\log_{1,b}(z)=\frac{\log(z)+2\pi i}{\pi/2}=\frac{2}{\pi}\log(z)+4i$ and further $\log_{1,b}(-1)=\frac{2}{\pi}\log(-1)+4i=\frac{2}{\pi}\pi i +4i=2i+4i=6i$.

Quote:So the problem is now turned to the properties of continuous application of logarithms with base e^pi/2 on -1?
Which in turn is calculated in each step via normal logarithms by formula:
log b (z) = (ln (z) + 2pik)/ ln (b)?
Yes, yes
bo198214 Wrote:
Quote:So the problem is now turned to the properties of continuous application of logarithms with base e^pi/2 on -1?
Which in turn is calculated in each step via normal logarithms by formula:
log b (z) = (ln (z) + 2pik)/ ln (b)?
Yes, yes

So now we need analytic expressions for that infinite taking of logarithms with base e^pi/2 of (-1) as a function of k.

Any ideas? I am searching as well.

if we put e^(pi/2) = e^((sqrt(pi)/sqrt(2))^(sgrt(pi)/sqrt(2) ) then

we can use Gamma(1/2) = sqrt(pi), so

e^(pi/2) = e^(Gamma^2 (1/2)/2)

and ln (e^pi/2) = Gamma^2 (1/2) /2 +- 2pik

next ln would be 2 lnGamma(1/2) - ln 2

next ln2+ ln*lnGamma(1/2) - lnln2 etc. ln2 and lnln2 will cancel out, so
in the end we will have lnln............ln (Gamma(1/2) when n-> infinity.
Ivars Wrote:next ln2+ ln*lnGamma(1/2) - lnln2 etc. ln2 and lnln2 will cancel out, so
in the end we will have lnln............ln (Gamma(1/2) when n-> infinity.

Actually it does not matter with which value (instead of -1) you start the iterations of $\log_b$ as long as you dont take some hyperpowers of $b$ for example $b^{b^b}$ would lead to 0 after 4 times application of the $\log_b$ and taking again $\log_b$ would give $-\infty$, so the formula would not work with this starting value.
bo198214 Wrote:
Ivars Wrote:next ln2+ ln*lnGamma(1/2) - lnln2 etc. ln2 and lnln2 will cancel out, so
in the end we will have lnln............ln (Gamma(1/2) when n-> infinity.

Actually it does not matter with which value (instead of -1) you start the iterations of $\log_b$ as long as you dont take some hyperpowers of $b$ for example $b^{b^b}$ would lead to 0 after 4 times application of the $\log_b$ and taking again $\log_b$ would give $-\infty$, so the formula would not work with this starting value.

So we have just limit log b log b log b log b ... log b (+-i*2pik) to be concerned with for branches in numerator. Right? But how do we get real part as well from this? because there is real value in denominator if we use ln to compute log b?

So we have lim k, n-> infintylog b log b log b ...... log b ( +-i*2pik/(pi/2+- i*2pi/k))

So the simplest starting value is then +-i*2pik/(pi/2+-i*2pik)?
Are these k in numerator and denominator the same or running independently ?

Ivars
Ivars Wrote:So we have just limit log b log b log b log b ... log b (+-i*2pik) to be concerned with for branches in numerator. Right?
...
So we have lim k, n-> infintylog b log b log b ...... log b ( +-i*2pik/(pi/2+- i*2pi/k))

So the simplest starting value is then +-i*2pik/(pi/2+-i*2pik)?
Are these k in numerator and denominator the same or running independently ?

I am not sure what you mean. The method is about repeated applying of $\log_{k,b}$ and not of $\log_b$. You can not pull out the $2\pi i k$.
bo198214 Wrote:I am not sure what you mean. The method is about repeated applying of $\log_{k,b}$ and not of $\log_b$. You can not pull out the $2\pi i k$.

May be I am trying to say something useless, but I am sure, that in the region where h(z) does not converge, and may be even in the region where it does converge( e.g h(sqrt(2))=2 or 4 ) e.g h(e^pi/2) You can not limit yourself to only one root of -1, but You have to use both separately, or information is lost. So the real way to get accurate results is to study both rotations anticlokwise (i) and clockwise (-i) independently. I am not sure computer programs handle it that way. I will try once more:

Knoebel (1981) gave the following series for

h(z) = 1+ lnz + 3^2/3!*(lnz)^2 .........

Now, if we take h(e^(pi/2) we know that answers will be +i, -i.

ln(e^pi/2) = pi/2+- 2*pi*i*k -> I stress +-, k >0.

If You put this in series above( I know they do not coverge, but they might give h(z) analytic continuation), you will get :

h(z,k) = SUM (( (n^n-1)/n!) * ( pi/2+-2pi*I*k)^n-1 or
h(z,k) = SUM (((n^n-1)/n!)* pi/2^(n-1)* (1+-4*I*k)^n-1

Therefore, if we get h(z) = i when k = 1, than we must get h(z) =-i when with k =-1. If k=0, we have a special case, as we can not get i in the series if k=0.

For k=2 and -2 the values of h(z) again may be complex conjugates , but may be not.

For k=3 and -3 again the imaginary values will definitely differ by sign because values for 1 and - 1 were complex conjugates.

So basically for each pair of k +- You will get 2 values of h(z) which will be complex conjugates, but sometimes without imaginary part when they converge. Then there will seem to be only 1 value, a real number.But not all cases.

As a conjecture, perhaps h (sqrt2) will have one real value at from ln(sqrt2) and another from ln(-sqrt2).
h(cube root 2) may have 3 values, h(4th root of 2) may have 4 converging values etc.
Ivars Wrote:As a conjecture, perhaps h (sqrt2) will have one real value at from ln(sqrt2) and another from ln(-sqrt2).
h(cube root 2) may have 3 values, h(4th root of 2) may have 4 converging values etc.

Ivars -

Hmm, I've not been in your discussion with Henryk (have my mind elsewhere these days) and I don't know, perhaps I'm missing the point.
What is the point of your question - besides, that we have conjugate solutions? If I recall my above graphs, for instance, then they show obviously conjugacy. And my graph for complex fixpoints for real bases b>e^(1/e)

[attachment=126]

can be continued for any number of solutions t and u, where imag(u)=some selection +- 2*k*pi (= beta in the graph). (though changing the k-factor only does not lead exactly to the same b, but this should not be seen as a problem). So there may be some problem, that I'm missing when I read your posts?

Gottfried

graph: alpha + I*beta = u
a +I*b = t = exp(u)
s = exp(u/t) = base-parameter
bo198214 Wrote:So how do you finally compute the $k$th branch of $h$ or $W$ with this formula?

I think the following two methods are possibly the standard branches of h:
$h_k(x) = e^{-W_k(-\log(x))} = \frac{W_k(-\log(x))}{-\log(x)}$
and much easier to compute in my opinion.

Andrew Robbins
Gottfried Wrote:Ivars -

Hmm, I've not been in your discussion with Henryk (have my mind elsewhere these days) and I don't know, perhaps I'm missing the point.
What is the point of your question - besides, that we have conjugate solutions? If I recall my above graphs, for instance, then they show obviously conjugacy. And my graph for complex fixpoints for real bases b>e^(1/e)

can be continued for any number of solutions t and u, where imag(u)=some selection +- 2*k*pi (= beta in the graph). (though changing the k-factor only does not lead exactly to the same b, but this should not be seen as a problem). So there may be some problem, that I'm missing when I read your posts?

You can not see what I can not express properly

Did You succeed in explaining why sqrt 2 tetrated might lead to both 2 and 4? Euler seems to have had an idea, and he also bothered with r=e^pi/2 in his article, but I can not read Latin good enough still.

May be what I am trying to say is:

We should define tetration generally with a help of some divergent series Euler style and then investigate all possible analytical developments of these series, including convergent ones in the region of convergence of h(z).

I have a gut feeling I can not get rid of that somewhere the +i and - i difference= ambiquity will pop out in such case clearly. And z=e^pi/2 should be the easiest place to investigate it properly. There has to be not 2 , but 4 values at all k which sometimes overlap, but in some places adding adding pik to i and pik to - i leads to different values.E.g.

ln(sqrt2) gives : ln sgrt2 +- 2 pi k =1/2ln2 +- 2 pi I k
ln(-sqrt 2) gives: ln sqrt 2+-pi k= 1/2ln 2 +- pi I k but k is not 0 as negative logarithms can only be complex, never real.

So totally you have : when k= 0-> 1 value

ln sqrt2=1/2 ln 2

when k=1 -> 4 values

ln sqrt2 = 1/2 ln 2 +- 2pi I -> 2 values
ln -sqrt2 = 1/2 ln 2 + - pi*I -> 2 different values

I have a feeling that tetration is the field where reducing this ambiquity early on and reaching definitions of tetration via convergent series or functions ( like W(z) ) looses important information on the way.

When You see e^i*pi/2 You think about rotation of 90 degrees. When i = h(e^pi/2) then e^h(e^pi/2) * pi/2 is also a rotation by 90 degress but involves infinite operation on real numbers. NO i. It is an identity- You can replace i with h(z) and -i with another branch of h(z) . And depending on the sign of h(z) it will be either rotation anticlockwise, or clockwise.

so h(e^pi/2)^2 = -1 but we clearly know that each h(e^pi/2) which is root of -1 is found via different branch, so they can not be replacable so easily .

May be I am terribly wrong, as I really can not nail the place where it really matters.
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text: Restricted Boltzmann machines (RBMs) are often used as building blocks in greedy learning of deep networks. However, training this simple model can be laborious. Traditional learning algorithms often converge only with the right choice of metaparameters that specify, for example, learning rate scheduling and the scale of the initial weights. They are also sensitive to specific data representation. An equivalent RBM can be obtained by flipping some bits and changing the weights and biases accordingly, but traditional learning rules are not invariant to such transformations. Without careful tuning of these training settings, traditional algorithms can easily get stuck or even diverge. In this letter, we present an enhanced gradient that is derived to be invariant to bit-flipping transformations. We experimentally show that the enhanced gradient yields more stable training of RBMs both when used with a fixed learning rate and an adaptive one.


text:
# How do you implicitly differentiate 2=e^ysiny-x^2y^3 ?

$y ' = \frac{2 x {y}^{3}}{{e}^{y} \cdot \cos y + {e}^{y} \cdot \sin y - 3 {x}^{2} {y}^{2}}$

#### Explanation:

From the given equation

$2 = {e}^{y} \sin y - {x}^{2} {y}^{3}$

Differentiate both sides of the equation with respect to $x$

$\frac{d}{\mathrm{dx}} \left(2\right) = \frac{d}{\mathrm{dx}} \left({e}^{y} \sin y - {x}^{2} {y}^{3}\right)$

$0 = {e}^{y} \cdot \frac{d}{\mathrm{dx}} \left(\sin y\right) + \sin y \cdot \frac{d}{\mathrm{dx}} \left({e}^{y}\right) - \left[{x}^{2} \cdot \frac{d}{\mathrm{dx}} \left({y}^{3}\right) + {y}^{3} \cdot \frac{d}{\mathrm{dx}} \left({x}^{2}\right)\right]$

$0 = {e}^{y} \cdot \left(\cos y\right) y ' + \sin y \cdot \left({e}^{y}\right) \cdot y ' - \left[{x}^{2} \left(3 {y}^{2}\right) y ' + \left({y}^{3}\right) \left(2 x\right)\right]$

$0 = {e}^{y} \cdot \left(\cos y\right) y ' + \sin y \cdot \left({e}^{y}\right) \cdot y ' - {x}^{2} \left(3 {y}^{2}\right) y ' - \left({y}^{3}\right) \left(2 x\right)$

$0 = \left[{e}^{y} \cdot \left(\cos y\right) + \sin y \cdot \left({e}^{y}\right) - {x}^{2} \left(3 {y}^{2}\right)\right] y ' - \left({y}^{3}\right) \left(2 x\right)$

$\left({y}^{3}\right) \left(2 x\right) = \left[{e}^{y} \cdot \left(\cos y\right) + \sin y \cdot \left({e}^{y}\right) - {x}^{2} \left(3 {y}^{2}\right)\right] y '$

$y ' = \frac{2 x {y}^{3}}{{e}^{y} \cdot \cos y + {e}^{y} \cdot \sin y - 3 {x}^{2} {y}^{2}}$

God bless....I hope the explanation is useful.


text:
# Distributed software for calculating $\pi$ or $e$

Is there a "SETI at Home" style initiative for calculating constants with distributed computing?

-


text:
# Chapter 1 - Whole Numbers - Summary Exercises - Whole Number Computation - Page 65: 47

1520 hours are spent painting each week.

#### Work Step by Step

To determine the total number of hours spent painting each week, you must first find the number of hours in a work week by multiplying the number of working days by the number of hours per day. $5 \times 8 = 40$ hours per week To find the total number of hours spent painting each week, multiply the number of hours per week by the number of painters. $40 \times 38 = 1520$ hours

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.