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2 edition of Semi-groups of operators and approximation found in the catalog.

Semi-groups of operators and approximation

Paul L. Butzer

# Semi-groups of operators and approximation

## by Paul L. Butzer

• 89 Want to read
• 9 Currently reading

Published by Springer in New York .
Written in English

Subjects:
• Banach spaces,
• Groups, Theory of,
• Numerical analysis,
• Integrals

• Edition Notes

Bibliography: p. [295]-313.

The Physical Object ID Numbers Series Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd. 145, Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd.145 Contributions Berens, Hubert, jt. author Pagination xi, 318 p. ; Number of Pages 318 Open Library OL21775353M

SIAM Journal on Mathematical Analysis > Vol Issue 2 > / accretive operators, contraction semi-groups, radiative transfer, splitting algorithms, transport equation. Existence and bounds for the half-moment entropy approximation to radiative transfer. Applied Mathematics Letters , Cited by: This book gives a systematic presentation of the theory of semi-groups of bounded linear operators in a Banach space and of their applications in theoretical questions of approximation. The material in it is based on the contributions of the authors and of their co-workers. Amnon Pazy, Semi-groups of linear operators and applications to partial differential equations, Department of Mathematics, University of Maryland, College Park, Md., Department of Mathematics, University of Maryland, Lecture Note, No. MR James D. Bjorken and Sidney D. Drell, Relativistic quantum fields, McGraw-Hill Book Co.

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### Semi-groups of operators and approximation by Paul L. Butzer Download PDF EPUB FB2

The aim of this book is to present a systematic treatment of semi groups of bounded linear operators on Banach spaces and their connec tions with approximation theoretical questions in a more classical setting as well Semi-groups of operators and approximation book within the setting of the theory of intermediate spaces.

The aim of this Semi-groups of operators and approximation book is to present a systematic treatment of semi groups of bounded linear operators on Banach spaces and their connec tions with approximation theoretical questions in a more classical setting as well as within the setting of the theory of intermediate by: COVID Resources.

Reliable information about the coronavirus (COVID) is available from the World Health Organization (current Semi-groups of operators and approximation book, international travel).Numerous and frequently-updated resource results are available from this ’s WebJunction has pulled together information and resources to assist library staff as they consider how to handle.

Get this from a library. Semi-groups of operators and approximation. [Paul Leo Butzer; Hubert Berens] -- In recent years important progress has been made in the study of semi-groups of operators from the viewpoint of approximation theory.

These advances have primarily been achieved by introducing the. semi groups of linear operators Download semi groups of linear operators or read online books in PDF, EPUB, Tuebl, and Mobi Format. Click Download or Read Online button to get semi groups of linear operators book now. This site is like a library, Use search box in.

Compactness of Markov and Schrödinger semi-groups: A probabilistic approach Takeda, Masayoshi, Tawara, Yoshihiro, and Tsuchida, Kaneharu, Osaka Journal of Mathematics, ; A note on the convergence of linear semi-groups of class (1,A) ABDELAZIZ, Nazar H., Hokkaido Mathematical Journal, Operator approximate biprojectivity of locally compact quantum Cited by: COVID campus closures: see options for getting or retaining Remote Access to subscribed contentCited by: 5.

Paul Leo Butzer is the author of Semi-Groups of Operators and Approximation ( avg rating, 0 ratings, 0 reviews, published ), Functional Analysis a. Part of the Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen book series (GL, volume ) Abstract In Semi-groups of operators and approximation book applications to the previous chapter, in particular in Sec.

we discussed the existence, uniqueness and explicit representation of the semi-group solution of the diffusion as well as Laplace equation for the by: 4. Approximation of Operator Semi-Groups THOMAS I. SEIDMAN Department of Mathematics, Carnegie-Mellon University, Pittsburgh, Pennsylvania Communicated by Tosio Kato Received Aug 1.

INTRODUCTION In his book [3], Yosida presents a. Semi-groups with power singularities. If in the previous example, then the integrals in (4) are divergent the generating operator for the corresponding semi-group may not have a resolvent for any, i.e.

it may have a spectrum equal to the entire complex Semi-groups of operators and approximation book, for large enough one can define for such operators functions which coincide with the functions in.

Linear Operators and Approximation / Lineare Operatoren und Approximation Book Subtitle Proceedings of the Conference held at the Oberwolfach Mathematical Research Institute, Black Forest, August 14–22, / Abhandlungen zur Tagung im Mathematischen Forschungsinstitut Oberwolfach, Schwarzwald, vom bis Brand: Birkhäuser Basel.

MATHEMATICS A RELATION BETWEEN SEMI-GROUPS AND SEQUENCES OF APPROXIMATION OPERATORS BY R. MARTINI 1) (Communicated by Prof. TIMMAN at the meeting of J ) 1. INTRODUCTION e a twice continuously differentiable real function defined on the closed interval [0, 1] of the real axis and let B.1 (n = 1, 2, ) be the n-th Cited by: 8.

A 'read' is counted each time someone views Semi-groups of operators and approximation book publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text.

Semi-Groups of Operators and Approximation. 点击放大图片 出版社: Springer. 作者: Butzer, Paul Leo; Berens, Hubert; 出版时间: 年01月01 日.

Approximation Theorems for Semi-Group Operators in Intermediate Spaces Article (PDF Available) in Semi-groups of operators and approximation book of the American Mathematical Society 70() September with. Theorems about the approximation of semi-groups play an essential role in the approximate solution of Cauchy problems.

Let, be Banach spaces; let, be operators defined and single-valued on, respectively, satisfying the assumptions of the fundamental theorem for the same type ; let be linear operators.

We derive several models in Physics of continuous media using Trotter theory of convergence of semi-groups of operators acting on variable spaces. Keywords: Asymptotic mathematical modeling, approximation of semi-groups in the sense of Trotter, Hilbert spaces of possible states with finite energy, transient problems, homogenization and Cited by: 3.

1 Operator semi-groups and their generator The theory of operator semi-groups originates from the study of the equation T(t+s) = T(t)T(s) T(0) = I () where T(t) is an operator-valued function taking values in the set of bounded linear operators acting on a suitable function space.

This problem was indepen. n widths in approximation theory Download n widths in approximation theory or read online books in PDF, EPUB, Tuebl, and Mobi Format. Click Download or Read Online button to get n widths in approximation theory book now. This site is like a library, Use search box in the widget to get ebook that you want.

In mathematics, the Lie product formula, named for Sophus Lie (), but also widely called the Trotter product formula, states that for arbitrary n × n real or complex matrices A and B, + = → ∞ (/ /), where e A denotes the matrix exponential of Lie–Trotter product formula (Trotter ) and the Trotter–Kato theorem extend this to certain unbounded linear operators A and B.

You seem to underestimate the amount of mathematical analysis used in numerical analysis. Semi-groups of operators are used in numerical PDEs, approximation theory is a standard part of courses in numerical analysis, and wavelets are arguably as much numerical analysis and mathematical analysis.

-- Jitse Niesen19 March (UTC) 4.(Rated B-class, High-importance):. Approximation theorems for operator semigroups Andr as B atkai Febru The central existence theorem of linear C 0-semigroups is usually proved by using approximation methods. The aim of this project is to survey existing approximation theorems and to show up their usefulness.

The central result. The analytical theory of one-parameter semi-groups deals with the ex-1 ponential function in inﬁnite dimensional function spaces. It is a natural generalization of the theorem of Stone on one-parameter groups of uni-tary operators in a Hilbert space.

In these lectures, we shall be concerned with the diﬀerentiabilityCited by: 6. of semigroups of operators. It also covers the cases 1 and 2. Contraction semigroups in Banach spaces The following account builds on Appendix 1 in the book of Lax and Phillips [LP67]. A semigroup of operators in a Banach space X is a family of operators G(t) ∈ B(X), parametrized by t∈ R+ and satisfyingFile Size: KB.

detailed proof may be found in Volume 1, Chapter 1 of the book by Cliﬀord and Preston [30], which is a standard reference for basic classical results and notation in semigroup theory. There is an obvious dual result involving right reversible semigroups and groups of left quotients.

Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide. Key words. Semigroup, rational approximation, resolvent series, exponential integrator. AMS subject classiﬁcations.

() 65F60, 65M15, 65M 1. Introduction and main results. In this note we will discuss approximations to semi-groups and to the operator functions that appear in exponential Runge–Kutta methods by resol-vent series.

holomorphic semi-groups of operators on a Banach space (, 4;1;4). These three papers are the basis for the most beautiful chaper in Hille's book on semi-groups of operators.

This chapter contains a basic genera-tion theorem, giving necessary and sufficient conditions for an operator to generate a semi-group holomorphic in aFile Size: 1MB. Taylor series are used to define functions and "operators" in diverse areas of mathematics.

In particular, this is true in areas where the classical definitions of functions break down. For example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as the matrix exponential or matrix logarithm.

Chapter 1 Examples of semigroups In this chapter we are going to describe the matrix valued function T(): R +. M n(C) which satis es the functional equation () discussed on Section We will see that, for A2M n(C), the continuous map R + 3t7!etA 2M n(C) satis es the functional equation and that etA forms a semigroup of matrices depending on t2RFile Size: KB.

DOWNLOAD NOW» The present book is based on lectures given by the author at the University of Tokyo during the past ten years. It is intended as a textbook to be studied by students on their own or to be used in a course on Functional Analysis, i.

e., the general theory of linear operators in function spaces together with salient features of its application to diverse fields of modern.

Chapter 8. Approximation Properties of Chlodowsky Variant of (P, Q) SzAsz-Mirakyan-Stancu Operators.-Chapter 9. Approximation Theorems for Positive Linear Operators Associated with Hermite and Laguerre Polynomials.-Chapter On Generalized Picard Integral Operators.-Chapter From Uniform to Statistical Convergence of Binomial-Type Operators I Semigroups, Monoids, and Groups 6 in Section I.6 and do so using cycles in a permutation group.

The multiplication table for group D∗ 4 is (this is Exercise I): ∗ I R R2 R3 T x Ty T1,3 T2,4 I I R R2 R3 T x Ty T1,3 T2,4 R R R2 R3 I T 2,4 T1,3 Tx Ty R2 R2 R3 I R T y Tx T2,4 T1,3 R3 R3 I R R2 T 1,3 T2,4 Ty Tx Tx Tx T2,4 Ty T1,3 I R 2 R R3 Ty Ty T1,3 Tx T2,4 R 2 I R3 R File Size: 93KB.

The present book deals with some basic problems of Approximation Theory: with properties of polynomials and splines, with approximation by poly- mials, splines, linear operators.

It also provides the necessary material ab out different function spaces. In some sense, this is a modern version of the corre sponding parts of the book of one of us (Lorentz [A]).

Butzer and H. Berens [], Semi-groups of operators and approximation, Springer-Verlag, Berlin. Zentralblatt MATH: Mathematical Reviews (MathSciNet): MR 7.

Paul L. Butzer and Hubert Berens, Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, BandSpringer-Verlag New York Inc.,New York, MR Paul R. Chernoff and Jerome A.

Goldstein, Admissible subspaces and the denseness of the intersection of the domains of semigroup generators, J. Functional Analysis Cited by: 9. Pade Approximants for Operators: Theory and Applications. 点击放大图片 出版社: Springer. 作者: Cuyt, A. 出版时间: 年06月01 日. 10位国际标准书号: 13位国际标准.

e-books in Group Theory category An Elementary Introduction to Group Theory by M. Charkani - AMS, The theory of groups is a branch of mathematics in which we study the concept of binaryoperations. Group theory has many applications in physics and chemistry, and is potentially applicable in any situation characterized by symmetry.

Under some restrictive assumptions about opacities, we show that the radiative transfer equations have the form $({{du} / {dt}}) + \mathcal{A}u + \mathcal{B}u = 0$ where $\mathcal{A}$ is m-accretive and $\mathcal{B}$ is Lipschitz.

Mathematically, this gives existence and uniqueness of the solution. We also show that the maximum principle by:. Lecture 3 Pdf SEMIGROUPS St ephane ATTAL Abstract This lecture is an introduction to the theory of Operator Semi-groups and its main ingredients: di erent types of continuity, associated gen-erator, dual and predual semigroups, Stone’s Theorem.

The lecture also starts with a complete introduction to the Bochner Size: KB.S E M I G R O U P S O F O P E R AT O R S LINEAR J K Semigroups of Operators: Theory and Applications Book of abstracts Kazimierz, Poland, September 30| October 5, Functional analysis and Semi-Groups E Hille ebook R S Phillips.

Many of the earliest books, particularly those dating back to the s and before, are now extremely scarce and increasingly expensive. operators exp holomorphic definition complex limit topology functional analysis Other readers will.


text:
# If the sun radiates energy at the rate of 4 x 1026 Js-1, what is the rate at which its mass is decreasing?

1. 5.54 x 109 kgs-1
2. 4.44 x 109 kgs-1
3. 3.44 x 109 kgs-1
4. 2.44 x 109 kgs-1

Option 2 : 4.44 x 109 kgs-1

## Detailed Solution

Concept:

• The equation that relates the mass and energy is given by Albert Einstein.
• According to the equation, the mass can be converted into equivalent energy and vice versa.

Einstein's equation is given by:

E = m c2

Where E = Energy, m = mass, c = speed of light.

Calculation:

Given that:

As from Einstein's mass-energy relation:

∴ E = m c2

E = 4 x 1026 Js-1, c = 3 × 10ms-1.

$$m = \frac{{4 \times {{10}^{26}}}}{{{{(3 \times {{10}^8})}^2}}} \Rightarrow \frac{{4 \times {{10}^{26}}}}{{(9 \times {{10}^{16}})}} = 4.44 \times {10^9}kg{s^{ - 1}}$$

format: plain


text:
Trace class pseudodifferential calculus with operator valued symbols and unusual index formulas
Séminaire Équations aux dérivées partielles (Polytechnique) (2000-2001), Talk no. 4, 14 p.

For several classes of pseudodifferential operators with operator-valued symbol analytic index formulas are found. The common feature is that usual index formulas are not valid for these operators. Applications are given to pseudodifferential operators on singular manifolds.

@article{SEDP_2000-2001____A4_0,
author = {Rozenblum, Grigori},
title = {Trace class pseudodifferential calculus with operator valued symbols and unusual index formulas},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2000-2001},
note = {talk:4},
zbl = {1057.58011},
language = {en},
url = {http://www.numdam.org/item/SEDP_2000-2001____A4_0}
}

Rozenblum, Grigori. Trace class pseudodifferential calculus with operator valued symbols and unusual index formulas. Séminaire Équations aux dérivées partielles (Polytechnique) (2000-2001), Talk no. 4, 14 p. http://www.numdam.org/item/SEDP_2000-2001____A4_0/

[1] B. Blackadar, K-theory for Operator Algebras, Springer-Verlag, NY,1986. | MR 859867 | Zbl 0597.46072

[2] A. Connes, Noncommutative Geometry, Academic Press, NY, 1994. | MR 1303779 | Zbl 0818.46076

[3] C.Dorschfeldt, B.-W. Schulze, Pseudo-differential operators with operator-valued symbols in the Mellin-edge-approach Ann. Global Anal. Geom., 12 (1994), 135–171. | MR 1277665 | Zbl 0827.35144

[4] B. Fedosov, Analytic formulas for the index of elliptic operators, Trans. Moscow Math. Soc., 30 (1974), 159–240. | MR 420731 | Zbl 0349.58006

[5] B. Fedosov, Analytic formulas for the index of an elliptic boundary value problem, Math. USSR Sbornik, 22 (1974), 61-90; 24 (1974), 511-535; 30 (1976), 341-359. | MR 341537 | Zbl 0323.58016

[6] B. Fedosov, B.-W. Schulze, On the index of elliptic operators on a cone, Schrödinger operators, Markov semigroups, wavelet analysis, operator algebras, 348–372, Akademie Verlag, Berlin, 1996. | MR 1409837 | Zbl 0865.35152

[7] B.Fedosov, B.-W. Schulze, N. Tarkhanov, On the index of elliptic operators on a wedge, J. Funct. Anal., 157 (1998), 164–209. | MR 1637925 | Zbl 0961.58010

[8] R. Melrose, P. Piazza, Analytic $K$-theory for manifolds with corners, Advances Math., 92 (1992), 1–27. | MR 1153932 | Zbl 0761.55002

[9] B. Plamenevsky, Algebras of pseudodifferential operators, Mathematics and its Applications (Soviet Series), 43, Kluwer Academic Publishers Group, Dordrecht, 1989. | MR 1026642 | Zbl 0679.47027

[10] B. Plamenevsky, G. Rozenblum, On the index of pseudodifferential operators with isolated singularities in the symbols. (Russian) Algebra i Analiz, 2 (1990), 5, 165–188. Translation in: Leningrad Math. J., 2, (1991), 1085–1110. | MR 1086450 | Zbl 0746.35061

[11] B. Plamenevsky, G. Rozenblum, Pseudodifferential operators with discontinuous symbols: $K$-theory and index formulas. (Russian), Funktsional. Anal. i Prilozhen., 26, (1992), 4, 45–56. Translation in: Functional Anal. Appl., 26 (1993), 266–275. | MR 1209943 | Zbl 0879.58073

[12] G. Rozenblum, Index formulae for pseudodifferential operators with discontinuous symbols, Ann. Global Anal. Geom., 15, (1997), 71–100. | MR 1443273 | Zbl 0894.58071

[13] E. Schrohe, J. Seiler, An analytic index formula for pseudodifferential operators on wedges, Preprint, Max-Planck-Institut für Mathematik, 96/172, 1996.

[14] B.-W. Schulze, Pseudodifferential operators on manifolds with singularities, 1991, North Holland, Amsterdam. | Zbl 0747.58003

[15] B.-W. Schulze, Pseudo-differential Boundary Value Problems, Conical Singularities, and Asymptotics, 1994, Academie-Verlag, Berlin. | MR 1282496 | Zbl 0810.35175

[16] B.-W. Schulze, Operator algebras with symbol hierarchies on manifolds with singularities, Preprint 99/30, Universitet Potsdam, 1999. | MR 1827174

[17] B.-W. Schulze, B. Sternin, V. Shatalov, Differential Equations on Singular Manifolds: Semiclassical Theory and Operator Algebra, 1999, Wiley, Weinheim. | MR 1673461 | Zbl 0946.58021

[18] B.-W. Schulze, B. Sternin, V. Shatalov,On the index of differential operators on manifolds with conical singularities, Ann. Global Anal. Geom., 16 (1998), 141–172. | MR 1622293 | Zbl 0914.58030

[19] V. Senichkin, Pseudodifferential operators on manifolds with edges, J. Math. Sci., 73 (1995), 711–747. | MR 1331194 | Zbl 0844.58074


text:
Next smallest dimension of Specht Module after $(n)$, $(1^n)$, $(n-1,1)$ and $(2,1^{n-2})$

In representation theory of $S_n$, we know that for $n \geq 9$, the only Specht modules $S^\alpha$ of dimension $f^\alpha < {n-1 \choose 2} - 1$ are:

$S^{(n)}$ and $S^{(1^n)}$ with dimension $1$, $S^{(n-1,1)}$ and $S^{(2,1^{n-2})}$ with dimension $n-1$.

By simple calculations, we know that the next smallest is of dimension ${n-1 \choose 2} - 1$ (it is calculated by considering $\alpha = (n-2,2)$ and $(2^2,1^{n-4})$). The question is:

Are $(n-2,2)$ and $(2^2,1^{n-4})$ the only Specht modules with dimension ${n-1 \choose 2} - 1$?

Further Question: The next smallest dimension after ${n-1 \choose 2} - 1$ is ${n-1 \choose 2}$. Are $(n-2,1^2)$ and $(3,1^{n-3})$ the only Specht modules with dimension ${n-1 \choose 2}$?

Thanks for any help!

-
I guess your other question, which is solved, is stronger than this one. –  Harry Huang Mar 6 '13 at 15:58
but that doesn't answer to this, isn't it? Correct me if I'm wrong :) –  terrylsc Mar 6 '13 at 23:42


text:
# REINFORCE calculating the log policy gradient for a continuous action space

I've noticed that when modelling a continuous action space, the default thing to do is to estimate a mean and a variance where each is parameterized by a neural network or some other model.

I also often see that it is one network $$\theta$$ models both. The REINFORCE objective can be written as

$$\nabla \mathcal{J}(\theta) = \mathbb{E}_{\pi} [\nabla_\theta \log \pi(a_t|s_t) * R_t]$$

For discrete action space this makes sense since the output of the network is determined by a softmax. However, if we explicitly model the output of the network as a Gaussian, then the gradient of the log likelihood is of a different form,

$$\pi_\theta(a_t|s_t) = Normal(\mu_\theta(s_t), \Sigma_\theta(s_t))$$

and the log is:

$$\log \pi_\theta(a_t | s_t) = -\frac{1}{2} (a_t-\mu_\theta)^\top \Sigma^{-1}_\theta(a_t-\mu_\theta) + \log 2 \pi \det({\Sigma_\theta})$$

In the slides provided here (slide 18): http://www0.cs.ucl.ac.uk/staff/d.silver/web/Teaching_files/pg.pdf

IF the variance is held constant, then we can solve this analytically:

$$\nabla \log \pi_\theta(a_t|s_t) = (a_t - \mu_\theta) \Sigma^{-1} \phi(s)$$

But, are things always modelled assuming a constant variance? If it's not constant then we have to account for the inverse of the covariance matrix as well as the determinant?

I've taken a look at code online and from what I've seen, most of them assume the variance is constant.

• I just wanted to confirm that reinforcement learning is well within the scope of our site, as well as that of ai.stackexchange.com; despite the impression that may have been given by a now-deleted comment. (I've undeleted this question even though you've asked it here.) – Scortchi - Reinstate Monica Feb 13 '19 at 14:45


text:
# 4. A spherical region of space of radius 0.1 m has a volumetric charge density ρυ-kr2...

###### Question:

4. A spherical region of space of radius 0.1 m has a volumetric charge density ρυ-kr2 where k = 4 × ttviyiteo everywhere Calculate the electric flux density (D) at r-0.02m Determine the distance r outside the sphere at which the magnitude of the electric flux density is equal to what you found in part (a). a. b.

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##### The graph below is the function f(z)Find limf(z)Find limf(c)Find limf(z)Find f( _ 1)
The graph below is the function f(z) Find lim f(z) Find lim f(c) Find lim f(z) Find f( _ 1)...
##### How does tuberculosis relate to communicable diseases?
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##### Consider the function f (r) 8r} centered at Find the third order Taylr polynomial and approximate 8(1.1)i Find the n-th order Taylor polynomial for the functionf(z) sin â‚¬centered aroundFind the general solution of the first order linear differential equationV (v(o) =3
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##### What risks does a bank take if it funds long-term loans with short-term deposits? (Risk management...
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##### Note: This is using Bayesian statistics (a) Suppose that in a population, the probability of having...
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text:
# LaTeX instructions

To use $\LaTeX$ on this blog, you first use a dollar sign and “latex” like this:

$latex x = \frac{ - b \pm \sqrt{b^2 - 4 a c} }{2 a} then you can close it with another dollar sign: $.

Putting it all together, the renderer will give you:

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

Basically, it’s like writing latex, except that every time you have to go into math mode, you have to type “latex” right after the first dollar sign.

## 4 thoughts on “LaTeX instructions”

1. Now, if only wordpress can make it so the images have a transparent background instead of a white background. They use PNGs for the images, so there really is no excuse on their part.

2. Yeah, it looks pretty atrocious in RSS readers without a white background. I had some problems with transparency in PNGs due to cross-platform issues (creating the transparency on a Mac wouldn’t read in Windows), so maybe they’re “playing it safe?” That was a few years ago, though….

3. foobar says:

$x = x$.

4. foobar says:

wow, it works 😀

$\frac{1}{n-1}\sum{(X_i-\overline{X})^2} = \frac{1}{n-1}(\sum{X_i^2}-n\overline{X}^2) = \frac{n}{n-1}(M_2-\overline{X}^2)$.

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text:
# The 35th International Symposium on Lattice Field Theory

18-24 June 2017
Palacio de Congresos
Home > Timetable > Session details > Contribution details

Andalucía II

# More on heavy tetraquarks in lattice QCD at almost physical pion mass

## Speakers

• Dr. Anthony FRANCIS

## Content

We report on our progress with the ongoing study of exotic, heavy tetraquark states, $qq’ \bar Q\bar Q’$. With regards to the previously reported $\bar Q\bar Q’=\bar b\bar b$ case, we have significantly increased our statistics by computing and saving more propagators on publicly available dynamical $n_f =2+1$ Wilson-Clover gauge configurations, generated by the PACS-CS collaboration, with pion masses $\simeq$164, 299 and 415 MeV. We extend our previous analysis to heavy quark components containing heavier than physical bottom quarks $\bar Q\bar Q’=\bar b’\bar b’$, charm and bottom quarks $\bar c\bar b$ and also only charm quarks $\bar c\bar c$. Throughout we employ NRQCD and Fermilab heavy quarks for the heavier than bottom, bottom and charm quarks. Using our previously established diquark-antidiquark and meson-meson operator basis we comment in particular on the dependence of the binding energy on the mass of the heavy quark component $\bar Q\bar Q$, with heavy quarks ranging from $m_Q=0.7…6.3\cdot m_b$. In the heavy flavor non-degenerate case, $\bar Q\bar Q’$, and especially for the tetraquark channel $ud\bar c\bar b$, we extend our work to utilize a $3\times 3$ GEVP to study the ground and threshold states, thereby enabling an identification also of finite volume effects in this channel. Finally, we present some initial work on the $\bar Q\bar Q’=\bar c\bar c$ system where a much larger operator basis is available in comparison to flavor combinations with NRQCD quarks.


text:
Synopsis

# Tin Gets Kinky

Physics 12, s58
The observation that tin nuclei suddenly increase in size when the number of neutrons they contain reaches a “magic” number helps test models of nucleon interactions.

Many atomic nuclei steadily swell in size as more neutrons are stuffed in, until the number of neutrons hits some “magic” value. At that point, there’s a jump in how fast their radii grow. These “kinks,” which reflect changes in how strongly neutrons and protons tug on each other as nuclear shells fill up, have been seen for just a handful of isotopes. Now, researchers have observed a nuclear kink for tin. This result helps test models of nucleon interactions, some of which predict that there should be no kink for tin.

Discussions about the origin of kinks largely focus on lead-208. This isotope is “doubly magic”—that is, its proton shells are also full (see 6 November 2017 Viewpoint)—causing it to have a spherical nucleus, which simplifies interpretations of its measured size. Tin-132 is also doubly magic, but the isotope has yet to come under the same experimental scrutiny.

Using a laser spectroscopy facility at CERN called COLLAPS, Christian Gorges of Darmstadt University of Technology in Germany, Liss Rodriguez of the French Institute of Nuclear Physics, and colleagues measured the radii of a number of tin isotopes. They found that the nuclear radius steadily increases from about 4.56 fm for tin-108 to 4.71 fm for tin-132. But from tin-132 to tin-134, they observe a significant jump in radius to 4.73 fm—a change roughly 60% greater than expected from the radius’ previous path—clearly signifying a kink.

Comparing their data to predictions from different theoretical models, the team found that their observation supports one specific model of the strong interaction between nucleons inside the nucleus. This model suggests that the kink arises from a reduction in the strength of pairing interactions among neutron pairs when tin turns doubly magic.

This research is published in Physical Review Letters.

–Christopher Crockett

Christopher Crockett is a freelance writer based in Arlington, Virginia.

Nuclear Physics

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Tests of the standard model of particle physics using nuclear isotopes are becoming increasingly precise but they have a way to go before they can confirm the existence of any new particles. Read More »


text:
# Why goal of PLA can ignore the norm of normal vector

Define hyperplane $$w*x+b=0$$, the goal of PLA(Perceptron Learning Algorithm) is minimizing the distance of misclassified points to the decision boundary, i.e. $$-\frac{1}{||w||}\sum_{i\in M} y_i(w*x_i + b)$$ But actual algorithm is $$\min_{w,b} -\sum_{i\in M} y_i(w*x_i + b)$$


text:
# Central tendency analysis on the cantilever beam example¶

In this example we perform a central tendency analysis of a random variable Y using the various methods available. We consider the cantilever beam example and show how to use the TaylorExpansionMoments and ExpectationSimulationAlgorithm classes.

[1]:

from __future__ import print_function
import openturns as ot

Create the random variable of interest Y=g(X).

[2]:

mean = [50.0, 1.0, 10.0, 5.0]
dim = len(mean)
sigma = ot.Point(dim, 1.0)
R = ot.IdentityMatrix(dim)
distribution = ot.Normal(mean, sigma, R)
X = ot.RandomVector(distribution)
X.setDescription(['E', 'F', 'L', 'I'])
f = ot.SymbolicFunction(['E', 'F', 'L', 'I'], ['F*L^3/(3*E*I)'])
Y = ot.CompositeRandomVector(f, X)
Y.setDescription('Y')

## Taylor expansion¶

Perform Taylor approximation to get the expected value of Y and the importance factors.

[3]:

taylor = ot.TaylorExpansionMoments(Y)
taylor_mean_fo = taylor.getMeanFirstOrder()
taylor_mean_so = taylor.getMeanSecondOrder()
taylor_cov = taylor.getCovariance()
taylor_if = taylor.getImportanceFactors()
print('model hessian calls number=', f.getHessianCallsNumber())
print('taylor mean first order=', taylor_mean_fo)
print('taylor variance=', taylor_cov)
print('taylor importance factors=', taylor_if)

model evaluation calls number= 1
model hessian calls number= 1
taylor mean first order= [1.33333]
taylor variance= [[ 2.0096 ]]
taylor importance factors= [E : 0.000353857, F : 0.884642, L : 0.079618, I : 0.0353857]

[4]:

taylor.drawImportanceFactors()

[4]:

We see that, at first order, the variable explains 88.5% of the variance of the output . On the other hand, the variable is not significant in the variance of the output: at first order, the random variable could be replaced by a constant with no change to the output variance.

## Monte-Carlo simulation¶

Perform a Monte Carlo simulation of Y to estimate its mean.

[5]:

algo = ot.ExpectationSimulationAlgorithm(Y)
algo.setMaximumOuterSampling(1000)
algo.setCoefficientOfVariationCriterionType('NONE')
algo.run()
print('model evaluation calls number=', f.getEvaluationCallsNumber())
expectation_result = algo.getResult()
expectation_mean = expectation_result.getExpectationEstimate()
print('monte carlo mean=', expectation_mean, 'var=', expectation_result.getVarianceEstimate())

model evaluation calls number= 1001
monte carlo mean= [1.39543] var= [0.00271142]

## Central dispersion analysis based on a sample¶

Directly compute statistical moments based on a sample of Y. Sometimes the probabilistic model is not available and the study needs to start from the data.

[6]:

Y_s = Y.getSample(1000)
y_mean = Y_s.computeMean()
y_stddev = Y_s.computeStandardDeviationPerComponent()
y_quantile_95p = Y_s.computeQuantilePerComponent(0.95)
print('mean=', y_mean, 'stddev=', y_stddev, 'quantile@95%', y_quantile_95p)

mean= [1.3887] stddev= [1.61762] quantile@95% [4.21421]

[7]:

graph = ot.KernelSmoothing().build(Y_s).drawPDF()
graph.setTitle("Kernel smoothing approximation of the output distribution")
graph

[7]:


text:
# How to solve a cubic equation

Consider the general cubic equation

$x^3+bx^2+cx+d=0.\quad\quad\quad(*)$

Let

$y = x + \omega \implies x = y - \omega.\quad\quad\quad(1)$

Substituting (1) into (*), we have

$(y-\omega)^3 + b(y-\omega)^2+c(y-\omega) + d = 0.$

This is a new cubic equation in $y$:

$y^3 + (b-3\omega)y^2 + (3\omega^2-2b\omega+c)y - \omega^3+b\omega^2-c\omega+d=0\quad\quad\quad(2)$

Let

$\omega = \frac{b}{3}\quad\quad\quad(3)$

so that (2) becomes

$y^3 + (c-\frac{b^2}{3})y+d-\frac{bc}{3}+\frac{2b^3}{27}=0.$

That is

$y^3+py=q\quad\quad\quad(4)$

where $p = c-\frac{b^2}{3}, q = -d+\frac{bc}{3}-\frac{2b^3}{27}$ (see Fig. 1).

Thus, (4) is the so called depressed cubic equation. A solution is readily available through Cardano’s formula (see “Through the Mind’s Eye“).

Once we know $y$, (1) gives

$x = y - \frac{b}{3}.$

Fig. 1

## One thought on “How to solve a cubic equation”

1. Pingback: How to solve a quartic equation |


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# Test: Statistics Foundation II - Ambitious

Double click on maths expressions to zoom
Question 1:   Find the mean value in the following data set:

$17,25,2,21,29,8,36,5,11,19$
$20.4$
$22.7$
$18.9$
$17.3$
Question 2:   Find the range in the following data set:

$115,36,118,-12,33,-10,98,106,124,119$
$124$
$134$
$136$
$112$
Question 3:   Bar chart below represents number of rainy days over a month in a group of towns:

How many towns are compared?
$41$
$37$
$43$
$25$
Question 4:   Using the bar chart from question 3, find the modal number of rainy days
$4$
$7$
$6$
$5$
Question 5:   Mean score of a basketball team in  $12$   games is   $93$  points. What is the total number of points scored?
$954$
$1012$
$1116$
$1206$
Please note, you have solved only half of the test. For the complete test get a Must Have account. Get started


text:
# Calculating elasticity between log term and non-log term [duplicate]

Given the following, I am asked to calculate the elasticity of wages with respect to age:

$$𝑙𝑛(𝑤_𝑖)=𝛽_1+𝛽_2𝑎𝑔𝑒_i+𝛽_3𝑎𝑔𝑒^2_𝑖+𝛽_4𝑌_𝑖+𝛽_5𝑇_𝑖+𝛽_6𝑀𝑎𝑟_𝑖+𝜖_𝑖$$

How does one calculate a full elasticity between a log variable and and non-log variable? Is the following attempt correct?

$$\frac{\partial ln(w)}{{\frac{\partial age}{age}}}=\frac{\partial ln(w)}{\partial age}age=(\beta_2+2\beta_3age)(age)$$


text:
# Kinematics (maybe)

1. Jan 25, 2010

### songoku

1. The problem statement, all variables and given/known data
A 10 metre ladder rests against a wall at $$\frac{\pi}{4}$$.If the ladder slips so that when its bottom moving at 0.02 m/s, how fast would the ladder be moving down the wall?

a. 0.02 m/s
b. 0.0025 m/s
c. 0.015 m/s
d. 0.12 m/s
e. 0.15 m/s

2. Relevant equations
not sure

3. The attempt at a solution
My answer is 0.02 m/s because the speed when the ladder falls down will be the same as when its bottom moves. Actually, I think the speed will be independent of the angle and at all condition, the speed will be the same. Am I right?

Thanks

Last edited: Jan 25, 2010
2. Jan 25, 2010

### ZeroNowhere

I assume you mean pi/4 radians, that is, 45 degrees ((pi/4)*(180/pi)), given the lack of units. In which case, it shouldn't be too complicated to draw up a diagram, containing the ladder both before and after motion (after 1 second, so with the bottom having moved 0.02m in the second ladder), in order to confirm the answer. Given the length of the ladder, you can calculate the height of it on the wall as well as the distance of the bottom from the wall, and then it should be fairly easy to derive one of the lengths on the second ladder (you already have the ladder's length), and then figure out the other through Pythagoras' Theorem.

As for whether it would be independent of the angle, if it happens to be with 45 degrees, repeat the calculations with a sufficiently different angle, like 70 degrees, between the ladder and wall, and find out if they remain the same.

3. Jan 25, 2010

### songoku

Hi ZeroNowhere

I've tried it and the answer for this question is 0.02 m/s. I tried it for 30 degrees and it came out different (0.0348 m/s). The question is I don't get the logic behind it because I think when the ladder moves 0.02 m to, say, left, then it also should move 0.02 m downward.

I also tried to find the velocity using trigonometry (tan) when the angle 30 degrees. Vx = 0.02 m/s, then Vy = Vx tan (30) = 0.0115 m/s, differs from when I used your method (find the vertical distance traveled by the ladder, then find the velocity). Why can't I use tan ?

Basically, I confuse why the speed depends to the angle and why tan doesn't work.

Thanks

4. Jan 26, 2010

### tiny-tim

Welcome to PF!

Hi ZeroNowhere! Welcome to PF!
songoku, why are you using angles?

Just follow ZeroNowhere's suggestion … use lengths and Pythagoras.
… you'll get y as a function of x, and then you can find how dy/dt depends on dx/dt.

5. Jan 27, 2010

### songoku

Re: Welcome to PF!

Hi tiny-tim
It just came across my mind. In parabolic motion, we can find the angle of projection using tan (angle) = (Vy)/(Vx). So I tried it here. Why doesn't it work?

This is what I've tried :
Let :
θ = angle between ladder and floor
a = horizontal distance between the bottom of the ladder and the wall = c cos θ
b = height of top of the ladder from wall = c sin θ

After the bottom moves x meters (further from the wall ) and the top falls y metres, we have:

c2=(a+x)2+(b-y)2

c2=(c cos θ+x)2+(c sin θ-y)2

c2=c2cos2θ+2cx cos θ+x2+c2sin2θ-2cy sin θ+y2

c2=c2+2cx cos θ+x2-2cy sin θ+y2

2cx cos θ+x2=2cy sin θ-y2

differentiate with respect to t

2c cos θ dx/dt + 2x dx/dt = 2c sin θ dy/dt - 2y dy/dt

Is this what you mean?

Thanks

6. Jan 30, 2010

### songoku

Sorry to bump up but can anyone help please?

Thanks

7. Jan 30, 2010

### tiny-tim

No, θ depends on t also, so you've left out all the terms with dθ/dt.

Can't you see that by introducing θ, you've only made it more complicated? … you're only interested in dx/dt and dy/dt, so what is the point of dragging θ into it … you'll only have to eliminate θ later.

Do what ZeroNowhere and I suggested … forget the angle, and just use Pythagoras to find y as a function of x, and then differentiate.

8. Jan 30, 2010

### songoku

Hi tiny-tim

Ok, I tried a new one :

c2 = (a+x)2 + (b-y)2

c2 = a2 + 2ax + x2 + b2 - 2by + y2

Differentiate with respect to t (a, b, and c are constants) :

0 = 0 + 2a dx/dt + 2x dx/dt + 0 - 2b dy/dt + 2y dy/dt

Is this what you mean?

9. Jan 30, 2010

### rl.bhat

In the problem it stated that the ladder is at π/4 in the beginning. Now the question is whether the ladder starts moving from that position or already in motion? To find dy/dt, you must know where dx/dt = 0.2 m/s. If the floor is frictionless, then only in the vertical position the ladder will not slip. Otherwise it will start slipping and KE of the center of mass of the ladder will increase. Consequently dx/dt will increase.

10. Jan 30, 2010

### songoku

Hi rl.bhat
I think it starts moving from that position. The ladder is at rest, then slips.

I don't get what "where dx/dt = 0.02 m/s" means. If dx/dt = 0.02 m/s, then the the bottom will move 0.02 m in 1 second. Is that your point?

I also don't get this. The question doesn't say about friction so I think it's frictionless and if the floor is frictionless, why the vertical position doesn't slip?

Thanks

11. Jan 31, 2010

### rl.bhat

If the ladder stars slipping from rest, it must accelerate to reach 0.02 m/s. At what distance from the starting point, the velocity of the foot of the ladder will be 0.02 m/s?
Now let us analyze the problem. Let us search for the origin of the acceleration.
Find the component of g along the ladder. It is g*sinθ.
Find its component along floor. It is g*sinθ*cosθ. Or acceleration a = g/2*sin2θ.
Now you can the purpose of setting the ladder at π/4. At that angle acceleration of the foot of the ladder is maximum and that is equal to g/2. After this point acceleration decrease with increase in x.
Now v^2 = 2*a*x. = 2*g*sinθ*cosθ*x = 2*g*(1 - cos^2θ)^1/2*cosθ*x
= 2*g*[1-{(x+a)/r}^2]^1/2*(x+a)/r*x
v is given and r is known. Find x. Distance of the foot of the ladder from the wall is
X = xo + x, where xo = r*sin(π/4). Then find Y and find dy/dt using the equation given by you in #8.

Last edited: Jan 31, 2010
12. Jan 31, 2010

### tiny-tim

Hi songoku!

(just got up :zzz: …)
Yup!

And so dy/dt = [(b - y)/(a + x)]dx/dt.

13. Jan 31, 2010

### songoku

Hi tiny-tim and rl.bhat

Now I want to ask the logic behind it. From the equation derived by rl.bhat or by myself (with tiny-tim and zeroNowhere advice, of course ) it can be seen that dy/dt is not always the same as dx/dt. I can't understand why because if I think about it, it seems that the top will fall down with the same speed as the bottom. I imagine a real ladder, if the bottom moves 0.5 m, I think the top will also move 0.5 m in the same time. What will cause the different in the distance traveled by the top and the bottom?

Thanks

14. Jan 31, 2010

### rl.bhat

OK.
Now initially x^2 + y^2 = r^2.
According to your suggestion, after the bottom moves through 0.5 m and top also move through 0.5 m, then
(x+0.5)^2 + (y-0.5)^2 must be equal to r^2.
Is it true? Check with any numerical value.

15. Feb 1, 2010

### songoku

Hi rl.bhat

I get it now. Thanks !! :

16. Feb 1, 2010

### rl.bhat

The problem is not phrased properly. It should be
A 10 metre ladder rests against a wall. If the ladder slips so that its bottom is moving at 0.02 m/s when it makes an angle π/4 with the floor, how fast would the ladder be moving down the wall?

17. Feb 1, 2010

### songoku

Hi rl.bat
Yes you're right, so we don't consider the acceleration. Thanks :)


text:
### 8 numbers problem

Time Limit: 1000 ms Memory Limit: 65536 KiB

#### Problem Description

I think almost every acmer will know the 8 numbers problem which is a very famous problem.The game begin from the initial state of a 3*3 matrix which makes up of 8 numbers(1-8) and a blank block(0).move the blank block with its adjacent block until reach the objective state.It is obvious that the blank block has four directions which it can move to when it is at the middle position,i.e.up,down,left,right.Also,it has two directions when it is at the corner of the matrix and three directions at other posintion.
Form example,the initial state of the matrix:
8 0 3
2 1 4
7 6 5

the objective state:
1 2 3
8 0 4
7 6 5

and we give a valid moving path:
   8 0 3     8 1 3     8 1 3     0 1 3      1 0 3      1 2 3
2 1 4  >  2 0 4  >  0 2 4  >  8 2 4   >  8 2 4   >  8 0 4
7 6 5     7 6 5     7 6 5     7 6 5      7 6 5      7 6 5

Moreover,the path with least steps is called the shortest path.And the 8 numbers is check whether there are the path from the initial state to the objective state and if it exists,give the shortest path.
And we all know huicpc229 is not very good at search,so he hasn't solved this problem now.But he has solved another easy problem. The problem is described as follow:
Give an initial state of the matrix,and give a sequence of moving. For every moving,if the blank block can move to the direction as the moving,move it,otherwise ignore this moving.And we want to know the final state of the matrix.

#### Input

The first line of the input is one integer t,the number of testcase.
For each testcase:
Three lines respond to the initial state of the matrix,and there will be three numbers on each of the three lines.
Follow by an interger m corresponding to the number of moving.
The next m line,every line contain only one character:
U: move the blank block up for one block.
D: move the blank block down for one block.
L: move the blank block left for one block.
R: move the blank block right for one block.

#### Output

For each testcase output the final state of the matrix for three lines as above.And there will be a blank space between every two numbers on the same line.And you should output one blank line after each testcase.

#### Sample Input

1
8 0 3
2 1 4
7 6 5
2
D
R

#### Sample Output

8 1 3
2 4 0
7 6 5

2009湖南大学校赛


text:
# Small fix for msvcrt locking code

Matt Lavin matt.lavin at gmail.com
Fri Aug 19 04:39:26 BST 2005

```This should fix many warnings being reported to the console when using
msvrt locking.  This is my very first time looking at or modifying the
source of bzr, so I could be way off base.  I couldn't figure out how to
create a nice patch file that I think I've seen on this list but here is
the diff

--- bzrlib\lock.py
+++ bzrlib\lock.py
@@ -168,6 +168,8 @@
LOCK_NB = 4
def unlock(self):
_msvc_unlock(self.f)
+                    self.f.close()
+                    self.f = None


text:
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# M17-03

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16 Sep 2014, 01:00
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Before a project started, there had been 20 male and 12 female employees in the finance department. If 5 more men were hired to work on the project, what was the percent increase in the ratio of male employees to female employees in the finance department?

A. 10%
B. 12%
C. 20%
D. 24%
E. 25%

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Posts: 53796

### Show Tags

16 Sep 2014, 01:00
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Official Solution:

Before a project started, there had been 20 male and 12 female employees in the finance department. If 5 more men were hired to work on the project, what was the percent increase in the ratio of male employees to female employees in the finance department?

A. 10%
B. 12%
C. 20%
D. 24%
E. 25%

Ratio before $$= \frac{20}{12}$$, ratio after $$= \frac{25}{12}$$. The percent increase = $$(\frac{\frac{25}{12} - \frac{20}{12}}{\frac{20}{12}})*100\% = \frac{5}{12}*\frac{12}{20}*100\% = 25\%$$.

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### Show Tags

12 Aug 2015, 01:09
we can also solve this by testing answer options.
Check for 20% (easy number to deal).
20/12 (M:F ratio before change) + 20% (20/12) = 24/12....but the ratio after increase is 25/12. hence C is ruled out.
A and B also ruled out as they will given even smaller ratio.

Lets check 25%
20/12 (ratio before change) + 25% (20/12) = 25/12...which is the M:F ratio after the change.
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16 Jan 2019, 01:16
Can I get an explanation for this question? Purchased GMAT Club Quant tests, but not really happy with the explanation provided. My quant is on the weak side, so a step by step explanation would be really ace people.

Math Expert
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Posts: 53796

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16 Jan 2019, 01:41
nust2017 wrote:
Can I get an explanation for this question? Purchased GMAT Club Quant tests, but not really happy with the explanation provided. My quant is on the weak side, so a step by step explanation would be really ace people.

Not sure what to elaborate there. The question tests ratios and percent increase/decrease formula (general formula for percent increase or decrease, (percent change): $$\text{Percent} = \frac{\text{Change}}{\text{Original}}*100$$).

Could you please specify what part is unclear in the solution? Thank you.
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Re: M17-03   [#permalink] 16 Jan 2019, 01:41
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# M17-03

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# $L^\infty$-Stability of IMEX-BDF2 Finite Volume Scheme for Convection-Diffusion Equation

1 RAPSODI - Reliable numerical approximations of dissipative systems
LPP - Laboratoire Paul Painlevé - UMR 8524, Inria Lille - Nord Europe
Abstract : In this paper, we propose a finite volume scheme for solving a two-dimensional convection-diffusion equation on general meshes. This work is based on a implicit-explicit (IMEX) second order method and it is issued from the seminal paper [2]. In the framework of MUSCL methods, we will prove that the local maximum property is guaranteed under an explicit Courant-Friedrichs-Levy condition and the classical hypothesis for the triangulation of the domain.
Keywords :
Document type :
Conference papers
Domain :
Complete list of metadata

Cited literature [14 references]

https://hal.archives-ouvertes.fr/hal-01574893
Contributor : Caterina Calgaro Connect in order to contact the contributor
Submitted on : Wednesday, August 16, 2017 - 6:47:32 PM
Last modification on : Wednesday, March 23, 2022 - 3:51:22 PM

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### Citation

Caterina Calgaro, Meriem Ezzoug. $L^\infty$-Stability of IMEX-BDF2 Finite Volume Scheme for Convection-Diffusion Equation. FVCA 2017: Finite Volumes for Complex Applications VIII - Methods and Theoretical Aspects, Jun 2017, Lille, France. pp.245-253, ⟨10.1007/978-3-319-57397-7_17⟩. ⟨hal-01574893⟩

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Menu Home Action My entries Defined browse Select Select in the list MathematicsPhysics Then Select Select in the list Then Select Select in the list Search

impulse

 Definition/Summary Force = impulse per time: $\mathbf{F}\ =\ d\mathbf{I}/dt$. For constant force, impulse = force times time: $\mathbf{I}\ =\ \mathbf{F}\,\Delta t$ (by comparison, work done = force "dot" distance: $W\ =\ \mathbf{F}\cdot \Delta\mathbf{s}$). For varying force, impulse is the integral of force over time: $\mathbf{I}\ =\ \int\mathbf{F}\,dt$ (and work done is the integral of force over distance: $W\ =\ \int\mathbf{F}\cdot d\mathbf{s}$). Newton's second law (force = rate of change of momentum: $\mathbf{F}\ =\ d(m\mathbf{v})/dt$) integrated over time becomes: impulse = total change of momentum: $\mathbf{I}\ =\ \int d(m\mathbf{v})/dt\,dt\ =\ \int d(m\mathbf{v})\ =\ \Delta(m\mathbf{v})$. Impulse is a vector, with the same dimensions as momentum: $ML/T$, and is measured in units of newton seconds ($N.s\text{, or }kg\,m\,s^{-1}$).

 Equations $$\mathbf{F}\ =\ \frac{d\mathbf{I}}{dt}$$ $$\mathbf{I}\ =\ \int\mathbf{F}\,dt$$ Impulse-momentum theorem: $$\mathbf{I}\ =\ \int\frac{d(m\mathbf{v})}{dt}\,dt\ =\ \int d(m\mathbf{v})\ =\ \Delta(m\mathbf{v})$$

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 Breakdown Physics > Classical Mechanics >> Newtonian Dynamics


text:
Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > DETAIL:

### Paper:

TR95-046 | 4th August 1995 00:00

#### On the Power of Circuits with Gates of Low L_1 Norms

TR95-046
Authors: Vince Grolmusz
Publication: 4th October 1995 16:36
Keywords:

Abstract:

We examine the power of Boolean functions with low L_1 norms in several
settings. In large part of the recent literature, the degree of a polynomial
which represents a Boolean function in some way was chosen to be the measure of the complexity of the Boolean function.
However, some functions with low communicational complexity (AND, OR, PARITY,
ID) have high degree, but small L_1 norms. So, in conjunction with
communication complexity, instead of the degree, the L_1 norm can be an important measure of
hardness. We conjecture that the randomized communication complexity of any
Boolean function is bounded by the polylogarithm of its L_1 norm. We can
prove only a weaker statement: we present a two-party, randomized, common-coin
communication protocol for computing functions with O(L^2_1\delta) bits of
communication, with error-probability of exp(-c\delta), (even with large
degree or exponential number of terms).
Then we present several applications of this theorem for circuit lower bounds
(both for bounded- and unbounded depth), and a decision-tree lower bound.

ISSN 1433-8092 | Imprint


text:
Opuscula Math. 35, no. 2 (2015), 191-197
http://dx.doi.org/10.7494/OpMath.2015.35.2.191

Opuscula Mathematica

# A new characterization of convex φ-functions with a parameter

Bartosz Micherda

Abstract. We show that, under some additional assumptions, all projection operators onto latticially closed subsets of the Orlicz-Musielak space generated by $$\Phi$$ are isotonic if and only if $$\Phi$$ is convex with respect to its second variable. A dual result of this type is also proven for antiprojections. This gives the positive answer to the problem presented in Opuscula Mathematica in 2012.

Keywords: Orlicz-Musielak space, convex function, isotonic operator, projection operator, antiprojection operator.

Mathematics Subject Classification: 41A65, 39B62, 46E30.

Full text (pdf)

1. G. Isac, On the order monotonicity of the metric projection operator, [in:] Approximation Theory, Wavelets and Applications, NATO ASI Series, Kluwer Acad. Publ., Dordrecht (1995), 365-379.
2. G. Isac, G. Lewicki, On the property of four elements in modular spaces, Acta Math. Hungar. 83 (1999) 4, 293-301.
3. G. Isac, A.B. Németh, Every generating isotone projection cone is latticial and correct, J. Math. Anal. Appl. 147 (1990) 1, 53-62.
4. G. Isac, A.B. Németh, Isotone projection cones in Euclidean spaces, Ann. Sci. Math. Québec 16 (1992) 1, 35-52.
5. G. Isac, L.E. Persson, On an inequality related to the isotonicity of the projection operator, J. Approx. Theory 86 (1996) 2, 129-143.
6. G. Isac, L.E. Persson, Inequalities related to isotonicity of projection and antiprojection operators, Math. Inequal. Appl. 1 (1998) 1, 85-97.
7. B. Micherda, The properties of four elements in Orlicz-Musielak spaces, Math. Inequal. Appl. 4 (2001) 4, 599-608.
8. B. Micherda, On the latticity of projection and antiprojection sets in Orlicz-Musielak spaces, Acta Math. Hungar. 119 (2008) 1-2, 165-180.
9. B. Micherda, A characterization of convex $$\varphi$$-functions, Opuscula Math. 32 (2012) 1, 171-178.
10. J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer-Verlag, Berlin, 1983.
11. M.M. Rao, Z.D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics 146, Marcel Dekker, Inc., New York, 1991.
• Bartosz Micherda
• University of Bielsko-Biała, Department of Mathematics, ul. Willowa 2, 43-309 Bielsko-Biała, Poland
• Communicated by Henryk Hudzik.
• Accepted: 2014-06-18.
• Published online: 2014-11-18.


text:
## Second Order Rxn

$\frac{d[R]}{dt}=-k[R]^{2}; \frac{1}{[R]}=kt + \frac{1}{[R]_{0}}; t_{\frac{1}{2}}=\frac{1}{k[R]_{0}}$

eduardomorales5
Posts: 77
Joined: Fri Aug 30, 2019 12:15 am

### Second Order Rxn

How would the graph of a second order reaction look like? What can you find with the second order equation ?

Samuel G Rivera - Discussion 4I
Posts: 61
Joined: Sat Aug 17, 2019 12:16 am
Been upvoted: 1 time

### Re: Second Order Rxn

The second order reaction has a positive slope on a graph with 1/[A] on the y axis and 1/time on the x axis

jisulee1C
Posts: 149
Joined: Thu Jul 25, 2019 12:17 am

### Re: Second Order Rxn

Once you plot the second order graph using 1/concentration versus time you can find slope which = k.
Attachments


text:
## Robust second-order accurate discretizations of the multi-dimensional Heaviside and Dirac delta functions.(English)Zbl 1153.65014

Summary: We present a robust second-order accurate method for discretizing the multi-dimensional Heaviside and the Dirac delta functions on irregular domains. The method is robust in the following ways: (1) integrations of source terms on a co-dimension one surface are independent of the underlying grid and therefore stable under perturbations of the domain’s boundary; (2) the method depends only on the function value of a level function, not on its derivatives.
We present the discretizations in tabulated form to make their implementations straightforward. We present numerical results in two and three spatial dimensions to demonstrate the second-order accuracy in the $$L^{1}$$-norm in the case of the solution of partial differential equations with singular source terms. In the case of evaluating the contribution of singular source terms on interfaces, the method is also second-order accurate in the $$L^{\infty }$$-norm.

### MSC:

 65D15 Algorithms for approximation of functions
Full Text:

### References:

 [1] Aslam, T., A partial differential equation approach to multidimensional extrapolation, J. comput. phys., 193, 349-355, (2004) · Zbl 1036.65002 [2] Brackbill, J.U.; Kothe, D.B.; Zemach, C., A continuum method for modelling surface tension, J. comput. phys., 100, 335-353, (1992) · Zbl 0775.76110 [3] D. Calhoun, P. Smereka. The numerical approximation of a delta function. Preprint , 2004. [4] Engquist, B.; Tornberg, A.K.; Tsai, R., Discretization of Dirac delta functions in level set methods, J. comput. phys., 207, 28-51, (2005) · Zbl 1074.65025 [5] Enright, D.; Fedkiw, R.; Ferziger, J.; Mitchell, I., A hybrid particle level set method for improved interface capturing, J. comput. phys., 183, 83-116, (2002) · Zbl 1021.76044 [6] Goodman, J.E.; O’Rourke, J., The handbook of discrete and computational geometry, (1997), CRC Press LL · Zbl 0890.52001 [7] Karma, A.; Rappel, W.-J., Quantitative phase-field modeling of dendritic growth in two and three dimensions, Phys. rev. E, 57, 4323-4349, (1997) · Zbl 1086.82558 [8] Kuhn, H.W., Some combinational lemmas in topology, IBM J. res. develop., 4, 508-524, (1960) · Zbl 0109.15603 [9] Mayo, A., The fast solution of poisson’s and the biharmonic equations on irregular regions, SIAM J. numer. anal., 21, 285-299, (1984) · Zbl 1131.65303 [10] Min, C., Simplicial isosurfacing in arbitrary dimension and codimension, J. comput. phys., 190, 295-310, (2003) · Zbl 1029.65016 [11] Min, C., Local level set method in high dimension and codimension, J. comput. phys., 200, 368-382, (2004) · Zbl 1086.65088 [12] Min, C.; Gibou, F., Geometric integration over irregular domains with application to level set methods, J. comput. phys., 226, 1432-1443, (2007) · Zbl 1125.65021 [13] Min, C.; Gibou, F., A second order accurate level set method on non-graded adaptive Cartesian grids, J. comput. phys., 225, 300-321, (2007) · Zbl 1122.65077 [14] Osher, S.; Fedkiw, R., Level set methods and dynamic implicit surfaces, (2002), Springer-Verlag New York, NY [15] Peskin, C., The immersed boundary method, Acta numer., 11, 479-517, (2002) · Zbl 1123.74309 [16] Sallee, J.F., The middle-cut triangulations of the n-cube, SIAM J. alg. disc. methods, 5, 407-419, (1984) · Zbl 0543.52004 [17] Sethian, J., Fast marching methods, SIAM rev., 41, 199-235, (1999) · Zbl 0926.65106 [18] Sethian, J.A., Level set methods and fast marching methods, (1999), Cambridge University Press Cambridge · Zbl 0929.65066 [19] Smereka, P., Semi-implicit level set methods for curvature and surface diffusion motion, J. sci. comput., 19, 439-456, (2003) · Zbl 1035.65098 [20] Smereka, P., The numerical approximation of a delta function with application to level set methods, J. comput. phys., 211, 77-90, (2006) · Zbl 1086.65503 [21] Sussman, M.; Puckett, E.G., A coupled level let and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows, J. comput. phys., 162, 301-337, (2000) · Zbl 0977.76071 [22] Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. comput. phys., 114, 146-159, (1994) · Zbl 0808.76077 [23] Tornberg, A.-K.; Engquist, B., Numerical approximations of singular source terms in differential equations, J. comput. phys., 200, 462-488, (2004) · Zbl 1115.76392 [24] Towers, J., Two methods for discretizing a delta function supported on a level set, J. comput. phys., 220, 915-931, (2007) · Zbl 1115.65028 [25] Towers, J., A convergence rate theorem for finite difference approximations to delta functions, J. comput. phys., 227, 6591-6597, (2008) · Zbl 1155.65016 [26] Tryggvason, G.; Bunner, B.; Esmaeeli, A.; Juric, D.; Al-Rawahi, N.; Tauber, W.; Han, J.; Nas, S.; Jan, Y.-J., A front-tracking method for the computations of multiphase flow, J. comput. phys., 169, 708-759, (2001) · Zbl 1047.76574 [27] Walén, J., On the approximation of singular source terms in differential equations, Numer. methods partial differ. equat., 15, 503-520, (1999)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
This, like the previous and probably the next few posts, is an attempt at understanding some of the ideas in Mumford’s Abelian Varieties.

Let ${A}$ be an abelian variety. Last time, I described a formula which allowed us to express the pull-back ${n_A^* \mathcal{L}}$ of a line bundle ${\mathcal{L} \in \mathrm{Pic}(A)}$ as

$\displaystyle n_A^*\mathcal{L} \simeq \mathcal{L}^{(n^2 + n)/2} \otimes (-1)^* \mathcal{L}^{(n^2 - n)/2}.$

This was a special case of the so-called “theorem of the cube,” which allowed us to express, for three morphisms ${f, g, h: X \rightarrow A}$, the pull-back

$\displaystyle (f + g + h)^* \mathcal{L}$

in terms of the various pull-backs of partial sums. Namely, we had the formula

$\displaystyle (f + g+ h)^* \mathcal{L} \simeq (f+g)^* \mathcal{L} \otimes (f + h)^* \mathcal{L} \otimes (g + h)^* \mathcal{L} \otimes f^* \mathcal{L}^{-1} \otimes g^* \mathcal{L}^{-1} \otimes h^{*} \mathcal{L}^{-1} .$

In this formula, we take ${f = \mathrm{Id} = 1_A}$ and ${g, h}$ constant maps at ${x,y \in A}$, respectively. Let ${T_x, T_y}$, etc. denote the translation maps on ${A}$. Then ${T_x, T_y, T_{x+y}}$ are the relevant sums, and we have

$\displaystyle T_{x+y}^* \mathcal{L} \simeq T_x^* \mathcal{L} \otimes T_y^* \mathcal{L} \otimes \mathcal{L}^{-1}.$

We get the theorem of the square:

Theorem 1 (Theorem of the square)Let ${A}$ be an abelian variety, ${\mathcal{L} \in \mathrm{Pic}(A)}$, and ${x, y \in A}$ points. Then

$\displaystyle T_{x+y}^* \mathcal{L} \otimes \mathcal{L} \simeq T_x^* \mathcal{L} \otimes T_y^* \mathcal{L}.$

In the rest of this post, I’ll describe some applications of this result, for instance the fact that abelian varieties are projective.

1. Translation-invariant line bundles

Let ${A}$ be an abelian variety. We will be interested in line bundles on ${A}$ which are translation-invariant, i.e. line bundles ${\mathcal{L}}$ such that ${T_x^* \mathcal{L} \simeq \mathcal{L}}$ for all ${x \in A}$.

Definition 2 The group of ${\mathcal{L} \in \mathrm{Pic}(A)}$ which are translation-invariant is ${\hat{A}}$.

In fact, ${\hat{A}}$ turns out to be not just a group, but an abelian variety itself. This is far from obvious, but it turns out to be a corollary of the (non-obvious) fact that line bundles on ${A}$ are themselves parametrized by a scheme, called the Picard scheme. The collection of translation-invariant bundles turns out to form the connected component at zero of the Picard scheme.

What are some examples of translation-invariant line bundles?

Example 1 Let ${\mathcal{L} \in \mathrm{Pic}(A)}$ be any line bundle. Then ${T_x^* \mathcal{L} \otimes \mathcal{L}^{-1}}$ is translation-invariant; this is a consequence of the theorem of the square.

One of the next goals is to show that any element of ${\hat{A}}$ can be obtained in the manner of the previous example.

Example 2 Let ${A}$ have dimension one, so ${A}$ is an elliptic curve. Then the translation-invariant line bundles correspond to divisors of total degree zero. In fact, by the theorem of the square, we find that the line bundle associated to ${(x) - (0)}$ for any ${x \in A}$ is in ${\hat{A}}$. The other direction will follow because any translation-invariant bundle is algebraically equivalent to zero, as we will see later, and consequently has degree zero.

Let ${m: A \times A \rightarrow A}$ be the multiplication map, and let ${\mathcal{L} \in \mathrm{Pic}(A)}$. Then the line bundle ${m^* \mathcal{L} \in \mathrm{Pic}(A \times A)}$ has the property that restriction to any fiber ${\left\{x \right\} \times A}$ is precisely ${T_x^* \mathcal{L}}$. If we think of ${m^* \mathcal{L}}$ as a family of line bundles on ${A}$, parametrized by ${\mathcal{L}}$, then to say that ${\mathcal{L} }$ is translation-invariant is a certain trivialty of this family. This is expressed in the following result.

Proposition 3 A line bundle ${\mathcal{L} \in \mathrm{Pic}(A)}$ belongs to ${\hat{A}}$ if and only if

$\displaystyle m^* \mathcal{L} \simeq \mathcal{L} \boxtimes \mathcal{L} \in \mathrm{Pic}(A \times A),$

for ${\mathcal{L}}$ the multiplication map.

In fact, this follows from a “rigidity lemma” on families of line bundles. If we consider the line bundle ${m^* \mathcal{L} \otimes (\mathcal{L} \boxtimes \mathcal{L})^{-1}}$, then we have seen that it is trivial on each vertical fiber, and similarly on each horizontal fiber. But the rigidity lemma (which is proved in Mumford’s book) states precisely that such a line bundle is trivial. The converse is easy, by restricting to the vertical fibers.

A consequence of this fact is that

$\displaystyle \mathcal{L}^{-1} \simeq (-1)_A^*\mathcal{L} \quad \text{for } \mathcal{L} \in \hat{A},$

because ${(-1)^* \mathcal{L} \otimes \mathcal{L}}$ is obtained by pulling back ${\mathcal{L} \boxtimes \mathcal{L} \simeq m^* \mathcal{L}}$ under the map

$\displaystyle d: A \rightarrow A \times A, \quad x \mapsto (x, -x).$

However, ${d^*(\mathcal{L} \boxtimes \mathcal{L}) \simeq d^* m^* ( \mathcal{L} ) \simeq \mathcal{O}_A}$ because ${m \circ d}$ is constant.

In particular, translation-invariant line bundles are very strongly non-ample: if ${\mathcal{L} \in \hat{A}}$ were ample, then ${(-1)^* \mathcal{L}}$ would be too (since ${(-1)_A}$ is an isomorphism). But, ${\mathcal{L}}$ and ${\mathcal{L}^{-1}}$ cannot both be ample unless ${A}$ is zero-dimensional. In fact, it turns out that ampleness is a polar opposite of translation-invariance: a line bundle ${\mathcal{L} \in \mathrm{Pic}(A)}$ is ample if and only if the set of ${x \in A}$ such that ${T_x^* \mathcal{L} \simeq \mathcal{L}}$ is finite, as it turns out.

2. A little intersection theory

Let ${A}$ be an abelian variety, and let ${D \subset A}$ be a divisor on ${A}$. We saw in the previous section that

$\displaystyle T_{x+y}^* D + D \sim T_x^* D + T_y^* D,$

where ${\sim}$ denotes linear equivalence, for any ${x, y \in A}$. In particular, we get

$\displaystyle T_x^* D + T_{-x}^* D \sim 2D.$

This means that if ${D}$ is any effective divisor on ${A}$, then the linear system ${|2D|}$ is base-point free: given ${y \in A}$, we can choose ${x}$ such that ${T_x^* D + T_{-x}^* D}$ does not contain ${y}$. In particular, ${2D}$ defines a map ${A \rightarrow \mathbb{P}^N}$ for some ${N}$. We’d like to choose ${D}$ to make this map an immersion, or at least finite. We can thus state:

Observation: On an abelian variety, there are a lot of line bundles generated by global sections (and thus a lot of maps to projective space.)

Suppose ${E}$ is any base-point free divisor, defining a map

$\displaystyle \phi_E: A \rightarrow \mathbb{P}^N.$

We’d like to know when ${\phi_E}$ is a finite morphism. By Zariski’s main theorem, finiteness is equivalent to quasi-finiteness. Consequently, ${\phi_E}$ fails to be finite precisely when there exists an irreducible curve ${C \subset A}$ such that ${\phi_E(C) = \ast}$. This means that every divisor in the linear system ${|E|}$ either contains ${C}$ or fails to intersect ${C}$. In the language of line bundles, any section of ${\mathcal{O}(E)}$ vanishing on a point of ${C}$ must vanish everywhere on ${C}$. By changing ${E}$ if necessary, we can assume that ${E \cap C = \emptyset}$.

Observation: The intersections ${C \cap T_x^* E}$ are either empty or all of ${C}$, for all ${x \in A}$. In other words, ${C}$ can intersect no translate of ${E}$.

In fact, the intersection number ${C. T_x^* E}$ for ${x \in A}$ is constant, and must thus be identically zero. Alternatively, one considers the family of line bundles ${\mathcal{O}(T_x^* E )|_{\widetilde{C}}}$ on the normalization of ${C}$. When ${x = 0}$, this line bundle is trivial; it follows that for all ${x}$, the degree of the line bundle is zero. Since ${E}$ is an effective divisor, this implies that ${C \cap T_x^* E = \emptyset}$ for all ${x}$ such that intersection is not ${C}$.

This is a pretty useful geometric argument: “wiggling” the divisor ${E}$ by a translation by ${x}$ can’t force it to intersect ${C}$. Motivated by this, we can deduce something seemingly stronger.

Observation: If ${\phi_E: A \rightarrow \mathbb{P}^N}$ contracts the irreducible curve ${C}$ to a point and ${C \cap E = \emptyset}$, then ${E}$ is invariant under translation by ${x-y}$ for ${x,y \in C}$.

In fact, if ${x, y \in C}$ and ${e \in E}$, then the translation ${T_{x-e}}$ has the property that ${T_{x-e}(E) \cap C \neq \emptyset}$ (because the intersection contains ${x}$). In particular, it contains ${y}$ by the previous observation: thus

$\displaystyle y - (x - e) \in E,$

which is what we wanted.

These two observations show that there are strong constraints on a base-point free divisor ${E}$ with the property that the induced map to ${\mathbb{P}^N}$ is not finite: the last observation shows that ${E}$ has to be invariant by a proper curve’s worth of translations. We thus find:

Proposition 4 Let ${E}$ be a divisor on ${A}$ contained in ${A \setminus U}$ for ${U}$ an open affine. If ${E}$ is base-point free, then the map ${\phi_E: A \rightarrow \mathbb{P}^N}$ is finite.

Proof: In fact, we can assume ${0 \in U}$. In this case, we find that the only translations which leave ${E}$ invariant have to be in ${U}$ (i.e., they have to send ${0}$ into something in ${U}$). This means that there can’t be a proper curve’s worth of translations leaving invariant ${E}$, and by the previous observations, ${\phi_E: A \rightarrow \mathbb{P}^N}$ can’t contract a curve. $\Box$

Finally, putting all this together, we may deduce:

Theorem 5 An abelian variety ${A}$ is projective.

Proof: It suffices to show that ${A}$ admits a finite map ${\phi_E: A \rightarrow \mathbb{P}^N}$ for a suitable divisor ${E}$, because the divisor ${E}$ is then the pull-back of the ample divisor on ${\mathbb{P}^N}$ by a finite map and is thus itself ample. In order to do this, we need to find a divisor on ${A}$ without base points and which is contained in ${A \setminus U}$ for ${U}$ an open affine. That’s easy: just start by picking an irreducible component of ${A \setminus U}$, and then take twice that for ${E}$. The analysis at the beginning of the section shows that ${E}$ is base-point free. $\Box$

In fact, using a refinement of these arguments, one can prove a characterization of ampleness:

Theorem 6 A line bundle ${\mathcal{L} \in \mathrm{Pic}(A)}$ associated to an effective divisor is ample if and only if the set of all ${x \in A}$ such that ${T_x^* \mathcal{L} \simeq \mathcal{L}}$ is finite.


text:
# [texhax] Trying to find if a listing is continued with listings package

Reinhard Kotucha reinhard.kotucha at web.de
Tue Dec 22 01:34:16 CET 2015

On 2015-12-20 at 09:50:10 +0000, Philip Taylor wrote:

> Reinhard Kotucha wrote:
>
> > Adrian is using listings.sty, hence I assume that he's using LaTeX.
> > LaTeX users should better use \@ifundefined instead of TeX primitives
> > or plain TeX macros.
>
> Adrian may well be using "listings.sty", but there is very little
> LaTeX-ish about his code :
>
> > \expandafter
> > \ifx
> >   \csname \@lst @SaveFirstNumber\endcsname
> >   \undefined
> >   it's a new listing
> > \else
> >   it's a continued listing
> > \fi
>
> so suggesting the use of \relax in this context would seem quite
> reasonable to me ...  And does \@undefined map to \relax in such a
> context, as it would need to if his test were to perform as intended ?

Hello Phil,
here is the definition (see texmf-dist/source/latex/ltdefns.ltx):

% \begin{macro}{\@ifundefined}
% \changes{LaTeX2e}{1993/11/23}{Redefined to remove a trailing \cs{fi}}
%    Check if first arg is undefined or \cs{relax} and execute second or
%    third arg depending,
%    \begin{macrocode}
\def\@ifundefined#1{%
\expandafter\ifx\csname#1\endcsname\relax
\expandafter\@firstoftwo
\else
\expandafter\@secondoftwo
\fi}
%    \end{macrocode}
% \end{macro}

So if you're using \@ifundefined exclusively you'll never run into the
\relax vs. \undefined problem.  If you don't need \csname \endcsname
in order to construct a control sequence name, you might think that
you can write

\ifx\csname\undefined

but the result is unpredictable.  Maybe a package you loaded before
did the same test with \@ifundefined and \csname is now mapped to
\relax.  It's much safer if LaTeX users use LaTeX code exclusively.

BTW, the fact that \csname \endcsname always sets the control sequence
to \relax, even if you are just teststing whether it exists at all, is
quite unfortunate.  It adds something to the hash table which can
never be removed and thus stays there forever.  This slows down table
lookups and blows up the string pool.  When I created huge files with
pgfplots I already had to increase the string pool.  In this case
however, I believe that there is a bug in one of related the packages.

Phil, we discussed this issue a few years ago and you forwarded a mail
from someone who explained *why* TeX cannot remove unused csnames from
the hash table.  Unfortunately I can't find this mail anymore.  Do you
remember it and can you forward it to me again?

Regards,
Reinhard

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text:
The usual change of variable technique takes the form

However, sometimes it is useful to relate the new variable and the old one implicitely:

$t$ may be explicitely expressed in terms of $x$, and this would lead to the same results, if we remember the derivative of the inverse function

but this may be much more awkward. It is just a question of ease