8,103 results
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2. A remark on a paper of P. B. Djakov and M. S. Ramanujan
- Author
-
Murat Yurdakul and Elif Uyanik
- Subjects
Unbounded operator ,Combinatorics ,symbols.namesake ,Monotone polygon ,Basis (linear algebra) ,General Mathematics ,Bounded function ,Operator (physics) ,symbols ,Sequence space ,Continuous linear operator ,Ramanujan's sum ,Mathematics - Abstract
Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We show that if there exists a continuous linear unbounded operator between l-K\"{o}the spaces, then there exists a continuous unbounded quasi-diagonal operator between them. Using this result, we study in terms of corresponding K\"{o}the matrices when every continuous linear operator between l-K\"{o}the spaces is bounded. As an application, we observe that the existence of an unbounded operator between l-K\"{o}the spaces, under a splitting condition, causes the existence of a common basic subspace.
- Published
- 2019
3. An unpublished paper ‘Über einige durch unendliche Reihen definirte Functionen eines complexen Argumentes’ by Adolf Hurwitz
- Author
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Nicola Oswald
- Subjects
History ,Pure mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Algebra ,symbols.namesake ,Continuation ,0103 physical sciences ,Functional equation ,symbols ,010307 mathematical physics ,0101 mathematics ,Dirichlet series ,Meromorphic function ,Mathematics - Abstract
In 1903, Epstein published his proof of meromorphic continuation and a functional equation for Dirichlet series associated with quadratic forms, now called Epstein zeta-functions. However, already in 1889 (or even earlier) Hurwitz was aware of these results as his mathematical diaries and some unpublished notes (in an almost final form) found in his estate at the ETH Zurich show. In this article we present and analyze Hurwitz's notes and compare his reasoning with Epstein's paper in detail.
- Published
- 2017
4. Corrigendum to the papers on Exceptional orthogonal polynomials: J. Approx. Theory 182 (2014) 29–58, 184 (2014) 176–208 and 214 (2017) 9–48
- Author
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Antonio J. Durán
- Subjects
Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Hilbert space ,Approx ,symbols.namesake ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Orthogonal polynomials ,symbols ,Analysis ,Mathematics - Abstract
We complete a gap in the proof that exceptional polynomials are complete orthogonal systems in the associated Hilbert spaces.
- Published
- 2020
5. Corrections to the paper 'The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces'
- Author
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Rovshan A. Bandaliev
- Subjects
Discrete mathematics ,Mathematics::Functional Analysis ,Sublinear function ,General Mathematics ,Mathematics::Classical Analysis and ODEs ,Lebesgue's number lemma ,Type (model theory) ,symbols.namesake ,Operator (computer programming) ,Mathematics - Classical Analysis and ODEs ,Ordinary differential equation ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,Standard probability space ,Lp space ,Variable (mathematics) ,Mathematics - Abstract
In this paper author was proved the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent. As an application we prove the boundedness of certain sublinear operators on the weighted variable Lebesgue space. Note that the proof of multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent don't contained any mistakes. But at the proving of the boundedness of certain sublinear operators on the weighted variable Lebesgue space Georgian colleagues discovered a small but significant error in this paper. This result is assigned as Theorem 5 in noted paper. In other words, sufficient conditions for general weights ensuring the validity of the two-weight strong type inequalities for some sublinear operator was found. In this theorem the inequality (9) isn't true. In this note we give the details of the correct argument. We presume that the reader is familiar with the contents and notation of our original paper. At the heart of our correction is the following Theorem which replaces Theorem 5.
- Published
- 2013
6. A Note on Recent Papers by Grafakos and Teschl, and Estrada
- Author
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Adam Nowak and Krzysztof Stempak
- Subjects
Hankel transform ,Partial differential equation ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Function (mathematics) ,Transplantation ,symbols.namesake ,Radial function ,Fourier transform ,Fourier analysis ,symbols ,Analysis ,Mathematics - Abstract
We indicate how recent results of Grafakos and Teschl (J Fourier Anal Appl 19:167–179, 2013), and Estrada (J Fourier Anal Appl 20:301–320, 2014), relating the Fourier transform of a radial function in $$\mathbb R^n$$ and the Fourier transform of the same function in $$\mathbb R^{n+2}$$ and $$\mathbb R^{n+1}$$ , respectively, are located within known results on transplantation for Hankel transforms.
- Published
- 2014
7. Corrigendum to the paper 'Adjoining an Order Unit to a Matrix Ordered Space'
- Author
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Anil Kumar Karn
- Subjects
Pure mathematics ,General Mathematics ,Mathematical analysis ,Operator theory ,Potential theory ,Theoretical Computer Science ,Strictly convex space ,symbols.namesake ,Matrix (mathematics) ,Fourier analysis ,Ordered space ,symbols ,Order (group theory) ,Unit (ring theory) ,Analysis ,Mathematics - Abstract
An error has been detected (and also corrected) in Theorem 2.8 of the paper entitled “Adjoining an Order Unit to a Matrix Ordered Space” (Positivity, (2005)9: 207–223; DOI 10.1007/s11117-003-2778-5). Accordingly, some of the results of the paper have been modified. Also, a notion of C*-matricially, Riesz normed spaces has been introduced.
- Published
- 2007
8. Einstein's First Published Paper
- Author
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Boris Iglewicz
- Subjects
Condensed Matter::Quantum Gases ,Statistics and Probability ,General Mathematics ,Proposition ,Epistemology ,symbols.namesake ,Graduate students ,Scientific method ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,symbols ,Calculus ,Data recording ,Statistics, Probability and Uncertainty ,Einstein ,Mathematics - Abstract
This article reviews Albert Einstein's first published paper, submitted for publication in 1900. At that time, Einstein was 21 and a recent college graduate. His paper uses modeling and least squares to analyze data in support of a scientific proposition. Einstein is shown to be well trained, for his day, in using statistics as a tool in his scientific research. This paper also shows his ability to make trivial arithmetic mistakes and some clumsiness in data recording. A major aim of this article is to help provide a better appreciation of Einstein as an active user of statistical arguments in this and other of his important publications.
- Published
- 2007
9. A remark on a paper of F. Luca and A. Sankaranarayanan
- Author
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Imre Kátai
- Subjects
Set (abstract data type) ,Discrete mathematics ,symbols.namesake ,Number theory ,Statement (logic) ,General Mathematics ,Ordinary differential equation ,Multiplicative function ,Zero (complex analysis) ,symbols ,Calculus ,Euler's totient function ,Mathematics - Abstract
We generalize a result of F. Luca and A. Sankaranarayanan by proving that the set of n for which ϕ(1) + + ϕ(n) is squareful is of zero density. A similar statement holds for σ (n) instead of ϕ(n) and for some other multiplicative functions.
- Published
- 2008
10. Correction to the paper 'on the curvature of a generalization of a contact metric manifolds'
- Author
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L. Di Terlizzi
- Subjects
Christoffel symbols ,Riemann curvature tensor ,symbols.namesake ,Mean curvature flow ,General Mathematics ,Mathematical analysis ,symbols ,Curvature form ,Sectional curvature ,Curvature ,Ricci curvature ,Scalar curvature ,Mathematics - Abstract
We considered in Example 3.1 of the paper [1] an S-structure on R2n+s . We concluded that when s > 1 this manifold cannot be of constant φ-sectional curvature. Unfortunately this result is wrong. In fact, essentially due to a sign mistake in defining the φ-structure and a consequent transposition of the elements of the φ-basis (3.2), some of the Christoffel’s symbols were incorrect. In the present rectification, using a more slendler tecnique, we prove that our manifold is of constant φ-sectional curvature −3s and then it is η-Einstein.
- Published
- 2009
11. A remark on a paper by Evans and Harris on the point spectra of Dirac operators
- Author
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Karl Michael Schmidt
- Subjects
Dirac measure ,General Mathematics ,Dirac (software) ,Dirac algebra ,Clifford analysis ,Dirac operator ,symbols.namesake ,Dirac spinor ,symbols ,Point (geometry) ,QA ,Eigenvalues and eigenvectors ,Mathematical physics ,Mathematics - Abstract
This paper presents a sufficient condition for a one-dimensional Dirac operator with a potential tending to infinity at infinity to have no eigenvalues. It also provides a quick proof (and suggests variations) of a related criterion given by Evans and Harris.
- Published
- 2001
12. On the Paper 'A Note on Spaces of Absolutely Convergent Fourier Transforms' by Björn G.Walther (this Issue). Letter to the Editor
- Author
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S. V. Kislyakov
- Subjects
Algebra ,symbols.namesake ,Letter to the editor ,Partial differential equation ,Fourier transform ,Fourier analysis ,Applied Mathematics ,General Mathematics ,Mathematics education ,symbols ,Absolute convergence ,Analysis ,Mathematics - Published
- 2014
13. A weighted uniform $L^{p}$--estimate of Bessel functions: A note on a paper of Guo
- Author
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Krzysztof Stempak
- Subjects
symbols.namesake ,Cylindrical harmonics ,Bessel process ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Struve function ,Bessel polynomials ,symbols ,Calculus ,Bessel function ,Lommel function ,Mathematics - Published
- 2000
14. Remarks on DiPerna’s paper 'Convergence of the viscosity method for isentropic gas dynamics'
- Author
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Gui-Qiang Chen
- Subjects
Discrete mathematics ,Isentropic process ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Vacuum state ,Finite difference method ,Euler equations ,Binary entropy function ,symbols.namesake ,Riemann hypothesis ,Compact space ,Mathematics Subject Classification ,symbols ,Mathematics - Abstract
Concerns have been voiced about the correctness of certain technical points in DiPerna’s paper (Comm. Math. Phys. 91 (1983), 1–30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods. In [Di], DiPerna found a global entropy solution of the isentropic Euler equations for the following exponents in the equation of state for the pressure: γ = 1 + 2/(2m+ 1), m ≥ 2 integer. (1) He divided his arguments into the following two steps. 1. Compactness framework Assume that a sequence of approximate solutions (ρ (x, t),m (x, t)), 0 ≤ t ≤ T , satisfies: (i). There exists a constant C(T ) > 0, independent of > 0, such that 0 ≤ ρ (x, t) ≤ C, |m (x, t)/ρ (x, t)| ≤ C; (ii). For all weak entropy pairs (η, q) of the isentropic Euler equations, the measure sequence η(ρ ,m )t + q(ρ ,m )x is contained in a compact subset of H −1 loc (R× [0, T ]). If γ satisfies (1), then the sequence (ρ (x, t),m (x, t)) is compact in Lloc(R× [0, T ]). The reason for the restriction on the number γ is that, in such a case, any weak entropy function is a polynomial function of the Riemann invariants (w, z). This is the key step in DiPerna’s arguments and is also his main contribution to the compensated compactness method in this aspect. Received by the editors May 16, 1996. 1991 Mathematics Subject Classification. Primary 35K55, 35L65; Secondary 76N15, 35L60, 65M06.
- Published
- 1997
15. Correction to the paper 'Existence of solutions for the Dirichlet problem with superlinear nonlinearities'
- Author
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Andrzej Rogowski and Andrzej Nowakowski
- Subjects
symbols.namesake ,Dirichlet kernel ,Dirichlet eigenvalue ,General Mathematics ,Dirichlet's principle ,Dirichlet boundary condition ,Mathematical analysis ,symbols ,Dirichlet L-function ,Dirichlet's energy ,General Dirichlet series ,Dirichlet series ,Mathematics - Published
- 2005
16. A remark to a paper of Kato and Ikebe
- Author
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Wolf von Wahl
- Subjects
Pure mathematics ,General Mathematics ,Mathematical analysis ,Algebraic geometry ,Sobolev space ,symbols.namesake ,Number theory ,Operator (computer programming) ,Square-integrable function ,symbols ,Order (group theory) ,Element (category theory) ,Schrödinger's cat ,Mathematics - Abstract
This paper deals with Schrodinger operators as they were treated by Kato - Ikebe [3]. It is shown that every element of the domain of definition of the adjoint of such an operator has locally square integrable distributional derivatives up to the order 2. For this the potential of the Schrodinger operator must fulfil a local Stummel condition; if the potential is only locally square integrable a somewhat weaker statement is possible for three dimensions (see remark 2 at the end of this paper).
- Published
- 1977
17. The greatest mathematical paper of all time
- Author
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A. J. Coleman
- Subjects
Weyl group ,Pure mathematics ,General Mathematics ,Cartan decomposition ,Killing form ,Kac–Moody algebra ,Affine Lie algebra ,Algebra ,symbols.namesake ,History and Philosophy of Science ,symbols ,Cartan matrix ,Lie theory ,Mathematics::Representation Theory ,E8 ,Mathematics - Abstract
Why do I think that Z.v.G.II was an epoch-making paper? (1) It was the paradigm for subsequent efforts to classify the possible structures for any mathematical object. Hawkins [15] documents the fact that Killing’s paper was the immediate inspiration for the work of Cartan, Molien, and Maschke on the structure of linearassociative algebras which culminated in Wedderburn’s theorems. Killing’s success was certainly an example which gave Richard Brauer the will to persist in the attempt to classify simple groups. (2) Weyl’s theory of the representation of semi-simple Lie groups would have been impossible without ideas, results, and methods originated by Killing in Z.v.G.II. Weyl’s fusion of global and local analysis laid the basis for the work of Harish-Chandra and the flowering of abstract harmonic analysis. (3) The whole industry of root systems evinced in the writings of I. Macdonald, V. Kac, R. Moody, and others started with Killing. For the latest see [21]. (4) The Weyl group and the Coxeter transformation are in Z.v.G.II. There they are realized not as orthogonal motions of Euclidean space but as permutations of the roots. In my view, this is the proper way to think of them for general Kac-Moody algebras. Further, the conditions for symmetrisability which play a key role in Kac’s book [17] are given on p. 21 of Z.v.G.II. (5) It was Killing who discovered the exceptional Lie algebra E8, which apparently is the main hope for saving Super-String Theory—not that I expect it to be saved! (6) Roughly one third of the extraordinary work of Elie Cartan was based more or less directly on Z.v.G.II.
- Published
- 1989
18. Book Review: The lost notebook and other unpublished papers
- Author
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Richard Askey
- Subjects
symbols.namesake ,Applied Mathematics ,General Mathematics ,symbols ,Ramanujan's sum ,Mathematics - Published
- 1988
19. Note on a paper of B. Grünbaum on acyclic colorings
- Author
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Gerd Wegner
- Subjects
Discrete mathematics ,symbols.namesake ,General Mathematics ,symbols ,Algebra over a field ,Arithmetic ,Notation ,Group theory ,Planar graph ,Mathematics - Abstract
The aim of this short note is to improve some recent results of B. Grunbaum by some remarks. We use Grunbaum's notations.
- Published
- 1973
20. An addendum to the paper ‘Dynamic response of an infinite plate subjected to a steadily moving transverse force’
- Author
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David H. Y. Yen and Clifford C. Chou
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,General Physics and Astronomy ,Addendum ,Transverse force ,Integral transform ,Physics::Fluid Dynamics ,symbols.namesake ,Classical mechanics ,Mach number ,Plate theory ,symbols ,Initial value problem ,Supersonic speed ,Mathematics - Abstract
The dynamic response of an infinite, elastic plate to a steadily moving transverse force is studied under the so-called improved plate theory. Analytic solutions for the subsonic and the intersonic cases are obtained using the method of integral transforms. The usual difficulty of the nonuniqueness of the steady-state solutions is circumvented by formulating the problems as initial value problems and then seeking the large time limits of the solutions of such initial value problems. The subsonic and intersonic solutions obtained here are similar to that obtained previously for the supersonic case. However, in contrast with the supersonic solution, which contains two families of Mach lines, the subsonic solution contains no Mach lines and the intersonic solution contains only one family of Mach lines.
- Published
- 1974
21. On positive eigenvalues of one-body schrödinger operators: Remarks on papers by agmon and simon
- Author
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K.-H. Jansen and H. Kalf
- Subjects
symbols.namesake ,Applied Mathematics ,General Mathematics ,symbols ,Eigenvalues and eigenvectors ,Schrödinger's cat ,Mathematical physics ,Mathematics - Published
- 1975
22. Note on a paper by C. C. Brown
- Author
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S. J. Bernau
- Subjects
Statistics and Probability ,Discrete mathematics ,symbols.namesake ,Sequence ,General Mathematics ,Hilbert space ,symbols ,Statistics, Probability and Uncertainty ,Lambda ,Analysis ,Self-adjoint operator ,Mathematics - Abstract
Let H be a real or complex Hilbert space and let L p(1≦p
- Published
- 1969
23. ON THE DISTRIBUTION OF THE CORRELATION COEFFICIENT IN SMALL SAMPLES. APPENDIX II TO THE PAPERS OF 'STUDENT' AND R. A. FISHER. A COOPERATIVE STUDY
- Author
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H. E. Soper, A. W. Young, A. Lee, Karl Pearson, and B. M. Cave
- Subjects
Statistics and Probability ,Distribution (number theory) ,Correlation coefficient ,Intraclass correlation ,Applied Mathematics ,General Mathematics ,Fisher transformation ,Correlation ratio ,Agricultural and Biological Sciences (miscellaneous) ,Spearman's rank correlation coefficient ,Pearson product-moment correlation coefficient ,symbols.namesake ,Statistics ,symbols ,Statistics, Probability and Uncertainty ,General Agricultural and Biological Sciences ,Mathematics - Published
- 1917
24. Lord Stanhope's papers on the Doctrine of Chances
- Author
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David R. Bellhouse
- Subjects
Theory of runs ,History ,Mathematics(all) ,General Mathematics ,media_common.quotation_subject ,De Moivre's formula ,Doctrine ,Duration of play ,Gambler's ruin ,symbols.namesake ,Duration (philosophy) ,GEORGE (programming language) ,symbols ,Criticism ,Doctrine of fluxions ,Special case ,Amateur ,Classics ,media_common ,Mathematics - Abstract
The Centre for Kentish Studies in Maidstone, Kent holds the mathematical manuscripts of Philip Stanhope (1714–1786), second Earl of Stanhope. He was an active and capable amateur mathematician. Although the manuscripts cover a wide range of mathematical topics, the current article focuses only on Stanhope's work in probability, where his interests appear to be on the theoretical rather than the applied side of the subject. His work, mainly derived from De Moivre's The Doctrine of Chances and Montmort's Essay d'analyse sur les jeux de hazard, touches on the major probability problems of the day. Among the notes on these two authors there is work that includes an alternate solution to the theory of runs and a simplified solution to a special case of the problem of the duration of play, related to the gambler's ruin problem. In addition, the manuscript collection contains Stanhope's transcription of an incorrect solution to the theory of runs by Thomas Bayes. There is also some correspondence with Sir Alexander Cuming that touches on George Berkeley's criticism of Isaac Newton's development of the calculus.
- Full Text
- View/download PDF
25. On the Characterizations of Wave Front Sets in Terms of the Short-Time Fourier Transform
- Author
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Stevan Pilipović and Bojan Prangoski
- Subjects
Wavefront ,General Mathematics ,010102 general mathematics ,Short paper ,Mathematical analysis ,Short-time Fourier transform ,02 engineering and technology ,01 natural sciences ,Sobolev space ,symbols.namesake ,020303 mechanical engineering & transports ,Fourier transform ,0203 mechanical engineering ,symbols ,0101 mathematics ,Mathematics - Abstract
© 2019, Pleiades Publishing, Ltd. It is well known that the classical and Sobolev wave fronts were extended to nonequivalent global versions by the use of the short-time Fourier transform. In this very short paper, we give complete characterizations of the former wave front sets in terms of the short-time Fourier transform.
- Published
- 2019
26. Extinction probabilities in branching processes: A note on holgate and Lakhani's paper
- Author
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D. J. Daley
- Subjects
Pharmacology ,Mathematical and theoretical biology ,Extinction ,General Mathematics ,General Neuroscience ,Immunology ,General Medicine ,Poisson distribution ,General Biochemistry, Genetics and Molecular Biology ,Combinatorics ,Branching (linguistics) ,symbols.namesake ,Computational Theory and Mathematics ,symbols ,General Agricultural and Biological Sciences ,General Environmental Science ,Branching process ,Mathematics - Abstract
Within the class of offspring distributions with given meanm>1 and probability of no offspringp o, the probabilityq of ultimate extinction in a Galton-Watson branching process starting from one individual satisfiesp 0
- Published
- 1969
27. A remark on my paper ?On the Saito-Kurokawa lifting?
- Author
-
I. I. Piatetski-Shapiro
- Subjects
Algebra ,symbols.namesake ,Automorphic L-function ,General Mathematics ,Artin L-function ,Eisenstein series ,Langlands–Shahidi method ,symbols ,Automorphic form ,Mathematics - Published
- 1984
28. Note on the paper of H. Amato und G. Mensch: Rank restriction on the quadratic form in indefinite quadratic programming
- Author
-
Götz Uebe
- Subjects
General Mathematics ,Quadratic function ,Management Science and Operations Research ,Isotropic quadratic form ,Legendre symbol ,Combinatorics ,Definite quadratic form ,Algebra ,symbols.namesake ,Quadratic form ,symbols ,Binary quadratic form ,Quadratic field ,Quadratic programming ,Software ,Mathematics - Published
- 1972
29. Observations on a paper by Rosenblum
- Author
-
S. Cater
- Subjects
Complex conjugate ,Applied Mathematics ,General Mathematics ,Hilbert space ,Uniform limit theorem ,Combinatorics ,symbols.namesake ,Operator (computer programming) ,Skew-Hermitian matrix ,Bounded function ,symbols ,Normal operator ,Complex number ,Mathematics - Abstract
M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.
- Published
- 1961
30. On the Zeros of Dirichlet L-Functions.II (With Corrections to Ön the Zeros of Dirichlet L-Functions.I' and the Subsequent Papers)
- Author
-
Akio Fujii
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Dirichlet L-function ,Dirichlet's energy ,Dirichlet eta function ,Class number formula ,symbols.namesake ,Dirichlet kernel ,Dirichlet's principle ,symbols ,General Dirichlet series ,Dirichlet series ,Mathematics - Published
- 1981
31. Phase portraits of separable quadratic systems and a bibliographical survey on quadratic systems
- Author
-
Jaume Llibre and Tao Li
- Subjects
Pure mathematics ,Class (set theory) ,Poincaré compactification ,Phase portrait ,General Mathematics ,010102 general mathematics ,Quadratic function ,01 natural sciences ,Separable space ,Quadratic system ,symbols.namesake ,Quadratic equation ,Separable system ,Poincaré conjecture ,symbols ,Compactification (mathematics) ,0101 mathematics ,Quadratic differential ,Mathematics - Abstract
Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we have two objectives. First we provide a brief bibliographical survey on the main results about quadratic systems. Here we do not consider the applications of these systems to many areas as in Physics, Chemist, Economics, Biology, … Second we characterize the new class of planar separable quadratic polynomial differential systems. For such class of systems we provide the normal forms which contain one parameter, and using the Poincare compactification and the blow up technique, we prove that there exist 10 non-equivalent topological phase portraits in the Poincare disc for the separable quadratic polynomial differential systems.
- Published
- 2021
32. Spectral cluster estimates for Schrödinger operators of relativistic type
- Author
-
Yannick Sire, Cheng Zhang, and Xiaoqi Huang
- Subjects
Applied Mathematics ,General Mathematics ,Eigenfunction ,Type (model theory) ,Wave equation ,Sobolev space ,Kernel (algebra) ,symbols.namesake ,Operator (computer programming) ,symbols ,Cluster (physics) ,Schrödinger's cat ,Mathematics ,Mathematical physics - Abstract
This paper is dedicated to L p bounds on eigenfunctions of a Schrodinger-type operator ( − Δ g ) α / 2 + V on closed Riemannian manifolds for critically singular potentials V. The operator ( − Δ g ) α / 2 is defined spectrally in terms of the eigenfunctions of − Δ g . We obtain also quasimodes and spectral clusters estimates. As an application, we derive Strichartz estimates for the fractional wave equation ( ∂ t 2 + ( − Δ g ) α / 2 + V ) u = 0 . The wave kernel techniques recently developed by Bourgain-Shao-Sogge-Yao [4] and Shao-Yao [27] play a key role in this paper. We construct a new reproducing operator with several local operators and some good error terms. Moreover, we shall prove that these local operators satisfy certain variable coefficient versions of the “uniform Sobolev estimates” by Kenig-Ruiz-Sogge [18] . These enable us to handle the critically singular potentials V and prove the quasimode estimates.
- Published
- 2021
33. On the singular value decomposition over finite fields and orbits of GU×GU
- Author
-
Robert M. Guralnick
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Unitary state ,Nilpotent matrix ,symbols.namesake ,Finite field ,Character (mathematics) ,Kronecker delta ,Singular value decomposition ,Linear algebra ,symbols ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GU m ( q ) × GU n ( q ) on M m × n ( q 2 ) (which is the analog of the singular value decomposition). The proof involves Kronecker’s theory of pencils and the Lang–Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick et al. (2020) where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form A A ∗ and A ∗ A over a finite field and A A ⊤ and A ⊤ A over arbitrary fields.
- Published
- 2021
34. On Lacunas in the Spectrum of the Laplacian with the Dirichlet Boundary Condition in a Band with Oscillating Boundary
- Author
-
Denis Borisov
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,Operator (physics) ,Mathematical analysis ,Spectrum (functional analysis) ,Boundary (topology) ,Function (mathematics) ,symbols.namesake ,Amplitude ,Dirichlet boundary condition ,symbols ,Flat band ,Laplace operator ,Mathematics - Abstract
In this paper, we consider the Laplace operator in a flat band whose lower boundary periodically oscillates under the Dirichlet boundary condition. The period and the amplitude of oscillations are two independent small parameters. The main result obtained in the paper is the absence of internal lacunas in the lower part of the spectrum of the operator for sufficiently small period and amplitude. We obtain explicit upper estimates of the period and amplitude in the form of constraints with specific numerical constants. The length of the lower part of the spectrum, in which the absence of lacunas is guaranteed, is also expressed explicitly in terms of the period function and the amplitude.
- Published
- 2021
35. A generalization of the Freidlin–Wentcell theorem on averaging of Hamiltonian systems
- Author
-
Yichun Zhu
- Subjects
Pure mathematics ,Girsanov theorem ,Weak convergence ,General Mathematics ,010102 general mathematics ,Identity matrix ,Differential operator ,01 natural sciences ,Hamiltonian system ,010101 applied mathematics ,symbols.namesake ,Matrix (mathematics) ,Compact space ,Wiener process ,symbols ,0101 mathematics ,Mathematics - Abstract
In this paper, we generalize the classical Freidlin-Wentzell’s theorem for random perturbations of Hamiltonian systems. In (Probability Theory and Related Fields 128 (2004) 441–466), M.Freidlin and M.Weber generalized the original result in the sense that the coefficient for the noise term is no longer the identity matrix but a state-dependent matrix and taking the drift term into consideration. In this paper, We generalize the result by adding a state-dependent matrix that converges uniformly to 0 on any compact sets as ϵ tends to 0 to a state-dependent noise and considering the drift term which contains two parts, the state-dependent mapping and a state-dependent mapping that converges uniformly to 0 on any compact sets as ϵ tends to 0. In the proof, we adapt a new way to prove the weak convergence inside the edge by constructing an auxiliary process and modify the proof in (Probability Theory and Related Fields 128 (2004) 441–466) when proving gluing condition.
- Published
- 2021
36. Logarithmic Potential and Generalized Analytic Functions
- Author
-
O.V. Nesmelova, Vladimir Gutlyanskiĭ, Vladimir Ryazanov, and A.S. Yefimushkin
- Subjects
Statistics and Probability ,Dirichlet problem ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Harmonic (mathematics) ,Unit disk ,Sobolev space ,Riemann hypothesis ,symbols.namesake ,Harmonic function ,symbols ,Neumann boundary condition ,Analytic function ,Mathematics - Abstract
The study of the Dirichlet problem in the unit disk 𝔻 with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin [31]. Later on, the known monograph of Vekua [48] has been devoted to boundary-value problems (only with Holder continuous data) for the generalized analytic functions, i.e., continuous complex valued functions h(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form 𝜕zh + ah + b $$ \overline{h} $$ = c ; where it was assumed that the complex valued functions a; b and c belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. The present paper is a natural continuation of our previous articles on the Riemann, Hilbert, Dirichlet, Poincar´e and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic, and the so-called A−harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here, we extend the corresponding results to the generalized analytic functions h : D → ℂ with the sources g : 𝜕zh = g ∈ Lp, p > 2 , and to generalized harmonic functions U with sources G : △U = G ∈ Lp, p > 2. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary-value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary-values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincar´e problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semilinear equations of mathematical physics in anisotropic and inhomogeneous media.
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- 2021
37. Finite Homogeneous Subspaces of Euclidean Spaces
- Author
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V. N. Berestovskiĭ and Yu. G. Nikonorov
- Subjects
Convex hull ,General Mathematics ,Archimedean solid ,Combinatorics ,symbols.namesake ,Polyhedron ,Metric space ,symbols ,Tetrahedron ,Mathematics::Metric Geometry ,Cube ,Isometry group ,Mathematics ,Regular polytope - Abstract
The paper is devoted to the study of the metric properties of regular and semiregular polyhedra in Euclidean spaces. In the first part, we prove that every regular polytope of dimension greater or equal than 4, and different from 120-cell in $$\mathbb {E}^4 $$ is such that the set of its vertices is a Clifford–Wolf homogeneous finite metric space. The second part of the paper is devoted to the study of special properties of Archimedean solids. In particular, for each Archimedean solid, its description is given as the convex hull of the orbit of a suitable point of a regular tetrahedron, cube or dodecahedron under the action of the corresponding isometry group.
- Published
- 2021
38. The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation
- Author
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Ihyeok Seo and Yoonjung Lee
- Subjects
symbols.namesake ,General Mathematics ,Open problem ,symbols ,Initial value problem ,Beta (velocity) ,Lambda ,Nonlinear Schrödinger equation ,Energy (signal processing) ,Mathematics ,Mathematical physics - Abstract
In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrodinger equation $$i\partial _{t}u+\Delta u=\lambda |x|^{-\alpha }|u|^{\beta }u$$ in $$H^1$$ . The well-posedness theory in $$H^1$$ has been intensively studied in recent years, but the currently known approaches do not work for the critical case $$\beta =(4-2\alpha )/(n-2)$$ . It is still an open problem. The main contribution of this paper is to develop the theory in this case.
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- 2021
39. Application of a fixed point theorem on infinite cartesian product to an infinite system of differential equations
- Author
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Marcel-Adrian Şerban
- Subjects
symbols.namesake ,System of differential equations ,General Mathematics ,Mathematical analysis ,symbols ,Fixed-point theorem ,Cartesian product ,Mathematics - Abstract
"In the paper Operators on infinite dimensional cartesian product, (Analele Univ. Vest Timişoara, Mat. Inform., 48 (2010), 253–263), by I. A. Rus and M. A. Şerban, the authors give a generalization of the Fibre contraction theorem on infinite dimensional cartesian product. In this paper we give an application of this abstract result to an infinite system of differential equations. "
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- 2021
40. Stability and collapse of the Lyapunov spectrum for Perron–Frobenius operator cocycles
- Author
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Anthony Quas and Cecilia González-Tokman
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Dense set ,Applied Mathematics ,General Mathematics ,Blaschke product ,Banach space ,Lyapunov exponent ,Fixed point ,symbols.namesake ,Unit circle ,symbols ,Invariant measure ,Mathematics ,Analytic function - Abstract
In this paper, we study random Blaschke products, acting on the unit circle, and consider the cocycle of Perron-Frobenius operators acting on Banach spaces of analytic functions on an annulus. We completely describe the Lyapunov spectrum of these cocycles. As a corollary, we obtain a simple random Blaschke product system where the Perron-Frobenius cocycle has infinitely many distinct Lyapunov exponents, but where arbitrarily small natural perturbations cause a complete collapse of the Lyapunov spectrum, except for the exponent 0 associated with the absolutely continuous invariant measure. That is, under perturbations, the Lyapunov exponents become 0 with multiplicity 1, and $-\infty$ with infinite multiplicity. This is superficially similar to the finite-dimensional phenomenon, discovered by Bochi \cite{Bochi-thesis}, that away from the uniformly hyperbolic setting, small perturbations can lead to a collapse of the Lyapunov spectrum to zero. In this paper, however, the cocycle and its perturbation are explicitly described; and further, the mechanism for collapse is quite different. We study stability of the Perron-Frobenius cocycles arising from general random Blaschke products. We give a necessary and sufficient criterion for stability of the Lyapunov spectrum in terms of the derivative of the random Blaschke product at its random fixed point, and use this to show that an open dense set of Blaschke product cocycles have hyperbolic Perron-Frobenius cocycles. In the final part, we prove a relationship between the Lyapunov spectrum of a single cocycle acting on two different Banach spaces, allowing us to draw conclusions for the same cocycles acting on $C^r$ functions spaces.
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- 2021
41. On Some Properties of the New Generalized Fractional Derivative with Non-Singular Kernel
- Author
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Khalid Hattaf
- Subjects
Lyapunov function ,Article Subject ,Non singular ,General Mathematics ,Science and engineering ,General Engineering ,Engineering (General). Civil engineering (General) ,01 natural sciences ,010305 fluids & plasmas ,Fractional calculus ,010101 applied mathematics ,symbols.namesake ,Exponential stability ,Kernel (statistics) ,0103 physical sciences ,QA1-939 ,symbols ,Applied mathematics ,TA1-2040 ,0101 mathematics ,Mathematics - Abstract
This paper presents some new formulas and properties of the generalized fractional derivative with non-singular kernel that covers various types of fractional derivatives such as the Caputo–Fabrizio fractional derivative, the Atangana–Baleanu fractional derivative, and the weighted Atangana–Baleanu fractional derivative. These new properties extend many recent results existing in the literature. Furthermore, the paper proposes some interesting inequalities that estimate the generalized fractional derivatives of some specific functions. These inequalities can be used to construct Lyapunov functions with the aim to study the global asymptotic stability of several fractional-order systems arising from diverse fields of science and engineering.
- Published
- 2021
42. New Computational Formulas for Special Numbers and Polynomials Derived from Applying Trigonometric Functions to Generating Functions
- Author
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Yilmaz Simsek and Neslihan Kilar
- Subjects
Catalan number ,Pure mathematics ,Bernoulli's principle ,symbols.namesake ,General Mathematics ,Factorial number system ,Euler's formula ,symbols ,Stirling number ,Trigonometric functions ,Type (model theory) ,Mathematics - Abstract
The aim of this paper is to apply trigonometric functions with functional equations of generating functions. Using the resulted new equations and formulas from this application, we obtain many special numbers and polynomials such as the Stirling numbers, Bernoulli and Euler type numbers, the array polynomials, the Catalan numbers, and the central factorial numbers. We then introduce combinatorial sums related to these special numbers and polynomials. Moreover, we gave some remarks that relates our new findings from this paper to the relations found in earlier studies.
- Published
- 2021
43. Stationary Wavelet with Double Generalised Rayleigh Distribution
- Author
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Hassan M. Aljohani
- Subjects
021103 operations research ,Article Subject ,Computer science ,Rayleigh distribution ,General Mathematics ,0211 other engineering and technologies ,General Engineering ,Wavelet transform ,Markov chain Monte Carlo ,02 engineering and technology ,Inverse problem ,Engineering (General). Civil engineering (General) ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Noise ,Wavelet ,Multicollinearity ,Gaussian noise ,QA1-939 ,symbols ,TA1-2040 ,0101 mathematics ,Algorithm ,Mathematics - Abstract
Statistics are mathematical tools applying scientific investigations, such as engineering and medical and biological analyses. However, statistical methods are often improved. Nowadays, statisticians try to find an accurate way to solve a problem. One of these problems is estimation parameters, which can be expressed as an inverse problem when independent variables are highly correlated. This paper’s significant goal is to interpret the parameter estimates of double generalized Rayleigh distribution in a regression model using a wavelet basis. It is difficult to use the standard version of the regression methods in practical terms, which is obtained using the likelihood. Since a noise level usually makes the result of estimation unstable, multicollinearity leads to various estimates. This kind of problem estimates that features of the truth are complicated. So it is reasonable to use a mixed method that combines a fully Bayesian approach and a wavelet basis. The usual rule for wavelet approaches is to choose a wavelet basis, where it helps to compute the wavelet coefficients, and then, these coefficients are used to remove Gaussian noise. Recovering data is typically calculated by inverting the wavelet coefficients. Some wavelet bases have been considered, which provide a shift-invariant wavelet transform, simultaneously providing improvements in smoothness, in recovering, and in squared-error performance. The proposed method uses combining a penalized maximum likelihood approach, a penalty term, and wavelet tools. In this paper, real data are involved and modeled using double generalized Rayleigh distributions, as they are used to estimate the wavelet coefficients of the sample using numerical tools. In practical applications, wavelet approaches are recommended. They reduce noise levels. This process may be useful since the noise level is often corrupted in real data, as a significant cause of most numerical estimation problems. A simulation investigation is studied using the MCMC tool to estimate the underlying features as an essential task statistics.
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- 2021
44. Fourier restriction in low fractal dimensions
- Author
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Bassam Shayya
- Subjects
Conjecture ,Measurable function ,Characteristic function (probability theory) ,General Mathematics ,Second fundamental form ,010102 general mathematics ,42B10, 42B20 (Primary), 28A75 (Secondary) ,0102 computer and information sciences ,Function (mathematics) ,Lebesgue integration ,01 natural sciences ,Measure (mathematics) ,Combinatorics ,symbols.namesake ,Hypersurface ,Mathematics - Classical Analysis and ODEs ,010201 computation theory & mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,0101 mathematics ,Mathematics - Abstract
Let $S \subset \Bbb R^n$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\Bbb R^n$. If $X$ contains a ball of each radius, then the problem of determining the range of exponents $(p,q)$ for which the estimate $\| Ef \|_{L^q(X)} \leq C \| f \|_{L^p(S)}$ holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set $X$: there is a number $0 < \alpha \leq n$ such that $|X \cap B_R| \leq c \, R^\alpha$ for all balls $B_R$ in $\Bbb R^n$ of radius $R \geq 1$. On the left-hand side of this estimate, we are integrating the function $|Ef(x)|^q$ against the measure $\chi_X dx$. Our approach consists of replacing the characteristic function $\chi_X$ of $X$ by an appropriate weight function $H$, and studying the resulting estimate in three different regimes: small values of $\alpha$, intermediate values of $\alpha$, and large values of $\alpha$. In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted H\"{o}lder-type inequality that holds for general non-negative Lebesgue measurable functions on $\Bbb R^n$, and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du-Zhang theorem in the range $0 < \alpha < n/2$., Comment: 31 pages. Minor revision
- Published
- 2021
45. Large Eddy Simulation and Flow Field Analysis of Car on the Bridge under Turbulent Crosswind
- Author
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Weitan Yin, Yongqi Ma, and Juyue Ding
- Subjects
Article Subject ,Computer simulation ,Turbulence ,General Mathematics ,Airflow ,General Engineering ,Reynolds number ,02 engineering and technology ,Engineering (General). Civil engineering (General) ,01 natural sciences ,Bridge (nautical) ,010305 fluids & plasmas ,symbols.namesake ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Range (aeronautics) ,0103 physical sciences ,QA1-939 ,symbols ,Environmental science ,TA1-2040 ,Mathematics ,Large eddy simulation ,Crosswind ,Marine engineering - Abstract
As more long-span bridges continue to be completed and opened to traffic, the safety of cars driving across the bridge has attracted more and more attention, especially when the car is suddenly affected by the crosswind, the car is likely to have direction deviation or even a rollover accident. In this paper, the large eddy simulation method is used to study the flow field characteristics and safety of the car on the bridge under the turbulent crosswind. The numerical simulation model is established by referring to the Donghai Bridge, and the correctness of the car model is validated by combining with the data of wind tunnel test. The influence of factors such as the porosity and height of the bridge guardrail and the Reynolds number of airflow on the flow field characteristics is analyzed. The study shows that, in order to ensure the safety of cars on the bridge, the bridge guardrail porosity should be small, 35.8% is more suitable, the guardrail height should be more suitable within the range of 1.5–1.625 meters, and the Reynolds number should not be 3.51e + 5. The research results of this paper will provide reference for the optimal design of bridge guardrail.
- Published
- 2021
46. Existence and Uniqueness of the Global L1 Solution of the Euler Equations for Chaplygin Gas
- Author
-
Zhen Wang, Tingting Chen, and Aifang Qu
- Subjects
Continuous function ,General Mathematics ,Weak solution ,010102 general mathematics ,General Physics and Astronomy ,Euler system ,Absolute continuity ,Lebesgue integration ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,symbols ,Local boundedness ,Applied mathematics ,Initial value problem ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
In this paper, we establish the global existence and uniqueness of the solution of the Cauchy problem of a one-dimensional compressible isentropic Euler system for a Chaplygin gas with large initial data in the space L loc 1 . The hypotheses on the initial data may be the least requirement to ensure the existence of a weak solution in the Lebesgue measurable sense. The novelty and also the essence of the difficulty of the problem lie in the fact that we have neither the requirement on the local boundedness of the density nor that which is bounded away from vacuum. We develop the previous results on this degenerate system. The method used is Lagrangian representation, the essence of which is characteristic analysis. The key point is to prove the existence of the Lagrangian representation and the absolute continuity of the potentials constructed with respect to the space and the time variables. We achieve this by finding a property of the fundamental theorem of calculus for Lebesgue integration, which is a sufficient and necessary condition for judging whether a monotone continuous function is absolutely continuous. The assumptions on the initial data in this paper are believed to also be necessary for ruling out the formation of Dirac singularity of density. The ideas and techniques developed here may be useful for other nonlinear problems involving similar difficulties.
- Published
- 2021
47. Effects of Southwest Airlines on Carrier Profits and Entry Probabilities
- Author
-
Junqiushi Ren
- Subjects
Estimation ,Counterfactual thinking ,050210 logistics & transportation ,Article Subject ,General Mathematics ,media_common.quotation_subject ,05 social sciences ,Stochastic game ,General Engineering ,Engineering (General). Civil engineering (General) ,NonStop ,Microeconomics ,symbols.namesake ,Nash equilibrium ,Service (economics) ,0502 economics and business ,QA1-939 ,symbols ,Economics ,Profitability index ,TA1-2040 ,050207 economics ,Mathematics ,media_common - Abstract
This paper studies the effects of Southwest Airlines, the largest low-cost carrier (LCC) in the U.S., on other carriers’ payoff functions and entry probabilities. A static entry game model is developed and estimated by viewing entry as an indicator of underlying profitability and making use of Nash Equilibrium. Results indicate that Southwest has a remarkable and negative impact on the payoffs of other carriers. This impact is firm-specific, with LCCs being more affected than full-service carriers (FSCs). Comparing the two service types, the results show that Southwest’s nonstop presence apparently imposes more downward pressure on opponents’ profits than its connecting presence. A counterfactual experiment is then conducted. Once Southwest is counterfactually removed, the probability of each carrier entering a market significantly changes. This paper examines Southwest’s impacts from a new perspective and extends literature on entry game estimation.
- Published
- 2021
48. On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equations
- Author
-
Thomas Y. Hou, De Huang, and Jiajie Chen
- Subjects
symbols.namesake ,Mathematics - Analysis of PDEs ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,FOS: Mathematics ,symbols ,Finite time ,Analysis of PDEs (math.AP) ,Mathematics ,Euler equations - Abstract
We present a novel method of analysis and prove finite time asymptotically self-similar blowup of the De Gregorio model \cite{DG90,DG96} for some smooth initial data on the real line with compact support. We also prove self-similar blowup results for the generalized De Gregorio model \cite{OSW08} for the entire range of parameter on $\mathbb{R}$ or $S^1$ for H\"older continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time asymptotically self-similar singularity into the problem of establishing the nonlinear stability of an approximate self-similar profile with a small residual error using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions to extract the damping effect from the linearized operator around the approximate self-similar profile and take into account cancellation among various nonlocal terms to establish stability analysis. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations., Comment: Added discussion in Section 2.3 and made some minor edits. Main paper 57 pages, Supplementary material 29 pages. In previous arXiv versions, the hyperlinks of the equation number in the main paper are linked to the supplementary material, which is fixed in this version
- Published
- 2021
49. On the pair correlations of powers of real numbers
- Author
-
Christoph Aistleitner and Simon Baker
- Subjects
11K06, 11K60 ,General Mathematics ,Modulo ,FOS: Physical sciences ,0102 computer and information sciences ,Lebesgue integration ,01 natural sciences ,Combinatorics ,symbols.namesake ,Pair correlation ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Algebra over a field ,Classical theorem ,Mathematical Physics ,Real number ,Mathematics ,Sequence ,Mathematics - Number Theory ,Probability (math.PR) ,010102 general mathematics ,Mathematical Physics (math-ph) ,010201 computation theory & mathematics ,symbols ,Martingale (probability theory) ,Mathematics - Probability - Abstract
A classical theorem of Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In the present paper we extend Koksma's theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every $x>1$ the pair correlations of the fractional parts of $(x^n)_{n=1}^{\infty}$ are asymptotically Poissonian. The proof is based on a martingale approximation method., Version 2: some minor changes. The paper will appear in the Israel Journal of Mathematics
- Published
- 2021
50. Concomitants of Generalized Order Statistics from Bivariate Cambanis Family of Distributions Under a General Setting
- Author
-
Haroon M. Barakat, M. A. Alawady, and M. A. Abd Elgawad
- Subjects
Recurrence relation ,General Mathematics ,010102 general mathematics ,Order statistic ,Extension (predicate logic) ,Bivariate analysis ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Joint probability distribution ,Statistics ,symbols ,0101 mathematics ,Fisher information ,Divergence (statistics) ,Mathematics - Abstract
In this paper, we study the concomitants of m-generalized order statistics (m-GOSs) and m-dual generalized order statistics (m-DGOSs) from bivariate Cambanis family with nonzero parameter values as an extension of several recent papers. Moreover, we derive some information measures, namely the Shannon entropy, Kullback–Leibler (KL) divergence and Fisher information number (FIN) for the concomitants of m-GOSs, when $$m>-1,$$ and record values, for $$m=-1.$$ Also, the joint distribution of concomitants of m-GOSs and record values for this family are studied. Besides, some useful recurrence relations between moments of concomitants are obtained. Finally, the ordinary order statistics (OOSs), record values and sequential order statistics (SOSs) as subclasses of m-GOSs, as well as the progressive type II censored order statistics (POSs) as a more general subclass of GOSs, are separately discussed.
- Published
- 2021
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