2,033 results
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2. On A.Ya. Khinchin's paper ‘Ideas of intuitionism and the struggle for a subject matter in contemporary mathematics’ (1926): A translation with introduction and commentary
- Author
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Lukas M. Verburgt and Olga Hoppe-Kondrikova
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History ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Victory ,Ignorance ,06 humanities and the arts ,0603 philosophy, ethics and religion ,01 natural sciences ,Epistemology ,Subject matter ,Formalism (philosophy of mathematics) ,Intuitionism ,060302 philosophy ,Calculus ,Ideology ,0101 mathematics ,Communism ,media_common ,Mathematics - Abstract
The translation into English of Aleksandr Yakovlevich Khinchin's (1894–1959) 1926 paper entitled ‘Ideas of intuitionism and the struggle for a subject matter in contemporary mathematics’ is made available for the first time. Here, Khinchin presented the famous foundational debate between L.E.J. Brouwer and David Hilbert of the 1920s in terms of a search for a mathematics with content. His main aim seems to have been to make intuitionism ideologically acceptable to his audience at the Communist Academy by means of the claim that insofar as Brouwer's intuitionism had a clear ‘subject matter’ and Hilbert's new program was a concession to intuitionism, the alleged victory of intuitionism not only implied the defeat of ‘empty’ formalism, but also showed the compatibility and affinity of Marxism with the newest developments in modern mathematics. This introduction provides a tentative exploration of the issue of what was tactical (or due to ideological pressure) and what was real scientific interest (or due to ignorance) (or what was both) in Khinchin's 1926 paper in the form of a detailed commentary, especially, on the tactical side of his presentation of the positions of Brouwer and Hilbert.
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- 2016
3. Corrigendum to the papers on Exceptional orthogonal polynomials: J. Approx. Theory 182 (2014) 29–58, 184 (2014) 176–208 and 214 (2017) 9–48
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Antonio J. Durán
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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.
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- 2020
4. Canonical factorization of rational matrix functions. A note on a paper by P. Dewilde
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Arthur E. Frazho, Marinus A. Kaashoek, and Mathematics
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Mathematics(all) ,General Mathematics ,Stein equation ,Rational matrix functions ,Operator theory ,Unitary state ,LU decomposition ,law.invention ,Algebra ,LTI system theory ,Riccati equation ,Unit circle ,Factorization ,law ,Canonical factorization ,Unitary matrix functions ,State space (physics) ,State space realization ,Toeplitz operators ,Mathematics - Abstract
A left canonical factorization theorem for rational matrix functions relative to the unit circle is presented. The result is a time invariant version of a recent strict LU factorization theorem for certain semi-separable operators, due to Dewilde (2012) [6]. Explicit formulas for the factors are also given. The theorem is proved, first by using the state space method of Dewilde (2012) [6], and next by using an operator theory approach. In both cases the main part of the proof concerns rational matrix functions that are unitary on the unit circle. © 2012 Royal Dutch Mathematical Society (KWG).
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- 2012
5. Winners of the 2016 Best Paper Award
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Erich Novak
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Statistics and Probability ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Applied Mathematics ,General Mathematics ,Mathematics education ,Mathematics - Published
- 2017
6. Some remarks on a paper by Liu and van Rooij
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Gerhard Racher
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Algebra ,Pure mathematics ,Mathematics(all) ,Function space ,General Mathematics ,Amenable group ,Translation invafiant operators ,Weak compactness ,Amenable groups ,Invariant (mathematics) ,Locally compact group ,Mathematics - Abstract
Complementing the work of T.-S. Liu and A.C.M. van Rooij we show that the existence of non-zero translation invariant operators between certain function spaces on a locally compact group implies its amenability.
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- 2007
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7. Thomas Müller-Gronbach, Klaus Ritter and Larisa Yaroslavtseva share the 2015 Best Paper Award
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Erich Novak
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Statistics and Probability ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,biology ,Applied Mathematics ,General Mathematics ,Larisa ,biology.organism_classification ,Classics ,Mathematics - Published
- 2016
8. Bernd Carl, Aicke Hinrichs, and Philipp Rudolph share the 2014 Best Paper Award
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Joseph F. Traub, Henryk Wozniakowski, Ian H. Sloan, Erich Novak, and Klaus Ritter
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Statistics and Probability ,Czech ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Applied Mathematics ,General Mathematics ,Banach space ,language ,Art history ,Kepler ,language.human_language ,Mathematics - Abstract
The Award Committee – Peter Kritzer, Johannes Kepler University Linz, Austria and Jan Vybiral, Charles University, Czech Republic – determined that the following paper exhibits exceptional merit and therefore awarded the prize to: Bernd Carl, Aicke Hinrichs, and Philipp Rudolph for their paper ‘‘Entropy numbers of convex hulls in Banach spaces and applications’’, which appeared in October, 2014. Vol. 30, pp. 555–587. The $3000 prize will be divided between the winners. Each author will also receive a plaque.
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- 2015
9. Dmitriy Bilyk, Lutz Kämmerer, Stefan Kunis, Daniel Potts, Volodya Temlyakov and Rui Yu share the 2012 Best Paper Award
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Joseph F. Traub, Henryk Wozniakowski, Ian H. Sloan, and Erich Novak
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Statistics and Probability ,Numerical Analysis ,Algebra and Number Theory ,Control and Optimization ,Operations research ,General Mathematics ,Applied Mathematics ,Art history ,Mathematics - Published
- 2013
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10. Shu Tezuka, Joos Heintz, Bart Kuijpers, and Andrés Rojas Paredes Share the 2013 Best Paper Award
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Ian H. Sloan, Joseph F. Traub, Henryk Wozniakowski, and Erich Novak
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Statistics and Probability ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Applied Mathematics ,General Mathematics ,Humanities ,Mathematics - Abstract
The Award Committee – Michael Gnewuch, University of Kaiserslautern, Germany and Friedrich Pillichshammer, Johannes Kepler University, Austria – determined that the following two papers exhibited exceptional merit and therefore awarded the prize to: ShuTezuka for his paper ‘‘On the discrepancy of generalizedNiederreiter sequences’’, which appeared in June–August, 2013. Vol. 29, pp. 240–247. Joos Heintz, Bart Kuijpers, and Andres Rojas Paredes for their paper ‘‘Software engineering and complexity in effective Algebraic Geometry’’, which appeared in February, 2013. Vol. 29, pp. 92–138. The $3000 prize will be divided between the winners. Each author will also receive a plaque.
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- 2014
11. Aicke Hinrichs, Simon Foucart, Alain Pajor, Holger Rauhut, Tino Ullrich win the 2010 Best Paper Award
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Joseph F. Traub, Henryk Wozniakowski, Erich Novak, and Ian H. Sloan
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Statistics and Probability ,Numerical Analysis ,Algebra and Number Theory ,Control and Optimization ,General Mathematics ,Applied Mathematics ,Art history ,Mathematics - Published
- 2011
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12. Frank Aurzada, Steffen Dereich, Michael Scheutzow, and Christian Vormoor win the 2009 best paper award
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Ian H. Sloan, Joseph F. Traub, Henryk Wozniakowski, and Erich Novak
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Statistics and Probability ,Numerical Analysis ,Algebra and Number Theory ,Control and Optimization ,General Mathematics ,Applied Mathematics ,Management ,Mathematics - Published
- 2010
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13. 2001 Best Paper Award
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Henryk Woźniakowski, Joseph F. Traub, and Harald Niederreiter
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Statistics and Probability ,Numerical Analysis ,Algebra and Number Theory ,Control and Optimization ,General Mathematics ,Applied Mathematics ,Mathematics education ,Mathematics - Published
- 2002
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14. J.Dick, F. Pillichshammer, and Y.Yomdin Share the 2005 Best Paper Award
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Joseph F. Traub, Harald Niederreiter, and Henryk Woźniakowski
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Statistics and Probability ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,business.industry ,Applied Mathematics ,General Mathematics ,Telecommunications ,business ,Management ,Mathematics - Published
- 2006
15. 2005 Best Paper Award Committee
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K. Sikorski and Luis Miguel Pardo
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Statistics and Probability ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Applied Mathematics ,General Mathematics ,GeneralLiterature_REFERENCE(e.g.,dictionaries,encyclopedias,glossaries) ,GeneralLiterature_MISCELLANEOUS ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,Management - Published
- 2005
16. Stefan Heinrich wins the 2004 Best Paper Award
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Joseph F. Traub, Harald Niederreiter, and Henryk Wozniakowski
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Statistics and Probability ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Operations research ,Applied Mathematics ,General Mathematics ,Management ,Mathematics - Published
- 2005
17. 2002 Best Paper Award Committee
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Fred J. Hickernell and Peter Mathé
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Statistics and Probability ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Applied Mathematics ,General Mathematics ,Management ,Mathematics - Published
- 2002
18. Weak independence of events and the converse of the Borel–Cantelli Lemma
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Csaba Biró and Israel R. Curbelo
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Discrete mathematics ,Pairwise independence ,Lemma (mathematics) ,Probability theory ,General Mathematics ,Converse ,Almost surely ,Mathematical proof ,Borel–Cantelli lemma ,Independence (probability theory) ,Mathematics - Abstract
The converse of the Borel–Cantelli Lemma states that if { A i } i = 1 ∞ is a sequence of independent events such that ∑ P ( A i ) = ∞ , then almost surely infinitely many of these events will occur. Erdős and Renyi proved that it is sufficient to weaken the condition of independence to pairwise independence. Later, several other weakenings of the condition appeared in the literature. The aim of this paper is to provide a collection of conditions, all of which imply that almost surely infinitely many of the events occur, and determine the complete implicational relationship between them. Many of these results are known, or follow from known results, however, they are not widely known among non-specialists. Yet, the results can be extremely useful for areas outside of probability theory, as evidenced by the original motivation of this paper emerging from infinite combinatorics. Our proofs are aimed to be accessible to a general mathematical audience.
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- 2022
19. Extrapolation of compactness on weighted spaces: Bilinear operators
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Stefanos Lappas, Tuomas Hytönen, Tuomas Hytönen / Principal Investigator, and Department of Mathematics and Statistics
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Pure mathematics ,General Mathematics ,COMMUTATORS ,Mathematics::Classical Analysis and ODEs ,Extrapolation ,Bilinear interpolation ,NORM INEQUALITIES ,47B38 (Primary), 42B20, 42B35, 46B70, 47H60 ,Space (mathematics) ,Multilinear Muckenhoupt weights ,01 natural sciences ,Rubio de Francia extrapolation ,Compact operators ,111 Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,0101 mathematics ,Lp space ,Mathematics ,Calderon-Zygmund operators ,Fractional integral operators ,010102 general mathematics ,Muckenhoupt weights ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,010101 applied mathematics ,Range (mathematics) ,Compact space ,Mathematics - Classical Analysis and ODEs ,Bounded function ,Fourier multipliers ,INTEGRAL-OPERATORS - Abstract
In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fr\'echet--Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method., Comment: v3: final version, incorporated referee comments, to appear in Indagationes Mathematicae, 27 pages
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- 2022
20. On the geometry of irreversible metric-measure spaces: Convergence, stability and analytic aspects
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Wei Zhao and Alexandru Kristály
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Pure mathematics ,Class (set theory) ,Applied Mathematics ,General Mathematics ,Stability (learning theory) ,Function (mathematics) ,Stability result ,Measure (mathematics) ,Metric (mathematics) ,Convergence (routing) ,Mathematics::Metric Geometry ,Mathematics::Differential Geometry ,Topology (chemistry) ,Mathematics - Abstract
The paper is devoted to the study of Gromov-Hausdorff convergence and stability of irreversible metric-measure spaces, both in the compact and noncompact cases. While the compact setting is mostly similar to the reversible case developed by J. Lott, K.-T. Sturm and C. Villani, the noncompact case provides various surprising phenomena. Since the reversibility of noncompact irreversible spaces might be infinite, it is motivated to introduce a suitable nondecreasing function that bounds the reversibility of larger and larger balls. By this approach, we are able to prove satisfactory convergence/stability results in a suitable – reversibility depending – Gromov-Hausdorff topology. A wide class of irreversible spaces is provided by Finsler manifolds, which serve to construct various model examples by pointing out genuine differences between the reversible and irreversible settings. We conclude the paper by proving various geometric and functional inequalities (as Brunn-Minkowski, Bishop-Gromov, log-Sobolev and Lichnerowicz inequalities) on irreversible structures.
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- 2022
21. Phase portraits of separable quadratic systems and a bibliographical survey on quadratic systems
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Jaume Llibre and Tao Li
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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.
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- 2021
22. Folded and contracted solutions of coupled classical dynamical Yang–Baxter and reflection equations
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Jasper V. Stokman
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Reflection formula ,Pure mathematics ,Integrable system ,Mathematics::Quantum Algebra ,General Mathematics ,Lie algebra ,Lie group ,Universal enveloping algebra ,Invariant (mathematics) ,Differential operator ,Representation theory ,Mathematics - Abstract
In this paper we give a concrete recipe how to construct triples of algebra-valued meromorphic functions on a complex vector space a satisfying three coupled classical dynamical Yang–Baxter equations and an associated classical dynamical reflection equation. Such triples provide the local factors of a consistent system of first order differential operators on a , generalising asymptotic boundary Knizhnik–Zamolodchikov–Bernard (KZB) equations. The recipe involves folding and contracting a -invariant and θ -twisted symmetric classical dynamical r -matrices along an involutive automorphism θ . In case of the universal enveloping algebra of a simple Lie algebra g we determine the subclass of Schiffmann’s classical dynamical r -matrices which are a -invariant and θ -twisted. The paper starts with a section highlighting the connections between asymptotic (boundary) KZB equations, representation theory of semisimple Lie groups, and integrable quantum field theories.
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- 2021
23. Spectral cluster estimates for Schrödinger operators of relativistic type
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Yannick Sire, Cheng Zhang, and Xiaoqi Huang
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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.
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- 2021
24. On the singular value decomposition over finite fields and orbits of GU×GU
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Robert M. Guralnick
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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.
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- 2021
25. Systems of quasilinear parabolic equations in Rn and systems of quadratic backward stochastic differential equations
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Sheung Chi Phillip Yam, Jens Frehse, and Alain Bensoussan
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Quadratic growth ,Applied Mathematics ,General Mathematics ,Open problem ,010102 general mathematics ,01 natural sciences ,Parabolic partial differential equation ,Domain (mathematical analysis) ,010104 statistics & probability ,Stochastic differential equation ,Quadratic equation ,Bounded function ,Applied mathematics ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
The objective of this paper is two-fold. The first objective is to complete the former work of Bensoussan and Frehse [2] . One big limitation of this paper was the fact that they are systems of PDE. on a bounded domain. One can expect solutions to be bounded, since one looks for smooth solutions. This is a very important property for the development of the method. It is true also that solutions which exist in a bounded domain may fail to exist on R n , because of the lack of bounds. We give conditions so that the results of [2] can be extended to R n . The second objective is to consider the BSDE (Backward stochastic differential equations) version of the system of PDE. This is the objective of a more recent work of Xing and Zitkovic [8] . They consider systems of BSDE with quadratic growth, which is a well-known open problem in the BSDE literature. Since the BSDE are Markovian, the problem is equivalent to the analytic one. However, because of this motivation the analytic problem is in R n and not on a bounded domain. Xing and Zitkovic developed a probabilistic approach. The connection between the analytic problem and the BSDE is not apparent. Our objective is to show that the analytic approach can be completely translated into a probabilistic one. Nevertheless probabilistic concepts are also useful, after their conversion into the analytic framework. This is in particular true for the uniqueness result.
- Published
- 2021
26. On additive and multiplicative decompositions of sets of integers with restricted prime factors, I. (Smooth numbers)
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Kálmán Győry, Lajos Hajdu, and András Sárközy
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Sequence ,Conjecture ,Mathematics - Number Theory ,General Mathematics ,Sieve (category theory) ,010102 general mathematics ,Multiplicative function ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Prime factor ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Unit (ring theory) ,Mathematics - Abstract
In Sarkozy (2001) the third author of this paper presented two conjectures on the additive decomposability of the sequence of ”smooth” (or ”friable”) numbers. Elsholtz and Harper (2015) proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of S -unit equations).
- Published
- 2021
27. Global solutions to systems of quasilinear wave equations with low regularity data and applications
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Dongbing Zha and Kunio Hidano
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Null condition ,Small data ,biology ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Symmetric case ,biology.organism_classification ,Wave equation ,01 natural sciences ,Global iteration ,010101 applied mathematics ,Nonlinear system ,Chen ,Initial value problem ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we study the Cauchy problem for systems of 3-D quasilinear wave equations satisfying the null condition with initial data of low regularity. In the radially symmetric case, we prove the global existence for every small data in H 3 × H 2 with a low weight. To achieve this goal, we will show how to extend the global iteration method first suggested by Li and Chen (1988) [32] to the low regularity case, which is also another purpose of this paper. Finally, we apply our result to 3-D nonlinear elastic waves.
- Published
- 2020
28. A new first-principles approach for the catenary
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George Victor McIlvaine
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Bernoulli's principle ,Perspective (geometry) ,Differential equation ,Physical constant ,General Mathematics ,Catenary ,Calculus ,Measure (mathematics) ,Mathematics - Abstract
This paper reports on findings relating to catenaries since the publication in Expositiones Mathematicae of Denzler and Hinz’s pioneering 1999 paper, Catenaria Vera – the True Catenary . New governing differential equations and explicit solutions are derived for the catenary in positive and negative radial potentials with physical constants incorporated in the derivations. In keeping with precedent by Denzler and Hinz, a measure of historical perspective is offered as homage to Gottfried Wilhelm Leibniz, Christiaan Huygens and Johann Bernoulli, the original first-solvers of the catenary.
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- 2020
29. On some universal Morse–Sard type theorems
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Alba Roviello, Adele Ferone, Mikhail V. Korobkov, Ferone, A., Korobkov, M. V., and Roviello, A.
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Uncertainty principle ,Dubovitskii-Federer theorems ,Near critical ,Morse-Sard theorem ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Algebraic geometry ,Morse code ,Sobolev-Lorentz mapping ,Holder mapping ,01 natural sciences ,law.invention ,Sobolev space ,Combinatorics ,law ,0103 physical sciences ,010307 mathematical physics ,Differentiable function ,Bessel potential space ,0101 mathematics ,Critical set ,Mathematics - Abstract
The classical Morse–Sard theorem claims that for a mapping v : R n → R m + 1 of class C k the measure of critical values v ( Z v , m ) is zero under condition k ≥ n − m . Here the critical set, or m-critical set is defined as Z v , m = { x ∈ R n : rank ∇ v ( x ) ≤ m } . Further Dubovitskiĭ in 1957 and independently Federer and Dubovitskiĭ in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the C k category. Here we formulate and prove a bridge theorem that includes all the above results as particular cases: namely, if a function v : R n → R d belongs to the Holder class C k , α , 0 ≤ α ≤ 1 , then for every q > m the identity H μ ( Z v , m ∩ v − 1 ( y ) ) = 0 holds for H q -almost all y ∈ R d , where μ = n − m − ( k + α ) ( q − m ) . Intuitively, the sense of this bridge theorem is very close to Heisenberg's uncertainty principle in theoretical physics: the more precise is the information we receive on measure of the image of the critical set, the less precisely the preimages are described, and vice versa. The result is new even for the classical C k -case (when α = 0 ); similar result is established for the Sobolev classes of mappings W p k ( R n , R d ) with minimal integrability assumptions p = max ( 1 , n / k ) , i.e., it guarantees in general only the continuity (not everywhere differentiability) of a mapping. However, using some N-properties for Sobolev mappings, established in our previous paper, we obtained that the sets of nondifferentiability points of Sobolev mappings are fortunately negligible in the above bridge theorem. We cover also the case of fractional Sobolev spaces. The proofs of the most results are based on our previous joint papers with J. Bourgain and J. Kristensen (2013, 2015). We also crucially use very deep Y. Yomdin's entropy estimates of near critical values for polynomials (based on algebraic geometry tools).
- Published
- 2020
30. Realizations of holomorphic and slice hyperholomorphic functions: The Krein space case
- Author
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Fabrizio Colombo, Irene Sabadini, and Daniel Alpay
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Quaternionic analysis ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Holomorphic function ,Hypercomplex analysis ,010103 numerical & computational mathematics ,Space (mathematics) ,Krein spaces ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Kernel (algebra) ,30G35, 47B32, 46C20, 47B50 ,Realizations of analytic functions ,FOS: Mathematics ,Slice hyperholomorphic functions ,State space (physics) ,0101 mathematics ,Quaternion ,Realization (systems) ,Mathematics - Abstract
In this paper we treat realization results for operator-valued functions which are analytic in the complex sense or slice hyperholomorphic over the quaternions. In the complex setting, we prove a realization theorem for an operator-valued function analytic in a neighborhood of the origin with a coisometric state space operator thus generalizing an analogous result in the unitary case. A main difference with previous works is the use of reproducing kernel Krein spaces. We then prove the counterpart of this result in the quaternionic setting. The present work is the first paper which presents a realization theorem with a state space which is a quaternionic Krein space and may open new avenues of research in hypercomplex analysis.
- Published
- 2020
31. Reflection positivity and Levin–Wen models
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Zhengwei Liu and Arthur Jaffe
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Property (philosophy) ,General Mathematics ,010102 general mathematics ,Boundary (topology) ,01 natural sciences ,Quantization (physics) ,Theoretical physics ,Reflection (mathematics) ,Chain (algebraic topology) ,0101 mathematics ,Variety (universal algebra) ,Algebraic number ,Link (knot theory) ,Mathematics - Abstract
We give a transparent algebraic formulation of our pictorial approach to the reflection positivity (RP), that we introduced in a previous paper. We apply this quantization to the 2 + 1 Levin–Wen model to obtain 1 + 1 anyonic/quantum spin chain theory on the boundary, possibly entangled in the bulk. The reflection positivity property has played a central role in both mathematics and physics, as well as providing a crucial link between the two subjects. In a previous paper we gave a new geometric approach to understanding reflection positivity in terms of pictures. Here we give a transparent algebraic formulation of our pictorial approach. We use insights from this translation to establish the reflection positivity property for the fashionable Levin–Wen models with respect both to vacuum and to bulk excitations. We believe these methods will be useful for understanding a variety of other problems.
- Published
- 2020
32. Null controllability of semi-linear fourth order parabolic equations
- Author
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K. Kassab, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)
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Null controllability ,Observability ,Global Carleman estimate ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Null (mathematics) ,Exact controllability ,01 natural sciences ,Parabolic partial differential equation ,Dirichlet distribution ,Domain (mathematical analysis) ,010101 applied mathematics ,Controllability ,symbols.namesake ,Linear and semi-linear fourth order parabolic equation ,Bounded function ,MSC : 35K35, 93B05, 93B07 ,Neumann boundary condition ,symbols ,[MATH]Mathematics [math] ,0101 mathematics ,Mathematics - Abstract
International audience; In this paper, we consider a semi-linear fourth order parabolic equation in a bounded smooth domain Ω with homogeneous Dirichlet and Neumann boundary conditions. The main result of this paper is the null controllability and the exact controllability to the trajectories at any time T > 0 for the associated control system with a control function acting at the interior.; Dans ce papier, on considère uneéquation parabolique semi-linéaire de quatrième ordre dans un domaine borné régulier Ω avec des conditions aux limites de type Dirichlet et Neumann homogènes. Le résultat principal de ce papier concerne la contrôlabilitéà zéro et la contrôlabilité exacte pour tout T > 0 du système de contrôle associé avec un contrôle agissantà l'interieur.
- Published
- 2020
33. Local uniqueness for vortex patch problem in incompressible planar steady flow
- Author
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Daomin Cao, Shusen Yan, Shuangjie Peng, and Yuxia Guo
- Subjects
Applied Mathematics ,General Mathematics ,Open problem ,010102 general mathematics ,Mathematical analysis ,Vorticity ,01 natural sciences ,Domain (mathematical analysis) ,Vortex ,010101 applied mathematics ,Flow (mathematics) ,Bounded function ,Stream function ,Uniqueness ,0101 mathematics ,ComputingMethodologies_COMPUTERGRAPHICS ,Mathematics - Abstract
We investigate a steady planar flow of an ideal fluid in a bounded simply connected domain and focus on the vortex patch problem with prescribed vorticity strength. There are two methods to deal with the existence of solutions for this problem: the vorticity method and the stream function method. A long standing open problem is whether these two entirely different methods result in the same solution. In this paper, we will give a positive answer to this problem by studying the local uniqueness of the solutions. Another result obtained in this paper is that if the domain is convex, then the vortex patch problem has a unique solution.
- Published
- 2019
34. Boundary value problems for the Brinkman system with L∞ coefficients in Lipschitz domains on compact Riemannian manifolds. A variational approach
- Author
-
Wolfgang L. Wendland and Mirela Kohr
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Weak solution ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Fixed-point theorem ,Riemannian manifold ,Lipschitz continuity ,01 natural sciences ,Dirichlet distribution ,Physics::Fluid Dynamics ,010101 applied mathematics ,Sobolev space ,Nonlinear system ,symbols.namesake ,symbols ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
The purpose of this paper is to show well-posedness results in L 2 -based Sobolev spaces for transmission, Dirichlet, Neumann, and mixed boundary value problems for the Brinkman system with L ∞ coefficients in Lipschitz domains on a compact Riemannian manifold of dimension m ≥ 2 . The Dirichlet, transmission, and mixed problems for the nonlinear Darcy-Forchheimer-Brinkman system with L ∞ coefficients are also analyzed. First, we focus on the well-posedness of linear transmission, Dirichlet and mixed boundary value problems for the Brinkman system with L ∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using a variational approach that reduces such a boundary value problem to a mixed variational formulation defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. Further, we show the equivalence between each boundary value problem for the Brinkman system with L ∞ coefficients and its mixed variational counterpart, and then the well posedness in L 2 -based Sobolev spaces by using the Necas-Babuska-Brezzi technique. The second goal of this paper is the construction of the Newtonian and layer potential operators for the Brinkman system with L ∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using the well-posedness results for the analyzed linear transmission problems. Various mapping properties of these operators are also obtained and used to describe the weak solutions of the Poisson problems with Dirichlet and Neumann conditions for the nonsmooth Brinkman system in terms of such potentials. Finally, we combine the well-posedness results of the Poisson problems of Dirichlet, transmission, and mixed type for the nonsmooth Brinkman system with a fixed point theorem in order to show the existence of a weak solution of the Poisson problem of Dirichlet, transmission, or mixed type for the (nonlinear) Darcy-Forchheimer-Brinkman system with L ∞ coefficients in L 2 -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds of dimension m ∈ { 2 , 3 } .
- Published
- 2019
35. Dynamics of time-periodic reaction-diffusion equations with compact initial support on R
- Author
-
Weiwei Ding and Hiroshi Matano
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Ode ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Bounded function ,Reaction–diffusion system ,Convergence (routing) ,Initial value problem ,Limit (mathematics) ,0101 mathematics ,Constant (mathematics) ,Mathematics - Abstract
This paper is concerned with the asymptotic behavior of bounded solutions of the Cauchy problem { u t = u x x + f ( t , u ) , x ∈ R , t > 0 , u ( x , 0 ) = u 0 , x ∈ R , where u 0 is a nonnegative bounded function with compact support and f is a rather general nonlinearity that is periodic in t and satisfies f ( ⋅ , 0 ) = 0 . In the autonomous case where f = f ( u ) , the convergence of every bounded solution to an equilibrium has been established by Du and Matano (2010). However, the presence of periodic forcing makes the problem significantly more difficult, partly because the structure of time periodic solutions is much less understood than that of steady states. In this paper, we first prove that any ω-limit solution is either spatially constant or symmetrically decreasing. Furthermore, we show that the set of ω-limit solutions either consists of a single time-periodic solution or it consists of multiple time-periodic solutions and heteroclinic connections among them. Next, under a mild non-degenerate assumption on the corresponding ODE, we prove that the ω-limit set is a singleton, which implies the solution converges to a time-periodic solution. Lastly, we apply these results to equations with bistable nonlinearity and combustion nonlinearity, and specify more precisely which time-periodic solutions can possibly be selected as the limit.
- Published
- 2019
36. Extensions of linear operators from hyperplanes and strong uniqueness of best approximation in L(X,W)
- Author
-
Paweł Wójcik
- Subjects
Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Banach space ,010103 numerical & computational mathematics ,Codimension ,Extension (predicate logic) ,01 natural sciences ,Projection (linear algebra) ,Operator (computer programming) ,Hyperplane ,Uniqueness ,0101 mathematics ,Analysis ,Subspace topology ,Mathematics - Abstract
The aim of this paper is to present some results concerning the problem of minimal projections and extensions. Let X be a reflexive Banach space and let Y be a closed subspace of X of codimension one. Let W be a finite-dimensional Banach space. We present a new sufficient condition under which any minimal extension of an operator A ∈ L ( Y , W ) is strongly unique. In this paper we show (in some circumstances) that if 1 λ ( Y , X ) , then a minimal projection from X onto Y is a strongly unique minimal projection. Moreover, we introduce and study a new geometric property of normed spaces. In this paper we also present a result concerning the strong unicity of best approximation.
- Published
- 2019
37. Convergence of boundary layers for the Keller–Segel system with singular sensitivity in the half-plane
- Author
-
Qianqian Hou and Zhi-An Wang
- Subjects
Plane (geometry) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Prandtl number ,Boundary (topology) ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Boundary layer ,symbols.namesake ,symbols ,Boundary value problem ,0101 mathematics ,Layer (object-oriented design) ,Degeneracy (mathematics) ,Mathematics - Abstract
Though the boundary layer formation in the chemotactic process has been observed in experiment (cf. [63] ), the mathematical study on the boundary layer solutions of chemotaxis models is just in its infant stage. Apart from the sophisticated theoretical tools involved in the analysis, how to impose/derive physical boundary conditions is a state-of-the-art in studying the boundary layer problem of chemotaxis models. This paper will proceed with a previous work [24] in one dimension to establish the convergence of boundary layer solutions of the Keller–Segel model with singular sensitivity in a two-dimensional space (half-plane) with respect to the chemical diffusion rate denoted by e ≥ 0 . Compared to the one-dimensional boundary layer problem, there are many new issues arising from multi-dimensions such as possible Prandtl type degeneracy, curl-free preservation and well-posedness of large-data solutions. In this paper, we shall derive appropriate physical boundary conditions and gradually overcome these barriers and hence establish the convergence of boundary layer solutions of the singular Keller–Segel system in the half-plane as the chemical diffusion rate vanishes. Specially speaking, we justify that the boundary layer converges to the outer layer (solution with e = 0 ) plus the inner layer as e → 0 , where both outer and inner layer profiles are precisely derived and well understood. By doing this, the structure of boundary layer solutions is clearly characterized. We hope that our results and methods can shed lights on the understanding of underlying mechanisms of the boundary layer patterns observed in the experiment for chemotaxis such as the work by Tuval et al. [63] , and open a new window in the future theoretical study of chemotaxis models.
- Published
- 2019
38. Stefan Kempisty (1892–1940)
- Author
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Małgorzata Terepeta, Lech Maligranda, and Izabela Jóźwik
- Subjects
XX-wieczna matematyka i matematycy w Europie ,Surface (mathematics) ,History ,funkcje quasi-ciagłe ,General Mathematics ,Mathematical analysis ,Surface area ,Mathematical Analysis ,Interval (mathematics) ,pole powierzchni ,Stefan Kempisty ,Matematisk analys ,Interval functions ,całka Kempistego ,funkcje przedziału ,Kempisty's integral ,Set theory ,badania bibliograficzne ,Quasi-continuous functions ,20th century mathematics and mathematicians in Europe ,Mathematics - Abstract
Stefan Kempisty był polskim matematykiem zajmujacym sie funkcjami zmiennej rzeczywistej, teoria mnogosci, całkami, funkcjami przedziału i teoria pola powierzchni. W 1919 roku obronił prace doktorska ,,O funkcjach nawpołciagłych" na Uniwersytecie Jagiellonskim w Krakowie, a promotorem był Kazimierz Zorawski. W grudniu 1924 roku habilitował sie na Uniwersytecie Warszawskim i w latach 1920--1939 pracował na Uniwersytecie Stefana Batorego w Wilnie. Opublikował ponad czterdziesci prac naukowych i trzy podreczniki z analizy rzeczywistej oraz jedna monografie. Reprezentował w swoich pracach i na seminariach szkołe warszawska. Nazwisko Kempistego w matematyce pojawia sie w zwiazku z definicja funkcji quasi-ciagłej, roznymi ciagłosciami funkcji wielu zmiennych, klasyfikacja funkcji Baire'a, Younga i Sierpinskiego, funkcjami przedziału oraz całkami Denjoy'a i Burkilla. Zainteresowalismy sie Kempistym ze wzgledu na jego dorobek naukowy, pewne niewyjasnione informacje osobiste oraz 125 rocznice urodzin przypadajaca w 2017 roku. Przedstawiamy wiec jego biografie, udział w konferencjach, sylwetki zony Eugenii i corki Marii oraz jego dorobek naukowy. Wszystkie informacje o nim pochodza z wielu zrodeł. Stefan Jan Kempisty was a Polish mathematician, working in the theory of real functions, set theory, integrals, interval functions and the thory of surface area. In 1919 he defended his Ph.D. thesis ``On semi-continuous functions" at the Jagiellonian University in Krakow under supervision of Kazimierz Zorawski. In December 1924 he did habilitation at the Warsaw University and from 1920 to 1939 he worked at the Stefan Batory University in Vilnius. He published over forty scientific papers, three textbooks and one monograph. He represented in his papers and on seminars the Warsaw school. Kempist's name in mathematics appears in connection with the definition of quasi-continuous functions, different kind of continuity of functions of several variables, the classification of Baire, Young and Sierpinski functions, interval functions and Denjoy and Burkill integrals. We took interest in Kempisty because of his academic achievements, some unexplained personal information, and his 125th birthday in 2017. So we present his biography, participation in conferences, the silhouette his wife Eugenia and daughter Maria, and his scientific achievements. All information about it comes from several sources. Validerad;2018;Nivå 1;2018-04-17 (andbra)
- Published
- 2019
39. Geodesic vector fields and Eikonal equation on a Riemannian manifold
- Author
-
Viqar Azam Khan and Sharief Deshmukh
- Subjects
Geodesic ,Eikonal equation ,Euclidean space ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Einstein manifold ,Riemannian manifold ,01 natural sciences ,Killing vector field ,Mathematics::Metric Geometry ,Vector field ,Mathematics::Differential Geometry ,0101 mathematics ,Ricci curvature ,Mathematics - Abstract
In this paper, we study the impact of geodesic vector fields (vector fields whose trajectories are geodesics) on the geometry of a Riemannian manifold. Since, Killing vector fields of constant lengths on a Riemannian manifold are geodesic vector fields, leads to the question of finding sufficient conditions for a geodesic vector field to be Killing. In this paper, we show that a lower bound on the Ricci curvature of the Riemannian manifold in the direction of geodesic vector field gives a sufficient condition for the geodesic vector field to be Killing. Also, we use a geodesic vector field on a 3-dimensional complete simply connected Riemannian manifold to find sufficient conditions to be isometric to a 3-sphere. We find a characterization of an Einstein manifold using a Killing vector field. Finally, it has been observed that a major source of geodesic vector fields is provided by solutions of Eikonal equations on a Riemannian manifold and we obtain a characterization of the Euclidean space using an Eikonal equation.
- Published
- 2019
40. Reproducing kernel orthogonal polynomials on the multinomial distribution
- Author
-
Robert C. Griffiths and Persi Diaconis
- Subjects
Numerical Analysis ,Stationary distribution ,Markov chain ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Poisson kernel ,010103 numerical & computational mathematics ,Kravchuk polynomials ,01 natural sciences ,Combinatorics ,symbols.namesake ,Kernel (statistics) ,Orthogonal polynomials ,symbols ,Test statistic ,Multinomial distribution ,0101 mathematics ,Analysis ,Mathematics - Abstract
Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Q n ( x , y ; N , p ) on the multinomial distribution which are sums of products of orthonormal polynomials in x and y of fixed total degree n = 0 , 1 , … , N . The Poisson kernel ∑ n = 0 N ρ n Q n ( x , y ; N , p ) arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new duplication formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The duplication formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The χ 2 cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.
- Published
- 2019
41. Balanced derivatives, identities, and bounds for trigonometric and Bessel series
- Author
-
Bruce C. Berndt, Sun Kim, Martino Fassina, and Alexandru Zaharescu
- Subjects
symbols.namesake ,Pure mathematics ,Series (mathematics) ,General Mathematics ,symbols ,Trigonometric functions ,Divisor (algebraic geometry) ,Trigonometry ,Upper and lower bounds ,Bessel function ,Gauss circle problem ,Ramanujan's sum ,Mathematics - Abstract
Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in an earlier paper, three of the present authors derived representations for certain sums of products of trigonometric functions as double series of Bessel functions [8] . These series are generalized in the present paper by introducing the novel notion of balanced derivatives, leading to further theorems. As we will see below, the regions of convergence in the unbalanced case are entirely different than those in the balanced case. From this viewpoint, it is remarkable that Ramanujan had the intuition to formulate entries that are, in our new terminology, “balanced”. If x denotes the number of products of the trigonometric functions appearing in our sums, in addition to proving the identities mentioned above, theorems and conjectures for upper and lower bounds for the sums as x → ∞ are established.
- Published
- 2022
42. Transfer operators and Hankel transforms between relative trace formulas, II: Rankin–Selberg theory
- Author
-
Yiannis Sakellaridis
- Subjects
Transfer (group theory) ,Pure mathematics ,Hecke algebra ,symbols.namesake ,Conjecture ,Trace (linear algebra) ,General Mathematics ,Poisson summation formula ,symbols ,Functional equation (L-function) ,Abelian group ,Fundamental lemma ,Mathematics - Abstract
The Langlands functoriality conjecture, as reformulated in the “beyond endoscopy” program, predicts comparisons between the (stable) trace formulas of different groups G 1 , G 2 for every morphism G 1 L → L G 2 between their L-groups. This conjecture can be seen as a special case of a more general conjecture, which replaces reductive groups by spherical varieties and the trace formula by its generalization, the relative trace formula. The goal of this article and its precursor [11] is to demonstrate, by example, the existence of “transfer operators” between relative trace formulas, which generalize the scalar transfer factors of endoscopy. These transfer operators have all properties that one could expect from a trace formula comparison: matching, fundamental lemma for the Hecke algebra, transfer of (relative) characters. Most importantly, and quite surprisingly, they appear to be of abelian nature (at least, in the low-rank examples considered in this paper), even though they encompass functoriality relations of non-abelian harmonic analysis. Thus, they are amenable to application of the Poisson summation formula in order to perform the global comparison. Moreover, we show that these abelian transforms have some structure — which presently escapes our understanding in its entirety — as deformations of well-understood operators when the spaces under consideration are replaced by their “asymptotic cones”. In this second paper we use Rankin–Selberg theory to prove the local transfer behind Rudnick's 1990 thesis (comparing the stable trace formula for SL 2 with the Kuznetsov formula) and Venkatesh's 2002 thesis (providing a “beyond endoscopy” proof of functorial transfer from tori to GL 2 ). As it turns out, the latter is not completely disjoint from endoscopic transfer — in fact, our proof “factors” through endoscopic transfer. We also study the functional equation of the symmetric-square L-function for GL 2 , and show that it is governed by an explicit “Hankel operator” at the level of the Kuznetsov formula, which is also of abelian nature. A similar theory for the standard L-function was previously developed (in a different language) by Jacquet.
- Published
- 2022
43. Comparison of probabilistic and deterministic point sets on the sphere
- Author
-
Peter J. Grabner and T. A. Stepanyuk
- Subjects
Unit sphere ,Numerical Analysis ,Sequence ,Applied Mathematics ,General Mathematics ,Existential quantification ,010102 general mathematics ,Probabilistic logic ,Sampling (statistics) ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Point (geometry) ,0101 mathematics ,Constant (mathematics) ,Analysis ,Mathematics - Abstract
In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (especially spherical t -designs) are better or as good as probabilistic ones like the jittered sampling model. We find asymptotic equalities for the discrete Riesz s -energy of sequences of well separated t -designs on the unit sphere S d ⊂ R d + 1 , d ≥ 2 . The case d = 2 was studied in Hesse (2009) and Hesse and Leopardi (2008). In Bondarenko et al., (2015) it was established that for d ≥ 2 , there exists a constant c d , such that for every N > c d t d there exists a well-separated spherical t -design on S d with N points. This paper gives results, based on recent developments that there exists a sequence of well separated spherical t -designs such that t and N are related by N ≍ t d .
- Published
- 2019
44. Critical points of the integral map of the charged three-body problem
- Author
-
Holger Waalkens, I. Hoveijn, and M. Zaman
- Subjects
Connection (fibred manifold) ,Pure mathematics ,Series (mathematics) ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,010103 numerical & computational mathematics ,Type (model theory) ,Three-body problem ,Infinity ,01 natural sciences ,Configuration space ,0101 mathematics ,media_common ,Mathematics - Abstract
This is the first in a series of three papers where we study the integral manifolds of the charged three-body problem. The integral manifolds are the fibers of the map of integrals. Their topological type may change at critical values of the map of integrals. Due to the non-compactness of the integral manifolds one has to take into account besides ‘ordinary’ critical points also critical points at infinity. In the present paper we concentrate on ‘ordinary’ critical points and in particular elucidate their connection to central configurations. In a second paper we will study critical points at infinity. The implications for the Hill regions, i.e. the projections of the integral manifolds to configuration space, are the subject of a third paper.
- Published
- 2019
45. On the relation of the spectral test to isotropic discrepancy and L-approximation in Sobolev spaces
- Author
-
Mathias Sonnleitner and Friedrich Pillichshammer
- Subjects
Statistics and Probability ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Isotropy ,Mathematical analysis ,Convex set ,Boundary (topology) ,010103 numerical & computational mathematics ,01 natural sciences ,Upper and lower bounds ,Spectral test ,Sobolev space ,Dimension (vector space) ,Unit cube ,0101 mathematics ,Mathematics - Abstract
This paper is a follow-up to the recent paper of Pillichshammer and Sonnleitner (2020) [12] . We show that the isotropic discrepancy of a lattice point set is at most d 2 2 ( d + 1 ) times its spectral test, thereby correcting the dependence on the dimension d and an inaccuracy in the proof of the upper bound in Theorem 2 of the mentioned paper. The major task is to bound the volume of the neighbourhood of the boundary of a convex set contained in the unit cube. Further, we characterize averages of the distance to a lattice point set in terms of the spectral test. As an application, we infer that the spectral test – and with it the isotropic discrepancy – is crucial for the suitability of the lattice point set for the approximation of Sobolev functions.
- Published
- 2021
46. Normal crossings singularities for symplectic topology
- Author
-
Mark McLean, Aleksey Zinger, and Mohammad Farajzadeh Tehrani
- Subjects
Pure mathematics ,Logarithm ,Divisor ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics - Symplectic Geometry ,0103 physical sciences ,FOS: Mathematics ,Symplectic Geometry (math.SG) ,53D05, 53D45, 14N35 ,Gravitational singularity ,010307 mathematical physics ,0101 mathematics ,Equivalence (formal languages) ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Symplectic sum ,Symplectic geometry ,Mathematics - Abstract
We introduce topological notions of normal crossings symplectic divisor and variety and establish that they are equivalent, in a suitable sense, to the desired geometric notions. Our proposed concept of equivalence of associated topological and geometric notions fits ideally with important constructions in symplectic topology. This partially answers Gromov's question on the feasibility of defining singular symplectic (sub)varieties and lays foundation for rich developments in the future. In subsequent papers, we establish a smoothability criterion for symplectic normal crossings varieties, in the process providing the multifold symplectic sum envisioned by Gromov, and introduce symplectic analogues of logarithmic structures in the context of normal crossings symplectic divisors., Comment: 65 pages, 4 figures; a number of typos fixed; the exposition has been significantly revised, fixing a technical error in the non-compact case in the process; this paper is now restricted to the simple normal crossings case; the arbitrary normal crossings case will be detailed in a followup paper
- Published
- 2018
47. On the mean field type bubbling solutions for Chern–Simons–Higgs equation
- Author
-
Shusen Yan and Chang-Shou Lin
- Subjects
General Mathematics ,010102 general mathematics ,Chern–Simons theory ,Structure (category theory) ,Type (model theory) ,01 natural sciences ,Mean field theory ,0103 physical sciences ,Higgs boson ,010307 mathematical physics ,Uniqueness ,0101 mathematics ,Parallelogram ,Mathematical physics ,Mathematics - Abstract
This paper is the second part of our comprehensive study on the structure of the solutions for the following Chern–Simons–Higgs equation: (0.1) { Δ u + 1 e 2 e u ( 1 − e u ) = 4 π ∑ j = 1 N δ p j , in Ω , u is doubly periodic on ∂ Ω , where Ω is a parallelogram in R 2 and e > 0 is a small parameter. In part 1 [29] , we proved the non-coexistence of different bubbles in the bubbling solutions and obtained an existence result for the Chern–Simons type bubbling solutions under some nearly necessary conditions. Mean field type bubbling solutions for (0.1) have been constructed in [27] . In this paper, we shall study two other important issues for the mean field type bubbling solutions: the necessary conditions for the existence and the local uniqueness. The results in this paper lay the foundation to find the exact number of solutions for (0.1) .
- Published
- 2018
48. Superconvergence of kernel-based interpolation
- Author
-
Robert Schaback
- Subjects
Numerical Analysis ,Applied Mathematics ,General Mathematics ,Open problem ,Hilbert space ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Positive-definite matrix ,Superconvergence ,Eigenfunction ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Spline (mathematics) ,FOS: Mathematics ,symbols ,Applied mathematics ,Mathematics - Numerical Analysis ,Boundary value problem ,0101 mathematics ,Spline interpolation ,Analysis ,Mathematics - Abstract
From spline theory it is well-known that univariate cubic spline interpolation, if carried out in its natural Hilbert space W 2 2 [ a , b ] and on point sets with fill distance h , converges only like O ( h 2 ) in L 2 [ a , b ] if no additional assumptions are made. But superconvergence up to order h 4 occurs if more smoothness is assumed and if certain additional boundary conditions are satisfied. This phenomenon was generalized in 1999 to multivariate interpolation in Reproducing Kernel Hilbert Spaces on domains Ω ⊂ R d for continuous positive definite Fourier-transformable shift-invariant kernels on R d . But the sufficient condition for superconvergence given in 1999 still needs further analysis, because the interplay between smoothness and boundary conditions is not clear at all. Furthermore, if only additional smoothness is assumed, superconvergence is numerically observed in the interior of the domain, but a theoretical foundation still is a challenging open problem. This paper first generalizes the “improved error bounds” of 1999 by an abstract theory that includes the Aubin–Nitsche trick and the known superconvergence results for univariate polynomial splines. Then the paper analyzes what is behind the sufficient conditions for superconvergence. They split into conditions on smoothness and localization, and these are investigated independently. If sufficient smoothness is present, but no additional localization conditions are assumed, it is numerically observed that superconvergence always occurs in the interior of the domain, and some supporting arguments are provided. If smoothness and localization interact in the kernel-based case on R d , weak and strong boundary conditions in terms of pseudodifferential operators occur. A special section on Mercer expansions is added, because Mercer eigenfunctions always satisfy the sufficient conditions for superconvergence. Numerical examples illustrate the theoretical findings.
- Published
- 2018
49. Generating new ideals using weighted density via modulus functions
- Author
-
Adam Kwela, Pratulananda Das, and Kumardipta Bose
- Subjects
Combinatorics ,Modulo operation ,General Mathematics ,010102 general mathematics ,Modulus ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper we extend the idea of weighted density of Balcerzak et al. (2015) by using a modulus function and introduce the idea f -density of weight g of subsets of ω ≔ { 0 , 1 , … } (at the same time extending the notion of f -density (Aizpuru et al., 2014)), which we name d g f where g : ω → [ 0 , ∞ ) satisfies g ( n ) → ∞ and n ∕ g ( n ) ↛ 0 and f is a modulus function. The aim of this paper is to show that we can get new ideals Z g ( f ) consisting of sets A ⊂ ω for which d g f ( A ) = 0 different from all the previously constructed ideals Z g of Balcerzak et al. (2015) and moreover they retain all the nice properties of the ideals Z g .
- Published
- 2018
50. Controllability for noninstantaneous impulsive semilinear functional differential inclusions without compactness
- Author
-
Ahmed Ibrahim, Donal O'Regan, Yong Zhou, and JinRong Wang
- Subjects
Pure mathematics ,Semigroup ,General Mathematics ,010102 general mathematics ,Banach space ,01 natural sciences ,010101 applied mathematics ,Controllability ,Compact space ,Operator (computer programming) ,Differential inclusion ,Piecewise ,Infinitesimal generator ,0101 mathematics ,Mathematics - Abstract
The first part of this paper considers the controllability for a functional semilinear differential inclusion governed by a family of operators { A ( t ) : t ∈ [ 0 , b ] } generating an evolution operator in a Banach space in the presence of noninstantaneous impulse effects. In the second part of this paper we study the controllability for a fractional noninstantaneous impulsive semilinear differential inclusion with delay, where the linear part is an infinitesimal generator of a C 0 − semigroup. Using a weakly convergent criterion in the space of piecewise continuous functions and weak topology theory (for weak sequentially closed graph operators) we establish sufficient conditions to guarantee controllability results. Examples are given to illustrate the abstract results.
- Published
- 2018
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