630 results
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2. Improving the performance of deep learning models using statistical features: The case study of COVID‐19 forecasting
- Author
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Hossein Abbasimehr, Reza Paki, and Aram Bahrini
- Subjects
2019-20 coronavirus outbreak ,Coronavirus disease 2019 (COVID-19) ,62‐07 ,General Mathematics ,Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) ,Context (language use) ,97r40 ,Machine learning ,computer.software_genre ,01 natural sciences ,Convolutional neural network ,Special Issue Paper ,0101 mathematics ,Combined method ,Mathematics ,Special Issue Papers ,business.industry ,Deep learning ,010102 general mathematics ,General Engineering ,deep learning ,COVID‐19 pandemic ,010101 applied mathematics ,hybrid methods ,Memory model ,Artificial intelligence ,business ,computer ,statistical features - Abstract
COVID-19 pandemic has affected all aspects of people's lives and disrupted the economy. Forecasting the number of cases infected with this virus can help authorities make accurate decisions on the interventions that must be implemented to control the pandemic. Investigation of the studies on COVID-19 forecasting indicates that various techniques such as statistical, mathematical, and machine and deep learning have been utilized. Although deep learning models have shown promising results in this context, their performance can be improved using auxiliary features. Therefore, in this study, we propose two hybrid deep learning methods that utilize the statistical features as auxiliary inputs and associate them with their main input. Specifically, we design a hybrid method of the multihead attention mechanism and the statistical features (ATT_FE) and a combined method of convolutional neural network and the statistical features (CNN_FE) and apply them to COVID-19 data of 10 countries with the highest number of confirmed cases. The results of experiments indicate that the hybrid models outperform their conventional counterparts in terms of performance measures. The experiments also demonstrate the superiority of the hybrid ATT_FE method over the long short-term memory model.
- Published
- 2021
3. The geometry of diagonal groups
- Author
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Peter J. Cameron, Cheryl E. Praeger, Csaba Schneider, R. A. Bailey, University of St Andrews. Pure Mathematics, University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra, and University of St Andrews. Statistics
- Subjects
Mathematics(all) ,South china ,Primitive permutation group ,General Mathematics ,Diagonal group ,T-NDAS ,Library science ,Group Theory (math.GR) ,O'Nan-Scott Theorem ,01 natural sciences ,Hospitality ,FOS: Mathematics ,NCAD ,Mathematics - Combinatorics ,QA Mathematics ,0101 mathematics ,Diagonal semilattice ,QA ,Cartesian lattice ,Mathematics ,business.industry ,20B05 ,Applied Mathematics ,010102 general mathematics ,Latin square ,Semilattice ,Latin cube ,010101 applied mathematics ,Hamming graph ,Research council ,Diagonal graph ,Combinatorics (math.CO) ,business ,Mathematics - Group Theory ,Partition - Abstract
Part of the work was done while the authors were visiting the South China University of Science and Technology (SUSTech), Shenzhen, in 2018, and we are grateful (in particular to Professor Cai Heng Li) for the hospitality that we received.The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Groups, representations and applications: new perspectives (supported by EPSRC grant no.EP/R014604/1), where further work on this paper was undertaken. In particular we acknowledge a Simons Fellowship (Cameron) and a Kirk Distinguished Visiting Fellowship (Praeger) during this programme. Schneider thanks the Centre for the Mathematics of Symmetry and Computation of The University of Western Australia and Australian Research Council Discovery Grant DP160102323 for hosting his visit in 2017 and acknowledges the support of the CNPq projects Produtividade em Pesquisa (project no.: 308212/2019-3) and Universal (project no.:421624/2018-3). Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not been studied in general. The main purpose of this paper is to describe and characterise such structures, which we call diagonal semilattices. Unlike the diagonal groups in the O'Nan-Scott theorem, which are defined over finite characteristically simple groups, our construction works over arbitrary groups, finite or infinite. A diagonal semilattice depends on a dimension m and a group T. For m=2, it is a Latin square, the Cayley table of T, though in fact any Latin square satisfies our combinatorial axioms. However, for m≥3, the group T emerges naturally and uniquely from the axioms. (The situation somewhat resembles projective geometry, where projective planes exist in great profusion but higher-dimensional structures are coordinatised by an algebraic object, a division ring.) A diagonal semilattice is contained in the partition lattice on a set Ω, and we provide an introduction to the calculus of partitions. Many of the concepts and constructions come from experimental design in statistics. We also determine when a diagonal group can be primitive, or quasiprimitive (these conditions turn out to be equivalent for diagonal groups). Associated with the diagonal semilattice is a graph, the diagonal graph, which has the same automorphism group as the diagonal semilattice except in four small cases with m
- Published
- 2022
4. Satisfiability in MultiValued Circuits
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Paweł M. Idziak and Jacek Krzaczkowski
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FOS: Computer and information sciences ,Computational complexity theory ,General Computer Science ,68Q17, 08A05, 08A70 (Primary) 68Q05, 68T27, 03B25, 08B05, 08B10 (Secondary) ,Boolean circuit ,General Mathematics ,010102 general mathematics ,circuit satisfiability ,Distributive lattice ,0102 computer and information sciences ,Computational Complexity (cs.CC) ,01 natural sciences ,Satisfiability ,Algebra ,Computer Science - Computational Complexity ,Monotone polygon ,010201 computation theory & mathematics ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Lie algebra ,0101 mathematics ,Time complexity ,solving equations ,Equation solving ,Mathematics - Abstract
Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebras $\mathbf A$ there is a polynomial time algorithm that decides if an equation over $\mathbf A$ has a solution. We are also looking for polynomial time algorithms that decide if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization for a very broad class of algebras from congruence modular varieties. This class includes most known and well-studied algebras such as groups, rings, modules (and their generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie algebras), lattices (and their extensions like Boolean algebras, Heyting algebras or other algebras connected with multi-valued logics including MV-algebras). This paper seems to be the first systematic study of the computational complexity of satisfiability of non-Boolean circuits and solving equations over finite algebras. The characterization results provided by the paper is given in terms of nice structural properties of algebras for which the problems are solvable in polynomial time., 50 pages
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- 2022
5. 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
6. Some significant remarks on multivalued Perov type contractions on cone metric spaces with a directed graph
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Aleksandra Sretenovic, Nicola Fabiano, Ana Savić, Stojan Radenović, and Nikola Mirkov
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Pure mathematics ,General Mathematics ,cone metric space ,010102 general mathematics ,multivalued mapping ,graphic contraction ,Directed graph ,common fixed point ,Fixed point ,Type (model theory) ,Mathematical proof ,directed graph ,01 natural sciences ,Cone (formal languages) ,c-sequence ,010101 applied mathematics ,Metric space ,QA1-939 ,0101 mathematics ,Contraction principle ,perov's type results ,Mathematics ,Complement (set theory) - Abstract
Using the approach of so-called c-sequences introduced by the fifth author in his recent work, we give much simpler and shorter proofs of multivalued Perov's type results with respect to the ones presented in the recently published paper by M. Abbas et al. Our proofs improve, complement, unify and enrich the ones from the recent papers. Further, in the last section of this paper, we correct and generalize the well-known Perov's fixed point result. We show that this result is in fact equivalent to Banach's contraction principle.
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- 2022
7. 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
8. Analyzing the Weyl Construction for Dynamical Cartan Subalgebras
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Elizabeth Gillaspy, Anna Duwenig, and Rachael Norton
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General Mathematics ,01 natural sciences ,Section (fiber bundle) ,Combinatorics ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,46L05, 22D25, 22A22 ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Twist ,Operator Algebras (math.OA) ,Mathematics::Representation Theory ,Quotient ,Mathematics ,Science & Technology ,Mathematics::Operator Algebras ,010102 general mathematics ,Spectrum (functional analysis) ,Mathematics - Operator Algebras ,Cartan subalgebra ,C-ASTERISK-ALGEBRAS ,Physical Sciences ,010307 mathematical physics ,EQUIVALENCE - Abstract
When the reduced twisted $C^*$-algebra $C^*_r(\mathcal{G}, c)$ of a non-principal groupoid $\mathcal{G}$ admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r(\mathcal{G}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid $\mathcal{S}$ of $\mathcal{G}$. In this paper, we study the relationship between the original groupoids $\mathcal{S}, \mathcal{G}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum $\mathfrak{B}$ of the Cartan subalgebra $C^*_r(\mathcal{S}, c)$. We then show that the quotient groupoid $\mathcal{G}/\mathcal{S}$ acts on $\mathfrak{B}$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly we show that, if the quotient map $\mathcal{G}\to\mathcal{G}/\mathcal{S}$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $2$-cocycle on $\mathcal{G}/\mathcal{S} \ltimes \mathfrak{B}$., 32 pages
- Published
- 2022
9. Order 3 symplectic automorphisms on K3 surfaces
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Alice Garbagnati and Yulieth Prieto Montañez
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Pure mathematics ,Endomorphism ,General Mathematics ,010102 general mathematics ,Lattice (group) ,Order (ring theory) ,Automorphism ,01 natural sciences ,Cohomology ,14J28, 14J50 ,Mathematics - Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Algebraic Geometry (math.AG) ,Quotient ,Mathematics ,Symplectic geometry - Abstract
The aim of this paper is to generalize results known for the symplectic involutions on K3 surfaces to the order 3 symplectic automorphisms on K3 surfaces. In particular, we will explicitly describe the action induced on the lattice $\Lambda_{K3}$, isometric to the second cohomology group of a K3 surface, by a symplectic automorphism of order 3; we exhibit the maps $\pi_*$ and $\pi^*$ induced in cohomology by the rational quotient map $\pi:X\dashrightarrow Y$, where $X$ is a K3 surface admitting an order 3 symplectic automorphism $\sigma$ and $Y$ is the minimal resolution of the quotient $X/\sigma$; we deduce the relation between the N\'eron--Severi group of $X$ and the one of $Y$. Applying these results we describe explicit geometric examples and generalize the Shioda--Inose structures, relating Abelian surfaces admitting order 3 endomorphisms with certain specific K3 surfaces admitting particular order 3 symplectic automorphisms., Comment: 28 pages. Version 2: this is the published version of the paper. The last section of the previous version (v1) was erased (the results are only stated) and it is now contained in arXiv:2209.10141
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- 2021
10. Quadratic Gorenstein Rings and the Koszul Property II
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Michael Stillman, Matthew Mastroeni, and Hal Schenck
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Pure mathematics ,Property (philosophy) ,Mathematics::Commutative Algebra ,General Mathematics ,Mathematics::Rings and Algebras ,010102 general mathematics ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,16. Peace & justice ,01 natural sciences ,010101 applied mathematics ,Quadratic equation ,FOS: Mathematics ,0101 mathematics ,Mathematics - Abstract
A question of Conca, Rossi, and Valla asks whether every quadratic Gorenstein ring $R$ of regularity three is Koszul. In a previous paper, we use idealization to answer their question, proving that in nine or more variables there exist quadratic Gorenstein rings of regularity three which are not Koszul. In this paper, we study the analog of the Conca-Rossi-Valla question when the regularity of $R$ is four or more. Let $R$ be a quadratic Gorenstein ring having $\mathrm{codim}\, R = c$ and $\mathrm{reg}\, R = r \ge 4$. We prove that if $c = r+1$ then $R$ is always Koszul, and for every $c \geq r+2$, we construct quadratic Gorenstein rings that are not Koszul, answering questions of Matsuda and Migliore-Nagel concerning the $h$-vectors of quadratic Gorenstein rings., Comment: v2 - Minor changes based on referee comments
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- 2021
11. Good Formal Structures for Flat Meromorphic Connections, III: Irregularity and Turning Loci
- Author
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Kiran S. Kedlaya
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Pure mathematics ,Mathematics - Complex Variables ,Divisor ,General Mathematics ,010102 general mathematics ,Fibration ,Codimension ,Lattice (discrete subgroup) ,14F10, 32C38, 14C20 ,01 natural sciences ,Blowing up ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,FOS: Mathematics ,Sheaf ,Complex Variables (math.CV) ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics ,Meromorphic function ,Resolution (algebra) - Abstract
Given a formal flat meromorphic connection over an excellent scheme over a field of characteristic zero, in a previous paper we established existence of good formal structures and a good Deligne-Malgrange lattice after suitably blowing up. In this paper, we reinterpret and refine these results by introducing some related structures. We consider the turning locus, which is the set of points at which one cannot achieve a good formal structure without blowing up. We show that when the polar divisor has normal crossings, the turning locus is of pure codimension 1 within the polar divisor, and hence of pure codimension 2 within the full space; this had been previously established by Andre in the case of a smooth polar divisor. We also construct an irregularity sheaf and its associated b-divisor, which measure irregularity along divisors on blowups of the original space; this generalizes another result of Andre on the semicontinuity of irregularity in a curve fibration. One concrete consequence of these refinements is a process for resolution of turning points which is functorial with respect to regular morphisms of excellent schemes; this allows us to transfer the result from schemes to formal schemes, complex analytic varieties, and nonarchimedean analytic varieties., 27 pages; v4: refereed version; some technical edits in 2.2
- Published
- 2021
12. Multivariate quasi-tight framelets with high balancing orders derived from any compactly supported refinable vector functions
- Author
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Bin Han and Ran Lu
- Subjects
FOS: Computer and information sciences ,Pure mathematics ,Information Theory (cs.IT) ,Computer Science - Information Theory ,General Mathematics ,010102 general mathematics ,Scalar (mathematics) ,42C40, 42C15, 41A25, 41A35, 65T60 ,010103 numerical & computational mathematics ,Spectral theorem ,Trigonometric polynomial ,01 natural sciences ,Hermitian matrix ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Spline (mathematics) ,Wavelet ,Factorization ,FOS: Mathematics ,0101 mathematics ,Vector-valued function ,Mathematics - Abstract
Generalizing wavelets by adding desired redundancy and flexibility, framelets (i.e., wavelet frames) are of interest and importance in many applications such as image processing and numerical algorithms. Several key properties of framelets are high vanishing moments for sparse multiscale representation, fast framelet transforms for numerical efficiency, and redundancy for robustness. However, it is a challenging problem to study and construct multivariate nonseparable framelets, mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices. Moreover, all the known multivariate tight framelets derived from spline refinable scalar functions have only one vanishing moment, and framelets derived from refinable vector functions are barely studied yet in the literature. In this paper, we circumvent the above difficulties through the approach of quasi-tight framelets, which behave almost identically to tight framelets. Employing the popular oblique extension principle (OEP), from an arbitrary compactly supported M-refinable vector function ϕ with multiplicity greater than one, we prove that we can always derive from ϕ a compactly supported multivariate quasi-tight framelet such that: (i) all the framelet generators have the highest possible order of vanishing moments; (ii) its associated fast framelet transform has the highest balancing order and is compact. For a refinable scalar function ϕ (i.e., its multiplicity is one), the above item (ii) often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived from ϕ satisfying item (i). We point out that constructing OEP-based quasi-tight framelets is closely related to the generalized spectral factorization of Hermitian trigonometric polynomial matrices. Our proof is critically built on a newly developed result on the normal form of a matrix-valued filter, which is of interest and importance in itself for greatly facilitating the study of refinable vector functions and multiwavelets/multiframelets. This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders. This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.
- Published
- 2021
13. On the singular value decomposition over finite fields and orbits of GU×GU
- Author
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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.
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- 2021
14. Metric Rectifiability of ℍ-regular Surfaces with Hölder Continuous Horizontal Normal
- Author
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Katrin Fässler, Daniela Di Donato, and Tuomas Orponen
- Subjects
0209 industrial biotechnology ,020901 industrial engineering & automation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Metric (mathematics) ,Mathematics::Metric Geometry ,Hölder condition ,02 engineering and technology ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Two definitions for the rectifiability of hypersurfaces in Heisenberg groups $\mathbb{H}^n$ have been proposed: one based on ${\mathbb{H}}$-regular surfaces and the other on Lipschitz images of subsets of codimension-$1$ vertical subgroups. The equivalence between these notions remains an open problem. Recent partial results are due to Cole–Pauls, Bigolin–Vittone, and Antonelli–Le Donne. This paper makes progress in one direction: the metric Lipschitz rectifiability of ${\mathbb{H}}$-regular surfaces. We prove that ${\mathbb{H}}$-regular surfaces in $\mathbb{H}^{n}$ with $\alpha $-Hölder continuous horizontal normal, $\alpha> 0$, are metric bilipschitz rectifiable. This improves on the work by Antonelli–Le Donne, where the same conclusion was obtained for $C^{\infty }$-surfaces. In $\mathbb{H}^{1}$, we prove a slightly stronger result: every codimension-$1$ intrinsic Lipschitz graph with an $\epsilon $ of extra regularity in the vertical direction is metric bilipschitz rectifiable. All the proofs in the paper are based on a new general criterion for finding bilipschitz maps between “big pieces” of metric spaces.
- Published
- 2021
15. A generalization of the Freidlin–Wentcell theorem on averaging of Hamiltonian systems
- Author
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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.
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- 2021
16. Simulations of nonlinear parabolic PDEs with forcing function without linearization
- Author
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Shko Ali Tahir and Murat Sari
- Subjects
010101 applied mathematics ,Nonlinear parabolic equations ,Nonlinear system ,Linearization ,Force function ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
This paper aims at producing numerical solutions of nonlinear parabolic PDEs with forcing term without any linearization. Since the linearization of nonlinear term leads to lose real features, without doing linearization, this paper focuses on capturing natural behaviour of the mechanism. Therefore we concentrate on analysis of the physical processes without losing their properties. To carry out this study, a backward differentiation formula in time and a spline method in space have been combined in leading to the discretized equation. This method leads to a very reliable alternative in solving the problem by conserving the physical properties of the nature. The efficiency of the present method are proved theoretically and illustrated by various numerical tests.
- Published
- 2021
17. Fekete-Szegö problem for starlike functions connected withk-Fibonacci numbers
- Author
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Serap Bulut
- Subjects
Combinatorics ,Subordination (linguistics) ,Fibonacci number ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Analytic function ,Mathematics - Abstract
In a recent paper, Sokół et al. [Applications of k-Fibonacci numbers for the starlike analytic functions, Hacet. J. Math. Stat. 44(1) (2015), 121{127] obtained an upper bound for the Fekete-Szegö functionalϕλwhenλ 2R of functions belong to the classSLkconnected withk-Fibonacci numbers. The main purpose of this paper is to obtain sharp bounds forϕλbothλ 2R andλ 2C.
- Published
- 2021
18. Maximal families of nodal varieties with defect
- Author
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REMKE NANNE KLOOSTERMAN
- Subjects
Surface (mathematics) ,Double cover ,Degree (graph theory) ,Plane (geometry) ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Combinatorics ,Mathematics - Algebraic Geometry ,Hypersurface ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,NODAL ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
In this paper we prove that a nodal hypersurface in P^4 with defect has at least (d-1)^2 nodes, and if it has at most 2(d-2)(d-1) nodes and d>6 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double cover of P^3 ramified along a surface of degree 2d with defect has at least d(2d-1) nodes. We construct the largest dimensional family of nodal degree d hypersurfaces in P^(2n+2) with defect for d sufficiently large., v2: A proof for the Ciliberto-Di Gennaro conjecture is added (Section 5); Some minor corrections in the other sections. v3: some minor corrections in the abstract v4: The proof for the Ciliberto-Di Gennaro conjecture has been modified; The paper is split into two parts, the complete intersection case will be discussed in a different paper
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- 2021
19. On Nilpotent Extensions of ∞-Categories and the Cyclotomic Trace
- Author
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Elden Elmanto and Vladimir Sosnilo
- Subjects
Trace (semiology) ,Pure mathematics ,Nilpotent ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
We do three things in this paper: (1) study the analog of localization sequences (in the sense of algebraic $K$-theory of stable $\infty $-categories) for additive $\infty $-categories, (2) define the notion of nilpotent extensions for suitable $\infty $-categories and furnish interesting examples such as categorical square-zero extensions, and (3) use (1) and (2) to extend the Dundas–Goodwillie–McCarthy theorem for stable $\infty $-categories that are not monogenically generated (such as the stable $\infty $-category of Voevodsky’s motives or the stable $\infty $-category of perfect complexes on some algebraic stacks). The key input in our paper is Bondarko’s notion of weight structures, which provides a “ring-with-many-objects” analog of a connective $\mathbb{E}_1$-ring spectrum. As applications, we prove cdh descent results for truncating invariants of stacks extending the work by Hoyois–Krishna for homotopy $K$-theory and establish new cases of Blanc’s lattice conjecture.
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- 2021
20. Entire Theta Operators at Unramified Primes
- Author
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Elena Mantovan and E. Eischen
- Subjects
Shimura variety ,Pure mathematics ,Mathematics - Number Theory ,Degree (graph theory) ,Mathematics::Number Theory ,General Mathematics ,Analytic continuation ,010102 general mathematics ,Modular form ,Automorphic form ,Differential operator ,Galois module ,01 natural sciences ,010101 applied mathematics ,Mathematics::Algebraic Geometry ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Mathematics::Representation Theory ,Signature (topology) ,Mathematics - Abstract
Starting with work of Serre, Katz, and Swinnerton-Dyer, theta operators have played a key role in the study of $p$-adic and $\bmod p$ modular forms and Galois representations. This paper achieves two main results for theta operators on automorphic forms on PEL-type Shimura varieties: 1) the analytic continuation at unramified primes $p$ to the whole Shimura variety of the $\bmod p$ reduction of $p$-adic Maass--Shimura operators {\it a priori} defined only over the $\mu$-ordinary locus, and 2) the construction of new $\bmod p$ theta operators that do not arise as the $\bmod p$ reduction of Maass--Shimura operators. While the main accomplishments of this paper concern the geometry of Shimura varieties and consequences for differential operators, we conclude with applications to Galois representations. Our approach involves a careful analysis of the behavior of Shimura varieties and enables us to obtain more general results than allowed by prior techniques, including for arbitrary signature, vector weights, and unramified primes in CM fields of arbitrary degree., Comment: Accepted for publication in IMRN. 42 pages
- Published
- 2021
21. Solving Bisymmetric Solution of a Class of Matrix Equations Based on Linear Saturated System Model Neural Network
- Author
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Feng Zhang
- Subjects
Normalization (statistics) ,Class (set theory) ,Article Subject ,Artificial neural network ,Computer science ,General Mathematics ,010102 general mathematics ,General Engineering ,Process (computing) ,Structure (category theory) ,Engineering (General). Civil engineering (General) ,01 natural sciences ,Backpropagation ,System model ,010101 applied mathematics ,Matrix (mathematics) ,QA1-939 ,Applied mathematics ,TA1-2040 ,0101 mathematics ,Mathematics - Abstract
In order to solve the complicated process and low efficiency and low accuracy of solving a class of matrix equations, this paper introduces the linear saturated system model neural network architecture to solve the bisymmetric solution of a class of matrix equations. Firstly, a class of matrix equations is constructed to determine the key problems of solving the equations. Secondly, the linear saturated system model neural network structure is constructed to determine the characteristic parameters in the process of bisymmetric solution. Then, the matrix equations is solved by using backpropagation neural network topology. Finally, the class normalization is realized by using the objective function of bisymmetric solution, and the bisymmetric solution of a class of matrix equations is realized. In order to verify the solving effect of the method in this paper, three indexes (accuracy, correction accuracy, and solving time) are designed in the experiment. The experimental results show that the proposed method can effectively reduce the solving time, can improve the accuracy and correction effect of the bisymmetric solution, and has high practicability.
- Published
- 2021
22. An index theorem for higher orbital integrals
- Author
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Xiang Tang, Peter Hochs, and Yanli Song
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Index (economics) ,General Mathematics ,01 natural sciences ,Mathematics::K-Theory and Homology ,0103 physical sciences ,FOS: Mathematics ,Representation Theory (math.RT) ,0101 mathematics ,Algebra over a field ,Operator Algebras (math.OA) ,Mathematics ,Group (mathematics) ,010102 general mathematics ,Mathematics - Operator Algebras ,Lie group ,K-Theory and Homology (math.KT) ,Elliptic operator ,Differential Geometry (math.DG) ,Mathematics - K-Theory and Homology ,Equivariant map ,010307 mathematical physics ,Atiyah–Singer index theorem ,Mathematics - Representation Theory - Abstract
Recently, two of the authors of this paper constructed cyclic cocycles on Harish-Chandra's Schwartz algebra of linear reductive Lie groups that detect all information in the $K$-theory of the corresponding group $C^*$-algebra. The main result in this paper is an index formula for the pairings of these cocycles with equivariant indices of elliptic operators for proper, cocompact actions. This index formula completely determines such equivariant indices via topological expressions., 40 pages; updates based on referee comments; expanded proof of Proposition 3.3
- Published
- 2021
23. The integrals and integral transformations connected with the joint vector Gaussian distribution
- Author
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N. F. Kako and V. S. Mukha
- Subjects
010302 applied physics ,Distribution (number theory) ,General Mathematics ,Gaussian ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,01 natural sciences ,symbols.namesake ,Computational Theory and Mathematics ,0103 physical sciences ,symbols ,0101 mathematics ,Joint (geology) ,Mathematics - Abstract
In many applications it is desirable to consider not one random vector but a number of random vectors with the joint distribution. This paper is devoted to the integral and integral transformations connected with the joint vector Gaussian probability density function. Such integral and transformations arise in the statistical decision theory, particularly, in the dual control theory based on the statistical decision theory. One of the results represented in the paper is the integral of the joint Gaussian probability density function. The other results are the total probability formula and Bayes formula formulated in terms of the joint vector Gaussian probability density function. As an example the Bayesian estimations of the coefficients of the multiple regression function are obtained. The proposed integrals can be used as table integrals in various fields of research.
- Published
- 2021
24. Third Hankel determinants for two classes of analytic functions with real coefficients
- Author
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Paweł Zaprawa and Young Jae Sim
- Subjects
010101 applied mathematics ,Algebra ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,Mathematics ,Analytic function - Abstract
In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H 2 , 2 {H_{2,2}} or H 3 , 1 {H_{3,1}} over different subclasses of 𝒮 {\mathcal{S}} . Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯 {\mathcal{T}} of typically real functions. The second object of our interest is 𝒦 ℝ ( i ) {\mathcal{K}_{\mathbb{R}}(i)} , the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.
- Published
- 2021
25. On curves with circles as their isoptics
- Author
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Waldemar Cieślak and Witold Mozgawa
- Subjects
Pure mathematics ,Class (set theory) ,Plane curve ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,Regular polygon ,02 engineering and technology ,Characterization (mathematics) ,Ellipse ,01 natural sciences ,Discrete Mathematics and Combinatorics ,0101 mathematics ,021101 geological & geomatics engineering ,Mathematics - Abstract
In the present paper we describe the family of all closed convex plane curves of class $$C^1$$ C 1 which have circles as their isoptics. In the first part of the paper we give a certain characterization of all ellipses based on the notion of isoptic and we give a geometric characterization of curves whose orthoptics are circles. The second part of the paper contains considerations on curves which have circles as their isoptics and we show the form of support functions of all considered curves.
- Published
- 2021
26. Limit theorems for linear random fields with tapered innovations. II: The stable case
- Author
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Vygantas Paulauskas and Julius Damarackas
- Subjects
Combinatorics ,010104 statistics & probability ,Number theory ,Random field ,General Mathematics ,010102 general mathematics ,Limit (mathematics) ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In the paper, we consider the limit behavior of partial-sum random field (r.f.) $$ \left.{S}_n\left({t}_1,{t}_2;\right)X\left(b\left(\mathbf{n}\right)\right)\right)={\sum}_{k=1}^{\left[{n}_1{t}_1\right]}{\sum}_{l=1}^{\left[{n}_2{t}_2\right]}{X}_{k,l}\left(b\left(\mathbf{n}\right)\right), $$ where $$ \left\{{X}_{k,l}\left(b\left(\mathbf{n}\right)\right)={\sum}_{i=0}^{\infty }{\sum}_{j=0}^{\infty }{c}_{i,j}{\upxi}_{k-i,l-j}\left(b\left(\mathbf{n}\right)\right),k,l\in \mathrm{\mathbb{Z}}\right\},n\ge 1, $$ is a family (indexed by n = (n1, n2), ni ≥ 1) of linear r.f.s with filter ci,j = aibj and innovations ξk,l(b(n)) having heavy-tailed tapered distributions with tapering parameter b(n) growing to infinity as n → ∞. In [V. Paulauskas, Limit theorems for linear random fields with tapered innovations. I: The Gaussian case, Lith. Math. J., 61(2):261–273, 2021], we considered the so-called hard tapering as b(n) grows relatively slowly and the limit r.f.s for appropriately normalized Sn(t1, t2;X(b(n))) are Gaussian. In this paper, we consider the case of soft tapering where b(n) grows more rapidly in comparison with the case of hard tapering and stable limit r.f.s.We consider cases where the sequences {ai} and {bj} are long-range, short-range, and negatively dependent.
- Published
- 2021
27. Counting tropical rational space curves with cross-ratio constraints
- Author
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Christoph Goldner
- Subjects
Pure mathematics ,Current (mathematics) ,Plane (geometry) ,General Mathematics ,010102 general mathematics ,Cross-ratio ,0102 computer and information sciences ,Algebraic geometry ,Space (mathematics) ,01 natural sciences ,Mathematics - Algebraic Geometry ,Number theory ,14N10, 14T05 ,010201 computation theory & mathematics ,FOS: Mathematics ,Tropical geometry ,Mathematics - Combinatorics ,Point (geometry) ,Combinatorics (math.CO) ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
This is a follow-up paper of arXiv:1805.00115, where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in arXiv:1509.07453 allowed us to use tropical geometry, and, in particular, a degeneration technique called floor diagrams. This correspondence theorem also holds in higher dimension. In the current paper, we introduce so-called cross-ratio floor diagrams and show that they allow us to determine the number of rational space curves that satisfy general positioned point and cross-ratio conditions. Moreover, graphical contributions are introduced which provide a novel and structured way of understanding multiplicities of floor decomposed curves in $\mathbb{R}^3$. Additionally, so-called condition flows on a tropical curve are used to reflect how conditions imposed on a tropical curve yield different types of edges. This concept is applicable in arbitrary dimension., 36 pages, 15 figures; fixed minor issues, added references
- Published
- 2021
28. Bivariate Sarmanov Phase-Type Distributions for Joint Lifetimes Modeling
- Author
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Hassan Abdelrahman and Khouzeima Moutanabbir
- Subjects
Statistics and Probability ,General Mathematics ,010102 general mathematics ,Phase (waves) ,Structure (category theory) ,Context (language use) ,Bivariate analysis ,Type (model theory) ,01 natural sciences ,010104 statistics & probability ,Distribution (mathematics) ,Range (statistics) ,Statistical physics ,0101 mathematics ,Marginal distribution ,Mathematics - Abstract
In this paper, we are interested in the dependence between lifetimes based on a joint survival model. This model is built using the bivariate Sarmanov distribution with Phase-Type marginal distributions. Capitalizing on these two classes of distributions’ mathematical properties, we drive some useful closed-form expressions of distributions and quantities of interest in the context of multiple-life insurance contracts. The dependence structure that we consider in this paper is based on a general form of kernel function for the Bivariate Sarmanov distribution. The introduction of this new kernel function allows us to improve the attainable correlation range.
- Published
- 2021
29. An improvement on Furstenberg’s intersection problem
- Author
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Han Yu
- Subjects
Combinatorics ,Intersection ,Applied Mathematics ,General Mathematics ,Bounded function ,010102 general mathematics ,Dimension (graph theory) ,Zero (complex analysis) ,0101 mathematics ,Invariant (mathematics) ,Dynamical system (definition) ,01 natural sciences ,Mathematics - Abstract
In this paper, we study a problem posed by Furstenberg on intersections between × 2 , × 3 \times 2, \times 3 invariant sets. We present here a direct geometrical counting argument to revisit a theorem of Wu and Shmerkin. This argument can be used to obtain further improvements. For example, we show that if A 2 , A 3 ⊂ [ 0 , 1 ] A_2,A_3\subset [0,1] are closed and × 2 , × 3 \times 2, \times 3 invariant respectively, assuming that dim A 2 + dim A 3 > 1 \dim A_2+\dim A_3>1 then A 2 ∩ ( u A 3 + v ) A_2\cap (uA_3+v) is sparse (defined in this paper) and has box dimension zero uniformly with respect to the real parameters u , v u,v such that u u and u − 1 u^{-1} are both bounded away from 0 0 .
- Published
- 2021
30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
- Author
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Zhiyue Zhang, Hüseyin Budak, Yu-Ming Chu, Necmettin Alp, Muhammad Ali, and [Belirlenecek]
- Subjects
Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,Type (model theory) ,Quantum calculus ,quantum calculus ,01 natural sciences ,Midpoint ,26d15 ,Hermite–Hadamard inequality ,QA1-939 ,26a51 ,Differentiable function ,0101 mathematics ,Quantum ,26d10 ,Mathematics ,media_common ,convex function ,hermite-hadamard inequality ,010102 general mathematics ,010101 applied mathematics ,Computer Science::Graphics ,q-integral ,Convex function - Abstract
The present paper aims to find some new midpoint-type inequalities for twice quantum differentiable convex functions. The consequences derived in this paper are unification and generalization of the comparable consequences in the literature on midpoint inequalities. © 2021 Muhammad Aamir Ali et al., published by De Gruyter. National Natural Science Foundation of China, NSFC: 11301127, 11601485, 11626101, 11701176, 11971241, 61673169 Funding information : The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485, 11971241). 2-s2.0-85105011594
- Published
- 2021
31. Existence on solutions of a class of casual differential equations on a time scale
- Author
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Yige Zhao
- Subjects
010101 applied mathematics ,Class (set theory) ,Scale (ratio) ,Casual ,Differential equation ,General Mathematics ,010102 general mathematics ,Applied mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we develop the theory of a class of casual differential equations on a time scale. An existence theorem for casual differential equations on a time scale is given under mixed Lipschitz and compactness conditions by the fixed point theorem in Banach algebra due to Dhage. Some fundamental differential inequalities on a time scale are also presented which are utilized to investigate the existence of extremal solutions. The comparison principle on casual differential equations on a time scale is established. Our results in this paper extend and improve some well-known results.
- Published
- 2021
32. Degrees of Enumerations of Countable Wehner-Like Families
- Author
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I. Sh. Kalimullin and M. Kh. Faizrahmanov
- Subjects
Statistics and Probability ,Class (set theory) ,Degree (graph theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Spectrum (topology) ,010305 fluids & plasmas ,Combinatorics ,0103 physical sciences ,Enumeration ,Countable set ,Family of sets ,0101 mathematics ,Turing ,computer ,Finite set ,computer.programming_language ,Mathematics - Abstract
This paper is a survey of results on countable families with natural degree spectra. These results were obtained by a modification of the methodology proposed by Wechner, who first found a family of sets with the spectrum consisting precisely of nonzero Turing degrees. Based on this method, many researchers obtained examples of families with other natural spectra. In addition, in this paper we extend these results and present new examples of natural spectra. In particular, we construct a family of finite sets with the spectrum consisting of exactly non-K-trivial degrees and also we find new sufficient conditions on $$ {\Delta}_2^0 $$ -degree a, which guarantees that the class {x : x ≰ a} is the degree spectrum of some family. Finally, we give a survey of our recent results on the degree spectra of α-families, where α is an arbitrary computable ordinal.
- Published
- 2021
33. Noncommutative Counting Invariants and Curve Complexes
- Author
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Ludmil Katzarkov and George Dimitrov
- Subjects
Intersection theory ,medicine.medical_specialty ,Functor ,Conjecture ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Quiver ,Type (model theory) ,01 natural sciences ,Combinatorics ,0103 physical sciences ,medicine ,010307 mathematical physics ,0101 mathematics ,Partially ordered set ,Commutative property ,Mathematics - Abstract
In our previous paper, viewing $D^b(K(l))$ as a noncommutative curve, where $K(l)$ is the Kronecker quiver with $l$-arrows, we introduced categorical invariants via counting of noncommutative curves. Roughly, these invariants are sets of subcategories in a given category and their quotients. The noncommutative curve-counting invariants are obtained by restricting the subcategories to be equivalent to $D^b(K(l))$. The general definition, however, defines a larger class of invariants and many of them behave properly with respect to fully faithful functors. Here, after recalling the definition, we focus on the examples and extend our studies beyond counting. We enrich our invariants with the following structures: the inclusion of subcategories makes them partially ordered sets and considering semi-orthogonal pairs of subcategories as edges amounts to directed graphs. It turns out that the problem for counting $D^b(A_k)$ in $D^b(A_n)$ has a geometric combinatorial parallel - counting of maps between polygons. Estimating the numbers counting noncommutative curves in $D^b({\mathbb P}^2)$ modulo the group of autoequivalences, we prove finiteness and that the exact determining of these numbers leads to a solution of Markov problem. Via homological mirror symmetry, this gives a new approach to this problem. Regarding the structure of a partially ordered set mentioned above, we initiate intersection theory of noncommutative curves focusing on the case of noncommutative genus zero. The above-mentioned structure of a directed graph (and related simplicial complex) is a categorical analogue of the classical curve complex, introduced by Harvey and Harrer. The paper contains pictures of the graphs in many examples and also presents an approach to Markov conjecture via counting of subgraphs in a graph associated with $D^b({{\mathbb{P}}}^2)$. Some of the results proved here were announced in a previous work.
- Published
- 2021
34. On the size of subsets of $$\mathbb{F}_p^n$$ without p distinct elements summing to zero
- Author
-
Lisa Sauermann
- Subjects
Mathematics - Number Theory ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Zero (complex analysis) ,Lattice (group) ,0102 computer and information sciences ,Infinity ,01 natural sciences ,Upper and lower bounds ,Prime (order theory) ,Combinatorics ,Integer ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Maximum size ,Combinatorics (math.CO) ,Number Theory (math.NT) ,0101 mathematics ,Constant (mathematics) ,media_common ,Mathematics - Abstract
Let us fix a prime $p$. The Erd\H{o}s-Ginzburg-Ziv problem asks for the minimum integer $s$ such that any collection of $s$ points in the lattice $\mathbb{Z}^n$ contains $p$ points whose centroid is also a lattice point in $\mathbb{Z}^n$. For large $n$, this is essentially equivalent to asking for the maximum size of a subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero. In this paper, we give a new upper bound for this problem for any fixed prime $p\geq 5$ and large $n$. In particular, we prove that any subset of $\mathbb{F}_p^n$ without $p$ distinct elements summing to zero has size at most $C_p\cdot \left(2\sqrt{p}\right)^n$, where $C_p$ is a constant only depending on $p$. For $p$ and $n$ going to infinity, our bound is of the form $p^{(1/2)\cdot (1+o(1))n}$, whereas all previously known upper bounds were of the form $p^{(1-o(1))n}$ (with $p^n$ being a trivial bound). Our proof uses the so-called multi-colored sum-free theorem which is a consequence of the Croot-Lev-Pach polynomial method. This method and its consequences were already applied by Naslund as well as by Fox and the author to prove bounds for the problem studied in this paper. However, using some key new ideas, we significantly improve their bounds., Comment: 11 pages
- Published
- 2021
35. A new obstruction for normal spanning trees
- Author
-
Max Pitz
- Subjects
Aleph ,Spanning tree ,General Mathematics ,010102 general mathematics ,Minor (linear algebra) ,Type (model theory) ,01 natural sciences ,Graph ,Combinatorics ,Mathematics::Logic ,Arbitrarily large ,Cardinality ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Connectivity ,05C83, 05C05, 05C63 ,Mathematics - Abstract
In a paper from 2001 (Journal of the LMS), Diestel and Leader offered a proof that a connected graph has a normal spanning tree if and only if it does not contain a minor from two specific forbidden classes of graphs, all of cardinality $\aleph_1$. Unfortunately, their proof contains a gap, and their result is incorrect. In this paper, we construct a third type of obstruction: an $\aleph_1$-sized graph without a normal spanning tree that contains neither of the two types described by Diestel and Leader as a minor. Further, we show that any list of forbidden minors characterising the graphs with normal spanning trees must contain graphs of arbitrarily large cardinality., Comment: 9 pages. arXiv admin note: text overlap with arXiv:2005.02833
- Published
- 2021
36. Systems of quasilinear parabolic equations in Rn and systems of quadratic backward stochastic differential equations
- Author
-
Sheung Chi Phillip Yam, Jens Frehse, and Alain Bensoussan
- Subjects
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
37. Wild Cantor actions
- Author
-
Ramón Barral Lijó, Hiraku Nozawa, Jesús A. Álvarez López, and Olga Lukina
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Closure (topology) ,Mathematics::General Topology ,Dynamical Systems (math.DS) ,16. Peace & justice ,Equicontinuity ,01 natural sciences ,Centralizer and normalizer ,Cantor set ,Group action ,Wreath product ,0103 physical sciences ,FOS: Mathematics ,Countable set ,2020: 37B05, 37E25, 20E08, 20E15, 20E18, 20E22, 22F05, 22F50 (Primary), 20F22, 57R30, 57R50 (Secondary) ,010307 mathematical physics ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics - Abstract
The discriminant group of a minimal equicontinuous action of a group $G$ on a Cantor set $X$ is the subgroup of the closure of the action in the group of homeomorphisms of $X$, consisting of homeomorphisms which fix a given point. The stabilizer and the centralizer groups associated to the action are obtained as direct limits of sequences of subgroups of the discriminant group with certain properties. Minimal equicontinuous group actions on Cantor sets admit a classification by the properties of the stabilizer and centralizer direct limit groups. In this paper, we construct new families of examples of minimal equicontinuous actions on Cantor sets, which illustrate certain aspects of this classification. These examples are constructed as actions on rooted trees. The acting groups are countable subgroups of the product or of the wreath product of groups. We discuss applications of our results to the study of attractors of dynamical systems and of minimal sets of foliations., 20 pages, 1 figure. The condition of finite generation in Thm 1.9 was replaced by countability. The proof of Thm 1.9 has been simplified. The notation used in 5 has been modified. Several minor corrections across the paper
- Published
- 2022
38. The moduli space of matroids
- Author
-
Oliver Lorscheid, Matthew Baker, and Dynamical Systems, Geometry & Mathematical Physics
- Subjects
Mathematics::Combinatorics ,Functor ,F-geometry ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,Tracts ,01 natural sciences ,Matroid ,Moduli space ,Combinatorics ,Matroids ,Mathematics - Algebraic Geometry ,Morphism ,010201 computation theory & mathematics ,Scheme (mathematics) ,Blueprints ,FOS: Mathematics ,Isomorphism ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics ,Initial and terminal objects - Abstract
In the first part of the paper, we clarify the connections between several algebraic objects appearing in matroid theory: both partial fields and hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are compatible with the respective matroid theories. Moreover, fuzzy rings are ordered blueprints and lie in the intersection of tracts with ordered blueprints; we call the objects of this intersection pastures. In the second part, we construct moduli spaces for matroids over pastures. We show that, for any non-empty finite set $E$, the functor taking a pasture $F$ to the set of isomorphism classes of rank-$r$ $F$-matroids on $E$ is representable by an ordered blue scheme $Mat(r,E)$, the moduli space of rank-$r$ matroids on $E$. In the third part, we draw conclusions on matroid theory. A classical rank-$r$ matroid $M$ on $E$ corresponds to a $\mathbb{K}$-valued point of $Mat(r,E)$ where $\mathbb{K}$ is the Krasner hyperfield. Such a point defines a residue pasture $k_M$, which we call the universal pasture of $M$. We show that for every pasture $F$, morphisms $k_M\to F$ are canonically in bijection with $F$-matroid structures on $M$. An analogous weak universal pasture $k_M^w$ classifies weak $F$-matroid structures on $M$. The unit group of $k_M^w$ can be canonically identified with the Tutte group of $M$. We call the sub-pasture $k_M^f$ of $k_M^w$ generated by ``cross-ratios' the foundation of $M$,. It parametrizes rescaling classes of weak $F$-matroid structures on $M$, and its unit group is coincides with the inner Tutte group of $M$. We show that a matroid $M$ is regular if and only if its foundation is the regular partial field, and a non-regular matroid $M$ is binary if and only if its foundation is the field with two elements. This yields a new proof of the fact that a matroid is regular if and only if it is both binary and orientable., 85 pages; some additional material, e.g. a new section 5.6; the terminology has been adapted to the usage in follow-up papers
- Published
- 2021
39. A free boundary problem arising from branching Brownian motion with selection
- Author
-
Sarah Penington, James Nolen, Éric Brunet, and Julien Berestycki
- Subjects
Mathematics(all) ,Interacting particle system ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Probability (math.PR) ,35R35, 35K55, 82C22 ,Boundary (topology) ,01 natural sciences ,Parabolic partial differential equation ,Constraint (information theory) ,Mathematics - Analysis of PDEs ,Free boundary problem ,FOS: Mathematics ,Uniqueness ,Limit (mathematics) ,0101 mathematics ,Mathematics - Probability ,Brownian motion ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
We study a free boundary problem for a parabolic partial differential equation in which the solution is coupled to the moving boundary through an integral constraint. The problem arises as the hydrodynamic limit of an interacting particle system involving branching Brownian motion with selection, the so-called Brownian bees model which is studied in a companion paper. In this paper we prove existence and uniqueness of the solution to the free boundary problem, and we characterise the behaviour of the solution in the large time limit., Comment: 53 pages
- Published
- 2021
40. McKay Quivers and Lusztig Algebras of Some Finite Groups
- Author
-
Ragnar-Olaf Buchweitz, Matthew Lewis, Colin Ingalls, and Eleonore Faber
- Subjects
General Mathematics ,Field (mathematics) ,Group Theory (math.GR) ,01 natural sciences ,Combinatorics ,Elementary algebra ,Symmetric group ,FOS: Mathematics ,Mathematics - Combinatorics ,0101 mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics ,Finite group ,05E10 16G20 16S35 16S37 20F55 20C30 ,010102 general mathematics ,Quiver ,Mathematics - Rings and Algebras ,010101 applied mathematics ,Clifford theory ,Rings and Algebras (math.RA) ,Combinatorics (math.CO) ,Mathematics - Group Theory ,Mathematics - Representation Theory ,Vector space ,Group ring - Abstract
We are interested in the McKay quiver $\Gamma(G)$ and skew group rings $A*G$, where $G$ is a finite subgroup of $\mathrm{GL}(V)$, where $V$ is a finite dimensional vector space over a field $K$, and $A$ is a $K-G$-algebra. These skew group rings appear in Auslander's version of the McKay correspondence. In the first part of this paper we consider complex reflection groups $G \subseteq \mathrm{GL}(V)$ and find a combinatorial method, making use of Young diagrams, to construct the McKay quivers for the groups $G(r,p,n)$. We first look at the case $G(1,1,n)$, which is isomorphic to the symmetric group $S_n$, followed by $G(r,1,n)$ for $r >1$. Then, using Clifford theory, we can determine the McKay quiver for any $G(r,p,n)$ and thus for all finite irreducible complex reflection groups up to finitely many exceptions. In the second part of the paper we consider a more conceptual approach to McKay quivers of arbitrary finite groups: we define the Lusztig algebra $\widetilde A(G)$ of a finite group $G \subseteq \mathrm{GL}(V)$, which is Morita equivalent to the skew group ring $A*G$. This description gives us an embedding of the basic algebra Morita equivalent to $A*G$ into a matrix algebra over $A$., Comment: v2: minor revision, final version to appear in Algebr. Represent. Theory
- Published
- 2021
41. Masur–Veech volumes, frequencies of simple closed geodesics, and intersection numbers of moduli spaces of curves
- Author
-
Vincent Delecroix, Elise Goujard, Peter Zograf, Anton Zorich, Groupe Sociétés, Religions, Laïcités (GSRL), Centre National de la Recherche Scientifique (CNRS)-École pratique des hautes études (EPHE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), École pratique des hautes études (EPHE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), École Pratique des Hautes Études (EPHE), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and ANR-19-CE40-0021,Phymath,physique mathématique(2019)
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Teichmüller space ,Surface (mathematics) ,Pure mathematics ,Geodesic ,General Mathematics ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,Dynamical Systems (math.DS) ,Algebraic geometry ,01 natural sciences ,Mathematics - Geometric Topology ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Combinatorics ,Mathematics - Dynamical Systems ,0101 mathematics ,Algebraic Geometry (math.AG) ,Quadratic differential ,Mathematics ,Meromorphic function ,010102 general mathematics ,Geometric Topology (math.GT) ,Mathematics::Geometric Topology ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Mapping class group ,Moduli space ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,Combinatorics (math.CO) ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,010307 mathematical physics - Abstract
We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space $\mathcal{Q}_{g,n}$ of genus $g$ meromorphic quadratic differentials with $n$ simple poles as polynomials in the intersection numbers of $\psi$-classes with explicit rational coefficients. The formulae obtained in this article result from lattice point counts involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli spaces of bordered hyperbolic surfaces with geodesic boundaries. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani via completely different approach. Furthermore, we prove that the density of the mapping class group orbit of any simple closed multicurve $\gamma$ inside the ambient set of integral measured laminations computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to $\gamma$ among all square-tiled surfaces in $\mathcal{Q}_{g,n}$. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case $n=0$. In particular, we compute the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus $g$ for small $g$ and we show that for large genera the separating closed geodesics are $\sqrt{\frac{2}{3\pi g}}\cdot\frac{1}{4^g}$ times less frequent., Comment: The current paper (as well as the companion paper arXiv:2007.04740) has grown from arxiv:1908.08611. The conjectures stated in arXiv:1908.08611 are proved by A. Aggarwal in arXiv:2004.05042
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- 2021
42. Stability Analysis and Existence of Solutions for a Modified SIRD Model of COVID-19 with Fractional Derivatives
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Farid Nouioua, Nacereddine Hammami, Bilal Basti, Noureddine Benhamidouche, Rabah Djemiat, and Imadeddine Berrabah
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Physics and Astronomy (miscellaneous) ,Coronavirus disease 2019 (COVID-19) ,General Mathematics ,Population ,Fixed-point theorem ,0102 computer and information sciences ,Stability result ,system ,01 natural sciences ,Stability (probability) ,Hadamard transform ,QA1-939 ,Computer Science (miscellaneous) ,Applied mathematics ,Quantitative Biology::Populations and Evolution ,Uniqueness ,0101 mathematics ,education ,Mathematics ,education.field_of_study ,pandemic ,010102 general mathematics ,existence ,COVID-19 ,fractional derivative ,uniqueness ,Fractional calculus ,010201 computation theory & mathematics ,Chemistry (miscellaneous) ,SIRD model - Abstract
This paper discusses and provides some analytical studies for a modified fractional-order SIRD mathematical model of the COVID-19 epidemic in the sense of the Caputo–Katugampola fractional derivative that allows treating of the biological models of infectious diseases and unifies the Hadamard and Caputo fractional derivatives into a single form. By considering the vaccine parameter of the suspected population, we compute and derive several stability results based on some symmetrical parameters that satisfy some conditions that prevent the pandemic. The paper also investigates the problem of the existence and uniqueness of solutions for the modified SIRD model. It does so by applying the properties of Schauder’s and Banach’s fixed point theorems.
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- 2021
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43. The universality of Hughes-free division rings
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Andrei Jaikin-Zapirain and UAM. Departamento de Matemáticas
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Group (mathematics) ,Matemáticas ,General Mathematics ,Existential quantification ,010102 general mathematics ,Universality (philosophy) ,General Physics and Astronomy ,Universal division ring of fractions ,Division (mathematics) ,01 natural sciences ,Combinatorics ,Crossed product ,0103 physical sciences ,Hughes-free division ring ,Division ring ,010307 mathematical physics ,0101 mathematics ,Locally indicable groups ,Mathematics - Abstract
Let E∗ G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to E∗ G-isomorphism, there exists at most one Hughes-free division E∗G-ring. However, the existence of a Hughes-free division E∗ G-ring DE∗G for an arbitrary locally indicable group G is still an open question. Nevertheless, DE∗G exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether DE∗G is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists DE[G] and it is universal. In Appendix we give a description of DE[G] when G is a RFRS group, This paper is partially supported by the Spanish Ministry of Science and Innovation through the grant MTM2017-82690-P and the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S4). I would like to thank Dawid Kielak and an anonymous referee for useful suggestions and comments
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- 2021
44. Quadruple Roman Domination in Trees
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Saeed Kosari, Jafar Amjadi, Nesa Khalili, Zheng Kou, and Guoliang Hao
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Vertex (graph theory) ,Physics and Astronomy (miscellaneous) ,Domination analysis ,General Mathematics ,Roman domination ,MathematicsofComputing_GENERAL ,Value (computer science) ,Minimum weight ,quadruple Roman domination ,0102 computer and information sciences ,01 natural sciences ,Upper and lower bounds ,Combinatorics ,Integer ,Computer Science (miscellaneous) ,QA1-939 ,0101 mathematics ,Mathematics ,010102 general mathematics ,Function (mathematics) ,trees ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,010201 computation theory & mathematics ,Chemistry (miscellaneous) ,Symmetry (geometry) - Abstract
This paper is devoted to the study of the quadruple Roman domination in trees, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. For any positive integer k, a [k]-Roman dominating function ([k]-RDF) of a simple graph G is a function from the vertex set V of G to the set {0,1,2,…,k+1} if for any vertex u∈V with f(u)<, k, ∑x∈N(u)∪{u}f(x)≥|{x∈N(u):f(x)≥1}|+k, where N(u) is the open neighborhood of u. The weight of a [k]-RDF is the value Σv∈Vf(v). The minimum weight of a [k]-RDF is called the [k]-Roman domination number γ[kR](G) of G. In this paper, we establish sharp upper and lower bounds on γ[4R](T) for nontrivial trees T and characterize extremal trees.
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- 2021
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45. Approximation of Endpoints for α—Reich–Suzuki Nonexpansive Mappings in Hyperbolic Metric Spaces
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Afrah An Abdou, Izhar Uddin, and Sajan Aggarwal
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Pure mathematics ,General Mathematics ,010102 general mathematics ,endpoint ,MathematicsofComputing_GENERAL ,Fixed-point theorem ,Fixed point ,01 natural sciences ,fixed point theorems ,010101 applied mathematics ,Metric space ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,α—Riech–Suzuki nonexpansive mapping ,Convergence (routing) ,Computer Science (miscellaneous) ,QA1-939 ,Computer Science::Programming Languages ,hyperbolic metric space ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics - Abstract
The concept of an endpoint is a relatively new concept compared to the concept of a fixed point. The aim of this paper is to perform a convergence analysis of M—iteration involving α—Reich–Suzuki nonexpansive mappings. In this paper, we prove strong and Δ—convergence theorems in a hyperbolic metric space. Thus, our results generalize and improve many existing results.
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- 2021
46. Existence and U-H-R Stability of Solutions to the Implicit Nonlinear FBVP in the Variable Order Settings
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Mohammed Said Souid, Mohammed K. A. Kaabar, Zailan Siri, Shahram Rezapour, Francisco Martínez, Sina Etemad, and Ahmed Refice
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Ulam–Hyers–Rassias stability ,Mathematics::Functional Analysis ,General Mathematics ,010102 general mathematics ,Fixed-point theorem ,variable-order operators ,implicit problem ,01 natural sciences ,Stability (probability) ,fixed point theorems ,010101 applied mathematics ,Nonlinear fractional differential equations ,piecewise constant functions ,Nonlinear system ,Computer Science (miscellaneous) ,QA1-939 ,Applied mathematics ,Order (group theory) ,Boundary value problem ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics ,Variable (mathematics) - Abstract
In this paper, the existence of the solution and its stability to the fractional boundary value problem (FBVP) were investigated for an implicit nonlinear fractional differential equation (VOFDE) of variable order. All existence criteria of the solutions in our establishments were derived via Krasnoselskii’s fixed point theorem and in the sequel, and its Ulam–Hyers–Rassias (U-H-R) stability is checked. An illustrative example is presented at the end of this paper to validate our findings.
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- 2021
47. Viscosity approximation method for solving variational inequality problem in real Banach spaces
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Godwin Chidi Ugwunnadi
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Sequence ,021103 operations research ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,Banach space ,Lipschitzian Mapping ,02 engineering and technology ,Nonexpansive Mapping ,Fixed point ,01 natural sciences ,Viscosity (programming) ,Fixed Point ,Variational inequality ,Strongly Accretive Mapping ,QA1-939 ,Applied mathematics ,Hierarchical Fixed Point Problems ,0101 mathematics ,Mathematics - Abstract
In this paper, we study the implicit and inertial-type viscosity approximation method for approximating a solution to the hierarchical variational inequality problem. Under some mild conditions on the parameters, we prove that the sequence generated by the proposed methods converges strongly to a solution of the above-mentioned problem in $q$-uniformly smooth Banach spaces. The results obtained in this paper generalize and improve many recent results in this direction.
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- 2021
48. Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials
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Vidya Venkateswaran, Jasper V. Stokman, Siddhartha Sahi, Algebra, Geometry & Mathematical Physics (KDV, FNWI), Quantum Matter and Quantum Information, KdV Other Research (FNWI), Faculty of Science, and KDV (FNWI)
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Weyl group ,Polynomial ,Pure mathematics ,Algebraic combinatorics ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,20C08 (Primary), 11F68, 22E50 (Secondary) ,Rational function ,01 natural sciences ,symbols.namesake ,Macdonald polynomials ,Gauss sum ,0103 physical sciences ,symbols ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Dirichlet series ,Mathematics - Representation Theory ,Mathematics - Abstract
Chinta and Gunnells introduced a rather intricate multi-parameter Weyl group action on rational functions on a torus, which, when the parameters are specialized to certain Gauss sums, describes the functional equations of Weyl group multiple Dirichlet series associated to metaplectic (n-fold) covers of algebraic groups. In subsequent joint work with Puskas, they extended this action to a "metaplectic" representation of the equal parameter affine Hecke algebra, which allowed them to obtain explicit formulas for the p-parts of these Dirichlet series. They have also verified by a computer check the remarkable fact that their formulas continue to define a group action for general (unspecialized) parameters. In the first part of paper we give a conceptual explanation of this fact, by giving a uniform and elementary construction of the "metaplectic" representation for generic Hecke algebras as a suitable quotient of a parabolically induced affine Hecke algebra module, from which the associated Chinta-Gunnells Weyl group action follows through localization. In the second part of the paper we extend the metaplectic representation to the double affine Hecke algebra, which provides a generalization of Cherednik's basic representation. This allows us to introduce a new family of "metaplectic" polynomials, which generalize nonsymmetric Macdonald polynomials. In this paper, we provide the details of the construction of metaplectic polynomials in type A; the general case will be handled in the sequel to this paper., 39 pages. Version 2 is a significant revision. Added second part introducing a new family of "metaplectic" polynomials, which generalize nonsymmetric Macdonald polynomials and metaplectic Iwahori-Whittaker functions. Title has been changed and the introduction has been expanded
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- 2021
49. A New Class of Plane Curves with Arc Length Parametrization and Its Application to Linear Analysis of Curved Beams
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Snježana Maksimović and Aleksandar Borković
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Basis (linear algebra) ,Plane curve ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Static analysis ,Space (mathematics) ,01 natural sciences ,Computer Science::Digital Libraries ,010101 applied mathematics ,analytical solution ,Bernoulli–Euler beam ,special functions ,Special functions ,Computer Science (miscellaneous) ,QA1-939 ,arc-length parametrization ,Development (differential geometry) ,0101 mathematics ,Sturm–Liouville differential equation ,Engineering (miscellaneous) ,Arc length ,Parametrization ,Mathematics - Abstract
The objective of this paper is to define one class of plane curves with arc-length parametrization. To accomplish this, we constructed a novel class of special polynomials and special functions. These functions form a basis of L2(R) space and some of their interesting properties are discussed. The developed curves are used for the linear static analysis of curved Bernoulli–Euler beam. Due to the parametrization with arc length, the exact analytical solution can be obtained. These closed-form solutions serve as the benchmark results for the development of numerical procedures. One such example is provided in this paper.
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- 2021
50. Hermite–Hadamard Inclusions for Co-Ordinated Interval-Valued Functions via Post-Quantum Calculus
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Jessada Tariboon, Sotiris K. Ntouyas, Hüseyin Budak, Muhammad Ali, and [Belirlenecek]
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Pure mathematics ,Physics and Astronomy (miscellaneous) ,General Mathematics ,Quantum calculus ,co-ordinated convexity ,quantum calculus ,01 natural sciences ,Interval valued ,Hadamard transform ,(p ,Hermite–Hadamard inequality ,Hermite–Hadamard inclusion ,Computer Science (miscellaneous) ,QA1-939 ,0101 mathematics ,interval-valued functions ,Mathematics ,Hermite polynomials ,010102 general mathematics ,Regular polygon ,(p, q)-integral ,Convex ,010101 applied mathematics ,Hermite-Hadamard inequality ,Chemistry (miscellaneous) ,Hermite-Hadamard inclusion ,q)-integral ,Midpoint Type Inequalities ,Symmetry (geometry) - Abstract
In this paper, the notions of post-quantum integrals for two-variable interval-valued functions are presented. The newly described integrals are then used to prove some new Hermite-Hadamard inclusions for co-ordinated convex interval-valued functions. Many of the findings in this paper are important extensions of previous findings in the literature. Finally, we present a few examples of our new findings. Analytic inequalities of this nature and especially the techniques involved have applications in various areas in which symmetry plays a prominent role. WOS:000677046700001 2-s2.0-85110868353
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- 2021
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