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2. Corrections to the paper 'The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces'
- Author
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Rovshan A. Bandaliev
- Subjects
Discrete mathematics ,Mathematics::Functional Analysis ,Sublinear function ,General Mathematics ,Mathematics::Classical Analysis and ODEs ,Lebesgue's number lemma ,Type (model theory) ,symbols.namesake ,Operator (computer programming) ,Mathematics - Classical Analysis and ODEs ,Ordinary differential equation ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,Standard probability space ,Lp space ,Variable (mathematics) ,Mathematics - Abstract
In this paper author was proved the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent. As an application we prove the boundedness of certain sublinear operators on the weighted variable Lebesgue space. Note that the proof of multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent don't contained any mistakes. But at the proving of the boundedness of certain sublinear operators on the weighted variable Lebesgue space Georgian colleagues discovered a small but significant error in this paper. This result is assigned as Theorem 5 in noted paper. In other words, sufficient conditions for general weights ensuring the validity of the two-weight strong type inequalities for some sublinear operator was found. In this theorem the inequality (9) isn't true. In this note we give the details of the correct argument. We presume that the reader is familiar with the contents and notation of our original paper. At the heart of our correction is the following Theorem which replaces Theorem 5.
- Published
- 2013
3. Remarks on DiPerna’s paper 'Convergence of the viscosity method for isentropic gas dynamics'
- Author
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Gui-Qiang Chen
- Subjects
Discrete mathematics ,Isentropic process ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Vacuum state ,Finite difference method ,Euler equations ,Binary entropy function ,symbols.namesake ,Riemann hypothesis ,Compact space ,Mathematics Subject Classification ,symbols ,Mathematics - Abstract
Concerns have been voiced about the correctness of certain technical points in DiPerna’s paper (Comm. Math. Phys. 91 (1983), 1–30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods. In [Di], DiPerna found a global entropy solution of the isentropic Euler equations for the following exponents in the equation of state for the pressure: γ = 1 + 2/(2m+ 1), m ≥ 2 integer. (1) He divided his arguments into the following two steps. 1. Compactness framework Assume that a sequence of approximate solutions (ρ (x, t),m (x, t)), 0 ≤ t ≤ T , satisfies: (i). There exists a constant C(T ) > 0, independent of > 0, such that 0 ≤ ρ (x, t) ≤ C, |m (x, t)/ρ (x, t)| ≤ C; (ii). For all weak entropy pairs (η, q) of the isentropic Euler equations, the measure sequence η(ρ ,m )t + q(ρ ,m )x is contained in a compact subset of H −1 loc (R× [0, T ]). If γ satisfies (1), then the sequence (ρ (x, t),m (x, t)) is compact in Lloc(R× [0, T ]). The reason for the restriction on the number γ is that, in such a case, any weak entropy function is a polynomial function of the Riemann invariants (w, z). This is the key step in DiPerna’s arguments and is also his main contribution to the compensated compactness method in this aspect. Received by the editors May 16, 1996. 1991 Mathematics Subject Classification. Primary 35K55, 35L65; Secondary 76N15, 35L60, 65M06.
- Published
- 1997
4. Rebuttal of Donnelly's paper on the spectral gap
- Author
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Antoine Henrot, Mark S. Ashbaugh, Richard S. Laugesen, Department of Mathematics, University of Missouri Columbia, University of Missouri [Columbia] (Mizzou), University of Missouri System-University of Missouri System, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Department of Mathematics [Urbana], University of Illinois at Urbana-Champaign [Urbana], University of Illinois System-University of Illinois System, CORIDA, and Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est
- Subjects
Discrete mathematics ,Sequence ,Conjecture ,General Mathematics ,010102 general mathematics ,Mathematics::History and Overview ,Mathematics::Spectral Theory ,01 natural sciences ,Domain (mathematical analysis) ,Computer Science::Computers and Society ,010101 applied mathematics ,symbols.namesake ,Physics::Popular Physics ,Dirichlet boundary condition ,Euclidean geometry ,symbols ,Calculus ,Convex body ,Quantitative Biology::Populations and Evolution ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Spectral gap ,0101 mathematics ,Mathematics ,Unit interval - Abstract
The spectral gap conjecture of M. van den Berg [2, formula (65)] asserts that λ2 − λ1 ≥ 3π for all convex euclidean domains of diameter 1, where λ1 and λ2 denote the first two eigenvalues of the Dirichlet Laplacian. Notice that equality holds for the 1-dimensional unit interval, which can be regarded also as a degenerate n-dimensional rectangular box. The gap estimate is conjectured to hold more generally for Schrodinger operators with convex potentials, under Dirichlet boundary conditions; see the work of S.-T. Yau and collaborators [9, 11]. This Schrodinger gap conjecture was proved some time ago in 1 dimension by R. Lavine [8], and more recently in all dimensions by B. Andrews and J. Clutterbuck [1]. The proof in this journal by H. Donnelly [3] of the original gap conjecture in 2 dimensions (for the Dirichlet Laplacian with zero potential) is not correct. The Editors of Mathematische Zeitschrift have asked us to describe the flaws in the proof, in order to clarify the state of the literature. Donnelly’s approach to the problem is a natural one: first perform a shape optimization to rule out a non-degenerate minimizing domain, and then analyze the spectral gap for a sequence of domains degenerating to an interval, with the help of results by D. Jerison [5]. (For some history on this approach, and on the gap conjecture more generally, see the report on the AIM meeting “Low Eigenvalues of Laplace and Schrodinger Operators” [10], especially page 12 of the open problems list.) The error lies in the proof of the shape optimization step, as we now explain. Donnelly wishes to prove that no minimizing domain can exist for
- Published
- 2011
5. Remark on Belyĭ's paper concerning Galois extensions of the maximal cyclotomic field with certain linear groups as Galois groups
- Author
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Hisashi Kojima
- Subjects
Discrete mathematics ,Pure mathematics ,12F10 ,Galois cohomology ,12F12 ,General Mathematics ,Fundamental theorem of Galois theory ,Galois group ,Abelian extension ,Galois module ,11R32 ,Differential Galois theory ,Embedding problem ,symbols.namesake ,symbols ,Galois extension ,Mathematics - Published
- 1991
6. Markov processes and related problems of analysis (selected papers)
- Author
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Gian-Carlo Rota
- Subjects
Discrete mathematics ,symbols.namesake ,Mathematics(all) ,General Mathematics ,symbols ,Markov process ,Mathematics - Published
- 1985
- Full Text
- View/download PDF
7. Lebesgue integration and measure, by Alan J. Weir. Pp xii, 281. £5·40 cloth, £23·40 paper. 1973 (Cambridge University Press)
- Author
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S. M. Edmonds
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Discrete mathematics ,symbols.namesake ,General Mathematics ,Weir ,symbols ,Lebesgue integration ,Measure (mathematics) ,Mathematics - Published
- 1977
8. A formula for generating weakly modular forms with weight 12
- Author
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Aykut Ahmet Aygunes
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Discrete mathematics ,symbols.namesake ,Pure mathematics ,Special solution ,General Mathematics ,Short paper ,Modular form ,Eisenstein series ,symbols ,Derivative ,Function (mathematics) ,Mathematics ,Möbius transformation - Abstract
In this short paper, generally, we define a family of functions fk depends on the Eisenstein series with weight 2k, for k ( N. More detail, by considering the function fk, we define a derivative formula for generating weakly modular forms with weight 12. As a result for this, we claim that this formula gives an advantage to find the special solutions of some differential equations.
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- 2016
9. Boolean algebra and its uses, by G. F. South. Pp vii, 102. £3·50 cloth, £1·50 paper. 1974. SBN 0 442 30006 9/30007 7 (Van Nostrana Reinhold)
- Author
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B. Collier
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Discrete mathematics ,symbols.namesake ,General Mathematics ,symbols ,Mathematics ,Boolean algebra - Published
- 1975
10. Integers represented as the sum of one prime, two squares of primes and powers of 2
- Author
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Haiwei Sun and Guangshi Lü
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Discrete mathematics ,Applied Mathematics ,General Mathematics ,Short paper ,MathematicsofComputing_GENERAL ,Prime number ,Prime (order theory) ,Algebra ,symbols.namesake ,Integer ,symbols ,Idoneal number ,Prime power ,Sphenic number ,Mathematics - Abstract
In this short paper we prove that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and 83 83 powers of 2 2 .
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- 2008
11. Notes to the Feit-Thompson conjecture
- Author
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Kaoru Motose
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Discrete mathematics ,Conjecture ,General Mathematics ,Legendre symbol ,20D05 ,11A07 ,cyclotomic polynomials ,Power residue symbol ,Combinatorics ,Feit–Thompson conjecture ,symbols.namesake ,symbols ,Odd paper ,Cyclotomic polynomial ,Mathematics - Abstract
We shall present partial solutions to the conjecture such that $(q^{p}-1)/(q-1)$ does not divide $(p^{q}-1)/(p-1)$ for distinct primes $p < q$.
- Published
- 2009
12. Nonlocal Games with Noisy Maximally Entangled States are Decidable
- Author
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Minglong Qin and Penghui Yao
- Subjects
Discrete mathematics ,Quantum Physics ,Computer Science::Computer Science and Game Theory ,General Computer Science ,General Mathematics ,ComputingMilieux_PERSONALCOMPUTING ,FOS: Physical sciences ,TheoryofComputation_GENERAL ,State (functional analysis) ,Special class ,Decidability ,symbols.namesake ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Fourier analysis ,Bipartite graph ,symbols ,High Energy Physics::Experiment ,Quantum Physics (quant-ph) ,Mathematics - Abstract
This paper considers a special class of nonlocal games $(G,\psi)$, where $G$ is a two-player one-round game, and $\psi$ is a bipartite state independent of $G$. In the game $(G,\psi)$, the players are allowed to share arbitrarily many copies of $\psi$. The value of the game $(G,\psi)$, denoted by $\omega^*(G,\psi)$, is the supremum of the winning probability that the players can achieve with arbitrarily many copies of preshared states $\psi$. For a noisy maximally entangled state $\psi$, a two-player one-round game $G$ and an arbitrarily small precision $\epsilon>0$, this paper proves an upper bound on the number of copies of $\psi$ for the players to win the game with a probability $\epsilon$ close to $\omega^*(G,\psi)$. Hence, it is feasible to approximately compute $\omega^*(G,\psi)$ to an arbitrarily precision. Recently, a breakthrough result by Ji, Natarajan, Vidick, Wright and Yuen showed that it is undecidable to approximate the values of nonlocal games to a constant precision when the players preshare arbitrarily many copies of perfect maximally entangled states, which implies that $\mathrm{MIP}^*=\mathrm{RE}$. In contrast, our result implies the hardness of approximating nonlocal games collapses when the preshared maximally entangled states are noisy. The paper develops a theory of Fourier analysis on matrix spaces by extending a number of techniques in Boolean analysis and Hermitian analysis to matrix spaces. We establish a series of new techniques, such as a quantum invariance principle and a hypercontractive inequality for random operators, which we believe have further applications., Comment: Supercedes arXiv:1904.08832, accepted by SIAM Journal of Computing
- Published
- 2021
13. An effective Chebotarev density theorem for families of number fields, with an application to $$\ell $$-torsion in class groups
- Author
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Lillian B. Pierce, Caroline L. Turnage-Butterbaugh, and Melanie Matchett Wood
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Discrete mathematics ,Mathematics - Number Theory ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Algebraic number field ,01 natural sciences ,Riemann hypothesis ,symbols.namesake ,Arbitrarily large ,Number theory ,Discriminant ,Field extension ,0103 physical sciences ,FOS: Mathematics ,symbols ,Torsion (algebra) ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Dedekind zeta function ,Mathematics - Abstract
We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal $L$-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal $L$-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of $L$-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree., Comment: 52 pages. This shorter version aligns with the published paper. Note that portions of Section 8 of the longer v1 have been developed as a separate paper with identifier arXiv:1902.02008
- Published
- 2019
14. On symmetric linear diffusions
- Author
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Liping Li and Jiangang Ying
- Subjects
Discrete mathematics ,Representation theorem ,Dirichlet form ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Disjoint sets ,01 natural sciences ,Dirichlet distribution ,010104 statistics & probability ,symbols.namesake ,Closure (mathematics) ,symbols ,Interval (graph theory) ,Countable set ,0101 mathematics ,Mathematics - Abstract
The main purpose of this paper is to explore the structure of local and regular Dirichlet forms associated with symmetric one-dimensional diffusions, which are also called symmetric linear diffusions. Let ( E , F ) (\mathcal {E},\mathcal {F}) be a regular and local Dirichlet form on L 2 ( I , m ) L^2(I,m) , where I I is an interval and m m is a fully supported Radon measure on I I . We shall first present a complete representation for ( E , F ) (\mathcal {E},\mathcal {F}) , which shows that ( E , F ) (\mathcal {E},\mathcal {F}) lives on at most countable disjoint “effective" intervals with an “adapted" scale function on each interval, and any point outside these intervals is a trap of the one-dimensional diffusion. Furthermore, we shall give a necessary and sufficient condition for C c ∞ ( I ) C_c^\infty (I) being a special standard core of ( E , F ) (\mathcal {E},\mathcal {F}) and shall identify the closure of C c ∞ ( I ) C_c^\infty (I) in ( E , F ) (\mathcal {E},\mathcal {F}) when C c ∞ ( I ) C_c^\infty (I) is contained but not necessarily dense in F \mathcal {F} relative to the E 1 1 / 2 \mathcal {E}_1^{1/2} -norm. This paper is partly motivated by a result of Hamza’s that was stated in a theorem of Fukushima, Oshima, and Takeda’s and that provides a different point of view to this theorem. To illustrate our results, many examples are provided.
- Published
- 2018
15. New criteria for Vandiver’s conjecture using Gauss sums – Heuristics and numerical experiments
- Author
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Georges Gras
- Subjects
Discrete mathematics ,Conjecture ,Mathematics::Number Theory ,General Mathematics ,Modulo ,010102 general mathematics ,Galois group ,Order (ring theory) ,Cyclotomic field ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Gauss sum ,symbols ,0101 mathematics ,Bernoulli number ,Mathematics ,Counterexample - Abstract
The link between Vandiver’s conjecture and Gauss sums is well known since the papers of Iwasawa (Symposia Mathematica, vol 15, Academic Press, pp 447–459, 1975), Thaine (Mich Math J 42(2):311–344, 1995; Trans Am Math Soc 351(12):4769–4790, 1999) and Angles and Nuccio (Acta Arith 142(3):199–218, 2010). This conjecture is required in many subjects and we shall give such examples of relevant references. In this paper, we recall our interpretation of Vandiver’s conjecture in terms of minus part of the torsion of the Galois group of the maximal abelian p-ramified pro-p-extension of the p-th cyclotomic field (Sur la p-ramification abelienne (1984) vol. 20, pp. 1–26). Then we provide a specific use of Gauss sums of characters of order p of $${\mathbb {F}}_\ell ^\times $$ and prove new criteria for Vandiver’s conjecture to hold (Theorem 2 (a) using both the sets of exponents of p-irregularity and of p-primarity of suitable twists of the Gauss sums, and Theorem 2 (b) which does not need the knowledge of Bernoulli numbers or cyclotomic units). We propose in §5.2 new heuristics showing that any counterexample to the conjecture leads to excessive constraints modulo p on the above twists as $$\ell $$ varies and suggests analytical approaches to evidence. We perform numerical experiments to strengthen our arguments in the direction of the very probable truth of Vandiver’s conjecture and to inspire future research. The calculations with their PARI/GP programs are given in appendices.
- Published
- 2020
16. Redheffer type bounds for Bessel and modified Bessel functions of the first kind
- Author
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Árpád Baricz and Khaled Mehrez
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Discrete mathematics ,Pure mathematics ,Hankel transform ,Cylindrical harmonics ,Bessel process ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Dirichlet eta function ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Bessel polynomials ,Struve function ,symbols ,Discrete Mathematics and Combinatorics ,Bessel's inequality ,0101 mathematics ,Bessel function ,Mathematics - Abstract
In this paper our aim is to show some new inequalities of the Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We also use some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research.
- Published
- 2018
17. The lattices of invariant subspaces of a class of operators on the Hardy space
- Author
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Zeljko Cuckovic and Bhupendra Paudyal
- Subjects
Discrete mathematics ,Pure mathematics ,Volterra operator ,Mathematics - Complex Variables ,General Mathematics ,010102 general mathematics ,Holomorphic function ,010103 numerical & computational mathematics ,Hardy space ,Reflexive operator algebra ,01 natural sciences ,Linear subspace ,symbols.namesake ,Operator (computer programming) ,Lattice (order) ,FOS: Mathematics ,symbols ,Complex Variables (math.CV) ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator., We deleted a proposition and a corollary from section 4, and simplified the proof of the main theorem. **The article has been published in Archiv der Mathematik**
- Published
- 2018
18. Fixed Point Theorems for Mappings Satisfying Weak Nonexpansivity Condition (Weak Contractivity Condition) into (from) Cartesian Products Normed Spaces
- Author
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Sahar Mohamed Ali Abou Bakr
- Subjects
Statistics and Probability ,Discrete mathematics ,Pure mathematics ,General Mathematics ,Banach space ,Fixed-point theorem ,Cartesian product ,Type (model theory) ,Fixed point ,Space (mathematics) ,symbols.namesake ,Monotone polygon ,symbols ,Mathematics ,Normed vector space - Abstract
This paper suggests new types of weak nonexpansive mappings defined from normed space X into its Cartesian product X × X, studies the main features of the fixed points for those mappings and extends the concept of (C)-contractivity condition introduced in some previous research papers. On other side, it introduces new types of contraction mappings with a mixed monotone property; the {a, b, c} M-first type and the {a, b, c} M-second type contractions, these types are defined from the Cartesian product space X × X into X, where X is a sequentially ordered Banach space, proves the existence of first-anti-second and second-anti-first couple fixed points of such types and generalizes some of the results given before.
- Published
- 2017
19. On embedding certain partial orders into the P-points under Rudin-Keisler and Tukey reducibility
- Author
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Dilip Raghavan and Saharon Shelah
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Boolean algebra (structure) ,010102 general mathematics ,Ultrafilter ,Natural number ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,symbols.namesake ,010201 computation theory & mathematics ,symbols ,Embedding ,Continuum (set theory) ,0101 mathematics ,Partially ordered set ,Continuum hypothesis ,Axiom ,Mathematics - Abstract
The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin’s axiom for σ \sigma -centered posets. In his 1973 paper he showed under this assumption that both ω 1 {\omega }_{1} and the reals can be embedded. Analogous results were obtained later for the coarser notion of Tukey reducibility. We prove in this paper that Martin’s axiom for σ \sigma -centered posets implies that the Boolean algebra P ( ω ) / FIN \mathcal {P}(\omega ) / \operatorname {FIN} equipped with its natural partial order can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility. Consequently, the continuum hypothesis implies that every partial order of size at most continuum embeds into the P-points under both notions of reducibility.
- Published
- 2017
20. Compatible adjacency relations for digital products
- Author
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Sang-Eon Han
- Subjects
Combinatorics ,Discrete mathematics ,symbols.namesake ,Automorphism group ,Digital image ,General Mathematics ,Product (mathematics) ,symbols ,Adjacency list ,Isomorphism ,Cartesian product ,Mathematics - Abstract
The present paper studies several types of compatible adjacency relations for digital products such as a $C$-compatible adjacency (or the $L_C$-property in \cite{H13}), an $S$-compatible adjacency in \cite{H19} (or the $L_S$-property in \cite{H13}), which can contribute to the study of product properties of digital spaces (or digital images). Furthermore, to study an automorphism group of a Cartesian product of two digital coverings which do not satisfy a radius $2$ local isomorphism, which remains open, the paper uses some properties of an ultra regular covering in \cite{H16}. By using this approach, we can substantially classify digital products. In particular, using a $C$-compatible adjacency (or the $L_C$-property), we address a product problem of a digital isomorphism (see Theorems 3.6 and 4.1).
- Published
- 2017
21. Degenerate abstract Volterra equations in locally convex spaces
- Author
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Marko Kostić
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Scalar (mathematics) ,Degenerate energy levels ,Volterra equations ,Equicontinuity ,01 natural sciences ,Volterra integral equation ,010101 applied mathematics ,symbols.namesake ,Locally convex topological vector space ,Resolvent operator ,symbols ,0101 mathematics ,Well posedness ,Mathematics - Abstract
In the paper under review, we analyze various types of degenerate abstract Volterra integrodifferential equations in sequentially complete locally convex spaces. From the theory of non-degenerate equations, it is well known that the class of (a,k)-regularized C-resolvent families provides an efficient tool for dealing with abstract Volterra integro-differential equations of scalar type. Following the approach of T.-J. Xiao and J. Liang [41]-[43], we introduce the class of degenerate exponentially equicontinuous (a,k)- regularized C-resolvent families and discuss its basic structural properties. In the final section of paper, we will look at generation of degenerate fractional resolvent operator families associated with abstract differential operators.
- Published
- 2017
22. New pathways and connections in number theory and analysis motivated by two incorrect claims of Ramanujan
- Author
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Arindam Roy, Atul Dixit, Bruce C. Berndt, and Alexandru Zaharescu
- Subjects
Discrete mathematics ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,Divisor function ,Divisor (algebraic geometry) ,Divergent series ,01 natural sciences ,Ramanujan's sum ,010101 applied mathematics ,symbols.namesake ,Identity (mathematics) ,Number theory ,symbols ,0101 mathematics ,Convergent series ,Mathematics - Abstract
The focus of this paper commences with an examination of three (not obviously related) pages in Ramanujan's lost notebook, pages 336, 335, and 332, in decreasing order of attention. On page 336, Ramanujan proposes two identities, but the formulas are wrong – each is vitiated by divergent series. We concentrate on only one of the two incorrect “identities,” which may have been devised to attack the extended divisor problem. We prove here a corrected version of Ramanujan's claim, which contains the convergent series appearing in it. The convergent series in Ramanujan's faulty claim is similar to one used by G.F. Voronoi, G.H. Hardy, and others in their study of the classical Dirichlet divisor problem. This now brings us to page 335, which comprises two formulas featuring doubly infinite series of Bessel functions, the first being conjoined with the classical circle problem initiated by Gauss, and the second being associated with the Dirichlet divisor problem. The first and fourth authors, along with Sun Kim, have written several papers providing proofs of these two difficult formulas in different interpretations. In this monograph, we return to these two formulas and examine them in more general settings. The aforementioned convergent series in Ramanujan's “identity” is also similar to one that appears in a curious identity found in Chapter 15 in Ramanujan's second notebook, written in a more elegant, equivalent formulation on page 332 in the lost notebook. This formula may be regarded as a formula for ζ ( 1 2 ) , and in 1925, S. Wigert obtained a generalization giving a formula for ζ ( 1 k ) for any even integer k ≥ 2 . We extend the work of Ramanujan and Wigert in this paper. The Voronoi summation formula appears prominently in our study. In particular, we generalize work of J.R. Wilton and derive an analogue involving the sum of divisors function σ s ( n ) . The modified Bessel functions K s ( x ) arise in several contexts, as do Lommel functions. We establish here new series and integral identities involving modified Bessel functions and modified Lommel functions. Among other results, we establish a modular transformation for an infinite series involving σ s ( n ) and modified Lommel functions. We also discuss certain obscure related work of N.S. Koshliakov. We define and discuss two new related classes of integral transforms, which we call Koshliakov transforms, because he first found elegant special cases of each.
- Published
- 2017
23. Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitrary linear functionals
- Author
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Peter Kritzer, Henryk Woźniakowski, and Friedrich Pillichshammer
- Subjects
Statistics and Probability ,Discrete mathematics ,Numerical Analysis ,Polynomial ,Control and Optimization ,Algebra and Number Theory ,Logarithm ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Hilbert space ,010103 numerical & computational mathematics ,01 natural sciences ,Exponential polynomial ,Exponential function ,Singular value ,symbols.namesake ,Tensor product ,Bounded function ,symbols ,0101 mathematics ,Mathematics - Abstract
We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals needed to obtain an e -approximation for the d -variate problem which is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function that depends polynomially on d and logarithmically on e − 1 . The corresponding unweighted problem was studied in Hickernell et al. (2020) with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on d and logarithmic dependence on e − 1 , we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential ( s , t ) -weak tractability with max ( s , t ) ≥ 1 . For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential ( s , t ) -weak tractability with max ( s , t ) 1 is left for future study. The paper uses some general results obtained in Hickernell et al. (2020) and Kritzer and Woźniakowski (2019).
- Published
- 2020
24. A scaling property of Farey fractions. Part II: convergence at rational points
- Author
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Matthias Kunik
- Subjects
Limit of a function ,Discrete mathematics ,Rational number ,General Mathematics ,Zero (complex analysis) ,Order (ring theory) ,Inverse ,Riemann zeta function ,Combinatorics ,symbols.namesake ,symbols ,Farey sequence ,Mathematics ,Prime number theorem - Abstract
The Farey sequence of order n consists of all reduced fractions a / b between 0 and 1 with positive denominator b less or equal to n. The sums of the inverse denominators 1 / b of the Farey fractions in prescribed intervals with rational bounds have a simple main term, but the deviations are determined by an interesting sequence of polygonal functions \(f_n\). In a former paper we obtained a limit function for \(n \rightarrow \infty \), which describes a scaling behaviour of the functions \(f_n\) in the vicinity of any fixed rational number a / b and which is independent of a / b. In this paper we prove that \(f_n(a/b)\) tends to zero for \(n \rightarrow \infty \) by using elementary representation formulas for \(f_n(a/b)\) as well as a variant of the prime number theorem. An application of this result immediately gives a global version of the scaling behaviour of the functions \(f_n\) around the rational numbers.
- Published
- 2016
25. Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems
- Author
-
Raffaella Servadei, Giovanni Molica Bisci, Alessio Fiscella, Fiscella, A, Molica Bisci, G, and Servadei, R
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,variational techniques ,010102 general mathematics ,Multiplicity (mathematics) ,integrodifferential operators ,01 natural sciences ,Dirichlet distribution ,Fractional Laplacian ,010101 applied mathematics ,Sobolev space ,symbols.namesake ,critical nonlinearities ,Operator (computer programming) ,Fractional Laplacian, critical nonlinearities, best fractional critical Sobolev constant, variational techniques, integrodifferential operators ,Bounded function ,best fractional critical Sobolev constant ,fractional Laplacian, critical nonlinearities, best fractional critical Sobolev constant, variational techniques, integrodifferential operators ,symbols ,Exponent ,0101 mathematics ,Bifurcation ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper we consider the following critical nonlocal problem { − L K u = λ u + | u | 2 ⁎ − 2 u in Ω u = 0 in R n ∖ Ω , where s ∈ ( 0 , 1 ) , Ω is an open bounded subset of R n , n > 2 s , with continuous boundary, λ is a positive real parameter, 2 ⁎ : = 2 n / ( n − 2 s ) is the fractional critical Sobolev exponent, while L K is the nonlocal integrodifferential operator L K u ( x ) : = ∫ R n ( u ( x + y ) + u ( x − y ) − 2 u ( x ) ) K ( y ) d y , x ∈ R n , whose model is given by the fractional Laplacian − ( − Δ ) s . Along the paper, we prove a multiplicity and bifurcation result for this problem, using a classical theorem in critical points theory. Precisely, we show that in a suitable left neighborhood of any eigenvalue of − L K (with Dirichlet boundary data) the number of nontrivial solutions for the problem under consideration is at least twice the multiplicity of the eigenvalue. Hence, we extend the result got by Cerami, Fortunato and Struwe in [14] for classical elliptic equations, to the case of nonlocal fractional operators.
- Published
- 2016
26. COMMON SOLUTION TO GENERALIZED MIXED EQUILIBRIUM PROBLEM AND FIXED POINT PROBLEM FOR A NONEXPANSIVE SEMIGROUP IN HILBERT SPACE
- Author
-
Mohammad Farid, K. R. Kazmi, and Behzad Djafari-Rouhani
- Subjects
Discrete mathematics ,Sequence ,Semigroup ,Generalization ,Iterative method ,General Mathematics ,010102 general mathematics ,Hilbert space ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Fixed point problem ,Scheme (mathematics) ,Variational inequality ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we introduce and study an explicit hybrid re- laxed extragradient iterative method to approximate a common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup in Hilbert space. Further, we prove that the sequence generated by the proposed iterative scheme converges strongly to the common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup. This common solu- tion is the unique solution of a variational inequality problem and is the optimality condition for a minimization problem. The results presented in this paper are the supplement, improvement and generalization of the previously known results in this area.
- Published
- 2016
27. DISCRETE MEASURES WITH DENSE JUMPS INDUCED BY STURMIAN DIRICHLET SERIES
- Author
-
DoYong Kwon
- Subjects
Discrete mathematics ,General Mathematics ,Sturmian word ,Lexicographical order ,Dirichlet distribution ,Combinatorics ,symbols.namesake ,Singular function ,Real-valued function ,Free monoid ,symbols ,Arithmetic function ,Dirichlet series ,Mathematics - Abstract
Let (s α (n)) n≥1 be the lexicographically greatest Sturmianword of slope α > 0. For a fixed σ > 1, we consider Dirichlet seriesof the form ν σ (α) :=P ∞n=1 s α (n)n −σ . This paper studies the singularproperties of the real function ν σ , and the Lebesgue-Stieltjes measurewhose distribution is given by ν σ . 1. IntroductionThroughout the paper, N(resp. N 0 ) denotes the set of positive (resp. non-negative) integers. We mean by ⌊·⌋ (resp. ⌈·⌉) the floor (resp. ceiling) function,and by A ∗ the set of finite words over the alphabet A, i.e., the free monoid gen-erated by A.For α ≥ 0, an arithmetic function s α : N→ N 0 is defined bys α (n) := ⌈αn⌉ −⌈α(n −1)⌉.Then s α := s α (1)s α (2)··· is an infinite word over the alphabet {⌈α⌉−1,⌈α⌉}.Now we set, for a fixed σ > 1,(1) ν σ (α) :=X ∞n=1 s α (n)n σ ,i.e., Dirichlet series whose coefficients are given by s α . From now on, we assumeσ > 1 unless otherwise stated explicitly. This real function ν σ : [0,∞) → Rwasfirstly considered in [3], and shown to be continuous at every irrational, whereasleft-continuous but not right-continuous at every rational. Furthermore, ν
- Published
- 2015
28. SUPERCYCLICITY OF JOINT ISOMETRIES
- Author
-
Mohammad Ansari, Karim Hedayatian, Bahram Khani-Robati, and Abbas Moradi
- Subjects
Discrete mathematics ,symbols.namesake ,Operator (computer programming) ,General Mathematics ,Bounded function ,Dimension (graph theory) ,Hilbert space ,symbols ,Isometry ,Tuple ,Space (mathematics) ,Separable space ,Mathematics - Abstract
Let H be a separable complex Hilbert space. A commut-ing tuple T = (T 1 ,...,T n ) of bounded linear operators on H is called aspherical isometry ifP ni=1 T ∗i T i = I. The tuple T is called a toral isom-etry if each T i is an isometry. In this paper, we show that for each n≥1there is a supercyclic n-tuple of spherical isometries on C n and there is nospherical or toral isometric tuple of operators on an infinite-dimensionalHilbert space. 1. IntroductionAnn-tuple ofoperatorsisafinite sequenceoflengthn ofcommutingboundedlinear operators T 1 ,T 2 ,...,T n acting on a Hilbert space H. For an n-tupleT = (T 1 ,T 2 ,...,T n ), if there exists an element x ∈ H such that orb(T,x) ={Sx : S ∈ F T } where F T = {T k 1 1 T k 2 2 ···T k n n : k i ≥ 0, i = 1,2,...,n}, isdense in H then x is called a hypercyclic vector for T, and T is said to be ahypercyclic n-tuple of operators. A vector x ∈ H is called a supercyclic vectorfor T if the set {λSx : S ∈ F T , λ ∈ C} is dense in H, and T is said to bea supercyclic n-tuple of operators. These definitions generalize the notions ofhypercyclicity and supercyclicity of a single operator to a tuple of operators.Hypercyclicity and supercyclicity of tuples of operators have been investigatedin ([3], [4], [6], [7]). On the other hand, spherical isometries are a consider-able part of tuples of operators. The authors in [5] proved that isometries onHilbert spaces with dimension more than one are not supercyclic. Recently,this fact has been proved for m-isometric operators which are a generalizationof isometric operators in some sense [2]. In this paper we see that sphericalisometries are not supercyclic on infinite-dimensional Hilbert spaces. Let A bea matrix we denote by A
- Published
- 2015
29. On Ahlfors–David regular weighted bounds for the extension operator associated to the circle
- Author
-
Tuomas Orponen
- Subjects
Discrete mathematics ,symbols.namesake ,Operator (computer programming) ,Fourier transform ,Mathematics - Classical Analysis and ODEs ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,Point (geometry) ,Extension (predicate logic) ,16. Peace & justice ,Mathematics - Abstract
This paper addresses the sharpness of a weighted $L^{2}$-estimate for the Fourier extension operator associated to the circle, obtained by J. Bennett, A. Carbery, F. Soria and A. Vargas in 2006. A point left open in their paper was the necessity of a certain $\log R$-factor in the bound. Here, I show that the factor is necessary for all $1/2$-Ahlfors-David regular weights on the circle, but it can be removed for $s$-Ahlfors-David regular weights with $s \neq 1/2$., 12 pages
- Published
- 2015
30. Theb-adic tent transformation for quasi-Monte Carlo integration using digital nets
- Author
-
Takashi Goda, Takehito Yoshiki, and Kosuke Suzuki
- Subjects
Discrete mathematics ,Numerical Analysis ,Polynomial ,Kernel (set theory) ,Applied Mathematics ,General Mathematics ,Lattice (group) ,Hilbert space ,Numerical Analysis (math.NA) ,Prime (order theory) ,Sobolev space ,symbols.namesake ,Rate of convergence ,FOS: Mathematics ,symbols ,Mathematics - Numerical Analysis ,Quasi-Monte Carlo method ,Analysis ,Mathematics - Abstract
In this paper we investigate quasi-Monte Carlo (QMC) integration using digital nets over Z b in reproducing kernel Hilbert spaces. The tent transformation (previously called baker’s transform) was originally used for lattice rules by Hickernell (2002) to achieve higher order convergence of the integration error for smooth non-periodic integrands, and later, has been successfully applied to digital nets over Z 2 by Cristea et al. (2007) and Goda (2015). The aim of this paper is to generalize the latter two results to digital nets over Z b for an arbitrary prime b . For this purpose, we introduce the b -adic tent transformation for an arbitrary positive integer b greater than 1, which is a generalization of the original (dyadic) tent transformation. Further, again for an arbitrary positive integer b greater than 1, we analyze the mean square worst-case error of QMC rules using digital nets over Z b which are randomly digitally shifted and then folded using the b -adic tent transformation in reproducing kernel Hilbert spaces. Using this result, for a prime b , we prove the existence of good higher order polynomial lattice rules over Z b among a smaller number of candidates as compared to the result by Dick and Pillichshammer (2007), which achieve almost the optimal convergence rate of the mean square worst-case error in unanchored Sobolev spaces of smoothness of arbitrary high order.
- Published
- 2015
31. Hilbert-Asai Eisenstein series, regularized products, and heat kernels
- Author
-
Serge Lang and Jay Jorgenson
- Subjects
Discrete mathematics ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Algebraic number field ,Space (mathematics) ,01 natural sciences ,Inversion (discrete mathematics) ,Matrix decomposition ,11F72 ,symbols.namesake ,Development (topology) ,0103 physical sciences ,Eisenstein series ,symbols ,0101 mathematics ,Heat kernel ,Axiom ,Mathematics ,11M36 - Abstract
In a famous paper, Asai indicated how to develop a theory of Eisenstein series for arbitrary number fields, using hyperbolic 3-space to take care of the complex places. Unfortunately he limited himself to class number 1. The present paper gives a detailed exposition of the general case, to be used for many applications. First, it is shown that the Eisenstein series satisfy the authors’ definition of regularized products satisfying the generalized Lerch formula, and the basic axioms which allow the systematic development of the authors’ theory, including the Cramér theorem. It is indicated how previous results of Efrat and Zograf for the strict Hilbert modular case extend to arbitrary number fields, for instance a spectral decomposition of the heat kernel periodized with respect to SL2 of the integers of the number field. This gives rise to a theta inversion formula, to which the authors’ Gauss transform can be applied. In addition, the Eisenstein series can be twisted with the heat kernel, thus encoding an infinite amount of spectral information in one item coming from heat Eisenstein series. The main expected spectral formula is stated, but a complete exposition would require a substantial amount of space, and is currently under consideration.
- Published
- 1999
32. Odd linking and bifurcation in gaps: the weakly indefinite case
- Author
-
Hans-Joerg Ruppen
- Subjects
Discrete mathematics ,General Mathematics ,Essential spectrum ,Multiplicity (mathematics) ,Lambda ,Schrödinger equation ,Periodic function ,Linear map ,symbols.namesake ,symbols ,Spectral gap ,Eigenvalues and eigenvectors ,Mathematics ,Mathematical physics - Abstract
In this paper, we consider nonlinear Schrödinger equations of the following type:−Δu(x)+ V(x)u(x) − q(x)|u(x)|σu(x) = λu(x), x ∈ ℝN, u ∈ H1(ℝN)\{0},where N ≥ 2 and σ > 0. We concentrate on situations where the potential function V appearing in the linear part of the equation is of Coulomb type; by this we mean potentials where the spectrum of the linear operator −Δ + V consists of an increasing sequence of eigenvalues λ1, λ2,… followed by an interval belonging to the essential spectrum.We study, for λ kept fixed inside a spectral gap or below λ1, the existence of multiple solution pairs, as well as the bifurcation behaviour of these solutions when λ approaches a point of the spectrum from the left-hand side. Our method proceeds by an analysis of critical points of the corresponding energy functional. To this end, we derive a new variational characterization of critical levelsc0 (λ) ≤ c1(λ) ≤ c2(λ) ≤ ⋯ corresponding to an infinite set of critical points.We derive such a multiplicity result even if some of the critical values cn(λ) coincide; this seems to be a major advantage of our approach. Moreover, the characterization of these values cn(λ) is suitable for an analysis of the bifurcation behaviour of the corresponding generalized solutions.The approach presented here is generic; for instance, it can be applied when V and q are periodic functions. Such generalizations are briefly described in this paper and will be the object of a forthcoming article.
- Published
- 2017
33. Inner product on B∗-algebras of operators on a free Banach space over the Levi-Civita field
- Author
-
José Aguayo, Miguel Nova, and Khodr Shamseddine
- Subjects
Discrete mathematics ,Pure mathematics ,Approximation property ,Nuclear operator ,General Mathematics ,Hilbert space ,Spectral theorem ,Operator theory ,Compact operator ,Compact operator on Hilbert space ,symbols.namesake ,symbols ,Operator norm ,Mathematics - Abstract
Let C be the complex Levi-Civita field and let c 0 ( C ) or, simply, c 0 denote the space of all null sequences z = ( z n ) n ∈ N of elements of C . The natural inner product on c 0 induces the sup-norm of c 0 . In a previous paper Aguayo et al. (2013), we presented characterizations of normal projections, adjoint operators and compact operators on c 0 . In this paper, we work on some B ∗ -algebras of operators, including those mentioned above; then we define an inner product on such algebras and prove that this inner product induces the usual norm of operators. We finish the paper with a characterization of closed subspaces of the B ∗ -algebra of all adjoint and compact operators on c 0 which admit normal complements.
- Published
- 2015
34. Generalized Orlicz-Lorentz sequence spaces and corresponding operator ideals
- Author
-
Antara Bhar and Manjul Gupta
- Subjects
Discrete mathematics ,Mathematics::Functional Analysis ,Pure mathematics ,General Mathematics ,Topological tensor product ,Lorentz transformation ,Banach space ,Finite-rank operator ,symbols.namesake ,symbols ,Interpolation space ,Dual polyhedron ,Birnbaum–Orlicz space ,Lp space ,Mathematics - Abstract
In this paper we introduce generalized or vector-valued Orlicz-Lorentz sequence spaces l p,q,M (X) on Banach space X with the help of an Orlicz function M and for different positive indices p and q. We study their structural properties and investigate cross and topological duals of these spaces. Moreover these spaces are generalizations of vector-valued Orlicz sequence spaces l M (X) for p = q and also Lorentz sequence spaces for M(x) = x q for q ≥ 1. Lastly we prove that the operator ideals defined with the help of scalar valued sequence spaces l p,q,M and additive s-numbers are quasi-Banach operator ideals for p < q and Banach operator ideals for p ≥ q. The results of this paper are more general than the work of earlier mathematicians, say A. Pietsch, M. Kato, L. R. Acharya, etc.
- Published
- 2014
35. Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections
- Author
-
J. R. Higgins, Gerhard Schmeisser, Paul L. Butzer, Maurice Dodson, R. L. Stens, and Paulo J. S. G. Ferreira
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,Poisson summation formula ,Mathematical proof ,Riemann zeta function ,Parseval's theorem ,symbols.namesake ,Riemann hypothesis ,Number theory ,Functional equation ,symbols ,Nyquist–Shannon sampling theorem ,Mathematics - Abstract
The present paper deals mainly with seven fundamental theorems of mathematical analysis, numerical analysis, and number theory, namely the generalized Parseval decomposition formula (GPDF), introduced 15 years ago, the well-known approximate sampling theorem (ASF), the new approximate reproducing kernel theorem, the basic Poisson summation formula, already known to Gaus, a newer version of the GPDF having a structure similar to that of the Poisson summation formula, namely, the Parseval decomposition–Poisson summation formula, the functional equation of Riemann’s zeta function, as well as the Euler–Maclaurin summation formula. It will in fact be shown that these seven theorems are all equivalent to one another, in the sense that each is a corollary of the others. Since these theorems can all be deduced from each other, one of them has to be proven independently in order to verify all. It is convenient to choose the ASF, introduced in 1963. The epilogue treats possible extensions to the more general contexts of reproducing kernel theory and of abstract harmonic analysis, using locally compact abelian groups. This paper is expository in the sense that it treats a number of mathematical theorems, their interconnections, their equivalence to one another. On the other hand, the proofs of the many intricate interconnections among these theorems are new in their essential steps and conclusions.
- Published
- 2014
36. An Iterative Method for Equilibrium, Variational Inequality, and Fixed Point Problems for a Nonexpansive Semigroup in Hilbert Spaces
- Author
-
Nguyen Thi Thu Thuy
- Subjects
Discrete mathematics ,Iterative method ,Semigroup ,General Mathematics ,Hilbert space ,Solution set ,Fixed point ,Lipschitz continuity ,symbols.namesake ,Monotone polygon ,Variational inequality ,symbols ,Applied mathematics ,Mathematics - Abstract
The purpose of this paper is to present a new iteration method based on the hybrid method in mathematical programming, extragradient method, and Mann’s method for finding a common element of the solution set of equilibrium problems, the solution set of variational inequality problems for a monotone, Lipschitz continuous mapping and the set of fixed points for a nonexpansive semigroup in Hilbert spaces. We obtain a strong convergence theorem for the sequences generated by this process. The results in this paper generalize and extend some well-known strong convergence theorems in the literature.
- Published
- 2014
37. On the generalization of density topologies on the real line
- Author
-
Jacek Hejduk and Renata Wiertelak
- Subjects
Discrete mathematics ,symbols.namesake ,Sequence ,Operator (computer programming) ,Generalization ,General Mathematics ,symbols ,Zero (complex analysis) ,Lebesgue integration ,Real line ,Measure (mathematics) ,Topology (chemistry) ,Mathematics - Abstract
The paper concerns the density points with respect to the sequences of intervals tending to zero in the family of Lebesgue measurable sets. It shows that for some sequences analogue of the Lebesgue density theorem holds. Simultaneously, this paper presents proof of theorem that for any sequence of intervals tending to zero a relevant operator ϕJ generates a topology. It is almost but not exactly the same result as in the category aspect presented in [WIERTELAK, R.: A generalization of density topology with respect to category, Real Anal. Exchange 32 (2006/2007), 273–286]. Therefore this paper is a continuation of the previous research concerning similarities and differences between measure and category.
- Published
- 2014
38. PROOFS OF CONJECTURES OF SANDON AND ZANELLO ON COLORED PARTITION IDENTITIES
- Author
-
Bruce C. Berndt and Roberta R. Zhou
- Subjects
Combinatorics ,Discrete mathematics ,symbols.namesake ,Colored ,General Mathematics ,symbols ,Partition (number theory) ,Theta function ,Remainder ,Mathematical proof ,Ramanujan's sum ,Mathematics - Abstract
In a recent systematic study, C. Sandon and F. Zanello of- fered 30 conjectured identities for partitions. As a consequence of their study of partition identities arising from Ramanujan's formulas for mul- tipliers in the theory of modular equations, the present authors in an earlier paper proved three of these conjectures. In this paper, we provide proofs for the remaining 27 conjectures of Sandon and Zanello. Most of our proofs depend upon known modular equations and formulas of Ra- manujan for theta functions, while for the remainder of our proofs it was necessary to derive new modular equations and to employ the process of duplication to extend Ramanujan's catalogue of theta function formulas.
- Published
- 2014
39. ASYMPTOTIC RUIN PROBABILITIES IN A GENERALIZED JUMP-DIFFUSION RISK MODEL WITH CONSTANT FORCE OF INTEREST
- Author
-
Di Bao and Qingwu Gao
- Subjects
Discrete mathematics ,Sequence ,Counting process ,General Mathematics ,Jump diffusion ,Actuarial notation ,symbols.namesake ,Wiener process ,Statistics ,symbols ,Almost surely ,Random variable ,Randomness ,Mathematics - Abstract
This paper studies the asymptotic behavior of the finite-timeruin probability in a jump-diffusion risk model with constant force of in-terest, upper tail asymptotically independent claims and a general count-ing arrival process. Particularly, if the claim inter-arrival times follow acertain dependence structure, the obtained result also covers the case ofthe infinite-time ruin probability. 1. IntroductionIn this paper, we consider the asymptotic ruin probabilities in a generalizedjump-diffusion risk model with constant force of interest, where the claim sizes{X i ,i≥ 1} are a sequence of nonnegative, but not necessarily independent,random variables (r.v.s) with distributions F i , i≥ 1, respectively, while theclaim arrival process {N(t),t≥ 0} is a general counting process, independentof {X i ,i≥ 1}. Hence, the aggregate claim amount up to time t≥ 0 isS(t) = N X (t)i=1 X i with S(t) = 0 if N(t) = 0. Assume that the total amount of premiums accu-mulated up to time t≥ 0, denoted by C(t), is a nonnegative and nondecreasingstochastic process with C(0) = 0 and C(t)
- Published
- 2014
40. Non-Separable and Planar Graphs
- Author
-
Hassler Whitney
- Subjects
Block graph ,Discrete mathematics ,Multidisciplinary ,Dense graph ,Applied Mathematics ,General Mathematics ,Symmetric graph ,Structure (category theory) ,Ear decomposition ,1-planar graph ,Graph ,Planar graph ,Separable space ,symbols.namesake ,Pathwidth ,Chordal graph ,Mathematical induction ,symbols ,Decomposition (computer science) ,Rank (graph theory) ,Mathematics ,MathematicsofComputing_DISCRETEMATHEMATICS ,Universal graph ,Forbidden graph characterization - Abstract
Introduction. In this paper the structure of graphs is studied by purely combinatorial methods. The concepts of rank and nullity are fundamental. The first part is devoted to a general study of non-separable graphs. Conditions that a graph be non-separable are given; the decomposition of a separable graph into its non-separable parts is studied; by means of theorems on circuits of graphs, a method for the construction of non-separable graphs is found, which is useful in proving theorems on such graphs by mathematical induction.
- Published
- 1992
41. Value distribution of L-functions concerning shared values and certain differential polynomials
- Author
-
Hong-Xun Yi, Fang Liu, and Xiao-Min Li
- Subjects
Discrete mathematics ,Gegenbauer polynomials ,Mathematics::Complex Variables ,Nevanlinna theory ,General Mathematics ,Discrete orthogonal polynomials ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,differential polynomials ,uniqueness theorems ,Classical orthogonal polynomials ,shared values ,symbols.namesake ,L-functions ,Difference polynomials ,Macdonald polynomials ,30D35 ,Orthogonal polynomials ,Wilson polynomials ,symbols ,Jacobi polynomials ,0101 mathematics ,Mathematics ,11M36 - Abstract
In this paper, we study a uniqueness question of meromorphic functions of certain differential polynomials that share a nonzero finite value or have the same fixed points with the same of L-functions. The results in this paper extend the corresponding results from Steuding [12, p. 152], Li [7], Fang [1] and Yang-Hua [14].
- Published
- 2017
42. A study of saturated tensor cone for symmetrizable Kac–Moody algebras
- Author
-
Merrick Brown and Shrawan Kumar
- Subjects
Discrete mathematics ,Weyl group ,Group (mathematics) ,Semigroup ,General Mathematics ,Cartan subalgebra ,symbols.namesake ,Borel subgroup ,Integer ,Mathematics::Quantum Algebra ,Lie algebra ,symbols ,Geometric invariant theory ,Mathematics::Representation Theory ,Mathematics - Abstract
Let $$\mathfrak {g}$$ be a symmetrizable Kac-Moody Lie algebra with the standard Cartan subalgebra $$\mathfrak {h}$$ and the Weyl group $$W$$ . Let $$P_+$$ be the set of dominant integral weights. For $$\lambda \in P_+$$ , let $$L(\lambda )$$ be the integrable, highest weight (irreducible) representation of $$\mathfrak {g}$$ with highest weight $$\lambda $$ . For a positive integer $$s$$ , define the saturated tensor semigroup as $$\begin{aligned} \Gamma _s:= \{(\lambda _1, \dots , \lambda _s,\mu )\in P_+^{s+1}: \exists \, N\ge 1 \,\text {with}\,L(N\mu )\subset L(N\lambda _1)\otimes \dots \otimes L(N\lambda _s)\}. \end{aligned}$$ The aim of this paper is to begin a systematic study of $$\Gamma _s$$ in the infinite dimensional symmetrizable Kac-Moody case. In this paper, we produce a set of necessary inequalities satisfied by $$\Gamma _s$$ . These inequalities are indexed by products in $$H^*(G^{\mathrm{min }}/B; \mathbb {Z})$$ for $$B$$ the standard Borel subgroup, where $$G^{\mathrm{min }}$$ is the ‘minimal’ Kac-Moody group with Lie algebra $$\mathfrak {g}$$ . The proof relies on the Kac-Moody analogue of the Borel-Weil theorem and Geometric Invariant Theory (specifically the Hilbert-Mumford index). In the case that $$\mathfrak {g}$$ is affine of rank 2, we show that these inequalities are necessary and sufficient. We further prove that any integer $$d>0$$ is a saturation factor for $$A^{(1)}_1$$ and 4 is a saturation factor for $$A^{(2)}_2$$ .
- Published
- 2014
43. Inductive topological Hausdorff dimensions and fibers of generic continuous functions
- Author
-
Richárd Balka
- Subjects
Discrete mathematics ,General Mathematics ,Dimension (graph theory) ,Banach space ,Hausdorff space ,Order (ring theory) ,Topology ,Combinatorics ,Metric space ,symbols.namesake ,Compact space ,Hausdorff dimension ,symbols ,Lebesgue covering dimension ,Mathematics - Abstract
In an earlier paper Buczolich, Elekes and the author introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. They proved that it is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space \(K\). The goal of this paper is to determine the Hausdorff dimension of the fibers of the generic continuous function from \(K\) to \(\mathbb {R}^n\). In order to do so, we define the \(n\)th inductive topological Hausdorff dimension, \(\dim _{t^nH} K\). Let \(\dim _H K,\,\dim _t K\) and \(C_n(K)\) denote the Hausdorff and topological dimension of \(K\) and the Banach space of the continuous functions from \(K\) to \(\mathbb {R}^n\). We show that \(\sup _{y\in \mathbb {R}^n} \dim _{H}f^{-1}(y) = \dim _{t^nH} K -n\) for the generic \(f \in C_n(K)\), provided that \(\dim _t K\ge n\), otherwise every fiber is finite. In order to prove the above theorem we give some equivalent definitions for the inductive topological Hausdorff dimensions, which can be interesting in their own right. Here we use techniques coming from the theory of topological dimension. We show that the supremum is actually attained on the left hand side of the above equation. We characterize those compact metric spaces \(K\) for which \(\dim _{H} f^{-1}(y)=\dim _{t^nH}K-n\) for the generic \(f\in C_n(K)\) and the generic \(y\in f(K)\). We also generalize a result of Kirchheim by showing that if \(K\) is self-similar and \(\dim _t K\ge n\) then \(\dim _{H} f^{-1}(y)=\dim _{t^nH}K-n\) for the generic \(f\in C_n(K)\) for every \(y\in {{\mathrm{int}}}f(K)\).
- Published
- 2014
44. Traces on operator ideals and related linear forms on sequence ideals (part I)
- Author
-
Albrecht Pietsch
- Subjects
Discrete mathematics ,Pure mathematics ,Ideal (set theory) ,Trace (linear algebra) ,Group (mathematics) ,General Mathematics ,Hilbert space ,Separable space ,symbols.namesake ,Fractional ideal ,symbols ,Commutative algebra ,Invariant (mathematics) ,Mathematics - Abstract
The Calkin theorem provides a one-to-one correspondence between all operator ideals A(H) over the separable infinite-dimensional Hilbert space H and all symmetric sequence ideals a(N) over the index set N≔{1,2,…}. The main idea of the present paper is to replace a(N) by the ideal z(N0) that consists of all sequences (αh) indexed by N0≔{0,1,2,…} for which (α0,α1,α1,…,αh,…,αh︷2hterms,…)∈a(N). This new kind of sequence ideals is characterized by two properties: (1) For (αh)∈z(N0) there is a non-increasing (βh)∈z(N0) such that ∣αh∣≤βh. (2) z(N0) is invariant under the operator S+:(α0,α1,α2,…)↦(0,α0,α1,…). Using this modification of the Calkin theorem, we simplify, unify, and complete earlier results of [4,5,7–9,13,14,19–21,25] The central theorem says that there are canonical isomorphisms between the linear spaces of all traces on A(H), all symmetric linear forms on a(N), and all 12S+-invariant linear forms on z(N0). In this way, the theory of linear forms on ideals of a non-commutative algebra that are invariant under the members of a non-commutative group is reduced to the theory of linear forms on ideals of a commutative algebra that are invariant under a single operator. It is hoped that the present approach deserves the rating “streamlined”. Our main objects are linear forms in the purely algebraic sense. Only at the end of this paper continuity comes into play, when the case of quasi-normed ideals is considered. We also sketch a classification of operator ideals according to the existence of various kinds of traces. Details will be discussed in a subsequent publication.
- Published
- 2014
45. Implicit iterative method for approximating a common solution of split equilibrium problem and fixed point problem for a nonexpansive semigroup
- Author
-
S. H. Rizvi and K.R. Kazmi
- Subjects
Discrete mathematics ,Nonexpansive semigroup ,Semigroup ,Iterative method ,General Mathematics ,Minimization problem ,Secondary 47J25 65J15 90C33 ,Hilbert space ,Averaged mapping ,Fixed point ,symbols.namesake ,Fixed point problem ,Implicit iterative method ,Variational inequality ,QA1-939 ,symbols ,Primary 65K15 ,Fixed-point problem ,Applied mathematics ,Equilibrium problem ,Split equilibrium problem ,Mathematics - Abstract
In this paper, we introduce and study an implicit iterative method to approximate a common solution of split equilibrium problem and fixed point problem for a nonexpansive semigroup in real Hilbert spaces. Further, we prove that the nets generated by the implicit iterative method converge strongly to the common solution of split equilibrium problem and fixed point problem for a nonexpansive semigroup. This common solution is the unique solution of a variational inequality problem and is the optimality condition for a minimization problem. Furthermore, we justify our main result through a numerical example. The results presented in this paper extend and generalize the corresponding results given by Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, Fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Math. Comput. Model. 48 (2008) 279–286] and Cianciaruso et al. [F. Cianciaruso, G. Marino, L. Muglia, Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert space, J. Optim. Theory Appl. 146 (2010) 491–509].
- Published
- 2014
- Full Text
- View/download PDF
46. A simple algorithm for sampling colourings of $G(n,d/n)$ up to Gibbs Uniqueness Threshold
- Author
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Charilaos Efthymiou
- Subjects
FOS: Computer and information sciences ,General Computer Science ,Discrete Mathematics (cs.DM) ,General Mathematics ,01 natural sciences ,Combinatorics ,Total variation ,symbols.namesake ,0103 physical sciences ,Uniqueness ,0101 mathematics ,QA ,Time complexity ,Mathematics ,Discrete mathematics ,Random graph ,010102 general mathematics ,Markov chain Monte Carlo ,Boltzmann distribution ,Primary 68R99, 68W25, 68W20 Secondary: 82B44 ,symbols ,Graph (abstract data type) ,010307 mathematical physics ,Constant (mathematics) ,MathematicsofComputing_DISCRETEMATHEMATICS ,Computer Science - Discrete Mathematics - Abstract
Approximate random $k$-colouring of a graph $G$ is a well studied problem in computer science and statistical physics. It amounts to constructing a $k$-colouring of $G$ which is distributed close to {\em Gibbs distribution} in polynomial time. Here, we deal with the problem when the underlying graph is an instance of Erd\H{o}s-R\'enyi random graph $G(n,d/n)$, where $d$ is a sufficiently large constant. We propose a novel efficient algorithm for approximate random $k$-colouring $G(n,d/n)$ for any $k\geq (1+\epsilon)d$. To be more specific, with probability at least $1-n^{-\Omega(1)}$ over the input instances $G(n,d/n)$ and for $k\geq (1+\epsilon)d$, the algorithm returns a $k$-colouring which is distributed within total variation distance $n^{-\Omega(1)}$ from the Gibbs distribution of the input graph instance. The algorithm we propose is neither a MCMC one nor inspired by the message passing algorithms proposed by statistical physicists. Roughly the idea is as follows: Initially we remove sufficiently many edges of the input graph. This results in a "simple graph" which can be $k$-coloured randomly efficiently. The algorithm colours randomly this simple graph. Then it puts back the removed edges one by one. Every time a new edge is put back the algorithm updates the colouring of the graph so that the colouring remains random. The performance of the algorithm depends heavily on certain spatial correlation decay properties of the Gibbs distribution., Comment: This paper is accepted for publication in SIAM Journal on Computing. This is the journal version of two papers of the author in SODA'12 and ESA'14
- Published
- 2016
47. Weighted local Orlicz-Hardy spaces on domains and their applications in inhomogeneous Dirichlet and Neumann problems
- Author
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Sibei Yang, Der-Chen Chang, Jun Cao, and Dachun Yang
- Subjects
Discrete mathematics ,Semigroup ,Applied Mathematics ,General Mathematics ,Order (ring theory) ,Muckenhoupt weights ,Type (model theory) ,Hardy space ,Omega ,Dirichlet distribution ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,symbols.namesake ,Mathematics - Classical Analysis and ODEs ,42B35 (Primary) 42B30, 42B20, 42B25, 35J25, 42B37, 47B38, 46E30 (Secondary) ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,Maximal function ,Mathematics - Abstract
Let $\Omega$ be either $\mathbb{R}^n$ or a strongly Lipschitz domain of $\mathbb{R}^n$, and $\omega\in A_{\infty}(\mathbb{R}^n)$ (the class of Muckenhoupt weights). Let $L$ be a second order divergence form elliptic operator on $L^2 (\Omega)$ with the Dirichlet or Neumann boundary condition, and assume that the heat semigroup generated by $L$ has the Gaussian property $(G_1)$ with the regularity of their kernels measured by $\mu\in(0,1]$. Let $\Phi$ be a continuous, strictly increasing, subadditive, positive and concave function on $(0,\infty)$ of critical lower type index $p_{\Phi}^-\in(0,1]$. In this paper, the authors introduce the "geometrical" weighted local Orlicz-Hardy spaces $h^{\Phi}_{\omega,\,r}(\Omega)$ and $h^{\Phi}_{\omega,\,z}(\Omega)$ via the weighted local Orlicz-Hardy spaces $h^{\Phi}_{\omega}(\mathbb{R}^n)$, and obtain their two equivalent characterizations in terms of the nontangential maximal function and the Lusin area function associated with the heat semigroup generated by $L$ when $p_{\Phi}^-\in(n/(n+\mu),1]$. As applications, the authors prove that the operators $\nabla^2{\mathbb G}_D$ are bounded from $h^{\Phi}_{\omega,\,r}(\Omega)$ to the weighted Orlicz space $L^{\Phi}_{\omega}(\Omega)$, and from $h^{\Phi}_{\omega,\,r}(\Omega)$ to itself when $\Omega$ is a bounded semiconvex domain in $\mathbb{R}^n$ and $p_{\Phi}^-\in(\frac{n}{n+1},1]$, and the operators $\nabla^2{\mathbb G}_N$ are bounded from $h^{\Phi}_{\omega,\,z}(\Omega)$ to $L^{\Phi}_{\omega}(\Omega)$, and from $h^{\Phi}_{\omega,\,z}(\Omega)$ to $h^{\Phi}_{\omega,\,r}(\Omega)$ when $\Omega$ is a bounded convex domain in $\mathbb{R}^n$ and $p_{\Phi}^-\in(\frac{n}{n+1},1]$, where ${\mathbb G}_D$ and ${\mathbb G}_N$ denote, respectively, the Dirichlet Green operator and the Neumann Green operator., Comment: This paper has been withdrawn by the authors
- Published
- 2013
48. Multiple-set split feasibility problems for a finite family of demicontractive mappings in Hilbert spaces
- Author
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Gang Wang and Li-Juan Qin
- Subjects
Set (abstract data type) ,Discrete mathematics ,Algebra ,symbols.namesake ,Iterative method ,Applied Mathematics ,General Mathematics ,Hilbert space ,symbols ,Mathematics - Abstract
In this paper, we introduce an iterative algorithm for solving the multiple-set split feasibility problems for a finite family demicontractive mappings in Hilbert spaces. The results presented in this paper improve and extend some recent corresponding results in (4), (7), (9), (10), (13), (14).
- Published
- 2013
49. Gigantic and small components in random distance graphs of special form
- Author
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A. R. Yarmukhametov
- Subjects
Random graph ,Discrete mathematics ,General Mathematics ,Distance-regular graph ,1-planar graph ,Geometric graph theory ,Planar graph ,Combinatorics ,symbols.namesake ,Graph power ,Random regular graph ,symbols ,Mathematics ,Forbidden graph characterization - Abstract
The paper [1] contained the proof of the theorem on the existence of a gigantic component in a random distance graph for the case in which the edge probability is equal to p = c/N , where c > 1. There it was also stated that “all the other vertices are contained in components of size o(N)”. In the present paper, we succeed in showing that “all the other vertices are contained in components of size O(lnN)”. This result is a significant step forward, because it is truly an analog of the Erdős–Renyi theorem for the classical model (see [2]). In the present paper, we consider the problem of the threshold probability of the existence of a gigantic component for a series of random distance graphs of special form. Set n = 4k, k ∈ N, and N = C n 1 and consider a complete distance graph GN = (VN , EN ) such that
- Published
- 2013
50. On the distribution of zeros of polynomials and analytic functions
- Author
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Sejong Chungnam and Younseok Choo
- Subjects
Discrete mathematics ,Gegenbauer polynomials ,General Mathematics ,Discrete orthogonal polynomials ,Algebra ,Classical orthogonal polynomials ,symbols.namesake ,Difference polynomials ,Wilson polynomials ,Hahn polynomials ,Orthogonal polynomials ,symbols ,Jacobi polynomials ,Mathematics - Abstract
This paper presents several results on the distribution of zeros of polynomials and analytic functions. The results of this paper include some existing ones as special cases. Mathematics Subject Classification: 30C10, 30C15
- Published
- 2013
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