3,217 results
Search Results
402. Study of Sensitivity of Parameters of Bernstein–Stancu Operators
- Author
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Vishnu Narayan Mishra and R. B. Gandhi
- Subjects
Discrete mathematics ,General Mathematics ,Uniform convergence ,010102 general mathematics ,General Physics and Astronomy ,General Chemistry ,01 natural sciences ,Bernstein polynomial ,Upper and lower bounds ,010101 applied mathematics ,Alpha (programming language) ,General Earth and Planetary Sciences ,Beta (velocity) ,Sensitivity (control systems) ,0101 mathematics ,General Agricultural and Biological Sciences ,Mathematics - Abstract
This paper is aimed at studying sensitivity of parameters $$\alpha $$ and $$\beta $$ appearing in the operators introduced by Stancu (Studia Universitatis Babes-Bolyai 14(2):31–45, 1969). Results are established on the behavior of the nodes used in Bernstein–Stancu polynomials and the nodes used in Bernstein polynomials and graphical presentations of them are generated. Alternate proof of uniform convergence of Bernstein–Stancu operators and an upper bound estimation are derived. It is also established that the parameters $$\alpha $$ and $$\beta $$ in Bernstein–Stancu polynomials can be used to get better approximation at a point $$x = \alpha /\beta $$ in [0, 1] to the Bernstein polynomials.
- Published
- 2019
403. On the Gitik–Shelah theorem
- Author
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Joanna Jureczko and Ryszard Frankiewicz
- Subjects
010101 applied mathematics ,Discrete mathematics ,Mathematics::Logic ,General Mathematics ,010102 general mathematics ,Mathematics::General Topology ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
The well-known Gitik–Shelah theorem (1989) touches the problem of existence isomorphisms between some quotient algebras. In this paper, we study a relation between the existence of such isomorphisms and the existence of so-called Kuratowski partitions of adequate Baire spaces. For this purpose, we give strictly combinatorial methods.
- Published
- 2019
404. Refinement of the Chowla–Erdős method and linear independence of certain Lambert series
- Author
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Daniel Duverney and Yohei Tachiya
- Subjects
010101 applied mathematics ,Discrete mathematics ,Lambert series ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Linear independence ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we refine the method of Chowla and Erdős on the irrationality of Lambert series and study a necessary condition for the infinite series ∑ θ ( n ) / q n {\sum\theta(n)/q^{n}} to be a rational number, where q is an integer with | q | > 1 {|q|>1} and θ is an arithmetic function with suitable divisibility and growth conditions. As applications of our main theorem, we give linear independence results for various kinds of Lambert series.
- Published
- 2019
405. On convergence of sequences of functions possessing closed graphs
- Author
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Tomasz Natkaniec and Waldemar Sieg
- Subjects
010101 applied mathematics ,Discrete mathematics ,General Mathematics ,010102 general mathematics ,Convergence (routing) ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In the first part of the paper we study the sets of boundedness and of convergence and divergence to infinity of sequences of real closed-graph functions. Generalization on ideal convergence of such sequences is discussed. Limits and ideal-limits of sequences of functions with closed graphs are considered in the last part of the article.
- Published
- 2019
406. A new type of generalized closed set via γ-open set in a fuzzy bitopological space
- Author
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Jayasree Chakaraborty, Birojit Das, Arnab Paul, Baby Bhattacharya, and Sree Anusha Ganapathiraju
- Subjects
Discrete mathematics ,Closed set ,Relation (database) ,Mathematics::General Mathematics ,General Mathematics ,Open set ,010103 numerical & computational mathematics ,Function (mathematics) ,(i, j)*-generalized fuzzy γ-irresolute function ,Type (model theory) ,01 natural sciences ,Fuzzy logic ,Bitopological space ,(i, j)*-fuzzy γ-open set ,010101 applied mathematics ,Set (abstract data type) ,(i, j)*-generalized fuzzy γ-continuous function ,(i, j)*-generalized fuzzy γ-closed set ,(i, j)*-γ-generalized fuzzy closed set ,0101 mathematics ,Mathematics - Abstract
This paper aims to present the notion of (i, j)*-fuzzy γ-open set in a fuzzy bitopological space as a parallel form of (i, j)-fuzzy γ-open set due to Tripathy and Debnath (2013) [17] and show that both of them are independent concepts. Then we extend our study to (i, j)*-generalized fuzzy γ-closed set and (i, j)*-γ-generalized fuzzy closed set. We show that (i, j)*-γ-generalized fuzzy closed set and (i, j)*-generalized fuzzy γ-closed set are also independent of each other in nature. Though every (i, j)*-fuzzy γ-closed set is a (i, j)*-generalized fuzzy γ-closed set but with (i, j)*-γ-generalized fuzzy closed set, the same relation is not linear. Similarly though every (i, j)*-fuzzy closed set is (i, j)*-γ-generalized fuzzy closed set but it is independent to (i, j)*-generalized fuzzy γ-closed set. Various properties related to (i, j)*-generalized fuzzy γ-closed set are also studied. Finally, (i, j)*-generalized fuzzy γ-continuous function and (i, j)*-generalized fuzzy γ-irresolute functions are introduced and interrelationships among them are established. We characterized these functions in different directions for different applications.
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- 2019
407. Formality and Kontsevich–Duflo type theorems for Lie pairs
- Author
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Mathieu Stiénon, Ping Xu, and Hsuan-Yi Liao
- Subjects
Mathematics - Differential Geometry ,Discrete mathematics ,General Mathematics ,Simple Lie group ,010102 general mathematics ,Gerstenhaber algebra ,Quasi-isomorphism ,Type (model theory) ,01 natural sciences ,Cohomology ,Differential Geometry (math.DG) ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Quantum Algebra (math.QA) ,010307 mathematical physics ,Nabla symbol ,Todd class ,Isomorphism ,0101 mathematics ,Mathematics - Abstract
$\newcommand{\poly}{_{\operatorname{poly}}^{\bullet}}\newcommand{\td}{(\operatorname{td}_{L/A}^{\nabla})^{\frac{1}{2}}}\newcommand{\cx}[1]{\operatorname{tot}\big(\Gamma(\Lambda^\bullet A^\vee)\otimes_R\mathcal{#1}\poly\big)}\newcommand{\cy}[1]{\mathbb{H}^\bullet_{\operatorname{CE}}(A,\mathcal{#1}\poly)}$Kontsevich's formality theorem states that there exists an $L_\infty$ quasi-isomorphism from the dgla $T\poly(M)$ of polyvector fields on a smooth manifold $M$ to the dgla $D\poly(M)$ of polydifferential operators on $M$, which extends the classical Hochschild--Kostant--Rosenberg map. In this paper, we extend Kontsevich's formality theorem to Lie pairs, a framework which includes a range of diverse geometric contexts such as complex manifolds, foliations, and $\mathfrak{g}$-manifolds. The spaces $\cx{T}$ and $\cx{D}$ associated with a Lie pair $(L,A)$ each carry an $L_\infty$ algebra structure canonical up to $L_\infty$ isomorphism. These two spaces serve as replacements for the spaces of polyvector fields and polydifferential operators, respectively. Their corresponding cohomology groups $\cy{T}$ and $\cy{D}$ admit canonical Gerstenhaber algebra structures. We establish the following formality theorem for Lie pairs: there exists an $L_\infty$ quasi isomorphism from $\cx{T}$ to $\cx{D}$ whose first Taylor coefficient is equal to $\operatorname{hkr}\circ\td$. Here $\td$ acts on $\cx{T}$ by contraction. Furthermore, we prove a Kontsevich--Duflo type theorem for Lie pairs: the Hochschild--Kostant--Rosenberg map twisted by the square root of the Todd class of the Lie pair $(L,A)$ is an isomorphism of Gerstenhaber algebras from $\cy{T}$ to $\cy{D}$. As applications, we establish formality theorems and Kontsevich--Duflo type theorems for complex manifolds, foliations, and $\mathfrak{g}$-manifolds. In the case of complex manifolds, we recover the Kontsevich--Duflo theorem of complex geometry., Comment: 55 pages, several typos corrected, some references added, some minor cosmetic changes in the presentation
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- 2019
408. Construction of (M, N)-hypermodule over (R, S)-hyperring
- Author
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Saeed Mirvakili, S. M. Anvariyeh, and Bijan Davvaz
- Subjects
Discrete mathematics ,hypermodule ,20n20 ,Mathematics::General Mathematics ,General Mathematics ,010103 numerical & computational mathematics ,hyperring ,01 natural sciences ,hypergroup ,16y99 ,010101 applied mathematics ,QA1-939 ,0101 mathematics ,Mathematics - Abstract
The aim of this paper is to introduce a new class of hyper-modules that may be called (M, N)-hypermodules over (R, S)-hyperrings. Then, we investigate some properties of this new class of hyperstructures. Since the main tools in the theory of hyperstructures are the fundamental relations, we give some results about them with respect to the fundamental relations.
- Published
- 2019
409. The variety of domination games
- Author
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Gašper Košmrlj, Tilen Marc, Boštjan Brešar, Máté Vizer, Balázs Patkós, Sandi Klavžar, Zsolt Tuza, Tanja Gologranc, and Csilla Bujtás
- Subjects
Discrete mathematics ,Computer Science::Computer Science and Game Theory ,Conjecture ,Domination analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Graph ,Continuation ,Dominator ,Bounded function ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Graph property ,Mathematics - Abstract
Domination game (Bresar et al. in SIAM J Discrete Math 24:979–991, 2010) and total domination game (Henning et al. in Graphs Comb 31:1453–1462 (2015) are by now well established games played on graphs by two players, named Dominator and Staller. In this paper, Z-domination game, L-domination game, and LL-domination game are introduced as natural companions of the standard domination games. Versions of the Continuation Principle are proved for the new games. It is proved that in each of these games the outcome of the game, which is a corresponding graph invariant, differs by at most one depending whether Dominator or Staller starts the game. The hierarchy of the five domination games is established. The invariants are also bounded with respect to the (total) domination number and to the order of a graph. Values of the three new invariants are determined for paths up to a small constant independent from the length of a path. Several open problems and a conjecture are listed. The latter asserts that the L-domination game number is not greater than 6 / 7 of the order of a graph.
- Published
- 2019
410. Lucas numbers as sums of two repdigits
- Author
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Alain Togbé, Florian Luca, and Chèfiath Adegbindin
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Discrete mathematics ,010104 statistics & probability ,Number theory ,Lucas number ,General Mathematics ,Ordinary differential equation ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we determine all Lucas numbers that are sums of two repdigits. The largest one is L14 = 843 = 66 + 777.
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- 2019
411. Polynomial Equivalence of the Problems 'Predicate Formulas Isomorphism and Graph Isomorphism'
- Author
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T. M. Kosovskaya and N. N. Kosovskii
- Subjects
Discrete mathematics ,General Mathematics ,010102 general mathematics ,Predicate (mathematical logic) ,01 natural sciences ,010305 fluids & plasmas ,First-order logic ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0103 physical sciences ,Isomorphism ,0101 mathematics ,Graph isomorphism ,Equivalence (formal languages) ,Mathematics - Abstract
The problem of isomorphism checking of two elementary conjunctions of predicate formulas is considered in this work. Such a problem appears while solving some Artificial Intelligence problems, admitting formalization by means of predicate calculus language. The exact definition of the concept of isomorphism of such formulas is given in this paper. However, isomorphic elementary conjunctions of predicate formulas are formulas that, with some substitution of variables instead of their arguments, coincide with the accuracy of the order of writing literals. Problems are described that, when solved, mean the necessity of testing formulas for isomorphism arises. Polynomial equivalence of this problem with the Graph Isomorphism (GI) problem is proved.
- Published
- 2019
412. p-Adic Multiwavelet Sets
- Author
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Debasis Haldar and Divya Singh
- Subjects
Set (abstract data type) ,Discrete mathematics ,Wavelet ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Computer Science::Numerical Analysis ,01 natural sciences ,Mathematics - Abstract
This paper contains a brief review of p-adic MRA theory along with the introduction of p-adic multiwavelet set. Here we have studied various properties of p-adic multiwavelet sets such as disjointness of their dilates, counting formula for the elements in a wavelet set etc. and proved some results analogous to real setting, in special cases.
- Published
- 2019
413. Axioms of Soft Logic
- Author
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Moshe Klein and Oded Maimon
- Subjects
Discrete mathematics ,General Mathematics ,Infinitesimal ,010102 general mathematics ,Dual number ,Term (logic) ,Type (model theory) ,01 natural sciences ,Algebraic operation ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Axiom ,Mathematics ,Real number ,p-adic number - Abstract
In this paper, we develop the foundation of a new mathematical language, which we term “Soft Logic”. This language enables us to present an extension of the number 0 from a singular point to a continuous line. We create a distinction between −0 and +0 and generate a new type of numbers, which we call ‘Bridge Numbers’ (BN): $${\boldsymbol{a}}\overline {\bf{0}} \bot {\boldsymbol{b}}\overline {\bf{1}} ,$$ where a, b are real numbers, “a” is the value on the $$\overline {\bf{0}} $$ axis, and “b” is the value on the $$\overline {\bf{1}} $$ axis. We proceed by defining arithmetic and algebraic operations on the Bridge Numbers, investigate their properties, and conclude by defining goals for further research. In the Attachment, we continue comparing our results with existing mathematical work on Infinitesimals, Dual numbers, and Nonstandard analysis. The research is a part of “Digital living 2030” project with Stanford University.
- Published
- 2019
414. On the paranormed space $\mathcal{M}_{u}(t)$ of double sequences
- Author
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Feyzi Başar and Hüsamettin Çapan
- Subjects
Discrete mathematics ,General Mathematics ,lcsh:Mathematics ,010103 numerical & computational mathematics ,matrix transformations ,lcsh:QA1-939 ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Transformation matrix ,Paranormed double sequence space ,Bounded function ,Dual spaces ,Beta (velocity) ,0101 mathematics ,Mathematics ,Normed vector space - Abstract
In this paper, we introduce the paranormed sequence space $\mathcal{M}_{u}(t)$ corresponding to the normed space $\mathcal{M}_{u}$ of bounded double sequences. We examine general topological properties of this space and determine its alpha-, beta- and gamma-duals. Furthermore, we characterize some classes of four-dimensional matrix transformations concerning this space and its dual spaces.
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- 2019
415. On fundamental units of real quadratic fields of class number 1
- Author
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Florian Luca and Andrej Dujella
- Subjects
Discrete mathematics ,General Mathematics ,010102 general mathematics ,Quadratic elds, class number, continued fractions ,01 natural sciences ,Upper and lower bounds ,Quadratic equation ,Base unit (measurement) ,Norm (mathematics) ,0103 physical sciences ,Quadratic field ,010307 mathematical physics ,0101 mathematics ,Class number ,Mathematics - Abstract
In this paper, we give a nontrivial lower bound for the fundamental unit of norm $$-1$$ of a real quadratic field of class number 1.
- Published
- 2019
416. Common fixed point theorems in fuzzy metric‐like spaces employing common property (E.A.)
- Author
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Phumin Sumalai, Mohd Hasan, Dhananjay Gopal, and Poom Kumam
- Subjects
Discrete mathematics ,Property (philosophy) ,General Mathematics ,010102 general mathematics ,General Engineering ,01 natural sciences ,Fuzzy logic ,Fuzzy metric space ,010101 applied mathematics ,Metric (mathematics) ,Common fixed point ,Common property ,0101 mathematics ,Mathematics - Abstract
The purpose of this paper is to utilize the property (E.A.) and the common property (E.A.) to prove some existence results on common fixed point for contractive mappings in fuzzy metric spaces which include fuzzy metric spaces of two types, namely, Kramosil and Michalek fuzzy metric spaces along with George and Veeramani fuzzy metric spaces. Our results generalize and extend several relevant common fixed point theorems from the literature. We also furnish an illustrative example. © 2011 Elsevier Ltd. All rights reserved.
- Published
- 2019
417. Power bounded weighted composition operators and power bounded below composition operators
- Author
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Hamzeh Keshavarzi
- Subjects
Discrete mathematics ,Mathematics::Complex Variables ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,05 social sciences ,Composition (combinatorics) ,01 natural sciences ,Unit disk ,Dirichlet space ,Power (physics) ,Alpha (programming language) ,Bounded function ,0502 economics and business ,0101 mathematics ,Algebra over a field ,Complex plane ,050203 business & management ,Mathematics - Abstract
In this paper, we characterize power bounded weighted composition operators on weighted Bergman spaces of strongly pseudoconvex bounded domains in $${\mathbb {C}}^n$$. Also, we introduce the notion of power bounded below operators, then, for $$\alpha >0$$, we characterize power bounded below composition operators on $${\mathcal {D}}_\alpha $$, the weighted Dirichlet space on the unit disk of the complex plane.
- Published
- 2019
418. Algebraic Structures of Constacyclic Codes Over Finite Chain Rings and Power Series Rings
- Author
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Habib Sharif, Marziyeh Beygi, and Shohreh Namazi
- Subjects
Power series ,Discrete mathematics ,Ring (mathematics) ,Formal power series ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,Hamming distance ,0102 computer and information sciences ,General Chemistry ,01 natural sciences ,Linear span ,Chain (algebraic topology) ,010201 computation theory & mathematics ,General Earth and Planetary Sciences ,Maximal ideal ,Generator matrix ,0101 mathematics ,General Agricultural and Biological Sciences ,Mathematics - Abstract
Let R be a chain ring with the maximal ideal $$\langle \gamma \rangle$$ . In this paper, we shall study $$(r_1+r_2\gamma )$$ -constacyclic codes of arbitrary length over R, where $$r_1, r_2$$ are units in R. We shall obtain the generators of these codes and their duals. Moreover, a minimal spanning set (and so a generator matrix) for a constacyclic code is obtained. We shall determine the minimum Hamming distance of a constacyclic code over the chain ring R. At last, we shall study some constacyclic codes over formal power series rings. The minimal spanning set for these codes are also established.
- Published
- 2019
419. On the Cegrell Classes Associated to a Positive Closed Current
- Author
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Mohamed Zaway
- Subjects
Discrete mathematics ,Domain of a function ,Current (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Open set ,01 natural sciences ,Omega ,010104 statistics & probability ,Operator (computer programming) ,Bounded function ,0101 mathematics ,Mathematics - Abstract
The aim of this paper is to study the operator (ddc▪)q ∧ T on some classes of plurisubharmonic (psh) functions, which are not necessary bounded, where T is a positive closed current of bidimension (q, q) on an open set Ω of ℂn. The author introduces two classes $${\cal F}_p^T\left( {\rm{\Omega }} \right)$$ and $${\cal E}_p^T\left( {\rm{\Omega }} \right)$$ and shows first that they belong to the domain of definition of the operator (ddc▪)q ∧ T. Then the author proves that all functions that belong to these classes are CT-quasi-continuous and that the comparison principle is valid for them.
- Published
- 2019
420. The N-star network evolution model
- Author
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István Fazekas, Attila Perecsényi, and Csaba Noszály
- Subjects
Statistics and Probability ,Star network ,Random graph ,Discrete mathematics ,General Mathematics ,010102 general mathematics ,Joins ,Preferential attachment ,01 natural sciences ,Power law ,Doob–Meyer decomposition theorem ,010104 statistics & probability ,Asymptotic power ,0101 mathematics ,Statistics, Probability and Uncertainty ,Unit (ring theory) ,Mathematics - Abstract
A new network evolution model is introduced in this paper. The model is based on cooperations of N units. The units are the nodes of the network and the cooperations are indicated by directed links. At each evolution step N units cooperate, which formally means that they form a directed N-star subgraph. At each step either a new unit joins the network and it cooperates with N − 1 old units, or N old units cooperate. During the evolution both preferential attachment and uniform choice are applied. Asymptotic power law distributions are obtained both for in-degrees and for out-degrees.
- Published
- 2019
421. Some fixed and common fixed point results in G-metric spaces which can't be obtained from metric spaces
- Author
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Kamaleldin Abodayeh and Wasfi Shatanawi
- Subjects
010101 applied mathematics ,Discrete mathematics ,Metric space ,021103 operations research ,General Mathematics ,0211 other engineering and technologies ,Common fixed point ,02 engineering and technology ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we use the concepts of (A;B)-weakly increasing mappings and altering distance functions to establish new contractive conditions for the pair of mappings in the setting of G-metric spaces. Many Fixed and common Fixed point results in the setting of G{metric spaces are formulated. Note that our new contractive conditions can't be reduces to contractive conditions in standard metric spaces.
- Published
- 2019
422. Topological Types of Algebraic Stacks
- Author
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Chang-Yeon Cho
- Subjects
Discrete mathematics ,Homotopy group ,Homotopy category ,Model category ,General Mathematics ,Homotopy ,010102 general mathematics ,Cofibration ,Topology ,Mathematics::Algebraic Topology ,01 natural sciences ,n-connected ,Derived algebraic geometry ,0103 physical sciences ,A¹ homotopy theory ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
The main goal of this paper is to set a foundation for homotopy theory of algebraic stacks under model category theory and to show how it can be applied in various contexts. It not only generalizes the étale homotopy theory to algebraic stacks but also provides more suitable framework for the homotopy theory in a broader context. Also, a new result that the profinite completion of pro-simplicial sets admits a right adjoint is provided and integrated with the foundational work to generalize Artin–Mazur’s comparison theorem from schemes to algebraic stacks in a formal way.
- Published
- 2019
423. Lattice rules with random n achieve nearly the optimal O(n−α−1∕2) error independently of the dimension
- Author
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Dirk Nuyens, Peter Kritzer, Frances Y. Kuo, and Mario Ullrich
- Subjects
Discrete mathematics ,Numerical Analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Upper and lower bounds ,Randomized algorithm ,Numerical integration ,Sobolev space ,Combinatorics ,Random variate ,Rate of convergence ,Lattice (order) ,Exponent ,0101 mathematics ,Analysis ,Mathematics - Abstract
We analyze a new random algorithm for numerical integration of d -variate functions over [ 0 , 1 ] d from a weighted Sobolev space with dominating mixed smoothness α ≥ 0 and product weights 1 ≥ γ 1 ≥ γ 2 ≥ ⋯ > 0 , where the functions are continuous and periodic when α > 1 ∕ 2 . The algorithm is based on rank-1 lattice rules with a random number of points n . For the case α > 1 ∕ 2 , we prove that the algorithm achieves almost the optimal order of convergence of O ( n − α − 1 ∕ 2 ) , where the implied constant is independent of the dimension d if the weights satisfy ∑ j = 1 ∞ γ j 1 ∕ α ∞ . The same rate of convergence holds for the more general case α > 0 by adding a random shift to the lattice rule with random n . This shows, in particular, that the exponent of strong tractability in the randomized setting equals 1 ∕ ( α + 1 ∕ 2 ) , if the weights decay fast enough. We obtain a lower bound to indicate that our results are essentially optimal. This paper is a significant advancement over previous related works with respect to the potential for implementation and the independence of error bounds on the problem dimension. Other known algorithms which achieve the optimal error bounds, such as those based on Frolov’s method, are very difficult to implement especially in high dimensions. Here we adapt a lesser-known randomization technique introduced by Bakhvalov in 1961. This algorithm is based on rank-1 lattice rules which are very easy to implement given the integer generating vectors. A simple probabilistic approach can be used to obtain suitable generating vectors.
- Published
- 2019
424. Uniqueness of Coxeter structures on Kac–Moody algebras
- Author
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Valerio Toledano Laredo and Andrea Appel
- Subjects
Pure mathematics ,Lie bialgebra ,General Mathematics ,Braid group ,Category O ,01 natural sciences ,symbols.namesake ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Category Theory (math.CT) ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics ,Discrete mathematics ,Weyl group ,Functor ,Quantum group ,010102 general mathematics ,Coxeter group ,Mathematics - Category Theory ,Monodromy ,symbols ,010307 mathematical physics ,Mathematics - Representation Theory - Abstract
Let g be a symmetrisable Kac-Moody algebra, and U_h(g) the corresponding quantum group. We showed in arXiv:1610.09744 and arXiv:1610.09741 that the braided quasi-Coxeter structure on integrable, category O representations of U_h(g) which underlies the R-matrix actions arising from the Levi subalgebras of U_h(g) and the quantum Weyl group action of the generalised braid group B_g can be transferred to integrable, category O representations of g. We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, R--matrices, and local monodromies. This extends, simplifies and strengthens a similar result of the second author valid when g is semisimple, and is used in arXiv:1512.03041 to describe the monodromy of the rational Casimir connection of g in terms of the quantum Weyl group operators of U_h(g). Our main tool is a refinement of Enriquez's universal algebras, which is adapted to the PROP describing a Lie bialgebra graded by the non-negative roots of g., Expanded Introduction and Sec. 5 to discuss convolution product (5.11), cosimplicial structure on basis elements (5.13) and module structure on coinvariants (5.15). Minor revisions in Sec. 7.1 (gradings), 7.4 (deformation DY modules), 9.7 (exposition), 15.7 (Drinfeld double) and 15.15 (rigidity for diagrammatic KM algebras). Final version, to appear in Adv. Math. 81 pages
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- 2019
425. On Bounds for Probabilities of Combinations of Events, the Jordan Formula, and the Bonferroni Inequalities
- Author
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Andrei N. Frolov
- Subjects
Discrete mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Conditional probability ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Bonferroni correction ,0103 physical sciences ,symbols ,0101 mathematics ,media_common ,Mathematics ,Corresponding conditional - Abstract
This paper presents a method for deriving optimal lower and upper bounds for probabilities and conditional probabilities (given a σ-field) for various combinations of events. The optimality is understood as the possibility that inequalities become equalities for some sets of events. New generalizations of the Jordan formula and the Bonferroni inequalities are obtained. The corresponding conditional versions of these results are also considered.
- Published
- 2019
426. Higher Summability and Discrete Weighted Muckenhoupt and Gehring Type Inequalities
- Author
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Ireneusz Kubiaczyk and Samir H. Saker
- Subjects
Discrete mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Type (model theory) ,Mathematical proof ,01 natural sciences ,010101 applied mathematics ,Completeness (order theory) ,0101 mathematics ,Reverse holder inequality ,media_common ,Weighted space ,Interpolation ,Mathematics - Abstract
In this paper, we prove some reverse discrete inequalities with weights of Muckenhoupt and Gehring types and use them to prove some higher summability theorems on a higher weighted space $l_{w}^{p}({\open N})$ form summability on the weighted space $l_{w}^{q}({\open N})$ when p>q. The proofs are obtained by employing new discrete weighted Hardy's type inequalities and their converses for non-increasing sequences, which, for completeness, we prove in our special setting. To the best of the authors' knowledge, these higher summability results have not been considered before. Some numerical results will be given for illustration.
- Published
- 2019
427. Refinement of Lower Bounds of the Chromatic Number of a Space with Forbidden One-Color Triangles
- Author
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A. E. Kupriyanov and A. V. Bobu
- Subjects
Discrete mathematics ,TheoryofComputation_COMPUTATIONBYABSTRACTDEVICES ,General Mathematics ,010102 general mathematics ,02 engineering and technology ,Space (mathematics) ,01 natural sciences ,020303 mechanical engineering & transports ,0203 mechanical engineering ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,Chromatic scale ,0101 mathematics ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics - Abstract
The present paper deals with estimates of the chromatic number of a space with forbidden one-color triangles. New lower bounds for the quantity under study, which are sharper than all bounds obtained so far, are presented.
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- 2019
428. Scalarizations for a set optimization problem using generalized oriented distance function
- Author
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C. S. Lalitha and Khushboo
- Subjects
Discrete mathematics ,021103 operations research ,Optimization problem ,Relation (database) ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,Solution set ,Signed distance function ,02 engineering and technology ,Operator theory ,01 natural sciences ,Potential theory ,Theoretical Computer Science ,Set (abstract data type) ,Bounded function ,0101 mathematics ,Analysis ,Mathematics - Abstract
The aim of this paper is to establish scalarizations for minimal and weak minimal solutions of a set optimization problem using generalized oriented distance function introduced by Crespi et al. (Math Methods Oper Res 63:87–106, 2006). The solution concepts are based on a partial set order relation on the family of nonempty bounded sets proposed by Karaman et al. (Positivity 22:783–802, 2018). Finally, we also provide existence results for minimal solutions and sufficient conditions for the solution sets to be closed.
- Published
- 2019
429. Common Fixed Points in Generalized Metric Spaces with a Graph
- Author
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Mustapha Kabil, Abdessamad Kamouss, and Karim Chaira
- Subjects
Discrete mathematics ,Article Subject ,lcsh:Mathematics ,General Mathematics ,010102 general mathematics ,Fixed-point theorem ,Fixed point ,lcsh:QA1-939 ,01 natural sciences ,Coincidence ,010101 applied mathematics ,Metric space ,Metric (mathematics) ,Graph (abstract data type) ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
The aim of this paper is to prove the existence and uniqueness of points of coincidence and common fixed points for a pair of self-mappings defined on generalized metric spaces with a graph. Our results improve and extend several recent results of metric fixed point theory.
- Published
- 2019
430. Uniformly distributed measures have big pieces of Lipschitz graphs locally
- Author
-
A. Dali Nimer
- Subjects
Discrete mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Metric Geometry ,0101 mathematics ,Variety (universal algebra) ,Lipschitz continuity ,01 natural sciences ,Measure (mathematics) ,Mathematics - Abstract
The study of uniformly distributed measures was crucial in Preiss' proof of his theorem on rectifiability of measures with positive density. It is known that the support of a uniformly distributed measure is an analytic variety. In this paper, we provide quantitative information on the rectifiability of this variety. Tolsa had already shown that $n$-uniform measures have Big Pieces of Lipschitz Graphs(BPLG) . Here, we prove that a uniformly distributed measure has BPLG locally.
- Published
- 2019
431. Interlacing polynomials and the veronese construction for rational formal power series
- Author
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Philip B. Zhang
- Subjects
Discrete mathematics ,Property (philosophy) ,Formal power series ,General Mathematics ,010102 general mathematics ,Interlacing ,0102 computer and information sciences ,01 natural sciences ,Identity (mathematics) ,Integer ,05A15, 13A02, 26C10, 52B20, 52B45 ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Descent (mathematics) ,Mathematics - Abstract
Fixing a positive integer $r$ and $0 \le k \le r-1$, define $f^{\langle r,k \rangle}$ for every formal power series $f$ as $ f(x) = f^{\langle r,0 \rangle}(x^r)+xf^{\langle r,1 \rangle}(x^r)+ \cdots +x^{r-1}f^{\langle r,r-1 \rangle}(x^r).$ Jochemko recently showed that the polynomial $U^{n}_{r,k}\, h(x) := \left( (1+x+\cdots+x^{r-1})^{n} h(x) \right)^{\langle r,k \rangle}$ has only nonpositive zeros for any $r \ge \deg h(x) -k$ and any positive integer $n$. As a consequence, Jochemko confirmed a conjecture of Beck and Stapledon on the Ehrhart polynomial $h(x)$ of a lattice polytope of dimension $n$, which states that $U^{n}_{r,0}\,h(x)$ has only negative, real zeros whenever $r\ge n$. In this paper, we provide an alternative approach to Beck and Stapledon's conjecture by proving the following general result: if the polynomial sequence $\left( h^{\langle r,r-i \rangle}(x)\right)_{1\le i \le r}$ is interlacing, so is $\left( U^{n}_{r,r-i}\, h(x) \right)_{1\le i \le r}$. Our result has many other interesting applications. In particular, this enables us to give a new proof of Savage and Visontai's result on the interlacing property of some refinements of the descent generating functions for colored permutations. Besides, we derive a Carlitz identity for refined colored permutations., Comment: 18 pages
- Published
- 2019
432. On the Discrete Criteria and Jørgensen Inequalities for SL(m, F̅((t)))
- Author
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Jinghua Yang
- Subjects
Discrete mathematics ,Mathematics::Commutative Algebra ,Inequality ,Mathematics::Operator Algebras ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Special linear group ,01 natural sciences ,Mathematics::Group Theory ,010104 statistics & probability ,0101 mathematics ,media_common ,Mathematics - Abstract
In this paper, the author gives the discrete criteria and Jorgensen inequalities of subgroups for the special linear group on F((t)) in two and higher dimensions.
- Published
- 2019
433. Asymptotic gcd and divisible sequences for entire functions
- Author
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Julie Wang and Ji Guo
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Entire function ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Let f f and g g be algebraically independent entire functions. We first give an estimate of the Nevanlinna counting function for the common zeros of f n − 1 f^n-1 and g n − 1 g^n-1 for sufficiently large n n . We then apply this estimate to study divisible sequences in the sense that f n − 1 f^n-1 is divisible by g n − 1 g^n-1 for infinitely many n n . For the first part of establishing our gcd estimate, we need to formulate a truncated second main theorem for effective divisors by modifying a theorem from a paper by Hussein and Ru and explicitly computing the constants involved for a blowup of P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 along a point with its canonical divisor and the pull-back of vertical and horizontal divisors of P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 .
- Published
- 2019
434. Unbalanced multi-drawing urn with random addition matrix
- Author
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Selmi Olfa and Aguech Rafik
- Subjects
Discrete mathematics ,General Mathematics ,0102 computer and information sciences ,Stochastic approximation ,01 natural sciences ,010104 statistics & probability ,Discrete time and continuous time ,010201 computation theory & mathematics ,Bounded function ,0101 mathematics ,Martingale (probability theory) ,Random variable ,Mathematics ,Central limit theorem - Abstract
In this paper, we consider a two color multi-drawing urn model. At each discrete time step, we draw uniformly at random a sample of m balls (m≥1) and note their color, they will be returned to the urn together with a random number of balls depending on the sample’s composition. The replacement rule is a 2 × 2 matrix depending on bounded discrete positive random variables. Using a stochastic approximation algorithm and martingales methods, we investigate the asymptotic behavior of the urn after many draws.
- Published
- 2019
435. End point results of generalized setvalued almost contractions in metric spaces endowed with a graph
- Author
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Nikhilesh Metiya, P. Saha, D. Khatua, and Binayak S. Choudhury
- Subjects
010101 applied mathematics ,Discrete mathematics ,Metric space ,End point ,Graph theoretic ,General Mathematics ,010102 general mathematics ,Graph (abstract data type) ,Graph theory ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we consider multivalued mappings satisfying two different inequalities and obtain end point results for such mappings in a metric space endowed with a graph. The main theorems are illustrated with an example. The line of research is setvalued analysis in the combined domain of graph theory and metric space. The methodology is a blending of graph theoretic and analytic methods.
- Published
- 2019
436. Cospectrality graphs of Smith graphs
- Author
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P Vesna Todorcevic and M Dragos Cvetkovic
- Subjects
Discrete mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,0101 mathematics ,01 natural sciences ,010305 fluids & plasmas ,Mathematics - Abstract
Graphs whose spectrum belongs to the interval [-2,2] are called Smith graphs. The structure of a Smith graph with a given spectrum depends on a system of Diophantine linear algebraic equations. We have established in [1] several properties of this system and showed how it can be simplified and effectively applied. In this way a spectral theory of Smith graphs has been outlined. In the present paper we introduce cospectrality graphs for Smith graphs and study their properties through examples and theoretical consideration. The new notion is used in proving theorems on cospectrality of Smith graphs. In this way one can avoid the use of the mentioned system of Diophantine linear algebraic equations.
- Published
- 2019
437. Best proximity points revisited
- Author
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Aleksandar Kostic
- Subjects
010101 applied mathematics ,Discrete mathematics ,Metric space ,General Mathematics ,010102 general mathematics ,Mathematics::General Topology ,Point (geometry) ,0101 mathematics ,Type (model theory) ,01 natural sciences ,Mathematics - Abstract
In this paper, using the concept of w-distance on a metric space, we prove some new best proximity point results for the mappings of Meir-Keeler type. As an application, we derive some recent best proximity point results of the aforementioned type.
- Published
- 2019
438. Packing Cycles Faster Than Erdos--Posa
- Author
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Meirav Zehavi, Amer E. Mouawad, Saket Saurabh, and Daniel Lokshtanov
- Subjects
Discrete mathematics ,000 Computer science, knowledge, general works ,General Mathematics ,010102 general mathematics ,Parameterized complexity ,02 engineering and technology ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,Packing problems ,010201 computation theory & mathematics ,Computer Science ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Graph algorithms ,0101 mathematics ,Undirected graph ,Mathematics - Abstract
The Cycle Packing problem asks whether a given undirected graph $G=(V,E)$ contains $k$ vertex-disjoint cycles. Since the publication of the classic Erdös--Pósa theorem in 1965, this problem received significant attention in the fields of graph theory and algorithm design. In particular, this problem is one of the first problems studied in the framework of parameterized complexity. The nonuniform fixed-parameter tractability of Cycle Packing follows from the Robertson--Seymour theorem, a fact already observed by Fellows and Langston in the 1980s. In 1994, Bodlaender showed that Cycle Packing can be solved in time $2^{\mathcal{O}(k^2)}\cdot |V|$ using exponential space. In the case a solution exists, Bodlaender's algorithm also outputs a solution (in the same time). It has later become common knowledge that Cycle Packing admits a $2^{\mathcal{O}(k\log^2k)}\cdot |V|$-time (deterministic) algorithm using exponential space, which is a consequence of the Erdös--Pósa theorem. Nowadays, the design of this algorithm is given as an exercise in textbooks on parameterized complexity. Yet, no algorithm that runs in time $2^{o(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$, beating the bound $2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$, has been found. In light of this, it seems natural to ask whetherthe $2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$ bound is essentially optimal. In this paper, we answer this question negatively by developing a $2^{\mathcal{O}(\frac{k\log^2k}{\log\log k})}\cdot |V|$-time (deterministic) algorithm for Cycle Packing. In the case a solution exists, our algorithm also outputs a solution (in the same time). Moreover, apart from beating the bound $2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$, our algorithm runs in time linear in $|V|$, and its space complexity is polynomial in the input size. publishedVersion
- Published
- 2019
439. Exact Value of the Nonmonotone Complexity of Boolean Functions
- Author
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Anna V. Mikhailovich and V. V. Kochergin
- Subjects
Discrete mathematics ,Basis (linear algebra) ,Markov chain ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,02 engineering and technology ,Function (mathematics) ,01 natural sciences ,020303 mechanical engineering & transports ,Monotone polygon ,0203 mechanical engineering ,0101 mathematics ,Boolean function ,Unit (ring theory) ,Realization (systems) ,Mathematics - Abstract
We study the complexity of the realization of Boolean functions by circuits in infinite complete bases containing all monotone functions with zero weight (cost of use) and finitely many nonmonotone functions with unit weight. The complexity of the realization of Boolean functions in the case where the only nonmonotone element of the basis is negation was completely described by A. A. Markov: the minimum number of negations sufficient for the realization of an arbitrary Boolean function f (the inversion complexity of the function f) is equal to ⌈log2(d(f) + 1)⌉, where d(f) is the maximum (over all increasing chains of sets of values of the variables) number of changes of the function value from 1 to 0. In the present paper, this result is generalized to the case of the computation of Boolean functions over an arbitrary basis B of prescribed form. It is shown that the minimum number of nonmonotone functions sufficient for computing an arbitrary Boolean function f is equal to ⌈log2(d(f)/D(B) +1)⌉, where D(B) = max d(ω); the maximum is taken over all nonmonotone functions ω of the basis B.
- Published
- 2019
440. On the Hodge Structure of Projective Hypersurfaces in Toric Varieties
- Author
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David A. Cox and Victor V. Batyrev
- Subjects
14F10 ,Pure mathematics ,General Mathematics ,Homogeneous coordinate ring ,01 natural sciences ,14C30 ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Projective space ,0101 mathematics ,14M25 ,Algebraic Geometry (math.AG) ,Mathematics ,Discrete mathematics ,Mathematics::Commutative Algebra ,010308 nuclear & particles physics ,Complex projective space ,010102 general mathematics ,14D07 ,Toric variety ,Cohomology ,Hypersurface ,Affine space ,Hodge structure - Abstract
This paper generalizes classical results of Griffiths, Dolgachev and Steenbrink on the cohomology of hypersurfaces in weighted projective spaces. Given a $d$-dimensional projective simplicial toric variety $P$ and an ample hypersurface $X$ defined by an polynomial $f$ in the homogeneous coordinate ring $S$ of $P$ (as defined in an earlier paper of the first author), we show that the graded pieces of the Hodge filtration on $H^d(P - X)$ are naturally isomorphic to certain graded pieces of $S/J(f)$, where $J(f)$ is the Jacobian ideal of $f$. We then discuss how this relates to the primitive cohomology of $X$. Also, if $T$ is the torus contained in $X$, then the intersection of $X$ and $T$ is an affine hypersurface in $T$, and we show how recent results of the second author can be stated using various ideals in the ring $S$. To prove our results, we must give a careful description (in terms of $S$) of $d$-forms and $(d-1)$-forms on the toric variety $P$. For completeness, we also provide a proof of the Bott-Steenbrink-Danilov vanishing theorem for simplicial toric varieties. Other topics considered in the paper include quasi-smooth hypersurfaces and $V$-submanifolds, the structure of the complement of $U$ when $P$ is represented as the quotient of an open subset $U$ of affine space, a generalization of the Euler exact sequence on projective space, and the relation between graded pieces of $R/J(f)$ and the moduli of ample hypersurfaces in $P$., Comment: 43 pages, LaTeX Version 2.09
- Published
- 1993
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- View/download PDF
441. New invariants and class number problem in real quadratic fields
- Author
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Hideo Yokoi
- Subjects
Discrete mathematics ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,11R29 ,01 natural sciences ,Class number formula ,11R11 ,Integer ,Quadratic form ,0103 physical sciences ,Class number problem ,Binary quadratic form ,Quadratic field ,0101 mathematics ,Stark–Heegner theorem ,Mathematics ,Fundamental unit (number theory) - Abstract
In recent papers [10, 11, 12, 13, 14], we defined some new ρ-invariants for any rational prime ρ congruent to 1 mod 4 and D-invariants for any positive square-free integer D such that the fundamental unit εD of real quadratic field Q(√D) satisfies NεD = –1, and studied relationships among these new invariants and already known invariants.One of our main purposes in this paper is to generalize these D-invariants to invariants valid for all square-free positive integers containing D with NεD = 1. Another is to provide an improvement of the theorem in [14] related closely to class number one problem of real quadratic fields. Namely, we provide, in a sense, a most appreciable estimation of the fundamental unit to be able to apply, as usual (cf. [3, 4, 5, 9, 12, 13]), Tatuzawa’s lower bound of L(l, XD) (Cf[7]) for estimating the class number of Q(√D) from below by using Dirichlet’s classical class number formula.
- Published
- 1993
442. Rosenthal’s Inequalities for Asymptotically Almost Negatively Associated Random Variables Under Upper Expectations
- Author
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Ning Zhang and Yuting Lan
- Subjects
Discrete mathematics ,Inequality ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Space (mathematics) ,01 natural sciences ,010104 statistics & probability ,Probability space ,Law of large numbers ,Negatively associated ,0101 mathematics ,Random variable ,Mathematics ,media_common - Abstract
In this paper, the authors generalize the concept of asymptotically almost negatively associated random variables from the classic probability space to the upper ex- pectation space. Within the framework, the authors prove some different types of Rosen- thal’s inequalities for sub-additive expectations. Finally, the authors prove a strong law of large numbers as the application of Rosenthal’s inequalities.
- Published
- 2018
443. On the law of the iterated logarithm for random exponential sums
- Author
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Bence Borda and István Berkes
- Subjects
Discrete mathematics ,Natural logarithm of 2 ,Logarithm ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Law of the iterated logarithm ,01 natural sciences ,Exponential function ,Iterated logarithm ,010104 statistics & probability ,Natural logarithm ,Logarithm of a matrix ,0101 mathematics ,Mathematics - Abstract
The asymptotic behavior of exponential sums ∑ k = 1 N exp ( 2 π i n k α ) \sum _{k=1}^N \exp ( 2\pi i n_k \alpha ) for Hadamard lacunary ( n k ) (n_k) is well known, but for general ( n k ) (n_k) very few precise results exist, due to number theoretic difficulties. It is therefore natural to consider random ( n k ) (n_k) , and in this paper we prove the law of the iterated logarithm for ∑ k = 1 N exp ( 2 π i n k α ) \sum _{k=1}^N \exp (2\pi i n_k \alpha ) if the gaps n k + 1 − n k n_{k+1}-n_k are independent, identically distributed random variables. As a comparison, we give a lower bound for the discrepancy of { n k α } \{n_k \alpha \} under the same random model, exhibiting a completely different behavior.
- Published
- 2018
444. Vector-valued modular forms on a three-dimensional ball
- Author
-
Riccardo Salvati Manni and Eberhard Freitag
- Subjects
Discrete mathematics ,Conjecture ,business.industry ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Modular form ,Modular design ,01 natural sciences ,Picard modular group ,0103 physical sciences ,vector valued modular forms ,010307 mathematical physics ,Ball (mathematics) ,0101 mathematics ,business ,Mathematics ,Siegel modular form - Abstract
In this paper we give a structure theorem for the module of vector valued modular forms in the case of a three dimensional ball with the action of the Picard modular group Γ 3 [ − 3 ] \Gamma _3 [\sqrt {-3}] . The corresponding modular variety of dimension 3 3 is a copy of the Segre cubic.
- Published
- 2018
445. Computing metric dimension of compressed zero divisor graphs associated to rings
- Author
-
M. Imran Bhat and Shariefuddin Pirzada
- Subjects
Discrete mathematics ,zero-divisor graph ,General Mathematics ,010102 general mathematics ,equivalence classes ,0102 computer and information sciences ,compressed zero-divisor graph ,01 natural sciences ,metric dimension ,Metric dimension ,13a99 ,010201 computation theory & mathematics ,QA1-939 ,05c78 ,05c12 ,0101 mathematics ,Zero divisor ,Mathematics - Abstract
For a commutative ring R with 1 ≠ 0, a compressed zero-divisor graph of a ring R is the undirected graph ΓE(R) with vertex set Z(RE) \ {[0]} = RE \ {[0], [1]} defined by RE = {[x] : x ∈ R}, where [x] = {y ∈ R : ann(x) = ann(y)} and the two distinct vertices [x] and [y] of Z(RE) are adjacent if and only if [x][y] = [xy] = [0], that is, if and only if xy = 0. In this paper, we study the metric dimension of the compressed zero divisor graph ΓE(R), the relationship of metric dimension between ΓE(R) and Γ(R), classify the rings with same or different metric dimension and obtain the bounds for the metric dimension of ΓE(R). We provide a formula for the number of vertices of the family of graphs given by ΓE(R×𝔽). Further, we discuss the relationship between metric dimension, girth and diameter of ΓE(R).
- Published
- 2018
446. Pretty good state transfer on 1-sum of star graphs
- Author
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Tong Mengdi, Gu Rui, and Hou Hailong
- Subjects
Discrete mathematics ,pretty good state transfer ,General Mathematics ,kronecker approximation theorem ,010103 numerical & computational mathematics ,0102 computer and information sciences ,State (functional analysis) ,Star (graph theory) ,01 natural sciences ,perfect state transfer ,Transfer (group theory) ,010201 computation theory & mathematics ,QA1-939 ,0101 mathematics ,05c50 ,Mathematics - Abstract
LetAbe the adjacency matrix of a graphGand supposeU(t) = exp(itA). We say that we have perfect state transfer inGfrom the vertexuto the vertexvat timetif there is a scalarγof unit modulus such thatU(t)eu=γ ev. It is known that perfect state transfer is rare. So C.Godsil gave a relaxation of this definition: we say that we have pretty good state transfer fromutovif there exists a complex numberγof unit modulus and, for each positive realϵthere is a timetsuch that ‖U(t)eu–γ ev‖ <ϵ. In this paper, the quantum state transfer on 1-sum of star graphsFk,lis explored. We show that there is no perfect state transfer onFk,l, but there is pretty good state transfer onFk,lif and only ifk=l.
- Published
- 2018
447. On the rate of convergence to zero of the measure of extremal sets in metric theory of transcendental numbers
- Author
-
Natalia Budarina
- Subjects
Discrete mathematics ,Degree (graph theory) ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,01 natural sciences ,Measure (mathematics) ,Rate of convergence ,Bounded function ,0103 physical sciences ,Metric (mathematics) ,010307 mathematical physics ,Transcendental number ,0101 mathematics ,Mathematics - Abstract
We investigate the question on the rate of convergence to zero of the measure of the set $$x\in \mathbb {R}$$ for which the inequality $$|P(x)|n$$ has a solution in integral polynomials of degree n and height bounded by $$Q\in \mathbb {N}$$ . In this paper, for the first time, we obtain an effective estimate for this rate of convergence to zero.
- Published
- 2018
448. Digital topological complexity numbers
- Author
-
Melih Is, Ismet Karaca, and Ege Üniversitesi
- Subjects
010101 applied mathematics ,Discrete mathematics ,Topological complexity ,digital topology ,Topological complexity number,digital topology,digital topological complexity number ,General Mathematics ,Topological complexity number ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,digital topological complexity number ,Mathematics - Abstract
WOS: 000451344700028, The intersection of topological robotics and digital topology leads to us a new workspace. In this paper we introduce the new digital homotopy invariant digital topological complexity number TC(X, kappa) for digital images and give some examples and results about it. Moreover, we examine adjacency relations in the digital spaces and observe how TC(X, kappa) changes when we take a different adjacency relation in the digital spaces., Scientific and Technological Research Council of TurkeyTurkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) [TUBITAK-2211-A], The second author was granted a fellowship by the Scientific and Technological Research Council of Turkey (TUBITAK-2211-A).
- Published
- 2018
449. Realizations and Factorizations of Positive Definite Kernels
- Author
-
Palle E. T. Jorgensen and Feng Tian
- Subjects
Statistics and Probability ,Discrete mathematics ,Spectral theory ,Stochastic process ,General Mathematics ,010102 general mathematics ,Hilbert space ,Positive-definite matrix ,Space (mathematics) ,01 natural sciences ,Measure (mathematics) ,010104 statistics & probability ,symbols.namesake ,Probability theory ,symbols ,0101 mathematics ,Statistics, Probability and Uncertainty ,Variety (universal algebra) ,Mathematics - Abstract
Given a fixed sigma-finite measure space $$\left( X,\mathscr {B},\nu \right) $$ , we shall study an associated family of positive definite kernels K. Their factorizations will be studied with view to their role as covariance kernels of a variety of stochastic processes. In the interesting cases, the given measure $$\nu $$ is infinite, but sigma-finite. We introduce such positive definite kernels $$K\left( \cdot ,\cdot \right) $$ with the two variables from the subclass of the sigma-algebra $$\mathscr {B}$$ whose elements are sets with finite $$\nu $$ measure. Our setting and results are motivated by applications. The latter are covered in the second half of the paper. We first make precise the notions of realizations and factorizations for K, and we give necessary and sufficient conditions for K to have realizations and factorizations in $$L^{2}\left( \nu \right) $$ . Tools in the proofs rely on probability theory and on spectral theory for unbounded operators in Hilbert space. Applications discussed here include the study of reversible Markov processes, and realizations of Gaussian fields, and their Ito-integrals.
- Published
- 2018
450. On $$\epsilon $$ ϵ -solutions for robust semi-infinite optimization problems
- Author
-
Gue Myung Lee and Jae Hyoung Lee
- Subjects
Discrete mathematics ,021103 operations research ,Optimization problem ,Semi-infinite ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,Regular polygon ,Duality (optimization) ,Robust optimization ,02 engineering and technology ,Operator theory ,01 natural sciences ,Potential theory ,Semi-infinite programming ,Theoretical Computer Science ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this paper, we consider semi-infinite optimization problems involving a convex objective function and infinitely many convex constraint functions with data uncertainty, and give its robust counterpart \(\mathrm{(RSIP)}\). Moreover, we consider approximate solutions (\(\epsilon \)-solutions) for \(\mathrm{(RSIP)}\). Using robust optimization approach (worst-case approach), we establish robust necessary optimality and robust sufficient theorems and give duality results for \(\epsilon \)-solutions for \(\mathrm{(RSIP)}\) under a closed convex cone constraint qualification. Moreover, an example is given to illustrate the obtained duality results.
- Published
- 2018
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