1,463 results
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2. Some comments on the paper: Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959
- Author
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Michelle Pierri and Donal O'Regan
- Subjects
Discrete mathematics ,Class (set theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,Differential systems ,01 natural sciences ,010101 applied mathematics ,Controllability ,Algebra ,0101 mathematics ,Differential (mathematics) ,Mathematics - Abstract
The abstract results and applications presented in “Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959, are not correct. Moreover, the class of differential control problems studied in [1] is not H-controllable.
- Published
- 2016
3. On a Paper by Barden
- Author
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A. V. Zhubr
- Subjects
Statistics and Probability ,Reduction (complexity) ,Discrete mathematics ,Combinatorics ,Applied Mathematics ,General Mathematics ,Simply connected space ,Bibliography ,Mathematics::Geometric Topology ,Mathematics - Abstract
It is shown that an approach earlier used by the author for classification of closed simply connected 6-manifolds (reduction to the problem of calculating certain bordism groups) can also be applied for easily obtaining the results by Barden (1965) on classification of closed simply connected 5-manifolds. Bibliography: 11 titles.
- Published
- 2004
4. Remarks on a paper by U. Zannier
- Author
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T. Toshimitsu
- Subjects
Power series ,Discrete mathematics ,Section (category theory) ,Integer ,Applied Mathematics ,General Mathematics ,Laurent series ,Functional equation ,Discrete Mathematics and Combinatorics ,Algebraic function ,Rational function ,Type (model theory) ,Mathematics - Abstract
Zannier proved that for a Laurent series f(x) satisfying the functional equation of type f(x m ) = P(x, f(x)), where \( P(x, y) \in {\Bbb C}(x)[y] \), if f(x) is not rational the set of such m consists of the powers of a single integer. He mentioned that the case f(x) = P(x, f(x m )) should be proved in a similar way. In this paper we first verify this statement and second we show a theorem which is useful for proving the transcendence of a Laurent series satisfying a certain functional equation. This theorem is a generalization of the result that a Laurent series which satisfiesf(x m ) = P(x, f(x)), where\( P(x, y) \in {\Bbb C}(x, y),\,m \geq 2 \)cannot represent an algebraic function unless it is rational (Ke. Nishioka [1], Zannier [8], Section 3).
- Published
- 2000
5. Some complements and corrections to my papers on the theory of attractors for abstract semigroups
- Author
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O. A. Ladyzhenskaya
- Subjects
Statistics and Probability ,Discrete mathematics ,Packing dimension ,Semigroup ,Applied Mathematics ,General Mathematics ,Hausdorff dimension ,Bounded function ,Minkowski–Bouligand dimension ,Dimension function ,Effective dimension ,Mathematics ,Additive group - Abstract
In Sec. 1 a correction is given of the estimate of the Hausdorff dimension and an estimate of the fractal dimension of a bounded subset of a Hilbert space, semiinvariant with respect to a flattening transformation. In Sec. 2 the results, proved by the author for semigroups with a continuous group parameter t∈R+≡[0, ∞), are carried over to the case when t runs through the semigroup ℑ+≡{t∈ℑ∣t⩾0} of some additive group ℑ⊂R=(−∞, ∞).
- Published
- 1992
6. Supplement to the paper 'the averaging operator with respect to a countable partition on a minimal symmetric ideal of the space L1(0, 1)'
- Author
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A. A. Mekler
- Subjects
Statistics and Probability ,Combinatorics ,Discrete mathematics ,Applied Mathematics ,General Mathematics ,Existential quantification ,Preorder ,Partition (number theory) ,Countable set ,Disjoint sets ,Absolute constant ,Mathematics - Abstract
Let A be a partition of the segment [0, 1] into a countable number of disjoint subsets of positive measure, let t∈L1(0,1), let Nt be the smallest rearrangement-invariant order ideal vector lattice in L1(0,1), containing t. In the paper one investigates the properties of the image E(Nt¦A) of the averaging operator with respect to A. In particular, one elucidates under what conditions there exists a function g, g∈L1(0,1), such that E(Nt¦A)⊂Ng. One formulates a generalization of the known Hardy-Littlewood inequality, namely Theorem E(t∣A)≺QE(t*∣A*), where ≺ is the Hardy-Littlewood preorder, t* and A* are the decreasing rearrangements of the function ¦t¦ and (in a special sense) of the partition A, while Q is an absolute constant, 1⩽Q⩽25. One formulates the problem of the smallest value of Q.
- Published
- 1988
7. Supplement to G. V. Kuz'mina's paper: 'On the problem of the maximum of the product of the conformal radii of nonoverlapping domains' (Zap. Nauchn. Sem. Lomi,100, 131–145, 1980)
- Author
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N. A. Lebedev
- Subjects
Statistics and Probability ,Discrete mathematics ,Applied Mathematics ,General Mathematics ,Product (mathematics) ,Conformal map ,Mathematics - Published
- 1984
8. Remark to a paper of J. A. Baker
- Author
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K. Lajkó
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Published
- 1978
9. Supplement to the Paper of Lupaş and Müller
- Author
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B. Boyanov
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Published
- 1970
10. Delone sets in ℝ3: Regularity Conditions
- Author
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N. P. Dolbilin
- Subjects
Statistics and Probability ,Discrete mathematics ,Euclidean space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Delone set ,01 natural sciences ,Identity (music) ,010305 fluids & plasmas ,Set (abstract data type) ,0103 physical sciences ,Homogeneous space ,Mathematics::Metric Geometry ,0101 mathematics ,Symmetry (geometry) ,Orbit (control theory) ,Link (knot theory) ,Mathematics - Abstract
A regular system is a Delone set in Euclidean space with a transitive group of symmetries or, in other words, the orbit of a crystallographic group. The local theory for regular systems, created by the geometric school of B. N. Delone, was aimed, in particular, to rigorously establish the “local-global-order” link, i.e., the link between the arrangement of a set around each of its points and symmetry/regularity of the set as a whole. The main result of this paper is a proof of the so-called 10R-theorem. This theorem asserts that identity of neighborhoods within a radius 10R of all points of a Delone set (in other words, an (r, R)-system) in 3D Euclidean space implies regularity of this set. The result was obtained and announced long ago independently by M. Shtogrin and the author of this paper. However, a detailed proof remains unpublished for many years. In this paper, we give a proof of the 10R-theorem. In the proof, we use some recent results of the author, which simplify the proof.
- Published
- 2020
11. Smooth Julia Sets
- Author
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V. S. Sekovanov
- Subjects
Statistics and Probability ,Discrete mathematics ,Mathematics::Dynamical Systems ,Fractal ,Mathematics::Complex Variables ,Applied Mathematics ,General Mathematics ,Chaotic ,Structure (category theory) ,Interval (mathematics) ,Julia set ,Complex plane ,Mathematics - Abstract
It is known that Julia sets, as a rule, have a fractal structure. In this paper, we give examples of smooth Julia sets, among them: a circle, a segment, an infinite interval, a straight line, and the complex plane. It is shown that the functions studied in the paper are chaotic on their Julia sets. The results obtained by analytical research are visualized using computer programs. The algorithms for constructing the Julia sets being considered are indicated.
- Published
- 2020
12. Quantum Markov States and Quantum Hidden Markov States
- Author
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Z. I. Bezhaeva and V. I. Oseledets
- Subjects
Statistics and Probability ,Discrete mathematics ,Markov chain ,Applied Mathematics ,General Mathematics ,Markov process ,Function (mathematics) ,State (functional analysis) ,Mathematical proof ,Tree (graph theory) ,symbols.namesake ,symbols ,Hidden Markov model ,Quantum ,Mathematics - Abstract
In a previous paper (Funct. Anal. Appl., 3 (2015), 205–209), we defined quantum Markov states. Here we recall this definition and present a proof of the results from that paper (which are given there without proofs). We give a definition of a quantum hidden Markov state generated by a function of a quantum Markov process and show how it is related to other definitions of such states. Our definitions work for quantum Markov fields on ℤN and on graphs. We consider an example with the Cayley tree.
- Published
- 2019
13. Computable Presentability of Countable Linear Orders
- Author
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A. N. Frolov
- Subjects
Statistics and Probability ,Set (abstract data type) ,Discrete mathematics ,Property (philosophy) ,Applied Mathematics ,General Mathematics ,Order (group theory) ,Countable set ,Natural number ,Order type ,Mathematics - Abstract
The main goal of this paper is to study algorithmic properties of countable linear orders by constructing effective presentations of these structures on the set of natural numbers. In 1991, C. Jockusch and R. Soare constructed a low linear order without computable presentations. Earlier, in 1989, R. Downey and M. Moses showed that each low discrete linear order has a computable copy. It is natural to ask for which order types of low presentations the existence of a computable presentation is sufficient. This question (namely, research program) was stated by R. Downey in 1998: Describe the order property P such that, for any low linear order L, P(L) implies the existence of a computable presentation of L. In this paper, we give a detailed review of the main results in this direction. These results are mostly obtained by the author or in co-authorship.
- Published
- 2021
14. An extension of the Hermite–Hadamard inequality for convex and s-convex functions
- Author
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Péter Kórus
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Regular polygon ,010103 numerical & computational mathematics ,Extension (predicate logic) ,01 natural sciences ,Iterated integrals ,Hermite–Hadamard inequality ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Convex function ,Mathematics - Abstract
The Hermite–Hadamard inequality was extended using iterated integrals by Retkes [Acta Sci Math (Szeged) 74:95–106, 2008]. In this paper we further extend the main results of the above paper for convex and also for s-convex functions in the second sense.
- Published
- 2019
15. Redheffer type bounds for Bessel and modified Bessel functions of the first kind
- Author
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Árpád Baricz and Khaled Mehrez
- Subjects
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
16. On p-convergent Operators on Banach Lattices
- Author
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Elroy D. Zeekoei and Jan Fourie
- Subjects
Unbounded operator ,Discrete mathematics ,Mathematics::Functional Analysis ,Approximation property ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Spectrum (functional analysis) ,Finite-rank operator ,Compact operator ,01 natural sciences ,Strictly singular operator ,010101 applied mathematics ,Pseudo-monotone operator ,0101 mathematics ,C0-semigroup ,Mathematics - Abstract
The notion of a p-convergent operator on a Banach space was originally introduced in 1993 by Castillo and Sanchez in the paper entitled “Dunford–Pettis-like properties of continuous vector function spaces”. In the present paper we consider the p-convergent operators on Banach lattices, prove some domination properties of the same and consider their applications (together with the notion of a weak p-convergent operator, which we introduce in the present paper) to a study of the Schur property of order p. Also, the notion of a disjoint p-convergent operator on Banach lattices is introduced, studied and its applications to a study of the positive Schur property of order p are considered.
- Published
- 2017
17. On the Enumeration of Hypermaps Which are Self-Equivalent with Respect to Reversing the Colors of Vertices
- Author
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M. A. Deryagina
- Subjects
Statistics and Probability ,Connected component ,Discrete mathematics ,Mathematics::Combinatorics ,Applied Mathematics ,General Mathematics ,Riemann surface ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Graph ,symbols.namesake ,Colored ,010201 computation theory & mathematics ,Enumeration ,symbols ,Bipartite graph ,Bibliography ,Reversing ,0101 mathematics ,Mathematics - Abstract
A map (S,G) is a closed Riemann surface S with embedded graph G such that S \G is the disjoint union of connected components, called faces, each of which is homeomorphic to an open disk. Tutte began a systematic study of maps in the 1960s and contemporary authors are actively developing it. In the present paper, after recalling the concept of a circular map introduced by the author and Mednykh, a relationship between bipartite maps and circular maps is demonstrated via the concept of the duality of maps. In this way an enumeration formula for the number of bipartite maps with a given number of edges is obtained. A hypermap is a map whose vertices are colored black and white in such a way that every edge connects vertices of different colors. The hypermaps are also known as dessins d’enfants (or Grothendieck’s dessins). A hypermap is self-equivalent with respect to reversing the colors of vertices if it is equivalent to the hypermap obtained by reversing the colors of its vertices. The main result of the present paper is an enumeration formula for the number of unrooted hypermaps, regardless of genus, which have n edges and are self-equivalent with respect to reversing the colors of vertices. Bibliography: 13 titles.
- Published
- 2017
18. Some weak specification properties and strongly mixing
- Author
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Jiandong Yin, Tao Wang, and Qi Yan
- Subjects
010101 applied mathematics ,Discrete mathematics ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Equivalence (formal languages) ,01 natural sciences ,Mathematics - Abstract
In this paper, the authors first construct a dynamical system which is strongly mixing but has no weak specification property. Then the authors introduce two new concepts which are called the quasi-weak specification property and the semi-weak specification property in this paper, respectively, and the authors prove the equivalence of quasi-weak specification property, semi-weak specification property and strongly mixing.
- Published
- 2017
19. Structure Graphs of Rings: Definitions and First Results
- Author
-
Aleksandar Lipkovski
- Subjects
Statistics and Probability ,Discrete mathematics ,Cayley graph ,Algebraic structure ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Directed graph ,01 natural sciences ,010101 applied mathematics ,Quadratic equation ,Vieta's formulas ,Branched covering ,0101 mathematics ,Commutative property ,Complex number ,Mathematics - Abstract
The quadratic Vieta formulas (x, y) ↦ (u, v) = (x + y, xy) in the complex geometry define a two-fold branched covering ℂ2 → ℂ2 ramified over the parabola u 2 = 4v. Thinking about topics considered in Arnold’s paper Topological content of the Maxwell theorem on multipole representation of spherical functions, I came to a very simple idea: in fact, these formulas describe the algebraic structure, i.e., addition and multiplication, of complex numbers. What if, instead of the field of complex numbers, we consider an arbitrary ring? Namely for an arbitrary ring A (commutative, with unity) consider the mapping Φ: A 2 → A 2 defined by the Vieta formulas (x, y) ↦ (u, v) = (x + y, xy). What kind of algebraic properties of the ring itself does this map reflect? At first, it is interesting to investigate the simplest finite rings A = ℤ m and A = ℤ k ×ℤ m . Recently, it has been very popular to consider graphs associated to rings (the zero-divisor graph, the Cayley graph, etc.). In the present paper, we study the directed graph defined by the Vieta mapping Φ.
- Published
- 2017
20. Sequential Analogues of the Lyapunov and Krein–Milman Theorems in Fréchet Spaces
- Author
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F. S. Stonyakin
- Subjects
Statistics and Probability ,Discrete mathematics ,Mathematics::Functional Analysis ,Dual space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Eberlein–Šmulian theorem ,Banach space ,Hahn–Banach theorem ,02 engineering and technology ,01 natural sciences ,Fréchet space ,Locally convex topological vector space ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Closed graph theorem ,0101 mathematics ,Open mapping theorem (functional analysis) ,Mathematics - Abstract
In this paper we develop the theory of anti-compact sets we introduced earlier. We describe the class of Frechet spaces where anti-compact sets exist. They are exactly the spaces that have a countable set of continuous linear functionals. In such spaces we prove an analogue of the Hahn–Banach theorem on extension of a continuous linear functional from the original space to a space generated by some anti-compact set. We obtain an analogue of the Lyapunov theorem on convexity and compactness of the range of vector measures, which establishes convexity and a special kind of relative weak compactness of the range of an atomless vector measure with values in a Frechet space possessing an anti-compact set. Using this analogue of the Lyapunov theorem, we prove the solvability of an infinite-dimensional analogue of the problem of fair division of resources. We also obtain an analogue of the Lyapunov theorem for nonadditive analogues of measures that are vector quasi-measures valued in an infinite-dimensional Frechet space possessing an anti-compact set. In the class of Frechet spaces possessing an anti-compact set, we obtain analogues of the Krein–Milman theorem on extreme points for convex bounded sets that are not necessarily compact. A special place is occupied by analogues of the Krein–Milman theorem in terms of extreme sequences introduced in the paper (the so-called sequential analogues of the Krein–Milman theorem).
- Published
- 2017
21. On K p,q -factorization of complete bipartite multigraphs
- Author
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Mingchao Li and Jian Wang
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Multigraph ,Prime number ,Complete bipartite graph ,Combinatorics ,Integer ,Factorization ,Bipartite graph ,Partition (number theory) ,lcsh:Q ,lcsh:Science ,Mathematics - Abstract
Let λK m,n be a complete bipartite multigraph with two partite sets having m and n vertices, respectively. A K p,q -factorization of λK m,n is a set of edge-disjoint K p,q -factors of λK m,n which partition the set of edges of λK m,n . When p = 1 and q is a prime number, Wang, in his paper [On K 1,q -factorization of complete bipartite graph, Discrete Math., 126: (1994), 359-364], investigated the K 1,q -factorization of K m,n and gave a sufficient condition for such a factorization to exist. In papers [K 1,k -factorization of complete bipartite graphs, Discrete Math., 259: 301-306 (2002),; K p,q -factorization of complete bipartite graphs, Sci. China Ser. A-Math., 47: (2004), 473-479], Du and Wang extended Wang’s result to the case that p and q are any positive integers. In this paper, we give a sufficient condition for λK m,n to have a K p,q -factorization. As a special case, it is shown that the necessary condition for the K p,q -factorization of λK m,n is always sufficient when p : q = k : (k + 1) for any positive integer k.
- Published
- 2017
22. On $$\varvec{n}$$ n -norm preservers and the Aleksandrov conservative $$\varvec{n}$$ n -distance problem
- Author
-
György Pál Gehér
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Order (ring theory) ,010103 numerical & computational mathematics ,01 natural sciences ,Surjective function ,Nonlinear system ,Transformation (function) ,Norm (mathematics) ,Distance problem ,Discrete Mathematics and Combinatorics ,Affine transformation ,0101 mathematics ,Unit (ring theory) ,Mathematics - Abstract
The goal of this paper is to point out that the results obtained in the recent papers (Chen and Song in Nonlinear Anal 72:1895–1901, 2010; Chu in J Math Anal Appl 327:1041–1045, 2007; Chu et al. in Nonlinear Anal 59:1001–1011, 2004a, J. Math Anal Appl 289:666–672, 2004b) can be seriously strengthened in the sense that we can significantly relax the assumptions of the main results so that we still get the same conclusions. In order to do this first, we prove that for $$n \ge 3$$ any transformation which preserves the n-norm of any n vectors is automatically plus-minus linear. This will give a re-proof of the well-known Mazur–Ulam-type result that every n-isometry is automatically affine ( $$n \ge 2$$ ) which was proven in several papers, e.g. in Chu et al. (Nonlinear Anal 70:1068–1074, 2009). Second, following the work of Rassias and Semrl (Proc Am Math Soc 118:919–925, 1993), we provide the solution of a natural Aleksandrov-type problem in n-normed spaces, namely, we show that every surjective transformation which preserves the unit n-distance in both directions ( $$n\ge 2$$ ) is automatically an n-isometry.
- Published
- 2017
23. On Fields of Definition of an Algebraic Curve
- Author
-
A. L. Smirnov
- Subjects
Statistics and Probability ,Algebraic cycle ,Discrete mathematics ,Polar curve ,Function field of an algebraic variety ,Stable curve ,Applied Mathematics ,General Mathematics ,Algebraic surface ,Real algebraic geometry ,Algebraic function ,Algebraic curve ,Mathematics - Abstract
The paper deals with geometric invariants of an algebraic curve such as the minimal number of crucial values of rational functions and the minimal transcendence degree of definition fields. The main question is if the difference of these two invariants is always equal to 3 for any curve with genus g > 0. For curves defined over an algebraic number field, a positive answer is given by Belyi’s theorem. In the paper, the main question is answered in the affirmative for some other cases.
- Published
- 2016
24. L p estimates of rough maximal functions along surfaces with applications
- Author
-
Abdulla M. Jarrah and Ahmad Al-Salman
- Subjects
Discrete mathematics ,Class (set theory) ,Pure mathematics ,General theorem ,Applied Mathematics ,General Mathematics ,Block (permutation group theory) ,Maximal function ,Singular integral ,Space (mathematics) ,Singular integral operators ,Mathematics - Abstract
In this paper, we study the Lp mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a consequence of such theorem, we establish the Lp boundedness of various maximal oscillatory singular integrals provided that their kernels belong to the natural space Llog L(Sn−1). Moreover, we highlight some additional results concerning operators with kernels in certain block spaces. The results in this paper substantially improve previously known results.
- Published
- 2016
25. To the History of the Appearance of the Notion of the ε-Entropy of an Authomorphism of a Lebesgue Space and (ε,T)-Entropy of a Dynamical System with Continuous Time
- Author
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D. Z. Arov
- Subjects
Statistics and Probability ,Discrete mathematics ,Dynamical systems theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Automorphism ,01 natural sciences ,Separable space ,Compact space ,0103 physical sciences ,Entropy (information theory) ,Standard probability space ,Ergodic theory ,010307 mathematical physics ,Invariant measure ,0101 mathematics ,Mathematics - Abstract
The paper is devoted to the master thesis on “information theory” which was written by the author in 1956–57. The topic was suggested by his advisor A. A. Bobrov (a student of A. Ya. Khinchin and A. N. Kolmogorov), and the thesis was written under the influence of lectures by N. I. Gavrilov (a student of I. G. Petrovskii) on the qualitative theory of differential equations, which included the statement of Birkhoff’s theorem for ergodic dynamical systems. In the thesis, the author used the concept of Shannon entropy in the study of ergodic dynamical systems f(p, t) in a separable compact metric space R with an invariant measure μ (where μ(R) = 1) and introduced the notion of the (ϵ, T)-entropy of a system as a quantitative characteristic of the degree of mixing. In the work, not only partitions of R were considered, but also partitions of the interval (−∞,∞) into subintervals of length T > 0. In particular, f(p, T) was regarded as an automorphism S of X = R, and the (ϵ, T)-entropy is essentially the e-entropy of S. But, despite some “oversights” in the definition of the (ϵ, T)-entropy and many years that have passed, the author decided to publish the corresponding chapter of the thesis in connection with the following: 1) There is a number of papers that refer to this work in the explanation of the history of the concept of Kolmogorov’s entropy. 2) Recently, B. M. Gurevich obtained new results on the ϵ-entropy hϵ(S), which show that for two ergodic automorphisms with equal finite entropies their ϵ-entropies also coincide for all ϵ, but, on the other hand, there are unexpected nonergodic automorphisms with equal finite entropies, but different ϵ-entropies for some ϵ. This shows that the concept of ϵ-entropy is of scientific value.
- Published
- 2016
26. Graph-Links: Nonrealizability, Orientation, and Jones Polynomial
- Author
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V. S. Safina and Denis Petrovich Ilyutko
- Subjects
Statistics and Probability ,Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Jones polynomial ,Bracket polynomial ,01 natural sciences ,Graph ,Combinatorics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,MathematicsofComputing_DISCRETEMATHEMATICS ,Writhe ,Mathematics - Abstract
The present paper is devoted to graph-links with many components and consists of two parts. In the first part of the paper we classify vertices of a labeled graph according to the component they belong to. Using this classification, we construct an invariant of graph-links. This invariant shows that the labeled second Bouchet graph generates a nonrealizable graph-link. In the second part of the work we introduce the notion of an oriented graph-link. We define a writhe number for the oriented graph-link and we get an invariant of oriented graph-links, the Jones polynomial, by normalizing the Kauffman bracket with the writhe number.
- Published
- 2016
27. Random Deviations of Ergodic Sums for the Pascal Adic Transformation in the Case of the Lebesgue Measure
- Author
-
Aleksei Minabutdinov
- Subjects
Statistics and Probability ,Discrete mathematics ,Class (set theory) ,Mathematics::Dynamical Systems ,Lebesgue measure ,Applied Mathematics ,General Mathematics ,Pascal (programming language) ,Measure (mathematics) ,Transformation (function) ,Ergodic theory ,computer ,Mathematics ,computer.programming_language - Abstract
The paper generalizes the results by E. Janvresse, T. de la Rue, and Y. Velenik on fluctuations in ergodic sums for the Pascal adic transformation in the case of the Lebesgue measure for a wide class of functions. In particular, we answer several questions from that paper.
- Published
- 2015
28. Two classes of operators with irreducibility and the small and compact perturbations of them
- Author
-
Yun Nan Zhang and Li Qiong Lin
- Subjects
Discrete mathematics ,Nuclear operator ,Applied Mathematics ,General Mathematics ,Finite-rank operator ,Spectral theorem ,Operator theory ,Compact operator ,Operator norm ,Compact operator on Hilbert space ,Mathematics ,Quasinormal operator - Abstract
This paper gives the concepts of finite dimensional irreducible operators ((FDI) operators) and infinite dimensional irreducible operators ((IDI) operators). Discusses the relationships of (FDI) operators, (IDI) operators and strongly irreducible operators ((SI) operators) and illustrates some properties of the three classes of operators. Some sufficient conditions for the finite-dimensional irreducibility of operators which have the forms of upper triangular operator matrices are given. This paper proves that every operator with a singleton spectrum is a small compact perturbation of an (FDI) operator on separable Banach spaces and shows that every bounded linear operator T can be approximated by operators in (ΣFDI)(X) with respect to the strong-operator topology and every compact operator K can be approximated by operators in (ΣFDI)(X) with respect to the norm topology on a Banach space X with a Schauder basis, where (ΣFDI)(X):= {T ∈ B(X): T = Σ =1 ⊕T i , T i ∈ (FDI), k ∈ ℕ}.
- Published
- 2015
29. Number of Jumps in Two-Sided First-Exit Problems for a Compound Poisson Process
- Author
-
Yi Lu, Can Jin, and Shuanming Li
- Subjects
Statistics and Probability ,Discrete mathematics ,Laplace transform ,General Mathematics ,010102 general mathematics ,Probability density function ,01 natural sciences ,Lévy process ,Exponential function ,010104 statistics & probability ,Compound Poisson process ,Applied mathematics ,Probability-generating function ,0101 mathematics ,Random variable ,Joint (geology) ,Mathematics - Abstract
In this paper, we study the joint Laplace transform and probability generating functions of two pairs of random variables: (1) the two-sided first-exit time and the number of claims by this time; (2) the two-sided smooth exit-recovery time and its associated number of claims. The joint transforms are expressed in terms of the so-called doubly-killed scale function that is defined in this paper. We also find explicit expressions for the joint density function of the two-sided first-exit time and the number of claims by this time. Numerical examples are presented for exponential claims.
- Published
- 2015
30. Almost Diagonal Matrices and Besov-Type Spaces Based on Wavelet Expansions
- Author
-
Markus Weimar
- Subjects
Discrete mathematics ,Sequence ,Function space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Order (ring theory) ,Context (language use) ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,Diagonal matrix ,Nabla symbol ,0101 mathematics ,Biorthogonal wavelet ,Analysis ,Mathematics - Abstract
This paper is concerned with problems in the context of the theoretical foundation of adaptive algorithms for the numerical treatment of operator equations. It is well-known that the analysis of such schemes naturally leads to function spaces of Besov type. But, especially when dealing with equations on non-smooth manifolds, the definition of these spaces is not straightforward. Nevertheless, motivated by applications, recently Besov-type spaces $$B^\alpha _{\Psi ,q}(L_p(\Gamma ))$$ on certain two-dimensional, patchwise smooth surfaces were defined and employed successfully. In the present paper, we extend this definition (based on wavelet expansions) to a quite general class of d-dimensional manifolds and investigate some analytical properties of the resulting quasi-Banach spaces. In particular, we prove that different prominent constructions of biorthogonal wavelet systems $$\Psi $$ on domains or manifolds $$\Gamma $$ which admit a decomposition into smooth patches actually generate the same Besov-type function spaces $$B^\alpha _{\Psi ,q}(L_p(\Gamma ))$$ , provided that their univariate ingredients possess a sufficiently large order of cancellation and regularity. For this purpose, a theory of almost diagonal matrices on related sequence spaces $$b^\alpha _{p,q}(\nabla )$$ of Besov type is developed.
- Published
- 2015
31. Ambiguities in One-Dimensional Discrete Phase Retrieval from Fourier Magnitudes
- Author
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Gerlind Plonka and Robert Beinert
- Subjects
Discrete mathematics ,Partial differential equation ,Applied Mathematics ,General Mathematics ,Structure (category theory) ,Space (mathematics) ,symbols.namesake ,Discrete-time signal ,Fourier transform ,Fourier analysis ,symbols ,Uniqueness ,Phase retrieval ,Analysis ,Mathematics - Abstract
The present paper is a survey aiming at characterizing all solutions of the discrete phase retrieval problem. Restricting ourselves to discrete signals with finite support, this problem can be stated as follows. We want to recover a complex-valued discrete signal $$\mathbf{x} :\mathbb {Z}\rightarrow \mathbb {C}$$ with support $$\{ 0, \ldots , N-1 \}$$ from the modulus of its discrete-time Fourier transform $$\widehat{x}(\omega )$$ . We will give a full classification of all trivial and nontrivial ambiguities of the discrete phase retrieval problem. In our classification, trivial ambiguities are caused either by signal shifts in space, by multiplication with a rotation factor $$\mathrm {e}^{\mathrm {i}\alpha }$$ , $$\alpha \in [-\pi , \pi )$$ , or by conjugation and reflection of the signal. Furthermore, we show that all nontrivial ambiguities of the finite discrete phase retrieval problem can be characterized by signal convolutions. In the second part of the paper, we examine the usually employed a priori conditions regarding their ability to reduce the number of ambiguities of the phase retrieval problem or even to ensure uniqueness. For the corresponding proofs we can employ our findings on the ambiguity classification. The considerations on the structure of ambiguities also show clearly the ill-posedness of the phase retrieval problem even in cases where uniqueness is theoretically shown.
- Published
- 2015
32. The Structure of Translation-Invariant Spaces on Locally Compact Abelian Groups
- Author
-
Marcin Bownik and Kenneth A. Ross
- Subjects
Discrete mathematics ,Pointwise ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Dimension function ,Second-countable space ,Linear subspace ,Euclidean geometry ,Locally compact space ,Abelian group ,Invariant (mathematics) ,Analysis ,Mathematics - Abstract
Let \(\Gamma \) be a closed co-compact subgroup of a second countable locally compact abelian (LCA) group \(G\). In this paper we study translation-invariant (TI) subspaces of \(L^2(G)\) by elements of \(\Gamma \). We characterize such spaces in terms of range functions extending the results from the Euclidean and LCA setting. The main innovation of this paper, which contrasts with earlier works, is that we do not require that \(\Gamma \) be discrete. As a consequence, our characterization of TI-spaces is new even in the classical setting of \(G=\mathbb {R}^n\). We also extend the notion of the spectral function in \(\mathbb {R}^n\) to the LCA setting. It is shown that spectral functions, initially defined in terms of \(\Gamma \), do not depend on \(\Gamma \). Several properties equivalent to the definition of spectral functions are given. In particular, we show that the spectral function scales nicely under the action of epimorphisms of \(G\) with compact kernel. Finally, we show that for a large class of LCA groups, the spectral function is given as a pointwise limit.
- Published
- 2015
33. Multivariate Estimates for the Concentration Functions of Weighted Sums of Independent, Identically Distributed Random Variables
- Author
-
Yu. S. Eliseeva
- Subjects
Statistics and Probability ,Independent and identically distributed random variables ,Discrete mathematics ,Multivariate statistics ,Applied Mathematics ,General Mathematics ,Probability (math.PR) ,Structure (category theory) ,Combinatorics ,FOS: Mathematics ,Bibliography ,Concentration function ,Random matrix ,Random variable ,Mathematics - Probability ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. The paper deals with the question about the behavior of the concentration function of the random variable $\sum\limits_{k=1}^{n}X_k a_k$ according to the arithmetic structure of vectors $a_k$. Recently, the interest to this question has increased significantly due to the study of distributions of eigenvalues of random matrices. In this paper we formulate and prove multidimensional generalizations of the results Eliseeva and Zaitsev (2012). They are also the refinements of the results of Friedland and Sodin (2007) and Rudelson and Vershynin (2009)., Comment: 13 pages
- Published
- 2014
34. 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
35. A STRONGLY CONVERGENT SHRINKING DESCENT-LIKE HALPERN’S METHOD FOR MONOTONE VARIATIONAL INEQUALITY AND FIXED POINT PROBLEMS
- Author
-
Nguyen Thi Thu Thuy
- Subjects
Discrete mathematics ,Iterative method ,General Mathematics ,Solution set ,Hilbert space ,Fixed point ,symbols.namesake ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Monotone polygon ,Fixed-point iteration ,Variational inequality ,symbols ,Countable set ,Applied mathematics ,Mathematics - Abstract
In this paper, we introduce a new iteration method based on the hybrid method in mathematical programming, the descent-like method and the Halpern’s iteration method for finding a common element of the solution set for a variational inequality and the set of common fixed points of a countably infinite family of nonexpansive mappings in Hilbert spaces. The result in this paper modifies and improves some well-known results in the literature for a more general problem.
- Published
- 2014
36. Hybrid Mann–Halpern Iteration Methods for Finding Fixed Points Involving Asymptotically Nonexpansive Mappings and Semigroups
- Author
-
Nguyen Thi Thu Thuy
- Subjects
Discrete mathematics ,symbols.namesake ,Iterative method ,Fixed-point iteration ,Semigroup ,General Mathematics ,Common fixed point ,Hilbert space ,symbols ,Applied mathematics ,Fixed point ,Mathematics - Abstract
In this paper, we introduce some new iteration methods combining the hybrid method in mathematical programming with Mann’s iterative method and the Halpern method for finding a fixed point of an asymptotically nonexpansive mapping and a common fixed point of an asymptotically nonexpansive semigroup in a Hilbert space. The main results in this paper modify and improve some well-known results in the literature.
- Published
- 2014
37. Existence of nontrivial solution for Schrödinger–Poisson systems with indefinite steep potential well
- Author
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Juntao Sun, Yuanze Wu, and Tsung-fang Wu
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,Lambda ,Poisson distribution ,01 natural sciences ,Omega ,010101 applied mathematics ,symbols.namesake ,symbols ,0101 mathematics ,Schrödinger's cat ,Mathematics - Abstract
In this paper, we study a class of nonlinear Schrodinger–Poisson systems with indefinite steep potential well: $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta u+V_{\lambda }(x)u+K(x)\phi u=|u|^{p-2}u &{} \text { in }\mathbb {R}^{3},\\ -\Delta \phi =K\left( x\right) u^{2} &{} \ \text {in }\mathbb {R}^{3}, \end{array} \right. \end{aligned}$$ where $$30$$ and $$ K(x)\ge 0$$ for all $$x\in \mathbb {R}^{3}$$ . We require that $$a\in C( \mathbb {R}^{3}) $$ is nonnegative and has a potential well $$\Omega _{a}$$ , namely $$a(x)\equiv 0$$ for $$x\in \Omega _{a}$$ and $$a(x)>0$$ for $$x\in \mathbb {R}^{3}\setminus \overline{\Omega _{a}}$$ . Unlike most other papers on this problem, we allow that $$b\in C(\mathbb {R}^{3}) $$ is unbounded below and sign-changing. By introducing some new hypotheses on the potentials and applying the method of penalized functions, we obtain the existence of nontrivial solutions for $$\lambda $$ sufficiently large. Furthermore, the concentration behavior of the nontrivial solution is also described as $$\lambda \rightarrow \infty $$ .
- Published
- 2017
38. Torsion Abelian RAI-Groups
- Author
-
Pham Thi Thu Thuy
- Subjects
Statistics and Probability ,Discrete mathematics ,Pure mathematics ,Torsion subgroup ,Mathematics::Commutative Algebra ,Applied Mathematics ,General Mathematics ,Elementary abelian group ,Rank of an abelian group ,Divisible group ,Non-abelian group ,Free abelian group ,Torsion (algebra) ,Condensed Matter::Strongly Correlated Electrons ,Abelian group ,Mathematics - Abstract
This paper is devoted to the study of Abelian afi-groups. A subgroup A of an Abelian group G is called its absolute ideal if A is an ideal of any ring on G. We will call an Abelian group an afi-group if all of its absolute ideals are fully invariant subgroups. In this paper, we will describe afi-groups in the class of fully transitive torsion groups (in particular, separable torsion groups) and divisible torsion groups.
- Published
- 2014
39. Entire functions sharing an entire function of smaller order with their difference operators
- Author
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Xiao-Min Li and Hong Xun Yi
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Entire function ,Calculus ,Order (group theory) ,Uniqueness ,Creating shared value ,Mathematics - Abstract
We study a uniqueness question of entire functions sharing an entire function of smaller order with their difference operators, and deal with a question posed by Liu and Yang. The results in this paper extend the corresponding results obtained by Liu-Yang and by Liu-Laine respectively. Examples are provided to show that the results in this paper, in a sense, are the best possible.
- Published
- 2014
40. Dawson’s chess revisited
- Author
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N. A. Ashrafi Payaman and Keivan Borna
- Subjects
Discrete mathematics ,Position (vector) ,If and only if ,Applied Mathematics ,General Mathematics ,Numerical analysis ,Mod ,ComputingMilieux_PERSONALCOMPUTING ,Algorithm ,Mathematics - Abstract
In this paper we prove that in a Quasi-Dawson’s Chess (a restricted version of Dawson’s Chess) playing on a 3 × d board, the first player is loser if and only if d (mod)5 = 1 or d (mod)5 = 2. Furthermore, we have designed two algorithms that are responsible for storing the results of Quasi-Dawson’s Chess games having less than d + 1 files and finding the strategy that leads to win, if there is a possibility of winning (by a wining position, we mean one from which one can win with best play). Moreover we show that the total complexity of our algorithms is O(d 2). Finally we have implemented our algorithm in C++ which admits the main results of the paper even for large values of d.
- Published
- 2013
41. Adelic Multiresolution Analysis, Construction of Wavelet Bases and Pseudo-Differential Operators
- Author
-
Andrei Khrennikov, V. M. Shelkovich, and Jan Harm van der Walt
- Subjects
Discrete mathematics ,Ring (mathematics) ,Operator (computer programming) ,Tensor product ,Wavelet ,Adele ring ,Applied Mathematics ,General Mathematics ,Multiresolution analysis ,Eigenfunction ,Differential operator ,Analysis ,Mathematics - Abstract
In our previous paper, the Haar multiresolution analysis (MRA) $\{V_{j}\}_{j\in \mathbb {Z}}$ in $L^{2}(\mathbb {A})$ was constructed, where $\mathbb {A}$ is the adele ring. Since $L^{2}(\mathbb {A})$ is the infinite tensor product of the spaces $L^{2}({\mathbb {Q}}_{p})$ , p=∞,2,3,…, the adelic MRA has some specific properties different from the corresponding finite-dimensional ones. Nevertheless, this infinite-dimensional MRA inherits almost all basic properties of the finite-dimensional case. In this paper we derive explicit formulas for bases in V j , $j\in \mathbb {Z}$ , and for the wavelet bases generated by the above-mentioned adelic MRA. In view of the specific properties of the adelic MRA, there arise some technical problems in the construction of wavelet bases. These problems were solved with the aid of the operator formalization of the process of generation of wavelet bases. We study the spectral properties of the fractional operator introduced by S. Torba and W.A. Zuniga-Galindo. We prove that the constructed wavelet functions are eigenfunctions of this fractional operator. This paper, as well as our previous paper, introduces new ideas to construct different infinite-dimensional MRAs. Our results can be used in the theory of adelic pseudo-differential operators and equations over the ring of adeles and in adelic models in physics.
- Published
- 2013
42. Stability of a conditional Cauchy equation on a set of measure zero
- Author
-
Jaeyoung Chung
- Subjects
Discrete mathematics ,Null set ,Applied Mathematics ,General Mathematics ,Discrete Mathematics and Combinatorics ,Direct consequence ,Cauchy's equation ,Measure (mathematics) ,Stability (probability) ,Stability theorem ,Mathematics - Abstract
In this paper, we prove the Hyers–Ulam stability theorem when $${f, g, h : \mathbb{R} \to \mathbb{R}}$$ satisfy $$|f(x + y) - g(x) - h(y)| \leq \epsilon$$ in a set $${\Gamma \subset \mathbb{R}^{2}}$$ of measure $${m(\Gamma) = 0}$$ , which refines a previous result in Chung (Aequat Math 83:313–320, 2012) and gives an affirmative answer to the question in the paper. As a direct consequence we obtain that if $${f, g, h : \mathbb{R} \to \mathbb{R}}$$ satisfy the Pexider equation $$f(x + y) - g(x) - h(y) = 0$$ in $${\Gamma}$$ , then the equation holds for all $${x, y \in \mathbb{R}}$$ . Using our method of construction of the set, we can find a set $${\Gamma \subset \mathbb{R}^{2n}}$$ of 2n-dimensional measure 0 and obtain the above result for the functions $${f, g, h : \mathbb{R}^{n} \to \mathbb{C}}$$ .
- Published
- 2013
43. Some results associated with the longest run in a strongly ergodic Markov chain
- Author
-
Xian Yuan Wu and Ya Zhe Zhang
- Subjects
Combinatorics ,Discrete mathematics ,Sequence ,Markov chain ,Applied Mathematics ,General Mathematics ,Ergodic theory ,Countable set ,Limit law ,State markov chain ,Type (model theory) ,Space (mathematics) ,Mathematics - Abstract
This paper discusses the asymptotic behaviors of the longest run on a countable state Markov chain. Let \(\left\{ {X_a } \right\}_{a \in Z_ + }\) be a stationary strongly ergodic reversible Markov chain on countablestate space S = {1, 2, ...}. Let T ⊂ S be an arbitrary finite subset of S. Denote by L n the length of the longest run of consecutive i’s for i ∈ T, that occurs in the sequence X 1, ..., X n . In this paper, we obtain a limit law and a week version of an Erdos-Renyi type law for L n . A large deviation result of L n is also discussed.
- Published
- 2013
44. On isotopies, parastrophies, and orthogonality of quasigroups
- Author
-
K. K. Shchukin
- Subjects
Statistics and Probability ,Discrete mathematics ,Pure mathematics ,Orthogonality ,Applied Mathematics ,General Mathematics ,Order (group theory) ,Quasigroup ,Mathematics - Abstract
In V. D. Belousov’s papers, some properties of parastrophies were studied and some relations between parastrophies of a given quasigroup were obtained. Also some invariants of a parastrophy were found. This article continues our paper with V. V. Gushan, in which minimal sets of parastrophy systems for quasigroups of order 6 were obtained and some questions about orthogonality of parastrophies of a given quasigroup were studied as well.
- Published
- 2013
45. On λ-compact operators
- Author
-
Antara Bhar and Manjul Gupta
- Subjects
Discrete mathematics ,Unbounded operator ,Pure mathematics ,Nuclear operator ,Approximation property ,Applied Mathematics ,General Mathematics ,Finite-rank operator ,Spectral theorem ,Operator theory ,Operator norm ,Compact operator on Hilbert space ,Mathematics - Abstract
Using the duality theory of sequence spaces, we study in this paper λ-compact operators defined on Banach spaces, corresponding to a sequence space λ. We show that these operators form a quasi-normed operator ideal under suitable restrictions on λ. We also study the relationships of these operators with λ-summing, λ-nuclear and quasi-λ-nuclear operators. The results of this paper generalize the earlier results proved by Sinha and Karn; and also Delgado, Pineiro and Serrano.
- Published
- 2013
46. Interval Reliability, Corrections and Developments of 'Reliability Measures of Semi-Markov Systems with General State Space'
- Author
-
Nikolaos Limnios
- Subjects
Statistics and Probability ,Discrete mathematics ,General state ,Continuation ,General Mathematics ,Markov systems ,Applied mathematics ,Renewal theory ,Interval (mathematics) ,Space (mathematics) ,Reliability (statistics) ,Mathematics - Abstract
This paper present the interval reliability of a semi-Markov systems in general state space in continuous and discrete-time cases. We get also, as a particular case the interval reliability for the alternating renewal process. This is a continuation of the paper Limnios (Methodol Comput Appl Probab 14(4):895–917, 2012) where the interval reliability is referred as interval availability and also Propositions 2.1 and 3.1 below replace Propositions 3.4 and 5.3 respectively.
- Published
- 2013
47. Nonlinear regularity models
- Author
-
Alexander D. Ioffe
- Subjects
Discrete mathematics ,Range (mathematics) ,Nonlinear system ,Metric space ,General Mathematics ,Numerical analysis ,Metric (mathematics) ,Applied mathematics ,Context (language use) ,Equivalence (measure theory) ,Software ,Mathematics ,Moduli - Abstract
The paper studies regularity properties of set-valued mappings between metric spaces. In the context of metric regularity, nonlinear models correspond to nonlinear dependencies of estimates of error bounds in terms of residuals. Among the questions addressed in the paper are equivalence of the corresponding concepts of openness and “pseudo-Holder” behavior, general and local regularity criteria with special emphasis on “regularity of order $$k$$ ”, for local settings, and variational methods to extimate regularity moduli in case of length range spaces. The majority of the results presented in the paper are new.
- Published
- 2013
48. Characterization of integrals with respect to arbitrary radon measures by the boundedness indices
- Author
-
A. V. Mikhalev, T.V. Rodionov, and V. K. Zakharov
- Subjects
Statistics and Probability ,Discrete mathematics ,Mathematics::Functional Analysis ,Riesz–Markov–Kakutani representation theorem ,Applied Mathematics ,General Mathematics ,Hausdorff space ,chemistry.chemical_element ,Radon ,Topological space ,Characterization (mathematics) ,Lebesgue integration ,symbols.namesake ,chemistry ,Bounded function ,Radon measure ,symbols ,Mathematics - Abstract
The problem of characterization of integrals as linear functionals is considered in the paper. It starts from the familiar results of F. Riesz (1909) and J. Radon (1913) on integral representation of bounded linear functionals by Riemann–Stieltjes integrals on a segment and by Lebesgue integrals on a compact in \( {\mathbb{R}^n} \), respectively. After works of J. Radon, M. Frechet, and F. Hausdorff the problem of characterization of integrals as linear functionals took the particular form of the problem of extension of Radon’s theorem from \( {\mathbb{R}^n} \) to more general topological spaces with Radon measures. This problem has turned out difficult and its solution has a long and rich history. Therefore, it may be naturally called the Riesz–Radon–Frechet problem of characterization of integrals. The important stages of its solution are connected with such mathematicians as S. Banach, S. Saks, S. Kakutani, P. Halmos, E. Hewitt, R. E. Edwards, N. Bourbaki, V. K. Zakharov, A. V. Mikhalev, et al. In this paper, the Riesz–Radon–Fr´echet problem is solved for the general case of arbitrary Radon measures on Hausdorff spaces. The solution is given in the form of a general parametric theorem in terms of a new notion of the boundedness index of a functional. The theorem implies as particular cases well-known results of the indicated authors characterizing Radon integrals for various classes of Radon measures and topological spaces.
- Published
- 2012
49. On the definition of B-points of a Borel charge on the real line
- Author
-
P. A. Mozolyako
- Subjects
Statistics and Probability ,Discrete mathematics ,Applied Mathematics ,General Mathematics ,Poisson kernel ,Charge (physics) ,Function (mathematics) ,symbols.namesake ,Bibliography ,symbols ,Borel set ,Real line ,Borel measure ,Mathematics - Abstract
Let μ be a Borel charge (i.e., a real Borel measure) on ℝ, and let $ {P_{(y)}}(t) = \frac{y}{\pi \left( {{y^2} + {t^2}} \right)},y > 0 $ , t ∈ ℝ, denote the Poisson kernel. Bourgain proved that for a nonnegative μ and for many points t ∈ ℝ, the variation of the function $ y \mapsto \left( \mu * {P_{ {(y)}}} \right)(x) $ on (0, 1] is finite. This is true, in particular, for so-called B-points x introduced in a previous author’s paper, In the present paper, we give new descriptions of B-points which are adjusted to some applications of this notion. Bibliography: 5 titles.
- Published
- 2012
50. Some further bounds for the Q-index of nested split graphs
- Author
-
C.M. da Fonseca, Slobodan K. Simić, Dejan V. Tošić, and Milica Anđelić
- Subjects
Statistics and Probability ,Discrete mathematics ,Spectral theory ,Simple graph ,Applied Mathematics ,General Mathematics ,Contrast (statistics) ,Mathematics::Spectral Theory ,Signless laplacian ,Combinatorics ,Indifference graph ,Chordal graph ,Cubic function ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The Q-index of a simple graph is the largest eigenvalue of its signless Laplacian, or Q-matrix. In our previous paper [1] we gave three lower and three upper bounds for the Q-index of nested split graphs, also known as threshold graphs. In this paper, we give another two upper bounds, which are expressed as solutions of cubic equations (in contrast to quadratics from [1]). Some computational results are also included.
- Published
- 2012
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