6,545 results
Search Results
2. Corrigendum to the papers on Exceptional orthogonal polynomials: J. Approx. Theory 182 (2014) 29–58, 184 (2014) 176–208 and 214 (2017) 9–48
- Author
-
Antonio J. Durán
- Subjects
Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Hilbert space ,Approx ,symbols.namesake ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Orthogonal polynomials ,symbols ,Analysis ,Mathematics - Abstract
We complete a gap in the proof that exceptional polynomials are complete orthogonal systems in the associated Hilbert spaces.
- Published
- 2020
3. Erratum to the paper 'L∞(L∞)-boundedness and convergence of DG(p)-solutions for nonlinear conservation laws with boundary conditions'
- Author
-
Christian Henke and Lutz Angermann
- Subjects
Conservation law ,Pure mathematics ,Lemma (mathematics) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Lebesgue integration ,Computational Mathematics ,Nonlinear system ,symbols.namesake ,Convergence (routing) ,symbols ,Boundary value problem ,Affine transformation ,Constant (mathematics) ,Mathematics - Abstract
In the paper (HA14), unfortunately, a computational error occurred in one estimate. Although the wrong estimate does not affect the main results, we want to present the necessary corrections. Essentially, Lemma 5.2 has to be corrected and, since it is used in the proof of Theorem 5.1, the proof of this theorem also requires an adaptation. (i) The corrected formulation of Lemma 5.2 is as follows. Lemma 5.2 For Lagrange finite elements with a shape-regular family of affine meshes { T n h } h>0 there is a constant C > 0 independent of q and h such that for all w ∈ Wh and q = 2m, m ∈N: CΛq−2 p (∇w,∇Ip h (wq−1))T ∫ T ‖∇w‖l2‖w‖ q−2 0,∞,T dx, ∀T ∈ T n h , (5.1) where Λp = ‖ ∑ndof i=1 |φi|‖0,∞,T is the Lebesgue constant.
- Published
- 2015
4. Corrigendum to the paper 'Adjoining an Order Unit to a Matrix Ordered Space'
- Author
-
Anil Kumar Karn
- Subjects
Pure mathematics ,General Mathematics ,Mathematical analysis ,Operator theory ,Potential theory ,Theoretical Computer Science ,Strictly convex space ,symbols.namesake ,Matrix (mathematics) ,Fourier analysis ,Ordered space ,symbols ,Order (group theory) ,Unit (ring theory) ,Analysis ,Mathematics - Abstract
An error has been detected (and also corrected) in Theorem 2.8 of the paper entitled “Adjoining an Order Unit to a Matrix Ordered Space” (Positivity, (2005)9: 207–223; DOI 10.1007/s11117-003-2778-5). Accordingly, some of the results of the paper have been modified. Also, a notion of C*-matricially, Riesz normed spaces has been introduced.
- Published
- 2007
5. Mellin analysis of partial differential equations in papers of B. Ziemian
- Author
-
Henryk Kołakowski
- Subjects
Pure mathematics ,Mellin transform ,symbols.namesake ,Partial differential equation ,General Mathematics ,Mellin inversion theorem ,symbols ,Ramanujan's master theorem ,Mathematics - Published
- 2000
6. Isometric Reflections on Banach Spaces after a Paper of A. Skorik and M. Zaidenberg
- Author
-
Julio Becerra Guerrero and Angel Rodríguez Palacios
- Subjects
Unit sphere ,Pure mathematics ,General Mathematics ,Mathematical analysis ,Hilbert space ,Banach space ,Surjective function ,symbols.namesake ,Reflection (mathematics) ,symbols ,Isometry ,46C15 ,Mathematics::Metric Geometry ,Identity component ,Mathematics ,Strong operator topology - Abstract
Let E be a real Banach space. A norm-one element e in E is said to be an isometric reflection vector if there exist a maximal subspace M of E and a linear isometry F : E → E fixing the elements of M and satisfying F (e) = −e. We prove that each of the conditions (i) and (ii) below implies that E is a Hilbert space. (i) There exists a nonrare subset of the unit sphere of E consisting only of isometric reflection vectors, (ii) There is an isometric reflection vector in E, the norm of E is convex transitive, and the identity component of the group of all surjective linear isometries on E relative to the strong operator topology is not reduced to the identity operator on E.
- Published
- 2000
7. Remark on Belyĭ's paper concerning Galois extensions of the maximal cyclotomic field with certain linear groups as Galois groups
- Author
-
Hisashi Kojima
- Subjects
Discrete mathematics ,Pure mathematics ,12F10 ,Galois cohomology ,12F12 ,General Mathematics ,Fundamental theorem of Galois theory ,Galois group ,Abelian extension ,Galois module ,11R32 ,Differential Galois theory ,Embedding problem ,symbols.namesake ,symbols ,Galois extension ,Mathematics - Published
- 1991
8. Hilbert space: compact operators and the trace theorem, by J. Retherford. Pp 131 £13.95 (paper), £27.95 (hard). 1993. ISBN 0- 521-42933-1, -41884-4 (Cambridge)
- Author
-
Nick Lord
- Subjects
Pure mathematics ,symbols.namesake ,General Mathematics ,Hilbert space ,symbols ,Trace theorem ,Compact operator ,Mathematics - Published
- 1994
9. Correction to the paper: NevanlinnaCartan theory over function fields and a Diophantine equation
- Author
-
Junjiro Noguchi
- Subjects
Algebra ,symbols.namesake ,Pure mathematics ,Diophantine set ,Diophantine geometry ,Applied Mathematics ,General Mathematics ,Diophantine equation ,symbols ,Function (mathematics) ,Legendre's equation ,Thue equation ,Mathematics - Published
- 1998
10. A remark to a paper of Kato and Ikebe
- Author
-
Wolf von Wahl
- Subjects
Pure mathematics ,General Mathematics ,Mathematical analysis ,Algebraic geometry ,Sobolev space ,symbols.namesake ,Number theory ,Operator (computer programming) ,Square-integrable function ,symbols ,Order (group theory) ,Element (category theory) ,Schrödinger's cat ,Mathematics - Abstract
This paper deals with Schrodinger operators as they were treated by Kato - Ikebe [3]. It is shown that every element of the domain of definition of the adjoint of such an operator has locally square integrable distributional derivatives up to the order 2. For this the potential of the Schrodinger operator must fulfil a local Stummel condition; if the potential is only locally square integrable a somewhat weaker statement is possible for three dimensions (see remark 2 at the end of this paper).
- Published
- 1977
11. The greatest mathematical paper of all time
- Author
-
A. J. Coleman
- Subjects
Weyl group ,Pure mathematics ,General Mathematics ,Cartan decomposition ,Killing form ,Kac–Moody algebra ,Affine Lie algebra ,Algebra ,symbols.namesake ,History and Philosophy of Science ,symbols ,Cartan matrix ,Lie theory ,Mathematics::Representation Theory ,E8 ,Mathematics - Abstract
Why do I think that Z.v.G.II was an epoch-making paper? (1) It was the paradigm for subsequent efforts to classify the possible structures for any mathematical object. Hawkins [15] documents the fact that Killing’s paper was the immediate inspiration for the work of Cartan, Molien, and Maschke on the structure of linearassociative algebras which culminated in Wedderburn’s theorems. Killing’s success was certainly an example which gave Richard Brauer the will to persist in the attempt to classify simple groups. (2) Weyl’s theory of the representation of semi-simple Lie groups would have been impossible without ideas, results, and methods originated by Killing in Z.v.G.II. Weyl’s fusion of global and local analysis laid the basis for the work of Harish-Chandra and the flowering of abstract harmonic analysis. (3) The whole industry of root systems evinced in the writings of I. Macdonald, V. Kac, R. Moody, and others started with Killing. For the latest see [21]. (4) The Weyl group and the Coxeter transformation are in Z.v.G.II. There they are realized not as orthogonal motions of Euclidean space but as permutations of the roots. In my view, this is the proper way to think of them for general Kac-Moody algebras. Further, the conditions for symmetrisability which play a key role in Kac’s book [17] are given on p. 21 of Z.v.G.II. (5) It was Killing who discovered the exceptional Lie algebra E8, which apparently is the main hope for saving Super-String Theory—not that I expect it to be saved! (6) Roughly one third of the extraordinary work of Elie Cartan was based more or less directly on Z.v.G.II.
- Published
- 1989
12. An operator valued function space integral: A sequel to Cameron and Storvick’s paper
- Author
-
D. L. Skoug and G. W. Johnson
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Multiple integral ,Integral representation theorem for classical Wiener space ,Mathematical analysis ,Riemann integral ,Riemann–Stieltjes integral ,Singular integral ,Fourier integral operator ,Volume integral ,symbols.namesake ,symbols ,Daniell integral ,Mathematics - Abstract
Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.
- Published
- 1971
13. Some remarks on a paper by W. N. Everitt
- Author
-
K. Daho and Heinz Langer
- Subjects
symbols.namesake ,Weight function ,Pure mathematics ,General Mathematics ,Operator (physics) ,Hilbert space ,symbols ,Space (mathematics) ,Mathematics - Abstract
Everitt has shown [1[, that for α ∊ [0, π/2] the undernoted problem (1.1–2) with an indefinite weight function r can be represented by a selfadjoint operator in a suitable Hilbert space. This result is extended to arbitrary α ∊ [0, π), replacing the Hilbert space in some cases by a Pontrjagin space with index one. The problem is also treated in the Krein space generated by the weight function r.
- Published
- 1977
14. Dedekind Sums and a Paper of G. H. Hardy
- Author
-
Bruce C. Berndt
- Subjects
symbols.namesake ,Pure mathematics ,General Mathematics ,Dedekind sum ,symbols ,Dedekind eta function ,Dedekind cut ,Mathematics ,Dedekind–MacNeille completion - Published
- 1976
15. Note on my paper 'a simple proof for von Neumann's minimax theorem'
- Author
-
I. Joó
- Subjects
Pure mathematics ,General Mathematics ,Minimax theorem ,symbols.namesake ,Von Neumann's theorem ,Parthasarathy's theorem ,Von Neumann algebra ,Calculus ,symbols ,Danskin's theorem ,Abelian von Neumann algebra ,Affiliated operator ,Analytic proof ,Mathematics - Published
- 1984
16. Comments on a paper of Brown and Guivarc'h : 'Espaces de Poisson des groupes de Lie'
- Author
-
Calvin C. Moore and Jonathan Rosenberg
- Subjects
symbols.namesake ,Pure mathematics ,General Mathematics ,symbols ,Poisson distribution ,Mathematics - Published
- 1975
17. Addendum and corrigendum to the paper 'Some applications of Zygmund's lemma to non-linear differential equations in Banach and Hilbert spaces' (Studia Math., 44 (1972), pp. 335-344)
- Author
-
T. M. Flett
- Subjects
Lemma (mathematics) ,Pure mathematics ,Nonlinear system ,symbols.namesake ,Differential equation ,General Mathematics ,Calculus ,Hilbert space ,symbols ,Addendum ,Mathematics - Published
- 1972
18. Derived Non-archimedean analytic Hilbert space
- Author
-
Mauro Porta, Jorge António, Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Pure mathematics ,Fiber (mathematics) ,General Mathematics ,010102 general mathematics ,Short paper ,Formal scheme ,Hilbert space ,Space (mathematics) ,01 natural sciences ,symbols.namesake ,Mathematics - Algebraic Geometry ,Mathematics::Category Theory ,0103 physical sciences ,Localization theorem ,FOS: Mathematics ,symbols ,010307 mathematical physics ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,0101 mathematics ,Algebraic Geometry (math.AG) ,Quotient ,Mathematics - Abstract
In this short paper we combine the representability theorem introduced in [17, 18] with the theory of derived formal models introduced in [2] to prove the existence representability of the derived Hilbert space RHilb(X) for a separated k-analytic space X. Such representability results relies on a localization theorem stating that if X is a quasi-compact and quasi-separated formal scheme, then the \infty-category Coh^+(X^rig) of almost perfect complexes over the generic fiber can be realized as a Verdier quotient of the \infty-category Coh^+(X). Along the way, we prove several results concerning the the \infty-categories of formal models for almost perfect modules on derived k-analytic spaces., 28 pages
- Published
- 2019
19. A formula for generating weakly modular forms with weight 12
- Author
-
Aykut Ahmet Aygunes
- Subjects
Discrete mathematics ,symbols.namesake ,Pure mathematics ,Special solution ,General Mathematics ,Short paper ,Modular form ,Eisenstein series ,symbols ,Derivative ,Function (mathematics) ,Mathematics ,Möbius transformation - Abstract
In this short paper, generally, we define a family of functions fk depends on the Eisenstein series with weight 2k, for k ( N. More detail, by considering the function fk, we define a derivative formula for generating weakly modular forms with weight 12. As a result for this, we claim that this formula gives an advantage to find the special solutions of some differential equations.
- Published
- 2016
20. On the Zeros of Dirichlet L-Functions.II (With Corrections to Ön the Zeros of Dirichlet L-Functions.I' and the Subsequent Papers)
- Author
-
Akio Fujii
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Dirichlet L-function ,Dirichlet's energy ,Dirichlet eta function ,Class number formula ,symbols.namesake ,Dirichlet kernel ,Dirichlet's principle ,symbols ,General Dirichlet series ,Dirichlet series ,Mathematics - Published
- 1981
21. Remark on the papers 'Certain properties of class functions and interpolation problems' and 'An analog of the poisson-jensen formula with a double integral'
- Author
-
V. P. Kabaila
- Subjects
Pure mathematics ,symbols.namesake ,Class (set theory) ,Number theory ,General Mathematics ,Ordinary differential equation ,Multiple integral ,Mathematical analysis ,Class function ,symbols ,Poisson distribution ,Mathematics ,Interpolation - Published
- 1973
22. Application of the Theory of Relative Cyclic Fields to both Cases of Fermat's Last Theorem (Second Paper)
- Author
-
H. S. Vandiver
- Subjects
Pure mathematics ,Fermat's little theorem ,Proofs of Fermat's little theorem ,Applied Mathematics ,General Mathematics ,Regular prime ,Fermat's theorem on sums of two squares ,Wieferich prime ,Fermat's factorization method ,symbols.namesake ,Fermat's theorem ,symbols ,Mathematics ,Fermat number - Published
- 1927
23. Remark to my Paper: Introduction to Von Neumann Algebras and Continuous Geometry
- Author
-
Israel Halperin
- Subjects
Algebra ,symbols.namesake ,Pure mathematics ,Von Neumann algebra ,General Mathematics ,symbols ,Tomita–Takesaki theory ,Abelian von Neumann algebra ,Affiliated operator ,Continuous geometry ,Von Neumann architecture ,Mathematics - Published
- 1962
24. A NOTE ON LOCALISED WEIGHTED INEQUALITIES FOR THE EXTENSION OPERATOR
- Author
-
Anthony Carbery, Jonathan Bennett, and Juan Antonio Barceló
- Subjects
Unit sphere ,Pure mathematics ,weighted inequalities ,Mathematics(all) ,Inequality ,General Mathematics ,media_common.quotation_subject ,Mathematical analysis ,Short paper ,Fourier extension operators ,symbols.namesake ,Fourier transform ,Norm (mathematics) ,symbols ,EQUATION ,media_common ,Mathematics - Abstract
We prove optimal radially weighted L-2-norm inequalities for the Fourier extension operator associated to the unit sphere in R-n. Such inequalities valid at all scales are well understood. The purpose of this short paper is to establish certain more delicate single-scale versions of these.
- Published
- 2008
25. An Existence Result for Stepanoff Almost-Periodic Differential Equations
- Author
-
S. Zaidman
- Subjects
Pure mathematics ,symbols.namesake ,Differential equation ,General Mathematics ,Ordinary differential equation ,Short paper ,Hilbert space ,symbols ,Sense (electronics) ,Mathematics - Abstract
In this short paper we present an existence (an unicity) result for a first order differential equation in Hilbert spaces with right-hand side almost-periodic in the sense of Stepanoff.
- Published
- 1971
26. Phase portraits of separable quadratic systems and a bibliographical survey on quadratic systems
- Author
-
Jaume Llibre and Tao Li
- Subjects
Pure mathematics ,Class (set theory) ,Poincaré compactification ,Phase portrait ,General Mathematics ,010102 general mathematics ,Quadratic function ,01 natural sciences ,Separable space ,Quadratic system ,symbols.namesake ,Quadratic equation ,Separable system ,Poincaré conjecture ,symbols ,Compactification (mathematics) ,0101 mathematics ,Quadratic differential ,Mathematics - Abstract
Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we have two objectives. First we provide a brief bibliographical survey on the main results about quadratic systems. Here we do not consider the applications of these systems to many areas as in Physics, Chemist, Economics, Biology, … Second we characterize the new class of planar separable quadratic polynomial differential systems. For such class of systems we provide the normal forms which contain one parameter, and using the Poincare compactification and the blow up technique, we prove that there exist 10 non-equivalent topological phase portraits in the Poincare disc for the separable quadratic polynomial differential systems.
- Published
- 2021
27. Generalized split null point of sum of monotone operators in Hilbert spaces
- Author
-
H. A. Abass, Olalwale K. Oyewole, Ojen Kumar Narain, Akindele Adebayo Mebawondu, and Kazeem Olalekan Aremu
- Subjects
47h09 ,Pure mathematics ,fixed point problem ,47j25 ,General Mathematics ,47j05 ,Hilbert space ,47h06 ,inertial iterative scheme ,symbols.namesake ,Monotone polygon ,firmly nonexpansive ,symbols ,generalized split monotone variational inclusion ,QA1-939 ,Null point ,Mathematics - Abstract
In this paper, we introduce a new type of a generalized split monotone variational inclusion (GSMVI) problem in the framework of real Hilbert spaces. By incorporating an inertial extrapolation method and an Halpern iterative technique, we establish a strong convergence result for approximating a solution of GSMVI and fixed point problems of certain nonlinear mappings in the framework of real Hilbert spaces. Many existing results are derived as corollaries to our main result. Furthermore, we present a numerical example to support our main result and propose an open problem for interested researchers in this area. The result obtained in this paper improves and generalizes many existing results in the literature.
- Published
- 2021
28. On the singular value decomposition over finite fields and orbits of GU×GU
- Author
-
Robert M. Guralnick
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Unitary state ,Nilpotent matrix ,symbols.namesake ,Finite field ,Character (mathematics) ,Kronecker delta ,Singular value decomposition ,Linear algebra ,symbols ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GU m ( q ) × GU n ( q ) on M m × n ( q 2 ) (which is the analog of the singular value decomposition). The proof involves Kronecker’s theory of pencils and the Lang–Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick et al. (2020) where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form A A ∗ and A ∗ A over a finite field and A A ⊤ and A ⊤ A over arbitrary fields.
- Published
- 2021
29. A generalization of the Freidlin–Wentcell theorem on averaging of Hamiltonian systems
- Author
-
Yichun Zhu
- Subjects
Pure mathematics ,Girsanov theorem ,Weak convergence ,General Mathematics ,010102 general mathematics ,Identity matrix ,Differential operator ,01 natural sciences ,Hamiltonian system ,010101 applied mathematics ,symbols.namesake ,Matrix (mathematics) ,Compact space ,Wiener process ,symbols ,0101 mathematics ,Mathematics - Abstract
In this paper, we generalize the classical Freidlin-Wentzell’s theorem for random perturbations of Hamiltonian systems. In (Probability Theory and Related Fields 128 (2004) 441–466), M.Freidlin and M.Weber generalized the original result in the sense that the coefficient for the noise term is no longer the identity matrix but a state-dependent matrix and taking the drift term into consideration. In this paper, We generalize the result by adding a state-dependent matrix that converges uniformly to 0 on any compact sets as ϵ tends to 0 to a state-dependent noise and considering the drift term which contains two parts, the state-dependent mapping and a state-dependent mapping that converges uniformly to 0 on any compact sets as ϵ tends to 0. In the proof, we adapt a new way to prove the weak convergence inside the edge by constructing an auxiliary process and modify the proof in (Probability Theory and Related Fields 128 (2004) 441–466) when proving gluing condition.
- Published
- 2021
30. Logarithmic Potential and Generalized Analytic Functions
- Author
-
O.V. Nesmelova, Vladimir Gutlyanskiĭ, Vladimir Ryazanov, and A.S. Yefimushkin
- Subjects
Statistics and Probability ,Dirichlet problem ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Harmonic (mathematics) ,Unit disk ,Sobolev space ,Riemann hypothesis ,symbols.namesake ,Harmonic function ,symbols ,Neumann boundary condition ,Analytic function ,Mathematics - Abstract
The study of the Dirichlet problem in the unit disk 𝔻 with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin [31]. Later on, the known monograph of Vekua [48] has been devoted to boundary-value problems (only with Holder continuous data) for the generalized analytic functions, i.e., continuous complex valued functions h(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form 𝜕zh + ah + b $$ \overline{h} $$ = c ; where it was assumed that the complex valued functions a; b and c belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. The present paper is a natural continuation of our previous articles on the Riemann, Hilbert, Dirichlet, Poincar´e and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic, and the so-called A−harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here, we extend the corresponding results to the generalized analytic functions h : D → ℂ with the sources g : 𝜕zh = g ∈ Lp, p > 2 , and to generalized harmonic functions U with sources G : △U = G ∈ Lp, p > 2. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary-value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary-values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincar´e problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semilinear equations of mathematical physics in anisotropic and inhomogeneous media.
- Published
- 2021
31. Stability and collapse of the Lyapunov spectrum for Perron–Frobenius operator cocycles
- Author
-
Anthony Quas and Cecilia González-Tokman
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Dense set ,Applied Mathematics ,General Mathematics ,Blaschke product ,Banach space ,Lyapunov exponent ,Fixed point ,symbols.namesake ,Unit circle ,symbols ,Invariant measure ,Mathematics ,Analytic function - Abstract
In this paper, we study random Blaschke products, acting on the unit circle, and consider the cocycle of Perron-Frobenius operators acting on Banach spaces of analytic functions on an annulus. We completely describe the Lyapunov spectrum of these cocycles. As a corollary, we obtain a simple random Blaschke product system where the Perron-Frobenius cocycle has infinitely many distinct Lyapunov exponents, but where arbitrarily small natural perturbations cause a complete collapse of the Lyapunov spectrum, except for the exponent 0 associated with the absolutely continuous invariant measure. That is, under perturbations, the Lyapunov exponents become 0 with multiplicity 1, and $-\infty$ with infinite multiplicity. This is superficially similar to the finite-dimensional phenomenon, discovered by Bochi \cite{Bochi-thesis}, that away from the uniformly hyperbolic setting, small perturbations can lead to a collapse of the Lyapunov spectrum to zero. In this paper, however, the cocycle and its perturbation are explicitly described; and further, the mechanism for collapse is quite different. We study stability of the Perron-Frobenius cocycles arising from general random Blaschke products. We give a necessary and sufficient criterion for stability of the Lyapunov spectrum in terms of the derivative of the random Blaschke product at its random fixed point, and use this to show that an open dense set of Blaschke product cocycles have hyperbolic Perron-Frobenius cocycles. In the final part, we prove a relationship between the Lyapunov spectrum of a single cocycle acting on two different Banach spaces, allowing us to draw conclusions for the same cocycles acting on $C^r$ functions spaces.
- Published
- 2021
32. Dynamical significance of generalized fractional integral inequalities via convexity
- Author
-
M. Zakarya, Kottakkaran Sooppy Nisar, Ahmed Morsy, Gauhar Rahman, Sabila Ali, Rana Safdar Ali, Sunil Dutt Purohit, and Shahid Mubeen
- Subjects
Pure mathematics ,Inequality ,Kernel (set theory) ,General Mathematics ,media_common.quotation_subject ,Mathematics::Classical Analysis and ODEs ,η2)-convex function ,generalized fractional inequalities ,Function (mathematics) ,Type inequality ,Type (model theory) ,hadamard inequality ,Convexity ,symbols.namesake ,fractional inequalities ,symbols ,QA1-939 ,wright generalized bessel function ,Convex function ,(η1 ,Bessel function ,Mathematics ,media_common - Abstract
The main goal of this paper is to develop the significance of generalized fractional integral inequalities via convex functions. We obtain the new version of fractional integral inequalities with the extended Wright generalized Bessel function acting as a kernel for the convex function, which deals with the Hermite-Hadamard type and trapezoid type inequalities. Moreover, we establish new mid-point type and trapezoid type integral inequalities for $ (\eta_{1}, \eta_{2}) $-convex function related to Hermite-Hadamard type inequality. We establish new version of integral inequality for $ (\eta_{1}, \eta_{2}) $-convex function related to Fejer type. The results discussed in this paper are a generalized version of many inequalities in literature.
- Published
- 2021
33. New Computational Formulas for Special Numbers and Polynomials Derived from Applying Trigonometric Functions to Generating Functions
- Author
-
Yilmaz Simsek and Neslihan Kilar
- Subjects
Catalan number ,Pure mathematics ,Bernoulli's principle ,symbols.namesake ,General Mathematics ,Factorial number system ,Euler's formula ,symbols ,Stirling number ,Trigonometric functions ,Type (model theory) ,Mathematics - Abstract
The aim of this paper is to apply trigonometric functions with functional equations of generating functions. Using the resulted new equations and formulas from this application, we obtain many special numbers and polynomials such as the Stirling numbers, Bernoulli and Euler type numbers, the array polynomials, the Catalan numbers, and the central factorial numbers. We then introduce combinatorial sums related to these special numbers and polynomials. Moreover, we gave some remarks that relates our new findings from this paper to the relations found in earlier studies.
- Published
- 2021
34. Sharp Hardy Identities and Inequalities on Carnot Groups
- Author
-
Guozhen Lu, Nguyen Lam, and Joshua Flynn
- Subjects
Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Statistical and Nonlinear Physics ,01 natural sciences ,03 medical and health sciences ,symbols.namesake ,0302 clinical medicine ,symbols ,030212 general & internal medicine ,0101 mathematics ,Carnot cycle ,Mathematics ,media_common - Abstract
In this paper we establish general weighted Hardy identities for several subelliptic settings including Hardy identities on the Heisenberg group, Carnot groups with respect to a homogeneous gauge and Carnot–Carathéodory metric, general nilpotent groups, and certain families of Hörmander vector fields. We also introduce new weighted uncertainty principles in these settings. This is done by continuing the program initiated by [N. Lam, G. Lu and L. Zhang, Factorizations and Hardy’s-type identities and inequalities on upper half spaces, Calc. Var. Partial Differential Equations 58 2019, 6, Paper No. 183; N. Lam, G. Lu and L. Zhang, Geometric Hardy’s inequalities with general distance functions, J. Funct. Anal. 279 2020, 8, Article ID 108673] of using the Bessel pairs introduced by [N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Math. Surveys Monogr. 187, American Mathematical Society, Providence, 2013] to obtain Hardy identities. Using these identities, we are able to improve significantly existing Hardy inequalities in the literature in the aforementioned subelliptic settings. In particular, we establish the Hardy identities and inequalities in the spirit of [H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 1997, 443–469] and [H. Brezis and M. Marcus, Hardy’s inequalities revisited. Dedicated to Ennio De Giorgi, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 1–2, 217–237] in these settings.
- Published
- 2021
35. Collectively fixed point theorems in noncompact abstract convex spaces with applications
- Author
-
Rong Li, Kai Zhang, and Haishu Lu
- Subjects
TheoryofComputation_MISCELLANEOUS ,collectively fixed point ,Computer Science::Computer Science and Game Theory ,Pure mathematics ,nash equilibrium ,General Mathematics ,generalized weak implicit inclusion problem ,Regular polygon ,Fixed-point theorem ,Fixed point ,symbols.namesake ,Intersection ,Nash equilibrium ,Generalized Pareto distribution ,abstract convex space ,QA1-939 ,symbols ,Product topology ,nonempty intersection theorem ,Finite set ,Mathematics - Abstract
In this paper, by using the KKM theory and the properties of $ \Gamma $-convexity and $ {\frak{RC}} $-mapping, we investigate the existence of collectively fixed points for a family with a finite number of set-valued mappings on the product space of noncompact abstract convex spaces. Consequently, as applications, some existence theorems of generalized weighted Nash equilibria and generalized Pareto Nash equilibria for constrained multiobjective games, some nonempty intersection theorems with applications to the Fan analytic alternative formulation and the existence of Nash equilibria, and some existence theorems of solutions for generalized weak implicit inclusion problems in noncompact abstract convex spaces are given. The results obtained in this paper extend and generalize many corresponding results of the existing literature.
- Published
- 2021
36. On the coercitive solvability of the non-linear Laplace-Beltrami equation in Hilbert space
- Subjects
Linear map ,Physics ,Pure mathematics ,symbols.namesake ,Nonlinear system ,Laplace–Beltrami operator ,General Mathematics ,Operator (physics) ,Hilbert space ,symbols ,Development (differential geometry) ,Space (mathematics) ,Differential operator - Abstract
The problem of separability of differential operators is considered for the first time in the works of V. N. Everitt and M. Hirz. Further development of this theory belongs to K. H. Boymatov, M. Otelbaev and their students. The main part of the published papers on this theory relates to linear operators. The nonlinear case was considered mainly when studied operator was a weak perturbation of the linear one. The case when the operator under study is not a weak perturbation of the linear operator is considered only in some works. The results obtained in this paper also relate to this little-studied case. The paper studies the coercive properties of the nonlinear Laplace-Beltrami operator in the space L2(R^n) $$L[u]=-\frac{1}{\sqrt{det g(x)}}\sum_{i,j=1}^n\frac{\partial}{\partial x_i}\left[\sqrt{det g(x)}g^{-1}(x)\frac{\partial u}{\partial x_j}\right]+V(x,u)u(x)$$, and proves its separability in this space by coercivity estimates. The operator under study is not a weak perturbation of the linear operator, i.e. it is strongly nonlinear. Based on the obtained coercive estimates and separability, the solvability of the nonlinear Laplace-Beltrami equation in the space L2(R^n) is studied
- Published
- 2021
37. On some additive problems of Goldbach’s type
- Subjects
Pure mathematics ,Mathematics::Number Theory ,General Mathematics ,Diophantine equation ,Function (mathematics) ,Riemann zeta function ,Riemann hypothesis ,symbols.namesake ,Exponential sum ,Goldbach's conjecture ,symbols ,Asymptotic formula ,Remainder ,Mathematics - Abstract
In this paper, we find an asymptotic formula with power-saving remainder term for the number of solutions of one non-linear ternary problem with primes. The proof is based on the "precise formula"for Chebyshev’s function involving the zeros of Riemann zeta function. In fact, a ternary problem "at each zero"is solved. I. M. Vinogradov’s solution of the ternary Goldbach problem (1937, see [1], [2]) opened the way of solving a wide class of problems of the above type. In 1938, he found a power-saving estimate (with respect to the length of the summation interval) for the mean value of the modulus of the exponential sum with primes (see [2], theorem 3, p.82; theorems 6 and 7, p.86). Starting at 1996, G.I.Arkhipov, K.Buriev and the author have obtained several results concerning the exceptional sets in some binary problems of Goldbach’s type. These results used both the tools of the theory of Diophantine approximations and the "precise formulas"from Riemann’s zeta function theory. At the same time, the method of estimating of linear sums with primes based on the measure theory was derived in the papers of G. L. Watson, D. Bruedern, R. D. Cook and A. Perelli.
- Published
- 2021
38. On the Relationship Between the Factorization Problem in the Wiener Algebra and the Truncated Wiener–Hopf Equation
- Author
-
A. F. Voronin
- Subjects
Reduction (complexity) ,symbols.namesake ,Pure mathematics ,Riemann problem ,Factorization ,General Mathematics ,Matrix function ,symbols ,Interval (mathematics) ,Wiener algebra ,Integral equation ,Mathematics ,Connection (mathematics) - Abstract
In this paper, we study the homogeneous vector Riemann boundary value problem (the factorization problem) from a new point of view. Namely, we reduce the Riemann problem to the truncated Wiener–Hopf equation (a convolution equation in a finite interval). We establish a connection between the problem of the factorization of a matrix function in the Wiener algebra of order two and the truncated Wiener–Hopf equation and obtain an explicit formula for this relationship. Note that the form of the matrix function considered in this paper differs from its most general form in the Wiener algebra; however, in this case, this is inessential. The truncated Wiener–Hopf equation is one of the most thoroughly studied Fredholm integral equations of the second kind. Therefore, the idea of the mentioned reduction can be expected to lead to new results in studying the factorization problem.
- Published
- 2020
39. Area‐Minimizing Currents mod 2 Q : Linear Regularity Theory
- Author
-
Jonas Hirsch, Camillo De Lellis, Salvatore Stuvard, and Andrea Marchese
- Subjects
Pure mathematics ,multiple valued functions, Dirichlet integral, regularity theory, area minimizing currents mod(p), minimal surfaces, linearization ,Generalization ,General Mathematics ,Dimension (graph theory) ,area minimizing currents mod(p) ,linearization ,minimal surfaces ,Dirichlet integral ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Mathematics - Analysis of PDEs ,Mod ,FOS: Mathematics ,49Q15, 49Q05, 49N60, 35B65, 35J47 ,0101 mathematics ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Codimension ,regularity theory ,symbols ,multiple valued functions ,Analysis of PDEs (math.AP) - Abstract
We establish a theory of $Q$-valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currents $\mathrm{mod}(p)$ when $p=2Q$, and to establish a first general partial regularity theorem for every $p$ in any dimension and codimension., 37 pages. First part of a two-papers work aimed at establishing a first general partial regularity theory for area minimizing currents modulo p, for any p and in any dimension and codimension. v3 is the final version, to appear on Comm. Pure Appl. Math
- Published
- 2020
40. On Bourgain’s Counterexample for the Schrödinger Maximal Function
- Author
-
Lillian B. Pierce
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,02 engineering and technology ,021001 nanoscience & nanotechnology ,01 natural sciences ,symbols.namesake ,symbols ,Maximal function ,0101 mathematics ,0210 nano-technology ,Schrödinger's cat ,Counterexample ,Mathematics - Abstract
This paper provides a rigorous derivation of a counterexample of Bourgain, related to a well-known question of pointwise a.e. convergence for the solution of the linear Schrödinger equation, for initial data in a Sobolev space. This counterexample combines ideas from analysis and number theory, and the present paper demonstrates how to build such counterexamples from first principles, and then optimize them.
- Published
- 2020
41. On Boundedness Property of Singular Integral Operators Associated to a Schrödinger Operator in a Generalized Morrey Space and Applications
- Author
-
Thanh-Nhan Nguyen, Xuan Truong Le, and Ngoc Trong Nguyen
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Property (philosophy) ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,General Physics and Astronomy ,Function (mathematics) ,Type (model theory) ,Space (mathematics) ,01 natural sciences ,Schrödinger equation ,010101 applied mathematics ,symbols.namesake ,Riesz transform ,Operator (computer programming) ,symbols ,0101 mathematics ,Schrödinger's cat ,Mathematics - Abstract
In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrodinger operator $${\cal L} = \Delta + {\bf{V}}$$ in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential V considered in this paper is a non-negative function satisfying the suitable reverse Holder’s inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schrodinger equations in the new Morrey space.
- Published
- 2020
42. Locally finiteness and convolution products in groupoids
- Author
-
Joseph Neggers, Hee Sik Kim, and In Ho Hwang
- Subjects
Pure mathematics ,moebius function ,General Mathematics ,interval value function ,lcsh:Mathematics ,above ,locally finite ,below ,groupoid ,lcsh:QA1-939 ,convolution product ,Riemann zeta function ,zeta function ,symbols.namesake ,Number theory ,Special functions ,Lattice (order) ,symbols ,transitive interval property ,Mathematics - Abstract
In this paper, we introduce a version of the Moebius function and other special functions on a particular class of intervals for groupoids, and study them to obtain results analogous to those obtained in the usual lattice, combinatorics and number theory setting, but of course much more general due to the viewpoint taken in this paper.
- Published
- 2020
43. Nψ,ϕ-type Quotient Modules over the Bidisk
- Author
-
Chang Hui Wu and Tao Yu
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Essential spectrum ,Hardy space ,Characterization (mathematics) ,Type (model theory) ,01 natural sciences ,symbols.namesake ,Compact space ,Compression (functional analysis) ,0103 physical sciences ,Quotient module ,symbols ,010307 mathematical physics ,0101 mathematics ,Quotient ,Mathematics - Abstract
Let H2(ⅅ2) be the Hardy space over the bidisk ⅅ2, and let Mψ,ϕ = [(ψ(z) − ϕ(w))2] be the submodule generated by (ψ(z) − ϕ(w))2, where ψ(z) and ϕ(w) are nonconstant inner functions. The related quotient module is denoted by Nψ,ϕ = H2(ⅅ2) ⊖ Mψ,ϕ. In this paper, we give a complete characterization for the essential normality of Nψ,ϕ. In particular, if ψ(z)= z, we simply write Mψ,ϕ and Nψ,ϕ as Mϕ and Nϕ respectively. This paper also studies compactness of evaluation operators L(0)∣nϕ and R(0)ϕnϕ, essential spectrum of compression operator Sz on Nϕ, essential normality of compression operators Sz and Sw on Nϕ.
- Published
- 2020
44. More about singular traces on simply generated operator ideals
- Author
-
Albrecht Pietsch
- Subjects
Large class ,Sequence ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Hilbert space ,Extension (predicate logic) ,Space (mathematics) ,01 natural sciences ,symbols.namesake ,Operator (computer programming) ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
During half a century, singular traces on ideals of Hilbert space operators have been constructed by looking for linear forms on associated sequence ideals. Only recently, the author was able to eliminate this auxiliary step by directly applying Banach’s version of the extension theorem; see (Integral Equ. Oper. Theory 91, 21, 2019 and 92, 7, 2020). Of course, the relationship between the new approach and the older ones must be investigated. In the first paper, this was done for $${\mathfrak {L}}_{1,\infty } (H)$$ . To save space, such considerations were postponed in the second paper, which deals with a large class of principal ideals, called simply generated. This omission will now be rectified.
- Published
- 2020
45. On Some Local Asymptotic Properties of Sequences with a Random Index
- Author
-
Yu. V. Yakubovich, O. V. Rusakov, and B. A. Baev
- Subjects
Rademacher distribution ,Hurst exponent ,Pure mathematics ,Fractional Brownian motion ,Stochastic process ,General Mathematics ,010102 general mathematics ,Poisson distribution ,01 natural sciences ,010305 fluids & plasmas ,Cox process ,symbols.namesake ,0103 physical sciences ,symbols ,0101 mathematics ,Telegraph process ,Random variable ,Mathematics - Abstract
Random sequences with random or stochastic indices controlled by a doubly stochastic Poisson process are considered in this paper. A Poisson stochastic index process (PSI-process) is a random process with the continuous time ψ(t) obtained by subordinating a sequence of random variables (ξj), j = 0, 1, …, by a doubly stochastic Poisson process Π1(tλ) via the substitution ψ(t) = $${{\xi }_{{{{\Pi }_{1}}(t\lambda )}}}$$ , t $$ \geqslant $$ 0, where the random intensity λ is assumed independent of the standard Poisson process Π1. In this paper, we restrict our consideration to the case of independent identically distributed random variables (ξj) with a finite variance. We find a representation of the fractional Ornstein–Uhlenbeck process with the Hurst exponent H ∈ (0, 1/2) introduced and investigated by R. Wolpert and M. Taqqu (2005) in the form of a limit of normalized sums of independent identically distributed PSI-processes with an explicitly given distribution of the random intensity λ. This fractional Ornstein–Uhlenbeck process provides a local, at t = 0, mean-square approximation of the fractional Brownian motion with the same Hurst exponent H ∈ (0, 1/2). We examine in detail two examples of PSI-processes with the random intensity λ generating the fractional Ornstein–Uhlenbeck process in the Wolpert and Taqqu sense. These are a telegraph process arising when ξ0 has a Rademacher distribution ±1 with the probability 1/2 and a PSI-process with the uniform distribution for ξ0. For these two examples, we calculate the exact and the asymptotic values of the local modulus of continuity for a single PSI-process over a small fixed time span.
- Published
- 2020
46. Vector-valued q-variational inequalities for averaging operators and the Hilbert transform
- Author
-
Tao Ma, Wei Liu, and Guixiang Hong
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Banach space ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,Variational inequality ,symbols ,010307 mathematical physics ,Hilbert transform ,0101 mathematics ,Martingale (probability theory) ,Mathematics - Abstract
Recently, the authors have established $$L^p$$ -boundedness of vector-valued q-variational inequalities for averaging operators which take values in the Banach space satisfying the martingale cotype q property in Hong and Ma (Math Z 286(1–2):89–120, 2017). In this paper, we prove that the martingale cotype q property is also necessary for the vector-valued q-variational inequalities, which was a question left open in the previous paper. Moreover, we also prove that the UMD property and the martingale cotype q property can be characterized in terms of vector valued q-variational inequalities for the Hilbert transform.
- Published
- 2020
47. Graphons, permutons and the Thoma simplex: three mod‐Gaussian moduli spaces
- Author
-
Ashkan Nikeghbali, Valentin Féray, Pierre-Loïc Méliot, Universität Zürich [Zürich] = University of Zurich (UZH), Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), University of Zurich, and Méliot, Pierre‐Loïc
- Subjects
Pure mathematics ,General Mathematics ,Gaussian ,340 Law ,610 Medicine & health ,0102 computer and information sciences ,01 natural sciences ,symbols.namesake ,510 Mathematics ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,FOS: Mathematics ,Limit (mathematics) ,0101 mathematics ,Concentration inequality ,ComputingMilieux_MISCELLANEOUS ,2600 General Mathematics ,Mathematics ,Central limit theorem ,Random graph ,Simplex ,Probability (math.PR) ,010102 general mathematics ,Observable ,10003 Department of Banking and Finance ,Moduli space ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,10123 Institute of Mathematics ,010201 computation theory & mathematics ,symbols ,Mathematics - Probability - Abstract
In this paper, we show how to use the framework of mod-Gaussian convergence in order to study the fluctuations of certain models of random graphs, of random permutations and of random integer partitions. We prove that, in these three frameworks, a generic homogeneous observable of a generic random model is mod-Gaussian under an appropriate renormalisation. This implies a central limit theorem with an extended zone of normality, a moderate deviation principle, an estimate of the speed of convergence, a local limit theorem and a concentration inequality. The universal asymptotic behavior of the observables of these models gives rise to a notion of mod-Gaussian moduli space., Comment: New version: the paper has been slightly shortened, and a few references were added. 52 pages, 13 figures
- Published
- 2020
48. Schrödinger Quantization of Infinite-Dimensional Hamiltonian Systems with a Nonquadratic Hamiltonian Function
- Author
-
N. N. Shamarov and Oleg G. Smolyanov
- Subjects
Hamiltonian mechanics ,Pure mathematics ,Lebesgue measure ,General Mathematics ,010102 general mathematics ,Convex set ,Space (mathematics) ,01 natural sciences ,Measure (mathematics) ,010305 fluids & plasmas ,Hamiltonian system ,symbols.namesake ,Fourier transform ,0103 physical sciences ,symbols ,0101 mathematics ,Hamiltonian (control theory) ,Mathematics - Abstract
According to a theorem of Andre Weil, there does not exist a standard Lebesgue measure on any infinite-dimensional locally convex space. Because of that, Schrodinger quantization of an infinite-dimensional Hamiltonian system is often defined using a sigma-additive measure, which is not translation-invariant. In the present paper, a completely different approach is applied: we use the generalized Lebesgue measure, which is translation-invariant. In implicit form, such a measure was used in the first paper published by Feynman (1948). In this situation, pseudodifferential operators whose symbols are classical Hamiltonian functions are formally defined as in the finite-dimensional case. In particular, they use unitary Fourier transforms which map functions (on a finite-dimensional space) into functions. Such a definition of the infinite-dimensional unitary Fourier transforms has not been used in the literature.
- Published
- 2020
49. Mappings with finite length distortion and prime ends on Riemann surfaces
- Author
-
Sergei Volkov and I Vladimir Ryazanov
- Subjects
Statistics and Probability ,Pure mathematics ,Series (mathematics) ,Generalization ,Applied Mathematics ,General Mathematics ,Riemann surface ,010102 general mathematics ,Boundary (topology) ,02 engineering and technology ,01 natural sciences ,Prime (order theory) ,010305 fluids & plasmas ,Sobolev space ,Distortion (mathematics) ,symbols.namesake ,020303 mechanical engineering & transports ,0203 mechanical engineering ,0103 physical sciences ,Euclidean geometry ,symbols ,0101 mathematics ,Mathematics - Abstract
The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings in the Sobolev classes (mappings with generalized derivatives) on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class of FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio-Ryazanov-Srebro-Yakubov at 2004 and then included in the known book of these authors at 2009 on the modern mapping theory. As was shown in the recent papers of Kovtonyuk-Petkov-Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes, because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasiisometries. We prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extensions to the boundary of the mappings with finite length distortion between domains on Riemann surfaces by Caratheodory prime ends. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to be the very simple condition on the integrability of the dilatations in the first power. The criteria for the continuous extension of the direct mappings to the boundary have a much more refined nature. One of such criteria is the existence of a majorant for the dilatation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation from its mean value over infinitesimal disks centered at boundary points. As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion to the closures of domains by Caratheodory prime ends are obtained.
- Published
- 2020
50. Dini–Lipschitz functions for the quaternion linear canonical transform
- Author
-
N. Safouane, Radouan Daher, Azzedine Achak, and A. Bouhlal
- Subjects
Pure mathematics ,symbols.namesake ,Fourier transform ,General Mathematics ,Computation ,symbols ,Image processing ,Equivalence (formal languages) ,Quaternion ,Singular integral operators ,Lipschitz continuity ,Interpolation theory ,Mathematics - Abstract
This paper is an exposition of some results on calculation of the K-functional which have number of applications of interpolation theory. In particular some recent problems in image processing and singular integral operators require the computation of suitable K-functionals. In this paper we will give some results concerning the equivalence of a K-functional and the modulus of smoothness constructed by the generalized Steklov function. We mention here that we have generalized the Steklov’s function for Fourier transform to quaternion linear canonical transform. This paper generalizes also Titchmarsh’s theorem for measurable sets from complex domain to hyper complex domain by using quaternion algebras, associated with the quaternion linear canonical transform.
- Published
- 2020
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.