1,377 results
Search Results
2. A remark to a paper of Kato and Ikebe
- Author
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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
3. The greatest mathematical paper of all time
- Author
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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
4. Note on my paper 'a simple proof for von Neumann's minimax theorem'
- Author
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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
5. 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
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V. P. Kabaila
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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
6. Logarithmic Potential and Generalized Analytic Functions
- Author
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O.V. Nesmelova, Vladimir Gutlyanskiĭ, Vladimir Ryazanov, and A.S. Yefimushkin
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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
7. New Computational Formulas for Special Numbers and Polynomials Derived from Applying Trigonometric Functions to Generating Functions
- Author
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Yilmaz Simsek and Neslihan Kilar
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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
8. On Boundedness Property of Singular Integral Operators Associated to a Schrödinger Operator in a Generalized Morrey Space and Applications
- Author
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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
9. Nψ,ϕ-type Quotient Modules over the Bidisk
- Author
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Chang Hui Wu and Tao Yu
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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
10. More about singular traces on simply generated operator ideals
- Author
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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
11. Vector-valued q-variational inequalities for averaging operators and the Hilbert transform
- Author
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Tao Ma, Wei Liu, and Guixiang Hong
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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
12. Dini–Lipschitz functions for the quaternion linear canonical transform
- Author
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N. Safouane, Radouan Daher, Azzedine Achak, and A. Bouhlal
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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
13. The Wiener Measure on the Heisenberg Group and Parabolic Equations
- Author
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S. V. Mamon
- Subjects
Statistics and Probability ,Pure mathematics ,Semigroup ,Stochastic process ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Markov process ,01 natural sciences ,Measure (mathematics) ,010305 fluids & plasmas ,Nilpotent ,symbols.namesake ,0103 physical sciences ,Path integral formulation ,Lie algebra ,symbols ,Heisenberg group ,0101 mathematics ,Mathematics - Abstract
In this paper, we study questions related to the theory of stochastic processes on Lie nilpotent groups. In particular, we consider the stochastic process on the Heisenberg group H3(ℝ) whose trajectories satisfy the horizontal conditions in the stochastic sense relative to the standard contact structure on H3 (ℝ). It is shown that this process is a homogeneous Markov process relative to the Heisenberg group operation. There was found a representation in the form of a Wiener integral for a one-parameter linear semigroup of operators for which the Heisenberg sublaplacian generated by basis vector fields of the corresponding Lie algebra L(H3) is producing. The main method of solving the problem in this paper is using the path integrals technique, which indicates the common direction of further development of the results.
- Published
- 2020
14. Titchmarsh’s theorem and some remarks concerning the right-sided quaternion Fourier transform
- Author
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Azzedine Achak, Radouan Daher, Aziz Bouhlal, and N. Safouane
- Subjects
Pure mathematics ,Quaternion algebra ,General Mathematics ,010102 general mathematics ,Space (mathematics) ,Lipschitz continuity ,01 natural sciences ,Square (algebra) ,Function of several real variables ,010101 applied mathematics ,Translation operator ,symbols.namesake ,Fourier transform ,symbols ,0101 mathematics ,Quaternion ,Mathematics - Abstract
This paper is based mainly on Titchmarsh’s theorem (Introduction to the theory of Fourier integrals. Clarendon Press, Oxford, 1937, Theorem 84) in the one-dimensional case. Abilov et al. (Comput Math Math Phys 48:2146, 2008) proved two useful estimates for the Fourier transform in the space of square integral multivariable functions on certain classes of functions characterized by the generalized continuity modulus, and these estimates are proved by Abilovs for only two variables, using a translation operator. The purpose of this paper is to study these estimates for Quaternion Fourier transforms, also the functions satisfy Lipschitz conditions of certain orders. Thus we study the Quaternion Fourier transforms of Lipschitz function in the functions space $$L^r({\mathbb {R}}^{2},{\mathcal {H}})$$, where $${\mathcal {H}}$$ a quaternion algebra which will be specified in due course.
- Published
- 2020
15. New solutions to Legendre’s incomplete elliptic integrals of the first and second kinds and p-elliptic integrals
- Author
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Prateek Pralhad Kulkarni
- Subjects
Pure mathematics ,Series (mathematics) ,Applied Mathematics ,General Mathematics ,Numerical analysis ,symbols.namesake ,Special functions ,Euler's formula ,symbols ,Elliptic integral ,Inverse trigonometric functions ,Gamma function ,Legendre polynomials ,Mathematics - Abstract
The elliptic integrals are of interest in various disciplines. While series solutions do exist for complete elliptic integrals, there are no deduced series solutions for Incomplete elliptic integrals, in terms of the special functions. This paper provides novel solutions of the Legendre forms of incomplete elliptic integrals of the first and second kinds in terms of the Euler’s gamma functions. The paper also proposes new solutions to inverse trigonometric functions, which has never been known till date.
- Published
- 2021
16. Existence and Regularity of Weak Solutions for $$\psi $$-Hilfer Fractional Boundary Value Problem
- Author
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J. Vanterler da C. Sousa, E. Capelas de Oliveira, and M. Aurora P. Pulido
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Pure mathematics ,General Mathematics ,010102 general mathematics ,Hilbert space ,01 natural sciences ,Fractional calculus ,010101 applied mathematics ,symbols.namesake ,symbols ,High Energy Physics::Experiment ,Integration by parts ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
In the present paper, we investigate the existence and regularity of weak solutions for $$\psi $$ -Hilfer fractional boundary value problem in $$\mathbb {C}^{\alpha ,\beta ;\psi }_{2}$$ and $$\mathcal {H}$$ (Hilbert space) spaces, using extension of the Lax–Milgram theorem. In this sense, to finalize the paper, we discuss the integration by parts for $$\psi $$ -Riemann–Liouville fractional integral and $$\psi $$ -Hilfer fractional derivative.
- Published
- 2021
17. Extremal decomposition of a multidimensional complex space for five domains
- Author
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Yaroslav Zabolotnii and I. V. Denega
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Statistics and Probability ,Pure mathematics ,Geometric function theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Unit circle ,Complex space ,Product (mathematics) ,Green's function ,0103 physical sciences ,Simply connected space ,Decomposition (computer science) ,symbols ,0101 mathematics ,Quadratic differential ,Mathematics - Abstract
The paper is devoted to one open extremal problem in the geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider the problem of the maximum of a product of inner radii of n non-overlapping domains containing points of a unit circle and the power γ of the inner radius of a domain containing the origin. The problem was formulated in 1994 in Dubinin’s paper in the journal “Russian Mathematical Surveys” in the list of unsolved problems and then repeated in his monograph in 2014. Currently, it is not solved in general. In this paper, we obtained a solution of the problem for five simply connected domains and power γ ∈ (1; 2:57] and generalized this result to the case of multidimensional complex space.
- Published
- 2019
18. Unbounded asymptotic equivalences of operator nets with applications
- Author
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Niyazi Anıl Gezer and Nazife Erkurşun-Özcan
- Subjects
Pure mathematics ,021103 operations research ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,02 engineering and technology ,Operator theory ,01 natural sciences ,Potential theory ,Theoretical Computer Science ,symbols.namesake ,Operator (computer programming) ,General theory ,Fourier analysis ,Lattice (order) ,symbols ,Equivalence relation ,0101 mathematics ,Mathematics::Representation Theory ,Martingale (probability theory) ,Analysis ,Mathematics - Abstract
Present paper deals with applications of asymptotic equivalence relations on operator nets. These relations are defined via unbounded convergences on vector lattices. Given two convergences $$\mathfrak {c}$$ and $$\mathfrak {d}$$ on a vector lattice, we study $$\mathfrak {d}$$ -asymptotic properties of operator nets formed by $$\mathfrak {c}$$ -continuous operators. Asymptotic equivalences are known to be useful and extremely important tools to study infinite behaviors of strongly convergent operator nets and continuous semigroups. After giving a general theory, paper focuses on $$\mathfrak {d}$$ -martingale and $$\mathfrak {d}$$ -Lotz–Rabiger nets.
- Published
- 2019
19. Grüss type and related integral inequalities in probability spaces
- Author
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László Horváth
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Inequality ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,MathematicsofComputing_NUMERICALANALYSIS ,Mathematics::Classical Analysis and ODEs ,Hilbert space ,Complex valued ,Type inequality ,Type (model theory) ,symbols.namesake ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,symbols ,Discrete Mathematics and Combinatorics ,Mathematics ,media_common - Abstract
In this paper we study Gruss type inequalities for real and complex valued functions in probability spaces. Some earlier Gruss type inequalities are extended and refined. Our approach leads to new integral inequalities which are interesting in their own right. As an application, we give a Gruss type inequality for normal operators in a Hilbert space. Similar results are obtained only for self-adjoint operators in earlier papers.
- Published
- 2018
20. On the Bessel–Wright Transform
- Author
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Ilyes Karoui, Lazhar Dhaouadi, and Ahmed Fitouhi
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,010103 numerical & computational mathematics ,Eigenfunction ,Differential operator ,01 natural sciences ,symbols.namesake ,Operator (computer programming) ,Schwartz space ,symbols ,Differentiable function ,0101 mathematics ,Invariant (mathematics) ,Bessel function ,Lommel function ,Mathematics - Abstract
In the present paper, we consider a class of second-order singular differential operators which generalize the well-known Bessel differential operator. The associated eigenfunctions are the Bessel–Wright functions. These functions can be obtained by the action of the Riemann–Liouville operator on the normalized Bessel functions. We introduce a Bessel–Wright transform with Bessel–Wright functions as kernel which is connected to the classical Bessel–Fourier transform via the dual of the Riemann–Liouville operator. The Bessel–Wright transform leaves invariant the Schwartz space and sends the set of functions indefinitely differentiable with compact support into the Paley–Wiener space. We conclude the paper by proving two variants of the inversion formulas.
- Published
- 2018
21. $$L^p$$ L p Sobolev Regularity for a Class of Radon and Radon-Like Transforms of Various Codimension
- Author
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Michael Greenblatt
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,020206 networking & telecommunications ,Resolution of singularities ,02 engineering and technology ,Codimension ,Surface (topology) ,01 natural sciences ,Measure (mathematics) ,Sobolev space ,Polyhedron ,symbols.namesake ,Fourier transform ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,0101 mathematics ,Oscillatory integral ,Analysis ,Mathematics - Abstract
In the paper (Greenblatt in J Funct Anal, https://doi.org/10.1016/j.jfa.2018.05.014 , 2018) the author proved $$L^p$$ Sobolev regularity results for averaging operators over hypersurfaces and connected them to associated Newton polyhedra. In this paper, we use rather different resolution of singularities techniques along with oscillatory integral methods applied to surface measure Fourier transforms to prove $$L^p$$ Sobolev regularity results for a class of averaging operators over surfaces which can be of any codimension.
- Published
- 2018
22. Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions. III. Hilbert Spaces and Universality
- Author
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Aparajita Dasgupta and Michael Ruzhansky
- Subjects
Pure mathematics ,CONVOLUTION ,General Mathematics ,Structure (category theory) ,Boundary (topology) ,Type (model theory) ,Universality ,01 natural sciences ,Mathematics - Spectral Theory ,symbols.namesake ,Mathematics - Analysis of PDEs ,Primary 46F05 ,Tensor (intrinsic definition) ,0103 physical sciences ,FOS: Mathematics ,DISTRIBUTIONS ,Secondary 22E30 ,0101 mathematics ,Spectral Theory (math.SP) ,Mathematics ,Hilbert spaces ,Sequence ,Applied Mathematics ,010102 general mathematics ,Hilbert space ,Universality (philosophy) ,Eigenfunction ,Sequence spaces ,Smooth functions ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Mathematics and Statistics ,Physics and Astronomy ,Komatsu classes ,symbols ,Tensor representations ,010307 mathematical physics ,Primary 46F05, Secondary 22E30 ,Analysis ,Analysis of PDEs (math.AP) - Abstract
In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our previous papers in this series. We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary., 23 pages
- Published
- 2021
23. The Computational Complexity of Plethysm Coefficients
- Author
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Christian Ikenmeyer and Nick Fischer
- Subjects
FOS: Computer and information sciences ,Pure mathematics ,Rank (linear algebra) ,Computational complexity theory ,Geometric complexity theory ,General Mathematics ,Computational Complexity (cs.CC) ,68Q17, 05E10 ,Matrix multiplication ,Theoretical Computer Science ,Computer Science - Computational Complexity ,Computational Mathematics ,symbols.namesake ,Computational Theory and Mathematics ,Kronecker delta ,FOS: Mathematics ,symbols ,Computer Science::Symbolic Computation ,Tensor ,Representation Theory (math.RT) ,F.1.3 ,Time complexity ,Discrete tomography ,Mathematics - Representation Theory ,Mathematics - Abstract
In two papers, Bürgisser and Ikenmeyer (STOC 2011, STOC 2013) used an adaption of the geometric complexity theory (GCT) approach by Mulmuley and Sohoni (Siam J Comput 2001, 2008) to prove lower bounds on the border rank of the matrix multiplication tensor. A key ingredient was information about certain Kronecker coefficients. While tensors are an interesting test bed for GCT ideas, the far-away goal is the separation of algebraic complexity classes. The role of the Kronecker coefficients in that setting is taken by the so-called plethysm coefficients: These are the multiplicities in the coordinate rings of spaces of polynomials. Even though several hardness results for Kronecker coefficients are known, there are almost no results about the complexity of computing the plethysm coefficients or even deciding their positivity.In this paper, we show that deciding positivity of plethysm coefficients is -hard and that computing plethysm coefficients is #-hard. In fact, both problems remain hard even if the inner parameter of the plethysm coefficient is fixed. In this way, we obtain an inner versus outer contrast: If the outer parameter of the plethysm coefficient is fixed, then the plethysm coefficient can be computed in polynomial time. Moreover, we derive new lower and upper bounds and in special cases even combinatorial descriptions for plethysm coefficients, which we consider to be of independent interest. Our technique uses discrete tomography in a more refined way than the recent work on Kronecker coefficients by Ikenmeyer, Mulmuley, and Walter (Comput Compl 2017). This makes our work the first to apply techniques from discrete tomography to the study of plethysm coefficients. Quite surprisingly, that interpretation also leads to new equalities between certain plethysm coefficients and Kronecker coefficients.
- Published
- 2020
24. Applications of a version of the de Rham lemma to the existence theory of a weak solution to the Maxwell–Stokes type equation
- Author
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Junichi Aramaki
- Subjects
010101 applied mathematics ,Lemma (mathematics) ,Pure mathematics ,Type equation ,symbols.namesake ,General Mathematics ,Weak solution ,Dirichlet boundary condition ,010102 general mathematics ,symbols ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we show the existence of a weak solution to the Maxwell–Stokes type equation with a potential satisfying the Dirichlet condition, under the hypothesis that the domain has no holes, using a version of the de Rham lemma that was proved in our previous paper. We also give the regularity of weak solutions.
- Published
- 2018
25. The correction term for the Riemann–Roch formula of cyclic quotient singularities and associated invariants
- Author
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José Ignacio Cogolludo-Agustín and Jorge Martín-Morales
- Subjects
Pure mathematics ,Riemann hypothesis ,symbols.namesake ,Dual graph ,Divisor ,General Mathematics ,Adjunction formula ,symbols ,Gravitational singularity ,Invariant (mathematics) ,Normal surface ,Quotient ,Mathematics - Abstract
This paper deals with the invariant $$R_X$$ called the RR-correction term, which appears in the Riemann–Roch and Numerical Adjunction Formulas for normal surface singularities. Typically, $$R_X=\delta ^\text {top}_X-\delta ^\text {an}_X$$ decomposes as difference of topological and analytical local invariants of its singularities. The invariant $$\delta ^\text {top}_X$$ is well understood and depends only on the dual graph of a good resolution. The purpose of this paper is to give a new interpretation for $$\delta ^\text {an}_X$$ , which in the case of cyclic quotient singularities can be explicitly computed via generic divisors. We also include two types of applications: one is related to the McKay decomposition of reflexive modules in terms of special reflexive modules in the context of the McKay correspondence. The other application answers two questions posed by Blache (Abh Math Semin Univ Hambg 65:307–340, 1995) on the asymptotic behavior of the invariant $$R_X$$ of the pluricanonical divisor.
- Published
- 2018
26. The Answer to a Problem Posed by Zhao and Ho
- Author
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Jing Lu, Kai Yun Wang, and Bin Zhao
- Subjects
Subcategory ,Pure mathematics ,Kolmogorov space ,Closed set ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Set (abstract data type) ,Negative - answer ,symbols.namesake ,010201 computation theory & mathematics ,Mathematics::Category Theory ,symbols ,0101 mathematics ,Construct (philosophy) ,Reflective subcategory ,Counterexample ,Mathematics - Abstract
Zhao and Ho asked in a recent paper that for each T0 space X, whether KB(X) (the set of all irreducible closed sets of X whose suprema exist) is the canonical k-bounded sobrification of X in the sense of Keimel and Lawson. In this paper, we construct a counterexample to give a negative answer. We also consider the subcategory Topκ of the category Top0 of T0 spaces, and prove that the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category Topκ.
- Published
- 2018
27. Continued fraction expansions for the Lambert $$\varvec{W}$$ W function
- Author
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Cristina B. Corcino, István Mező, and Roberto B. Corcino
- Subjects
Pure mathematics ,Integral representation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,symbols.namesake ,Lambert W function ,symbols ,Discrete Mathematics and Combinatorics ,Fraction (mathematics) ,0101 mathematics ,Principal branch ,Mathematics - Abstract
In the first part of the paper we give a new integral representation for the principal branch of the Lambert W function. Then we deduce two continued fraction expansions for this branch. At the end of the paper we study the numerical behavior of the approximants of these expansions.
- Published
- 2018
28. Minimal Complex Surfaces with Levi–Civita Ricci-flat Metrics
- Author
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Kefeng Liu and Xiaokui Yang
- Subjects
0209 industrial biotechnology ,Pure mathematics ,Class (set theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,02 engineering and technology ,Link (geometry) ,Riemannian geometry ,01 natural sciences ,Connection (mathematics) ,symbols.namesake ,Continuation ,020901 industrial engineering & automation ,Complex geometry ,symbols ,Curvature form ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics - Abstract
This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli–Chern class on compact complex manifolds, and proved that the (1, 1) curvature form of the Levi–Civita connection represents the first Aeppli–Chern class which is a natural link between Riemannian geometry and complex geometry. In this paper, we study the geometry of compact complex manifolds with Levi–Civita Ricci-flat metrics and classify minimal complex surfaces with Levi–Civita Ricci-flat metrics. More precisely, we show that minimal complex surfaces admitting Levi–Civita Ricci-flat metrics are Kahler Calabi–Yau surfaces and Hopf surfaces.
- Published
- 2018
29. 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
30. The lattices of invariant subspaces of a class of operators on the Hardy space
- Author
-
Zeljko Cuckovic and Bhupendra Paudyal
- Subjects
Discrete mathematics ,Pure mathematics ,Volterra operator ,Mathematics - Complex Variables ,General Mathematics ,010102 general mathematics ,Holomorphic function ,010103 numerical & computational mathematics ,Hardy space ,Reflexive operator algebra ,01 natural sciences ,Linear subspace ,symbols.namesake ,Operator (computer programming) ,Lattice (order) ,FOS: Mathematics ,symbols ,Complex Variables (math.CV) ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator., We deleted a proposition and a corollary from section 4, and simplified the proof of the main theorem. **The article has been published in Archiv der Mathematik**
- Published
- 2018
31. Kreĭn space representation and Lorentz groups of analytic Hilbert modules
- Author
-
Yue Wu, Michio Seto, and Rongwei Yang
- Subjects
Pure mathematics ,General Mathematics ,Lorentz transformation ,010102 general mathematics ,Hilbert space ,Hardy space ,Congruence relation ,01 natural sciences ,Lorentz group ,symbols.namesake ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Abelian group ,Analytic function ,Mathematics - Abstract
This paper aims to introduce some new ideas into the study of submodules in Hilbert spaces of analytic functions. The effort is laid out in the Hardy space over the bidisk H2(D2). A closed subspace M in H2(D2) is called a submodule if z i M ⊂ M (i = 1, 2). An associated integral operator (defect operator) C M captures much information about M. Using a Kreĭn space indefinite metric on the range of C M , this paper gives a representation of M. Then it studies the group (called Lorentz group) of isometric self-maps of M with respect to the indefinite metric, and in finite rank case shows that the Lorentz group is a complete invariant for congruence relation. Furthermore, the Lorentz group contains an interesting abelian subgroup (called little Lorentz group) which turns out to be a finer invariant for M.
- Published
- 2018
32. Central Limit Theorem for Lipschitz–Killing Curvatures of Excursion Sets of Gaussian Random Fields
- Author
-
Sreekar Vadlamani and Marie Kratz
- Subjects
Statistics and Probability ,Pure mathematics ,Random field ,Series (mathematics) ,Stochastic process ,General Mathematics ,Gaussian ,010102 general mathematics ,Mathematical analysis ,Excursion ,Lipschitz continuity ,01 natural sciences ,Gaussian random field ,010104 statistics & probability ,symbols.namesake ,symbols ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics ,Central limit theorem - Abstract
Our interest in this paper is to explore limit theorems for various geometric functionals of excursion sets of isotropic Gaussian random fields. In the past, asymptotics of nonlinear functionals of Gaussian random fields have been studied [see Berman (Sojourns and extremes of stochastic processes, Wadsworth & Brooks, Monterey, 1991), Kratz and Leon (Extremes 3(1):57–86, 2000), Kratz and Leon (J Theor Probab 14(3):639–672, 2001), Meshenmoser and Shashkin (Stat Probab Lett 81(6):642–646, 2011), Pham (Stoch Proc Appl 123(6):2158–2174, 2013), Spodarev (Chapter in modern stochastics and applications, volume 90 of the series Springer optimization and its applications, pp 221–241, 2013) for a sample of works in such settings], the most recent addition being (Adler and Naitzat in Stoch Proc Appl 2016; Estrade and Leon in Ann Probab 2016) where a central limit theorem (CLT) for Euler integral and Euler–Poincare characteristic, respectively, of the excursions set of a Gaussian random field is proven under some conditions. In this paper, we obtain a CLT for some global geometric functionals, called the Lipschitz–Killing curvatures of excursion sets of Gaussian random fields, in an appropriate setting.
- Published
- 2017
33. On the $$\mathbb {K}$$ K -Riemann integral and Hermite–Hadamard inequalities for $$\mathbb {K}$$ K -convex functions
- Author
-
Andrzej Olbryś
- Subjects
Mathematics(all) ,Pure mathematics ,Hermite polynomials ,Mathematics::Complex Variables ,Generalization ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Riemann integral ,01 natural sciences ,Convexity ,010101 applied mathematics ,symbols.namesake ,Hadamard transform ,symbols ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Convex function ,Mathematics - Abstract
In the present paper we introduce a notion of the \(\mathbb {K}\)-Riemann integral as a natural generalization of a usual Riemann integral and study its properties. The aim of this paper is to extend the classical Hermite–Hadamard inequalities to the case when the usual Riemann integral is replaced by the \(\mathbb {K}\)-Riemann integral and the convexity notion is replaced by \(\mathbb {K}\)-convexity.
- Published
- 2017
34. On the space of delta m-subharmonic functions
- Author
-
H. Hawari and M. Zaway
- Subjects
Hessian matrix ,Pure mathematics ,Subharmonic ,Weak convergence ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Banach space ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Plurisubharmonic function ,Norm (mathematics) ,symbols ,Hessian operator ,0101 mathematics ,Mathematics ,Vector space - Abstract
The aim of this paper is to introduce the vector space δFm(Ω). Equipping this space by a norm defined using the Hessian measure, we prove that it is a Banach space and that the class Fm(Ω) is closed in it. Moreover we show that the topology defined by this norm is stronger than the convergence in Capm-capacity. At the end of this paper we prove a relationship between weak convergence and the convergence in capacity Capm,T in the class δSHmloc (Ω).
- Published
- 2016
35. Poincaré duality for spaces with isolated singularities
- Author
-
Mathieu Klimczak
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Algebraic geometry ,01 natural sciences ,57P10, 55N33, 55P62 ,symbols.namesake ,Formalism (philosophy of mathematics) ,Number theory ,0103 physical sciences ,symbols ,Gravitational singularity ,Mathematics - Algebraic Topology ,010307 mathematical physics ,0101 mathematics ,Poincaré duality ,Mathematics - Abstract
In this paper we assign, under reasonable hypothesis, to each pseudomanifold with isolated singularities a rational Poincar\'e duality space. These spaces are constructed with the formalism of intersections spaces defined by Markus Banagl and are indeed related to them in the even dimensional case., Comment: New version of the paper formerly known as "Spatialization of self dual complexes for spaces with isolated singularities". Final version to appear in manuscripta mathematica
- Published
- 2016
36. DG Poisson algebra and its universal enveloping algebra
- Author
-
Xingting Wang, Jiafeng Lü, and Guangbin Zhuang
- Subjects
Pure mathematics ,Algebraic structure ,General Mathematics ,010102 general mathematics ,Universal enveloping algebra ,Mathematics - Rings and Algebras ,010103 numerical & computational mathematics ,Poisson distribution ,01 natural sciences ,symbols.namesake ,Tensor product ,Differential geometry ,Rings and Algebras (math.RA) ,FOS: Mathematics ,symbols ,Homological algebra ,Universal property ,0101 mathematics ,Poisson algebra ,Mathematics - Abstract
In this paper, we introduce the notions of differential graded (DG) Poisson algebra and DG Poisson module. Let $A$ be any DG Poisson algebra. We construct the universal enveloping algebra of $A$ explicitly, which is denoted by $A^{ue}$. We show that $A^{ue}$ has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over $A$ is isomorphic to the category of DG modules over $A^{ue}$. Furthermore, we prove that the notion of universal enveloping algebra $A^{ue}$ is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra., Comment: Accepted by Science China Mathematics
- Published
- 2016
37. Demicompactness Results for Strongly Continuous Semigroups, Generators and Resolvents
- Author
-
Asrar Elleuch, Hedi Benkhaled, and Aref Jeribi
- Subjects
Pure mathematics ,Mathematics::Operator Algebras ,Semigroup ,Generator (category theory) ,General Mathematics ,010102 general mathematics ,Hilbert space ,Banach space ,01 natural sciences ,Bounded operator ,010101 applied mathematics ,symbols.namesake ,Product (mathematics) ,Bounded function ,symbols ,0101 mathematics ,Mathematics ,Resolvent - Abstract
Let $$(U(t))_ {t\ge 0}$$ be a strongly continuous semigroup of bounded linear operators on a Banach space X and B be a bounded operator on X. In this paper, we develop some aspects of the theory of semigroup for which U(t)B (respectively, BU(t), BU(t)B) is demicompact for some (respectively, every) $$t>0$$ . In addition, we study the demicompactness of similar, subspace and product semigroups. We also investigate the demicompactness of the resolvent. We close this paper by giving some conditions guaranteeing the demicompactness of a generator of a strongly continuous semigroup in a Hilbert space.
- Published
- 2018
38. Conformality on Semi-Riemannian Manifolds
- Author
-
Cornelia-Livia Bejan and Şemsi Eken
- Subjects
Pure mathematics ,Curvature of Riemannian manifolds ,Riemannian submersion ,General Mathematics ,010102 general mathematics ,Geodesic map ,Mathematical analysis ,Harmonic map ,Conformal map ,Riemannian geometry ,01 natural sciences ,030218 nuclear medicine & medical imaging ,03 medical and health sciences ,symbols.namesake ,0302 clinical medicine ,Ricci-flat manifold ,symbols ,Mathematics::Differential Geometry ,0101 mathematics ,Exponential map (Riemannian geometry) ,Mathematics - Abstract
We introduce here the notion of conformal semi-Riemannian map between semi-Riemannian manifolds aiming to unify and generalize two geometric concepts. The first one is studied by Garcia-Rio and Kupeli (namely, semi-Riemannian map between semi-Riemannian manifolds). The second notion is defined by Aahin (namely, conformal Riemannian map between Riemannian manifolds) as an extension of the notion of Riemannian map introduced by Fischer. We support the main notion of this paper with several classes of examples, e.g. semi-Riemanninan submersions (see O'Neill's book and Falcitelli, Ianus and Pastore's book) and isometric immersions between semi-Riemannian manifolds. As a tool, we use the screen distributions (specific in semi-Riemannian geometry) of Duggal and Bejancu's book to obtain some characterizations and to give a semi-Riemannian version of Fischer's (resp. Aahin's) results, using the new map introduced here. We study the generalized eikonal equation and at the end relate the main notion of the paper with harmonicity.
- Published
- 2015
39. On the Convolution of a Finite Number of Analytic Functions Involving a Generalized Srivastava–Attiya Operator
- Author
-
Janusz Sokół, Ravinder Krishna Raina, and Poonam Sharma
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Convolution power ,01 natural sciences ,Convexity ,Circular convolution ,Riemann zeta function ,Convolution ,symbols.namesake ,Operator (computer programming) ,symbols ,0101 mathematics ,Finite set ,Mathematics ,Analytic function - Abstract
The present paper gives several subordination results involving a generalized Srivastava–Attiya operator (defined below). Among the results presented in this paper include also a sufficiency condition for the convexity of the convolution of certain functions and a sharp result relating to the convolution structure. We also mention various useful special cases of the main results including those which are related to the Zeta function.
- Published
- 2015
40. A study of variations of pseudoconvex domains via Kähler-Einstein metrics
- Author
-
Young-Jun Choi
- Subjects
Pure mathematics ,Mathematics::Complex Variables ,General Mathematics ,Mathematical analysis ,Dimension (graph theory) ,Triviality ,Domain (mathematical analysis) ,symbols.namesake ,Kähler–Einstein metric ,Pseudoconvexity ,Bounded function ,symbols ,Einstein ,Pseudoconvex function ,Mathematics - Abstract
This paper is a sequel to Choi (Math Ann 362(1–2):121–146, 2015) in Math. Ann. In that paper we studied the subharmonicity of Kahler–Einstein metrics on strongly pseudoconvex domains of dimension greater than or equal to 3. In this paper, we study the variations Kahler–Einstein metrics on bounded strongly pseudoconvex domains of dimension 2. In addition, we discuss the previous result with general bounded pseudoconvex domain and local triviality of a family of bounded strongly pseudoconvex domains.
- Published
- 2015
41. On the Semicircular Law of Large-Dimensional Random Quaternion Matrices
- Author
-
Zhidong Bai, Jiang Hu, and Yanqing Yin
- Subjects
Statistics and Probability ,Lemma (mathematics) ,Pure mathematics ,General Mathematics ,Gaussian ,010102 general mathematics ,01 natural sciences ,Hermitian matrix ,010104 statistics & probability ,symbols.namesake ,Law ,symbols ,Multiplication ,0101 mathematics ,Statistics, Probability and Uncertainty ,Quaternion ,Commutative property ,Eigenvalues and eigenvectors ,Symplectic geometry ,Mathematics - Abstract
It is well known that the Gaussian symplectic ensemble is defined on the space of \(n\times n\) quaternion self-dual Hermitian matrices with Gaussian random elements. There is a huge body of literature regarding this kind of matrices based on the exact known form of the density function of the eigenvalues (see Erdős in Russ Math Surv 66(3):507–626, 2011; Erdős in Probab Theory Relat Fields 154(1–2):341–407, 2012; Erdős et al. in Adv Math 229(3):1435–1515, 2012; Knowles and Yin in Probab Theory Relat Fields, 155(3–4):543–582, 2013; Tao and Vu in Acta Math 206(1):127–204, 2011; Tao and Vu in Electron J Probab 16(77):2104–2121, 2011). Due to the fact that multiplication of quaternions is not commutative, few works about large-dimensional quaternion self-dual Hermitian matrices are seen without normality assumptions. As in natural, we shall get more universal results by removing the Gaussian condition. For the first step, in this paper, we prove that the empirical spectral distribution of the common quaternion self-dual Hermitian matrices tends to the semicircular law. The main tool to establish the universal result is given as a lemma in this paper as well.
- Published
- 2015
42. Kähler–Einstein metrics on Fano manifolds
- Author
-
Gang Tian
- Subjects
Pure mathematics ,symbols.namesake ,General Mathematics ,Stability (learning theory) ,symbols ,Mathematics::Differential Geometry ,Fano plane ,Einstein ,Mathematics::Symplectic Geometry ,Manifold ,Mathematics ,Scalar curvature - Abstract
This is an expository paper on Kahler metrics of positive scalar curvature. It is for my Takagi Lectures at RIMS in November of 2013. In this paper, I first discuss the Futaki invariants, the K-stability and its relation to the K-energy. Next I will outline my work in 2012 on the existence of Kahler–Einstein metrics on K-stable Fano manifolds. Finally, I will present S. Paul’s work on stability of pairs with some modifications of mine.
- Published
- 2014
43. A New Extension of Generalized Hermite Matrix Polynomials
- Author
-
Ayman Shehata and Lalit Mohan Upadhyaya
- Subjects
Pure mathematics ,Gegenbauer polynomials ,General Mathematics ,Discrete orthogonal polynomials ,Mathematics::Classical Analysis and ODEs ,Classical orthogonal polynomials ,Algebra ,symbols.namesake ,Difference polynomials ,Orthogonal polynomials ,Wilson polynomials ,Hahn polynomials ,symbols ,Jacobi polynomials ,Mathematics - Abstract
The Hermite matrix polynomials have been generalized in a number of ways, and many of these generalizations have been shown to be important tools in applications. In this paper, we introduce a new generalization of the Hermite matrix polynomials and present the recurrence relations and the expansion of these new generalized Hermite matrix polynomials. We also give new series expansions of the matrix functions \(\exp (xB)\), \(\sin (xB)\), \(\cos (xB)\), \(\cosh (xB)\) and \(\sinh (xB)\) in terms of these generalized Hermite matrix polynomials and thus prove that many of the seemingly different generalizations of the Hermite matrix polynomials may be viewed as particular cases of the two-variable polynomials introduced here. The generalized Chebyshev and Legendre matrix polynomials have also been introduced in this paper in terms of these generalized Hermite matrix polynomials.
- Published
- 2014
44. Analytic Detection in Homotopy Groups of Smooth Manifolds
- Author
-
I. S. Zubov
- Subjects
Statistics and Probability ,Pure mathematics ,Fundamental group ,Homotopy group ,Riemann surface ,Applied Mathematics ,General Mathematics ,Holomorphic function ,General Medicine ,Central series ,Hopf invariant ,symbols.namesake ,Linear differential equation ,symbols ,Element (category theory) ,Mathematics - Abstract
In this paper, for the mapping of a sphere into a compact orientable manifold S n → M , n ⩾ 1 , we solve the problem of determining whether it represents a nontrivial element in the homotopy group of the manifold π n ( M ) πn(M ). For this purpose, we consistently use the theory of iterated integrals developed by K.-T. Chen. It should be noted that the iterated integrals as repeated integration were previously meaningfully used by Lappo-Danilevsky to represent solutions of systems of linear differential equations and by Whitehead for the analytical description of the Hopf invariant for mappings f : S 2 n - 1 → S n , n ⩾ 2 . We give a brief description of Chen’s theory, representing Whitehead’s and Haefliger’s formulas for the Hopf invariant and generalized Hopf invariant. Examples of calculating these invariants using the technique of iterated integrals are given. Further, it is shown how one can detect any element of the fundamental group of a Riemann surface using iterated integrals of holomorphic forms. This required to prove that the intersection of the terms of the lower central series of the fundamental group of a Riemann surface is a unit group.
- Published
- 2022
45. Character varieties with Zariski closures of $$\mathrm{GL}_n$$ GL n -conjugacy classes at punctures
- Author
-
Emmanuel Letellier
- Subjects
Weyl group ,Pure mathematics ,Trace (linear algebra) ,General Mathematics ,General Physics and Astronomy ,Unipotent ,Algebra ,symbols.namesake ,Conjugacy class ,Character (mathematics) ,Intersection homology ,Filtration (mathematics) ,symbols ,Mathematics::Representation Theory ,Stack (mathematics) ,Mathematics - Abstract
In the paper [20], we gave a conjectural formula for the mixed Hodge polynomials of character varieties with generic semisimple conjugacy classes at punctures and we prove a formula for the \(E\)-polynomial. We also proved that these character varieties are irreducible [21]. In this paper, we extend the results of [20, 21] to character varieties with Zariski closures of arbitrary generic conjugacy classes at punctures working with intersection cohomology. We also study Weyl group action on the intersection cohomology of the partial resolutions of these character varieties and give a conjectural formula for the two-variable polynomials that encode the trace of the elements of the Weyl group on the subquotients of the weight filtration. Finally, we compute the generating function of the stack count of character varieties with Zariski closure of unipotent regular conjugacy class at punctures.
- Published
- 2014
46. ON M-Projectively Flat LP-Sasakian Manifolds
- Author
-
Füsun Özen Zengin
- Subjects
Pure mathematics ,Riemann curvature tensor ,General Mathematics ,Cosmological constant ,Space (mathematics) ,Manifold ,Sasakian manifold ,symbols.namesake ,Ricci-flat manifold ,symbols ,Mathematics::Differential Geometry ,Algebra over a field ,Mathematics::Symplectic Geometry ,Ricci curvature ,Mathematics - Abstract
In the present paper, we study the nature of LP-Sasakian manifolds admitting the M-projective curvature tensor. It is examined whether this manifold satisfies the condition W(X, Y ).R = 0. Moreover, it is proved that, in the M-projectively flat LP-Sasakian manifolds, the conditions R(X, Y ).R = 0 and R(X, Y ).S = 0 are satisfied. In the last part of the paper, an M-projectively flat space-time is introduced, and some properties of this space are obtained.
- Published
- 2014
47. On a Sum of Modified Bessel Functions
- Author
-
Tibor K. Pogány and Árpád Baricz
- Subjects
Pure mathematics ,General Mathematics ,Open problem ,Mathematics::Classical Analysis and ODEs ,Monotonic function ,Type (model theory) ,Convexity ,Modified Bessel functions ,Concentration bounds ,Functional inequalities ,symbols.namesake ,Mathematics - Classical Analysis and ODEs ,symbols ,Point (geometry) ,39B62, 33C10, 33C15 ,Mathematics - Probability ,Bessel function ,Mathematics - Abstract
In this paper we consider a sum of modified Bessel functions of the first kind of which particular case is used in the study of Kanter's sharp modified Bessel function bound for concentrations of some sums of independent symmetric random vectors. We present some monotonicity and convexity properties for that sum of modified Bessel functions of the first kind, as well as some Tur\'an type inequalities, lower and upper bounds. Moreover, we point out an error in Kanter's paper [Ka] and at the end of the paper we pose an open problem, which may be of interest for further research., Comment: 8 pages
- Published
- 2013
48. Degeneracy of Holomorphic Curves into Algebraic Varieties II
- Author
-
Jörg Winkelmann, Junjiro Noguchi, and Katsutoshi Yamanoi
- Subjects
Pure mathematics ,General Mathematics ,Degenerate energy levels ,Holomorphic function ,Algebraic variety ,Finite morphism ,Algebra ,symbols.namesake ,Fourier transform ,Morphism ,symbols ,Algebraic number ,Degeneracy (mathematics) ,Mathematics - Abstract
In our former paper (Noguchi et al. in J. Math. Pures Appl. 88:293–306, 2007) we proved an algebraic degeneracy of entire holomorphic curves into a variety X which carries a finite morphism to a semi-abelian variety, but which is not isomorphic to a semi-abelian variety by itself. The finiteness condition of the morphism is necessary in general by example. In this paper we improve that finiteness condition under an assumption such that some open subset of non-singular points of X is of log-general type, and simplify the proof in (Noguchi et al. in J. Math. Pures Appl. 88:293–306, 2007), which was rather involved. As a corollary it implies that every entire holomorphic curve $f:\mathbb{C} \to V$ into an algebraic variety V with $\bar{q}(V)\geq\dim V=\bar {\kappa}(V)$ is algebraically degenerate, which is due to Winkelmann (dimV=2) (Winkelmann in Ann. Inst. Fourier 61:1517–1537, 2011) and Lu–Winkelmann (Lu and Winkelmann in Forum Math. 24:399–418, 2012).
- Published
- 2013
49. Spectral problems in Lipschitz domains
- Author
-
M. S. Agranovich
- Subjects
Statistics and Probability ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Banach space ,Boundary (topology) ,Mathematics::Spectral Theory ,Lipschitz continuity ,Dirichlet distribution ,symbols.namesake ,Lipschitz domain ,Bounded function ,symbols ,Boundary value problem ,Sobolev spaces for planar domains ,Mathematics - Abstract
The paper is devoted to spectral problems for strongly elliptic second-order systems in bounded Lipschitz domains. We consider the spectral Dirichlet and Neumann problems and three problems with spectral parameter in conditions at the boundary: the Poincare–Steklov problem and two transmission problems. In the style of a survey, we discuss the main properties of these problems, both self-adjoint and non-self-adjoint. As a preliminary, we explain several facts of the general theory of the main boundary value problems in Lipschitz domains. The original definitions are variational. The use of the boundary potentials is based on results on the unique solvability of the Dirichlet and Neumann problems. In the main part of the paper, we use the simplest Hilbert L2-spaces Hs, but we describe some generalizations to Banach spaces Hsp of Bessel potentials and Besov spaces Bsp at the end of the paper.
- Published
- 2013
50. The variational Poisson cohomology
- Author
-
Alberto De Sole, Victor G. Kac, Massachusetts Institute of Technology. Department of Mathematics, and Kac, Victor
- Subjects
Pure mathematics ,General Mathematics ,FOS: Physical sciences ,Lie superalgebra ,Mathematics::Algebraic Topology ,01 natural sciences ,universal lie superalgebra and lie conformal superalgebra ,symbols.namesake ,linearly closed ,Vertex operator algebra ,Mathematics::K-Theory and Homology ,0103 physical sciences ,bi-hamiltonian pde ,FOS: Mathematics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematical Physics ,Mathematics ,Exact sequence ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,poisson vertex algebra ,lie conformal algebra ,generalized variational complex ,linearly closed differential field ,basic and variational poisson cohomology ,variational polyvector field ,010308 nuclear & particles physics ,010102 general mathematics ,Mathematical Physics (math-ph) ,Superalgebra ,Cohomology ,Lie conformal algebra ,Linear algebra ,symbols ,37K10 (Primary) 37K30, 17B80 (Secondary) 37K10 (Primary) 37K30, 17B80 (Secondary) 37K10 (Primary) 37K30, 17B80 (Secondary) ,Exactly Solvable and Integrable Systems (nlin.SI) ,Hamiltonian (quantum mechanics) ,Mathematics - Representation Theory - Abstract
It is well known that the validity of the so called Lenard–Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson cohomology complexes, and basic and reduced Lie conformal algebra and Poisson vertex algebra cohomology complexes, by making use of the corresponding universal Lie superalgebra or Lie conformal superalgebra. The most relevant are certain subcomplexes of the basic and reduced Poisson vertex algebra cohomology complexes, which we identify (non-canonically) with the generalized de Rham complex and the generalized variational complex. In the second part of the paper we compute the cohomology of the generalized de Rham complex, and, via a detailed study of the long exact sequence, we compute the cohomology of the generalized variational complex for any quasiconstant coefficient Hamiltonian operator with invertible leading coefficient. For the latter we use some differential linear algebra developed in the Appendix., National Science Foundation (U.S.), ERC (Advanced Grant)
- Published
- 2013
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