Fourier transforms and bent functions on faithful actions of finite abelian groups

Let G be a finite abelian group acting faithfully on a finite set X. The G-bentness and G-perfect nonlinearity of functions on X are studied by Poinsot and co-authors (Discret Appl Math 157:1848–1857, 2009; GESTS Int Trans Comput Sci Eng 12:1–14, 2005) via Fourier transforms of functions on G. In this paper we introduce the so-called \(G\)-dual set \(\widehat{X}\) of X, which plays the role similar to the dual group \(\widehat{G}\) of G, and develop a Fourier analysis on X, a generalization of the Fourier analysis on the group G. Then we characterize the bentness and perfect nonlinearity of functions on X by their own Fourier transforms on \(\widehat{X}\). Furthermore, we prove that the bentness of a function on X can be determined by its distance from the set of G-linear functions. As direct consequences, many known results in Logachev et al. (Discret Math Appl 7:547–564, 1997), Carlet and Ding (J Complex 20:205–244, 2004), Poinsot (2009), Poinsot et al. (2005) and some new results about bent functions on G are obtained. In order to explain the theory developed in this paper clearly, examples are also presented.     

On the Bessel Heat Equation Related to the Bessel Diamond Operator

In this article, we study the equation

$\frac{\partial }{\partial t}u(x,t)=c^{2}\Diamond _{B}^{k}u(x,t)$

with the initial condition u(x,0)=f(x) for x∈? _n ⁺ . The operator ? _B ^k is named to be Bessel diamond operator iterated k-times and is defined by

$\Diamond _{B}^{k}=\bigl[(B_{x_{1}}+B_{x_{2}}+\cdots +B_{x_{p}})^{2}-(B_{x_{p+1}}+\cdots +B_{x_{p+q}})^{2}\bigr]^{k},$

where k is a positive integer, p+q=n, \(B_{x_{i}}=\frac{\partial ^{2}}{\partial x_{i}^{2}}+\frac{2v_{i}}{x_{i}}\frac{\partial }{\partial x_{i}},\) 2v _i =2α _i +1,\(\;\alpha _{i}>-\frac{1}{2}\), x _i >0, i=1,2,…,n, and n is the dimension of the ? _n ⁺ , u(x,t) is an unknown function of the form (x,t)=(x ₁,…,x _n ,t)∈? _n ⁺ ×(0,∞), f(x) is a given generalized function and c is a positive constant (see Levitan, Usp. Mat. 6(2(42)):102–143, 1951; Y?ld?r?m, Ph.D. Thesis, 1995; Y?ld?r?m and Sar?kaya, J. Inst. Math. Comput. Sci. 14(3):217–224, 2001; Y?ld?r?m, et al., Proc. Indian Acad. Sci. (Math. Sci.) 114(4):375–387, 2004; Sar?kaya, Ph.D. Thesis, 2007; Sar?kaya and Y?ld?r?m, Nonlinear Anal. 68:430–442, 2008, and Appl. Math. Comput. 189:910–917, 2007). We obtain the solution of such equation, which is related to the spectrum and the kernel, which is so called Bessel diamond heat kernel. Moreover, such Bessel diamond heat kernel has interesting properties and also related to the kernel of an extension of the heat equation.

     

On the Reconstruction of Conductivity of a Bordered Two-dimensional Surface in ℝ<Superscript>3</Superscript> from Electrical Current Measurements on Its Boundary

An electrical potential U on a bordered real surface X in ?³ with isotropic conductivity function σ>0 satisfies the equation d(σ d ^c U)| _X =0, where \(d^{c}= i(\bar{ \partial }-\partial )\), \(d=\bar{ \partial }+\partial \) are real operators associated with a complex (conformal) structure on X induced by the Euclidean metric of ?³. This paper gives an exact reconstruction of the conductivity function σ on X from the Dirichlet-to-Neumann mapping U| _bX →σ d ^c U| _bX . This paper extends to the case of Riemann surfaces the reconstruction schemes of R. Novikov (Funkt. Anal. Prilozh. 22(4):11–22, 1988) and of A. Bukhgeim (J. Inv. Ill-posed Probl. 16:19–34, 2008), given for the case X??². The paper extends and corrects the statements of Henkin and Michel (J. Geom. Anal. 18:1033–1052, 2008), where the inverse boundary value problem on the Riemann surfaces was first considered.     

The Reich–Zaslavski property and fixed points of non-self multivalued mappings

Let (X, d) be a metric space, Y be a nonempty subset of X, and let \(T:Y \rightarrow P(X)\) be a non-self multivalued mapping. In this paper, by a new technique we study the fixed point theory of multivalued mappings under the assumption of the existence of a bounded sequence \((x_n)_n\) in Y such that \(T^nx_n\subseteq Y,\) for each \(n \in \mathbb {N}\). Our main result generalizes fixed point theorems due to Matkowski (Diss. Math. 127, 1975), W?grzyk (Diss. Math. (Rozprawy Mat.) 201, 1982), Reich and Zaslavski (Fixed Point Theory 8:303–307, 2007), Petru?el et al. (Set-Valued Var. Anal. 23:223–237, 2015) and provides a solution to the problems posed in Petru?el et al. (Set-Valued Var. Anal. 23:223–237, 2015) and Rus and ?erban (Miskolc Math. Notes 17:1021–1031, 2016).     

Every 3-connected $$\{K_{1,3},N_{1,2,3}\}$$-free graph is Hamilton-connected

A graph G is \(\{X,Y\}\)-free if it contains neither X nor Y as an induced subgraph. Pairs of connected graphs X, Y such that every 3-connected \(\{X,Y\}\)-free graph is Hamilton-connected have been investigated recently in (2002, 2000, 2012). In this paper, it is shown that every 3-connected \(\{K_{1,3},N_{1,2,3}\}\)-free graph is Hamilton-connected, where \(N_{1,2,3}\) is the graph obtained by identifying end vertices of three disjoint paths of lengths 1, 2, 3 to the vertices of a triangle.     

Nonlinear functions and difference sets on group actions

There are many generalizations of the classical Boolean bent functions. Let G, H be finite groups and let X be a finite G-set. G-perfect nonlinear functions from X to H have been studied in several papers. They are generalizations of perfect nonlinear functions from G itself to H. By introducing the concept of a (G, H)-related difference family of X, we obtain a characterization of G-perfect nonlinear functions on X in terms of a (G, H)-related difference family. When G is abelian, we prove that there is a normalized G-dual set \(\widehat{X}\) of X, and characterize a G-difference set of X by the Fourier transform on a normalized G-dual set \({{\widehat{X}}}\). We will also investigate the existence and constructions of G-perfect nonlinear functions and G-bent functions. Several known results (IEEE Trans Inf Theory 47(7):2934–2943, 2001; Des Codes Cryptogr 46:83–96, 2008; GESTS Int Trans Comput Sci Eng 12:1–14, 2005; Linear Algebra Appl 452:89–105, 2014) are direct consequences of our results.     

An Assmus–Mattson theorem for codes over commutative association schemes

We prove an Assmus–Mattson-type theorem for block codes where the alphabet is the vertex set of a commutative association scheme (say, with s classes). This in particular generalizes the Assmus–Mattson-type theorems for \(\mathbb {Z}_4\)-linear codes due to Tanabe (Des Codes Cryptogr 30:169–185, 2003) and Shin et al. (Des Codes Cryptogr 31:75–92, 2004), as well as the original theorem by Assmus and Mattson (J Comb Theory 6:122–151, 1969). The weights of a code are s-tuples of non-negative integers in this case, and the conditions in our theorem for obtaining t-designs from the code involve concepts from polynomial interpolation in s variables. The Terwilliger algebra is the main tool to establish our results.     

On the maximum TSP with <Emphasis Type="Italic">γ</Emphasis>-parameterized triangle inequality

The maximum TSP with γ-parameterized triangle inequality is defined as follows. Given a complete graph G = (V, E, w) in which the edge weights satisfy w(uv) ≤ γ · (w(ux) + w(xv)) for all distinct nodes \({u,x,v \in V}\), find a tour with maximum weight that visits each node exactly once. Recently, Zhang et al. (Theor Comput Sci 411(26–28):2537–2541, 2010) proposed a \({\frac{\gamma+1}{3\gamma}}\)-approximation algorithm for \({\gamma\in\left[\frac{1}{2},1\right)}\). In this paper, we show that the approximation ratio of Kostochka and Serdyukov’s algorithm (Upravlyaemye Sistemy 26:55–59, 1985) is \({\frac{4\gamma+1}{6\gamma}}\), and the expected approximation ratio of Hassin and Rubinstein’s randomized algorithm (Inf Process Lett 81(5):247–251, 2002) is \({\frac{3\gamma+\frac{1}{2}}{4\gamma}-O\left(\frac{1}{\sqrt{n}}\right)}\), for \({\gamma\in\left[\frac{1}{2},+\infty\right)}\). These improve the result in Zhang et al. (Theor Comput Sci 411(26–28):2537–2541, 2010) and generalize the results in Hassin and Rubinstein and Kostochka and Serdyukov (Inf Process Lett 81(5):247–251, 2002; Upravlyaemye Sistemy 26:55–59, 1985).     

Isomorphisms,definable relations,and Scott families for integral domains and commutative semigroups

In the present article, we prove the following four assertions: (1) For every computable successor ordinal α, there exists a Δ _α ⁰ -categorical integral domain (commutative semigroup) which is not relatively Δ _α ⁰ -categorical (i.e., no formally Σ _α ⁰ Scott family exists for such a structure). (2) For every computable successor ordinal α, there exists an intrinsically Σ _α ⁰ -relation on the universe of a computable integral domain (commutative semigroup) which is not a relatively intrinsically Σ _α ⁰ -relation. (3) For every computable successor ordinal α and finite n, there exists an integral domain (commutative semigroup) whose Δ _α ⁰ -dimension is equal to n. (4) For every computable successor ordinal α, there exists an integral domain (commutative semigroup) with presentations only in the degrees of sets X such that Δ _α ⁰ (X) is not Δ _α ⁰ . In particular, for every finite n, there exists an integral domain (commutative semigroup) with presentations only in the degrees that are not n-low.     

The Volume Growth of Hyper-Kähler Manifolds of Type <Emphasis Type="Italic">A</Emphasis><Subscript>∞</Subscript>

We study the volume growth of hyper-Kähler manifolds of type A _∞ constructed by Anderson–Kronheimer–LeBrun (Commun. Math. Phys. 125:637–642, 1989) and Goto (Geom. Funct. Anal. 4(4):424–454, 1994). These are noncompact complete 4-dimensional hyper-Kähler manifolds of infinite topological type. These manifolds have the same topology, but the hyper-Kähler metrics depend on the choice of parameters. By taking a certain parameter, we show that there exists a hyper-Kähler manifold of type A _∞ whose volume growth is r ^α for each 3<α<4.     

Constructing New Braided <Emphasis Type="Italic">T</Emphasis>-Categories over Weak Hopf Algebras

Let Aut _{weak Hopf} (H) denote the set of all automorphisms of a weak Hopf algebra H with bijective antipode in the sense of Böhm et al. (J Algebra 221:385–438, 1999) and let G be a certain crossed product group Aut _{weak Hopf} (H)×Aut _{weak Hopf} (H). The main purpose of this paper is to provide further examples of braided T-categories in the sense of Turaev (1994, 2008). For this, we first introduce a class of new categories \( _{H}{\mathcal {WYD}}^{H}(\alpha, \beta)\) of weak (α, β)-Yetter-Drinfeld modules with α, β?∈?Aut _{weak Hopf} (H) and we show that the category \({\mathcal WYD}(H) =\{{}_{H}\mathcal {WYD}^{H}(\alpha, \beta)\}_{(\alpha , \beta )\in G}\) becomes a braided T-category over G, generalizing the main constructions by Panaite and Staic (Isr J Math 158:349–365, 2007). Finally, when H is finite-dimensional we construct a quasitriangular weak T-coalgebra WD(H)?=?{WD(H)_{(α, β)}}_{(α, β)?∈?G} in the sense of Van Daele and Wang (Comm Algebra, 2008) over a family of weak smash product algebras \(\{\overline{H^{*cop}\# H_{(\alpha,\beta)}}\}_{(\alpha , \beta)\in G}\), and we obtain that \({\mathcal {WYD}}(H)\) is isomorphic to the representation category of the quasitriangular weak T-coalgebra WD(H).     

Derivatives for smooth representations of <Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, ℝ) and <Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, ℂ)

The notion of derivatives for smooth representations of GL(n, ? _p ) was defined in [BZ77]. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations in [Sah89] and called the “adduced” representation. In this paper we define derivatives of all orders for smooth admissible Fréchet representations of moderate growth. The real case is more problematic than the p-adic case; for example, arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation.In the companion paper [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations.We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89, Sah90, SaSt90, GS13a].     

Toeplitz Operators with Uniformly Continuous Symbols

Let T _f be a Toeplitz operator on the Segal–Bargmann space or the standard weighted Bergman space over a bounded symmetric domain \({\Omega \subset {\bf C}^n}\) with possibly unbounded symbol f. Combining recent results in Bauer et al. (J. Funct. Anal. 259:57–78, 2010), Bauer et al. (J. reine angew. Math. doi: 10.1515/crelle-2015-0016), Issa (Integr. Equ. Oper. Theory 70:569–582, 2011) we show that in the case of uniformly continuous symbols f with respect to the Euclidean metric on C ⁿ and the Bergman metric on \({\Omega}\), respectively, the operator T _f is bounded if and only if f is bounded. Moreover, T _f is compact if and only if f vanishes at the boundary of \({\Omega.}\) This observation substantially extends a result in Coburn (Indiana Univ. Math. J. 23:433–439, 1973).     

Generalized growth and best approximation of entire functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-norm in several complex variables

The paper deals with the characterization of generalized order and generalized type of entire functions in several complex variables in terms of the coefficients of the development with respect to the sequence of extremal polynomials and the best L ^p -approximation and interpolation errors, 0 < p ≤ ∞, on a compact set K with respect to the set

$K_r = \left\{z \in \mathbb{C}^n, {\rm exp} (V_K (z)) \leq r\right\}$

where V _K is the Siciak extremal function of a L-regular compact set K or V _K is the pluricomplex Green function with a pole at infinity. It has been noticed that in the study of growth of entire functions, the set K _r has not been used so extensively in comparison to disk. Our results apply satisfactorily for slow growth in \({\mathbb{C}^n}\) , replacing the circle \({\{z \in \mathbb{C}; |z| = r\}}\) by the set K _r and improve and extend various results of Harfaoui (Int J Maths Math Sci 2010:1–15, 2010), Seremeta (Am Math Soc Transl 88(2):291–301, 1970), Shah (J Approx Theory 19:315–324, 1977) and Vakarchuk and Zhir (Ukr Math J 54(9):1393–1401, 2002).

     

A new unicity theorem and Erdös’ problem for polarized semi-abelian varieties

In 1988 Erdös asked if the prime divisors of x ⁿ ? 1 for all n = 1, 2, … determine the given integer x; the problem was affirmatively answered by Corrales-Rodrigáñez and Schoof (J Number Theory 64:276–290, 1997) [but a solution could also be deduced from an earlier result of Schinzel (Bull Acad Polon Sci 8:307–309, 2007)] together with its elliptic version. Analogously, Yamanoi (Forum Math 16:749–788, 2004) proved that the support of the pulled-back divisor f ^* D of an ample divisor on an abelian variety A by an algebraically non-degenerate entire holomorphic curve f : C → A essentially determines the pair (A, D). By making use of the main theorem of Noguchi (Forum Math 20:469–503, 2008) we here deal with this problem for semi-abelian varieties; namely, given two polarized semi-abelian varieties (A ₁, D ₁), (A ₂, D ₂) and algebraically non-degenerate entire holomorphic curves f _i : C → A _i , i = 1, 2, we classify the cases when the inclusion \({{\rm{Supp}}\, f_1^*D_1\subset {\rm{Supp}}\, f_2^* D_2}\) holds. We shall remark in §5 that these methods yield an affirmative answer to a question of Lang formulated in 1966. Our answer is more general and more geometric than the original question. Finally, we interpret the main result of Corvaja and Zannier (Invent Math 149:431–451, 2002) to provide an arithmetic counterpart in the toric case.     

On the Optimality of Gluing over Scales

We show that for every α>0, there exist n-point metric spaces (X,d) where every “scale” admits a Euclidean embedding with distortion at most α, but the whole space requires distortion at least \(\varOmega (\sqrt {\alpha\log n})\). This shows that the scale-gluing lemma (Lee in SODA’05: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 92–101, Society for Industrial and Applied Mathematics, Philadelphia, 2005) is tight and disproves a conjecture stated there. This matching upper bound was known to be tight at both endpoints, i.e., when α=Θ(1) and α=Θ(log?n), but nowhere in between.More specifically, we exhibit n-point spaces with doubling constant λ requiring Euclidean distortion \(\varOmega (\sqrt{\log\lambda\log n})\), which also shows that the technique of “measured descent” (Krauthgamer et al. in Geom. Funct. Anal. 15(4):839–858, 2005) is optimal. We extend this to L _p spaces with p>1, where one requires distortion at least Ω((log?n)^1/q (log?λ)^1?1/q ) when q=max?{p,2}, a result which is tight for every p>1.     

Martingale Approximation and Optimality of Some Conditions for the Central Limit Theorem

Let (X _i ) be a stationary and ergodic Markov chain with kernel Q and f an L ² function on its state space. If Q is a normal operator and f=(I?Q)^1/2 g (which is equivalent to the convergence of \(\sum_{n=1}^{\infty}\frac{\sum_{k=0}^{n-1}Q^{k}f}{n^{3/2}}\) in L ²), we have the central limit theorem [cf. (Derriennic and Lin in C.R. Acad. Sci. Paris, Sér. I 323:1053–1057, 1996; Gordin and Lif?ic in Third Vilnius conference on probability and statistics, vol. 1, pp. 147–148, 1981)]. Without assuming normality of Q, the CLT is implied by the convergence of \(\sum_{n=1}^{\infty}\frac{\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}}{n^{3/2}}\), in particular by \(\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}=o(\sqrt{n}/\log^{q}n)\), q>1 by Maxwell and Woodroofe (Ann. Probab. 28:713–724, 2000) and Wu and Woodroofe (Ann. Probab. 32:1674–1690, 2004), respectively. We show that if Q is not normal and f∈(I?Q)^1/2 L ², or if the conditions of Maxwell and Woodroofe or of Wu and Woodroofe are weakened to \(\sum_{n=1}^{\infty}c_{n}\frac{\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}}{n^{3/2}}<\infty\) for some sequence c _n ↘0, or by \(\|\sum_{k=0}^{n-1}Q^{k}f\|_{2}=O(\sqrt{n}/\log n)\), the CLT need not hold.     

O $$\varvec{}$$-orm preservers ad the Aleksadrov coservative $$\varvec{}$$-distace problem

The goal of this paper is to point out that the results obtained in the recent papers (Chen and Song in Nonlinear Anal 72:1895–1901, 2010; Chu in J Math Anal Appl 327:1041–1045, 2007; Chu et al. in Nonlinear Anal 59:1001–1011, 2004a, J. Math Anal Appl 289:666–672, 2004b) can be seriously strengthened in the sense that we can significantly relax the assumptions of the main results so that we still get the same conclusions. In order to do this first, we prove that for \(n \ge 3\) any transformation which preserves the n-norm of any n vectors is automatically plus-minus linear. This will give a re-proof of the well-known Mazur–Ulam-type result that every n-isometry is automatically affine (\(n \ge 2\)) which was proven in several papers, e.g. in Chu et al. (Nonlinear Anal 70:1068–1074, 2009). Second, following the work of Rassias and ?emrl (Proc Am Math Soc 118:919–925, 1993), we provide the solution of a natural Aleksandrov-type problem in n-normed spaces, namely, we show that every surjective transformation which preserves the unit n-distance in both directions (\(n\ge 2\)) is automatically an n-isometry.     

Set-valued Brownian motion

Brownian motions, martingales, and Wiener processes are introduced and studied for set valued functions taking values in the subfamily of compact convex subsets of arbitrary Banach spaces X. The present paper is an application of the paper (Labuschagne et al. in Quaest Math 30(3):285–308, 2007) in which an embedding result is obtained which considers also the ordered structure of the family of compact convex subsets of a Banach space X and of Grobler and Labuschagne (J Math Anal Appl 423(1):797–819, 2015; J Math Anal Appl 423(1):820–833, 2015) in which these processes are considered in f-algebras. Moreover, in the space of continuous functions defined on a Stonian space, a direct Levy’s result follows.