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.     

A generalized extension theorem for linear codes

We prove that an [n, k, d] _q code \({\mathcal{C}}\) with gcd(d, q) = 1 is extendable if \({\sum_{i \not\equiv 0,d}A_i < (q-1)q^{k-2}}\), where A _i denotes the number of codewords of \({\mathcal{C}}\) with weight i. This is a generalization of extension theorems for linear codes by Hill and Lizak (Proceedings of the IEEE International Symposium on Information Theory, Whistler, Canada, 1995) and by Landjev and Rousseva (Probl. Inform. Transm. 42: 319–329, 2006).     

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].     

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.     

Dependence structure and test of independence for some well-known bivariate distributions

In this paper, we study the dependence structure of some bivariate distribution functions based on dependence measures of Kochar and Gupta (Biometrika 74(3):664–666, 1987) and Shetty and Pandit (Stat Methods Appl 12:5–17, 2003) and then compare these measures with Spearman’s rho and Kendall’s tau. Moreover, the empirical power of the class of distribution-free tests introduced by Kochar and Gupta (1987) and Shetty and Pandit (2003) is computed based on exact and asymptotic distribution of U-statistics. Our results are obtained from simulation work in some continuous bivariate distributions for the sample of sizes \(n=6,8,15,20\) and 50. Also, we apply some examples to illustrate the results. Finally, we compare the common estimators of dependence parameter based on empirical MSE.     

Congruences of two-variable <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis>-functions of congruent modular forms of different weights

Vatsal (Duke Math J 98(2):397–419, 1999) proved that there are congruences between the p-adic L-functions (constructed by Mazur and Swinnerton-Dyer in Invent Math 25:1–61, 1974) of congruent modular forms of the same weight under some conditions. On the other hand, Kim (J Number Theory 144: 188–218, 2014), the second author, constructed two-variable p-adic L-functions of modular forms attached to imaginary quadratic fields generalizing Hida’s work (Invent Math 79:159–195, 1985), and the novelty of his construction was that it works whether p is an ordinary prime or not. In this paper, we prove congruences between the two-variable p-adic L-functions (of the second author) of congruent modular forms of different but congruent weights under some conditions when p is a nonordinary prime for the modular forms. This result generalizes the work of Emerton et al. (Invent Math 163(3): 523–580, 2006), who proved similar congruences between the p-adic L-functions of congruent modular forms of congruent weights when p is an ordinary prime.     

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).     

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.     

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).     

On the reflexivity of $$\mathcal {P}_{w}(^{n}E;F)$$

In this paper we prove that if E and F are reflexive Banach spaces and G is a closed linear subspace of the space \(\mathcal {L}_{K}(E;F)\) of all compact linear operators from E into F, then G is either reflexive or non-isomorphic to a dual space. This result generalizes (Israel J Math 21:38-49, 1975, Theorem 2) and gives the solution to a problem posed by Feder (Ill J Math 24:196-205, 1980, Problem 1). We also prove that if E and F are reflexive Banach spaces, then the space \(\mathcal {P}_{w}(^{n}E;F)\) of all n-homogeneous polynomials from E into F which are weakly continuous on bounded sets is either reflexive or non-isomorphic to a dual space.     

Twisted homology for the mirabolic nilradical

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 [AGS] derivatives of all orders were defined for smooth admissible Fréchet representations (of moderate growth).A key ingredient of this definition is the functor of twisted coinvariants with respect to the nilradical of the mirabolic subgroup. In this paper we prove exactness of this functor and compute it on a certain class of representations. This implies exactness of the highest derivative functor, and allows to compute highest derivatives of all monomial representations.In [AGS] these results are applied to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations.     

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.     

Robust return risk measures

In this paper we provide an axiomatic foundation to Orlicz risk measures in terms of properties of their acceptance sets, by exploiting their natural correspondence with shortfall risk Föllmer and Schied (Stochastic finance. De Gruyter, Berlin, 2011), thus paralleling the characterization in Weber (Math Financ 16:419–442, 2006). From a financial point of view, Orlicz risk measures assess the stochastic nature of returns, in contrast to the common use of risk measures to assess the stochastic nature of a position’s monetary value. The correspondence with shortfall risk leads to several robustified versions of Orlicz risk measures, and of their optimized translation invariant extensions (Rockafellar and Uryasev in J Risk 2:21–42, 2000, Goovaerts et al. in Insur Math Econ 34:505–516, 2004), arising from an ambiguity averse approach as in Gilboa and Schmeidler (J Math Econ 18:141–153, 1989), Maccheroni et al. (Econometrica 74:1447–1498, 2006), Chateauneuf and Faro (J Math Econ 45:535–558, 2010), or from a multiplicity of Young functions. We study the properties of these robust Orlicz risk measures, derive their dual representations, and provide some examples and applications.     

Palindromic widths of nilpotent and wreath products

We prove that the nilpotent product of a set of groups A ₁,…,A _s has finite palindromic width if and only if the palindromic widths of A _i ,i=1,…,s,are finite. We give a new proof that the commutator width of F _n ?K is infinite, where F _n is a free group of rank n≥2 and K is a finite group. This result, combining with a result of Fink [9] gives examples of groups with infinite commutator width but finite palindromic width with respect to some generating set.     

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).     

Construction of cyclotomic codebooks nearly meeting the Welch bound

Ding and Feng (IEEE Trans Inform Theory 52(9):4229–4235, 2006, IEEE Trans Inform Theory 53(11):4245–4250, 2007) constructed series of (N, K) codebooks which meet or nearly meet the Welch bound \({\sqrt{\frac{N-K}{(N-1)K}}}\) by using difference set (DS) or almost difference set (ADS) in certain finite abelian group respectively. In this paper, we generalize the cyclotomic constructions considered in (IEEE Trans Inform Theory 52(9):4229–4235, 2006, IEEE Trans Inform Theory 53(11):4245–4250, 2007) and (IEEE Trans Inform Theory 52(5), 2052–2061, 2006) to present more series of codebooks which nearly meet the Welch bound under looser conditions than ones required by DS and ADS.     

Symmetries on the Lattice of <Emphasis Type="Italic">k</Emphasis>-Bounded Partitions

In 2002, Suter [25] identified a dihedral symmetry on certain order ideals in Young’s lattice and gave a combinatorial action on the partitions in these order ideals. Viewing this result geometrically, the order ideals can be seen to be in bijection with the alcoves in a 2- fold dilation in the geometric realization of the affine symmetric group. By considering the m-fold dilation we observe a larger set of order ideals in the k-bounded partition lattice that was considered by Lapointe, Lascoux, and Morse [14] in the study of k-Schur functions. We identify the order ideal and the cyclic action on it explicitly in a geometric and combinatorial form.     

Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence

The Richardson variety X _α ^γ in the Grassmannian is defined to be the intersection of the Schubert variety X ^γ and opposite Schubert variety X _α . We give an explicit Gröbner basis for the ideal of the tangent cone at any T-fixed point of X _α ^γ , thus generalizing a result of Kodiyalam-Raghavan (J. Algebra 270(1):28–54, 2003) and Kreiman-Lakshmibai (Algebra, Arithmetic and Geometry with Applications, 2004). Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the bounded RSK (BRSK). We use the Gröbner basis result to deduce a formula which computes the multiplicity of X _α ^γ at any T-fixed point by counting families of nonintersecting lattice paths, thus generalizing a result first proved by Krattenthaler (Sém. Lothar. Comb. 45:B45c, 2000/2001; J. Algebr. Comb. 22:273–288, 2005).     

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.

     

Arc-Transitive Pentavalent Graphs of Square-Free Order

Hua et al. (Discrete Math 311, 2259–2267, 2011) and Yang et al. (Discrete Math. 339, 522–532, 2016) classify arc-transitive pentavalent graphs of order 2pq and of order 2pqr (with p, q, r distinct odd primes), respectively. In this paper, we extend their results by giving a classification of arc-transitive pentavalent graphs of any square-free order.