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.     

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).     

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).     

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).     

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.     

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].     

A Nonlinear Age-Structured Model of Population Dynamics with Inherited Properties

In this paper, we present some results regarding existence and uniqueness of solution on L ^p -spaces, 1 < p < + ∞, to a nonlinear initial boundary value problem originally proposed by Lebowitz and Rubinow (J Math Biol 1:17–36, 1974) to model an age-structured cell population with inherited properties. Our results complete those obtained by Garcia-Falset (Math Meth Appl Sci 34:1658–1666, 2011).     

Uniform Sublevel Radon-like Inequalities

This paper is concerned with establishing uniform weighted L ^p –L ^q estimates for a class of operators generalizing both Radon-like operators and sublevel set operators. Such estimates are shown to hold under general circumstances whenever a sublevel-type inequality is satisfied by certain associated measures (the inequality is of the sort studied by Oberlin (Math. Proc. Camb. Philos. Soc. 129(3):517–526, 2000), relating measures of parallelepipeds to powers of their Euclidean volumes). These ideas lead to previously unknown, weighted affine-invariant estimates for Radon-like operators as well as new L ^p -improving estimates for degenerate Radon-like operators with folding canonical relations which satisfy an additional curvature condition of Greenleaf and Seeger (J. Reine Angew. Math. 455:35–56, 1994) for FIOs (building on the ideas of Sogge (Invent. Math. 104(2):349–376, 1991) and Mockenhaupt et al. (J. Am. Math. Soc. 6(1):65–130, 1993)); these new estimates fall outside the range of estimates which are known to hold in the generality of the FIO context.     

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).     

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.     

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).

     

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.     

Refined Bounds on the Number of Connected Components of Sign Conditions on a Variety

Let \({\textnormal {R}}\) be a real closed field, \(\mathcal{P},\mathcal{Q} \subset {\textnormal {R}}[X_{1},\ldots,X_{k}]\) finite subsets of polynomials, with the degrees of the polynomials in \(\mathcal{P}\) (resp., \(\mathcal{Q}\)) bounded by d (resp., d ₀). Let \(V \subset {\textnormal {R}}^{k}\) be the real algebraic variety defined by the polynomials in \(\mathcal{Q}\) and suppose that the real dimension of V is bounded by k′. We prove that the number of semi-algebraically connected components of the realizations of all realizable sign conditions of the family \(\mathcal{P}\) on V is bounded by

$\sum_{j=0}^{k'}4^j{s +1\choose j}F_{d,d_0,k,k'}(j),$

where \(s = \operatorname {card}\mathcal{P}\), and

$F_{d,d_0,k,k'}(j)=\binom{k+1}{k-k'+j+1} (2d_0)^{k-k'}d^j \max\{2d_0,d \}^{k'-j}+2(k-j+1).$

In case 2d ₀≤d, the above bound can be written simply as

$\sum_{j = 0}^{k'} {s+1 \choose j}d^{k'} d_0^{k-k'} O(1)^{k}= (sd)^{k'} d_0^{k-k'} O(1)^k$

(in this form the bound was suggested by Matousek 2011). Our result improves in certain cases (when d ₀?d) the best known bound of

$\sum_{1 \leq j \leq k'}\binom{s}{j} 4^{j} d(2d-1)^{k-1}$

on the same number proved in Basu et al. (Proc. Am. Math. Soc. 133(4):965–974, 2005) in the case d=d ₀.

The distinction between the bound d ₀ on the degrees of the polynomials defining the variety V and the bound d on the degrees of the polynomials in \(\mathcal{P}\) that appears in the new bound is motivated by several applications in discrete geometry (Guth and Katz in arXiv:1011.4105v1 [math.CO], 2011; Kaplan et al. in arXiv:1107.1077v1 [math.CO], 2011; Solymosi and Tao in arXiv:1103.2926v2 [math.CO], 2011; Zahl in arXiv:1104.4987v3 [math.CO], 2011).     

Stark points and the Hida–Rankin <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis>-function

This article is devoted to the elliptic Stark conjecture formulated by Darmon (Forum Math Pi 3:e8, 2015), which proposes a formula for the transcendental part of a p-adic avatar of the leading term at \(s=1\) of the Hasse–Weil–Artin L-series \(L(E,\varrho _1\otimes \varrho _2,s)\) of an elliptic curve \(E/\mathbb {Q}\) twisted by the tensor product \(\varrho _1\otimes \varrho _2\) of two odd 2-dimensional Artin representations, when the order of vanishing is two. The main ingredient of this formula is a \(2\times 2\) p-adic regulator involving the p-adic formal group logarithm of suitable Stark points on E. This conjecture was proved by Darmon (Forum Math Pi 3:e8, 2015) in the setting where \(\varrho _1\) and \(\varrho _2\) are induced from characters of the same imaginary quadratic field K. In this note, we prove a refinement of this result that was discovered experimentally by Darmon (Forum Math Pi 3:e8, 2015, [Remark 3.4]) in a few examples. Namely, we are able to determine the algebraic constant up to which the main theorem of Darmon (Forum Math Pi 3:e8, 2015) holds in a particular setting where the Hida–Rankin p-adic L-function associated to a pair of Hida families can be exploited to provide an alternative proof of the same result. This constant encodes local and global invariants of both E and K.     

Domination problem for AM-compact abstract Uryson operators

The Boolean algebra of fragments of a positive abstract Uryson operator recently was described in M. Pliev (Positivity, doi:10.1007/s11117-016-0401-9, 2016). Using this result, we prove a theorem of domination for AM-compact positive abstract Uryson operators from a Dedekind complete vector lattice E to a Banach lattice F with an order continuous norm.     

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.     

Clique-Coloring Claw-Free Graphs

A clique is defined as a complete subgraph maximal under inclusion and having at least two vertices. A k-clique-coloring of a graph G is an assignment of k colors to the vertices of G such that no clique of G is monochromatic. Bacsó et al. (SIAM J Discrete Math 17:361–376, 2004) noted that the clique-coloring number is unbounded even for the line graphs of complete graphs. In this paper, we prove that a claw-free graph with maximum degree at most 7, except an odd cycle longer than 3, has a 2-clique-coloring by using a decomposition theorem of Chudnovsky and Seymour (J Combin Theory Ser B 98:839–938, 2008) and the limitation of the degree 7 is necessary since the line graph of \(K_{6}\) is a graph with maximum degree 8 but its clique-coloring number is 3 by the Ramsey number \(R(3,3)=6\). In addition, we point out that, if an arbitrary line graph of maximum degree at most d is m-clique-colorable (\(m\ge 3\)), then an arbitrary claw-free graph of maximum degree at most d is also m-clique-colorable.     

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.     

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.