Authors:Sergio Conti; Adriana Garroni; Stefan Müller Pages: 3 - 17 Abstract: We consider target-space homogenization for energies defined on partitions parametrized by a discrete lattice \(\mathcal {B}\subset \mathbb {R}^N\) . For a small \(\sigma >0\) , the variable is a piecewise constant function taking values in \(\sigma \mathcal {B}\) , and the energy depends on the jumps and their orientation. In the limit as \(\sigma \rightarrow 0\) we obtain a homogenized functional defined on functions of bounded variation. This result is relevant in the study of dislocation structures in plastically deformed crystals. We review recent literature on the topic and propose our limiting effective energy as a continuum model for strain-gradient plasticity. PubDate: 2017-03-01 DOI: 10.1007/s40574-016-0083-z Issue No:Vol. 10, No. 1 (2017)

Authors:Giuseppe Molteni Pages: 19 - 28 Abstract: Let \(\psi _{\mathbb {{{K}}}}\) be the Chebyshev function of a number field \(\mathbb {{{K}}}\) , and let \(\psi ^{(1)}_\mathbb {{{K}}}(x):=\int _{0}^{x}\psi _\mathbb {{{K}}}(t)\,\mathrm {d}t\) and \(\psi ^{(2)}_\mathbb {{{K}}}(x):=2\int _{0}^{x}\psi ^{(1)}_\mathbb {{{K}}}(t)\,\mathrm {d}t\) . Assuming the truth of Riemann’s hypothesis for Dedekind’s zeta function of \(\mathbb {{{K}}}\) it is possible to prove explicit bounds for \( \psi _\mathbb {{{K}}}(x) - x \) , \( \psi ^{(1)}_\mathbb {{{K}}}(x) - \tfrac{x^2}{2} \) and \( \psi ^{(2)}_\mathbb {{{K}}}(x) - \tfrac{x^3}{3} \) . These results can be used to prove the existence of many ideals with a small norms and to produce an extremely efficient algorithm for the computation of the residue of Dedekind’s zeta function. PubDate: 2017-03-01 DOI: 10.1007/s40574-016-0084-y Issue No:Vol. 10, No. 1 (2017)

Authors:Alberto Perelli Pages: 29 - 53 Abstract: This is an expanded version of the author’s lecture at the XX Congresso U.M.I., held in Siena in September 2015. After a brief review of L-functions, we turn to the classical converse theorems of Hamburger, Hecke and Weil, and to some later developments. Finally we present several results on converse theorems in the framework of the Selberg class of L-functions. PubDate: 2017-03-01 DOI: 10.1007/s40574-016-0085-x Issue No:Vol. 10, No. 1 (2017)

Authors:A. Figalli Pages: 55 - 74 Abstract: In this note we discuss a new recent approach, based on transportation techniques, to obtain universality results in random matrix theory. PubDate: 2017-03-01 DOI: 10.1007/s40574-016-0087-8 Issue No:Vol. 10, No. 1 (2017)

Authors:Andrea Malchiodi Pages: 75 - 97 Abstract: We consider a class of Liouville equations motivated by uniformization problems in a singular setting, as well as from models in Mathematical Physics. We will study existence from a variational point of view, using suitable improvements of the Moser-Trudinger inequality. These improvements allow to study the concentration properties of conformal volume, and to reduce the problem to studying explicit finite-dimensional objects. PubDate: 2017-03-01 DOI: 10.1007/s40574-016-0092-y Issue No:Vol. 10, No. 1 (2017)

Authors:Giovanna Carnovale; Iulian I. Simion Pages: 99 - 112 Abstract: A conjecture of De Concini Kac and Procesi provides a bound on the minimal possible dimension of an irreducible module for quantized enveloping algebras at an odd root of unity. We pose the problem of the existence of modules whose dimension equals this bound. We show that this question cannot have a positive answer in full generality and discuss variants of this question. PubDate: 2017-03-01 DOI: 10.1007/s40574-016-0094-9 Issue No:Vol. 10, No. 1 (2017)

Authors:R. E. Caflisch; F. Gargano; M. Sammartino; V. Sciacca Pages: 113 - 141 Abstract: We consider the Euler- \(\alpha \) regularization of the Birkhoff–Rott equation and compare its solutions with the dynamics of the non regularized vortex-sheet. For a flow induced by an infinite array of planar vortex-sheets we analyze the complex singularities of the solutions.Through the singularity tracking method we show that the regularized solution has several complex singularities that approach the real axis. We relate their presence to the formation of two high-curvature points in the vortex sheet during the roll-up phenomenon. PubDate: 2017-03-01 DOI: 10.1007/s40574-016-0097-6 Issue No:Vol. 10, No. 1 (2017)

Authors:Marzia Bisi Pages: 143 - 158 Abstract: We review the main results on some basic kinetic models for wealth distribution in a simple market economy, with interaction rules involving random variables to take into account effects due to market risks. Then, we investigate in more detail long time behavior of a model which includes the taxation phenomenon and the redistribution of collected wealth according to proper criterions. Finally, we propose a new class of kinetic equations in which agent’s trading propensity varies according to the personal amount of wealth. PubDate: 2017-03-01 DOI: 10.1007/s40574-016-0099-4 Issue No:Vol. 10, No. 1 (2017)

Authors:Julie Decaup; Adrien Dubouloz Abstract: We classify closed curves isomorphic to the affine line in the complement of a smooth rational projective plane conic Q. Over a field of characteristic zero, we show that up to the action of the subgroup of the Cremona group of the plane consisting of birational endomorphisms restricting to biregular automorphisms outside Q, there are exactly two such lines: the restriction of a smooth conic osculating Q at a rational point and the restriction of the tangent line to Q at a rational point. In contrast, we give examples illustrating the fact that over fields of positive characteristic, there exist exotic closed embeddings of the affine line in the complement of Q. We also determine an explicit set of birational endomorphisms of the plane whose restrictions generates the automorphism group of the complement of Q over a field of arbitrary characteristic. PubDate: 2017-03-18 DOI: 10.1007/s40574-017-0119-z

Authors:Francesco Bastianelli Abstract: This survey retraces the author’s talk at the Workshop Birational geometry of surfaces, Rome, January 11–15, 2016. We consider various birational invariants extending the notion of gonality to projective varieties of arbitrary dimension, and measuring the failure of a given projective variety to satisfy certain rationality properties, such as being uniruled, rationally connected, unirational, stably rational or rational. Then we review a series of results describing these invariants for various classes of projective surfaces. PubDate: 2017-01-27 DOI: 10.1007/s40574-017-0116-2

Authors:Mauro Maccioni Abstract: I investigate on the number t of real eigenvectors of a real symmetric tensor. In particular, given a homogeneous polynomial f of degree d in 3 variables, I prove that t is greater or equal than \(2c+1\) , if d is odd, and t is greater or equal than \(\max (3,2c+1)\) , if d is even, where c is the number of ovals in the zero locus of f. About binary forms, I prove that t is greater or equal than the number of real roots of f. Moreover, the above inequalities are sharp for binary forms of any degree and for cubic and quartic ternary forms. PubDate: 2017-01-09 DOI: 10.1007/s40574-016-0112-y

Authors:Marco Franciosi; Sönke Rollenske Abstract: Extending the description of canonical rings from Reid (J Fac Sci Univ Tokyo Sect IA Math 25(1):75–92, 1978) we show that every Gorenstein stable Godeaux surface with torsion of order at least 3 is smoothable. PubDate: 2017-01-05 DOI: 10.1007/s40574-016-0114-9

Authors:Lazhar Dhaouadi Abstract: In the first part of this paper, we give a sufficient condition for a particular case of the symmetric moment problem to be determinate using standards methods of q-Bessel Fourier analysis. This condition it cannot be deduced from any other classical criterion of determinacy. In the second part, we study the q-Strum–Liouville equation in the non-real case and we elaborate an analogue of the well known theorem due to Hermann Weyl concerning the Strum–Liouville equation. This emphasizes the connection between the moment problem associated to a particular class of orthonormal polynomials \((P_n)\) and the uniqueness of solution which belong to the \(L^2\) space. The third part is devoted to the study of the q-Strum–Liouville equation in the real case and the behavior of solutions at infinity, which give more information about this type of orthonormal polynomials. PubDate: 2017-01-04 DOI: 10.1007/s40574-016-0115-8

Authors:H. E. Darwish; A. Y. Lashin; B. F. Hassan Pages: 483 - 493 Abstract: The purpose of the present paper is to derive some inclusion properties and argument estimates of certain classes of meromorphic functions in the punctured unit disc, which are defined by means of Bessel function. Furthermore, the integral preserving properties in a sector are investigated. PubDate: 2016-12-01 DOI: 10.1007/s40574-016-0063-3 Issue No:Vol. 9, No. 4 (2016)

Authors:Soshal Saini; Uaday Singh Pages: 495 - 504 Abstract: Fourier series and operators based on it are of great importance in both theoretical and applied mathematics because they can be considered as representation of a function or signal. In this paper, we estimate the degree of approximation of \(f \in Lip(\omega (t),p)\) -class and its conjugate \(\tilde{f},\) by matrix means of their trigonometric Fourier series by relaxing the conditions on \(\omega (t)\) imposed by the earlier researchers. We also discuss some results which are analogous to our results. PubDate: 2016-12-01 DOI: 10.1007/s40574-016-0064-2 Issue No:Vol. 9, No. 4 (2016)

Authors:Mohammad Shafiee Pages: 505 - 511 Abstract: The set of all linear transformations on \(\mathbb {R}^{2m}\) , preserving its b-symplectic structure, is a Lie group. This group has two components and its identity component contains the unitary group \(U(m-1)\) as maximal compact subgroup. We obtain a unitary representation for the identity component. At last some results on embedding by this transformations are presented. PubDate: 2016-12-01 DOI: 10.1007/s40574-016-0068-y Issue No:Vol. 9, No. 4 (2016)

Authors:Sekhar Jyoti Baishya Pages: 527 - 531 Abstract: Given a group G, let \({{\mathrm{Cent}}}(G)\) denote the set of distinct centralizers of elements of G. The group G is called an n-centralizer group if \( {{\mathrm{Cent}}}(G) =n\) and a primitive n-centralizer group if \( {{\mathrm{Cent}}}(G) = {{\mathrm{Cent}}}(\frac{G}{Z(G)}) =n\) . In this paper, we characterize the finite 9-centralizer and the primitive 9-centralizer groups. PubDate: 2016-12-01 DOI: 10.1007/s40574-016-0070-4 Issue No:Vol. 9, No. 4 (2016)

Authors:C. Selvaraj; R. Saravanan Abstract: In this paper, we introduce and study the notions of Gorenstein n-FP-injective and Gorenstein n-flat complexes, which are special cases of Gorenstein FP-injective and Gorenstein flat complexes respectively. We prove that over a left n-coherent ring R the class of all Gorenstein n-FP-injective (resp., Gorenstein n-flat) complexes is injectively (resp., projectively) resolving and we discuss the relationship between Gorenstein n-FP-injective and Gorenstein n-flat complexes. PubDate: 2016-12-08 DOI: 10.1007/s40574-016-0113-x

Authors:Ilwoo Cho Abstract: In this paper, we generalize classical Hecke algebras \(\mathcal {H}(G_{p})\) over the generalized linear groups \(G_{p} = GL_{2}(\mathbb {Q}_{p})\) induced by the p-adic number fields \(\mathbb {Q}_{p}\) , for primes p. For a given group \(G_{p},\) construct a suitable semigroup \(W^{*}\) -dynamical system \((M, \sigma (G_{p}), \pi ),\) where M is a fixed von Neumann algebra, and \( \pi \) is a semigroup-action of the \(\sigma \) -algebra \(\sigma (G_{p})\) of \( G_{p}\) acting on M. By constructing the corresponding crossed product \( W^{*}\) -algebra \(M \times _{\pi } \sigma (G_{p})\) generated by \((M, \sigma (G_{p}), \pi ),\) we study free probability on the \(W^{*}\) -subalgebra \(\mathcal {H}_{M}(G_{p})\) of \(M \times _{\pi } \sigma (G_{p})\) . One can understand our von Neumann algebra \(\mathcal {H}_{M}(G_{p})\) as a generalized \(*\) -algebra over both M and a Hecke algebra \(\mathcal {H} (G_{p}),\) for a prime p. PubDate: 2016-12-02 DOI: 10.1007/s40574-016-0111-z