Authors:Aida Kh. Asgarova Pages: 1 - 6 Abstract: Assume X is a compact Hausdorff space and C(X) is the space of real-valued continuous functions on X. A version of the Stone–Weierstrass theorem states that a closed subalgebra \(A\subset C(X)\) , which contains a nonzero constant function, coincides with the whole space C(X) if and only if A separates points of X. In this paper, we generalize this theorem to the case in which two subalgebras of C(X) are involved. PubDate: 2018-04-01 DOI: 10.1007/s40316-017-0081-2 Issue No:Vol. 42, No. 1 (2018)

Authors:Philippe Charron; Bernard Helffer; Thomas Hoffmann-Ostenhof Pages: 7 - 29 Abstract: In 1956, Pleijel gave his celebrated theorem showing that the inequality in Courant’s theorem on the number of nodal domains is strict for large eigenvalues of the Laplacian. This was a consequence of a stronger result giving an asymptotic upper bound for the number of nodal domains of the eigenfunction as the eigenvalue tends to \(+\infty \) . A similar question occurs naturally for the case of the Schrödinger operator. The first significant result has been obtained recently by the first author for the case of the harmonic oscilllator. The purpose of this paper is to consider more general potentials which are radial. We will analyze either the case when the potential tends to \(+\infty \) or the case when the potential tends to zero, the considered eigenfunctions being associated with the eigenvalues below the essential spectrum. PubDate: 2018-04-01 DOI: 10.1007/s40316-017-0078-x Issue No:Vol. 42, No. 1 (2018)

Authors:Jean-Marie De Koninck; Imre Kátai Pages: 31 - 47 Abstract: We show that some sequences of real numbers involving sharp normal numbers or non-Liouville numbers are uniformly distributed modulo 1. In particular, we prove that if \(\tau (n)\) stands for the number of divisors of n and \(\alpha \) is a binary sharp normal number, then the sequence \((\alpha \tau (n))_{n\ge 1}\) is uniformly distributed modulo 1 and that if g(x) is a polynomial of positive degree with real coefficients and whose leading coefficient is a non-Liouville number, then the sequence \((g(\tau (\tau (n))))_{n \ge 1}\) is also uniformly distributed modulo 1. PubDate: 2018-04-01 DOI: 10.1007/s40316-017-0080-3 Issue No:Vol. 42, No. 1 (2018)

Authors:Andrew Fiori Pages: 49 - 78 Abstract: In this paper we study the integral structure of lattices over finite extensions of \(\mathbb {Z}_p\) which arise from restriction or transfer from a lattice over a finite extension. We describe explicitly the structure of the resulting lattices. Special attention is given to the case of lattices whose quadratic forms arise from Hermitian forms. Then, in the case of Hermitian lattices where the final lattice is over \(\mathbb {Z}_p\) we focus on the problem of computing the local densities. PubDate: 2018-04-01 DOI: 10.1007/s40316-017-0083-0 Issue No:Vol. 42, No. 1 (2018)

Authors:Maria Rosaria Pati Pages: 107 - 125 Abstract: Let K be a p-adic field. Restricting to the case of no intermediate extensions, we obtain formulæ counting the number of (totally and wildly) ramified extensions of degree \(p^4\) of K up to K-isomorphism and in particular, we count the number of isomorphism classes of extensions for which the Galois closure has a prescribed Galois group. The principal tool used is a result, proved in Del Corso et al. (On wild extensions of a p-adic field, arXiv:1601.05939v1), which states that there is a one-to-one correspondence between the isomorphism classes of extensions of degree \(p^k\) of K having no intermediate extensions and the irreducible H-sub-modules of dimension k of \(F^*{/}{F^*}^p\) , where F is the composite of certain fixed normal extensions of K and H is its Galois group over K. PubDate: 2018-04-01 DOI: 10.1007/s40316-016-0076-4 Issue No:Vol. 42, No. 1 (2018)

Authors:Nipen Saikia; Jubaraj Chetry Pages: 127 - 132 Abstract: For any non-negative integer n and non-zero integer r, let \(p_r(n)\) denote Ramanujan’s general partition function. In this paper, we prove many infinite families of congruences modulo 7 for the general partition function \(p_r(n)\) for negative values of r by using q-identities. PubDate: 2018-04-01 DOI: 10.1007/s40316-017-0084-z Issue No:Vol. 42, No. 1 (2018)

Authors:Rita Gitik Abstract: We prove that the free product of two finitely presented locally tame groups is locally tame and describe many examples of tame subgroups of finitely presented groups. We also include some open problems related to tame subgroups. PubDate: 2018-04-21 DOI: 10.1007/s40316-018-0102-9

Authors:Daniel Delbourgo; Qin Chao Abstract: We completely describe \(\hbox {K}_{1}({\mathbb {Z}}_p[\![{\mathcal {G}}_{\infty }]\!])\) and its localisations by using an infinite family of p-adic congruences, where \({\mathcal {G}}_{\infty }\) is any solvable p-adic Lie group of dimension 3. This builds on earlier work of Kato when \(\hbox {dim}({\mathcal {G}}_{\infty })=2\) , and of the first named author and Lloyd Peters when \({\mathcal {G}}_{\infty } \cong {\mathbb {Z}}_p^{\times }\ltimes {\mathbb {Z}}_p^d\) with a scalar action of \({\mathbb {Z}}_p^{\times }\) . The method exploits the classification of 3-dimensional p-adic Lie groups due to González-Sánchez and Klopsch, as well as the fundamental ideas of Kakde, Burns, etc. in non-commutative Iwasawa theory. PubDate: 2018-04-16 DOI: 10.1007/s40316-018-0100-y

Authors:Tadashi Ochiai; Kazuma Shimomoto Abstract: The aim of this article is to establish the specialization method on characteristic ideals for finitely generated torsion modules over a complete local normal domain R that is module-finite over \({\mathcal {O}}[[x_1,\ldots ,x_d]]\) , where \({\mathcal {O}}\) is the ring of integers of a finite extension of the field of p-adic integers \({\mathbb {Q}}_p\) . The specialization method is a technique that recovers the information on the characteristic ideal \({\text {char}}_R (M)\) from \({\text {char}}_{R/I}(M/IM)\) , where I varies in a certain family of nonzero principal ideals of R. As applications, we prove Euler system bound over Cohen–Macaulay normal domains by combining the main results in Ochiai (Nagoya Math J 218:125–173, 2015) and then we prove one of divisibilities of the Iwasawa main conjecture for two-variable Hida deformations generalizing the main theorem obtained in Ochiai (Compos Math 142:1157–1200, 2006). PubDate: 2018-02-20 DOI: 10.1007/s40316-018-0099-0

Authors:Bernadette Faye; Florian Luca; Pieter Moree Abstract: We consider the family of Lucas sequences uniquely determined by \(U_{n+2}(k)=(4k+2)U_{n+1}(k) -U_n(k),\) with initial values \(U_0(k)=0\) and \(U_1(k)=1\) and \(k\ge 1\) an arbitrary integer. For any integer \(n\ge 1\) the discriminator function \(\mathcal {D}_k(n)\) of \(U_n(k)\) is defined as the smallest integer m such that \(U_0(k),U_1(k),\ldots ,U_{n-1}(k)\) are pairwise incongruent modulo m. Numerical work of Shallit on \(\mathcal {D}_k(n)\) suggests that it has a relatively simple characterization. In this paper we will prove that this is indeed the case by showing that for every \(k\ge 1\) there is a constant \(n_k\) such that \({\mathcal D}_{k}(n)\) has a simple characterization for every \(n\ge n_k\) . The case \(k=1\) turns out to be fundamentally different from the case \(k>1\) . PubDate: 2018-02-12 DOI: 10.1007/s40316-017-0097-7

Authors:André Boivin; Paul M. Gauthier; Myrto Manolaki Abstract: In this paper we study some questions related to the zero sets of harmonic and real analytic functions in \({\mathbb {R}}^N\) . We introduce the notion of analytic uniqueness sequences and, as an application, we show that the zero set of a non-constant real analytic function on a domain always has empty fine interior. We also prove that, for a certain category of sets \(E\subset {\mathbb {R}}^N\) (containing the finely open sets), each function f defined on E is the restriction of a real analytic (respectively harmonic) function on an open neighbourhood of E if and only if f is “analytic (respectively harmonic) at each point” of E. PubDate: 2018-02-01 DOI: 10.1007/s40316-018-0098-1

Authors:Kevin Buzzard; Alan Lauder Pages: 213 - 219 Abstract: We report on a computation of holomorphic cuspidal modular forms of weight one and small level (currently level at most 1500) and classification of them according to the projective image of their attached Artin representations. The data we have gathered, such as Fourier expansions and projective images of Hecke newforms and dimensions of space of forms, is available in both Magma and Sage readable formats on a webpage created in support of this project. PubDate: 2017-10-01 DOI: 10.1007/s40316-016-0072-8 Issue No:Vol. 41, No. 2 (2017)

Authors:Antoine Métras Abstract: We consider the Dirichlet eigenvalue problem on a polytope. We use the Rellich identity to obtain an explicit formula expressing the Dirichlet eigenvalue in terms of the Neumann data on the faces of the polytope of the corresponding eigenfunction. The formula is particularly simple for polytopes admitting an inscribed ball tangent to all the faces. Our result could be viewed as a generalization of similar identities for simplices recently found by Christianson (Equidistribution of Neumann data mass on simplices and a simple inverse problem, ArXiv e-prints, 2017, Equidistribution of Neumann data mass on triangles. ArXiv e-prints, 2017). PubDate: 2017-11-06 DOI: 10.1007/s40316-017-0096-8

Authors:Jay Jorgenson; Anna-Maria von Pippich; Lejla Smajlović Abstract: We develop two applications of the Kronecker’s limit formula associated to elliptic Eisenstein series: A factorization theorem for holomorphic modular forms, and a proof of Weil’s reciprocity law. Several examples of the general factorization results are computed, specifically for certain moonshine groups, congruence subgroups, and, more generally, non-compact subgroups with one cusp. In particular, we explicitly compute the Kronecker limit function associated to certain elliptic fixed points for a few small level moonshine groups. PubDate: 2017-10-31 DOI: 10.1007/s40316-017-0094-x

Authors:Eudes Leite de Lima; Henrique Fernandes de Lima Abstract: We obtain a sharp estimate to the scalar curvature of stochastically complete hypersurfaces immersed with constant mean curvature in a locally symmetric Riemannian space obeying standard curvature constraints (which includes, in particular, a Riemannian space with constant sectional curvature). For this, we suppose that these hypersurfaces satisfy a suitable Okumura-type inequality recently introduced by Meléndez (Bull Braz Math Soc 45:385–404, 2014), which is a weaker hypothesis than to assume that they have two distinct principal curvatures. Our approach is based on the equivalence between stochastic completeness and the validity of the weak version of the Omori–Yau’s generalized maximum principle, which was established by Pigola et al. (Proc Am Math Soc 131:1283–1288, 2002; Mem Am Math Soc 174:822, 2005). PubDate: 2017-10-28 DOI: 10.1007/s40316-017-0095-9

Authors:Thong Nguyen Quang Do Abstract: Greenberg’s well known conjecture, (GC) for short, asserts that the Iwasawa invariants \(\lambda \) and \(\mu \) associated to the cyclotomic \({\mathbb {Z}}_p\) -extension of any totally real number field F should vanish. In his foundational 1976 paper, Greenberg has shown two necessary and sufficient conditions for (GC) to hold, in two seemingly opposite cases, when p is undecomposed, resp. totally decomposed in F. In this article we present an encompassing approach covering both cases and resting only on “ genus formulas ”, that is (roughly speaking) on formulas which express the order of the Galois (co-)invariants of certain modules along the cyclotomic tower. These modules are akin to class groups, and in the end we obtain several unified criteria, which naturally contain the particular conditions given by Greenberg. PubDate: 2017-10-20 DOI: 10.1007/s40316-017-0093-y

Authors:Maia Fraser; Leonid Polterovich; Daniel Rosen Abstract: For certain contact manifolds admitting a 1-periodic Reeb flow we construct a conjugation-invariant norm on the universal cover of the contactomorphism group. With respect to this norm the group admits a quasi-isometric monomorphism of the real line. The construction involves the partial order on contactomorphisms and symplectic intersections. This norm descends to a conjugation-invariant norm on the contactomorphism group. As a counterpoint, we discuss conditions under which conjugation-invariant norms for contactomorphisms are necessarily bounded. PubDate: 2017-10-16 DOI: 10.1007/s40316-017-0092-z

Authors:A. Iacobucci; S. Olla; G. Stoltz Abstract: We study the exponential convergence to the stationary state for nonequilibrium Langevin dynamics, by a perturbative approach based on hypocoercive techniques developed for equilibrium Langevin dynamics. The Hamiltonian and overdamped limits (corresponding respectively to frictions going to zero or infinity) are carefully investigated. In particular, the maximal magnitude of admissible perturbations are quantified as a function of the friction. Numerical results based on a Galerkin discretization of the generator of the dynamics confirm the theoretical lower bounds on the spectral gap. PubDate: 2017-10-06 DOI: 10.1007/s40316-017-0091-0

Authors:V. Nestoridis Abstract: We give a simple proof that the notions of Domain of Holomorphy and Weak Domain of Holomorphy are equivalent. This proof is based on a combination of Baire’s Category Theorey and Montel’s Theorem. We also obtain generalizations by demanding that the non-extentable functions belong to a particular class of functions \(X=X({\varOmega })\subset H({\varOmega })\) . We show that the set of non-extendable functions not only contains a \(G_{\delta }\) -dense subset of \(X({\varOmega })\) , but it is itself a \(G_{\delta }\) -dense set. We give an example of a domain in \(\mathbb {C}\) which is a \(H({\varOmega })\) -domain of holomorphy but not a \(A({\varOmega })\) -domain of holomorphy. PubDate: 2017-09-21 DOI: 10.1007/s40316-017-0089-7

Authors:Sergei Lanzat; Frol Zapolsky Abstract: We construct an embedding of the full braid group on \(m+1\) strands \(B_{m+1}\) , \(m \ge 1\) , into the contact mapping class group of the contactization \(Q \times S^1\) of the \(A_m\) -Milnor fiber Q. The construction uses the embedding of \(B_{m+1}\) into the symplectic mapping class group of Q due to Khovanov and Seidel, and a natural lifting homomorphism. In order to show that the composed homomorphism is still injective, we use a partially linearized variant of the Chekanov–Eliashberg dga for Legendrians which lie above one another in \(Q \times {\mathbb {R}}\) , reducing the proof to Floer homology. As corollaries we obtain a contribution to the contact isotopy problem for \(Q \times S^1\) , as well as the fact that in dimension 4, the lifting homomorphism embeds the symplectic mapping class group of Q into the contact mapping class group of \(Q \times S^1\) . PubDate: 2017-07-01 DOI: 10.1007/s40316-017-0085-y