Normes cyclotomiques naïves et unités logarithmiques

We compute the \({\mathbb {Z}}\)-rank of the subgroup \(\widetilde{E}_K =\bigcap _{n\in {\mathbb {N}}} N_{K_n/K}(K_n^\times )\) of elements of the multiplicative group of a number field K that are norms from every finite level of the cyclotomic \({\mathbb {Z}}_\ell \)-extension \(K^c\) of K. Thus we compare its \(\ell \)-adification \({\mathbb {Z}}_\ell \otimes _{\mathbb {Z}}\widetilde{E}_K\) with the group of logarithmic units \(\widetilde{\varepsilon }_K\). By the way we point out an easy proof of the Gross–Kuz’min conjecture for \(\ell \)-undecomposed extensions of abelian fields.     

Interpretable groups in Mann pairs

In this paper, we study an algebraically closed field \(\Omega \) expanded by two unary predicates denoting an algebraically closed proper subfield k and a multiplicative subgroup \(\Gamma \). This will be a proper expansion of algebraically closed field with a group satisfying the Mann property, and also pairs of algebraically closed fields. We first characterize the independence in the triple \((\Omega , k, \Gamma )\). This enables us to characterize the interpretable groups when \(\Gamma \) is divisible. Every interpretable group H in \((\Omega ,k, \Gamma )\) is, up to isogeny, an extension of a direct sum of k-rational points of an algebraic group defined over k and an interpretable abelian group in \(\Gamma \) by an interpretable group N, which is the quotient of an algebraic group by a subgroup \(N_1\), which in turn is isogenous to a cartesian product of k-rational points of an algebraic group defined over k and an interpretable abelian group in \(\Gamma \).     

A generalization of zero divisor graphs associated to commutative rings

Let R be a commutative ring with a nonzero identity element. For a natural number n, we associate a simple graph, denoted by \(\Gamma ^n_R\), with \(R^n\backslash \{0\}\) as the vertex set and two distinct vertices X and Y in \(R^n\) being adjacent if and only if there exists an \(n\times n\) lower triangular matrix A over R whose entries on the main diagonal are nonzero and one of the entries on the main diagonal is regular such that \(X^TAY=0\) or \(Y^TAX=0\), where, for a matrix \(B, B^T\) is the matrix transpose of B. If \(n=1\), then \(\Gamma ^n_R\) is isomorphic to the zero divisor graph \(\Gamma (R)\), and so \(\Gamma ^n_R\) is a generalization of \(\Gamma (R)\) which is called a generalized zero divisor graph of R. In this paper, we study some basic properties of \(\Gamma ^n_ R\). We also determine all isomorphic classes of finite commutative rings whose generalized zero divisor graphs have genus at most three.     

An <Emphasis Type="Italic">A</Emphasis>-invariant subspace for bipartite distance-regular graphs with exactly two irreducible <Emphasis Type="Italic">T</Emphasis>-modules with endpoint 2, both thin

Let \(\Gamma \) denote a bipartite distance-regular graph with vertex set X, diameter \(D \ge 4\), and valency \(k \ge 3\). Let \({{\mathbb {C}}}^X\) denote the vector space over \({{\mathbb {C}}}\) consisting of column vectors with entries in \({{\mathbb {C}}}\) and rows indexed by X. For \(z \in X\), let \({{\widehat{z}}}\) denote the vector in \({{\mathbb {C}}}^X\) with a 1 in the z-coordinate, and 0 in all other coordinates. Fix a vertex x of \(\Gamma \) and let \(T = T(x)\) denote the corresponding Terwilliger algebra. Assume that up to isomorphism there exist exactly two irreducible T-modules with endpoint 2, and they both are thin. Fix \(y \in X\) such that \(\partial (x,y)=2\), where \(\partial \) denotes path-length distance. For \(0 \le i,j \le D\) define \(w_{ij}=\sum {{\widehat{z}}}\), where the sum is over all \(z \in X\) such that \(\partial (x,z)=i\) and \(\partial (y,z)=j\). We define \(W=\mathrm{span}\{w_{ij} \mid 0 \le i,j \le D\}\). In this paper we consider the space \(MW=\mathrm{span}\{mw \mid m \in M, w \in W\}\), where M is the Bose–Mesner algebra of \(\Gamma \). We observe that MW is the minimal A-invariant subspace of \({{\mathbb {C}}}^X\) which contains W, where A is the adjacency matrix of \(\Gamma \). We show that \(4D-6 \le \mathrm{dim}(MW) \le 4D-2\). We display a basis for MW for each of these five cases, and we give the action of A on these bases.     

On color-preserving automorphisms of Cayley graphs of odd square-free order

An automorphism \(\alpha \) of a Cayley graph \(\mathrm{Cay}(G,S)\) of a group G with connection set S is color-preserving if \(\alpha (g,gs) = (h,hs)\) or \((h,hs^{-1})\) for every edge \((g,gs)\in E(\mathrm{Cay}(G,S))\). If every color-preserving automorphism of \(\mathrm{Cay}(G,S)\) is also affine, then \(\mathrm{Cay}(G,S)\) is a Cayley color automorphism (CCA) graph. If every Cayley graph \(\mathrm{Cay}(G,S)\) is a CCA graph, then G is a CCA group. Hujdurovi? et al. have shown that every non-CCA group G contains a section isomorphic to the non-abelian group \(F_{21}\) of order 21. We first show that there is a unique non-CCA Cayley graph \(\Gamma \) of \(F_{21}\). We then show that if \(\mathrm{Cay}(G,S)\) is a non-CCA graph of a group G of odd square-free order, then \(G = H\times F_{21}\) for some CCA group H, and \(\mathrm{Cay}(G,S) = \mathrm{Cay}(H,T)\mathbin {\square }\Gamma \).     

Fourier Spectrum Characterizations of $$H^{p}$$ Spaces on Tubes Over Cones for $$1\le p \le \infty $$1≤p≤∞

We give Fourier spectrum characterizations of functions in the Hardy \(H^p\) spaces on tubes for \(1\le p \le \infty .\) For \(F\in L^p(\mathbb {R}^n), \) we show that F is the non-tangential boundary limit of a function in a Hardy space, \(H^{p}(T_\Gamma ),\) where \(\Gamma \) is an open cone of \(\mathbb {R}^n\) and \(T_\Gamma \) is the related tube in \(\mathbb {C}^n,\) if and only if the classical or the distributional Fourier transform of F is supported in \(\Gamma ^*,\) where \(\Gamma ^*\) is the dual cone of \(\Gamma .\) This generalizes the results of Stein and Weiss for \(p=2\) in the same context, as well as those of Qian et al. in one complex variable for \(1\le p\le \infty .\) Furthermore, we extend the Poisson and Cauchy integral representation formulas to the \(H^p\) spaces on tubes for \(p\in [1, \infty ]\) and \(p\in [1,\infty ),\) with, respectively, the two types of representations.     

A method to find generators of a semi-simple Lie group via the topology of its flag manifolds

In this paper we continue to develop the topological method to get semigroup generators of semi-simple Lie groups. Consider a subset \(\Gamma \subset G\) that contains a semi-simple subgroup \(G_{1}\) of G. If one can show that \( \Gamma \) does not leave invariant a contractible subset on any flag manifold of G, then \(\Gamma \) generates G if \(\mathrm {Ad}\left( \Gamma \right) \) generates a Zariski dense subgroup of the algebraic group \(\mathrm {Ad}\left( G\right) \). The proof is reduced to check that some specific closed orbits of \(G_{1}\) in the flag manifolds of G are not trivial in the sense of algebraic topology. Here, we consider three different cases of semi-simple Lie groups G and subgroups \(G_{1}\subset G\).     

Triviality of Iwasawa module associated to some abelian fields of prime conductors

Let p be an odd prime number and \(\ell \) an odd prime number dividing \(p-1\). We denote by \(F=F_{p,\ell }\) the real abelian field of conductor p and degree \(\ell \), and by \(h_F\) the class number of F. For a prime number \(r \ne p,\,\ell \), let \(F_{\infty }\) be the cyclotomic \(\mathbb {Z}_r\)-extension over F, and \(M_{\infty }/F_{\infty }\) the maximal pro-r abelian extension unramified outside r. We prove that \(M_{\infty }\) coincides with \(F_{\infty }\) and consequently \(h_F\) is not divisible by r when r is a primitive root modulo \(\ell \) and r is smaller than an explicit constant depending on p.     

Comparison of graphs associated to a commutative Artinian ring

Let R be a commutative ring with \(1\ne 0\) and the additive group \(R^+\). Several graphs on R have been introduced by many authors, among zero-divisor graph \(\Gamma _1(R)\), co-maximal graph \(\Gamma _2(R)\), annihilator graph AG(R), total graph \( T(\Gamma (R))\), cozero-divisors graph \(\Gamma _\mathrm{c}(R)\), equivalence classes graph \(\Gamma _\mathrm{E}(R)\) and the Cayley graph \(\mathrm{Cay}(R^+ ,Z^*(R))\). Shekarriz et al. (J. Commun. Algebra, 40 (2012) 2798–2807) gave some conditions under which total graph is isomorphic to \(\mathrm{Cay}(R^+ ,Z^*(R))\). Badawi (J. Commun. Algebra, 42 (2014) 108–121) showed that when R is a reduced ring, the annihilator graph is identical to the zero-divisor graph if and only if R has exactly two minimal prime ideals. The purpose of this paper is comparison of graphs associated to a commutative Artinian ring. Among the results, we prove that for a commutative finite ring R with \(|\mathrm{Max}(R)|=n \ge 3\), \( \Gamma _1(R) \simeq \Gamma _2(R)\) if and only if \(R\simeq \mathbb {Z}^n_2\); if and only if \(\Gamma _1(R) \simeq \Gamma _\mathrm{E}(R)\). Also the annihilator graph is identical to the cozero-divisor graph if and only if R is a Frobenius ring.     

Relative Difference Sets and Hadamard Matrices from Perfect Quaternionic Arrays

Let \(G=\mathbf{C}_{n_1}\times \cdots \times \mathbf{C}_{n_m}\) be an abelian group of order \(n=n_1\dots n_m\), where each \(\mathbf{C}_{n_t}\) is cyclic of order \(n_t\). We present a correspondence between the (4n, 2, 4n, 2n)-relative difference sets in \(G\times Q_8\) relative to the centre \(Z(Q_8)\) and the perfect arrays of size \(n_1\times \dots \times n_m\) over the quaternionic alphabet \(Q_8\cup qQ_8\), where \(q=(1+i+j+k)/2\). In view of this connection, for \(m=2\) we introduce new families of relative difference sets in \(G\times Q_8\), as well as new families of Williamson and Ito Hadamard matrices with G-invariant components.     

Directed graphs of inner translations of semigroups

A mapping \(\alpha :S\rightarrow S\) is called a Cayley function if there exist an associative operation \(\mu :S\times S\rightarrow S\) and an element \(a\in S\) such that \(\alpha (x)=\mu (a,x)\) for every \(x\in S\). The aim of the paper is to give a characterization of Cayley functions in terms of their directed graphs. This characterization is used to determine which elements of the centralizer of a permutation on a finite set are Cayley functions. The paper ends with a number of problems.     

The distortion dimension of $$\mathbb $$-rank 1 lattices

Let \(X=G/K\) be a symmetric space of noncompact type and rank \(k\ge 2\). We prove that horospheres in X are Lipschitz \((k-2)\)-connected if their centers are not contained in a proper join factor of the spherical building of X at infinity. As a consequence, the distortion dimension of an irreducible \(\mathbb {Q}\)-rank-1 lattice \(\Gamma \) in a linear, semisimple Lie group G of \(\mathbb R\)-rank k is \(k-1\). That is, given \(m< k-1\), a Lipschitz m-sphere S in (a polyhedral complex quasi-isometric to) \(\Gamma \), and a \((m+1)\)-ball B in X (or G) filling S, there is a \((m+1)\)-ball \(B'\) in \(\Gamma \) filling S such that \({{\mathrm{vol}}}B'\sim {{\mathrm{vol}}}B\). In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension \(k-1\).     

Vanishing theorems for constructible sheaves on abelian varieties over finite fields

Let \(\kappa \) be a field, finitely generated over its prime field, and let k denote an algebraically closed field containing \(\kappa \). For a perverse \(\overline{\mathbb {Q}}_\ell \)-adic sheaf \(K_0\) on an abelian variety \(X_0\) over \(\kappa \), let K and X denote the base field extensions of \(K_0\) and \(X_0\) to k. Then, the aim of this note is to show that the Euler–Poincare characteristic of the perverse sheaf K on X is a non-negative integer, i.e. \(\chi (X,K)=\sum _\nu (-1)^\nu \dim _{\overline{\mathbb {Q}}_\ell }(H^\nu (X,K))\ge 0\). This generalizes the result of Franecki and Kapranov [9] for fields of characteristic zero. Furthermore we show that \(\chi (X,K)=0\) implies K to be translation invariant. This result allows to considerably simplify the proof of the generic vanishing theorems for constructible sheaves on complex abelian varieties of [11]. Furthermore it extends these vanishing theorems to constructible sheaves on abelian varieties over finite fields.     

The Terwilliger algebra of the incidence graph of the Hamming graph

Let \(\varGamma \) be a distance-semiregular graph on Y, and let \(D^Y\) be the diameter of \(\varGamma \) on Y. Let \(\varDelta \) be the halved graph of \(\varGamma \) on Y. Fix \(x \in Y\). Let T and \(T'\) be the Terwilliger algebras of \(\varGamma \) and \(\varDelta \) with respect to x, respectively. Assume, for an integer i with \(1 \le 2i \le D^Y\) and for \(y,z \in \varGamma _{2i}(x)\) with \(\partial _{\varGamma }(y,z)=2\), the numbers \(|\varGamma _{2i-1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) and \(|\varGamma _{2i+1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) depend only on i and do not depend on the choice of y, z. The first goal in this paper is to show the relations between T-modules of \(\varGamma \) and \(T'\)-modules of \(\varDelta \). Assume \(\varGamma \) is the incidence graph of the Hamming graph H(D, n) on the vertex set Y and the set \({\mathcal {C}}\) of all maximal cliques. Then, \(\varGamma \) satisfies above assumption and \(\varDelta \) is isomorphic to H(D, n). The second goal is to determine the irreducible T-modules of \(\varGamma \). For each irreducible T-module W, we give a basis for W the action of the adjacency matrix on this basis and we calculate the multiplicity of W.     

Non-triviality conditions for integer-valued polynomial rings on algebras

Let D be a commutative domain with field of fractions K and let A be a torsion-free D-algebra such that \(A \cap K = D\). The ring of integer-valued polynomials on A with coefficients in K is \( Int _K(A) = \{f \in K[X] \mid f(A) \subseteq A\}\), which generalizes the classic ring \( Int (D) = \{f \in K[X] \mid f(D) \subseteq D\}\) of integer-valued polynomials on D. The condition on \(A \cap K\) implies that \(D[X] \subseteq Int _K(A) \subseteq Int (D)\), and we say that \( Int _K(A)\) is nontrivial if \( Int _K(A) \ne D[X]\). For any integral domain D, we prove that if A is finitely generated as a D-module, then \( Int _K(A)\) is nontrivial if and only if \( Int (D)\) is nontrivial. When A is not necessarily finitely generated but D is Dedekind, we provide necessary and sufficient conditions for \( Int _K(A)\) to be nontrivial. These conditions also allow us to prove that, for D Dedekind, the domain \( Int _K(A)\) has Krull dimension 2.     

Lengths of words in transformation semigroups generated by digraphs

Given a simple digraph D on n vertices (with \(n\ge 2\)), there is a natural construction of a semigroup of transformations \(\langle D\rangle \). For any edge (a, b) of D, let \(a\rightarrow b\) be the idempotent of rank \(n-1\) mapping a to b and fixing all vertices other than a; then, define \(\langle D\rangle \) to be the semigroup generated by \(a \rightarrow b\) for all \((a,b) \in E(D)\). For \(\alpha \in \langle D\rangle \), let \(\ell (D,\alpha )\) be the minimal length of a word in E(D) expressing \(\alpha \). It is well known that the semigroup \(\mathrm {Sing}_n\) of all transformations of rank at most \(n-1\) is generated by its idempotents of rank \(n-1\). When \(D=K_n\) is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate \(\ell (K_n,\alpha )\), for any \(\alpha \in \langle K_n\rangle = \mathrm {Sing}_n\); however, no analogous non-trivial results are known when \(D \ne K_n\). In this paper, we characterise all simple digraphs D such that either \(\ell (D,\alpha )\) is equal to Howie–Iwahori’s formula for all \(\alpha \in \langle D\rangle \), or \(\ell (D,\alpha ) = n - \mathrm {fix}(\alpha )\) for all \(\alpha \in \langle D\rangle \), or \(\ell (D,\alpha ) = n - \mathrm {rk}(\alpha )\) for all \(\alpha \in \langle D\rangle \). We also obtain bounds for \(\ell (D,\alpha )\) when D is an acyclic digraph or a strong tournament (the latter case corresponds to a smallest generating set of idempotents of rank \(n-1\) of \(\mathrm {Sing}_n\)). We finish the paper with a list of conjectures and open problems.     

Integral Cayley graphs over dihedral groups

In this paper, we give a necessary and sufficient condition for the integrality of Cayley graphs over the dihedral group \(D_n=\langle a,b\mid a^n=b^2=1,bab=a^{-1}\rangle \). Moreover, we also obtain some simple sufficient conditions for the integrality of Cayley graphs over \(D_n\) in terms of the Boolean algebra of \(\langle a\rangle \), from which we find infinite classes of integral Cayley graphs over \(D_n\). In particular, we completely determine all integral Cayley graphs over the dihedral group \(D_p\) for a prime p.     

Generalized Bernstein Operators on the Classical Polynomial Spaces

We study generalizations of the classical Bernstein operators on the polynomial spaces \(\mathbb {P}_{n}[a,b]\), where instead of fixing \(\mathbf {1}\) and x, we reproduce exactly \(\mathbf {1}\) and a polynomial \(f_1\), strictly increasing on [a, b]. We prove that for sufficiently large n, there always exist generalized Bernstein operators fixing \(\mathbf {1}\) and \(f_1\). These operators are defined by non-decreasing sequences of nodes precisely when \(f_1^\prime > 0\) on (a, b), but even if \(f_1^\prime \) vanishes somewhere inside (a, b), they converge to the identity.     

Polynomial partitioning for several sets of varieties

We give a new, systematic proof for a recent result of Larry Guth and thus also extend the result to a setting with several families of varieties: For any integer \(D\ge 1\) and any collection of sets \(\Gamma _1,\ldots ,\Gamma _j\) of low-degree k-dimensional varieties in \(\mathbb {R}^n\), there exists a non-zero polynomial \(p\in \mathbb {R}[X_1,\ldots ,X_n]\) of degree at most D, so that each connected component of \(\mathbb {R}^n{\setminus }Z(p)\) intersects \(O(jD^{k-n}|\Gamma _i|)\) varieties of \(\Gamma _i\), simultaneously for every \(1\le i\le j\). For \(j=1\), we recover the original result by Guth. Our proof, via an index calculation in equivariant cohomology, shows how the degrees of the polynomials used for partitioning are dictated by the topology, namely, by the Euler class being given in terms of a top Dickson polynomial.     

<Emphasis Type="Italic">b</Emphasis>-Generalized Skew Derivations on Lie Ideals

Let R be a non-commutative prime ring, Z(R) its center, Q its right Martindale quotient ring, C its extended centroid, \(F\ne 0\) an b-generalized skew derivation of R, L a non-central Lie ideal of R, \(0\ne a\in R\) and \(n\ge 1\) a fixed integer. In this paper, we prove the following two results:

1. 1.
   If R has characteristic different from 2 and 3 and \(a[F(x),x]^n=0\), for all \(x\in L\), then either there exists an element \(\lambda \in C\), such that \(F(x)=\lambda x\), for all \(x\in R\) or R satisfies \(s_4(x_1,\ldots ,x_4)\), the standard identity of degree 4, and there exist \(\lambda \in C\) and \(b\in Q\), such that \(F(x)=bx+xb+\lambda x\), for all \(x\in R\).
    
2. 2.
   If \(\mathrm{{char}}(R)=0\) or \(\mathrm{{char}}(R) > n\) and \(a[F(x),x]^n\in Z(R)\), for all \(x\in R\), then either there exists an element \(\lambda \in C\), such that \(F(x)=\lambda x\), for all \(x\in R\) or R satisfies \(s_4(x_1,\ldots ,x_4)\).