Rigidity theorems for complete $$\lambda $$ λ -hypersurfaces

Archiv der Mathematik - In this article, we study hypersurfaces $$\Sigma \subset {\mathbb {R}}^{n+1}$$ with constant weighted mean curvature, also known as $$\lambda $$ -hypersurfaces. Recently,...     

Restriction semigroups and $$\lambda $$-Zappa-Szép products

The aim of this paper is to study \(\lambda \)-semidirect and \(\lambda \)-Zappa-Szép products of restriction semigroups. The former concept was introduced for inverse semigroups by Billhardt, and has been extended to some classes of left restriction semigroups. The latter was introduced, again in the inverse case, by Gilbert and Wazzan. We unify these concepts by considering what we name the scaffold of a Zappa-Szép product \(S\bowtie T\) where S and T are restriction. Under certain conditions this scaffold becomes a category. If one action is trivial, or if S is a semilattice and T a monoid, the scaffold may be ordered so that it becomes an inductive category. A standard technique, developed by Lawson and based on the Ehresmann-Schein-Nambooripad result for inverse semigroups, allows us to define a product on our category. We thus obtain restriction semigroups that are \(\lambda \)-semidirect products and \(\lambda \)-Zappa-Szép products, extending the work of Billhardt and of Gilbert and Wazzan. Finally, we explicate the internal structure of \(\lambda \)-semidirect products.     

On difference sets with small $$\lambda $$ λ

Journal of Algebraic Combinatorics - In a 1989 paper, Arasu (Arch Math 53:622–624, 1989) used an observation about multipliers to show that no (352, 27, 2) difference set...     

Distribution of Artin–Schreier extensions

The article at hand contains exact asymptotic formulas for the distribution of conductors of elementary abelian p-extensions of global function fields of characteristic p. As a consequence for the distribution of discriminants, this leads to an exact asymptotic formula for simple cyclic extensions and an interesting lower bound for noncyclic elementary abelian extensions.     

$$I_0$$ and combinatorics at $$\lambda ^+$$λ+

We investigate the compatibility of \(I_0\) with various combinatorial principles at \(\lambda ^+\), which include the existence of \(\lambda ^+\)-Aronszajn trees, square principles at \(\lambda \), the existence of good scales at \(\lambda \), stationary reflections for subsets of \(\lambda ^{+}\), diamond principles at \(\lambda \) and the singular cardinal hypothesis at \(\lambda \). We also discuss whether these principles can hold in \(L(V_{\lambda +1})\).     

Semidirect products and wreath products of strongly π-inverse monoids

In this paper we determine the necessary and sufficient conditions for the semidirect products and the wreath products of two monoids to be strongly -inverse. Furthermore, we determine the least group congruence on a strongly -inverse monoid, and we give some important isomorphism theorems.     

Bruck–Reilly extensions of direct products of monoids and completely (0-)simple semigroups

We say that a class of monoids satisfies the property ℘ if every monoid in that class that admits a finitely presented Bruck–Reilly extension is finitely generated. We show that completely (0-)simple semigroups satisfy ℘, and that the direct product of two monoids in a class that satisfy ℘ also satisfies ℘ subject to a certain condition on the endomorphisms of the direct product. As a consequence of this result we obtain a new class of bands and a new class of completely regular semigroups that satisfy property ℘.     

Artin–Schreier extensions of the rational function field

In this paper we study the arithmetic of Artin–Schreier extensions of $\mathbb {F}_{q}(T)$ . We determine the integral closure of $\mathbb {F}_{q}[T]$ in Artin–Schreier extension of $\mathbb {F}_{q}(T)$ . We also investigate the average values of the $L$ -functions of orders of Artin–Schreier extensions and study the average values of ideal class numbers when $p=3$ in detail.     

Complete classification of finite semigroups for which the inverse monoid of local automorphisms is a $$\varDelta $$ Δ -semigroup

Semigroup Forum - A semigroup S is called a $$\varDelta $$ -semigroup if the lattice of its congruences forms a chain relative to the inclusion. A local automorphism of a semigroup S is defined as...     

Quantitative weakly compact sets and Banach-Saks sets in $$\ell _1$$ ℓ 1

Archiv der Mathematik - In this paper, we show a quantitative version of the theorem stating that relatively weakly compact sets in $$\ell _1$$ coincide with those having the Banach-Saks property....     

On $$\gamma $$ γ - and local $$\gamma $$ γ -vectors of the interval subdivision

Journal of Algebraic Combinatorics - We show that the $$\gamma $$ -vector of the interval subdivision of a simplicial complex with a nonnegative and symmetric h-vector is nonnegative. In...     

Spectra of elements in operator space tensor products of $$\hbox {C}^*$$ C ∗ -algebras

Positivity - Given two operator spaces E and F, injectivity of the canonical map from the $$\lambda $$ -tensor product $$E\bigotimes ^\lambda F$$ into the operator space injective tensor product...     

Hidden q-analogues of Ramanujan-like $$\pi $$ π -series

The Ramanujan Journal - By examining known q-series identities, we derive q-analogues of several $$\pi $$ -related infinite series, including some of Ramanujan’s series for $$1/\pi $$ .     

Infinite time extensions of Kleene’s $${\mathcal{O}}$$

Using infinite time Turing machines we define two successive extensions of Kleene’s O{\mathcal{O}} and characterize both their height and their complexity. Specifically, we first prove that the one extension—which we will call O⁺{\mathcal{O}^{+}}—has height equal to the supremum of the writable ordinals, and that the other extension—which we will call O⁺⁺{\mathcal{O}}^{++}—has height equal to the supremum of the eventually writable ordinals. Next we prove that O⁺{\mathcal{O}^+} is Turing computably isomorphic to the halting problem of infinite time Turing computability, and that O⁺⁺{\mathcal{O}^{++}} is Turing computably isomorphic to the halting problem of eventual computability.     

Schreier theory for singular extensions of categorical groups and homotopy classification∗ ∗

If G is a categorical group, a G-module is defined to be a braided categorical group (A c) together with an action of G on (A,c). In this work we define the notions of singular extension of G by the G-module (A,c) and of 1-cocycle of G with coefficients in (A,c) and we obtain, first, a bijection between the set of equivalence classes of singular extensions of G by (Ac) and the set of equivalence classes of 1-cocycles. Next, we associate to any G-module (Ac) a Kan fibration of simplicial sets ?: Ner(GAc)) → Ner(G)and then we show that there is a bijection between the set of equivalence classes of singular extensions of G by (A,c) and Γ[Ner(G,A,c/Ner(G)]the set of fibre homotopy classes of cross-sections of the fibration ?.     

On the Study of Processes of $$\sum (H)$$ and $$\sum _\mathrm{s}(H)$$∑s(H) Classes

In papers by Yor, a remarkable class \(({\varSigma })\) of submartingales is introduced, which, up to technicalities, are submartingales \((X_{t})_{t\ge 0}\) whose increasing process is carried by the times t such that \(X_{t}=0\). These submartingales have several applications in stochastic analysis: for example, the resolution of Skorokhod embedding problem, the study of Brownian local times and the study of zeros of continuous martingales. The submartingales of class \(({\varSigma })\) have been extensively studied in a series of articles by Nikeghbali (part of them in collaboration with Najnudel, some others with Cheridito and Platen). On the other hand, stochastic calculus has been extended to signed measures by Ruiz de Chavez (Le théorème de Paul Lévy pour des mesures signées. Séminaire de probabilités (Strasbourg). Springer, Berlin, 1984) and Beghdadi-Sakrani (Calcul stochastique pour les mesures signées. Séminaire de probabilités (Strasbourg). Springer, Berlin, 2003). In Eyi Obiang et al. (Stochastics 86(1):70–86, 2014), the authors of the present paper have extended the notion of submartingales of class \(({\varSigma })\) to the setting of Ruiz de Chavez (1984) and Beghdadi-Sakrani (2003), giving two different classes of stochastic processes named classes \(\sum (H)\) and \(\sum _\mathrm{s}(H)\) where from tools of the theory of stochastic calculus for signed measures, the authors provide general frameworks and methods for dealing with processes of these classes. In this work, we first give some formulas of multiplicative decomposition for processes of these classes. Afterward, we shall establish some representation results allowing to recover any process of one of these classes from its final value and the last time it visited the origin.