On $${\mathcal {I}}_{

Tóth  János T.  Filip  Ferdinánd  Bukor  József  Zsilinszky  László 《Periodica Mathematica Hungarica》2021,82(2):125-135

Z∗$$ {\mathcal{Z}}^{\ast } $$-Semilocal Modules and the Proper Class ℛS$$ \mathrm{\mathcal{R}}\mathcal{S} $$

Ukrainian Mathematical Journal - Over an arbitrary ring, a module M is said to be $$ {\mathcal{Z}}^{\ast } $$-semilocal if every submodule U of M has a $$ {\mathcal{Z}}^{\ast } $$ -supplement V in...     

On Utiyama’s Theme Through “{\mathcal {A}} -Invariance”

The term ?? ${\mathcal {A}}$ -invariance?? refers to the invariance of our results, with respect to the ??arithmetic?? employed, viz. to an appropriate algebra sheaf ${\mathcal {A}}$ . This, combined with the categorical notion of ??adjunction??, particularized here with the homological, in nature Hom-? adjunction, affords the classical perspective of Utiyama, pertaining to the characteristic type of field interactions. Yet, all this, without any ??space-time?? support, in the classical sense of the term (: smooth manifolds), but, just based on the ??functorial?? character of ADG (acronym of ??Abstract Differential Geometry??) and the aforementioned two fundamental principles (:?? ${\mathcal {A}}$ -invariance?? and ??adjunction??).     

Uniqueness of the Group Measure Space Decomposition for Popa’s {\mathcal{HT}} Factors

We prove that if ${\Gamma\curvearrowright (X, \mu)}$ is a free ergodic rigid (in the sense of Popa in Ann Math 163:809–889, 2006) probability measure preserving action of a group Γ with positive first ${\ell^2}$ -Betti number, then the II₁ factor ${L^{\infty}(X)\rtimes\Gamma}$ has a unique group measure space Cartan subalgebra, up to unitary conjugacy. We deduce that many ${\mathcal{HT}}$ factors, including the II₁ factors associated with the usual actions ${\Gamma\curvearrowright \mathbb{T^2}}$ and ${\Gamma\curvearrowright}$ ${{\rm SL}_2(\mathbb R)/{\rm SL}_2(\mathbb Z)}$ , where Γ is a non-amenable subgroup of ${{\rm SL}_2(\mathbb Z)}$ , have a unique group measure space decomposition.     

Beurling’s theorem for SL(2,$${\mathbb{R}}$$)

We prove Beurling’s theorem for the full group SL(2,). This is the master theorem in the quantitative uncertainty principle as all the other theorems of this genre follow from it.     

Cartan-Dieudonné Theorem for {\mathcal {A}}-Modules

Like the classical Cartan-Dieudonné theorem, the sheaf-theoretic version shows that A{\mathcal {A}}-isometries on a convenient A{\mathcal {A}}-module E{\mathcal {E}} of rank n can be decomposed in at most n orthogonal symmetries (reflections) with respect to non-isotropic hyperplanes. However, the coefficient sheaf of \mathbb C{\mathbb {C}}-algebras A{\mathcal {A}} is assumed to be a PID \mathbb C{\mathbb {C}}-algebra sheaf and, if (E,f){(\mathcal {E},\phi)} is a pairing with f{\phi} a non-degenerate A{\mathcal {A}}-bilinear morphism, we assume that E{\mathcal {E}} has nowhere-zero (local) isotropic sections; but, for Riemannian sheaves of A{\mathcal {A}}-modules, this is not necessarily required.     

{mathcal {A}}-Transvections and Witt��s Theorem in Symplectic {mathcal {A}}-Modules

Building on prior joint work by Mallios and Ntumba, we study transvections (J. Dieudonné), a theme already important from the classical theory, in the realm of Abstract Geometric Algebra, referring herewith to symplectic A{mathcal A}-modules. A characterization of A{mathcal A}-transvections, in terms of A{mathcal A}-hyperplanes (Theorem 1.4), is given together with the associated matrix definition (Corollary 1.5). By taking the domain of coefficients A{mathcal A} to be a PID-algebra sheaf, we also consider the analogue of a form of the classical Witt’s extension theorem, concerning A{mathcal A}-symplectomorphisms defined on appropriate Lagrangian sub-A{mathcal A}-modules (Theorem 2.3 and 2.4).     

Perelman’s $${\bar{\lambda}}$$ -invariant and collapsing via geometric characteristic splittings

Any closed, orientable, smooth, nonpositively curved manifold M is known to admit a geometric characteristic splitting, analogous to the JSJ decomposition in three dimensions. We show that when this splitting consists of pieces which are Seifert fibered or pieces each of whose fundamental group has non-trivial centre, then M collapses with bounded curvature and has zero Perelman [`(l)]{\bar{\lambda}} -invariant.     

On the Hodge–Newton filtration for p-divisible {\mathcal{O}} -modules

The notions Hodge–Newton decomposition and Hodge–Newton filtration for F-crystals are due to Katz and generalize Messing’s result on the existence of the local-étale filtration for p-divisible groups. Recently, some of Katz’s classical results have been generalized by Kottwitz to the context of F-crystals with additional structures and by Moonen to μ-ordinary p-divisible groups. In this paper, we discuss further generalizations to the situation of crystals in characteristic p and of p-divisible groups with additional structure by endomorphisms.     

The Aronszajn–Donoghue Theory for Rank One Perturbations of the $$\mathcal{H}_{-2} {\text{-Class}}$$

A singular rank one perturbation of a self-adjoint operator A in a Hilbert space is considered, where and but with the usual A–scale of Hilbert spaces. A modified version of the Aronszajn-Krein formula is given. It has the form where F denotes the regularized Borel transform of the scalar spectral measure of A associated with . Using this formula we develop a variant of the well known Aronszajn–Donoghue spectral theory for a general rank one perturbation of the class.Submitted: March 14, 2002 Revised: December 15, 2002     

A note on κ-uniform points of G^{\mathcal{LUC}}

Let G be a Hausdorff, non-compact, locally compact topological group. We show that, for every infinite cardinal number κ with κ≤κ(G), the set of all κ-uniform points of $G^{\mathcal{LUC}}$ is a closed, two-sided ideal of $G^{\mathcal{LUC}}$ .     

Products of Functions in $$H^1_{\rho }({{\mathcal {X}}})$$ and $${\mathrm {BMO}}_{\rho }({{\mathcal {X}}})$$BMOρ(X) over RD-Spaces and Applications to Schrödinger Operators

Let \(({{\mathcal {X}}},d,\mu )\) be an RD-space, \(H^1_{\rho }({{\mathcal {X}}})\), and \({\mathrm {BMO}}_{\rho }({{\mathcal {X}}})\) be, respectively, the local Hardy space and the local BMO space associated with an admissible function \(\rho \). Under an additional assumption that there exists a specific generalized approximation of the identity, the authors prove that the product \(f\times g\) of \(f\in H^1_{\rho }({{\mathcal {X}}})\) and \(g\in {\mathrm {BMO}}_{\rho }({{\mathcal {X}}})\), viewed as a distribution, can be written into a sum of two bounded bilinear operators, respectively, from \(H^1_{\rho }({{\mathcal {X}}})\times {\mathrm {BMO}}_{\rho } ({{\mathcal {X}}})\) into \(L^1({{\mathcal {X}}})\) and from \(H^1_{\rho }({{\mathcal {X}}}) \times {\mathrm {BMO}}_{\rho } ({{\mathcal {X}}})\) into \(H^{\log }({{\mathcal {X}}})\), which is of wide generality. The authors also give out four applications of this result to Schrödinger operators, respectively, over different underlying spaces, where three of these applications are new.     

Generalization of Selberg’s $$ frac{3}{{16}} $$ theorem and affine sieve

An analogue of the well-known frac316 frac{3}{{16}} lower bound for the first eigenvalue of the Laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non-elementary subgroup L of SL(2,Z). The proof in the case that the Hausdorff of the limit set of L is bigger than frac12 frac{1}{2} is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting. In the case that delta is less than frac12 frac{1}{2} we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function. These “spectral gaps” are then applied to sieving problems on orbits of such groups.     

A new optimal quaternary sequence family of length 2(2 n − 1) obtained from the orthogonal transformation of Families {\mathcal{B}} and {\mathcal{C}}

In this paper, a general orthogonal transformation on the optimal quaternary sequence Families ${\mathcal{B}}$ and ${\mathcal{C}}$ is presented. Consequently, the known optimal Family ${\mathcal{D}}$ and a new optimal Family ${\mathcal{E}}$ are produced in a uniform method. In contrast to the known optimal Family ${\mathcal{D}}$ , the new Family ${\mathcal{E}}$ has the same parameters such as the sequence length 2(2 ⁿ ? 1), the family size 2 ⁿ , and the maximal nontrivial correlation value ${2^{\frac{n+1}{2}}+2}$ , where n is a positive integer, but with a different correlation function.     

On Nikol’skii Inequalities for Domains in $${\mathbb {R}}^d$$

Nikol’skii inequalities for various sets of functions, domains, and weights will be discussed. Much of the work is dedicated to the class of algebraic polynomials of total degree n on a bounded convex domain D. That is, we study \(\sigma := \sigma (D,d)\) for which

$$\begin{aligned} \Vert P\Vert _{L_q(D)}\le c n^{\sigma (\frac{1}{p}-\frac{1}{q})}\Vert P\Vert _{L_p(D)},\quad 0<p\le q\le \infty , \end{aligned}$$

where P is a polynomial of total degree n. We use geometric properties of the boundary of D to determine \(\sigma (D,d)\) with the aid of comparison between domains. Computing the asymptotics of the Christoffel function of various domains is crucial in our investigation. The methods will be illustrated by the numerous examples in which the optimal \(\sigma (D,d)\) will be computed explicitly.

     

$${mathcal {W}}_infty $$ W ∞ -transport with discrete target as a combinatorial matching problem

Archiv der Mathematik - In this short note, we show that given a cost function c, any coupling $$pi $$ of two probability measures where the second is a discrete measure can be associated to a...     

A proof of the Anderson–Badawi $$ \mathbf{rad}\varvec{(I)}^{\varvec{n}} \varvec{\subseteq } {\varvec{I}}$$ formula for $$\varvec{n}$$n-absorbing ideals

In [1], Anderson and Badawi conjectured that \(\mathrm{rad}(I)^n \subseteq I\) for every n-absorbing ideal I of a commutative ring. In this article, we prove their conjecture. We also prove related conjectures for radical ideals.     

$${\overline{1}}$$ 1 ¯ -Evaluating linear functionals on function spaces

Positivity - Let C(X) denotes the Riesz algebra of all real valued continuous functions on a Tychonoff space X, and let L be a vector subspace of C(X). A linear functional $$H:L\rightarrow {\mathbb...     

Lagrangian submanifolds in the homogeneous nearly Kähler $${\mathbb {S}}^3 \times {\mathbb {S}}^3$$

In this paper, we investigate Lagrangian submanifolds in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\). We introduce and make use of a triplet of angle functions to describe the geometry of a Lagrangian submanifold in \(\mathbb {S}^3 \times \mathbb {S}^3\). We construct a new example of a flat Lagrangian torus and give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature. As a corollary of our main result, we obtain that the radius of a round Lagrangian sphere in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\) can only be \(\frac{2}{\sqrt{3}}\) or \(\frac{4}{\sqrt{3}}\).