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...     

Covers ℘ for Abstract Regular Polytopes \mathcal {Q} such that \mathcal{Q}=\mathcal{P}/\mathbf{Z}_{p}^{k}

One key problem in the theory of abstract polytopes is the so-called amalgamation problem. In its most general form, this is the problem of characterising the polytopes with given facets  $\mathcal {K}$ and vertex figures ?. The first step in solving it for particular  $\mathcal{K}$ and ? is to find the universal such polytope, which covers all the others. This article explains a construction that may be attempted on an arbitrary polytope ?, which often yields an infinite family of finite polytopes covering ? and sharing its facets and vertex figures. The existence of such an infinite family proves that the universal polytope is infinite; alternatively, the construction can produce an explicit example of an infinite polytope of the desired type. An algorithm for attempting the construction is explained, along with sufficient conditions for it to work. The construction is applied to a few  $\mathcal{K}$ and ? for which it was previously not known whether or not the universal polytope was infinite, or for which only a finite number of finite polytopes was previously known. It is conjectured that the construction is quite broadly applicable.     

The Gieseker-Petri divisor in {\mathcal{M}_g} for g ≤ 13

The Gieseker-Petri locus GP _g is defined as the locus inside ${\mathcal{M}_g}$ consisting of curves which violate the Gieseker-Petri Theorem. It is known that GP _g has always some divisorial components and it has been conjectured that it is of pure codimension 1 inside ${\mathcal{M}_g}$ . We prove that this holds true for genus up to 13.     

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

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

An affine open covering of {\mathcal{M}_g} for g ≤ 5

We prove that the moduli space ${\mathcal{M}_g}$ of smooth curves of genus g is the union of g?1 affine open subsets for every g with 2 ?? g ?? 5, as predicted by an intriguing conjecture of Eduard Looijenga.     

$${\mathcal{L}_1}$$-spaces with the Radon-Nikodým property containing reflexive subspaces

We show that for every sequence \({(p_n)_{n\in\mathbb{N}}}\) with 1 ≤ p _n  ≤ 2 there exists an \({\mathcal{L}_1}\) -space with the Radon-Nikodým containing an isomorphic copy of \({\ell_1(\ell_{p_n})}\) .     

The best polynomial approximations of entire transcendental functions with generalized growth order in Banach spaces \mathcal{E}_p^{\prime}(G) and {\mathcal{E}_p}(G) , p ? 1

A theorem of the Hadamard type for entire transcendental functions f, which have a generalized ??-order of growth ?? _?? (f), has been obtained. This theorem connects the values $ \widetilde{M}\left( {f,r} \right)\;\left( {r > 1} \right) $ and the coefficients a _n (f) $ \left( {n \in {\mathbb{Z}_{+} }} \right) $ of the expansion of f in Faber series in a finite domain D whose boundary ?? belongs to the Al??per class. This result is the extension of a result obtained by M. N. Sheremeta onto a simply connected domain. The necessary and sufficient conditions for an analytic function $ f \in \mathcal{E}_p^{\prime}(G) $ or $ f \in {\mathcal{E}_p}(G)\;\left( {1 \leqslant p \leqslant \infty } \right) $ to be entire transcendental with a generalized ??-order of growth ?? _?? (f) are obtained. These conditions include the best polynomial approximations of the function f and determine the rate of their convergence to zero, as the degree of polynomials increases.     

The Best Constant in the Embedding of WN,1(ℝN) into L ∞ ( ℝ N ) $L^{\infty }(\mathbb {R} ^{N})$

Potential Analysis - We compute the best constant in the embedding of $W^{N,1}(\mathbb {R} ^{N})$ into $L^{\infty }(\mathbb {R} ^{N})$ , extending a result of Humbert and Nazaret in dimensions one...     

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.     

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     

System Representations for the Paley–Wiener Space $$\mathcal {PW}_{\pi }^2$$

In this paper we study the approximation of stable linear time-invariant systems for the Paley–Wiener space \(\mathcal {PW}_{\pi }^2\), i.e., the set of bandlimited functions with finite \(L^2\)-norm, by convolution sums. It is possible to use either, the convolution sum where the time variable is in the argument of the bandlimited impulse response, or the convolution sum where the time variable is in the argument of the function, as an approximation process. In addition to the pointwise and uniform convergence behavior, the convergence behavior in the norm of the considered function space, i.e. the \(L^2\)-norm in our case, is important. While it is well-known that both convolution sums converge uniformly on the whole real axis, the \(L^2\)-norm of the second convolution sum can be divergent for certain functions and systems. We show that the there exist an infinite dimensional closed subspace of functions and an infinite dimensional closed subspace of systems, such that for any pair of function and system from these two sets, we have norm divergence.     

与群PSL2(\mathcal{R})相关的交叉积{\mathcal{R}(\mathcal{A}},α)的一点注记

在上半复平面$\mathbb{H}$上给定双曲测度$dxdy/y^{2}$, 群$G={\rm PSL}_{2}(\mathbb{R})$ 在$\mathbb{H}$上的分式线性作用导出了$G$在Hilbert空间$L^{2}(\mathbb{H}, dxdy/y^{2})$上的酉表示$\alpha$. 证明了交叉积 $\mathcal{R}(\mathcal{A}, \alpha)$是$\mathrm{I}$型von Neumann代数, 其中$\mathcal{A}= \{M_{f}:f\in L^{\infty}(\mathbb{H},dxdy/y^{2} )\}$. 具体地, 交叉积代数$\mathcal{R}(\mathcal{A}, \alpha)$与von Neumann代数$\mathcal{B}(L^{2}(P, \nu))\overline{\otimes}\mathcal{L}_{K}$是*-同构的, 其中$\mathcal{L}_{K}$是$G$中子群 $K$的左正则表示生成的群von Neumann代数.     

Classical Limit of the Scaled Elliptic Algebra \mathcal{A} ħ,η (\widetilde{\mathfrak{s}\mathfrak{l}}_2)

The classical limit of the scaled elliptic algebra $\mathcal{A}$ ?,η ( $\widetilde{\mathfrak{s}\mathfrak{l}}_2$ ) is investigated. The limiting Lie algebra is described in two equivalent ways: as a central extension of the algebra of generalized automorphic sl2 valued functions on a strip and as an extended algebra of decreasing automorphic sl2 valued functions on the real line. A bialgebra structure and an infinite-dimensional representation in the Fock space are studied. The classical limit of elliptic algebra $\mathcal{A}$ q,p ( $\widetilde{\mathfrak{s}\mathfrak{l}}_2$ ) is also briefly presented.     

Метод многопараметрической интерполяции и теоремы вложения пространств БесоваB_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)

In this paper we determine the method of multi-parameter interpolation and the scales of Lebesgue spaces $B_{\vec p} \left[ {0,2\pi } \right)$ and Besov spaces $B_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)$ , which are generalizations of the Lorentz spacesL _pq [0, 2π) and Besov spacesB _pq ^α [0, 2π). We also prove imbedding theorems.     

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.     

Lattice-theoretic characterizations of the group classes {\mathfrak{N}^{k-1}\mathfrak{A}} and {\mathfrak{N}^k} for k?�� 2

Let ${2\leq k\in \mathbb{N}}$ . Recently, Costantini and Zacher obtained a lattice-theoretic characterization of the classes ${\mathfrak{N}^k}$ of finite soluble groups with nilpotent length at most k. It is the aim of this paper to give a lattice-theoretic characterization of the classes ${\mathfrak{N}^{k-1}\mathfrak{A}}$ of finite groups with commutator subgroup in ${\mathfrak{N}^{k-1}}$ ; in addition, our method also yields a new characterization of the classes ${\mathfrak{N}^k}$ . The main idea of our approach is to use two well-known theorems of Gaschütz on the Frattini and Fitting subgroups of finite groups.     

关于$\mathcal{F}$-S-可补子群

设$\mathcal{F}$是一个群类. 群$G$的子群$H$称为在$G$中$\mathcal{F}$-S-可补的，如果存在$G$的一个子群$K$,使得$G=HK$且$K/K\cap{H_G}\in\mathcal{F}$, 其中$H_G=\bigcap_{g\in G}H^g$是包含在$H$中的$G$的最大正规子群．本文利用子群的$\mathcal{F}$-S-可补性, 给出了有限群的可解性, 超可解性和幂零性的一些新的刻画. 应用这些结果, 我们可以得到一系列推论, 其中包括有关已知的著名结果.     

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.     

Subdirectly irreducible \mathcal{RGC} -commutative right H-semigroups

A semigroup S is said to be ℛ-commutative if, for all elements a,b∈S, there is an element x∈S ¹ such that ab=bax. A semigroup S is called a generalized conditionally commutative (briefly, -commutative) semigroup if it satisfies the identity aba ²=a ² ba. An ℛ-commutative and -commutative semigroup is called an -commutative semigroup. A semigroup S is said to be a right H-semigroup if every right congruence of S is a congruence of S. In this paper we characterize the subdirectly irreducible semigroups in the class of -commutative right H-semigroups. Research supported by the Hungarian NFSR grant No T029525.     

The variety generated by {\mathbb {A}(\mathcal {T})}– two counterexamples

We show that \({\mathcal {V}(\mathbb {A}(\mathcal {T}))}\) does not have definable principal subcongruences or bounded Maltsev depth. When the Turing machine \({\mathcal {T}}\) halts, \({\mathcal {V}(\mathbb {A}(\mathcal {T}))}\) is an example of a finitely generated semilattice based (and hence congruence \({\wedge}\)-semidistributive) variety with only finitely many subdirectly irreducible members, all finite. This is the first known example of a variety with these properties that does not have definable principal subcongruences or bounded Maltsev depth.