Über gerade unimodulare Gitter der Dimension 32, III

Let A_j, B_j be complex B-spaces, j = 0, 1, A_θ and B_θ–the complex-interpolation spaces generated by the couples (A₀, A₁) and (B₀, B₁), resp., by CALDERON'S/LIONS'S method. Let T: A₀ ∧ A₁ → B₀ → B₁ be an operator satisfying some conditions such as continuity, estimates etc. in terms of the norms of A_j, B_j (j = 0, 1). We consider the question which one of these properties is inherited to T when A₀ → A₁ and B₀ → B₁ are equipped with the norm of A_θ and B_θ.     

Radius preserving (semi)regularities in Banach algebras

Abstract

In this note we investigate regularities (semiregularities) R and S in a Banach algebra A satisfying S ? R and the corresponding spectra σ_S and σ_R Satisfying

sup {|λ| : λ ∈ σ_R(a)} = sup{|λ| : λ ∈ σ_S (a)}     

Basic Propositional Calculus I

We present an axiomatization for Basic Propositional Calculus BPC and give a completeness theorem for the class of transitive Kripke structures. We present several refinements, including a completeness theorem for irreflexive trees. The class of intermediate logics includes two maximal nodes, one being Classical Propositional Calculus CPC, the other being E₁, a theory axiomatized by T → ⊥. The intersection CPC ∩ E₁ is axiomatizable by the Principle of the Excluded Middle A V ∨ ?A. If B is a formula such that (T → B) → B is not derivable, then the lattice of formulas built from one propositional variable p using only the binary connectives, is isomorphically preserved if B is substituted for p. A formula (T → B) → B is derivable exactly when B is provably equivalent to a formula of the form ((T → A) → A) → (T → A).     

Characterization of varietal covering and stem groups

Let A _m be the variety of abelian groups of exponent m, where mis a square free positive integer. It is shown that every group G has an A _m -covering group, which generalizes the work of I. Schur (1904) and M.R. Jones (1973). Also, similar to the work of Yamazaki (1964) in the variety of abelian groups, if 1→ A → H → G → 1 is an A _m -stem extension, then we show that H is a homomorphic image of an A _m -covering group of G. Finally,in this variety some results of Hall's type will be proved.     

Continuous Linear Extension of Ultradifferentiable Functions of Beurling Type

For a weight function ω and a closed set A ? ?^N let ?_(ω)(A) denote the space of all ω-Whitney jets of Beurling type on A. It is shown that for each closed set A ? ?^N there exists an ω-extension operator EA: ?_(ω)(A) → ?_(ω)(?^N) if and only if ω is a (DN)-function (see MEISE and TAYLOR [18], 3.3). Moreover for a fixed compact set K ? ?^N there exists an ω-extension operator E_K: ?_(ω)(K) → ?_(ω)(?^N) if and only if the Fréchet space ?_(ω)(K) satisfies the property (DN) (see Vogt [29], 1.1.).     

Simplicity of Skew Group Rings of Abelian Groups

Necessary and sufficient conditions for simplicity of a general skew group ring A ?_σ G are not known. In this article, we show that a skew group ring A ?_σ G, of an abelian group G, is simple if and only if its centre is a field and A is G-simple. As an application, we show that a transformation group (X, G), where X is a compact Hausdorff space acted upon by an abelian group G, is minimal and faithful if and only if its associated skew group algebra C(X) ?_σ G is simple.     

Bilinear isometries on subspaces of continuous functions

Let A and B be strongly separating linear subspaces of C₀(X) and C₀(Y), respectively, and assume that ?A ≠ ?? (?A stands for the set of generalized peak points for A) and ?B ≠ ??. Let T: A × B → C₀(Z) be a bilinear isometry. Then there exist a nonempty subset Z₀ of Z, a surjective continuous mapping h: Z₀ → ?A × ?B and a norm‐one continuous function a: Z₀ → K such that T (f, g)(z) = a (z)f (π_x (h (z))g (π_y (h (z)) for all z ∈ Z₀ and every pair (f, g) ∈ A × B. These results can be applied, for example, to non‐unital function algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The finite simple groups with at most two fusion classes of every order

Elements a,b of a group G are said to be fused if a = b^σ and to be inverse-fused if a =(b^-1)^σ for some σ ? Aut(G). The fusion class of a ? G is the set {a^σ | σ ? Aut(G)}, and it is called a fusion class of order i if a has order iThis paper gives a complete classification of the finite nonabelian simple groups G for which either (i) or (ii) holds, where:

(i) G has at most two fusion classes of order i for every i (23 examples); and

(ii) any two elements of G of the same order are fused or inversenfused.

The examples in case (ii) are: A₅, A₆,L₂(7),L₂(8), L₃(4), Sz(8), M₁₁ and M₂₃An application is given concerning isomorphisms of Cay ley graphs.     

A Tauberian theorem related to Weyl's criterion

Let {x_n} be a sequence of real numbers and let a(n) be a sequence of positive real numbers, with A(N) = Σ_n=1^Na(n). Tsuji has defined a notion of a(n)-uniform distribution mod 1 which is related to the problem of determining those real numbers t₀ for which A(N)^?1 Σ_n=1^Na(n)e^?it₀x_n → 0 as N → ∞. In case f(s) = Σ_n=1^∞a(n)e^?sx_n, s = σ + it, is analytic in the right half-plane 0 < σ, and satisfies a certain smoothness condition as σ → 0 +, we show that f(σ)^?1f(σ + it₀) → 0 as σ → 0 + if and only if A(N)^?1 Σ_n=1^Na(n)e^?it₀x_n → 0 as N → ∞.     

Sharp Estimate of Global Coulomb Gauge

Let A be a W^{1, 2} -connection on a principal SU(2) -bundle P over a compact 4 -manifold M whose curvature F_A satisfies . Our main result is the existence of a global section σ : M → P with finite singularities on M such that the connection form σ^*A satisfies the Coulomb equation d^*(σ^*A) = 0 and admits a sharp estimate . Here ℒ^{4, ∞} is a new function space we introduce in this paper that satisfies L⁴(M) ⊊ ℒ^{4, ∞}(M) ⊊ L^{4 − ϵ}(M) for all ϵ > 0 . More precisely, ℒ^{4, ∞}(M) is the collection of measurable function u such that , where L^{4, ∞} is the classical Lorentz space and s_u is the L⁴ -integrability radius function associated to u defined by Briefly speaking, we achieve the estimate of by showing that σ^*A is effectively L⁴ -integrable away from controllably many points on M . © 2020 Wiley Periodicals LLC     

On finite groups with generalized σ-subnormal Schmidt subgroups

Let G be a finite group and σ?=?{σ_i|i∈I} some partition of the set of all primes. A subgroup A of G is said to be generalized σ-subnormal in G if A?=??L,T?, where L is a modular subgroup and T is a σ-subnormal subgroup of G. In this paper, we prove that if every Schmidt subgroup of G is generalized σ-subnormal in G, then the commutator subgroup G^′ of G is σ-nilpotent.     

On the condition numbers associated with the polar factorization of a matrix

We are interested in the calculation of explicit formulae for the condition numbers of the two factors of the polar decomposition of a full rank real or complex m × n matrix A, where m ≥ n. We use a unified presentation that enables us to compute such condition numbers in the Frobenius norm, in cases where A is a square or a rectangular matrix subjected to real or complex perturbations. We denote by σ₁ (respectively σ _n) the largest (respectively smallest) singular value of A, and by K(A) = σ₁/σ _n the generalized condition number of A. Our main results are that the absolute condition number of the Hermitian polar factor is √2(1 + K(A)²)^1/2/(1 + K(A)) and that the absolute condition number of the unitary factor of a rectangular matrix is 1/σ _n. Copyright © 2000 John Wiley & Sons, Ltd.     

On Weak Crossed Products of Weak Hopf Algebras

Let H be a weak Hopf algebra in the sense of Böhm et al. (J Algebra 221:385–438, 1999) measuring an algebra A. Let A# _σ H be a weak crossed product with σ invertible. Then in this paper we first give some conditions for A# _σ H to be a weak Hopf algebra. Next the spectral sequence for Ext will be constructed which yields an estimate for the global dimension of A# _σ H in terms of the corresponding data for H and A. Furthermore, we will investigate when A???A# _σ H becomes a separable extension. Finally, we prove that if H and its dual H* are both semisimple, then the finitistic dimension of A# _σ H is equal to that of A.     

MULTIPLICATIVELY CLOSED SETS OF IDEALS AND RESIDUAL DIVISION

Let R be a ring (commutative with identity), let Γ be a multiplicatively closed set of finitely generated nonzero ideals of R, for an ideal I of R let I _Γ = ∪ {I : G; G ∈ Γ}, and for an R-algebra A such that GA ≠ (0) for all G ∈ Γ let A ^Γ = ∪ {A : _T GA; G ∈ Γ}, where T is the total quotient ring of A. Then I _Γ is an ideal in R, I → I _Γ is a semi-prime operation (on the set I of ideals I of R) that satisfies a cancellation law, and it is a prime operation on I if and only if R = R ^Γ. Also, A ^Γ is an R-algebra and A → A ^Γ is a closure operation on any set A = {A; A is an R-algebra, R ? A, and if C is a ring between R and A, then regular elements in C remain regular in A}. Finally, several results are proved concerning relations between the ideals I _Γ and (IA)_ΓA and between the R-algebras R ^Γ and A ^Γ.

     

Toeplitz matrices and determinants with Fisher-Hartwig symbols

If A and B are self-adjoint operators, this paper shows that A and B have order isomorphic invariant subspace lattices if and only if there are Borel subsets E and F of σ(A) and σ(B), respectively, whose complements have spectral measure zero, and there is a bijective function φ: E → F such that (i) Δ is a Borel subset of E if and only if φ(Δ) is a Borel subset of F; (ii) a Borel subset Δ of E has A-spectral measure zero if and only if φ(Δ) has B-spectral measure zero; (iii) B is unitarily equivalent to φ(A). If A is any self-adjoint operator, there is an associated function κ_A : $\text{N}$ ∪ {∞} → ($\text{N}$ ∪ {0, ∞}) × {0,1} defined in this paper. If $\text{F}$ denotes the collection of all functions from $\text{N}$ ∪ {∞} into ($\text{N}$ ∪ {0,∞}) × {0,1}, then $\text{F}$ is a parameter space for the isomorphism classes of the invariant subspace lattices of self-adjoint operators. That is, two self-adjoint operators A and B have isomorphic invariant subspace lattices if and only if κ_A = κ_B. The paper ends with some comments on the corresponding problem for normal operators.     

A note about topologically transitive cylindrical cascades

Letσ: Σ → Σ be a topologically mixing shift of finite type. Letβ: Σ → ? be a continuous function, and σ _β be the skew-product ofσ byβ. Assume that σ _β has a positive semi-orbit that reaches any upper height, and any lower height. Then, arbitrarilyC ⁰-close toβ there exists a Hölder mapβ′: Σ → ? such that the skew-product $\sigma _{\beta ^\prime } $ ofσ byβ′ is topologically transitive.     

Extendible elements of the alternating groups

An element?σ?of A_n , the Alternating group of degree n, is extendible in S_n , the Symmetric group of degree n, if there exists a subgroup H of S_n but not A_n whose intersection with A_n is the cyclic group generated by σ. A simple number-theoretic criterion, in terms of the cycle-decomposition, for an element of A_n to be extendible in S_n is given here.     

Generators of C0‐groups with smallest spectra

Let T = {T (t)}_{t ∈?} be a C₀‐group on a Banach space X with generator A. Under what conditions the assumption σ (A) = {0} implies that A = 0? This is called “A = 0” problem. In this paper we present some results related to this problem. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The expansion of a semigroup and a Riesz basis criterion

Problems on the expansion of a semigroup and a criterion for being a Riesz basis are discussed in the present paper. Suppose that A is the generator of a C₀ semigroup on a Hilbert space and σ(A)=σ₁(A)∪σ₂(A) with σ₂(A) is consisted of isolated eigenvalues distributed in a vertical strip. It is proved that if σ₂(A) is separated and for each λ∈σ₂(A), the dimension of its root subspace is uniformly bounded, then the generalized eigenvectors associated with σ₂(A) form an L-basis. Under different conditions on the Riesz projection, the expansion of a semigroup is studied. In particular, a simple criterion for the generalized eigenvectors forming a Riesz basis is given. As an application, a heat exchanger problem with boundary feedback is investigated. It is proved that the heat exchanger system is a Riesz system in a suitable state Hilbert space.