and Applied Analysis 3 of positive real numbers. By S 2 − X , we denote the space of all sequences defined over X, ‖·, ·‖ . Now we define the following sequence spaces: W ( M,Δ, p, ‖, ·, ‖ ⎧ ⎪⎨ ⎪⎩ x ∈ S 2 −X : ∀ε > 0 { n ∈ : ∞ ∑ k 1 ank [ M (∥ ∥ ∥ ∥ Δxk − L ρ , z ∥ ∥ ∥ ∥ )]pk ≥ ε } ∈ I for some ρ > 0, L ∈ X and each z ∈ X ⎫ ⎪⎬ ⎪⎭ , W 0 ( A,M,Δ, p, ‖, ·, ‖ ⎧ ⎪⎨ ⎪⎩ x ∈ S 2 −X : ∀ε > 0 { n ∈ : ∞ ∑...

Abstract Let $$w \in F_2$$ w ? F 2 be a word and let m n two positive integers. We say that finite group G has the $$w_{m,n}$$ m , n -property if however set M of...

Let X be a real linear space of vectors x with a norm ‖x‖X , W ⊂ X, W 6= ∅ and V ⊂ X, V 6= ∅. Let L be a subspace in X of dimension dim L ≤ n, n ≥ 0 and M = M(z) := z + L be a shift of the subspace L by an arbitrary vector z ∈ X. If M ∩ V 6= ∅, then we denote by E(x, M ∩ V )X := inf y∈Mn∩V ‖x− y‖X , the best approximation of the vector x ∈ X by M ∩ V , and by E(W,M ∩ V )X := sup x∈W E(x,M ∩ V )X ,

Orthogonal polynomials pn(W ; x) for exponential weights W 2 = e−2Q on a nite or in nite interval I , have been intensively studied in recent years. We discuss e orts of the authors to extend and unify some of the theory; our deepest result is the bound |pn(W ; x)|W (x)|(x − a−n)(x − an)|6C; x∈ I with C independent of n and x. Here a±n are the Mhaskar–Rahmanov–Sa numbers for Q and Q must satisf...

AbstractLet W be a non-empty subset of a free group. The automorphism of a group G is said to be a marginal automorphism, if for all x in G,x^−1alpha(x) in W^*(G), where W^*(G) is the marginal subgroup of G.In this paper, we give necessary and sufficient condition for a purelynon-abelian p-group G, such that the set of all marginal automorphismsof G forms an elementary abelian p-group.

Let the special linear group G := SL 2 act regularly on a Q-factorial variety X. Consider a maximal torus T ⊂ G and its normalizer N ⊂ G. We prove: If U ⊂ X is a maximal open N-invariant subset admitting a good quotient U → U/ /N with a divisorial quotient space, then the intersection W (U) of all translates g·U is open in X and admits a good quotient W (U) → W (U)/ /G with a divisorial quotien...

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A singular masa A in a II1 factor N is defined by the property that any unitary w ∈ N for which A = wAw∗ must lie in A. A strongly singular masa A is one that satisfies the inequality ‖EA − EwAw∗‖∞,2 ≥ ‖w − EA(w)‖2 for all unitaries w ∈ N , where EA is the conditional expectation of N onto A, and ‖ · ‖∞,2 is defined for bounded maps φ : N → N by sup{‖φ(x)‖2 : x ∈ N, ‖x‖ ≤ 1}. Strong singularity...

In this note, we establish $L^p$-bounds for the semigroup $e^{-tq^w(x,D)}$, $t \ge 0$, generated by a quadratic differential operator $q^w(x,D)$ on $\mathbb{R}^n$ that is Weyl quantization of complex-valued form $q$ defined phase space $\mathbb{R}^{2n}$ with non-negative real part $\operatorname{Re} q 0$ and trivial singular space. Specifically, show $e^{-tq^w(x,D)}$ bounded from $L^p(\mathbb{R...

A singular masa A in a II1 factor N is defined by the property that any unitary w ∈ N for which A = wAw∗ must lie in A. A strongly singular masa A is one that satisfies the inequality ‖EA − EwAw∗‖∞,2 ≥ ‖w − EA(w)‖2 for all unitaries w ∈ N , where EA is the conditional expectation of N onto A, and ‖ · ‖∞,2 is defined for bounded maps φ : N → N by sup{‖φ(x)‖2 : x ∈ N, ‖x‖ ≤ 1}. Strong singularity...