Small into-isomorphism from L∞ (A, μ) into L∞(B, υ)

In this paper we shall assert that if T is an isomorphism of L_∞(Ω₁, A, μ) into L_∞(Ω₂, B, υ) satisfying the condition ‖T‖·‖T ^?1‖?1+? for ?∈ $\left( {0,\frac{1}{5}} \right)$ , then $\frac{T}{{\parallel T\parallel }}$ is close to an isometry with an error less than 6ε in some conditions.     

On products of <Emphasis Type="Italic">k</Emphasis> atoms

Let H be an atomic monoid. For let denote the set of all with the following property: There exist atoms (irreducible elements) u ₁, …, u _k , v ₁, …, v _m ∈ H with u ₁· … · u _k = v ₁ · … · v _m . We show that for a large class of noetherian domains satisfying some natural finiteness conditions, the sets are almost arithmetical progressions. Suppose that H is a Krull monoid with finite cyclic class group G such that every class contains a prime (this includes the multiplicative monoids of rings of integers of algebraic number fields). We show that, for every , max which settles Problem 38 in [4]. Authors’ addresses: W. Gao, Center for Combinatorics, Nankai University, Tianjin 300071, P.R. China; A. Geroldinger, Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universit?t Graz, Heinrichstra?e 36, 8010 Graz, Austria     

Interassociates of the Free Commutative Semigroup on n Generators

The interassociates of the free commutative semigroup on n generators, for n > 1, are identified. For fixed n, let (S, ·) denote this semigroup. We show that every interassociate can be written in the form , depending only on a n-tuple . Next, if and are isomorphic interassociates of (S, ·) such that , for x_ii and x_j in the generating set of S, then . Moreover, if and only if is a permutation of .     

Toeplitz condition numbers as an <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> interpolation problem

The condition numbers CN(T) = ∥T∥ · ∥T⁻¹∥ of Toeplitz and analytic n × n matrices T are studied. It is shown that the supremum of CN(T) over all such matrices with ∥T∥ ≤ 1 and the given minimum of eigenvalues r = min |λ_i| > 0 behaves as the corresponding supremum over all n × n matrices (i.e., as (Kronecker)), and this equivalence is uniform in n and r. The proof is based on a use of the Sarason-Sz.Nagy-Foias commutant lifting theorem. Bibliography: 2 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 173–179.     

On <Emphasis Type="Italic">N</Emphasis>-Summations,II

Given any R-semimodule M equipped with a semitopology we construct an N-protosummation for M. If satisfies certain properties, then a similar construction leads to an unconditional N-summation for M, that is an N-summation for M equipped with the trivial prenorm MD over the N-summation (D^N,_D) for D. Conversely any N-protosummation on M gives rise to a topology . If both and satisfy a certain separation property, then and form a Galois connection. Dedicated to my friend and collegue Nico Pumplün on the occasion of his 70th birthdayMathematics Subject Classifications (2000) 16Y60, 54A05.     

Extremes of Shepp statistics for the Wiener process

Define , where W(·) is a standard Wiener process. We study the maximum of Y up to time T: and de termine an asymptotic expression for when u→ ∞. Further we establish the limiting Gumbel distribution of M _T when T→ ∞ and present the corresponding normalization sequence.      

Relative widths of function classes of L 2(T) defined by a linear differential operator in L q (T)

In this paper, the smallest number M which makes the equality $$ K_n (W_2^{L_r } (T),MW_2^{L_r } (T),L_2 (T)) = d_n (W_2^{L_r } (T),L_2 (T)) $$ valid, is established and the asymptotic order of $$ K_n (W_2^{L_r } (T),W_2^{L_r } (T),L_q (T)),1 \leqslant q \leqslant \infty $$ , is obtained, where $ W_2^{L_r } $ (T) is a periodic smooth function class which is determined by a linear differential operator, K _n (·, ·, ·) and d _n (·, ·) are the relative width and the width in the sense of Kolmogorov, respectively.     

Constructing metrics on a 2-torus with a partially prescribed stable norm

A result of Bangert states that the stable norm associated to any Riemannian metric on the 2-torus T ² is strictly convex. We demonstrate that the space of stable norms associated to metrics on T ² forms a proper dense subset of the space of strictly convex norms on ${{\mathbb R}^2}$ . In particular, given a strictly convex norm || · ||_∞ on ${{\mathbb R}^2}$ we construct a sequence ${\langle {\| \cdot \|}_j \rangle_{j=1}^{\infty}}$ of stable norms that converge to || · ||_∞ in the topology of compact convergence and have the property that for each r > 0 there is an ${N \equiv N(r)}$ such that || · || _j agrees with || · ||_∞ on ${{\mathbb Z}^2 \cap \{(a,b) : a^2 + b^2 \leq r \}}$ for all j ≥ N. Using this result, we are able to derive results on multiplicities which arise in the minimum length spectrum of 2-tori and in the simple length spectrum of hyperbolic tori.     

Old and New Morrey Spaces with Heat Kernel Bounds

Given p ∈ [1,∞) and λ ∈ (0, n), we study Morrey space of all locally integrable complex-valued functions f on such that for every open Euclidean ball B ⊂ with radius r_B there are numbers C = C(f ) (depending on f ) and c = c(f,B) (relying upon f and B) satisfying

Invariant subspaces for operators in a general II<Subscript>1</Subscript>-factor

Let ℳ be a von Neumann factor of type II₁ with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ _T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace affiliated with ℳ, such that the Brown measure of is concentrated on B and the Brown measure of is concentrated on ℂ∖B. Moreover, is T-hyperinvariant and the trace of is equal to μ _T(B). In particular, if T∈ℳ has a Brown measure which is not concentrated on a singleton, then there exists a non-trivial, closed, T-hyperinvariant subspace. Furthermore, it is shown that for every T∈ℳ the limit exists in the strong operator topology, and the projection onto is equal to 1_[0,r](A), for every r>0. Supported by The Danish National Research Foundation.     

Some remarks on spectral representation for symmetric stable processes

For the symmetric α-stable stochastic process X={X_t∶t∈T} with reproducing kernel space H(X) ? L_α constructed in § 1 we define the following parameters: $\alpha _0 = \sup {\mathbf{ }}\{ \beta \in (0.2]:{\mathbf{ }}\mathcal{H}\mathcal{X}$ embeds isometrically into some L^β}, containsl _β ⁿ 's uniformly}. In §2 we show that for α₀ > α the stochastic process X admits the representation $$X_t = \smallint Y_t (w){\mathbf{ }}Z_\alpha (dw),{\mathbf{ }}t \in T,$$ where {Y_t∶t∈T} itself is a symmetric stable process and Z_α is a symmetric α-stable independently scattered random measure. We show also how some properties of the stochastic process {X_t∶t∈T} depend on the corresponding properties of the process {Y_t∶t∈T}.     

Extinction profile of the logarithmic diffusion equation

Let ${N \geq 3}$ and u be the solution of u _t = Δ log u in ${\mathbb{R}^N \times (0, T)}$ with initial value u ₀ satisfying ${B_{k_1}(x, 0) \leq u_{0} \leq B_{k_2}(x, 0)}$ for some constants k ₁ > k ₂ > 0 where ${B_k(x, t) = 2(N - 2)(T - t)_{+}^{N/(N - 2)}/(k + (T - t)_{+}^{2/(N - 2)}|x|^{2})}$ is the Barenblatt solution for the equation and ${u_0 - B_{k_0} \in L^{1}(\mathbb{R}^{N})}$ for some constant k ₀ > 0 if ${N \geq 4}$ . We give a new different proof on the uniform convergence and ${L^1(\mathbb{R}^N)}$ convergence of the rescaled function ${\tilde{u}(x, s) = (T - t)^{-N/(N - 2)}u(x/(T - t)^{-1/(N - 2)}, t), s = -{\rm log}(T - t)}$ , on ${\mathbb{R}^N}$ to the rescaled Barenblatt solution ${\tilde{B}_{k_0}(x) = 2(N - 2)/(k_0 + |x|^{2})}$ for some k ₀ > 0 as ${s \rightarrow \infty}$ . When ${N \geq 4, 0 \leq u_0(x) \leq B_{k_0}(x, 0)}$ in ${\mathbb{R}^N}$ , and ${|u_0(x) - B_{k_0}(x, 0)| \leq f \in L^{1}(\mathbb{R}^{N})}$ for some constant k ₀ > 0 and some radially symmetric function f, we also prove uniform convergence and convergence in some weighted L ¹ space in ${\mathbb{R}^N}$ of the rescaled solution ${\tilde{u}(x, s)}$ to ${\tilde{B}_{k_0}(x)}$ as ${s \rightarrow \infty}$ .     

Dimension of the Torelli group for Out(F<Subscript>n</Subscript>)

Let be the kernel of the natural map Out(F_n)→GL_n(ℤ). We use combinatorial Morse theory to prove that has an Eilenberg–MacLane space which is (2n-4)-dimensional and that is not finitely generated (n≥3). In particular, this shows that the cohomological dimension of is equal to 2n-4 and recovers the result of Krstić–McCool that is not finitely presented. We also give a new proof of the fact, due to Magnus, that is finitely generated.     

Scattering Matrix for Magnetic Potentials with Coulomb Decay at Infinity

We consider the Schrödinger operator H in the space $ L_{2}(\mathbb{R}^{d})$ with a magnetic potential A(x) decaying as $ \vert x\vert^{-1} $ at infinity and satisfying the transversal gauge condition <A(x), x > = 0. Our goal is to study properties of the scattering matrix S() associated to the operator H. In particular, we find the essential spectrum _ess of S() in terms of the behaviour of A(x) at infinity. It turns out that _ess(S()) is normally a rich subset of the unit circle $\mathbb{T}$ or even coincides with $\mathbb{T}$. We find also the diagonal singularity of the scattering amplitude (of the kernel of S() regarded as an integral operator). In general, the singular part S₀ of the scattering matrix is a sum of a multiplication operator and of a singular integral operator. However, if the magnetic field decreases faster than $ \vert x\vert^{-1} $ for d 3 (and the total magnetic flux is an integer times 2 for dd = 2), then this singular integral operator disappears. In this case the scattering amplitude has only a weak singularity (the diagonal Dirac function is neglected) in the forward direction and hence scattering is essentially of short-range nature. Moreover, we show that, under such assumptions, the absolutely continuous parts of the operators S() and S₀ are unitarily equivalent. An important point of our approach is that we consider S() as a pseudodifferential operator on the unit sphere and find an explicit expression of its principal symbol in terms of A(x). Another ingredient is an extensive use (for d 3) of a special gauge adapted to a magnetic potential A(x).     

Trivectors yielding spreads in PG(5, 2)

Given an alternating trilinear form ${T\in {\rm Alt}(\times^{3}V_{6})}$ on V ₆ = V(6, 2) let ${\mathcal{L}_{T}}$ denote the set of those lines ${\langle a, b \rangle}$ in ${{\rm PG}(5,2)=\mathbb{P}V_{6}}$ which are T-singular, satisfying, that is, T(a, b, x) = 0 for all ${x\in {\rm PG}(5, 2).}$ If ${\mathcal{L}_{21}}$ is a Desarguesian line-spread in PG(5, 2) it is shown that ${\mathcal{L}_{T}=\mathcal{L}_{21}}$ for precisely three choices T ₁,T ₂,T ₃ of T, which moreover satisfy T ₁ + T ₂ + T ₃ = 0. For ${T\in\mathcal{T}:=\{T_{1},T_{2},T_{3}\}}$ the ${\mathcal{G}_{T}}$ -orbits of flats in PG(5, 2) are determined, where ${\mathcal{G}_{T}\cong {\rm SL}(3,4).2}$ denotes the stabilizer of T under the action of GL(6, 2). Further, for a representative U of each ${\mathcal{G}_{T}}$ -orbit, the T-associate U ^# is also determined, where by definition $$U^{\#}=\{v\in {\rm PG}(5,2)\, |\, T(u_{1},u_{2},v) = 0\, \,{\rm for\,all }\, \, u_{1},u_{2}\in U\}$$ .     

Estimations of Solutions of the Sturm– Liouville Equation with Respect to a Spectral Parameter

This paper is concerned with estimations of solutions of the Sturm–Liouville equation $$\big(p(x)y'(x)\big)'+\Big(\mu^2 -2i\mu d(x)-q(x)\Big)\rho(x)y(x)=0, \ \ x\in[0,1],$$ ( p ( x ) y ' ( x ) ) ' + ( μ 2 - 2 i μ d ( x ) - q ( x ) ) ρ ( x ) y ( x ) = 0 , x ∈ [ 0 , 1 ] , where ${\mu\in\mathbb{C}}$ μ ∈ C is a spectral parameter. We assume that the strictly positive function ${\rho\in L_{\infty}[0,1]}$ ρ ∈ L ∞ [ 0 , 1 ] is of bounded variation, ${p\in W^1_1[0,1]}$ p ∈ W 1 1 [ 0 , 1 ] is also strictly positive, while ${d\in L_1[0,1]}$ d ∈ L 1 [ 0 , 1 ] and ${q\in L_1[0,1]}$ q ∈ L 1 [ 0 , 1 ] are real functions. The main result states that for any r > 0 there exists a constant c _r such that for any solution y of the Sturm–Liouville equation with μ satisfying ${|{\rm Im}\, \mu|\leq r}$ | Im μ | ≤ r , the inequality ${\|y(\cdot,\mu)\|_C\leq c_r\|y(\cdot,\mu)\|_{L_1}}$ ∥ y ( · , μ ) ∥ C ≤ c r ∥ y ( · , μ ) ∥ L 1 is true. We apply our results to a problem of vibrations of an inhomogeneous string of length one with damping, modulus of elasticity and potential, rewritten in an operator form. As a consequence, we obtain that the operator acting on a certain energy Hilbert space is the generator of an exponentially stable C ₀-semigroup.     

Operators with bounded conjugation orbits

For a bounded invertible operator on a complex Banach space let be the set of operators in for which Suppose that and is in A bound is given on in terms of the spectral radius of the commutator. Replacing the condition in by the weaker condition as for every 0$">, an extension of the Deddens-Stampfli-Williams results on the commutant of is given.

     

Microwave spectrum of pentafluorobenzonitrile

The absorption spectrum of pentafluorobenzonitrile has been investigated in the frequency range of 18–26·5 GHz using a 100 KHz stark modulated microwave spectrometer. The analysis of the spectrum is based on the rigid asymmetric rotor theory. The rotational constants obtained are A=1026·82±0·3 MHz, B=776·34±0·1 MHz, C=442·06±0·1 MHz and the asymmetry parameterχ=+0·1433. The inertial defect is I _o ?I _a ?I _b =0·081 amu Ų. The bond distances ared _CF=1·328 Å andd _CN=1·157 Å. The results are in good agreement with the assumed planarity of the molecule and the normal values of bond distances.     

Jordan isomorphisms of nest algebras

Let and be two nest algebras. A Jordan isomorphism from onto is a bijective linear map such that for every . In this note, we prove that every Jordan isomorphism of nest algebras is of the form or and then is, in fact, an isomorphism or an anti-isomorphism.

     

A central limit theorem for convex sets

We show that there exists a sequence for which the following holds: Let K⊂ℝⁿ be a compact, convex set with a non-empty interior. Let X be a random vector that is distributed uniformly in K. Then there exist a unit vector θ in ℝⁿ, t₀∈ℝ and σ>0 such that