Weakly Birkhoff recurrent switching signals, almost sure and partial stability of linear switched dynamical systems

Let {A₁,…,AK}⊂Cd×d be arbitrary K matrices, where K and d both ?2. For any 0<Δ<∞, we denote by the set of all switching sequences u=(λ.,t.):N→{1,…,K}×R₊ satisfying tj−tj−1?Δ and

     

Jordan maps on triangular algebras

Let T be a triangular algebra and R′ be an arbitrary ring. Suppose that M:T→R^′ and M^∗:R^′→T are surjective maps such that

     

Involutions and representations for reduced quantum algebras

In the context of deformation quantization, there exist various procedures to deal with the quantization of a reduced space Mred. We shall be concerned here mainly with the classical Marsden-Weinstein reduction, assuming that we have a proper action of a Lie group G on a Poisson manifold M, with a moment map J for which zero is a regular value. For the quantization, we follow Bordemann et al. (2000) [6] (with a simplified approach) and build a star product red_? on Mred from a strongly invariant star product ? on M. The new questions which are addressed in this paper concern the existence of natural ^∗-involutions on the reduced quantum algebra and the representation theory for such a reduced ^∗-algebra.We assume that ? is Hermitian and we show that the choice of a formal series of smooth densities on the embedded coisotropic submanifold C=J−1(0), with some equivariance property, defines a ^∗-involution for red_? on the reduced space. Looking into the question whether the corresponding ^∗-involution is the complex conjugation (which is a ^∗-involution in the Marsden-Weinstein context) yields a new notion of quantized modular class.We introduce a left (C^∞(M)?λ?,?)-submodule and a right (C^∞(Mred)?λ?,red_?)-submodule of C^∞(C)?λ?; we define on it a C^∞(Mred)?λ?-valued inner product and we establish that this gives a strong Morita equivalence bimodule between C^∞(Mred)?λ? and the finite rank operators on . The crucial point is here to show the complete positivity of the inner product. We obtain a Rieffel induction functor from the strongly non-degenerate ^∗-representations of (C^∞(Mred)?λ?,red_?) on pre-Hilbert right D-modules to those of (C^∞(M)?λ?,?), for any auxiliary coefficient ^∗-algebra D over C?λ?.     

Normalized Leonard pairs and Askey-Wilson relations

Let V denote a vector space with finite positive dimension, and let (A, A^∗) denote a Leonard pair on V. As is known, the linear transformations A, A^∗ satisfy the Askey-Wilson relations

     

Lieb-Thirring type inequalities and Gagliardo-Nirenberg inequalities for systems

This paper is devoted to inequalities of Lieb-Thirring type. Let V be a nonnegative potential such that the corresponding Schrödinger operator has an unbounded sequence of eigenvalues (λi(V))_i∈N^∗. We prove that there exists a positive constant C(γ), such that, if γ>d/2, then

(∗)     

Existence and regularity to an elliptic equation with logarithmic nonlinearity

We study the nonlinear elliptic problem −Δu=χ{u>0}(logu+λf(x,u)) in Ω⊂Rn with u=0 on ∂Ω. The function is nondecreasing, sublinear and fu is continuous. For every λ>0, we obtain a maximal solution uλ?0 and prove its global regularity . There is a constant λ^∗ such that uλ vanishes on a set of positive measure for 0<λ<λ^∗, and uλ>0 for λ>λ^∗. If f is concave, for λ>λ^∗ we characterize uλ by its stability.     

Positive solutions for semilinear elliptic equations with singular forcing terms

We consider the existence of solutions to the semilinear elliptic problem

(∗)_κ     

The lack of strict convexity and the validity of the Comparison Principle for a simple class of minimizers

Let w^∗(x)=a⋅x+b be an affine function in R^N, Ω⊂R^N, L:R^N→R be convex and w be a local minimizer of

     

An analytic approach to purely nonlocal Bellman equations arising in models of stochastic control

Given a bounded domain Ω⊂Rd and two integro-differential operators L¹, L² of the form we study the fully nonlinear Bellman equation

(0.1)     

A spectral radius formula for the Fourier transform on compact groups and applications to random walks

Let G be a compact group, not necessarily abelian, let ? be its unitary dual, and for f∈L¹(G), let fn?f∗?∗f denote n-fold convolution of f with itself and f? the Fourier transform of f. In this paper, we derive the following spectral radius formula

     

Isomorphisms between C-ternary algebras

In this paper, we prove the Hyers-Ulam-Rassias stability of homomorphisms in C^∗-ternary algebras and of derivations on C^∗-ternary algebras for the following Cauchy-Jensen additive mappings:

(0.1)     

On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields

Let m?−1 be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equations

X³−mX²Y−(m+3)XY²−Y³=λ     

Properties of the Holmes space

The following properties of the Holmes space H are established:

(i)

H has the Metric Approximation Property (MAP).

(ii)

The w^∗-closure of the set of extreme points of the unit ball BH^∗ of the dual space H^∗ is the whole ball BH^∗.

A family of compact subsets X⊂U of the Urysohn space is described such that the Lipschitz-free space F(X) has a finite-dimensional decomposition and is not complemented in H.     

Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A^∗ : V → V that satisfy (i) and (ii) below:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

We call such a pair a Leonard pair on V. Let X denote the set of linear transformations X : V → V such that the matrix representing X with respect to the basis (i) is tridiagonal and the matrix representing X with respect to the basis (ii) is tridiagonal. We show that X is spanned by

     

Three-dimensional time-varying nonlinear systems containing a Hamilton system

In this paper the following three-dimensional nonlinear system is considered:

     

A note on unbounded strongly measure zero subgroups of the Baer-Specker group

We show that it is consistent with ZFC that there exist:

(1)

An unbounded (with respect to _?∗) and strongly measure zero subgroup of ZN, but without the Menger property.

(2)

An unbounded (with respect to _?∗) and strongly measure zero subgroup of ZN with the Menger property which does not have the Rothberger property.

This answers the last two problems which remained from a classification project of Machura and Tsaban.     

Nonlinear conditions for weighted composition operators between Lipschitz algebras

Let T:Lip₀(X)→Lip₀(Y) be a surjective map between pointed Lipschitz ^∗-algebras, where X and Y are compact metric spaces. On the one hand, we prove that if T satisfies the non-symmetric norm ^∗-multiplicativity condition: