Spectral theory for perturbed Krein Laplacians in nonsmooth domains

We study spectral properties for HK,Ω, the Krein-von Neumann extension of the perturbed Laplacian −Δ+V defined on , where V is measurable, bounded and nonnegative, in a bounded open set Ω⊂Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1,r, r>1/2. In particular, in the aforementioned context we establish the Weyl asymptotic formula

     

High dimensional random sections of isotropic convex bodies

We study two properties of random high dimensional sections of convex bodies. In the first part of the paper we estimate the central section function for random F∈Gn,k and K⊂Rn a centrally symmetric isotropic convex body. This partially answers a question raised by V.D. Milman and A. Pajor (see [V.D. Milman, A. Pajor, Isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in: Lecture Notes in Math., vol. 1376, Springer, 1989, p. 88]). In the second part we show that every symmetric convex body has random high dimensional sections F∈Gn,k with outer volume ratio bounded by

     

Distance to spaces of continuous functions

We link here distances between iterated limits, oscillations, and distances to spaces of continuous functions. For a compact space K, a uniformly bounded set H of the space of real-valued continuous functions C(K), and ε?0, we say that H ε-interchanges limits with K, if the inequality

     

On an extension of the Blaschke-Santaló inequality and the hyperplane conjecture

Let K be a symmetric convex body and K^○ its polar body. Call

     

On the spectral radius of positive operators on Banach sequence spaces

Let K₁,…,K_n be (infinite) non-negative matrices that define operators on a Banach sequence space. Given a function f:[0,∞)×…×[0,∞)→[0,∞) of n variables, we define a non-negative matrix and consider the inequality

     

Interior estimates for solutions of a fourth order nonlinear partial differential equation

Let be a locally strongly convex hypersurface, given by the graph of a convex function xn+1=f(x₁,…,xn) defined in a convex domain Ω⊂Rn. M is called a α-extremal hypersurface, if f is a solution of

     

Isomorphic properties of intersection bodies

We study isomorphic properties of two generalizations of intersection bodies - the class of k-intersection bodies in Rn and the class of generalized k-intersection bodies in Rn. In particular, we show that all convex bodies can be in a certain sense approximated by intersection bodies, namely, if K is any symmetric convex body in Rn and 1≤k≤n−1 then the outer volume ratio distance from K to the class can be estimated by

     

Concentration of mass on the Schatten classes

Let 1?p?∞ and be the unit ball of the Schatten trace class of matrices on Cn or on Rn, normalized to have Lebesgue measure equal to one. We prove that

     

A Robertson-type uncertainty principle and quantum Fisher information

Let A₁,…,A_N be complex self-adjoint matrices and let ρ be a density matrix. The Robertson uncertainty principle

     

Spectral analysis of a family of second-order elliptic operators with nonlocal boundary condition indexed by a probability measure

Let D⊂Rd be a bounded domain and let

     

Noncommutative transforms and free pluriharmonic functions

In this paper, we study free pluriharmonic functions on noncommutative balls γ[Bn(H)], γ>0, and their boundary behavior. These functions have the form

     

Norm estimates of almost Mathieu operators

We estimate the norm of the almost Mathieu operator , regarded as an element in the rotation C^*-algebra . In the process, we prove for every λ∈R and the inequality

     

Integer cells in convex sets

Every convex body K in Rⁿ has a coordinate projection PK that contains at least cells of the integer lattice PZⁿ, provided this volume is at least one. Our proof of this counterpart of Minkowski's theorem is based on an extension of the combinatorial density theorem of Sauer, Shelah and Vapnik-Chervonenkis to Zⁿ. This leads to a new approach to sections of convex bodies. In particular, fundamental results of the asymptotic convex geometry such as the Volume Ratio Theorem and Milman's duality of the diameters admit natural versions for coordinate sections.     

Adams' inequalities for bi-Laplacian and extremal functions in dimension four

Let Ω⊂R⁴ be a smooth oriented bounded domain, be the Sobolev space, and be the first eigenvalue of the bi-Laplacian operator Δ². Then for any α: 0?α<λ(Ω), we have

     

The topology of the monodromy map of a second order ODE

We consider the following question: given A∈SL(2,R), which potentials q for the second order Sturm-Liouville problem have A as its Floquet multiplier? More precisely, define the monodromy map μ taking a potential q∈L²([0,2π]) to , the lift to the universal cover of SL(2,R) of the fundamental matrix map ,

     

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

     

Jensen's inequality for the lower semicontinuous quasiconvex envelope and relaxation of multidimensional control problems

Assume that K⊂Rnm is a convex body with o∈int(K) and is a function with f|K∈C⁰(K,R) and f|(Rnm?K)≡+∞. We show that its lower semicontinuous quasiconvex envelope

     

The Bourgain ?-index of mixed Tsirelson space

Suppose that is a sequence of regular families of finite subsets of such that contains all singletons, and (θ_n)_n=1^∞ is a nonincreasing null sequence in (0,1). The mixed Tsirelson space is the completion of c₀₀ with respect to the implicitly defined norm