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

     

Heisenberg inequalities for Jacobi transforms

The classical Heisenberg uncertainty principle states that for f∈L²(R),

     

Global branch from the second eigenvalue for a semilinear Neumann problem in a ball

Let (n?3) be a ball, and let f∈C³. We are concerned with the Neumann problem

     

Equivalence of some integrals and modulus of smoothness

Let f∈C(R). We are interested in lower and upper bounds of the integrals

     

Iterates of a Berezin-type transform

Let B be the open unit ball of Rn and dV denote the Lebesgue measure on Rn normalized so that the measure of B equals 1. Suppose f∈L¹(B,dV). The Berezin-type transform of f is defined by

     

Bernstein widths of Hardy-type operators in a non-homogeneous case

Let I=[a,b]⊂R, let 1<p?q<∞, let u and v be positive functions with u∈Lp^′(I), v∈Lq(I) and let be the Hardy-type operator given by

     

Weak type estimates on certain Hardy spaces for smooth cone type multipliers

Let ?n∈C^∞(Rd?{0}) be a non-radial homogeneous distance function of degree n∈N satisfying ?n(tξ)=tn?n(ξ). For f∈S(Rd+1) and δ>0, we consider convolution operator associated with the smooth cone type multipliers defined by

     

Schatten-von Neumann properties in the Weyl calculus

Let Opt(a), for t∈R, be the pseudo-differential operator

     

A singular Sturm-Liouville equation under homogeneous boundary conditions

Given α>0 and f∈L²(0,1), we are interested in the equation

     

On small fractional parts of polynomials

Text

We prove that for any real polynomial f(x)∈R[x] the set

     

Positive solutions of reaction diffusion equations with super-linear absorption: Universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions

Consider classical solutions u∈C²(Rⁿ×(0,∞))∩C(Rⁿ×[0,∞)) to the parabolic reaction diffusion equation

     

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 ,

     

Notes on the Duren-Leung conjecture

For the logarithmic coefficients γn of a univalent function f(z)=z+a₂z²+?∈S, the well-known de Branges' theorem shows that

     

Uniqueness and nonuniqueness of nodal radial solutions of sublinear elliptic equations in a ball

The following Dirichlet problem

(1.1)

is considered, where , N≥2, KC²[0,1] and K(r)>0 for 0≤r≤1, , sf(s)>0 for s≠0. Assume moreover that f satisfies the following sublinear condition: f(s)/s>f^′(s) for s≠0. A sufficient condition is derived for the uniqueness of radial solutions of (1.1) possessing exactly k−1 nodes, where . It is also shown that there exists KC^∞[0,1] such that (1.1) has three radial solutions having exactly one node in the case N=3.     

Interior components of a tile associated to a quadratic canonical number system

Let be a root of the polynomial p(x)=x²+4x+5. It is well known that the pair (p(x),{0,1,2,3,4}) forms a canonical number system, i.e., that each x∈Z[α] admits a finite representation of the shape x=a₀+a₁α+?+a?α? with ai∈{0,1,2,3,4}. The set T of points with integer part 0 in this number system

     

Accretive operators and Cassels inequality

Let a,b>0 and let Z∈M_n(R) such that Z lies into the operator ball of diameter [aI,bI]. Then for all positive definite A∈M_n(R),