Strong subtournaments containing a given vertex in regular multipartite tournaments

If x is a vertex of a digraph D, then we denote by d⁺(x) and d⁻(x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is defined by

     

A functional equation characterizing the second derivative

Consider an operator T:C²(R)→C(R) and isotropic maps A₁,A₂:C¹(R)→C(R) such that the functional equation

     

A transform involving Chebyshev polynomials and its inversion formula

We define a functional analytic transform involving the Chebyshev polynomials Tn(x), with an inversion formula in which the Möbius function μ(n) appears. If s∈C with Re(s)>1, then given a bounded function from [−1,1] into C, or from C into itself, the following inversion formula holds:

     

Asymptotic properties of generalized Laguerre orthogonal polynomials

In the present paper we deal with the polynomials L_n^(α,M,N) (x) orthogonal with respect to the Sobolev inner product

     

Regularity results for a class of obstacle problems under nonstandard growth conditions

We prove regularity results for minimizers of functionals in the class , where is a fixed function and f is quasiconvex and fulfills a growth condition of the type

L⁻¹|z|^p(x)?f(x,ξ,z)?L(1+|z|^p(x)),     

Characterizing the derivative and the entropy function by the Leibniz rule

Consider an operator T:C¹(R)→C(R) satisfying the Leibniz rule functional equation

     

The chain rule as a functional equation

We consider operators T from C¹(R) to C(R) satisfying the “chain rule”

     

Periodic solutions of Duffing equations with semi-quadratic potential and singularity

In this paper, we deal with the existence of periodic solutions of the second order differential equations x^″+g(x)=p(t) with singularity. We prove that the given equation has at least one periodic solution when g(x) has singularity at origin, satisfies

     

Initial blow-up of solutions of semilinear parabolic inequalities

We study classical nonnegative solutions u(x,t) of the semilinear parabolic inequalities

     

On the stability of limit cycles for planar differential systems

We consider a planar differential system , , where P and Q are C¹ functions in some open set U⊆R², and . Let γ be a periodic orbit of the system in U. Let f(x,y):U⊆R²→R be a C¹ function such that

     

Square-free values of the Carmichael function

We obtain an asymptotic formula for the number of square-free values among p−1, for primes p?x, and we apply it to derive the following asymptotic formula for L(x), the number of square-free values of the Carmichael function λ(n) for 1?n?x,

     

A mean value theorem for cubic fields

Let r(n) denote the number of integral ideals of norm n in a cubic extension K of the rationals, and define and Δ(x)=S(x)−αx where α is the residue of the Dedekind zeta function ζ(s,K) at 1. It is shown that the abscissa of convergence of

     

Optimal bounds on the Kuramoto-Sivashinsky equation

In this paper, we consider solutions u(t,x) of the one-dimensional Kuramoto-Sivashinsky equation, i.e.

     

Continuity modulus of stochastic homeomorphism flows for SDEs with non-Lipschitz coefficients

Let Xt(x) solve the following Itô-type SDE (denoted by EQ.(σ,b,x)) in Rd

     

Convergence theorems for fixed points of uniformly continuous generalized Φ-hemi-contractive mappings

Let E be a real normed linear space, K be a nonempty subset of E and be a uniformly continuous generalized Φ-hemi-contractive mapping, i.e., , and there exist x^∗∈F(T) and a strictly increasing function , Φ(0)=0 such that for all x∈K, there exists j(x−x^∗)∈J(x−x^∗) such that

〈Tx−x^∗,j(x−x^∗)〉?‖x−x^∗‖2−Φ(‖x−x^∗‖).     

Packing equal squares into a large square

Let s(x) denote the maximum number of non-overlapping unit squares which can be packed into a large square of side length x. Let W(x)=x²−s(x) denote the “wasted” area, i.e., the area not covered by the unit squares. In this note we prove that

     

Diophantine equations for classical continuous orthogonal polynomials

Let A, B, C denote rational numbers with AB ≠ 0 and m > n ≥ 3 arbitrary rational integers. We study the Diophantine equation AP_m(x) + Bp_n(y) = C, in x, y ? , where {Pk(x)}I is one of the three classical continuous orthogonal polynomial families, i.e. Laguerre polynomials, Jacobi polynomials (including Gegenbauer, Legendre or Chebyshev polynomials) and Hermite polynomials. We prove that with exception of the Chebyshev polynomials for all such polynomial families there are at most finitely many solutions (x, y) ? ² provided n > 4. The tools are besides the criterion [3], a theorem of Szeg— [14] on monotonicity of stationary points of polynomials which satisfy a second order Sturm-Liouville differential equation,

     

Uniformly continuous superposition operators in the Banach space of Hölder functions

Let I,J⊂R be intervals. One of the main results says that if a superposition operator H generated by a two place ,

H(φ)(x):=h(x,φ(x)),     

Compactness in kinetic transport equations and hypoellipticity

We establish improved hypoelliptic estimates on the solutions of kinetic transport equations, using a suitable decomposition of the phase space. Our main result shows that the relative compactness in all variables of a bounded family of nonnegative functions fλ(x,v)∈L¹ satisfying some appropriate transport relation