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,

     

An inequality for sums of binary digits, with application to Takagi functions

Let ?(x)=2inf{|x−n|:n∈Z}, and define for α>0 the function

     

Front formation and motion in quasilinear parabolic equations

This paper deals with the singular limit for

L?u:=ut−Fx(u,?ux)−?⁻¹g(u)=0,     

Regularization of a three-element functional equation

In this paper we study the three-element functional equation

$(V\Phi )(z) \equiv \Phi (iz) + \Phi ( - iz) + G(z)\Phi \left( {\frac{1}{z}} \right) = g(z), z \in R,$

, subject to

$R: = \{ z:\left| z \right| < 1, \left| {\arg z} \right| < \frac{\pi }{4}\} .$

We assume that the coefficients G(z) and g(z) are holomorphic in R and their boundary values G ⁺(t) and g ⁺(t) belong to H(Γ), G(t)G(t ^?1) = 1. We seek for solutions Φ(z) in the class of functions holomorphic outside of \(\bar R\) such that they vanish at infinity and their boundary values Φ^?(t) also belong to H(Γ). Using the method of equivalent regularization, we reduce the problem to the 2nd kind integral Fredholm equation.

     

Extensions of the Scherk-Kemperman Theorem

Let Γ=(V,E) be a reflexive relation with a transitive automorphism group. Let F be a finite subset of V containing a fixed element v. We prove that the size of Γ(F) (the image of F) is at least

|F|+|Γ(v)|−|Γ⁻(v)∩F|.     

An inequality for ratios of gamma functions

Let Γ(x) denote Euler's gamma function. The following inequality is proved: for y>0 and x>1 we have

     

Semiclassical analysis of low and zero energy scattering for one-dimensional Schrödinger operators with inverse square potentials

This paper studies the scattering matrix S(E;?) of the problem

−?²ψ^″(x)+V(x)ψ(x)=Eψ(x)     

Oscillation and nonoscillation criteria for second order quasilinear difference equations

New oscillation and nonoscillation theorems are obtained for the second order quasilinear difference equation

Δ(|Δxn−1|^ρ−1Δxn−1)+pn|xn|^ρ−1xn=0,     

Symmetry and specializability in the continued fraction expansions of some infinite products

Let f(x)∈Z[x]. Set f₀(x)=x and, for n?1, define fn(x)=f(fn−1(x)). We describe several infinite families of polynomials for which the infinite product

     

Compactly supported solutions to stationary degenerate diffusion equations

We consider non-negative solutions of the semilinear elliptic equation in Rn with n?3:

−Δu=a(x)uq+b(x)up,     

Linearized oscillation of odd order nonlinear neutral delay differential equations (I)

In this paper, we proved that the odd order nonlinear neutral delay differential equation

[x(t)−p(t)g(x(t−τ))^](n)+q(t)h(x(t−σ))=0     

Quantum Dynamical coBoundary Equation for finite dimensional simple Lie algebras

For a finite dimensional simple Lie algebra g, the standard universal solution R(x)∈Uq(g)^⊗2 of the Quantum Dynamical Yang-Baxter Equation quantizes the standard trigonometric solution of the Classical Dynamical Yang-Baxter Equation. It can be built from the standard R-matrix and from the solution F(x)∈Uq(g)^⊗2 of the Quantum Dynamical coCycle Equation as . F(x) can be computed explicitly as an infinite product through the use of an auxiliary linear equation, the ABRR equation.Inspired by explicit results in the fundamental representation, it has been conjectured that, in the case where g=sl(n+1)(n?1) only, there could exist an element M(x)∈Uq(sl(n+1)) such that the dynamical gauge transform RJ of R(x) by M(x),

RJ=M₁⁻¹(x)M₂(xqh₁⁾⁻¹R(x)M₁(xqh₂)M₂(x),     

Stability of the preservation of the equality of distance

We study the stability problem for mappings satisfying the equation

‖f(x−y)‖=‖f(x)−f(y)‖.     

Numerical radius and zero pattern of matrices

Let A be an n×n complex matrix and r be the maximum size of its principal submatrices with no off-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting ‖A‖ be the Frobenius norm of A, we show that

|〈Ax,x〉^|2?(1−1/2r−1/2n)‖A^‖2.     

Partial Order Embeddings with Convex Range

A careful study is made of embeddings of posets which have a convex range. We observe that such embeddings share nice properties with the homomorphisms of more restrictive categories; for example, we show that every order embedding between two lattices with convex range is a continuous lattice homomorphism. A number of posets are considered; for one of the simplest examples, we prove that every product order embedding σ : ?^? → ?^? with convex range is of the form

$$ \sigma(x)(n)=\left( (x\circ g_{\sigma})+y_{\sigma}\right)(n) ~~~~\text{if}~ n\in K_{\sigma}, $$

(1)

and σ(x)(n) = y _σ (n) otherwise, for all x ∈ ?^?, where K _σ ? ?, g _σ : K _σ → ? is a bijection and y _σ ∈ ?^?. The most complex poset examined here is the quotient of the lattice of Baire measurable functions, with codomain of the form ? ^I for some index set I, modulo equality on a comeager subset of the domain, with its ‘natural’ ordering.

     

On a nonlinear eigenvalue problem in ODE

In this paper we shall study the following variant of the logistic equation with diffusion:

−du^″(x)=g(x)u(x)−u²(x)     

On low discrepancy sequences and low discrepancy ergodic transformations of the multidimensional unit cube

In this paper we describe a third class of low discrepancy sequences. Using a lattice Γ ? ? ^s , we construct Kronecker-like and van der Corput-like ergodic transformations T _1,Γ and T _2,Γ of [0, 1) ^s . We prove that for admissible lattices Γ, (T _ν,Γ ⁿ (x))_n≥0 is a low discrepancy sequence for all x ∈ [0, 1) ^s and ν ∈ {1, 2}. We also prove that for an arbitrary polyhedron P ? [0, 1) ^s , for almost all lattices Γ ∈ L _s = SL(s,?)/SL(s, ?) (in the sense of the invariant measure on L _s ), the following asymptotic formula

$\# \{ 0 \le n < N:T_{v,\Gamma }^n(x) \in P\} = NvolP + O({(\ln N)^{s + \varepsilon }}),N \to \infty$

holds with arbitrary small ? > 0, for all x ∈ [0, 1) ^s , and ν ∈ {1, 2}.

     

Study of the solutions to a model porous medium equation with variable exponent of nonlinearity

The authors of this paper study the Dirichlet problem of the following equation

ut−div(|u|^ν(x,t)∇u)=f−|u|^p(x,t)−1u.     

Interlacing properties and the Schur-Szegő composition

Each degree n polynomial in one variable of the form (x+1)(x ^n?1+c ₁ x ^n?2+???+c _n?1) is representable in a unique way as a Schur-Szeg? composition of n?1 polynomials of the form (x+1) ^n?1(x+a _i ), see Kostov (2003), Alkhatib and Kostov (2008) and Kostov (Mathematica Balkanica 22, 2008). Set $\sigma _{j}:=\sum _{1\leq i_{1}<\cdots <i_{j}\leq n-1}a_{i_{1}}\cdots a_{i_{j}}$ . The eigenvalues of the affine mapping (c ₁,…,c _n?1)?(σ ₁,…,σ _n?1) are positive rational numbers and its eigenvectors are defined by hyperbolic polynomials (i.e. with real roots only). In the present paper we prove interlacing properties of the roots of these polynomials.