Representing the GCD as linear combination in non-PID rings

We prove the following fact: If finitely many elements p ₁,p ₂,…,p _n of a unique factorization domain are given such that the greatest common divisor of each pair (p _i ,p _j ) can be expressed as a linear combination of p _i and p _j , then the greatest common divisor of all the p _i ’s can also be expressed as a linear combination of p ₁,…,p _n . We prove an analogous statement in general commutative rings.     

Integral complete 4-partite graphs

A graph G is called integral if all the roots of the characteristic polynomial P(G;x) are integers. In the paper the first known integral complete 4-partite graph K_p1,p2,p3,p4, where p₁<p₂<p₃<p₄, is constructed.     

Line graph eigenvalues and line energy of caterpillars

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d?3 and n?2(d-1). Let p=[p₁,p₂,…,p_d-1] with p₁?1,p₂?1,…,p_d-1?1 such that

p₁+p₂+?+p_d-1=n-d+1.     

The maximal solution of a restricted subadditive inequality in numerical analysis

For a fixed non-negative integerp, letU _2p = {U _2p (n)},n ≥ 0, denote the sequence that is defined by the initial conditionsU _2p (0) =U _2p (1) =U _2p (2) = =U _2p (2p) = 1 and the restricted subadditive recursion $$U_{2p} (n + 2p + 1) = \mathop {\min }\limits_{0 \leqslant l \leqslant p} (U_{2p} (n + l) + U_{2p} (n + 2p - l)),n \geqslant 0$$ U _2p is of importance in the theory of sequential search for simple real zeros of real valued continuous 2p-th derivatives In this paper, several closed form expressions forU _2p (n), n > 2p, are determined, thereby providing insight into the structure ofU _2p Two of the properties thus illuminated are (a) the existence of exactlyp + 1 limit points (1 + 1/(p + 1 +i), 0 ≤i ≤p) of the associated sequence {U _2p (n + 1)/U _2p (n)},n ≥ 0, and (b) the relevance toU _2p of the classic number theoretic function ord     

p-Adic dynamical systems of Chebyshev polynomials

We study the behaviour of the iterates of the Chebyshev polynomials of the first kind in p-adic fields. In particular, we determine in the field of complex p-adic numbers for p > 2, the periodic points of the p-th Chebyshev polynomial of the first kind. These periodic points are attractive points. We describe their basin of attraction. The classification of finite field extensions of the field of p-adic numbers ? _p , enables one to locate precisely, for any integer ν ≥ 1, the ν-periodic points of T _p : they are simple and the nonzero ones lie in the unit circle of the unramified extension of ? _p , (p > 2) of degree ν. This generalizes a result, stated by M. Zuber in his PhD thesis, giving the fixed points of T _p in the field ? _p , (p > 2). As often happens, we consider separately the case p = 2. Also, if the integer n ≥ 2 is not divisible by p, then any fixed point w of T _n is indifferent in the field of p-adic complex numbers and we give for p ≥ 3, the p-adic Siegel disc around w.     

A priori estimates for one-dimensional singular integral operators with continuous coefficients

Let the contour γ consist of a finite number of simple closed pairwise nonintersecting curves, satisfying a Lyapunov condition, let S be the operator of singular integration in spacel _p , (γ) (1 <p < ∞), and leta (t), b (t) εC (γ) 1 <p ₁. <p < ∞. The necessary and sufficient condition for A = aI+ bS to be a Φ-operator in space L_p(γ) is that, for all?ε L_p(γ), ∥?∥_p ? const (∥ A?_p + ∥ ? ∥_p1), where ∥?∥_p = ∥?∥L_p (γ).     

On some special codes over Fp + uFp + uFp

We define a new map between codes over F_p + uF_p + u²F_p and F_p which is different to that defined in [2]. It is proved that the image of the linear cyclic code over the commutative ring F_p + uF_p + u²F_p with length n under this map is a distance-invariant quasi-cyclic code of index p² with length p²n over F_p. Moreover, it is proved that, if (n, p) = 1, then every code with length p²n over F_p which is the image of a linear (1 − u²)-cyclic code with length n over F_p + uF_p + u²F_p under this map is permutation equivalent to a quasi-cyclic code of index p².     

Some bounds on the p-domination number in trees

Let p be a positive integer and G=(V,E) a graph. A subset S of V is a p-dominating set if every vertex of V-S is dominated at least p times, and S is a p-dependent set of G if the subgraph induced by the vertices of S has maximum degree at most p-1. The minimum cardinality of a p-dominating set a of G is the p-domination number γ_p(G) and the maximum cardinality of a p-dependent set of G is the p-dependence number β_p(G). For every positive integer p?2, we show that for a bipartite graph G, γ_p(G) is bounded above by (|V|+|Y_p|)/2, where Y_p is the set of vertices of G of degree at most p-1, and for every tree T, γ_p(T) is bounded below by β_p-1(T). Moreover, we characterize the trees achieving equality in each bound.     

Best constants in Sobolev inequalities for compact manifolds of nonpositive curvature

Let (M,g) be a smooth compact Riemannian n-manifold, n ≥ 2, let p(1, n) real, and let H₁^p (M) be the standard Sobolev space of order p. By the Sobolev embedding theorem, H₁^p(M) ⊂ L^p* (M) where p^* = np/(n - p). Classically, this leads to some Sobolev inequality (I_p¹), and then to some Sobolev inequality (I_p^p) where each term in (I_p¹) is elevated to the power p. Long standing questions were to know if the optimal versions with respect to the first constant of (I_p¹) and (I_p^p) do hold. Such questions received an affirmative answer by Hebey-Vaugon for p = 2, and on what concerns (I_p¹), by Aubin for two-dimensional manifolds and for manifolds of constant sectional curvature. Recently, Druet proved that for p > 2, and p² < n, the optimal version of (I_p^p) is false if the scalar curvature of g is positive somewhere, while for p > 1, the optimal version of (I_p^p) does hold on flat torii and compact hyperbolic spaces. We prove here that the optimal version of (I_p^p), p > 1, does hold for compact manifolds of nonpositive sectional curvature in any dimension where the Cartan-Hadamard conjecture is true. In particular, since the Cartan-Hadmard conjecture is true in dimensions 2, 3, and 4, the optimal version of (I_p^p) does hold on any compact manifold of nonpositive sectional curvature of dimension 2, 3, or 4.     

On two-dimensional gas expansion for pressure-gradient equations of Euler system

In this paper we establish the existence of global continuous solutions of gas expansion into a vacuum for the two-dimensional pressure-gradient equations in gas dynamics. Under irrotational condition, By hodograph transformation, the flow is governed by the equation (p−p²_v)p_uu+2p_up_vp_uv+(p−p²_u)p_vv=0, which can be further reduced to a inhomogeneous linearly degenerate system of three equations. Then the problem of the expansion of a wedge of gas into a vacuum is solved in the same way.     

Isoperimetric type inequalities for harmonic functions

For 0<p<+∞ let hp be the harmonic Hardy space and let bp be the harmonic Bergman space of harmonic functions on the open unit disk U. Given 1?p<+∞, denote by ‖⋅bp_‖ and ‖⋅hp_‖ the norms in the spaces bp and hp, respectively. In this paper, we establish the harmonic hp-analogue of the known isoperimetric type inequality ‖fb2p_‖?‖fhp_‖, where f is an arbitrary holomorphic function in the classical Hardy space Hp. We prove that for arbitrary p>1, every function f∈hp satisfies the inequality

‖fb2p_‖?ap‖fhp_‖,     

Characterization of elements of polynomials in ��S

Given the discrete space of natural numbers, we characterize the elements of polynomials evaluated on the points of ???. We establish these results by proving the characterization in a far more general setting. Let S be a discrete set which is a semigroup under two operations ? and +. Let g(z ₁,z ₂,??,z _k ) be any polynomial and p ₁,p ₂,??,p _k be elements of ??S. We provide a sufficient condition that a set A?S is a member of g(p ₁,p ₂,??,p _k ) and use it to characterize the members of g(p ₁,p ₂,??,p _k ) if each p _i is an idempotent in (??S,+).     

On the Lp-Theory of C0-Semigroups Associated with Second-Order Elliptic Operators, II

We study positive C₀-semigroups on L_p associated with second-order uniformly elliptic divergence-type operators with singular lower-order terms, subject to a wide class of boundary conditions. We obtain an interval (p_min, p_max) in the L_p-scale where these semigroups can be defined, including the case 2∉(p_min, p_max). We present an example showing that the result is optimal. We also show that the semigroups are analytic with angles of analyticity and spectra of the generators independent of p, for the whole range of p where the semigroups are defined.     

On density of interpolation points,a Kadec-type theorem,and Saff's principle of contamination inL p-approximation

Letf ∈L _p (I) and denote byB _n,p (f) the polynomial of bestL _p-approximation tof of degreen (1<p<∞,I=[?1,1], the norm is weightedL _p-norm with an arbitrary positive weight). Extending a result proved by Saff and Shekhtman forp=2 we show that for every 1<p<∞ andf ∈L _p (I) (not a polynomial) points of sign change of the error functionf-B _n,p (f) are dense inI asn→∞.     

Stably average shadowable homoclinic classes

Let p be a hyperbolic periodic saddle of a diffeomorphism of f on a closed smooth manifold M, and let H_f(p) be the homoclinic class of f containing p. In this paper, we show that if H_f(p) is locally maximal and every hyperbolic periodic point in H_f(p) is uniformly far away from being nonhyperbolic, and H_f(p) has the average shadowing property, then H_f(p) is hyperbolic.     

On a problem of Erdős

Let s≥2 be an integer. Denote by f ₁(s) the least integer so that every integer l>f ₁(s) is the sum of s distinct primes. Erd?s proved that f ₁(s)<p ₁+p ₂+?+p _s +Cslogs, where p _i is the ith prime and C is an absolute constant. In this paper, we prove that f ₁(s)=p ₁+p ₂+?+p _s +(1+o(1))slogs=p ₂+p ₃+?+p _s+1+o(slogs). This answers a question posed by P. Erd?s.     

On the multiplicative structure of odd perfect numbers

Starting with Euler's theorem that any odd perfect number n has the form n = p^ep_i^2e_i … p_k^2e_k, where p, p₁,…,p_k are distinct odd primes and p ≡ e ≡ 1 (mod 4), we show that extensive subsets of these numbers (so described) can be eliminated from consideration. A typical result says: if p^e, p_i^2e_i,…,p_r^2e_r are all of the prime-power divisors of such an n with p ≡ p_i ≡ 1 (mod 4), then the ordered set {e₁,…,e_r} contains an even number or odd number of odd numbers according as e ≡ pore ≡ p (mod 8).     

Complemented subspaces of (l 2⊕l 2⊕…) p (1<p<∞) with an unconditional basis

It is proved that every infinite dimensional complemented subspace of (l ₂⊕l ₂⊕…) _p (1<p<∞) with an unconditional basis is isomorphic to one of the following four spaces:l ₂,l _p,l ₂⊕l _p, (l ₂⊕l ₂⊕…) _p .     

Upper bounds for p-divisibility of sets of Bernoulli numbers

Kummer's criterion asserts that a prime p divides the class number of the pield of pth roots of unity if and only if p divides the numerator of at least one of the Bernoulli numbers B₂, B₄,…,B_p?3. We partition the set {B₂,B₄,…,B_p?3} into certain “divisions” and prove that up to a higher order term at most one-half of the B_2k in a given division can be divisible by p.