Time-discretized steady compressible Navier–Stokes equations with inflow and outflow boundaries

The time-discretized steady compressible Navier–Stokes equations in n-dimensional bounded domains with the velocity specified only at the inflow boundary are considered. The existence and uniqueness of L ^p solutions are proved for p > n. For time-discretized steady flows, results of Kweon and Kellogg and of Kweon and Song are extended in a manner that allows for more general domains and for density-dependent viscosity coefficients. Moreover, we only require p > n which is a critical barrier in the previous works.     

On multiplicative functions which are additive on sums of primes

Let ${f : \mathbb{N} \to \mathbb{C}}$ be a multiplicative function satisfying f(p ₀) ≠ 0 for at least one prime number p ₀, and let k ≥ 2 be an integer. We show that if the function f satisfies f(p ₁ + p ₂ + . . . + p _k ) = f(p ₁) + f(p ₂) + . . . + f(p _k ) for any prime numbers p ₁, p ₂, . . . ,p _k then f must be the identity f(n) = n for each ${n \in \mathbb{N}}$ . This result for k = 2 was established earlier by Spiro, whereas the case k = 3 was recently proved by Fang. In the proof of this result for k ≥ 6 we use a recent result of Tao asserting that every odd number greater than 1 is the sum of at most five primes.     

On a Diophantine inequality involving prime powers

Suppose that N is a sufficiently large real number. In this paper it is proved that if 1 < c < 108/53, c ≠ 2, then the Diophantine inequality ${| p^c_1 + p^c_2+ p^c_3 +p^c_4+ p^c_5 - N| < \log^{-1}N}$ is solvable in primes p ₁, p ₂, p ₃, p ₄, p ₅. This result constitutes an improvement upon that of Zhai and Cao for the range 1 < c < 81/40, c ≠ 2.     

Remark on the completeness of an exponential type sequence

B. J. Birch [1] proved that all sufficiently large integers can be expressed as a sum of pairwise distinct terms of the form p ^a q ^b , where p, q are given coprime integers greater than 1. Subsequently, Davenport pointed out that the exponent b can be bounded in terms of p and q. N. Hegyvári [3] gave an effective version of this bound. In this paper, we improve this bound by reducing one step.     

Small point sets of PG(n, p 3h ) intersecting each line in 1 mod p h points

The main result of this paper is that point sets of PG(n, q), q = p ^3h , p ≥ 7 prime, of size < 3(q ^n-1 + 1)/2 intersecting each line in 1 modulo ${\sqrt[3] q}$ points (these are always small minimal blocking sets with respect to lines) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p ³), p ≥ 7 prime, of size < 3(p ^3(n-1) + 1)/2 with respect to lines are always linear.     

On the global convergence of a generalized iterative procedure for the minisum location problem with ℓ p distances for p > 2

This paper presents a procedure to solve the classical location median problem where the distances are measured with ? _p -norms with p > 2. In order to do that we consider an approximated problem. The global convergence of the sequence generated by this iterative scheme is proved. Therefore, this paper closes the still open question of giving a modification of the Weiszfeld algorithm that converges to an optimal solution of the median problem with ? _p norms and ${p \in (2, \infty)}$ . The paper ends with a computational analysis of the different provided iterative schemes.     

Dead-core rates for the fast diffusion equation with a spatially dependent strong absorption

This paper deals with the dead-core rates problem for the fast diffusion equation with a spatially dependent strong absorption $$u_t=(u^{m})_{xx}-x^{q}u^p, \quad(x,t)\in(0,1)\times(0,\infty),$$ where 0 < p < m < 1 and ?1 < q < 0. By using the self-similar transformation technique and the Zelenyak method, we proved that the temporal dead-core rate is non-self-similar.     

Improving integrability via absolute summability: a general version of Diestel’s Theorem

A classical result by J. Diestel establishes that the composition of a summing operator with a (strongly measurable) Pettis integrable function gives a Bochner integrable function. In this paper we show that a much more general result is possible regarding the improvement of the integrability of vector valued functions by the summability of the operator. After proving a general result, we center our attention in the particular case given by the \((p,\sigma )\)-absolutely continuous operators, that allows to prove a lot of special results on integration improvement for selected cases of classical Banach spaces—including C(K), \(L^p\) and Hilbert spaces—and operators—p-summing, (q, p)-summing and p-approximable operators.     

Small point sets of PG(n, q 3) intersecting each k-subspace in 1 mod q points

The main result of this paper is that point sets of PG(n, q ³), q = p ^h , p ≥ 7 prime, of size less than 3(q ^3(n?k) + 1)/2 intersecting each k-space in 1 modulo q points (these are always small minimal blocking sets with respect to k-spaces) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p ³), p ≥ 7 prime, of size less than 3(p ^3(n?k) + 1)/2 with respect to k-spaces are linear. We also give a classification of small linear blocking sets of PG(n, q ³) which meet every (n ? 2)-space in 1 modulo q points.     

Harnack estimates for non-negative weak solutions of a class of singular parabolic equations

We prove forward, backward and elliptic Harnack type inequalities for non-negative local weak solutions of singular parabolic differential equations of type $$u_t={\rm div}{\bf A}(x, t, u, Du)$$ where A satisfies suitable structure conditions and a monotonicity assumption. The prototype is the parabolic p?Laplacian with 1 < p < 2. By using only the structure of the equation and the comparison principle, we generalize to a larger class of equations the estimates first proved by Bonforte et al. (Adv. Math. 224, 2151–2215, 2010) for the model equation.     

Carleson and Vanishing Carleson Measures on Radial Trees

We extend a discrete version of an extension of Carleson’s theorem proved in [5] to a large class of trees that have certain radial properties. We introduce the geometric notion of s-vanishing Carleson measure on such a tree T (with s ≥ 1) and give several characterizations of such measures. Given a measure σ on T and p ≥ 1, let L ^p (σ) denote the space of functions g defined on T such that |g| ^p is integrable with respect to σ and let L ^p (? T) be the space of functions f defined on the boundary of T such that |f| ^p is integrable with respect to the representing measure of the harmonic function 1.We prove the following extension of the discrete version of a classical theorem in the unit disk proved by Power. A finite measure σ on T is an s-vanishing Carleson measure if and only if for any real number p > 1, the Poisson operator P : L ^p (? T) → L ^sp (σ) is compact. Characterizations of weak type for the case p = 1 and in terms of the tree analogue of the extended Poisson kernel are also given. Finally, we show that our results hold for homogeneous trees whose forward probabilities are radial and whose backward probabilities are constant, as well as for semihomogeneous trees.     

On two (p,q)-analogues of the Laplace transform

Two (p, q)-Laplace transforms are introduced and their relative properties are stated and proved. Applications are made to solve some (p, q)-linear difference equations.     

Universal rigidity of bar frameworks via the geometry of spectrahedra

A bar framework (G, p) in dimension r is a graph G whose nodes are points \(p^1,\ldots ,p^n\) in \(\mathbb {R}^r\) and whose edges are line segments between pairs of these points. Two frameworks (G, p) and (G, q) are equivalent if each edge of (G, p) has the same (Euclidean) length as the corresponding edge of (G, q). A pair of non-adjacent vertices i and j of (G, p) is universally linked if \(||p^i-p^j||\) = \(||q^i-q^j||\) in every framework (G, q) that is equivalent to (G, p). Framework (G, p) is universally rigid iff every pair of non-adjacent vertices of (G, p) is universally linked. In this paper, we present a unified treatment of the universal rigidity problem based on the geometry of spectrahedra. A spectrahedron is the intersection of the positive semidefinite cone with an affine space. This treatment makes it possible to tie together some known, yet scattered, results and to derive new ones. Among the new results presented in this paper are: (1) The first sufficient condition for a given pair of non-adjacent vertices of (G, p) to be universally linked. (2) A new, weaker, sufficient condition for a framework (G, p) to be universally rigid thus strengthening the existing known condition. An interpretation of this new condition in terms of the Strong Arnold Property is also presented.     

On Siegel modular forms of level p and their properties mod p

Using theta series we construct Siegel modular forms of level p which behave well modulo p in all cusps. This construction allows us to show (under a mild condition) that all Siegel modular forms of level p and weight 2 are congruent mod p to level one modular forms of weight p + 1; in particular, this is true for Yoshida lifts of level p.     

The Classification of Naturally Graded p-Filiform Leibniz Algebras

In the present article the classification of n-dimensional naturally graded p-filiform (1 ≤ p ≤ n ? 4) Leibniz algebras is obtained. A splitting of the set of naturally graded Leibniz algebras into the families of Lie and non Lie Leibniz algebras by means of characteristic sequences (isomorphism invariants) is proved.     

The Euler-Jacobi-Lie integrability theorem

This paper addresses a class of problems associated with the conditions for exact integrability of systems of ordinary differential equations expressed in terms of the properties of tensor invariants. The general theorem of integrability of the system of n differential equations is proved, which admits n ? 2 independent symmetry fields and an invariant volume n-form (integral invariant). General results are applied to the study of steady motions of a continuum with infinite conductivity.     

On <Emphasis Type="Italic">r</Emphasis>-recognition by prime graph of <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(3) where <Emphasis Type="Italic">p</Emphasis> is an odd prime

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) =  r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then B _p (3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then \({G\cong B_p(3)}\) or C _p (3). Also if Γ(G) = Γ(B ₃(3)), then \({G\cong B_3(3), C_3(3), D_4(3)}\), or \({G/O_2(G)\cong {\rm Aut}(^2B_2(8))}\). As a corollary, the main result of the above paper is obtained.     

On special zeros of p-adic L-functions of Hilbert modular forms

Let E be a modular elliptic curve over a totally real number field F. We prove the weak exceptional zero conjecture which links a (higher) derivative of the p-adic L-function attached to E to certain p-adic periods attached to the corresponding Hilbert modular form at the places above p where E has split multiplicative reduction. Under some mild restrictions on p and the conductor of E we deduce the exceptional zero conjecture in the strong form (i.e. where the automorphic p-adic periods are replaced by the $\mathcal {L}$ -invariants of E defined in terms of Tate periods) from a special case proved earlier by Mok. Crucial for our method is a new construction of the p-adic L-function of E in terms of local data.     

Embedding an Edge-colored K(a (p); λ, μ) into a Hamiltonian Decomposition of K(a (p+r); λ, μ)

Let K(a ^(p); λ, μ) be a graph with p parts, each part having size a, in which the multiplicity of each pair of vertices in the same part (in different parts) is λ (μ, respectively). In this paper we consider the following embedding problem: When can a graph decomposition of K(a ^(p); λ, μ) be extended to a Hamiltonian decomposition of K(a ^(p+r); λ, μ) for r > 0? A general result is proved, which is then used to solve the embedding problem for all ${r \geq \frac{\lambda}{\mu a}+ \frac{p-1}{a-1}}$ . The problem is also solved when r is as small as possible in two different senses, namely when r = 1 and when ${r = \frac{\lambda}{\mu a} - p+1}$ .     

A remark on Gelfand–Graev characters for finite simple groups

The famous Gelfand–Graev character of a group of Lie type G is a multiplicity free character of shape ν ^G , where ν is a suitable degree 1 character of a Sylow p-subgroup and p is the defining characteristic of G. We show that, for an arbitrary non-abelian simple group G, if ν is a linear character of a Sylow p-subgroup of G such that ν ^G is multiplicity free, then G is isomorphic to either a group of Lie type in defining characteristic p, or to a group PSL(2, q), where either p = q + 1, or p = 2 and q + 1 or q ? 1 is a 2-power.