Double Arrays,Triple Arrays and Balanced Grids with <Emphasis Type="Italic">v</Emphasis>=<Emphasis Type="Italic">r</Emphasis>+<Emphasis Type="Italic">c</Emphasis> −1

In Theorem 6.1 of McSorley et al. [3] it was shown that, when v=r+c−1, every triple array TA(v,k,λ_rr,λ_cc,k:r× c) is a balanced grid BG(v,k,k:r × c). Here we prove the converse of this Theorem. Our final result is: Let v=r+c−1. Then every triple array is a TA(v,k,c−k,r−k,k:r× c) and every balanced grid is a BG(v,k,k:r× c), and they are equivalent.Communicated by: J.D. Key     

On blocks with <Emphasis Type="Italic">K</Emphasis>(<Emphasis Type="Italic">B</Emphasis>)−<Emphasis Type="Italic">L</Emphasis>(<Emphasis Type="Italic">B</Emphasis>)=1

We use the method of local representation and original method of Brauer to study the block with K(B)−L(B)=1, and get some properties on the defect group and the structure of this kind of blocks. Then, we show that K(B) conjecture holds for this kind of blocks.     

Difference sets with <Emphasis Type="Italic">n</Emphasis> = 5<Emphasis Type="Italic"> p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>

Let D be a (v, k, λ)-difference set in an abelian group G, and (v, 31) = 1. If n = 5p ^r with p a prime not dividing v and r a positive integer, then p is a multiplier of D. In the case 31|v, we get restrictions on the parameters of such difference sets D for which p may not be a multiplier.      

On the equation Δ<Emphasis Type="Italic">u</Emphasis> + <Emphasis Type="Italic">q</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)<Emphasis Type="Italic">u</Emphasis> = 0

Sufficient conditions for the blow-up of nontrivial generalized solutions of the interior Dirichlet problem with homogeneous boundary condition for the homogeneous elliptic-type equation Δu + q(x)u = 0, where either q(x) ≠ const or q(x) = const= λ > 0, are obtained. A priori upper bounds (Theorem 4 and Remark 6) for the exact constants in the well-known Sobolev and Steklov inequalities are established.     

On 2 – (<Emphasis Type="Italic">n</Emphasis><Superscript>2</Superscript>, 2<Emphasis Type="Italic">n</Emphasis>, 2<Emphasis Type="Italic">n</Emphasis>–1) designs with three intersection numbers

The simple incidence structure , formed by the points and the unordered pairs of distinct parallel lines of a finite affine plane of order n > 4, is a 2 – (n ²,2n,2n–1) design with intersection numbers 0,4,n. In this paper, we show that the converse is true, when n ≥ 5 is an odd integer. Supported by M.I.U.R., Università di Palermo.     

On Analytic Campanato and <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>, <Emphasis Type="Italic">s</Emphasis>) Spaces

We investigate the relation between analytic Campanato spaces \(\mathcal {AL}_{p,s}\) and the spaces F(p, q, s), characterize the bounded and compact Riemann–Stieltjes operators from \(\mathcal {AL}_{p,s}\) to \(F(p,p-s-1,s)\). We also describe the corona theorem and the interpolating sequences for the class \(F(p,p-2,s)\), which is the Möbius invariant subspace of the analytic Besov type spaces \(B_p(s)\).     

On the regularity of scalar <Emphasis Type="Italic">p</Emphasis>-Harmonic functions in <Emphasis Type="Italic">n</Emphasis> dimensions with 2 ≤ <Emphasis Type="Italic">p</Emphasis> < <Emphasis Type="Italic">n</Emphasis>

Using Tilli’s technique [Cal Var 25(3):395–401, 2006], we shall give a new proof of the regularity of the local minima of the functional

$J\left( u\right) =\int\limits_{\Omega } \left\vert \partial u\right\vert^{p}\,dx$

with Ω a domain of class C ^{0, 1} in \({\mathbb{R}^{n}}\) and 2 ≤ p < n.

     

Loop subgroups of <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">r</Emphasis></Subscript> and the image of their stabilizer subgroups in GL<Subscript><Emphasis Type="Italic">r</Emphasis></Subscript>(ℤ)

For a representative class of subgroups of F _r , the image of their stabilizer subgroup under the action of Aut(F _r ) in GL _r (ℤ) is calculated.     

On common values of <Emphasis Type="Italic">φ</Emphasis>(<Emphasis Type="Italic">n</Emphasis>) and <Emphasis Type="Italic">σ</Emphasis>(<Emphasis Type="Italic">m</Emphasis>). I

We show, conditional on a uniform version of the prime k-tuples conjecture, that there are x/(log x)^1+o(1) numbers not exceeding x common to the ranges of φ and σ. Here φ is Euler’s totient function and σ is the sum-of-divisors function.     

Asymptotic Properties of <Emphasis Type="Italic">QML</Emphasis> Estimation of Multivariate Periodic <Emphasis Type="Italic">CCC</Emphasis> − <Emphasis Type="Italic">GARCH</Emphasis> Models

In this paper, we explore some probabilistic and statistical properties of constant conditional correlation (CCC) multivariate periodic GARCH models (CCC ? PGARCH for short). These models which encompass some interesting classes having (locally) long memory property, play an outstanding role in modelling multivariate financial time series exhibiting certain heteroskedasticity. So, we give in the first part some basic structural properties of such models as conditions ensuring the existence of the strict stationary and geometric ergodic solution (in periodic sense). As a result, it is shown that the moments of some positive order for strictly stationary solution of CCC ? PGARCH models are finite.Upon this finding, we focus in the second part on the quasi-maximum likelihood (QML) estimator for estimating the unknown parameters involved in the models. So we establish strong consistency and asymptotic normality (CAN) of CCC ? PGARCH models.     

On <Emphasis Type="Italic">α</Emphasis>-regular archimedean <Emphasis Type="Italic">f</Emphasis>-rings

For a commutative ring A with identity, and for infinite cardinals α as well as the symbol ∞, which indicates the situation in which there are no cardinal restrictions, one defines A to be α-regular if for each subset D of A, with |D| < α and de = 0, for any two distinct d, e ∈ D, there is an s ∈ A such that d ² s = d, for each d ∈ D, and if xd = 0, for each d ∈ D, then xs = 0     

On the Strong <Emphasis Type="Italic">n</Emphasis>-partite Tournaments with Exactly Two Cycles of Length <Emphasis Type="Italic">n</Emphasis> − 1

Gutin and Rafiey (Australas J. Combin. 34 (2006), 17-21) provided an example of an n-partite tournament with exactly n ? m + 1 cycles of length of m for any given m with 4 ≤ m ≤ n, and posed the following question. Let 3 ≤ m ≤ n and n ≥ 4. Are there strong n-partite tournaments, which are not themselves tournaments, with exactly n ? m + 1 cycles of length m for two values of m? In the same paper, they showed that this question has a negative answer for two values n ? 1 and n. In this paper, we prove that a strong n-partite tournament with exactly two cycles of length n ? 1 must contain some given multipartite tournament as subdigraph. As a corollary, we also show that the above question has a negative answer for two values n ? 1 and any l with 3 ≤ l ≤ n and l ≠ n ? 1.     

On <Emphasis Type="Italic">g</Emphasis>-<Emphasis Type="Italic">α</Emphasis>-irresolute functions

We introduce the so-called g-α-irresolute functions in generalized topological spaces. We obtain some properties and several characterizations of this type of functions.     

Volume Comparison of Conformally Compact Manifolds with Scalar Curvature <Emphasis Type="Italic">R</Emphasis> ≥ −<Emphasis Type="Italic">n</Emphasis> (<Emphasis Type="Italic">n</Emphasis> − 1)

In this paper, we use the normalized Ricci–DeTurk flow to prove a stability result for strictly stable conformally compact Einstein manifolds. As an application, we show a local volume comparison of conformally compact manifolds with scalar curvature R ≥ ?n (n ? 1) and also the rigidity result when certain relative volume is zero.     

Derivatives for smooth representations of <Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, ℝ) and <Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, ℂ)

The notion of derivatives for smooth representations of GL(n, ? _p ) was defined in [BZ77]. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations in [Sah89] and called the “adduced” representation. In this paper we define derivatives of all orders for smooth admissible Fréchet representations of moderate growth. The real case is more problematic than the p-adic case; for example, arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation.In the companion paper [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations.We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89, Sah90, SaSt90, GS13a].     

A theorem of the alternatives for the equation |<Emphasis Type="Italic">Ax</Emphasis>| − |<Emphasis Type="Italic">B</Emphasis>||<Emphasis Type="Italic">x</Emphasis>| = <Emphasis Type="Italic">b</Emphasis>

A theorem of the alternatives for the equation \({|Ax|-|B||x|=b\ (A,B\in{\mathbb{R}}^{n\times n},\, b\in{\mathbb{R}}^n)}\) is proved and several consequences are drawn. In particular, a class of matrices A, B is identified for which the equation has exactly 2 ⁿ solutions for each positive right-hand side b.     

Bilocal automorphisms of <Emphasis Type="Italic">T</Emphasis><Subscript>∞</Subscript>(<Emphasis Type="Italic">F</Emphasis>)

We prove that if F is a field such that |F| > 2, then every bilocal automorphism of T _∞(F) - the algebra of ? × ? upper triangular matrices over F, is an automorphism.     

On nonregular ideals and <Emphasis Type="Italic">z</Emphasis>°-ideals in <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">X</Emphasis>)

The spaces X in which every prime z°-ideal of C(X) is either minimal or maximal are characterized. By this characterization, it turns out that for a large class of topological spaces X, such as metric spaces, basically disconnected spaces and one-point compactifications of discrete spaces, every prime z°-ideal in C(X) is either minimal or maximal. We will also answer the following questions: When is every nonregular prime ideal in C(X) a z°-ideal? When is every nonregular (prime) z-ideal in C(X) a z°-ideal? For instance, we show that every nonregular prime ideal of C(X) is a z°-ideal if and only if X is a ?-space (a space in which the boundary of any zeroset is contained in a zeroset with empty interior).     

On the Crossing Number of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> □ <Emphasis Type="Italic">P</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

Crossing numbers of graphs are in general very difficult to compute. There are several known exact results on the crossing number of the Cartesian products of paths, cycles or stars with small graphs. In this paper we study cr(K_m □ P_n), the crossing number of the Cartesian product K_m □ P_n. We prove that for m ≥ 3,n ≥ 1 and cr(K_m □ P_n)≥ (n − 1)cr(K_m+2 − e) + 2cr(K_m+1). For m≤ 5, according to Klešč, Jendrol and Ščerbová, the equality holds. In this paper, we also prove that the equality holds for m = 6, i.e., cr(K₆ □ P_n) = 15n + 3. Research supported by NFSC (60373096, 60573022).     

Characterization of <Emphasis Type="Italic">α</Emphasis>1 and <Emphasis Type="Italic">α</Emphasis>2-matrices

This paper deals with some properties of α1-matrices and α2-matrices which are subclasses of nonsingular H-matrices. In particular, new characterizations of these two subclasses are given, and then used for proving algebraic properties related to subdirect sums and Hadamard products.