Characterizations of the Vertex Operator Algebras ${V_{L}^{T}}$ and VLO${V_{L}^{O}}$

In this paper, we give characterizations of the rational vertex operator algebras \({V_{L}^{T}}\) and \({V_{L}^{O}}\), where L is the root lattice of type A ₁, T is the tetrahedral group, and O is the octahedral group. By these two characterizations, the classification of rational VOAs of central charge 1 is reduced to the characterization of \({V_{L}^{I}}\) where I is the icosahedral group.     

Dual affine connections on a quadratic hyperband distribution in a projective-metric space and their applications

In this paper, we develop fundamentals of the dual theory of quadratic hyperband distributions H of m-dimensional line elements in a projective-metric space K _n (m < n ? 1). In particular, we show that, on a dual normalized distribution H, there are induced two dual affine connections and indicate some applications of these connections to the geometry of m-webs on H.     

Maxima and minima of independent and non-identically distributed bivariate Gaussian triangular arrays

In this paper, joint limit distributions of maxima and minima on independent and non-identically distributed bivariate Gaussian triangular arrays is derived as the correlation coefficient of ith vector of given nth row is the function of i/n. Furthermore, second-order expansions of joint distributions of maxima and minima are established if the correlation function satisfies some regular conditions.     

Orthogonal Polynomials Associated with Equilibrium Measures on $\mathbb {R}$

Let K be a non-polar compact subset of \(\mathbb {R}\) and μ _K denote the equilibrium measure of K. Furthermore, let P _n (?;μ _K ) be the n-th monic orthogonal polynomial for μ _K . It is shown that \(\|P_{n}\left (\cdot ; \mu _{K}\right )\|_{L^{2}(\mu _{K})}\), the Hilbert norm of P _n (?;μ _K ) in L ²(μ _K ), is bounded below by Cap(K) ⁿ for each \(n\in \mathbb {N}\). A sufficient condition is given for\(\left (\|P_{n}\left (\cdot ;\mu _{K}\right )\|_{L^{2}(\mu _{K})}/\text {Cap}(K)^{n}\right )_{n=1}^{\infty }\) to be unbounded. More detailed results are presented for sets which are union of finitely many intervals.     

The Real Powers of the Convolution of a Gamma Distribution and a Bernoulli Distribution

In this paper, we essentially compute the set of x,y>0 such that the mapping \(z\longmapsto(1-r+re^{z})^{x}(\frac{\lambda}{\lambda-z})^{y}\) is a Laplace transform. If X and Y are two independent random variables which have respectively Bernoulli and Gamma distributions, we denote by μ the distribution of X+Y. The above problem is equivalent to finding the set of x>0 such that μ ^*x exists.     

On the boundedness of generalized solutions of higher-order nonlinear elliptic equations with data from an Orlicz–Zygmund class

In the present paper, a 2mth-order quasilinear divergence equation is considered under the condition that its coefficients satisfy the Carathéodory condition and the standard conditions of growth and coercivity in the Sobolev space W^m,p(Ω), Ω ? Rⁿ, p > 1. It is proved that an arbitrary generalized (in the sense of distributions) solution u ∈ W₀^m,p (Ω) of this equation is bounded if m ≥ 2, n = mp, and the right-hand side of this equation belongs to the Orlicz–Zygmund space L(log L)^n?1(Ω).     

Expectiles,Omega Ratios and Stochastic Ordering

In this paper we introduce the expectile order, defined by X ≤ _e Y if e _α (X) ≤e _α (Y) for each α ∈ (0, 1), where e _α denotes the α-expectile. We show that the expectile order is equivalent to the pointwise ordering of the Omega ratios, and we derive several necessary and sufficient conditions. In the case of equal means, the expectile order can be easily characterized by means of the stop-loss transform; in the more general case of different means we provide some sufficient conditions. In contrast with the more common stochastic orders such as ≤ _{s t} and ≤ _{c x} , the expectile order is not generated by a class of utility functions and is not closed with respect to convolutions. As an illustration, we compare the ≤ _{s t} , ≤ _{i c x} and ≤ _e orders in the family of Lomax distributions and compare Lomax distributions fitted to real world data of natural disasters in the U.S. caused by different sources of weather risk like storms or floods.     

Hardy and Carleson Measure Spaces Associated with Operators on Spaces of Homogeneous Type

Let (X, d, μ) be a metric measure space with doubling property. The Hardy spaces associated with operators L were introduced and studied by many authors. All these spaces, however, were first defined by L ²(X) functions and finally the Hardy spaces were formally defined by the closure of these subspaces of L ²(X) with respect to Hardy spaces norms. A natural and interesting question in this context is to characterize the closure. The purpose of this paper is to answer this question. More precisely, we will introduce \({CMO}_{L}^{p}(X)\), the Carleson measure spaces associated with operators L, and characterize the Hardy spaces associated with operators L via \(({CMO}_{L}^{p}(X))'\), the distributions of \({CMO}_{L}^{p}(X)\).     

A new class of metric divergences on probability spaces and its applicability in statistics

The classI _f β, βε(0, ∞], off-divergences investigated in this paper is defined in terms of a class of entropies introduced by Arimoto (1971,Information and Control,19, 181–194). It contains the squared Hellinger distance (for β=1/2), the sumI(Q ₁‖(Q ₁+Q ₂)/2)+I(Q ₂‖(Q ₁+Q ₂)/2) of Kullback-Leibler divergences (for β=1) and half of the variation distance (for β=∞) and continuously extends the class of squared perimeter-type distances introduced by Österreicher (1996,Kybernetika,32, 389–393) (for βε (1, ∞]). It is shown that\((I_{f_\beta } (Q_1 ,Q_2 ))^{\min (\beta ,1/2)}\) are distances of probability distributionsQ ₁,Q ₂ for β ε (0, ∞). The applicability of\(I_{f_\beta }\)-divergences in statistics is also considered. In particular, it is shown that the\(I_{f_\beta }\)-projections of appropriate empirical distributions to regular families define distribution estimates which are in the case of an i.i.d. sample of size'n consistent. The order of consistency is investigated as well.     

Knauf’s degree and monodromy in planar potential scattering

We consider Hamiltonian systems on (T*?², dq ∧ dp) defined by a Hamiltonian function H of the “classical” form H = p ²/2 + V(q). A reasonable decay assumption V(q) → 0, ‖q‖ → ∞, allows one to compare a given distribution of initial conditions at t = ?∞ with their final distribution at t = +∞. To describe this Knauf introduced a topological invariant deg(E), which, for a nontrapping energy E > 0, is given by the degree of the scattering map. For rotationally symmetric potentials V(q) = W(‖q‖), scattering monodromy has been introduced independently as another topological invariant. In the present paper we demonstrate that, in the rotationally symmetric case, Knauf’s degree deg(E) and scattering monodromy are related to one another. Specifically, we show that scattering monodromy is given by the jump of the degree deg(E), which appears when the nontrapping energy E goes from low to high values.     

<Emphasis Type="Italic">LJ</Emphasis>-spaces

In this paper LJ-spaces are introduced and studied. They are a common generalization of Lindelöf spaces and J-spaces researched by E. Michael. A space X is called an LJ-space if, whenever {A, B} is a closed cover of X with A ∩ B compact, then A or B is Lindelöf. Semi-strong LJ-spaces and strong LJ-spaces are also defined and investigated. It is demonstrated that the three spaces are different and have interesting properties and behaviors.     

Pure Gaussian limit distributions of trigonometric series with bounded gaps

This paper is mainly concerned with the limit distribution of \((\cos 2\pi n_{1}x+\cdots +\cos 2\pi n_{N}x)/\sqrt{N}\) on the unit interval when the increasing sequence {n _k } has bounded gaps, i.e., 1≤n _k+1?n _k =O(1). By Bobkov–Götze [4], it was proved that the limiting variance must be less than 1/2 in this case. They proved that the centered Gaussian distribution with variance 1/4 together with mixtures of Gaussian distributions belonging to a huge class can be limit distributions. In this paper it is proved that any Gaussian distribution with variance less than 1/2 can be a limit distribution.     

Cartesian Closed Subcategories of $CONT_{\ll }^{{\kern 8pt}\ast }$

Let CONT _? be the category of continuous domains and Scott continuous mappings that preserve the way-below relation on domains. Let ω-ALG _? be the full subcategory of CONT _? consisting of all countably based algebraic domains, and F I N be the category of finite posets and monotone mappings. The main result proved in this paper is that F I N is the largest Cartesian closed full subcategory of ω-ALG _?. On the other hand, it is shown that the algebraic L-domains form a Cartesian closed full subcategory of ALG _?.     

Generalized 3-edge-connectivity of Cartesian product graphs

The generalized k-connectivity κ _k (G) of a graph G was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized k-edge-connectivity which is defined as λ _k (G) = min{λ(S): S ? V (G) and |S| = k}, where λ(S) denotes the maximum number l of pairwise edge-disjoint trees T ₁, T ₂, …, T _l in G such that S ? V (T _i ) for 1 ? i ? l. In this paper we prove that for any two connected graphs G and H we have λ ₃(G □ H) ? λ ₃(G) + λ ₃(H), where G □ H is the Cartesian product of G and H. Moreover, the bound is sharp. We also obtain the precise values for the generalized 3-edge-connectivity of the Cartesian product of some special graph classes.     

Lie n-derivatio ns o n $ {\mathcal {}}$-subspace lattice algebras

Let \(\mathcal {L}\) be a \(\mathcal {J}\)-subspace lattice on a Banach space X over the real or complex field \(\mathbb {F}\) with dimX ≥ 3 and let n ≥ 2 be an integer. Suppose that dimK ≠ 2 for every \(K\in \mathcal {J}{(\mathcal L)}\) and \(L: \text {Alg}\, \mathcal {L}\rightarrow \text {Alg}\,\mathcal {L}\) is a linear map. It is shown that L satisfies \({\sum }_{i=1}^{n}p_{n} (A_{1}, \ldots , A_{i-1}, L(A_{i}), A_{i+1}, \ldots , A_{n})=0\) whenever p _n (A ₁,A ₂,…,A _n ) = 0 for \(A_{1},A_{2},\ldots ,A_{n}\in \text {Alg}\,\mathcal {L}\) if and only if for each \(K\in \mathcal {J}(\mathcal {L})\), there exists a bounded linear operator \(T_{K}\in \mathcal {B}(K)\), a scalar λ _K and a linear functional \(h_{K}: \text {Alg}\,\mathcal {L}\rightarrow \mathbb {F}\) such that L(A)x = (T _K A ? A T _K + λ _K A + h _K (A)I)x for all x ∈ K and all \(A\in \text {Alg}\,\mathcal {L}\). Based on this result, a complete characterization of linear n-Lie derivations on \(\text {Alg}\,\mathcal {L}\) is obtained.     

Information of orderα and typeβ

Information I_α ^β (Q/P) of orderα and typeβ is introduced and it is shown that for every fixedβ, this information is a monotonic increasing function ofα. It is also shown that information of orderα and type 1 is non-negative when\(\sum\limits_{k = 1}^N { q_k } \geqslant \sum\limits_{k = 1}^N { p_k } \), where (q ₁,q ₂ …,q _N) and (p ₁,p ₂, …,p _N) are generalised probability distributions for Q and P respectively.     

Moduli of Continuity of Harmonic Quasiregular Mappings in $\mathbb{B}^n$

We prove that ω _u (δ)?≤?Cω _f (δ), where \(u : \overline{\mathbb{B}^n} \rightarrow \mathbb{R}^n\) is the harmonic extension of a continuous map \(f:\mathbb{S}^{n-1}\rightarrow\mathbb{R}^n\), if u is a K-quasiregular map. Here C is a constant depending only on n, ω _f and K and ω _h denotes the modulus of continuity of h.     

Binomial transforms and integer partitions into parts of <Emphasis Type="Italic">k</Emphasis> different magnitudes

A relationship between the general linear group of degree n over a finite field and the integer partitions of n into parts of k different magnitudes was investigated recently by the author. In this paper, we use a variation of the classical binomial transform to derive a new connection between partitions into parts of k different magnitudes and another finite classical group, namely the symplectic group Sp. New identities involving the number of partitions of n into parts of k different magnitudes are introduced in this context.     

Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight

In this paper,the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w(x) = |x|~(2α)e~(-(x~4+tx~2)),x ∈ R,where α is a constant larger than -1/2 and t is any real number. They consider this problem in three separate cases:(i) c -2,(ii) c =-2,and(iii) c -2,where c := t N~(-1/2) is a constant,N = n + α and n is the degree of the polynomial. In the first two cases,the support of the associated equilibrium measure μ_t is a single interval,whereas in the third case the support of μ_t consists of two intervals. In each case,globally uniform asymptotic expansions are obtained in several regions. These regions together cover the whole complex plane. The approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou(1993).     

Extremal behavior of recurrent random sequences

A process Y _n , n ≥ 1, satisfying the stochastic recurrent equation Y _n = A _n Y _n?1 + B _n , n ≥ 1, Y ₀ ≥ 0, is studied in the paper; here (A _n , B _n ), n ≥ 1, are independent identically distributed pairs of nonnegative random variables. The cases when the values A _n have a lognormal and log-Laplace distributions are considered. The tail index κ (for a stationary distribution) and the extremal index ? are studied. In the lognormal case, κ is determined and some useful properties of ? are established. In the log-Laplace case the both characteristics are obtained in explicit form.