Asymptotic <Emphasis Type="Italic">T</Emphasis><Subscript><Emphasis Type="Italic">u</Emphasis></Subscript>-Toeplitzness of weighted composition operators on <Emphasis Type="Italic">H</Emphasis><Superscript>2</Superscript>

Given a unilateral forward shift S acting on a complex, separable, innite dimensional Hilbert space H, an asymptotically S-Toeplitz operator is a bounded linear operator T on H satisfying that {S* ⁿ TS ⁿ } is convergent with respect to one of the topologies commonly used in the algebra of bounded linear operators on H. In this paper, we study the asymptotic T _u -Toeplitzness of weighted composition operators on the Hardy space H², where u is a nonconstant inner function.     

Curvature identities for principle <Emphasis Type="Italic">T</Emphasis><Superscript>1</Superscript>-bundles over almost Hermitian manifolds

We show that the identities R ₁, R ₂ and R ₃ for an almost Hermitian structure S on the base of the canonical principal T ¹-bundle are equivalent to their contact analogs for the induced almost contact metric structure S ^# on the total space of this bundle. We prove that the canonical connection of the canonical principal T ¹-bundle over a Hermitian or a quasi-Kähler manifold of class R ₃ is normal. We also prove that that the canonical connection of the canonical principal T ¹-bundle over a Vaisman-Gray manifold M of class R ₃ is normal if and only if the Lee vector of M belongs to the center of the adjoint K-algebra.     

<Emphasis Type="Italic">T</Emphasis>-Coloring of Certain Networks

Given a graph G and a finite set T of non-negative integers containing zero, a T-coloring of G is a non-negative integer function f defined on V(G) such that \(|f(x)-f(y)|\not \in T\) whenever \((x,y)\in E(G)\). The span of T-coloring is the difference between the largest and smallest colors, and the T-span of G is the minimum span over all T-colorings f of G. The edge span of a T-coloring is the maximum value of \(|f(x)-f(y)|\) over all edges \((x,y)\in E(G)\), and the T-edge span of G is the minimum edge span over all T-colorings f of G. In this paper, we compute T-span and T-edge span of crown graph, circular ladder and mobius ladder, generalized theta graph, series-parallel graph and wrapped butterfly network.     

The existence of eigenvalues for operators acting in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(<Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We present conditions that allow us to prove the existence of eigenvalues and characteristic values for operator F(D) ? C(λ): L ²(R ^m ) → L ²(R ^m ), where F(D) is a pseudo-differential operator with a symbol F(iξ) and C(λ): L ²(R ^m ) → L ²(R ^m ) is a linear continuous operator.     

Economic design of an adaptive <Emphasis Type="Italic">T</Emphasis><Superscript>2</Superscript> control chart

Varying three parameters of the Hotelling's T² control chart, namely, the sample size, the sampling interval length, and the action limit, results in a great improvement over the traditional T² control chart particularly for the small process shift. This paper presents the economic model for designing the variable parameters T² chart. In the economic design, a cost function is constructed, involving the cost of sampling and testing, the cost of false alarm, the cost to detect and remove the assignable cause, and the cost when the process is operating out-of-control. Also, a heuristic method of finding optimal values of adaptive design parameters for minimizing the cost function is presented. Variable and fixed parameters T² control charts are compared with respect to the expected loss per unit time. Furthermore, the effects of the initial sample number (used for estimating the parameters of F-distribution) upon the operating cost and adaptive design parameters are examined.     

Processes of <Emphasis Type="Italic">r</Emphasis><Superscript><Emphasis Type="Italic">t</Emphasis><Emphasis Type="Italic">h</Emphasis></Superscript> largest

For integers n ≥ r, we treat the rth largest of a sample of size n as an \(\mathbb {R}^{\infty }\)-valued stochastic process in r which we denote as M^(r). We show that the sequence regarded in this way satisfies the Markov property. We go on to study the asymptotic behavior of M^(r) as r → ∞, and, borrowing from classical extreme value theory, show that left-tail domain of attraction conditions on the underlying distribution of the sample guarantee weak limits for both the range of M^(r) and M^(r) itself, after norming and centering. In continuous time, an analogous process Y^(r) based on a two-dimensional Poisson process on \(\mathbb {R}_{+}\times \mathbb {R}\) is treated similarly, but we note that the continuous time problems have a distinctive additional feature: there are always infinitely many points below the rth highest point up to time t for any t >?0. This necessitates a different approach to the asymptotics in this case.     

2-Recognizability by prime graph of <Emphasis Type="Italic">PSL</Emphasis>(2, <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>)

Let G be a finite group and let Γ(G) be the prime graph of G. Assume p prime. We determine the finite groups G such that Γ(G) = Γ(PSL(2, p ²)) and prove that if p ≠ 2, 3, 7 is a prime then k(Γ(PSL(2, p ²))) = 2. We infer that if G is a finite group satisfying |G| = |PSL(2, p ²)| and Γ(G) = Γ(PSL(2, p ²)) then G ? PSL(2, p ²). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications are also considered of this result to the problem of recognition of finite groups by element orders.     

<Emphasis Type="Italic">M</Emphasis><Superscript><Emphasis Type="Italic">X</Emphasis></Superscript>/<Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">c</Emphasis> Queue with catastrophes and state-dependent control at idle time

We consider an M ^X /M/c queue with catastrophes and state-dependent control at idle time. Properties of the queues which terminate when the servers become idle are first studied. Recurrence, equilibrium distribution, and equilibrium queue-size structure are studied for the case of resurrection and no catastrophes. All of these properties and the first effective catastrophe occurrence time are then investigated for the case of resurrection and catastrophes. In particular, we obtain the Laplace transform of the transition probability for the absorbing M ^X /M/c queue.     

Integral Modular Categories of Frobenius-Perron Dimension <Emphasis Type="Italic">pq</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Integral modular categories of Frobenius-Perron dimension pq ⁿ , where p and q are primes, are considered. It is already known that such categories are group-theoretical in the cases of 0 ≤ n ≤ 4. In the general case we determine that these categories are either group-theoretical or contain a Tannakian subcategory of dimension q ⁱ for i > 1. We then show that all integral modular categories \(\mathcal {C}\) with \(\text {FPdim}(\mathcal {C})=pq^{5}\) are group-theoretical, and, if in addition p < q, all with \(\text {FPdim}(\mathcal {C})=pq^{6}\) or pq ⁷ are group-theoretical. In the process we generalize an existing criterion for an integral modular category to be group-theoretical.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic"> p</Emphasis></Superscript>-multipliers for the Hilbert space valued functions on the Heisenberg group

It is well known that if m is an L ^p -multiplier for the Fourier transform on \({\mathbb{R}^n}\) , (1 < p < ∞) then there exists a pseudomeasure σ such that T _m f = σ * f . A similar problem is discussed for the L ^p ?Fourier multipliers for \({\mathcal{H}}\) -valued functions on the Heisenberg group, where \({\mathcal{H}}\) is a separable Hilbert space.     

Intersections of Primary Subgroups in Nonsoluble Finite Groups Isomorphic to <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(2<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>)

Given a finite group G with socle isomorphic to L _n (2 ^m ), we describe (up to conjugacy) all ordered pairs of primary subgroups A and B in G such that A ∩ B ^g ≠ 1 for all g ∈ g.     

Influence of <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-supplemented subgroups on the structure of <Emphasis Type="Italic">p</Emphasis>-modular subgroups

A subgroup K of G is M _p -supplemented in G if there exists a subgroup B of G such that G = KB and TB < G for every maximal subgroup T of K with |K: T| = p ^α. We study the structure of the chief factor of G by using M _p -supplemented subgroups and generalize the results of Monakhov and Shnyparkov by involving the relevant results about the p-modular subgroup O ^p (G) of G.     

On unconditional basis property in the space <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">R</Emphasis>) of a system of Franklin functions with vanishing means

We define a general Franklin system of functions on R with vanishing means, generated by an admissible sequence T. A necessary and sufficient condition on T is found for the corresponding general Franklin system of functions on R with vanishing means to be an unconditional basis in the space H ¹(R).     

Some <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Rigidity Results for Complete Manifolds with Harmonic Curvature

Let (M ⁿ , g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R m? the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R m? goes to zero uniformly at infinity if for \(p\geq \frac n2\), the L ^p -norm of R m? is finite. Moreover, If R is positive, then (M ⁿ , g) is compact. As applications, we prove that (M ⁿ , g) is isometric to a spherical space form if for \(p\geq \frac n2\), R is positive and the L ^p -norm of R m? is pinched in [0, C ₁), where C ₁ is an explicit positive constant depending only on n, p, R and the Yamabe constant. We give an isolation theorem of the trace-free Ricci curvature tensor of compact locally conformally flat Riemannian n-manifolds with constant positive scalar curvature, which extends Theorem 1 of Hebey and M. Vaugon (J. Geom. Anal. 6, 531–553, 1996). This result is sharp, and we can precisely characterize the case of equality. In particular, when n = 4, we recover results by Gursky (Indiana Univ. Math. J. 43, 747–774, 1994; Ann. Math. 148, 315–337, 1998).     

Largest <Emphasis Type="Italic">H</Emphasis>-eigenvalue of uniform <Emphasis Type="Italic">s</Emphasis>-hypertrees

The k-uniform s-hypertree G = (V,E) is an s-hypergraph, where 1 ≤ s ≤ k - 1; and there exists a host tree T with vertex set V such that each edge of G induces a connected subtree of T. In this paper, some properties of uniform s-hypertrees are establised, as well as the upper and lower bounds on the largest H-eigenvalue of the adjacency tensor of k-uniform s-hypertrees in terms of the maximal degree Δ. Moreover, we also show that the gap between the maximum and the minimum values of the largest H-eigenvalue of k-uniform s-hypertrees is just Θ(Δ ^s/k ).     

Periodic Groups Saturated with the Groups <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

Given an indexing set I and a finite field K_α for each α ∈ I, let ? = {L₂(K_α) | α ∈ I} and \(\mathfrak{N} = \{ SL_2 (K_\alpha )|\alpha \in I\}\). We prove that each periodic group G saturated with groups in \(\Re (\mathfrak{N})\) is isomorphic to L₂(P) (respectively SL₂(P)) for a suitable locally finite field P.     

<Emphasis Type="Italic">E</Emphasis><Superscript>*</Superscript>-dense <Emphasis Type="Italic">E</Emphasis>-semigroups

We investigate E^*-dense semi\-groups as analogues of E-densesemigroupsfor semigroups with zero. We give a characterisation theorem forE^*-dense semigroups whose idempotents form a *-rectangularband. The construction methods of generalised Rees matrix semigroupsare employed to provide examples and illustrations. Our results areanalogous to those of Weipoltshammer for E-dense semigroups.     

Limit theorems for inverse process <Emphasis Type="Italic">T</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> of Hawkes process

We consider linear Hawkes process N _t and its inverse process T _n . The limit theorems for N _t are well known and studied by many authors. In this paper, we study the limit theorems for T _n . In particular, we investigate the law of large numbers, the central limit theorem and the large deviation principle for T _n . The main tool of the proof is based on immigration-birth representation and the observations on the relation between N _t and T _n .     

The dual space of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> of a vector measure

For a vector measure ν having values in a real or complex Banach space and \({p \in}\) [1, ∞), we consider L ^p (ν) and \({L_{w}^{p}(\nu)}\), the corresponding spaces of p-integrable and scalarly p-integrable functions. Given μ, a Rybakov measure for ν, and taking q to be the conjugate exponent of p, we construct a μ-Köthe function space E _q (μ) and show it is σ-order continuous when p > 1. In this case, for the associate spaces we prove that L ^p (ν) ^×  = E _q (μ) and \({E_q(\mu)^\times = L_w^p(\nu)}\). It follows that \({L_p (\nu) ^{**} = L_w^p (\nu)}\). We also show that L ¹ (ν) ^×  may be equal or not to E _∞ (μ).