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## You are looking at 1 - 100 of 11,017 items

## Abstract

Consider the sequence *s* of the signs of the coefficients of a real univariate polynomial *P* of degree *d*. Descartes’ rule of signs gives compatibility conditions between *s* and the pair (*r*
^{+}
*,r*
^{−}), where *r*
^{+} is the number of positive roots and *r*
^{−} the number of negative roots of *P*. It was recently asked if there are other compatibility conditions, and the answer was given in the form of a list of incompatible triples (*s*; *r*
^{+}
*,r*
^{−}) which begins at degree *d* = 4 and is known up to degree 8. In this paper we raise the question of the compatibility conditions for *i*-th derivative of *P*. We prove that up to degree 5, there are no other compatibility conditions than the Descartes conditions, the above recent incompatibilities for each *i*, and the trivial conditions given by Rolle’s theorem.

## Abstract

Let *l,m,r* be fixed positive integers such that 2*l*, 3*lm*, *l > r* and 3 | *r*. In this paper, using the BHV theorem on the existence of primitive divisors of Lehmer numbers, we prove that if min{*rlm*
^{2} − 1*,*(*l* − *r*)*lm*
^{2} + 1} *>* 30, then the equation (*rlm*
^{2} − 1)^{x} + ((*l* − *r*)*lm*
^{2} + 1)^{y} = (*lm*)^{z} has only the positive integer solution (*x,y,z*) = (1*,*1*,*2).

## Abstract

In 1975 C. F. Chen and C. H. Hsiao established a new procedure to solve initial value problems of systems of linear differential equations with constant coefficients by Walsh polynomials approach. However, they did not deal with the analysis of the proposed numerical solution. In a previous article we study this procedure in case of one equation with the techniques that the theory of dyadic harmonic analysis provides us. In this paper we extend these results through the introduction of a new procedure to solve initial value problems of differential equations with not necessarily constant coefficients.

## Abstract

Let *n*. Further, let*p*(*z*) ≡ *z*
^{n}
*p*(1*/z*). In this paper we obtain some inequalites in this direction for polynomials that belong to this class and have all their coefficients in any sector of opening *γ*, where 0 *γ < π*. Our results generalize and sharpen several of the known results in this direction, including those of Govil and Vetterlein [3], and Rahman and Tariq [12]. We also present two examples to show that in some cases the bounds obtained by our results can be considerably sharper than the known bounds.

## Abstract

In this paper, we prove that if *X* is a space with a regular *G*
_{δ}-diagonal and *X*
^{2} is star Lindelöf then the cardinality of *X* is at most 2^{c}. We also prove that if *X* is a star Lindelöf space with a symmetric *g*-function such that *g*
^{2}(*n, x*): *n* ∈ *ω*} = {*x*} for each *x* ∈ *X* then the cardinality of *X* is at most 2^{c}. Moreover, we prove that if *X* is a star Lindelöf Hausdorff space satisfying *Hψ*(*X*) = *κ* then *e*(*X*) ^{2κ}; and if *X* is Hausdorff and *we*(*X*) = *Hψ*(*X*) = *κ*subset of a space then *e*(*X*) ^{κ}. Finally, we prove that under *V* = *L* if *X* is a first countable DCCC normal space then *X* has countable extent; and under MA+¬CH there is an example of a first countable, DCCC and normal space which is not star countable extent. This gives an answer to the Question 3.10 in *Spaces with property* (*DC*(*ω*
_{1})), *Comment. Math. Univ. Carolin.*, **58(1)** (2017), 131-135.

## Abstract

Fejes Tóth [] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the *square* of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.

## Abstract

Let *H*
_{n} be the *n*-th harmonic number and let *v*
_{n} be its denominator. It is known that *v*
_{n} is even for every integer *H*
_{n} and prove that for any integer *n*, *v*
_{n} = *e*
^{n(1+o(1))}. In addition, we obtain some results of the logarithmic density of harmonic numbers.

## Abstract

We verify an upper bound of Pach and Tóth from 1997 on the midrange crossing constant. Details of their

## Abstract

Let 0 *< γ*
_{1}
*< γ*
_{2}
*<* ··· ⩽ ··· be the imaginary parts of non-trivial zeros of the Riemann zeta-function. In the paper, we consider the approximation of analytic functions by shifts of the Hurwitz zeta-function *ζ*(*s* + *iγ*
_{k}
*h, α*), *h >* 0, with parameter *α* such that the set {log(*m* + *α*): *m* ∈ *γ*
_{k}} is applied.

## Abstract

We study certain subgroups of the full group of Hopf algebra automorphisms of twisted tensor biproducts.

## Abstract

In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class *A _{p}* by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝ

*are obtained.*

^{n}## Abstract

Let *X* be a Hilbert *C**-module over a *C**-algebra *B*. In this paper we introduce two classes of operator algebras on the Hilbert *C**-module *X* called operator algebras with property *B* and *X*. Some of our results generalize the previous results. Also we investigate some properties of these classes of operator algebras.

## Abstract

Let *m* ≠ 0, ±1 and *n* ≥ 2 be integers. The ring of algebraic integers of the pure fields of type *n* = 2, 3,4. It is well known that for *n* = 2, an integral basis of the pure quadratic fields can be given parametrically, by using the remainder of the square-free part of m modulo 4. Such characterisation of an integral basis also exists for cubic and quartic pure fields, but for higher degree pure fields there are only results for special cases.

In this paper we explicitly give an integral basis of the field *n*.

## Abstract

Two classes of trigonometric sums about integer powers of secant function are evaluated that are closely related to Jordan's totient function.

## Abstract

We provide a new proof of Hua's result that every sufficiently large integer *N* ≡ 5 (mod 24) can be written as the sum of the five prime squares. Hua's original proof relies on the circle method and uses results from the theory of *L*-functions. Here, we present a proof based on the transference principle first introduced in[5]. Using a sieve theoretic approach similar to ([10]), we do not require any results related to the distributions of zeros of *L*- functions. The main technical difficulty of our approach lies in proving the pseudo-randomness of the majorant of the characteristic function of the *W*-tricked primes which requires a precise evaluation of the occurring Gaussian sums and Jacobi symbols.

## Abstract

For each even classical pretzel knot *P*(2*k*_{1} + 1, 2*k*_{2} + 1, 2*k*_{3}), we determine the character variety of irreducible SL (2, ℂ)-representations, and clarify the steps of computing its A-polynomial.

## Abstract

We present a technique to construct Cohen–Macaulay graphs from a given graph; if this graph fulfills certain conditions. As a consequence, we characterize Cohen–Macaulay paths.

## Abstract

We prove that, for any cofinally Polish space *X*, every locally finite family of non-empty open subsets of *X* is countable. It is also established that Lindelöf domain representable spaces are cofinally Polish and domain representability coincides with subcompactness in the class of *σ*-compact spaces. It turns out that, for a topological group *G* whose space has the Lindelöf Σ-property, the space *G* is domain representable if and only if it is Čech-complete. Our results solve several published open questions.

## Abstract

Let *N* be a positive integer, *N* is called an *α-Korselt number* (equivalently *α* is said an *N-Korselt base*) if *α*
_{2}
*p* − *α*
_{1} divides *α*
_{2}
*N* − *α*
_{1} for every prime divisor *p* of *N.* By the *Korselt set* of *N* over *N* is an *α*-Korselt number.

In this paper we determine explicitly for a given prime number *q* and an integer *l* ∈ ℕ \{0, 1}, the set *q ^{l}* -Korselt bases in ℚ and others in ℤ. The case of

*l*= 2.

Moreover, we show that each nonzero rational *α* is an *N*-Korselt base for infinitely many numbers *N* = *q ^{l}* where

*q*is a prime number and

*l*∈ ℕ.

## Abstract

Sufficient conditions on associated parameters *p, b* and *c* are obtained so that the generalized and “normalized” Bessel function *u _{p}*(

*z*) =

*u*(

_{p,b,c}*z*) satisfies the inequalities ∣(1 + (

*zu*″

*(*

_{p}*z*)/

*u*′

*(*

_{p}*z*)))

^{2}− 1∣ < 1 or ∣((

*zu*

_{p}(

*z*))′/

*u*(

_{p}*z*))

^{2}− 1∣ < 1. We also determine the condition on these parameters so that $$-(4(p+(b+1)/2)/c){u}_{p}^{\text{'}}(x)\prec \sqrt{1+z}$$. Relations between the parameters

*μ*and

*p*are obtained such that the normalized Lo

__mmel__function of first kind

*h*(

_{μ,p}*z*) satisfies the subordination $$1+(z{h}_{\mu ,p}^{\text{'}\text{'}}(z)/{h}_{\mu ,q}^{\text{'}}(z))\prec \sqrt{1+z}$$. Moreover, the properties of Alexander transform of the function

*h*(

_{μ,p}*z*) are discussed.

## Abstract

In this paper, we obtain necessary as well as sufficient conditions for exponential rate of decrease of the variance of the best linear unbiased estimator (BLUE) for the unknown mean of a stationary sequence possessing a spectral density. In particular, we show that a necessary condition for variance of BLUE to decrease to zero exponentially is that the spectral density vanishes on a set of positive Lebesgue measure in any vicinity of zero.

## Abstract

We prove completeness, interpolation, decidability and an omitting types theorem for certain multi-dimensional modal logics where the states are not abstract entities but have an inner structure. The states will be sequences. Our approach is algebraic addressing varieties generated by complex algebras of Kripke semantics for such logics. The algebras dealt with are common cylindrification free reducts of cylindric and polyadic algebras. For finite dimensions, we show that such varieties are finitely axiomatizable, have the super amalgamation property, and that the subclasses consisting of only completely representable algebras are elementary, and are also finitely axiomatizable in first order logic. Also their modal logics have an *N P* complete satisfiability problem. Analogous results are obtained for infinite dimensions by replacing finite axiomatizability by finite schema axiomatizability.

## Abstract

The Pell sequence *P _{n}* = 2

*P*

_{n}_{−1}+

*P*

_{n}_{−2}with initial condition

*P*

_{0}= 0,

*P*

_{1}= 1 and its associated Pell-Lucas sequence

*Q*

_{0}

*=*2,

*Q*

_{1}= 2. Here we show that 6 is the only perfect number appearing in these sequences. This paper continues a previous work that searched for perfect numbers in the Fibonacci and Lucas sequences.

## Abstract

We record an implication between a recent result due to Li, Pratt and Shakan and large gaps between arithmetic progressions.

## Abstract

We study the discrete time risk process modelled by the skip-free random walk and derive results connected to the ruin probability and crossing a fixed level for this type of process. We use the method relying on the classical ballot theorems to derive the results for crossing a fixed level and compare them to the results known for the continuous time version of the risk process. We generalize this model by adding a perturbation and, still relying on the skip-free structure of that process, we generalize the previous results on crossing the fixed level for the generalized discrete time risk process. We further derive the famous Pollaczek-Khinchine type formula for this generalized process, using the decomposition of the supremum of the dual process at some special instants of time.

## Abstract

In this paper, it has been investigated that how various stronger notions of sensitivity like 𝓕-sensitive, multi-𝓕-sensitive, (𝓕_{1}, 𝓕_{2})-sensitive, etc., where 𝓕, 𝓕_{1}, 𝓕_{2} are Furstenberg families, are carried over to countably infinite product of dynamical systems having these properties and vice versa. Similar results are also proved for induced hyperspaces.

## Abstract

We prove: For all natural numbers n and real numbers *x* ∈ [0, π] we have

The sign of equality holds if and only if *n* = 2 and *x* = 4π/5.

## Abstract

Let Vect (ℝℙ^{1}) be the Lie algebra of smooth vector fields on ℝℙ^{1}. In this paper, we classify ^{1}) to ^{1}) with coefficients in

## Abstract

To a branched cover *f* between orientable surfaces one can associate a certain *branch datum*
*f* such that *how many* these *f*'s exist, but one must of course decide what restrictions one puts on such *f*’s, and choose an equivalence relation up to which one regards them. As it turns out, quite a few natural choices for this relation are possible. In this short note we carefully analyze all these choices and show that the number of actually distinct ones is only three. To see that these three choices are indeed different from each other we employ Grothendieck's *dessins d'enfant.*

## Abstract

The aim of this paper is to study the congruences on abundant semigroups with quasi-ideal adequate transversals. The good congruences on an abundant semigroup with a quasi-ideal adequate transversal *S*° are described by the equivalence triple abstractly which consists of equivalences on the structure component parts *I, S*° and Λ. Also, it is shown that the set of all good congruences on this kind of semigroup forms a complete lattice.

## Abstract

We prove that in the category of firm acts over a firm semigroup monomorphisms co-incide with regular monomorphisms and we give an example of a non-injective monomorphism in this category. We also study conditions under which monomorphisms are injective and we prove that the lattice of subobjects of a firm act over a firm semigroup is isomorphic to the lattice unitary subacts of that act.

## Abstract

Let {*P _{n}*}

_{n≥0}be the sequence of Padovan numbers defined by

*P*

_{0}= 0,

*P*

_{1}= 1,

*P*

_{2}= 1, and

*P*

_{n}_{+3}=

*P*

_{n}_{+1}+

*P*for all

_{n}*n*≥ 0. In this paper, we find all integers

*c*admitting at least two representations as a difference between a Padovan number and a power of 3.

## Abstract

By making use of the pre-Schwarzian norm given by

*α*and type

*β*. We also employ an entirely new method to generalize and extend the results of Theorems 1, 2 and 3 in . Finally, some inequalities concerning the norm of the pre-Schwarzian derivative for Dziok-Srivastava operator are also considered.

## Abstract

For *β* an ordinal, let PEA_{β} (SetPEA_{β}) denote the class of polyadic equality (set) algebras of dimension *β*. We show that for any infinite ordinal *α*, if *n* <* ω*, the *n*-neat reduct of _{n} (regardless of the representability of *f _{a}*(

*a*) ≠ 0 and

_{∞,ω}. Various notions of representability (such as ‘satisfying the Lyndon conditions’, weak and strong) are lifted from the level of atom structures to that of atomic algebras and are further characterized via special neat embeddings. As a sample, we show that the class of atomic CA

_{n}s satisfying the Lyndon conditions coincides with the class of atomic algebras in

**ElS**

_{c}

**Nr**

_{n}

**CA**

_{ω}, where

**El**denotes ‘elementary closure’ and

**S**

_{c}is the operation of forming complete subalgebras.

## Abstract

Let {*X _{n}*:

*n*≧ 1} be a sequence of dependent random variables and let {

*w*: 1 ≦

_{nk}*k*≦

*n, n*≧ 1} be a triangular array of real numbers. We prove the almost sure version of the CLT proved by Peligrad and Utev [7] for weighted partial sums of mixing and associated sequences of random variables, i.e.

## Abstract

It is shown that if *N*(*R*) is a Lie ideal of *R* (respectively Jordan ideal and *R* is 2-torsion-free), then *N*(*R*) is an ideal. Also, it is presented a characterization of Noetherian NR rings with central idempotents (respectively with the commutative set of nilpotent elements, the Abelian unit group, the commutative commutator set).

## Abstract

In this paper we establish the boundedness of commutators of sublinear operators in weighted grand Morrey spaces. The sublinear operators under consideration contain integral operators such as Hardy-Littlewood and fractional maximal operators, Calderón-Zygmund operators, potential operators etc. The operators and spaces are defined on quasi-metric measure spaces with doubling measure.

## Abstract

In this article, the eigenvalues and eigenvectors of positive binomial operators are presented. The results generalize the previously obtained ones related to Bernstein operators. Illustrative examples are supplied.

## Abstract

A group *G* is called metahamiltonian if all its non-abelian subgroups are normal. The aim of this paper is to investigate the structure of uncountable groups of cardinality ℵ in which all proper subgroups of cardinality ℵ are metahamiltonian. It is proved that such a group is metahamiltonian, provided that it has no simple homomorphic images of cardinality ℵ. Furthermore, the behaviour of elements of finite order in uncountable groups is studied in the second part of the paper.

## Abstract

We study some mathematical properties of a new generator of continuous distributions called the Odd Nadarajah-Haghighi (ONH) family. In particular, three special models in this family are investigated, namely the ONH gamma, beta and Weibull distributions. The family density function is given as a linear combination of exponentiated densities. Further, we propose a bivariate extension and various characterization results of the new family. We determine the maximum likelihood estimates of ONH parameters for complete and censored data. We provide a simulation study to verify the precision of these estimates. We illustrate the performance of the new family by means of a real data set.

## Abstract

The intersection of all maximal right ideals of a near-ring *N* is called the quasi-radical of *N*. In this paper, first we show that the quasi-radical of the zero-symmetric near-ring of polynomials *R*
_{0}[*x*] equals to the set of all nilpotent elements of *R*
_{0}[*x*], when *R* is a commutative ring with Nil (*R*)^{2} = 0. Then we show that the quasi-radical of *R*
_{0}[*x*] is a subset of the intersection of all maximal left ideals of *R*
_{0}[*x*]. Also, we give an example to show that for some commutative ring R the quasi-radical of *R*
_{0}[*x*] coincides with the intersection of all maximal left ideals of *R*
_{0}[*x*]. Moreover, we prove that the quasi-radical of *R*
_{0}[*x*] is the greatest quasi-regular (right) ideal of it.

## Abstract

Let *G* be a finite group and *H* a subgroup of *G.* We say that *H* is an *ℌ*-subgroup of *G* if *N _{G}* (

*H*) ∩

*H*≤

^{g}*H*for all

*g*∈

*G*;

*H*is called weakly

*ℌ*-embedded in

*G*if

*G*has a normal subgroup

*K*such that

*H*=

^{G}*HK*and

*H*∩

*K*is an

*ℌ*-subgroup of

*G*, where

*H*is the normal clousre of

^{G}*H*in

*G*, i. e.,

*H*= 〈

^{G}*H*|

^{g}*g*∈

*G*〉. In this paper, we study the

*p*-nilpotence of a group

*G*under the assumption that every subgroup of order

*d*of a Sylow

*p*-subgroup

*P*of

*G*with 1 <

*d*< |

*P*| is weakly

*ℌ*-embedded in

*G*. Many known results related to

*p*-nilpotence of a group

*G*are generalized.

## Abstract

Let *R* be a discrete valuation ring, *P* ∈*R*[*X*] a monic irreducible polynomial, and *K* the quotient field of *R*. We give in this paper a lower bound for the *P* over *R* in terms of the degrees of the monic irreducible factors of the reduction of *P* modulo *P* generates a power basis for the integral closure of R in the field extension of *K* defined by *P*.

## Abstract

In this article, we study the class of rings in which every regular locally principal ideal is projective called *LPP*-rings. We investigate the transfer of this property to various constructions such as direct products, amalgamation of rings, and trivial ring extensions. Our aim is to provide examples of new classes of commutative rings satisfying the above-mentioned property.

## Abstract

In this paper we study the uniform approximation of functions by a generalization of the Picard and Gauss-Weierstrass operators of max-product type in exponential weighted spaces. We estimate the rate of approximation in terms of a suitable modulus of continuity. We extend and improve previous results.

## Abstract

This paper deals with the existence and uniqueness of weak solution of a problem which involves a class of *A*-harmonic elliptic equations of nonhomogeneous type. Under appropriate assumptions on the function *f*, our main results are obtained by using Browder Theorem.

## Abstract

In this paper, we introduce a new three-parameter generalized version of the Gompertz model called the odd log-logistic Gompertz (OLLGo) distribution. It includes some well-known lifetime distributions such as Gompertz (Go) and odd log-logistic exponential (OLLE) as special sub-models. This new distribution is quite flexible and can be used effectively in modeling survival data and reliability problems. It can have a decreasing, increasing and bathtub-shaped failure rate function depending on its parameters. Some mathematical properties of the new distribution, such as closed-form expressions for the density, cumulative distribution, hazard rate function, the *k*th order moment, moment generating function and the quantile measure are provided. We discuss maximum likelihood estimation of the OLLGo parameters as well as three other estimation methods from one observed sample. The flexibility and usefulness of the new distribution is illustrated by means of application to a real data set.

## Abstract

Let *R* be an *IF* ring, or be a ring such that each right *R*-module has a monomorphic flat envelope and the class of flat modules is coresolving. We firstly give a characterization of copure projective and cotorsion modules by lifting and extension diagrams, which implies that the classes of copure projective and cotorsion modules have some balanced properties. Then, a relative right derived functor is introduced to investigate copure projective and cotorsion dimensions of modules. As applications, some new characterizations of *QF* rings, perfect rings and noetherian rings are given.

## Abstract

In 1971 Onnewer and Waterman establish a sufficient condition which guarantees uniform convergence of Vilenkin-Fourier series of continuous function. In this paper we consider different classes of functions of generalized bounded oscillation and in the terms of these classes there are established sufficient conditions for uniform convergence of Cesàro means of negative order.

## Abstract

In , a universal linear algebraic model was proposed for describing homogeneous conformal geometries, such as the spherical, Euclidean, hyperbolic, Minkowski, anti-de Sitter and Galilei planes (). This formalism was independent from the underlying field, providing an extension and general approach to other fields, such as finite fields. Some steps were taken even for the characteristic 2 case.

In this article, we undertake the study of the characteristic 2 case in more detail. In particular, the concept of virtual quadratic spaces is used (), and a similar result is achieved for finite fields of characteristic 2 as for other fields. Some differences from the non-characteristic 2 case are also pointed out.

## Abstract

In this note connections between root extensions of monoids and some finiteness conditions on monoids are studied, giving new proofs and generalizing results of Etingof, Malcolmson and Okoh for domains. In the same spirit, results of Jedrzejewicz and Zielinski on root-closed extensions of domains are generalized and sharpened to monoids. Using the same methods, a criterion for being a completely integrally closed domain is generalized to monoids.

## Abstract

In this paper first, we prove some new generalizations of Hermite-Hadamard type inequalities for the convex function *f* and for (*s, m*)-convex function *f* in the second sense in conformable fractional integral forms. Second, by using five new integral identities, we present some new Riemann-Liouville fractional trapezoid and midpoint type inequalities. Third, using these results, we present applications to *f*-divergence measures. At the end, some new bounds for special means of different positive real numbers and new error estimates for the trapezoidal and midpoint formula are provided as well. These results give us the generalizations of the earlier results.

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and hyperbolic spaces, under some regularity assumptions. Suppose that in any of these spaces there is a pair of closed convex sets of class *C*
_{+}
^{2} with interior points, different from the whole space, and the intersections of any congruent copies of these sets are centrally symmetric (provided they have non-empty interiors). Then our sets are congruent balls. Under the same hypotheses, but if we require only central symmetry of small intersections, then our sets are either congruent balls, or paraballs, or have as connected components of their boundaries congruent hyperspheres (and the converse implication also holds).

Under the same hypotheses, if we require central symmetry of all compact intersections, then either our sets are congruent balls or paraballs, or have as connected components of their boundaries congruent hyperspheres, and either *d* ≥ 3, or *d* = 2 and one of the sets is bounded by one hypercycle, or both sets are congruent parallel domains of straight lines, or there are no more compact intersections than those bounded by two finite hypercycle arcs (and the converse implication also holds).

We also prove a dual theorem. If in any of these spaces there is a pair of smooth closed convex sets, such that both of them have supporting spheres at any of their boundary points *S*
_{d} for Sd of radius less than π/2- and the closed convex hulls of any congruent copies of these sets are centrally symmetric, then our sets are congruent balls.

In this article, a new four-parameter model is introduced which can be used in mod- eling survival data and fatigue life studies. Its failure rate function can be increasing, decreasing, upside down and bathtub-shaped depending on its parameters. We derive explicit expressions for some of its statistical and mathematical quantities. Some useful characterizations are presented. Maximum likelihood method is used to estimate the model parameters. The censored maximum likelihood estimation is presented in the general case of the multi-censored data. We demonstrate empirically the importance and exibility of the new model in modeling a real data set.

In this paper, we proved theorems which give the conditions that special operator nets on a predual of von Neumann algebras are strongly convergent under the Markov case. Moreover, we investigate asymptotic stability and existence of a lower-bound function for such nets.

Reconstruction theorems for martingales with respect to regular filtration are proved provided that the majorant of the martingale satisfies some specified condition. The ob-tained results are applied to obtain formulas for restoration of coeffcients for multiple Haar series.

For fixed integers *n*(= 0) and μ, the number of ways in which a moving particle taking a horizontal step with probability p and a vertical step with probability *q*, touches the line *Y* = *n*+*μX* for the first time, have been counted. The concept has been applied to obtain various probability distributions in independent and Markov dependent trials.

We discuss the weakly compact subsets of direct sum cones for the upper, lower and symmetric topologies and investigate the *X*-topologies of the weak upper, lower and sym-metric compact subsets of direct sum cones on product cones.

Any sequence of 4-dimensional cubes of total volume not greater than 1/8 can be online packed into the unit cube.

In this paper we formulate and prove that there exists the first positive eigenvalue of the eigenvalue problem with oblique derivative for the Laplace-Beltrami operator on the unit sphere. The firrst eigenvalue plays a major role in studying the asymptotic behaviour of solutions of oblique derivative problems in cone-like domains. Our work is motivated by the fact that the precise solutions decreasing rate near the boundary conical point is dependent on the first eigenvalue.

In this paper, we introduce comatrix group corings and define the generalized Galois group corings. Then we give the generalized Galois group coring Structure Theorem.

In present paper, we give some new reverses of the Young type inequalities which were established by X. Hu and J. Xue [7] via Kantorovich constant. Then we apply these inequalities to establish corresponding inequalities for the Hilbert-Schmidt norm and the trace norm.

Let *L*(*f*, *r*) denote the length of the closed curve which is the image of |*z*| = *r* < 1 under the mapping *w* = *f*(*z*). We establish some sufficient conditions for *L*(*f*, *r*) to be bounded and for *f*(*z*) to in the classes of strongly close-to-convex function of order *α* and to be strongly Bazilevič function of type *β* of order *α*. Moreover, we prove an inequality connected with the Fejér-Riesz's inequality.