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For a large class of arithmetic functions *f*, it is possible to show that, given an arbitrary integer κ ≤ 2, the string of inequalities *f*(*n* + 1) < *f*(*n* + 2) < … < *f*(*n* + κ) holds for in-finitely many positive integers *n*. For other arithmetic functions *f*, such a property fails to hold even for κ = 3. We examine arithmetic functions from both classes. In particular, we show that there are only finitely many values of *n* satisfying σ_{2}(*n* − 1) < σ_{2} < σ_{2}(*n* + 1), where σ_{2}(*n*) = ∑_{d|n}
*d*
^{2}. On the other hand, we prove that for the function *f*(*n*) := ∑_{p|n}
*p*
^{2}, we do have *f*(*n* − 1) < *f*(*n*) < *f*(*n* + 1) in finitely often.

In this paper, a fourth-order nonlinear difference equation is considered. By making use of the critical point theory, we establish various sets of sufficient conditions for the existence and nonexistence of solutions for Neumann boundary value problem and give some new results. Results obtained generalize and complement the existing ones.

In this paper, we consider the system

^{N}with smooth boundary, λ,

*μ*> 0,

*p*,

*q*,

*r*1,

*r*2,

*α*and

*β*are continuous functions on

*μ*> 0, there exists λ

_{*}sufficiently small, and λ* large enough such that for any λ ∈ (0; λ

_{*}) ∪ (λ*, ∞), the above system has a nontrivial weak solution. The proof relies on some variational arguments based on the Ekeland’s variational principle and some adequate variational methods.

In order to give an excellent description of income distributions, although a large number of functional forms have been proposed, but the four-parameter generalized beta model of the second kind (GB2), introduced by J. B. McDonald [18], is now widely acknowledged which is including many other models as special or limiting cases.One of the fundamentals of statistical inference is the estimation problem of a function of unknown parameter in a probability distribution and computing the variance of the estimator or approximating it by lower bounds.In this paper, we consider two famous lower bounds for the variance of any unbiased estimator, which are Bhattacharyya and Kshirsagar bounds. We obtain the general forms of the Bhattacharyya and Kshirsagar matrices in the GB2 distribution. In addition, we compare different Bhattacharyya and Kshirsagar bounds for the variance of any unbiased estimator of some parametric functions such as mode, mean, skewness and kurtosis in GB2 distribution and conclude that in each case, which bound is better to use. The results of this paper can be useful for researchers trying to find the accuracy of the estimators.

More than two centuries ago Malfatti (see [9]) raised and solved the following problem (the so-called Malfatti’s construction problem): Construct three circles into a triangle so that each of them touches the two others from outside moreover touches two sides of the triangle too. It is an interesting fact that nobody investigated this problem on the hyperbolic plane, while the case of the sphere was solved simultaneously with the Euclidean case. In order to compensate this shortage we solve the following exercise: *Determine three cycles of the hyperbolic plane so that each of them touches the two others moreover touches two of three given cycles of the hyperbolic plane*.

Hirano [On annihilator ideals of a polynomial ring over a noncommutative ring, *J. Pure Appl. Algebra*, **168** (2002), 45–52] studied relations between the set of annihilators in a ring *R* and the set of annihilators in a polynomial extension *R*[*x*] and introduced quasi-Armendariz rings. In this paper, we give a sufficient condition for a ring *R* and a monoid *M* such that the monoid ring *R*[*M*] is quasi-Armendariz. As a consequence we show that if *R* is a right APP-ring, then *R*[*x*]=(*x*
^{n}) and hence the trivial extension *T*(*R*,*R*) are quasi-Armendariz. They allow the construction of rings with a non-zero nilpotent ideal of arbitrary index of nilpotency which are quasi-Armendariz.

We introduce Kurosh elements in division rings based on the idea of a conjecture of Kurosh. Using this, we generalize a result of Faith in {xc[3]} and of Herstein in {xc[6]}.

This paper concerns the existence of mild solutions for some fractional Cauchy problem with nonlocal conditions in the α-norm. The linear part of the equations is assumed to generate an analytic compact bounded semigroup, and the nonlinear part satisfies some Lipschitz conditions with respect to the fractional power norm of the linear part. By using a fixed point theorem of Sadovskii, we establish some existence results which generalize ones in the case of fractional order derivative.

In recent years, slash and skew slash distributions have been employed, as flexible models, in various fields. In this paper, we study several properties of these distributions in both univariate and multivariate cases. Some recurrence relations for the probability density functions are derived and the behavior of reliability measures, such as hazard rate and mean residual life, associated to these distributions are investigated.

A ring *R* is called *NLI* (rings whose nilpotent elements form a Lie ideal) if for each *a* ∈ *N*(*R*) and *b* ∈ *R*, *ab* − *ba* ∈ *N*(*R*). Clearly, *NI* rings are *NLI*. In this note, many properties of *NLI* rings are studied. The main results we obtain are the following: (1) *NLI* rings are directly finite and left min-abel; (2) If *R* is a *NLI* ring, then (a) *R* is a strongly regular ring if and only if *R* is a Von Neumann regular ring; (b) *R* is (weakly) exchange if and only if *R* is (weakly) clean; (c) *R* is a reduced ring if and only if *R* is a *n*-regular ring; (3) If *R* is a *NLI* left *MC*2 ring whose singular simple left modules are *Wnil*-injective, then *R* is reduced.

A recently published paper [6] considered the total graph of commutative ring *R*. In this paper, we compute Wiener, hyper-Wiener, reverse Wiener, Randić, Zagreb, *ABC* and *GA* indices of zero-divisor graph.

The classification of Bruce and Gaffney respectively Gibson and Hobbs for simple plane curve singularities respectively simple space curve singularities is characterized in terms of invariants. This is the basis for the implementation of a classifier in the computer algebra system singular.

The maximal Orlicz spaces such that the mixed logarithmic means of multiple Walsh-Fourier series for the functions from these spaces converge in measure and in norm are found.

Let *V* be the 2-dimensional column vector space over a finite field

*q*is necessarily a power of a prime number) and let ℙ

_{q}be the projective line over

*GL*

_{2}(

*q*), for

*q*≠ 3, and

*SL*

_{2}(

*q*) acting on

*V*− {0} have the strict EKR property and

*GL*

_{2}(3) has the EKR property, but it does not have the strict EKR property. Also, we show that

*GL*

_{n}(

*q*) acting on

*PSL*

_{2}(

*q*) acting on ℙ

_{q}, where

*q*≡ −1 (mod 4), has a clique of size

*q*+ 1.

We study basic properties of the generalized ideal transforms *D*
_{I} (*M*, *N*) and the set of associated primes of the modules *R*
^{i}
*D*
_{I} (*M*, *N*).

Let *G* be a finite group. A subgroup *H* of *G* is called an

*G*if

*N*

_{G}(

*H*) ∩

*H*

^{g}≤

*H*for all

*g*∈

*G*. A subgroup

*H*of

*G*is called a weakly

*G*if there exists a normal subgroup

*K*of

*G*such that

*G*=

*HK*and

*H*∩

*K*is an

*G*. In this article, we investigate the structure of a group

*G*in which every subgroup with order

*p*

^{m}of a Sylow

*p*-subgroup

*P*of

*G*is a weakly

*G*, where

*m*is a fixed positive integer. Our results improve and extend the main results of Skiba [13], Jaraden and Skiba [11], Guo and Wei [8], Tong-Veit [15] and Li et al. [12].

A new generalization of the logarithmic series distribution has been obtained as a limiting case of the zero-truncated Mishra’s [10] generalized negative binomial distribution (GNBD). This distribution has an advantage over the Mishra’s [9] quasi logarithmic series distribution (QLSD) as its moments appear in compact forms unlike the QLSD. This makes the estimation of parameters easier by the method of moments. The first four moments of this distribution have been obtained and the distribution has been fitted to some well known data-sets to test its goodness of fit.

An upper bound is given on the size of a *k*-fan-free 3-graph, and an infinite family reaching this bound is also described.

Applying certain convexity arguments we investigate the existence of a classical solution for a Dirichlet problem for which the Euler action functional is not necessarily differentiable in the sense of Gâteaux.

The aim of this paper is to investigate the semifields of order *q*
^{4} over a finite field of order *q*, *q* an odd prime power, admitting a Klein 4-group of automorphisms.

The main aim of this paper is to investigate (*H*
_{p}, *L*
_{p}) and (*H*
_{p}, *L*
_{p,∞}) type inequalities for maximal operators of Riesz logarithmic means of one-dimensional Vilenkin—Fourier series.

We consider a Riesz decomposition theorem for super-polyharmonic functions satisfying certain growth condition on surface integrals in the punctured unit ball. We give a condition that super-polyharmonic functions *u* have the bound

*u*in ℝ

^{n}.

In the following text we prove that in a generalized shift dynamical system (*X*
^{Г}, *σ*
_{φ}) for infinite countable Г and discrete *X* with at least two elements the following statements are equivalent:

- the dynamical system (
*X*^{Г},*σ*_{φ}) is chaotic in the sense of Devaney - the dynamical system (
*X*^{Г},*σ*_{φ}) is topologically transitive - the map
*φ*: Г → Г is one to one without any periodic point.

*X*with at least two elements (

*X*

^{Г},

*σ*

_{φ}) is exact Devaney chaotic, if and only if

*φ*: Г → Г is one to one and

*φ*: Г → Г has niether periodic points nor

*φ*-backwarding infinite sequences.

Let *C* be a class of some finitely presented left *R*-modules. A left *R*-module *M* is called *C*-injective, if Ext_{R}
^{1}(*C*, *M*) = 0 for each *C* ∈ *C*. A right *R*-module *M* is called C-flat, if Tor_{1}
^{R}(*M*, *C*) = 0 for each *C* ∈ *C*. A ring *R* is called *C*-coherent, if every *C* ∈ *C* is 2-presented. A ring *R* is called *C*-semihereditary, if whenever 0 → *K* → *P* → *C* → 0 is exact, where *C* ∈ *C* and *P* is finitely generated projective and *K* is finitely generated, then *K* is also projective. A ring *R* is called *C*-regular, if whenever *P*/*K* ∈ *C*, where *P* is finitely generated projective and *K* is finitely generated, then *K* is a direct summand of *P.* Using the concepts of *C*-injectivity and *C*-flatness of modules, we present some characterizations of *C*-coherent rings, *C*-semihereditary rings, and *C*-regular rings.

A general class of linear and positive operators dened by nite sum is constructed. Some of their approximation properties, including a convergence theorem and a Voronovskaja-type theorem are established. Next, the operators of the considered class which preserve exactly two test functions from the set {*e*
_{0}, *e*
_{1}, *e*
_{2}} are determined. It is proved that the test functions *e*
_{0} and *e*
_{1} are preserved only by the Bernstein operators, the test functions *e*
_{0} and *e*
_{2} only by the King operators while the test functions *e*
_{1} and *e*
_{2} only by the operators recently introduced by P. I. Braica, O. T. Pop and A. D. Indrea in [4].

The beta generalized half-normal distribution is commonly used to model lifetimes. We propose a new wider distribution called the beta generalized half-normal geometric distribution, whose failure rate function can be decreasing, increasing or upside-down bathtub. Its density function can be expressed as a linear combination of beta generalzed half-normal density functions. We derive quantile function, moments and generating unction. We characterize the proposed distribution using a simple relationship between wo truncated moments. The method of maximum likelihood is adapted to estimate the model parameters and its potentiality is illustrated with an application to a real fatigue data set. Further, we propose a new extended regression model based on the logarithm of the new distribution. This regression model can be very useful for the analysis of real data and provide more realistic fits than other special regression models.

The purpose of this paper is to revise von Neumann’s characterizations of selfadjoint operators among symmetric ones. In fact, we do not assume that the underlying Hilbert space is complex, nor that the corresponding operator is densely defined, moreover, that it is closed. Following Arens, we employ algebraic arguments instead of the geometric approach of von Neumann using the Cayley transform.

In the paper we give some remarks on the article of Janet Mills. In particular, the proof of Lemma 1.2 (in her work) is incorrect, and so the proof of Theorem 3.5 is not valid, too. Using different methods we show the mentioned theorem. Moreover, we find a new equivalent condition to the statements in Theorem 3.5. In particular, an explicit definition of a new class of orthodox semigroups is introduced.

The distance of two-dimensional samples is studied. The distance of two samples is based on the optimal matching method. Simulation results are obtained when the samples are drawn from normal and uniform distributions.

By [12], a ring *R* is *left APP* if *R* has the property that “the left annihilator of a principal ideal is pure as a left ideal”. Equivalently, *R* is a left APP-ring if *R* modulo the left annihilator of any principal left ideal is flat. Let *R* be a ring, (*S*, ≦) a strictly totally ordered commutative monoid and *ω*: *S* → End(*R*) a monoid homomorphism. Following [16], we show that, when *R* is a (*S*, *ω*)-weakly rigid and (*S*, *ω*)-Armendariz ring, then the skew generalized power series ring *R*[[*S*
^{≦}, *ω*]] is right APP if and only if *r*
_{R}(*A*) is *S*-indexed left *s*-unital for every *S*-indexed generated right ideal *A* of *R*. We also show that when *R* is a (*S, ω*)-strongly Armendariz ring and *ω*(*S*) ⫅ Aut(*R*), then the ring *R*[[*S*
^{≦}, *ω*]] is left APP if and only if *ℓ*
_{R}(∑_{a}
_{∈}
_{A} ∑_{s}
_{∈}
_{S}
*Rω*
_{s}(*a*)) is *S*-indexed right *s*-unital, for any *S*-indexed subset *A* of *R*. In particular, when *R* is Armendariz relative to *S*, then *R*[[*S*
^{≦}]] is right APP if and only if *r*
_{R}(*A*) is *S*-indexed left *s*-unital, for any *S*-indexed generated right ideal *A* of *R*.

Let *P*
_{r} denote an almost-prime with at most *r* prime factors, counted according to multiplicity. In this paper we show that the inequality

*p*such that

*p*+ 2 =

*P*

_{4}.

Given a covering of the plane by closed unit discs

*A*and

*B*in the region doubly covered by

*A*and

*B*is

*d*, then the length of this path is at most

*d*+

*O*(1).

It is shown that in a packing of open circular discs with radii not exceeding 1, any two points lying outside the circles at distance *d* from one another can be connected by a path traveling outside the circles and having length at most

*E*

^{n}and two points outside the balls at distance

*d*from one another, the length of the shortest path connecting the two points and avoiding the balls is

*d*+

*O*(

*d*/

*n*) as

*d*and

*n*approaches infinity.

In this study, we investigate approximation properties and obtain Voronovskaja type results for complex modified Szász-Mirakjan operators. Also, we estimate the exact orders of approximation in compact disks and prove that the complex modified Szász-Mirakjan operators attached to an analytic function preserve the univalence, starlikeness, convexity and spirallikeness in the unit disk.

In this paper, the authors investigate the linearization problems associated with two families of generalized Lauricella polynomials of the first and second kinds. By means of their multiple integral representations, it is shown how one can linearize the product of two different members of each of these two families of the generalized Lauricella polynomials. Upon suitable specialization of the main results presented in this paper, the corresponding integral representations are deduced for such familiar classes of multivariable hypergeometric polynomials as (for example) the Lauricella polynomials *F*
_{A}
^{(r)} in *r* variables, the Appell polynomials *F*
_{2} in two variables and the multivariable Laguerre polynomials. Each of these integral representations, which are derived as special cases of the main results in this paper, may also be viewed as a linearization relationship for the product of two different members of the associated family of multivariable hypergeometric polynomials.

We give a characterization of all those commutative groups which admit at least one absolutely nonmeasurable homomorphism into the real line (or into the one-dimensional torus). These are exactly those commutative groups (*G*, +) for which the quotient group *G*/*G*
_{0} is uncountable, where *G*
_{0} denotes the torsion subgroup of *G*.

Let *b* = 2, 5, 10 or 17 and *t* > 0. We study the existence of *D*(−1)-quadruples of the form {1, *b*, *c*, *d*} in the ring *b*, *c*} is a *D*(−1)-triple in *c* is an integer. As a consequence of this result, we show that for *t* ∉ {1, 4, 9, 16} there does not exist a subset of *b*, *c*, *d*} with the property that the product of any two of its distinct elements diminished by 1 is a square of an element in

Let *N*(*k*, *l*) be the smallest positive integer such that any set of *N*(*k*, *l*) points in the plane, no three collinear, contains both a convex *k*-gon and a convex *l*-gon with disjoint convex hulls. In this paper, we prove that *N*(3, 4) = 7, *N*(4, 4) = 9, *N*(3, 5) = 10 and *N*(4, 5) = 11.

In this paper we analyse the natural permutation module of an affine permutation group. For this the regular module of an elementary Abelian *p*-group is described in detail. We consider the inequivalent permutation modules coming from nonconjugate complements. We prove their strong structural similarity well exceeding the fact that they have equal Brauer characters.

The purpose of this paper is to investigate the relations of incomparability between so called convergence classes of the permutations of ℕ. The convergence class of any permutation *p* of ℕ, denoted by Σ(*p*), is defined to be the family of all real series Σ*a*
_{n} such that both Σ*a*
_{n} and Σ*a*
_{p(n)} are convergent. A permutation *p* of ℕ is called a divergent permutation if there exists a conditionally convergent real series Σ*a*
_{n} such that the *p*-rearranged series Σ*a*
_{p(n)} is divergent.It is proved that for every divergent permutation *p* of ℕ there exists a family *p*) of divergent permutations of ℕ such that card *p*) = *q* ∈ *p*) the family Σ(*q*) is a proper subset of Σ(*p*) and, furthermore, Σ(*q*
_{1})\Σ(*q*
_{2}) ≠ ∅ and Σ(*q*
_{2})\Σ(*q*
_{1}) ≠ ∅ whenever *q*
_{1}; *q*
_{2} ∈ *p*) are different. Permutations *q*
_{1}, *q*
_{2} of ℕ satisfying the above relations are called the incomparable permutations.This result, like many other results of the paper, is given in more general context resulting from the more subtle discussion on the subfamilies of

In this paper, we introduce *inclusion ideals I*(*H*) associated to a special class of *non uniform hypergraphs H*(*gC; ɛ; d*), namely, the *uniformly increasing hypergraphs*. We discuss some algebraic properties of the inclusion ideals. In particular, we give an upper bound of the *Castelnouvo-Mumford regularity* of the *special dual ideal I*
^{[*]}(*H*).

Csáki and Vincze have defined in 1961 a discrete transformation *T* which applies to simple random walks and is measure preserving. In this paper, we are interested in ergodic and asymptotic properties of *T*. We prove that *T* is exact: ∩_{k≧1}
*σ*(*T*
^{k}(*S*)) is trivial for each simple random walk *S* and give a precise description of the lost information at each step *k*. We then show that, in a suitable scaling limit, all iterations of *T* “converge” to the corresponding iterations of the continuous Lévy transform of Brownian motion.

The purpose of this note is to show by constructing counterexamples that two conjectures of Móri and Székely for the Borel-Cantelli lemma are false.

A new model, in terms of finite bipartite graphs, of the free pseudosemilattice is presented. This will then be used to obtain several results about the variety **SPS** of all strict pseudosemilattices: (i) an identity basis for **SPS** is found, (ii) **SPS** is shown to be inherently non-finitely based, (iii) **SPS** is shown to have no irredundant identity basis, and (iv) **SPS** is shown to have no covers and to be ∩-prime in the lattice of all varieties of pseudosemilattices. Some applications to e-varieties of locally inverse semigroups are also derived.

Let *K* ⊂ ℝ^{2} be an *o*-symmetric convex body, and *K** its polar body. Then we have |*K*| · |*K**| ≧ 8, with equality if and only if *K* is a parallelogram. (|·| denotes volume). If *K* ⊂ ℝ^{2} is a convex body, with *o* ∈ int *K*, then |*K*| · |*K**| ≧ 27/4, with equality if and only if *K* is a triangle and *o* is its centroid. If *K* ⊂ ℝ^{2} is a convex body, then we have |*K*| · |[(*K* − *K*)/2)]*| ≧ 6, with equality if and only if *K* is a triangle. These theorems are due to Mahler and Reisner, Mahler and Meyer, and to Eggleston, respectively. We show an analogous theorem: if *K* has *n*-fold rotational symmetry about *o*, then |*K*| · |*K**| ≧ *n*
^{2} sin^{2}(*π*/*n*), with equality if and only if *K* is a regular *n*-gon of centre *o*. We will also give stability variants of these four inequalities, both for the body, and for the centre of polarity. For this we use the Banach-Mazur distance (from parallelograms, or triangles), or its analogue with similar copies rather than affine transforms (from regular *n*-gons), respectively. The stability variants are sharp, up to constant factors. We extend the inequality |*K*| · |*K**| ≧ *n*
^{2} sin^{2}(*π*/*n*) to bodies with *o* ∈ int *K*, which contain, and are contained in, two regular *n*-gons, the vertices of the contained *n*-gon being incident to the sides of the containing *n*-gon. Our key lemma is a stability estimate for the area product of two sectors of convex bodies polar to each other. To several of our statements we give several proofs; in particular, we give a new proof for the theorem of Mahler-Reisner.

We give optimal bounds for Kloosterman sums that arise in the estimation of Fourier coefficients of Siegel modular forms of genus 2.

Recently Khan and Orhan have proved that an ordinary (single) sequence is *A*-strongly convergent if and only if it is *A*-statistically convergent and *A*-uniformly integrable. In this paper we consider the similar problem for multidimensional sequences when *A* is a multivariable-to-single matrix. We also study the same question when A is a multivariable-to-multivariable matrix.

Let *ν* be a positive Borel measure on ℝ̄_{+}:= [0;∞) and let *p*: ℝ̄_{+} → ℝ̄_{+} be a weight function which is locally integrable with respect to *ν*. We assume that *f*: ℝ̄_{+} → ℂ be a locally integrable function with respect to *p dν*, and define its weighted averages by *t*, where *P*(*t*) > 0. We prove necessary and sufficient conditions under which the finite limit

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

We construct an infinite dimensional quasi-polyadic equality algebra

For two locally compact groups *G* and *H*, we show that if *L*
^{1}(*G*) is strictly inner amenable, then *L*
^{1}(*G* × *H*) is strictly inner amenable. We then apply this result to show that there is a large class of locally compact groups *G* such that *L*
^{1}(*G*) is strictly inner amenable, but *G* is not even inner amenable.

We study some properties of representable or *I*-stable local homology modules *H*
_{i}
^{I}
(*M*) where *M* is a linearly compact module. By duality, we get some properties of good or at local cohomology modules *H*
_{I}
^{i}
(*M*) of A. Grothendieck.

The sub-bifractional Brownian motion, which is a quasi-helix in the sense of Kahane, is presented. The upper classes of some of its increments are characterized by an integral test.

Let {*X*
_{n}}_{n∈ℕ} be a sequence of i.i.d. random variables in ℤ^{d}. Let *S*
_{k} = *X*
_{1} + … + *X*
_{k} and *Y*
_{n}(*t*) be the continuous process on [0, 1] for which *Y*
_{n}(*k/n*) = *S*
_{k}/*n*
^{1/2} for *k* = 1, … *n* and which is linearly interpolated elsewhere. The paper gives a generalization of results of ([2]) on the weak limit laws of *Y*
_{n}(*t*) conditioned to stay away from some small sets. In particular, it is shown that the diffusive limit of the random walk meander on ℤ^{d}: *d* ≧ 2 is the Brownian motion.

Characterizations of the Amoroso distribution based on a simple relationship between two truncated moments are presented. A remark regarding the characterization of certain special cases of the Amoroso distribution based on hazard function is given. We will also point out that a sub-family of the Amoroso family is a member of the generalized Pearson system.

We compute the fundamental group of various spaces of Desargues configurations in complex projective spaces: planar and non-planar configurations, with a fixed center and also with an arbitrary center.

In this paper, we have used Eryilmaz’s (2008) multi-colour Pólya urn model to obtain joint distributions of runs of *t*-types of exact lengths (*k*
_{1}, *k*
_{2}, …, *k*
_{t}), at least lengths (*k*
_{1}, *k*
_{2}, …, *k*
_{t}), non-overlapping runs of lengths (*k*
_{1}, *k*
_{2}, … *k*
_{t}) and overlapping runs of lengths (*k*
_{1}, *k*
_{2}, … *k*
_{t}) when counting of runs is done in a circular setup. We have also derived joint distributions of longest runs of various types under similar conditions. Distributions of runs have found applications in fields of reliability of consecutive-*k*-out-of *n*: *F* system, consecutive *k*-out-of-*r*-from *n*: *F* system, start-up demonstration test, molecular biology, radar detection, time sharing systems and quality control. The literature is profound in discussion of marginal distribution and joint distribution of runs of various types under linear and circular setup using techniques like urn model with balls of two or more colours, probability generating function and compounding discrete distribution with suitable beta functions. Through this paper for first time effort been made to discuss joint distributions of runs of various lengths and types using Multi-colour urn model.

Suppose *X* is a locally convex space, *Y* is a topological vector space and λ(*X*)^{βY} is the *β*-dual of some *X* valued sequence space *λ*(*X*). When *λ*(*X*) is *c*
_{0}(*X*) or *l*
_{∞}(*X*), we have found the largest *M* ⊂ 2^{λ(X)} for which (*A*
_{j}) ∈ λ(*X*)^{βY} if and only if Σ
_{j=1}
^{∞}
*A*
_{j}(*x*
_{j}) converges uniformly with respect to (*x*
_{j}) in any *M* ∈ *M*. Also, a remark is given when *λ*(*X*) is *l*
_{p}(*X*) for 0 < *p* < + ∞.

Let *G* be a finite group and *H* a subgroup of *G*. *H* is said to be *S*-quasinormal in *G* if *HP* = *PH* for all Sylow subgroups *P* of *G*. Let *H*
_{sG} be the subgroup of *H* generated by all those subgroups of *H* which are *S*-quasinormal in *G* and *H*
^{sG} the intersection of all *S*-quasinormal subgroups of *G* containing *H*. The symbol |*G*|_{p} denotes the order of a Sylow *p*-subgroup of *G*. We prove the followingTheorem A. *Let G be a finite group and p a prime dividing* |*G*|. *Then G is p-supersoluble if and only if for every cyclic subgroup H of*
*Ḡ* (*G*) *of prime order or order* 4 (*if p* = 2*)*, *Ḡ*
*has a normal subgroup T such that*
*H*
^{sḠ}
*and*
*H*∩*T*=*H*
_{sḠ}∩*T*.Theorem B. *A soluble finite group G is p-supersoluble if and only if for every* 2-*maximal subgroup E of G such that O*
_{p′} (*G*) ≦ *E and* |*G: E*| *is not a power of p, G has an S-quasinormal subgroup T with cyclic Sylow p-subgroups such that E*
^{sG} = *ET and* |*E* ∩ *T*|_{p} = |*E*
_{sG} ∩ *T*|_{p}.Theorem C. *A finite group G is p-soluble if for every* 2-*maximal subgroup E of G such that O*
_{p′} (*G*) ≦ *E and* |*G: E*| *is not a power of p, G has an S-quasinormal subgroup T such that E*
^{sG} = *ET and* |*E* ∩ *T*
_{p} = |*E*
_{sG} ∩ *T*|_{p}.

We introduce the concept of nil-McCoy rings to study the structure of the set of nilpotent elements in McCoy rings. This notion extends the concepts of McCoy rings and nil-Armendariz rings. It is proved that every semicommutative ring is nil-McCoy. We shall give an example to show that nil-McCoy rings need not be semicommutative. Moreover, we show that nil-McCoy rings need not be right linearly McCoy. More examples of nil-McCoy rings are given by various extensions. On the other hand, the properties of *α*-McCoy rings by considering the polynomials in the skew polynomial ring *R*[*x*; *α*] in place of the ring *R*[*x*] are also investigated. For a monomorphism *α* of a ring *R*, it is shown that if *R* is weak *α*-rigid and *α*-reversible then *R* is *α*-McCoy.

Suppose that *A* is either the Banach algebra *L*
^{1}(*G*) of a locally compact group *G*, or measure algebra *M*(*G*), or other algebras (usually larger than *L*
^{1}(*G*) and *M*(*G*)) such as the second dual, *L*
^{1}(*G*)**, of *L*
^{1}(*G*) with an Arens product, or *LUC*(*G*)* with an Arenstype product. The left translation invariant closed convex subsets of *A* are studied. Finally, we obtain necessary and sufficient conditions for *LUC*(*G*)* to have 1-dimensional left ideals.

Let *K* be a finite field and let *X** be an affine algebraic toric set parameterized by monomials. We give an algebraic method, using Gröbner bases, to compute the length and the dimension of *C*
_{X*} (*d*), the parameterized affine code of degree *d* on the set *X**. If *Y* is the projective closure of *X**, it is shown that *C*
_{X*} (*d*) has the same basic parameters that *C*
_{Y} (*d*), the parameterized projective code on the set *Y*. If *X** is an affine torus, we compute the basic parameters of *C*
_{X}* (*d*). We show how to compute the vanishing ideals of *X** and *Y*.

The smallest monoid containing a 2-testable semigroup is defined to be a 2-*testable monoid*. The well-known Brandt monoid *B*
_{2}
^{1}
of order six is an example of a 2-testable monoid. The finite basis problem for 2-testable monoids was recently addressed and solved. The present article continues with the investigation by describing all monoid varieties generated by 2-testable monoids. It is shown that there are 28 such varieties, all of which are finitely generated and precisely 19 of which are finitely based. As a comparison, the sub-variety lattice of the monoid variety generated by the monoid *B*
_{2}
^{1}
is examined. This lattice has infinite width, satisfies neither the ascending chain condition nor the descending chain condition, and contains non-finitely generated varieties.