Abstract: The paper has a form of a survey and consists of three parts. It is focused on the relationship between the many-sorted theory, which leads to logical geometry and one-sorted theory, which is based on the important model-theoretic concepts. Our aim is to show that both approaches go in parallel and there are bridges which allow to transfer results, notions and problems back and forth. Thus, an additional freedom in choosing an approach appear.

Abstract: We give a complete list of the Lebesgue-Jordan decomposition of Boolean and monotone stable distributions and a complete list of the mode of them. They are not always unimodal.

Abstract: We prove a sufficient condition under which a semigroup admits no finite identity basis. As an application, it is shown that the identities of the Kauffman monoid \({\mathcal{K}_n}\) are nonfinitely based for each \({n \geq 3}\). This result holds also for the case when \({\mathcal{K}_n}\) is considered as an involution semigroup under either of its natural involutions.

Abstract: Here we present very general fractional representation formulae for a function in terms of the fractional Riemann-Liouville integrals of different orders of the function and its ordinary derivatives under initial conditions.

Abstract: We study here the relative cohomology and the Gauss-Manin connections associated to an isolated singularity of a function on a manifold with boundary, i.e. with a fixed hyperplane section. We prove several relative analogs of classical theorems obtained mainly by E. Brieskorn and B. Malgrange, concerning the properties of the Gauss-Manin connection as well as its relations with the Picard-Lefschetz monodromy and the asymptotics of integrals of holomorphic forms along the vanishing cycles. Finally, we give an application in isochore deformation theory, i.e. the deformation theory of boundary singularities with respect to a volume form. In particular we

Abstract: Abstract A class of (right) Rickart rings, called strong, is isolated. In particular, every Rickart *-ring is strong. It is shown in the paper that every strong Rickart ring R admits a binary operation which turns R into a right normal band having an upper bound property with respect to its natural order ≤; such bands are known as right normal skew nearlattices. The poset (R, ≤) is relatively orthocomplemented; in particular, every initial segment in it is orthomodular. The order ≤ is actually a version of the so called right-star order. The one-sided star orders are well-investigated for matrices and recently have been generalized to bounded linear Hilbert space operators and to abstract Rickart *-rings. The paper demonstrates that they can successfully be treated also in Rickart rings without involution.

Abstract: Abstract The paper deals with the monounary algebras for which the second centralizer equals the first centralizer. We describe Green’s relations on the semigroup C, where C is the centralizer of such algebra.

Abstract: Abstract This is a survey article about the study of the links of some complex hypersurface singularities in ℂ3 . We study the links of simple singularities, simple elliptic singularities and cusp singularities, and the canonical contact structures on them. It is known that each singularity link is diffeomorphic to a compact quotient of a 3-dimensional Lie group SU (2), Nil3 or Sol3 , respectively. Moreover, the canonical contact structure is equivalent to the contact structure invariant under the action of each Lie group. We show a new proof of this fact using the moment polytope of S5 . Our proof gives a new aspect to the relation between simple elliptic singularities and cusp singularities, and visualizes how the singularity links are embedded in S5 as codimension two contact submanifolds.

Abstract: Abstract A duchain complex of W. Dwyer and D. Kan is a common extension of the notions of a chain complex and a cochain complex. Given a square commutative diagram of duchain complexes, the lifting-extension problem asks whether there exists a diagonal map making the two resulting triangles commute. Duchain complexes have a model category structure, and hence a lift exists if the left vertical map is a cofibration, the right vertical map is a fibration, and one of them is a weak equivalence. We show that it is possible to replace the two conditions above, by a countably infinite, bigraded, family of conditions which guarantee the existence of a lift.

Abstract: In this paper we obtain some new fixed point theorems in generalized metric space for maps satisfying an implicit relation. The obtained results unify, generalize, enrich, complement and extend a multitude of related fixed point theorems from metric spaces to generalized metric spaces.

Abstract: Abstract We establish necessary and sufficient conditions for a parameter depending sequence (Ln,λ)n≥1 of positive linear operators such that (Ln,λ)n≥1 converges in the strong operator topology to its limit operator. Some applications of our theorem are also presented.

Abstract: Abstract Let K be any field and L be any lattice. In this note we show that L is a sublattice of annihilators in an associative and commutative K-algebra. If L is finite, then our algebra will be finite dimensional over K.

Abstract: Abstract We study singularities of smooth mappings F̄ of ℝ2n into symplectic space (ℝ2n , ω̇) by their isotropic liftings to the corresponding symplectic tangent bundle (Tℝ2n,w). Using the notion of local solvability of lifting as a generalized Hamiltonian system, we introduce new symplectic invariants and explain their geometric meaning. We prove that a basic local algebra of singularity is a space of generating functions of solvable isotropic mappings over F̄ endowed with a natural Poisson structure. The global properties of this Poisson algebra of the singularity among the space of all generating functions of isotropic liftings are investigated. The solvability criterion of generalized Hamiltonian systems is a strong method for various geometric and algebraic investigations in a symplectic space. We illustrate this by explicit classification of solvable systems in codimension one.

Abstract: Abstract In the present paper, we investigate the existence, uniqueness and continuous dependence of mild solutions of an impulsive neutral integro-differential equations with nonlocal condition in Banach spaces. We use Banach contraction principle and the theory of fractional power of operators to obtain our results.

Abstract: Abstract The applications of q-calculus in the approximation theory is a very interesting area of research in the recent years, several new q-operators were introduced and their behaviour were discussed by many researchers. This paper is the extension of the paper [15], in which Durrmeyer type generalization of q-Baskakov-Stancu type operators were discussed by using the concept of q-integral operators. Here, we propose to study the Stancu variant of q-Baskakov-Stancu type operators. We establish an estimate for the rate of convergence in terms of modulus of continuity and weighted approximation properties of these operators.

Abstract: Abstract The notions of a soft general algebra and a soft subalgebra are introduced and studied. The operations on them such as a restricted intersection, an extended intersection, a restricted union, a ∧-intersection, a ∨-union and a cartesian product are established.

Abstract: Abstract This paper is concerned with the existence of at least one solution of the nonlinear 2k-th order BVP. We use the Mountain Pass Lemma to get an existence result for the problem, whose linear part depends on several parameters.

Abstract: Abstract We use the method of algebraic restrictions to classify symplectic U7, U8 and U9 singularities. We use discrete symplectic invariants to distinguish symplectic singularities of the curves. We also give the geometric description of symplectic classes

Abstract: Abstract We consider here some notions and results about sets of generators of finite algebras motivated by the case of finite groups. We illustrate these notions by simple examples closed to lattices and semigroups. Next, we examine these notions in the case of groups with operators.

Abstract: Abstract We deal with the monomial difference n!F(y) - △ny F(x) assuming that its norm is majorized by some function depending upon the variable y.

Abstract: Abstract In this paper, we study the asymptotic behavior of the solutions of the one-dimensional Cauchy problem in Timoshenko system with thermal effect. The heat conduction is given by the type III theory of Green and Naghdi. We prove that the dissipation induced by the heat conduction alone is strong enough to stabilize the system, but with slow decay rate. To show our result, we transform our system into a first order system and, applying the energy method in the Fourier space, we establish some pointwise estimates of the Fourier image of the solution. Using those pointwise estimates, we prove the decay estimates of the solution and show that those decay estimates are very slow and, in the case of nonequal wave speeds, are of regularity-loss type. This paper solves the open problem stated in [10] and shows that the stability of the solution holds without any additional mechanical damping term.

Abstract: Abstract In this paper, we study weakly idempotent lattices with an additional interlaced operation. We characterize interlacity of a weakly idempotent semilattice operation, using the concept of hyperidentity and prove that a weakly idempotent bilattice with an interlaced operation is epimorphic to the superproduct with negation of two equal lattices. In the last part of the paper, we introduce the concepts of a non-idempotent Plonka function and the weakly Plonka sum and extend the main result for algebras with the well known Plonka function to the algebras with the non-idempotent Plonka function. As a consequence, we characterize the hyperidentities of the variety of weakly idempotent lattices, using non-idempotent Plonka functions, weakly Plonka sums and characterization of cardinality of the sets of operations of subdirectly irreducible algebras with hyperidentities of the variety of weakly idempotent lattices. Applications of weakly idempotent bilattices in multi-valued logic is to appear.

Abstract: Here we derive very general multivariate Ostrowski and Gruss type inequalities for several functions by involving harmonic polynomials. Estimates are with respect to all basic norms. We give applications. 2010 AMS Subject Classi…cation: 26B20, 26B40, 26D10, 26D15.

Abstract: Abstract We examine the conditional regularity of the solutions of Navier-Stokes equations in the entire three-dimensional space under the assumption that the data are axially symmetric. We show that if positive part of the radial component of velocity satisfies a weighted Serrin type condition and in addition angular component satisfies some condition, then the solution is regular.

Abstract: Abstract An extended generalization of recent result of Kikina and Kikina (2011) has been established through the notions of weak compatibility and the property E.A., under an implicit-type relation and restricted orbital completeness of the space. The result of this paper also extends and generalizes that of Imdad and Ali (2007).

Abstract: Given a five dimensional space endowed with a Cartan distribution, the abnormal geodesics form another five dimensional space with a cone structure. Then it is shown, if the cone structure is regarded as a control system, then, the space of abnormal geodesics of the cone structure is naturally identified with the original space. In this paper, we provide an exposition on the duality by abnormal geodesics in a wider framework, namely, in terms of quotients of control systems and sub-Riemannian pseudo-product structures. Also we consider the controllability of cone structures and describe the constrained Hamiltonian equations on normal and abnormal geodesics.

Abstract: Abstract Every collection of n (arbitrary-oriented) unit squares can be packed translatively into any equilateral triangle of side length 2:3755· √n.

Abstract: Abstract Let Xn be an affine variety of dimension n and Yn be a quasi-projective variety of the same dimension. We prove that for a quasi-finite polynomial mapping ƒ : Xn → Yn, every non-empty component of the set Yn\ ƒ (Xn) is closed and it has dimension greater or equal to n μ(ƒ), where μ(ƒ) is a geometric degree of ƒ. Moreover, we prove that generally, if ƒ : Xn → Yn is any polynomial mapping, then either every non-empty component of the set is of dimension ≥ n μ(ƒ) or ƒ contracts a subvariety of dimension ≥ n μ(ƒ) + 1.