Inequality of Petrović and Giaccardi for Convex Function of Higher Order

In this paper we use Lidstone polynomials to prove further generalization of Giaccardi generalization of the well-known Petrovis inequality.     

Hulls for Various Kinds of α-Completeness in Archimedean Lattice-Ordered Groups

Within Archimedean -groups, and with an infinite cardinal or , we consider X-hulls where X stands for any of the following classes of -groups: -projectable; laterally -complete; boundedly laterally -complete; conditionally -complete; combinations of the preceding, together with divisibility and/or relative uniform completeness. All these hulls exist, and may be obtained by iterated adjunction of the required extra elements, within the essential hull. When the -groups is relatively -complemented one step in the iteration suffices for several crucial properties. We derive from the above a considerable number of equations involving combinations of these hull operators.     

On the Operator Semirings of a Γ-Semiring

In this paper we introduce the notion of operator semirings of a -semiring to study -semirings. It is shown that the lattices of all left (right) ideals (two-sided ideals) of a -semiring and its right (respectively left) operator semiring are isomorphic. This has many applications to characterize various -semirings.AMS Subject Classification (2000): 16Y60, 16Y99     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.     

(φ,ξ,η,λ)‐Structures of Parabolic Type

We consider (,,,)structures of parabolic type on hypersurfaces of dual spaces and study the rank of the affinor . We consider almost contact metric structures of parabolic type of the first kind on hypersurfaces of 4dimensional dual metric space. We study the properties of these structures and give examples of normal, integrable, and Sasakian parabolic structures.     

Explicit Formulas for Green’s Functions on the Annulus and on the Möbius Strip

Given a fixed point free antianalytic involution k of a domain G in thecomplex plane, bounded by a finite number of analytic curves, k-invariant Greensfunctions are defined on G. The Lindelöfs principle is extended to k-invariantGreens functions. When G is the annulus, k-invariant Greens functions areobtained in the explicit form. Since the factorization of the annulus by the group kgenerated by k produces a Möbius strip, the respective result helped us to obtain explicitforms for Greens functions on the Möbius strip.     

Radical Classes of Lattice-Ordered Groups vs. Classes of Compact Spaces

For a given class T of compact Hausdorff spaces, let Y(T) denote the class of -groups G such that for each gG, the Yosida space Y(g) of g belongs to T. Conversely, if R is a class of ;-groups, then T(R) stands for the class of all spaces which are homeomorphic to a Y(g) for some gGR. The correspondences TY(T) and RT(R) are examined with regard to several closure properties of classes. Several sections are devoted to radical classes of -groups whose Yosida spaces are zero-dimensional. There is a thorough discussion of hyper-projectable -groups, followed by presentations on Y(e.d.), where e.d. denotes the class of compact extremally disconnected spaces, and, for each regular uncountable cardinal , the class Y(disc), where disc stands for the class of all compact -disconnected spaces. Sample results follow. Every strongly projectable -group lies in Y(e.d.). The -group G lies in Y(e.d.) if and only if for each gG Y(g) is zero-dimensional and the Boolean algebra of components of g, comp(g), is complete. Corresponding results hold for Y(disc). Finally, there is a discussion of Y(F), with F standing for the class of compact F-spaces. It is shown that an Archimedean -group G is in Y(F) if and only if, for each pair of disjoint countably generated polars P and Q, G=P +Q .     

A Note on π-Engel Conditions

In this paper we show that the weakly -Engel conditions are closely related to the existance of normal -complements; while the -Engel conditions are closely related to the -nilpotent groups.AMS Subject Classification (2000): 20D20     

Some remarks on the product of two C α-compact subsets

For a cardinal , we say that a subset B of a space X is C -compact in X if for every continuous function is a compact subset of . If B is a C-compact subset of a space X, then (B, X) denotes the degree of C -compactness of B in X. A space X is called -pseudocompact if X is C -compact into itself. For each cardinal , we give an example of an -pseudocompact space X such that X × X is not pseudocompact: this answers a question posed by T. Retta in Some cardinal generalizations of pseudocompactness Czechoslovak Math. J. 43 (1993), 385–390. The boundedness of the product of two bounded subsets is studied in some particular cases. A version of the classical Glicksberg's Theorem on the pseudocompactness of the product of two spaces is given in the context of boundedness. This theorem is applied to several particular cases.     

Study of Noetherian Γ-Semirings via Operator Semirings

In this paper we study Noetherian -semirings and obtain Cohens theorem for a special class of -semirings. Weak primary decomposition theorem for a particular type of -semirings is also obtained.Presently Lecturer in Mathematics, University of Burdwan, GOLAPBAG, W.B. INDIA.     

Generalizations of Perfect, Semiperfect, and Semiregular Rings

For a ring R and a right R-module M, a submodule N of M is said to be -small in M if, whenever N + X = M with M/X singular, we have X = M. If there exists an epimorphism p: P M such that P is projective and Ker(p) is -small in P, then we say that P is a projective -cover of M. A ring R is called -perfect (resp., -semiperfect, -semiregular) if every R-module (resp., simple R-module, cyclically presented R-module) has a projective -cover. The class of all -perfect (resp., -semiperfect, -semiregular) rings contains properly the class of all right perfect (resp., semiperfect, semiregular) rings. This paper is devoted to various properties and characterizations of -perfect, -semiperfect, and -semiregular rings. We define (R) by (R)/Soc(R_R) = Jac(R/Soc(R_R)) and show, among others, the following results:

Metric and Complete Metric σ-Frames

A -frame is a lattice in which countable joins exist and binary meets distribute over countable joins. In this paper, the category MFrm, of metric -frames, is introduced, and it is shown to be equivalent to the category MLFrm _u, of metric Lindelöf frames.Finally, it is shown that the complete metric -frames are exactly the cozero parts of complete metric Lindelöf frames.     

On θ-Principal Extension and H-Closedness

A type of extensions called the -extention of topological spaces and their -equivalence and -trace systems are introduced, which ultimately characterize H-closed -extensions of a Hausdorff topological space. Also, the notion of -principal extensions is defined. A typical -principal extension consisting of certain grills on a Hausdorff space is constructed, and finally, some characterizations of H-closedness of a Hausdorff space are obtained.AMS Subject Classification (2000) 54D30 54D99     

Hyperbolic Geometry with Clifford Algebra

The Clifford algebra in D. Hestenes formulation is used to study hyperbolic geometry and some interesting theorems are obtained. The computational power of this formulation is fully revealed by the ease of extending old results and discovering new ones. An important new result is the formulas on the area and perimeter of a convex polygon, based on extending Gauss equalities.     

Fine Analysis of the Quasi-Orderings on the Power Set

We pursue the fine analysis of the quasi-orderings and on the power set of a quasi-ordering (Q,). We set X Y if every xX is majorized in by some yY, and X Y if every yY is minorized in by some xX. We show that both these quasi-orderings are -wqo if and only if the original quasi-ordering is ( )-wqo. For this holds also restricted to finite subsets, thus providing an example of a finitary operation on quasi-orderings which does not preserve wqo but preserves bqo.     

Bounds for optimalα-β binary trees

In this paper we consider a special kind of binary trees where each right edge is associated with a positive number and each left edge with a positive number( ). Given, and the number of nodesn, an optimal tree is one which minimizes the total weighted path length. An algorithm for constructing an optimal tree for given, , n is presented, based on which bounds for balances and total weighted path lengths of optimal trees are derived.     

Functional Approach of Large Deviations in General Spaces

Let X be a topological space, ( ) a net of Borel probability measures on X, and (t) a net in ]0,[ converging to 0. Let be a set of continuous functions such that for all x X that can be suitably distinguished by some continuous functions from any closed set not containing contains such a distinguishing function. Assuming that exists for all , we give a sufficient condition in order that ( ₎ satisfies a large deviation principle with powers (t₎ and not necessary tight rate function. When X is completely regular (not necessary Hausdorff), this condition is also necessary, and so strictly weaker than exponential tightness; this allows us to strengthen Brycs theorem in various ways. We give the general form of a rate function in terms of . A Prohorov-type theorem with a weaker notion than exponential tightness is obtained, which improves known results.     

Improvement of the Delsarte Bound for τ-Designs When It Is Not the Best Bound Possible

In this paper the problem for improvement of the Delsarte bound for -designs is investigated. Two main results are presented. Firstly, necessary and sufficient conditions for improving the bound are proved. We define test functions with the property that they are negative if and only if the Delsarte bound D(phmmat;, ) can be improved by linear programming. Then we investigate the infinite polynomial metric spaces and give exact intervals, when the Delsarte bound is not the best linear programming bound possible. Secondly, we derive a new bound for the infinite PMS. Analytical forms of the extremal polynomials of degree + 2 for non-antipodal PMS and of degree + 3 for antipodal PMS are given. The new bound is investigated in different asymptotical processes for infinite PMS. When and n grow simultaneously to infinity our bound is better than Delsarte bound.     

On <Emphasis Type="Italic">L</Emphasis><Subscript>α,ω</Subscript> complete extensions of complete theories of Boolean algebras

For a complete first order theory of Boolean algebras T which has nonisomorphic countable models, we determine the first limit ordinal = (T) such that We show that for some and for all other Ts, A nonprincipal ideal I of B is almost principal, if a is a principal ideal of B} is a maximal ideal of B. We show that the theory of Boolean algebras with an almost principal ideal has complete extensions and characterize them by invariants similar to the Tarskis invariants.Mathematics Subject Classification (2000): Primary 03C15, Secondary 03C35, 06E05Revised version: 2 February 2004