A maschke type theorem for weak Hopf algebras

In this paper, we give a necessary and sufficient condition for a comodule algebra over a weak Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the Hopf algebra setting. Also, from these results, we deduce a version of Maschke＇s Theorem for （H, B）-Hopf modules associated with a weak Hopf algebra H and a right H-comodule algebra B.     

A structure theorem for groups of weak characteristic p-type

We prove a structure theorem for groups of weak characteristic p-type. We use this result to explain how to identify an "end-node" of a diagram for a group.     

Weak buildings of spherical type

In this article constructions are given for weak buildings of spherical type. The buildings are constructed as point-line geometries according to a pattern found within the appropriate Coxeter complexes. In all the examples constructed here there is at least one point of the geometry on no lines of more than two points. We show that this is true in all weak spherical buildings, and discuss to what further extent all weak buildings follow the given construction.     

弱模余代数的结构定理

设H为弱Hopf代数,C为弱右H-模余代数,令C=C/C·ker L.利用Smash余积来研究弱模余代数上的结构定理,并给出了C与C×H作为余代数同构的条件.     

The structure theorem for weak module coalgebras

Let H be a weak Hopf algebra, let C be a weak right H-module coalgebra, and let $ \bar C = {C \mathord{\left/ {\vphantom {C C}} \right. \kern-0em} C} \cdot Ker \sqcap ^L $ . We prove a structure theorem for weak module coalgebras, namely, C is isomorphic as a weak right H-module coalgebra to a weak smash coproduct $ \bar C $ × H defined on a k-space $$ \{ \Sigma c_{(0)} \otimes h_2 \varepsilon (c_{( - 1)} h_1 )|c \in C,h \in H\} $$ if there exists a weak right H-module coalgebra map ?: C → H.     

A Maschke type theorem for weak group entwined modules and applications

Let π be a discrete group. Given a weak π-entwining structure \({(A,C)_{\pi - \psi }}\) and α ∈ π, we give the necessary and sufficient conditions for the forgetful functor \({F^{(\alpha )}}\) from the category \(U_A^{\pi - C}(\psi )\) of right \({(A,C)_{\pi - \psi }}\) -modules to the category \({M_{{A_\alpha }}}\) of right \({A_\alpha }\) -modules to be separable. This leads to a generalized notion of integrals. The results are applied to weak Doi-Hopf π-modules and to weak entwining modules.     

A characterization theorem for weak varieties

In this paper I define the notion of mixed product of partial algebras. It is a common generalization of two operators: weak subalgebras and products of partial algebras. I show that a class of similar partial algebras is a weak variety if and only if it is closed under the operators of mixed products and closed homomorphic images.     

The structure theorem and duality theorem for endomorphism algebras of weak Hopf algebras

With an aim of exploring homological algebra for weak Hopf modules, this paper investigates the HOM-functor and presents the structure theorem for endomorphism algebras of weak two-sided (A,H)-Hopf modules, and gives the duality theorem for weak “big” smash products.     

Spanning point-line geometries in buildings of spherical type

We consider the point-line geometries that arise as a shadow space in a spherical building with a diagram of type A_n, B_n, C_n, D_n or E_n, and determine in which cases the geometry is spanned by the set of points on an apartment. It turns out that this happens precisely in the cases corresponding to a minimal weight.     

The structure theorem of endomorphism algebras for weak Doi-Hopf modules

The paper is concerned with endomorphism algebras for weak Doi-Hopf modules. Under the condition “weak Hopf-Galois extensions”, we present the structure theorem of endomorphism algebras for weak Doi-Hopf modules, which extends Theorem 3.2 given by Schneider in [1]. As applications of the structure theorem, we obtain the Kreimer-Takeuchi theorem (see Theorem 1.7 in [2]) and the Nikshych duality theorem (see Theorem 3.3 in [3]) in the case of weak Hopf algebras, respectively.     

Noncommutative Riesz theorem and weak Burnside type theorem on twisted conjugacy

The paper consists of two parts. In the first part, we prove a noncommutative analog of the Riesz(— Markov—Kakutani) theorem on representation of functionals on an algebra of continuous functions by regular measures on the underlying space. In the second part, using this result, we prove a weak version of a Burnside type theorem on twisted conjugacy for arbitrary discrete groups.     

A weak existence theorem and weak permanence properties for strong liftings

On a locally compact space with a first-uncountable basis for its topology every positive Radon measure is majorized by an upper integral which admits a strong lifting. The set of B-continuous measures on any space that are majorized by an upper integral with strong lifting forms a band.Supported by NSF-Grant-20541     

A theorem about snakes for weak Cartesian systems

It is shown that Karlin's theorem about snakes remains true for polynomials in any weak Cartesian system of continuous functions on a compact set without any change in the statement, and the existence part is valid for arbitrary weak Chebyshev systems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 315–321, March, 1995.     

A weak duality theorem for stochastic linear programming

Summary A linear programming problem is said to be stochastic if one or more of the coefficients in the objective function or the system of constraints or resource availabilities is known only by its probability distribution. A distinction is usually made between two related approaches to stochastic linear programming, the active and passive approach respectively. An extension of the duality theorem of non-stochastic or deterministic programming problem has been attempted in this paper in the area of stochastic linear programming in its two approaches. The method of proof is based on the idea that since the parameter space defined by a stochastic linear programme is the topological product of the real line with itself, it forms a first countable topological space. Using a set of distinct and selected points in the parameter space the concepts of feasibility, optimality and duality are extended to stochastic linear programming problems of arbitrary dimensionality. Based on the non-singular regions of the parameter space of a stochastic linear programming problem the theorem utilizes the conditions of convergence of the sequence of distinct and selected points in the parameter space to a limit point and thereby generalizes the duality theorem in the stochastic case. Furthermore it is shown that the regions of feasibility of the active and passive approaches of stochastic linear programming may be different, so that on this basis it may be possible to establish some inequality relations for the optimal solutions defined for the respective feasible regions.

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Zusammenfassung Ein lineares Programmproblem wird stochastisch genannt, wenn ein oder mehrere Koeffizienten der Zielfunktion oder des Systems der Beschränkungen oder der verfügbaren Ressourcen nur durch ihre Wahrscheinlichkeitsverteilung bekannt sind. Gewöhnlich wird zwischen zwei verwandten Verfahren für das stochastische lineare Programmieren unterschieden, dem aktiven und dem passiven Verfahren.In der vorliegenden Arbeit wird versucht, das für nichtstochastische oder deterministische Programmprobleme gültige Dualitätstheorem unter Berücksichtigung beider Verfahrensweisen auf den Bereich des stochastischen linearen Programmierens auszudehnen. Der Beweis gründet sich auf den Gedanken, daß der Parameterraum einen abzählbaren topologischen Raum bildet, da er — durch ein stochastisches lineares Programm definiert — das topologische Produkt der reellen Achse mit sich selbst ist. Unter Benutzung einer Menge von verschiedenen und ausgewählten Punkten im Parameterraum werden die Begriffe der Zulässigkeit, der Optimalität und der Dualität auf stochastische lineare Programmprobleme beliebiger Dimension ausgedehnt. Auf der Grundlage nichtsingulärer Bereiche des Parameterraumes eines stochastischen linearen Programmproblems benutzt das Theorem die Bedingungen für die Konvergenz einer Folge verschiedener und ausgewählter Punkte des Parameterraumes nach einem Grenzpunkt und verallmeinert damit das Dualitätstheorem für den stochastischen Fall. Weiter wird gezeigt, daß die Zulässigkeitsbereiche der aktiven und passiven Verfahren des stochastischen linearen Programmierens verschieden sein können, so daß es möglich sein kann, gewisse Ungleichungen für die optimalen Lösungen aufzustellen, die für die entsprechenden zulässigen Bereiche definiert sind.

     

A weak version of Rolle's theorem

We investigate the fields with the property that any polynomial over the field which splits in the field has a derivative which also splits.

     

A separation theorem for the weak s-convex orders

The present paper extends to higher degrees the well-known separation theorem decomposing a shift in the increasing convex order into a combination of a shift in the usual stochastic order followed by another shift in the convex order. An application in decision making under risk is provided to illustrate the interest of the result.     

A weak limit theorem for generalized multifractional Brownian motion

We provide a result on an approximation to the generalized multifractional Brownian motion in the space of continuous functions on [0, 1]. The construction of this approximation is based on the Poisson process.     

A weak Brun—Titchmarsh theorem for multiplicative functions

An important theorem of Shiu gives a (precise) bound for the average of values of multiplicative functions, of a certain class, over ‘short’ intervals. Here we obtain, by simple means, the above result of same qualitative order.