Chains and Trees: ‘Strong’–‘Weak’ Order in Job Scheduling

We present a summary of recent NP-hardness and polynomial time solvability results for the distinction between strong and weak precedence for chains and trees in scheduling. We distinguish between chains and proper trees which are not chains, and demonstrate that the strong-weak precedence distinction for chains is not inclusive with regards to NP-hardness, and conjecture that the same holds for strong-weak tree precedence. The objective is to show that different interpretations for chain and tree order relations in scheduling might have far reaching computational implications.     

A Categorical Definition of Semidirect Products

For any punctured category, a definition of a semidirect product and its dual counterpart, a semidirect sum, is given. Several examples are studied, among which are semidirect products of commutative Banach algebras and locally compact topological groups, and semidirect sums of compact Hausdorff spaces with basepoint. Also, further applications to commutative Banach algebras are given.     

Transferring estimates for zonal convolution operators

_p - , . Ad- .     

Lipschitz and darbo conditions for the superposition operator in ideal spaces

Summary In this paper we give necessary and sufficient conditions for the superposition operator Fx(s)=f(s, x(s)) to satisfy a Lipschitz condition Fx₁ - Fx₂kx₁ - x₂ or a Darbo condition (FN)k(N) in ideal spaces of measurable functions, where is the Hausdorff measure of noncompactness. Moreover, we characterize a large class of spaces in which the above mentioned two conditions are equivalent.

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Sunto In questo lavoro diamo delle condizioni necessarie e sufficienti perchè l'operatore di sovrapposizione Fx(s)=f (s, x(s)) soddisfi alla condizione di Lipschitz Fx₁–Fx₂ kx₁–x₂ o quella di Darbo (FN)k(N) in spazi ideali di funzioni misurabili, ove è la misura di non compattezza di Hausdorff. Inoltre, caratterizziamo un'ampia classe di spazi in cui le suddette due condizioni sono equivalenti.

     

Нормаляный поридок аддитявных арифметичыских функций на множестве дсдвинутых” пностых чясел

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On a problem of Tandori

(r _k ) - , +1 –1 1/2. =( _i ) , 0< _{1 p} ... _n ... . (a _i )M. (a _i ) . , [2], .     

On the almost everywhere convergence of double Fourier series of integrable functions

, . . Q _k [0,2],k=1,2, — . F(x, y)L(T), T=[0, 2]², G(x, y)L(T) , G(x,y)=F(x,y) Q=Q ₁ ×Q ₂ - .     

Foundational Isomorphisms in Continuous Differentiation Theory

We provide an elementary proof of existence for the Foundational Isomorphism in each of the categories of convergence spaces, compactly generated topological spaces and sequential convergence spaces. This isomorphism embodies the germ of differentiation and its inverse the germ of integration.     

Invariant Subspaces for Positive Operators Acting on a Banach Space with Markushevich Basis

We introduce weak quasinilpotence for operators. Then, by substituting Markushevich basis and weak quasinilpotence at a nonzero vector for Schauder basis and quasinilpotence at a nonzero vector, respectively, we answer a question on the invariant subspaces of positive operators in [3].     

Nonstandard analysis of ordered sets

We introduce a nonstandard approach to the study of ordered setsX based on a classification of the elements of the ordered set *X into three types, upward, downward, and lateral, which may be thought of dynamically as arising from the possibilities of upward, downward, and lateral motion withinX. Initial applications include the characterization thatX has no infinite diverse subset iff *X has no lateral elements, a result subsequently exploited in work on the interval topology and order-compatibility, where we give a nonstandard proof of Naito's result that ifX has no infinite diverse subset, it has a unique order-compatible topology. We also describe how the completion of a nonempty linearly ordered setX may be obtained as a quotient of *X.     

Sur les puissances de l’idéal fondamental de l’anneau de Witt

We reformulate a recent result of Arason and Elman by giving a very simple presentation of the powers of the fundamental ideal of the Witt ring of a field of characteristic 2.

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Sur les puissances de lidéal fondamental de lanneau de Witt

Nous reformulons un résultat récent de Arason et Elman en donnant une présentation très simple des puissances de lidéal fondamental de lanneau de Witt dun corps de caractéristique 2.

     

Noncommutative differential geometry on the quantum SU(2), I: An algebraic viewpoint

We compute the cyclic homology of the coordinate ring A(SL_q(2)) of the quantum algebraic group SL _q (2). We observe a degeneration of the noncommutative de Rham complex. The results are also verified from the point of view of Connes' noncommutative differential geometry.     

The degree of maps between certain 6-manifolds

For manifolds M,M of the form S² e⁴ e⁶ we compute the homomorphisms H_*M H_*M between homology groups which are realizable by a map F: M M.     

Complements to ‘The Bruhat order on symmetric varieties’

We give several complements to the paper The Bruhat order on symmetric varieties. Our main result shows that the partial order on the set of twisted involutions in the Weyl groupW, which was introduced in the earlier paper, agrees with the partial order on induced by the usual Bruhat order onW.R. W. Richardson died on 15 June. 1993.     

Constraint decomposition algorithms in global optimization

Many global optimization problems can be formulated in the form min{c(x, y): x X, y Y, (x, y) Z, y G} where X, Y are polytopes in ^p , ⁿ , respectively, Z is a closed convex set in ^p+n, while G is the complement of an open convex set in ⁿ . The function c: ^p+n is assumed to be linear. Using the fact that the nonconvex constraints depend only upon they-variables, we modify and combine basic global optimization techniques such that some new decomposition methods result which involve global optimization procedures only in ⁿ . Computational experiments show that the resulting algorithms work well for problems with smalln.     

One Machine Scheduling Problem with Fuzzy Random Due-Dates

In many classical or basic scheduling models, jobs processing times and due-dates are crisp values. Recently, they have been formulated as uncertain values in some more actual models. That is the introduction of fuzziness. However, in a real situation of decision making, there exists uncertainty that can not be described only by fuzziness. In this paper, we propose an n-job, one machine scheduling model, where due-dates for jobs are fuzzy random variables. In the model, jobs processing times are crisp, and we assign satisfaction levels to jobs completion times according to membership functions. They are non-increasing functions, but their support positions depend upon the expected due-dates, which are exponentially distributed random variables.     

A new example of ‘independence’ and ‘white noise’

Summary We examine the notion of free independence according to Voiculescu. This form of independence is used for defining free white noise or process with stationary and freely independent increments. We prove a general limit theorem giving the combinatorics of infinitely freely divisible states and thus of free white noises with the help of admissible partitions. We realize the free analogues of the Wiener process and of the Poisson process as processes on the full Fock space ofL ² (—).     

Generalized logic and the representation of rings

Following the progression towards weaker logics, a number of authors have considered the notion of a sheaf over a quantale or, equivalently, a quantale valued set. In this paper, we use ideas from enriched category theory to motivate the definition of a quantic sheaf. Given a localic subquantale of Q, a quantic sheaf over Q gives a sheaf in the usual sense. As an application, we derive a series of sheaf representations for commutative rings including the familiar Pierce representation.     

La conjecture de Novikov pour les feuilletages hyperboliques

Nous définissons la notion de bolicité pour les feuilletages, qui est une notion plus faible que l'hyperbolicité de Gromov, et nous démontrons la conjecture de Novikov pour les feuilletages boliques à base compacte dont le groupoï de d'holonomie est séparé en établissant l'injectivité de l'application de Baum–Connes. Ce résultat généralise celui de Kasparov et Skandalis obtenu dans le cas des groupes boliques.We define the notion of bolicity for foliations, which is a weaker notion than Gromov's hyperbolicity, and we prove the Novikov conjecture for foliations with compact base and whose holonomy groupoid is Hausdorff, by showing that the Baum–Connes map is injective. This result generalizes that of Kasparov and Skandalis in the case of bolic groups.     

On term by term dyadic differentiability of Walsh series

. . . . : {ja _j },j=1,2,... — , f(x) , , f ^[1](x) — f .