Network manipulation algorithm based on inexact alternating minimization

In this paper, we present a network manipulation algorithm based on an alternating minimization scheme from Nesterov (Soft Comput 1–12, 2020). In our context, the alternative process mimics the natural behavior of agents and organizations operating on a network. By selecting starting distributions, the organizations determine the short-term dynamics of the network. While choosing an organization in accordance with their manipulation goals, agents are prone to errors. This rational inattentive behavior leads to discrete choice probabilities. We extend the analysis of our algorithm to the inexact case, where the corresponding subproblems can only be solved with numerical inaccuracies. The parameters reflecting the imperfect behavior of agents and the credibility of organizations, as well as the condition number of the network transition matrix have a significant impact on the convergence of our algorithm. Namely, they turn out not only to improve the rate of convergence, but also to reduce the accumulated errors. From the mathematical perspective, this is due to the induced strong convexity of an appropriate potential function.

     

Category equivalences of clones

This note was inspired by McKenzie's recent paper [8] whose main result is a characterization of categorical equivalence (for short category equivalence) between varieties. Here we use this result to define category equivalent clones and to describe all clones category equivalent to maximal clones. If one is interested in maximal clones only up to category equivalence it is sufficient to consider very simple maximal clones, essentially the same as considered in Jablonski's early papers about maximal clones [6]. The results we have found encourage us in our feeling that category equivalence could be a useful tool in clone theory.Dedicated to the memory of Alan Day.Presented by J. Sichler.     

Hybrid extension of Schottky space

Part of this work was written while the author was feliow of the Graduiertenkolleg Geometrie und Mathematische Physik at the University of Bochum, Germany.     

Partial regularity for certain classes of polyconvex functionals related to nonlinear elasticity

Consider the variational integral where Ω⊂ℝ ⁿ andp≥n≥2. H: (0, ∞)→[0, ∞) is a smooth convex function such that . We approximateJ by a sequence of regularized functionalsJ _δ whose minimizers converge strongly to anJ-minimizing function and prove partial regularity results forJ _δ-minimizers.     

Random rotations of the Wiener path

Summary Let (W, H, ) be an abstract Wiener space and letR(w) be a strongly measurable random variable with values in the set of isometries onH. Suppose that Rh is smooth in the Sobolev sense and that it is a quasi-nilpotent operator onH for everyhH. It is shown that (R(w)h) is again a Gaussian (0, |h| _H ² )-random variable. Consequently, if (e _i ,i)W ^* is a complete, orthonormal basis ofH, then defines a measure preserving transformation, a rotation, onW. It is also shown that if for some strongly measurable, operator valued (onH) random variableR, (R(w+k)h) is (0, |h| _H ² )-Gaussian for allk, hH, thenR is an isometry and Rh is quasi-nilpotent for allHH. The relation between the stochastic calculi for these Wiener pathsw and , as well as the conditions of the inverbibility of the map are discussed and the problem of the absolute continuity of the image of the Wiener measure under Euclidean motion on the Wiener space (i.e. composed with a shift) is studied.The research of the second author was supported by the Fund for the Promotion of Research at the TechnionDedicated to the memory of Albert Badrikian     

A Heuristic Decomposition Algorithm for Scheduling Problems on Mixed Graphs

We consider a scheduling problem where a set of n jobs has to be processed on a set of m machines and arbitrary precedence constraints between operations are given. Moreover, for any two operations i and j values a _ij >0 and a _ji >0 may be given where a _ij is the minimal difference between the starting times of operations i and j when operation i is processed first. Often, the objective is to minimize the makespan but we consider also arbitrary regular criteria. Even the special cases of the classical job shop problem J//C_max belong to the set of NP-hard problems. Therefore, approximation or heuristic algorithms are necessary to handle large-dimension problems. Based on the mixed graph model we give a heuristic decomposition algorithm for such a problem, i.e. the initial problem is partitioned into subproblems that can be solved exactly or approximately with a small error bound. These subproblems are obtained by a relaxation of a subset of the set of undirected edges of the mixed graph. The subproblems are successively solved and a proportion of the results obtained for one subproblem is kept for further subproblem definitions. Numerical results of the algorithm presented here are given.     

Classification of bicovariant differential calculi on quantum groups of type A,B, C and D

Under the assumptions thatq is not a root of unity and that the differentialsdu _j ⁱ of the matrix entries span the left module of first order forms, we classify bicovariant differential calculi on quantum groupsA _n–1 ,B _n ,C _n andD _n . We prove that apart one dimensional differential calculi and from finitely many values ofq, there are precisely2n such calculi on the quantum groupA _n–1 =SL _q (n) forn3. All these calculi have the dimensionn ². For the quantum groupsB _n ,C _n andD _n we show that except for finitely manyq there exist precisely twoN ²-dimensional bicovariant calculi forN3, whereN=2n+1 forB _n andN=2n forC _n ,D _n . The structure of these calculi is explicitly described and the corresponding ad-invariant right ideals of ker are determined. In the limitq1 two of the 2n calculi forA _n–1 and one of the two calculi forB _n ,C _n andD _n contain the ordinary classical differential calculus on the corresponding Lie group as a quotient.     

On a topological characterization of the real line

M.C. Zdun [17] asked whether a subset S of R² such that R × S is homeomorphic to R² must be homeomorphic to R, all these sets being endowed with the usual topologies. We show that the answer is affirmative.     

?*-extensions of tori, higher chow groups and applications to incidence equivalence relations for algebraic cycles

LetX be a smooth projective variety over an algebraically closed fieldk. We repeat Bloch's construction of aG _m -biextension (torseur)E over CH _hom ^p (X)×CH _hom ^q (X) forp+q=dim(X)+1. First we show that in characteristic zeroE comes via pullback from the Poincaré biextension over the corresponding product of intermediate Jacobians which has been conjectured by Bloch and Murre. Then the relations betweenE and various equivalence relations for algebraic cycles are studied. In particular we reprove Murre's theorem stating that Griffiths' conjecture holds for codimension 2 cycles, i.e. every 2-codimensional cycle which is algebraically and incidence equivalent to zero has torsion Abel-Jacobi invariant.