Monoid Properties as Invariants of Toposes of Monoid Actions

Applied Categorical Structures - We systematically investigate, for a monoid M, how topos-theoretic properties of $${{,mathrm{mathbf {PSh}},}}(M)$$ , including the properties of being atomic,...     

Descent Theory for Schemes

We give a complete characterization of the class of quasi-compact morphisms of schemes that are stable effective descent morphisms for the SCHEMES-indexed category given by quasi-coherent sheaves of modules.     

The Homology of Partial Monoid Actions and Petri Nets

The aim of this paper is to study the homology theory of partial monoid actions and apply it to computing the homology groups of mathematical models for concurrency. We study the Baues–Wirsching homology groups of a small category associated with a partial monoid action on a set. We prove that these groups can be reduced to the Leech homology groups of the monoid. For a trace monoid with a partial action on a set, we build a complex of free Abelian groups for computing the homology groups of this small category. It allows us to solve the problem posed by the author on the construction of an algorithm to computing the homology groups of elementary Petri nets. We describe the algorithm and give examples of computing the homology groups.     

Descent Theory and Morita Theory for Ultrametric Banach Modules

In this paper we consider ultrametric Banach modules over commutative ultrametric Banach algebras with unit. We study the descent problem along a morphism f: R S of such algebras and show that descent morphisms coincide with weak retracts. We give further conditions for having an effective descent morphism or for having a Morita equivalence between the corresponding categories of ultrametric Banach modules.     

Finiteness Results in Descent Theory

It is shown that a -curve of genus g and with stable reduction (in some generalized sense)at every finite place outside a finite set S can be definedover a finite extension L of its field of moduli K dependingonly on g, S and K. Furthermore, there exist L-models that inheritall places of good and stable reduction of the original curve(except possibly for finitely many exceptional places dependingon g, K and S). This descent result yields this moduli formof the Shafarevich conjecture: given g, K and S as above, onlyfinitely many K-points on the moduli space M_g correspond to-curves of genus g and with good reduction outside S. Other applications to arithmetic geometry,like a modular generalization of the Mordell conjecture, aregiven.     

Another Approach to Topological Descent Theory

In the category T op of topological spaces and continuous functions, we prove that surjective maps which are descent morphisms with respect to the class E of continuous bijections are exactly the descent morphisms, providing a new characterization of the latter in terms of subfibrations E(X) of the basic fibration given by T op/X which are, essentially, complete lattices. Also effective descent morphisms are characterized in terms of effective morphisms with respect to continuous bijections. For classes E satisfying suitable conditions, we show that the class of effective descent morphisms coincides with the one of effective E-descent morphisms.     

Character Formulas for q-Rook Monoid Algebras

The q-rook monoid R _n(q) is a semisimple (q)-algebra that specializes when q 1 to [R _n], where R _n is the monoid of n × n matrices with entries from {0, 1} and at most one nonzero entry in each row and column. We use a Schur-Weyl duality between R _n(q) and the quantum general linear group to compute a Frobenius formula, in the ring of symmetric functions, for the irreducible characters of R _n(q). We then derive a recursive Murnaghan-Nakayama rule for these characters, and we use Robinson-Schensted-Knuth insertion to derive a Roichman rule for these characters. We also define a class of standard elements on which it is sufficient to compute characters. The results for R _n(q) specialize when q = 1 to analogous results for R _n.     

On Semihereditary and p.p. Monoid Rings

No Abstract. June 30, 1998     

On the Convergence of Descent Algorithms

It is proved that any cluster point of a sequence defined by a steepest descent algorithm in a general normed vector space is a critical point. The function is just assumed to be continuously differentiable. The class of algorithms we consider encompasses several choices such as the Cauchy steplength and the Curry steplength.     

On the Monoid of All Partial Order-Preserving Extensive Transformations

A partial transformation α on an n-element chain X _n is called order-preserving if x ≤ y implies xα ≤yα for all x, y in the domain of α and it is called extensive if x ≤ xα for all x in the domain of α. The set of all partial order-preserving extensive transformations on X _n forms a semiband POE _n . We determine the maximal subsemigroups as well as the maximal subsemibands of POE _n .     

Tachyon Effective Actions in String Theory

We discuss the derivation of string effective actions for tachyons and massless modes within the sigma-model approach. We mostly consider the open string case, demonstrating that the renormalized partition function of the boundary sigma model gives the effective action for the massless vector and tachyon in the derivative expansion. We give a manifestly gauge-invariant definition of Z(T,A) in the non-Abelian Neveu–Schwarz–Ramond (NSR) open string theory and verify that its derivative reproduces the tachyon beta function in a particular scheme. We also comment on the derivation of similar actions for tachyons and massless modes in the closed bosonic and the NSR string theories.     

Comparing Semigroup and Monoid Presentations for Finite Monoids

 It is known that for any finite group G given by a finite group presentation there exists a finite semigroup presentation for G of the same deficiency, i.e. satisfying . It is also known that the analogous statement does not hold for all finite monoids. In this paper we give a necessary and sufficient condition for a finite monoid M, given by a finite monoid presentation, to have a finite semigroup presentation of the same deficiency.     

Band Monoid Languages Revisited

We describe the atoms of the varieties of band monoid languages as disjoint unions of concatenations of words and free monoids. We present a new proof of a solution of the word problem for the free band due to Siekman and Szabo.     

On Early Stopping in Gradient Descent Learning

In this paper we study a family of gradient descent algorithms to approximate the regression function from reproducing kernel Hilbert spaces (RKHSs), the family being characterized by a polynomial decreasing rate of step sizes (or learning rate). By solving a bias-variance trade-off we obtain an early stopping rule and some probabilistic upper bounds for the convergence of the algorithms. We also discuss the implication of these results in the context of classification where some fast convergence rates can be achieved for plug-in classifiers. Some connections are addressed with Boosting, Landweber iterations, and the online learning algorithms as stochastic approximations of the gradient descent method.     

On Primitive Idempotents of the Strong Endomorphism Monoid of a Graph

51.IntroductionandPreIiminariesThemonoidofendomorphismsofagraph,inparticular,thatofstrongendomorphismsofagraph,hasbeentheobjectofresearchesinthetheoryofsemigroupsforquitesometime(cf.Llj-Lloj).LlJandL2jcanserveasasurvey.Theaimoftheseresearchesistocon-tributetothealgebraicanalysisofgraphs.Thesubjectyieldssomeinterestsincetheresultsoftheseresearchesmayopenvastpossibilitiesforapplicationsofsemigrouptheorytographtheo-ry.InL1]andL3j,ithasbeenprovedthatforfinitegraphG,sEnd(G),themonoidofstrongen…     

Comparing Semigroup and Monoid Presentations for Finite Monoids

 It is known that for any finite group G given by a finite group presentation there exists a finite semigroup presentation for G of the same deficiency, i.e. satisfying . It is also known that the analogous statement does not hold for all finite monoids. In this paper we give a necessary and sufficient condition for a finite monoid M, given by a finite monoid presentation, to have a finite semigroup presentation of the same deficiency. (Received 17 April 2001; in revised form 15 September 2001)