Freeoids: a semi-abstract view on endomorphism monoids of relatively free algebras

A freeoid over a (normally, infinite) set of variables X is defined to be a pair (W, E), where W is a superset of X, and E is a submonoid of W ^W containing just one extension of every mapping X → W. For instance, if W is a relatively free algebra over a set of free generators X, then the pair F(W) := (W, End(W)) is a freeoid. In the paper, the kernel equivalence and the range of the transformation F are characterized. Freeoids form a category; it is shown that the transformation F gives rise to a functor from the category of relatively free algebras to the category of freeoids which yields a concrete equivalence of the first category to a full subcategory of the second one. Also, the concept of a model of a freeoid is introduced; the variety generated by a free algebra W is shown to be concretely equivalent to the category of models of F(W). The sets X, W, and the algebras W may generally be many-sorted.     

Rational curves of degree at most 9 on a general quintic threefold

ABSTRACT.

We prove the following form of the Clemens conjecture in low degree. Let d ≤ 9, and let F be a general quintic threefold in P ⁴. Then (1) the Hilbert scheme of rational, smooth and irreducible curves of degree d on F is finite, nonempty, and reduced; moreover, each curve is embedded in F with normal bundle (?1) ⊕ (?1), and in P ⁴ with maximal rank. (2) On F, there are no rational, singular, reduced and irreducible curves of degree d, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3) On F, there are no connected, reduced and reducible curves of degree d with rational components.     

A GENERALIZATION OF AUSLANDER–BUCHSBAUM AND BASS FORMULAS

Abstract

Given a contravariant functor F : 𝒞 → 𝒮ets for some category 𝒞, we say that F (𝒞) (or F) is generated by a pair (X, x) where X is an object of 𝒞 and x ∈ F(X) if for any object Y of 𝒞 and any y ∈ F(Y), there is a morphism f : Y → X such that F(f)(x) = y. Furthermore, when Y = X and y = x, any f : X → X such that F(f)(x) = x is an automorphism of X, we say that F is minimally generated by (X, x). This paper shows that if the ring R is left noetherian, then there exists a minimal generator for the functor ?xt (?, M) : ? → 𝒮ets, where M is a left R-module and ? is the class (considered as full subcategory of left R-modules) of injective left R-modules.     

Torsion Elements in Incidence Algebras

Abstract

Let I(X, F) denote the incidence algebra of the locally finite partially ordered set, X, defined over the field, F. This paper considers when the set of torsion elements of I(X, F) forms a group. If F is of finite characteristic, it is shown that the torsion elements of I(X, F) form a group if and only if X is bounded, while if F is of characteristic 0, the torsion elements of the incidence algebra form a group if and only if X is an antichain.     

Deciding finiteness for matrix groups over function fields

LetF be a field andt an indeterminate. In this paper we consider aspects of the problem of deciding if a finitely generated subgroup of GL(n,F(t)) is finite. WhenF is a number field, the analysis may be easily reduced to deciding finiteness for subgroups of GL(n,F), for which the results of [1] can be applied. WhenF is a finite field, the situation is more subtle. In this case our main results are a structure theorem generalizing a theorem of Weil and upper bounds on the size of a finite subgroup generated by a fixed number of generators with examples of constructions almost achieving the bounds. We use these results to then give exponential deterministic algorithms for deciding finiteness as well as some preliminary results towards more efficient randomized algorithms. Supported in part by NSF DMS Awards 9404275 and Presidential Faculty Fellowship.     

Universal bases and greedy algorithms for anisotropic function classes

We suggest a three-step strategy to find a good basis (dictionary) for non-linear m-term approximation. The first step consists of solving an optimization problem of finding a near best basis for a given function class F, when we optimize over a collection D of bases (dictionaries). The second step is devoted to finding a universal basis (dictionary) D _u ∈ D for a given pair (F, D) of collections: F of function classes and D of bases (dictionaries). This means that D_u provides near optimal approximation for each class F from a collection F. The third step deals with constructing a theoretical algorithm that realizes near best m-term approximation with regard to D _u for function classes from F. In this paper we work this strategy out in the model case of anisotropic function classes and the set of orthogonal bases. The results are positive. We construct a natural tensor-product-wavelet-type basis and prove that it is universal. Moreover, we prove that a greedy algorithm realizes near best m-term approximation with regard to this basis for all anisotropic function classes.     

Additive Maps Preserving Moore–Penrose Inverses of Matrices on Symmetric Matrix Spaces

Suppose F is a field of characteristic not 2. Let M_n F and S_n F be the n × n full matrix space and symmetric matrix space over F, respectively. All additive maps from S_n F to S_n F (respectively, M_n F) preserving Moore–Penrose inverses of matrices are characterized. We first characterize all additive Moore–Penrose inverse preserving maps from S_n F to M_n F, and thereby, all additive Moore–Penrose inverse preserving maps from S_n F to itself are characterized by restricting the range of these additive maps into the symmetric matrix space.     

Idempotency of Extensions via the Bicompletion

Let Top ₀ be the category of topological T ₀-spaces, QU ₀ the category of quasi-uniform T ₀-spaces, T : QU ₀ → Top ₀ the usual forgetful functor and K : QU ₀ → QU ₀ the bicompletion reflector with unit k : 1 → K. Any T-section F : Top ₀ → QU ₀ is called K-true if KF = FTKF, and upper (lower) K-true if KF is finer (coarser) than FTKF. The literature considers important T-sections F that enjoy all three, or just one, or none of these properties. It is known that T(K,k)F is well-pointed if and only if F is upper K-true. We prove the surprising fact that T(K,k)F is the reflection to Fix(TkF) whenever it is idempotent. We also prove a new characterization of upper K-trueness. We construct examples to set apart some natural cases. In particular we present an upper K-true F for which T(K,k)F is not idempotent, and a K-true F for which the coarsest associated T-preserving coreflector in QU ₀ is not stable under K. We dedicate this paper to the memory of Sérgio de Ornelas Salbany (1941–2005).     

On discrete α-unimodality

A continuous composition semigroup of probability generating functionsF≔(F _t ,t≥0) and the corresponding multiplication ⊙ _F of van Harnet al. (1982,Z. Wahrsch. Verw. Gebiete,61, 97–118) are used to introduce the concept of [F; α]-unimodality which generalizes the discrete α-unimodality due to Abouammoh (1987,Statist. Neerlandica,41, 239–244) and Alamatsaz (1993,Statist. Neerlandica,47, 245–252). We offer various characterizations and other properties of [F;α]-unimodality. Notably, several convolution results are presented. Moreover, we explore the relationship between [F;α]-unimodality and the concepts of discrete self-decomposability and stability. Finally, lower bounds for variances of [F;α]-monotone and [F;α]-unimodal random variables are derived and some examples are also mentioned. Research supported by Grant SS024 of the Research Center of Kuwait University.     

Minimal Generators in Varieties

Given a variety V, the free V-algebra F 1 on one generator represents the canonical underlying functor from V to the category Set of sets. Hence one might ask, whether F 1 is, in some sense, a `canonical' generator of V. To make this question precise the notion of `minimal varietal generator' is introduced. It is shown that in many (though not all) varieties F 1 is a generator of this kind, and often even the unique one. The question whether every variety has a (unique) minimal varietal generator, remains on open problem.     

On the Grigorchuk–Kurchanov conjecture

 Let denote the free group of rank 2g. An automorphism φ? Aut(F _{2 g} ) is generating if N _a φ (N _b ) = F _{2 g} , where N _a is the normal closure of and N _b is defined analogously. We present a characterization of generating automorphisms in Aut(F ₂) and observe that there exists a unique (up to equivalence) epimorphism F ₂→Z×Z: this is a particular case of the Grigorchuk–Kurchanov conjecture. This leads to further investigations for splitting homomorphisms for the pairs (F _{2 g} , F _g) and (G _g, F _g) where G _g denotes the fundamental group of a closed orientable surface of genus g and a reformulation of the Poincaré and Grigorchuk–Kurchanov conjectures is derived. Received: 1 October 2001     

On First Range Times of Linear Diffusions

In this paper we consider first range times (with randomised range level) of a linear diffusion on R. Inspired by the observation that the exponentially randomised range time has the same law as a similarly randomised first exit time from an interval, we study a large family of non-negative 2-dimensional random variables (X,X′) with this property. The defining feature of the family is F^c(x,y)=F^c(x+y,0), ∀ x ≥ 0, y ≥ 0, where F^c(x,y):=P (X > x, X′ > y) We also explain the Markovian structure of the Brownian local time process when stopped at an exponentially randomised first range time. It is seen that squared Bessel processes with drift are serving hereby as a Markovian element.     

Graph colourings with forbidden k-coloured subgraphs

Abstract

For a set F of graphs and a natural number k, an (F, k)-colouring of a graph G is a proper colouring of V (G) such that no subgraph of G isomorphic to an element of F is coloured with at most k colours. Equivalently, if P is the class of all graphs that do not contain an element of F as a subgraph, a χP,k colouring of G is a proper colouring such that the union of at most k colour classes induces a graph in P. The smallest number of colours in such a colouring of G, if it exists, is denoted by χP,k (G). We give some general results on χP,k-colourings and investigate values of χP,k (G) for some choices of P and classes of graphs G.     

Elements of Small Orders in K2F Ⅱ

In "Elements of small orders in K2(F)" (Algebraic K-Theory, Lecture Notes in Math., 966, 1982, 1-6.), the author investigates elements of the form {a, Φn(a)} in the Milnor group K2F of a field F, where Φn(x) is the n-th cyclotomic polynomial. In this paper, these elements are generalized. Applying the explicit formulas of Rosset and Tate for the transfer homomorphism for K2, the author proves some new results on elements of small orders in K2F.     

Integrability and closest approximation representations

Summary Suppose U is a set,F is a field of subsets of U, andp _AB is the set of all real-valued bounded finitely additive functions defined onF. This paper consists of two main parts. In the first, a previously given (seeRiv. Math. Univ. Parma, (3),2 (1973), pp. 251–276) notion of a subset ofp _AB defined by certain closure properties and called a C-set, is considered, and those C-sets that are linear spaces are characterized. Now, suppose γ is a function whose domain isF and whose range is a collection of number sets with bounded union. The set,J _γ, of all elements ofp _AB with respect to which γ is integrable, for refinements of subdivisions, is a C-set and a linear space (seeRend. Sem. Mat. Univ. Padova,52 (1974), pp. 1–24). The second part of this paper concerns, for μ inp _AB and nonnegative-valued, a representation of the element ofJ _γ closest to μ with respect to variation norm. Entrata in Redazione il 14 giugno 1977.     

Determination of a basis for linear least square approximation

A vector of datapoints,Y, which is the sum of a deterministic component,F, and a stochastic (error) component,E, is considered. If is the result of smoothingY in the least square sense, thenE can be split into two residual vectors:E= – F – ( – Y). It is shown in this paper, that, under the assumption that the errors have a zero mean, there exists a simple linear relationship between the expected mean square values of these residuals and the expected variance of the errors (var (E)). If a rough estimate of var (E) is available, the derived relationship may prove useful in determining that particular member of the class of functions considered which best approximates the (unknown) datapoints ofF.     

Higher Power Residue Codes

For certain primeslandp, and characters χ:F*_p→F*_l², we construct codesWof lengthp+ 1 overF_l²which are linear overF_l, but not overF_l², and which are invariant under a monomial action of the group SL(2,p). We consider the cases of cubic and quartic characters in detail and use theWto construct linear codes overF_lin these cases.     

0-Minimal Quasi-ideals of Generalized Linear Transformation Semigroups

Abstract

For vector spaces V and W over a field F, L _F (V, W) denotes the set of all linear transformations α : V → W, and for a cardinal number k > 0, let L _F (V, W, k) be the set of all α ∈ L _F (V, W) of rank less than k. For θ ∈ L _F (W, V), let (L _F (V, W, k), θ) denote the semigroup L _F (V, W, k) under the operation ? defined by α ? β = αθβ for all α, β ∈ L _F (V, W, k). In this paper, all 0-minimal quasi-ideals of the semigroup (L _F (V, W, k), θ) are completely characterized. It is also shown from this characterization that every nonzero semigroup (L _F (V, W, k), θ) always has a 0-minimal quasi-ideal.     

Infinite dimensional linear groups with many G — invariant subspaces

Let F be a field, A be a vector space over F, GL(F, A) be the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dim _F (BFG/B) is finite. A subspace B is called almost G-invariant, if dim _F (B/Core _G (B)) is finite. In the current article, we study linear groups G such that every subspace of A is either nearly G-invariant or almost G-invariant in the case when G is a soluble p-group where p = char F.