On the arithmetic of tame monoids with applications to Krull monoids and Mori domains

Let H be an atomic monoid (e.g., the multiplicative monoid of a noetherian domain). For an element b∈H, let ω(H,b) be the smallest  N∈N₀∪{∞} having the following property: if  n∈N and  a₁,…,a_n∈H are such that b divides  a₁⋅…⋅a_n, then b already divides a subproduct of a₁⋅…⋅a_n consisting of at most N factors. The monoid H is called tame if . This is a well-studied property in factorization theory, and for various classes of domains there are explicit criteria for being tame. In the present paper, we show that, for a large class of Krull monoids (including all Krull domains), the monoid is tame if and only if the associated Davenport constant is finite. Furthermore, we show that tame monoids satisfy the Structure Theorem for Sets of Lengths. That is, we prove that in a tame monoid there is a constant M such that the set of lengths of any element is an almost arithmetical multiprogression with bound M.     

On the arithmetic of Krull monoids with infinite cyclic class group

Let H be a Krull monoid with infinite cyclic class group G and let G_P⊂G denote the set of classes containing prime divisors. We study under which conditions on G_P some of the main finiteness properties of factorization theory-such as local tameness, the finiteness and rationality of the elasticity, the structure theorem for sets of lengths, the finiteness of the catenary degree, and the existence of monotone and near monotone chains of factorizations-hold in H. In many cases, we derive explicit characterizations.     

Atoms of the set of numerical semigroups with fixed Frobenius number

Given a positive integer g, we denote by F(g) the set of all numerical semigroups with Frobenius number g. The set (F(g),∩) is a semigroup. In this paper we study the generators of this semigroup.     

Rings of matrix invariants in positive characteristic

Denote by R_n,m the ring of invariants of m-tuples of n×n matrices (m,n?2) over an infinite base field K under the simultaneous conjugation action of the general linear group. When char(K)=0, Razmyslov (Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974) 723) and Procesi (Adv. Math. 19 (1976) 306) established a connection between the Nagata-Higman theorem and the degree bound for generators of R_n,m. We extend this relationship to the case when the base field has positive characteristic. In particular, we show that if 0<char(K))?n, then R_n,m is not generated by its elements whose degree is smaller than m. A minimal system of generators of R_2,m is determined for the case char(K)=2: it consists of 2^m+m−1 elements, and the maximum of their degrees is m. We deduce a consequence indicating that the theory of vector invariants of the special orthogonal group in characteristic 2 is not analogous to the case char(K)≠2. We prove that the characterization of the R_n,m that are complete intersections, known before when char(K)=0, is valid for any infinite K. We give a Cohen-Macaulay presentation of R_2,4, and analyze the difference between the cases char(K)=2 and char(K)≠2.     

Hodge decomposition of Alexander invariants

Multivariable Alexander invariants of algebraic links calculated in terms of algebro-geometric invariants (polytopes and ideals of quasiadjunction). The relations with log-canonical divisors, the multiplier ideals and a semicontinuity property of polytopes of quasiadjunction are discussed. Received: 8 February 2001 / Revised version: 1 December 2001     

New developments in the theory of Gröbner bases and applications to formal verification

We present foundational work on standard bases over rings and on Boolean Gröbner bases in the framework of Boolean functions. The research was motivated by our collaboration with electrical engineers and computer scientists on problems arising from formal verification of digital circuits. In fact, algebraic modelling of formal verification problems is developed on the word-level as well as on the bit-level. The word-level model leads to Gröbner basis in the polynomial ring over Z/2ⁿ while the bit-level model leads to Boolean Gröbner bases. In addition to the theoretical foundations of both approaches, the algorithms have been implemented. Using these implementations we show that special data structures and the exploitation of symmetries make Gröbner bases competitive to state-of-the-art tools from formal verification but having the advantage of being systematic and more flexible.     

On the direct image of intersections in exact homological categories

Given a regular epimorphism f:X?Y in an exact homological category C, and a pair (U,V) of kernel subobjects of X, we show that the quotient (f(U)∩f(V))/f(U∩V) is always abelian. When C is nonpointed, i.e. only exact protomodular, the translation of the previous result is that, given any pair (R,S) of equivalence relations on X, the difference mappingδ:Y/f(R∩S)?Y/(f(R)∩f(S)) has an abelian kernel relation. This last result actually holds true in any exact Mal'cev category. Setting Y=X/T, this result says that the difference mapping determined by the inclusion T∪(R∩S)?(T∪R)∩(T∪S) has an abelian kernel relation, which casts a new light on the congruence distributive property.     

The structure of spaces of quasianalytic functions of Roumieu type

It is shown that spaces of quasianalytic ultradifferentiable functions of Roumieu type ℰ_{w}(Ω), on an open convex set , satisfy some new (Ω) -type linear topological invariants. Some consequences for the splitting of short exact sequences of these spaces as well as for the structure of the spaces are derived. In particular, Fréchet quotients of ℰ_{w}(Ω) have property (), while dual Fréchet quotients have property () of Vogt. The work of P. Domański was supported by Committee of Scientific Research (KBN), Poland, grant P03A 022 25.     

Total Least-Squares regularization of Tykhonov type and an ancient racetrack in Corinth

In this contribution a variation of Golub/Hansen/O’Leary’s Total Least-Squares (TLS) regularization technique is introduced, based on the Hybrid APproximation Solution (HAPS) within a nonlinear Gauss-Helmert Model. By applying a traditional Lagrange approach to a series of iteratively linearized Gauss-Helmert Models, a new iterative scheme has been found that, in practice, can generate the Tykhonov regularized TLS solution, provided that some care is taken to do the updates properly.The algorithm actually parallels the standard TLS approach as recommended in some of the geodetic literature, but unfortunately all too often in combination with erroneous updates that would still show convergence, although not necessarily to the (unregularized) TLS solution. Here, a key feature is that both standard and regularized TLS solutions result from the same computational framework, unlike the existing algorithms for Tykhonov-type TLS regularization.The new algorithm is then applied to a problem from archeology. There, both the radius and the center-point coordinates of a circle have to be determined, of which only a small part of the arc had been surveyed in-situ, thereby giving rise to an ill-conditioned set of equations. According to the archaeologists involved, this circular arc served as the starting line of a racetrack in the ancient Greek stadium of Corinth, ca. 500 BC. The present study compares previous estimates of the circle parameters with the newly developed “Regularized TLS Solution of Tykhonov type.”     

Étale groupoids and their quantales

We establish close and previously unknown relations between quantales and groupoids. In particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic étale groupoids and their quantales, which are given a rather simple characterization and here are called inverse quantal frames. We show that the category of inverse quantal frames is equivalent to the category of complete and infinitely distributive inverse monoids, and as a consequence we obtain a (non-functorial) correspondence between these and localic étale groupoids that generalizes more classical results concerning inverse semigroups and topological étale groupoids. This generalization is entirely algebraic and it is valid in an arbitrary topos. As a consequence of these results we see that a localic groupoid is étale if and only if its sublocale of units is open and its multiplication map is semiopen, and an analogue of this holds for topological groupoids. In practice we are provided with new tools for constructing localic and topological étale groupoids, as well as inverse semigroups, for instance via presentations of quantales by generators and relations. The characterization of inverse quantal frames is to a large extent based on a new quantale operation, here called a support, whose properties are thoroughly investigated, and which may be of independent interest.     

Weak subintegral closure of ideals

We describe some basic facts about the weak subintegral closure of ideals in both the algebraic and complex-analytic settings. We focus on the analogy between results on the integral closure of ideals and modules and the weak subintegral closure of an ideal. We start by giving a new geometric interpretation of the Reid–Roberts–Singh criterion for when an element is weakly subintegral over a subring. We give new characterizations of the weak subintegral closure of an ideal. We associate with an ideal I of a ring A an ideal I_>, which consists of all elements of A such that v(a)>v(I), for all Rees valuations v of I. The ideal I_> plays an important role in conditions from stratification theory such as Whitney's condition A and Thom's condition Af and is contained in every reduction of I. We close with a valuative criterion for when an element is in the weak subintegral closure of an ideal. For this, we introduce a new closure operation for a pair of modules, which we call relative weak closure. We illustrate the usefulness of our valuative criterion.     

Tropical representation of Weyl groups associated with certain rational varieties

Starting from certain rational varieties blown-up from N(P¹), we construct a tropical, i.e., subtraction-free birational, representation of Weyl groups as a group of pseudo-isomorphisms of the varieties. We develop an algebro-geometric framework of τ-functions as defining functions of exceptional divisors on the varieties. In the case where the corresponding root system is of affine type, our construction yields a class of (higher order) q-difference Painlevé equations and its algebraic degree grows quadratically.     

Gaps in the number of generators of monomial Togliatti systems

Let $I_{d,n}?k[x_0,?,x_n]$ be a minimal monomial Togliatti system of forms of degree d. In [4], Mezzetti and Miró-Roig proved that the minimal number of generators $\mu (I_{d,n})$ of $I_{d,n}$ lies in the interval $[2n+1,\left(\begin{array}{c}n+d?1\\ n?1\end{array}\right)]$. In this paper, we prove that for $n\ge 4$ and $d\ge 3$, the integer values in $[2n+3,3n?1]$ cannot be realized as the number of minimal generators of a minimal monomial Togliatti system. We classify minimal monomial Togliatti systems $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{d,n})=2n+2$ or 3n (i.e. with the minimal number of generators reaching the border of the non-existence interval). Finally, we prove that for $n=4$, $d\ge 3$ and $\mu \in [9,\left(\begin{array}{c}d+3\\ 3\end{array}\right)]?\{11\}$ there exists a minimal monomial Togliatti system $I_{d,n}?k[x_0,?,x_n]$ of forms of degree d with $\mu (I_{n,d})=\mu$.     

The splitting of exact sequences of PLS-spaces and smooth dependence of solutions of linear partial differential equations

We investigate the splitting of short exact sequences of the form

0→X→Y→E→0,     

Graph spectra in Computer Science

In this paper, we shall give a survey of applications of the theory of graph spectra to Computer Science. Eigenvalues and eigenvectors of several graph matrices appear in numerous papers on various subjects relevant to information and communication technologies. In particular, we survey applications in modeling and searching Internet, in computer vision, data mining, multiprocessor systems, statistical databases, and in several other areas. Some related new mathematical results are included together with several comments on perspectives for future research. In particular, we claim that balanced subdivisions of cubic graphs are good models for virus resistent computer networks and point out some advantages in using integral graphs as multiprocessor interconnection networks.