Finitely additive random walks on infinitely generated free groups

We consider any purely finitely additive probability measure supported on the generators of an infinitely generated free group and the Markov strategy with stationary transition probability . As well as for the case of random walks (with countably additive transition probability) on finitely generated free groups, we prove that all bounded sets are transient. Finally, we consider any finitely additive measure (supported on the group generators) and we prove that the classification of the state space depends only on the continuous part of .     

Loeb measure,finitely additive measures,and a theorem of Banach

The Loeb measure construction from nonstandard analysis is applied to two theorems in standard measure theory. In both cases the essential simplification offered by the approach is the ability to work with a σ-additive measure space, even if the hypotheses only guarantee finite additivity. The key to this simplification is the property of -saturated nonstandard models, that any finitely additive measure on an internal algebra extends immediately to a σ-additive measure.      

Finitely additive,modular, and probability functions on pre-Semirings

In this paper, we define finitely additive, probability and modular functions over semiring-like structures. We investigate finitely additive functions with the help of complemented elements of a semiring. We also generalize some classical results in probability theory such as the law of total probability, Bayes’ theorem, the equality of parallel systems, and Poincaré’s inclusion-exclusion theorem. While we prove that modular functions over a couple of known semirings are almost constant, we show it is possible to define many different modular functions over some semirings such as bottleneck algebras and the semiring (Id(D),+,?), where D is a Dedekind domain.     

A combinatorial approach to coarse geometry

Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one of them can be transformed to the other by contiguous factorizations of bonding maps and by taking infinite subsequences. This embedding can be realized by either Rips complexes or analogs of Roe?s anti-?ech approximations of spaces.In this model coarse n-connectedness of K={K₁→K₂→?} means that for each k there is m>k such that the bonding map from Kk to Km induces trivial homomorphisms of all homotopy groups up to and including n.The asymptotic dimension being at most n means that for each k there is m>k such that the bonding map from Kk to Km factors (up to contiguity) through an n-dimensional complex.Property A of G. Yu is equivalent to the condition that for each k and for each ?>0 there is m>k such that the bonding map from |Kk| to |Km| has a contiguous approximation g:|Kk|→|Km| which sends simplices of |Kk| to sets of diameter at most ?.     

Co-generators for algebras over fields and commutative applications

Let R be an algebra over a field k. We say that an ideal I of R is co-generated by a set F of k-linear functionals R → k if I is the largest ideal of R that is contained in the kernels of all the functionals in F;. We will state sufficient conditions for an ideal of a free k-algebra to be co-generated by finitely many functionals. We then get as a corollary that every ideal of the polynomial algebra over a field in finitely many variables is finitely co-generated. This is a known result, but the way we construct the co-generators in the general case, leads to that we get a tight bound on how many co-generators are needed for such ideals.     

Equational theory of idempotent algebras

A finitely based equational class of idempotent algebras of type <m, n>,m, n≥2, is two-based. More generally, any finitely based equational class of idempotent algebras of type <m ₁, ..., m_k> withm _i≥2 andk≥2 isk-based.     

Some Examples of Affine Monoid Algebras

We construct a finitely generated monoid S with a zero element such that for every field K the Jacobson radical of the monoid algebra K[S] is a sum of nilpotent ideals but is not nilpotent. Moreover, the contracted monoid algebra K ₀[S] is a monomial algebra.

If K is a field of characteristic p > 0, then we construct a finitely presented group H _p such that the Jacobson radical J of the group algebra K[H _p ] is a sum of nilpotent ideals, but is not nilpotent. Moreover, K[H _p ]/J is a domain.     

Determinantal varieties over truncated polynomial rings

We study components and dimensions of higher-order determinantal varieties obtained by considering generic m×n (m?n) matrices over rings of the form F[t]/(t^k), and for some fixed r, setting the coefficients of powers of t of all r×r minors to zero. These varieties can be interpreted as spaces of (k−1)th order jets over the classical determinantal varieties; a special case of these varieties first appeared in a problem in commuting matrices. We show that when r=m, the varieties are irreducible, but when r<m, these varieties are reducible. We show that when r=2<m (any k), there are exactly ⌊k/2⌋+1 components, which we determine explicitly, and for general r<m, we show there are at least ⌊k/2⌋+1 components. We also determine the components explicitly for k=2 and 3 for all values of r (for k=3 for all but finitely many pairs of (m,n)).     

On finitely embedded rings

A result of Ginn and Moss asserts that a left and right noetherian ring with essential right socle is left and right artinian. There are examples of right finitely embedded rings with ACC on left and right annihilators which are not artinian. Motivated by this, it was shown by Faith that a commutative, finitely embedded ring with ACC on annihilators (and square-free socle) is artinian (quasi-Frobenius). A ring R is called right minsymmetric if, whenever k R is a simple right ideal of R, then R k is also simple. In this paper we show that a right noetherian right minsymmetric ring with essential right socle is right artinian. As a consequence we show that a ring is quasi-Frobenius if and only if it is a right and left mininjective, right finitely embedded ring with ACC on right annihilators. This extends the known work in the artinian case, and also extends Faith's result to the non-commutative case.     

Extension theorems of Hahn-Banach type for nonlinear disjointly additive functionals and operators in Lebesgue spaces

Let (Ω, τ, M) be a nonatomic separable finite measure space. Every continuous functional N on L^p(m), 1 ? p < ∞, which is disjointly additive in the sense N(u + v) = N(u) + N(v) whenever uv = 0, is known to be representable by an integral with a nonlinear Caratheodory kernel. Such functionals share several regularity properties with continuous linear functionals. Here we study the question of whether every continuous, disjointly additive functional defined on a closed subspace of L^p(m) possesses an extension to L^p(m) with these same properties. This question has applications to the study of nonlinear functionals on Sobolev spaces. It is shown that for a class of subspaces, including those of finite codimension, such an extension always exists, but there are also closed subspaces not possessing this extension property. Analogous results are obtained for disjointly additive mappings from closed subspaces of L^p(m) into L¹(m) and for functionals defined on subspaces of L^∞(m). The techniques depend heavily on the utilization of Lyapunov vector measures.     

Some categorical properties of the functors O τ and O R of weakly additive functionals

In the paper, the spaces of weakly additive τ-smooth and Radon functionals are investigated. It is proved that the functors of weakly additive τ-smooth and Radon functionals weakly preserve the density of Tychonoff spaces, and the functor of weakly additive τ-smooth functionals forms a monad in the category of Tychonoff spaces and their continuous mappings. Examples and remarks are given showing that these functors fail to satisfy certain Shchepin normality conditions. Problems having positive solutions for normal functors are presented.     

A mean value theorem for discriminants of abelian extensions of a number field

Let k be an algebraic number field and let N(k,C?;m) denote the number of abelian extensions K of k with G(K/k)≅C?, the cyclic group of prime order ?, and the relative discriminant D(K/k) of norm equal to m. In this paper, we derive an asymptotic formula for _∑m?XN(k,C?;m) using the class field theory and a method, developed by Wright. We show that our result is identical to a result of Cohen, Diaz y Diaz and Olivier, obtained by methods of classical algebraic number theory, although our methods allow for a more elegant treatment and reduce a global calculation to a series of local calculations.     

Enumerating K best paths in length order in DAGs

We address the problem of finding the K best paths connecting a given pair of nodes in a directed acyclic graph (DAG) with arbitrary lengths. One of the main results in this paper is the proof that a tree representing the kth shortest path is obtained by an arc exchange in one of the previous (k − 1) trees (each of which contains a previous best path). An O(m + K(n + log K)) time and O(K + m) space algorithm is designed to explicitly determine the K shortest paths in a DAG with n nodes and m arcs. The algorithm runs in O(m + Kn) time using O(K + m) space in DAGs with integer length arcs. Empirical results confirming the superior performance of the algorithm to others found in the literature for randomly generated graphs are reported.     

Finitely strictly singular operators between James spaces

An operator between Banach spaces is said to be finitely strictly singular if for every ε>0 there exists n such that every subspace E⊆X with dimE?n contains a vector x such that ‖Tx‖<ε‖x‖. We show that, for 1?p<q<∞, the formal inclusion operator from Jp to Jq is finitely strictly singular. As a consequence, we obtain that the strictly singular operator with no invariant subspaces constructed by C. Read is actually finitely strictly singular. These results are deduced from the following fact: if k?n then every k-dimensional subspace of Rn contains a vector x with ‖x?_∞‖=1 such that xmi=i(−1) for some m₁<?<mk.     

Finitely semisimple spherical categories and modular categories are self-dual

We show that every essentially small finitely semisimple k-linear additive spherical category for which k=End(1) is a field, is equivalent to its dual over the long canonical forgetful functor. This includes the special case of modular categories. In order to prove this result, we show that the universal coend of the spherical category, with respect to the long forgetful functor, is self-dual as a Weak Hopf Algebra.     

Properties of connected graphs having minimum degree distance

The degree distance of a connected graph, introduced by Dobrynin, Kochetova and Gutman, has been studied in mathematical chemistry. In this paper some properties of graphs having minimum degree distance in the class of connected graphs of order n and size m≥n−1 are deduced. It is shown that any such graph G has no induced subgraph isomorphic to P₄, contains a vertex z of degree n−1 such that G−z has at most one connected component C such that |C|≥2 and C has properties similar to those of G.For any fixed k such that k=0,1 or k≥3, if m=n+k and n≥k+3 then the extremal graph is unique and it is isomorphic to K₁+(K_1,k+1∪(n−k−3)K₁).     

Positivity of Riesz functionals and solutions of quadratic and quartic moment problems

We employ positivity of Riesz functionals to establish representing measures (or approximate representing measures) for truncated multivariate moment sequences. For a truncated moment sequence y, we show that y lies in the closure of truncated moment sequences admitting representing measures supported in a prescribed closed set K⊆Rn if and only if the associated Riesz functional Ly is K-positive. For a determining set K, we prove that if Ly is strictly K-positive, then y admits a representing measure supported in K. As a consequence, we are able to solve the truncated K-moment problem of degree k in the cases: (i) (n,k)=(2,4) and K=R²; (ii) n?1, k=2, and K is defined by one quadratic equality or inequality. In particular, these results solve the truncated moment problem in the remaining open cases of Hilbert's theorem on sums of squares.     

On p-adic vector measure spaces

For a separating algebra R of subsets of a set X, E a complete Hausdorff non-Archimedean locally convex space and m:R→E a bounded finitely additive measure, we study some of the properties of the integrals with respect to m of scalar-valued functions on X. The concepts of convergence in measure, with respect to m, and of m-measurable functions are introduced and several results concerning these notions are given.     

A Non-Existence Theorem for Almost Split Sequences

Let k be a field, Q a quiver with countably many vertices and I an ideal of kQ such that kQ/I is a spectroid. In this note, we prove that there is no almost split sequence ending at an indecomposable not finitely presented representation of the bound quiver (Q, I). We then get that an indecomposable representation M of (Q, I) is the ending term of an almost split sequence if and only if it is finitely presented and not projective. The dual results are also true.