Extensions of Isometric Dual Representations of Semigroups

Let T be a dual representation of a suitable subsemigroup Sof a locally compact abelian group G by isometries on a dualBanach space X=(X_*)^*. It is shown that (X, T) can be extendedto a dual representation of G on a dual Banach space Y containingX, and that this extension can be done in a canonical way. Inthe case of a representation by *-monomorphisms of a von Neumannalgebra, the extension is a representation of G by *-automorphismsof a von Neumann algebra.     

A linear algorithm for the domination number of a series-parallel graph

A vertex u in an undirected graph G = (V, E) is said to dominate all its adjacent vertices and itself. A subset D of V is a dominating set in G if every vertex in G is dominated by a vertex in D, and is a minimum dominating set in G if no other dominating set in G has fewer vertices than D. The domination number of G is the cardinality of a minimum dominating set in G.The problem of determining, for a given positive integer k and an undirected graph G, whether G has a dominating set D in G satisfying ¦D¦ ≤ k, is a well-known NP-complete problem. Cockayne have presented a linear time algorithm for finding a minimum dominating set in a tree. In this paper, we will present a linear time algorithm for finding a minimum dominating set in a series-parallel graph.     

Some results on embeddings of algebras, after de Bruijn and McKenzie

In 1957, N.G. de Bruijn showed that the symmetric group Sym(Ω) on an infinite set Ω contains a free subgroup on 2^card(Ω) generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality card(Ω), the group Sym(Ω) contains a coproduct of 2^card(Ω) copies of A, not only in the variety of all groups, but in any variety of groups to which A belongs. His key lemma is here generalized to an arbitrary variety of algebras V, and formulated as a statement about functors Set V. From this one easily obtains analogs of the results stated above with “group” and Sym(Ω) replaced by “monoid” and the monoid Self(Ω) of endomaps of Ω, by “associative K-algebra” and the K-algebra End_K (V) of endomorphisms of a K-vector-space V with basis Ω, and by “lattice” and the lattice Equiv(Ω) of equivalence relations on Ω. It is also shown, extending another result from de Bruijn's 1957 paper, that each of Sym(Ω), Self(Ω) and End_K(V) contains a coproduct of 2^card(Ω) copies of itself.That paper also gave an example of a group of cardinality 2^card(Ω) that was not embeddable in Sym(Ω), and R. McKenzie subsequently established a large class of such examples. Those results are shown here to be instances of a general property of the lattice of solution sets in Sym(Ω) of sets of equations with constants in Sym(Ω). Again, similar results - this time of varying strengths - are obtained for Self(Ω), End_K(V), and Equiv(Ω), and also for the monoid Rel(Ω) of binary relations on Ω.Many open questions and areas for further investigation are noted.     

Algebra Forms with $$d^{N} = 0$$ on Quantum Plane. Generalized Clifford Algebra Approach

We construct a q-analog of exterior calculus with a differential d satisfying d ^N = 0, where N ≥ 2 and q is a primitive Nth root of unity, on a noncommutative space and introduce a notion of a q-differential k-form. A noncommutative space we consider is a reduced quantum plane. Our construction of a q-analog of exterior calculus is based on a generalized Clifford algebra with four generators and on a graded q-differential algebra. We study the structure of the algebra of q-differential forms on a reduced quantum plane and show that the first order calculus induced by the differential d is a coordinate calculus. The explicit formulae for partial derivatives of this first order calculus are found.     

Submanifolds Of Maximal Nullity In Symmetric Spaces

A non-totally-geodesic submanifold of relative nullity n — 1 in a symmetric space M is a cylinder over one of the following submanifolds: a surface F ² of nullity 1 in a totally geodesic submanifold N³ ? M locally isometric to S ²(c) × ? or H ²(c) × ?; a submanifold F ^k+1 spanned by a totally geodesic submanifold F ^k(c) of constant curvature moving by a special curve in the isometry group of M; a submanifold F ^k+l of nullity k in a flat totally geodesic submanifold of M; a curve.     

On the lattice formed by all normal subloops of a finite loop

All normal subloops of a loopG form a modular latticeL _n (G). It is shown that a finite loopG has a complemented normal subloop lattice if and only ifG is a direct product of simple subloops. In particular,L _n (G) is a Boolean algebra if and only if no two isomorphic factors occurring in a decomposition ofG are abelian groups. The normal subloop lattice of a finite loop is a projective geometry if and only ifG is an elementary abelianp-group for some primep.     

A just basis

An old problem of P. Erdös and P. Turán asks whether there is a basisA of order 2 for which the number of representationsn=a+a′, a,a′∈A is bounded. Erd?s conjectured that such a basis does not exist. We answer a related finite problem and find a basis for which the number of representations is bounded in the square mean. Writing σ (n)=|{(a, a ^t ) ∈A ²:a+a′=n}| we prove that there exists a setA of nonnegative integers that forms a basis of order 2 (that is,s(n)≥1 for alln), and satisfies ∑_{n ? N} σ(N)² = O(N).     

Topological vector spaces, compacta, and unions of subspaces

In the first two sections, we study when a σ-compact space can be covered by a point-finite family of compacta. The main result in this direction concerns topological vector spaces. Theorem 2.4 implies that if such a space L admits a countable point-finite cover by compacta, then L has a countable network. It follows that if f is a continuous mapping of a σ-compact locally compact space X onto a topological vector space L, and fibers of f are compact, then L is a σ-compact space with a countable network (Theorem 2.10). Therefore, certain σ-compact topological vector spaces do not have a stronger σ-compact locally compact topology.In the last, third section, we establish a result going in the orthogonal direction: if a compact Hausdorff space X is the union of two subspaces which are homeomorphic to topological vector spaces, then X is metrizable (Corollary 3.2).     

Circuit preserving edge maps

It is proved that any one-to-one edge map f from a 3-connected graph G onto a graph G′, G and G′ possibly infinite, satisfying f(C) is a circuit in G′ whenever C is a circuit in G is induced by a vertex isomorphism. This generalizes a result of Whitney which hypothesizes f(C) is a circuit in G′ if and only if C is a circuit in G.     

Multivalued contractions

LetE be a Banach space,C a closed convex subset ofE, F a multivalued contraction fromC to itself with closed values. Ifx ₀ is a fixed point and ifF(x ₀) is not a singleton, then there exists a fixed pointx ₁ ofF which is different fromx ₀. We prove also that there is in the Euclidean space ?² a multivalued contraction with compact connected values having a nonconnected set of fixed points.     

On Partial Tilting Complexes

Let R be a commutative noetherian ring and A a noetherian R-algebra. Let P ^? ∈ 𝒦^b(𝒫 _A ) with Hom_𝒦(Mod-A)(P ^?, P ^?[i]) = 0 for i > 0. We will provide a sufficient condition for P ^? to be a direct summand of a silting complex. Also, in case Hom_𝒦(Mod-A)(P ^?, P ^?[i]) = 0 for i ≠ 0, we will provide a sufficient condition for P ^? to be a direct summand of a tilting complex.     

Symetries isometriques applications a la dualite des espaces reticules

We show that ifE is a non-reflexive Banach lattice, there exists for everyn a dual of finite even order ofE which contins isometicallyl _n ^/l . We show that itE is a Banach lattice which is isometric to the dual of a Banach spaceX, then the order intervals are σ (E, X)-compact. We prove then that under various conditions, a Banach lattice which is a dual as a Banach space, is a dual as a Banach lattice. In particular, this is true when the predual ofE is unique.      

Derived equivalences and Gorenstein algebras

We introduce a notion of Gorenstein R-algebras over a commutative Gorenstein ring R. Then we provide a necessary and sufficient condition for a tilting complex over a Gorenstein R-algebra A to have a Gorenstein R-algebra B as the endomorphism algebra and a construction of such a tilting complex. Furthermore, we provide an example of a tilting complex over a Gorenstein R-algebra A whose endomorphism algebra is not a Gorenstein R-algebra.     

Zeta Functions for Curves and Log Canonical Models

The topological zeta function and Igusa's local zeta functionare respectively a geometrical invariant associated to a complexpolynomial f and an arithmetical invariant associated to a polynomialf over a p-adic field. When f is a polynomial in two variables we prove a formula forboth zeta functions in terms of the so-called log canonicalmodel of f^-1{0} in A². This result yields moreover a conceptualexplanation for a known cancellation property of candidate polesfor these zeta functions. Also in the formula for Igusa's localzeta function appears a remarkable non-symmetric ‘q-deformation’of the intersection matrix of the minimal resolution of a Hirzebruch-Jungsingularity. 1991 Mathematics Subject Classification: 32S5011S80 14E30 (14G20)     

Total Colorings Of Degenerate Graphs

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, such that no two adjacent or incident elements receive the same color. A graph G is s-degenerate for a positive integer s if G can be reduced to a trivial graph by successive removal of vertices with degree ≤s. We prove that an s-degenerate graph G has a total coloring with Δ+1 colors if the maximum degree Δ of G is sufficiently large, say Δ≥4s+3. Our proof yields an efficient algorithm to find such a total coloring. We also give a lineartime algorithm to find a total coloring of a graph G with the minimum number of colors if G is a partial k-tree, that is, the tree-width of G is bounded by a fixed integer k.     

Remainders in compactifications of topological groups

In this paper, we consider the following question: when does a topological group G have a Hausdorff compactification bG with a remainder belonging to a given class of spaces? We extend the results of A.V. Arhangel'skii by showing that if a remainder of a non-locally compact topological group G has a countable open point-network or a locally Gδ-diagonal, then G and the compactification bG of G are separable and metrizable.     

Bounds on the number of switchings for trajectories of piecewise analytic vector fields

We study the trajectories of systems $\text{x}\text{?}\text{= X(x)}$, where X is a continuous “extendably piecewise analytic” vector field, i.e., a continuous vector field X such that the domain of ? admits a locally finite partition $\text{I}$ into sets such that for each A ∈ $\text{I}$ there is a vector field X_A which is analytic on a neighborhood of the closure of A and whose restriction to A coincides with that of X. We prove that the trajectories are piecewise analytic, with a priori bounds on the number of switchings for all trajectories that stay in a fixed compact set and whose duration does not exceed a fixed number T. This result implies the existence of a regular synthesis for optimal control problems with a strictly convex Lagrangian, and a linear dynamics with polyhedral constraints on the controls.     

Planar G2 Hermite interpolation with some fair,C-shaped curves

G² Hermite data consists of two points, two unit tangent vectors at those points, and two signed curvatures at those points. The planar G² Hermite interpolation problem is to find a planar curve matching planar G² Hermite data. In this paper, a C-shaped interpolating curve made of one or two spirals is sought. Such a curve is considered fair because it comprises a small number of spirals. The C-shaped curve used here is made by joining a circular arc and a conic in a G² manner. A curve of this type that matches given G² Hermite data can be found by solving a quadratic equation. The new curve is compared to the cubic Bézier curve and to a curve made from a G² join of a pair of quadratics. The new curve covers a much larger range of the G² Hermite data that can be matched by a C-shaped curve of one or two spirals than those curves cover.     

Onn-th commutators with nilpotent or regular values in rings

Then-th commutator for a,b in a ringR is defined inductively as follows: [a,b]₁=[a,b]=ab−ba and[a,b] _n=[[a,b]₋₁,b]. We characterize the ringsR without non-zero nil right ideals in which[a,b] _nis nilpotent or regular for alla,b∈R. We also examine the case whereR is a semiprime ring with involution in which[t ₁, t₂]_nis nilpotent or regular for all tracest ₁,t₂∈R.     

The Recognition of Double Euler Trails in Series-Parallel Networks

Given a series-parallel network (network, for short)N, its dual networkN′ is given by interchanging the series connection and the parallel connection of networkN. We usually use a series-parallel graph to represent a network. LetG[N] andG[N′] be graph representations ofNandN′, respectively. A sequence of edgese₁, e₂,…,e_kis said to form a common trail on (G[N], G[N′]) if it is a trail on bothG[N] andG[N′]. If a common trail covers all of the edges inG[N] andG[N′], it is called adouble Euler trail.However, there are many different graph representations for a network. We say that a networkNhas a double Euler trail (DET) if there is a common Euler trail for someG[N] and someG[N′]. Finding a DET in a network is essential for optimizing the layout area of a complementary CMOS functional cell. Maziasz and Hayes (IEEE Trans. Computer-Aided Design9(1990), 708–719) gave a linear time algorithm for solving the layout problem in fixedG[N] andG[N′] and an exponential algorithm for finding the optimal cover in a network without fixing graph representations. In this paper, we study properties of subnetworks of a DET network. According to these properties, we propose an algorithm that automatically generates the rules for composition of trail cover classes. On the basis of these rules, a linear time algorithm for recognizing DET networks is presented. Furthermore, we also give a necessary and sufficient condition for the existence of a double Euler circuit in a network.