Topology of the Semigroup of Singular Endomorphisms

The topological interpretations of some of the algebraic properties of the semigroup Sn of singular endomorphisms of an n-dimensional vector space over K are discussed here. Since Sn is known to be an idempotent generated regular semigroup, we pay more attention to the topological properties of the set En of idempotents in Sn. The local structure of En is shown to be that of a C^infinity-manifold and of a finite-dimensional vector bundle over the Grassmann manifolds. The topology of the biorder relations and sandwich sets are also discussed.     

The Monoid of the Random Graph

We investigate properties of the endomorphism monoid of the countable random graph R. We show that End(R) is not regular and is not generated by its idempotents. The Rees order on the idempotents of End(R) has 2^N0 many minimal elements. We also prove that the order type of Q is embeddable in the Rees order of End(R).     

The Minding formula and its applications

We apply the Minding Formula for geodesic curvature and the Gauss-Bonnet Formula to calculate the total Gaussian curvature of certain 2-dimensional open complete branched Riemannian manifolds, the M\cal M surfaces. We prove that for an M\cal M surface, the total curvature depends only on its Euler characteristic and the local behaviour of its metric at ends and branch points. Then we check that many important surfaces, such as complete minimal surfaces in \Bbb Rⁿ{\Bbb R}^n with finite total curvature, complete constant mean curvature surfaces in hyperbolic 3-space H³ (–1) with finite total curvature, are actually branch point free M\cal M surfaces. Therefore as corollaries we give simple proofs of some classical theorems such as the Chern-Osserman theorem for complete minimal surfaces in \Bbb Rⁿ{\Bbb R}^n with finite total curvature. For the reader's convenience, we also derive the Minding Formula.     

On Partitional Labelings of Graphs

The notion of partitional graphs, a subclass of sequential graphs, is introduced, and the cartesian product of a partitional graph and K ₂ is shown to be partitional. Every sequential graph is harmonious and felicitous. The partitional property of some bipartite graphs including the n-dimensional cube Q _n is studied, and thus this paper extends what was known about the sequentialness, harmoniousness and felicitousness of such graphs.     

Finitely Presented, Coherent, and Ultrasimplicial Ordered Abelian Groups

We study notions such as finite presentability and coherence, for partially ordered abelian groups and vector spaces. Typical results are the following: (i) A partially ordered abelian group G is finitely presented if and only if G is finitely generated as a group, G⁺ is well-founded as a partially ordered set, and the set of minimal elements of G⁺\ {0} is finite. (ii) Torison-free, finitely presented partially ordered abelian groups can be represented as subgroups of some Zⁿ, with a finitely generated submonoid of (Z⁺)ⁿ as positive cone. (iii) Every unperforated, finitely presented partially ordered abelian group is Archimedean. Further, we establish connections with interpolation. In particular, we prove that a divisible dimension group G is a directed union of simplicial subgroups if and only if every finite subset of G is contained into a finitely presented ordered subgroup.     

The dimension of the Cartesian product of partial orders

If P and Q are partial orders, then the dimension of the cartesian product P × Q does not exceed the sum of the dimensions of P and Q. There are several known sufficient conditions for this bound to be attained. On the other hand, the only known lower bound for the dimension of a cartesian product is the trivial inequality dim (P × Q)⩾ max{dim P, dim Q}. In particular, if P has dimension n, we know only that n⩽dim(P×P)⩽2n. In this paper, we show that for each n⩾3, the crown S_n⁰ is an n-dimensional partial order for which dim(S_n⁰×S_n⁰)=2n−2. No example for which dim(P×Q<dimP+dimQ−2 is known.     

Some Relations Between Packing Premeasure and Packing Measure

Let K be a compact subset of Rⁿ, 0 s n. Let , P^s denote s-dimensional packing premeasure andmeasure, respectively. We discuss in this paper the relationbetween and P^s. We prove:if , then ; and if , then for any > 0, there exists a compact subset F of K such that and P^s(F) P^s(K) – .1991 Mathematics Subject Classification 28A80, 28A78.     

Precise Asymptotic Formulas for Semilinear Eigenvalue Problems

. We consider the nonlinear Sturm-Liouville problem¶¶-u"(t) = | u(t) | ^p-1u(t) - lu(t), t ? I :=(0,1), u(0) = u(1) = 0 -u'(t) = \mid u(t)\mid^{p-1}u(t) - \lambda u(t), t \in I :=(0,1), u(0) = u(1) = 0 ,¶¶ where p > 1 and l ? R \lambda \in {\bf R} is an eigenvalue parameter. To investigate the global L²-bifurcation phenomena, we establish asymptotic formulas for the n-th bifurcation branch l = l_n (a) \lambda = \lambda_n (\alpha) with precise remainder term, where a \alpha is the L² norm of the eigenfunction associated with l \lambda .     

Isoperimetric numbers of graph bundles

The main aim of this paper is to give some upper and lower bounds for the isoperimetric numbers of graph coverings or graph bundles, with exact values in some special cases. In addition, we show that the isoperimetric number of any covering graph is not greater than that of the base graph. Mohar's theorem for the isoperimetric number of the cartesian product of a graph and a complete graph can be extended to a more general case: The isoperimetric numberi(G × K _2n) of the cartesian product of any graphG and a complete graphK _2n on even vertices is the minimum of the isoperimetric numberi(G) andn, and it is also a sharp lower bound of the isoperimetric numbers of all graph bundles over the graphG with fiberK _2n. Furthermore, ifn 2i(G) then the isoperimetric number of any graph bundle overG with fibreK _n is equal to the isoperimetric numberi(G) ofG. Partially supported by The Ministry of Education, Korea.     

On the Resolution of the Heat Equation with Discontinuous Coefficients

We give in this work some results about the existence and uniqueness with optimal regularity for solutions of a parabolic equation in nondivergence form in L^q(0,T;L^p(Omega)) where 1 < p,q < infinity in two cases. We use Lamberton's results (cf. [9]) in the first case and Dore-Venni's results (cf. [6]) in the second case.     

A Cantor-Bernstein type theorem for effect algebras

We prove that if E¹ and E² are &sgr;-complete effect algebras such that E¹ is a factor of E² and E² is a factor of E¹, then E¹ and E² are isomorphic.     

The hyper-Zagreb index and some graph operations

Let G be a simple connected graph. The Hyper-Zagreb index is defined as \(\textit{HM}(G)=\sum _{uv\in E_{G}}(d_{G}(u)+d_{G}(v))^2\). In this paper some exact expressions for the hyper-Zagreb index of graph operations containing cartesian product and join of n graphs, splice, link and chain of graphs will be presented. We also apply these results to some graphs to chemical and general interest, such as \(C_4\) nanotube, rectangular grid, prism, complete n-partite graph.     

Successive Determination and Verification of Polytopes by their X-Rays

It is shown that each convex polytope P in ^d can be verifiedby ([d/(d–k)] + 1) k-dimensional X-rays. This means thatP is uniquely determined by these X-rays and the choice of thedirection of each X-ray depends only on P. Examples are constructedto show that in general this number cannot be reduced. Further,it is shown that each convex polytope P in ³ can be successivelydetermined by only two one-dimensional X-rays. This means thatP is uniquely determined by one X-ray taken in an arbitrarydirection together with another whose direction depends onlyon the first X-ray. The results extend those for the case d= 2 of Giering and of Edelsbrunner and Skiena.     

Gorenstein injective dimension and Tor-depth of modules

The main aim of this paper is to obtain a dual result to the now well known Auslander-Bridger formula for G-dimension. We will show that if R is a complete Cohen-Macaulay ring with residue field k, and M is a non-injective h-divisible Ext-finite R-module of finite Gorenstein injective dimension such that for each i 3 1i \geq 1 Extⁱ (E,M) = 0 for all indecomposable injective R-modules E 1 E(k)E \neq E(k), then the depth of the ring is equal to the sum of the Gorenstein injective dimension and Tor-depth of M. As a consequence, we get that this formula holds over a d-dimensional Gorenstein local ring for every nonzero cosyzygy of a finitely generated R-module and thus in particular each such n^th cosyzygy has its Tor-depth equal to the depth of the ring whenever n 3 dn \geq d.     

Efficient Testing of Large Graphs

Let P be a property of graphs. An e\epsilon -test for P is a randomized algorithm which, given the ability to make queries whether a desired pair of vertices of an input graph G with n vertices are adjacent or not, distinguishes, with high probability, between the case of G satisfying P and the case that it has to be modified by adding and removing more than en²\epsilon n^2 edges to make it satisfy P. The property P is called testable, if for every e\epsilon there exists an e\epsilon -test for P whose total number of queries is independent of the size of the input graph. Goldreich, Goldwasser and Ron [8] showed that certain individual graph properties, like k-colorability, admit an e\epsilon -test. In this paper we make a first step towards a complete logical characterization of all testable graph properties, and show that properties describable by a very general type of coloring problem are testable. We use this theorem to prove that first order graph properties not containing a quantifier alternation of type ``"$\forall \exists ' are always testable, while we show that some properties containing this alternation are not.     

L p Norms of Eigenfunctions in the Completely Integrable Case

The eigenfunctions e^iál,x? e^{i\langle\lambda,x\rangle} of the Laplacian on a flat torus have uniformly bounded L^p norms. In this article, we prove that for every other quantum integrable Laplacian, the L^p norms of the joint eigenfunctions blow up at least at the rate || j_k || L^p 3 C(e)l_k^{[(p-2)/(4p)]-e} \| \varphi_k \| L^{p} \geq C(\epsilon)\lambda_{k}^{{p-2\over4p}-\epsilon} when p > 2. This gives a quantitative refinement of our recent result [TZ1] that some sequence of eigenfunctions must blow up in L^p unless (M,g) is flat. The better result in this paper is based on mass estimates of eigenfunctions near singular leaves of the Liouville foliation.     

On paraconvexity of graphs of continuous functions

The concept of paraconvexity of a subsetP E of a normed spaceE was first introduced by E. Michael. Roughly speaking, it consists of a controlled weakening of the convexity assumption forP, where the control is guaranteed via some parameter [0, 1). In this paper, we consider the case whenP is a subset of some (n+1)-dimensional Euclidean spaceE andP is the graph of some continuous functionf:V , whereV E is some convexn-dimensional subset ofE. Our key result is that paraconvexity of such a setP follows from the paraconvexity of sections ofP by two-dimensional planes, orthogonal toV. As an application, we prove a selection theorem for graph-valued mappings whose values have Lipschitzian (with a fixed constant) or monotone two-dimensional sections.Supported in part by the Ministry of Science and Technology of the Republic of Slovenia Research Grant No. P1-0214-101-93.Supported in part by G. Soros International Science Foundation.     

On the Topology of Singularities of the Set of Supporting Hyperplanes of a Smooth Submanifold in an Affine Space

Many new universal relations are obtained between the Eulernumbers of manifolds of singular supporting hyperplanes of anarbitrary generic smooth closed k-dimensional submanifold inRⁿ where n 7 or k = 1. These relations are applied to Barner-convexcurves in an odd-dimensional space Rⁿ. A universal (nontrivial)linear relation is established between the numbers of singularsupporting hyperplanes of various types but of the same totalmultiplicity n of tangency with a given generic smooth closedconnected Barner-convex curve in Rⁿ. The coefficients of thisrelation are defined by Catalan numbers.     

Long Snakes in Powers of the Complete Graph with an Odd Number of Vertices

In [5] Abbott and Katchalski ask if there exists a constantc < 0 such that for every d 2 there is a snake (cycle withoutchords) of length at least c3^d in the product of d copies ofthe complete graph K₃. We show that the answer to the abovequestion is positive, and that in general for any odd integern there is a constant c_n such that for every d 2 there is asnake of length at least c_n n^d in the product of d copies ofthe complete graph K_n.     

On the Eigenfunction Expansion of a Piecewise Smooth Function

For eigenfunction expansions of piecewise smooth functionsassociated with an elliptic operator of order m in R² theconvergence is proved.