Homeomorphism classification of complex projective complete intersections of dimensions 5, 6 and 7

In this paper we prove that two complete intersections X_n(d){X_n(\underline{d})} and X_n(d￠){X_n(\underline{d}^\prime)} are homeomorphic if and only if they have the same total degree, Pontrjagin classes and Euler characteristics, provided n = 5, 6, 7. This extends earlier result of Fang and Klaus (Manuscr Math 90:139–147, 1996).     

Generating Random Vectors in (ℤ/<Emphasis Type="Italic">p</Emphasis>ℤ)<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> via an Affine Random Process

This paper considers some random processes of the form X _n+1=T X _n +B _n (mod p) where B _n and X _n are random variables over (ℤ/pℤ) ^d and T is a fixed d×d integer matrix which is invertible over the complex numbers. For a particular distribution for B _n , this paper improves results of Asci to show that if T has no complex eigenvalues of length 1, then for integers p relatively prime to det (T), order (log p)² steps suffice to make X _n close to uniformly distributed where X ₀ is the zero vector. This paper also shows that if T has a complex eigenvalue which is a root of unity, then order p ^b steps are needed for X _n to get close to uniformly distributed for some positive value b≤2 which may depend on T and X ₀ is the zero vector.     

Diffeomorphism classification of smooth weighted complete intersections

Xn(d1, . . . , dr-1, dr; w) and Xn(e1, . . . , er-1, dr; w) are two complex odd-dimensional smooth weighted complete intersections defined in a smooth weighted hypersurface Xn+r-1(dr; w). We prove that they are diffeomorphic if and only if they have the same total degree d, the Pontrjagin classes and the Euler characteristic, under the following assumptions: the weights w = (ω0, . . . , ωn+r) are pairwise relatively prime and odd, νp(d/dr) ≥ 2n+1/ 2(p-1) + 1 for all primes p with p(p-1) ≤ n + 1, where νp(d/dr) satisfies d/dr =Ⅱp prime pνp (d/dr).     

Minimal Blocking Sets in PG(n, 2) and Covering Groups by Subgroups

In this article we prove that a set of points B of PG(n, 2) is a minimal blocking set if and only if ?B? = PG(d, 2) with d odd and B is a set of d + 2 points of PG(d, 2) no d + 1 of them in the same hyperplane. As a corollary to the latter result we show that if G is a finite 2-group and n is a positive integer, then G admits a ? _n+1-cover if and only if n is even and G? (C ₂) ⁿ , where by a ? _m -cover for a group H we mean a set 𝒞 of size m of maximal subgroups of H whose set-theoretic union is the whole H and no proper subset of 𝒞 has the latter property and the intersection of the maximal subgroups is core-free. Also for all n < 10 we find all pairs (m,p) (m > 0 an integer and p a prime number) for which there is a blocking set B of size n in PG(m,p) such that ?B? = PG(m,p).     

Homotopy type of smooth weighted complete intersections

If the total degree d has no prime divisors less than(n+3)/2,then we prove that the homotopy type of complex odd dimensional smooth weighted complete intersection Xn(d;w) is determined by the dimension n,the total degree d,the Euler characteristic and the Kervaire invariant,provided that the weights w =(ω0,...,ωn+r) is pairwise relatively prime.     

Classes of Commutative Rings Defined by Special Conditions

ABSTRACT

In this article, we are mainly concerned with (n, d)-Krull rings, i.e., rings in which each n-presented prime ideal has height at most d. Precisely, we show that weakly n-Von Neumann regular rings are (n ? 1, 0)-Krull rings. Also, we prove that (n, d)-Krull property is not local property and that R is an (n, d)-Krull ring if and only if dim(R _P ) ≤ d for each n-presented prime ideal P of R. Finally, we construct a class of (2, d)-Krull rings which are neither (2, d ? 1)-Krull rings (for d = 1) nor (1, d)-Krull rings for d = 0,1.     

On the average distance property and certain energy integrals

One of our main results is the following: LetX be a compact connected subset of the Euclidean spaceR ⁿ andr(X, d ₂) the rendezvous number ofX, whered ₂ denotes the Euclidean distance inR ⁿ . (The rendezvous numberr(X, d ₂) is the unique positive real number with the property that for each positive integern and for all (not necessarily distinct)x ₁,x ₂,...,x _n inX, there exists somex inX such that .) Then there exists some regular Borel probability measure μ₀ onX such that the value of ∫ _X d ₂(x, y)dμ₀ (y) is independent of the choicex inX, if and only ifr(X, d ₂) = sup_μ ∫ _X ∫ _X d ₂(x, y)dμ(x)dμ(y), where the supremum is taken over all regular Borel probability measures μ onX.     

Friabilité des valeurs d’un polynôme

Let F(X) be an absolutely irreducible polynomial in \mathbbZ [X₁,..., X_n]{\mathbb{Z} [X_{1},\dots, X_{n}]}, with degree d. We prove that, for any δ < 4/3, for any sufficiently large x, there exists a positive density of integral n-tuples m = (m ₁, . . . , m _n ) in the hypercube max |m _i | ≤ x such that every prime divisor of F(m) is smaller than x ^d–δ . This result is improved when F satisfies some geometrical hypotheses.     

EXAMPLES OF FCR-ALGEBRAS

Abstract

Let R be a prime ring and 𝒰(R) its group of units. We prove that if 𝒰(R) satisfies a group identity and 𝒰(R) generates R,then either R is a domain or R is isomorphic to the algebra of n × n matrices over a finite field of order d. Moreover the integers n and d depend only on the group identity satisfed by 𝒰(R). This result has been recently proved by C. H. Liu and T. K. Lee (Liu,C. H.; Lee,T. K. Group identities and prime rings generated by units. Comm. Algebra (to appear)) and here we present a new different proof.     

Two-point concentration in random geometric graphs

A random geometric graph G _n is constructed by taking vertices X ₁,…,X _n ∈ℝ ^d at random (i.i.d. according to some probability distribution ν with a bounded density function) and including an edge between X _i and X _j if ‖X _i -X _j ‖ < r where r = r(n) > 0. We prove a conjecture of Penrose ([14]) stating that when r=r(n) is chosen such that nr ^d = o(lnn) then the probability distribution of the clique number ω(G _n ) becomes concentrated on two consecutive integers and we show that the same holds for a number of other graph parameters including the chromatic number χ(G _n ). The author was partially supported by EPSRC, the Department of Statistics, Bekkerla-Bastide fonds, Dr. Hendrik Muller’s Vaderlandsch fonds, and Prins Bernhard Cultuurfonds.     

Completeness of hyperspaces of compact subsets of quasi-metric spaces

We study the hyperspace K ₀(X) of non-empty compact subsets of a Smyth-complete quasi-metric space (X, d). We show that K ₀(X), equipped with the Hausdorff quasi-pseudometric H _d forms a (sequentially) Yoneda-complete space. Moreover, if d is a T ₁ quasi-metric, then the hyperspace is algebraic, and the set of all finite subsets forms a base for it. Finally, we prove that K ₀(X), H _d ) is Smyth-complete if (X, d) is Smyth-complete and all compact subsets of X are d ⁻¹-precompact.     

Degenerations of the moduli spaces of vector bundles on curves II (generalized Gieseker moduli spaces)

LetX ₀ be a projective curve whose singularity is one ordinary double point. We construct a birational modelG(n, d) of the moduli spaceU(n, d) of stable torsion free sheaves in the case (n, d)= 1, such that G(n, d) has normal crossing singularities and behaves well under specialization i.e. if a smooth projective curve specializes toX ₀, then the moduli space of stable vector bundles of rankn and degreed onX specializes toG(n, d). This generalizes an earlier work of Gieseker in the rank two case.     

Uniformity over primes of tamely ramified splittings

We fix a prime p and let f(X) vary over all monic integer polynomials of fixed degree n. Given any possible shape of a tamely ramified splitting of p in an extension of degree n, we prove that there exists a rational function φ(X)∈ℚ(X) such that the density of the monic integer polynomials f(X) for which the splitting of p has the given shape in ℚ[X]/f(X) is φ(p) (here reducible polynomials can be neglected). As a corollary, we prove that, for p≥n, the density of irreducible monic polynomials of degree n in ℤ _p [X] is the value at p of a rational function φ _n (X)∈ℚ(X). All rational functions involved are effectively computable. Received: 15 September 1998 / Revised version: 21 October 1999     

From an Elementary Result on Real Closed Fields to Positive Operators on Ordered Fields

Let K be a (commutative totally) ordered field, let K[X ₁,…, X _n ] be the K-vector space of the polynomials with n variables.

An operator T (i.e., an endomorphism of K[X ₁,…,X _n ] into itself) is said to be “positive” if the image of every positive polynomial is a positive polynomial, where a positive polynomial is a polynomial which takes only non-negative values. First we prove that in ?[X], the sum of the derivatives of a positive polynomial is a positive polynomial too. Then we give what we believe to be a good framework to prove that this result remains true for every ordered field and we propose generalizations.     

Wijsman and Hit-and-Miss Topologies of Quasi-Metric Spaces

The relationship between the Wijsman topology and (proximal) hit-and-miss topologies is studied in the realm of quasi-metric spaces. We establish the equivalence between these hypertopologies in terms of Urysohn families of sets. Our results generalize well-known theorems and provide easier proofs. In particular, we prove that for a quasi-pseudo-metrizable space (X,T) the Vietoris topology on the set P ₀(X) of all nonempty subsets of X is the supremum of all Wijsman topologies associated with quasi-pseudo-metrics compatible with T. We also show that for a quasi-pseudo-metric space (X,d) the Hausdorff extended quasi-pseudo-metric is compatible with the Wijsman topology on P ₀(X) if and only if d ^–1 is hereditarily precompact.     

The energy of unitary cayley graphs

A graph G of order n is called hyperenergetic if E(G)>2n-2, where E(G) denotes the energy of G. The unitary Cayley graph X_n has vertex set Z_n={0,1,2,…,n-1} and vertices a and b are adjacent, if gcd(a-b,n)=1. These graphs have integral spectrum and play an important role in modeling quantum spin networks supporting the perfect state transfer. We show that the unitary Cayley graph X_n is hyperenergetic if and only if n has at least two prime factors greater than 2 or at least three distinct prime factors. In addition, we calculate the energy of the complement of unitary Cayley graph and prove that is hyperenergetic if and only if n has at least two distinct prime factors and n≠2p, where p is a prime number. By extending this approach, for every fixed , we construct families of k hyperenergetic non-cospectral integral circulant n-vertex graphs with equal energy.     

Relations between ΛBV and BV(p(n) ↑∞) Classes of Functions

The base radical class L _b(X), generated by a class X was introduced in [12]. It consists of those rings whose nonzero homomorphic images have nonzero accessible subrings in X. When X is homomorphically closed, L _b(X) is the lower radical class defined by X, but otherwise X may not be contained in L _b(X). We prove that for a hereditary radical class L with semisimple class S(R), L _b(S(R)) is the class of strongly R-semisimple rings if and only if R is supernilpotent or subidempotent. A number of further examples of radical classes of the form L _b(X) are discussed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Geometry of Periodic Modules over Tame Concealed and Tubular Algebras

Let A be a tame concealed or tubular algebra and d the dimension-vector of a periodic module with respect to the action of the Auslander–Reiten translation. We prove that the affine variety mod _A (d) of all A-modules of dimension-vector d is a normal complete intersection. Moreover, we show that a module M in mod _A (d) is nonsingular if and only if Ext _A ²(M,M)=0.     

The norm theorem for quadratic forms over a field of characteristic 2

ABSTRACT

In this paper we prove that a finite group G is isomorphic to the finite simple group L _n (q) with n ≥ 3 if and only if they have the same set of order of solvable subgroups.

     

Topological entropy zero and asymptotic pairs

Let T be a continuous map on a compact metric space (X, d). A pair of distinct points x, y ∈ X is asymptotic if lim _n→∞ d(T ⁿ x, T ⁿ y) = 0. We prove the following four conditions to be equivalent: 1. h _top(T) = 0; 2. (X, T) has a (topological) extension (Y,S) which has no asymptotic pairs; 3. (X, T) has a topological extension (Y ′, S′) via a factor map that collapses all asymptotic pairs; 4. (X, T) has a symbolic extension (i.e., with (Y ′, S′) being a subshift) via a map that collapses asymptotic pairs. The maximal factors (of a given system (X, T)) corresponding to the above properties do not need to coincide.