Koszul Complexes and Symmetric Forms over the Punctured Affine Space

Let X be a regular separated scheme of finite Krull dimensionand let be the punctured affine n-space over X. We show that the total graded Witt ring of is a free graded module over the totalgraded Witt ring of X with two generators 1 and . The secondgenerator satisfies the equation ² = 1 when n = 1 and ² = 0when n 2. 2000 Mathematics Subject Classification 11E81, 19G12.     

Rational Points on a Class of Superelliptic Curves

A famous Diophantine equation is given by y^k=(x+1)(x+2)...(x+m). (1) For integers k2 and m2, this equation only has the solutionsx = –j (j = 1, ..., m), y = 0 by a remarkable result ofErds and Selfridge [9] in 1975. This put an end to the old questionof whether the product of consecutive positive integers couldever be a perfect power (except for the obviously trivial cases).In a letter to D. Bernoulli in 1724, Goldbach (see [7, p. 679])showed that (1) has no solution with x0 in the case k = 2 andm = 3. In 1857, Liouville [18] derived from Bertrand's postulatethat for general k2 and m2, there is no solution with x0 ifone of the factors on the right-hand side of (1) is prime. Byuse of the Thue–Siegel theorem, Erds and Siegel [10] provedin 1940 that (1) has only trivial solutions for all sufficientlylarge kk₀ and all m. This was closely related to Siegel's earlierresult [30] from 1929 that the superelliptic equation y^k=f(x) has at most finitely many integer solutions x, y under appropriateconditions on the polynomial f(x). The ineffectiveness of k₀was overcome by Baker's method [1] in 1969 (see also [2]). In 1955, Erds [8] managed to re-prove the result jointly obtainedwith Siegel by elementary methods. A refinement of Erds' ideasfinally led to the above-mentioned theorem as follows.     

On the Edge Reconstruction of Graphs Embedded in Surfaces II

In this paper, we prove the following theorems. (i) Let G bea graph of minimum degree 5. If G is embeddable in a surface and satisfies (–5)|V(G)|+6()0, then G is edge reconstructible.(ii) Any graph of minimum degree 4 that triangulates a surfaceis edge reconstructible. (iii) Any graph which triangulatesa surface of characteristic 0 is edge reconstructible.     

On the Length of Snakes in Powers of Complete Graphs

The conjecture stated in an earlier paper by the author thatthere is a constant (independent from both n and k) such that n^d–1 holds for everyn 2 and d 2, where is the length of the longest snake (cycle without chords) in the Cartesianproduct of d copies of thecomplete graph K_n, is proved.     

On the Total Coloring of Graphs Embeddable in Surfaces

The paper shows that any graph G with the maximum degree (G) 8, which is embeddable in a surface of Euler characteristic() 0, is totally ((G)+2)-colorable. In general, it is shownthat any graph G which is embeddable in a surface and satisfiesthe maximum degree (G) (20/9) (3–())+1 is totally ((G)+2)-colorable.     

On the Compactness and Approximation Numbers of Hardy-Type Integral Operators in Lorentz Spaces

We characterize the mapping properties such as the boundedness,compactness and measure of noncompactness for those real weightfunctions , , u0, v0, for which the Hardy-type integral operatorof the form acts from to , when the parameters are restricted to the range 1 < max (r,s) min (p, q) < and the kernel k(x, y) 0 satisfies theOinarov condition (see (2) below). For the case k(x, y) = 1,we obtain lower and upper estimates of the approximation numbers,extending the result of [5].     

The Representation of Some Integers as a Subset Sum

Let A N. The cardinality (the sum of the elements) of A willbe denoted by |A| ((A)). Let m N and p be a prime. Let A {1, 2,...,p}. We prove thefollowing results. If |A| [(p+m–2)/m]+m, then for every integer x such that0 x p – 1, there is B A such that |B| = m and (B) x mod p. Moreover, the bound is attained. If |A| [(p+m–2)/m]+m!, then there is B A such that |B| 0 mod m and (B) = (m – 1)!p. If |A| [(p + 1)/3]+29, then for every even integer x such that4p s x p(p + 170)/48, there is S A such that x = (S). In particular,for every even integer a 2 such that p 192a – 170, thereare an integer j 0 and S A such that (S) = a^j+1.     

Logistic Type Equations on RN by a Squeezing Method Involving Boundary Blow-Up Solutions

We study, on the entire space R^N(N 1), the diffusive logisticequation u_t–u=u–u^p, u0 (1.1) and its generalizations. Here p > 1 is a constant. Problem(1.1) plays an important role in understanding various populationmodels and some other problems in applied mathematics. When = 1 and p = 2, it is also known as the Fisher equation andKPP equation, due to the pioneering works of Fisher [8] andKolmogoroff, Petrovsky and Piscounoff [18].     

On the extension of real regular embedding

Let X be a real nonsingular affine algebraic variety of dimensionk. It is proved that any two regular (algebraic) embeddingsX ⁿ are regularly equivalent, provided that n 4k + 2.     

Characterising Completely Positive Elementary Operators

Let S be an elementary operator on a C*-algebra with minimallength l(S)1. Then S is completely positive if and only if Sis n-positive for some natural number n(l(S)–1)/2. 1991Mathematics Subject Classification 47B47.     

Maximal Convex Subgroups of the Automorphism Group of a Doubly Transitive Chain

We characterise all maximal convex l-subgroups of A(), the automorphismgroup of a doubly transitive chain of countable coterminality.We then determine, up to isomorphism, all doubly transitiveactions of the l-group A(). Finally, assuming the ContinuumHypothesis, we show that there exists an unbounded chain ofproper prime subgroups of A().     

The quaternion group as a subgroup of the sphere braid groups

Let n 3. We prove that the quaternion group of order 8 is realisedas a subgroup of the sphere braid group B_n(²) if and only ifn is even. If n is divisible by 4, then the commutator subgroupof B_n(²) contains such a subgroup. Further, for all n 3, B_n(²)contains a subgroup isomorphic to the dicyclic group of order4n.     

On Ramanujan's Double Inequality for the Gamma Function

Ramanujan presented (without proof) the following double inequalityfor the gamma function: .Recently, Karatsuba established that these inequalities holdfor x 1. We show that this can be slightly improved: the inequalitieshold for all x 0, even if we replace 1/100 by where f(x) = (1/)³[(x + 1)(e/x)^x]⁶–8x³–4x²–x.Moreover, and 1/30 are the best possible constant terms. 2000Mathematics Subject Classification 33B15 (primary); 26D15, 26D20(secondary).     

The Dimension Subgroup Conjecture

For each n 4 there exists a finite 2-group G_n such that itsnth dimension subgroup does not coincide with its nth lowercentral subgroup. This settles the dimension subgroup conjecturenegatively for all n4.     

Prime Divisors of Shifted Factorials

For any positive integer n we let P(n) be the largest primefactor of n. We improve and generalize several results of P.Erds and C. Stewart on P(n!+1). In particular, we show thatlim sup_ninfin; P(n!+1)/n2.5, which improves their lower boundof lim sup_ninfin; P(n!+1)/n2. 2000 Mathematics Subject Classification11A05, 11A07, 11J86.     

On the direct product conjecture for sigma invariants

We show that the direct product conjecture for ⁿ(G; ), whereG is the direct product of two groups of type FP_n, holds forn = 3 and give counterexamples for n 4. Previously, counter-exampleswere known only for a related conjecture involving the homotopical-invariants, where the conjecture already fails for n 3.     

Exact Sequences of Semistable Vector Bundles on Algebraic Curves

Let X be a smooth complex projective curve of genus g 1. Ifg 2, then assume further that X is either bielliptic or withgeneral moduli. Fix integers r, s, a, b with r > 1, s >1 and as br. Here we prove the existence of an exact sequence [formula] of semistable vector bundles on X with rk(H) = r, rk(Q) = s,deg(H) = a and deg(Q) = b. 1991 Mathematics Subject Classification14H60.     

Stability of the Picard Bundle

Let X be a non-singular algebraic curve of genus g 2, n 2an integer, a line bundle over X of degree d > 2n(g –1) with (n,d) = 1 and M the moduli space of stable bundles ofrank n and determinant over X. It is proved that the Picardbundle W is stable with respect to the unique polarisation ofM. 2000 Mathematics Subject Classification 14H60, 14J60.     

Normal Transversality and Uniform Bounds

Let A be a commutative ring. A graded A-algebra U = _n0 U_n isa standard A-algebra if U₀ = A and U = A[U₁] is generated asan A-algebra by the elements of U₁. A graded U-module F = _n0F_nis a standard U-module if F is generated as a U-module by theelements of F₀, that is, F_n = U_nF₀ for all n 0. In particular,F_n = U₁F_n–1 for all n 1. Given I, J, two ideals of A,we consider the following standard algebras: the Rees algebraof I, R(I) = _n0Iⁿtⁿ = A[It] A[t], and the multi-Rees algebraof I and J, R(I, J) = _n0(_p+q=nI^pJ^qu^pv^q) = A[Iu, Jv] A[u, v].Consider the associated graded ring of I, G(I) = R(I) A/I =_n0Iⁿ/Iⁿ⁺¹, and the multi-associated graded ring of I and J,G(I, J) = R(I, J) A/(I+J) = _n0(_p+q=nI^pJ^q/(I+J)I^pJ^q). We canalways consider the tensor product of two standard A-algebrasU = _p0U_p and V = _q0V_q as a standard A-algebra with the naturalgrading U V = _n0(_p+q=nU_p V_q). If M is an A-module, we havethe standard modules: the Rees module of I with respect to M,R(I; M) = _n0IⁿMtⁿ = M[It] M[t] (a standard R(I)-module), andthe multi-Rees module of I and J with respect to M, R(I, J;M) = _n0(_p+q=nI^pJ^qMu^pv^q) = M[Iu, Jv] M[u, v] (a standard R(I,J)-module). Consider the associated graded module of M withrespect to I, G(I; M) = R(I; M) A/I = _n0IⁿM/Iⁿ⁺¹M (a standardG(I)-module), and the multi-associated graded module of M withrespect to I and J, G(I, J; M) = R(I, J; M) A/(I+J) = _n0(_p+q=nI^pJ^qM/(I+J)I^pJ^qM)(a standard G(I, J)-module). If U, V are two standard A-algebras,F is a standard U-module and G is a standard V-module, thenF G = _n0(_p+q=nF_p G_q) is a standard U V-module. Denote by :R(I) R(J; M) R(I, J; M) and :R(I, J; M) R(I+J;M) the natural surjective graded morphisms of standard RI) R(J)-modules. Let :R(I) R(J; M) R(I+J; M) be . Denote by :G(I) G(J; M) G(I, J; M) and :G(I, J; M) G(I+J; M) the tensor productof and by A/(I+J); these are two natural surjective gradedmorphisms of standard G(I) G(J)-modules. Let :G(I) G(J; M) G(I+J; M) be . The first purpose of this paper is to prove the following theorem.     

A Sharp Comparison Result Concerning Schrodinger Heat Kernels

Let G₀ and G be the heat kernels of and –V respectively.Under some sharp conditions on V, a proof is given that G(x,t;y,0)/G₀(x,t;y,0)can be bounded from above and below by two positive constants.This largely improves the well-known bounds G(x,t;y,0)/G₀(x,t;y,0) t (0,T]. The result answers an open question of V. Liskevichand Yu. Semenov, in the case where V 0. A sharp global boundin time is also obtained when V 0. 2000 Mathematics SubjectClassification 35K05.