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.     

Binomial Sums and Functions of Exponential Type

Let (a_n)_n0 be a sequence of complex numbers, and, for n0, let A number of results are proved relating the growth of the sequences(b_n) and (c_n) to that of (a_n). For example, given p0, if b_n= O(n^p and for all > 0,then a_n=0 for all n > p. Also, given 0 < p < 1, then for all > 0 if and onlyif . It is further shown that, given rß > 1, if b_n,c_n=O(rßⁿ), then a_n=O(ⁿ),where , thereby proving a conjecture of Chalendar, Kellay and Ransford. The principal ingredientsof the proogs are a Phragmén-Lindelöf theorem forentire functions of exponential type zero, and an estimate forthe expected value of e^(X), where X is a Poisson random variable.2000 Mathematics Subject Classification 05A10 (primary), 30D15,46H05, 60E15 (secondary).     

Lie Powers of Modules for Groups of Prime Order

Let L(V) be the free Lie algebra on a finite-dimensional vectorspace V over a field K, with homogeneous components Lⁿ(V) forn 1. If G is a group and V is a KG-module, the action of Gextends naturally to L(V), and the Lⁿ(V) become finite-dimensionalKG-modules, called the Lie powers of V. In the decompositionproblem, the aim is to identify the isomorphism types of indecomposableKG-modules, with their multiplicities, in unrefinable directdecompositions of the Lie powers. This paper is concerned withthe case where G has prime order p, and K has characteristicp. As is well known, there are p indecomposables, denoted hereby J₁,...,J_p, where J_r has dimension r. A theory is developedwhich provides information about the overall module structureof LV) and gives a recursive method for finding the multiplicitiesof J₁,...,J_p in the Lie powers Lⁿ(V). For example, the theoryyields decompositions of L(V) as a direct sum of modules isomorphiceither to J₁ or to an infinite sum of the form J_r J_{p-1} J_{p-1} ... with r 2. Closed formulae are obtained for the multiplicitiesof J₁,..., J_p in Lⁿ(J_p and Lⁿ(J_{p-1). For r < p-1, the indecomposableswhich occur with non-zero multiplicity in Lⁿ(J_r) are identifiedfor all sufficiently large n. 2000 Mathematical Subject Classification:17B01, 20C20.     

Realisations of Non v-Sufficient Jets with value in Rp, p > 1

The purpose of this paper is to show that if a jet cue oJ^r(n,p), n p, p > 1, is not v-sufficient in C^r+1, there existsan infinite sequence (f_i)_iN^* of realisations of o with mutuallynon-homeomorphic germs of varieties . Bochnak and Kuo [2, 5] showed it when p = 1 and thought thatthe same argument slightly modified can be used in the casep 2 [7, p. 225]. But when n p + 2, p > 1, we have to proceeddifferently. Moreover, it is necessary to prove separately theresult when n = p and n = p + 1. About C⁰-sufriciency and p> 1, Brodersen [3, p. 168] showed a similar theorem.     

ON THE INTEGRAL GALOIS MODULE STRUCTURE OF CYCLIC EXTENSIONS OF p-ADIC FIELDS

We strengthen results of Miyata on the integral Galois modulestructure of totally ramified cyclic Kummer extensions K ofdegree pⁿ of a p-adic field k. Let c₁(K/k) be the first ramificationnumber of K/k, and let c(K/k) be the least non-negative residueof c₁(K/k) modulo pⁿ. Suppose that K is of the form k() with^pn k and val _K(–1)>0, (val _K(–1), p)= 1. Thenthe valuation ring of K is free over its associated order ifc(K/k) divides p^m–1 for some m with 1mn; the converseholds if n= 2; and is a Hopf order (or a Gorenstein order)if and only if c(K/k) = pⁿ–1.     

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.     

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.     

On the Greatest Prime Divisor of the Sum of Two Squares of Primes

One of the most famous theorems in number theory states thatthere are infinitely many positive prime numbers (namely p =2 and the primes p 1 mod4) that can be represented in the formx²₁+x²₂, where x₁ and x₂ are positive integers. In a recentpaper, Fouvry and Iwaniec [2] have shown that this statementremains valid even if one of the variables, say x₂, is restrictedto prime values only. In the sequel, the letter p, possiblywith an index, is reserved to denote a positive prime number.As p²₁=p²₂ = p is even for p₁, p₂ > 2, it is reasonable toconjecture that the equation p²₁=p²₂ = 2p has an infinity ofsolutions. However, a proof of this statement currently seemsfar beyond reach. As an intermediate step in this direction,one may quantify the problem by asking what can be said aboutlower bounds for the greatest prime divisor, say P(N), of thenumbers p²₁=p²₂, where p₁, p₂ N, as a function of the realparameter N 1. The well-known Chebychev–Hooley methodcombined with the Barban–Davenport–Halberstam theoremalmost immediately leads to the bound P(N) N^1–, if N N_o(); here, denotes some arbitrarily small fixed positivereal number. The first estimate going beyond the exponent 1has been achieved recently by Dartyge [1, Théorème1], who showed that P(N) N^10/9–. Note that Dartyge'sproof provides the more general result that for any irreduciblebinary form f of degree d 2 with integer coefficients the greatestprime divisor of the numbers |f(p₁, p₂)|, p₁, p₂ N, exceedsN^_d–, where _d = 2 – 8/(d = 7). We in particular wantto point out that Dartyge does not make use of the specificfeatures provided by the form x²₁+x²₂. By taking advantage ofsome special properties of this binary form, we are able toimprove upon the exponent ₂ = 10/9 considerably.     

Ganea's Conjecture on Lusternik-Schnirelmann Category

A series of complexes Q_p indexed by all primes p is constructedwith catQ_p=2 and catQ_pxSⁿ=2 for either n2 or n=1 and p=2. Thisdisproves Ganea's conjecture on Lusternik–Schnirelmann(LS) category. 1991 Mathematics Subject Classification 55M30.     

A Graph-Theoretic Approach to the Unique Midset Property of Metric Spaces

A metric space X has the unique midset property if there isa topology-preserving metric d on X such that for every pairof distinct points x, y there is one and only one point p suchthat d(x, p) = d(y, p). The following are proved. (1) The discretespace with cardinality n has the unique midset property if andonly if n 2, 4 and n c, where c is the cardinality of thecontinuum. (2) If D is a discrete space with cardinality notgreater than c, then the countable power D^N of D has the uniquemidset property. In particular, the Cantor set and the spaceof irrational numbers have the unique midset property. A finite discrete space with n points has the unique midsetproperty if and only if there is an edge colouring of the completegraph K_n such that for every pair of distinct vertices x, ythere is one and only one vertex p such that (xp) = (yp). Letump(K_n) be the smallest number of colours necessary for sucha colouring of K_n. The following are proved. (3) For each k 0, ump(K_2k+1) = k. (4) For each k 3, k ump(K_2k) 2k–1.     

On a Theorem of Childs on Normal Bases of Rings of Integers: Addendum

Let p 3 be a prime number, F be a number field with _p F^x,and K = F(_p). In a previous paper, the author proved, undersome assumption on p and F, that an unramified cyclic extensionN/F of degree p has a normal integral basis if and only if thepushed-up extension NK/K has a normal integral basis. This addendumshows that the assertion holds without the above-mentioned assumption.     

Moments of Some Stopping Rules

Let X_n for n1 be independent random variables with . Set . Define T_k,c,m=inf{nm:|k!S_k,n|>cn^k/2}.We study critical values c_k,p for k2 and p>0, such that for c<c_k,p and all m, and for c>c_k,p and all sufficientlylarge m. In particular, c_1,1=c_2,1=1, c_3,1=2 and c_4,1=3 undercertain moment conditions on X₁, when X_n are identically distributed.We also investigate perturbed stopping rules of the form T_h,m=inf{nm:h(S_1,n/n^1/2)<_nor >_n} for continuous functions h and random variables _naand _nb with a<b. Related stopping rules of the Wiener processare also considered via the Uhlenbeck process.     

The Coherence of One-Relator Groups with Torsion and the Hanna Neumann Conjecture

A conjecture is proposed, bounding the number of cycles withlabel Wⁿ in a labeled directed graph. Some partial results towardsthis conjecture are established. As a consequence, it is provedthat a₁, a₂,...|Wⁿ is coherent for n 4. Furthermore, it iscoherent for n 2, provided that the strengthened Hanna Neumannconjecture holds. 2000 Mathematics Subject Classification 20F06,05C38.     

Complete Manifolds with Sectional Curvature Bounded Below and Large Volume Growth

This paper provides a proof that an n-dimensional complete openRiemannian manifold M with sectional curvature K_M –1is diffeomorphic to a Euclidean n-space Rⁿ if the volume growthof geodesic balls in M is close to that of the balls in an n-dimensionalhyperbolic space Hⁿ(–1) of sectional curvature –1.     

Contractions et Hyperdistributions a Spectre de Carleson

Let =(_n)_n1 be a log concave sequence such that lim inf_n+n/n^c>0for some c>0 and ((log _n)/n)_n1 is nonincreasing for some<1/2. We show that, if T is a contraction on the Hilbertspace with spectrum a Carleson set, and if ||T^–n||=O(_n)as n tends to + with _n11/(n log _n)=+, then T is unitary. Onthe other hand, if _n11/(n log _n)<+, then there exists a (non-unitary)contraction T on the Hilbert space such that the spectrum ofT is a Carleson set, ||T^–n||=O(_n) as n tends to +, andlim sup_n+||T^–n||=+.     

The eigenvalues of limits of radial Toeplitz operators

Let A² be the Bergman space on the unit disk. A bounded operatorS on A² is called radial if Szⁿ = _n zⁿ for all n 0, where _nis a bounded sequence of complex numbers. We characterize theeigenvalues of radial operators that belong to the Toeplitzalgebra.     

The Hausdorff Dimension of Julia Sets of Meromorphic Functions II

A family of transcendental meromorphic functions, f_p(z), p N is considered. It is shown that, if p 6, then the Hausdorffdimension of the Julia set of f_p satisfies dim J(f_p) 1/p, for0 < < 1/6^p, and dim J(f_p) 1–(30 ln ln p/ln p),for p^4p–1/10⁵ ln p < < p^4p–1/10⁴ ln p. Theseresults are used elsewhere to show that, for each d (0, 1),there exists a transcendental meromorphic function for whichdim J(f) = d.     

A Necessary and Sufficient Condition for a Linear Differential System to be Strongly Monotone

In order to present the results of this note, we begin withsome definitions. Consider a differential system [formula] where IR is an open interval, and f(t, x), (t, x)IxRⁿ, is acontinuous vector function with continuous first derivativesf_r/x_s, r, s=1, 2, ..., n. Let D_xf(t, x), (t, x)IxRⁿ, denote the Jacobi matrix of f(t,x), with respect to the variables x₁, ..., x_n. Let x(t, t₀,x₀), tI(t₀, x₀) denote the maximal solution of the system (1)through the point (t₀, x₀)IxRⁿ. For two vectors x, yRⁿ, we use the notations x>y and x>>yaccording to the following definitions: [formula] An nxn matrix A=(a_rs) is called reducible if n2 and there existsa partition [formula] (p1, q1, p+q=n) such that [formula] The matrix A is called irreducible if n=1, or if n2 and A isnot reducible. The system (1) is called strongly monotone if for any t₀I, x₁,x₂Rⁿ [formula] holds for all t>t₀ as long as both solutions x(t, t₀, x_i),i=1, 2, are defined. The system is called cooperative if forall (t, x)IxRⁿ the off-diagonal elements of the nxn matrix D_xf(t,x) are nonnegative. 1991 Mathematics Subject Classification34A30, 34C99.     

OPTIMAL INTEGRABILITY CONDITION FOR THE LOG-SOBOLEV INEQUALITY

Let M denote a connected complete Riemannian manifold (possiblywith a convex boundary), the Riemannian distance function froma fixed point and V C² (M) such that dµ_V e^V d xis a probability measure. For any K 0, we prove that K/2 isthe infimum over all > 0 such that Ric_M – Hess_V –Kand imply the log-Sobolevinequality for the Dirichlet form µ_V(| f |²).