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.     

Sandwiching C0-Semigroups

Let T = {T(t)}_t0 be a C₀-semigroup on a Banach space X. Thefollowing results are proved. (i) If X is separable, there exist separable Hilbert spacesX₀ and X₁, continuous dense embeddings j₀:X₀ X and j₁:X X₁,and C₀-semigroups T₀ and T₁ on X₀ and X₁ respectively, suchthat j₀ T₀(t) = T(t) j₀ and T₁(t) j₁ = j₁ T(t) for all t 0. (ii) If T is -reflexive, there exist reflexive Banach spacesX₀ and X₁ , continuous dense embeddings j:D(A²) X₀, j₀:X₀ X, j₁:X X₁, and C₀-semigroups T₀ and T₁ on X₀ and X₁ respectively,such that T₀(t) j = j T(t), j₀ T₀(t) = T(t) j₀ and T(t) j₁ = j₁ T(t) for all t 0, and such that (A₀) = (A) = (A₁),where A_k is the generator of T_k, k = 0, Ø, 1.     

Necessary and Sufficient Conditions for Exponential Stability and Ultimate Boundedness of Systems Governed by Stochastic Partial Differential Equations

Consider the following infinite dimensional stochastic evolutionequation over some Hilbert space H with norm |·|: It is proved that under certain mild assumptions, the strongsolution X_t(x₀)VHV*, t 0, is mean square exponentially stableif and only if there exists a Lyapunov functional (·,·):HxR₊R¹ which satisfies the following conditions: (i)c₁|x|²–k₁e^–µ₁t(x,t)c₂|x|²+k₂+k₂e^–µ₂t; (ii) L(x,t)–c₃(x,t)+k₃e^–µ₃t, xV, t0; where L is the infinitesimal generator of the Markov processX_t and c_i, k_i, µ_i, i = 1, 2, 3, are positive constants.As a by-product, the characterization of exponential ultimateboundedness of the strong solution is established as the nulldecay rates (that is, µ_i = 0) are considered.     

Systems of Diagonal Equations Over p-Adic Fields

Let K be a p-adic field, and consider the system F = (F₁,...,F_R)of diagonal equations (1) with coefficients in K. It is an interesting problem in numbertheory to determine when such a system possesses a nontrivialK-rational solution. In particular, we define *(k, R, K) tobe the smallest natural number such that any system of R equationsof degree k in N variables with coefficients in K has a nontrivialK-rational solution provided only that N*(k, R, K). For example,when k = 1, ordinary linear algebra tells us that *(1, R, K)= R + 1 for any field K. We also define *(k, R) to be the smallestinteger N such that *(k, R, Q_p) N for all primes p.     

Asymptotics for Fractional Nonlinear Heat Equations

The Cauchy problem is studied for the nonlinear equations withfractional power of the negative Laplacian where (0,2), with critical = /n and sub-critical (0,/n)powers of the nonlinearity. Let u₀ L^1,a L C, u₀(x) 0 in Rⁿ, = . The case of not small initial data is of interest. It is proved that the Cauchy problemhas a unique global solution u C([0,); L L^1,a C) and the largetime asymptotics are obtained.     

Elliptic Genera for Z/k-Manifolds I

Using Z/k-manifolds we give a geometric interpretation of thering homomorphism from cobordismwith Z/k coefficients to elliptic cohomology with Z/k coefficients,induced by the map of spectra MSO Ell associated to the naturaltransformation of cohomology theories MSO* Ell*.     

Strong Weighted Mean Summability and Kuttner's Theorem

In this paper we study sequence spaces that arise from the conceptof strong weighted mean summability. Let q = (q_n) be a sequenceof positive terms and set Q_n = ⁿ_k=1q_k. Then the weighted meanmatrix M_q = (a_nk) is defined by if kn, a_nk=0 if k>n. It is well known that M_q defines a regular summability methodif and only if Q_n. Passing to strong summability, we let 0<p<.Then , are the spaces of all sequences that are strongly M_q-summablewith index p to 0, strongly M_q-summable with index p and stronglyM_q-bounded with index p, respectively. The most important specialcase is obtained by taking M_q = C₁, the Cesàro matrix,which leads to the familiar sequence spaces w₀(p), w(p) and w(p), respectively, see [4, 21]. We remark that strong summabilitywas first studied by Hardy and Littlewood [8] in 1913 when theyapplied strong Cesàro summability of index 1 and 2 toFourier series; orthogonal series have remained the main areaof application for strong summability. See [32, 6] for furtherreferences. When we abstract from the needs of summability theory certainfeatures of the above sequence spaces become irrelevant; forinstance, the q_k simply constitute a diagonal transform. Hence,from a sequence space theoretic point of view we are led tostudy the spaces     

5-Torsion Points on Curves of Genus 2

Let C be a smooth proper curve of genus 2 over an algebraicallyclosed field k. Fix a Weierstrass point in C(k) and identifyC with its image in its Jacobian J under the Albanese embeddingthat uses as base point. For any integer N1, we write J_N forthe group of points in J(k) of order dividing N and for the subset of J_N of points oforder N. It follows from the Riemann–Roch theorem thatC(k)J₂ consists of the Weierstrass points of C and that C(k) and C(k) are empty (see [3]). The purpose of this paper is to study curvesC with C(k) non-empty.     

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].     

Some Remarks on the Cone of Completely Positive Maps between Von Neumann Algebras

The close relationship between the notions of positive formsand representations for a C*-algebra A is one of the most basicfacts in the subject. In particular the weak containment ofrepresentations is well understood in terms of positive forms:given a representation of A in a Hilbert space H and a positiveform on A, its associated representation is weakly containedin (that is, ker ker ) if and only if belongs to the weak*closure of the cone of all finite sums of coefficients of .Among the results on the subject, let us recall the followingones. Suppose that A is concretely represented in H. Then everypositive form on A is the weak* limit of forms of the typex ^k_{i=1 i}, x_i with the _i in H; moreover if A is a von Neumannsubalgebra of (H) and is normal, there exists a sequence (_i)_{i 1} in H such that (x) = _{i 1 i}, x_i for all x.     

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||=+.     

A Strong Law for the Largest Nearest-Neighbour Link between Random Points

Suppose that X₁, X₂, X₃, ... are independent random points inR^d with common density f, having compact support with smoothboundary , with f| continuous. Let Rⁿ_{i, k} denote the distancefrom X_i to its kth nearest neighbour amongst the first n points,and let M_{n, k} = max_in Rⁿ_{i, k}. Let denote the volume of theunit ball. Then as n , , almost surely If instead the points lie in a compact smooth d-dimensionalRiemannian manifold K, then nM^d_{n, k}/log n (min_Kf)^–1,almost surely.     

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.     

Acyclic Colourings of Planar Graphs with Large Girth

A proper vertex-colouring of a graph is acyclic if there areno 2-coloured cycles. It is known that every planar graph isacyclically 5-colourable, and that there are planar graphs withacyclic chromatic number _a = 5 and girth g = 4. It is provedhere that a planar graph satisfies _a 4 if g 5 and _a 3 ifg 7.     

Faithful Representations of Free Products

In 1940 Nisnevi published the following theorem [3]. Let (G) be a family of groups indexed by some set and (F) a family of fields of the same characteristic p0. Iffor each the group G has a faithful representation of degreen over F then the free product* G has a faithful representationof degree n+1 over some field of characteristic p. In [6] Wehrfritzextended this idea. If (G) GL(n, F) is a family of subgroupsfor which there exists ZGL(n, F) such that for all the intersectionGF.1_n=Z, then the free product of the groups *_ZG with Z amalgamatedvia the identity map is isomorphic to a linear group of degreen over some purely transcendental extension of F. Initially, the purpose of this paper was to generalize theseresults from the linear to the skew-linear case, that is, togroups isomorphic to subgroups of GL(n, D) where the D are divisionrings. In fact, many of the results can be generalized to ringswhich, although not necessarily commutative, contain no zero-divisors.We have the following.     

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 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.     

Support Varieties of Non-Restricted Modules over Lie Algebras of Reductive Groups

Let G be a connected semisimple group over an algebraicallyclosed field K of characteristic p>0, and g=Lie (G). Fixa linear function g* and let Z_g() denote the stabilizer of in g. Set N_p(g)={xg|x^[p]=0}. Let C(g) denote the category offinite-dimensional g-modules with p-character . In [7], Friedlanderand Parshall attached to each MOb(C(g)) a Zariski closed, conicalsubset V_g(M)N_p(g) called the support variety of M. Suppose thatG is simply connected and p is not special for G, that is, p2if G has a component of type B_n, C_n or F₄, and p3 if G has acomponent of type G₂. It is proved in this paper that, for anynonzero MOb(C(g)), the support variety V_g(M) is contained inN_p(g)Z_g(). This allows one to simplify the proof of the Kac–Weisfeilerconjecture given in [18].     

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.