Let f be a non-zero cusp form with real Fourier coefficients a(n) (n ≥ 1) of positive real weight k and a unitary multiplier system v on a subgroup Γ ⊂ SL2(ℝ) that is finitely generated and of Fuchsian type of the first kind. Then, it is known that the sequence (a(n))(n ≥ 1) has infinitely many sign changes. In this short note, we generalise the above result to the case of entire modular integrals of non-positive integral weight k on the group Γ0*(N) (N ∈ ℕ) generated by the Hecke congruence subgroup Γ0(N) and the Fricke involution provided that the associated period functions are polynomials.

We recall that a polynomial f(X) ∈ K[X] over a field K is called stable if all its iterates are irreducible over K. We show that almost all monic quadratic polynomials f(X) ∈ ℤ[X] are stable over ℚ. We also show that the presence of squares in so-called critical orbits of a quadratic polynomial f(X) ∈ ℤ[X] can be detected by a finite algorithm; this property is closely related to the stability of f(X). We also prove there are no stable quadratic polynomials over finite fields of characteristic 2 but they exist over some infinite fields of characteristic 2.

Let T be an algebraic torus over ℚ such that T(ℝ) is compact. Assuming the generalized Riemann hypothesis, we give a lower bound for the size of the class group of T modulo its n-torsion in terms of a small power of the discriminant of the splitting field of T. As a corollary, we obtain an upper bound on the n-torsion in that class group. This generalizes known results on the structure of class groups of complex multiplication fields.

For a sensor network, a tractable spatially dependent node deployment model is presented with the property that the density is inversely proportional to the sink distance. A stochastic model is formulated to examine message advancements under greedy routing in such a sensor network. The aim of this work is to demonstrate that an inhomogeneous Poisson process can be used to model a sensor network with spatially dependent node density. Symmetric elliptic integrals and asymptotic approximations are used to describe the random behaviour of hops. Types of dependence that affect hop advancements are examined. We observe that the dependence between successive jumps in a multi-hop path is captured by including only the previous forwarding node location. We include a simple uncoordinated sleep scheme, and observe that the complexity of the model is reduced when sufficiently many nodes are asleep. All expressions involving multi-dimensional integrals are derived and evaluated with quasi-Monte Carlo integration methods based on Halton sequences and recently developed lattice rules. An importance sampling function is derived to speed up the quasi-Monte Carlo methods. The ensuing results agree extremely well with simulations.

Paucity is established for a system of diagonal diophantine equations, in which the degrees are the odd numbers in ascending order.

We develop Weyl differencing and Hua-type lemmata for a class of multidimensional exponential sums. We then apply our estimates to bound the number of variables required to establish an asymptotic formula for the number of solutions of a system of diophantine equations arising from the study of linear spaces on hypersurfaces. For small values of the degree and dimension, our results are superior to those stemming from the author’s earlier work on Vinogradov’s mean value theorem.

For each poset H whose Hasse diagram is a tree of height k, we show that the largest size of a family of subsets of [n]={1,. . ., n} not containing H as an induced subposet is asymptotic to . This extends a result of Bukh [1], which in turn generalizes several known results including Sperner's theorem.

Let M be a closed 4-manifold with a free circle action. If the orbit manifold N3 satisfies an appropriate fibering condition, then we show how to represent a cone in H2(M; ℝ) by symplectic forms. This generalizes earlier constructions by Thurston, Bouyakoub and Fernández et al. In the case that M is the product 4-manifold S1 × N, our construction complements our previous results and allows us to determine completely the symplectic cone of such 4-manifolds.

We consider a class of dynamical systems of the form du/dt + Bu + F(u) = b on a Hilbert space H where the self-adjoint linear operator B is positive with a strictly positive first eigenvalue and b = b0 + b1 such that (b0, Bv) = 0 for all v ∈ H. Given two solutions u and v, we set u − v = w and show that if u(t) → 0 and v(t) → 0 as t → ∞, then in fact eventually w(t) → 0 at an exponential rate. We apply these results to the two-dimensional Navier–Stokes equations (NSEs), the three-dimensional hyperviscous NSEs and the three-dimensional NS-α equations on bounded domains and also establish stability in the sense of Lyapunov; for these systems we assume a condition on b1 to impose decaying turbulence. We also show for the case of decaying turbulence that Leray solutions of the three-dimensional NSEs on bounded domains eventually become regular in addition to decaying to zero. In particular, they eventually satisfy the conditions needed for the abstract stability results.

We consider non-negative solutions of the Cauchy problem for quasilinear parabolic equations ut = Δum + f(u), where m > 1 and f(ξ) is a positive function in ξ > 0 satisfying f(0) = 0 and a blow-up condition

We show that if ξm+2/N /(−log ξ)β = O(f(ξ)) as ξ ↓ 0 for some 0 < β < 2/(mN + 2), one of the following holds: (i) all non-trivial solutions blow up in finite time; (ii) every non-trivial solution with an initial datum u0 having compact support exists globally in time and grows up to ∞ as t → ∞: limtt→∞ inf|x|<Ru(x, t) = ∞ for any R > 0. Moreover, we give a condition on f such that (i) holds, and show the existence of f such that (ii) holds.

We prove that any invariant strong Kähler structure with torsion (SKT structure) on a flag manifold M = G/K of a semi-simple compact Lie group G is Kähler. As an application we describe invariant generalized Kähler structures on M.

We give characterizations of the positive Borel measures μ on so that the weighted Hardy–Sobolev spaces are imbedded in Lq(dμ), for a range of s > 0, 0 < p, q < + ∞, q ≠ p, where w is a doubling weight in the unit sphere of ℂn.

An adequate semigroup S is called ample if ea = a(ea)* and ae = (ae)†a for all a ∈ S and e ∈ E(S). Inverse semigroups are exactly those ample semigroups that are regular. After obtaining some characterizations of finite ample semigroups, it is proved that semigroup algebras of finite ample semigroups have generalized triangular matrix representations. As applications, the structure of the radicals of semigroup algebras of finite ample semigroups is obtained. In particular, it is determined when semigroup algebras of finite ample semigroup are semiprimitive.

We study the motion of a rigid body with a cavity filled with a viscous liquid. The main objective is to investigate the well-posedness of the coupled system formed by the Navier–Stokes equations describing the motion of the fluid and the ordinary differential equations for the motion of the rigid part. To this end, appropriate function spaces and operators are introduced and analysed by considering a completely general three-dimensional bounded domain. We prove the existence of weak solutions using the Galerkin method. In particular, we show that if the initial velocity is orthogonal, in a certain sense, to all rigid velocities, then the velocity of the system decays exponentially to zero as time goes to infinity. Then, following a functional analytic approach inspired by Kato's scheme, we prove the existence and uniqueness of mild solutions. Finally, the functional analytic approach is extended to obtain the existence and uniqueness of strong solutions for regular data.

We obtain an integral representation of an energy for structured deformations of continua in the space of functions of bounded variation, as a first step to the study of asymptotic models for thin defective crystalline structures, where phenomena as slips, vacancies and dislocations prevent the effectiveness of classical theories.

Scott conjectured in [6] that the class of graphs with no induced subdivision of a given graph is χ-bounded. We verify his conjecture for maximal triangle-free graphs.

We are interested in the cubic-to-tetragonal phase transition in a shape memory alloy. We consider geometrically linear elasticity. In this framework, Dolzmann and Müller have shown that the only stress-free configurations are (locally) twins (i.e. laminates of just two of the three martensitic variants). However, configurations with arbitrarily small elastic energy are not necessarily close to these twins. The formation of a microstructure allows all three martensitic variants to be mixed at arbitrary volume fractions. We take an interfacial energy into account and establish a (local) lower bound on elastic plus interfacial energy in terms of the martensitic volume fractions. The introduction of an interfacial energy introduces a length scale and, thus, together with the linear dimensions of the sample, a non-dimensional parameter. Our lower Ansatz-free bound has optimal scaling in this parameter. It is the scaling predicted by a reduced model introduced and analysed by Kohn and Müller with the purpose of describing the microstructure near an interface between austenite and twinned martensite. The optimal construction features branching of the martensitic twins when approaching this interface.