New necessary and sufficient conditions are given for the quantization of a class of periodic second-order non-homogeneous ordinary differential equations in the complex plane. The problem is studied from the viewpoint of complex oscillation theory first developed in works by Bank and Laine and Gundersen and Steinbart. We show that, when a solution is complex non-oscillatory (finite exponent of convergence of zeros), then the solution, which can be written as special functions, must degenerate. This gives a necessary and sufficient condition when the Lommel function has finitely many zeros in every branch, and this is a type of quantization for the non-homogeneous differential equation. The degenerate solutions are polynomial/rational-type functions, which are of independent interest. In particular, this shows that complex non-oscillatory solutions of this class of differential equations are equivalent to the subnormal solutions considered in a recent paper by Chiang and Yu. In addition to the asymptotics of special functions, the other main idea that we apply in our proof is a classical result by Wright that gives precise asymptotic locations of large zeros of a functional equation.

We study the asymptotic behaviour of the equilibrium configurations of a nonlinearly elastic thin rod as the diameter of the cross-section tends to zero. Convergence results are established assuming physical growth conditions for the elastic energy density and suitable scalings of the applied loads that correspond at the limit to different rod models: the constrained linear theory, the analogue of the von Kármán plate theory for rods, and the linear theory.

Carleson's corona theorem is used to obtain two results on cyclicity of singular inner functions in weighted Bergman-type spaces on the unit disk. Our method of proof requires no regularity conditions on the weights.

Although in recent years bone piezoelectricity has been normally neglected, lately a new interest has appeared to show the importance of bone piezoelectricity in wet bone's complex response to loading. Here we numerically study a problem, including a strain-adaptive bone remodelling and the piezoelectricity. Its variational formulation leads to a coupled system composed of two linear variational equations for displacements and electric potential, and a parabolic variational inequality for the apparent density. Fully discrete approximations are now introduced by using the finite element method to approximate spatial variable and the explicit Euler scheme to discretise time derivatives. Some a priori error estimates are proved and the linear convergence of the algorithm is deduced under additional regularity conditions. Finally, some one- and two-dimensional numerical simulations are described to show the accuracy of the proposed algorithm and the behaviour of the solution.

This paper is concerned with the stability and instability of standing waves for the quasilinear Schrödinger equation of the formwhich has been derived in many models from mathematical physics. We find the exact threshold depending upon the interplay of quasilinear and nonlinear terms that separates stability and instability. More precisely, we prove that for α ∈ and odd p ∈ , when , the standing wave is stable, and when (where for N ≥ 3 and 2 α ċ 2* = +∞ for N = 2), the standing wave is strongly unstable. Our results show that the quasilinear term 2 α(△|φ|2α)|φ|2α−2φ makes the standing waves more stable, which is consistent with the physical phenomena.

This paper is devoted to the study of the low Mach number limit for the isentropic Euler system with axisymmetric initial data without swirl. In the first part of the paper we analyze the problem corresponding to the subcritical regularities, that is ${H}^{s} $ with $s\gt \frac{5}{2} $. Taking advantage of the Strichartz estimates and using the special structure of the vorticity we show that the lifespan ${T}_{\varepsilon } $ of the solutions is bounded below by $\log \log \log \frac{1}{\varepsilon } $, where $\varepsilon $ denotes the Mach number. Moreover, we prove that the incompressible parts converge to the solution of the incompressible Euler system when the parameter $\varepsilon $ goes to zero. In the second part of the paper we address the same problem but for the Besov critical regularity ${ B}_{2, 1}^{\frac{5}{2} } $. This case turns out to be more subtle because of at least two features. The first one is related to the Beale–Kato–Majda criterion which is not known to be valid for rough regularities. The second one concerns the critical aspect of the Strichartz estimate ${ L}_{T}^{1} {L}^{\infty } $ for the acoustic parts $(\nabla {\Delta }^{- 1} \mathrm{div} \hspace{0.167em} {v}_{\varepsilon } , {c}_{\varepsilon } )$: it scales in the space variables like the space of the initial data.

In this paper, we revisit the quantile mechanics approach, which was introduced by Steinbrecher and Shaw (Steinbrecher, G. & Shaw, W. T. (2008) Quantile mechanics. European. J. Appl. Math.19, 87–112). Our objectives are (i) to derive the method of trimmed moments (MTM) estimators for the parameters of gamma and Student's t distributions, and (ii) to examine their large- and small-sample statistical properties. Since trimmed moments are defined through the quantile function of the distribution, quantile mechanics seems like a natural approach for achieving objective (i). To accomplish the second goal, we rely on the general large sample results for MTMs, which were established by Brazauskas et al. (Brazauskas, V., Jones, B. L. & Zitikis, R. (2009) Robust fitting of claim severity distributions and the method of trimmed moments. J. Stat. Plan. Inference139, 2028–2043), and then use Monte Carlo simulations to investigate small-sample behaviour of the newly derived estimators. We find that, unlike the maximum likelihood method, which usually yields fully efficient but non-robust estimators, the MTM estimators are robust and offer competitive trade-offs between robustness and efficiency. These properties are essential when one employs gamma or Student's t distributions in such outlier-prone areas as insurance and finance.

In this note, we study an invariant associated with the zeros of the moment map generated by an action form, the infinitesimal index. This construction will be used to study the compactly supported equivariant cohomology of the zeros of the moment map and to give formulas for the multiplicity index map of a transversally elliptic operator.

This paper concerns arithmetic families of $\varphi $-modules over reduced affinoid spaces. For such a family, we first prove that the slope polygons are lower semicontinuous around any rigid point. We further prove that if the slope polygons are locally constant around a rigid point, then around this point, the family has a global slope filtration after base change to some extended Robba ring.

We examine the effects of pure additive noise on spatially extended systems with quadratic nonlinearities. We develop a general multi-scale theory for such systems and apply it to the Kuramoto–Sivashinsky equation as a case study. We first focus on a regime close to the instability onset (primary bifurcation), where the system can be described by a single dominant mode. We show analytically that the resulting noise in the equation describing the amplitude of the dominant mode largely depends on the nature of the stochastic forcing. For a highly degenerate noise, in the sense that it is acting on the first stable mode only, the amplitude equation is dominated by a pure multiplicative noise, which in turn induces the dominant mode to undergo several critical state transitions and complex phenomena, including intermittency and stabilisation, as the noise strength is increased. The intermittent behaviour is characterised by a power-law probability density and the corresponding critical exponent is calculated rigorously by making use of the first-passage properties of the amplitude equation. On the other hand, when the noise is acting on the whole subspace of stable modes, the multiplicative noise is corrected by an additive-like term, with the eventual loss of any stabilised state. We also show that the stochastic forcing has no effect on the dominant mode dynamics when it is acting on the second stable mode. Finally, in a regime which is relatively far from the instability onset so that there are two unstable modes, we observe numerically that when the noise is acting on the first stable mode, both dominant modes show noise-induced complex phenomena similar to the single-mode case.

Linear differential algebraic groups (LDAGs) measure differential algebraic dependencies among solutions of linear differential and difference equations with parameters, for which LDAGs are Galois groups. Differential representation theory is a key to developing algorithms computing these groups. In the rational representation theory of algebraic groups, one starts with and tori to develop the rest of the theory. In this paper, we give an explicit description of differential representations of tori and differential extensions of irreducible representation of . In these extensions, the two irreducible representations can be non-isomorphic. This is in contrast to differential representations of tori, which turn out to be direct sums of isotypic representations.

Given a quasi-hereditary algebra , we present conditions which guarantee that the algebra obtained by grading by its radical filtration is Koszul and at the same time inherits the quasi-hereditary property and other good Lie-theoretic properties that might possess. The method involves working with a pair consisting of a quasi-hereditary algebra and a (positively) graded subalgebra . The algebra arises as a quotient of by a defining ideal of . Along the way, we also show that the standard (Weyl) modules for have a structure as graded modules for . These results are applied to obtain new information about the finite dimensional algebras (e.g., the -Schur algebras) which arise as quotients of quantum enveloping algebras. Further applications, perhaps the most penetrating, yield results for the finite dimensional algebras associated with semisimple algebraic groups in positive characteristic . These results require, at least at present, considerable restrictions on the size of .

In a recent paper, Gross and Reeder study the arithmetic properties of discrete Langlands parameters for semi-simple -adic groups, and they conjecture that a special class of these – the simple wild parameters – should correspond to -packets consisting of simple supercuspidal representations. We provide a construction of this correspondence, and show that the simple wild -packets satisfy many expected properties. In particular, they admit a description in terms of the Langlands dual group, and contain a unique generic element for a fixed Whittaker datum. Moreover, we prove their stability on an open subset of the regular semi-simple elements, and show that they satisfy a natural compatibility with respect to unramified base-change.

We study analogues of Tate’s conjecture on homomorphisms for abelian varieties when the ground field is finitely generated over an algebraic closure of a finite field. Our results cover the case of abelian varieties without non-trivial endomorphisms.

Let be a field complete with respect to a discrete valuation whose residue field is perfect of characteristic . We prove that every smooth, projective, geometrically irreducible curve of genus one defined over with a non-zero divisor of degree a power of has a solvable point over .

In a recent paper Moshe Jarden (Diamonds in torsion of Abelian varieties, J. Inst. Math. Jussieu9(3) (2010), 477–480) proposed a conjecture, later named the Kuykian conjecture, which states that if $A$ is an Abelian variety defined over a Hilbertian field $K$, then every intermediate field of $K({A}_{\mathrm{tor} } )/ K$ is Hilbertian. We prove that the conjecture holds for Galois extensions of $K$ in $K({A}_{\mathrm{tor} } )$.

We study the arithmetic of abelian varieties over where is an arbitrary field. The main result relates Mordell–Weil groups of certain Jacobians over to homomorphisms of other Jacobians over . Our methods also yield completely explicit points on elliptic curves with unbounded rank over and a new construction of elliptic curves with moderately high rank over .