Research in the Mathematical Sciences最新文献

We investigate geometric properties of a special kind of evolutes, so-called horocyclic evolutes, of smooth curves in hyperbolic plane from the viewpoint of duality. To do that, we first review the basic notions of (spacelike) frontals in hyperbolic plane, which developed by the first and the third authors by using basic Legendrian duality theorem developed by the second author. Moreover, two kinds of horocyclic evolutes are defined and the relationship between these two different evolutes are studied. As results, they are Legendrian dual to each other.

Let (F: {mathbb {R}}^2rightarrow {mathbb {R}}^2) be a polynomial mapping. We consider the image of the compositions (F^k) of F. We prove that under some condition then the image of the iterated map (F^k) is stable when k is large.

We introduced an (tilde{mathcal {A}})-invariant for quasi-ordinary parameterizations, and we consider it to describe quasi-ordinary surfaces with one generalized characteristic exponent admitting a countable moduli.

This is a survey article, in which we explore how the presence of singularities affects the geometry and topology of complex projective hypersurfaces.

The purpose of this paper is to present an algebraic theoretical basis for the study of (omega )-Hamiltonian vector fields defined on a symplectic vector space ((V,omega )) with respect to coordinates that are not necessarily symplectic. We introduce the concepts of (omega )-symplectic and (omega )-semisymplectic groups, and describe some of their properties that may not coincide with the classical context. We show that the Lie algebra of such groups is a useful tool in the recognition and construction of (omega )-Hamiltonian vector fields.

In this paper we present a constructive method to characterize ideals of the local ring ({mathscr {O}}_{{mathbb {C}}^n,0}) of germs of holomorphic functions at (0in {mathbb {C}}^n) which arise as the moduli ideal (langle f,{mathfrak {m}}, j(f)rangle ), for some (fin {mathfrak {m}}subset {mathscr {O}}_{{mathbb {C}}^n,0}). A consequence of our characterization is an effective solution to a problem dating back to the 1980s, called the Reconstruction Problem of the hypersurface singularity from its moduli algebra. Our results work regardless of whether the hypersurface singularity is isolated or not.

We prove an exact formula for the variance of the divisor function over short intervals in ({mathcal {A}}:= {mathbb {F}}_q [T]), where q is a prime power; and for correlations of the form (d(A) d(A+B)), where we average both A and B over certain intervals in ({mathcal {A}}). We also obtain an exact formula for correlations of the form (d(KQ+N) d (N)), where Q is prime and K and N are averaged over certain intervals with ({{,textrm{deg},}}N le {{,textrm{deg},}}Q -1 le {{,textrm{deg},}}K); and we demonstrate that (d(KQ+N)) and d(N) are uncorrelated. We generalize our results to (sigma _z) defined by (sigma _z (A):= sum _{E mid A} |A |^z) for all monics (A in {mathcal {A}}). Our approach is to use the orthogonality relations of additive characters on ({mathbb {F}}_q) to translate the problems to ones involving the ranks of Hankel matrices over ({mathbb {F}}_q). We prove several results regarding the rank and kernel structure of these matrices, thus demonstrating their number-theoretic properties. We also discuss extending our method to other divisor sums, such as those involving (d_k).

Abstract

We establish a new version of Siegel’s lemma over a number field k, providing a bound on the maximum of heights of basis vectors of a subspace of (k^N) , (N ge 2) . In addition to the small-height property, the basis vectors we obtain satisfy certain sparsity condition. Further, we produce a nontrivial bound on the heights of all the possible subspaces generated by subcollections of these basis vectors. Our bounds are absolute in the sense that they do not depend on the field of definition. The main novelty of our method is that it uses only linear algebra and does not rely on the geometry of numbers or the Dirichlet box principle employed in the previous works on this subject.

Abstract

Let (rho :Grightarrow {{,textrm{GL},}}_2(K)) be a continuous representation of a compact group G over a complete discretely valued field K with ring of integers (mathcal {O}) and uniformiser (pi ) . We prove that ({{,textrm{tr},}}rho ) is reducible modulo (pi ^n) if and only if (rho ) is reducible modulo (pi ^n) . More precisely, there exist characters (chi _1,chi _2 :Grightarrow (mathcal {O}/pi ^nmathcal {O})^times ) such that (det (t - rho (g))equiv (t-chi _1(g))(t-chi _2(g))pmod {pi ^n}) for all (gin G) , if and only if there exists a G-stable lattice (Lambda subseteq K^2) such that (Lambda /pi ^nLambda ) contains a G-invariant, free, rank one (mathcal {O}/pi ^nmathcal {O}) -submodule. Our result applies in the case that (rho ) is not residually multiplicity-free, in which case it answers a question of Bellaïche and Chenevier (J Algebra 410:501–525, 2014, pp. 524). As an application, we prove an optimal version of Ribet’s lemma, which gives a condition for the existence of a G-stable lattice (Lambda ) that realises a non-split extension of (chi _2) by (chi _1) .

In 1920, P. A. MacMahon generalized the (classical) notion of divisor sums by relating it to the theory of partitions of integers. In this paper, we extend the idea of MacMahon. In doing so, we reveal a wealth of divisibility theorems and unexpected combinatorial identities. Our initial approach is quite different from MacMahon and involves rational function approximation to MacMahon-type generating functions. One such example involves multiple q-harmonic sums