Research in the Mathematical Sciences最新文献

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

In this paper, by virtue of unfolding theory in singularity theory, we investigate the singularities of five special surfaces generated by a regular curve lying on a spacelike hypersurface in Lorentz–Minkowski 4-space. Using two kinds of extended Lorentzian Darboux frames along the curve as tools, five new invariants are obtained to characterize the singularities of five special surfaces and their geometric meanings are discussed in detail. In addition, some dual relationships between a normal curve of the original curve and five surfaces are revealed under the meanings of Legendrian duality.

We give an explicit and computationally efficient construction of harmonic weak Maass forms which map to weight 2 newforms under the (xi )-operator. Our work uses a new non-analytic completion of the Kleinian (zeta )-function from the theory of Abelian functions.

In image processing, problems of separation and reconstruction of missing pixels from incomplete digital images have been far more advanced in past decades. Many empirical results have produced very good results; however, providing a theoretical analysis for the success of algorithms is not an easy task, especially, for inpainting and separating multi-component signals. In this paper, we propose two main algorithms based on (l_1) constrained and unconstrained minimization for separating N distinct geometric components and simultaneously filling in the missing part of the observed image. We then present a theoretical guarantee for these algorithms using compressed sensing technique, which is based on a principle that each component can be sparsely represented by a suitably chosen dictionary. Those sparsifying systems are extended to the case of general frames instead of Parseval frames which have been typically used in the past. We finally prove that the method does indeed succeed in separating point singularities from curvilinear singularities and texture as well as inpainting the missing band contained in curvilinear singularities and texture.

We prove that the real-valued electric potential (q in L^{max (2,3 n /5)}(Omega )) of the Dirichlet Laplacian (-Delta +q) acting in a bounded domain (Omega subset mathbb {R}^n), (n ge 3), is uniquely determined by the asymptotics of the eigenpairs formed by the eigenvalues and the boundary observation of the normal derivative of the eigenfunctions.