Research in the Mathematical Sciences最新文献

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.

The Erdős-Hajnal conjecture is one of the most classical and well-known problems in extremal and structural combinatorics dating back to 1977. It asserts that in stark contrast to the case of a general n-vertex graph, if one imposes even a little bit of structure on the graph, namely by forbidding a fixed graph H as an induced subgraph, instead of only being able to find a polylogarithmic size clique or an independent set, one can find one of polynomial size. Despite being the focus of considerable attention over the years, the conjecture remains open. In this paper, we improve the best known lower bound of (2^{Omega (sqrt{log n})}) on this question, due to Erdős and Hajnal from 1989, in the smallest open case, namely when one forbids a (P_5), the path on 5 vertices. Namely, we show that any (P_5)-free n-vertex graph contains a clique or an independent set of size at least (2^{Omega (log n)^{2/3}}). We obtain the same improvement for an infinite family of graphs.

The zeros of the nth D’Arcais polynomial, also known in combinatorics as the Nekrasov–Okounkov polynomial, dictate the vanishing properties of the nth Fourier coefficients of all complex powers x of the Dedekind (eta )-function. In this paper, we prove that these coefficients are non-vanishing for (vert x vert > kappa , (n-1)) and (kappa approx 9.7225). Numerical computations imply that 9.72245 is a lower bound for (kappa ). This significantly improves previous results by Kostant, Han, and Heim–Neuhauser. The polynomials studied in this paper include Chebyshev polynomials of the second kind, 1-associated Laguerre polynomials, Hermite polynomials, and polynomials associated with overpartitions and plane partitions.