Newton–Hensel Interpolation Lifting

The main result of this paper is a new version of Newton-Hensel lifting that relates to interpolation questions. It allows one to lift polynomials in ℤ[x] from information modulo a prime number p ≠ 2 to a power p^k for any k, and its originality is that it is a mixed version that not only lifts the coefficients of the polynomial but also its exponents. We show that this result corresponds exactly to a Newton--Hensel lifting of a system of 2t generalized equations in 2t unknowns in the ring of p-adic integers ℤ_p. Finally, we apply our results to sparse polynomial interpolation in ℤ[x].     

牛顿运动定律和机翼举力

将空气简化为理想流体，由牛顿运动定律推出气体做曲线运动时的切向伯努利方程和法向压强梯度表达式，并解释了机翼举力。     

On the set covering polytope: II. Lifting the facets with coefficients in {0, 1, 2}

In an earlier paper (Mathematical Programming 43 (1989) 57–69) we characterized the class of facets of the set covering polytope defined by inequalities with coefficients equal to 0, 1 or 2. In this paper we connect that characterization to the theory of facet lifting. In particular, we introduce a family of lower dimensional polytopes and associated inequalities having only three nonzero coefficients, whose lifting yields all the valid inequalities in the above class, with the lifting coefficients given by closed form expressions.The research underlying this report was supported by Grant ECS-8601660 of the National Science Foundation, Contract N00014-85-K-0198 with the Office of Naval Research, and Grant AFOSR-870292 of the Air Force Office of Scientific Research.     

Order Properties of the Space of Finitely Additive Transition Functions

The basic order properties, as well as some metric and algebraic properties, are studied of the set of finitely additive transition functions on an arbitrary measurable space, as endowed with the structure of an ordered normed algebra, and some connections are revealed with the classical spaces of linear operators, vector measures, and measurable vector-valued functions. In particular, the question is examined of splitting the space of transition functions into the sum of the subspaces of countably additive and purely finitely additive transition functions.     

Minimization of a quasi-concave function over an efficient set

We introduce a projective approach for studying symmetric travelling salesman polytopes (STSPs). Thesymmetric travelling salesman polytope STSP(V) (resp.,Hamiltonian path polytope HP(V)) is the convex hull of incidence vectors of all Hamiltonian cycles (resp., paths) on the complete undirected graph with node setV. For any nodeh V, HP(V) is aprojection of STSP(V {h}). We show that HP(V) and STSP(V {h}) are isomorphic, and HP(V) is of full dimension minus one. By this projective approach, we obtain generalclique-lifting results, all based on simple conditions, for deriving large new classes of STSP facets. These results apply to all known non-trivial STSP facets, and generalize clique-lifting results of Maurras (1975), Grötschel and Padberg (1979) and Naddef and Rinaldi (1988).This research is supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada to the first author.     

Constructions of primitive formally dual pairs having subsets with unequal sizes

The concept of formal duality was proposed by Cohn, Kumar and Schürmann, which reflects a remarkable symmetry among energy‐minimizing periodic configurations. This formal duality was later translated into a purely combinatorial property by Cohn, Kumar, Reiher and Schürmann, where the corresponding combinatorial objects were called formally dual pairs. Motivated by this surprising application on the energy minimization problem, we focus on the algebraic constructions of primitive formally dual pairs. It is worth noting that almost all known examples of primitive formally dual pairs satisfy that the two subsets have the same size. Indeed, prior to this work, there was only one known example derived from computer search, which had subsets with unequal sizes in ${\mathbb{Z}}_2\times {\mathbb{Z}}_4^2$. Inspired by this example, we propose a lifting construction framework and a recursive construction framework, which generate new primitive formally dual pairs from known ones. As an application, for $m\ge 2$, we obtain $m+1$ pairwise inequivalent primitive formally dual pairs in ${\mathbb{Z}}_2\times {\mathbb{Z}}_4^{2m}$, which have subsets with unequal sizes.     

Construction of Biorthogonal Multiwavelets Using the Lifting Scheme

The lifting scheme has been found to be a flexible method for constructing scalar wavelets with desirable properties. Here it is extended to the construction of multiwavelets. It is shown that any set of compactly supported biorthogonal multiwavelets can be obtained from the Lazy matrix filters with a finite number of lifting steps. As an illustration of the general theory, compactly supported biorthogonal multiwavelets with optimum time–frequency resolution are constructed. In addition, experimental results of applying these multiwavelets to image compression are presented.     

Lifting Galois sections along torsors

The cuspidalization conjecture, which is a consequence of Grothendieck's section conjecture, asserts that for any smooth hyperbolic curve X over a finitely generated field k of characteristic 0 and any non empty Zariski open , every section of lifts to a section of . We consider in this article the problem of lifting Galois sections to the intermediate quotient introduced by Mochizuki 10 . We show that when and is an union of torsion sub‐packets every Galois section actually lifts to . One of the main tools in the proof is the construction of torus torsors and over X and the geometric interpretation .     

A compact C0 discontinuous Galerkin method for Kirchhoff plates

A compact C⁰ discontinuous Galerkin (CCDG) method is developed for solving the Kirchhoff plate bending problems. Based on the CDG (LCDG) method for Kirchhoff plate bending problems, the CCDG method is obtained by canceling the term of global lifting operator and enhancing the term of local lifting operator. The resulted CCDG method possesses the compact stencil, that is only the degrees of freedom belonging to neighboring elements are connected. The advantages of CCDG method are: (1) CCDG method just requires C⁰ finite element spaces; (2) the stiffness matrix is sparser than CDG (LCDG) method; and (3) it does not contain any parameter which can not be quantified a priori compared to C⁰ interior penalty (IP) method. The optimal order error estimates in certain broken energy norm and H¹‐norm for the CCDG method are derived under minimal regularity assumptions on the exact solution with the help of some local lower bound estimates of a posteriori error analysis. Some numerical results are included to verify the theoretical convergence orders. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1265–1287, 2015