Recipe theorem for the Tutte polynomial for matroids,renormalization group-like approach

Using a quantum field theory renormalization group-like differential equation, we give a new proof of the recipe theorem for the Tutte polynomial for matroids. The solution of such an equation is in fact given by some appropriate characters of the Hopf algebra of isomorphic classes of matroids, characters which are then related to the Tutte polynomial for matroids. This Hopf algebraic approach also allows to prove, in a new way, a matroid Tutte polynomial convolution formula appearing in [W. Kook, V. Reiner, D. Stanton, A convolution formula for the Tutte polynomial, J. Combin. Theory Ser. B 76 (1999) 297–300] and [G. Etienne, M. Las Vergnas, External and internal elements of a matroid basis, Discrete Math. 179 (1998) 111–119].     

Descents, Quasi-Symmetric Functions, Robinson-Schensted for Posets, and the Chromatic Symmetric Function

We investigate an apparent hodgepodge of topics: a Robinson-Schensted algorithm for (3 + 1)-free posets, Chung and Graham's G-descent expansion of the chromatic polynomial, a quasi-symmetric expansion of the path-cycle symmetric function, and an expansion of Stanley's chromatic symmetric function X _G in terms of a new symmetric function basis. We show how the theory of P-partitions (in particular, Stanley's quasi-symmetric function expansion of the chromatic symmetric function X _G ) unifies them all, subsuming two old results and implying two new ones. Perhaps our most interesting result relates to the still-open problem of finding a Robinson-Schensted algorithm for (3 + 1)-free posets. (Magid has announced a solution but it appears to be incorrect.) We show that such an algorithm ought to respect descents, and that the best partial algorithm so far—due to Sundquist, Wagner, and West—respects descents if it avoids a certain induced subposet.     

New estimates for the Laplacian,the div–curl,and related Hodge systems

We establish new estimates for the Laplacian, the div–curl system, and more general Hodge systems in arbitrary dimension, with an application to minimizers of the Ginzburg–Landau energy. To cite this article: J. Bourgain, H. Brezis, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

混合新几何体Γ_(-p，i)K的不等式

近来,Lutwak,Yang和Zhang提出了新几何体Γ_(-p)K的概念,获得一些结论.本文引入混合新几何体Γ_(-p,i)K的概念,即新几何体是它的特殊类,得到了混合新几何体的一些性质,并建立了相关的不等式.     

Shorter tours by nicer ears: 7/5-Approximation for the graph-TSP, 3/2 for the path version,and 4/3 for two-edge-connected subgraphs

We prove new results for approximating the graph-TSP and some related problems. We obtain polynomial-time algorithms with improved approximation guarantees. For the graph-TSP itself, we improve the approximation ratio to 7=5. For a generalization, the minimum T-tour problem, we obtain the first nontrivial approximation algorithm, with ratio 3=2. This contains the s-t-path graph-TSP as a special case. Our approximation guarantee for finding a smallest 2-edge-connected spanning subgraph is 4=3. The key new ingredient of all our algorithms is a special kind of ear-decomposition optimized using forest representations of hypergraphs. The same methods also provide the lower bounds (arising from LP relaxations) that we use to deduce the approximation ratios.     

Nonanticipative duality,relaxations, and formulations for chance-constrained stochastic programs

We propose two new Lagrangian dual problems for chance-constrained stochastic programs based on relaxing nonanticipativity constraints. We compare the strength of the proposed dual bounds and demonstrate that they are superior to the bound obtained from the continuous relaxation of a standard mixed-integer programming (MIP) formulation. For a given dual solution, the associated Lagrangian relaxation bounds can be calculated by solving a set of single scenario subproblems and then solving a single knapsack problem. We also derive two new primal MIP formulations and demonstrate that for chance-constrained linear programs, the continuous relaxations of these formulations yield bounds equal to the proposed dual bounds. We propose a new heuristic method and two new exact algorithms based on these duals and formulations. The first exact algorithm applies to chance-constrained binary programs, and uses either of the proposed dual bounds in concert with cuts that eliminate solutions found by the subproblems. The second exact method is a branch-and-cut algorithm for solving either of the primal formulations. Our computational results indicate that the proposed dual bounds and heuristic solutions can be obtained efficiently, and the gaps between the best dual bounds and the heuristic solutions are small.     

On the a priori estimates for the Euler, the Navier-Stokes and the quasi-geostrophic equations

We prove new a priori estimates for the 3D Euler, the 3D Navier-Stokes and the 2D quasi-geostrophic equations by the method of similarity transforms.     

Ninth-order,explicit, two-step methods for second-order inhomogeneous linear IVPs

In the present paper, the second-order linear inhomogeneous initial value problems (LI-IVP) with constant coefficients is of our interest. Two-step hybrid methods (ie, of Numerov-type) are considered for addressing this problem. Thus, a special set of order conditions is given and solved for derivation a low cost new method. Actually, we manage to save two stages per step. This is crucial as shown in various numerical tests where our new proposal outperforms standard methods in the relevant literature.     

Nonlocal representation of the sl(2, R) algebra for the Chazy equation

A demonstration of how the point symmetries of the Chazy equation become nonlocal symmetries for the reduced equation is discussed. Moreover we construct an equivalent third-order differential equation which is related to the Chazy equation under a generalized transformation, and find the point symmetries of the Chazy equation are generalized symmetries for the new equation. With the use of singularity analysis and a simple coordinate transformation we construct a solution for the Chazy equation which is given by a right Painlevé series. The singularity analysis is applied to the new third-order equation and we find that it admits two solutions, one given by a left Painlevé series and one given by a right Painlevé series where the leading-order behaviors and the resonances are explicitly those of the Chazy equation.     

An implementable algorithm for the optimal design centering,tolerancing, and tuning problem

An implementable master algorithm for solving optimal design centering, tolerancing, and tuning problems is presented. This master algorithm decomposes the original nondifferentiable optimization problem into a sequence of ordinary nonlinear programming problems. The master algorithm generates sequences with accumulation points that are feasible and satisfy a new optimality condition, which is shown to be stronger than the one previously used for these problems.This research was sponsored by the National Science Foundation (RANN), Grant No. ENV-76-04264, and by the Joint Services Electronic Program, Contract No. F44620-76-C-0100.     

Fixed-Size Quadruples for a New, Hardware-Oriented Representation of the 4D Clifford Algebra

Clifford algebra (geometric algebra) offers a natural and intuitive way to model geometry in fields as robotics, machine vision and computer graphics. This paper proposes a new representation based on fixed-size elements (quadruples) of 4D Clifford algebra and demonstrates that this choice leads to an algorithmic simplification which in turn leads to a simpler and more compact hardware implementation of the algebraic operations. In order to prove the advantages of the new, quadruple-based representation over the classical representation based on homogeneous elements, a coprocessing core supporting the new fixed-size Clifford operands, namely Quad-CliffoSor (Quadruple-based Clifford coprocesSor) was designed and prototyped on an FPGA board. Test results show the potential to achieve a 23× speedup for Clifford products and a 33× speedup for Clifford sums and differences compared to the same operations executed by a software library running on a general-purpose processor.     

Configuration spaces,bistellar moves,and combinatorial formulae for the first Pontryagin class

The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces and the local combinatorial formula obtained by the author in 2004. The latter formula is based on the notion of a universal local formula introduced by the author and on the usage of bistellar moves. We give a brief sketch for the first formula and a rather detailed exposition for the second one. For the second formula, we also succeed to simplify it by providing a new simpler algorithm for decomposing a cycle in the graph of bistellar moves of two-dimensional combinatorial spheres into a linear combination of elementary cycles.     

Fundamental Constructions for Coalgebras, Corings, and Comodules

We study the various categories of corings, coalgebras, and comodules from a categorical perspective. Emphasis is given to the question which properties of these categories can be seen as instances of general categorical resp. algebraic results. However, we also obtain new results concerning the existence of limits and of factorizations of morphisms.      

A new geometrical construction for the near hexagon with parameters $(s,t,T_2)=(2,5, \lbrace 1, 2 \rbrace)$

We provide purely geometrical proofs for the uniqueness of the near hexagon with parameters $(s,t,T_2)=(2, 5, \{1, 2\})$. As a by-product we find a new construction for the near hexagon. Generalizing this new construction, we obtain an infinite class $\mathcal{C}$ of incidence structures.In [4] it will be proved that all members of $\mathcal{C}$ are near polygons.     

New congruences modulo 2, 4, and 8 for the number of tagged parts over the partitions with designated summands

The Ramanujan Journal - Recently, Lin introduced two new partition functions $$\hbox {PD}_{\mathrm{t}}(n)$$ and $$\hbox {PDO}_{\mathrm{t}}(n)$$, which count the total number of tagged parts over...     

Literacy, Matheracy, and Technocracy: A Trivium for Today

This article focuses on the relations between mathematics and mathematics education on the one hand and human behavior, societal models, and power on the other. Based on a critical analysis of school systems and of mathematical thinking, its history and its sociopolitical implications, anew concept of curriculum is suggested, organized in 3 strands: literacy, matheracy, and technoracy. This new concept sees education and scholarship as pursuing a major, comprehensive goal of building up a new civilization that rejects arrogance, inequity, and bigotry. Because the development of mathematics has been intertwined with all forms of human behavior in the history of human- kind, it is relevant to discuss mathematics and mathematics education with this major goal in mind.     

Practical criteria for positive-definite matrix,M-matrix and Hurwitz matrix

All the conventional criterions to verify if a matrix is positive definite, Hurwitz or an M-matrix, such as Sylvester condition and Hurwitz theorem, require us to compute n determinants. This paper develops a new and simple criterion, based on a new type of Gaussian elimination process. In contrast to the traditional criterions, the new criterion only computes one determinate. Thus, the computing cost is greatly reduced. An illustrative example is given to show the application of the new criterion in the stability analysis of Hopfied-type neural networks with time delays.     

Exact Solvability Conditions for the Cauchy Problem for Systems of First-Order Linear Functional-Differential Equations Determined by (σ1, σ2, ... , σ n ; τ)-Positive Operators

We obtain new sufficient conditions under which the Cauchy problem for a system of linear functional-differential equations is uniquely solvable for arbitrary forcing terms. The conditions established are unimprovable in a certain sense.     

Bounds for the Perron root, singularity/nonsingularity conditions, and eigenvalue inclusion sets

Using a unified approach based on the monotonicity property of the Perron root and its circuit extension, a series of exact two-sided bounds for the Perron root of a nonnegative matrix in terms of paths in the associated directed graph is obtained. A method for deriving the so-called mixed upper bounds is suggested. Based on the upper bounds for the Perron root, new diagonal dominance type conditions for matrices are introduced. The singularity/nonsingularity problem for matrices satisfying such conditions is analyzed, and the associated eigenvalue inclusion sets are presented. In particular, a bridge connecting Gerschgorin disks with Brualdi eigenvalue inclusion sets is found. Extensions to matrices partitioned into blocks are proposed.     

KKT conditions for weak compact convex sets,theorems of the alternative,and optimality conditions

We present several equivalent conditions for the Karush–Kuhn–Tucker conditions for weak^? compact convex sets. Using them, we extend several existing theorems of the alternative in terms of weak^? compact convex sets. Such extensions allow us to express the KKT conditions and hence necessary optimality conditions for more general nonsmooth optimization problems with inequality and equality constraints. Furthermore, several new equivalent optimality conditions for optimization problems with inequality constraints are obtained.