A Matrix Model for the Topological String II: The Spectral Curve and Mirror Geometry

In a previous paper, we presented a matrix model reproducing the topological string partition function on an arbitrary given toric Calabi–Yau manifold. Here, we compute the spectral curve of our matrix model and thus provide a matrix model derivation of the large volume limit of the BKMP “remodeling the B-model” conjecture, the claim that Gromov–Witten invariants of any toric Calabi–Yau threefold coincide with the spectral invariants of its mirror curve.     

模糊判断矩阵排序向量的最小偏差法

基于积型模糊一致性判断矩阵和模糊判断矩阵的排序向量之间的偏差,通过建立并求解一个规划模型,得到模糊判断矩阵排序向量的最小偏差法,并研究了它的一些性质.最后通过两个算例说明了方法的可行性与有效性.     

Topological geometrodynamics,Part I: General theory

The recent status of topological geometrodynamics (TGD) is reviewed. One can end up with TGD either by starting from the energy problem of general relativity or from the need to generalize hadronic or superstring models. The basic principle of the theory is `Do not quantize!' meaning that quantum physics is reduced to Kähler geometry and spinor structure of the infinite-dimensional space of 3-surfaces in 8-dimensional space H=M⁴₊×CP₂ with physical states represented by classical spinor fields. General coordinate invariance implies that classical theory becomes an exact part of the quantum theory and configuration space geometry and that space-time surfaces are generalized Bohr orbits. The uniqueness of the infinite-dimensional Kähler geometric existence fixes imbedding space and the dimension of the space-time highly uniquely and implies that superconformal and supercanonical symmetries acting on the lightcone boundary δM⁴₊×CP₂ are cosmologies symmetries.The work with the p-adic aspects of TGD, the realization of the possible role of quaternions and octonions in the formulation of quantum TGD, the discovery of infinite primes, and TGD inspired theory of consciousness encouraged the vision about TGD as a generalized number theory. The vision leads to a considerable generalization of TGD and to an extension of the symmetries of the theory to include superconformal and Super-Kac-Moody symmetries associated with the group P×SU(3)×U(2)_ew (P denotes the Poincaré group) acting as the local symmetries of the theory. Quantum criticality, which can be seen as a prediction of the theory, fixes the value spectrum for the coupling constants of the theory.The proper mathematical and physical interpretation of the p-adic numbers has remained a long-lasting challenge. Both TGD inspired theory of consciousness and the vision about physics as a generalized number theory suggest that p-adic space-time regions obeying p-adic counterparts of the field equations are geometric correlates of mind in the sense that they provide cognitive representations for the physics in the real space-time regions representing matter. Evolution identified as a gradual increase of the infinite p-adic prime characterizing the entire Universe is basic prediction of the theory.S-matrix elements can be identified as Glebsch–Gordan coefficients between interacting and free Super-Kac-Moody algebra representations and it is now possible to give Feynmann rules for the S-matrix in the approximation that elementary particles correspond to the so-called CP₂ type extremals.     

Topological regular variation: I. Slow variation

Motivated by the Category Embedding Theorem, as applied to convergent automorphisms (Bingham and Ostaszewski (in press) [11]), we unify and extend the multivariate regular variation literature by a reformulation in the language of topological dynamics. Here the natural setting are metric groups, seen as normed groups (mimicking normed vector spaces). We briefly study their properties as a preliminary to establishing that the Uniform Convergence Theorem (UCT) for Baire, group-valued slowly-varying functions has two natural metric generalizations linked by the natural duality between a homogenous space and its group of homeomorphisms. Each is derivable from the other by duality. One of these explicitly extends the (topological) group version of UCT due to Bajšanski and Karamata (1969) [4] from groups to flows on a group. A multiplicative representation of the flow derived in Ostaszewski (2010) [45] demonstrates equivalence of the flow with the earlier group formulation. In companion papers we extend the theory to regularly varying functions: we establish the calculus of regular variation in Bingham and Ostaszewski (2010) [13] and we extend to locally compact, σ-compact groups the fundamental theorems on characterization and representation (Bingham and Ostaszewski (2010) [14]). In Bingham and Ostaszewski (2009) [15], working with topological flows on homogeneous spaces, we identify an index of regular variation, which in a normed-vector space context may be specified using the Riesz representation theorem, and in a locally compact group setting may be connected with Haar measure.     

Generalized voronoi diagrams for moving a ladder. I: Topological analysis

Given a bounded open subset Ω of the plane whose boundary is the union of finitely many polygons, and a real number d > 0, a manifold FP (the [free placements]) may be defined as the set of placements of a closed oriented line-segment B (a [ladder]) of length d inside Ω. FP is a three-dimensional manifold. A [Voronoi complex] in this manifold, a two-dimensional cell complex, is defined by analogy with the classical geometric construction in the plane; within this complex a one-dimensional subcomplex N, called the skeleton, is defined. It is shown that every component of FP contains a unique component of N, and canonical motions are given to move the ladder to placements within N. In this way, general motion planning is reduced to searching in a suitable representation of N as a (combinatorial) graph. Efficient construction of N is described in a companion paper.     

Matrix Mechanics of the Relativistic Point Particle and String in Clifford Space

We resolve the space-time canonical variables of the relativistic point particle into inner products of Weyl spinors with components in a Clifford algebra and find that these spinors themselves form a canonical system with generalized Poisson brackets. For N particles, the inner products of their Clifford coordinates and momenta form two N × N Hermitian matrices X and P which transform under a U(N) symmetry in the generating algebra. This is used as a starting point for defining matrix mechanics for a point particle in Clifford space. Next we consider the string. The Lorentz metric induces a metric and a scalar on the world sheet which we represent by a Jackiw–Teitelboim term in the action. The string is described by a polymomenta canonical system and we find the wave solutions to the classical equations of motion for a flat world sheet. Finally, we show that the \({SL(2.\mathbb{C})}\) charge and space-time momentum of the quantized string satisfy the Poincaré algebra.     

Further Discussions on Induced Bias Matrix Model for the Pair-Wise Comparison Matrix

The inconsistency issue of pairwise comparison matrices has been an important subject in the study of the analytical network process. Most inconsistent elements can efficiently be identified by inducing a bias matrix only based on the original matrix. This paper further discusses the induced bias matrix and integrates all related theorems and corollaries into the induced bias matrix model. The theorem of inconsistency identification is proved mathematically using the maximum eigenvalue method and the contradiction method. In addition, a fast inconsistency identification method for one pair of inconsistent elements is proposed and proved mathematically. Two examples are used to illustrate the proposed fast identification method. The results show that the proposed new method is easier and faster than the existing method for the special case with only one pair of inconsistent elements in the original comparison matrix.     

Topological duality and lattice expansions,I: A topological construction of canonical extensions

The two main objectives of this paper are (a) to prove purely topological duality theorems for semilattices and bounded lattices, and (b) to show that the topological duality from (a) provides a construction of canonical extensions of bounded lattices. In previously known dualities for semilattices and bounded lattices, the dual spaces are compact 0-dimensional spaces with additional algebraic structure. For example, semilattices are dual to 0-dimensional compact semilattices. Here we establish dual categories in which the spaces are characterized purely in topological terms, with no additional algebraic structure. Thus the results can be seen as generalizing Stone’s duality for distributive lattices rather than Priestley’s. The paper is the first of two parts. The main objective of the sequel is to establish a characterization of lattice expansions, i.e., lattices with additional operations, in the topological setting built in this paper.     

On the Transfer Matrix for the Two-Dimensional Ising Model

The eigenvalue problem for the transfer matrix of the two-dimensional Ising lattice, with free boundaries perpendicular to the transfer direction, is solved. The mathematical mechanism which produces the asymptotic degeneracy of the eigenvalue spectrum below the bulk transition temperature is different from the case with cyclic boundary conditions. The results are used to calculate the two-spin correlation function between two spins in the same edge of the lattice, and between two spins in opposite edges.     

Going Off the Grid: Iterative Model Selection for Biclustered Matrix Completion

We consider the problem of performing matrix completion with side information on row-by-row and column-by-column similarities. We build upon recent proposals for matrix estimation with smoothness constraints with respect to row and column graphs. We present a novel iterative procedure for directly minimizing an information criterion to select an appropriate amount of row and column smoothing, namely, to perform model selection. We also discuss how to exploit the special structure of the problem to scale up the estimation and model selection procedure via the Hutchinson estimator, combined with a stochastic Quasi-Newton approach. Supplementary material for this article is available online.     

灰色系统模型病态矩阵的双正则化算法

灰色系统模型矩阵会存在病态问题.为消除其病态性,基于病态矩阵的双正则化方法,建立了正则化灰色系统模型中灰参数求解的表达式,给出了其导出方式;提出了正则参数α的选择原则.从而避免了灰参数求解过程中矩阵的病态问题.数值试验分析说明,灰色系统模型的双正则化算法是正确和适用的.     

拟-A-Proper映射的拓扑度

拓扑度理论为研究非线性方程多解问题提供了强有力的工具。迄今拓扑度方法已成为非线性泛函分析中最基本的方法之一(关于非线性泛函分析的发展近况请见田方增[1])。 本文提出很广泛的拟-A-proper映射类,用A-proper映射([5])逼近拟-A-proper映射的方法(类似[2])建立了后者的拓扑度;证明出两类映射拓扑度的性质几乎全同;讨论     

A Matrix of a Growth Optimization Model for the Production Activity of Cooperating Enterprises

This paper deals with direct and indirect growth rates of co-operating enterprises and with the corresponding matrix of growth. Its purpose is to determine through indirect growth rates the growth structure of co-operating enterprises and to construct, on the basis of the matrix of growth, a model which enables us to follow and program them. In other words, by means of the matrix of growth or the model which is based on it we can estimate the future development of co-operating enterprises. The conditions of optimality have been determined through linear programming, whereby account has been taken of the upper and the lower limit of development of the enterprise, the available funds and the efficiency of investments.     

Deriving the DEA frontier for two-stage processes

Data envelopment analysis (DEA) is a method for measuring the efficiency of peer decision making units (DMUs). Recently DEA has been extended to examine the efficiency of two-stage processes, where all the outputs from the first stage are intermediate measures that make up the inputs to the second stage. The resulting two-stage DEA model provides not only an overall efficiency score for the entire process, but as well yields an efficiency score for each of the individual stages. Due to the existence of intermediate measures, the usual procedure of adjusting the inputs or outputs by the efficiency scores, as in the standard DEA approach, does not necessarily yield a frontier projection. The current paper develops an approach for determining the frontier points for inefficient DMUs within the framework of two-stage DEA.     

A Topological Criterion for the Existence of Half-Bound States

The following theorem is proved: if (M⁴ⁿ⁺¹,g) is a completeRiemannian manifold and M is an oriented hypersurface partitioningM and with non-zero signature, then the spectrum of the Hodge–deRhamLaplacian is [0,]. This result is obtained by a new Callias-typeindex. This new formula links half-bound harmonic forms (thatis, nearly L² but not in L²) with the signature of .     

Matrix probing: A randomized preconditioner for the wave-equation Hessian

This paper considers the problem of approximating the inverse of the wave-equation Hessian, also called normal operator, in seismology and other types of wave-based imaging. An expansion scheme for the pseudodifferential symbol of the inverse Hessian is set up. The coefficients in this expansion are found via least-squares fitting from a certain number of applications of the normal operator on adequate randomized trial functions built in curvelet space. It is found that the number of parameters that can be fitted increases with the amount of information present in the trial functions, with high probability. Once an approximate inverse Hessian is available, application to an image of the model can be done in very low complexity. Numerical experiments show that randomized operator fitting offers a compelling preconditioner for the linearized seismic inversion problem.     

求弹性半平面问题基本解的一个直接方法

本文利用镜像法[2]和直接积分导出了弹性半平面问题的基本解.这个推导方法比Gladwell用镜像法和富氏变换要直观.这个基本解对于用边界元法分析半平面域具有任意外形孔洞的应力集中问题将起着十分重要的作用.     

Unifying Impossibility Theorems: A Topological Approach

In this note a topological approach is developed which permits us to prove Arrow′s impossibility theorem in the social choice theory using a topological technique close to that of Chichilnisky.     

A Generalized Topological Recursion for Arbitrary Ramification

The Eynard–Orantin topological recursion relies on the geometry of a Riemann surface S and two meromorphic functions x and y on S. To formulate the recursion, one must assume that x has only simple ramification points. In this paper, we propose a generalized topological recursion that is valid for x with arbitrary ramification. We justify our proposal by studying degenerations of Riemann surfaces. We check in various examples that our generalized recursion is compatible with invariance of the free energies under the transformation ${(x, y) \mapsto (y, x)}$ , where either x or y (or both) have higher order ramification, and that it satisfies some of the most important properties of the original recursion. Along the way, we show that invariance under ${(x, y) \mapsto (y, x)}$ is in fact more subtle than expected; we show that there exists a number of counterexamples, already in the case of the original Eynard–Orantin recursion, that deserve further study.