Topological Formula Research Articles

Index theorem for homogeneous spaces of Lie groups

We study index theory for homogeneous spaces associated to an almost connected Lie group in terms of the topological and analytic aspects. For the topological aspect, we obtain a topological formula as a result of the Riemann–Roch formula for proper cocompact actions of the Lie group, inspired by the work of Paradan and Vergne. For the analytic aspect, we apply heat kernel methods to obtain a local index formula representing the higher indices of equivariant elliptic operators with respect to a proper, cocompact action of the Lie group.

الفضاءات القابلة للتفكك والفضاءات غير القابلة للتفكك عبر فضاء القرب

اهتم كثيرا من الباحثين في دراسة اغلب المشكلات بعد تحويلها الى صيغ رياضية داخل فضاء القرب لسهولة توظيف هذاالفضاء مع المشكلات الحياتية المختلفة التي تواجهنا. لذلك يصب اهتمام البحث في ايجاد صفة جديدة الى فضاءات القرب باستخدام خواص القرب وخواص العناقيد وربطها معا للحصول على نقاط تراكمية جديدة اطلقنا عليها اسم نقاط كثه ومجموعة كل هذه النقاط اطلقنا عليها اسم مجموعة كثه حيث تم تكوين اساس رياضي متماسك الى الصيغة التبلوجية التي نبحث عنها والتي اطلقنا عليها التفكك واذا كانالفضاء لا يحمل هذه الصفة نطلق عليه فضاء غير قابل للتفكك.

Deriving the Hawking Temperature of (Massive) Global Monopole Spacetime via a Topological Formula.

In this work, we study the Hawking temperature of the global monopole spacetime (non-spherical symmetrical black hole) based on the topological method proposed by Robson, Villari, and Biancalana (RVB). By connecting the Hawking temperature with the topological properties of black holes, the Hawking temperature of the global monopole spacetime can be obtained by the RVB method. We also discuss the Hawking temperature in massive gravity, and find that the effect of the mass term cannot be ignored in the calculation of the Hawking temperature; the corrected Hawking temperature in massive gravity can be derived by adding an integral constant, which can be determined by the standard definition.

The Abel map for surface singularities II. Generic analytic structure

We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respect to a fixed topological type), under the condition that the link is a rational homology sphere. The list of analytic invariants includes: the geometric genus, the cohomology of certain natural line bundles, the cohomology of their restrictions on effective cycles (supported on the exceptional curve of a resolution), the cohomological cycle of natural line bundles, the multivariable Hilbert and Poincaré series associated with the divisorial filtration, the analytic semigroup, the maximal ideal cycle.The first part contains the definition of ‘generic structure’ based on the work of Laufer [14]. The second technical ingredient is the Abel map developed in [21].The results can be compared with certain parallel statements from the Brill–Noether theory and from the theory of Abel map associated with projective smooth curves (see e.g. [1] and [6]), though the tools and machineries are very different.

Topological asymptotic analysis for tumor identification problem

This work is concerned with the problem of identifying the shape, size and location of a small embedded tumor from measured temperature on the skin surface. The temperature distribution in the biological tissue is governed by the Pennes model equation. The proposed approach is based on the Kohn–Vogelius formulation and the topological sensitivity analysis method. The ill-posed geometric inverse problem is reformulated as a topology optimization. The temperature field perturbation, caused by the presence of a small anomaly, is analyzed and estimated. A topological asymptotic formula, describing the variation of the considered Kohn–Vogelius type functional with respect to the presence of a small anomaly is derived.

Localization of the action in AdS/CFT

We derive a simple formula for the action of any supersymmetric solution to minimal gauged supergravity in the AdS4/CFT3 correspondence. Such solutions are equipped with a supersymmetric Killing vector, and we show that the holographically renormalized action may be expressed entirely in terms of the weights of this vector field at its fixed points, together with certain topological data. In this sense, the classical gravitational partition function localizes in the bulk. We illustrate our general formula with a number of explicit examples, in which exact dual field theory computations are also available, which include supersymmetric Taub-NUT and Taub-bolt type spacetimes, as well as black hole solutions. Our simple topological formula also allows us to write down the action of any solution, provided it exists.

Topological formula of the loop expansion of the colored Jones polynomials

We give a formula of the loop expansion of the colored Jones polynomials based on homological representation of braid groups. This gives a direct proof of the Melvin-Morton-Rozansky conjecture, and a connection between entropy of braids and quantum representations.

Perturbation Sensitivity Analysis and Dynamic Topology Optimization for Heat Conduction Structure

Topology optimization is a fundamental problem for steady heat conduction. Due to the unidirectional treatment for the intermediate density elements in the traditional topology optimization, we propose a Bi-Directional Interpolation Model (BDIM) that intermediate density elements are judged by a threshold, and establish optimization model. Besides, considering the generation of new hole boundary in the process of heat conduction enhancement topology optimization, we study the perturbation sensitivity analysis with respect to new hole, and present an adaptive dynamic boundary method for the new boundary in the optimization process. Furthermore, based on the domain perturbation technique and Lebesgue differential theory, the topological derivative formulas with different objective functions subjected to three kinds of boundary conditions are derived for the control system of Poisson equation. Finally, a number of numerical examples are presented to demonstrate the feasibility and effectiveness of the proposed method for designing the heat conduction structure.

A symbol calculus for foliations

The classical Getzler rescaling theorem of [15] is extended to the transverse geometry of foliations. More precisely, a Getzler rescaling calculus, [15], as well as a Block–Fox calculus of asymptotic pseudodifferential operators (A‰ \Psi DOs), [10], is constructed for all transversely spin foliations. This calculus applies to operators of degree m globally times degree \ell in the leaf directions, and is thus an appropriate tool for a better understanding of the index theory of transversely elliptic operators on foliations [13]. The main result is that the composition of A‰ \Psi DOs is again an A \Psi ‰DO, and includes a formula for the leading symbol. Our formula is more complicated due to its wide generality but its form is essentially the same, and it simplifies notably for Riemannian foliations. In short, we construct an asymptotic pseudodifferential calculus for the “leaf space” of any foliation. Applications will be derived in [5,6] where we give a Getzler-like proof of a local topological formula for the Connes–Chern character of the Connes–Moscovici spectral triple of [20], as well as the (semi-finite) spectral triple given in [5], yielding an extension of the seminal Atiyah–Singer L^2 covering index theorem, [2], to coverings of “leaf spaces” of foliations.

Immersions associated with holomorphic germs

A holomorphic germ \Phi: ( \mathbb C^2, 0) \to ( \mathbb C^3, 0) , singular only at the origin, induces at the links level an immersion of S^3 into S^5 . The regular homotopy type of immersions S^3\looparrowright S^5 are determined by their Smale invariant, defined up to a sign ambiguity. In this paper we fix a sign of the Smale invariant and we show that for immersions induced by holomorphic gems the sign-refined Smale invariant \Omega is the negative of the number of cross caps appearing in a generic perturbation of \Phi . Using the algebraic method we calculate \Omega for some families of singularities, among others the A-D-E quotient singularities. As a corollary, we obtain that the regular homotopy classes which admit holomorphic representatives are exactly those, which have non-positive sign-refined Smale invariant. This answers a question of Mumford regarding exactly this correspondence. We also determine the sign ambiguity in the topological formulae of Hughes–Melvin and Ekholm–Szűcs connecting the Smale invariant with (singular) Seifert surfaces. In the case of holomorphic realizations of Seifert surfaces, we also determine their involved invariants in terms of holomorphic geometry.

Dimensionality of hypercube clusters

We investigate clusters of hypercubes in d-dimensional space as a function of the number of vertices, N, and number of cluster shells, L. The number of links, vertices, and exterior vertices exhibit ‘magic number’ characteristics versus L, as the dimension of the space changes. Starting with only the spatial coordinates, we create an adjacency and distance matrix that facilitates the calculation of topological indices, including the Wiener, hyper-Wiener, reverse Wiener, Szeged, Balaban, and Kirchhoff indices. Some known topological formulas for hypercubes when L = 1 are experimentally verified. The asymptotic limits with N of the topological indices are shown to exhibit power law behavior whose exponent changes with d and type of topological index. The asymptotic graph energy is linear with N, whose slope changes with d, and in 2D agrees numerically with previous calculations. Also, the thermodynamic properties such as entropy, free energy, and enthalpy of these lattices show logarithmic behavior with increasing N. The hypercubic clusters are projected onto 3D space when the dimensionality $$d>3$$ .

The asymptotic value of the zeroth-order Randić index and sum-connectivity index for trees

The zeroth-order Randić index and sum–connectivity index are two indices based on the vertex degrees. They appeared in the topological formula for the total π-electron energy of conjugated molecules and attracted a lot of attention in recent years. Let Tn be the set of trees of order n. Suppose each tree in Tn is equally likely. We get that for almost every tree, the zeroth-order Randić index is among (r1 ± ε)n and the sum–connectivity index is among (r2 ± ε)n, where r1, r2 are some constants and ε is any positive real number.

Inverse acoustic scattering by solid obstacles: topological sensitivity and its preliminary application

A to pological sensitivity approach is developed for the imaging of penetrable solid obstacles embedded in a fluid background medium by way of inverse acoustic scattering. To this end, an asymptotic form for the scattered field caused by a nucleating spherical solid obstacle in the reference fluid medium is derived within the boundary integral equation framework, where the required limiting behaviour of the acceleration field inside the perturbation is established based on solutions of two simplified fluid–solid interaction problems. With this result, the equivalence of acoustic scattering by fluid and solid scatterers in terms of asymptotic behaviour is observed and numerically validated. The direct and adjoint-field topological sensitivity expressions for transient and time-harmonic excitations are obtained accordingly. The utility of the proposed method as a tool for preliminary obstacle reconstruction and bulk modulus characterization is illustrated through numerical examples and an exploratory experimental study. On the basis of adjusted topological sensitivity formulas for fluid reference domains containing pre-existing solid inhomogeneities, an iterative 3D obstacle reconstruction approach is established and its performance with synthetic data is demonstrated.

INDEX AND SOLVABILITY OF UNCOUPLED CIRCUITS: A CHARACTERIZATION WITHOUT RESTRICTIONS ON THEIR PASSIVITY, TOPOLOGY OR CONTROLLING STRUCTURE

In this paper, we extend the topological formulae of Maxwell and Kirchhoff, characterizing the non-singularity of node-admittance and loop-impedance matrices, to mixed problems, that is, to circuits combining admittance and impedance descriptions or, in nonlinear cases, involving both voltage- and current-controlled resistors. By means of this mixed formula we analyze the index of differential-algebraic models of nonlinear uncoupled circuits in a very broad setting, namely, without assumptions on their topology, their passivity or the controlling variables for nonlinear resistors. In particular, our approach allows for a characterization of index two circuits in topologically degenerate settings, which had been so far elusive in the non-passive context. As a byproduct we address the unique solvability of mixed resistive circuits, a problem which also arises in connection to the so-called DC-solvability condition of dynamic circuits. For the sake of brevity, we discuss in less detail how to extend the analysis to problems with mixed descriptions in reactive devices.

Siegel modular varieties and the Eisenstein cohomology of PGL2g+1

We use the twisted topological trace formula developed in Weselmann (Compos. Math. 148:65–120, 2012) to understand liftings from symplectic to general linear groups. We analyse the lift from SP2g to PGL2g+1 over the ground field \({\mathbb{Q}}\) in further detail, and we get a description of the image of this lift of the L2 cohomology of SP2g (which is related to the intersection cohomology of the Shimura variety attached to GSp2g) in terms of the Eisenstein cohomology of the general linear group, whose building constituents are cuspidal representations of Levi groups. This description may be used to understand endoscopic and CAP-representations of the symplectic group.

Phase-Field Simulation of Interface Effect during Grain Nucleation

In this paper, a phase field model based on Ginzburg-Landau theory is used to analyze the topological phenomena during grain growth. The simulation results show that two topological transformations exist during the grain growth—Neighbor Switching and Grain Annihilation; and we have found different kinds of topological events during the disappearance of a grain: direct vanishing of trilateral grain and pentagonal grain, as well as neighbor switching，which are right with classical topological theory and Euler formula. The simulation results are similar with experiments.

The Atiyah-Singer index formula for subelliptic operators on contact manifolds. Part I

The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their symbolic calculus is noncommutative, and is closely related to analysis on the Heisenberg group. For a hypoelliptic differential operator in the Heisenberg calculus on a contact manifold we construct a symbol class in the K-theory of a noncommutative C * -algebra that is associated to the algebra of symbols. There is a canonical map from this analytic K-theory group to the ordinary cohomology of the manifold, which gives a de Rham class to which the Atiyah-Singer formula can be applied. We prove that the index formula holds for these hypoelliptic operators. Our methods derive from Connes' tangent groupoid proof of the index theorem.

The Atiyah-Singer index formula for subelliptic operators on contact manifolds. Part II

The Atiyah-Singer index theorem is a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their symbolic calculus is noncommutative, and involves analysis on the Heisenberg group. For a hypoelliptic differential operator in the Heisenberg calculus on a contact manifold we construct a symbol class in the K-theory of a noncommutative C*-algebra of symbols. There is a canonical map from this analytic K-theory group to the deRham cohomology of the manifold, which gives a class to which the Atiyah-Singer formula can be applied. We prove that the index formula holds for these hypoelliptic operators. Our methods derive from Connes' tangent groupoid proof of the index theorem.

Topographical distance matrices for porous arrays

Thetopographical Wienerindex iscalculated fortwo-dimensional graphs describing porous arrays, including bee honeycomb. For tiling in the plane, we model hexagonal, triangular, and square arrays and compare with topological formulas for the Wiener index derived from the distance matrix. The normalized Wiener indices of C4 ,T 13, and O(4), for hexagonal, triangular, and square arrays are 0.993, 0.995, and 0.985,respectively,indicatingthatthearrayshavesmallerbondlengthsnearthecenter of the array, since these contribute more to the Wiener index. The normalized Perron root (the first eigenvalue, λ1), calculated from distance/distance matrices describes an order parameter, φ = λ1/n, where φ = 1 for a linear graph and n is the order of the matrix. This parameter correlates with the convexity of the tessellations. The distri- butions of the normalized distances for nearest neighbor coordinates are determined from the porous arrays. The distributions range from normal to skewed to multimodal depending on the array. These results introduce some new calculations for 2D graphs of porous arrays.

The Generalized Topological Formula for Transfer Function Generation by Two-Graph Tree Enumeration

In this paper the topological approach for transfer function generation by two-graph tree enumeration is presented. The generalized topological formula with homogeneous parameters is proved for all the circuit functions, and a simple representation of the four types of controlled sources by admittances is proposed, that allows a uniform treatment of the entire circuit in terms of admittances. To implement our procedure, the rules to automatically generate the two graphs and to enumerate the common spanning trees are presented. Some simplifications in the circuit and in the two graph structure before tree generation and a graph representation on levels, improve the efficiency of the tree enumeration procedure. An illustrative example is presented and a reduced symbolic form of the network function is obtained by some simplifications after generation.