Topological Theorem Research Articles

Convexity in partial cubes: The hull number

We prove that the combinatorial optimization problem of determining the hull number of a partial cube is NP-complete. This makes partial cubes the minimal graph class for which NP-completeness of this problem is known and improves earlier results in the literature.On the other hand we provide a polynomial-time algorithm to determine the hull number of planar partial cube quadrangulations.Instances of the hull number problem for partial cubes described include poset dimension and hitting sets for interiors of curves in the plane.To obtain the above results, we investigate convexity in partial cubes and obtain a new characterization of these graphs in terms of their lattice of convex subgraphs. This refines a theorem of Handa. Furthermore we provide a topological representation theorem for planar partial cubes, generalizing a result of Fukuda and Handa about tope graphs of rank 3 oriented matroids.

Decomposition of Pseudo-effect Algebras and the Hammer–Sobczyk Theorem

We prove an algebraic and a topological decomposition theorem for complete pseudo-D-lattices (i.e. lattice-ordered pseudo-effect algebras). As a consequence, we obtain a Hammer–Sobczyk type decomposition theorem for group-valued modular measures defined on pseudo-D-lattices and compactness of the range of every \(\mathbb {R}^{n} \)-valued σ-additive modular measure on a σ-complete pseudo-D-lattice.

Raychaudhuri equation and singularity theorems in Finsler spacetimes

The Raychaudhuri equation and its consequences for chronality are studied in the context of Finsler spacetimes. It is proved that the notable singularity theorems of Lorentzian geometry extend to the Finslerian domain. Indeed, so do the theorems by Hawking, Penrose, Hawking and Penrose, Geroch, Gannon, Tipler and Kriele, and also the Topological Censorship theorem and so on. It is argued that the notable results in causality theory connected to achronal sets, future sets, domains of dependence, limit curve theorems, length functional, Lorentzian distance and geodesic connectedness, extend to the Finslerian domain. Results concerning the spacetime asymptotic structure, horizons differentiability and conformal transformations are also included.

A novel plant-station and inter-plant-station topology analysis method

Summary In view of the requirement for instantaneity on power system topology analysis, a novel power system topology analysis method is proposed in this paper. For the plant-station topology method, the nodes that represent the circuit breakers are numbered according to the graph theory. On this basis, combining the main wiring characteristics and the switch status information, the plant-station topology is analyzed using the topological base theorem. When the switch status changes, the affected part of topology is modified using the radius search method, so that reforming of the topology is avoided and fast update of topology is guaranteed. When dealing with inter-plant-station topology, the incidence matrix notation method is used to judge the connectivity of the graph and mark the tree branches and link branches in the network, thus conducting static topology analysis of the power network. And then, when the network topology changes, the property of the changing branch is determined with the help of the loop matrix. On this basis, the local network topology and the loop matrix are updated by means of the broken circle method and the radius search method. Calculation examples verify the advantage of the proposed methods over traditional methods in calculation speed. With the proposed topology analysis methods, fast tracking of the switch status can be realized. Copyright © 2015 John Wiley & Sons, Ltd.

A topological theorem and correlations, within the context of stochastic evolution

A topological theorem, that proves that any d-dimensional lattice is equivalent to a one-dimensional one, allows to write the evolution equations as a function of only one spatial coordinate. Stochastic and continuum deterministic evolution equations, are derived from a set of discrete stochastic evolution equations. The evolution equations of the dynamical variables and correlations are obtained for processes that evolve non-Markovianly. Some relatively simple examples are given in order to illustrate the procedures.

Cofinality spectrum theorems in model theory, set theory, and general topology

We connect and solve two long-standing open problems in quite different areas: the model-theoretic question of whether S O P 2 SOP_2 is maximal in Keisler’s order, and the question from general topology/set theory of whether p = t \mathfrak {p} = \mathfrak {t} , the oldest problem on cardinal invariants of the continuum. We do so by showing these problems can be translated into instances of a more fundamental problem which we state and solve completely, using model-theoretic methods.

Optimal bounds for the colored Tverberg problem

We prove a “Tverberg type” multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of B´ar´any & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.

Topological index theorem on the lattice through the spectral flow of staggered fermions

We investigate numerically the spectral flow introduced by Adams for the staggered Dirac operator on realistic (quenched) gauge configurations. We obtain clear numerical evidence that the definition works as expected: there is a clear separation between crossings near and far away from the origin, and the topological charge defined through the crossings near the origin agrees, for most configurations, with the one defined through the near-zero modes of large taste-singlet chirality of the staggered Dirac operator. The crossings are much closer to the origin if we improve the Dirac operator used in the definition, and they move towards the origin as we decrease the lattice spacing.

Obstacles for splitting multidimensional necklaces

The well-known “necklace splitting theorem” of Alon (1987) asserts that every k k -colored necklace can be fairly split into q q parts using at most t t cuts, provided k ( q − 1 ) ≤ t k(q-1)\leq t . In a joint paper with Alon et al. (2009) we studied a kind of opposite question. Namely, for which values of k k and t t is there a measurable k k -coloring of the real line such that no interval has a fair splitting into 2 2 parts with at most t t cuts? We proved that k > t + 2 k>t+2 is a sufficient condition (while k > t k>t is necessary). We generalize this result to Euclidean spaces of arbitrary dimension d d , and to arbitrary number of parts q q . We prove that if k ( q − 1 ) > t + d + q − 1 k(q-1)>t+d+q-1 , then there is a measurable k k -coloring of R d \mathbb {R}^d such that no axis-aligned cube has a fair q q -splitting using at most t t axis-aligned hyperplane cuts. Our bound is of the same order as a necessary condition k ( q − 1 ) > t k(q-1)>t implied by Alon’s 1987 work. Moreover, for d = 1 , q = 2 d=1,q=2 we get exactly the result of the 2009 work. Additionally, we prove that if a stronger inequality k ( q − 1 ) > d t + d + q − 1 k(q-1)>dt+d+q-1 is satisfied, then there is a measurable k k -coloring of R d \mathbb {R}^d with no axis-aligned cube having a fair q q -splitting using at most t t arbitrary hyperplane cuts. The proofs are based on the topological Baire category theorem and use algebraic independence over suitably chosen fields.

Plasmonic metamaterials for ultra-sensitive sensing: topological darkness

Extinction of the reflection of the p-polarized wave by the surface of a semi-infinite transparent medium is a well-known phenomenon exploited in Brewster angle microscopy. For absorbing media, however, described by a complex refractive index of refraction, the Brewster extinction condition does not exist anymore since the real and imaginary parts of the reflection coefficient cannot be canceled simultaneously. The exact zero reflection condition can be restored if a thin film of suitable complex index is deposited on top of a semi-infinite substrate. In such case, whatever the accuracy of the experimental setup, the existence of a true perfect cancelation of the reflection is guaranteed by the Jordan topological theorem. In the immediate vicinity of the zero reflection point, the phase of the reflected wave undergoes an extremely steep variation that enables ultra-sensitive sensing. The existence of a topological darkness condition and its application to the detection of dilute biological molecules were first demonstrated on plasmonic meta-surfaces made of two-dimensional arrays of gold nanopillars deposited on a glass substrate by electron beam lithography. The effective complex refractive index of the homogenized equivalent film indeed satisfies the topological darkness condition. In this paper, we show that several plasmonic systems prepared by the physical chemistry methods of colloids exhibit topological darkness and may provide good substrates for ultra-sensitive sensing at lower cost.

Analysis of gauge-equivariant complexes and a topological index theorem for gauge-invariant families

We continue our study of gauge equivariant K-theory. We thus study the analysis of complexes endowed with the action of a family of compact Lie groups and their index in gauge equivariant K-theory. We introduce various index functions, including an axiomatic one, and show that all index functions coincide. As an application, we prove a topological index theorem for a family D = (D b ) b∈B of gauge-invariant elliptic operators on a G-bundle X → B, where G → B is a locally trivial bundle of compact groups, with typical fiber G. More precisely, one of our main results states that a-ind(D) = t-ind(D) ∈ K 0 (X), that is, the equality of the analytic index and of the topological index of the family D in the gauge-equivariant K-theory groups of X. The analytic index ind a (D) is defined using analytic properties of the family D and is essentially the difference of the kernel and cokernel K G -classes of D. The topological index is defined purely in terms of the principal symbol of D.

Pattern equivariant cohomology and theorems of Kesten and Oren

In 1966, Harry Kesten settled the Erdős–Szüsz conjecture on the local discrepancy of irrational rotations. His proof made heavy use of continued fractions and Diophantine analysis. In this paper, we give a purely topological proof Kesten's theorem (and Oren's generalization of it) using the pattern equivariant cohomology of aperiodic tiling spaces.

Spatiality of countably presentable locales (proved with the Baire category theorem)

The first part of the paper presents a generalization of the well-known Baire category theorem. The generalization consists in replacing the dense open sets of the original formulation by dense UCO sets, where UCO means union of closed and open. This topological theorem is exactly what is needed to prove in the second part of the paper the locale-theoretic result that locales whose frame of opens has a countable presentation (countably many generators and countably many relations) are spatial. This spatiality theorem does not require choice.

A decomposition theorem for Herman maps

In 1980s, Thurston established a topological characterization theorem for postcritically finite rational maps. In this paper, a decomposition theorem for a class of postcritically infinite branched covering termed Herman map is developed. It's shown that every Herman map can be decomposed along a stable multicurve into finitely many Siegel maps and Thurston maps, such that the combinations and rational realizations of these resulting maps essentially dominate the original one. This result is motivated by a non-expanding version of McMullen's problem, and Thurston's theory on characterization of rational maps. It enables us to prove a Thurston-type theorem for rational maps with Herman rings.

-INVARIANT AND MODULAR FORMS

We show that theAtiyah–Patodi–Singer reduced η-invariant of the twisted Dirac operator on a closed 4m − 1-dimensional spin manifold, with the twisted bundle being the Witten bundle appearing in the theory of elliptic genus, is a meromorphic modular form of weight 2m up to an integral q-series. We prove this result by combining our construction of certain modular characteristic forms associated to a generalized Witten bundle on spin c -manifolds with a deep topological theorem due to Hopkins.

A topological theorem and its application to the superlinear equations including impulses for the existence of periodic solutions

In this paper,we prove the existence and multiplicity of 2π periodic solutions for a class of nonconservative super-linear equations including periodic impulses with one non-homeomorphic impulse in one period.By applying the method of the phase plane analysis,we consider the fixed points problem for the Poincare mapping of the original equation after introducing the angle variable for impulses.Note that the Poincare mapping is not homeomorphic due to the impulses,we give a new topological fixed point theorem to obtain the fixed points for the mapping instead of the Poincare-Birkhoff fixed point theorem.

Topological Blount's theorem of odd-parity superconductors

From the group theoretical ground, the Blount's theorem prohibits the existence of line nodes for odd-parity superconductors (SCs) in the presence of spin-orbit coupling (SOC). We study the topological stability of line nodes under inversion symmetry. From the topological point of view, we renovate the stability condition of line nodes, in which we not only generalize the original statement, but also establish the relation to zero-energy surface flat dispersions. The topological instability of line nodes in odd-parity SCs implies not the absence of bulk line nodes but the disappearance of the corresponding zero-energy surface flat dispersions due to surface Rashba SOC, which gives an experimental means to distinguish line nodes in odd-parity SCs from those in other SCs.

Tverberg plus constraints

Many of the strengthenings and extensions of the topological Tverberg theorem can be derived with surprising ease directly from the original theorem: For this, we introduce a proof technique that combines a concept of ‘Tverberg unavoidable subcomplexes’ with the observation that Tverberg points that equalize the distance from such a subcomplex can be obtained from maps to an extended target space. Thus, we obtain simple proofs for many variants of the topological Tverberg theorem, such as the colored Tverberg theorem of Živaljević and Vrećica (‘The colored Tverberg's problem and complexes of injective functions’, J. Combin. Theory, Ser. A 61 (1992) 309–318). We also get a new strengthened version of the generalized van Kampen–Flores theorem by Sarkaria (‘A generalized van Kampen–Flores theorem’, Proc. Amer. Math. Soc. 11 (1991) 559–565) and Volovikov (‘On the van Kampen–Flores theorem’, Math. Notes ( 5 ) 59 (1996) 477–481), an affine version of their ‘ j-wise disjoint’ Tverberg theorem, and a topological version of Soberón's result (‘Equal coefficients and tolerance in coloured Tverberg partitions’, Proceeding of the 29th Annual Symposium on Computational Geometry (SoCG) (ACM, Rio de Janeiro, 2013), 91–96) on Tverberg points with equal barycentric coordinates.

DIFFERENTIAL INCLUSIONS, NON-ABSOLUTELY CONVERGENT INTEGRALS AND THE FIRST THEOREM OF COMPLEX ANALYSIS

In the theory of complex valued functions of a complex variable arguably the first striking theorem is that pointwise differentiability implies C∞ regularity. As mentioned in Ahlfors [Ah 78] there have been a number of studies [Po 61], [Pl 59] proving this theorem without use of complex integration but at the cost of considerably more complexity. In this note we will use the theory of non-absolutely convergent integrals to firstly give a very short proof of this result without complex integration and secondly (in combination with some elements of the theory of elliptic regularity) provide a far reaching generalization. One of the first and most striking theorems about the analysis of complex valued functions of a complex variable is that merely from considering the class of pointwise complex differentiable functions on an open set we instantly find ourself in the category of C∞ functions. Theorem 1. Given open set Ω ⊂ C. Suppose f : Ω → C is a complex differentiable at every point. Then f is C∞ on Ω. Typically Theorem 1 is proved via the method of complex integration. The first step is to prove that the integral of a differentiable function over the boundary of a rectangle inside a ball is zero, this was first proved by Goursat [Go 01]. The existence of an anti-derivative is then concluded, Cauchy’s integral formula follows and it is shown that you can differentiate through the integral of Cauchy’s integral formula infinitely many times and hence the function is C∞. On the first paragraph of page 101 of Ahlfors’s standard text [Ah 78], he writes that many important properties of analytic functions are difficult to prove without use of complex integration. Ahlfors states that only recently1 it has been possible to prove continuity of the gradient (or the existence of higher gradients) without the use of complex integration. He refers to articles of Plunkett [Pl 59], and Porcelli and Connell [Po 61] both of which rely on a topological theorem of Whyburn [Wh 58]. Ahlfors notes that both these proofs are much more complicated than the original proof. It turns out that generalizations of Goursat’s theorem have a long history and one line of generalization provides an alternative proof of the theorem that in essence does not require complex integration. This line of research was started by Montel [Mo 07] and further developed by Looman [Lo 23] and Menchoff [Me 36] (their theorem receives a very clear exposition in Saks [Sa 37]) and later by Tolstov [To 42]. Although not explicitly stated in these works the method of proof was essentially to construct a Denjoy type integral to integrate the divergence of a differentiable vector field [Pr 12]. The explicit application of the theory of non-absolutely convergent integrals appears to have first been made in [Ju-No 90] by Jurkat and Nonnenmacher, they used a kind of Perron integral in the plane to prove a generalization of Goursat’s theorem due to Besicovitch [Be 31]. The purpose of this note is firstly to use the theory of non-absolutely convergent integrals to provide the shortest proof of Theorem 1, the proof we provide is also independent of the theory of complex line integrals. Secondly by rephrasing Theorem 1 in terms of differential inclusions 2010 Mathematics Subject Classification. 30A99,26A39,35J47.

On paranorm BVσ I-convergent sequence spaces defined by an Orlicz function

In this article we introduce and study \(_{0}BV^I_\sigma(M,p)\), \(BV^I_{\sigma}(M,p)\) and \(_{\infty}BV^I_{\sigma}(M,p)\) sequence spaces where \(p=(p_{k})\) is the sequence of strictly positive real numbers with the help of \(BV_\sigma\) space [see [23]] and an Orlicz function \(M\). We study some topological and algebraic properties and decompostion theorem. Further we prove some inclusion relations related to these new spaces.