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Existential graphs on the plane constitute a two-dimensional representation of classical logic, in which a Jordan curve stands for the negation of its inside. In this paper we propose a program to develop existential Alpha graphs, which correspond to propositional logic, on various surfaces. The geometry of each manifold determines the possible Jordan curves on it, leading to diverse interpretations of negation. This may open a way for appointing a \natural" logic to any surface. [ABSTRACT FROM AUTHOR]

A slow manifold is a low-dimensional invariant manifold to which trajectories nearby are rapidly attracted on the way to the equilibrium point. The exact computation of the slow manifold simplifies the model without sacrificing accuracy on the slow time scales of the system. The Maas-Pope intrinsic low-dimensional manifold (ILDM) [Combust. Flame 88, 239 (1992)] is frequently used as an approximation to the slow manifold. This approximation is based on a linearized analysis of the differential equations and thus neglects curvature. We present here an efficient way to calculate an approximation equivalent to the ILDM. Our method, called functional equation truncation (FET), first develops a hierarchy of functional equations involving higher derivatives which can then be truncated at second-derivative terms to explicitly neglect the curvature. We prove that the ILDM and FET-approximated (FETA) manifolds are identical for the one-dimensional slow manifold of any planar system. In higher-dimensional spaces, the ILDM and FETA manifolds agree to numerical accuracy almost everywhere. Solution of the FET equations is, however, expected to generally be faster than the ILDM method. [ABSTRACT FROM AUTHOR]

Characterizing the statistically significant centrality of nodes is one of the main goals of multiplex networks. However, current centrality measures for node rankings focus only on either random walks or on the topological structure of the network. A pressing challenge is how to measure centrality of nodes in multiplex networks, depending both on network topology and on the biased types of random walks, such as the biased walks dealing with the properties of each node separately at each layer, or the biased walks considering instead one or even more intrinsically multiplex properties of the arrival node. In the paper, considering these two aspects, we propose a mathematical framework based on topologically biased random walk, called topologically biased multiplex PageRank, which allows to calculate centrality and accordingly rank nodes in multiplex networks. In particular, depending on the nature of biases and the interaction of nodes between different layers, we distinguish additive, multiplicative and combined cases of topologically biased multiplex PageRank. Each case by tuning the bias parameters reflects how the centrality ranking of a node in one layer affects the ranking its replica can gain in the other layers, and captures the extent to which the walkers preferentially visit hubs or poorly connected nodes. Experiments on two real-world multiplex networks show that the topologically biased multiplex PageRank outperforms both its corresponding unbiased case and the current ranking methods, and it can efficiently capture the significantly top-ranked nodes in multiplex networks by means of a proper tuning of the biases in the walks. [ABSTRACT FROM AUTHOR]

It is shown that the stereographic coordinates on the (1 + 3)-dimensional de Sitter spacetime induce artificial singularities without physical motivation in the parametric equations of all the null or time-like geodesics on this manifold. [ABSTRACT FROM AUTHOR]

Abstract: A new method for the design of inhomogeneous materials and block structures with complex physicomechanical properties is stated. This approach based on the method of a block element can conveniently complement different numerical approaches used for these purposes and reveal the boundary problem solution properties, which either are complicated or cannot be analyzed by other methods. It is proven that a crucial role in the construction of this new approach is played by packed block elements as topological manifolds with a boundary, which can form similar new topological manifolds as a result of conjugation. It is demonstrated how homeomorphisms, which represent the mappings of topological manifolds of block elements onto real number spaces and further enable the construction of quotient topologies, which “glue” together both block elements and boundary problem solution fragments on block elements as carriers, are constructed for these purposes. [ABSTRACT FROM AUTHOR]

This article covers some technical aspects about the product topology which are usually not given much of a thought in mathematics and standard literature like [7] and [6], not even by Bourbaki in [4]. Let {Ti}i∈I be a family of topological spaces. The prebasis of the product space T = ∏i∈I Ti is defined in [5] as the set of all π−1i(V) with i ∈ I and V open in Ti. Here it is shown that the basis generated by this prebasis consists exactly of the sets ∏i∈I Vi with Vi open in Ti and for all but finitely many i ∈ I holds Vi = Ti. Given I = {a} we have T ≅ Ta, given I = {a, b} with a≠ b we have T ≅ Ta ×Tb. Given another family of topological spaces {Si}i∈I such that Si ≅ Ti for all i ∈ I, we have S = ∏i∈I Si ≅ T. If instead Si is a subspace of Ti for each i ∈ I, then S is a subspace of T. These results are obvious for mathematicians, but formally proven here by means of the Mizar system [3], [2]. [ABSTRACT FROM AUTHOR]

We study two inverse problems on a globally hyperbolic Lorentzian manifold (M, g). The problems are:Passive observations in spacetime: consider observations in an open set V⊂M. The light observation set corresponding to a point source at q∈M is the intersection of V and the light-cone emanating from the point q.Let W⊂M be an unknown open, relatively compact set. We show that under natural causality conditions, the family of light observation sets corresponding to point sources at points q∈W determine uniquely the conformal type of W.Active measurements in spacetime: we develop a new method for inverse problems for non-linear hyperbolic equations that utilizes the non-linearity as a tool. This enables us to solve inverse problems for non-linear equations for which the corresponding problems for linear equations are still unsolved. To illustrate this method, we solve an inverse problem for semilinear wave equations with quadratic non-linearities. We assume that we are given the neighborhood V of the time-like path μ and the source-to-solution operator that maps the source supported on V to the restriction of the solution of the wave equation to V. When M is 4-dimensional, we show that these data determine the topological, differentiable, and conformal structures of the spacetime in the maximal set where waves can propagate from μ and return back to μ. [ABSTRACT FROM AUTHOR]

The first Zagreb eccentricity index E1(G) of a graph G is defined as the sum of squares of the eccentricities of the vertices. In this paper some bounds for the first Zagreb eccentricity index and first Zagreb degree eccentricity index are computed. [ABSTRACT FROM AUTHOR]

We construct examples of principal groupoids that have weak containment but are not amenable, thus answering questions by Claire Anantharaman-Delaroche and Rufus Willett. [ABSTRACT FROM AUTHOR]

Abstract: We consider AKSZ constructions of BV actions for closed topological membranes, and their dimensional reductions to topological string sigma‐models. Two inequivalent AKSZ constructions for topological membranes on G2‐manifolds are proposed, in each of which the two existing topological membrane theories appear as different gauge fixed versions. Their dimensional reductions give new AKSZ constructions for the topological A‐model, which on further dimensional reduction gives an AKSZ formulation of supersymmetric quantum mechanics. We show that the two AKSZ membrane models originate through worldvolume dimensional reduction of a single AKSZ threebrane theory, which gives the standard 2‐Courant bracket as the underlying derived bracket. Double dimensional reduction of the twisted topological threebrane theory on a circle yields the standard Courant sigma‐model for string theory with NS–NS flux. [ABSTRACT FROM AUTHOR]

In the paper, we examine local aspects of chaos for self-functions on topological manifolds. For this purpose, issues related to focal entropy points and homoclinic points are used. We consider also distortion of dynamical systems which is a theoretical analogue to wave interference. [ABSTRACT FROM AUTHOR]

The aim of this work is to study translation surfaces in a Minkowski 3-space with density. Translation surfaces in are defined as the two generating curves which lie in orthogonal planes. They have actually three different possible parametrizations according to the intersecting straight line of the two planes. We completely classify all translation surfaces with zero weighted mean curvature in with density by solving the second-order non-linear ODE with some smooth functions. [ABSTRACT FROM AUTHOR]

Inspired by the analogous result in the algebraic setting ( Theorem 1 ) we show ( Theorem 2 ) that the product M × R P n of a closed and orientable topological manifold M with the n -dimensional real projective space cannot be embedded into R P m + n + 1 for all even n > m . [ABSTRACT FROM AUTHOR]

Let ( V , W ; F ) be a weakly reducible, unstabilized, Heegaard splitting of genus at least three in an orientable, irreducible 3-manifold M . Then M o d ( M , F ) naturally acts on the disk complex D ( F ) as a group action. In this article, we prove if F is topologically minimal and its topological index is two, then the orbit of any element of D ( F ) for this group action consists of infinitely many elements. Moreover, we prove there are at most two elements of D ( F ) whose orbits are finite if the genus of F is three. [ABSTRACT FROM AUTHOR]

Kodaira fibred surfaces are remarkable examples of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic of the use of topological methods in the study of moduli spaces of surfaces and higher dimensional complex algebraic varieties, and their compactifications. Our tour through algebraic surfaces and their moduli (with results valid also for higher dimensional varieties) deals with fibrations, questions on monodromy and factorizations in the mapping class group, old and new results on Variation of Hodge Structures, especially a recent answer given (in joint work with Dettweiler) to a long standing question posed by Fujita. In the landscape of our tour, Galois coverings, deformations and rigid manifolds (there are by the way rigid Kodaira fibrations), projective classifying spaces, the action of the absolute Galois group on moduli spaces, stand also in the forefront. These questions lead to interesting algebraic surfaces, for instance remarkable surfaces constructed from VHS, surfaces isogenous to a product with automorphisms acting trivially on cohomology, hypersurfaces in Bagnera-de Franchis varieties, Inoue-type surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we present a new piecewise parametric approach to obtain smooth surfaces from any given orientable 2-manifold polynomial control mesh. Our approach provides C 2 continuous Bi-Cubic Bézier patches that are guaranteed to be stitched with G 1 continuity regardless of the underlying mesh topology. A high level summary of the approach can be given as a two-step process: (1) Starting from an orientable 2-manifold mesh we apply vertex-insertion, which is remeshing algorithm of Catmull–Clark subdivision, to break that surface into a set of all quadrilateral patches. (2) for each such quadrilateral patch, we construct a Bi-Cubic Bézier patch, where the positions of the 16 control points are obtained from the limit surface of the Doo–Sabin subdivision scheme. This approach, when combined with topological modeling, provides a framework for real-time interactive modeling of smooth manifold meshes with arbitrary topology. Based on this approach, we have developed an interactive modeling system that allows users to work with smooth surfaces in real-time by opening or closing holes, creating handles, and combining and disconnecting surfaces. The paper has three main contributions: (1) We guarantee that our method will produce only smooth manifold meshes with C 2 continuous Bi-Cubic Bézier patches. (2) We guarantee visual smoothness for any arbitrary topological structure by providing geometric G 1 continuity in boundaries of patches. (3) We demonstrate that smooth surfaces with arbitrary topology can be created interactively in real-time using parametric patches. [ABSTRACT FROM AUTHOR]

The second author and Hara introduced the notion of an essential tribranched surface that is a generalisation of the notion of an essential embedded surface in a 3-manifold. We show that any 3-manifold for which the fundamental group has at least rank four admits an essential tribranched surface. [ABSTRACT FROM AUTHOR]

Given a finite set of points in Rn and a radius parameter, we study the Čech, Delaunay-Čech, Delaunay (or alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four complexes are simple-homotopy equivalent by a sequence of simplicial collapses, which are explicitly described by a single discrete gradient field. [ABSTRACT FROM AUTHOR]

In this paper, some new results, concerned with the geodesic convex hull and geodesic convex combination, are given on Hadamard manifolds. An S-KKM theorem on a Hadamard manifold is also given in order to generalize the KKM theorem. As applications, a Fan-Browder-type fixed point theorem and a fixed point theorem for the a new mapping class are proved on Hadamard manifolds. [ABSTRACT FROM AUTHOR]

We show that for any cohomogeneity-one continuous action of a compact connected Lie group G on a closed topological manifold the equivariant cohomology equipped with its canonical H ⁎ ( B G ) -module structure is Cohen–Macaulay. The proof relies on the structure theorem for these actions recently obtained by Galaz-García and Zarei. We generalize in this way our previous result concerning smooth actions. [ABSTRACT FROM AUTHOR]

Using the [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a neutrosophic crisp manifold as a new topological structure of manifold via neutrosophic crisp set. Therefore, we study some new topological concepts and some metric distances on a neutrosophic crisp manifold. [ABSTRACT FROM AUTHOR]

The paper provides a review of science, technology, engineering, and mathematics (STEM) education policy research completed thus far and continues by calling for STEM analysis that understands policy processes as governance rather than government. Instead of a fixed and easily locatable governmental structure by which one can analyze the nature of governing, governance is comprised of an always-shifting topological manifold. Using such theory, STEM education policy must be analyzed for its context and text. Using such analysis could suggest that the current policy directives in STEM education, including those laid out in the Every Student Succeeds Act of 2015, are more related to an increase in the application of market logics to education rather than supportive of specific actions related to STEM teaching and learning. [ABSTRACT FROM AUTHOR]

The object of the present paper is to studyϕ-pseudo symmetric andϕ-pseudo Ricci symmetric LP-Sasakian manifolds with respect to Levi-Civita connections and quarter-symmetric non-metric connections. We obtain a necessary and sufficient condition for a ϕ-pseudo symmetric LP-Sasakian manifold with respect to a quarter symmetric non-metric connection to beϕ-pseudo symmetric LP-Sasakian manifold with respect to a Levi-Civita connection. [ABSTRACT FROM AUTHOR]

Two new actinide metal-organic frameworks were constructed by using a tripodal flexible ligand tris (2-carboxyethyl) isocyanurate (H 3 tci) under hydrothermal condition. The combination of H 3 tci and uranyl nitrate hexahydrate in aqueous solution leads to the isolation of [(UO 2 ) 2 (H 2 O) 4 ] 0.5 (tci) 2 (UO 2 ) 4 (OH) 4 ·18H 2 O ( 1 ), which contains two distinct UO 2 2+ coordination environments. Four uranyl cations, linked through μ 3 -OH respectively, result in the edge-sharing ribbons. Then, the layer structure is constructed by U-O clusters linked through other eight-coordinated uranyl unions, giving rise to a porous structure in the space. Topological analysis reveals that complex 1 belongs to a (4, 8)-connected net with a schläfli symbol of (3 4. 2 6. 3) 2 (3 4. 4 6. 5 6. 6 8. 7 3. 8). Th 3 (tci) 2 O 2 (OH) 2 (H 2 O) 3 ·12H 2 O ( 2 ) generated by the reaction of H 3 tci and thorium nitrate tetrahydrate, possesses nine-fold coodinated Th(IV) centers with a monocapped square antiprismatic geometry. The hexamers “Th 6 O 4 (OH) 4 ” motifs are connected together by the carboxylate groups, showing a three-dimensional structures. Complex 2 takes on an 8-connected architecture and the point symbol is (4 24. 6 4 ). [ABSTRACT FROM AUTHOR]

Discovery of exotic phases of matter from the topologically non-trivial systems not only makes the research on topological materials more interesting but also enriches our understanding of the fascinating physics of such materials. Pb0.6Sn0.4Te was recently shown to be a topological crystalline insulator. Here, we show that by forming a mesoscopic point-contact using a normal non-superconducting elemental metal on the surface of Pb0.6Sn0.4Te, a superconducting phase is created locally in a confined region under the point-contact. This happens when the bulk of the sample remains to be nonsuperconducting, and the superconducting phase emerges as a nano-droplet under the point-contact. The superconducting phase shows a high transition temperature Tc that varies for different pointcontacts and falls in a range between 3.7K and 6.5 K. Therefore, this Letter presents the discovery of a superconducting phase on the surface of a topological crystalline insulator, and the discovery is expected to shed light on the mechanism of induced superconductivity in topologically non-trivial systems in general. [ABSTRACT FROM AUTHOR]

Reference metrics are used to define the differential structure on multicube representations of manifolds, i.e., they provide a simple and practical way to define what it means globally for tensor fields and their derivatives to be continuous. This paper introduces a general procedure for constructing reference metrics automatically on multicube representations of manifolds with arbitrary topologies. The method is tested here by constructing reference metrics for compact, orientable two-dimensional manifolds with genera between zero and five. These metrics are shown to satisfy the Gauss–Bonnet identity numerically to the level of truncation error (which converges toward zero as the numerical resolution is increased). These reference metrics can be made smoother and more uniform by evolving them with Ricci flow. This smoothing procedure is tested on the two-dimensional reference metrics constructed here. These smoothing evolutions (using volume-normalized Ricci flow with DeTurck gauge fixing) are all shown to produce reference metrics with constant scalar curvatures (at the level of numerical truncation error). [ABSTRACT FROM AUTHOR]

By proving that several new complexes of embedded disks are highly connected, we obtain several new homological stability results. Our main result is homological stability for topological chiral homology on an open manifold with coefficients in certain partial framed E n -algebras. Using this, we prove a special case of a conjecture of Vakil and Wood on homological stability for complements of closures of particular strata in the symmetric powers of an open manifold and we prove that the bounded symmetric powers of closed manifolds satisfy homological stability rationally. [ABSTRACT FROM AUTHOR]

Let X be an (n + 1)-dimensional manifold, ∆ be a one-dimensional foliation on X, and p : X → X/∆ be a quotient map. We will say that a leaf ω of ∆ is special whenever the space of leaves X/∆ is not Hausdorff at ω. We present necessary and sufficient conditions for the map p : X → X/∆ to be a locally trivial fibration under assumptions that all leaves of ∆ are non-compact and the family of all special leaves of ∆ is locally finite. [ABSTRACT FROM AUTHOR]

Let Y be a Gromov-Hausdorff limit of complete Riemannian n-manifolds with Ricci curvature bounded from below. A point in Y is called k-regular, if its tangent is unique and is isometric to a k-dimensional Euclidean space. By Cheeger-Colding and Colding-Naber, there is k > 0 such that the set of all k-regular point R has a full renormalized measure. An open problem is if R = Ø for all l < k? The main result in this paper asserts that if R ≠ Ø, then Y is a one-dimensional topological manifold. Our result improves Honda's result that under the assumption that 1 ⩽ dim( Y) < 2. [ABSTRACT FROM AUTHOR]

This paper presents an interactive technique to optimise the shape and topology of three dimensional trusses. The presented study uses evolutionary solver and finite element analysis to evaluate the fitness function. The problem considered for this optimisation is a long span steel truss that serves as the main supporting system of a modern irrigation procedure (centre pivot irrigation system (CPIS)). The problem has been encoded in a graphical programming interface for achieving the discrete optimisation of this truss. The design space and node connectivity has been constrained so that the evolving truss topology is simplified. The objective function is to minimise the weight of truss while satisfying the strength and serviceability criteria. The routine generates several truss geometries within the discrete design space and the fitness of each shape is evaluated by integrating the routine with structural analysis software. Evolutionary solver has been employed to iterate the variables logically so that the optimum solution is obtained. Consequently, a simple cost effective solution for CPIS has been presented that can be easily fabricated in local environment. [ABSTRACT FROM AUTHOR]

Let v be a continuous flow with arbitrary singularities on a compact surface. Then we show that if v is non-wandering, then v is topologically equivalent to a C∞ flow such that Per(v) is open, there are no exceptional orbits, and that P... Sing(v) = {x ∈ M | ω(x) υ α(x) ⊆ Sing(v)}, where P is the union of non-closed proper orbits and ... is the disjoint union symbol. Moreover, v is non-wandering if and only if ... ⊇ M - Sing(v), where LD is the union of locally dense orbits and Ā is the closure of a subset A ⊆ M. On the other hand, v is topologically transitive if and only if v is non-wandering such that int(Per(v) ... Sing(v)) = ø and M - (P... Sing(v)) is connected, where intA is the interior of a subset A ⊆ M. In addition, we construct a smooth flow on T² with ... = ... = T². [ABSTRACT FROM AUTHOR]

The Hubbard and spinless fermion chains are paradigms of strongly correlated systems, very well understood using the Bethe ansatz, density matrix renormalization group (DMRG), and field theory/renormalization group (RG) methods. They have been applied to one-dimensional materials and have provided important insights for understanding higher-dimensional cases. Recently, an interacting fermion model has been introduced, with possible applications to topological materials. It has a single Majorana fermion operator on each lattice site and interactions with the shortest possible range that involve four sites. We present a thorough analysis of the phase diagram of this model in one dimension using field-theory/RG and DMRG methods. It includes a gapped supersymmetric region and a gapless phase with coexisting Luttinger liquid and Ising degrees of freedom. In addition to a first-order transition, three critical points occur: tricritical Ising, Lifshitz, and a generalization of the commensurate-incommensurate transition. We also survey various gapped phases of the system that arise when the translation symmetry is broken by dimerization and find both trivial and topological phases with 0, 1, and 2 Majorana zero modes bound to the edges of the chain with open boundary conditions. [ABSTRACT FROM AUTHOR]

In this article, we prove the topological minimality of unions of several almost orthogonal planes of arbitrary dimensions. A particular case was proved in [13], where we proved the Almgren minimality (which is a weaker property than the topological minimality) of the union of two almost orthogonal two-dimensional planes. On the one hand, the topological minimality is almost always proved by variations of calibration methods, but in this article, we give a continuous family topological minimal sets, hence calibrations cannot apply. The advantage of a set being topological minimal (compared with Almgren minimal) is that its product with Rn stays topological minimal. This leads also to finding minimal sets which are unions of non transversal (hence far from almost orthogonal) planes. Notice that the regularity for higher dimensional minimal sets is much less clear than those of dimension 2, hence more efforts are needed for the present higher dimensional case. [ABSTRACT FROM AUTHOR]

We study the set vol(M,G) of volumes of all representations ρ: π1M → G, where M is a closed oriented 3-manifold and G is either Iso+H3 or IsoeSL2(ℝ). By various methods, including relations between the volume of representations and the Chern-Simons invariants of flat connections, and recent results of surfaces in 3-manifolds, we prove that any 3-manifold M with positive Gromov simplicial volume has a finite cover M with vol (M, Iso+H3) ≠ {0}, and that any non-geometric 3-manifold M containing at least one Seifert piece has a finite cover M with vol (M, IsoeSL2(ℝ)) ≠ {0}. We also find 3-manifolds M with positive simplicial volume but vol(M,Iso+H3) = {0}, and non-trivial graph manifolds M with vol (M, IsoeSL2(ℝ)) ≠ {0}, proving that it is in general necessary to pass to some finite covering to guarantee that vol(M,G) ≠ {0}. Besides we determine vol(M,G) when M supports the Seifert geometry. [ABSTRACT FROM AUTHOR]

Factorization homology theories of topological manifolds, after Beilinson, Drinfeld, and Lurie, are homology-type theories for topological n-manifolds whose coefficient systems are n-disk algebras or n-disk stacks. In this work, we prove a precise formulation of this idea, giving an axiomatic characterization of factorization homology with coefficients in n-disk algebras in terms of a generalization of the Eilenberg-Steenrod axioms for singular homology. Each such theory gives rise to a kind of topological quantum field theory, for which observables can be defined on general n-manifolds and not only closed n-manifolds. For n-disk algebra coefficients, these field theories are characterized by the condition that global observables are determined by local observables in a strong sense. Our axiomatic point of view has a number of applications. In particular, we give a concise proof of the non-abelian Poincaré duality of Salvatore, Segal, and Lurie. We present some essential classes of calculations of factorization homology, such as for free n-disk algebras and enveloping algebras of Lie algebras, several of which have a conceptual meaning in terms of Koszul duality. [ABSTRACT FROM AUTHOR]

This paper begins the classification of topological actions on manifolds by compact, connected, Lie groups beyond the circle group. It treats multiaxial topological actions of unitary and symplectic groups without the dimension restrictions used in earlier works by M. Davis and W. C. Hsiang on differentiable actions. The general results are applied to give detailed calculations for topological actions homotopically modeled on standard multiaxial representation spheres. In the present topological setting, Schubert calculus of complex Grassmannians surprisingly enters in the calculations, yielding a profusion of "fake" representation spheres compared with the paucity in the previously studied smooth setting. [ABSTRACT FROM AUTHOR]

We prove that if a paratopological group G is a continuous image of an arbitrary product of regular Lindelöf Σ-spaces, then it is ℝ-factorizable and has countable cellularity. If in addition, G is regular, then it is totally ω-narrow and satisfies celω(G) ≤ ω, and the Hewitt-Nachbin completion of G is again an ℝ-factorizable paratopological group. [ABSTRACT FROM AUTHOR]

In this paper, we declare the equivalence between the principal component analysis and the nearest q-flat in the least square sense by showing that, for given m data points, the linear manifold with nearest distance is identical to the linear manifold with largest variance. Furthermore, from this observation, we give a new simpler proof for the approach to find the nearest q-flat. [ABSTRACT FROM AUTHOR]

The article looks at the topological aspects and analog of the house with two rooms creation by mathematician R. H. Bing. Topics discussed include approach to the 3-dimensional Poincaré conjecture, carving of Bing's house from a solid 3-dimensional cube, the mathematical equation, elementary collapse, and contractible dimensional complex.

In this article we study the fractional Schrödinger equations (-Δ)αu + V(x)u = f(x, u) in ℝN, where 0 < α < 1, N ≥ 2, (-Δ)α stands for the fractional Laplacian of order α. First by using Morse theory in combination with local linking arguments, we prove the existence of at least two nontrivial solutions. Next we prove that the problem has k distinct pairs of solutions by using the Clark theorem. [ABSTRACT FROM AUTHOR]

This article shows that the Gibbs function topological manifold $$G(p,T,x_{1}, x_{2},\ldots ,x_{C})$$ at the thermodynamic equilibrium is always two-dimensional (2D). This means that the set of values $$G$$ , regardless of the number of independent components $$C$$ , creates a 2-D surface. Based on a state with zero degrees of freedom as a reference state, it was shown that the state of a thermodynamic equilibrium is represented by a graph on such a 2-D surface. In the equilibrium state, graph edges that connect points corresponding to individual degrees of freedom have a minimal length. [ABSTRACT FROM AUTHOR]

In complex systems with at least three independent components, one-phase normal states may transform into exotic states. The former are represented by a non-branching tree, while the latter are represented by a branching tree. The transformation takes place through a non-congruent two-phase equilibrium. Until recently, researchers using this process were able to obtain stable quasicrystals with three, four, or more components. It therefore seemed justified to suppose that exotic states constituted quasicrystals. In 2000, however, Tsai's team discovered two stable binary quasicrystals formed through a congruent process. Virtually no reports on other stable binary quasicrystals have been obtained since that discovery despite considerable effort on the part of researchers. The graph-based representation of equilibrium states rules out the existence of exotic one-phase equilibria (i.e., stable quasicrystals) in binary systems. A question arises: What types of systems did Tsai discover?. [ABSTRACT FROM AUTHOR]

The first author conjectured in 1990 (see [18]) that for any simply-connected elliptic space, the total dimension of the rational homotopy does not exceed that of its rational cohomology. Our main purpose in this paper is to investigate the following: does the Hilali conjecture holds for the configuration spaces of a rationally elliptic and simply connected topological space when it already holds for the space itself. We will prove that this statement is true for closed manifolds. [ABSTRACT FROM AUTHOR]

In Appendix A of his article on rational functions, Segal proved homological stability for configuration spaces with a stability slope of 1/2. This was later improved to a slope of 1 by Randal-Williams if one works with rational coefficients and manifolds of dimension at least 3. In this note we prove that the stability slope of 1 holds even with Z[1/2] coefficients, and clarify some aspects of Segal’s proof for topological manifolds. [ABSTRACT FROM AUTHOR]

Considering point transitive generalized shit dynamical system (X⌈, σφ) for discrete X with at least two elements and infinite ⌈, we prove that X is countable and ⌈ has at most 2ϰ0 elements. Then, we find a transitive point of the dynamical system (ℕℕxℤ, στ) for τ : ℕ x ℤ → ℕ x ℤ with τ (n, m) = (n, m + 1) and show that point transitive (X⌈, σφ), for infinite countable ⌈, is a factor of (ℕℕ x ℤ, στ). [ABSTRACT FROM AUTHOR]

ECH (embedded contact homology) capacities give obstructions to symplectically embedding one symplectic four-manifold with boundary into another. We compute the ECH capacities of a large family of symplectic four-manifolds with boundary, called ‘concave toric domains’. Examples include the (nondisjoint) union of two ellipsoids in ${\mathbb R} ^4$. We use these calculations to find sharp obstructions to certain symplectic embeddings involving concave toric domains. For example: (1) we calculate the Gromov width of every concave toric domain; (2) we show that many inclusions of an ellipsoid into the union of an ellipsoid and a cylinder are ‘optimal’; and (3) we find a sharp obstruction to ball packings into certain unions of an ellipsoid and a cylinder. [ABSTRACT FROM AUTHOR]

Certain remnants of a quantum spacetime foam can be modeled by a distribution of defects embedded in a flat classical spacetime. The presence of such spacetime defects affects the propagation of elementary particles. In this article, we show explicitly that both topology and differential structure of the defects are important for the particle motion. Specifically, we consider three types of spacetime defects which are described by the same topological manifold R x (RP³ - {point}) but which are not diffeomorphic to each other. We investigate the propagation of a massless scalar field over the three different manifolds and find different solutions of the Klein-Gordon equation. [ABSTRACT FROM AUTHOR]