Topological Theorem Research Articles

Success concepts for causal discovery

Existing causal discovery algorithms are often evaluated using two success criteria, one that is typically unachievable and the other which is too weak for practical purposes. The unachievable criterion—uniform consistency—requires that a discovery algorithm identify the correct causal structure at a known sample size. The weak but achievable criterion— pointwise consistency—requires only that one identify the correct causal structure in the limit. We investigate two intermediate success criteria—decidability and progressive solvability—that are stricter than mere consistency but weaker than uniform consistency. To do so, we review several topological theorems characterizing which discovery problems are decidable and/or progressively solvable. These theorems apply to any problem of statistical model selection, but in this paper, we apply the theorems only to selection of causal models. We show, under several common modeling assumptions, that there is no uniformly consistent procedure for identifying the direction of a causal edge, but there are statistical decision procedures and progressive solutions. We focus on linear models in which the error terms are either non-Gaussian or contain no Gaussian components; the latter modeling assumption is novel to this paper. We focus especially on which success criteria remain feasible when confounders are present.

Perfect Matching in Random Graphs is as Hard as Tseitin

We study the complexity of proving that a sparse random regular graph on an odd number of vertices does not have a perfect matching, and related problems involving each vertex being matched some pre-specified number of times. We show that this requires proofs of degree $\Omega(n / \log n)$ in the Polynomial Calculus (over fields of characteristic $\ne 2$) and Sum-of-Squares proof systems, and exponential size in the bounded-depth Frege proof system. This resolves a question by Razborov asking whether the Lov\'asz-Schrijver proof system requires $n^\delta$ rounds to refute these formulas for some $\delta > 0$. The results are obtained by a worst-case to average-case reduction of these formulas relying on a topological embedding theorem which may be of independent interest.

Asymptotically holomorphic methods for infinitely renormalizable unimodal maps

AbstractThe purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ( $r>3$ ) unimodal maps that are infinitely renormalizable of bounded type. Here we prove a version of the Fatou–Julia–Sullivan theorem and a topological straightening theorem in this setting. In particular, these maps do not have wandering domains and their Julia sets are locally connected.

Vanishing Homology of Warped Product Submanifolds in Complex Space Forms and Applications

In this paper, we prove the nonexistence of stable integral currents in compact oriented warped product pointwise semi-slant submanifold Mn of a complex space form M˜(4ϵ) under extrinsic conditions which involve the Laplacian, the squared norm gradient of the warped function, and pointwise slant functions. We show that i-the homology groups of Mn are vanished. As applications of homology groups, we derive new topological sphere theorems for warped product pointwise semi-slant submanifold Mn, in which Mn is homeomorphic to a sphere Sn if n≥4 and if n=3, then M3 is homotopic to a sphere S3 under the assumption of extrinsic conditions. Moreover, the same results are generalized for CR-warped product submanifolds.

A note on the topological stability theorem from $${{\,\textrm{RCD}\,}}$$ spaces to Riemannian manifolds

A note on the topological stability theorem from $${{\,\textrm{RCD}\,}}$$ spaces to Riemannian manifolds

Synthesis of Stabilizing Control of a Quadcopter Based on Linear Matrix Inequalities

The relevance of the work is due to the widespread introduction of unmanned aerial vehicles, including quadcopters, in various areas of both civil and military applications. A review of various methods of controlling quadcopters, considering their features as nonlinear objects of high dimensionality, is performed. The work is devoted to stabilizing a quadcopter on a complex trajectory defined by functional coordinate relationships in 3D space. A nonlinear dynamic model of the quadcopter in a coupled coordinate system is constructed. The quadcopter control is based on a combination of two control actions. When solving the inverse dynamics problem, program control provides motion along a given trajectory. Stabilization of motion along the desired trajectory is provided by phase coordinate feedback. The stabilizing regulator ratios are found by the modal control method based on the solution of a linear matrix inequality using a linearized model. The found feedback ratios provide the required degree of stability of the closed-loop system, ensuring the quadcopter robustness to parametric perturbations. The legitimacy of this approach to the synthesis of control of a nonlinear system is substantiated by the topological equivalence theorem for the nonlinear system and the linearized model in that the nonlinear system has stable or unstable manifolds, which are analogs of the stable or unstable spaces of the linearized system. The results of computational experiments to estimate the error in reproducing a given quadcopter trajectory are presented. A simulation of the quadcopter behavior was performed, and the trajectory reproduction error was calculated to confirm the effectiveness of the synthesized stabilizing control and the optimal control based on the Pontryagin maximum principle. According to this criterion, the stabilizing control synthesized based on linear matrix inequalities is more effective for the quadcopter. Computational experiments were performed using the MATLAB application software package.

Topological Censorship in Spacetimes Compatible with $$\Lambda > 0$$

Currently available topological censorship theorems are meant for gravitationally isolated black hole spacetimes with cosmological constant $$\Lambda =0$$ or $$\Lambda <0$$ . Here, we prove a topological censorship theorem that is compatible with $$\Lambda >0$$ and which can be applied to whole universes containing possibly multiple collections of black holes. The main assumption in the theorem is that distinct black hole collections eventually become isolated from one another at late times, and the conclusion is that the regions near the various black hole collections have trivial fundamental group, in spite of there possibly being nontrivial topology in the universe.

A Topological sphere theorem for contact CR-warped product submanifolds of an odd-dimensional unit sphere

AbstractIn this paper, we investigate topological sphere theorems for compact minimal contact CR-submanifolds of odd dimensional unit sphere. We show that if an inequality involving the warping function and the scalar curvature of the fibers is satisfied, a compact minimal contact CR-warped product submanifold of the odd dimensional unit sphere is homeomorphic to the sphere. In particular case, for 5-dimensional unit sphere, we show that a 4-dimensional compact minimal contact CR-warped product submanifold is homeomorphic to a sphere if ∣∇lnf∣2 &lt; 1 is satisfied. By using Bonnet-Myers’s theorem we give a result about fundamental group and by Leung’s theorem we obtain a result about homology groups of a contact CR-warped submanifold.

A topological fluctuation theorem

Fluctuation theorems specify the non-zero probability to observe negative entropy production, contrary to a naive expectation from the second law of thermodynamics. For closed particle trajectories in a fluid, Stokes theorem can be used to give a geometric characterization of the entropy production. Building on this picture, we formulate a topological fluctuation theorem that depends only by the winding number around each vortex core and is insensitive to other aspects of the force. The probability is robust to local deformations of the particle trajectory, reminiscent of topologically protected modes in various classical and quantum systems. We demonstrate that entropy production is quantized in these strongly fluctuating systems, and it is controlled by a topological invariant. We demonstrate that the theorem holds even when the probability distributions are non-Gaussian functions of the generated heat.

Local and global Hopf bifurcation analysis of an age-infection HIV dynamics model with cell-to-cell transmission

In this study, we formulate an age-infection human immunodeficiency viruses (HIV) model incorporating cell-to-cell transmission and virus-to-cell infection. Our main goal is to investigate the local and global Hopf bifurcations of the model. We apply integrated semi-group, Hopf bifurcation theory and topological degree theorem. The theoretical analysis indicates that local and global bifurcation can occur from the positive steady state with respect to bifurcation parameters. These bifurcations are crucial for clinical treatment and experimental of HIV infections within the host. Because of the periodic oscillation, higher (or lower) virus load detected at a moment does not indicate the same high (or lower) load as time goes to infinity. At the same time, the appearance of periodic solution caused by time delay τ also explain the fluctuation of viral load in plasma. Finally, we presented some numerical simulations to validate our theoretical results.

Contact Join-semilattices

Contact algebra is one of the main tools in region-based theory of space. In \cite{dmvw1, dmvw2,iv,i1} it is generalized by dropping the operation Boolean complement. Furthermore we can generalize contact algebra by dropping also the operation meet. Thus we obtain structures, called contact join-semilattices (CJS) and structures, called distributive contact join-semilattices (DCJS). We obtain a set-theoretical representation theorem for CJS and a relational representation theorem for DCJS. As corollaries we get also topological representation theorems. We prove that the universal theory of CJS and of DCJS is the same and is decidable.

Some unusual wormholes in general relativity.

In this short review, we present some recently obtained traversable wormhole models in the framework of general relativity (GR) in four and six dimensions that somehow widen our common ideas on wormhole existence and properties. These are, first, rotating cylindrical wormholes, asymptotically flat in the radial direction and existing without exotic matter. The topological censorship theorems are not violated due to lack of asymptotic flatness in all spatial directions. Second, these are cosmological wormholes constructed on the basis of the Lemaître-Tolman-Bondi solution. They connect two copies of a closed Friedmann world filled with dust, or two otherwise distant parts of the same Friedmann world. Third, these are wormholes obtained in six-dimensional GR, whose one entrance is located in 'our' asymptotically flat world with very small extra dimensions while the other 'end' belongs to a universe with large extra dimensions and therefore different physical properties. The possible observable features of such wormholes are briefly discussed. This article is part of the theme issue 'The future of mathematical cosmology, Volume 1'.

Continuation theorems for weakly inward Kakutani and Strongly inward acyclic maps

In this paper we begin by presenting a general Leray-Schauder alternative and a topological transversality theorem for Kakutani (upper semicontinuous maps with nonempty convex compact values) compact weakly inward maps. Then with some observations and extra assumptions we present a Leray-Schauder alternative and a topological transversality theorem for acyclic (upper semicontinuous maps with nonempty acyclic compact values) compact strongly inward maps.

On a topological Ramsey theorem

AbstractWe introduce natural strengthenings of sequential compactness, the r-Ramsey property for each natural number $r\geq 1$ . We prove that metrizable compact spaces are r-Ramsey for all r and give examples of compact spaces that are r-Ramsey but not $(r+1)$ -Ramsey for each $r\geq 1$ (assuming Continuum Hypothesis (CH) for all $r&gt;1$ ). Productivity of the r-Ramsey property is considered.

Anyon condensation: coherent states, symmetry enriched topological phases, Goldstone theorem, and dynamical rearrangement of symmetry

Although the mathematics of anyon condensation in topological phases has been studied intensively in recent years, a proof of its physical existence is tantamount to constructing an effective Hamiltonian theory. In this paper, we concretely establish the physical foundation of anyon condensation by building the effective Hamiltonian and the Hilbert space, in which we explicitly construct the vacuum of the condensed phase as the coherent states that are the eigenstates of the creation operators creating the condensate anyons. Along with this construction, which is analogous to Laughlin’s construction of wavefunctions of fractional quantum hall states, we generalize the Goldstone theorem in the usual spontaneous symmetry breaking paradigm to the case of anyon condensation. We then prove that the condensed phase is a symmetry enriched (protected) topological phase by directly constructing the corresponding symmetry transformations, which can be considered as a generalization of the Bogoliubov transformation.

The existence of periodic solutions to doubly degenerate Allen-Cahn equation with Neumann boundary condition

This work is concerned with the periodic solution of a doubly degenerate Allen-Cahn equation with nonlocal terms associated with Neumann boundary conditions. Firstly, we define a new associated auxiliary problem. Secondly, the topological degree theorem is applied to prove the existence of a limit point to the auxiliary problem, where this limit point represents a nontrivial nonnegative time-periodic solution of the main studied problem. It is observed that the topological degree theorem technique plays an important role in proving the desired results. Furthermore, this technique can be applied to other similar equations with homogeneous Dirichlet or Neumann boundary conditions.

A new study on the existence and stability to a system of coupled higher-order nonlinear BVP of hybrid FDEs under the $ p $-Laplacian operator

&lt;abstract&gt;&lt;p&gt;In this paper, we study a general system of fractional hybrid differential equations with a nonlinear $ \phi_p $-operator, and prove the existence of solution, uniqueness of solution and Hyers-Ulam stability. We use the Caputo fractional derivative in this system so that our system is more general and complex than other nonlinear systems studied before. To establish the results, Green functions are used to transform the considered hybrid boundary problem into a system of fractional integral equations. Then, with the help of the topological degree theorem, we derive some sufficient conditions that ensure the existence and uniqueness of solutions for the proposed system. Finally, an example is presented to show the validity and correctness of the obtained results.&lt;/p&gt;&lt;/abstract&gt;

Structure of Chern-Simons scattering amplitudes from topological equivalence theorem and double-copy

We study the mechanism of topological mass-generation for 3d Chern-Simons (CS) gauge theories, where the CS term can retain the gauge symmetry and make gauge boson topologically massive. Without CS term the 3d massless gauge boson has a single physical transverse polarization state, while adding the CS term converts it into a massive physical polarization state and conserves the total physical degrees of freedom. We formulate the mechanism of topological mass-generation at S-matrix level. For this, we propose and prove a Topological Equivalence Theorem (TET) which connects the N-point scattering amplitude of the gauge boson’s physical polarization states ( {A}_{mathrm{P}}^a ) to that of the transverse polarization states ( {A}_{mathrm{T}}^a ) under high energy expansion. We present a general 3d power counting method on the leading energy dependence of the scattering amplitudes in both topologically massive Yang-Mills (TMYM) and topologically massive gravity (TMG) theories. With these, we uncover a general energy cancellation mechanism for N -gauge boson scattering amplitudes which predicts the cancellation E4→ E4−N at tree level. Then, we compute the 4-gauge boson amplitudes of {A}_{mathrm{P}}^a -states and {A}_{mathrm{T}}^a -states, with which we explicitly demonstrate the TET and establish such energy cancellations for N = 4. We further extend the double-copy approach to reconstruct the massive 4-graviton amplitude of TMG from the massive 4-gauge boson amplitude of TMYM. With these, we uncover striking large energy cancellations in the 4-graviton amplitude: E12→ E1, and establish its correspondence to the leading energy cancellation E4→ E0in the 4-gauge boson amplitude of TMYM.

Using Leray-Schauder topological degree to solve a linear diffusion parabolic equation with periodic initial conditions

Throughout this manuscript, we show time periodic solutions to a linear diffusion parabolic equation with Diriclet condition. Based on the topological degree theorem, we prove a time periodic solutions of the system such that we found the fixed point when the domain of the solution is sufficiently small.

On two-player games with pure strategies on intervals $ [a, \; b] $ and comparisons with the two-player, two-strategy matrix case

&lt;p style='text-indent:20px;'&gt;We consider games of two-players with utility functions which are not necessarily linear on the product of convex and compact intervals of &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \mathcal{R}^2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. An issue is how far an analogy can be drawn with two-player, two-strategy matrix games with linear utility functions, where [0, 1] registers probabilities and equilibria are at the intersection of reaction functions. Now, the idea of &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \delta $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; functions is exploited to construct mixed strategies to look for Nash equilibria (NE). "Reaction" functions are constructed and results are obtained graphically. They are related to topological theorems on NE. The games chosen make specific points in relation to existence conditions and properties of solutions. It is a distinguishing feature that an interval [a, b] now registers both pure and mixed strategies. For NE a choice has to be justified. Also "reaction" functions are more complicated and their intersection does not guarantee an equilibrium.&lt;/p&gt;