Theorem States Research Articles

Kernel methods for center manifold approximation and a weak data-based version of the Center Manifold Theorem

For dynamical systems with a non hyperbolic equilibrium, it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to the equilibrium point and to obtain meaningful predictions of its behavior by analyzing a reduced order system on the so-called center manifold.Since the center manifold is usually not known, good approximation methods are important as the center manifold theorem states that the stability properties of the origin of the reduced order system are the same as those of the origin of the full order system.In this work, we establish a data-based version of the center manifold theorem that works by considering an approximation in place of an exact manifold. Also the error between the approximated and the original reduced dynamics is quantified.We then use an apposite data-based kernel method to construct a suitable approximation of the manifold close to the equilibrium, which is compatible with our general error theory. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used as a training set. The method is tested on different examples which show promising performance and good accuracy.

Monsky’s Theorem / O Teorema de Monsky

The main objective of this paper is to prove Monsky’s Theorem, that provides a beautiful application of the 2-adic valuation in order to solve a plane geometry problem. This theorem states that given any dissection of a square into finitely many nonoverlapping triangles of equal area the number of triangles must be even. In order to prove this statement, we will need some previous results from Combinatorial Topology and Algebra.

Akiba's logic of indeterminacy

ABSTRACT A standard approach to indeterminacy treats ‘determinately’ and ‘indeterminately’ as modal operators. Determinacy behaves like necessity; indeterminacy like contingency. This raises two questions. What is the appropriate modal system for these operators? And how should we interpret that system? Ken Akiba has developed an account of ontic indeterminacy that interprets possible worlds as worldly precisifications. He argues that this account vindicates S4 as the logic of indeterminacy. In this paper I explore one significant and surprising consequence of this view. I do this in two stages. First, I prove a technical result, which I call the Infinite Indeterminacy Theorem. Roughly put, this theorem states that, at any given precisification w, either every instance of indeterminacy admits of never-ending higher-order indeterminacy, or else there's indeterminacy in an infinite amount of distinct atomic formulas. In slogan form: either all indeterminacy is infinitely ascending or else there's infinitely wide indeterminacy. Second, I unpack the metaphysical consequences of this technical result (under its intended interpretation) and assess their plausibility. I conclude that these consequences commit Akiba's theory to highly contentious – and arguably untenable – views about the metaphysics of indeterminacy and related matters.

Divergent Fourier series in function spaces near L1[0;1

Bochkariev's theorem states that for any uniformly bounded orthonormal system Φ, there is a Lebesgue integrable function such that the Fourier series of it with respect to the system Φ diverges on the set of positive measure. In this paper, we extended Bochkariev's theorem for some class of variable exponent Lebesgue spaces. We characterized the class of variable exponent Lebesgue spaces Lp(⋅)[0;1], 1<p(x)<∞ a.e. on [0;1], such that above mentioned Bochkarev's theorem is valid.

On length measures of planar closed curves and the comparison of convex shapes

In this paper, we revisit the notion of length measures associated to planar closed curves. These are a special case of area measures of hypersurfaces which were introduced early on in the field of convex geometry. The length measure of a curve is a measure on the circle \(\mathbb {S}^1\) that intuitively represents the length of the portion of curve which tangent vector points in a certain direction. While a planar closed curve is not characterized by its length measure, the fundamental Minkowski–Fenchel–Jessen theorem states that length measures fully characterize convex curves modulo translations, making it a particularly useful tool in the study of geometric properties of convex objects. The present work, that was initially motivated by problems in shape analysis, introduces length measures for the general class of Lipschitz immersed and oriented planar closed curves, and derives some of the basic properties of the length measure map on this class of curves. We then focus specifically on the case of convex shapes and present several new results. First, we prove an isoperimetric characterization of the unique convex curve associated to some length measure given by the Minkowski–Fenchel–Jessen theorem, namely that it maximizes the signed area among all the curves sharing the same length measure. Second, we address the problem of constructing a distance with associated geodesic paths between convex planar curves. For that purpose, we introduce and study a new distance on the space of length measures that corresponds to a constrained variant of the Wasserstein metric of optimal transport, from which we can induce a distance between convex curves. We also propose a primal-dual algorithm to numerically compute those distances and geodesics, and show a few simple simulations to illustrate the approach.

Physical limitations of the Hohenberg–Mermin–Wagner theorem

The Hohenberg–Mermin–Wagner (HMW) theorem states that infrared (IR) fluctuations prevent long-range order which breaks continuous symmetries in two dimensions (2D), at finite temperatures. We note that the theorem becomes physically effective for superconductivity (SC) only for astronomical sample sizes, so it does not prevent 2D SC in practice. We systematically explore the sensitivity of the magnetic and SC versions of the theorem to finite-size and disorder effects. For magnetism, finite-size effects, disorder, and perpendicular coupling can all restore the order parameter at a non-negligible value of T c equally well, making the physical reason for finite T c sample-dependent. For SC, an alternative version of the HMW theorem is presented, in which the temperature cutoff is set by Cooper pairing, in place of the Fermi energy in the standard version. It still allows 2D SC at 2–3 times the room temperature when the interaction scale is large and Cooper pairs are small, the case with high-T c SC in the cuprates. Thus IR fluctuations do not prevent 2D SC at room temperatures in samples of any reasonable size, by any known version of the HMW argument. A possible approach to derive mechanism-dependent upper bounds for SC T c is pointed out.

Strongly budget balanced auctions for multi-sided markets

In two-sided markets, Myerson and Satterthwaite's impossibility theorem states that one can not maximize the gain-from-trade while also satisfying truthfulness, individual-rationality and no deficit. Attempts have been made to circumvent Myerson and Satterthwaite's result by attaining approximately-maximum gain-from-trade: the double-sided auction of McAfee (1992) [35] is truthful and has no deficit — it is weakly-budget-balanced, and the one by Segal-Halevi et al.'s (2016) [38] additionally has no surplus — it is strongly-budget-balanced. They consider two categories of agents — buyers and sellers, where each trade set is composed of a single buyer and a single seller.The practical complexity of applications in areas such as supply chain requires one to look beyond two-sided markets. Common requirements are for: buyers trading with multiple sellers of different or identical items, buyers trading with sellers through transporters and mediators, and sellers trading with multiple buyers. We attempt to address these settings.We generalize Segal-Halevi et al. (2016)'s [38] strongly-budget-balanced double-sided auction setting to a multilateral market where each trade set is composed of any number of agent categories. Our generalization refines the notion of competition in multi-sided auctions by introducing the concepts of external competition and trade reduction. We also show an obviously-truthful implementation of our auction using multiple ascending prices.

Abrasive Waterjet (AWJ) Forces-Potential Indicators of Machining Quality.

The necessity of monitoring the abrasive waterjet (AWJ) processes increases with the spreading of this tool into the machining processes. The forces produced on the workpiece during the abrasive waterjet machining can yield some valuable information. Therefore, a special waterjet-force measuring device designed and produced in the past has been used for the presented research. It was tested during the AWJ cutting processes, because they are the most common and the best described up-to-date AWJ applications. Deep studies of both the cutting process and the respective force signals led to the decision that the most appropriate indication factor is the tangential-to-normal force ratio (TNR). Three theorems concerning the TNR were formulated and investigated. The first theorem states that the TNR strongly depends on the actual-to-limit traverse speed ratio. The second theorem claims that the TNR relates to the cutting-to-deformation wear ratio inside the kerf. The third theorem states that the TNR value changes when the cutting head and the respective jet axis are tilted so that a part of the jet velocity vector projects into the traverse speed direction. It is assumed that the cutting-to-deformation wear ratio increases in a certain range of tilting angles of the cutting head. This theorem is supported by measured data and can be utilized in practice for the development of a new method for the monitoring of the abrasive waterjet cutting operations. Comparing the tilted and the non-tilted jet, we detected the increase of the TNR average value from 1.28 ± 0.16 (determined for the declination angle 20° and the respective tilting angle 10°) up to 2.02 ± 0.25 (for the declination angle 30° and the respective tilting angle of 15°). This finding supports the previously predicted and published assumptions that the tilting of the cutting head enables an increase of the cutting wear mode inside the forming kerf, making the process more efficient.

The Hairy Ball problem is PPAD-complete

The Hairy Ball Theorem states that every continuous tangent vector field on an even-dimensional sphere must have a zero. We prove that the associated computational problem of (a) computing an approximate zero is PPAD-complete, and (b) computing an exact zero is FIXP-hard. We also consider the Hairy Ball Theorem on toroidal instead of spherical domains and show that the approximate problem remains PPAD-complete. On a conceptual level, our PPAD-membership results are particularly interesting, because they heavily rely on the investigation of multiple-source variants of End-of-Line, the canonical PPAD-complete problem. Our results on these new End-of-Line variants are of independent interest and provide new tools for showing membership in PPAD. In particular, we use them to provide the first full proof of PPAD-completeness for the Imbalance problem defined by Beame et al. in 1998.

Correction to: Tropical varieties for exponential sums

A correction to this paper has been published: https ://doi.org/10.1007/s00208-019-01808-5 A subsidiary result (Theorem 1.1, on the undecidability of determining membership in the amoeba of $$1-x-y$$ ) from our original article had an erroneous proof. We give a correct proof. The statement of Theorem 1.1 remains correct. We thank Alexander Rashkovskii for pointing out the error in our original proof. We also correct a small typo in our main theorem on univariate exponential sums (Theorem 1.5).

Application of neural networks in industrial production

The article describes an algorithm for the synthesis of neural networks for controlling the gyrostabilizer. The neural network acts as an observer of the state vector. The role of such an observer is to provide feedback to the gyrostabilizer, which is illustrated in the article. Gyrostabilizer is a gyroscopic device designed to stabilize individual objects or devices, as well as to determine the angular deviations of objects. Gyrostabilizer systems will be more widely used, as they provide an effective means of motion control with a number of significant advantages for various designs. The article deals in detail with the issue of specific stage features of classical algorithms: selecting the network architecture, training the neural network, and verifying the results of feedback control. In recent years, neural networks have become an increasingly powerful tool in scientific computing. The universal approximation theorem states that a neural network can be constructed to approximate any given continuous function with the required accuracy. The back propagation algorithm also allows effectively optimizing the parameters when training a neural network. Due to the use of graphics processors, it is possible to perform efficient calculations for scientific and engineering tasks. The article presents the optimal configuration of the neural network, such as the depth of memory, the number of layers and neurons in these layers, as well as the functions of the activation layer. In addition, it provides data on dynamic systems to improve neural network training. An optimal training scheme is also provided.

Revisiting the Time-Domain and Frequency-Domain Definitions of Capacitance

The capacitance is a characteristic function of an electrical energy storage device that relates the applied voltage on the device to the accumulated electric charge. It is inconsistently taken in some studies as a multiplicative function in the time domain [i.e., q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> (t)=c <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> (t) ×v <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> (t)], and in others as a multiplicative function in the frequency domain [i.e., Q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> (s)=C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> (s) × V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> (s) derived from the definition of admittance I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> (s)/V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> (s) = s C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> (s)], despite the fact that the capacitance is time- and frequency-dependent. However, the convolution theorem states that multiplication of functions in the time domain is equivalent to a convolution operation in the frequency domain, and vice versa. In this work, we revisit and compare the two outlined definitions of capacitance for an ideal capacitor and for a lossy fractional-order capacitor. Although c <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> (t) = C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> (s) = C for an ideal constant capacitor, we show that this is not the case for fractional-order capacitors which exhibit frequency-dispersed impedance, memory effects, and nonexponential relaxation functions. This fact is crucial in the accurate modeling and characterization of supercapacitors and batteries. For these devices, and for being consistent with measurements using conventional impedance analyzers, it is recommended to apply the integral convolution definition in the time domain which reverts to the multiplicative definition in the frequency domain.

Corrigendum to: The asymptotic profile of an eta-theta quotient related to entanglement entropy in string theory

This corrigendum serves to correct the article [1, Theorem 1.1]. In doing so, we correct the proof and statement of Theorem 3.7, and see that one may disregard Proposition 3.4, Lemma 3.5 and Proposition 3.6.

Corrigenda to “Cohen-Macaulay bipartite graphs in arbitrary codimension”

A misuse of terminology has occurred in the statement and proof of Theorem 4.1 in our paper [Proc. Amer. Math. Soc. 143 (2015), pp. 1981–1989] which caused some justifiable misinterpretation of this result. To recover this result we provide a new definition and give the statement of our result in terms of this definition. The proof of the new version is an improvement of the old proof. The effect of the new definition on further relevant results in our paper has been adopted in a remark.

Anti-invariant submanifolds of a normal paracontact metric manifold

In this paper, anti-invariant submanifolds of a normal paracontact metric manifold are studied and characterizing the submanifold with respect to covariant derivative of the second fundamental form of anti-invariant submanifold. Furthermore, some special cases are also discussed and we give a non-trivial example which satisfies the statements of theorems.

Bounds for d-distinct partitions

Euler's identity and the Rogers-Ramanujan identities are perhaps the most famous results in the theory of partitions. According to them, 1-distinct and 2-distinct partitions of n are equinumerous with partitions of n into parts congruent to ±1 modulo 4 and partitions of n into parts congruent to ±1 modulo 5, respectively. Furthermore, their generating functions are modular functions up to multiplication by rational powers of q. For d ≥ 3, however, there is neither the same type of partition identity nor modularity for d-distinct partitions. Instead, there are partition inequalities and mock modularity related with d-distinct partitions. For example, the Alder-Andrews Theorem states that the number of d-distinct partitions of n is greater than or equal to the number of partitions of n into parts which are congruent to ±1 (mod d+3). In this note, we present the recent developments of generalizations and analogs of the Alder-Andrews Theorem and establish asymptotic lower and upper bounds for the d-distinct partitions. Using the asymptotic relations and data obtained from computation, we propose a conjecture on a partition inequality that gives an upper bound for d-distinct partitions. Specifically, for d ≥ 4, the number of d-distinct partitions of n is less than or equal to the number of partitions of n into parts congruent to ±1 (mod m), where m ≤ 2dπ^2 / [3 log^2 (d)+6 log d] .

Learnability and robustness of shallow neural networks learned by a performance-driven BP and a variant of PSO for edge decision-making

It may not be easy to implement complex AI models in edge devices without strong computing capacity (e.g. GPU). The Universal Approximation Theorem states that a shallow neural network (SNN) can represent any nonlinear function. In this paper, we focus on the learnability and robustness of SNNs, obtained by a greedy tight force heuristic algorithm [a Performance Driven Back-Propagation (PDBP)] and a loose force meta-heuristic algorithm [a variant of particle swarm optimization (VPSO)]. From the engineering prospective, all sensors are well justified for a specific task. Hence, all sensor readings should be strongly correlated to the target, and the structure of an SNN should depend on the dimensions of a problem space. The key findings of the research are summarized as follows: (1) The number of hidden neurons of an SNN depends on the nonlinearity of the training data, and the number of hidden neurons up to the dimension number of a problem space could be enough; (2) The learnability of SNNs, produced by error-driven PDBP, is always better than that of SNNs, optimized by error-driven VPSOs; (3) The performances of SNNs, obtained by PDBPs and VPSOs, do not change much for different training rates; and (4) Comparing with other classic machine learning algorithms, such as C4.5, NB and NN in literature, the SNNs, obtained by accuracy-driven PDBPs, win for all tested data sets, and the improvement percentage is up to 32.86%. Hence, the research could provide valued guidance for the implementation of edge intelligence.

On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvature

A consequence of the Cheeger-Gromoll splitting theorem states that for any open manifold $(M,x)$ of nonnegative Ricci curvature, if all the minimal geodesic loops at $x$ that represent elements of $\pi_1(M,x)$ are contained in a bounded ball, then $\pi_1(M,x)$ is virtually abelian. We generalize the above result: if these minimal representing geodesic loops of $\pi_1(M,x)$ escape from any bounded metric balls at a sublinear rate with respect to their lengths, then $\pi_1(M,x)$ is virtually abelian.

The structure of claw-free binary matroids

A simple binary matroid is called claw-free if none of its rank-3 flats are independent sets. These objects can be equivalently defined as the sets E of points in PG(n−1,2) for which E∩P is not a basis of P for any plane P, or as the subsets X of F2n containing no linearly independent triple x,y,z for which x+y,y+z,x+z,x+y+z∉X.We prove a decomposition theorem that exactly determines the structure of all claw-free matroids. The theorem states that claw-free matroids either belong to one of three particular basic classes of claw-free matroids, or can be constructed from these basic classes using a certain ‘join’ operation.

Gröbner bases for staged trees

In this article we consider the problem of finding generators of the toric ideal associated to a combinatorial object called a staged tree. Our main theorem states that toric ideals of staged trees that are balanced and stratified are generated by a quadratic Gr\"obner basis whose intial terms are squarefree. The proof of this result is based on Sullivant's toric fiber product construction.