Geometric Idea Research Articles

The Data from the Lunar Laser Ranging Project Contradict Einstein’s Genius Spacetime Idea

What happens to a laser beam that points from the bottom of a train (at rest) to a point in the ceiling when the train is moving very fast. Will the beam still hit this point, or will it hit the ceiling behind this point? In the years around 1900 scientists were convinced that photons get a lateral momentum in the direction of movement, because they are particles. But R. Feynman concluded that a mirror emits new photons and is therefore a light source and together with Einstein’s second STR-postulate the laser beam should hit the ceiling behind this point. This can be tested with the return rates of photons from a mirror on the Moon. The results show clearly with an error of probability < 10-80 that photons do not get a lateral momentum but arrive at that location where the Earth was 2.55 seconds before. Besides the detection of an additional velocity of Earth in the Universe this article proves that Einstein’s geometric spacetime idea is wrong because the physical basis for that is wrong

Актуальные вопросы в грузовой и коммерческой работе на железнодорожных станциях

Purpose: To consider the issue of interaction between carriers and counterparties using new approaches aimed at attracting customers to the transport services market, with the creation of a single mechanism based on digitalization. Methods: A number of works explore the issues of the location of warehouse areas. In them, the task is solved by various methods, such as the heuristic definition of the “center of gravity” for enterprises, the multi-criteria Choquet integral, the method of hierarchy analysis consisting in pairwise comparisons, as well as the author’s methods of economic and geographical modeling of transport systems, reflecting the geometric idea of the approach in transport and logistics research with the differentiation of “areas of influence”. Results: The necessity of creating a single mechanism of interaction between the carrier and the counterparty is indicated. The implementation of the set aspects is played by the nature of transportation, taking into account the nomenclature of goods. Preference should be directed to mass categories. In solving the set goals, it is possible to use the method of linear programming (equality). The necessity of conducting an additional study of the conjuncture of an economic entity has been identified. Practical significance: The necessity of creating a single digital platform for interaction of participants in the transportation process is shown, the implementation of which will speed up the process of solving problems aimed at using new tools that contribute to improving the work of an economic entity. The presented schemes of the location of warehouses can be recommended for practical use.

Some Topological Approaches for Generalized Rough Sets via Ideals

The idea of neighborhood systems is induced from the geometric idea of “near,” and it is primitive in the topological structures. Now, the idea of neighborhood systems has been extensively applied in rough set theory. The master contribution of this manuscript is to generate various topologies by means of the concepts of j -adhesion neighborhoods and ideals. Then, we define a new rough set model derived from these topologies and discussed main features. We show that these topologies are finer than those given in the previous ones under arbitrary binary relations. In addition, we elucidate that these topologies are finer than those topologies initiated based on different neighborhoods and ideals under reflexive relations. Several examples are provided to validate that our model is better than the previous ones.

Unions, intersections and a one-shot quantum joint typicality lemma

A fundamental tool to prove inner bounds in classical network information theory is the so-called ‘conditional joint typicality lemma’. In addition to the lemma, one often uses unions and intersections of typical sets in the inner bound arguments without so much as giving them a second thought. These arguments do not work in the quantum setting. This bottleneck shows up in the fact that so-called ‘simultaneous decoders’, as opposed to ‘successive cancellation decoders’, are known for very few channels in quantum network information theory. Another manifestation of this bottleneck is the lack of so-called ‘simultaneous smoothing’ theorems for quantum states. In this paper, we overcome the bottleneck by proving for the first time a one-shot quantum joint typicality lemma with robust union and intersection properties. To do so we develop two novel tools in quantum information theory, which may be of independent interest. The first tool is a simple geometric idea called tilting, which increases the angles between a family of subspaces in orthogonal directions. The second tool, called smoothing and augmentation, is a way of perturbing a multipartite quantum state such that the partial trace over any subset of registers does not increase the operator norm much. Our joint typicality lemma allows us to construct simultaneous quantum decoders for many multiterminal quantum channels. It provides a powerful tool to extend many results in classical network information theory to the one-shot quantum setting.

Categorical mirror symmetry on cohomology for a complex genus 2 curve

Motivated by observations in physics, mirror symmetry is the concept that certain manifolds come in pairs X and Y such that the complex geometry on X mirrors the symplectic geometry on Y. It allows one to deduce symplectic information about Y from known complex properties of X. Strominger-Yau-Zaslow [61] described how such pairs arise geometrically as torus fibrations with the same base and related fibers, known as SYZ mirror symmetry. Kontsevich [43] conjectured that a complex invariant on X (the bounded derived category of coherent sheaves) should be equivalent to a symplectic invariant of Y (the Fukaya category, see [9], [29], [49], [1]). This is known as homological mirror symmetry. In this project, we first use the construction of “generalized SYZ mirrors” for hypersurfaces in toric varieties following Abouzaid-Auroux-Katzarkov [6], in order to obtain X and Y as manifolds. The complex manifold is the genus 2 curve Σ2 (so of general type c1<0) as a hypersurface in its Jacobian torus. Its generalized SYZ mirror is a Landau-Ginzburg model (Y,v0) equipped with a holomorphic function v0:Y→C which we put the structure of a symplectic fibration on. We then describe an embedding of a full subcategory of DbCoh(Σ2) into a cohomological Fukaya-Seidel category of Y as a symplectic fibration. While our fibration is one of the first nonexact, non-Lefschetz fibrations to be equipped with a Fukaya category, the main geometric idea in defining it is the same as in Seidel's construction for Fukaya categories of Lefschetz fibrations in [55] and in Abouzaid-Seidel [3].

Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums

We prove non-trivial bounds for bilinear forms with hyper-Kloosterman sums with characters modulo a prime $q$ which, for both variables of length $M$, are non-trivial as soon as $M\geq q^{3/8+\delta}$ for any $\delta>0$. This range, which matches Burgess's range, is identical with the best results previously known only for simpler exponentials of monomials. The proof combines refinements of the analytic tools from our previous paper and new geometric methods. The key geometric idea is a comparison statement that shows that even when the "sum-product" sheaves that appear in the analysis fail to be irreducible, their decomposition reflects that of the "input" sheaves, except for parameters in a high-codimension subset. This property is proved by a subtle interplay between \'etale cohomology in its algebraic and diophantine incarnations. We prove a first application concerning the first moment of a family of $L$-functions of degree $3$.

Fast algorithms for chiral fermions in 2 dimensions

In lattice QCD simulations the formulation of the theory in lattice should be chiral in order that symmetry breaking happens dynamically from interactions. In order to guarantee this symmetry on the lattice one uses overlap and domain wall fermions. On the other hand high computational cost of lattice QCD simulations with overlap or domain wall fermions remains a major obstacle of research in the field of elementary particles. We have developed the preconditioned GMRESR algorithm as fast inverting algorithm for chiral fermions in U(1) lattice gauge theory. In this algorithm we used the geometric multigrid idea along the extra dimension.The main result of this work is that the preconditioned GMRESR is capable to accelerate the convergence 2 to 12 times faster than the other optimal algorithms (SHUMR) for different coupling constant and lattice 32x32. Also, in this paper we tested it for larger lattice size 64x64. From the results of simulations we can see that our algorithm is faster than SHUMR. This is a very promising result that this algorithm can be adapted also in 4 dimension.

Van Hileova teorija geometrijskog mišljenja u nastavi matematike

Although the twentieth century was marked research in algebra, the emergence of the theory of geometric thinking encourages interest in geometric content. With increasing interest in geometric idea is important to research about the nature of geometrical reasoning and how it develops. This paper summarizes the theoretical framework for the study of geometrical reasoning van Hiele model of thinking in geometry. First, we give van Hiele theory of geometrical reasoning, levels of understanding of spatial relationships and geometric phase of learning content. Second, we present and analyze the interpretation of van Hiele theories. Then, we give a brief overview of research based on van Hiele theory. The paper uses the methods of theoretical analysis. Based on the researched literature we conclude that still need a lot of research on deep process of development and learning of geometrical reasoning.

Probing the active galactic nucleus unified model torus properties in Seyfert galaxies

We studied the physical parameters of a sample comprising of all Spitzer/IRS public spectra of Seyfert galaxies in the mid-infrared (5.2-38$\mu$m range) under the active galactic nuclei (AGN) unified model. We compare the observed spectra with $\sim10^6$ CLUMPY model spectral energy distributions, which consider a torus composed of dusty clouds. We find a slight difference in the distribution of line-of-sight inclination angle, $i$, requiring larger angles for Seyfert 2 (Sy2) and a broader distribution for Seyfert 1 (Sy1). We found small differences in the torus angular width, $\sigma$, indicating that Sy1 may host a slightly narrower torus than Sy2. The torus thickness, together with the bolometric luminosities derived, suggest a very compact torus up to $\sim$6 pc from the central AGN. The number of clouds along the equatorial plane, $N$, as well the index of the radial profile, $q$, are nearly the same for both types. These results imply that the torus cloud distribution is nearly the same for type 1 and type 2 objects. The torus mass is almost the same for both types of activity, with values in the range of $M_{tor}\sim$10$^{4}-$10$^{7}\rm M_{\odot}$. The main difference appears to be related to the clouds' intrinsic properties: type 2 sources present higher optical depths $\tau_V$. The results presented here reinforce the suggestion that the classification of a galaxy may depend also on the intrinsic properties of the torus clouds rather than simply on their inclination. This is in contradiction with the simple geometric idea of the unification model.

Halfspace type theorems for self-shrinkers

In this short paper we extend the classical Hoffman-Meeks Halfspace Theorem to self-shrinkers, that is: "Let $P $ be a hyperplane passing through the origin. The only properly immersed self-shrinker $\Sigma$ contained in one of the closed half-space determined by $P$ is $\Sigma = P$." Our proof is geometric and uses a catenoid type hypersurface discovered by Kleene-Moller. Also, using a similar geometric idea, we obtain that the only complete self-shrinker properly immersed in an closed cylinder $ \overline{B ^{k+1} (R)} \times \mathbb{R}^{n-k}\subset \mathbb R^{n+1}$, for some $k\in \{1, \ldots ,n\}$ and radius $R$, $R \leq \sqrt{2k}$, is the cylinder $\mathbb S ^k (\sqrt{2k}) \times \mathbb{R}^{n-k}$. We also extend the above results for $\lambda -$hypersurfaces.

ConvexLAR: An Extension of Least Angle Regression

The least angle regression (LAR) was proposed by Efron, Hastie, Johnstone and Tibshirani in the year 2004 for continuous model selection in linear regression. It is motivated by a geometric argument and tracks a path along which the predictors enter successively and the active predictors always maintain the same absolute correlation (angle) with the residual vector. Although it gains popularity quickly, its extensions seem rare compared to the penalty methods. In this expository article, we show that the powerful geometric idea of LAR can be generalized in a fruitful way. We propose a ConvexLAR algorithm that works for any convex loss function and naturally extends to group selection and data adaptive variable selection. After simple modification, it also yields new exact path algorithms for certain penalty methods such as a convex loss function with lasso or group lasso penalty. Variable selection in recurrent event and panel count data analysis, Ada-Boost, and Gaussian graphical model is reconsidered from the ConvexLAR angle. Supplementary materials for this article are available online.

Enhanced low-rank representation via sparse manifold adaption for semi-supervised learning

Constructing an informative and discriminative graph plays an important role in various pattern recognition tasks such as clustering and classification. Among the existing graph-based learning models, low-rank representation (LRR) is a very competitive one, which has been extensively employed in spectral clustering and semi-supervised learning (SSL). In SSL, the graph is composed of both labeled and unlabeled samples, where the edge weights are calculated based on the LRR coefficients. However, most of existing LRR related approaches fail to consider the geometrical structure of data, which has been shown beneficial for discriminative tasks. In this paper, we propose an enhanced LRR via sparse manifold adaption, termed manifold low-rank representation (MLRR), to learn low-rank data representation. MLRR can explicitly take the data local manifold structure into consideration, which can be identified by the geometric sparsity idea; specifically, the local tangent space of each data point was sought by solving a sparse representation objective. Therefore, the graph to depict the relationship of data points can be built once the manifold information is obtained. We incorporate a regularizer into LRR to make the learned coefficients preserve the geometric constraints revealed in the data space. As a result, MLRR combines both the global information emphasized by low-rank property and the local information emphasized by the identified manifold structure. Extensive experimental results on semi-supervised classification tasks demonstrate that MLRR is an excellent method in comparison with several state-of-the-art graph construction approaches.

Rectifiability of Self-Contracted Curves in the Euclidean Space and Applications

It is hereby established that, in Euclidean spaces of finite dimension, bounded self-contracted curves have finite length. This extends the main result of Daniilidis et al. (J. Math. Pures Appl. 94:183–199, 2010) concerning continuous planar self-contracted curves to any dimension, and dispenses entirely with the continuity requirement. The proof borrows heavily from a geometric idea of Manselli and Pucci (Geom. Dedic. 38:211–227, 1991) employed for the study of regular enough curves, and can be seen as a nonsmooth adaptation of the latter, albeit a nontrivial one. Applications to continuous and discrete dynamical systems are discussed: continuous self-contracted curves appear as generalized solutions of nonsmooth convex foliation systems, recovering a hidden regularity after reparameterization, as a consequence of our main result. In the discrete case, proximal sequences (obtained through implicit discretization of a gradient system) give rise to polygonal self-contracted curves. This yields a straightforward proof for the convergence of the exact proximal algorithm, under any choice of parameters.

A Simply Typed λ-Calculus of Forward Automatic Differentiation

We present an extension of the simply typed λ-calculus with pushforward operators. This extension is motivated by the desire to incorporate forward automatic differentiation, which is an important technique in numeric computing, into functional programming. Our calculus is similar to Ehrhard and Regnierʼs differential λ-calculus, but is based on the differential geometric idea of pushforward rather than derivative. We prove that, like the differential λ-calculus, our calculus can be soundly interpreted in differential λ-categories.

Finding the Form of an Irregular Meshed Steel and Glass Shell Based on Construction Constraints

In the context of the search for an efficient structural shape to cover the Dutch Maritime Museum courtyard in Amsterdam, Netherlands, the authors briefly discuss the driving design factors that influenced the earliest glass roof coverings. The trends that emerged during the late 20th and early 21st century in the design of skeletal steel glass shells are exposed. These design developments range from sculptural to geometric and structural intentions. The discussion of the competition design development of the Dutch Maritime Museum steel glass shell roof shows the quest for a structurally efficient catenary form based on a poetic geometric idea. This paper presents a construction-driven design methodology that slightly adapts the numerical form found catenary shape with the objective of achieving planarity in all the triangulated, four-sided and five-sided mesh faces. The challenge of facet planarity is gracefully solved by an analytical origami approach and presented. This approach is compared with finding the Maxwell reciprocal network diagram. The final faceted shape shows elegance and structural efficiency.

A Geometrical Explanation of Stein Shrinkage

Shrinkage estimation has become a basic tool in the analysis of high-dimensional data. Historically and conceptually a key development toward this was the discovery of the inadmissibility of the usual estimator of a multivariate normal mean. This article develops a geometrical explanation for this inadmissibility. By exploiting the spherical symmetry of the problem it is possible to effectively conceptualize the multidimensional setting in a two-dimensional framework that can be easily plotted and geometrically analyzed. We begin with the heuristic explanation for inadmissibility that was given by Stein [In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954--1955, Vol. I (1956) 197--206, Univ. California Press]. Some geometric figures are included to make this reasoning more tangible. It is also explained why Stein's argument falls short of yielding a proof of inadmissibility, even when the dimension, $p$, is much larger than $p=3$. We then extend the geometric idea to yield increasingly persuasive arguments for inadmissibility when $p\geq3$, albeit at the cost of increased geometric and computational detail.

Drilling for Polygons

We solve the problem of designing a simple device that uses rotary motion to drill a hole with a cross-section that is a regular polygon with an odd number of sides: the main idea is to use a polygonal trammel and a family of rotors. By using different rotors, one can produce a hole with a cross-section that is in any proportion of the trammel size from zero to exactly one. The key geometric idea is a result about the envelope of an edge of a triangle that rotates so that the other two edges maintain tangential contact with two fixed circles.

Positive topological entropy of Reeb flows on spherizations

AbstractLet M be a closed manifold whose based loop space Ω (M) is “complicated”. Examples are rationally hyperbolic manifolds and manifolds whose fundamental group has exponential growth. Consider a hypersurface Σ in T*M which is fiberwise starshaped with respect to the origin. Choose a function H : T*M → ℝ such that Σ is a regular energy surface of H, and let ϕt be the restriction to Σ of the Hamiltonian flow of H.Theorem 1. The topological entropy of ϕt is positive.This result has been known for fiberwise convex Σ by work of Dinaburg, Gromov, Paternain, and Paternain–Petean on geodesic flows. We use the geometric idea and the Floer homological technique from [19], but in addition apply the sandwiching method. Theorem 1 can be reformulated as follows.Theorem 1'. The topological entropy of any Reeb flow on the spherization SM of T*M is positive.For q ∈ M abbreviate Σq = Σ ∩ Tq*M. The following corollary extends results of Morse and Gromov on the number of geodesics between two points.Corollary 1. Given q ∈ M, for almost every q′ ∈ M the number of orbits of the flow ϕt from Σq to Σq′ grows exponentially in time.In the lowest dimension, Theorem 1 yields the existence of many closed, orbits.Corollary 2. Let M be a closed surface different from S2, ℝP2, the torus and the Klein bottle. Then ϕt carries a horseshoe. In particular, the number of geometrically distinct closed orbits grows exponentially in time.

Mesh parameterization based on one-step inverse forming

A novel planar mesh parameterization algorithm via flattening or unfolding is proposed by introducing a so-called one-step inverse approach (IA) based on physical plastic deformation of metal materials. As opposed to methods based on geometric idea, the algorithm is more suitable for CAD models, e.g. rapid prediction of blank shape in sheet metal forming for auto-body panels. Benefiting from proper pre-processing steps, rapid simulation of sheet metal forming and optimal initial solution guess, the proposed algorithm flattens meshes onto a plane with lower area distortion and minimizes the angular distortion. Some numerical examples of results generated by the new parameterization method are provided, and the corresponding analyses are given as well, such as distributions of strain, stress, thickness distribution and formability, etc.

Iterated conditional expectations

Consider the probability space ( [ 0 , 1 ) , B , λ ) , where B is the Borel σ-algebra on [ 0 , 1 ) and λ the Lebesgue measure. Let f = 1 [ 0 , 1 / 2 ) and g = 1 [ 1 / 2 , 1 ) . Then for any ε > 0 there exists a finite sequence of sub- σ-algebras G j ⊂ B ( j = 1 , … , N ) , such that putting f 0 = f and f j = E ( f j − 1 | G j ) , j = 1 , … , N , we have ‖ f N − g ‖ ∞ < ε ; here E ( ⋅ | G j ) denotes the operator of conditional expectation given σ-algebra G j . This is a particular case of a surprising result by Cherny and Grigoriev (2007) [1] in which f and g are arbitrary equidistributed bounded random variables on a nonatomic probability space. The proof given in Cherny and Grigoriev (2007) [1] is very complicated. The purpose of this note is to give a straightforward analytic proof of the above mentioned result, motivated by a simple geometric idea, and then show that the general result is implied by its special case.