A Geometric Framework for Optimal Surface Design

1992 ◽  
Author(s):  
Frank C. Park
Keyword(s):  
Harmonic Maps ◽  
Engineering Application ◽  
Physical Space ◽  
Geometric Approach ◽  
Geometric Framework ◽  
Surface Design ◽  
Optimal Surface ◽  
Engineering Constraints ◽  
Minimum Distortion ◽  
Special Case

Abstract We present a Riemannian geometric framework for variational approaches to geometric design. Optimal surface design is regarded as a special case of the more general problem of finding a minimum distortion mapping between Riemannian manifolds. This geometric approach emphasizes the coordinate-invariant aspects of the problem, and engineering constraints are naturally embedded by selecting a suitable metric in the physical space. In this context we also present an engineering application of the theory of harmonic maps.

scholarly journals On the smallest singular value of symmetric random matrices

2021 ◽  
pp. 1-22
Author(s):  
Vishesh Jain ◽  
Ashwin Sah ◽  
Mehtaab Sawhney
Keyword(s):  
Random Matrices ◽  
Symmetric Matrix ◽  
Geometric Approach ◽  
Random Variable ◽  
Symmetric Matrices ◽  
Common Denominator ◽  
Geometric Framework ◽  
Small Constant ◽  
Special Case ◽  
Smallest Singular Value

Abstract We show that for an $n\times n$ random symmetric matrix $A_n$ , whose entries on and above the diagonal are independent copies of a sub-Gaussian random variable $\xi$ with mean 0 and variance 1, \begin{equation*}\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] \le O_{\xi}(\epsilon^{1/8} + \exp(\!-\Omega_{\xi}(n^{1/2}))) \quad \text{for all } \epsilon \ge 0.\end{equation*} This improves a result of Vershynin, who obtained such a bound with $n^{1/2}$ replaced by $n^{c}$ for a small constant c, and $1/8$ replaced by $(1/8) - \eta$ (with implicit constants also depending on $\eta > 0$ ). Furthermore, when $\xi$ is a Rademacher random variable, we prove that \begin{equation*}\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] \le O(\epsilon^{1/8} + \exp(\!-\Omega((\!\log{n})^{1/4}n^{1/2}))) \quad \text{for all } \epsilon \ge 0.\end{equation*} The special case $\epsilon = 0$ improves a recent result of Campos, Mattos, Morris, and Morrison, which showed that $\mathbb{P}[s_n(A_n) = 0] \le O(\exp(\!-\Omega(n^{1/2}))).$ Notably, in a departure from the previous two best bounds on the probability of singularity of symmetric matrices, which had relied on somewhat specialized and involved combinatorial techniques, our methods fall squarely within the broad geometric framework pioneered by Rudelson and Vershynin, and suggest the possibility of a principled geometric approach to the study of the singular spectrum of symmetric random matrices. The main innovations in our work are new notions of arithmetic structure – the Median Regularized Least Common Denominator (MRLCD) and the Median Threshold, which are natural refinements of the Regularized Least Common Denominator (RLCD)introduced by Vershynin, and should be more generally useful in contexts where one needs to combine anticoncentration information of different parts of a vector.

scholarly journals A geometric framework for modeling dynamic decisions among arbitrarily many alternatives

2018 ◽  
Author(s):  
Peter D. Kvam
Keyword(s):  
Context Effects ◽  
Geometric Approach ◽  
Stopping Rules ◽  
Decision Space ◽  
Geometric Framework ◽  
Continuous Response ◽  
Similarity Relations ◽  
Preferential Choice ◽  
Dynamic Decisions ◽  
Choice Set

Multiple-choice and continuous-response tasks pose a number of challenges for models of the decision process, from empirical challenges like context effects to computational demands imposed by choice sets with a large number of outcomes. This paper develops a general framework for constructing models of the cognitive processes underlying both inferential and preferential choice among any arbitrarily large number of alternatives. This geometric approach represents the alternatives in a choice set along with a decision maker's beliefs or preferences in a ``decision space,'' simultaneously capturing the support for different alternatives along with the similarity relations between the alternatives in the choice set. Support for the alternatives (represented as vectors) shifts over time according to the dynamics of the belief / preference state (represented as a point) until a stopping rule is met (state crosses a hyperplane) and the corresponding selection is made. I review stopping rules that guarantee optimality in multi-alternative inferential choice, minimizing response time for a desired level of accuracy, as well as methods for constructing the decision space, which can result in context effects when applied to preferential choice.

scholarly journals Geometry of variational methods: dynamics of closed quantum systems

2020 ◽  
Vol 9 (4) ◽  
Author(s):  
Lucas Hackl ◽  
Tommaso Guaita ◽  
Tao Shi ◽  
Jutho Haegeman ◽  
Eugene Demler ◽  
...  
Keyword(s):  
Variational Methods ◽  
Time Evolution ◽  
Coherent States ◽  
Geometric Approach ◽  
Quantum Systems ◽  
Imaginary Time ◽  
Excitation Spectra ◽  
Spectral Functions ◽  
Geometric Framework ◽  
Gaussian States

We present a systematic geometric framework to study closed quantum systems based on suitably chosen variational families. For the purpose of (A) real time evolution, (B) excitation spectra, (C) spectral functions and (D) imaginary time evolution, we show how the geometric approach highlights the necessity to distinguish between two classes of manifolds: Kähler and non-Kähler. Traditional variational methods typically require the variational family to be a Kähler manifold, where multiplication by the imaginary unit preserves the tangent spaces. This covers the vast majority of cases studied in the literature. However, recently proposed classes of generalized Gaussian states make it necessary to also include the non-Kähler case, which has already been encountered occasionally. We illustrate our approach in detail with a range of concrete examples where the geometric structures of the considered manifolds are particularly relevant. These go from Gaussian states and group theoretic coherent states to generalized Gaussian states.

A Computational Geometric Approach for Motion Generation of Spatial Linkages With Sphere and Plane Constraints

2018 ◽  
Vol 11 (1) ◽  
Author(s):  
Xiangyun Li ◽  
Q. J. Ge ◽  
Feng Gao
Keyword(s):  
Geometric Approach ◽  
Geometric Constraints ◽  
Dimensional Synthesis ◽  
Motion Generation ◽  
Geometric Framework ◽  
Kinematic Chains ◽  
Kinematic Mapping ◽  
Spherical Linkages ◽  
Spatial Displacements ◽  
The Given

This paper studies the problem of spatial linkage synthesis for motion generation from the perspective of extracting geometric constraints from a set of specified spatial displacements. In previous work, we have developed a computational geometric framework for integrated type and dimensional synthesis of planar and spherical linkages, the main feature of which is to extract the mechanically realizable geometric constraints from task positions, and thus reduce the motion synthesis problem to that of identifying kinematic dyads and triads associated with the resulting geometric constraints. The proposed approach herein extends this data-driven paradigm to spatial cases, with the focus on acquiring the point-on-a-sphere and point-on-a-plane geometric constraints which are associated with those spatial kinematic chains commonly encountered in spatial mechanism design. Using the theory of kinematic mapping and dual quaternions, we develop a unified version of design equations that represents both types of geometric constraints, and present a simple and efficient algorithm for uncovering them from the given motion.

THE PARASUPERSYMMETRIC SPINNING PARTICLE AND TOPOLOGICAL GAUGE FIELDS

1992 ◽  
Vol 07 (15) ◽  
pp. 3493-3514 ◽  
Author(s):  
G. P. KORCHEMSKY
Keyword(s):  
Wave Function ◽  
Gauge Field ◽  
Polynomial Algebra ◽  
Physical Space ◽  
Symmetry Algebra ◽  
Massless Particle ◽  
Gauge Fields ◽  
Spinning Particle ◽  
Chern Simons ◽  
Special Case

The first quantization of the D-dimensional relativistic spinning particle with the action invariant under reparametrizations and local worldline parasupersymmetric transformations is performed. The corresponding symmetry algebra is not of the Lie kind and is known as a polynomial algebra. It is found that the physical space of the massless particle is described by the strength tensors of Abelian antisymmetric gauge fields and topological gauge fields. In the special case D = 3 the Abelian Chern–Simons gauge field appears as a wave function of the particle. The generalizations to a massive parasupersymmetric spinning particle are presented.

THE MODULI SPACE OF SUPERCONFORMAL INSTANTONS IN SIGMA MODELS

1991 ◽  
Vol 06 (19) ◽  
pp. 1787-1796 ◽  
Author(s):  
M. I. MONASTYRSKY ◽  
S. M. NATANZON
Keyword(s):  
Sigma Models ◽  
Moduli Space ◽  
Physical Space ◽  
Target Space ◽  
Topological Invariants ◽  
New Approach ◽  
Special Case

A new approach to instantons in supersymmetrical 2-dimensional sigma models is discussed. In this approach superinstantons are characterized as the superconformal maps of a physical space into the isotopic (target) space. We consider a special case of the supersphere with punctures. New topological invariants as the number of the so-called fermionic points appear in this case. We also analyze the structure of the moduli space of superinstantons within this framework.

scholarly journals Rotational-monad metaphysics

2021 ◽  
Author(s):  
Roman Kremen
Keyword(s):  
Three Dimensional ◽  
Physical Space ◽  
Discrete Element ◽  
Physical Reality ◽  
Foundations Of Mathematics ◽  
Important Special Case ◽  
Infinite Dimensional ◽  
The One ◽  
Special Case ◽  
Mathematics And Physics

The article presents a metaphysical concept, in the main tesa of which the simplest discrete element of physical reality is constituted — designated as a protomonad — which forms the basis of spatial forms of materiality, including space itself as such. The genesis of the protomonad is clarified by a certain way interpreted rotation of the spiritual essence, which in itself does not have an extension, and both the indicated essence and its rotation have a metaphysical order, and the dimension of physical space finds a rational interpretation through the characteristics of metaphysical rotation. The semantic aspects of complex mathematical constructs are considered that convey the semantics of rotations, and quite reasonably proposed by some mathematicians as the unified foundations of mathematics and physics, where the properties of constructs act as a mathematically strict co-proof of the validity of the concept. The meaning of number is explained as a method of restriction on infinite pre-physical multiplicity, and finite natural multiplicities are the result of such restrictions; the most important special case is the three-dimensional spatial metric given in the experiment, which appears as a restriction of an infinite-dimensional metaphysical space. The so-called principle of genetic inheritance is formulated, which makes it possible to remove the dialectical opposition between the one and the multiple and illustrates the categories of time and space as dialectical oppositions.

scholarly journals The geometry of the space of Cauchy data of nonlinear PDEs

2013 ◽  
Vol 11 (11) ◽  
Author(s):  
Giovanni Moreno
Keyword(s):  
Infinite Order ◽  
Geometric Approach ◽  
Cauchy Data ◽  
Nonlinear Pdes ◽  
General Notion ◽  
Geometric Framework ◽  
Transversality Conditions ◽  
Jet Bundles ◽  
First Order ◽  
Definition Of

AbstractFirst-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways — for instance, by means of Grassmann bundles. In this paper we generalize it by means of flag bundles, and develop the corresponding theory for higher-oder and infinite-order jet bundles. We show that this is a natural geometric framework for the space of Cauchy data for nonlinear PDEs. As an example, we derive a general notion of transversality conditions in the Calculus of Variations.

scholarly journals ON THE SHAPES OF ELEMENTARY DOMAINS OR WHY MANDELBROT SET IS MADE FROM ALMOST IDEAL CIRCLES?

2008 ◽  
Vol 23 (22) ◽  
pp. 3613-3684 ◽  
Author(s):  
V. DOLOTIN ◽  
A. MOROZOV
Keyword(s):  
Phase Transition ◽  
Forest Structure ◽  
Qualitative Difference ◽  
Three Dimensional ◽  
Geometric Approach ◽  
Mandelbrot Set ◽  
The Family ◽  
Mandelbrot Sets ◽  
Special Case

Direct look at the celebrated "chaotic" Mandelbrot Set (in Fig. 1) immediately reveals that it is a collection of almost ideal circles and cardioids, unified in a specific forest structure. In the paper arXiv:hep-th/0501235, a systematic algebro-geometric approach was developed to the study of generic Mandelbrot sets, but emergency of nearly ideal circles in the special case of the family x2 + c was not fully explained. In the present, paper, the shape of the elementary constituents of Mandelbrot Set is explicitly calculated, and difference between the shapes of root and descendant domains (cardioids and circles respectively) is explained. Such qualitative difference persists for all other Mandelbrot sets: descendant domains always have one less cusp than the root ones. Details of the phase transition between different Mandelbrot sets are explicitly demonstrated, including overlaps between elementary domains and dynamics of attraction/repulsion regions. Explicit examples of three-dimensional sections of Universal Mandelbrot Set are given. Also a systematic small-size approximation is developed for evaluation of various Feigenbaum indices.

scholarly journals A geometric approach to scaling individual distributions to macroecological patterns

2018 ◽  
Author(s):  
Nao Takashina ◽  
Buntarou Kusumoto ◽  
Yasuhiro Kubota ◽  
Evan P. Economo
Keyword(s):  
Point Processes ◽  
Geometric Model ◽  
Species Abundance ◽  
Geometric Approach ◽  
Distribution Patterns ◽  
Single Equation ◽  
Species Range ◽  
Geometric Framework ◽  
Individual Level ◽  
Species Area

Understanding macroecological patterns across scales is a central goal of ecology and a key need for conservation biology. Much research has focused on quantifying and understanding macroecological patterns such as the species-area relationship (SAR), the endemic-area relationship (EAR) and relative species abundance curve (RSA). Understanding how these aggregate patterns emerge from underlying spatial pattern at individual level, and how they relate to each other, has both basic and applied relevance. To address this challenge, we develop a novel spatially explicit geometric framework to understand multiple macroecological patterns, including the SAR, EAR, RSA, and their relationships, using theory of point processes. The geometric approach provides a theoretical framework to derive SAR, EAR, and RSA from species range distributions and the pattern of individual distribution patterns therein. From this model, various well-documented macroecological patterns are recovered, including the tri-phasic SAR on a log-log plot with its asymptotic slope, and various RSAs (e.g., Fisher's logseries and the Poisson lognormal distribution). Moreover, this approach can provide new insights such as a single equation describing the RSA at an arbitrary spatial scale, and explicit forms of the EAR with its asymptotic slope. The theory, which links spatial distributions of individuals and species with macroecological patterns, is ambiguous with regards to the mechanism(s) responsible for the statistical properties of individual distributions and species range sizes. However, our approach can be connected to mechanistic models that make such predictions about lower-level patterns and be used to scale them up to aggregate patterns, and therefore is applicable to many ecological questions. We demonstrate an application of the geometric model to scaling issue of beta diversity.

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