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projection functions, area measures and the Alesker-Fourier transform

JOURNAL OF FUNCTIONAL ANALYSIS 2017年第6期273卷 2026-2069页

作者： Dorrek, Felix Schuster, Franz E. Vienna Univ Technol Inst Discrete Math & Geometry Wiedner Hauptstr 8-10-1047 A-1040 Vienna Austria

Dual to Koldobsky's notion of j-intersection bodies, the class of j-projection bodies is introduced, generalizing Minkowski's classical notion of projection, bodies of convex bodies. A Fourier analytic characterization of j-projection bodies in terms of their area measures of order j is obtained. In turn, this yields an equivalent characterization of j-projection bodies involving Alesker's Fourier type transform on translation invariant smooth spherical valuations, As applications of these results, several basic properties of j-projection bodies are established and new non-trivial examples are constructed. (C) 2017 Elsevier Inc. All rights reserved.

关键词： projection functions Area measures Valuations Alesker-Fourier transform

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THE HETEROGENEITY OF VENTRAL TEGMENTAL AREA NEURONS: projection functions IN A MOOD-RELATED CONTEXT

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NEUROSCIENCE 2014年 282卷 101-108页

作者： Walsh, J. J. Han, M. H. Icahn Sch Med Mt Sinai Dept Pharmacol & Syst Therapeut New York NY 10029 USA Icahn Sch Med Mt Sinai Grad Sch Biomed Sci Neurosci Program New York NY 10029 USA Icahn Sch Med Mt Sinai Friedman Brain Inst Dept Neurosci New York NY 10029 USA

The ventral tegmental area (VTA) in the brain's reward circuitry is composed of a heterogeneous population of dopamine, GABA, and glutamate neurons that play important roles in mediating mood-related functions including depression. These neurons project to different brain regions, including the nucleus accumbens (NAc), the medial prefrontal cortex (mPFC), and the amygdala. The functional understanding of these projection pathways has been improved since the extensive use of advanced techniques such as viral-mediated gene transfer, cell-type-specific neurophysiology and circuit-probing optogenetics. In this article, we will discuss the recent progress in understanding these VTA projection-specific functions, focusing on mood-related disorders. This article is part of a Special Issue entitled: Ventral Tegmentum & Dopamine. Published by Elsevier Ltd. on behalf of IBRO.

关键词： ventral tegmental area projection functions animal models mood disorders neural circuits depression

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Completeness of a relational calculus for program schemes

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THEORETICAL COMPUTER SCIENCE 2001年第1-2期254卷 543-556页

作者： Frias, MF Maddux, RD Iowa State Univ Dept Math Ames IA 50011 USA

The relational calculus,MU was presented in Willem-Paul de Roever's dissertation as a framework for describing and proving properties of programs. MU is axiomatized by de oever in stages. The next-to-last stage is the calculus MU2, namely MU without the recursive p-operator. Its axioms include typed versions of Tarski's axioms for the calculus of relations, together with axioms for the projection functions. For,MU there is, in addition, an axiom expressing the least-fixed-point property of terms containing the mu -operator, and Scott's induction rule. Thus,MU2 is a calculus for nonrecursive program schemes. Around 1976 David Park conjectured that de Roever's axiomatization for MU2 is complete. In this paper, we confirm Park's conjecture. (C) 2001 Elsevier Science B.V. All rights reserved.

关键词： relational calculus completeness recursive program schemes projection functions

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SPECIFICATION AND VERIFICATION OF DIGITAL-SYSTEMS USING HIGHER-ORDER PREDICATE LOGIC

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IEE PROCEEDINGS-E COMPUTERS AND DIGITAL TECHNIQUES 1986年第5期133卷 242-254页

作者： HANNA, FK DAECHE, N Electronic Engineering Laboratories University of Kent Canterbury UK

The paper describes how higher-order predicate logic may be used to specify both the structure and the behaviour of a digital system, and to reason about their interrelationship. The overall approach is named VERITAS; the paper concentrates particularly on describing its methodological aspects. The behaviour of a system is specified by a predicate on the analogue waveforms at the ports of the system. In general, behavioural specifications are partial. The internal structure of a system is defined by a set of projection functions that yield its component parts, together with a set of equations describing their interconnections. Reasoning about the behavioural properties of digital systems is carried out within the framework of an axiomatic theory that describes relevant properties of arithmetic, time, waveforms and structures. The logic is embedded within a programming language, MV, whose data types include signature, term and derivation. This allows inferencing to be carried out computationally, which in turn guarantees its correctness.

关键词： arithmetic digital systems specification computer testing higher-order predicate logic programming language projection functions axiomatic theory analogue waveforms Performance evaluation and testing structures waveforms MV VERITAS time verification

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A simple and effective inverse projection scheme for void distribution control in topology optimization

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STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION 2009年第4期39卷 359-371页

作者： Almeida, S. R. M. Paulino, G. H. Silva, E. C. N. Univ Illinois Dept Civil & Environm Engn Newmark Lab Urbana IL 61801 USA Univ Fed Goias Escola Engn Civil BR-74605220 Goiania Go Brazil Univ Sao Paulo Dept Mechatron & Mech Syst Engn Escola Politecn BR-05508900 Sao Paulo Brazil

The ability to control both the minimum size of holes and the minimum size of structural members are essential requirements in the topology optimization design process for manufacturing. This paper addresses both requirements by means of a unified approach involving mesh-independent projection techniques. An inverse projection is developed to control the minimum hole size while a standard direct projection scheme is used to control the minimum length of structural members. In addition, a heuristic scheme combining both contrasting requirements simultaneously is discussed. Two topology optimization implementations are contributed: one in which the projection (either inverse or direct) is used at each iteration;and the other in which a two-phase scheme is explored. In the first phase, the compliance minimization is carried out without any projection until convergence. In the second phase, the chosen projection scheme is applied iteratively until a solution is obtained while satisfying either the minimum member size or minimum hole size. Examples demonstrate the various features of the projection-based techniques presented.

关键词： Topology optimization projection functions Direct projection Inverse projection Combined projection Two-phase optimization

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Topology optimization with multiple phase projection

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2009年第1-4期199卷 123-135页

作者： Guest, James K. Johns Hopkins Univ Dept Civil Engn Baltimore MD 21218 USA

Topology optimization is a powerful tool capable of generating new solutions to engineering design problems. While these designs may offer optimal performance in a computational setting, it is not uncommon for them to be impractical or unrealizable from engineering or fabrication points-of-view. This challenge motivates the present work. A methodology is proposed for restricting the minimum length scale of each material phase used in the design. The technique allows a designer to, for example, prescribe a minimum allowable length scale of structural members (solid phase) as well as the minimum allowable length scale on holes (void phase). The proposed approach utilizes the Heaviside projection Method (HPM) to continuum topology optimization. Each material phase is associated with a design variable field that is projected onto element space using regularized Heaviside functions. The fields are independently projected and the resulting distributions are assembled to yield topology. The technique is demonstrated on classic topology optimization problems including minimum compliance, heat conduction, and compliant mechanisms. Solutions are shown to be near-discrete topologies satisfying minimum length scale criterion placed on each phase. Control over length scale is achieved implicitly and therefore the technique does not require additional constraints. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Minimum length scale Manufacturability Filters projection functions Heaviside projection Method SIMP

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Honeycomb Wachspress finite elements for structural topology optimization

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STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION 2009年第6期37卷 569-583页

作者： Talischi, Cameron Paulino, Glaucio H. Le, Chau H. Univ Illinois Dept Civil & Environm Engn Newman Lab Urbana IL 61801 USA

Traditionally, standard Lagrangian-type finite elements, such as linear quads and triangles, have been the elements of choice in the field of topology optimization. However, finite element meshes with these conventional elements exhibit the well-known "checkerboard" pathology in the iterative solution of topology optimization problems. A feasible alternative to eliminate such long-standing problem consists of using hexagonal (honeycomb) elements with Wachspress-type shape functions. The features of the hexagonal mesh include two-node connections (i.e. two elements are either not connected or connected by two nodes), and three edge-based symmetry lines per element. In contrast, quads can display one-node connections, which can lead to checkerboard;and only have two edge-based symmetry lines. In addition, Wachspress rational shape functions satisfy the partition of unity condition and lead to conforming finite element approximations. We explore the Wachspress-type hexagonal elements and present their implementation using three approaches for topology optimization: element-based, continuous approximation of material distribution, and minimum length-scale through projection functions. Examples are presented that demonstrate the advantages of the proposed element in achieving checkerboard-free solutions and avoiding spurious fine-scale patterns from the design optimization process.

关键词： Topology optimization Checkerboard Wachspress interpolation functions Continuous approximation of material distribution projection functions

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Novel finite particle formulations based on projection methodologies

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS 2011年第11-12期65卷 1376-1388页

作者： Asprone, D. Auricchio, F. Reali, A. Univ Pavia Dept Struct Mech I-27100 Pavia Italy Univ Naples Federico 2 Dept Struct Engn Naples Italy

Particle methods are among the most studied meshless approaches, with applications ranging from solid mechanics to fluid-dynamics and thermo-dynamics. The objective of the present paper is to analyze the behavior of finite particle formulations based on projection methodologies, investigating in particular how these approaches behave in approximating 1D and 2D second-order problems. Moreover, the issue of choosing suitable projection functions is discussed and 1D and 2D numerical tests, showing the second-order accuracy of the methods under investigation, are performed. Copyright (C) 2010 John Wiley & Sons, Ltd.

关键词： meshless methods particle methods projection functions second-order accuracy smoothed particle hydrodynamics 2D Poisson problem

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Nakajima's problem for general convex bodies

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 2009年第1期137卷 255-263页

作者： Hug, Daniel Univ Karlsruhe TH Inst Algebra & Geometrie Fak Math KIT D-76128 Karlsruhe Germany

For a convex body K subset of R-n, the kth projection function of K assigns to any k-dimensional linear subspace of R-n the k-volume of the orthogonal projection of K to that subspace. Let K and K-0 be convex bodies in Rn, and let K0 be centrally symmetric and satisfy a weak regularity assumption. Let i, j is an element of N be such that 1 <= i <= j <= n-2 with (i, j) not equal (1, n-2). Assume that K and K0 have proportional ith projection functions and proportional jth projection functions. Then we show that K and K-0 are homothetic. In the particular case where K-0 is a Euclidean ball, we thus obtain characterizations of Euclidean balls as convex bodies having constant i-brightness and constant j-brightness. This special case solves Nakajima's problem in arbitrary dimensions and for general convex bodies for most indices (i, j).

关键词： projection functions SURFACE-AREA BRIGHTNESS WIDTH SETS

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Contour features using morphological transforms

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ELECTRONICS LETTERS 1998年第1期34卷 56-57页

作者： Zois, EN Anastassopoulos, V Univ Patras Dept Phys Elect Lab Patras 26100 Greece

A feature extraction method is proposed for contour or graph information retrieval. The procedure is based on morphologically processing the projection functions of the contour. The image of the contour is processed not only as a whole, but also divided into sub-blocks. The level of image partitioning may vary depending on the degree of feature decomposition required and the achieved discriminative capabilities. Simulation results are provided.

关键词： image recognition transforms feature extraction method Integral transforms morphological transforms feature decomposition projection functions feature extraction Pattern recognition Optical information, image and video signal processing graph information retrieval image partitioning Computer vision and image processing techniques mathematical morphology contour features

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