Optimization of Utility Functions in an Admissible Space of Higher Dimension

2016 ◽  
pp. 102-113
Author(s):  
German Almanza ◽  
Victor M. Carrillo ◽  
Cely C. Ronquillo
Keyword(s):  
Graduate Students ◽  
Linear Algebra ◽  
Algebraic Topology ◽  
Morse Theory ◽  
Utility Functions ◽  
Higher Dimension ◽  
Differential Topology ◽  
Several Variables

S. Smale published a paper where announce a theorem which optimize a several utility functions at once (cf. Smale, 1975) using Morse Theory, this is a very abstract subject that require high skills in Differential Topology and Algebraic Topology. Our goal in this paper is announce the same theorems in terms of Calculus of Manifolds and Linear Algebra, those subjects are more reachable to engineers and economists whom are concern with maximizing functions in several variables. Moreover, the elements involved in our theorems are accessible to graduate students, also we putting forward the results we consider economically relevant.

scholarly journals Detection of the homotopy type of an object using differential invariants of an approximating map

2019 ◽  
Vol 43 (4) ◽  
pp. 611-617
Author(s):  
S.V. Kurochkin
Keyword(s):  
Neural Network ◽  
Data Analysis ◽  
Homotopy Type ◽  
Morse Theory ◽  
Object Representation ◽  
Differential Invariants ◽  
Topological Data Analysis ◽  
Higher Dimension ◽  
Differential Topology ◽  
Open Questions

A method of topological data analysis is proposed that allows one to find out the homotopy type of the object under study. Unlike mature and widely used methods based on persistent homologies, our method is based on computing differential invariants of some map associated with an approximating map. Differential topology tools and the analogy with the main result in Morse theory are used. The approximating map can be constructed in the usual way using a neural network or otherwise. The method allows one to identify the homotopy type of an object in the plane because the number of circles in the homotopy equivalent object representation as a wedge is expressed through the degree of some map associated with the approximating map. The performance of the algorithm is illustrated by examples from the MNIST database and transforms thereof. Generalizations and open questions relating to a higher-dimension case are discussed.

Review Lecture Periods of integrals and topology of algebraic varieties

1984 ◽  
Vol 391 (1801) ◽  
pp. 231-254
Keyword(s):  
Nineteenth Century ◽  
Singular Point ◽  
Algebraic Topology ◽  
Major Theme ◽  
Algebraic Varieties ◽  
Hodge Structures ◽  
Algebraic Functions ◽  
Several Variables ◽  
And Topology ◽  
Satisfactory Theory

A major theme of nineteenth century mathematics was the study of integrals of algebraic functions of one variable. This culminated in Riemann’s introduction of the surfaces that bear his name and analysis of periods of integrals on cycles on the surface. The creation of a correspondingly satisfactory theory for functions of several variables had to wait on the development of algebraic topology and its application by Lefschetz to algebraic varieties. These results were refined by Hodge’s theory of harmonic integrals. A closer analysis of Hodge structures by P. A. Griffiths and P. Deligne in recent years has led to unexpectedly strong restrictions on the topology of the variety and to a diversity of other applications. This advance is closely linked to the study of variation of integrals under deformations, particularly in the neighbourhood of a singular point.

scholarly journals PyTrilinos: Recent Advances in the Python Interface to Trilinos

2012 ◽  
Vol 20 (3) ◽  
pp. 311-325 ◽  
Author(s):  
William F. Spotz
Keyword(s):  
Linear Algebra ◽  
Utility Functions ◽  
Ease Of Use ◽  
Linear Solution ◽  
Dense Linear Algebra ◽  
Multilevel Preconditioners ◽  
Front End ◽  
Nonlinear Solvers ◽  
Sparse Linear Algebra ◽  
Python Package

PyTrilinos is a set of Python interfaces to compiled Trilinos packages. This collection supports serial and parallel dense linear algebra, serial and parallel sparse linear algebra, direct and iterative linear solution techniques, algebraic and multilevel preconditioners, nonlinear solvers and continuation algorithms, eigensolvers and partitioning algorithms. Also included are a variety of related utility functions and classes, including distributed I/O, coloring algorithms and matrix generation. PyTrilinos vector objects are compatible with the popular NumPy Python package. As a Python front end to compiled libraries, PyTrilinos takes advantage of the flexibility and ease of use of Python, and the efficiency of the underlying C++, C and Fortran numerical kernels. This paper covers recent, previously unpublished advances in the PyTrilinos package.

scholarly journals Particulate Exotica

2021 ◽  
Author(s):  
Fan Zhang
Keyword(s):  
Algebraic Topology ◽  
Complete Information ◽  
Differential Topology ◽  
Four Dimensions ◽  
Recent Advances ◽  
Obvious Reason ◽  
Microscopic World

Recent advances in differential topology single out four-dimensions as being special, allowing for vast varieties of exotic smoothness (differential) structures, distinguished by their handlebody decompositions, even as the coarser algebraic topology is fixed. Should the spacetime we reside in takes up one of the more exotic choices, and there is no obvious reason why it shouldn't, apparent pathologies would inevitably plague calculus-based physical theories assuming the standard vanilla structure, due to the non-existence of a diffeomorphism and the consequent lack of a suitable portal through which to transfer the complete information regarding the exotic physical dynamics into the vanilla theories. An obvious plausible consequence of this deficiency would be the uncertainty permeating our attempted description of the microscopic world. We tentatively argue here, that a re-inspection of the key ingredients of the phenomenological particle models, from the perspective of exotica, could possibly yield interesting insights. Our short and rudimentary discussion is qualitative and speculative, because the necessary mathematical tools have only just began to be developed.

5. Flavours of topology

2019 ◽  
pp. 90-114
Author(s):  
Richard Earl
Keyword(s):  
19Th Century ◽  
Set Theory ◽  
Algebraic Topology ◽  
General Topology ◽  
Major Theme ◽  
Henri Poincaré ◽  
Combinatorial Topology ◽  
Differential Topology ◽  
French Mathematician ◽  
Modern Mathematics

From the mid-19th century, topological understanding progressed on various fronts. ‘Flavours of topology’ considers other areas such as differential topology, algebraic topology, and combinatorial topology. Geometric topology concerned surfaces and grew out of the work of Euler, Möbius, Riemann, and others. General topology was more analytical and foundational in nature; Hausdorff was its most significant progenitor and its growth mirrored other fundamental work being done in set theory. The chapter introduces the hairy ball theorem, and the work of great French mathematician and physicist Henri Poincaré, which has been rigorously advanced over the last century, making algebraic topology a major theme of modern mathematics.

An Introduction to Quantum Computing

2006 ◽  
Author(s):  
Phillip Kaye ◽  
Raymond Laflamme ◽  
Michele Mosca
Keyword(s):  
Quantum Computing ◽  
Graduate Students ◽  
Prior Knowledge ◽  
Linear Algebra ◽  
Spectral Decomposition ◽  
Tensor Products ◽  
Vector Spaces ◽  
Computer Engineering ◽  
Physical Sciences ◽  
Inner Products

This concise, accessible text provides a thorough introduction to quantum computing - an exciting emergent field at the interface of the computer, engineering, mathematical and physical sciences. Aimed at advanced undergraduate and beginning graduate students in these disciplines, the text is technically detailed and is clearly illustrated throughout with diagrams and exercises. Some prior knowledge of linear algebra is assumed, including vector spaces and inner products. However, prior familiarity with topics such as tensor products and spectral decomposition is not required, as the necessary material is reviewed in the text.

scholarly journals RIEMANNIAN GEOMETRICAL OPTICS: SURFACE WAVES IN DIFFRACTIVE SCATTERING

2000 ◽  
Vol 12 (06) ◽  
pp. 849-872 ◽  
Author(s):  
E. DE MICHELI ◽  
G. MONTI BRAGADIN ◽  
G. A. VIANO
Keyword(s):  
Algebraic Topology ◽  
Morse Theory ◽  
Riemannian Geometry ◽  
Diffraction Theory ◽  
Saddle Point Method ◽  
Asymptotic Approximations ◽  
Homotopy Classes ◽  
Manifold With Boundary ◽  
Diffractive Scattering ◽  
The Cauchy Problem

The geometrical diffraction theory, in the sense of Keller, is here reconsidered as an obstacle problem in the Riemannian geometry. The first result is the proof of the existence and the analysis of the main properties of the "diffracted rays", which follow from the non-uniqueness of the Cauchy problem for geodesics in a Riemannian manifold with boundary. Then, the axial caustic is here regarded as a conjugate locus, in the sense of the Riemannian geometry, and the results of the Morse theory can be applied. The methods of the algebraic topology allow us to introduce the homotopy classes of diffracted rays. These geometrical results are related to the asymptotic approximations of a solution of a boundary value problem for the reduced wave equation. In particular, we connect the results of the Morse theory to the Maslov construction, which is used to obtain the uniformization of the asymptotic approximations. Then, the border of the diffracting body is the envelope of the diffracted rays and, instead of the standard saddle point method, use is made of the procedure of Chester, Friedman and Ursell to derive the damping factors associated with the rays which propagate along the boundary. Finally, the amplitude of the diffracted rays when the diffracting body is an opaque sphere is explicitly calculated.

scholarly journals Fluctuations in Overlapping Generations Economies

2009 ◽  
Vol 9 (1) ◽  
Author(s):  
Mich Tvede
Keyword(s):  
Linear Algebra ◽  
Implicit Function Theorem ◽  
Overlapping Generations ◽  
Utility Functions ◽  
The State ◽  
Implicit Function ◽  
Endogenous Fluctuations ◽  
Pure Exchange ◽  
Sunspot Equilibria ◽  
Dense Set

In the present paper stationary pure-exchange overlapping generations economies with L goods per date and M consumers per generation are considered. It is shown that for an open and dense set of utility functions there exist endowment vectors such that N-cycles exist for N less than or equal to L+1 and L less than or equal to M. The approach to existence of endogenous fluctuations is basic in the sense that the prime ingredients are the implicit function theorem and linear algebra. Moreover it is sketched how the approach can be applied to show that for an open and dense set of utility functions there exist endowment vectors such that sunspot equilibria, where prices at every date only depends on the state at that date, exist.

Mathematical analysis in examples and tasks. Part 2

2020 ◽  
Author(s):  
Galina Zhukova ◽  
Margarita Rushaylo
Keyword(s):  
Field Theory ◽  
Graduate Students ◽  
Mathematical Analysis ◽  
Differential Calculus ◽  
Advanced Mathematics ◽  
Analytic Geometry ◽  
Several Variables ◽  
Background Material ◽  
Basic Concepts ◽  
Independent Work

The purpose of the textbook is to help students to master basic concepts and research methods used in mathematical analysis. In part 2 of the proposed cycle of workshops on the following topics: analytic geometry in space; differential calculus of functions of several variables; local, conditional, global extrema of functions of several variables; multiple, curvilinear and surface integrals; elements of field theory; numerical, power series, Fourier series; applications to the analysis and solution of applied problems. These topics are studied in universities, usually in the second semester in the discipline "Mathematical analysis" or the course "Higher mathematics", "Mathematics". For the development of each topic the necessary theoretical and background material, reviewed a large number of examples with detailed analysis and solutions, the options for independent work. For self-training and quality control of the acquired knowledge in each section designed exercises and tasks with answers and guidance. It is recommended that teachers, students and graduate students studying advanced mathematics.

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