One-Parameter Groups of Linear Transformations

(1994). A Trace Inequality for Unitary Matrices. The American Mathematical Monthly: Vol. 101, No. 5, pp. 453-455.

Qualitative Problems in Matrix Theory

Adequate predimension inequalities in differential fields

Three conservation laws are associated with the dynamics of Hamiltonian systems with symmetry: The total energy, the momentum map associated with the symmetry group, and the symplectic structure are invariant under the flow. Discrete time approximations of Hamiltonian flows typically do not share these properties unless specifically designed to do so. We develop explicit conservation conditions for a general class of algorithms on Lie groups. For the rigid body these conditions lead to a single-step algorithm that exactly preserves the energy, spatial momentum, and symplectic form. For homogeneous nonlinear elasticity, we find algorithms that conserve angular momentum and either the energy or the symplectic form.

The material presented below amounts roughly to the so-called fundamental theorems of Lie and their implications concerning Lie algebras, Lie subgroups and subalgebras. By straightforward and fairly elementary steps, we shall extend the concept of Lie group to include groups admitting coordinate systems in which the functions defining ab, the group product of elements a, b, possess continuous first order derivatives in the coordinates of a and satisfy a Lipschitz condition in those of b. That such groups are equivalent to classical (i.e. analytical) Lie groups was announced in 1936 by van Kampen, although apparently van Kampen published nothing by way of proof beyond a certain decisive uniqueness theorem concerning systems of ordinary differential equations. (This is contained in our theorem (12.1); the proof given below is essentially that of van Kampen [3]). It is conceivable that the condition relative to coordinates of b could be weakened or dispensed with entirely. For, the only part the condition in question plays in our development is to make it certain that the inverse of a certain mapping-the mapping into canonical coordinates-is single-valued. In this connection mention should be made of the paper [1J of Garrett Birkhoff in which it is shown that the existence of continuous first order derivatives of ab with respect to the coordinates of both the a's and the b's is sufficient for defining a Lie group. We shall consider only groups with a finite number of real parameters (whereas the groups considered by Birkhoff are not necessarily finite dimensional). This makes it possible to use the standard existence theorems for systems of ordinary real differential equations. We make no use whatever of the theory of partial differential equations. We maintain throughout a purely local point of view in the sense that we consider only what happens in the neighborhood of the identity. Historical notes. The proof of the uniqueness of square roots (8.5) is due to Claude Chevalley (unpublished). The proof given below of Lie's theorem that a linear system of vector fields which is closed under commutation defines a Lie group is, we believe, due to van der Waerden [4]. We have supplied a number of preliminary lemmas which make the proof rigorously applicable to the case in which the vector fields are only assumed to possess continuous first order derivatives. The theorem in ?17 that every local subgroup of a Lie group is a Lie group is due essentially to E. Cartan [2]. 481

Previous article Next article Complementarity Theorems for Linear ProgrammingA. C. WilliamsA. C. Williamshttps://doi.org/10.1137/1012015PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Frank Eugene Clark, Mathematical Notes: Remark on the Constraint Sets in Linear Programming, Amer. Math. Monthly, 68 (1961), 351–352 MR1531192 0109.38204 CrossrefISIGoogle Scholar[2] A. J. Goldman, Resolution and separation theorems for polyhedral convex setsLinear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N. J., 1956, 41–51, in [4] MR0089113 0072.37505 Google Scholar[3] A. J. Goldman and , A. W. Tucker, Theory of linear programmingLinear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N.J., 1956, 53–97, in [4] MR0101826 0072.37601 Google Scholar[4] H. W. Kuhn and , A. W. Tucker, Linear Inequalities and Related Systems, Princeton University Press, Princeton, 1956 0072.37502 Google Scholar[5] A. C. Williams, Boundedness relations for linear constraint sets, Linear Algebra and Appl., 3 (1970), 129–141 10.1016/0024-3795(70)90009-1 MR0266612 0201.22003 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Sparse solutions to an underdetermined system of linear equations via penalized Huber loss6 November 2020 | Optimization and Engineering, Vol. 22, No. 3 Cross Ref Dynamic Non-diagonal Regularization in Interior Point Methods for Linear and Convex Quadratic Programming26 February 2019 | Journal of Optimization Theory and Applications, Vol. 181, No. 3 Cross Ref Necessary and Sufficient Conditions for Noiseless Sparse Recovery via Convex Quadratic SplinesMustafa Ç Pinar12 February 2019 | SIAM Journal on Matrix Analysis and Applications, Vol. 40, No. 1AbstractPDF (416 KB)A labeling algorithm for the sensitivity ranges of the assignment problemApplied Mathematical Modelling, Vol. 35, No. 10 Cross Ref Bibliography15 August 2011 Cross Ref Sensitivity Analysis and Dynamic Programming15 February 2011 Cross Ref Bibliography Cross Ref Condition measures and properties of the central trajectory of a linear programMathematical Programming, Vol. 83, No. 1-3 Cross Ref New characterizations of ℓ1 solutions to overdetermined systems of linear equationsOperations Research Letters, Vol. 16, No. 3 Cross Ref Limiting behavior of weighted central paths in linear programmingMathematical Programming, Vol. 65, No. 1-3 Cross Ref Stability of linearly constrained convex quadratic programsJournal of Optimization Theory and Applications, Vol. 64, No. 1 Cross Ref Marginal values in mixed integer linear programmingMathematical Programming, Vol. 44, No. 1-3 Cross Ref A theory of linear inequality systemsLinear Algebra and its Applications, Vol. 106 Cross Ref Boundedness relations in linear semi-infinite programmingAdvances in Applied Mathematics, Vol. 8, No. 1 Cross Ref A Variable-Complexity Norm Maximization ProblemO. L. Mangasarian and T. -H. Shiau17 July 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 7, No. 3AbstractPDF (698 KB)Simple computable bounds for solutions of linear complementarity problems and linear programs26 February 2009 Cross Ref On polyhedral extension of some LP theoremsMathematical Programming, Vol. 30, No. 1 Cross Ref Polyhedral extensions of some theorems of linear programmingMathematical Programming, Vol. 24, No. 1 Cross Ref Optimal simplex tableau characterization of unique and bounded solutions of linear programsJournal of Optimization Theory and Applications, Vol. 35, No. 1 Cross Ref Projection and Restriction Methods in Geometric Programming and Related Problems Cross Ref Representation of Convex Sets Cross Ref Projection and restriction methods in geometric programming and related problemsJournal of Optimization Theory and Applications, Vol. 26, No. 1 Cross Ref The complementary unboundedness of dual feasible solution sets in convex programmingMathematical Programming, Vol. 12, No. 1 Cross Ref Theorems on the dimension of convex setsLinear Algebra and its Applications, Vol. 12, No. 1 Cross Ref On the primal and dual constraint sets in geometric programmingJournal of Mathematical Analysis and Applications, Vol. 32, No. 3 Cross Ref Volume 12, Issue 1| 1970SIAM Review History Submitted:23 December 1968Published online:18 July 2006 InformationCopyright © 1970 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1012015Article page range:pp. 135-137ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

We introduce poly-symplectic extension of Souriau Lie groups thermodynamics based on higher-order model of statistical physics introduced by Ingarden. This extended model could be used for small data analytics and machine learning on Lie groups. Souriau geometric theory of heat is well adapted to describe density of probability (maximum entropy Gibbs density) of data living on groups or on homogeneous manifolds. For small data analytics (rarified gases, sparse statistical surveys, …), the density of maximum entropy should consider higher order moments constraints (Gibbs density is not only defined by first moment but fluctuations request 2nd order and higher moments) as introduced by Ingarden. We use a poly-sympletic model introduced by Christian Günther, replacing the symplectic form by a vector-valued form. The poly-symplectic approach generalizes the Noether theorem, the existence of moment mappings, the Lie algebra structure of the space of currents, the (non-)equivariant cohomology and the classification of G-homogeneous systems. The formalism is covariant, i.e., no special coordinates or coordinate systems on the parameter space are used to construct the Hamiltonian equations. We underline the contextures of these models, and the process to build these generic structures. We also introduce a more synthetic Koszul definition of Fisher Metric, based on the Souriau model, that we name Souriau-Fisher metric. This Lie groups thermodynamics is the bedrock for Lie group machine learning providing a full covariant maximum entropy Gibbs density based on representation theory (symplectic structure of coadjoint orbits for Souriau non-equivariant model associated to a class of co-homology).

The main objective of this article is to review the major progress that has been made on controllability of dynamical systems over the past number of years. Controllability is one of the fundamental concepts in the mathematical control theory. This is a qualitative property of dynamical control systems and is of particular importance in control theory. A systematic study of controllability was started at the beginning of sixties in the last century, when the theory of controllability based on the description in the form of state space for both time-invariant and time-varying linear control systems was worked out. Roughly speaking, controllability generally means, that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It should be mentioned, that in the literature there are many different definitions of controllability, which strongly depend on a class of dynamical control systems and on the other hand on the form of admissible controls. Controllability problems for different types of dynamical systems require the application of numerous mathematical concepts and methods taken directly from differential geometry, functional analysis, topology, matrix analysis and theory of ordinary and partial differential equations and theory of difference equations. In the paper we use mainly state-space models of dynamical systems, which provide a robust and universal method for studying controllability of various classes of systems. Controllability plays an essential role in the development of modern mathematical control theory. There are various important relationships between controllability, stability and stabilizability of linear both finite-dimensional and infinite-dimensional control systems. Controllability is also strongly related to the theory of realization and so called minimal realization and canonical forms for linear time-invariant control systems such as the Kalmam canonical form, the Jordan canonical form or the Luenberger canonical form. It should be mentioned, that for many dynamical systems there exists a formal duality between the concepts of controllability and observability. Moreover, controllability is strongly connected with the minimum energy control problem for many classes of linear finite dimensional, infinite dimensional dynamical systems, and delayed systems both deterministic and stochastic. Finally, it is well known, that controllability concept has many important applications not only in control theory and systems theory, but also in such areas as industrial and chemical process control, reactor control, control of electric bulk power systems, aerospce engineering and recently in quantum systems theory.

This paper develops fast algorithms for the construction of a circulant modulated rate process to match with the two primary traffic statistical functions: rate distribution f(x) and autocorrelation R(/spl tau/). Using existing modeling techniques, f(x) has to be limited to certain forms such as Gaussian or binomial; R(/spl tau/) can only consist of one or two exponential terms which are often real exponentials rather than complex. In reality, these two functions are collected from real traffic traces and generally expressed in a very complicated form. We only consider the traffic whose correlation function can be approximated by the sum of complex exponentials. Our emphasis is placed on the algorithm design for matching complicated R(/spl tau/) in traffic modeling. The typical CPU time for traffic modeling with R(/spl tau/) consisting of five or six complex exponential terms is found to be in the range of a few minutes by the proposed algorithms. Our study further shows an excellent agreement between the original traffic traces and the sequences generated by the matched analytical model. The selection of the measurement-window in traffic statistics collection for queueing performance analysis is also discussed.

We present a new approach to exponential functions on time scales and to timescale analogues of ordinary differential equations. We describe in detail the Cayley-exponential function and associated trigonometric and hyperbolic functions. We show that the Cayley-exponential is related to implicit midpoint and trapezoidal rules, similarly as delta and nabla exponential functions are related to Euler numerical schemes. Extending these results on any Pad\'e approximants, we obtain Pad\'e-exponential functions. Moreover, the exact exponential function on time scales is defined. Finally, we present applications of the Cayley-exponential function in the q-calculus and suggest a general approach to dynamic systems on Lie groups.

The design of offshore power stations (for wind, wave or tidal energy generation) requires assessing their environmental impact. In particular, it is of the major importance to predict the impact of the generated subsea noise on the marine fauna, especially sea mammals and fishes. Here, the noise propagation is modelled with the Helmholtz equation and numerically solved using a Partition of Unity Method (PUM). The aim is simulating the underwater sound propagation of multiple non-impulsive sources. The output of the simulations consists of spatial distributions of the sound pressure level. The mathematical model at hand considers the most relevant aspects involved in environmental underwater acoustics. Specifically, Helmholtz equation allows accounting for the most important wave phenomena: absorption, interference, reflection, refraction and diffraction. For instance, the acoustic absorption produced by seawater is represented by the imaginary part of the wavenumber. In addition, a non-uniform wavenumber that depends on the salinity, temperature and depth is considered. The model is completed with a set of boundary conditions providing a specific treatment for the reflective properties of the sea bottom and surface. The input noise is also introduced as a boundary condition. Finally, Perfectly Matched Layers (PMLs) are placed at the lateral artificial boundaries of the domain to avoid spurious reflections. The numerical strategy is based on a PUM enriched with plane waves. The plane wave functions are combined with the classical polynomial shape functions (hat functions, a partition of unity preserving the continuity of the approximation space among elements). The choice of plane waves provides two main advantages. On the one hand, since the enriching functions satisfy the governing equation and include a priori knowledge of the solution, they mitigate the pollution error that is intrinsic to the solutions obtained with standard polynomial approximations. On the other hand, they allow using coarser meshes with a larger element size. This leads to a drastic reduction in the number of degrees of freedom. Therefore, the method is well suited for solving the Helmholtz equation in large domains (from hundreds of meters to kilometers) compared to the characteristic wavelength (from centimeters to meters). However, since the plane waves are described by complex exponential functions, the computation of the elemental matrices entails the integral of highly-oscillatory functions. This increases the requirements involved in the integration step and makes the standard Gauss-Legendre rules lose their competitiveness. In the 2D version of the tool, we overcome this drawback by implementing an existing semi-analytical rule. In the 3D version, we develop a novel efficient rule to integrate highly oscillatory functions over tetrahedra. The integrand is expressed as the product of a non-oscillatory part and a complex exponential function. The rule is designed to be exact, except round-off errors, for integrals with a polynomial non-oscillatory part, which is the case of the Helmholtz equation solved with the PUM enriched with plane waves. To conclude, we present several examples that assess and illustrate the capabilities of the tool, including sea water absorption, homogeneous or heterogeneous media, seabeds with non-uniform transmission coefficient, and single or multiple sources. El diseño de centrales de producción eléctrica offshore (para la captación de energía eólica, mareomotriz o asociada al oleaje) requiere evaluar su impacto ambiental. En particular, es de suma importancia predecir el impacto del ruido submarino generado sobre la fauna marina, especialmente sobre los mamíferos marinos y especies de peces. En este trabajo la propagación del ruido se modela mediante la ecuación de Helmholtz, y se resuelve numéricamente usando un método de partición de unidad (Partition of Unity Method, PUM). El objetivo es simular la propagación del sonido submarino proveniente de múltiples fuentes no impulsivas. El resultado de las simulaciones consiste en distribuciones espaciales del nivel de presión sonora. El modelo matemático que nos ocupa considera los aspectos más relevantes relacionados con la acústica ambiental subacuática. Específicamente, la ecuación de Helmholtz permite tener en cuenta los fenómenos ondulatorios más relevantes: absorción, interferencia, reflexión, refracción y difracción. Por ejemplo, la absorción acústica producida por el agua de mar está representada por la parte imaginaria del número de onda. Además, se considera un número de onda no uniforme que depende de la salinidad, temperatura y profundidad. El modelo se completa con un conjunto de condiciones de contorno que proporcionan un tratamiento específico de la reflexión del fondo y la superficie marinos. El ruido de entrada también se introduce como condición de contorno. Finalmente, se disponen "Perfectly Matched Layers" (PML) en los límites artificiales laterales del dominio para evitar reflexiones espurias. La estrategia numérica se basa en un PUM enriquecido con ondas planas. Las funciones de onda planas se combinan con las funciones clásicas de forma polinomial ("hat functions", una partición de unidad que preserva la continuidad del espacio de aproximación entre elementos). La elección de ondas planas ofrece dos ventajas principales. Por un lado, dado que las funciones de enriquecimiento satisfacen la ecuación de gobierno e incluyen conocimiento a priori de la solución, mitigan "polution error" intrínseco a las soluciones obtenidas con aproximaciones polinomiales estándar. Por otro lado, permiten utilizar mallas más gruesas con un tamaño de elemento mayor. Esto conduce a una reducción drástica del número de grados de libertad. Por lo tanto, el método es muy adecuado para resolver la ecuación de Helmholtz en dominios grandes (de cientos de metros a kilómetros) en comparación con la longitud de onda característica (de centímetros a metros). Sin embargo, dado que las ondas planas se describen mediante funciones exponenciales complejas, el cálculo de las matrices elementales implica integrar funciones altamente oscilatorias. Esto incrementa los requisitos involucrados en el paso de integración e implica que las reglas tipo Gauss-Legendre estándar pierdan competitividad. En la versión 2D de la herramienta, superamos este inconveniente implementando una regla semianalítica existente. En la versión 3D, desarrollamos una nueva regla eficiente en la integración de funciones altamente oscilatorias sobre tetraedros. El integrando se expresa como el producto de una parte no oscilatoria y una función exponencial compleja. La regla está diseñada para ser exacta, salvo errores de redondeo, para integrales con parte polinomial no oscilatoria, que es el caso de la ecuación de Helmholtz resuelta con un PUM enriquecido con ondas planas. Para concluir, presentamos varios ejemplos que evalúan e ilustran las capacidades de la herramienta, incluida la absorción producida por agua salina, medios homogéneos o heterogéneos, fondos marinos con coeficiente de transmisión no uniforme, y fuentes únicas o múltiples.

In this chapter, we introduce the reader to the homology theory for algebras known as cyclic homology. As we shall see in the next section, cyclic homology may be viewed as the “linearization” of K-theory, in the same sense in which the matrix ring M n (R) is the “linearization” of the general linear group GL(n, R). For motivation, it is useful to think of the case where the ring R is ℝ or ℂ. Then GL(n, R)is a Lie group, and the space BGL(n, R)+ giving rise to the higher K-groups is by its construction an H-space whose homology agrees with the homology of this Lie group (never mind for the moment that we are forgetting the topology!). On the other hand, one of the basic principles of Lie theory is that the best way to study Lie groups (which are “non-linear” objects) is often by way of their Lie algebras (which can be studied using linear algebra). For example, it is a famous result that for any compact Lie group G with Lie algebra g, the topological cohomology H●(G;ℝ) (here we are just thinking of G as a space and ignoring the group structure) is canonically isomorphic to the Lie algebra cohomology H●(g;ℝ), which can at least in principle be computed using only finite-dimensional linear algebra. This suggests that some construction with the Lie algebra M n (R) of GL(n, R)ought to yield a reasonable approximation to the K-theory of R, at least in the limit as n → ∞. While the actual construction of cyclic homology and of the Chern character which relates it to K-theory is a bit roundabout, this philosophy turns out to be basically correct.KeywordsExact SequenceCommutative RingHomology TheoryChern CharacterCyclic HomologyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Hilbert [13] suggested in connection with his fifth problem that, while the theory of differential equations provides elegant and powerful techniques for solving functional equations, the inherent differentiability assumptions are typically unnatural (see [4]). Such shortcomings can sometimes be overcome by appealing to regularity theorems of the type "continuity implies differentiability" (see e.g. [4], and w of [1]) and of the type "measurability implies continuity" beginning with Fr~chet [9] and continued by scorses of others (see e.g. [4], [5], [16] and [19] for references to further such results). The theory of distribution of Laurent Schwartz has also been used, perhaps first by Feny5 [8], (see also [2], [6], [11], [12], [20] and [25]) to partially overcome this deficiency. The main aim of the present paper is to introduce another distributional method, based on approximating distributional derivatives by differences, which allows one to find the distributional solutions to distributional analogues of many functional equations by reduction to (distributional) differential equations. This method, when combined with regularity theorems, allows one to find all the Continuous (or even locally Lebesgue integrable) solutions of certain functional equations. References to some of the vast literature concerning many of the functional equations considered herein can be found in [1] and [3]. While many of our results concerning functional equations may be considered as bu"t slight generalizations of known results, our main aim is to illustrate a method of solution. A novelty of our approach is that it permits "local" consideration. Unless otherwise indicated, our notation and terminology is that of Rudin [24]. The set of all Schwartz test function on R n will be denoted by 79~. We let D~ denote the set of Schwartz distributions on R n. All measure-theoretic language refers to Lebesgue measure.

The complete first-order theories of the exponential differential equations of semiabelian varieties are given. It is shown that these theories also arise from an amalgamation-with-predimension construction in the style of Hrushovski. The theories include necessary and sufficient conditions for a system of equations to have a solution. The necessary conditions generalize Ax’s differential fields version of Schanuel’s conjecture to semiabelian varieties. There is a purely algebraic corollary, the “Weak CIT” for semiabelian varieties, which concerns the intersections of algebraic subgroups with algebraic varieties.

By using exponential functions and the theory of ordinary differential equations, the classical theorem about the existence of a Jordan canonical form of a square matrix is proved.