Academic literature on the topic 'Weyl functional calculus'
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Journal articles on the topic "Weyl functional calculus"
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Jefferies, Brian, and Alan McIntosh. "The Weyl calculus and Clifford analysis." Bulletin of the Australian Mathematical Society 57, no. 2 (April 1998): 329–41. http://dx.doi.org/10.1017/s0004972700031695.
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Kim, Joonil. "Marcinkiewicz multiplier theorem for the Weyl functional calculus." Mathematische Zeitschrift 258, no. 2 (April 25, 2007): 271–90. http://dx.doi.org/10.1007/s00209-007-0164-x.
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Jefferies, Brian, and Bernd Straub. "Lacunas in the Support of the Weyl Calculus for Two Hermitian Matrices." Journal of the Australian Mathematical Society 75, no. 1 (August 2003): 85–124. http://dx.doi.org/10.1017/s1446788700003499.
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AbstractThe connection between Clifford analysis and the Weyl functional calculus for a d-tuple of bounded selfadjoint operators is used to prove a geometric condition due to J. Bazer and D. H. Y. Yen for a point to be in the support of the Weyl functional calculus for a pair of hermitian matrices. Examples are exhibited in which the support has gaps.
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Ricker, Werner J. "The Weyl functional calculus and two-by-two selfadjoint matrices." Bulletin of the Australian Mathematical Society 55, no. 2 (April 1997): 321–25. http://dx.doi.org/10.1017/s0004972700033980.
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Let D be a (2 × 2) matrix with distinct eigenvalues λ1 and λ2. There is a basic and well known functional equation which provides a formula for constructing the matrix g (D), for any ℂ-valued function g defined on a subset of ℂ containing {λ1,λ2}, namely .This equation is used to give a direct and transparent proof of the following fact due to Anderson: A pair of (2 × 2) selfadjoint matrices A1 and A2 commute if and only if the Weyl functional calculus of the pair (A1,A2), which is a matrix-valued distribution, has order zero (that is, is a measure).
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Eydenberg, Michael. "The Weyl Correspondence as a Functional Calculus for Non-Commuting Operators." Rocky Mountain Journal of Mathematics 39, no. 5 (October 2009): 1467–96. http://dx.doi.org/10.1216/rmj-2009-39-5-1467.
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Le Merdy, Christian. "Two Results About H∞ Functional Calculus on Analytic umd Banach Spaces." Journal of the Australian Mathematical Society 74, no. 3 (June 2003): 351–78. http://dx.doi.org/10.1017/s1446788700003360.
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AbstractLet X be a Banach space with the analytic UMD property, and let A and B be two commuting sectorial operators on X which admit bounded H∞ functional calculi with respect to angles θ1 and θ2 satisfying θ1 + θ2 > π. It was proved by Kalton and Weis that in this case, A + B is closed. The first result of this paper is that under the same conditions, A + B actually admits a bounded H∞ functional calculus. Our second result is that given a Banach space X and a number 1 ≦ p < ∞, the derivation operator on the vector valued Hardy space Hp (R; X) admits a bounded H∞ functional calculus if and only if X has the analytic UMD property. This is an ‘analytic’ version of the well-known characterization of UMD by the boundedness of the H∞ functional calculus of the derivation operator on vector valued Lp-spaces Lp (R; X) for 1 < p < ∞ (Dore-Venni, Hieber-Prüss, Prüss).
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PAVLOVIĆ, DUšKO. "Categorical logic of names and abstraction in action calculi." Mathematical Structures in Computer Science 7, no. 6 (December 1997): 619–37. http://dx.doi.org/10.1017/s0960129597002296.
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Milner's action calculus implements abstraction in monoidal categories, so that familiar λ-calculi can be subsumed together with the π-calculus and the Petri nets. Variables are generalised to names, which allow only a restricted form of substitution.In the present paper, the well-known categorical semantics of the λ-calculus is generalised to the action calculus. A suitable functional completeness theorem for symmetric monoidal categories is proved: we determine the conditions under which the abstraction is definable. Algebraically, the distinction between the variables and the names boils down to the distinction between the transcendental and the algebraic elements. The former lead to polynomial extensions, like, for example, the ring ℤ[x]; the latter lead to algebraic extensions like ℤ[√2] or ℤ[i].Building upon the work of P. Gardner, we introduce action categories, and show that they are related to the static action calculus in exactly the same way as cartesian closed categories are related to the λ-calculus. Natural examples of this structure arise from allegories and cartesian bicategories. On the other hand, the free algebras for any commutative Moggi monad form an action category. The general correspondence of action calculi and Moggi monads will be worked out in a sequel to this work.
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Doust, Ian, and Qiu Bozhou. "The spectral theorem for well-bounded operators." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 54, no. 3 (June 1993): 334–51. http://dx.doi.org/10.1017/s1446788700031827.
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AbstractWell-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.
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SCHMITT, T. "FUNCTIONALS OF CLASSICAL FIELDS IN QUANTUM FIELD THEORY." Reviews in Mathematical Physics 07, no. 08 (November 1995): 1249–301. http://dx.doi.org/10.1142/s0129055x95000463.
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Many methods of modern quantum field theory rely heavily on functionals of classical fields; this notion is however problematic whenever anticommuting fields are present. We propose a calculus for such functionals which avoids the use of auxiliary Grassmann algebras, and which relies on an infinite-dimensional version of Berezin-Leites supermanifold theory. We begin by studying “functional power series expansions” without growth conditions; this already allows to make e.g. the Yang-Mills action functional with fermionic, anticommuting matter fields a well-defined mathematical object. We introduce analytical conditions on power series which enable us to substitute them into each other, and we globalize them to superfunctionals. Also, infinitesimal transformation laws of the fields are discussed.
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Jacobson, David K., Tanvi P. Honap, Cara Monroe, Justin Lund, Brett A. Houk, Anna C. Novotny, Cynthia Robin, Elisabetta Marini, and Cecil M. Lewis. "Functional diversity of microbial ecologies estimated from ancient human coprolites and dental calculus." Philosophical Transactions of the Royal Society B: Biological Sciences 375, no. 1812 (October 5, 2020): 20190586. http://dx.doi.org/10.1098/rstb.2019.0586.
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Human microbiome studies are increasingly incorporating macroecological approaches, such as community assembly, network analysis and functional redundancy to more fully characterize the microbiome. Such analyses have not been applied to ancient human microbiomes, preventing insights into human microbiome evolution. We address this issue by analysing published ancient microbiome datasets: coprolites from Rio Zape ( n = 7; 700 CE Mexico) and historic dental calculus ( n = 44; 1770–1855 CE, UK), as well as two novel dental calculus datasets: Maya ( n = 7; 170 BCE-885 CE, Belize) and Nuragic Sardinians ( n = 11; 1400–850 BCE, Italy). Periodontitis-associated bacteria ( Treponema denticola , Fusobacterium nucleatum and Eubacterium saphenum ) were identified as keystone taxa in the dental calculus datasets. Coprolite keystone taxa included known short-chain fatty acid producers ( Eubacterium biforme, Phascolarctobacterium succinatutens ) and potentially disease-associated bacteria ( Escherichia , Brachyspira) . Overlap in ecological profiles between ancient and modern microbiomes was indicated by similarity in functional response diversity profiles between contemporary hunter–gatherers and ancient coprolites, as well as parallels between ancient Maya, historic UK, and modern Spanish dental calculus; however, the ancient Nuragic dental calculus shows a distinct ecological structure. We detected key ecological signatures from ancient microbiome data, paving the way to expand understanding of human microbiome evolution. This article is part of the theme issue ‘Insights into health and disease from ancient biomolecules’.
Dissertations / Theses on the topic "Weyl functional calculus"
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Gunturk, Kamil Serkan. "Covariant Weyl quantization, symbolic calculus, and the product formula." Texas A&M University, 2003. http://hdl.handle.net/1969.1/3963.
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A covariant Wigner-Weyl quantization formalism on the manifold that uses
pseudo-differential operators is proposed. The asymptotic product formula that leads
to the symbol calculus in the presence of gauge and gravitational fields is presented.
The new definition is used to get covariant differential operators from momentum
polynomial symbols. A covariant Wigner function is defined and shown to give
gauge-invariant results for the Landau problem. An example of the covariant Wigner
function on the 2-sphere is also included.
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Mustapha, Sami. "Sous-ellipticité, interpolation réelle et calcul de Weyl-Hormander." Paris 6, 1994. http://www.theses.fr/1994PA066205.
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Cette these est constituee de quatre articles et d'une note consacres a la sous-ellipticite ; au calcul s(m,g), aux distances sous-elliptiques et a l'interpolation reelle. Le but de cette these est double: en premier lieu introduire et etudier une notion de sous-ellipticite adaptee au cadre du calcul de weyl-hormander. Les techniques utilisees sont basees sur l'interpolation reelle et les techniques de semi-groupe et conduisent a une bonne generalisation des resultats classiques. Le second objectif de cette these est l'investigation de la conjecture suivante: soient deux operateurs differentiels, sous-elliptiques du second ordre a,b et soient d et les distances qui leurs correspondent, alors il est naturel de conjecturer l'equivalence des estimations d# #b et b#b a#. Les methodes developpees dans cette these permettent d'obtenir de nouveaux resultats concernant cette conjecture
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Khan, Mumtaz Ahmad, and Bhagwat Swaroop Sharma. "A study of three variable analogues of certain fractional integral operators." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/95821.
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The paper deals with a three variable analogues of certain fractional integral operators introduced by M. Saigo. Resides giving three variable analogues of earlier known fractional integral operators of one variable as a special cases of newly defined operators, the paper establishes certain results in the form of theorems including integration by parts.
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Raffoul, Raed Wissam Mathematics & Statistics Faculty of Science UNSW. "Functional calculus and coadjoint orbits." 2007. http://handle.unsw.edu.au/1959.4/43693.
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Let G be a compact Lie group and let π be an irreducible representation of G of highest weight λ. We study the operator-valued Fourier transform of the product of the j-function and the pull-back of ?? by the exponential mapping. We show that the set of extremal points of the convex hull of the support of this distribution is the coadjoint orbit through ?? + ??. The singular support is furthermore the union of the coadjoint orbits through ?? + w??, as w runs through the Weyl group. Our methods involve the Weyl functional calculus for noncommuting operators, the Nelson algebra of operants and the geometry of the moment set for a Lie group representation. In particular, we re-obtain the Kirillov-Duflo correspondence for compact Lie groups, independently of character formulae. We also develop a "noncommutative" version of the Kirillov character formula, valid for noncentral trigonometric polynomials. This generalises work of Cazzaniga, 1992.
Books on the topic "Weyl functional calculus"
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Behrndt, Jussi. Boundary Value Problems, Weyl Functions, and Differential Operators. Cham: Springer Nature, 2020.
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Mann, Peter. The Stationary Action Principle. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0007.
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This crucial chapter focuses on the stationary action principle. It introduces Lagrangian mechanics, using first-order variational calculus to derive the Euler–Lagrange equation, and the inverse problem is described. The chapter then considers the Ostrogradsky equation and discusses the properties of the extrema using the second-order variation to the action. It then discusses the difference between action functions (of Dirichlet boundary conditions) and action functionals of the extremal path. The different types of boundary conditions (Dirichlet vs Neumann) are elucidated. Topics discussed include Hessian conditions, Douglas’s theorem, the Jacobi last multiplier, Helmholtz conditions, Noether-type variation and Frenet–Serret frames, as well as concepts such as on shell and off shell. Actions of non-continuous extremals are examined using Weierstrass–Erdmann corner conditions, and the action principle is written in the most general form as the Hamilton–Suslov principle. Important applications of the Euler–Lagrange formulation are highlighted, including protein folding.
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Horing, Norman J. Morgenstern. Schwinger Action Principle and Variational Calculus. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.003.0004.
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Chapter 4 introduces the Schwinger Action Principle, along with associated particle and potential sources. While the methods described here originally arose in the relativistic quantum field theory of elementary particle physics, they have also profoundly advanced our understanding of non-relativistic many-particle physics. The Schwinger Action Principle is a quantum-mechanical variational principle that closely parallels the Hamilton Principle of Least Action of classical mechanics, generalizing it to include the role of quantum operators as generalized coordinates and momenta. As such, it unifies all aspects of quantum theory, incorporating Hamilton equations of motion for those operators and the Heisenberg equation, as well as producing the canonical equal-time commutation/anticommutation relations. It yields dynamical coupled field equations for the creation and annihilation operators of the interacting many-body system by variational differentiation of the Hamiltonian with respect to the field operators. Also, equations for the development of matrix elements (underlying Green’s functions) are derived using variations with respect to particle and potential “sources” (and coupling strength). Variational calculus, involving impressed potentials, c-number coordinates and fields, also quantum operator coordinates and fields, is discussed in full detail. Attention is given to the introduction of fermion and boson particle sources and their use in variational calculus.
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Horing, Norman J. Morgenstern. Quantum Statistical Field Theory. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.001.0001.
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The methods of coupled quantum field theory, which had great initial success in relativistic elementary particle physics and have subsequently played a major role in the extensive development of non-relativistic quantum many-particle theory and condensed matter physics, are at the core of this book. As an introduction to the subject, this presentation is intended to facilitate delivery of the material in an easily digestible form to students at a relatively early stage of their scientific development, specifically advanced undergraduates (rather than second or third year graduate students), who are mathematically strong physics majors. The mechanism to accomplish this is the early introduction of variational calculus with particle sources and the Schwinger Action Principle, accompanied by Green’s functions, and, in addition, a brief derivation of quantum mechanical ensemble theory introducing statistical thermodynamics. Important achievements of the theory in condensed matter and quantum statistical physics are reviewed in detail to help develop research capability. These include the derivation of coupled field Green’s function equations of motion for a model electron-hole-phonon system, extensive discussions of retarded, thermodynamic and non-equilibrium Green’s functions, and their associated spectral representations and approximation procedures. Phenomenology emerging in these discussions includes quantum plasma dynamic, nonlocal screening, plasmons, polaritons, linear electromagnetic response, excitons, polarons, phonons, magnetic Landau quantization, van der Waals interactions, chemisorption, etc. Considerable attention is also given to low-dimensional and nanostructured systems, including quantum wells, wires, dots and superlattices, as well as materials having exceptional conduction properties such as superconductors, superfluids and graphene.
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Antognazza, Maria Rosa, ed. The Oxford Handbook of Leibniz. Oxford University Press, 2013. http://dx.doi.org/10.1093/oxfordhb/9780199744725.001.0001.
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The extraordinary breadth and depth of Leibniz’s intellectual vision commands ever increasing attention. As more texts gradually emerge from seemingly bottomless archives, new facets of his contribution to an astonishing variety of fields come to light. This volume provides a uniquely comprehensive, systematic, and up-to-date appraisal of Leibniz’s thought thematically organized around its diverse but interrelated aspects. Discussion of his philosophical system naturally takes place of pride. A cluster of original essays revisit his logic, metaphysics, epistemology, philosophy of nature, moral and political philosophy, and philosophy of religion. The scope of the volume, however, goes beyond that of a philosophical collection to embrace all the main features of Leibniz’s thought and activity. Contributions are offered on Leibniz as a mathematician (including not only his calculus but also determinant theory, symmetric functions, the dyadic, the analysis situs, probability and statistics); on Leibniz as a scientist (physics and also optics, cosmology, geology, physiology, medicine, and chemistry); on his technical innovations (the calculating machine and the technology of mining, as well as other discoveries); on his work as an ‘intelligencer’ and cultural networker, as jurist, historian, editor of sources and librarian; on his views on Europe’s political future, religious toleration, and ecclesiastical reunification; on his proposals for political, administrative, economic, and social reform. In so doing, the volume serves as a unique cross-disciplinary point of contact for the many domains to which Leibniz contributed. By assembling leading specialists on all these topics, it offers the most rounded picture of Leibniz’s endeavors currently available.
Book chapters on the topic "Weyl functional calculus"
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Hausmann, Daniel, and Lutz Schröder. "Quasipolynomial Computation of Nested Fixpoints." In Tools and Algorithms for the Construction and Analysis of Systems, 38–56. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-72016-2_3.
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AbstractIt is well-known that the winning region of a parity game with n nodes and k priorities can be computed as a k-nested fixpoint of a suitable function; straightforward computation of this nested fixpoint requires $$\mathcal {O}(n^{\frac{k}{2}})$$ O ( n k 2 ) iterations of the function. Calude et al.’s recent quasipolynomial-time parity game solving algorithm essentially shows how to compute the same fixpoint in only quasipolynomially many iterations by reducing parity games to quasipolynomially sized safety games. Universal graphs have been used to modularize this transformation of parity games to equivalent safety games that are obtained by combining the original game with a universal graph. We show that this approach naturally generalizes to the computation of solutions of systems of any fixpoint equations over finite lattices; hence, the solution of fixpoint equation systems can be computed by quasipolynomially many iterations of the equations. We present applications to modal fixpoint logics and games beyond relational semantics. For instance, the model checking problems for the energy $$\mu $$ μ -calculus, finite latticed $$\mu $$ μ -calculi, and the graded and the (two-valued) probabilistic $$\mu $$ μ -calculus – with numbers coded in binary – can be solved via nested fixpoints of functions that differ substantially from the function for parity games but still can be computed in quasipolynomial time; our result hence implies that model checking for these $$\mu $$ μ -calculi is in $$\textsc {QP}$$ QP . Moreover, we improve the exponent in known exponential bounds on satisfiability checking.
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Barthe, Gilles, Raphaëlle Crubillé, Ugo Dal Lago, and Francesco Gavazzo. "On the Versatility of Open Logical Relations." In Programming Languages and Systems, 56–83. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-44914-8_3.
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AbstractLogical relations are one among the most powerful techniques in the theory of programming languages, and have been used extensively for proving properties of a variety of higher-order calculi. However, there are properties that cannot be immediately proved by means of logical relations, for instance program continuity and differentiability in higher-order languages extended with real-valued functions. Informally, the problem stems from the fact that these properties are naturally expressed on terms of non-ground type (or, equivalently, on open terms of base type), and there is no apparent good definition for a base case (i.e. for closed terms of ground types). To overcome this issue, we study a generalization of the concept of a logical relation, called open logical relation, and prove that it can be fruitfully applied in several contexts in which the property of interest is about expressions of first-order type. Our setting is a simply-typed $$\lambda $$ λ -calculus enriched with real numbers and real-valued first-order functions from a given set, such as the one of continuous or differentiable functions. We first prove a containment theorem stating that for any collection of real-valued first-order functions including projection functions and closed under function composition, any well-typed term of first-order type denotes a function belonging to that collection. Then, we show by way of open logical relations the correctness of the core of a recently published algorithm for forward automatic differentiation. Finally, we define a refinement-based type system for local continuity in an extension of our calculus with conditionals, and prove the soundness of the type system using open logical relations.
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Alvarez-Picallo, Mario, and Jean-Simon Pacaud Lemay. "Cartesian Difference Categories." In Lecture Notes in Computer Science, 57–76. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-45231-5_4.
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AbstractCartesian differential categories are categories equipped with a differential combinator which axiomatizes the directional derivative. Important models of Cartesian differential categories include classical differential calculus of smooth functions and categorical models of the differential $$\lambda $$ λ -calculus. However, Cartesian differential categories cannot account for other interesting notions of differentiation such as the calculus of finite differences or the Boolean differential calculus. On the other hand, change action models have been shown to capture these examples as well as more “exotic” examples of differentiation. However, change action models are very general and do not share the nice properties of a Cartesian differential category. In this paper, we introduce Cartesian difference categories as a bridge between Cartesian differential categories and change action models. We show that every Cartesian differential category is a Cartesian difference category, and how certain well-behaved change action models are Cartesian difference categories. In particular, Cartesian difference categories model both the differential calculus of smooth functions and the calculus of finite differences. Furthermore, every Cartesian difference category comes equipped with a tangent bundle monad whose Kleisli category is again a Cartesian difference category.
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deLaubenfels, Ralph. "Well-posedness on a larger space; Generalized solutions." In Existence Families, Functional Calculi and Evolution Equations, 55–59. Berlin, Heidelberg: Springer Berlin Heidelberg, 1994. http://dx.doi.org/10.1007/bfb0073407.
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deLaubenfels, Ralph. "The solution space of an operator and automatic well-posedness." In Existence Families, Functional Calculi and Evolution Equations, 24–37. Berlin, Heidelberg: Springer Berlin Heidelberg, 1994. http://dx.doi.org/10.1007/bfb0073405.
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Miana, Pedro J. "Integrated Exponential Functions, the Weyl Fractional Calculus and the Laplace Transform." In Difference Equations, Discrete Dynamical Systems and Applications, 223–36. Berlin, Heidelberg: Springer Berlin Heidelberg, 2016. http://dx.doi.org/10.1007/978-3-662-52927-0_17.
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Freire, J. L., E. Freire, and A. Blanco. "On Recursive Functions and Well–Founded Relations in the Calculus of Constructions." In Lecture Notes in Computer Science, 69–80. Berlin, Heidelberg: Springer Berlin Heidelberg, 2005. http://dx.doi.org/10.1007/11556985_12.
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Ullrich, Peter. "„Niemand kann erklären, was eine Funktion ist“." In Auch wenn A falsch ist, kann B wahr sein. Was wir aus Fehlern lernen können, 259–68. WTM-Verlag Münster, 2019. http://dx.doi.org/10.37626/ga9783959871143.0.17.
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Starting from the quote from Hermann Weyl given in the title a ramble is undertaken through the development of the notion of function with special emphasis on the question whether the values are associated following a law. On the one hand, this shows a success story of the interplay of this notion and of infinitesimal calculus. On the other hand, one finds impressive examples of overgeneralizations. Classification: C30, D70, E40, I20, I30, M10. Keywords: notion of function, functional laws, overgeneralization.
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Tsarava, Katerina, Spyros T. Halkidis, Pantelis Venardos, and George Stephanides. "Teaching Basic Calculus Using SAGE." In Advances in Educational Marketing, Administration, and Leadership, 276–94. IGI Global, 2013. http://dx.doi.org/10.4018/978-1-4666-4233-1.ch014.
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This chapter presents an attempt to review basic calculus concepts to high school students with the help of a Computer Algebra System, namely SAGE. A review lesson on limits and derivatives as well as an introduction of the bisection method for finding roots of continuous functions is presented. The evaluation of the lesson by the students is analyzed. The aim of this chapter is to examine the power of SAGE in reviewing basic calculus concepts, presenting the advantages and disadvantages of SAGE compared to other Computer Algebra Systems, as well as the benefit from using a computer system in making concepts such as the squeeze theorem for computing limits of functions more clear.
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Coquand, Catarina. "A realizability interpretation of Martin-Löf’s type Theory." In Twenty Five Years of Constructive Type Theory. Oxford University Press, 1998. http://dx.doi.org/10.1093/oso/9780198501275.003.0007.
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The notion of computability predicate developed by Tail gives a powerful method for proving normalization for various λ-calculi. There are actually two versions of this method: a typed and an untyped approach. In the former approach we define a computability predicate over well-typed terms. This technique was developed by Tail (1967) in order to prove normalization for Gödel’s system T. His method was extended by Girard (1971) for his System F and by Martin-Löf (1971) for his type theory. The second approach is similar to Kleene’s realizability interpretation (Kleene 1945), but in this approach formulas are realized by (not necessarily well-typed) λ-terms rather than Gödel numbers. This technique was developed by Tail (1975) where he uses this method to obtain a simplified proof of normalization for Girard’s system F. The method was later rediscovered independently by several others, for instance Mitchell (1986). There are two ways of defining typed λ-calculus: we have either equality as conversion (on raw terms) or equality as judgments. It is more difficult to show in the theory with equality as judgments that every well-typed term has a normal form of the same type. We can find different approaches to how to solve this problem in Altenkirch (1993), Coquand (1991), Goguen (1994) and Martin-Löf (1975). There are several papers addressing normalization for calculi with equality as untyped conversion; two relevant papers showing normalization are Altenkirch’s (1994) proof for system F using the untyped method and Werner’s (1994) proof for the calculus of constructions with inductive types using the other method. Another difference is the predicative and impredicative formulations of λ-calculi where it is more intricate to prove normalization for the impredicative ones. The aim of this paper is to give a simple argument of normalization for a theory with “real” dependent types, i.e. a theory in which we can define types by recursion on the natural numbers (like Nn). For this we choose to study the simplest theory that has such types, namely the N, Π, U-fragment of Martin-Löf’s polymorphic type theory (Martin-Löf 1972) which contains the natural numbers, dependent functions and one universe. This theory has equality as conversion and is predicative.
Conference papers on the topic "Weyl functional calculus"
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Agrawal, Om P., Md Mehedi Hasan, and X. W. Tangpong. "A Numerical Scheme for a Class of Parametric Problem of Fractional Variational Calculus." In ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/detc2011-48768.
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Fractional derivatives (FDs) or derivatives of arbitrary order have been used in many applications, and it is envisioned that in future they will appear in many functional minimization problems of practical interest. Since fractional derivatives have such property as being non-local, it can be extremely challenging to find analytical solutions for fractional parametric optimization problems, and in many cases, analytical solutions may not exist. Therefore, it is of great importance to develop numerical methods for such problems. This paper presents a numerical scheme for a linear functional minimization problem that involves FD terms. The FD is defined in terms of the Riemann-Liouville definition; however, the scheme will also apply to Caputo derivatives, as well as other definitions of fractional derivatives. In this scheme, the spatial domain is discretized into several subdomains and 2-node one-dimensional linear elements are adopted to approximate the solution and its fractional derivative at point within the domain. The fractional optimization problem is converted to an eigenvalue problem, the solution of which leads to fractional orthogonal functions. Convergence study of the number of elements and error analysis of the results ensure that the algorithm yields stable results. Various fractional orders of derivative are considered and as the order approaches the integer value of 1, the solution recovers the analytical result for the corresponding integer order problem.
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Coopmans, Calvin, Ivo Petra´sˇ, and YangQuan Chen. "Analogue Fractional-Order Generalized Memristive Devices." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-86861.
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Memristor is a new electrical element which has been predicted and described in 1971 by Leon O. Chua and for the first time realized by HP laboratory in 2008. Chua proved that memristor behavior could not be duplicated by any circuit built using only the other three elements (resistor, capacitor, inductor), which is why the memristor is truly fundamental. Memristor is a contraction of memory resistor, because that is exactly its function: to remember its history. The memristor is a two-terminal device whose resistance depends on the magnitude and polarity of the voltage applied to it and the length of time that voltage has been applied. The missing element—the memristor, with memristance M—provides a functional relation between charge and flux, dφ = Mdq. In this paper, for the first time, the concept of (integer-order) memristive systems is generalized to non-integer order case using fractional calculus. We also show that the memory effect of such devices can be also used for an analogue implementation of the fractional-order operator, namely fractional-order integral and fractional-order derivatives. This kind of operators are useful for realization of the fractional-order controllers. We present theoretical description of such implementation and we proposed the practical realization and did some experiments as well.
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Costarella, Marianna, Lucilla Monteleone, Roberto Steindler, and Stefano Maria Zuccaro. "Physical and Psychical Conditions Decline of Older People With Age, Measured by Functional Reach Test and by Mini Mental State Examination." In ASME 2008 9th Biennial Conference on Engineering Systems Design and Analysis. ASMEDC, 2008. http://dx.doi.org/10.1115/esda2008-59055.
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There are several tests to value the psychophysical characteristics of older people and, among all, the most suitable to this aim are here considered the Functional Reach (FR) test, as an index of the aptitude to maintain balance in upright position, and the Mini Mental State Examination (MMSE), as a global index of cognitive abilities. The sample of older people we have analysed concerns 50 healthy subjects divided into three groups according to the age (15 from 55 to 64 years old, 19 from 65 to 74 years old, and 16 more than 75 years old); they underwent a FR test, which consists first in the measurement of the anthropometric characteristics, then in the execution of the test itself, and finally in the study of the upright posture carried out analysing the Centre of Pressure (COP) trend; they underwent as well a MMSE to value the main areas of the cognitive function concerning the space-temporal orientation, the short-term memory, the attention ability, the calculus ability and the praxis-constructive ability. The results of these tests show, according to the age, a loss both of the physical performances (FR, FR related to height, and COP displacement), and of the cognitive abilities (MMSE); however, in all cases, the only significant changes are those between the first and the other two groups of age. A comparison between the results of male and female subjects inside the three groups, although the results of the males are generally superior to the female ones, is never significant; moreover, the differences of the FR tests, in particular, are completely not significant if compared to the height of the subjects. Finally, a comparison between FR and MMSE shows a quicker decline of the physical performances with regard to the cognitive ones.
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Kudish, I. I. "Rough Contacts Modeled by Nonlinear Coatings." In ASME/STLE 2007 International Joint Tribology Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/ijtc2007-44148.
Full textAbstract:
A number of experimental studies [1–3] revealed that the normal displacement in a contact of rough surfaces due to asperities presence is a nonlinear function of local pressure and it can be approximated by a power function of pressure. Originally, a linear mathematical model accounting for surface roughness of elastic solids in contact was introduced by I. Shtaerman [4]. He assumed that the effect of asperities present in a contact of elastic solids can be essentially replaced by the presence of a thin coating simulated by an additional normal displacement of solids’ surfaces proportional to a local pressure. Later, a similar but nonlinear problem formulation that accounted for the above mentioned experimental fact was proposed by L. Galin. In a series of papers this problem was studied by numerical and asymptotic methods [5–9]. The present paper has a dual purpose: to analyze the problem analytically and to provide some asymptotic and numerical solutions. The results presented below provide an overview of the results obtained on the topic and published by the author earlier in the journals hardly accessible to the international tribological community (such as Russian and mathematical journals) and, therefore, mostly unknown by tribologists. A number of recent publications on contacts of rough elastic solids supports the view that these results are still of value to the specialists involved in nanotribology. The existence and uniqueness of a solution of a contact problem for elastic bodies with rough (coated) surfaces is established based on the variational inequalities approach. Four different equivalent formulations of the problem including three variational ones were considered. A comparative analysis of solutions of the contact problem for different values of initial parameters (such as the indenter shape, parameters characterizing roughness, elastic parameters of the substrate material) is done with the help of calculus of variations and the Zaremba-Giraud principle of maximum for harmonic functions [10,11]. The results include the relations between the pressure and displacement distributions for rough and smooth solids as well as the relationships for solutions of the problems for rough solids with fixed and free contact boundaries. For plane and axially symmetric cases some asymptotic and numerical solutions are presented.
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Murphey, Carey, Kenneth Ruddy, and Kate Ruddy. "Making the Case for the Use of Effective Tension in Casing Design to Eliminate Approximations and Wall Thickness Limitations." In SPE Thermal Integrity and Design Symposium. SPE, 2021. http://dx.doi.org/10.2118/203862-ms.
Full textAbstract:
Abstract Existing plastic limit load equations for casing design include approximations that can result in overly conservative (and costly) well designs and ignore forces that may prove critical for assured integrity in complex operations. The use of effective tension in place of physical tension can simultaneously simplify casing design and eliminate existing approximations and wall thickness limitations. Effective tension has not yet been widely adopted because a rigorous derivation based on axiomatic mechanics and calculus does not exist in the current body of literature. This paper presents a thorough derivation for effective tension in terms of working stresses, with tensile, radial, and tangential stresses all receiving proper treatment. Unlike Barlow’s equation, which is routinely used for casing selection, the analytical result presented here allows fully plastic limit loads to be established for tubulars of any wall thickness, without introducing any approximations. Determining plastic limit loads is a crucial component of designing for well integrity, especially with complex operational loads where classical load equations are too conservative. The methods of transformation detailed in this paper facilitate the use of effective tension, which is often the more efficient primary tension load variable for pipe design, as opposed to physical tension. The rigorous model can also be used to derive strains as functions of loads, which is required for some design problems such as seal effectiveness. Additionally, the formulas can be used to dynamically simulate load changes and dimensional changes of pipe when it deforms, such as expandable casing, deformable ball seats, or when approaching rupture pressure. The elimination of approximations in the equations presented in this paper is significant in today’s well design climate, where integrity is demanded yet excessive cost is not tolerated. The derivation provides a foundation upon which existing casing design can be improved by eliminating wall thickness limitations and dependencies on approximations. This foundation will allow new design techniques to be developed that were not previously achievable due to the inefficiencies inherent in the use of physical tension.