Metric Cotype

We introduce the notion of metric cotype, a property of metric spaces related to a property of normed spaces, called Rademacher cotype. Apart from settling a long standing open problem in metric geometry, this property is used to prove the following dichotomy: A family of metric spaces F is either almost universal (i.e., contains any finite metric space with any distortion > 1), or there exists α > 0, and arbitrarily large n-point metrics whose distortion when embedded in any member of F is at least Ω((log n)^α). The same property is also used to prove strong non-embeddability theorems of L_q into L_p, when q > max{2,p}. Finally we use metric cotype to obtain a new type of isoperimetric inequality on the discrete torus

Hilbert's thesis : some considerations about formalizations of mathematics

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1982.MICROFICHE COPY AVAILABLE IN ARCHIVES AND HUMANITIESBibliography: leaves 175-176.by Lon A. Berk.Ph.D

Rings of quantum integrals for generalised Calogero–Moser problems

The rings of quantum integrals for generalised Calogero–Moser problems are studied in the special case when all the parameters are integers. The problem is reduced to the description of the rings of polynomials satisfying a certain quasi-invariance property (quasi-invariants). The quasi-invariants of dihedral groups are fully described. It is shown that they form a free module over invariants generated by m-harmonic polynomials. The m-harmonic polynomials for general Coxeter group are introduced and investigated. For the non-Coxeter generalisations of Calogero–Moser problems related to the systems An(m), Cn+1(m, l), the rings of quantum integrals are considered. The Poincare series for the quasi-invariants of two-dimensional deformations are computed. It is shown that the rings of quasi-invariants are Gorenstein like in the Coxeter case

Solving linear partial differential equations via semidefinite optimization

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2001.Includes bibliographical references (p. 49-51).Using recent progress on moment problems, and their connections with semidefinite optimization, we present in this thesis a new methodology based on semidefinite optimization, to obtain a hierarchy of upper and lower bounds on both linear and nonlinear functionals defined on solutions of linear partial differential equations. We apply the proposed methods to examples of PDEs in one and two dimensions with very encouraging results. We also provide computational evidence that the semidefinite constraints are critically important in improving the quality of the bounds, that is without them the bounds are weak.by Constantine Caramanis.S.M

The Necessary Structure of the All-pervading Aether: Discrete or Continuous? Simple or Symmetric?

In this book I investigate the necessary structure of the aether – the stuff that fills the whole universe. Some of my conclusions are. 1. There is an enormous variety of structures that the aether might, for all we know, have. 2. Probably the aether is point-free. 3. In that case, it should be distinguished from Space-time, which is either a fiction or a construct. 4. Even if the aether has points, we should reject the orthodoxy that all regions are grounded in points by summation. 5. If the aether is point-free but not continuous, its most likely structure has extended atoms that are not simples. 6. Space-time is symmetric if and only if the aether is continuous. 7. If the aether is continuous, we should reject the standard interpretation of General Relativity, in which geometry determines gravity. 8. Contemporary physics undermines an objection to discrete aether based on scale invariance, but does not offer much positive support

Aspectos aritméticos y geométricos del problema decimoséptimo de Hilbert para gérmenes analíticos

Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEProQuestunpu

Gemischte Volumina, gemischte Ehrhart-Theorie und deren Anwendungen in tropischer Geometry und Gestaengekonfigurationsproblemen

The aim of this thesis is the discussion of mixed volumes, their interplay with algebraic geometry, discrete geometry and tropical geometry and their use in applications such as linkage configuration problems. Namely we present new technical tools for mixed volume computation, a novel approach to Ehrhart theory that links mixed volumes with counting integer points in Minkowski sums, new expressions in terms of mixed volumes of combinatorial quantities in tropical geometry and furthermore we employ mixed volume techniques to obtain bounds in certain graph embedding problems.Ziel dieser Arbeit ist die Diskussion gemischter Volumina, ihres Zusammenspiels mit der algebraischen Geometrie, der diskreten Geometrie und der tropischen Geometrie sowie deren Anwendungen im Bereich von Gestaenge-Konfigurationsproblemen. Wir praesentieren insbesondere neue Methoden zur Berechnung gemischter Volumina, einen neuen Zugang zur Ehrhart Theorie, welcher gemischte Volumina mit der Enumeration ganzzahliger Punkte in Minkowski-Summen verbindet, neue Formeln, die kombinatorische Groessen der tropischen Geometrie mithilfe gemischter Volumina beschreiben, und einen neuen Ansatz zur Verwendung gemischter Volumina zur Loesung eines Einbettungsproblems der Graphentheorie

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2

Formal methods and digital systems validation for airborne systems

This report has been prepared to supplement a forthcoming chapter on formal methods in the FAA Digital Systems Validation Handbook. Its purpose is as follows: to outline the technical basis for formal methods in computer science; to explain the use of formal methods in the specification and verification of software and hardware requirements, designs, and implementations; to identify the benefits, weaknesses, and difficulties in applying these methods to digital systems used on board aircraft; and to suggest factors for consideration when formal methods are offered in support of certification. These latter factors assume the context for software development and assurance described in RTCA document DO-178B, 'Software Considerations in Airborne Systems and Equipment Certification,' Dec. 1992