Author: Gabriel Debs
Publisher: American Mathematical Soc.
ISBN: 0821839713
Category : Mathematics
Languages : en
Pages : 134
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One of the aims of this work is to investigate some natural properties of Borel sets which are undecidable in $ZFC$. The authors' starting point is the following elementary, though non-trivial result: Consider $X \subset 2omega\times2omega$, set $Y=\pi(X)$, where $\pi$ denotes the canonical projection of $2omega\times2omega$ onto the first factor, and suppose that $(\star)$: Any compact subset of $Y$ is the projection of some compact subset of $X$. If moreover $X$ is $\mathbf{\Pi 0 2$ then $(\star\star)$: The restriction of $\pi$ to some relatively closed subset of $X$ is perfect onto $Y$ it follows that in the present case $Y$ is also $\mathbf{\Pi 0 2$. Notice that the reverse implication $(\star\star)\Rightarrow(\star)$ holds trivially for any $X$ and $Y$. But the implication $(\star)\Rightarrow (\star\star)$ for an arbitrary Borel set $X \subset 2omega\times2omega$ is equivalent to the statement $\forall \alpha\in \omegaomega, \, \aleph 1$ is inaccessible in $L(\alpha)$. More precisely The authors prove that the validity of $(\star)\Rightarrow(\star\star)$ for all $X \in \varSigma0 {1+\xi+1 $, is equivalent to $\aleph \xi \aleph 1$. $ZFC$, derive from $(\star)$ the weaker conclusion that $Y$ is also Borel and of the same Baire class as $X$. This last result solves an old problem about compact covering mappings. In fact these results are closely related to the following general boundedness principle Lift$(X, Y)$: If any compact subset of $Y$ admits a continuous lifting in $X$, then $Y$ admits a continuous lifting in $X$, where by a lifting of $Z\subset \pi(X)$ in $X$ we mean a mapping on $Z$ whose graph is contained in $X$. The main result of this work will give the exact set theoretical strength of this principle depending on the descriptive complexity of $X$ and $Y$. The authors also prove a similar result for a variation of Lift$(X, Y)$ in which continuous liftings are replaced by Borel liftings, and which answers a question of H. Friedman. Among other applications the authors obtain a complete solution to a problem which goes back to Lusin concerning the existence of $\mathbf{\Pi 1 1$ sets with all constituents in some given class $\mathbf{\Gamma $ of Borel sets, improving earlier results by J. Stern and R. Sami. Borel sets (in $ZFC$) of a new type, involving a large amount of abstract algebra. This representation was initially developed for the purposes of this proof, but has several other applications.
Author: Matthias Baaz
Publisher: Cambridge University Press
ISBN: 1139498436
Category : Mathematics
Languages : en
Pages : 541
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This volume commemorates the life, work and foundational views of Kurt Gödel (1906–78), most famous for his hallmark works on the completeness of first-order logic, the incompleteness of number theory, and the consistency - with the other widely accepted axioms of set theory - of the axiom of choice and of the generalized continuum hypothesis. It explores current research, advances and ideas for future directions not only in the foundations of mathematics and logic, but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology and the history of science. The discussion is supplemented by personal reflections from several scholars who knew Gödel personally, providing some interesting insights into his life. By putting his ideas and life's work into the context of current thinking and perceptions, this book will extend the impact of Gödel's fundamental work in mathematics, logic, philosophy and other disciplines for future generations of researchers.
Author: Dominic Verity
Publisher: American Mathematical Soc.
ISBN: 0821841424
Category : Mathematics
Languages : en
Pages : 208
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The primary purpose of this work is to characterise strict $\omega$-categories as simplicial sets with structure. The author proves the Street-Roberts conjecture in the form formulated by Ross Street in his work on Orientals, which states that they are exactly the ``complicial sets'' defined and named by John Roberts in his handwritten notes of that title (circa 1978). On the way the author substantially develops Roberts' theory of complicial sets itself and makes contributions to Street's theory of parity complexes. In particular, he studies a new monoidal closed structure on the category of complicial sets which he shows to be the appropriate generalisation of the (lax) Gray tensor product of 2-categories to this context. Under Street's $\omega$-categorical nerve construction, which the author shows to be an equivalence, this tensor product coincides with those of Steiner, Crans and others.
Author: Gail Letzter
Publisher: American Mathematical Soc.
ISBN: 0821841319
Category : Mathematics
Languages : en
Pages : 104
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Book Description
This paper studies quantum invariant differential operators for quantum symmetric spaces in the maximally split case. The main results are quantum versions of theorems of Harish-Chandra and Helgason: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential operators and the ring of invariants of a certain Laurent polynomial ring under an action of the restricted Weyl group. Moreover, the image of the center under this map is the entire invariant ring if and only if the underlying irreducible symmetric pair is not of four exceptional types. In the process, the author finds a particularly nice basis for the quantum invariant differential operators that provides a new interpretation of difference operators associated to Macdonald polynomials.
Author: Ron Donagi
Publisher: American Mathematical Soc.
ISBN: 0821840924
Category : Mathematics
Languages : en
Pages : 104
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Let $X$ be a smooth elliptic fibration over a smooth base $B$. Under mild assumptions, the authors establish a Fourier-Mukai equivalence between the derived categories of two objects, each of which is an $\mathcal{O} DEGREES{\times}$ gerbe over a genus one fibration which is a twisted form
Author: Yuanhua Wang
Publisher: American Mathematical Soc.
ISBN: 0821841661
Category : Mathematics
Languages : en
Pages : 104
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The purpose of this paper is to establish the spinor genus theory of quadratic forms over global function fields in characteristic 2. The first part of the paper computes the integral spinor norms and relative spinor norms. The second part of the paper gives a complete answer to the integral representations of one quadratic form by another with more than four variables over a global function field in characteristic 2.
Author: Alessandra Celletti
Publisher: American Mathematical Soc.
ISBN: 0821841696
Category : Mathematics
Languages : en
Pages : 150
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KAM theory is a powerful tool apt to prove perpetual stability in Hamiltonian systems, which are a perturbation of integrable ones. The smallness requirements for its applicability are well known to be extremely stringent. A long standing problem, in this context, is the application of KAM theory to ``physical systems'' for ``observable'' values of the perturbation parameters. The authors consider the Restricted, Circular, Planar, Three-Body Problem (RCP3BP), i.e., the problem of studying the planar motions of a small body subject to the gravitational attraction of two primary bodies revolving on circular Keplerian orbits (which are assumed not to be influenced by the small body). When the mass ratio of the two primary bodies is small, the RCP3BP is described by a nearly-integrable Hamiltonian system with two degrees of freedom; in a region of phase space corresponding to nearly elliptical motions with non-small eccentricities, the system is well described by Delaunay variables. The Sun-Jupiter observed motion is nearly circular and an asteroid of the Asteroidal belt may be assumed not to influence the Sun-Jupiter motion. The Jupiter-Sun mass ratio is slightly less than 1/1000. The authors consider the motion of the asteroid 12 Victoria taking into account only the Sun-Jupiter gravitational attraction regarding such a system as a prototype of a RCP3BP. for values of mass ratios up to 1/1000, they prove the existence of two-dimensional KAM tori on a fixed three-dimensional energy level corresponding to the observed energy of the Sun-Jupiter-Victoria system. Such tori trap the evolution of phase points ``close'' to the observed physical data of the Sun-Jupiter-Victoria system. As a consequence, in the RCP3BP description, the motion of Victoria is proven to be forever close to an elliptical motion. The proof is based on: 1) a new iso-energetic KAM theory; 2) an algorithm for computing iso-energetic, approximate Lindstedt series; 3) a computer-aided application of 1)+2) to the Sun-Jupiter-Victoria system. The paper is self-contained but does not include the ($\sim$ 12000 lines) computer programs, which may be obtained by sending an e-mail to one of the authors.
Author: Tao Mei
Publisher: American Mathematical Soc.
ISBN: 0821839802
Category : Mathematics
Languages : en
Pages : 78
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The author gives a systematic study of the Hardy spaces of functions with values in the noncommutative $Lp$-spaces associated with a semifinite von Neumann algebra $\mathcal{M .$ This is motivated by matrix valued Harmonic Analysis (operator weighted norm inequalities, operator Hilbert transform), as well as by the recent development of noncommutative martingale inequalities. in this paper noncommutative Hardy spaces are defined by noncommutative Lusin integral function, and it isproved that they are equivalent to those defined by noncommutative Littlewood-Paley G-functions. The main results of this paper include: (i) The analogue in the author's setting of the classical Fefferman duality theorem between $\mathcal{H 1$ and $\mathrm{BMO $. (ii) The atomic decomposition of theauthor's noncommutative $\mathcal{H 1.$ (iii) The equivalence between the norms of the noncommutative Hardy spaces and of the noncommutative $Lp$-spaces $(1
Author: Raphael Ponge
Publisher: American Mathematical Soc.
ISBN: 0821841483
Category : Mathematics
Languages : en
Pages : 150
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This memoir deals with the hypoelliptic calculus on Heisenberg manifolds, including CR and contact manifolds. In this context the main differential operators at stake include the Hormander's sum of squares, the Kohn Laplacian, the horizontal sublaplacian, the CR conformal operators of Gover-Graham and the contact Laplacian. These operators cannot be elliptic and the relevant pseudodifferential calculus to study them is provided by the Heisenberg calculus of Beals-Greiner andTaylor.
Author: Bruce C. Berndt
Publisher: American Mathematical Soc.
ISBN: 082183973X
Category : Mathematics
Languages : en
Pages : 110
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Book Description
Sir Arthur Conan Doyle's famous fictional detective Sherlock Holmes and his sidekick Dr. Watson go camping and pitch their tent under the stars. During the night, Holmes wakes his companion and says, ``Watson, look up at the stars and tell me what you deduce.'' Watson says, ``I see millions of stars, and it is quite likely that a few of them are planets just like Earth. Therefore there may also be life on these planets.'' Holmes replies, ``Watson, you idiot. Somebody stole ourtent.'' When seeking proofs of Ramanujan's identities for the Rogers-Ramanujan functions, Watson, i.e., G. N. Watson, was not an ``idiot.'' He, L. J. Rogers, and D. M. Bressoud found proofs for several of the identities. A. J. F. Biagioli devised proofs for most (but not all) of the remaining identities.Although some of the proofs of Watson, Rogers, and Bressoud are likely in the spirit of those found by Ramanujan, those of Biagioli are not. in particular, Biagioli used the theory of modular forms. Haunted by the fact that little progress has been made into Ramanujan's insights on these identities in the past 85 years, the present authors sought ``more natural'' proofs. Thus, instead of a missing tent, we have had missing proofs, i.e., Ramanujan's missing proofs of his forty identities for theRogers-Ramanujan functions. in this paper, for 35 of the 40 identities, the authors offer proofs that are in the spirit of Ramanujan. Some of the proofs presented here are due to Watson, Rogers, and Bressoud, but most are new. Moreover, for several identities, the authors present two or threeproofs. For the five identities that they are unable to prove, they provide non-rigorous verifications based on an asymptotic analysis of the associated Rogers-Ramanujan functions. This method, which is related to the 5-dissection of the generating function for cranks found in Ramanujan's lost notebook, is what Ramanujan might have used to discover several of the more difficult identities. Some of the new methods in this paper can be employed to establish new identities for the Rogers-Ramanujanfunctions.
