This note tries to show that a re-examination of a first course in analysis, using the more sophisticated tools and approaches obtained in later stages, can be a real fun for experts, advanced students, etc. We start by going to the extreme, namely w
e present two proofs of the Extreme Value Theorem: the programmer proof that suggests a method (which is practical in down-to-earth settings) to approximate, to any required precision, the extreme values of the given function in a metric space setting, and an abstract space proof (the level-set proof) for semicontinuous functions defined on compact topological spaces. Next, in the intermediate part, we consider the Intermediate Value Theorem, generalize it to a wide class of discontinuous functions, and re-examine the meaning of the intermediate value property. The trek reaches the final frontier when we discuss the Uniform Continuity Theorem, generalize it, re-examine the meaning of uniform continuity, and find the optimal delta of the given epsilon. Have fun!
In this paper we explain how Morse theory for the Yang-Mills functional can be used to prove an analogue, for surface groups, of the Atiyah-Segal theorem. Classically, the Atiyah-Segal theorem relates the representation ring R(Gamma) of a compact Lie
group $Gamma$ to the complex K-theory of the classifying space $BGamma$. For infinite discrete groups, it is necessary to take into account deformations of representations, and with this in mind we replace the representation ring by Carlssons deformation $K$--theory spectrum $K (Gamma)$ (the homotopy-theoretical analogue of $R(Gamma)$). Our main theorem provides an isomorphism in homotopy $K_*(pi_1 Sigma)isom K^{-*}(Sigma)$ for all compact, aspherical surfaces $Sigma$ and all $*>0$. Combining this result with work of Tyler Lawson, we obtain homotopy theoretical information about the stable moduli space of flat unitary connections over surfaces.
We revisit Atiyah and Botts study of Morse theory for the Yang-Mills functional over a Riemann surface, and establish new formulas for the minimum codimension of a (non-semi-stable) stratum. These results yield the exact connectivity of the natural m
ap (C_{min} E)//G(E) --> Map^E (M, BU(n)) from the homotopy orbits of the space of central Yang-Mills connections to the classifying space of the gauge group G(E). All of these results carry over to non-orientable surfaces via Ho and Lius non-orientable Yang-Mills theory. A somewhat less detailed version of this paper (titled On the Yang-Mills stratification for surfaces) will appear in the Proceedings of the AMS.
We compute the homotopy type of the moduli space of flat, unitary connections over aspherical surfaces, after stabilizing with respect to the rank of the underlying bundle. Over the orientable surface M^g, we show that this space has the homotopy typ
e of the infinite symmetric product of M^g, generalizing a well-known fact for the torus. Over a non-orientable surface, we show that this space is homotopy equivalent to a disjoint union of two tori, whose common dimension corresponds to the rank of the first (co)homology group of the surface. Similar calculations are provided for products of surfaces, and show a close analogy with the Quillen-Lichtenbaum conjectures in algebraic K-theory. The proofs utilize Tyler Lawsons work in deformation K-theory, and rely heavily on Yang-Mills theory and gauge theory.
Motivated by the fact that a polarized ${}^3$He nucleus behaves as an `effective neutron target, we examine manifestations of neutron electromagnetic polarizabilities in elastic Compton scattering from the Helium-3 nucleus. We calculate both unpolari
zed and double-polarization observables using chiral perturbation theory to next-to-leading order (${mathcal O}(e^2 Q)$) at energies, $omega leq m_{pi}$, where $m_{pi}$ is the pion mass. Our results show that the unpolarized differential cross section can be used to measure neutron electric and magnetic polarizabilities, while two double-polarization observables are sensitive to different linear combinations of the four neutron spin polarizabilities. [Note added in 2018] The qualitative conclusions and analytic formulae presented in this paper are correct, but several of the numerical results are wrong: see the erratum posted as arXiv:1804.01206 for further details. A full suite of corrected numerical results for cross sections and asymmetries can be found in Margaryan et al., arXiv:1804.00956. They can also be obtained as an interactive Mathematica notebook by emailing [email protected].
Collaborative work on unstructured or semi-structured documents, such as in literature corpora or source code, often involves agreed upon templates containing metadata. These templates are not consistent across users and over time. Rule-based parsing
of these templates is expensive to maintain and tends to fail as new documents are added. Statistical techniques based on frequent occurrences have the potential to identify automatically a large fraction of the templates, thus reducing the burden on the programmers. We investigate the case of the Project Gutenberg corpus, where most documents are in ASCII format with preambles and epilogues that are often copied and pasted or manually typed. We show that a statistical approach can solve most cases though some documents require knowledge of English. We also survey various technical solutions that make our approach applicable to large data sets.
Many applications use sequences of n consecutive symbols (n-grams). Hashing these n-grams can be a performance bottleneck. For more speed, recursive hash families compute hash values by updating previous values. We prove that recursive hash families
cannot be more than pairwise independent. While hashing by irreducible polynomials is pairwise independent, our implementations either run in time O(n) or use an exponential amount of memory. As a more scalable alternative, we make hashing by cyclic polynomials pairwise independent by ignoring n-1 bits. Experimentally, we show that hashing by cyclic polynomials is is twice as fast as hashing by irreducible polynomials. We also show that randomized Karp-Rabin hash families are not pairwise independent.
Increasingly, business projects are ephemeral. New Business Intelligence tools must support ad-lib data sources and quick perusal. Meanwhile, tag clouds are a popular community-driven visualization technique. Hence, we investigate tag-cloud views wit
h support for OLAP operations such as roll-ups, slices, dices, clustering, and drill-downs. As a case study, we implemented an application where users can upload data and immediately navigate through its ad hoc dimensions. To support social networking, views can be easily shared and embedded in other Web sites. Algorithmically, our tag-cloud views are approximate range top-k queries over spontaneous data cubes. We present experimental evidence that iceberg cuboids provide adequate online approximations. We benchmark several browser-oblivious tag-cloud layout optimizations.
General conservation equations are derived for 2D dense granular flows from the Euler equation within the Boussinesq approximation. In steady flows, the 2D fields of granular temperature, vorticity and stream function are shown to be encoded in two s
calar functions only. We checked such prediction on steady surface flows in a rotating drum simulated through the Non-Smooth Contact Dynamics method. This result is non trivial because granular flows are dissipative and therefore not necessarily compatible with Euler equation. Finally, we briefly discuss some possible ways to predict theoretically these two functions using statistical mechanics.
We establish a connection between certain unique models, or equivalently unique functionals, for representations of p-adic groups and linear characters of their corresponding Hecke algebras. This allows us to give a uniform evaluation of the image of
spherical and Iwahori-fixed vectors in the unramified principal series for this class of models. We provide an explicit alternator expressionfor the image of the spherical vectors under these functionals in terms of the representation theory of the dual group.
