258 research outputs found
Tutorial on machine learning and data mining
The ability to learn from observations and to modify our understanding of the world based on experience is an essential aspect of intelligent behavior. Machine learning has thus become an important sub-area of artificial intelligence research and has yielded a number of interesting results about how it is possible to build computer systems that learn as well as insights into the process of learning. In addition, many new technologies have been developed which have been applied in the area of automatic learning from large databases.
Searching large databases for hidden relationships is the focus of the new area known as data mining, which is being applied increasingly to search large databases for bidden nuggets of information.
In this tutorial we will quickly introduce the related fields of machine learning and data mining. The tutorial will run for two hours. The first hour will be an introduction to machine learning, and the second will be a more in depth look at induction and data mining and how these fields extend from machine learning
Asymptotics for Sobolev orthogonal polynomials for exponential weights
38 pages, no figures.-- MSC2000 codes: 42C05, 33C25.MR#: MR2164139 (2006c:41040)Zbl#: Zbl 1105.42016^aLet , and let , x\in \mbox{\smallbf R}. Let \psi \in L_{\infty }(\mbox{\smallbf R}) be positive on a set of positive measure. For , one may form Sobolev orthonormal polynomials , associated with the Sobolev inner product ( f,g) =\int_{\mbox{\scriptsize\bf R}}fg( \psi W) ^{2}+\lambda \int_{\mbox{\scriptsize\bf R}}f^{\prime }g^{\prime }W^{2}. We establish strong asymptotics for the in terms of the ordinary orthonormal polynomials for the weight , on and off the real line. More generally, we establish a close asymptotic relationship between and for exponential weights on a real interval , under mild conditions on . The method is new and will apply to many situations beyond that treated in this paper.The work by F. Marcellan has been supported by Dirección General de Investigación (Ministerio de Ciencia y Technología) of Spain under grant BFM
2003-06335-C03-07, as well as NATO Collaborative grant PST.CLG 979738. J. Geronimo and D. Lubinsky, respectively, acknowledge support by NSF grants DMS-0200219 and DMS-0400446.Publicad
Non-normality of continued fraction partial quotients modulo q
It is well known that almost all real numbers (in the sense of Lebesgue measure) are normal to base q where q ≥ 2 is any integer base
Finite dimensional quantizations of the (q,p) plane : new space and momentum inequalities
We present a N-dimensional quantization a la Berezin-Klauder or frame
quantization of the complex plane based on overcomplete families of states
(coherent states) generated by the N first harmonic oscillator eigenstates. The
spectra of position and momentum operators are finite and eigenvalues are
equal, up to a factor, to the zeros of Hermite polynomials. From numerical and
theoretical studies of the large behavior of the product of non null smallest positive and largest eigenvalues, we infer
the inequality (resp. ) involving, in suitable
units, the minimal () and maximal () sizes of
regions of space (resp. momentum) which are accessible to exploration within
this finite-dimensional quantum framework. Interesting issues on the
measurement process and connections with the finite Chern-Simons matrix model
for the Quantum Hall effect are discussed
Ewens measures on compact groups and hypergeometric kernels
On unitary compact groups the decomposition of a generic element into product
of reflections induces a decomposition of the characteristic polynomial into a
product of factors. When the group is equipped with the Haar probability
measure, these factors become independent random variables with explicit
distributions. Beyond the known results on the orthogonal and unitary groups
(O(n) and U(n)), we treat the symplectic case. In U(n), this induces a family
of probability changes analogous to the biassing in the Ewens sampling formula
known for the symmetric group. Then we study the spectral properties of these
measures, connected to the pure Fisher-Hartvig symbol on the unit circle. The
associated orthogonal polynomials give rise, as tends to infinity to a
limit kernel at the singularity.Comment: New version of the previous paper "Hua-Pickrell measures on general
compact groups". The article has been completely re-written (the presentation
has changed and some proofs have been simplified). New references added
Theory of random matrices with strong level confinement: orthogonal polynomial approach
Strongly non-Gaussian ensembles of large random matrices possessing unitary
symmetry and logarithmic level repulsion are studied both in presence and
absence of hard edge in their energy spectra. Employing a theory of polynomials
orthogonal with respect to exponential weights we calculate with asymptotic
accuracy the two-point kernel over all distance scale, and show that in the
limit of large dimensions of random matrices the properly rescaled local
eigenvalue correlations are independent of level confinement while global
smoothed connected correlations depend on confinement potential only through
the endpoints of spectrum. We also obtain exact expressions for density of
levels, one- and two-point Green's functions, and prove that new universal
local relationship exists for suitably normalized and rescaled connected
two-point Green's function. Connection between structure of Szeg\"o function
entering strong polynomial asymptotics and mean-field equation is traced.Comment: 12 pages (latex), to appear in Physical Review
Some extremal functions in Fourier analysis, III
We obtain the best approximation in , by entire functions of
exponential type, for a class of even functions that includes
, where , and , where . We also give periodic versions of these results where the
approximating functions are trigonometric polynomials of bounded degree.Comment: 26 pages. Submitte
Introduction to Random Matrices
These notes provide an introduction to the theory of random matrices. The
central quantity studied is where is the integral
operator with kernel 1/\pi} {\sin\pi(x-y)\over x-y} \chi_I(y). Here
and is the characteristic function
of the set . In the Gaussian Unitary Ensemble (GUE) the probability that no
eigenvalues lie in is equal to . Also is a tau-function
and we present a new simplified derivation of the system of nonlinear
completely integrable equations (the 's are the independent variables)
that were first derived by Jimbo, Miwa, M{\^o}ri, and Sato in 1980. In the case
of a single interval these equations are reducible to a Painlev{\'e} V
equation. For large we give an asymptotic formula for , which is
the probability in the GUE that exactly eigenvalues lie in an interval of
length .Comment: 44 page
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