59,661 research outputs found
A multi-variable version of the completed Riemann zeta function and other -functions
We define a generalisation of the completed Riemann zeta function in several
complex variables. It satisfies a functional equation, shuffle product
identities, and has simple poles along finitely many hyperplanes, with a
recursive structure on its residues. The special case of two variables can be
written as a partial Mellin transform of a real analytic Eisenstein series,
which enables us to relate its values at pairs of positive even points to
periods of (simple extensions of symmetric powers of the cohomology of) the CM
elliptic curve corresponding to the Gaussian integers. In general, the totally
even values of these functions are related to new quantities which we call
multiple quadratic sums.
More generally, we cautiously define multiple-variable versions of motivic
-functions and ask whether there is a relation between their special values
and periods of general mixed motives. We show that all periods of mixed Tate
motives over the integers, and all periods of motivic fundamental groups (or
relative completions) of modular groups, are indeed special values of the
multiple motivic -values defined here.Comment: This is the second half of a talk given in honour of Ihara's 80th
birthday, and will appear in the proceedings thereo
Sub-computable Boundedness Randomness
This paper defines a new notion of bounded computable randomness for certain
classes of sub-computable functions which lack a universal machine. In
particular, we define such versions of randomness for primitive recursive
functions and for PSPACE functions. These new notions are robust in that there
are equivalent formulations in terms of (1) Martin-L\"of tests, (2) Kolmogorov
complexity, and (3) martingales. We show these notions can be equivalently
defined with prefix-free Kolmogorov complexity. We prove that one direction of
van Lambalgen's theorem holds for relative computability, but the other
direction fails. We discuss statistical properties of these notions of
randomness
Note on improvement precision of recursive function simulation in floating point standard
An improvement on precision of recursive function simulation in IEEE floating
point standard is presented. It is shown that the average of rounding towards
negative infinite and rounding towards positive infinite yields a better result
than the usual standard rounding to the nearest in the simulation of recursive
functions. In general, the method improves one digit of precision and it has
also been useful to avoid divergence from a correct stationary regime in the
logistic map. Numerical studies are presented to illustrate the method.Comment: DINCON 2017 - Conferencia Brasileira de Dinamica, Controle e
Aplicacoes - Sao Jose do Rio Preto - Brazil. 8 page
Diophantine sets of polynomials over algebraic extensions of the rationals
Let L be a recursive algebraic extension of Q. Assume that, given alpha is an element of L, we can compute the roots in L of its minimal polynomial over Q and we can determine which roots are Aut(L)-conjugate to alpha. We prove that there exists a pair of polynomials that characterizes the Aut(L)-conjugates of alpha, and that these polynomials can be effectively computed. Assume furthermore that L can be embedded in R, or in a finite extension of Q(p) (with p an odd prime). Then we show that subsets of L[X](k) that are recursively enumerable for every recursive presentation of L[X], are diophantine over L[X]
A recursive scheme for computing autocorrelation functions of decimated complex wavelet subbands
This paper deals with the problem of the exact computation of the autocorrelation function of a real or complex discrete wavelet subband of a signal, when the autocorrelation function (or Power Spectral Density, PSD) of the signal in the time domain (or spatial domain) is either known or estimated using a separate technique. The solution to this problem allows us to couple time domain noise estimation techniques to wavelet domain denoising algorithms, which is crucial for the development of blind wavelet-based denoising techniques. Specifically, we investigate the Dual-Tree complex wavelet transform (DT-CWT), which has a good directional selectivity in 2-D and 3-D, is approximately shift-invariant, and yields better denoising results than a discrete wavelet transform (DWT). The proposed scheme gives an analytical relationship between the PSD of the input signal/image and the PSD of each individual real/complex wavelet subband which is very useful for future developments. We also show that a more general technique, that relies on Monte-Carlo simulations, requires a large number of input samples for a reliable estimate, while the proposed technique does not suffer from this problem
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
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