5,327 research outputs found
Survey on counting special types of polynomials
Most integers are composite and most univariate polynomials over a finite
field are reducible. The Prime Number Theorem and a classical result of
Gau{\ss} count the remaining ones, approximately and exactly.
For polynomials in two or more variables, the situation changes dramatically.
Most multivariate polynomials are irreducible. This survey presents counting
results for some special classes of multivariate polynomials over a finite
field, namely the the reducible ones, the s-powerful ones (divisible by the
s-th power of a nonconstant polynomial), the relatively irreducible ones
(irreducible but reducible over an extension field), the decomposable ones, and
also for reducible space curves. These come as exact formulas and as
approximations with relative errors that essentially decrease exponentially in
the input size.
Furthermore, a univariate polynomial f is decomposable if f = g o h for some
nonlinear polynomials g and h. It is intuitively clear that the decomposable
polynomials form a small minority among all polynomials. The tame case, where
the characteristic p of Fq does not divide n = deg f, is fairly
well-understood, and we obtain closely matching upper and lower bounds on the
number of decomposable polynomials. In the wild case, where p does divide n,
the bounds are less satisfactory, in particular when p is the smallest prime
divisor of n and divides n exactly twice. The crux of the matter is to count
the number of collisions, where essentially different (g, h) yield the same f.
We present a classification of all collisions at degree n = p^2 which yields an
exact count of those decomposable polynomials.Comment: to appear in Jaime Gutierrez, Josef Schicho & Martin Weimann
(editors), Computer Algebra and Polynomials, Lecture Notes in Computer
Scienc
Tame Decompositions and Collisions
A univariate polynomial f over a field is decomposable if f = g o h = g(h)
for nonlinear polynomials g and h. It is intuitively clear that the
decomposable polynomials form a small minority among all polynomials over a
finite field. The tame case, where the characteristic p of Fq does not divide n
= deg f, is fairly well-understood, and we have reasonable bounds on the number
of decomposables of degree n. Nevertheless, no exact formula is known if
has more than two prime factors. In order to count the decomposables, one wants
to know, under a suitable normalization, the number of collisions, where
essentially different (g, h) yield the same f. In the tame case, Ritt's Second
Theorem classifies all 2-collisions.
We introduce a normal form for multi-collisions of decompositions of
arbitrary length with exact description of the (non)uniqueness of the
parameters. We obtain an efficiently computable formula for the exact number of
such collisions at degree n over a finite field of characteristic coprime to p.
This leads to an algorithm for the exact number of decomposable polynomials at
degree n over a finite field Fq in the tame case
Selectron production at an e-e- linear collider with transversely polarized beams
We study selectron production at an e-e- linear collider. With the help of
transverse beam polarizations, we define CP sensitive observables in the
production process e- e- --> selectron_L selectron_R. This process proceeds via
t-channel and u-channel exchange of neutralinos, and is sensitive to CP
violation in the neutralino sector. We present numerical results and estimate
the significances to which the CP sensitive observables can be measured.Comment: 14 page
The Thermal Evolution of the Postshock Layer in Pregalactic Clouds
We re-examine the thermal evolution of the postshock layer in primordial gas
clouds. Comparing the time scales, we find that the evolutionary paths of
postshock regions in primordial gas clouds can be basically understood in terms
of the diagram drawn in the ionization degree vs temperature plane. The results
obtained from the diagram are independent of the density in the case that we do
not include photodissociation and photoionization. We also argue that the
diagram is not only relevant to the case of the steady postshock flow, but also
to the isochorically cooling gas.Comment: 15pages, tar gzipped, 5 figures, PTP TeX (PTP style files are in
http://www2.yukawa.kyoto-u.ac.jp/~ptpwww/ptptex-eng.ptp.html). Progress of
Theoretical Physics, in pres
Two generalizations of the Boltzmann equation
We connect two different generalizations of Boltzmann's kinetic theory by
requiring the same stationary solution. Non-extensive statistics can be
produced by either using corresponding collision rates nonlinear in the
one-particle densities or equivalently by using nontrivial energy composition
rules in the energy conservation constraint. Direct transformation formulas
between key functions of the two approaches are given.Comment: Talk given at the 3rd NEXT-Sigma-Phi Conference, Crete, Aug.2005,
revtex, 10 page, 2 fig
A Monte-Carlo generator for statistical hadronization in high energy e+e- collisions
We present a Monte-Carlo implementation of the Statistical Hadronization
Model in e+e- collisions. The physical scheme is based on the statistical
hadronization of massive clusters produced by the event generator Herwig within
the microcanonical ensemble. We present a preliminary comparison of several
observables with measurements in e+e- collisions at the Z peak. Although a fine
tuning of the model parameters is not carried out, a general good agreement
between its predictions and data is found.Comment: 19 pages, 28 figures, 6 tables. v2: added sections on comparison
between the Statistical Hadronization Model and the Cluster Model and on the
interplay between Herwig cluster splitting algorithm and Statistical
Hadronization Model predictions. Fixed typos and references added. Version
accepted for publication in EPJ
Low pressure gas flow analysis through an effusive inlet using mass spectrometry
A mass spectrometric method for analyzing flow past and through an effusive inlet designed for use on the tethered satellite and other entering vehicles is discussed. Source stream concentrations of species in a gaseous mixture are determined using a calibration of measured mass spectral intensities versus source stream pressure for standard gas mixtures and pure gases. Concentrations are shown to be accurate within experimental error. Theoretical explanations for observed mass discrimination effects as they relate to the various flow situations in the effusive inlet and the experimental apparatus are discussed
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