10,461 research outputs found
Sensitivity of Markov chains for wireless protocols
Network communication protocols such as the IEEE 802.11 wireless protocol are currently best modelled as Markov chains. In these situations we have some protocol parameters , and a transition matrix from which we can compute the steady state (equilibrium) distribution and hence final desired quantities , which might be for example the throughput of the protocol. Typically the chain will have thousands of states, and a particular example of interest is the Bianchi chain defined later. Generally we want to optimise , perhaps subject to some constraints that also depend on the Markov chain. To do this efficiently we need the gradient of with respect to , and therefore need the gradient of and other properties of the chain with respect to . The matrix formulas available for this involve the so-called fundamental matrix, but are there approximate gradients available which are faster and still sufficiently accurate? In some cases BT would like to do the whole calculation in computer algebra, and get a series expansion of the equilibrium with respect to a parameter in . In addition to the steady state , the same questions arise for the mixing time and the mean hitting times. Two qualitative features that were brought to the Study Group’s attention were:
* the transition matrix is large, but sparse.
* the systems of linear equations to be solved are generally singular and need some additional normalisation condition, such as is provided by using the fundamental matrix.
We also note a third highly important property regarding applications of numerical linear algebra:
* the transition matrix is asymmetric.
A realistic dimension for the matrix in the Bianchi model described below is 8064×8064, but on average there are only a few nonzero entries per column. Merely storing such a large matrix in dense form would require nearly 0.5GBytes using 64-bit floating point numbers, and computing its LU factorisation takes around 80 seconds on a modern microprocessor. It is thus highly desirable to employ specialised algorithms for sparse matrices. These algorithms are generally divided between those only applicable to symmetric matrices, the most prominent being the conjugate-gradient (CG) algorithm for solving linear equations, and those applicable to general matrices. A similar division is present in the literature on numerical eigenvalue problems
Revisiting the Complexity of Stability of Continuous and Hybrid Systems
We develop a framework to give upper bounds on the "practical" computational
complexity of stability problems for a wide range of nonlinear continuous and
hybrid systems. To do so, we describe stability properties of dynamical systems
using first-order formulas over the real numbers, and reduce stability problems
to the delta-decision problems of these formulas. The framework allows us to
obtain a precise characterization of the complexity of different notions of
stability for nonlinear continuous and hybrid systems. We prove that bounded
versions of the stability problems are generally decidable, and give upper
bounds on their complexity. The unbounded versions are generally undecidable,
for which we give upper bounds on their degrees of unsolvability
A formally verified proof of the prime number theorem
The prime number theorem, established by Hadamard and de la Vall'ee Poussin
independently in 1896, asserts that the density of primes in the positive
integers is asymptotic to 1 / ln x. Whereas their proofs made serious use of
the methods of complex analysis, elementary proofs were provided by Selberg and
Erd"os in 1948. We describe a formally verified version of Selberg's proof,
obtained using the Isabelle proof assistant.Comment: 23 page
The Archimedean trap: Why traditional reinforcement learning will probably not yield AGI
After generalizing the Archimedean property of real numbers in such a way as to make it adaptable to non-numeric structures, we demonstrate that the real numbers cannot be used to accurately measure non-Archimedean structures. We argue that, since an agent with Artificial General Intelligence (AGI) should have no problem engaging in tasks that inherently involve non-Archimedean rewards, and since traditional reinforcement learning rewards are real numbers, therefore traditional reinforcement learning probably will not lead to AGI. We indicate two possible ways traditional reinforcement learning could be altered to remove this roadblock
Root Refinement for Real Polynomials
We consider the problem of approximating all real roots of a square-free
polynomial . Given isolating intervals, our algorithm refines each of them
to a width of or less, that is, each of the roots is approximated to
bits after the binary point. Our method provides a certified answer for
arbitrary real polynomials, only considering finite approximations of the
polynomial coefficients and choosing a suitable working precision adaptively.
In this way, we get a correct algorithm that is simple to implement and
practically efficient. Our algorithm uses the quadratic interval refinement
method; we adapt that method to be able to cope with inaccuracies when
evaluating , without sacrificing its quadratic convergence behavior. We
prove a bound on the bit complexity of our algorithm in terms of the degree of
the polynomial, the size and the separation of the roots, that is, parameters
exclusively related to the geometric location of the roots. Our bound is near
optimal and significantly improves previous work on integer polynomials.
Furthermore, it essentially matches the best known theoretical bounds on root
approximation which are obtained by very sophisticated algorithms. We also
investigate the practical behavior of the algorithm and demonstrate how closely
the practical performance matches our asymptotic bounds.Comment: This is a substantially extended version of the conference paper
"Efficient Real Root Approximation", appeared at the 36th International
Symposium on Symbolic and Algebraic Computation (ISSAC 2011
Pi Visits Manhattan
Is it possible to draw a circle in Manhattan, using only its discrete network
of streets and boulevards? In this study, we will explore the construction and
properties of circular paths on an integer lattice, a discrete space where the
distance between two points is not governed by the familiar Euclidean metric,
but the Manhattan or taxicab distance, a metric linear in its coordinates. In
order to achieve consistency with the continuous ideal, we need to abandon
Euclid's very original definition of the circle in favour of a parametric
construction. Somewhat unexpectedly, we find that the Euclidean circle's
defining constant can be recovered in such a discrete setting.Comment: 18 pages, 7 figure
Semi-stable models of modular Curves and some arithmetic applications
In this paper, we compute the semi-stable models of modular curves
for odd primes and compute the Arakelov self-intersection numbers of
the relative dualising sheaves for these models. We give two arithmetic
applications of our computations. In particular, we give an effective version
of the Bogomolov conjecture following the strategy outlined by Zhang and find
the stable Faltings heights of the arithmetic surfaces corresponding to these
modular curves.Comment: 28 pages, 14 figure
Tamagawa numbers of polarized algebraic varieties
Let be an ample metrized invertible sheaf on
a smooth quasi-projective algebraic variety defined over a number field.
Denote by the number of rational points in having -height . We consider the problem of a geometric and arithmetic
interpretation of the asymptotic for as in
connection with recent conjectures of Fujita concerning the Minimal Model
Program for polarized algebraic varieties.
We introduce the notions of -primitive varieties and -primitive fibrations. For -primitive varieties over we
propose a method to define an adelic Tamagawa number which
is a generalization of the Tamagawa number introduced by Peyre for
smooth Fano varieties. Our method allows us to construct Tamagawa numbers for
-Fano varieties with at worst canonical singularities. In a series of
examples of smooth polarized varieties and singular Fano varieties we show that
our Tamagawa numbers express the dependence of the asymptotic of on the choice of -adic metrics on .Comment: 54 pages, minor correction
Low degree polynomial equations: arithmetic, geometry and topology
These are the notes of my lectures at the 1996 European Congress of
Mathematicians. {} Polynomials appear in mathematics frequently, and we all
know from experience that low degree polynomials are easier to deal with than
high degree ones. It is, however, not clear that there is a well defined class
of "low degree" polynomials. For many questions, polynomials behave well if
their degree is low enough, but the precise bound on the degree depends on the
concrete problem. {} It turns out that there is a collection of basic questions
in arithmetic, algebraic geometry and topology all of which give the same class
of "low degree" polynomials. The aim of this lecture is to explain these
properties and to provide a survey of the known results.Comment: Main changes are: some errors in sections 3.3 and 4.3 are corrected.
5.5 is newly added AMSTeX 2.
Reference Frame Fields based on Quantum Theory Representations of Real and Complex Numbers
A quantum theory representations of real (R) and complex (C) numbers is given
that is based on states of single, finite strings of qukits for any base k > 1.
Both unary representations and the possibility that qukits with k a prime
number are elementary and the rest composite are discussed. Cauchy sequences of
qukit string states are defined from the arithmetic properties. The
representations of R and C, as equivalence classes of these sequences, differ
from classical kit string state representations in two ways: the freedom of
choice of basis states, and the fact that each quantum theory representation is
part of a mathematical structure that is itself based on the real and complex
numbers. These aspects enable the description of 3 dimensional frame fields
labeled by different k values, different basis or gauge choices, and different
iteration stages. The reference frames in the field are based on each R and C
representation where each frame contains representations of all physical
theories as mathematical structures based on the R and C representation.
Approaches to integrating this with physics are described. It is observed that
R and C values of physical quantities, matrix elements, etc. which are viewed
in a frame as elementary and featureless, are seen in a parent frame as
equivalence classes of Cauchy sequences of qukit string states.Comment: Extensively revised and expanded, title changed. Submitted to
Conference proceedings, NSF Conference, Tyler TX, Sept. 200
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