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Topic: Sigma-finite

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hdm5 Any finite cover is an expansion of a free finite cover with the same fibre and binding groups, and we aim for classification up to conjugacy within the automorphism group of this free cover. We will concentrate on finite covers of countable $\aleph_{0}$-categorical structures, and we often describe structures entirely by their automorphism groups, without reference to any particular language (in the $\aleph_{0}$-categorical case this is justified by the Ryll-Nardzewski theorem). Finite covers whose kernels are finite are analysed in Section 4. www.amsta.leeds.ac.uk /Pure/preprints/hdm/hdm5   (5874 words)

Measure (mathematics) - Wikipedia, the free encyclopedia Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set. It turns out that in general, finitely additive measures are connected with notions such as Banach limits, the dual of L This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. en.wikipedia.org /wiki/Measure_(mathematics)   (931 words)

ISMEAR, SIGMA, FERWE, FERDO tag For the finite temperature LDA SIGMA determines the width of the smearing in eV. Large SIGMA values lead to a similar error as the MP scheme, but in contrast to the MP scheme there is no way to figure out, how large the error due to the smearing is. Therefore the method of MP seems to be superior to the GS method. SIGMA should be as large as possible keeping the difference between the free energy and the total energy (i.e. cms.mpi.univie.ac.at /vasp/vasp/node117.html   (599 words)

National Taiwan University: Model NTU GCM (T42 L13) 1995 At a particular sigma level the frictional drag on the atmosphere from breaking gravity waves depends on the projection of the wind on the surface wind and on the Froude number, which in turn is a function of the reference height, the atmospheric density, the Brunt-Vaisalla frequency, and the wind shear. The model allows partial or total cloud cover in seven vertical sigma layers (layers 3 to 9), while the top two layers and the bottom three layers are specified as being cloudless. The drag at the surface is dependent on sub-gridscale orographic variance (see Orography), and it is parameterized by means of a reference height which is defined to be twice the local standard deviation of the surface heights. www-pcmdi.llnl.gov /projects/modeldoc/amip/30ntu.html   (1705 words)

group.txt The sign of a permutation sigma is (-1)^n where n is a finite positive integer such that there exist n transpositions whose composition equals sigma. For example, the permutation sigma: 1, 2, 3 to 1, 2, 3 defined by sigma(1) = 2, sigma(2) = 3, and sigma(3) = 1 is equivalent to the composition of two transpositions. If a group G has n elements where n is a positive integer, then G is a finite group with order n. www.math.harvard.edu /~knill/sofia/data/group.txt   (4457 words)

PROPSND A tableau from sigma is the union of a sequence of finite tableaux constructed by application of (ii) and (iii). Deductions from premises ------------------------ The finite tableaux (with premises) from sigma is defined by induction: ------------------------------------------ (i) An atomic tableau is a finite tableau from sigma. Let sigma be an infinite set of propositions such that every finite subset of sigma is satisfiable. www.eecis.udel.edu /~chester/courses/604hw/PROPSND   (905 words)

Introduction 'Finite' and 'Svm' are standard set-theoretic predicates, which we assume to have been defined prior to the introduction of the theory displayed: 'Finite(f)' states that f is finite, and 'Svm(f)' states that f is a single-valued map. Since, as we shall see, a finite string representation is available for each hereditarily finite set, we can arrange such sets in order of the length of their string representations. Since the formulae used to speak of infinite sets are the same as those used for finite sets, and since much the same axioms are assumed for sets of both kinds, many of the properties deduced for infinite sets stand in analogy to the more directly visible properties of finite sets. www.settheory.com /intro.html   (18848 words)

File Finite.ML (Isabelle_17-Sep-2000) finite A; finite B ] \ \ ==> setsum g (A Un B) + setsum g (A Int B) = setsum g A + setsum g B"; by (etac finite_induct 1); by (Simp_tac 1); by (asm_full_simp_tac (simpset() addsimps (thms "plus_ac0") @ [Int_insert_left, insert_absorb]) 1); qed "setsum_Un_Int"; Goal "[ finite A; finite B ] \ \ ==> fold f (fold f e B) A = fold f (fold f e (A Int B)) (A Un B)"; by (etac finite_induct 1); by (Simp_tac 1); by (asm_simp_tac (simpset() addsimps [fold_insert, fold_commute, Int_insert_left, insert_absorb]) 1); qed "fold_nest_Un_Int"; Goal "[ finite A; x ~: A ] ==> fold f e (insert x A) = f x (fold f e A)"; by (asm_simp_tac (simpset() addsimps [lemma]) 1); by (rtac some_equality 1); by (auto_tac (claset() addIs [finite_imp_foldSet], simpset() addcongs [conj_cong] addsimps [symmetric fold_def, fold_equality])); qed "fold_insert"; Goal "finite A ==> ALL e. www.dcs.shef.ac.uk /~matt/lax/browser_info/HOL/Finite.ML.html   (1421 words)

PlanetMath: $\sigma$-finite -finite if the total space is the union of a finite or countable family of sets of finite measure, i.e. planetmath.org /encyclopedia/SigmaFinite.html   (71 words)

Annotated Bibliography on the Range of Vector Measures If mu is an atomless finite-dimensional vector measure (not necessary finite), then the range of mu is convex, the closure of the range does not contain a line and each compact extreme face of the closure of the range is contained in the range. The fact that the range of an atomless finitely additive mu is the interval [0,1] is deduced as a corollary. Lyapunov (1940) proved that the range of an atomless finite vector-valued measure is closed and convex. www.math.gatech.edu /~hill/publications/annotated.html   (5995 words)

LN07.txt Sigma must be a subset of Gamma, and Gamma must include a "blank" symbol 'b' that is not in Sigma. Given a function f : Sigma* -> Sigma*, we say that f is "computable" if there exists a Turing Machine M that computes f (with the conventions given above). Also, note that we have given "labels" to the different states that indicate the purpose of each state: this makes it much easier to keep track of what is happening while you are designing a Turing Machine. www.cs.toronto.edu /~fpitt/2000W/CSC366/LN07.txt   (1045 words)

week2 The finite automaton is a good model for a computer with an extremely small memory. In fact, the possible moves of consuming one input symbol are modeling by the following: $$\Delta^i (\Delta (\Delta^j (q, e \circ \cdots \circ e), \sigma), e \circ \cdots \circ e).$$ To form the equivalent NFA $M'$ without $e$-moves we need to introduce the concept of $e$-closure of a state or a set of states. In this week's lecture, we study the simplest model - the finite state machine or the finite automaton. www.mcs.drexel.edu /~rboyer/courses/mcs559/fall99/week2   (947 words)

92-186 In particular some {\it finite size conditions} are discussed, together with their implications for the Gibbs measures and for the approach to equilibrium of Glauber dynamics in {\it arbitrarily large } volumes. A finite subset A of the cubic lattice is said to be connected if for any two sites x and y in A there is a sequence of nearest neighbor sites $x_o,\,x_1,\,..x_n$ in A connecting x to y i.e. It is understood that outside $\Lambda$ the configurations $\sigma$ and $\eta$ agree with $\tau$.\bigskip {\bf Remark} The above version of the Gibbs sampling is different from the one employed by Holley [H2], Aizenman and Holley [AH] and Strook and Zegarlinski [SZ]. www.ma.utexas.edu /mp_arc/papers/92-186   (5211 words)

ftgt.txt Let $\Sigma$ be an algebraic number field which is a splitting field of some $f \in \mathbb Q[X]$ (so the extension $\mathbb Q \leq \Sigma$ is normal). To each subgroup $H$ of $G$ we associate the subfield $H^\dagger$ of $\Sigma$ consisting of the elements of $\Sigma$ which are fixed by each element of $H$. At this stage we assume that we have proved that if $G$ is the Galois group of finite normal field extension inside $\mathbb C$, then $G$ is the degree of the extension. www.bath.ac.uk /~masgcs/math0037/ftgt.txt   (833 words)

file:///c:/usr/yfdeng/amsdeng/teach/ams310/Bayesian.html For samples from infinite populations the variance of this distribution is \sigma^2/n; For samples from a finite populations of size N is \sigma^2/n (N-n)/(N-1). Definitions: finite population of random sample: A set of observations x1, x2,..., xn constitute a random samples of size "n" from a finite pupolation of size "N", if it's chosen so that each subset of n of the N elements of the population has the same probability of being selected. If a random sample of size n is taken from a population having the mean \mu and the variance \sigma^2, then \bar{X} is a random variable whose distribution has the mean \mu. www.ams.sunysb.edu /~deng/teach/ams310/regression.html   (247 words)

CMSC 451 Selected Lecture Notes Example of a NFA, Nondeterministic Finite Automata given by a) Machine Definition M = (Q, sigma, delta, q0, F) b) Equivalent regular expression c) Equivalent state transition diagram and example tree of states for input string 0100011 and an equivalent DFA, Deterministic Finite Automata for the first 3 states. L=phi r = phi 3) An element of the input alphabet, sigma, is in the regular expression r = a where a is an element of sigma. Sigma star is all possible strings over the alphabet sigma. www.cs.umbc.edu /~squire/cs451_lect.html   (10942 words)

Theory Finite_Set (Isabelle repository version) *} proof - assume "finite B" thus "!!A. A ⊆ B ==> finite A" proof induct case empty thus ?case by simp next case (insert x F A) have A: "A ⊆ insert x F" and r: "A - {x} ⊆ F ==> finite (A - {x})". finite A; a ∈ A ] ==> setsum f A = f a + setsum f (A - {a}) finite A ==> (setprod f A = (0::'a)) = (∃x∈A. www.cse.unsw.edu.au /~kleing/Isabelle-Library/Finite_Set.html   (6051 words)

uneven_sigma_1.html (a model top at a finite value of sigma doesn't make a whole lot of physical sense, and is not recommended). Otherwise, the top of the model is at finite sigma. Returns values of "A and B" appropriate for a particular unevenly spaced sigma coordinate model www.gfdl.noaa.gov /~ih/documentation/uneven_sigma_1.html   (142 words)

PlanetMath: Radon-Nikodym theorem Cross-references: properties, signed measure, theorem, function, finite, measurable function, absolutely continuous, measurable space, measures , which is nonnegative and finite, such that for each This function is unique (any other function satisfying these conditions is equal to planetmath.org /encyclopedia/RadonNikodymTheorem.html   (90 words)

Listing of intr.tex file \end{enumerate} When taking a random sample of size n from a (nondegenerate) normal distribution with mean $\mu$ and finite variance $\sigma^2$, $(n-1)S^2/\sigma^2$ has the Chisquare distribution with $v=n-1$ degrees of freedom (d.f.) and probability density function (pdf) \[f(x;v)=\frac{1}{2^{v/2}{\Gamma(\frac{v}{2})}}x^{v/2}e^{-x/2}\mbox{ for $x>0$}\] and the random variable $\sqrt{n}(\bar X -\mu)/S$ has Student's t distribution with $v=n-1$ d.f. \section{Introduction} Let $X_1,\cdots,X_n$ be a random sample of size n from a random variable $X$ where $X$ has finite variance $\sigma^2$ and let $S^2=\frac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar X)^2$ be the sample variance. Our proof does not use Stirling's approximation to the gamma function, but uses the recursive property of the gamma function, $\Gamma(x+1)=x\Gamma(x)$, elementary properties of convergent sequences, and the fact that $E(S) \leq \sigma$. www.math.utep.edu /temp/Epoch/nasaepoch/node60.html   (214 words)

pathTheory.sig finite p1 ∧ (last p1 = first p2) ⇒ (okpath R (plink p1 p2) = okpath R p1 ∧ okpath R p2) [okpath_pmap] Theorem - ∀R f g p. finite (pconcat p1 lab p2) = finite p1 ∧ finite p2 [finite_plink] Theorem - ∀p1 p2. finite (plink p1 p2) = finite p1 ∧ finite p2 [finite_pmap] Theorem - ∀f g p. www.cs.utah.edu /~swalton/hol98/sigobj/pathTheory.sig   (2533 words)

oct12 "Today I will engage in one of my favorite pasttime, which is to try and pass as an algebraic geometer." Algebraic varieties: Def'n: A {\em rational polyhedral cone} is some $\sigma \subset \R^n$ defined be finitely many inequalities, $ The elements of $A_\Sigma$ in $A_\Sigma'$ will be precisely those where the $i^{\rm th}$ coordinate is divisible by $\rho_i$; thus they will be the ones in which $ The invariant functions on a variety are the functions on a quotient; thus $X_\Sigma=X_{\Sigma'}/G$. To avoid pathologies, we'll assume that $\Sigma$ is $n$-dimensional (i.e., not contained in some hypersurface). www.math.uchicago.edu /~msmukler/oct12   (1027 words)

actions-report.html The same conclusion is true if the algebra L\Sigma is finite dimensional or if D\Sigma is of constant rank The one-dimensional example.x = 1 shows that the ARC does not imply that the reachable set from, must be open, or even that it must be a neighborhood of,, as is true for the accessible set. Examples of locally finitely generated involutive \Phi are finite dimensional Lie algebras of vector fields (just take for the Yi's a basis of this Lie algebra), as well as involutive sets of vector fields for which the distribution D(\Phi) 30 has constant rank (if {Y1(,),. * The span of \Phi is finite dimensional. www.math.rutgers.edu /~sontag/actions-report.html   (14799 words)

01-258 Choosing $\Lambda$ finite such that $\Sigma\subset\Lambda\subset L$, we may define ${\cal H}_a={\cal H}_{\Lambda a}=\otimes_{x\in\Lambda\cap R_a}{\cal H}_x$, ${\cal H}={\cal H}_\Lambda=\otimes_{x\in\Lambda}{\cal H}_x$, and study the {\it finite system} defined by a {\it density matrix} $\psi(t)=\psi_\Lambda(t)$ on ${\cal H}_\Lambda$ and a (self-adjoint) {\it Hamiltonian} $H=H_\Lambda$ on ${\cal H}_\Lambda$. We let $L$ be the disjoint union $L=\cup_{a\ge0}R_a$, where $R_0=\Sigma$ is finite and the $R_a$ with $a>0$ are infinite. For {\it finite range} interactions, $h$ is a well defined element of ${\cal A}$ and independent of $\Lambda$ for sufficiently large $\Lambda$. www.ma.utexas.edu /mp_arc/papers/01-258   (2067 words)

paper.ltx The complex exponential term in equation~(\ref{eqn:px}) simply represents a shift of a $t$-trace according to the slope $\sigma$ and the trace separation $x$. In the case of a constant slope, equation~(\ref{eqn:pde}) has the simple general solution \begin{equation} \label{eqn:plane} P(t,x) = f(t - \sigma x)\;, \end{equation} where $f(t)$ is an arbitrary waveform. As expected, the phases match the exact line at low frequencies, and the accuracy of the approximation increases with the length of the filter. sepwww.stanford.edu /sep/sergey/pwd/paper.ltx   (5370 words)

BRICS Logic and Semantics Seminar Then there exists a finite collection H \subseteq Q[x] of polynomials, such that for any n, and for any linear vector space U \subseteq V_{k,n} which is closed under the natural action of the symmetric group S_n, there exists a polynomial p \in H such that the vector space dimension of U is p(n). In categorical terms, a specie of structures is a functor from the category of finite sets and bijections to itself. The intended interpretation of a proposition of the calculus is an object of a $\mu$-bicomplete category, that is, a category with finite products and finite coproducts, initial algebras and final coalgebras of functors that are recursively constructible out of these operations. www.brics.dk /Activities/LogSem/index.html   (6185 words)

On Consistency of Bayes Procedures For a wide class of statistical models with $\sigma $-finite priors, sufficient conditions for consistency of Bayes procedures for almost all values of the parameter are given. For a more restrictive class of invariant statistical models with some $\sigma $-finite priors, we give sufficient conditions for consistency of Bayes procedures for all values of the parameter. Particular cases of these invariant statistical models include the location model on a locally compact Polish group, rotation invariant families on the unit sphere in $\mathbf{R}^k$, and the scale-location model in $\mathbf{R}^k$. epubs.siam.org /sam-bin/dbq/article/97961   (138 words)

WWW interactive multipurpose server Sigma, calcule sommes des séries ou sommes finies de tous genres. wims.unice.fr /wims/wims.cgi?...&lang=en&module=tool/analysis/sigma.en   (4387 words)

hbar_4.23.99.txt For finite sigma_p/p, P decreases as w_0 increases (detuning). For a sigma_b=0 antihydrogen beam, laser power of 100 kW and sigma_p/p=0, we have an excitation probability P of 58% independent of w_0. www.ps.uci.edu /~markm/antihydrogen/hbar_4.23.99.txt   (954 words)

Recherche en Prévision Numérique: Model RPN NWP-D40P29 (T63 L23) 1993 The semi-Lagrangian numerics are described by Ritchie (1991) [1], and the finite element discretization by Beland and Beaudoin (1985) [6]. Following Monin-Obukhov similarity theory, the surface turbulent momentum, sensible heat, and moisture fluxes are expressed as bulk formulae, with drag and transfer coefficients that are functions of surface roughness length (see Surface Characteristics) and of stability (expressed as a bulk Richardson number computed between level sigma = 0.99 and the surface). The surface solar absorption is determined from surface albedos, and longwave emission from the Planck equation with prescribed surface emissivities (see Surface Characteristics). www-pcmdi.llnl.gov /projects/modeldoc/amip1/30rpn.html   (1797 words)

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