scholarly journals Gaussoids are two-antecedental approximations of Gaussian conditional independence structures

Author(s):  
Tobias Boege

AbstractThe gaussoid axioms are conditional independence inference rules which characterize regular Gaussian CI structures over a three-element ground set. It is known that no finite set of inference rules completely describes regular Gaussian CI as the ground set grows. In this article we show that the gaussoid axioms logically imply every inference rule of at most two antecedents which is valid for regular Gaussians over any ground set. The proof is accomplished by exhibiting for each inclusion-minimal gaussoid extension of at most two CI statements a regular Gaussian realization. Moreover we prove that all those gaussoids have rational positive-definite realizations inside every ε-ball around the identity matrix. For the proof we introduce the concept of algebraic Gaussians over arbitrary fields and of positive Gaussians over ordered fields and obtain the same two-antecedental completeness of the gaussoid axioms for algebraic and positive Gaussians over all fields of characteristic zero as a byproduct.

2010 ◽  
Vol 06 (03) ◽  
pp. 579-586 ◽  
Author(s):  
ARNO FEHM ◽  
SEBASTIAN PETERSEN

A field K is called ample if every smooth K-curve that has a K-rational point has infinitely many of them. We prove two theorems to support the following conjecture, which is inspired by classical infinite rank results: Every non-zero Abelian variety A over an ample field K which is not algebraic over a finite field has infinite rank. First, the ℤ(p)-module A(K) ⊗ ℤ(p) is not finitely generated, where p is the characteristic of K. In particular, the conjecture holds for fields of characteristic zero. Second, if K is an infinite finitely generated field and S is a finite set of local primes of K, then every Abelian variety over K acquires infinite rank over certain subfields of the maximal totally S-adic Galois extension of K. This strengthens a recent infinite rank result of Geyer and Jarden.


2021 ◽  
Vol 2021 ◽  
pp. 1-22
Author(s):  
Sourav Shil ◽  
Hemant Kumar Nashine

In this work, the following system of nonlinear matrix equations is considered, X 1 + A ∗ X 1 − 1 A + B ∗ X 2 − 1 B = I  and  X 2 + C ∗ X 2 − 1 C + D ∗ X 1 − 1 D = I , where A , B , C ,  and  D are arbitrary n × n matrices and I is the identity matrix of order n . Some conditions for the existence of a positive-definite solution as well as the convergence analysis of the newly developed algorithm for finding the maximal positive-definite solution and its convergence rate are discussed. Four examples are also provided herein to support our results.


2020 ◽  
pp. 21-52
Author(s):  
Jared Warren

What are linguistic conventions? This chapter begins by noting and setting aside philosophical accounts of social conventions stemming from Lewis’s influential treatment. It then criticizes accounts that see conventions as explicit stipulations. From there the chapter argues that conventions are syntactic rules of inference, arguing that there are scientific reasons to posit these rules as part of our linguistic competence and that we need to include both bilateralist and open-ended inference rules for a full account. The back half of the chapter aims to naturalize inference rule-following by providing functionalist-dispositionalist approaches to our attitudes, inference, and inference-rule–following, addressing Kripkenstein’s arguments and several other concerns along the way.


1969 ◽  
Vol 33 (4) ◽  
pp. 560-564 ◽  
Author(s):  
Raymond M. Smullyan

The real importance of cut-free proofs is not the elimination of cuts per se, but rather that such proofs obey the subformula principle. In this paper we accomplish this latter objective in a different manner.In the usual formulations of Gentzen systems, there is only one axiom scheme; all the other postulates are inference rules. By contrast, we consider here some Gentzen type axiom systems for propositional logic and Quantification Theory in which there is only one inference rule; all the other postulates are axiom schemes. This admits of an unusually elegant axiomatization of logic.


1992 ◽  
Vol 57 (3) ◽  
pp. 912-923 ◽  
Author(s):  
Vladimir V. Rybakov

AbstractAn algorithm recognizing admissibility of inference rules in generalized form (rules of inference with parameters or metavariables) in the intuitionistic calculus H and, in particular, also in the usual form without parameters, is presented. This algorithm is obtained by means of special intuitionistic Kripke models, which are constructed for a given inference rule. Thus, in particular, the direct solution by intuitionistic techniques of Friedman's problem is found. As a corollary an algorithm for the recognition of the solvability of logical equations in H and for constructing some solutions for solvable equations is obtained. A semantic criterion for admissibility in H is constructed.


2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Naglaa M. El-Shazly

In this paper necessary and sufficient conditions for the matrix equation to have a positive definite solution are derived, where , is an identity matrix, are nonsingular real matrices, and is an odd positive integer. These conditions are used to propose some properties on the matrices , . Moreover, relations between the solution and the matrices are derived.


1998 ◽  
Vol 09 (06) ◽  
pp. 723-757 ◽  
Author(s):  
MASANORI MORISHITA ◽  
TAKAO WATANABE

We study the asymptotic distribution of S-integral points on affine homogeneous spaces in the light of the Hardy–Littlewood property introduced by Borovoi and Rudnick. We introduce the S-Hardy–Littlewood property for affine homogeneous spaces defined over an algebraic number field and a finite set S of places of the base field. We work with the adelic harmonic analysis on affine algebraic groups over a number field to determine the asymptotic density of S-integral points under congruence conditions. We give some new examples of strongly or relatively S-Hardy–Littlewood homogeneous spaces over number fields. As an application, we prove certain asymptotically uniform distribution property of integral points on an ellipsoid defined by a totally positive definite tenary quadratic form over a totally real number field.


1987 ◽  
Vol 107 ◽  
pp. 25-47
Author(s):  
Yoshiyuki Kitaoka

Let M be a quadratic lattice with positive definite quadratic form over the ring of rational integers, M’ a submodule of finite index, S a finite set of primes containing all prime divisors of 2[M: M’] and such that Mp is unimodular for p ∉ S. In [2] we showed that there is a constant c such that for every lattice N with positive definite quadratic form and every collection (fp)p∊s of isometries fp: NP → MP there is an isometry f: N → M satisfyingf ≡ fp mod M′p for every p |[M: M],f(Np) is private in Mp for every p ∉ S,provided the minimum of N ≥ c and rank M ≥ 3 rank N + 3.


1977 ◽  
Vol 1 (1) ◽  
pp. 195-230
Author(s):  
Antoni Kreczmar

In the present paper we investigate algorithmic properties of fields. We prove that axioms of formally real fields for the field R of reals and axioms of fields of characteristic zero for the field C of complex numbers, give the complete characterization of algorithmic properties. By Kfoury’s theorem programs which define total functions over R or C are effectively equivalent to loop-free programs. Examples of programmable and nonprogrammable functions and relations over R and C are given. In the case of ordered reals the axioms of Archimedean ordered fields completely characterize algorithmic properties. We show how to use the equivalent version of Archimed’s axiom (the exhaustion rule) in order to prove formally the correctness of some iterative numerical algorithms.


2004 ◽  
Vol 2004 (25) ◽  
pp. 1315-1327
Author(s):  
R. R. Khazal ◽  
M. M. Chawla

For the solution of symmetric linear systems, the classical Cholesky method has proved to be difficult to parallelize. In the present paper, we first describe an elimination variant of Cholesky method to produce a lower triangular matrix which reduces the coefficient matrix of the system to an identity matrix. Then, this elimination method is combined with the partitioning method to obtain a parallel Cholesky algorithm. The total serial arithmetical operations count for the parallel algorithm is of the same order as that for the serial Cholesky method. The present parallel algorithm could thus perform withefficiencyclose to 1 if implemented on a multiprocessor machine. We also discuss theexistenceof the parallel algorithm; it is shown that for a symmetric and positive definite system, the presented parallel Cholesky algorithm is well defined and will run to completion.


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