Independence Theorem Research Articles

INVARIANT KEISLER MEASURES FOR $\omega $ -CATEGORICAL STRUCTURES

Abstract A recent article of Chernikov, Hrushovski, Kruckman, Krupinski, Moconja, Pillay, and Ramsey finds the first examples of simple structures with formulas which do not fork over the empty set but are universally measure zero. In this article we give the first known simple $\omega $ -categorical counterexamples. These happen to be various $\omega $ -categorical Hrushovski constructions. Using a probabilistic independence theorem from Jahel and Tsankov, we show how simple $\omega $ -categorical structures where a formula forks over $\emptyset $ if and only if it is universally measure zero must satisfy a stronger version of the independence theorem.

Integration over facet-simple polytopes

AbstractWe present an elementary approach for the computation of integrals of the form $$\int _{\mathcal {P}} f^{(n)}(\textbf{s} \cdot \textbf{x})\,\textbf{dx}$$ ∫ P f ( n ) ( s · x ) dx over polytopes $$\mathcal {P}$$ P , where $$f: \mathbb {C}\rightarrow \mathbb {C}$$ f : C → C is analytic. The proof is based on an independence theorem on exponential functions over the field of rational functions and needs only simple facts from the theory of polyhedra. In particular we present an explicit formula for generalized facet-simple polytopes. Here a convex polytope is called facet-simple if each of its facets is simple and a set of points is called a generalized facet-simple polytope if it is a finite union of n-dimensional facet-simple convex polytopes such that any two distinct members are either disjoint or intersect in a common facet.

Relations between values of arithmetic Gevrey series, and applications to values of the Gamma function

We investigate the relations between the rings E, G and D of values taken at algebraic points by arithmetic Gevrey series of order either −1 (E-functions), 0 (analytic continuations of G-functions) or 1 (renormalization of divergent series solutions at ∞ of E-operators) respectively. We prove in particular that any element of G can be written as multivariate polynomial with algebraic coefficients in elements of E and D, and is the limit at infinity of some E-function along some direction. This prompts to defining and studying the notion of mixed functions, which generalizes simultaneously E-functions and arithmetic Gevrey series of order 1. Using natural conjectures for arithmetic Gevrey series of order 1 and mixed functions (which are analogues of a theorem of André and Beukers for E-functions) and the conjecture D∩E=Q‾ (but not necessarily all these conjectures at the same time), we deduce a number of interesting Diophantine results such as an analogue for mixed functions of Beukers' linear independence theorem for values of E-functions, the transcendence of the values of the Gamma function and its derivatives at all non-integral algebraic numbers, the transcendence of Gompertz constant as well as the fact that Euler's constant is not in E.

Existence and linear independence theorem for linear fractional differential equations with constant coefficients

Abstract We consider the l-th order linear fractional differential equations with constant coefficients. Here l ∈ ℕ {l\in\mathbb{N}} is the ceiling for the highest derivative of order α, l - 1 < α ≤ l {l-1<\alpha\leq l} . If β i < α {\beta_{i}<\alpha} are the other derivatives, the existing theory requires α - max ⁡ { β i } ≥ l - 1 {\alpha-\max\{\beta_{i}\}\geq l-1} for the existence of l linearly independent solutions. Thus, at most one derivative may have order greater than one, but all other derivatives must be between zero and one. We remove this essential restriction and construct l linearly independent solutions. With this aim, we remodel the series approaches and elaborate the multi-sum fractional series method in order to obtain the existence and linear independence results. We consider both Riemann–Liouville or Caputo fractional derivatives.

A Model in Which Well-Orderings of the Reals First Appear at a Given Projective Level, Part II

We consider the problem of the existence of well-orderings of the reals, definable at a certain level of the projective hierarchy. This research is motivated by the modern development of descriptive set theory. Given n≥3, a finite support product of forcing notions similar to Jensen’s minimal-Δ31-real forcing is applied to define a model of set theory in which there exists a good Δn1 well-ordering of the reals, but there are no Δn−11 well-orderings of the reals (not necessarily good). We conclude that the existence of a good well-ordering of the reals at a certain level n≥3 of the projective hierarchy is strictly weaker than the existence of a such well-ordering at the previous level n−1. This is our first main result. We also demonstrate that this independence theorem can be obtained on the basis of the consistency of ZFC− (that is, a version of ZFC without the Power Set axiom) plus ‘there exists the power set of ω’, which is a much weaker assumption than the consistency of ZFC usually assumed in such independence results obtained by the forcing method. This is our second main result. Further reduction to the consistency of second-order Peano arithmetic PA2 is discussed. These are new results in such a generality (with n≥3 arbitrary), and valuable improvements upon earlier results. We expect that these results will lead to further advances in descriptive set theory of projective classes.

Co-theory of sorted profinite groups for PAC structures

We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding absolute Galois groups of structures, so they can be interpreted in some monster model with an additional predicate. Third, we prove the “Weak Independence Theorem” for pseudo-algebraically closed (PAC) substructures of an ambient structure with no finite cover property (nfcp) and the property [Formula: see text]. Fourth, we describe Kim-dividing in these PAC substructures and show several results related to the SOPnhierarchy. Fifth, we characterize the algebraic closure in PAC structures.

Теоремы существования и единственности для одной бесконечной системы нелинейных алгебраических уравнений

We study a infinite system of algebraic equations with monotone nonlinearities and with an infinite Toeplitz type matrix. This system has applications in discrete problems of the dynamical theory of 𝑝-adic open-closed strings, the kinetic theory of gases, and mathematical biology. Under certain restrictions on the nonlinearities and on the corresponding Toeplitz matrix, it is possible to prove existence and uniqueness theorems for a nontrivial solution in the class of bounded sequences. The main tool for proving the uniqueness theorem for a nontrivial solution is an auxiliary independent theorem on the asymptotic behavior of a nonnegative nontrivial and bounded solution on ±∞. At the end of the paper, specific applied examples of nonlinearities and the corresponding matrix are given to illustrate the importance of the results obtained.

Independence over arbitrary sets in NSOP1 theories

We study Kim-independence over arbitrary sets. Assuming that forking satisfies existence, we establish Kim's lemma for Kim-dividing over arbitrary sets in an NSOP1 theory. We deduce symmetry of Kim-independence and the independence theorem for Lascar strong types.

Modified homotopy perturbation approach for the system of fractional partial differential equations: A utility of fractional Wronskian

In this paper, generalized aspects of least square homotopy perturbation are presented to treat the system of nonlinear fractional partial differential equations (FPDEs), and the method is called as generalization of least square homotopy perturbation (GLSHP). The concept of fractional partial Wronskian is introduced to detect the linear independence of functions depending on more than one variable through Caputo fractional calculus. Definition of fractional partial Wronskian is provided along with linear independence theorem. It is found that solutions converge more rapidly through GLSHP in comparison to classical fractional homotopy perturbation method (FHPM). Outcomes of this generalization are validated by taking examples from nonlinear fractional wave equations.

Infinite combinatorics in mathematical biology

Is it possible to apply infinite combinatorics and (infinite) set theory in theoretical biology? We do not know the answer yet but in this article we try to present some techniques from infinite combinatorics and set theory that have been used over the last decades in order to prove existence results and independence theorems in algebra and that might have the flexibility and generality to be also used in theoretical biology. In particular, we will introduce the theory of forcing and an algebraic construction technique based on trees and forests using infinite binary sequences. We will also present an overview of the theory of circular codes. Such codes had been found in the genetic information and are assumed to play an important role in error detecting and error correcting mechanisms during the process of translation. Finally, examples and constructions of infinite mixed circular codes using binary sequences hopefully show some similarity between these theories - a starting point for future applications.

On Kim-independence

We study NSOP _1 theories. We define Kim-independence , which generalizes non-forking independence in simple theories and corresponds to non-forking at a generic scale. We show that Kim-independence satisfies a version of Kim’s lemma, local character, symmetry, and an independence theorem, and that moreover these properties individually characterize NSOP _1 theories. We describe Kim-independence in several concrete theories and observe that it corresponds to previously studied notions of independence in Frobenius fields and vector spaces with a generic bilinear form.

ABOUT INSUFFICIENCY OF GRAMMARS FOR SEMANTIC VALIDITY OF COMPUTER PROGRAMS AND ONTOLOGIES AS AN ALTERNATIVE APPROACH

The Rice theorem proves that semantic properties of computer programs are not decidable. In this paper it will be proved that grammars are not a sufficient mean to provide semantic validity on computer programs, first as a corollary of the Rice theorem and then as an independent theorem. Once that is proved that they are not sufficient, the use of ontologies are presented as a viable alternative to grammars and the additional benefits they offer. Finally, an implementation of a programming language is presented which is not based in any grammar, but in an ontology.

Homomorphisms and principal congruences of bounded lattices. III. The Independence Theorem

A new result of G. Czedli states that for an ordered set P with at least two elements and a group G, there exists a bounded lattice L such that the ordered set of principal congruences of L is isomorphic to P and the automorphism group of L is isomorphic to G.

Cognitive, neural and endocrine functioning during late pregnancy: An ERP study

We study Kim-independence over arbitrary sets. Assuming that forking satisfies existence, we establish Kim's lemma for Kim-dividing over arbitrary sets in an NSOP1 theory. We deduce symmetry of Kim-independence and the independence theorem for Lascar strong types.

Graded version of local cohomology with respect to a pair of ideals

In this paper, we prove some well-known results on local cohomology with respect to a pair of ideals in graded version, such as the Independence theorem, Lichtenbaum-Harshorne vanishing theorem, Basic finiteness and vanishing theorem, among others. In addition, we present a generalized version of the Melkersson theorem regarding the Artinianness of modules, and a result concerning Artinianness of local cohomology modules.

Permutation modules of profinite groups

Given an arbitrary profinite group G and a commutative domain R, we define the notion of permutation RG-module which generalizes the known notion from the representation theory of profinite groups. We establish an independence theorem of such a module as an R-module over a ring of scalars.

Dedekind’s linear independence theorem over commutative semirings

In this paper, Dedekind’s theorem on the linear independence of homomorphisms of commutative semirings is studied. Furthermore, this theorem is extended to the case of linear independence of compositions of homomorphisms and powers of a derivation. The results obtained in this paper generalize and develop previous results for fields.

An independence theorem for ordered sets of principal congruences and automorphism groups of bounded lattices

An independence theorem for ordered sets of principal congruences and automorphism groups of bounded lattices

Court-day crowds in colonial Virginia

ABSTRACTFor a generation, legal historians investigating colonial Virginia have emphasized the dramaturgy of court day. According to the dramaturgical school of interpretation, administrative and judicial activities of county court officials amounted to theatrical performances that simultaneously enforced economic order and stabilized traditional social relationships. Such interpretations assume a large audience routinely attended county courts to observe legal dramas. Often, however, only a small number of persons can be documented as present during court day. The independence theorem from probability theory suggests that the number of documentable attendees is a useful and easily calculated estimate for actual total crowd size. If so, some Virginia court sessions were attended by hundreds of people, while others drew only a few participants. A variety of factors apparently inhibited court attendance in older Virginia counties. By contrast, in newer frontier counties, mid-eighteenth-century revisions of court calendars produced heavy attendance at court day. Regardless of the number of people in attendance, any Virginia county court could still effectively enforce credit contracts.

AN INDEPENDENCE THEOREM FOR NTP2 THEORIES

Abstract We establish several results regarding dividing and forking in NTP2 theories. We show that dividing is the same as array-dividing. Combining it with existence of strictly invariant sequences we deduce that forking satisfies the chain condition over extension bases (namely, the forking ideal is S1, in Hrushovski’s terminology). Using it we prove an independence theorem over extension bases (which, in the case of simple theories, specializes to the ordinary independence theorem). As an application we show that Lascar strong type and compact strong type coincide over extension bases in an NTP2 theory. We also define the dividing order of a theory—a generalization of Poizat’s fundamental order from stable theories—and give some equivalent characterizations under the assumption of NTP2. The last section is devoted to a refinement of the class of strong theories and its place in the classification hierarchy.