Independence Theorem Research Articles

Path-Independence of Work Done Theorem Is Invalid in Center-Bound Force Fields

Abstract: The notion of work done, and the corresponding to it concept of potential energy, was incompletely defined making the path independence theorem of work done by center-bound force fields invalid for other than radial/conservative forces. Hence nonradial effects along equipotential surfaces, whose presence was suggested by experiments, can exist. New, mathematically complete representation of work done by center-bound force fields (generated by a single source) is offered.

Stable group theory and approximate subgroups

We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group G G , we show that a finite subset X X with | X X − 1 X | / | X | |X X ^{-1}X |/ |X| bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of G G . We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.

Enumeration of matchings in families of self-similar graphs

The number of matchings of a graph G is an important graph parameter in various contexts, notably in statistical physics (dimer–monomer model). Following recent research on graph parameters of this type in connection with self-similar, fractal-like graphs, we study the asymptotic behavior of the number of matchings in families of self-similar graphs that are constructed by a very general replacement procedure. Under certain conditions on the geometry of the graphs, we are able to prove that the number of matchings generally follows a doubly exponential growth. The proof depends on an independence theorem for the number of matchings that has been used earlier to treat the special case of Sierpiński graphs. We also further extend the technique to the matching-generating polynomial (equivalent to the partition function for the dimer–monomer model) and provide a variety of examples.

An Asymptotic Independence Theorem for the Number of Matchings in Graphs

Let z(G) be the number of matchings (independent edge subsets) of a graph G. For a set M of edges and/or vertices, the ratio $$r_{G}(M) = z(G \setminus M)/z(G)$$represents the probability that a randomly picked matching of G does not contain an edge or cover a vertex that is an element of M. We provide estimates for the quotient $$r_{G}(A \cup B)/(r_{G}(A)r_G(B))$$, depending on the sizes of the disjoint sets A and B, their distance and the maximum degree of the underlying graph G. It turns out that this ratio approaches 1 as the distance of A and B tends to ∞, provided that the size of A and B and the maximum degree are bounded, showing asymptotic independence. We also provide an application of this theorem to an asymptotic enumeration problem related to the dimer-monomer model from statistical physics.

Burnside-type problems, theorems on height, and independence

This review paper is devoted to some questions related to investigations of bases in PI-algebras. The central point is generalization and refinement of the Shirshov height theorem, of the Amitsur–Shestakov hypothesis, and of the independence theorem. The paper is mainly inspired by the fact that these topics shed some light on the analogy between structure theory and constructive combinatorial reasoning related to the “microlevel,” to relations in algebras and straightforward calculations. Together with the representation theory of monomial algebras, height and independence theorems are closely connected with combinatorics of words and of normal forms, as well as with properties of primary algebras and with combinatorics of matrix units. Another aim of this paper is an attempt to create a kind of symbolic calculus of operators defined on records of transformations.

Hilbert’s Independence Theorem and the Proof of the Second Law of Thermodynamics

Abstract Hamilton taking time and velocity as co-ordinates of optics induced Jacobi to take these as co-ordinates of dynamics too. Considering Hilbert’s paper „Transition of the Method of the independent integral to double Integrals“ the force dxdt double integral can be presented by an action integral thus avoiding the splitting of classic and action integrals.

Local cohomology on diagrams of schemes

We define local cohomology on a diagram of schemes, and discuss basic properties of it. In particular, we prove the independence theorem and the flat base change for local cohomology on diagrams of schemes. We also introduce the notion of G-localness of a G-scheme. It is an equivariant analogue of localness of a scheme. As an application, we prove that Cohen–Macaulay property is inherited by the affine geometric quotient under the action of a linearly reductive group scheme over a field. This generalizes the special case (that the ring in question contains a field) of the theorem of Hochster and Eagon on the Cohen–Macaulay property of an invariant subring under the action of a finite group.

Abstract interpretation and types for systems biology

Abstract interpretation is a theory of abstraction that has been introduced for the analysis of programs. In particular, it has proved useful for organizing the multiple semantics of a given programming language in a hierarchy corresponding to different detail levels, and for defining type systems for programming languages and program analyzers in software engineering. In this paper, we investigate the application of these concepts to systems biology formalisms. More specifically, we consider the Systems Biology Markup Language SBML, and the Biochemical Abstract Machine BIOCHAM with its differential, stochastic, discrete and boolean semantics. We first show how all of these different semantics, except the differential one, can be formally related by simple Galois connections. Then we define three type systems: one for checking or inferring the functions of proteins in a reaction model, one for checking or inferring the activation and inhibition effects of proteins in a reaction model, and another one for checking or inferring the topology of compartments or locations. We show that the framework of abstract interpretation elegantly applies to the formalization of these further abstractions, and to the implementation of linear or quadratic time type checking as well as type inference algorithms. Furthermore, we show a theorem of independence of the graph of activation and inhibition effects from the kinetic expressions in the reaction model, under general conditions. Through some examples, we show that the analysis of biochemical models by type inference provides accurate and useful information. Interestingly, such a mathematical formalization of the abstractions commonly used in systems biology already provides some guidelines for the extensions of biochemical reaction rule languages.

The integer points in a plane curve

Bombieri and Pila gave sharp estimates for the number of integer points $(m,n)$ on a given arc of a curve $y = F(x)$, enlarged by some size parameter $M$, for algebraic curves and for transcendental analytic curves. The transcendental case involves the maximum number of intersections of the given arc by algebraic curves of bounded degree. We obtain an analogous result for functions $F(x)$ of some class $C^k$ that satisfy certain differential inequalities that control the intersection number. We allow enlargement by different size parameters $M$ and $N$ in the $x$- and $y$-directions, and we also estimate integer points close to the curve, with $$\left|n - NF ( {m\over M} )| \leq \delta,$$ for $\delta$ sufficiently small in terms of $M$ and $N$. As an appendix we obtain a determinant mean value theorem which is a quantitative version of a linear independence theorem of Pólya.

Eikonal Representation of Thermodynamics

Potentials are integrals governed by a minimum principle. Taking into account the principle of least time the dependence of the position from time arising therefrom and called the geodesic slope excludes the position as a variable of potentials. Instead of using time und position as variables Hilbert's Independent Theorem presents an integral with respect to time whose integrand contains the geodesic slope. The Gibbs thermodynamics suspending the dependence of time is based on differential theorems. From the fact, however, that the Erdmann-WeierstraB comer conditions arising from Hilbert's theorem are in agreement with the relevant thermodynamic conditions of Gibbs we conclude that Hilbert's Independent Integral presents the action of thermodynamics. The thermodynamic action integral is dependent on the lower limit, the least action, and the higher limit presenting the actual state. Because of the reversibility every state between both limits is available. With the reduction of the higher limit into the past we return to the history of all events that happened in the course of milliards of years. The split of this recording into characteristic components is possible with the help of suitable analytical methods. These reveals the history and the origin of thermodynamic compounds. The genetic fingerprint is a well known example.

Time Dependence of Thermodynamic Potentials

Integrals being dependent on the limits only are denominated potentials. According to Hilbert’s Independence Theorem potentials are given by the limits of an action integral. Whereas the upper limit presents the actual action the lower limit concerns the least action. The action can be a constant, the least action can be zero or can be given by a finite value. As to thermodynamics the latter is true as found by Planck.

N-Simple theories

The main topic of this paper is the investigation of generalized amalgamation properties for simple theories. That is, we are trying to answer the question of when a simple theory has the property of n-dimensional amalgamation, where two-dimensional amalgamation is the Independence Theorem for simple theories. We develop the notions of strong n-simplicity and n-simplicity for 1≤ n≤ ω, where both “1-simple” and “strongly 1-simple” are the same as “simple”. For strong n-simplicity, we present examples of simple unstable theories in each subclass and prove a characteristic property of strong n-simplicity in terms of strong n-dividing, a strengthening of the dependence relation called dividing in simple theories. We prove a strong three-dimensional amalgamation property for strongly 2-simple theories, and, under an additional assumption, a strong ( n+1)-dimensional amalgamation property for strongly n-simple theories. In the last section of the paper we comment on why strong n-simplicity is called strong.

Lagrange and Hamilton Integrals and their Connection

Abstract The object of the following is the change from Lagrange line integrals of single particles to Lagrange line integrals of systems of particles under the condition of the first law. The first problem is the historical variational principle as imagined by Bernoulli, whereas the second problem concerns the built up of thermodynamic systems from elementary integrals. Since thermodynamics disregards mechanisms governing the connection or the interaction of particles the built up of thermodynamic systems is possible if Lagrange integrals exhibiting addition theorems between the limits are taken into account. Hilbert’s Independence Theorem involving the transition from the Lagrange to the Hamilton potential proved to be a suitable method to turn over from the mathematical Lagrange potentials to the Hamilton representation as used in physics. Caratheodory, basing his “Complete Figures of Calculus of Variation” on Hilberts conception has given the variational principle that minimizes the connection of the elementary Lagrange integrals being connected between the limits. The extension of thermodynamics to integrals composed from components showing a connection between the limits yields important thermodynamic properties.

Logic-based subsumption architecture

We describe a logic-based AI architecture based on Brooks' subsumption architecture. In this architecture, we axiomatize different layers of control in First-Order Logic (FOL) and use independent theorem provers to derive each layer's outputs given its inputs. We implement the subsumption of lower layers by higher layers using nonmonotonic reasoning principles. In particular, we use circumscription to make default assumptions in lower layers, and nonmonotonically retract those assumptions when higher layers draw new conclusions. We also give formal semantics to our approach. Finally, we describe layers designed for the task of robot control and a system that we have implemented that uses this architecture for the control of a Nomad 200 mobile robot. Our system combines the virtues of using the represent-and-reason paradigm and the behavioral-decomposition paradigm. It allows multiple goals to be serviced simultaneously and reactively. It also allows high-level tasks and is tolerant to different changes and elaborations of its knowledge in runtime. Finally, it allows us to give more commonsense knowledge to robots. We report on several experiments that empirically show the feasibility of using fully expressive FOL theorem provers for robot control with our architecture and the benefits claimed above.

Some remarks on indiscernible sequences

AbstractWe prove a property of generic homogeneity of tuples starting an infinite indiscernible sequence in a simple theory and we use it to give a shorter proof of the Independence Theorem for Lascar strong types. We also characterize the relation of starting an infinite indiscernible sequence in terms of coheirs.

On the independence theorem of related structures for modular (arguesian) lattices

LetDbe a finite distributive lattice with more than one element and letGbe a finite group. We prove that there exists a modular (arguesian) lattice M such that the congruence lattice ofMis isomorphic toDand the automorphism group ofMis isomorphic toG.

Very simple theories without forking

We prove Vaught's conjecture for minimal trivial simple theories satisfying the generalized independence theorem.

Simple generic structures

A study of smooth classes whose generic structures have simple theory is carried out in a spirit similar to Hrushovski (Ann. Pure Appl. Logic 62 (1993) 147; Simplicity and the Lascar group, preprint, 1997) and Baldwin–Shi (Ann. Pure Appl. Logic 79 (1) (1996) 1). We attach to a smooth class 〈 K 0,≺〉 of finite L -structures a canonical inductive theory T Nat, in an extension-by-definition of the language L . Here T Nat and the class of existentially closed models of ( T Nat) ∀= T +, EX( T +), play an important role in description of the theory of the 〈 K 0,≺〉-generic. We show that if M is the 〈 K 0,≺〉-generic then M∈ EX( T +). Furthermore, if this class is an elementary class then Th( M)= Th( EX( T +)). The investigations by Hrushovski (preprint, 1997) and Pillay (Forking in the category of existentially closed structures, preprint, 1999), provide a general theory for forking and simplicity for the nonelementary classes, and using these ideas, we show that if 〈 K 0,≺〉, where ≺∈{⩽,⩽ ∗} , has the joint embedding property and is closed under the Independence Theorem Diagram then EX( T +) is simple. Moreover, we study cases where EX( T +) is an elementary class. We introduce the notion of semigenericity and show that if a 〈 K 0,≺〉-semigeneric structure exists then EX( T +) is an elementary class and therefore the L -theory of 〈 K 0,≺〉-generic is near model complete. By this result we are able to give a new proof for a theorem of Baldwin and Shelah (Trans. AMS 349 (4) (1997) 1359). We conclude this paper by giving an example of a generic structure whose (full) first-order theory is simple.

Prelogical Relations

We study a weakening of the notion of logical relations, called prelogical relations , that has many of the features that make logical relations so useful as well as further algebraic properties including composability. The basic idea is simply to require the reverse implication in the definition of logical relations to hold only for pairs of functions that are expressible by the same lambda term. Prelogical relations are the minimal weakening of logical relations that gives composability for extensional structures and simultaneously the most liberal definition that gives the Basic Lemma. Prelogical predicates (i.e., unary prelogical relations) coincide with sets that are invariant under Kripke logical relations with varying arity as introduced by Jung and Tiuryn, and prelogical relations are the closure under projection and intersection of logical relations. These conceptually independent characterizations of prelogical relations suggest that the concept is rather intrinsic and robust. The use of prelogical relations gives an improved version of Mitchell's representation independence theorem which characterizes observational equivalence for all signatures rather than just for first-order signatures. Prelogical relations can be used in place of logical relations to give an account of data refinement where the fact that prelogical relations compose explains why stepwise refinement is sound.

Prelogical Relations

We study a weakening of the notion of logical relations, called prelogical relations, that has many of the features that make logical relations so useful as well as further algebraic properties including composability. The basic idea is simply to require the reverse implication in the definition of logical relations to hold only for pairs of functions that are expressible by the same lambda term. Prelogical relations are the minimal weakening of logical relations that gives composability for extensional structures and simultaneously the most liberal definition that gives the Basic Lemma. Prelogical predicates (i.e., unary prelogical relations) coincide with sets that are invariant under Kripke logical relations with varying arity as introduced by Jung and Tiuryn, and prelogical relations are the closure under projection and intersection of logical relations. These conceptually independent characterizations of prelogical relations suggest that the concept is rather intrinsic and robust. The use of prelogical relations gives an improved version of Mitchell's representation independence theorem which characterizes observational equivalence for all signatures rather than just for first-order signatures. Prelogical relations can be used in place of logical relations to give an account of data refinement where the fact that prelogical relations compose explains why stepwise refinement is sound.