Parallel Theorem Research Articles

A combinatorial proof of a formula for the Lucas-Narayana polynomials

In 2020, Bennett, Carrillo, Machacek and Sagan gave a polynomial generalization of the Narayana numbers and conjectured that these polynomials have positive integer coefficients for 1≤k≤n and for n≥1. In 2020, Sagan and Tirrell used a powerful algebraic method to prove this conjecture (in fact, they extend and prove the conjecture for more than just the type A case). In this paper we give a combinatorial proof of a formula satisfied by the Lucas-Narayana polynomials described by Bennett et al. This gives a combinatorial proof that these polynomials have positive integer coefficients. A corollary of our main result establishes a parallel theorem for the FiboNarayana numbers Nn,k,F, providing a combinatorial proof of the conjecture that these are positive integers for n≥1.

Limits of the Ap constants

The constant [w]A∞ of a weight w in the Muckenhoupt class A∞=⋃p>1Ap can be seen as limit of the [w]Ap constant as p tends to infinity in virtue of a theorem proved by Sbordone and Wik in [21]. A parallel theorem holds true if we look at weights v in the Gehring class RH1=⋃q>1RHq, being A∞=RH1. In this note we prove that corresponding theorems hold true, in dimension one, for the smallest limit classes A1 and RH∞. We show that the constants [w]A1 and [v]RH∞ can be obtained from the constants [w]Ap and [v]RHq, passing to the limits respectively for p→1 and q→∞. Our method relies on some recent sharp integrability results which highlight some new symmetries between the classes. Furthermore we provide a simplified proof in R for the original theorems in A∞ and RH1.

Development Identification Method of Inertia Properties for Heavy Truck Engine Based on MIMS Test Rig

A new development for the accurate measurement of the inertia parameters for heavy truck engine is presented. It is specifically intended for measuring the inertia properties of vehicle bodies, and it has the potential to be applied to the measurement of the properties of vehicle bodies, such as the vehicle powertrain, engine, and gearbox. This paper, based on CATARC MIMS test rig, develops an accurate measuring method to identify inertia parameters of heavy truck engine. Firstly corresponding tests are carried out and the lever principle and moments of inertia parallel theorem are employed to calculate and analyze the test results, which leads to the accurate value of inertia parameters. Secondly the accuracy of proposed method is verified through the calibration system. As a result the method shows high accuracy, which provides an experimental basis for accurate inertia parameters measurement of heavy truck engine.

Experimental investigation and performance analysis of inertia properties measurement for heavy truck cab

An experimental investigation and performance analysis of inertia properties measurement for heavy truck cab is presented. This method is specifically intended for measuring the inertia properties of irregular rigid bodies, and it has the potential to be applied to the measurement of the inertia properties of vehicle bodies, such as the cab, engine, and gearbox. This article, based on CATARC moment of inertia measurement system test rig, develops an accurate measuring method to identify inertia parameters of heavy truck cab. First, corresponding tests are carried out, and the lever principle and moments of inertia parallel theorem are employed to calculate and analyse the test results, which leads to the accurate value of inertia parameters. Second, the performance analysis of the proposed method is verified through the measurement error analysis. As a result, the proposed method shows high accuracy, which provides an experimental basis for accurate inertia parameter measurement of heavy truck cab.

Algebraic graph transformations with inheritance and abstraction

In this paper, we propose a new approach to inheritance and abstraction in the context of algebraic graph transformation by providing a suitable categorial framework which reflects the semantics of class-based inheritance in software engineering. Inheritance is modelled by a type graph T that comes equipped with a partial order. Typed graphs are arrows with codomain T which preserve graph structures up to inheritance. Morphisms between typed graphs are “down typing” graph morphisms: An object of class t can be mapped to an object of a subclass of t. Abstract classes are modelled by a subset of vertices of the type graph. We prove that this structure is an adhesive HLR category, i.e. pushouts along extremal monomorphisms are “well-behaved”. This infers validity of classical results such as the Local Church–Rosser Theorem, the Parallelism Theorem, and the Concurrency Theorem.

INFLUENCE OF TEST CONDITIONS ON THE STIFFNESS OF RAT FRACTURE CALLUS

Rats are often used for experimental studies on fracture healing. Mostly, the rats were sacrificed after a certain time point and the fracture calli were investigated via biomechanical torsion testing, μCT and histology [Strube, 2008]. The easiest way to perform ex vivo torsion testing is to fix the long bone at the distal end in all six degrees-of-freedom (DOF), to fix the proximal end in all DOF except torsion, to apply a torsion moment to the proximal end and to measure the torsion angle. However, this might lead to an error in the measured torsional stiffness due to the parallel axis theorem if the specimen's axis of gravity is shifted or tilted with reference to the fixed axis of rotation during testing. As a result, significant differences in the fracture healing outcome between two animal groups might be undetected. The aim of this study was to quantify this error in the torsional stiffness via finite element (FE) voxel models of rat fracture calli after 35 days of healing.

Design of Automatic Measuring Instrument of Irregular Objects’ Centroid

By utilizing the parallel axis theorem of moment of inertia and the graphic geometric transformation theory as well as the method of rotating and moving axis, the calculation formula of irregular objects’ centroid was derived. And on the basis of a single axis air bearing table which was used for measuring the moment of inertia of objects around a fixed axis, the complete instrument of measuring the irregular objects’ centroid was designed. The objects’ centroid and moment of inertia could be measured synchronously. Through the analysis of the influence on measuring accuracy caused by measuring factors’ errors, the method of improving the measuring accuracy of instrument was derived. The measuring instrument meets the requirement of measuring the irregular objects’ centroid precisely.

The automorphism tower of a centerless group without Choice

For a centerless group G, we can define its automorphism tower. We define G α : G 0 = G, G α+1 = Aut(G α ) and for limit ordinals $${G^{\delta}=\bigcup_{\alpha<\delta}G^{\alpha}}$$ . Let τ G be the ordinal when the sequence stabilizes. Thomas’ celebrated theorem says $${\tau_{G}<(2^{|G|})^{+}}$$ and more. If we consider Thomas’ proof too set theoretical (using Fodor’s lemma), we have here a more direct proof with little set theory. However, set theoretically we get a parallel theorem without the Axiom of Choice. Moreover, we give a descriptive set theoretic approach for calculating an upper bound for τ G for all countable groups G (better than the one an analysis of Thomas’ proof gives). We attach to every element in G α , the αth member of the automorphism tower of G, a unique quantifier free type over G (which is a set of words from $${G*\langle x\rangle}$$ ). This situation is generalized by defining “(G, A) is a special pair”.

Parallelism and Concurrency in Adhesive High-Level Replacement Systems with Negative Application Conditions

The goal of this paper is the generalization of parallelism and concurrency results for adhesive High-Level Replacement (HLR) systems to adhesive HLR systems with negative application conditions. These conditions restrict the application of a rule by expressing that a specific structure should not be present before or after applying the rule to a certain context. Such a condition influences thus each rule application or transformation and therefore changes significantly the properties of the replacement system. The effect of negative application conditions on parallelism and concurrency in the replacement system is described in the generalization of the following results, formulated already for adhesive HLR systems without negative application conditions: Local Church-Rosser Theorem, Parallelism Theorem and Concurrency Theorem. These important generalized results will support the development of formal analysis techniques for adhesive HLR systems with negative application conditions.

Reducing Complexity Assumptions for Statistically-Hiding Commitment*

We revisit the following question: what are the minimal assumptions needed to construct statistically-hiding commitment schemes? Naor et al. show how to construct such schemes based on any one-way permutation. We improve upon this by showing a construction based on any approximable preimage-size one-way function. These are one-way functions for which it is possible to eciently approximate the number of pre-images of a given output. A special case is the class of regular one-way functions where all points in the image of the function have the same (known) number of pre-images.We also prove two additional results related to statistically-hiding commitment. First, we prove a (folklore) parallel composition theorem showing, roughly speaking, that the statistical hiding property of any such commitment scheme is amplified exponentially when multiple independent parallel executions of the scheme are carried out. Second, we show a compiler which transforms any commitment scheme which is statistically hiding against an honest-but-curious receiver into one which is statistically hiding even against a malicious receiver.

Adhesive DPO Parallelism for Monic Matches

This paper presents indispensable technical results of a general theory that will allow to systematically derive from a given reduction system a behavioral congruence that respects concurrency. The theory is developed in the setting of adhesive categories and is based on the work by Ehrig and König on borrowed contexts; the latter are an instance of relative pushouts, which have been proposed by Leifer and Milner. In order to lift the concurrency theory of dpo rewriting to borrowed contexts we will study the special case of dpo rewriting with monic matches in adhesive categories: more specifically we provide a generalized Butterfly Lemma together with a Local Church Rosser and Parallelism theorem.

복합진자 모형의 뉴튼.오일러 알고리즘 비교

The Primary type of swinging motion in human movement is that which is characteristic of a pendulum. The two types of pendulums are identified as simple and compound. A simple pendulum consist of a small body suspended by a relatively long cord. Its total mass is contained within the bob. The cord is not considered to have mass. A compound pendulum, on the other hand, is any pendulum such as the human body swinging by hands from a horizontal bar. Therefore a compound pendulum depicts important motions that are harmonic, periodic, and oscillatory. In this paper one discusses and compares two algorithms of Newton's method(F = m a) and Euler's method (M = <TEX>$I{\times}{\alpha}$</TEX>) in compound pendulum. Through exercise model such as human body with weight(m = 50 kg), body length(L = 1.5m), and center of gravity (<TEX>$L_c$</TEX> = 0.4119L) from proximal end swinging by hands from a horizontal bar, one finds kinematic variables(angle displacement / velocity / acceleration), and simulates kinematic variables by changing body lengths and body mass. BSP by Clauser et al.(1969) & Chandler et al.(1975) is used to find moment of inertia of the compound pendulum. The radius of gyration about center of gravity (CoG) is <TEX>$k_c\;=\;K_c{\times}L$</TEX> (단, k= radius of gyration, K= radius of gyration /segment length), and then moment of inertia about center of gravity(CoG) becomes <TEX>$I_c\;=\;m\;k_c^2$</TEX>. Finally, moment of inertia about Z-axis by parallel theorem becomes <TEX>$I_o\;=\;I_c\;+\;m\;k^2$</TEX>. The two-order ordinary differential equations of models are solved by ND function of numeric analysis method in Mathematica5.1. The results are as follows; First, The complexity of Newton's method is much more complex than that of Euler's method Second, one could be find kinematic variables according to changing body lengths(L = 1.3 / 1.7 m) and periods are increased by body length increment(L = 1.3 / 1.5 / 1.7 m). Third, one could be find that periods are not changing by means of changing mass(m = 50 / 55 / 60 kg). Conclusively, one is intended to meditate the possibility of applying a compound pendulum to sports(balling, golf, gymnastics and so on) necessary swinging motions. Further improvements to the study could be to apply Euler's method to real motions and one would be able to develop the simulator.

The parallel logic programming system in the FGCS project and its future directions

In the fifth generation computer systems (FGCS) project, a parallel logic programming language, KL1, was adopted as the project's kernel language. It was not only used to determine architectures of highly parallel machines called parallel inference machines (PIMs) consisting of about 1000 element processors but also used as a system description language to develop basic software such as a parallel operating system (PIMOS), and symbolic processing and knowledge processing application systems such as knowledge description languages, a parallel theorem prover, and a protein sequence analysis program. It achieved great success in exploiting of parallelism involved in several important application systems. The prototype of the FGCS attained a linear speed-up that was proportional to the number of processing elements (PEs) for the application systems we had targeted. The MGTP parallel theorem prover was one of such application systems, and can prove theorems based on full first-order logic. Thus, it indicates the possibility of designing a new practical knowledge representation language whose expressive power will be much greater than that of conventional ones. In the FGCS follow-on project, KL1 and its programming system were ported to Unix-based stock parallel machines. This new system called KLIC is expected to greatly extend the use of highly parallel systems.

STRATEGY PARALLELISM IN AUTOMATED THEOREM PROVING

Automated theorem provers use search strategies. Unfortunately, there is no unique strategy which is uniformly successful on all problems. This motivates us to apply different strategies in parallel, in a competitive manner. In this paper, we discuss properties, problems, and perspectives of strategy parallelism in theorem proving. We develop basic concepts like the complementarity and the overlap value of strategy sets. Some of the problems such as initial strategy selection and run-time strategy exchange are discussed in more detail. The paper also contains the description of an implementation of a strategy parallel theorem prover (p-SETHEO) and an experimental evaluation.

Parallel cooperative propositional theorem proving

A parallel satisfiability testing algorithm called Parallel Modoc is presented. Parallel Modoc is based on Modoc, which is based on propositional Model Elimination with an added capability to prune away certain branches that cannot lead to a successful subrefutation. The pruning information is encoded in a partial truth assignment called an autarky. Parallel Modoc executes multiple instances of Modoc as separate processes and allows processes to cooperate by sharing lemmas and autarkies as they are found. When a Modoc process finds a new autarky or a new lemma, it makes the information available to other Modoc processes via a “blackboard”. Combining autarkies generally is not straightforward because two autarkies found by two separate processes may have conflicting assignments. The paper presents an algorithm to combine two arbitrary autarkies to form a larger autarky. Experimental results show that for many of the formulas, Parallel Modoc achieves speedup greater than the number of processors. Formulas that could not be solved in an hour by Modoc were often solved by Parallel Modoc in the order of minutes and, in some cases, in seconds.

Morera theorems via microlocal analysis

We prove Morera theorems for curves in the plane using microlocal analysis. The key is that microlocal smoothness of functions is reflected by smoothness of their Morera integrals on curvestheir Radon transforms. Parallel support theorems for the associated Radon transforms follow from our arguments by a simple correspondence.

GRAPH GRAMMARS WITH NEGATIVE APPLICATION CONDITIONS

In each graph-grammar approach it is defined how and under which conditions graph productions can be applied to a given graph in order to obtain a derived graph. The conditions under which productions can be applied are called application conditions. Although the generative power of most of the known general graph-grammar approaches is sufficient to generate any recursively enumerable set of graphs, it is often convenient to have specific application conditions for each production. Such application conditions, on the one hand, include context conditions like the existence or non-existence of nodes, edges, or certain subgraphs in the given graph as well as embedding restrictions concerning the morphisms from the left-hand side of the production to the given graph. In this paper, the concept of application conditions introduced by Ehrig and Habel is restricted to contextual conditions, especially negative ones. In addition to the general concept, we state local confluence and the Parallelism Theorem for derivations with application conditions. Finally we study context-free graph grammars with application conditions with respect to their generative power.

Parallel Gradient Distribution in Unconstrained Optimization

A parallel version is proposed for a fundamental theorem of serial unconstrained optimization. The parallel theorem allows each of $k$ parallel processors to use simultaneously a different algorithm, such as a descent, Newton, quasi-Newton, or conjugate gradient algorithm. Each processor can perform one or many steps of a serial algorithm on a portion of the gradient of the objective function assigned to it, independently of the other processors. Eventually a synchronization step is performed which, for differentiable convex functions, consists of taking a strong convex combination of the $k$ points found by the $k$ processors. A more general synchronization step, applicable to convex as well as nonconvex functions, consists of taking the best point found by the $k$ processors or any point that is better. The fundamental result that we establish is that any accumulation point of the parallel algorithm is stationary for the nonconvex case and is a global solution for the convex case. Computational testing on the Thinking Machines CM-5 multiprocessor indicates a speedup of the order of the number of processors employed.

Algebraic high-level net transformation systems

The concept of algebraic high-level net transformation systems combines two important lines of research recently introduced in the literature:algebraic high-level nets(AHL-nets for short) andhigh-level replacement systems(HLR-systems for short). In both cases a categorical formulation of the corresponding theory has turned out to be highly important and is also a good basis for the integration of these concepts in this paper.AHL-nets combine Petri nets with algebraic specifications and provide a powerful specification technique for distributed systems including data types and processes.HLR-systems are transformation systems for high-level structures such as graphs, hypergraphs, algebraic specifications and different kinds of Petri nets. The theory of HLRsystems - formulated already in a categorical framework - is applied in this paper to AHLnets. Thus we obtain AHL-net transformation systems as an instantiation of HLR-systems to AHL-nets. This allows us to build up AHL-nets from basic components and to transform the net structure using rules or productions in the sense of graph grammars. This concept is illustrated by extending the well-known example of ‘dining philosophers’. We are able to show that AHL-net-transformation systems satisfy several important compatibility properties. On the one hand we obtain a local Church-Rosser and Parallelism Theorem, which is well-known for graph grammars and has recently been generalized to HLR-systems. This allows us to analyse concurrency in AHL-nets not only on the token level but also on the level of transformations of the net structure. On the other hand, we consider the ‘fusion’ and ‘union’ constructions for high-level structures, motivated by corresponding concepts for high-level Petri nets in the literature, and we show compatibility of these constructions with derivations of HLR-systems in general and AHL-nettransformations in particular. This means compatibility of vertical and horizontal structuring in terms of software development.

THE CLAUSE-DIFFUSION METHODOLOGY FOR DISTRIBUTED DEDUCTION

This paper describes a methodology for parallel theorem proving in a distributed environment, called deduction by Clause-Diffusion. This methodology utilizes parallelism at the search level, by having concurrent, asynchronous deductive processes searching in parallel the search space of the problem. The search space is partitioned among the processes by distributing the clauses and by subdividing certain classes of inferences. The processes communicate by exchanging data. Policies for distributing the clauses and for scheduling inference and communication steps complete the picture. A distributed derivation is made of the collection of the derivations computed by the concurrent deductive processes and it halts successfully as soon as one of them does. While the Clause-Diffusion methodology applies to theorem proving in general, it has been designed to provide solutions to the problems in the parallelization of contraction-based strategies, such as rewriting-based methods. We identify backward contraction, i.e. the task of maintaining clauses reduced in a dynamically changing data base, as the main obstacle in parallel theorem proving with contraction. In parallel implementations of contraction-based strategies in shared memory, this difficulty appears as a write-bottleneck, which we have termed the backward contraction bottleneck. The Clause-Diffusion approach avoids this problem by adopting a mostly distributed memory and distributed global contraction schemes. We conclude by reporting some of our results with an implementation of Clause-Diffusion.