Terms Of Calculus Research Articles

PHYSARUM SPATIAL LOGIC

Plasmodium of Physarum polycephalum is a large single cell capable for distributed sensing, information processing, decentralized decision-making and collective action. In the paper, we interpret basic features of the plasmodium foraging behavior in terms of process calculus and spatial logic and show that this behavior could be regarded as one of the natural implementations of spatial logic without modal operators.

Generalized variational calculus in terms of multi-parameters fractional derivatives

In this paper, we briefly introduce two generalizations of work presented a few years ago on fractional variational formulations. In the first generalization, we consider the Hilfer’s generalized fractional derivative that in some sense interpolates between Riemann–Liouville and Caputo fractional derivatives. In the second generalization, we develop a fractional variational formulation in terms of a three parameter fractional derivative. We develop integration by parts formulas for the generalized fractional derivatives which are key to developing fractional variational calculus. It is shown that many derivatives used recently and their variational formulations can be obtained by setting different parameters to different values. We also define fractional generalized momenta and provide fractional Hamiltonian formulations in terms of the new generalized derivatives. An example is presented to show applications of the formulations presented here. Some possible extensions of this research are also discussed.

A calculus of multiary sequent terms

Multiary sequent terms were originally introduced as a tool for proving termination of permutative conversions in cut-free sequent calculus. This work develops the language of multiary sequent terms into a term calculus for the computational (Curry-Howard) interpretation of a fragment of sequent calculus with cuts and cut-elimination rules. The system, called generalized multiary λ-calculus , is a rich extension of the λ-calculus where the computational content of the sequent calculus format is explained through an enlarged form of the application constructor. Such constructor exhibits the features of multiarity (the ability to form lists of arguments) and generality (the ability to prescribe a kind of continuation). The system integrates in a modular way the multiary λ-calculus and an isomorphic copy of the λ-calculus with generalized application, Λ J (in particular, natural deduction is captured internally up to isomorphism). In addition, the system: (i) comes with permutative conversion rules, whose role is to eliminate the new features of application; (ii) is equipped with reduction rules — either the μ-rule, typical of the multiary setting, or rules for cut-elimination, which enlarge the ordinary β-rule. This article establishes the metatheory of the system, with emphasis on the role of the μ-rule, and including a study of the interaction of reduction and permutative conversions.

Gauss’ quadratic reciprocity theorem and mathematical fruitfulness

This paper presents an account of the fruitfulness of new mathematical calculi in terms of their relationship to existing mathematical methods which is suggested by Carl Friedrich Gauss. This is done by considering some remarks that Gauss made explaining the fruitfulness of new calculi. These can be clarified in the context of his own (very fruitful) theory of congruences, which is considered as a case study for this alternative account. Such an account has the benefit of not being dependent on a particular metaphysical view in the philosophy of mathematics.

Dimensional and correlation effects of charged excitons in low-dimensional semiconductors

In this paper, we investigate the existence of bound trion states in fractional dimensional nanostructures, in terms of variational calculus. We start with trial states, then we refine the result with the help of the Hartree–Fock approximation and finally we use a partial basis expansion. We show that Hartree–Fock significantly underestimates the trion binding energy and that the correlation energy is comparable with the trion binding energy. Furthermore we calculate the binding energies of positive and negative trions restricted to a large subspace of functions, which we expect to span the low-lying eigenstates of the full Hamiltonian. We find that the difference between the positive and negative trion binding energies varies very little for the electron–hole mass fractions me/mh = σ ∊ [0.8; 1.0] and that the difference between the positive and negative trion energies grows as the dimension decreases. Finally, we compare a cylindrical effective-mass model of a typical carbon nanotube, with a fractional dimensional model with D = 1.71. We find very good agreement between the trion binding energies predicted by the two models.

Soundness and principal contexts for a shallow polymorphic type system based on classical logic

In this paper we investigate how to adapt the well-known notion of ML-style polymorphism (shallow polymorphism) to a term calculus based on a Curry-Howard correspondence with classical sequent calculus, namely, theX i -calculus. We show that the intuitive approach is unsound, and pinpoint the precise nature of the problem. We define a suitably refined type system, and prove its sou ndness. We then define a notion of principal contextsfor the type system, and provide an algorithm to compute these, which is proved to be sound and complete with respect to the type system. In the process, we formalise and prove correctness of generic unification, which generalises Robinson’s unification to shallow-poly morphic types.

McCarthy variations in a modal key

We take a fresh look at some major strands in John McCarthy's work from a logician's perspective. First, we re-analyze circumscription in dynamic logics of belief change under hard and soft information. Next, we re-analyze the regression method in the Situation Calculus in terms of update axioms for dynamic–epistemic temporal logics. Finally, we draw some general methodological comparisons between ‘Logical AI’ and practices in modal logic, pointing at some creative tensions.

Gelfand-Zetlin Polytopes and Flag Varieties

I construct a correspondence between the Schubert cycles on the variety of complete flags in ℂn and some faces of the Gelfand–Zetlin polytope associated with the irreducible representation of SLn(ℂ) with a strictly dominant highest weight. The construction is motivated by the geometric presentation of Schubert cells using Demazure modules due to Bernstein–Gelfand–Gelfand [3]. The correspondence between the Schubert cycles and faces is then used to interpret the classical Chevalley formula in Schubert calculus in terms of the Gelfand–Zetlin polytopes. The whole picture resembles the picture for toric varieties and their polytopes.

A process calculus for Mobile Ad Hoc Networks

We present the ω -calculus, a process calculus for formally modeling and reasoning about Mobile Ad Hoc Wireless Networks (MANETs) and their protocols. The ω -calculus naturally captures essential characteristics of MANETs, including the ability of a MANET node to broadcast a message to any other node within its physical transmission range (and no others), and to move in and out of the transmission range of other nodes in the network. A key feature of the ω -calculus is the separation of a node’s communication and computational behavior, described by an ω -process, from the description of its physical transmission range, referred to as an ω -process interface. Our main technical results are as follows. We give a formal operational semantics of the ω -calculus in terms of labeled transition systems and show that the state reachability problem is decidable for finite-control ω -processes. We also prove that the ω -calculus is a conservative extension of the π -calculus, and that late bisimulation equivalence (appropriately lifted from the π -calculus to the ω -calculus) is a congruence. Congruence results are also established for a weak version of late bisimulation equivalence, which abstracts away from two types of internal actions: τ -actions, as in the π -calculus, and μ -actions, signaling node movement. We additionally define a symbolic semantics for the ω -calculus extended with the mismatch operator, along with a corresponding notion of symbolic bisimulation equivalence, and establish congruence results for this extension as well. Finally, we illustrate the practical utility of the calculus by developing and analyzing formal models of a leader election protocol for MANETs and the AODV routing protocol.

Lie Conformal Algebra Cohomology and the Variational Complex

We find an interpretation of the complex of variational calculus in terms of the Lie conformal algebra cohomology theory. This leads to a better understanding of both theories. In particular, we give an explicit construction of the Lie conformal algebra cohomology complex, and endow it with a structure of a g-complex. On the other hand, we give an explicit construction of the complex of variational calculus in terms of skew-symmetric poly-differential operators.

Ureteroscopic Laser Lithotripsy versus Ballistic Lithotripsy for Treatment of Ureteric Stones: A Prospective Comparative Study

Objectives: This study was undertaken to compare the results of laser (Ho:YAG) and pneumatic (ballistic) intracorporeal lithotripsy for ureteric calculi in terms of efficacy, safety and complications. Methods: 55 patients having ureteric calculus were randomly allocated into pneumatic lithotripsy (PL) and laser lithotripsy (LL) groups. Swiss lithoclast was used for PL (3 atm pressure and 12 Hz frequency) and the VersaPulse PowerSuite was used for LL. Appropriate statistical tests were applied. Results: 30 patients (34 stones) were treated with LL and 25 patients (25 stones) with PL. Both groups were comparable in profile. Mean lithotripsy time was 24.03 ± 9.51 min in the LL group and 19.80 ± 4.44 min in the PL group (p = 0.027). The immediate stone clearance rate was higher in the LL group (p = 0.001), but it was comparable at 4 weeks (p = 0.097). Stone migration occurred in 16% of cases in the PL group. No major complication was observed in either group. Conclusions: We conclude that both laser and pneumatic energies are effective and safe for intracorporeal lithotripsy. Laser lithotripsy takes more time but provides earlier stone-free status.

The logic of message-passing

Message-passing is a key ingredient of concurrent programming. The purpose of this paper is to describe the equivalence between the proof theory, the categorical semantics, and term calculus of message-passing. In order to achieve this we introduce the categorical notion of a linear actegory and the related polycategorical notion of a poly-actegory. Not surprisingly the notation used for the term calculus borrows heavily from the (synchronous) π -calculus. The cut-elimination procedure for the system provides an operational semantics.

Comparison of Sapphire and Germanium Fibers for Erbium : Yag Lithotripsy

We studied the sapphire and germanium fibers to determine which optical fiber best transmits Erbium:YAG laser for intracorporeal lithotripsy. Human calculi were ablated with an Erbium:YAG laser in contact mode using two fibers. Optical outputs at the distal end of fibers were measured before and after laser lithotripsy. Upon the irradiation on the calculus with the 50 mJ and 100 mJ pulse energy, the output energy at the distal end of germanium fiber declined to approximately 50% of the input energy. For the sapphire fiber, the output energy at the distal end remained unchanged with 100 mJ input energy; however the output energy had dropped to 50% for 200 mJ input energy. In order to examine how the types of target tissue affect the fiber damage, the sapphire fiber was tested for the irradiation on soft tissue and water as well. No energy decline was observed during soft tissue and water irradiation. We also characterized ablation craters with both optical fibers. Both fibers produced similar craters on calculi in terms of depth and diameter. Sapphire fibers are better suited than germanium fibers for Erbium:YAG lithotripsy in terms of the fiber damage.

Kant’s Theory of Arithmetic: A Constructive Approach?

Kant’s theory of arithmetic is not only a central element in his theoretical philosophy but also an important contribution to the philosophy of arithmetic as such. However, modern mathematics, especially non-Euclidean geometry, has placed much pressure on Kant’s theory of mathematics. But objections against his theory of geometry do not necessarily correspond to arguments against his theory of arithmetic and algebra. The goal of this article is to show that at least some important details in Kant’s theory of arithmetic can be picked up, improved by reconstruction and defended under a contemporary perspective: the theory of numbers as products of rule following construction presupposing successive synthesis in time and the theory of arithmetic equations, sentences or “formulas”—as Kant says—as synthetic a priori. In order to do so, two calculi in terms of modern mathematics are introduced which formalise Kant’s theory of addition as a form of synthetic operation.

Pict correctness revisited

The Pict programming language is an implementation of the π -calculus in which executions of π -calculus terms are specified via an abstract machine. An important property of any concurrent programming language implementation is the fair execution of threads. After defining fairness for the π -calculus, we show that Pict abstract machine executions implement fair π -calculus executions. We also give new proofs of soundness and liveness for the Pict abstract machine.

A new interpretation for the dynamic behaviour of complex fluids at the sol–gel transition using the fractional calculus

We propose to analyse power law shear stress relaxation modulus observed at the sol–gel transition (SGT) in many gelling systems in terms of fractional calculus. We show that the critical gel (gel at SGT) can be associated to a single fractional element and the gel in the post-SGT state to a fractional Kelvin–Voigt model. In this case, it is possible to give a physical interpretation to the fractional derivative order. It is associated to the power law exponent of the shear modulus related to the fractal dimension of the critical gel. A preliminary experimental application to silica alkoxide-based systems is given.

A compositional Petri net translation of general π -calculus terms

Abstract We propose a finite structural translation of possibly recursive π -calculus terms into Petri nets. This is achieved by using high-level nets together with an equivalence on markings in order to model entering into recursive calls, which do not need to be guarded. We view a computing system as consisting of a main program ( π -calculus term) together with procedure declarations (recursive definitions of π -calculus identifiers). The control structure of these components is represented using disjoint high-level Petri nets, one for the main program and one for each of the procedure declarations. The program is executed once, while each procedure can be invoked several times (even concurrently), each such invocation being uniquely identified by structured tokens which correspond to the sequence of recursive calls along the execution path leading to that invocation.

On the reducibility of failure theories for composite materials

Abstract This paper reports on the initiation of an effort to address three ontological aspects of failure criteria as they are applicable to composite materials applications. The first attempts to clarify “what a criterion really is” from the perspective of physical objectivity. The second attempts to clarify “how a criterion relates with another” in terms of their main semantic attributes. The third attempts to identify “how one criterion is related to another” in terms of their syntactic representation in the context of term calculus especially from the perspective of polynomial association with the strain energy density. Under the first aspect the non-physical nature of a theory is exposed when it is used as a criterion and is evaluated from its semantic perspective. The human role on the formation of any failure criterion is shown to have a foundationally subjective character, thus, rendering the corresponding criterion as non-objective. Under the second aspect the creation of classification ontology in terms of the semantic projections of failure criteria in their structural heritage and usage contexts is given. The common attributes of failure criteria are utilized to identify the bases of the attribute space that can be used to enable their ontological classification. Web ontology software is utilized for visual interpretation of the ontological context. Under the third aspect the syntactic reducibility of the various known failure theories to one another is addressed. Thermodynamically founded strain energy density expansion leads to the creation of various generalized strain energy based criteria. Two theorems are proposed and proved to show that under specific conditions failure criteria and theories of certain polynomial structure are constrained renderings of criteria based on strain energy density. Examples are given to demonstrate how a critical energy density criterion represents the least upper bound in the partial ordered set of many well known failure criteria.

Verification of Ptime Reducibility for system F Terms: Type Inference in Dual Light Affine Logic

In a previous work Baillot and Terui introduced Dual light affine logic (DLAL) as a variant of Light linear logic suitable for guaranteeing complexity properties on lambda calculus terms: all typable terms can be evaluated in polynomial time by beta reduction and all Ptime functions can be represented. In the present work we address the problem of typing lambda-terms in second-order DLAL. For that we give a procedure which, starting with a term typed in system F, determines whether it is typable in DLAL and outputs a concrete typing if there exists any. We show that our procedure can be run in time polynomial in the size of the original Church typed system F term.

The BFKL pomeron calculus in the dipole approach

In this paper we continue to pursue a goal of finding an effective theory for high energy interaction in QCD based on the colour dipole approach, for which the BFKL pomeron calculus gives a low energy limit. The key problem, that we try to solve in this paper is the probabilistic interpretation of the BFKL pomeron calculus in terms of the colourless dipoles and their interactions. We demonstrate that the BFKL pomeron calculus has two equivalent descriptions: (i) one is the generating functional which gives a clear probabilistic interpretation of the processes of high energy scattering and also provides a Hamiltonian-like description of the system of interacting dipoles; (ii) the second is the Langevin equation with a specific noise term which is rather complicated. We found that at high energies this Langevin equation can be reduced to the Langevin equation for directed percolation in the momentum space if the impact parameter is large, namely, b ≫ 1 / k , where k is the transverse momentum of a dipole. Unfortunately, this simplified form of Langevin equation is not applicable for summation of pomeron loops, where one integrates over all possible values of impact parameter. We show that the BFKL pomeron calculus with two vertices (splitting P → P + P and merging P + P → P of pomerons) can be interpreted as a system of colourless dipoles with two processes: the decay of one dipole into two and the merging of two dipoles into one dipole. However, a number of assumptions we have to make on the way to simplify the noise term in the Langevin equation and/or to apply the probabilistic interpretation, therefore, we can consider both of these approaches in the present form only as the QCD motivated models.