Terms Of Calculus Research Articles

Supervised data extraction from transformer representation of Lambda-terms

The object of this research is the process of compiler optimization, as it is essential in modern software development, particularly in functional programming languages like Lambda Calculus. Optimization strategies directly impact interpreter and compiler performance, influencing resource efficiency and program execution. While functional programming compilers have garnered less attention regarding optimization efforts than their object-oriented counterparts, Lambda Calculus’s complexity poses unique challenges. Bridging this gap requires innovative approaches like leveraging machine learning techniques to enhance optimization strategies. This study focuses on leveraging machine learning to bridge the optimization gap in functional programming, particularly within the context of Lambda Calculus. This study delves into the extraction features from Lambda terms related to reduction strategies by applying machine learning. Previous research has explored various approaches, including analyzing reduction step complexities and using sequence analysis Artificial Neural Networks (ANNs) with simplified term representation. This research aims to develop a methodology for extracting comprehensive term data and providing insights into optimal reduction priorities by employing Large Language Models (LLMs). Tasks were set to generate embeddings from Lambda terms using LLMs, train ANN models to predict reduction steps, and compare results with simplified term representations. This study employs a sophisticated blend of machine learning algorithms and deep learning models as a method of analyzing and predicting optimal reduction paths in Lambda Calculus terms. The result of this study is a method that showed improvement in determining the number of reduction steps by using embeddings. Conclusions: The findings of this research offer significant implications for further advancements in compiler and interpreter optimization. This study paves the way for future research to enhance compiler efficiency by demonstrating the efficacy of employing LLMs to prioritize normalization strategies. Using machine learning in functional programming optimization opens avenues for dynamic optimization strategies and comprehensive analysis of program features.

Meaning and identity of proofs in a bilateralist setting: A two-sorted typed Lambda-calculus for proofs and refutations

Abstract In this paper, I will develop a $\lambda $-term calculus, $\lambda ^{2Int}$, for a bi-intuitionistic logic and discuss its implications for the notions of sense and denotation of derivations in a bilateralist setting. Thus, I will use the Curry–Howard correspondence, which has been well-established between the simply typed $\lambda $-calculus and natural deduction systems for intuitionistic logic, and apply it to a bilateralist proof system displaying two derivability relations, one for proving and one for refuting. The basis will be the natural deduction system of Wansing’s bi-intuitionistic logic 2Int, which I will turn into a term-annotated form. Therefore, we need a type theory that extends to a two-sorted typed $\lambda $-calculus. I will present such a term-annotated proof system for 2Int and prove a Dualization Theorem relating proofs and refutations in this system. On the basis of these formal results, I will argue that this gives us interesting insights into questions about sense and denotation as well as synonymy and identity of proofs from a bilateralist point of view.

Node Replication: Theory And Practice

We define and study a term calculus implementing higher-order node replication. It is used to specify two different (weak) evaluation strategies: call-by-name and fully lazy call-by-need, that are shown to be observationally equivalent by using type theoretical technical tools.

Computational Skills in Solving Application Problems Involving Basic Differentiation Rules in Differential Calculus: An Explanatory Sequential Study

This study aimed to identify the level of computational skills and the challenges of students in solving application problems using basic differentiation rules in differential calculus. This study employed a mixed method explanatory-sequential design, which involves collecting and analyzing quantitative data first, followed by the collection and analysis of qualitative data. In the quantitative phase of this research, a simple random sampling method was utilized to administer a modified questionnaire (problem-solving examination type) to 50 calculus students. In the qualitative phase, purposive sampling was used to administer semi-structured in-depth interviews (IDIs) to a sample of 6 participants. Mean and thematic analysis with document analysis were utilized to examine the information that helped researchers identify problem about the subject matter. The study shows an overall high level of computational skills in basic differentiation, which means that the computational skills of students are often manifested. However, the computational skills of students in differential calculus in terms of chain rule are low, which is interpreted as rarely manifested. With this, this research had undergone an in-depth analysis of the challenges of the students in solving application problems using chain rule. The results reveal 3 challenges why the students’ computational skills in terms of the chain rule are low: the complexity of the composition of the chain rule, a lack of practice and exposure in using the chain rule, and uncertainty regarding its application. Effective teaching strategies are essential for breaking down complex concepts and enhancing students' computational skills in basic differentiation rules in calculus.

Fractional Linear Reservoir Model as Elementary Hydrologic Response Function

This paper presents a fractional linear reservoir model as the elementary response function of hydrologic systems corresponding to the classical linear reservoir model and tests its applicability to rainfall–runoff modeling. To this end, we formulate a fractional linear reservoir model in terms of fractional calculus following the same procedure as the classical linear reservoir model and, at the simplest level, compare its performance of rainfall–runoff modeling with the linear and nonlinear reservoir models. The impulse response function of a fractional linear reservoir model, a probability density function (PDF) following the Mittag–Leffler distribution, shows nonlinearity due to its time-variant behavior compared to that of a linear reservoir model. In traditional linear hydrologic system theory, the lag and route version of a fractional linear reservoir model produces the fast-rising and slow-recession of runoff hydrographs, implying the mixed response of linear and nonlinear reservoir models to rainfall. So, a fractional linear reservoir model could be considered a fundamental tool to effectively reflect the nonlinearity of rainfall–runoff phenomena within the framework of the linear hydrologic system theory. In this respect, the fractional order of the storage relationship specifying a fractional linear reservoir model can be viewed as a kind of parameter to quantify the heterogeneity of runoff generation within a river basin.

Working Temperature’s Influence on the Lifepo4 Battery Generated Power

Abstract The battery needed to store electricity is an essential element of a photovoltaic energy production system. The place where this system element is located, during its operation, will thermally influence the way the battery works. The specialized literature presents studies on the influence of temperature on the electrical working capacity of the battery, but through this scientific paper we want to study a concrete case of a LiFePO4 battery and quantify the influence of temperature differences in the working space on the differences in the energy capacities generated by the same battery. The experimental setup allowed recording the temperature of the space where the battery is placed for a long time, so we ensured that the temperature of the battery is approximately equal to that of the working space, just like in a real case of a photovoltaic system. The monitoring of the electric power of the battery in operation was carried out only during its discharge, by imposing a constant discharge current simultaneously with the recording of the voltage at the battery terminals throughout this cycle. At the end of the experiments, we associated the average temperatures during the discharge cycles with the average powers discharged by the battery in each thermal frame. The quantification was achieved by relating the power differences generated to the temperature differences of the different thermal situations experienced. The importance of this study resides into the highlighting of importance to maintain an optimal temperature for the space where the batteries are stored, having as technical effect the maintaining the power’s functioning optimal rate of these. The design of a photovoltaic system must be oriented to a long term calculus regarding power’s rate depending by battery’s working space temperature.

Rensets and Renaming-Based Recursion for Syntax with Bindings Extended Version

We introduce renaming-enriched sets (rensets for short), which are algebraic structures axiomatizing fundamental properties of renaming (also known as variable-for-variable substitution) on syntax with bindings. Rensets compare favorably in some respects with the well-known foundation based on nominal sets. In particular, renaming is a more fundamental operator than the nominal swapping operator and enjoys a simpler, equationally expressed relationship with the variable-freshness predicate. Together with some natural axioms matching properties of the syntactic constructors, rensets yield a truly minimalistic characterization of lambda -calculus terms as an abstract datatype—one involving an infinite set of unconditional equations, referring only to the most fundamental term operators: the constructors and renaming. This characterization yields a recursion principle, which (similarly to the case of nominal sets) can be improved by incorporating Barendregt’s variable convention. When interpreting syntax in semantic domains, our renaming-based recursor is easier to deploy than the nominal recursor. Our results have been validated with the proof assistant Isabelle/HOL.

Joint discrete and continuous matrix distribution modeling

In this paper, we introduce a bivariate distribution on arising from a single underlying Markov jump process. The marginal distributions are phase-type and discrete phase-type distributed, respectively, which allow for flexible behavior for modeling purposes. We show that the distribution is dense in the class of distributions on and derive some of its main properties, all explicit in terms of matrix calculus. Furthermore, we develop an effective EM algorithm for the statistical estimation of the distribution parameters. In the last part of the paper, we apply our methodology to an insurance dataset, where we model the number of claims and the mean claim sizes of policyholders, which is seen to perform favorably. An additional consequence of the latter analysis is that the total loss size in the entire portfolio is captured substantially better than with independent phase-type models.

Fractional Modeling of Non-Newtonian Casson Fluid between Two Parallel Plates

In this manuscript, fractional modeling of non-Newtonian Casson fluid squeezed between two parallel plates is performed under the influence of magneto-hydro-dynamic and Darcian effects. The Casson fluid model is fractionally transformed through mixed similarity transformations. As a result, partial differential equations (PDEs) are transformed to a fractional ordinary differential equation (FODE). In the current modeling, the continuity equation is satisfied while the momentum equation of the integral order Casson fluid is recovered when the fractional parameter is taken as α = 1 . A modified homotopy perturbation algorithm is used for the solution and analysis of highly nonlinear and fully fractional ordinary differential equations. Obtained solutions and errors are compared with existing integral order results from the literature. Graphical analysis is also performed at normal and radial velocity components for different fluid and fractional parameters. Analysis reveals that a few parameters are showing different behavior in a fractional environment as compared to existing integer-order cases from the literature. These findings affirm the importance of fractional calculus in terms of more generalized analysis of physical phenomena.

Nichols Algebras and Quantum Principal Bundles

Abstract We introduce a general framework for associating to a homogeneous quantum principal bundle a Yetter–Drinfeld module structure on the cotangent space of the base calculus. The holomorphic and anti-holomorphic Heckenberger–Kolb calculi of the quantum Grassmannians are then presented in this framework. This allows us to express the calculi in terms of the corresponding Nichols algebras. The extension of this result to all irreducible quantum flag manifolds is then conjectured.

Beyond the trade-offs on Facebook: the underlying mechanisms of privacy choices

The theory of privacy calculus in terms of the trade-offs between benefits and risks is believed to explain people’s willingness to disclose private information online. However, the phenomenon of privacy paradox, referring to the preference-behavior inconsistency, misfits the risk–benefit analysis. The phenomenon of privacy paradox matters because it reflects an illusion of personal control over privacy choices. The anomaly of privacy paradox is perhaps attributed to cognitive heuristics and biases in making privacy decisions. We consider the stability-instability of privacy choices is better used to explain the underlying mechanisms of paradoxical relationship. A rebalanced trade-off, referring to the embeddedness of “bridging” and “bonding” social support in privacy calculus, is derived to develop the risk–benefit paradigms to explain the underlying mechanisms. In this study we address the underlying mechanisms of privacy choices in terms of self-disclosure and user resistance. To test the hypotheses (or mechanisms) of the research model, we developed the instrument by modifying previous scales. A general sample of 311 experienced Facebook users was collected via online questionnaire survey. From the empirical results, perceived benefits based on information support rather than emotion support can motivate self-disclosure willingness. In contrast, privacy risks rather than privacy concerns inhibit the willingness to disclose private information. The risk–benefit paradigms instead of the imbalanced trade-offs help to explain the instability of privacy choices where privacy calculus sticks with the stability view. Implications for the theory and practice of privacy choices are discussed accordingly.

A solving method for two-dimensional homogeneous system of fuzzy fractional differential equations

<abstract> <p>The purpose of this study is to extend and determine the analytical solution of a two-dimensional homogeneous system of fuzzy linear fractional differential equations with the Caputo derivative of two independent fractional orders. We extract two possible solutions to the coupled system under the definition of strongly generalized $ H $-differentiability, uncertain initial conditions and fuzzy constraint coefficients. These potential solutions are determined using the fuzzy Laplace transform. Furthermore, we extend the concept of fuzzy fractional calculus in terms of the Mittag-Leffler function involving triple series. In addition, several important concepts, facts, and relationships are derived and proved as property of boundedness. Finally, to grasp the considered approach, we solve a mathematical model of the diffusion process using proposed techniques to visualize and support theoretical results.</p></abstract>

Comparison of the Efficacy of Silodosin vs Tamsulosin, as a Medical Expulsive Therapy, in the Management of Distal Ureteric Calculus in Terms of Stone Clearance Rate

Comparison of the Efficacy of Silodosin vs Tamsulosin, as a Medical Expulsive Therapy, in the Management of Distal Ureteric Calculus in Terms of Stone Clearance Rate

Bell-Type Inequalities from the Perspective of Non-Newtonian Calculus

AbstractA class of quantum probabilities is reformulated in terms of non-Newtonian calculus and projective arithmetic. The model generalizes spin-1/2 singlet state probabilities discussed in Czachor (Acta Physica Polonica:139 70–83, 2021) to arbitrary spins s. For $$s\rightarrow \infty$$ s → ∞ the formalism reduces to ordinary arithmetic and calculus. Accordingly, the limit “non-Newtonian to Newtonian” becomes analogous to the classical limit of a quantum theory.

Penalised likelihood methods for phase-type dimension selection

Abstract Phase-type distributions are dense in the class of distributions on the positive real line, and their flexibility and closed-form formulas in terms of matrix calculus allow fitting models to data in various application areas. However, the parameters are in general non-identifiable, and hence the dimension of two similar models may be very different. This paper proposes a new method for selecting the dimension of phase-type distributions via penalisation of the likelihood function. The penalties are in terms of the Green matrix, from which it is possible to extract the contributions of each state to the overall mean. Since representations with higher dimensions are penalised, a parsimony effect is obtained. We perform a numerical study with randomly generated phase-type samples to illustrate the effectiveness of the proposed procedure, and also apply the technique to the absolute log-returns of EURO STOXX 50 and Bitcoin prices.

How can a language have double-passives but lack antipassives?

Passivization in Turkish may target both internal and external argument positions, for passives of unaccusatives, unergatives and transitives are possible in the language. Interestingly, Turkish also has double passives where the passive morpheme is iterated and both argument positions are saturated. However, the language does not allow for antipassives, where only the internal argument position of a transitive predicate is saturated by passivization. Importantly, under plausible compositional analyses, antipassives are fully predicted, contrary to the fact. Hence, the unavailability of antipassives in a language that has double passives (and passives of unaccusatives) remains as a puzzle, which this paper sets out to solve. In particular, we argue that antipassives are ruled out on the grounds that the passive morpheme is visible to the case calculus in terms of Dependent Case Theory but has to remain caseless throughout the derivation. We show that this account also correctly predicts that internal argument positions that happen to be lexical case positions can never be targeted by passivization. Finally, we provide a fine-grained analysis of internal argument suppression accounting for the aspectual restrictions it has been reported to have, and compare our work to the existing proposals.

Proof Nets for the Lambek Calculus with One Division and a Negative-Polarity Modality for Weakening

In this paper, we introduce a variant of the Lambek calculus allowing empty antecedents. This variant uses two connectives: the left division and a unary modality that occurs only with negative polarity and allows weakening in antecedents of sequents. We define the notion of a proof net for this calculus, which is similar to those for the ordinary Lambek calculus and multiplicative linear logic. We prove that a sequent is derivable in the calculus under consideration if and only if there exists a proof net for it. Thus, we establish a derivability criterion for this calculus in terms of the existence of a graph with certain properties. The size of the graph is bounded by the length of the sequent.

Parameter Identification and the Finite-Time Combination–Combination Synchronization of Fractional-Order Chaotic Systems with Different Structures under Multiple Stochastic Disturbances

This paper researches the issue of the finite-time combination-combination (C-C) synchronization (FTCCS) of fractional order (FO) chaotic systems under multiple stochastic disturbances (SD) utilizing the nonsingular terminal sliding mode control (NTSMC) technique. The systems we considered have different characteristics of the structures and the parameters are unknown. The stochastic disturbances are considered parameter uncertainties, nonlinear uncertainties and external disturbances. The bounds of the uncertainties and disturbances are unknown. Firstly, we are going to put forward a new FO sliding surface in terms of fractional calculus. Secondly, some suitable adaptive control laws (ACL) are found to assess the unknown parameters and examine the upper bound of stochastic disturbances. Finally, combining the finite-time Lyapunov stability theory and the sliding mode control (SMC) technique, we propose a fractional-order adaptive combination controller that can achieve the finite-time synchronization of drive-response (D-R) systems. In this paper, some of the synchronization methods, such as chaos control, complete synchronization, projection synchronization, anti-synchronization, and so forth, have become special cases of combination-combination synchronization. Examples are presented to verify the usefulness and validity of the proposed scheme via MATLAB.

A fine-grained computational interpretation of Girard’s intuitionistic proof-nets

This paper introduces a functional term calculus, called pn, that captures the essence of the operational semantics of Intuitionistic Linear Logic Proof-Nets with a faithful degree of granularity, both statically and dynamically. On the static side, we identify an equivalence relation on pn-terms which is sound and complete with respect to the classical notion of structural equivalence for proof-nets. On the dynamic side, we show that every single (exponential) step in the term calculus translates to a different single (exponential) step in the graphical formalism, thus capturing the original Girard’s granularity of proof-nets but on the level of terms. We also show some fundamental properties of the calculus such as confluence, strong normalization, preservation of β-strong normalization and the existence of a strong bisimulation that captures pairs of pn-terms having the same graph reduction.

Nonlocal fractal calculus based analyses of electrical circuits on fractal set

PurposeThe purpose of this paper is to present the analyses of electrical circuits with arbitrary source terms defined on middle b cantor set by means of nonlocal fractal calculus and to evaluate the appropriateness of such unconventional calculus.Design/methodology/approachThe nonlocal fractal integro-differential equations describing RL, RC, LC and RLC circuits with arbitrary source terms defined on middle b cantor set have been formulated and solved by means of fractal Laplace transformation. Numerical simulations based on the derived solutions have been performed where an LC circuit has been studied by means of Lagrangian and Hamiltonian formalisms. The nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been derived and the local fractal calculus-based ones have been revisited.FindingsThe author has found that the LC circuit defined on a middle b cantor set become a physically unsound system due to the unreasonable associated Hamiltonian unless the local fractal calculus has been applied instead.Originality/valueFor the first time, the nonlocal fractal calculus-based analyses of electrical circuits with arbitrary source terms have been performed where those circuits with order higher than 1 have also been analyzed. For the first time, the nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been proposed. The revised contradiction free local fractal calculus-based Lagrangian and Hamiltonian equations have been presented. A comparison of local and nonlocal fractal calculus in terms of Lagrangian and Hamiltonian formalisms have been made where a drawback of the nonlocal one has been pointed out.