Standard Calculus Research Articles

Predefined‐time tracking control of multiplicative systems

ABSTRACTThis article describes a control approach for obtaining predefined‐time robust tracking in multiplicative systems despite positive, bounded, and unknown multiplicative disturbances. The proposed approach is distinguished by imposing predefined‐time convergence, a topic previously studied in conventional calculus in the context of multiplicative systems. Multiplicative calculus is recognized as a beneficial tool that complements standard calculus by simplifying the modeling and comprehension of numerous processes. Simulations are carried out to illustrate that the given control strategy enforces convergence before a predefined time instant and, while inducing robustness against system uncertainties. The findings of this article pave the way for further research into predefined‐time synchronization of multiplicative oscillator systems, which would bring promising implications for data encryption and secure communication.

On the adaptation of the Lagrange formalism to continuous time stochastic optimal control: A Lagrange-Chow redux

We show how the classical Lagrangian approach to solving constrained optimization problems from standard calculus can be extended to solve continuous time stochastic optimal control problems. Connections to mainstream approaches such as the Hamilton-Jacobi-Bellman equation and the stochastic maximum principle are drawn. Our approach is linked to the stochastic maximum principle, but more direct and tied to the classical Lagrangian principle, avoiding the use of backward stochastic differential equations in its formulation. Using infinite dimensional functional analysis, we formalize and extend the approach first outlined in Chow (1992) within a rigorous mathematical setting using infinite dimensional functional analysis. We provide examples that demonstrate the usefulness and effectiveness of our approach in practice. Further, we demonstrate the potential for numerical applications facilitating some of our key equations in combination with Monte Carlo backward simulation and linear regression, therefore illustrating a completely different and new avenue for the numerical application of Chow's methods.

Research on Solving Flexible Job Shop Scheduling Problem Based on Improved GWO Algorithm SS-GWO

As an important branch of production scheduling, the flexible job shop scheduling problem (FJSP) is a typical NP-hard problem. Researchers have adopted many intelligent algorithms to solve the FJSP problem, nonetheless, the task of dynamically adapting its essential parameters during the computational process is a significant challenge, resulting in the solution efficiency and quality failing to meet the production requirements. To this end, this paper proposes an adaptive gray wolf fast optimization algorithm (SS-GWO), which adopts the gray wolf algorithm (GWO) as the basic optimization method, and the algorithm adaptively selects the global search or local search according to the degree of agglomeration of individuals. Firstly, a non-linear convergence factor strategy is employed to control the global exploration and local exploitation capabilities of the algorithm at different stages. This enhances optimization precision and accelerates convergence speed, achieving a dynamic balance between the two. Secondly, the spiral search mechanism of Whale Optimization Algorithm is used in GWO to improve the exploration capability of Gray Wolf Optimization Algorithm. Finally, the effectiveness of SS-GWO model is verified by comparison experiments. The comparison demonstrates the superiority of SS-GWO over the other five state-of-the-art algorithms in solving the 22 classical benchmark test functions. SS-GWO is applied to solve FJSP by means of the standard test function bandimarte calculus. The optimal solution and performance of SS-GWO for solving FJSP are compared with other algorithms. The experimental results show that the SS-GWO algorithm has good optimization performance, and the maximum completion time is reduced by 19% and 37% compared with that of IGWO and GWO, respectively, and the proposed SS-GWO algorithm achieves a better solution effect on flexible job shop scheduling instances, which can satisfy the actual production scheduling needs.

Dynamic Separation Logic

This paper introduces a dynamic logic extension of separation logic. The assertion language of separation logic is extended with modalities for the five types of the basic instructions of separation logic: simple assignment, look-up, mutation, allocation, and de-allocation. The main novelty of the resulting dynamic logic is that it allows to combine different approaches to resolving these modalities. One such approach is based on the standard weakest precondition calculus of separation logic. The other approach introduced in this paper provides a novel alternative formalization in the proposed dynamic logic extension of separation logic. The soundness and completeness of this axiomatization has been formalized in the Coq theorem prover.

Strong Stationarity for Optimal Control Problems with Non-smooth Integral Equation Constraints: Application to a Continuous DNN

Motivated by the residual type neural networks (ResNet), this paper studies optimal control problems constrained by a non-smooth integral equation associated to a fractional differential equation. Such non-smooth equations, for instance, arise in the continuous representation of fractional deep neural networks (DNNs). Here the underlying non-differentiable function is the ReLU or max function. The control enters in a nonlinear and multiplicative manner and we additionally impose control constraints. Because of the presence of the non-differentiable mapping, the application of standard adjoint calculus is excluded. We derive strong stationary conditions by relying on the limited differentiability properties of the non-smooth map. While traditional approaches smoothen the non-differentiable function, no such smoothness is retained in our final strong stationarity system. Thus, this work also closes a gap which currently exists in continuous neural networks with ReLU type activation function.

An intersection-theoretic proof of the Harer–Zagier fomula

We provide an intersection-theoretic formula for the Euler characteristic of the moduli space of smooth curves. This formula reads purely in terms of Hodge integrals and, as a corollary, the standard calculus of tautological classes gives a new short proof of the Harer-Zagier formula. Our result is based on the Gauss-Bonnet formula, and on the observation that a certain parametrisation of the $\Omega$-class - the Chern class of the universal $r$-th root of the twisted log canonical bundle - provides the Chern class of the log tangent bundle to the moduli space of smooth curves. Being $\Omega$-classes by now employed in many enumerative problems, mostly recently found and at times surprisingly different from each other, we dedicate some work to produce an extensive list of their general properties: extending existing ones, finding new ones, and writing down some only known to the experts.

A Multiplicative Calculus Approach to Solve Applied Nonlinear Models

Problems such as population growth, continuous stirred tank reactor (CSTR), and ideal gas have been studied over the last four decades in the fields of medical science, engineering, and applied science, respectively. Some of the main motivations were to understand the pattern of such issues and how to obtain the solution to them. With the help of applied mathematics, these problems can be converted or modeled by nonlinear expressions with similar properties. Then, the required solution can be obtained by means of iterative techniques. In this manuscript, we propose a new iterative scheme for computing multiple roots (without prior knowledge of multiplicity m) based on multiplicative calculus rather than standard calculus. The structure of our scheme stands on the well-known Schröder method and also retains the same convergence order. Some numerical examples are tested to find the roots of nonlinear equations, and results are found to be competent compared with ordinary derivative methods. Finally, the new scheme is also analyzed by the basin of attractions that also supports the theoretical aspects.

An operational calculus generalization of Ramanujan's master theorem

We give a formal extension of Ramanujan's master theorem using operational methods. The resulting identity transforms the computation of an integral of a product on the half-line to the computation of a Laplace transform. Since the identity is purely formal, we show consistency of this operational approach with various standard calculus results, followed by several examples, including a Feynman diagram evaluation, to illustrate the power of the extension. We then briefly discuss the connection between Ramanujan's master theorem and identities of Hardy and Carr before extending the latter identities in the same way we extended Ramanujan's. Finally, we generalize our results, producing additional interesting identities as a corollary.

Strong stationarity for a highly nonsmooth optimization problem with control constraints

<p style='text-indent:20px;'>This paper deals with a control constrained optimization problem governed by a nonsmooth elliptic PDE in the presence of a non-differentiable objective. The nonsmooth non-linearity in the state equation is locally Lipschitz continuous and directionally differentiable, while one of the nonsmooth terms appearing in the objective is convex. Since these mappings are not necessarily Gâteaux-differentiable, the application of standard adjoint calculus is excluded. Based on their limited differentiability properties, we derive a strong stationary optimality system, i.e., an optimality system which is equivalent to the purely primal optimality condition saying that the directional derivative of the reduced objective in feasible directions is nonnegative.</p>

Emergent dynamics of various Cucker-Smale type models with a fractional derivative.

In this paper, we demonstrate emergent dynamics of various Cucker-Smale type models, especially standard Cucker-Smale (CS), thermodynamic Cucker-Smale (TCS), and relativistic Cucker-Smale (RCS) with a fractional derivative in time variable. For this, we adopt the Caputo fractional derivative as a widely used standard fractional derivative. We first introduce basic concepts and previous properties based on fractional calculus to explain its unusual aspects compared to standard calculus. Thereafter, for each proposed fractional model, we provide several sufficient frameworks for the asymptotic flocking of the proposed systems. Unlike the flocking dynamics which occurs exponentially fast in the original models, we focus on the flocking dynamics that occur slowly at an algebraic rate in the fractional systems.

Integral transform approach to mimetic discrete calculus

We introduce an integral transform that maps differential equations and special functions of standard continuous calculus onto finite difference equations and deformed special functions of mimetic discrete calculus, or h-calculus. We show that our procedure leads to insightful reformulations of several problems in mathematics and physics where discrete equations play a significant role, such as in solving finite difference equations, in applying discrete versions of integral transforms, such as the h-Laplace transform, in solving master equations of stochastic physics, in developing a discrete version of H theory of multiscale complex hierarchical phenomena and in finding lattice Green’s functions for describing quantum charge transport through phase coherent systems. We believe that our integral transform technique, or mimetic map, will help systematize the connections through analogy between discrete calculus and standard continuous calculus.

Non-linearity in cancer dose-response: The role of exposure duration

ContextAn apparent deviation from nonlinearity in cancer dose-response was reported for various carcinogens. In particular, some studies hypothesized that in mesothelioma, the exposure-response relationship can be modelled as a power function with exponent from 0.6 to 1. However, various factors can affect the shape of the dose-response, producing the apparent supralinear trend. ObjectiveTo develop a mathematical model that would demonstrate a relationship of mesothelioma lifetime risk and exposure duration, with various assumptions about a hazard rate function. MethodsTwo different hazard rate functions – the Peto model and the two-stage clonal expansion (TSCE) model – were considered. The analytical formulas for lifetime risk were developed, with and without a lifetable correction. Standard calculus methods were applied to test the shape of the lifetime risk curve. ResultsFor both Peto and TSCE models, it was shown that mesothelioma lifetime risk changes supralinearly with duration; the exponent of the power function was ranging from 0.68 to 0.89. However, the dose-response curve by cumulative exposure is close to linearity and is linear if the exposure duration would be constant. The model has been tested for chrysotile asbestos cohorts, with a good agreement demonstrated with published mesothelioma excess mortality (R=0.88, p<0.0041). ConclusionFor mesothelioma, the observed deviation from linearity in the dose-response relationship can be potentially explained by the impact of a change in the duration of exposure. In a meta-analysis, this deviation can be eliminated by standardizing the mortality data for various cohorts by duration of exposure. Short AbstractAn apparent deviation from nonlinearity in cancer dose-response was reported for various carcinogens. We applied two different hazard rate equations – the Peto model and the two-stage clonal expansion (TSCE) model – to pleural mesothelioma mortality. The analytical formulas for lifetime risk were developed. For both the Peto and TSCE models, it was shown that mesothelioma lifetime risk changes supralinearly with duration. However, the dose-response curve for cumulative exposure is close to linearity.

Internal Combustion Engines: Modeling Internal Temperature as a Function of Time

Just like any thermodynamic system, combustion engines must be cooled to eliminate friction due to heat. Without proper cooling, internal components, such as connecting rods, rod bearings, and pistons can be severely damaged due to thermal expansion, leading to severe damage to the engine block or outright catastrophic failure. Modern engines are cooled using coolant, which flows through internal passageways within the engine block to pull heat away from the system. The use of coolant and external components, such as a water pump, radiator, and thermostat allow an engine to efficiently warm to standard operating temperature and remain at said temperature. Using standard calculus and differential equations, the model for the idle coolant temperature increase inside of an engine block as it approaches a steady state can be obtained. For overall application purposes, this model includes an arbitrary number of inlets and exits for coolant to flow through. This model can also be applied to air cooled or electric systems, such as small engines or electrical motors which produce a quantifiable amount of heat.

Characteristic Classes of Homogeneous Essential Isolated Determinantal Varieties

The (homogeneous) Essentially Isolated Determinantal Variety is the natural generalization of generic determinantal variety, and is fundamental example to study non-isolated singularities. In this paper we study the characteristic classes on these varieties. We give explicit formulas of their Chern-Schwartz-MacPherson classes and Chern-Mather classes via standard Schubert calculus. As corollaries we obtain formulas for their (generic) sectional Euler characteristics, characteristic cycles and polar classes.

Optimization and learning with nonlocal calculus

&lt;p style='text-indent:20px;'&gt;Nonlocal models have recently had a major impact in nonlinear continuum mechanics and are used to describe physical systems/processes which cannot be accurately described by classical, calculus based "local" approaches. In part, this is due to their multiscale nature that enables aggregation of micro-level behavior to obtain a macro-level description of singular/irregular phenomena such as peridynamics, crack propagation, anomalous diffusion and transport phenomena. At the core of these models are &lt;i&gt;nonlocal&lt;/i&gt; differential operators, including nonlocal analogs of the gradient/Hessian. This paper initiates the use of such nonlocal operators in the context of optimization and learning. We define and analyze the convergence properties of nonlocal analogs of (stochastic) gradient descent and Newton's method on Euclidean spaces. Our results indicate that as the nonlocal interactions become less noticeable, the optima corresponding to nonlocal optimization converge to the "usual" optima. At the same time, we argue that nonlocal learning is possible in situations where standard calculus fails. As a stylized numerical example of this, we consider the problem of non-differentiable parameter estimation on a non-smooth translation manifold and show that our &lt;i&gt;nonlocal&lt;/i&gt; gradient descent recovers the unknown translation parameter from a non-differentiable objective function.&lt;/p&gt;

An Integro-differential-delay Operator Approach to Transformations of Linear Differential Time-delay Systems

In this paper, we further develop the study of rings of integro-differential-delay operators considered as noncommutative polynomial algebras satisfying standard calculus identities. Within the algebraic analysis approach, we show that transformations and reductions of linear differential time-delay systems can be interpreted as homomorphisms and isomorphisms of finitely presented left modules over an algebra of integro-differential-delay operators. In particular, we show how Fiagbedzi-Pearson's transformation can be found again and generalized. This transformation maps the solutions of a first-order differential linear system with state and input delays to the solutions of a purely state-space linear system. Fiagbedzi-Pearson's transformation reduces to the well-known Artstein's reduction when the system has no state delay and yields an isomorphism of the solution spaces.

Taylor Series for $\frac{1}{1+x^2}$ at $x=a$

We determine a formula for all coefficients of the Taylor series for the function $ \frac{1}{1+x^2} $ centered at $ x=a $ for any real number $ a $ along with the radius of convergence of the series. We utilize complex numbers to extend a standard calculus technique involving geometric series, but we write the coefficients using real numbers only.

Quasiconvexity in the fractional calculus of variations: Characterization of lower semicontinuity and relaxation

Based on recent developments in the theory of fractional Sobolev spaces, an interesting new class of nonlocal variational problems has emerged in the literature. These problems, which are the focus of this work, involve integral functionals that depend on Riesz fractional gradients instead of ordinary gradients and are considered subject to a complementary-value condition.With the goal of establishing a comprehensive existence theory, we provide a full characterization for the weak lower semicontinuity of these functionals under suitable growth assumptions on the integrands. In doing so, we surprisingly identify quasiconvexity, which is intrinsic to the standard vectorial calculus of variations, as the natural notion also in the fractional setting. In the absence of quasiconvexity, we determine a representation formula for the corresponding relaxed functionals, obtained via partial quasiconvexification outside the region where complementary values are prescribed. Thus, in contrast to classical results, the relaxation process induces a structural change in the functional, turning the integrand from a homogeneous into an inhomogeneous one. Our proofs rely crucially on an inherent relation between classical and fractional gradients, which we extend to Sobolev spaces, enabling us to transition between the two settings.

Forced to improve: open book and open internet assessment in vector calculus

Calculus is traditionally tested, at least in part, by a closed book written examination taken under secure conditions. In March 2020, two-thirds of the way through a vector calculus course, our institution went into lockdown and all teaching and assessment had to shift online. While some teachers insisted that remotely written non-proctored examinations remain closed book, we accepted that this would result in inequity, with some students certainly choosing to flout that rule should it be imposed, to their advantage over their classmates. Driven by circumstance, we were forced to design a vector calculus examination that would be written open book and ‘open internet’ with access to online calculators and computer graphing packages. Proficiency at vector calculus is exhibited far more in the interpretation of information and effective decision-making than in the computational solving of integrals. In this article, we shall present and discuss our experiences, address the challenge of non-proctored examinations and argue for the merit of open book examinations in an otherwise standard vector calculus course.

Safe equilibrium and crack growth in inhomogeneous materials as a variational problem

The variational principle of safe equilibrium for inhomogeneous elastic cracked bodies is formulated. Using the standard calculus of variations, we show that the crack remains in safe equilibrium as long as the maximum energy reduction rate of the virtually growing crack is negative. The crack starts to grow in the direction of the maximum energy reduction rate when the latter becomes zero. This energetic criterion implies the criteria proposed by He and Hutchinson (1989). As an application we use this criterion to predict the growth direction of an interface crack in a bimaterial.