Calculus Of Order Research Articles

On general tempered fractional calculus with Luchko kernels

In this paper, we construct the n-fold ψ-fractional integrals and derivatives and study their properties. This construction is purely based on the approach proposed by Luchko (2021). The fundamental theorems of fractional calculus are formulated and proved for the proposed n-fold ψ-fractional integrals and derivatives. On the other hand, a suitable generalization of the Luchko condition is presented to discuss the ψ-tempered fractional calculus of arbitrary order. We introduce an important class of kernels that satisfy this condition. For the ψ-tempered fractional integrals and derivatives of arbitrary order, two fundamental theorems are proven, along with a relation between Riemann–Liouville and Caputo derivatives. Finally, Cauchy problems for the fractional differential equations with the ψ-tempered fractional derivatives are solved.

Notion of quadratic variation in Banach spaces

This article introduces the notion of global quadratic variation which is based on a notion weaker than the usual tensor topology appearing in the literature of stochastic processes in Banach separable spaces. It is based on convergence in the weak-star topology of approximating sequences. This calculus pursues in the infinite dimensional space the stochastic calculus via regularization introduced by Russo and Vallois for real valued processes and developed in Banach space by Di Girolami and Russo. This article in particular focuses on the relation with classical calculus in order to compare our global quadratic variation with the classical existing notions of quadratic variations in the literature.

On topology changes in quantum field theory and quantum gravity

Two singularity theorems can be proven if one attempts to let a Lorentzian cobordism interpolate between two topologically distinct manifolds. On the other hand, Cartier and DeWitt-Morette have given a rigorous definition for quantum field theories (QFTs) by means of path integrals. This paper uses their results to study whether QFTs can be made compatible with topology changes. We show that path integrals over metrics need a finite norm for the latter and for degenerate metrics, this problem can sometimes be resolved with tetrads. We prove that already in the neighborhood of some cuspidal singularities, difficulties can arise to define certain QFTs. On the other hand, we show that simple QFTs can be defined around conical singularities that result from a topology change in a simple setup. We argue that the ground state of many theories of quantum gravity will imply a small cosmological constant and, during the expansion of the universe, will cause frequent topology changes. Unfortunately, it is difficult to describe the transition amplitudes consistently due to the aforementioned problems. We argue that one needs to describe QFTs by stochastic differential equations, and in the case of gravity, by Regge calculus in order to resolve this problem.

Advancing viscoelastic material modeling : Tackling time-dependent behavior with fractional calculus

A combination of inherent unique stress-strain response features, Viscoelastic Materials (VM) provide an integral part in many different fields of engineering. Although practical, these materials’ highly complex time-dependent methods according to non-linear loading scenarios are impossible to model using conventional viscoelastic models such as Maxwell and Kelvin-Voigt. The present study introduces a framework that pushes over traditional Maxwell and Kelvin-Voigt approaches by employing fractional calculus in order to enhance the prediction of VM performance. The mathematical representation makes use of the Caputo fractional derivative for expressing an artificial viscoelastic polymer’s non-linear and time-dependent responses. Dynamic Mechanical Analysis (DMA) and Stress Relaxation Tests (SRT) proved the polymer possessed 1500 MPa fractional modulus and 0.65 fractional order, respectively. The resulting model involved more significant computing resources, but contrasting testing indicated that it accurately depicted stress relaxation and dynamic responses. As mentioned, the technique integrates mathematical and actual viscoelasticity for industrial uses while offering a precise basis for advanced material analysis.

A Mechanical Picture of Fractal Darcy’s Law

The main goal of this manuscript is to generalize Darcy’s law from conventional calculus to fractal calculus in order to quantify the fluid flow in subterranean heterogeneous reservoirs. For this purpose, the inherent features of fractal sets are scrutinized. A set of fractal dimensions is incorporated to describe the geometry, morphology, and fractal topology of the domain under study. These characteristics are known through their Hausdorff, chemical, shortest path, and elastic backbone dimensions. Afterward, fractal continuum Darcy’s law is suggested based on the mapping of the fractal reservoir domain given in Cartesian coordinates xi into the corresponding fractal continuum domain expressed in fractal coordinates ξi by applying the relationship ξi=ϵ0(xi/ϵ0)αi−1, which possesses local fractional differential operators used in the fractal continuum calculus framework. This generalized version of Darcy’s law describes the relationship between the hydraulic gradient and flow velocity in fractal porous media at any scale including their geometry and fractal topology using the αi-parameter as the Hausdorff dimension in the fractal directions ξi, so the model captures the fractal heterogeneity and anisotropy. The equation can easily collapse to the classical Darcy’s law once we select the value of 1 for the alpha parameter. Several flow velocities are plotted to show the nonlinearity of the flow when the generalized Darcy’s law is used. These results are compared with the experimental data documented in the literature that show a good agreement in both high-velocity and low-velocity fractal Darcian flow with values of alpha equal to 0<α1<1 and 1<α1<2, respectively, whereas α1=1 represents the standard Darcy’s law. In that way, the alpha parameter describes the expected flow behavior which depends on two fractal dimensions: the Hausdorff dimension of a porous matrix and the fractal dimension of a cross-section area given by the intersection between the fractal matrix and a two-dimensional Cartesian plane. Also, some physical implications are discussed.

The Flipped Classroom: Enhancing Students’ Learning in Teaching Calculus

Educators have faced difficulties in efficiently teaching within the present regular school system. State, university, and college officials have been pressed to shift from face-to-face classes to teaching methodologies that address the constraints of the traditional method of instruction. Digital natives, also known as modern educators, spend more class time connecting with students and filling learning gaps with technological tools. To assist the development of learning, they began using interactive movies, interactive in-class activities, and video conference technologies. This includes utilizing a Flipped Classroom. This effort determined the effect of employing the flipped classroom technique in teaching Calculus in order to solve the issues experienced in teaching Mathematics during the new normal. The study included 20 first-year Electrical and Mechanical Engineering students who were enrolled in the second semester of the academic year 2019-2020 students whose problem-solving abilities and Calculus ability were assessed using a researcher-created test. A One-Group Pretest-Posttest Experimental Design was used by the researcher. The experimental study used four sessions of video instruction during pandemic season before the online class interactions. The researchers feel that flipped classrooms are appropriate for teaching Calculus since slow learners can watch the video repeatedly until they understand the material. As a result, assignments in the classroom focused on collaborative learning. As seen by the posttest findings, the use of the flipped classroom technique helped students enhance their problem-solving skills and mastery of solving Calculus issues. Thus, the flipped classroom method was found to be helpful in improving students' problem-solving skills and performance in Calculus among Guimaras State College's College of Engineering and Industrial Technology students.

General Nonlocal Probability of Arbitrary Order.

Using the Luchko's general fractional calculus (GFC) and its extension in the form of the multi-kernel general fractional calculus of arbitrary order (GFC of AO), a nonlocal generalization of probability is suggested. The nonlocal and general fractional (CF) extensions of probability density functions (PDFs), cumulative distribution functions (CDFs) and probability are defined and its properties are described. Examples of general nonlocal probability distributions of AO are considered. An application of the multi-kernel GFC allows us to consider a wider class of operator kernels and a wider class of nonlocality in the probability theory.

Various fractional-order type operators and some of their implications to certain normalized functions analytic in the open unit disc

Abstract In this scientific research note, certain necessary-basic information in relation to some types of the operators specified by fractional calculus of arbitrary order will be firstly introduced, various argument properties of certain analytic functions specified by those fractional-order type operators will be then determined, and a number of special consequences of those extensive properties will be also pointed out.

Multi-Kernel General Fractional Calculus of Arbitrary Order

An extension of the general fractional calculus (GFC) of an arbitrary order, proposed by Luchko, is formulated. This extension is also based on a multi-kernel approach, in which the Laplace convolutions of different Sonin kernels are used. The proposed multi-kernel GFC of an arbitrary order is also considered for the case of intervals (a,b) where −∞<a<b≤∞. Examples of multi-kernel general fractional operators of arbitrary orders are proposed.

A polyanalytic functional calculus of order 2 on the 𝑆-spectrum

The Fueter theorem provides a two step procedure to build an axially monogenic function, i.e. a null-solution of the Cauchy-Riemann operator inR4\mathbb {R}^4, denoted byD\mathcal {D}. In the first step a holomorphic function is extended to a slice hyperholomorphic function, by means of the so-called slice operator. In the second step a monogenic function is built by applying the Laplace operatorΔ\Deltain four real variables to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e.Δ=D¯D\Delta = \mathcal {\overline {D}} \mathcal {D}to split the previous procedure. From this splitting we get a class of functions that lies between the set of slice hyperholomorphic functions and the set of axially monogenic functions: the set of axially polyanalytic functions of order 2, i.e. null-solutions ofD2\mathcal {D}^2. We show an integral representation formula for this kind of functions. The formula obtained is fundamental to define the associated functional calculus on theSS-spectrum.

Fractional boundary value problem in complex domain

Fractional calculus of complex order in complex domain has emerged as a brand-new area of study. Over the past few years, fractional boundary value problems (FBVP) in real variables have been extensively studied but there are few attempts on these types of problems in complex variables. In this study, the existence and uniqueness for the solutions of fractional differential equation (FDE) in complex domain with boundary conditions is examined. We established the existence of solutions using the Krasnoselskii fixed point theorem; however, the uniqueness result is proved by applying the Banach contraction principle. To explain our findings, an illustrative example is presented. The special cases of the derived findings are equivalent to the theorems that already exist.

Heat‐conduction in a semi‐infinite fractal bar using advanced Yang‐Fourier transforms

The main objective of this article is to use local fractional calculus (Calculus of arbitrary order) and the analytical Advanced Yang‐Fourier transforms method to solve the one‐dimensional fractal heat‐conduction problem in a fractal semi‐infinite bar.

Combined metagenomic and archaeobotanical analyses on human dental calculus: A cross-section of lifestyle conditions in a Copper Age population of central Italy

Multidisciplinary analyses on ancient dental calculus revealed the possibility to reconstruct habits and diet of ancient human populations, investigate individual health status, as well as provide information on past environments. In the present study we applied both metagenomic and microscopic analysis on ancient human dental calculus in order to obtain a cross-section of the life conditions in a population of central Italy belonging to the Copper Age culture of Rinaldone (IV millennium BCE). The metagenomic profile suggested an agricultural subsistence and a dietary regimen particularly enriched in complex carbohydrates with low soluble fiber. Even bacterial functional profile seems to indicate an almost exclusive carbohydrates intake that could have favoured the occurrence of nutritional stress in the individuals. Exploring the diversity of the plant food consumed, we detected direct evidence of cereals such as wheat and/or barley, and found signals of the use of leaf vegetables, thus providing additional information on human/environment relationship. The presence of oral pathogens, even if at low abundance (<0.1%), can be related to the high consumption of carbohydrates and finds correspondence with the palaeopathological evidence. In conclusion, starting from very minute amounts of ancient dental calculus, our molecular and microscopic analysis jointly provided complementary data in support of past life condition reconstruction in ancient human populations.

Density estimates for jump diffusion processes

We consider a real-valued diffusion process with a linear jump term driven by a Poisson point process and we assume that the jump amplitudes have a centered density with finite moments. We show upper and lower estimates for the density of the solution in the case that the jump amplitudes follow a Gaussian or Laplacian law. The proof of the lower bound uses a general expression for the density of the solution in terms of the convolution of the density of the continuous part and the jump amplitude density. The upper bound uses an upper tail estimate in terms of the jump amplitude distribution and techniques of the Malliavin calculus in order to bound the density by the tails of the solution. We also extend the lower bounds to the multidimensional case.

Graph-matrix modeling of production systems as a basis for managing the production capacity of metal working enterprises

Development of information systems and technologies makes it possible to use complex models for information support of management decisions. The most particular interest shows the theory of graphs and matrix calculus in order to assess the production capacity of an industrial enterprise. The study of the authors is devoted to development of methodological approaches to assessing of production capacity based on graph-matrix models. The essence of these models lies in the fact that a manufacturing enterprise is depersonalized by perceiving it as a kind of production system, consisting of production links that add up to a production chain and form a kind of production network, which is represented as a model of the production cycle. In order to perform the appropriate calculations, the specified graph model is linked to the matrix model, which takes into account the main parameters of the production system: technological connections; assortment structure of products; direct consumption coefficients; production capacity of the links. Suggested approach determines the flexibility and broad analytical capabilities of the proposed models for calculating of production capacities. The basis for theoretical and methodological researches became the methods of analysis and synthesis, principles of consistency and complexity, graph theory and matrix calculus. The approbation of the proposed graph-matrix models of production systems was carried out using the examples of a metal working enterprise. The theoretical and practical significance of the study lies in the fact that the proposed graph-matrix model makes it possible to ensure effective management of production facilities by taking into account the mobility of the product assortment structure and complex technological links.

General Fractional Calculus: Multi-Kernel Approach

For the first time, a general fractional calculus of arbitrary order was proposed by Yuri Luchko in 2021. In Luchko works, the proposed approaches to formulate this calculus are based either on the power of one Sonin kernel or the convolution of one Sonin kernel with the kernels of the integer-order integrals. To apply general fractional calculus, it is useful to have a wider range of operators, for example, by using the Laplace convolution of different types of kernels. In this paper, an extended formulation of the general fractional calculus of arbitrary order is proposed. Extension is achieved by using different types (subsets) of pairs of operator kernels in definitions general fractional integrals and derivatives. For this, the definition of the Luchko pair of kernels is somewhat broadened, which leads to the symmetry of the definition of the Luchko pair. The proposed set of kernel pairs are subsets of the Luchko set of kernel pairs. The fundamental theorems for the proposed general fractional derivatives and integrals are proved.

A New Model for Determining Production Capacity

Traian Alexandru Buda, EThis paper introduces a new model for computing the production capacity in a manufacturing system or in a supply chain. Solving the production capacity problem means to be able to answer the following question: how many parts, from each product, can be produced by a given manufacturing system in a given time span considering the product mix and a multi-stage Bill of Materials? The proposed model is able to determine the production capacity and the loading level per resource for a manufacturing system using as inputs the Bill of Materials, Routing file, time span and product mix. The novelty brought by this method consists in the adoption of the matrix calculus in order to manipulate the inputs to obtain the requested outputs. The opportunity of such a model is that it offers the complete view on the capacity problem with a full range of answers: the available and required capacity at resource and finished product level and the loading level for each resource. Also the facile implementation and integration in ERP systems is a vital point. The use of such a model is in the investment process, middle and long-term capacity planning and client order confirmation process. The model aims to solveany type of manufacturing system.

Graph Calculus and the Disconnected-Boundary Schwinger-Dyson Equations of Quartic Tensor Field Theories

Tensor field theory (TFT) focuses on quantum field theory aspects of random tensor models, a quantum-gravity-motivated generalisation of random matrix models. The TFT correlation functions have been shown to be classified by graphs that describe the geometry of the boundary states, the so-called boundary graphs. These graphs can be disconnected, although the correlation functions are themselves connected. In a recent work, the Schwinger-Dyson equations for an arbitrary albeit connected boundary were obtained. Here, we introduce the multivariable graph calculus in order to derive the missing equations for all correlation functions with disconnected boundary, thus completing the Schwinger-Dyson pyramid for quartic melonic (‘pillow’-vertices) models in arbitrary rank. We first study finite group actions that are parametrised by graphs and build the graph calculus on a suitable quotient of the monoid algebra mathcal {A}[G] corresponding to a certain function space mathcal {A} and to the free monoid G in finitely many graph variables; a derivative of an element of mathcal {A}[G] with respect to a graph yields its corresponding group action on mathcal {A}. The present result and the graph calculus have three potential applications: the non-perturbative large-N limit of tensor field theories, the solvability of the theory by using methods that generalise the topological recursion to the TFT setting and the study of ‘higher dimensional maps’ via Tutte-like equations. In fact, we also offer a term-by-term comparison between Tutte equations and the present Schwinger-Dyson equations.

Proof Compression and NP Versus PSPACE II

We upgrade [3] to a complete proof of the conjecture NP = PSPACE that is known as one of the fundamental open problems in the mathematical theory of computational complexity; this proof is based on [2]. Since minimal propositional logic is known to be PSPACE complete, while PSPACE to include NP, it suffices to show that every valid purely implicational formula ρ has a proof whose weight (= total number of symbols) and time complexity of the provability involved are both polynomial in the weight of ρ. As is [3], we use proof theoretic approach. Recall that in [3] we considered any valid ρ in question that had (by the definition of validity) a “short” tree-like proof π in the Hudelmaier-style cutfree sequent calculus for minimal logic. The “shortness” means that the height of π and the total weight of different formulas occurring in it are both polynomial in the weight of ρ. However, the size (= total number of nodes), and hence also the weight, of π could be exponential in that of ρ. To overcome this trouble we embedded π into Prawitz’s proof system of natural deductions containing single formulas, instead of sequents. As in π, the height and the total weight of different formulas of the resulting tree-like natural deduction ∂1 were polynomial, although the size of ∂1 still could be exponential, in the weight of ρ. In our next, crucial move, ∂1 was deterministically compressed into a “small”, although multipremise, dag-like deduction ∂ whose horizontal levels contained only mutually different formulas, which made the whole weight polynomial in that of ρ. However, ∂ required a more complicated verification of the underlying provability of ρ. In this paper we present a nondeterministic compression of ∂ into a desired standard dag-like deduction ∂0 that deterministically proves ρ in time and space polynomial in the weight of ρ.2 Together with [3] this completes the proof of NP = PSPACE.Natural deductions are essential for our proof. Tree-to-dag horizontal compression of π merging equal sequents, instead of formulas, is (possible but) not sufficient, since the total number of different sequents in π might be exponential in the weight of ρ – even assuming that all formulas occurring in sequents are subformulas of ρ. On the other hand, we need Hudelmaier’s cutfree sequent calculus in order to control both the height and total weight of different formulas of the initial tree-like proof π, since standard Prawitz’s normalization although providing natural deductions with the subformula property does not preserve polynomial heights. It is not clear yet if we can omit references to π even in the proof of the weaker result NP = coNP.

Further results on advanced robust iterative learning control and modeling of robotic systems

Recently, calculus of general order [Formula: see text] has attracted attention in scientific literature, where fractional operators are often used for control issues and the modeling of the dynamics of complex systems. In this work, some attention will be devoted to the problem of viscous friction in robotic joints. The calculus of general order and the calculus of variations are utilized for the modeling of viscous friction which is extended to the fractional derivative of the angular displacement. In addition, to solve the output tracking problem of a robotic manipulator with three DOFs with revolute joints in the presence of model uncertainties, robust advanced iterative learning control (AILC) is introduced. First, a feedback linearization procedure of a nonlinear robotic system is applied. Then, the proposed intelligent feedforward-feedback AILC algorithm is introduced. The convergence of the proposed AILC scheme is established in the time domain in detail. Finally, simulations on the given robotic arm system confirm the effectiveness of the robust AILC method.