Version Of Calculus Research Articles

On the Generalization of Tempered-Hilfer Fractional Calculus in the Space of Pettis-Integrable Functions

We propose here a general framework covering a wide range of fractional operators for vector-valued functions. We indicate to what extent the case in which assumptions are expressed in terms of weak topology is symmetric to the case of norm topology. However, taking advantage of the differences between these cases, we emphasize the possibly less-restrictive growth conditions. In fact, we present a definition and a serious study of generalized Hilfer fractional derivatives. We propose a new version of calculus for generalized Hilfer fractional derivatives for vector-valued functions, which generalizes previously studied cases, including those for real functions. Note that generalized Hilfer fractional differential operators in terms of weak topology are studied here for the first time, so our results are new. Finally, as an application example, we study some n-point boundary value problems with just-introduced general fractional derivatives and with boundary integral conditions expressed in terms of fractional integrals of the same kind, extending all known cases of studies in weak topology.

MULTIPLE SOLITONS, BIFURCATIONS, CHAOTIC PATTERNS AND FISSION/FUSION, ROGUE WAVES SOLUTIONS OF TWO-COMPONENT EXTENDED (2+1)-D ITÔ CALCULUS SYSTEM

The exploration of nonlinear phenomena entails the representation of intricate systems with space-time variables, and across this line, Itô calculus, as the stochastic calculus version of the change pertaining to the variables formula and chain rules, involves the second derivative of f, coming from the property that Brownian motion has non-zero quadratic variation. To this end, the extended ([Formula: see text])-dimensional two-component Itô equation, as an applicable mathematical tool employed in this study for enhancing our understanding of the complex dynamics inherent in multidimensional physical systems, serves the purpose of modeling and understanding dynamic phenomena pervading various disciplines. For modeling complex phenomena, fractional differential equations (FDEs), ordinary differential equations (ODEs), partial differential equations (PDEs) as well as the other ones provide benefits in terms of accuracy and tractability. Accordingly, our study provides the analysis of the two-component nonlinear extended ([Formula: see text])-dimensional Itô equation using the Hirota bilinear method to derive multiple soliton solutions, including novel variations along with their dispersion coefficients, which shed light into the intriguing attributes of the Itô equation. The investigation further encompasses diverse soliton types, such as the general first-order soliton, second-order soliton with fission and bifurcation, third-order soliton, and fourth-order soliton with fission and bifurcation. Besides these, the study also explores the rogue wave and lump solutions by varying parameters across distinct planes. Consequently, these results validate the characteristics and utility of the two-component nonlinear extended ([Formula: see text])-dimensional Itô equation and its relevance to related systems. The novel findings based on the Itô calculus systems revealed, through the analyses, theoretical and experimental aspects in combination with the graphical presentation of the parameter effects on solitons in line with the analyses obtained. These have enhanced the understanding of the dynamics of intricate attributes governed by the two-component nonlinear extended ([Formula: see text])-dimensional Itô equation, particularly concerning chaotic patterns, fission and bifurcation soliton nonlinear complexities.

BCI-Algebras and Related Logics

Kabzinski in [6] first introduced an extension of BCI-logic that is isomorphic to BCI-algebras. Kashima and Komori in [7] gave a Gentzen-style sequent calculus version of this logic as well as another sequent calculus which they proved to be equivalent. They used the second to prove decidability of the word problem for BCI-algebras. The decidability proof relies on cut elimination for the second system, this paper provides a fuller and simpler proof of this. Also supplied is a new decidability proof and proof finding algorithm for their second extension of BCI-logic and so for BCI-algebras.

Alternative Multilattice Logics: An Approach Based on Monosequent and Indexed Monosequent Calculi

Two new multilattice logics called submultilattice logic and indexed multilattice logic are introduced as a monosequent calculus and an indexed monosequent calculus, respectively. The submultilattice logic is regarded as a monosequent calculus version of Shramko’s original multilattice logic, which is also known as the logic of logical multilattices. The indexed multilattice logic is an extension of the submultilattice logic, and is regarded as the logic of multilattices. A completeness theorem with respect to a lattice-valued semantics is proved for the submultilattice logic, and a completeness theorem with respect to a multilattice-valued semantics is proved for the indexed multilattice logic.

Comparing the orthogonal and unitary functor calculi

The orthogonal and unitary calculi give a method to study functors from the category of real or complex inner product spaces to the category of based topological spaces. We construct functors between the calculi from the complexification-realification adjunction between real and complex inner product spaces. These allow for movement between the versions of calculi, and comparisons between the Taylor towers produced by both calculi. We show that when the inputted orthogonal functor is weakly polynomial, the Taylor tower of the functor restricted through realification and the restricted Taylor tower of the functor agree up to weak equivalence. We further lift the homotopy level comparison of the towers to a commutative diagram of Quillen functors relating the model categories for orthogonal calculus and the model categories for unitary calculus.

The effects of different versions of a gateway STEM course on student attitudes and beliefs

BackgroundSubstantial research has been conducted focusing on student outcomes in mathematics courses in order to better understand the ways in which these outcomes depend on the underlying instructional methodologies found in the courses. From 2009 to 2014, the Mathematical Association of America (MAA) studied Calculus I instruction in United States (US) colleges and universities in the Characteristics of Successful Programs of College Calculus (CSPCC). One aspect of this study attempted to understand the impact of these courses on student experience.ResultsIn this paper, we describe results from an examination of the effect of course structure on students’ attitudes and beliefs across different versions of Calculus I at a large research university in the USA. To do this, we implemented a follow-up study of the national MAA study of calculus programs in part to identify potential relationships between various course structures and changes in attitudes and beliefs during the course. We compare our results both internally across these course structures and to the national data set.ConclusionsWe find that the statistically significant changes measured in confidence and enjoyment exhibit differences across the different calculus implementations and that these changes are statistically independent of the underlying student academic backgrounds as shown by standardized test scores and high school GPA. This suggests that these observed changes in attitudes and beliefs relate to the experience in our varied course structures and not to the academic characteristics of students as they enter the course. In addition to our findings, we show how this national study can be used locally to study effects of courses on student affective traits.

A Trajectory Calculus for Qualitative Spatial Reasoning Using Answer Set Programming

AbstractSpatial information is often expressed using qualitative terms such as natural language expressions instead of coordinates; reasoning over such terms has several practical applications, such as bus routes planning. Representing and reasoning on trajectories is a specific case of qualitative spatial reasoning that focuses on moving objects and their paths. In this work, we propose two versions of a trajectory calculus based on the allowed properties over trajectories, where trajectories are defined as a sequence of non-overlapping regions of a partitioned map. More specifically, if a given trajectory is allowed to start and finish at the same region, 6 base relations are defined (TC-6). If a given trajectory should have different start and finish regions but cycles are allowed within, 10 base relations are defined (TC-10). Both versions of the calculus are implemented as ASP programs; we propose several different encodings, including a generalised program capable of encoding any qualitative calculus in ASP. All proposed encodings are experimentally evaluated using a real-world dataset. Experiment results show that the best performing implementation can scale up to an input of 250 trajectories for TC-6 and 150 trajectories for TC-10 for the problem of discovering a consistent configuration, a significant improvement compared to previous ASP implementations for similar qualitative spatial and temporal calculi.

SOLOW MODELS ON TIME SCALES

We introduce a general Solow model on time scales and derive a nonlinear first-order dynamic equation that describes such a model. We first assume that there is neither technological development nor a change in the population. We present the Cobb– Douglas production function on time scales and use it to give the solution for the equation that describes the model. Next, we provide several applications of the generalized Solow model. Finally, we generalize our work by allowing technological development and population growth. The presented results not only unify the continuous and the discrete Solow models but also extend them to other cases “in between”, e.g., a quantum calculus version of the Solow model. Finally it is also noted that our results even generalize the classical continuous and discrete Solow models since we allow the savings rate, the depreciation factor of goods, the growth rate of the population, and the technological growth rates to be functions of time rather than taking constant values as in the classical Solow models.

Harmony in a sequent setting: a reply to Tennant

In my Steinberger 2009 I argued that Neil Tennant’s Harmony requirement is untenable because of its failure to account for the standard quantifier rules.1 Instead of justifying the customary rules for the existential and universal quantifiers, Tennant’s account appears to sanction only wholly unrestricted – and so patently disharmonious – quantifier rules. In his characteristically thoughtful response Tennant 2010, Tennant offers a sequent calculus version of his Harmony requirement that rules out such pathological would-be quantifiers. While I agree with Tennant that recasting his Harmony requirement in the sequent format as he proposes blocks the said disharmonious quantifier rules, I submit that Tennant’s revamped Harmony requirement nevertheless misses the mark. I present two objections to substantiate my claim. First, I show that the crucial additional assumption underlying Tennant’s sequent calculus-based account – what I call the admissibility assumption – is excessively strong: so strong, in fact, that it renders otiose the core of Tennant’s original account. Second, I argue that the admissibility assumption, as a global requirement on deductive systems, is ill-suited for the purposes of codifying the intuitive notion of harmony. Fortunately, though, as I will demonstrate, we can dispense with the admissibility assumption altogether.

Measures from Dixmier traces and zeta functions

For L ∞ -functions on a (closed) compact Riemannian manifold, the noncommutative residue and the Dixmier trace formulation of the noncommutative integral are shown to equate to a multiple of the Lebesgue integral. The identifications are shown to continue to, and be sharp at, L 2 -functions. For functions strictly in L p , 1 ⩽ p < 2 , symmetrised noncommutative residue and Dixmier trace formulas must be introduced, for which the identification is shown to continue for the noncommutative residue. However, a failure is shown for the Dixmier trace formulation at L 1 -functions. It is shown the noncommutative residue remains finite and recovers the Lebesgue integral for any integrable function while the Dixmier trace expression can diverge. The results show that a claim in the monograph [J.M. Gracia-Bondía, J.C. Várilly, H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser Adv. Texts, Birkhäuser, Boston, 2001], that the equality on C ∞ -functions between the Lebesgue integral and an operator-theoretic expression involving a Dixmier trace (obtained from Connes' Trace Theorem) can be extended to any integrable function, is false. The results of this paper include a general presentation for finitely generated von Neumann algebras of commuting bounded operators, including a bounded Borel or L ∞ functional calculus version of C ∞ results in IV.2. δ of [A. Connes, Noncommutative Geometry, Academic Press, New York, 1994].

Partial Fractions in Calculus, Number Theory, and Algebra

In second-semester calculus, most students view partial fractions as either a miracle or a torture, sometimes both. Yet, students rarely understand why rational functions can always be written in terms of partial fractions. The rationale comes from abstract algebra, but a simple analogue in the natural numbers reveals many of the intricacies. If you have an interest in examining partial fractions beyond accepting their existence for use in a technique of integration, read on. We begin by recalling the general theorem on the existence of partial fractions in calculus, after which we investigate the corresponding version in the natural numbers. Then we move to a discussion of the calculus version: rational functions over the reals. Finally, we generalize the theorem to rational functions in one variable over an arbitrary field. (For a discussion of the general case of Euclidean domains, see Packard and Wilson [4].)

The Goodwillie Calculus of Functors

The purpose of this meeting was to introduce the participants to calculus of functors, a theory aimed at “approximating” functors in algebra and topology. The focus of the talks was on two related branches of the theory, homotopy and manifold calculus. T he organizers, Thomas Goodwillie and Randy McCarthy, scheduled 16 talks. The first was given by Goodwillie who introduced the main ideas and outlined the plan for the rest of the meeting. In talks 2–5, speakers explained the most important terminology, techniques, and results used in both versions of the theory. In talks 6–8, participants learned about manifold calculus and how it applies to spaces of knots (talk 8), while talks 9–15 dealt with homotopy calculus. In the last four of those lectures, some applications, elaborations on results established in previous talks, and different versions of certain ideas and proofs encountered so far were given. Goodwillie explored some connections between the two versions of calculus of functors (as they apply to the embedding and identity functors) and gave the concluding remarks in the last talk. It should be noted that a third version of the theory, orthogonal calculus, was not discussed due to time constraints, but Goodwillie explained some of its most salient features in an extra evening session. Other evening events took place as well, such as Rainer Vogt's elaboration of an aspect of his talk and informal research reports given by some participants.

Propositional mixed logic: its syntax and semantics

In this paper, we present a propositional logic (called mixed logic) containing disjoint copies of minimal, intuitionistic and classical logics. We prove a completeness theorem for this logic with respect to a Kripke semantics. We establish some relations between mixed logic and minimal, intuitionistic and classical logics. We present at the end a sequent calculus version for this logic.

A formalization of Sambins's normalization for GL

AbstractSambin [6] proved the normalization theorem (Hauptsatz) for GL, the modal logic of provability, in a sequent calculus version called by him GLS. His proof does not take into account the concept of reduction, commonly used in normalization proofs. Bellini [1], on the other hand, gave a normalization proof for GL using reductions. Indeed, Sambin's proof is a decision procedure which builds cut‐free proofs. In this work we formalize this procedure as a recursive function and prove its recursiveness in an arithmetically formalizable way, concluding that the normalization of GL can be formalized in PA. MSC: 03F05, 03B35, 03B45.

Regge calculus and observations. I. Formalism and applications to radial motion and circular orbits

A method is developed for tracing geodesics of particles and light rays through Regge calculus space-times. In the flat interiors of the blocks, the geodesics are straight lines, and at the boundaries between blocks they are refracted according to an “extremal distance” law. The method is then tested for the Regge calculus version of the Schwarzschild space-time. For radial motion, it gives good agreement with the analytic solution, provided the blocks are sufficiently small. Exactly circular orbits are impossible in Regge calculus, but it is shown how to construct “approximately circular” ones. These examples are in preparation for an investigation of more general orbits and more general space-times, using Regge calculus.