Retention of differential and integral calculus: a case study of a university student in physical chemistry

This paper reports a study on retention of differential and integral calculus concepts of a second-year student of physical chemistry at a Danish university. The focus was on what knowledge the student retained 14 months after the course and on what effect beliefs about mathematics had on the retention. We argue that if a student can quickly reconstruct the knowledge, given a few hints, this is just as good as retention. The study was conducted using a mixed method approach investigating students’ knowledge in three worlds of mathematics. The results showed that the student had a very low retention of concepts, even after hints. However, after completing the calculus course, the student had successfully used calculus in a physical chemistry study programme. Hence, using calculus in new contexts does not in itself strengthen the original calculus learnt; they appeared as disjoint bodies of knowledge.     

Positivity Theorem for a Stochastic Delay Equation on a Manifold

We consider a nondegenerated stochastic delay manifold on a compact manifold. We show that we can apply Malliavin calculus in order to show that its law has a strictly positive density.     

Tensor calculus: unlearning vector calculus

Tensor calculus is critical in the study of the vector calculus of the surface of a body. Indeed, tensor calculus is a natural step-up for vector calculus. This paper presents some pitfalls of a traditional course in vector calculus in transitioning to tensor calculus. We show how a deeper emphasis on traditional topics such as the Jacobian can serve as a bridge for vector calculus into tensor calculus.     

On some functional calculus of closed operators in a Banach space

We develop a functional calculus of closed operators in a Banach space based on the class of functions in the form 1/g, where g belongs to the class R[a, b] introduced by M. G. Krein. We prove continuity, stability, uniqueness, spectral mapping, and inverse operator theorems and describe some other properties of the considered calculus.     

Liouville在分数阶微积分概念方面的研究

分数阶微积分的概念是以整数阶微积分理论研究为基础,而分数阶微积分概念的建立经历了漫长的过程.探析此过程中数学家在研究分数阶微积分理论方面的贡献,进而整理Liouville在分数阶微积分概念方面的研究,进一步概括分数阶微积分第一定义的由来以及为后续相关研究奠定的坚实基础.     

Cohomology of Sheaves of Frechet Algebras and Spectral Theory

We propose a holomorphic functional calculus for a noncommutative operator family generating a supernilpotent Lie subalgebra. This calculus extends Taylor's holomorphic functional calculus.     

Nearest interval,triangular and trapezoidal approximation of a fuzzy number preserving ambiguity

The ambiguity was introduced to simplify the task of representing and handling of fuzzy numbers. We find the nearest real interval, nearest triangular (symmetric) fuzzy number, nearest trapezoidal (symmetric) fuzzy number of a fuzzy number, with respect to average Euclidean distance, preserving the ambiguity. A simpler and elementary method, to avoid the Karush–Kuhn–Tucker theorem and the laborious calculus associated with it and to prove the continuity is used. We give algorithms for calculus and several examples. The approximations are discussed in relation to data aggregation.     

On the existence in a problem of the calculus of variations without convexity assumptions

Summary We obtain new sufficient conditions for the existence in a problem of the calculus of variations without convexity assumptions.     

一类非线性微分方程的算符解法

利用Mikusinski算符演算的方法,对一类非线性微分方程进行算符解法,给出这一类方法的级数形式解.     

Calculus of ordered operators on a factor algebra

We consider an algebra of operators in a Banach scale and its factor algebra modulo some power of a small parameter. On the factor algebra we construct an induced calculus of functions of ordered operators.     

PSPACE complexity of modal logic <Emphasis Type="Italic">K D</Emphasis>45<Subscript>n</Subscript>

We study sequent calculus for multi-modal logic K D45_n and its complexity. We introduce a loop-check free sequent calculus. Loop-check is eliminated by using the marked modal operator □_i^•, which is used as an alternative to sequents with histories ([8], [3], [5]). All inference rules are invertible or semi-invertible. To get this, we use or branches beside common and branches. We prove the equivalence between known sequent calculus and our newly introduced efficient sequent calculus. We concentrate on the complexity analysis of the introduced sequent calculus for multi-modal logic K D45_n. We prove that the space complexity of the given calculus is polynomial (O(l ³)). We show the maximum height of the constructed derivation tree that leads to the reduction of the time and space complexity. We present a decision algorithm for multi-modal logic K D45_n and some nontrivial examples to improve the introduced loop-check free sequent calculus.     

The obrechkoff integral transform: properties and relation to a generalized fractional calculus

This survey is devoted to one of the most general Laplace-type integral transforms, the so-called Obrechkoff integral transform, introduced and studied for the first time by Obrechkoff[25]. It has been modified by Dimovski [5],[6] and used as a basis of a Mikusinski-type operational calculus for the hyper-Bessel differential operators of arbitrary order. Later, in a series of papers Dimovski and Kiryakova [8],[9],[10] have found operational properties, complex and real inversion formulas, Abel-type theorems for the Obrechkoff transform. This theory has been further developed by Kiryakova [16],[17],[18] using the tools of the Meijer's G-functions and of the fractional calculus. Namely, a new definition as a G-transform has been given for the Obrechkoff transform. The hyper-Bessel operators themselves, have given rise to a new generalized fractional calculus and further extensive use of the G-functions. Many other generalized differentiation and integration operators happen to be special cases in this calculus, too. Special cases of the Obrechkoff transform have been "rediscovered" later by many authors. We give examples how their results could be derived from the general ones surveyed here.     

A coding method for a sequent calculus of propositional logic

The paper deals with a coding method for a sequent calculus of the propositional logic. The method is based on the sequent calculus. It allows us to determine if a formula is derivable in the calculus without constructing a derivation tree. The main advantage of the coding method is its compactness in comparison with derivation trees of the sequent calculus. The coding method can be used as a decision procedure for the propositional logic.     

Instantaneous rate of change: a numerical approach

The calculus reform movement has encouraged numerical and graphical approaches to functions in addition to the more traditional analytical approach. While valiant efforts have been made to use these other approaches in newer calculus curricula, more numerical approaches should be introduced. Research on student learning in calculus indicates that particular numerical approaches hold promise for students' learning of instantaneous rate of change. Numerical approaches involving the average rate of change over successively smaller intervals can be used to obtain the instantaneous rate of change for a given function at a given value of x. These approaches can help students appreciate the fundamental relationship between average and instantaneous rates of change. They can also be used to obtain general expressions for the derivative of most elementary functions. Standard computer spreadsheet programs facilitate this process and make numerical approaches a more viable option for calculus instruction. These are underutilized resources for instruction in calculus, even in reform or other new calculus curricula.     

Lambek calculus with a unit and one division

In this paper we present a substitution that reduces the derivability in the Lambek calculus with a unit and one division to the derivability in the Lambek calculus with one division permitting empty antecedents. Using this substitution, we establish the existence of an algorithm checking the derivability in the Lambek calculus with a unit and one division in polynomial time.     

An algebra generated by multiplicative discrete convolution operators

We consider a Banach algebra generated by multiplicative discrete convolution operators. We construct a symbolic calculus for this algebra and in terms of this calculus we describe criteria for the Noetherian property of operators and obtain a formula for their index.     

A Functional Calculus for n-Tuples of Noncommuting Operators

We employ the notion of slice monogenic functions to define a new functional calculus for an n-tuple of not necessarily commuting operators. This calculus is consistent with the Riesz-Dunford calculus for a single operator. Received: October, 2007. Accepted: February, 2008.     

Non-differentiable variational principles

We develop a calculus of variations for functionals which are defined on a set of non-differentiable curves. We first extend the classical differential calculus in a quantum calculus, which allows us to define a complex operator, called the scale derivative, which is the non-differentiable analogue of the classical derivative. We then define the notion of extremals for our functionals and obtain a characterization in term of a generalized Euler-Lagrange equation. We finally prove that solutions of the Schrödinger equation can be obtained as extremals of a non-differentiable variational principle, leading to an extended Hamilton's principle of least action for quantum mechanics. We compare this approach with the scale relativity theory of Nottale, which assumes a fractal structure of space-time.     

Bicomplex holomorphic functional calculus

In this paper we introduce and study a functional calculus for bicomplex linear bounded operators. The study is based on the decomposition of bicomplex numbers and of linear operators using the two nonreal idempotents. We show that, due to the presence of zero divisors in the bicomplex numbers, the spectrum of a bounded operator is unbounded. We therefore introduce a different spectrum (called reduced spectrum) which is bounded and turns out to be the right tool to construct the bicomplex holomorphic functional calculus. Finally we provide some properties of the calculus.     

The impact of taking a college pre-calculus course on students’ college calculus performance

Poor performance on placement exams keeps many US students who pursue a STEM (science, technology, engineering, mathematics) career from enrolling directly in college calculus. Instead, they must take a pre-calculus course that aims to better prepare them for later calculus coursework. In the USA, enrollment in pre-calculus courses in two- and four-year colleges continues to grow, and these courses are well-populated with students who already took pre-calculus in high school. We examine student performance in college calculus, using regression discontinuity to estimate the effects of taking college pre-calculus or not, in a national US sample of 5507 students at 132 institutions. We find that students who take college pre-calculus do not earn higher calculus grades.