Testing sets for properness of polynomial mappings

In this paper we characterize testing sets for properness of polynomial mappings . We also study the set of points at which such mappings are not proper. As the first application we give a proof of the Complementary Conjecture of McKay and Wang (Conjecture 12 in [16]). The second application is an answer to the problem of Kraft- Russell about a characterization of The third application is (given in [13]) a solution of the Problem of Van den Essen and Shpilrain about endomorphism of the ring The fourth application is a theorem about extensions of affine varieties. Received: 27 February 1998 / in revised form: 10 January 1999     

Brouwer's degree without properness

A degree deg(f,y) is defined for every continuous function , which possesses all the properties of Brouwer's degree provided that y is restricted to the complement of some closed set A(f) of “asymptotic” values. Sufficient conditions are given for A(f) to be nowhere dense. It is also shown that, in the opposite direction, A(f) having nonempty interior has a direct impact on the solutions of f(x)=y, which cannot be discovered by degree arguments.     

A residue calculus approach to the uniform Artin-Rees lemma

The uniform Artin-Rees lemma has been proved by C. Huneke using algebraic methods. We give a new proof for this result in the analytic setting, using residue calculus and a method involving complexes of Hermitian vector bundles. We also have to introduce a type of product of complexes of vector bundles, which may be applicable in the solution of other division problems with respect to product ideals.     

Malliavin calculus for Markov chains using perturbations of time

In this article, we develop a Malliavin calculus associated to a time-continuous Markov chain with finite state space. We apply it to get a criterion of density for solutions of stochastic differential equation involving the Markov chain and also to compute greeks.     

Formulating dynamic strategies using decision calculus

This paper presents an applied methodology to assist managers in strategically setting prices and allocating resources over the product, brand, or adoption (diffusion) life cycle. While substantial theoretical work has been achieved in this area in the management science and operations research disciplines, approaches which can be implemented as managerial tools are generally lacking. Our methodology, which has been applied as a PC-based decision support tool in the telecommunications industry, marries the dynamic optimization literature with decision calculus approaches offering sufficient flexibility for applied contexts. We discuss the approach and describe its use to derive optimal price and advertising policies. Strategies not commonly suggested using traditional formulations are also presented.     

The index calculus method using non-smooth polynomials

We study a generalized version of the index calculus method for the discrete logarithm problem in , when , is a small prime and . The database consists of the logarithms of all irreducible polynomials of degree between given bounds; the original version of the algorithm uses lower bound equal to one. We show theoretically that the algorithm has the same asymptotic running time as the original version. The analysis shows that the best upper limit for the interval coincides with the one for the original version. The lower limit for the interval remains a free variable of the process. We provide experimental results that indicate practical values for that bound. We also give heuristic arguments for the running time of the Waterloo variant and of the Coppersmith method with our generalized database.

     

Bunny hops: using multiplicities of zeroes in calculus for graphing

Students learn a lot of material in each mathematics course they take. However, they are not always able to make meaningful connections between content in successive mathematics courses. This paper reports on a technique to address a common topic in calculus I courses (intervals of increase/decrease and concave up/down) while also making use of students’ pre-existing knowledge about the behaviour of functions around zeroes based on multiplicities.     

Two crucial examples related to properness of weakly relaxed delayed controls

In this paper, we deal with the problem of finding a proper relaxation procedure (in the sense that relaxed minimizers can be approximated with ordinary controls) for optimal control problems involving delays in the control variables. For systems involving commensurate delays it is well known that a ‘strong’ model provides a proper relaxation procedure. For the noncommensurate case, however, the question of how to properly relax delayed controls has remained open and a natural candidate has been a modified version of the strong model, which we call the ‘weak’ procedure, first introduced by Warga. In this paper, we show by means of two examples that this procedure may fail to be proper for systems with either commensurate or noncommensurate delays. It is expected that these examples will provide some insight into the problem of finding a proper relaxation procedure applicable to the general case.     

Malliavin calculus in abstract Wiener space using infinitesimals

Using infinitesimals, we develop Malliavin calculus on spaces which result from the classical Wiener space by replacing with any abstract Wiener space .We start from a Brownian motion b on a Loeb probability space Ω with values in the Banach space is the standard part of a ∗finite-dimensional Brownian motion B. Then we define iterated Itô integrals as standard parts of internal iterated Itô integrals. The integrator of the internal integrals is B and the values of the integrands are multilinear forms on , where is a ∗finite-dimensional linear space over between the Hilbert space and its ∗-extension .In the first part we prove a chaos decomposition theorem for L²-functionals on Ω that are measurable with respect to the σ-algebra generated by b. This result yields a chaos decomposition of L²-functionals with respect to the Wiener measure on the standard space of -valued continuous functions on [0,1]. In the second part we define the Malliavin derivative and the Skorohod integral as standard parts of internal operators defined on ∗finite-dimensional spaces. In an application we use the transformation rule for finite-dimensional Euclidean spaces to study time anticipating and non-anticipating shifts of Brownian motion by Bochner integrals (Girsanov transformations).     

On using cubic spline for the solution of problems in calculus of variations

Two different approaches based on cubic B-spline are developed to approximate the solution of problems in calculus of variations. Both direct and indirect methods will be described. It is known that, when using cubic spline for interpolating a function g∈C⁴[a,b] on a uniform partition with the step size h, the optimal order of convergence derived is O(h⁴). In Zarebnia and Birjandi (J. Appl. Math. 1–10, 2012) a non-optimal O(h²) method based on cubic B-spline has been used to solve the problems in calculus of variations. In this paper at first we will obtain an optimal O(h⁴) indirect method using cubic B-spline to approximate the solution. The convergence analysis will be discussed in details. Also a locally superconvergent O(h⁶) indirect approximation will be describe. Finally the direct method based on cubic spline will be developed. Some test problems are given to demonstrate the efficiency and applicability of the numerical methods.     

Experience with using counterexamples in an introductory calculus class

The paper describes a personal experience with using counterexamples as a pedagogical strategy in the teaching and learning of an introductory calculus course at a university of technology.     

Hedging lookback and partial lookback options using Malliavin calculus

The paper considers a Black and Scholes economy with constant coefficients. A contingent claim is said to be simple if the payoff at maturity is a function of the value of the underlying security at maturity. To replicate a simple contingent claim one uses so called delta-hedging, and the well-known strategy is derived from Itô calculus and the theory of partial differentiable equations. However, hedging path-dependent options require other tools since the price processes, in general, no longer have smooth stochastic differentials. It is shown how Malliavin calculus can be used to derive the hedging strategy for any kind of path-dependent options, and in particular for lookback and partial lookback options.     

Normal convergence using Malliavin calculus with applications and examples

We prove the chain rule in the more general framework of the Wiener–Poisson space, allowing us to obtain the so-called Nourdin–Peccati bound. From this bound, we obtain a second-order Poincaré-type inequality that is useful in terms of computations. For completeness we survey these results on the Wiener space, the Poisson space, and the Wiener–Poisson space. We also give several applications to central limit theorems with relevant examples: linear functionals of Gaussian subordinated fields (where the subordinated field can be processes like fractional Brownian motion or the solution of the Ornstein–Uhlenbeck SDE driven by fractional Brownian motion), Poisson functionals in the first Poisson chaos restricted to infinitely many “small” jumps (particularly fractional Lévy processes), and the product of two Ornstein–Uhlenbeck processes (one in the Wiener space and the other in the Poisson space). We also obtain bounds for their rate of convergence to normality.     

Subdifferential calculus for convex operators

Well-known scalar results on the Subdifferential of composite functions are extended to the framework of ordered topological vector spaces. This is done by using the sandwich theorem for convex operators which is derived from an extension of the Hahn-Banach theorem.     

Individualized additional instruction for calculus

College students enrolling in the calculus sequence have a wide variance in their preparation and abilities, yet they are usually taught from the same lecture. We describe another pedagogical model of Individualized Additional Instruction (IAI) that assesses each student frequently and prescribes further instruction and homework based on the student's performance. Our study compares two calculus classes, one taught with mandatory remedial IAI and the other without. The class with mandatory remedial IAI did significantly better on comprehensive multiple-choice exams, participated more frequently in classroom discussion and showed greater interest in theorem-proving and other advanced topics.     

Residue calculus for c-holomorphic functions

In this paper we introduce Coleff–Herrera residue currents defined by systems of c-holomorphic functions and prove a Lelong–Poincaré and a Cauchy-type formula as well as the transformation law for these currents.