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We develop a nonanticipative calculus for functionals of a continuous semimartingale, using an extension of the Itô formula to path-dependent functionals which possess certain directional derivatives. The construction is based on a pathwise derivative, introduced by Dupire, for functionals on the space of right-continuous functions with left limits. We show that this functional derivative admits a suitable extension to the space of square-integrable martingales. This extension defines a weak derivative which is shown to be the inverse of the Itô integral and which may be viewed as a nonanticipative \"lifting\" of the Malliavin derivative. These results lead to a constructive martingale representation formula for Itô processes. By contrast with the Clark—Haussmann—Ocone formula, this representation only involves nonanticipative quantities which may be computed pathwise.

We generalize the notions of the fractional g-integral and g-derivative by introducing variable lower limit of integration. We discuss some properties and their relations. Finally, we give a q-Taylor-like formula which includes fractional q-derivatives of the function.

We study certain spectral properties and the invariant subspaces for some classes of integration operators of Volterra type on the Fock space.

In this paper we establish global Lp(ℝn)-regularity properties of Fourier integral operators. The orders of decay of the amplitude are determined for operators to be bounded on Lp(ℝn), 1 < p < ∞, as well as to be bounded from Hardy space H¹(ℝn) to L¹(ℝn). This extends local Lp-regularity properties of Fourier integral operators, as well as results of global L²(ℝn) boundedness, to the global setting of Lp(ℝn). Global boundedness in weighted Sobolev spaces ${\\mathrm{W}}_{\\mathrm{s}}^{\\mathrm{\\sigma },\\mathrm{p}}\\left({\\mathrm{\\mathbb{R}}}^{\\mathrm{n}}\\right)$ is also established, and applications to hyperbolic partial differential equations are given.

We show that a general class of active scalar equations, including porous media and certain magnetostrophic turbulence models, admits non-unique weak solutions in the class of bounded functions. The proof is based upon the method of convex integration recently implemented for equations of fluid dynamics.

Despite the promise of technology in education, many practicing teachers face several challenges when trying to effectively integrate technology into their classroom instruction. Additionally, while national statistics cite a remarkable improvement in access to computer technology tools in schools, teacher surveys show consistent declines in the use and integration of computer technology to enhance student learning. This article reports on primary technology integration barriers that mathematics teachers identified when using technology in their classrooms. Suggestions to overcome some of these barriers are also provided.

We study multivariate integration for a weighted Korobov space of periodic infinitely many times differentiable functions for which the Fourier coefficients decay exponentially fast. The weights are defined in terms of two non-decreasing sequences a={ai}\\mathbf {a}=\\{a_i\\} and b={bi}\\mathbf {b}=\\{b_i\\} of numbers no less than one and a parameter ω∈(0,1)\\omega \\in (0,1). Let e(n,s)e(n,s) be the minimal worst-case error of all algorithms that use nn function values in the ss-variate case. We would like to check conditions on a\\mathbf {a}, b\\mathbf {b} and ω\\omega such that e(n,s)e(n,s) decays exponentially fast, i.e., for some q∈(0,1)q\\in (0,1) and p>0p>0 we have e(n,s)=O(qnp)e(n,s)=\\mathcal {O}(q^{\\,n^{\\,p}}) as nn goes to infinity. The factor in the O\\mathcal {O} notation may depend on ss in an arbitrary way. We prove that exponential convergence holds iff B:=∑i=1∞1/bi>∞B:=\\sum _{i=1}^\\infty 1/b_i>\\infty independently of a\\mathbf {a} and ω\\omega. Furthermore, the largest pp of exponential convergence is 1/B1/B. We also study exponential convergence with weak, polynomial and strong polynomial tractability. This means that e(n,s)≤C(s)qnpe(n,s)\\le C(s)\\,q^{\\,n^{\\,p}} for all nn and ss and with logC(s)=exp⁡(o(s))\\log \\,C(s)=\\exp (o(s)) for weak tractability, with a polynomial bound on logC(s)\\log \\,C(s) for polynomial tractability, and with uniformly bounded C(s)C(s) for strong polynomial tractability. We prove that the notions of weak, polynomial and strong polynomial tractability are equivalent, and hold iff B>∞B>\\infty and aia_i are exponentially growing with ii. We also prove that the largest (or the supremum of) pp for exponential convergence with strong polynomial tractability belongs to [1/(2B),1/B][1/(2B),1/B].

In this paper, we consider the infinite-dimensional integration problem on weighted reproducing kernel Hilbert spaces with norms induced by an underlying function space decomposition of analysis of variance type. The weights model the relative importance of different groups of variables. We present new randomized multilevel algorithms to tackle this integration problem and prove upper bounds for their randomized error. Furthermore, we provide in this setting the first nontrivial lower error bounds for general randomized algorithms, which, in particular, may be adaptive or nonlinear. These lower bounds show that our multilevel algorithms are optimal. Our analysis refines and extends the analysis provided in [F. J. Hickernell, T. Müller-Gronbach, B. Niu, and K. Ritter, J. Complexity, 26 (2010), pp. 229–254], and our error bounds improve substantially on the error bounds presented there. As an illustrative example, we discuss the unanchored Sobolev space and employ randomized quasi–Monte Carlo multilevel algorithms based on scrambled polynomial lattice rules.

This article reviews research on the achievement outcomes of three types of approaches to improving elementary mathematics: mathematics curricula, computer-assisted instruction (CAI), and instructional process programs. Study inclusion requirements included use of a randomized or matched control group, a study duration of at least 12 weeks, and achievement measures not inherent to the experimental treatment. Eighty-seven studies met these criteria, of which 36 used random assignment to treatments. There was limited evidence supporting differential effects of various mathematics text-books. Effects of CAI were moderate. The strongest positive effects were found for instructional process approaches such as forms of cooperative learning, classroom management and motivation programs, and supplemental tutoring programs. The review concludes that programs designed to change daily teaching practices appear to have more promise than those that deal primarily with curriculum or technology alone.

In this paper we study multivariate integration for a weighted Korobov space for which the Fourier coefficients of the functions decay exponentially fast. This implies that the functions of this space are infinitely times differentiable. Weights of the Korobov space monitor the influence of each variable and each group of variables. We show that there are numerical integration rules which achieve an exponential convergence of the worst-case integration error. We also investigate the dependence of the worst-case error on the number of variables ss, and show various tractability results under certain conditions on weights of the Korobov space. Tractability means that the dependence on ss is never exponential, and sometimes the dependence on ss is polynomial or there is no dependence on ss at all.