Mathematics Commons^™

An Investigation Of Secondary Teachers’ Knowledge Of Rate Of Change In The Context Of Teaching A Standards-Based Curriculum, Jihwa Noh

Dissertations

This study investigated teachers' mathematical content knowledge and pedagogical content knowledge with respect to rate of change in the context of teaching a Standards-based high school mathematics curriculum that emphasizes rate of change as a central theme, the Core-Plus Mathematics Project (CPMP) materials. A framework was designed to provide a comprehensive guide for analyzing different aspects of rate of change knowledge incorporating existing frameworks relative to rate of change, NCTM recommendations described in Curriculum and Evaluation Standards for School Mathematics andPrinciples and Standards for School Mathematics (NCTM, 2000), and research related to pedagogical understanding of rate of change.

Data …

Optimal Control Of Semilinear Evolution Inclusions Via Discrete Approximations, Boris S. Mordukhovich, Dong Wang

Mathematics Research Reports

This paper studies a Mayer type optimal control problem with general endpoint constraints for semilinear unbounded evolution inclusions in reflexive and separable Banach spaces. First, we construct a sequence of discrete approximations to the original optimal control problem for evolution inclusions and prove that optimal solutions to discrete approximation problems uniformly converge to a given optimal solution for the original continuous-time problem. Then, based on advanced tools of generalized differentiation, we derive necessary optimality conditions for discrete-time problems under fairly general assumptions. Combining these results with recent achievements of variational analysis in infinite-dimensional spaces, we establish new necessary optimality conditions …

Variational Stability And Marginal Functions Via Generalized Differentiation, Boris S. Mordukhovich, Nguyen Mau Nam

Mathematics Research Reports

Robust Lipschitzian properties of set-valued mappings and marginal functions play a crucial role in many aspects of variational analysis and its applications, especially for issues related to variational stability and optimizatiou. We develop an approach to variational stability based on generalized differentiation. The principal achievements of this paper include new results on coderivative calculus for set-valued mappings and singular subdifferentials of marginal functions in infinite dimensions with their extended applications to Lipschitzian stability. In this way we derive efficient conditions ensuring the preservation of Lipschitzian and related properties for set-valued mappings under various operations, with the exact bound/modulus estimates, as …

Of Parts And Wholes: Self-Similarity And Synecdoche In Science, Culture And Literature, Dirk Vanderbeke

Reconstruction: Studies in Contemporary Culture

[Editors' Introduction]

The discussion here centers on how self-similarity, or parts resembling a whole, has been a recurring aspect in the most diverse fields of human thinking from antiquity to the present. Primary examples are drawn from physics, biology, cybernetics, alchemy, philosophy, myth and, of course, language and literature. The author argues that self-similar patterns are one of the persistent ways in which the human mind structures its experience and knowledge of the world, and then examines some of the questions that arise when an almost ubiquitous concept, structure or linguistic and literary phenomenon resurfaces in a new guise in …

Reconstruction Of Partially Conductive Cracks Using Boundary Data, David Mccune, Janine Haugh

Mathematical Sciences Technical Reports (MSTR)

This paper develops an algorithm for finding one or more non-insulated, pair-wise disjoint, linear cracks in a two dimensional region using boundary measurements.

Subgradients Of Distance Functions At Out-Of-Set Points, Boris S. Mordukhovich, Nguyen Mau Nam

Mathematics Research Reports

This paper deals with the classical distance function to closed sets and its extension to the case of set-valued mappings. It has been well recognized that the distance functions play a crucial role in many aspects of variational analysis, optimization, and their applications. One of the most remarkable properties of even the classical distance function is its intrinsic nonsmoothness, which requires the usage of generalized differential constructions for its study and applications. In this paper we present new results in theser directions using mostly the generalized differential constructions introduced earlier by the first author, as well as their recent modifications. …

On The Dressing Method For The Generalised Zakharov-Shabat System, Rossen Ivanov

Articles

The dressing procedure for the Generalised Zakharov-Shabat system is well known for systems, related to sl(N) algebras. We extend the method, constructing explicitly the dressing factors for some systems, related to orthogonal and symplectic Lie algebras. We consider ’dressed’ fundamental analytical solutions with simple poles at the prescribed eigenvalue points and obtain the corresponding Lax potentials, representing the soliton solutions for some important nonlinear evolution equations.

Reconstruction Of An Unknown Boundary Portion From Cauchy Data In N-Dimensions, Kurt M. Bryan, Lester Caudill

Mathematical Sciences Technical Reports (MSTR)

We consider the inverse problem of determining the shape of some inacces­ sible portion of the boundary of a region in n dimensions from Cauchy data for the heat equation on an accessible portion of the boundary. The inverse problem is quite ill-posed, and nonlinear. We develop a Newton-like algorithm for solving the problem, with a simple and efficient means for computing the required derivatives, develop methods for regularizing the process, and provide computational examples

Determining The Length Of A One-Dimensional Bar, Natalya Yarlikina, Holly Walrath

Mathematical Sciences Technical Reports (MSTR)

In this paper we examine the inverse problem of determining the length of a one-dimensional bar from thermal measurements (temperature and heat flux) at one end of the bar (the "accessible" end); the other inaccessible end of the bar is assumed to be moving. We develop two different approaches to estimating the length of the bar, and show how one approach can also be adapted to find unknown boundary conditions at the inaccessible end of the bar.

Asymptotic Solutions Of Semilinear Stochastic Wave Equations, Pao-Liu Chow

Mathematics Research Reports

Large-time asymptotic properties of solutions to a class of semilinear stochastic wave equations with damping in a bounded domain are considered. First an energy inequality and the exponential bound for a linear stochastic equation are established. Under appropriate conditions, the existence theorem for a unique global solution is given. Next the questions of bounded solutions and the exponential stability of an equilibrium solution, in mean-square and the almost sure sense, are studied. Then, under some sufficient conditions, the existence of a unique invariant measure is proved. Two examples are presented to illustrate some applications of the theorems.

A Liouville-Gelfand Equation For K-Hessian Operators, Jon T. Jacobsen

All HMC Faculty Publications and Research

In this paper we establish existence and multiplicity results for a class of fully nonlinear elliptic equations of k-Hessian type with exponential nonlinearity. In particular, we characterize the precise dependence of the multiplicity of solutions with respect to both the space dimension and the value of k. The choice of exponential nonlinearity is motivated by the classical Liouville-Gelfand problem from combustible gas dynamics and prescribed curvature problems.

Absolutely Continuous Representations And A Kaplansky Density Theorem For Free Semigroup Algebras, Kenneth R. Davidson, Jiankui Li, David R. Pitts

Department of Mathematics: Faculty Publications

We introduce notions of absolutely continuous functionals and representations on the non-commutative disk algebra An. Absolutely continuous functionals are used to help identify the type L part of the free semigroup algebra associated to a ∗-extendible represen- tation . A ∗-extendible representation of An is regular if the absolutely continuous part coincides with the type L part. All known examples are regular. Absolutely continuous func- tionals are intimately related to maps which intertwine a given ∗-extendible representation with the left regular representation. A simple application of these ideas extends reflexivity and hyper-reflexivity results. Moreover the use of absolute continuity is …

Optimal Control Of Delay Systems With Differential And Algebraic Dynamic Constraints, Boris S. Mordukhovich, Lianwen Wang

Mathematics Research Reports

This paper concerns constrained dynamic optimization problems governed by delay control systems whose dynamic constraints are described by both delay-differential inclusions and linear algebraic equations. This is a new class of optimal control systems that, on one hand, may be treated as a specific type of variational problems for neutral functional-differential inclusions while, on the other hand, is related to a special class of differential-algebraic systems with a general delay-differential inclusion and a linear constraint link between "slow" and "fast" variables. We pursue a two-hold goal: to study variational stability for this class of control systems with respect to discrete …

Optimal Boundary Control Of Hyperbolic Equations With Pointwise State Constraints, Boris S. Mordukhovich, Jean-Pierre Raymond

Mathematics Research Reports

In this paper we consider dynamic optimization problems for hyperbolic systems with boundary controls and pointwise state constraints. In contrast to parabolic dynamics, such systems have not been sufficiently studied in the literature. The reason is the lack of regularity in the case of hyperbolic dynamics. We present necessary optimality conditions for both Neumann and Dirichlet boundary control problems and discuss differences and relationships between them.

Interacting Near-Solutions Of A Hamiltonian System, Gregory S. Spradlin

Publications

A Hamiltonian system with a superquadratic potential is examined. The system is asymptotic to an autonomous system. The difference between the Hamiltonian system and the “problem at infinity,” the autonomous system, may be large, but decays exponientially. The existence of a nontrivial solution homoclinic to zero is proven. Many results of this type rely on a monotonicity condition on the nonlinearity, not assumed here, which makes the problem resemble in some sense the special case of homogeneous (power) nonlinearity. The proof employs variational, minimax arguments. In some similar results requiring the monotonicity condition, solutions inhabit a manifold homeomorphic to the …

Computing Isotypic Projections With The Lanczos Iteration, David K. Maslen, Michael E. Orrison, Daniel N. Rockmore

Dartmouth Scholarship

When the isotypic subspaces of a representation are viewed as the eigenspaces of a symmetric linear transformation, isotypic projections may be achieved as eigenspace projections and computed using the Lanczos iteration. In this paper, we show how this approach gives rise to an efficient isotypic projection method for permutation representations of distance transitive graphs and the symmetric group.

Finite Horizon Riemann Structures And Ergodicity, Victor J. Donnay, Charles Pugh

Mathematics Faculty Research and Scholarship

In this paper we show that any surface in R-3 can be modified by gluing on small 'focusing caps' so that its geodesic flow becomes ergodic. A new concept, finite horizon cap geometry, is what makes the construction work.

A Computational Model For Martensitic Thin Films With Compositional Fluctuation, Pavel Bělík, Mitchell Luskin

Faculty Authored Articles

We develop a computational model for the martensitic first-order structural phase transformation in a single crystal thin film, and we use this model to study the effect of spatial compositional fluctuation, spatial temporal noise, and the loss of stability of the metastable phase at temperatures sufficiently far from the transformation temperature.

Examples Of Cayley 4-Manifolds, Weiqing Gu, Christopher Pries '03

All HMC Faculty Publications and Research

We determine several families of so-called Cayley 4-dimensional manifolds in the real Euclidean 8-space. Such manifolds are of interest because Cayley 4-manifolds are supersymmetric cycles that are candidates for representations of fundamental particles in String Theory. Moreover, some of the examples of Cayley manifolds discovered in this paper may be modified to construct explicit examples in our current search for new holomorphic invariants for Calabi-Yau 4-folds and for the further development of mirror symmetry.

We apply the classic results of Harvey and Lawson to find Cayley manifolds which are graphs of functions from the set of quaternions to itself. We …

Discrete-Time Approximations Of Stochastic Delay Equations: The Milstein Scheme, Yaozhong Hu, Salah-Eldin A. Mohammed, Feng Yan

Articles and Preprints

In this paper, we develop a strong Milstein approximation scheme for solving stochastic delay differential equations (SDDE's). The scheme has convergence order 1. In order to establish the scheme, we prove an infinite-dimensional Itô formula for "tame" functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus and the anticipating stochastic analysis of Nualart and Pardoux. Given the non-anticipating nature of the SDDE, the use of anticipating calculus methods appears to be novel.

Semilinear Equations With Discrete Spectrum, Alfonso Castro

All HMC Faculty Publications and Research

This is an overview of the solvability of semilinear equations where the linear part has discrete spectrum. Semilinear elliptic and hyperbolic equations, as well as Hammerstein integral equations, are used as motivating examples. The presentation is intended to be accessible to non experts.

An Existence Result For A Class Of Sublinear Semipositone Systems, Alfonso Castro, C. Maya, Ratnasingham Shivaji

All HMC Faculty Publications and Research

We consider the existence of positive solutions for the system

-Δu_i = λ[f_i(u₁,u₂,...,u_m) - h_i]; Ω

u_i = 0; ∂Ω

where λ > 0 is a parameter, Δ is the Laplacian operator, Ω is a bounded domain in Rⁿ; n ≥ 1 with a smooth boundary ∂Ω, f_i are C¹ functions satisfying f₁(0,0,...,0) = 0, lim z→∞ f_i(z,z,...,z) = ∞ and lim z→∞ f_i(z,z,...,z)/z = 0, and h_i are nonnegative continuous functions in Ω for i = 1,2,...,m. …

Integral Transforms, Convolution Products, And First Variations, Bong Jin Kim, Byoung Soo Kim, David Skough

Department of Mathematics: Faculty Publications

We establish the various relationships that exist among the integral transform F_α,βF, the convolution product (F∗G)α, and the first variation δF for a class of functionals defined on K[0,T], the space of complex-valued continuous functions on [0,T] which vanish at zero.

Discrete Approximations And Necessary Optimality Conditions For Functional-Differential Inclusions Of Neutral Type, Boris S. Mordukhovich, Lianwen Wang

Mathematics Research Reports

This paper deals with necessary optimality conditions for optimal control systems governed by constrained functional-differential inclusions of neutral type. While some results are available for smooth control systems governed by neutral functional-differential equations, we are not familiar with any results for neutral functional-differential inclusions, even with smooth cost functionals in the absence of endpoint constraints. Developing the method of discrete approximations and employing advanced tools of generalized differentiation, we conduct a variational analysis of neutral functional-differential inclusions and obtain new necessary optimality conditions of both Euler-Lagrange and Hamiltonian types.

Quasioptimality Of Some Spectral Mixed Methods, Jay Gopalakrishnan, Leszek Demkowicz

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we construct a sequence of projectors into certain polynomial spaces satisfying a commuting diagram property with norm bounds independent of the polynomial degree. Using the projectors, we obtain quasioptimality of some spectralmixed methods, including the Raviart–Thomas method and mixed formulations of Maxwell equations. We also prove some discrete Friedrichs type inequalities involving curl.

Analysis Of A Multigrid Algorithm For Time Harmonic Maxwell Equations, Jay Gopalakrishnan, Joseph E. Pasciak, Leszek Demkowicz

Mathematics and Statistics Faculty Publications and Presentations

This paper considers a multigrid algorithm suitable for efficient solution of indefinite linear systems arising from finite element discretization of time harmonic Maxwell equations. In particular, a "backslash" multigrid cycle is proven to converge at rates independent of refinement level if certain indefinite block smoothers are used. The method of analysis involves comparing the multigrid error reduction operator with that of a related positive definite multigrid operator. This idea has previously been used in multigrid analysis of indefinite second order elliptic problems. However, the Maxwell application involves a nonelliptic indefinite operator. With the help of a few new estimates, the …

A Characterization Of Hybridized Mixed Methods For Second Order Elliptic Problems, Bernardo Cockburn, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we give a new characterization of the approximate solution given by hybridized mixed methods for second order self-adjoint elliptic problems. We apply this characterization to obtain an explicit formula for the entries of the matrix equation for the Lagrange multiplier unknowns resulting from hybridization. We also obtain necessary and sufficient conditions under which the multipliers of the Raviart–Thomas and the Brezzi–Douglas–Marini methods of similar order are identical.

On The Empirical Balanced Truncation For Nonlinear Systems, Marissa Condon, Rossen Ivanov

Articles

Novel constructions of empirical controllability and observability gramians for nonlinear systems for subsequent use in a balanced truncation style of model reduction are proposed. The new gramians are based on a generalisation of the fundamental solution for a Linear Time-Varying system. Relationships between the given gramians for nonlinear systems and the standard gramians for both Linear Time-Invariant and Linear Time-Varying systems are established as well as relationships to prior constructions proposed for empirical gramians. Application of the new gramians is illustrated through a sample test-system.

Uniqueness Theorems In Bioluminescence Tomography, Ge Wang, Yi Li, Ming Jiang

Mathematics and Statistics Faculty Publications

Motivated by bioluminescent imaging needs for studies on gene therapy and other applications in the mouse models, a bioluminescence tomography (BLT) system is being developed in the University of Iowa. While the forward imaging model is described by the well-known diffusion equation, the inverse problem is to recover an internal bioluminescent source distribution subject to Cauchy data. Our primary goal in this paper is to establish the solution uniqueness for BLT under practical constraints despite the ill-posedness of the inverse problem in the general case. After a review on the inverse source literature, we demonstrate that in the general case …

The Dual Spectral Set Conjecture, Steen Pedersen

Mathematics and Statistics Faculty Publications

Suppose that Λ = (aZ + b) ∪ (cZ + d) where a, b, c, d are real numbers such that a ≠ 0 and c ≠ 0. The union is not assumed to be disjoint. It is shown that the translates Ω + λ, λ is an element of Λ, tile the real line for some bounded measurable set Ω if and only if the exponentials e_λ(x) = e^i2πλx, λ is an element of Λ, form an orthogonal basis for some bounded measurable set Ω'.