An Overview Of The Geometry And Combinatorics Of The Macdonald Polynomial And Q-T Catalan Number, 2010 University of Pennsylvania

#### An Overview Of The Geometry And Combinatorics Of The Macdonald Polynomial And Q-T Catalan Number, Tian Liang

*Publicly Accessible Penn Dissertations*

We will explore the combinatorial and geometric properties related to the Macdonald polynomials and the diagonal harmonics. We have the combinatorial Macdonald polynomial formula that ts the dening conditions directly. The shue conjecture gives an elegant expression of the Frobenius series of the diagonal harmonics. While the geometric properties of the Hilbert scheme and schemes over it provides explanations from a dierent perspective. We use examples to show that these two approaches arrive at the same goal.

Factorizations In The Irreducible Characters Of Compact Semisimple Lie Groups, 2010 University of Pennsylvania

#### Factorizations In The Irreducible Characters Of Compact Semisimple Lie Groups, Andrew Rupinski

*Publicly Accessible Penn Dissertations*

Our goal is to describe factorizations of the characters of irreducible representations of compact semisimple Lie groups. It is well-known that for a given Lie group G of rank n, the Virtual Representation Ring R(G) with the operations of tensor product, direct sum, and direct difference is isomorphic to a polynomial ring with integer coefficients and number of generators equal to n. As such, R(G) is a Unique Factorization Domain and thus, viewing a given representation of G as an element of this ring, it makes sense to ask questions about how a representation factors. Using various approaches ...

An Iterated Pseudospectral Method For Functional Partial Differential Equations, 2010 Boise State University

#### An Iterated Pseudospectral Method For Functional Partial Differential Equations, J. Mead, B. Zubik-Kowal

*Barbara Zubik-Kowal*

Chebyshev pseudospectral spatial discretization preconditioned by the Kosloff and Tal-Ezer transformation [10] is applied to hyperbolic and parabolic functional equations. A Jacobi waveform relaxation method is then applied to the resulting semi-discrete functional systems, and the result is a simple system of ordinary differential equations d/dtUk+1(t) = MαUk+1(t)+f(t,U kt). Here Mα is a diagonal matrix, k is the index of waveform relaxation iterations, U kt is a functional argument computed from the previous iterate and the function f, like the matrix Mα, depends on the process of semi-discretization. This waveform relaxation splitting has ...

Noncommutative Topology And The World’S Simplest Index Theorem, 2010 Dartmouth College

#### Noncommutative Topology And The World’S Simplest Index Theorem, Erik Van Erp

*Open Dartmouth: Faculty Open Access Articles*

In this article we outline an approach to index theory on the basis of methods of noncommutative topology. We start with an explicit index theorem for second-order differential operators on 3-manifolds that are Fredholm but not elliptic. This low-brow index formula is expressed in terms of winding numbers. We then proceed to show how it is derived as a special case of an index theorem for hypoelliptic operators on contact manifolds. Finally, we discuss the noncommutative topology that is employed in the proof of this theorem. The article is intended to illustrate that noncommutative topology can be a powerful tool ...

Scalable Probabilistic Databases With Factor Graphs And Mcmc, 2010 University of Massachusetts - Amherst

#### Scalable Probabilistic Databases With Factor Graphs And Mcmc, Michael Wick, Andrew Mccallum, Gerome Miklau

*Andrew McCallum*

Probabilistic databases play a crucial role in the management and understanding of uncertain data. However, incorporating probabilities into the semantics of incomplete databases has posed many challenges, forcing systems to sacrifice modeling power, scalability, or restrict the class of relational algebra formula under which they are closed. We propose an alternative approach where the underlying relational database always represents a single world, and an external factor graph encodes a distribution over possible worlds; Markov chain Monte Carlo (MCMC) inference is then used to recover this uncertainty to a desired level of fidelity. Our approach allows the efficient evaluation of arbitrary ...

Explicit And Implicit Methods In Solving Differential Equations, 2010 University of Connecticut - Storrs

#### Explicit And Implicit Methods In Solving Differential Equations, Timothy Bui

*Honors Scholar Theses*

Differential equations are equations that involve an unknown function and derivatives. Euler's method are efficient methods to yield fairly accurate approximations of the actual solutions. By manipulating such methods, one can find ways to provide good approximations compared to the exact solution of parabolic partial differential equations and nonlinear parabolic differential equations.

A Predictive Model For Secondary Rna Structure Using Graph Theory And A Neural Network., 2010 East Tennessee State University

#### A Predictive Model For Secondary Rna Structure Using Graph Theory And A Neural Network., Denise Renee Koessler

*Electronic Theses and Dissertations*

In this work we use a graph-theoretic representation of secondary RNA structure found in the database RAG: RNA-As-Graphs. We model the bonding of two RNA secondary structures to form a larger structure with a graph operation called merge. The resulting data from each tree merge operation is summarized and represented by a vector. We use these vectors as input values for a neural network and train the network to recognize a tree as RNA-like or not based on the merge data vector.

The network correctly assigned a high probability of RNA-likeness to trees identified as RNA-like in the RAG database ...

Total Domination Dot Critical And Dot Stable Graphs., 2010 East Tennessee State University

#### Total Domination Dot Critical And Dot Stable Graphs., Stephanie Anne Marie Mcmahon

*Electronic Theses and Dissertations*

Two vertices are said to be identifed if they are combined to form one vertex whose neighborhood is the union of their neighborhoods. A graph is total domination dot-critical if identifying any pair of adjacent vertices decreases the total domination number. On the other hand, a graph is total domination dot-stable if identifying any pair of adjacent vertices leaves the total domination number unchanged. Identifying any pair of vertices cannot increase the total domination number. Further we show it can decrease the total domination number by at most two. Among other results, we characterize total domination dot-critical trees with total ...

The Hamiltonian Index Of Graphs, 2010 Butler University

#### The Hamiltonian Index Of Graphs, Zhi-Hong Chen, Yi Hong, Jian-Liang Lin, Zhi-Sui Tao

*Zhi-Hong Chen*

The Hamiltonian index of a graph GG is defined as h(G)=min{m:Lm(G) is Hamiltonian}.h(G)=min{m:Lm(G) is Hamiltonian}. In this paper, using the reduction method of Catlin [P.A. Catlin, A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12 (1988) 29–44], we constructed a graph sourceH̃^(m)(G) from GG and prove that if h(G)≥2h(G)≥2, then h(G) = min{m : H̃^(m)(G) has a spanning Eulerian subgraph}.

Spanning Trails Containing Given Edges, 2010 Butler University

#### Spanning Trails Containing Given Edges, Zhi-Hong Chen, Weiqi Luo, Wei-Guo Chen

*Zhi-Hong Chen*

A graph GG is Eulerian-connected if for any uu and vv in V(G)V(G), GG has a spanning (u,v)(u,v)-trail. A graph GG is edge-Eulerian-connected if for any e′e′ and e″e″ in E(G)E(G), GG has a spanning (e′,e″)(e′,e″)-trail. For an integer r⩾0r⩾0, a graph is called rr-Eulerian-connected if for any X⊆E(G)X⊆E(G) with |X|⩽r|X|⩽r, and for any View the MathML sourceu,v∈V(G), GG has a spanning (u,v)(u,v)-trail TT such that X ...

Spanning Eulerian Subgraphs In Claw-Free Graphs, 2010 Butler University

#### Spanning Eulerian Subgraphs In Claw-Free Graphs, Zhi-Hong Chen, Hong-Jian Lai, Weiqi Luo, Yehomg Shao

*Zhi-Hong Chen*

A graph is claw-free if it has no induced K 1,3, subgraph. A graph is essential 4-edge-connected if removing at most three edges, the resulting graph has at most one component having edges. In this note, we show that every essential 4-edge-connected claw free graph has a spanning Eulerian subgraph with maximum degree at most 4.

Collapsible Graphs And Reductions Of Line Graphs, 2010 Butler University

#### Collapsible Graphs And Reductions Of Line Graphs, Zhi-Hong Chen, Peter C.B. Lam, Wai-Chee Shiu

*Zhi-Hong Chen*

A graph GG is collapsible if for every even subset X⊆V(G)X⊆V(G), GG has a subgraph ΓΓ such that G−E(Γ)G−E(Γ) is connected and the set of odd-degree vertices of ΓΓ is XX. A graph obtained by contracting all the non-trivial collapsible subgraphs of GG is called the reduction of GG. In this paper, we characterize graphs of diameter two in terms of collapsible subgraphs and investigate the relationship between the line graph of the reduction and the reduction of the line graph. Our results extend former results in [H.-J. Lai ...

The Shallow Water Equations In Lagrangian Coordinates, 2010 Boise State University

#### The Shallow Water Equations In Lagrangian Coordinates, J. L. Mead

*Jodi Mead*

Recent advances in the collection of Lagrangian data from the ocean and results about the well-posedness of the primitive equations have led to a renewed interest in solving flow equations in Lagrangian coordinates. We do not take the view that solving in Lagrangian coordinates equates to solving on a moving grid that can become twisted or distorted. Rather, the grid in Lagrangian coordinates represents the initial position of particles, and it does not change with time. However, using Lagrangian coordinates results in solving a highly nonlinear partial differential equation. The nonlinearity is mainly due to the Jacobian of the coordinate ...

Towards Regional Assimilation Of Lagrangian Data: The Lagrangian Form Of The Shallow Water Reduced Gravity Model And Its Inverse, 2010 Boise State University

#### Towards Regional Assimilation Of Lagrangian Data: The Lagrangian Form Of The Shallow Water Reduced Gravity Model And Its Inverse, J. L. Mead, A. F. Bennett

*Jodi Mead*

Variational data assimilation for Lagrangian geophysical fluid dynamics is introduced. Lagrangian coordinates add numerical difficulties into an already difficult subject, but also offer certain distinct advantages over Eulerian coordinates. First, float position and depth are defined by linear measurement functionals. Second, Lagrangian or ‘comoving’ open domains are conveniently expressed in Lagrangian coordinates. The attraction of such open domains is that they lead to well-posed prediction problems [Bennett and Chua (1999)] and hence efficient inversion algorithms. Eulerian and Lagrangian solutions of the inviscid forward problem in a doubly periodic domain, with North Atlantic mesoscales, are compared and found to be in ...

An Iterated Pseudospectral Method For Functional Partial Differential Equations, 2010 Boise State University

#### An Iterated Pseudospectral Method For Functional Partial Differential Equations, J. Mead, B. Zubik-Kowal

*Jodi Mead*

Chebyshev pseudospectral spatial discretization preconditioned by the Kosloff and Tal-Ezer transformation [10] is applied to hyperbolic and parabolic functional equations. A Jacobi waveform relaxation method is then applied to the resulting semi-discrete functional systems, and the result is a simple system of ordinary differential equations d/dtUk+1(t) = MαUk+1(t)+f(t,U kt). Here Mα is a diagonal matrix, k is the index of waveform relaxation iterations, U kt is a functional argument computed from the previous iterate and the function f, like the matrix Mα, depends on the process of semi-discretization. This waveform relaxation splitting has ...

Assimilation Of Simulated Float Data In Lagrangian Coordinates, 2010 Boise State University

#### Assimilation Of Simulated Float Data In Lagrangian Coordinates, J. L. Mead

*Jodi Mead*

We implement an approach for the accurate assimilation of Lagrangian data into regional general ocean circulation models. The forward model is expressed in Lagrangian coordinates and simulated float data are incorporated into the model via four dimensional variational data assimilation. We show that forward solutions computed in Lagrangian coordinates are reliable for time periods of up to 100 days with phase speeds of 1 m/s and deformation radius of 35 km. The position and depth of simulated floats are assimilated into the viscous, Lagrangian shallow water equations. The weights for the errors in the model and data are varied ...

The 1905 Einstein Equation In A General Mathematical Analysis Model Of Quasars, 2010 DePaul University and Columbia College Chicago

#### The 1905 Einstein Equation In A General Mathematical Analysis Model Of Quasars, Byron E. Bell

*Byron E. Bell*

Profiles Of Large Combinatorial Structures, 2010 University of Pennsylvania

#### Profiles Of Large Combinatorial Structures, Michael T. Lugo

*Publicly Accessible Penn Dissertations*

We derive limit laws for random combinatorial structures using singularity analysis of generating functions. We begin with a study of the Boltzmann samplers of Flajolet and collaborators, a useful method for generating large discrete structures at random which is useful both for providing intuition and conjecture and as a possible proof technique. We then apply generating functions and Boltzmann samplers to three main classes of objects: permutations with weighted cycles, involutions, and integer partitions. Random permutations in which each cycle carries a multiplicative weight $\sigma$ have probability $(1-\gamma)^\sigma$ of having a random element be in a cycle of ...

On Directionally Dependent Subdifferentials, 2010 Technical University of Varna, Bulgaria

#### On Directionally Dependent Subdifferentials, Ivan Ginchev, Boris S. Mordukhovich

*Mathematics Research Reports*

In this paper directionally contextual concepts of variational analysis, based on dual-space constructions similar to those in [4, 5], are introduced and studied. As an illustration of their usefulness, necessary and also sufficient optimality conditions in terms of directioual subdifferentials are established, and it is shown that they can be effective in the situations where known optimality conditions in terms of nondirectional subdifferentials fail.

Certain Two-Parameter Representations Of The Lie Algebra Sl(2,C), 2010 University at Albany, State University of New York

#### Certain Two-Parameter Representations Of The Lie Algebra Sl(2,C), Scott Sidoli

*Mathematics and Statistics*

Classical Lie algebras, like *sl*(2,**C**) can be represented using differential operators that act on polynomial space. These operators will take a different form when they are used on the space of polynomials of several variables and when the differentials are taken to be of higher order. We recall some known realizations and discuss possible deformations. In our two-parameter case we describe decomposition into indecomposable components.