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Articles 1 - 9 of 9
Full-Text Articles in Mathematics
Multi-Trace Matrix Models From Noncommutative Geometry, Hamed Hessam
Multi-Trace Matrix Models From Noncommutative Geometry, Hamed Hessam
Electronic Thesis and Dissertation Repository
Dirac ensembles are finite dimensional real spectral triples where the Dirac operator is allowed to vary within a suitable family of operators and is assumed to be random. The Dirac operator plays the role of a metric on a manifold in the noncommutative geometry context of spectral triples. Thus, integration over the set of Dirac operators within a Dirac ensemble, a crucial aspect of a theory of quantum gravity, is a noncommutative analog of integration over metrics.
Dirac ensembles are closely related to random matrix ensembles. In order to determine properties of specific Dirac ensembles, we use techniques from random …
On The Spatial Modelling Of Biological Invasions, Tedi Ramaj
On The Spatial Modelling Of Biological Invasions, Tedi Ramaj
Electronic Thesis and Dissertation Repository
We investigate problems of biological spatial invasion through the use of spatial modelling. We begin by examining the spread of an invasive weed plant species through a forest by developing a system of partial differential equations (PDEs) involving an invasive weed and a competing native plant species. We find that extinction of the native plant species may be achieved by increasing the carrying capacity of the forest as well as the competition coefficient between the species. We also find that the boundary conditions exert long-term control on the biomass of the invasive weed and hence should be considered when implementing …
Automorphism-Preserving Color Substitutions On Profinite Graphs, Michal Cizek
Automorphism-Preserving Color Substitutions On Profinite Graphs, Michal Cizek
Electronic Thesis and Dissertation Repository
Profinite groups are topological groups which are known to be Galois groups. Their free product was extensively studied by Luis Ribes and Pavel Zaleskii using the notion of a profinite graph and having profinite groups act freely on such graphs. This thesis explores a different approach to study profinite groups using profinite graphs and that is with the notion of automorphisms and colors. It contains a generalization to profinite graphs of the theorem of Frucht (1939) that shows that every finite group is a group of automorphisms of a finite connected graph, and establishes a profinite analog of the theorem …
Reduction Of L-Functions Of Elliptic Curves Modulo Integers, Félix Baril Boudreau
Reduction Of L-Functions Of Elliptic Curves Modulo Integers, Félix Baril Boudreau
Electronic Thesis and Dissertation Repository
Let $\mathbb{F}_q$ be a finite field of size $q$, where $q$ is a power of a prime $p \geq 5$. Let $C$ be a smooth, proper, and geometrically connected curve over $\mathbb{F}_q$. Consider an elliptic curve $E$ over the function field $K$ of $C$ with nonconstant $j$-invariant. One can attach to $E$ its $L$-function $L(T,E/K)$, which is a generating function that contains information about the reduction types of $E$ at the different places of $K$. The $L$-function of $E/K$ was proven to be a polynomial in $\mathbb{Z}[T]$.
In 1985, Schoof devised an algorithm to compute the zeta function of an …
The Design And Implementation Of A High-Performance Polynomial System Solver, Alexander Brandt
The Design And Implementation Of A High-Performance Polynomial System Solver, Alexander Brandt
Electronic Thesis and Dissertation Repository
This thesis examines the algorithmic and practical challenges of solving systems of polynomial equations. We discuss the design and implementation of triangular decomposition to solve polynomials systems exactly by means of symbolic computation.
Incremental triangular decomposition solves one equation from the input list of polynomials at a time. Each step may produce several different components (points, curves, surfaces, etc.) of the solution set. Independent components imply that the solving process may proceed on each component concurrently. This so-called component-level parallelism is a theoretical and practical challenge characterized by irregular parallelism. Parallelism is not an algorithmic property but rather a geometrical …
Polynomial Identities Of Algebras With Actions: A Unified Combinatorial Approach, Mayecxiliana Cardenas Montoya
Polynomial Identities Of Algebras With Actions: A Unified Combinatorial Approach, Mayecxiliana Cardenas Montoya
Electronic Thesis and Dissertation Repository
Fix a field k. In this thesis, we establish different criteria for when a given (associative or Lie) k-algebra is a PI-algebra.
Fix a unitary associative k-algebra R. Our focus is on associative k-algebras A endowed with an R-module action ρ:R\to End_k(A) with the property that ρ(R) is finite-dimensional. Our main result asserts that if such an algebra is endowed with an R-module action that is `compatible’ with the multiplicative structure of A, then A is a PI-algebra if (and only if), for some positive integer d and all a_1,...,a_d in A, the product a_1···a_d is a linear combination of …
Towards A Generalization Of Fulton's Intersection Multiplicity Algorithm, Ryan Sandford
Towards A Generalization Of Fulton's Intersection Multiplicity Algorithm, Ryan Sandford
Electronic Thesis and Dissertation Repository
In this manuscript we generalize Fulton's bivariate intersection multiplicity algorithm to a partial intersection multiplicity algorithm in the n-variate setting. We extend this generalization of Fulton's algorithm to work at any point, rational or not, using the theory of regular chains. We implement these algorithms in Maple and provide experimental testing. The results indicate the proposed algorithm often outperforms the existing standard basis-free intersection multiplicity algorithm in Maple, typically by one to two orders of magnitude. Moreover, we also provide some examples where the proposed algorithm outperforms intersection multiplicity algorithms which rely on standard bases, indicating the proposed algorithm is …
On The Geometry Of Multi-Affine Polynomials, Junquan Xiao
On The Geometry Of Multi-Affine Polynomials, Junquan Xiao
Electronic Thesis and Dissertation Repository
This work investigates several geometric properties of the solutions of the multi-affine polynomials. Chapters 1, 2 discuss two different notions of invariant circles. Chapter 3 gives several loci of polynomials of degree three. A locus of a complex polynomial p(z) is a minimal, with respect to inclusion, set that contains at least one point of every solution of the polarization of the polynomial. The study of such objects allows one to improve upon know results about the location of zeros and critical points of complex polynomials, see for example [22] and [24]. A complex polynomial has many loci. It is …
An Implementation Of Integrated Information Theory In Resting-State Fmri, Idan E. Nemirovsky
An Implementation Of Integrated Information Theory In Resting-State Fmri, Idan E. Nemirovsky
Electronic Thesis and Dissertation Repository
Integrated Information Theory (IIT) is a framework developed to explain consciousness, arguing that conscious systems consist of interacting elements that are integrated through their causal properties. In this study, we present the first application of IIT to functional magnetic resonance imaging (fMRI) data and investigate whether its principal metric, Phi, can meaningfully quantify resting-state cortical activity patterns. Data was acquired from 17 healthy subjects who underwent sedation with propofol, a short acting anesthetic. Using PyPhi, a software package developed for IIT, we thoroughly analyze how Phi varies across different networks and throughout sedation. Our findings indicate that variations in Phi …