Algebraic Geometry Commons^™

Geometry Of Optimal Control For Control-Affine Systems, Jeanne N. Clelland, Christopher G. Moseley, George R. Wilkens

University Faculty Publications and Creative Works

Motivated by the ubiquity of control-affine systems in optimal control theory, we investigate the geometry of point-affine control systems with metric structures in dimensions two and three. We compute local isometric invariants for point-affine distributions of constant type with metric structures for systems with 2 states and 1 control and systems with 3 states and 1 control, and use Pontryagin's maximum principle to find geodesic trajectories for homogeneous examples. Even in these low dimensions, the behavior of these systems is surprisingly rich and varied.

Geometry Of Optimal Control For Control-Affine Systems, Jeanne N. Clelland, Christopher G. Moseley, George R. Wilkens

University Faculty Publications and Creative Works

Motivated by the ubiquity of control-affine systems in optimal control theory, we investigate the geometry of point-affine control systems with metric structures in dimensions two and three. We compute local isometric invariants for point-affine distributions of constant type with metric structures for systems with 2 states and 1 control and systems with 3 states and 1 control, and use Pontryagin's maximum principle to find geodesic trajectories for homogeneous examples. Even in these low dimensions, the behavior of these systems is surprisingly rich and varied.

Results On Containments And Resurgences, With A Focus On Ideals Of Points In The Plane, Annika Denkert

Department of Mathematics: Dissertations, Theses, and Student Research

Let K be an algebraically closed field and I ⊆ R=K[P^N] a nontrivial homogeneous ideal. We can describe ordinary powers I^r and symbolic powers I^(m) of I. One question that has been of interest over the past couple of years is that of when we have containment of I^(m) in I^r. Bocci and Harbourne defined the resurgence of I as rho(I)=sup_m,r{m/r | I^(m) is not contained in I^r}. Hence in particular I^(m) ⊆ I^r whenever m/r is at least …

Geometric Study Of The Category Of Matrix Factorizations, Xuan Yu

Department of Mathematics: Dissertations, Theses, and Student Research

We study the geometry of matrix factorizations in this dissertation.
It contains two parts. The first one is a Chern-Weil style
construction for the Chern character of matrix factorizations; this
allows us to reproduce the Chern character in an explicit,
understandable way. Some basic properties of the Chern character are
also proved (via this construction) such as functoriality and that
it determines a ring homomorphism from the Grothendieck group of
matrix factorizations to its Hochschild homology. The second part is
a reconstruction theorem of hypersurface singularities. This is
given by applying a slightly modified version of Balmer's tensor
triangular geometry …

On Cyclic Fixed Points Of Spectra, Marcel Bökstedt, Robert R. Bruner, Sverre Lunøe-Nielsen, John Rognes

Mathematics Faculty Research Publications

For a finite ��-group �� and a bounded below ��-spectrum �� of finite type mod ��, the ��-equivariant Segal conjecture for �� asserts that the canonical map ��^��→��^ℎ��, from ��-fixed points to ��-homotopy fixed points, is a ��-adic equivalence. Let ��_(��^��) be the cyclic group of order ��^��. We show that if the ��_��-equivariant Segal conjecture holds for a ��_(��^��)-spectrum ��, as well as for each of its geometric fixed point spectra Φ^(��_(��^��))(��) for 0<��<��, then the ��_(��^��)-equivariant Segal conjecture holds for ��. Similar results also hold for weaker forms of the Segal conjecture, asking only that the canonical map induces an equivalence in sufficiently high degrees, on homotopy groups with suitable finite coefficients.

Projected Surface Finite Elements For Elliptic Equations, Necibe Tuncer

Applications and Applied Mathematics: An International Journal (AAM)

In this article, we define a new finite element method for numerically approximating solutions of elliptic partial differential equations defined on “arbitrary” smooth surfaces S in RN+1. By “arbitrary” smooth surfaces, we mean surfaces that can be implicitly represented as level sets of smooth functions. The key idea is to first approximate the surface S by a polyhedral surface Sh, which is a union of planar triangles whose vertices lie on S; then to project Sh onto S. With this method, we can also approximate the eigenvalues and eigenfunctions of th Laplace-Beltrami operator on these “arbitrary” surfaces.

Chip Firing Games And Riemann-Roch Properties For Directed Graphs, Joshua Z. Gaslowitz

HMC Senior Theses

The following presents a brief introduction to tropical geometry, especially tropical curves, and explains a connection to graph theory. We also give a brief summary of the Riemann-Roch property for graphs, established by Baker and Norine (2007), as well as the tools used in their proof. Various generalizations are described, including a more thorough description of the extension to strongly connected directed graphs by Asadi and Backman (2011). Building from their constructions, an algorithm to determine if a directed graph has Row Riemann-Roch Property is given and thoroughly explained.

Split Abelian Chief Factors And First Degree Cohomology For Lie Algebras, Jorg Feldvoss, Salvatore Siciliano, Thomas Weigel

University Faculty and Staff Publications

In this paper we investigate the relation between the multiplicities of split abelian chief factors of finite-dimensional Lie algebras and first degree cohomology. In particular, we obtain a characterization of modular solvable Lie algebras in terms of the vanishing of first degree cohomology or in terms of the multiplicities of split abelian chief factors. The analogues of these results are well known in the modular representation theory of finite groups. An important tool in the proof of these results is a refinement of a non-vanishing theorem of Seligman for the first degree cohomology of non-solvable finite-dimensional Lie algebras in prime …

On Toric Symmetry Of P1 X P2, Olivia D. Beckwith

HMC Senior Theses

Toric varieties are a class of geometric objects with a combinatorial structure encoded in polytopes. P1 x P2 is a well known variety and its polytope is the triangular prism. Studying the symmetries of the triangular prism and its truncations can lead to symmetries of the variety. Many of these symmetries permute the elements of the cohomology ring nontrivially and induce nontrivial relations. We discuss some toric symmetries of P1 x P2, and describe the geometry of the polytope of the corresponding blowups, and analyze the induced action on the cohomology ring. We exhaustively compute the toric symmetries of P1 …

Symbolic Powers Of Ideals In K[PN], Michael Janssen

Department of Mathematics: Dissertations, Theses, and Student Research

Let I ⊆ k[P^N] be a homogeneous ideal and k an algebraically closed field. Of particular interest over the last several years are ideal containments of symbolic powers of I in ordinary powers of I of the form I^(m) ⊆ I^r, and which ratios m/r guarantee such containment. A result of Ein-Lazarsfeld-Smith and Hochster-Huneke states that, if I ⊆ k[P^N], where k is an algebraically closed field, then the symbolic power I^(Ne) is contained in the ordinary power I^e, and thus, whenever …

Propeller, Joel Kahn

The Transdisciplinary STEAM+ Journal

This image is based on several different algorithms interconnected within a single program in the language BASIC-256. The fundamental structure involves a tightly wound spiral working outwards from the center of the image. As the spiral is drawn, different values of red, green and blue are modified through separate but related processes, producing the changing appearance. Algebra, trigonometry, geometry, and analytic geometry are all utilized in overlapping ways within the program. As with many works of algorithmic art, small changes in the program can produce dramatic alterations of the visual output, which makes lots of variations possible.

Whitney's 2-Isomorphism Theorem For Hypergraphs, Eric Anthony Taylor

Theses Digitization Project

This study will examine a fundamental theorem from graph theory: Whitney's 2-Isomorphism Theorem. Whitney's 2-Isomorphism theorem characterizes when two graphs have isomorphic cycle matroids.

Behavior Of Solutions For Bernoulli Initial-Value Problems, Carlos Marcelo Sardan

Theses Digitization Project

The purpose of this project is to investigate blow-up properties of solutions for specific initial-value problems that involve Bernoulli Ordinary Differential Equations (ODE's). The objective is to find conditions on the coefficients and on the initial-values that lead to unbounded growth of solutions in finite time.

Set Theoretic Approach To Algebraic Structures In Mathematics - A Revelation, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors bring out how sets in algebraic structure can be used to construct most generalized algebraic structures, like set linear algebra/vector space, set ideals in rings and semigroups. This sort of study is not only innovative but infact very helpful in cases instead of working with a large data we can work with a considerably small data. Thus instead of working with a vector space or a linear algebra V over a field F we can work with a subset in V and a needed subset in F, this can save both time and economy. The concept …

Subset Non Associative Topological Spaces, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The concept of non associative topological space is new and innovative. In general topological spaces are defined as union and intersection of subsets of a set X. In this book authors for the first time define non associative topological spaces using subsets of groupoids or subsets of loops or subsets of groupoid rings or subsets of loop rings. This study leads to several interesting results in this direction.

Over hundred problems on non associative topological spaces using of subsets of loops or groupoids is suggested at the end of chapter two. Also conditions for these non associative subset topological spaces …

Subset Groupoids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors introduce the new notion of constructing non associative algebraic structures using subsets of a groupoid. Thus subset groupoids are constructed using groupoids or loops. Even if we use subsets of loops still the algebraic structure we get with it is only a groupoid. However we can get a proper subset of it to be a subset loop which will be isomorphic with the loop which was used in the construction of the subset groupoid. To the best of the authors’ knowledge this is the first time non associative algebraic structures are constructed using subsets. We get …

Subset Semirings, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors study the new notion of the algebraic structure of the subset semirings using the subsets of rings or semirings. This study is innovative and interesting for the authors feel giving algebraic structure to collection of sets is not a new study, for when set theory was introduced such study was in vogue. But a systematic development of constructing algebraic structures using subsets of a set is absent, except for the set topology and in the construction of Boolean algebras. The authors have explored the study of constructing subset algebraic structures like semigroups, groupoids, semirings, non commutative …

Subset Polynomial Semirings And Subset Matrix Semirings, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors introduce the notion of subset polynomial semirings and subset matrix semirings. The study of algebraic structures using subsets were recently carried out by the authors. Here we define the notion of subset row matrices, subset column matrices and subset m × n matrices. Study of this kind is developed in chapter two of this book. If we use subsets of a set X; say P(X), the power set of the set X....

Hence if P(X) is replaced by a group or a semigroup we get the subset matrix to be only a subset matrix semigroup. If …

Subset Non Associative Semirings, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book for the first time we introduce the notion of subset non associative semirings. It is pertinent to keep on record that study of non associative semirings is meager and books on this specific topic is still rare. Authors have recently introduced the notion of subset algebraic structures. The maximum algebraic structure enjoyed by subsets with two binary operations is just a semifield and semiring, even if a ring or a field is used. In case semigroups or groups are used still the algebraic structure of the subset is only a semigroup. To construct a subset non associative …

Subset Interval Groupoids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The study of groupoids is meager and we have recently introduced the new notion of subset groupoids and have studied them. It is interesting to keep on record that interval groupoids have been studied by us in 2010. Further when the subsets of a loop are taken they also form only a subset groupoid and not a subset loop. Thus we do not have the concept of subset interval loop they only form a subset interval groupoid. Special elements like subset interval zero divisors, subset interval idempotents and subset interval units are studied. Concept of subset interval groupoid homomorphism is …

Algebraic Structures Using Subsets, Florentin Smarandache, W.B Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The study of subsets and giving algebraic structure to these subsets of a set started in the mid 18th century by George Boole. The first systematic presentation of Boolean algebra emerged in 1860s in papers written by William Jevons and Charles Sanders Peirce. Thus we see if P(X) denotes the collection of all subsets of the set X, then P(X) under the op erations of union and intersection is a Boolean algebra. Next the subsets of a set was used in the construction of topological spaces. We in this book consider subsets of a semigroup or a group or a …

Algebraic Structures Using [0,N), Florentin Smarandache, Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book authors for the first time introduce a new method of building algebraic structures on the interval [0, n). This study is interesting and innovative. However, [0, n) is a semigroup under product, × modulo n and a semigroup under min or max operation. Further [0, n) is a group under addition modulo n. We see [0, n) under both max and min operation is a semiring. [0, n) under + and × is not in general a ring. We define S = {[0, n), +, ×} to be a pseudo special ring as the distributive law is …