Algebraic Geometry Commons^™

Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces (Corrected), Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

Two Riemann surfaces S₁ and S₂ with conformal G-actions have topologically equivalent actions if there is a homeomorphism h : S₁ -> S₂ which intertwines the actions. A weaker equivalence may be defined by comparing the representations of G on the spaces of holomorphic q-differentials H^q(S₁) and H^q(S₂). In this note we study the differences between topological equivalence and H^q equivalence of prime cyclic actions, where S₁/G and S₂/G have genus zero.

A Counterexample For Lightning Flash Modules Over E(E1,E2), David Benson, Robert R. Bruner

Mathematics Faculty Research Publications

We give a counterexample to Theorem 5 in Section 18.2 of Margolis’ book, “Spectra and the Steenrod Algebra” and make remarks about the proofs of some later theorems in the book that depend on it. The counterexample is a module which does not split as a sum of lightning flash modules and free modules.

Convexity Of Neural Codes, Robert Amzi Jeffs

HMC Senior Theses

An important task in neuroscience is stimulus reconstruction: given activity in the brain, what stimulus could have caused it? We build on previous literature which uses neural codes to approach this problem mathematically. A neural code is a collection of binary vectors that record concurrent firing of neurons in the brain. We consider neural codes arising from place cells, which are neurons that track an animal's position in space. We examine algebraic objects associated to neural codes, and completely characterize a certain class of maps between these objects. Furthermore, we show that such maps have natural geometric implications related to …

Complements To Classic Topics Of Circles Geometry, Florentin Smarandache, Ion Patrascu

Branch Mathematics and Statistics Faculty and Staff Publications

We approach several themes of classical geometry of the circle and complete them with some original results, showing that not everything in traditional math is revealed, and that it still has an open character. The topics were chosen according to authors’ aspiration and attraction, as a poet writes lyrics about spring according to his emotions.

Trilobic Vibrant Systems, Florentin Smarandache, Mircea Eugen Selariu

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Sisteme Vibrante Trilobice, Florentin Smarandache, Mircea Eugen Selariu

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Probleme De Geometrie Și Trigonometrie, Compilate Și Rezolvate, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Euclid Squares On Infinite Planes, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

In this book for the first time the authors study the new type of Euclid squares in various planes like real plane, complex plane, dual number plane, special dual like number plane and special quasi dual number plane. There are six such planes and they behave distinctly. From the study it is revealed that each type of squares behave in a different way depending on the plane. We define several types of algebraic structures on them. Such study is new, innovative and interesting. However for some types of squares; one is not in a position to define product. Further under …

Continuous Dependence Of Solutions Of Equations On Parameters, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

It is shown under very general conditions that the solutions of equations depend continuously on the coefficients or parameters of the equations. The standard examples are solutions of monic polynomial equations and the eigenvalues of a matrix. However, the proof methods apply to any finite map T : Cⁿ -> Cⁿ.

A Kleinian Approach To Fundamental Regions, Joshua L. Hidalgo

Electronic Theses, Projects, and Dissertations

This thesis takes a Kleinian approach to hyperbolic geometry in order to illustrate the importance of discrete subgroups and their fundamental domains (fundamental regions). A brief history of Euclids Parallel Postulate and its relation to the discovery of hyperbolic geometry be given first. We will explore two models of hyperbolic $n$-space: $U^n$ and $B^n$. Points, lines, distances, and spheres of these two models will be defined and examples in $U^2$, $U^3$, and $B^2$ will be given. We will then discuss the isometries of $U^n$ and $B^n$. These isometries, known as M\"obius transformations, have special properties and turn out to be …

Some Properties Of The Harmonic Quadrilateral, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this article, we review some properties of the harmonic quadrilateral related to triangle simedians and to Apollonius circles.

Groupoids Of Type I And Ii Using [0, N), Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Study of algebraic structures built using [0, n) happens to be one of an interesting and innovative research. Here in this book authors define non associative algebraic structures using the interval [0, n). Here we define two types of groupoids using [0, n) both of them are of infinite order. It is an open conjecture to find whether these new class of groupoids satisfy any of the special identities like Moufang identity or Bol identity or Bruck identity or so on. We know on [0, n) we cannot build rings only pseudo rings, however in this book we use these …

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.

The Geometry Of Homological Triangles, Florentin Smarandache, Ion Patrascu

Branch Mathematics and Statistics Faculty and Staff Publications

This book is addressed to students, professors and researchers of geometry, who will find herein many interesting and original results. The originality of the book The Geometry of Homological Triangles consists in using the homology of triangles as a “filter” through which remarkable notions and theorems from the geometry of the triangle are unitarily passed. Our research is structured in seven chapters, the first four are dedicated to the homology of the triangles while the last ones to their applications. In the first chapter one proves the theorem of homological triangles (Desargues, 1636), one survey the remarkable pairs of homological …

Centric Cardinal Sine Function, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

According to any standard dictionary, the word "cardinal" is synonymous with "principal", "essential", "fundamental". In centric mathematics (CM), or ordinary mathematics, cardinal is, on the one hand, a number equal to a number of finite aggregate, called the power of the aggregate, and on the other hand, known as the sine cardinal sinc(x) or cosine cardinal cosc(x), is a special function defined by the centric circular function (CCF). sin(x) and cos(x) are commonly used in undulatory physics (see Figure 1) and whose graph, the graph of cardinal sine, which is called as "Mexican hat" (sombrero) because of its shape.

Integer Optimization And Computational Algebraic Topology, Bala Krishnamoorthy

Systems Science Friday Noon Seminar Series

We present recently discovered connections between integer optimization, or integer programming (IP), and homology. Under reasonable assumptions, these results lead to efficient solutions of several otherwise hard-to-solve problems from computational topology and geometric analysis. The main result equates the total unimodularity of the boundary matrix of a simplicial complex to an algebraic topological condition on the complex (absence of relative torsion), which is often satisfied in real-life applications . When the boundary matrix is totally unimodular, the problem of finding the shortest chain homologous under Z (ring of integers) to a given chain, which is inherently an integer program, can …

Complete Graphs Whose Topological Symmetry Groups Are Polyhedral, Erica Flapan, Blake Mellor, Ramin Naimi

Pomona Faculty Publications and Research

We determine for which m the complete graph K_m has an embedding in S³ whose topological symmetry group is isomorphic to one of the polyhedral groupsA₄, A₅ or S₄.

Interval Semigroups, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book we introduce the notion of interval semigroups using intervals of the form [0, a], a is real. Several types of interval semigroups like fuzzy interval semigroups, interval symmetric semigroups, special symmetric interval semigroups, interval matrix semigroups and interval polynomial semigroups are defined and discussed. This book has eight chapters. The main feature of this book is that we suggest 241 problems in the eighth chapter. In this book the authors have defined 29 new concepts and illustrates them with 231 examples. Certainly this will find several applications. The authors deeply acknowledge Dr. Kandasamy for the proof reading …

Problems With And Without … Problems!, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

This book is addressed to College honor students, researchers, and professors. It contains 136 original problems published by the author in various scientific journals around the world. The problems could be used to preparing for courses, exams, and Olympiads in mathematics. Many of these have a generalized form. For each problem we provide a detailed solution.

I was a professeur coopérant between 1982-1984, teaching mathematics in French language at Lycée Sidi EL Hassan Lyoussi in Sefrou, Province de Fès, Morocco. I used many of these problems for selecting and training, together with other Moroccan professors, in Rabat city, of the …

Rethinking Geometrical Exactness, Marco Panza

MPP Published Research

A crucial concern of early modern geometry was fixing appropriate norms for deciding whether some objects, procedures, or arguments should or should not be allowed into it. According to Bos, this is the exactness concern. I argue that Descartes’s way of responding to this concern was to suggest an appropriate conservative extension of Euclid’s plane geometry (EPG). In Section 2, I outline the exactness concern as, I think, it appeared to Descartes. In Section 3, I account for Descartes’s views on exactness and for his attitude towards the most common sorts of constructions in classical geometry. I also explain in …

Fan Cohomology And Its Application To Equivariant K-Theory Of Toric Varieties, Suanne Au

Department of Mathematics: Dissertations, Theses, and Student Research

Mu-Wan Huang, Mark Walker and I established an explicit formula for the equivariant K-groups of affine toric varieties. We also recovered a result due to Vezzosi and Vistoli, which expresses the equivariant K-groups of a smooth toric variety in terms of the K-groups of its maximal open affine toric subvarieties. This dissertation investigates the situation when the toric variety X is neither affine nor smooth. In many cases, we compute the Čech cohomology groups of the presheaf K_q^T on X endowed with a topology. Using these calculations and Walker's Localization Theorem for equivariant K-theory, we give explicit formulas …

Fiber Bundle Description Of Flow And Nonlinear Hydrodynamics On Circles, Andrei Ludu

Publications

We introduce a differential geometry description of the path lines, stream lines and particles contours in hydrodynamics. We present a generalized form of a Korteweg-de Vries type of equation for the exterior of a circle. Nonlinearities from the boundary conditions, surface tension and the Euler equations are taken into account, but the flow is considered inviscid and irrotational. For the circular case we describe the traveling waves shapes, solitons and the particles trajectories.

N-Linear Algebra Of Type Ii, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

This book is a continuation of the book n-linear algebra of type I and its applications. Most of the properties that could not be derived or defined for n-linear algebra of type I is made possible in this new structure: n-linear algebra of type II which is introduced in this book. In case of n-linear algebra of type II, we are in a position to define linear functionals which is one of the marked difference between the n-vector spaces of type I and II. However all the applications mentioned in n-linear algebras of type I can be appropriately extended to …

The Local Gromov–Witten Invariants Of Configurations Of Rational Curves, Dagan Karp, Chiu-Chu Melissa Liu, Marcos Mariño

All HMC Faculty Publications and Research

We compute the local Gromov–Witten invariants of certain configurations of rational curves in a Calabi–Yau threefold. These configurations are connected subcurves of the “minimal trivalent configuration”, which is a particular tree of ℙ¹’s with specified formal neighborhood. We show that these local invariants are equal to certain global or ordinary Gromov–Witten invariants of a blowup of ℙ³ at points, and we compute these ordinary invariants using the geometry of the Cremona transform. We also realize the configurations in question as formal toric schemes and compute their formal Gromov–Witten invariants using the mathematical and physical theories of the …

Intrinsic Linking And Knotting Of Graphs In Arbitrary 3–Manifolds, Erica Flapan, Hugh Howards, Don Lawrence, Blake Mellor

Pomona Faculty Publications and Research

We prove that a graph is intrinsically linked in an arbitrary 3–manifold M if and only if it is intrinsically linked in S³. Also, assuming the Poincaré Conjecture, we prove that a graph is intrinsically knotted in M if and only if it is intrinsically knotted in S³.

Linked Exact Triples Of Triangulated Categories And A Calculus Of T-Structures, Michael Berg

Mathematics, Statistics and Data Science Faculty Works

We introduce a new formalism of exact triples of triangulated categories arranged in certain types of diagrams. We prove that these arrangements are well-behaved relative to the process of gluing and ungluing t-structures defined on the indicated categories and we connect our con. structs to· a problem (from number theory) involving derived categories. We also briefly address a possible connection with a result of R. Thomason.

A Constructive Proof Of Ky Fan's Generalization Of Tucker's Lemma, Timothy Prescott '02, Francis E. Su

All HMC Faculty Publications and Research

We present a proof of Ky Fan's combinatorial lemma on labellings of triangulated spheres that differs from earlier proofs in that it is constructive. We slightly generalize the hypotheses of Fan's lemma to allow for triangulations of Sⁿ that contain a flag of hemispheres. As a consequence, we can obtain a constructive proof of Tucker's lemma that holds for a more general class of triangulations than the usual version.

Differentials In The Homological Homotopy Fixed Point Spectral Sequence, Robert R. Bruner, John Rognes

Mathematics Faculty Research Publications

We analyze in homological terms the homotopy fixed point spectrum of a T–equivariant commutative S–algebra R. There is a homological homotopy fixed point spectral sequence with E^2_(s,t) = H^(−s)_(gp) (��;H_t(R;��_p)), converging conditionally to the continuous homology H^c_(s+t)(R^(h��);��_p) of the homotopy fixed point spectrum. We show that there are Dyer–Lashof operations β^ϵQ^i acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the E^(2r)–term of the spectral sequence there are 2r other classes in the E^(2r)–term (obtained mostly by Dyer–Lashof operations …

Orthogonal Curvilinear Coordinates, Lindsey J. Bromenshenkel

Honors Capstones

Capstone submitted as a graduation requirement for the BSU Honors Program.

The Birational Isomorphism Types Of Smooth Real Elliptic Curves, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

In this note we determine all birational isomorphism types of real elliptic curves and show that it is the same as the orbit space of smooth cubic real curves in real projective space under linear projective equivalence. There are two families, each depending polynomially on a real parameter in a open subinterval of R. We further show that the complexification of a real elliptic curve has exactly two real forms. Thus the real elliptic curves come in pairs which are isomorphic over C. Finally, the map taking a real elliptic curve to its j-invariant maps the two families …