Algebraic Geometry Commons^™

Mordell-Weil Theorem And The Rank Of Elliptical Curves, Hazem Khalfallah

Theses Digitization Project

The purpose of this thesis is to give a detailed group theoretic proof of the rank formula in a more general setting. By using the proof of Mordell-Weil theorem, a formula for the rank of the elliptical curves in certain cases over algebraic number fields can be obtained and computable.

Primary Decomposition Of Ideals In A Ring, Sola Oyinsan

Theses Digitization Project

The concept of unique factorization was first recognized in the 1840s, but even then, it was still fairly believed to be automatic. The error of this assumption was exposed largely through attempts to prove Pierre de Fermat's, 1601-1665, last theorem. Once mathematicians discovered that this property did not always hold, it was only natural for them to try to search for the strongest available alternative. Thus began the attempt to generalize unique factorization. Using the ascending chain condition on principle ideals, we will show the conditions under which a ring is a unique factorization domain.

Collected Papers Vol. 1, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

The Elementary Theory Of Normed Linear Spaces And Linear Functionals, Suresh Eswarthasan

Honors Capstone Projects - All

Abstract not Included

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 …

Toric Modular Forms And Nonvanishing Of L-Functions, Lev A. Borisov, Paul E. Gunnells

Paul Gunnells

In a previous paper \cite{BorGunn}, we defined the space of toric forms $\TTT(l)$, and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group Γ1(l). In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that L(f,1)≠0. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols.

Injective Modules And Prime Ideals Of Universal Enveloping Algebras, Jorg Feldvoss

University Faculty and Staff Publications

In this paper we study injective modules over universal enveloping algebras of finite-dimensional Lie algebras over fields of arbitrary characteristic. Most of our results are dealing with fields of prime characteristic but we also elaborate on some of their analogues for solvable Lie algebras over fields of characteristic zero. It turns out that analogous results in both cases are often quite similar and resemble those familiar from commutative ring theory.

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.

Neutrosophic Rings, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book we define the new notion of neutrosophic rings. The motivation for this study is two-fold. Firstly, the classes of neutrosophic rings defined in this book are generalization of the two well-known classes of rings: group rings and semigroup rings. The study of these generalized neutrosophic rings will give more results for researchers interested in group rings and semigroup rings. Secondly, the notion of neutrosophic polynomial rings will cause a paradigm shift in the general polynomial rings. This study has to make several changes in case of neutrosophic polynomial rings. This would give solutions to polynomial equations for …

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

Mathematics 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.

Fuzzy Interval Matrices, Neutrosophic Interval Matrices And Their Applications, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The new concept of fuzzy interval matrices has been introduced in this book for the first time. The authors have not only introduced the notion of fuzzy interval matrices, interval neutrosophic matrices and fuzzy neutrosophic interval matrices but have also demonstrated some of its applications when the data under study is an unsupervised one and when several experts analyze the problem. Further, the authors have introduced in this book multiexpert models using these three new types of interval matrices. The new multi expert models dealt in this book are FCIMs, FRIMs, FCInMs, FRInMs, IBAMs, IBBAMs, nIBAMs, FAIMs, FAnIMS, etc. Illustrative …

Some Neutrosophic Algebraic Structures And Neutrosophic N-Algebraic Structures, 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 neutrosophic algebraic structures for groups, loops, semigroups and groupoids and also their neutrosophic N-algebraic structures. One is fully aware of the fact that many classical theorems like Lagrange, Sylow and Cauchy have been studied only in the context of finite groups. Here we try to shift the paradigm by studying and introducing these theorems to neutrosophic semigroups, neutrosophic groupoids, and neutrosophic loops. We have intentionally not given several theorems for semigroups and groupoid but have given several results with proof mainly in the case of neutrosophic loops, biloops …

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 …

Where Does It All End? Boundaries Beyond Euclidean Space, Jonathan Thompson

Inquiry: The University of Arkansas Undergraduate Research Journal

No abstract provided.

A Template Functional-Gage Design Using Parameter-File Table In Autodesk Inventor, Cheng Lin, Moustafa Moustafa

Engineering Technology Faculty Publications

A systematic approach using Autodesk Inventor to design the functional gages of Geometric Dimensioning & Tolerancing (GD&T) is presented. The gages can be used to check straightness, angularity, perpendicularity, parallelism, and position tolerances of a part when geometric tolerances are specified with Maximum Material Condition (MMC). Four steps are proposed to accomplish the task: (1) creation of two-dimensional (2-D) initial template files, (2) generation of hierarchical folders for the template files, (3) creation a 3-D gage model from a specific template file, and (4) dimensioning and generation of the gage drawing. Results show that, by following this approach, students can …

The Closed Topological Vertex Via The Cremona Transform, Jim Bryan, Dagan Karp

All HMC Faculty Publications and Research

We compute the local Gromov-Witten invariants of the "closed vertex", that is, a configuration of three rational curves meeting in a single triple point in a Calabi-Yau threefold. The method is to express the local invariants of the vertex in terms of ordinary Gromov-Witten invariants of a certain blowup of CP^3 and then to compute those invariants via the geometry of the Cremona transformation.

Fuzzy And Neutrosophic Analysis Of Women With Hiv/Aids, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Fuzzy theory is one of the best tools to analyze data, when the data under study is an unsupervised one, involving uncertainty coupled with imprecision. However, fuzzy theory cannot cater to analyzing the data involved with indeterminacy. The only tool that can involve itself with indeterminacy is the neutrosophic model. Neutrosophic models are used in the analysis of the socio-economic problems of HIV/AIDS infected women patients living in rural Tamil Nadu. Most of these women are uneducated and live in utter poverty. Till they became seriously ill they worked as daily wagers. When these women got admitted in the hospital …

Applications Of Bimatrices To Some Fuzzy And Neutrosophic Models, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

Graphs and matrices play a vital role in the analysis and study of several of the real world problems which are based only on unsupervised data. The fuzzy and neutrosophic tools like fuzzy cognitive maps invented by Kosko and neutrosophic cognitive maps introduced by us help in the analysis of such real world problems and they happen to be mathematical tools which can give the hidden pattern of the problem under investigation. This book, in order to generalize the two models, has systematically invented mathematical tools like bimatrices, trimatrices, n-matrices, bigraphs, trigraphs and n-graphs and describe some of its properties. …

N-Algebraic Structures And S-N-Algebraic Structures, Florentin Smarandache, Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

In this book, for the first time we introduce the notions of Ngroups, N-semigroups, N-loops and N-groupoids. We also define a mixed N-algebraic structure. We expect the reader to be well versed in group theory and have at least basic knowledge about Smarandache groupoids, Smarandache loops, Smarandache semigroups and bialgebraic structures and Smarandache bialgebraic structures. The book is organized into six chapters. The first chapter gives the basic notions of S-semigroups, S-groupoids and S-loops thereby making the book self-contained. Chapter two introduces N-groups and their Smarandache analogues. In chapter three, Nloops and Smarandache N-loops are introduced and analyzed. Chapter four …

Introduction To Bimatrices, Florentin Smarandache, W.B. Vasantha Kandasamy, K. Ilanthenral

Branch Mathematics and Statistics Faculty and Staff Publications

Matrix theory has been one of the most utilised concepts in fuzzy models and neutrosophic models. From solving equations to characterising linear transformations or linear operators, matrices are used. Matrices find their applications in several real models. In fact it is not an exaggeration if one says that matrix theory and linear algebra (i.e. vector spaces) form an inseparable component of each other. The study of bialgebraic structures led to the invention of new notions like birings, Smarandache birings, bivector spaces, linear bialgebra, bigroupoids, bisemigroups, etc. But most of these are abstract algebraic concepts except, the bisemigroup being used in …

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 …

On The Behavior Of The Algebraic Transfer, Robert R. Bruner, Lê M. Hà, Nguyễn H. V Hưng

Mathematics Faculty Research Publications

Let Tr_k : ��_2 (⊗ over GL_k) PH_i(B��_k) → Ext^(k,k+i)_A(��_2,��_2) be the algebraic transfer, which is defined by W. Singer as an algebraic version of the geometrical transfer tr_k : π_∗^S((B��_k)_+) → π_∗^S(S^0). It has been shown that the algebraic transfer is highly nontrivial and, more precisely, that Tr_k is an isomorphism for k = 1,2,3. However, Singer showed that Tr_5 is not an epimorphism. In this paper, we prove that Tr_4 does not detect the non zero element g_s ∈ Ext^(4,12·2^s)_A(��_2,��_2) for every s ≥ 1. As a consequence, the localized (Sq^0)^(−1)Tr_4 given by inverting the squaring operation Sq^0 …

Fuzzy Relational Maps And Neutrosophic Relational Maps, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

The aim of this book is two fold. At the outset the book gives most of the available literature about Fuzzy Relational Equations (FREs) and its properties for there is no book that solely caters to FREs and its applications. Though we have a comprehensive bibliography, we do not promise to give all the possible available literature about FRE and its applications. We have given only those papers which we could access and which interested us specially. We have taken those papers which in our opinion could be transformed for neutrosophic study. The second importance of this book is that …

Quantization With Knowledge Base Applied To Geometrical Nesting Problem, Grzegorz Chmaj, Leszek Koszalka

Electrical & Computer Engineering Faculty Research

Nesting algorithms deal with placing two-dimensional shapes on the given canvas. In this paper a binary way of solving the nesting problem is proposed. Geometric shapes are quantized into binary form, which is used to operate on them. After finishing nesting they are converted back into original geometrical form. Investigations showed, that there is a big influence of quantization accuracy for the nesting effect. However, greater accuracy results with longer time of computation. The proposed knowledge base system is able to strongly reduce the computational time.

Basic Neutrosophic Algebraic Structures And Their Application To Fuzzy And Neutrosophic Models, Florentin Smarandache, W.B. Vasantha Kandasamy

Branch Mathematics and Statistics Faculty and Staff Publications

Study of neutrosophic algebraic structures is very recent. The introduction of neutrosophic theory has put forth a significant concept by giving representation to indeterminates. Uncertainty or indeterminacy happen to be one of the major factors in almost all real-world problems. When uncertainty is modeled we use fuzzy theory and when indeterminacy is involved we use neutrosophic theory. Most of the fuzzy models which deal with the analysis and study of unsupervised data make use of the directed graphs or bipartite graphs. Thus the use of graphs has become inevitable in fuzzy models. The neutrosophic models are fuzzy models that permit …

Determining Error Bars In Measurements Of Ultrashort Laser Pulses, Ziyang Wang, Erik Zeek, Rick Trebino, Paul Kvam

Department of Math & Statistics Faculty Publications

We present a simple and automatic method for determining the uncertainty in the retrieved intensity and phase versus time (and frequency) due to noise in a frequency-resolved optical-gating trace, independent of noise source. It uses the ‘‘bootstrap’’ statistical method and also yields an automated method for phase blanking (omitting the phase when the intensity is too low to determine it).

The Cohomology Of The Steendrod Algebra And Representations Of The General Linear Groups, Nguyen H. V. Hu'ng

Mathematics Research Reports

Let Tr_k be the algebraic transfer that maps from the coinvariants of certain GL_k-representation to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer tr_k : pi_*^S((B[doublestrike V]_k)_+) --> pi_*^S(S^0). It has been shown that the algebraic transfer is highly nontrivial, more precisely, that Tr_k is an isomorphism for k = 1, 2, 3 and that T_r = ⊕_k(Tr_k) is a homomorphism of algebras.

In this paper, we first recognize the phenomenon that if we start from any degree d, and apply Sq^0 repeatedly at most (k- 2) …

Consensus-Halving Via Theorems Of Borsuk-Ulam And Tucker, Forrest W. Simmons, Francis E. Su

All HMC Faculty Publications and Research

In this paper we show how theorems of Borsuk-Ulam and Tucker can be used to construct a consensus-halving: a division of an object into two portions so that each of n people believes the portions are equal. Moreover, the division takes at most n cuts, which is best possible. This extends prior work using methods from combinatorial topology to solve fair division problems. Several applications of consensus-halving are discussed.

The Connective K-Theory Of Finite Groups, Robert R. Bruner, John Greenlees

Mathematics Faculty Research Publications

This paper is devoted to the connective K homology and cohomology of finite groups G. We attempt to give a systematic account from several points of view. In Chapter 1, following Quillen [50, 51], we use the methods of algebraic geometry to study the ring ku^*(BG) where ku denotes connective complex K-theory. We describe the variety in terms of the category of abelian p-subgroups of G for primes p dividing the group order. As may be expected, the variety is obtained by splicing that of periodic complex K-theory and that of integral ordinary homology, however the way these parts fit …