Algebra Commons^™

A Survey On Varieties Generated By Small Semigroups And A Companion Website, João Araújo, João Pedro Araújo, Peter J. Cameron, Edmond W. H. Lee, Jorge Raminhos

Mathematics Faculty Articles

This paper presents new findings on varieties generated by small semigroups and groups, and offers a survey of existing results. A companion website is provided which hosts a computational system integrating automated reasoning tools, finite model builders, SAT solvers, and GAP. This platform is a living guide to the literature. In addition, the first complete and justified list of identity bases for all varieties generated by a semigroup of order up to 4 is provided as supplementary material. The paper concludes with an extensive list of open problems.

⊕-Supplemented Semimodules, Ahmed H. Alwan

Al-Bahir Journal for Engineering and Pure Sciences

In this paper, ⊕-Supplemented Semimodules are defined as generalizations of ⊕-Supplemented modules. Let S be a semiring. An S-semimodule A is named a ⊕-supplemented semimodule, if every subsemimodule of A has a supplement which is a direct summand of A. In this paper, we investigate some properties of ⊕-supplemented semimodules besides generalize certain results on ⊕-supplemented modules to semimodules.

Certain Invertible Operator-Block Matrices Induced By C*-Algebras And Scaled Hypercomplex Numbers, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

The main purposes of this paper are (i) to enlarge scaled hypercomplex structures to operator-valued cases, where the operators are taken from a C*-subalgebra of an operator algebra on a separable Hilbert space, (ii) to characterize the invertibility conditions on the operator-valued scaled-hypercomplex structures of (i), (iii) to study relations between the invertibility of scaled hypercomplex numbers, and that of operator-valued cases of (ii), and (iv) to confirm our invertibility of (ii) and (iii) are equivalent to the general invertibility of (2×2)-block operator matrices.

Math 115: College Algebra For Pre-Calculus, Seth Lehman

Open Educational Resources

OER course syllabus for Math 115, College Algebra, at Queens College

Many-Valued Coalgebraic Logic: From Boolean Algebras To Primal Varieties, Alexander Kurz, Wolfgang Poiger

Engineering Faculty Articles and Research

We study many-valued coalgebraic logics with primal algebras of truth-degrees. We describe a way to lift algebraic semantics of classical coalgebraic logics, given by an endofunctor on the variety of Boolean algebras, to this many-valued setting, and we show that many important properties of the original logic are inherited by its lifting. Then, we deal with the problem of obtaining a concrete axiomatic presentation of the variety of algebras for this lifted logic, given that we know one for the original one. We solve this problem for a class of presentations which behaves well with respect to a lattice structure …

The Gamma-Signless Laplacian Adjacency Matrix Of Mixed Graphs, Omar Alomari, Mohammad Abudayah, Manal Ghanem

Theory and Applications of Graphs

The α-Hermitian adjacency matrix H_α of a mixed graph X has been recently introduced. It is a generalization of the adjacency matrix of unoriented graphs. In this paper, we consider a special case of the complex number α. This enables us to define an incidence matrix of mixed graphs. Consequently, we define a generalization of line graphs as well as a generalization of the signless Laplacian adjacency matrix of graphs. We then study the spectral properties of the gamma-signless Laplacian adjacency matrix of a mixed graph. Lastly, we characterize when the signless Laplacian adjacency matrix of …

Generating Polynomials Of Exponential Random Graphs, Mohabat Tarkeshian

Electronic Thesis and Dissertation Repository

The theory of random graphs describes the interplay between probability and graph theory: it is the study of the stochastic process by which graphs form and evolve. In 1959, Erdős and Rényi defined the foundational model of random graphs on n vertices, denoted G(n, p) ([ER84]). Subsequently, Frank and Strauss (1986) added a Markov twist to this story by describing a topological structure on random graphs that encodes dependencies between local pairs of vertices ([FS86]). The general model that describes this framework is called the exponential random graph model (ERGM).

In the past, determining when a probability distribution has strong …

A Vector-Valued Trace Formula For Finite Groups, Miles Chasek

Electronic Theses and Dissertations

We derive a trace formula that can be used to study representations of a finite group G induced from arbitrary representations of a subgroup Γ. We restrict our attention to finite-dimensional representations over the field of complex numbers. We consider some applications and examples of our trace formula, including a proof of the well-known Frobenius reciprocity theorem.

Efficient And Secure Digital Signature Algorithm (Dsa), Nissa Mehibel, M'Hamed Hamadouche

Emirates Journal for Engineering Research

The digital signature is used to ensure the integrity of messages as well as the authentication and non-repudiation of users. Today it has a very important role in information security. Digital signature is used in various fields such as e-commerce and e-voting, health, internet of things (IOT). Many digital signature schemes have been proposed, depending on the computational cost and security level. In this paper, we analyzed a recently proposed digital signature scheme based on the discrete logarithm problem (DLP). Our analysis shows that the scheme is not secure against the repeated random number attack to determine the secret keys …

One Theorem, Two Ways: A Case Study In Geometric Techniques, John B. Little

Journal of Humanistic Mathematics

If the three sides of a triangle ABΓ in the Euclidean plane are cut by points H on AB, Θ on BΓ, and K on ΓA cutting those sides in same ratios:

AH : HB = BΘ : ΘΓ = ΓK : KA,

then Pappus of Alexandria proved that the triangles ABΓ and HΘK have the same centroid (center of mass). We present two proofs of this result: an English translation of Pappus's original synthetic proof and a modern algebraic proof making use of Cartesian coordinates and vector concepts. Comparing the two methods, we can see that while the algebraic …

Algebraic And Integral Closure Of A Polynomial Ring In Its Power Series Ring, Joseph Swanson

All Dissertations

Let R be a domain. We look at the algebraic and integral closure of a polynomial ring, R[x], in its power series ring, R[[x]]. A power series α(x) ∈ R[[x]] is said to be an algebraic power series if there exists F (x, y) ∈ R[x][y] such that F (x, α(x)) = 0, where F (x, y) ̸ = 0. If F (x, y) is monic, then α(x) is said to be an integral power series. We characterize the units of algebraic and integral power series. We show that the only algebraic power series with infinite radii of convergence are …

The G_2-Hitchin Component Of Triangle Groups: Dimension And Integer Points, Hannah E. Downs

Doctoral Dissertations

The image of $\PSL(2,\reals)$ under the irreducible representation into $\PSL(7,\reals)$ is contained in the split real form $G_{2}^{4,3}$ of the exceptional Lie group $G_{2}$. This irreducible representation therefore gives a representation $\rho$ of a hyperbolic triangle group $\Gamma(p,q,r)$ into $G_{2}^{4,3}$, and the \textit{Hitchin component} of the representation variety $\Hom(\Gamma(p,q,r),G_{2}^{4,3})$ is the component of $\Hom(\Gamma(p,q,r),G_{2}^{4,3})$ containing $\rho$.

This thesis is in two parts: (i) we give a simple, elementary proof of a formula for the dimension of this Hitchin component, this formula having been obtained earlier in [Alessandrini et al.], \citep{Alessandrini2023}, as part of a wider investigation using Higgs bundle techniques, …

An Exploration Of Absolute Minimal Degree Lifts Of Hyperelliptic Curves, Justin A. Groves

Doctoral Dissertations

For any ordinary elliptic curve E over a field with non-zero characteristic p, there exists an elliptic curve E over the ring of Witt vectors W(E) for which we can lift the Frobenius morphism, called the canonical lift. Voloch and Walker used this theory of canonical liftings of elliptic curves over Witt vectors of length 2 to construct non-linear error-correcting codes for characteristic two. Finotti later proved that for longer lengths of Witt vectors there are better lifts than the canonical. He then proved that, more generally, for hyperelliptic curves one can construct a lifting over …

An Introduction To The Algebra Revolution, Art Bardige

Numeracy

Bardige, Art. 2022. The Algebra Revolution: How Spreadsheets Eliminate Algebra 1 to Transform Education; (Bookbaby) 135 pp. UNSPSC 55111505.

The Algebra Revolution: How Spreadsheets Eliminate Algebra 1 to Transform Education argues that Algebra 1 can be eliminated by teaching mathematics through spreadsheets. Such a change would eliminate the greatest roadblock to student achievement.

Interpolation Problems And The Characterization Of The Hilbert Function, Bryant Xie

Mathematical Sciences Undergraduate Honors Theses

In mathematics, it is often useful to approximate the values of functions that are either too awkward and difficult to evaluate or not readily differentiable or integrable. To approximate its values, we attempt to replace such functions with more well-behaving examples such as polynomials or trigonometric functions. Over the algebraically closed field C, a polynomial passing through r distinct points with multiplicities m1, ..., mr on the affine complex line in one variable is determined by its zeros and the vanishing conditions up to its mi − 1 derivative for each point. A natural question would then be to consider …

Schur Analysis And Discrete Analytic Functions: Rational Functions And Co-Isometric Realizations, Daniel Alpay, Dan Volok

Mathematics, Physics, and Computer Science Faculty Articles and Research

We define and study rational discrete analytic functions and prove the existence of a coisometric realization for discrete analytic Schur multipliers.

Pairings In A Ring Spectrum-Based Bousfield-Kan Spectral Sequence, Jonathan Toledo

Dissertations, Theses, and Capstone Projects

Bousfield and Kan traditionally formulated their homotopy spectral sequence over a simplicial set X resolved with respect to a ring R. By considering an adequate category of ring spectra, one can take a ring spectrum E, create from it a functor of a triple on the category of simplicial sets, and build a cosimplicial simplicial set EX. The homotopy spectral sequence can then be formed over such cosimplicial spaces by a similar construction to the original. Pairings can be established on these spectral sequences, and, for nice enough spaces, these pairings on the E₂-terms coincide with certain …

Representations From Group Actions On Words And Matrices, Joel T. Anderson

Master's Theses

We provide a combinatorial interpretation of the frequency of any irreducible representation of Sn in representations of Sn arising from group actions on words. Recognizing that representations arising from group actions naturally split across orbits yields combinatorial interpretations of the irreducible decompositions of representations from similar group actions. The generalization from group actions on words to group actions on matrices gives rise to representations that prove to be much less transparent. We share the progress made thus far on the open problem of determining the irreducible decomposition of certain representations of Sm × Sn arising from group actions on matrices.

An Algebraic Characterization Of Total Input Strictly Local Functions, Dakotah Lambert, Jeffrey Heinz

Proceedings of the Society for Computation in Linguistics

This paper provides an algebraic characteriza- tion of the total input strictly local functions. Simultaneous, noniterative rules of the form A→B/C D, common in phonology, are defin- able as functions in this class whenever CAD represents a finite set of strings. The algebraic characterization highlights a fundamental con- nection between input strictly local functions and the simple class of definite string languages, as well as connections to string functions stud- ied in the computer science literature, the def- inite functions and local functions. No effec- tive decision procedure for the input strictly local maps was previously available, but one arises …

Discrete Wiener Algebra In The Bicomplex Setting, Spectral Factorization With Symmetry, And Superoscillations, Daniel Alpay, Izchak Lewkowicz, Mihaela Vajiac

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper we present parallel theories on constructing Wiener algebras in the bicomplex setting. With the appropriate symmetry condition, the bicomplex matrix valued case can be seen as a complex valued case and, in this matrix valued case, we make the necessary connection between bicomplex analysis and complex analysis with symmetry. We also write an application to superoscillations in this case.