Graduate Student Seminar
The Graduate Student Seminar is a venue that any graduate student can use to present their work, to practice for an upcoming “official” talk, or just to discuss mathematical concepts they find interesting. We also invite Faculty to introduce their research to Graduate Students.
If you are interested in giving a talk, please contact Zamzam Qazi, Barkat Mian, James Torrent, Anuj Verma and Shunzhe Zhang.
Past and upcoming Seminars
Seminars in Fall 2024
9/20/2024
Speaker: Zamzam Qazi
Title: Parameter Estimation of a new class of non-symmetric copulas, and POMP Modeling using Weighted Average Iterated Filtering
Abstract:
My work consists of two parts. First, I consider parameter estimation for a new copula family CΛ(u, v). Asymptotic distributions of parameter estimators are provided. Consider a simple random sample from bivariate distribution with the copula CΛ(u, v), then copula based stationary Markov chain. The central limit theorem is provided for eigen function estimators of Longla and Hamadou (2023), and asymptotic normatliy of the MLE is established in each of the set ups. Special properties of the functions Φk are used in the proof, along with the mixing structure of the Markov chains. Simulation study of sine, cosine and FGM copulas is provided, showing the comparisons between the proposed and the other estimators. The second part is about parameter estimation of POMP models using the algorithm, ”Weighted Average Iterated Filtering” and p-generalized Gaussian Smoothing. The algorithm is effective for estimating numerous types of infectious disease models such as Covid-19, Influenza or various other diseases. First, generalization of Stein’s identity for normal distribution to p-generalized Gaussian distribution is introduced, enabling more flexible perturbation with different tail behaviors. Building upon these gradient approximations, a novel weighted average algorithm for maximizing likelihood through the two-time-scale stochastic approximation is introduced. I show the integration of the algorithm into iterated filtering framework, relaxing the requirement for a bounded variance of the two-timescale stochastic approximation. Initially, p-generalised Gaussian smoothing is applied through Weighted Average Iterated filtering in two toy problems: a linear ou2 model and a nonlinear Gompertz model. Subsequently, the potential of this technique is demonstrated in fitting a real example of a more complex cholera model, incorporating a highly nonlinear structure with discrete population dynamics, seasonality, and extra-demographic stochasticity.
Seminars in Fall 2023
11/6/2023
Speaker: Barkat Mian
Title: On Planar Brownian motion singularly tilted through a point potential
Abstract:
We will discuss a special family of two-dimensional diffusions, defined over a finite time interval [0, T]. These diffusions have transition density functions that are given by the integral kernels of the semigroup corresponding to the two-dimensional Schrodinger operator with a point potential at the origin. Although, in a few ways, our processes of interest are closely related to two-dimensional Brownian motion, they have a singular drift pointing in the direction of the origin that is strong enough to enable the possibility of visiting there with positive probability. Our main focus is on characterizing a local time process at the origin for these diffusions analogous to that for a one-dimensional Brownian motion.
11/2/2023
Speaker: Apoorva Mate
Title: Pełczyński’s Property (V) on Positive Tensor Product on Banach Lattices
Abstract:
Click here to read the abstract.
10/11/2023
Speaker: Mous-Abou Hamadou
Title: Estimation problems for some perturbations of the independence copula
Abstract:
This work provides central limit theorems for estimators of parameters based on functions of Markov chains generated by some perturbations of the independence copula. We provide a study of maximum likelihood estimators and confidence intervals for copula parameters for several families of copulas introduced in Longla (2023). We investigate the particular case Markov chains generated by cosine copulas, sine-cosine copulas and the extended Farlie-Gumbel-Morgenstern copula family. Some tests of independence are proposed. A simulation study is provided for the three copula families. A comparative study of the introduced estima- tors and other known estimators such as that of Longla and Peligrad(2021) and MLE is proposed, showing advantages of the proposed work.
Speaker: Jaime Palacios
Title: Partitions into sums of two squares
9/25/2023
Posters presentation
Presenter: Apoorva Mate
Title: Pelcyński’s Property (V) on Positive Tensor Products of Banach Lattices
Presenter: Gauree Kiran Wathodkar:
Title: Partition regularity in commutative rings
9/22/2023
Speaker: Joe Lopez
Title: Questioning techniques
Abstract:
Whether we are tutors, supplemental instruction leaders, teaching assistants, or the instructor on record, everything we do to help our students learn new material is based on the questions we want students to be able to answer. In particular, we ask ourselves what kinds of questions we should be able to answer, and what are the questions in between we need to ask to break down the material. In this presentation, we will analyze questions and questioning techniques. We will discuss the traits of good and bad questions, the golden question, how to respond when a student says I do not know, questioning techniques. We will also engage in different types of questioning technique scenarios that have occurred in the past.
9/1/2023
Speaker: Mathias Muia
Title: Dependence And Mixing for Perturbations of Copula-Based Markov Chains
Abstract:
In this prospectus defense, two different types of copula perturbations are presented. One that modifies existing copulas by adding a factor term to their structure and another that adds some random noise to the underlying random variables. New copula families are brought into the investigation, expanding the scope of perturbation techniques beyond what has been previously considered in the literature. We further discuss the impact of perturbations on dependence and mixing properties of Markov chains. Results are provided for mixing coefficients \beta_n, \psi_n and \phi_n. Moreover, we explore the impact of marginals on mixing considering a case of a Markov chain with Bernoulli marginals. The Maximum likelihood approach is applied to derive estimators of model parameters. Asymptotic distributions of parameter estimators are provided. A simulation study showcases the performance of different estimators for the Bernoulli parameter of the marginal distribution.
Seminars in Spring 2023
5/5/2023
Speaker: Dr. Theophile Bougna
Title: A session on Industry jobs and opportunities
Abstract:
Dr. Theophile Bougna, is an economist in the Development Impact Evaluation team ´
of the World Bank. His current research focuses on transport, urban and spatial economics
with a particular interest on measuring and explaining changes in the spatial distribution of
economic activity. His research has been published in Regional Sciences and Urban Economics and featured in VoxEU. Prior to joining the World Bank, he worked as an Analyst and Statistician at the Cameroon National Office of Statistics (INS), and Consultant for
the Cameroon National Association for Family Welfare (CAMNAFAW-IPPF). Theophile received his Ph.D. degree in Economics from Universite du Quebec a Montreal (UQAM), where ´
he taught several economics courses and was a research assistant for the Canada Research
Chair in Regional Impacts of Globalization. He also holds a Masters degree in Economics
from the Centre d’Etudes et de Recherche sur le Developpement International (CERDI – ´
France), a Bachelor of Engineering in Statistics from E.N.S.E.A (Cote d’Ivoire), and a Bach- ˆ
elor in Economics and Management from Universite de Dschang (Cameroon). You can find ´
his list of publication via the link
https://www.worldbank.org/en/about/people/t/theophile-bougna
4/14/2023
Speaker: Chathurika Abeykoon
Title: Got My Doctorate, What next?
Abstract:
Landing a permanent academic job is not an easy task. With the increase in the
number of PhDs graduating globally, we must compete with hundreds of applicants
contesting for a single position. Along with your prospectus or dissertation stress,
preparing the documents, searching, and applying for jobs, and then getting ready for
interviews are hectic. Even if you wish to move outside academia, you will need to start
your preparations early. Mine was also a rough and rugged journey and I’m happy to
share what I did, my experience, and how I succeeded. This talk will share tips to help
you during the academic job search process to feel more confident in your job search.
3/24/2023
Speaker: Mahmud Hasan
Title: Error analysis of generative adversarial network
Abstract:
I will present error estimation on the generative adversarial network (GAN) which is
the minimax problem of two neural networks developed by Ian Goodfellow who was a
research scientist and director of Google Brain and AI, at Apple. Most of the research
on GAN was on application based where the generalized method needs to be understood
for its error bound and convergence rate ability. The estimation of the error helps in
the analysis of more realistic data. The error bound is developed and explained by a
class of functions combined with the neural network structure of the discriminator and
generator class. The new class of functions is uniformly bounded, VC type concerning
an envelope and VC dimension that helps to get a Talagarnd Inequality. We found a
tight convergence rate for the error of GAN after applying Talagrand inequality and
Borel Cantelli lemma. The error bound and convergence rate was generalized for the
existing error estimation of GAN and we obtain a better convergence rate.
2/17/2023
Speaker: Page Thorn
Title: The Vector Lattice Tensor Product of Ideals
Abstract:
In this brief talk, I introduce Archimedean Riesz spaces, also known as vector
lattices, and their Archimedean Riesz space tensor product as defined by Fremlin. I define
what it means to be an ideal in a Riesz space, then provide a counterexample to the statement
“the Fremlin tensor product of ideals is an ideal.” Specifically, there exist an ideal I in an
Archimedean Riesz space E such that the Fremlin tensor product of I with itself is not an
ideal in the Fremlin tensor product of E with itself.
Seminars in Fall 2022
11/18/2022
Speaker: Kanchana Gamladdalage
Title: The Independent Bondage Number of Planar
Graphs with Girth Conditions
Abstract: For a finite graph G, a vertex set D of G is said to be a dominating set of G, if
every vertex v ∈ V (G) − D has a neighbor in D. Further, if D is independent as well, we
say D is an independent dominating set. Define γi(G) to be the minimum cardinality among
all independent dominating sets of G. The independent bondage number of G denoted by
bi(G), is defined as min{|B| : B ⊂ E(G) such that γi(G − B) > γi(G)}. In this talk, we
show that bi(G) ≤ 5 and bi(G) ≤ 6 for planar graphs of δ(G) ≥ 3 with g(G) ≥ 5 and
g(G) ≥ 4 respectively. Furthermore, we present some upper bounds for the independent
bondage number with other girth conditions
10/14/2022
Speaker: Dr. Emily Pitts Donahoe
Title: Teaching Statements that Work.
How to craft effective and engaging teaching statements that present your pedagogy in ways that are both authentic to you and appealing to hiring committees.
09/23/2022
Speaker: Dr. Sandra Spiroff
Title: Insight to future after Ph.D
Seminars in Spring 2022
04/22/2022
Speaker: Jacob Derricks
Title: Distribution of the Number of Prime Factors of an Integer
Abstract: The distribution of the prime numbers has always intrigued number
theorists. As our understanding of this distribution has evolved, so too have our methods of analyzing the related arithmetic functions. If we let ω(n) denote the number of distinct prime divisors of a natural number n, then the celebrated Erdős-Kac Theorem states that the values of ω(n) are normally distributed (satisfying a central limit theorem as n varies). This result is considered the beginning of Probabilistic Number Theory. We present a modern proof of the Erdős-Kac Theorem using a moment based argument due to Granville and Soundararajan, which we explain in full detail. We also use similar techniques to study the variance of ω(n), refining a classical result of Turán.
03/31/2022
Speaker: Joe Lopez
Title: A survey of Morse Homology
Abstract: One method to conduct data analysis is inspired by Morse Homology. In this talk, we will briefly survey topics in algebraic topology we need to proceed. Afterwards, we will discuss Morse theory, and then dive into Morse Homology. We will verify the Morse inequality relating critical points of a Morse function to the Betti number of a manifold. Then we will examine, depending on time, other topics in Morse Homology.
03/25/2022
Speaker: Stephan Roberts
Title:The Postgraduate Job Market
Abstract: Stephan Roberts is a mathematician at Esports Technologies. Stephan completed his Ph.D. in mathematics at the University of Mississippi in 2018 under the supervision of Professor Gerard Buskes. Since then, he has pursued a position in the gambling industry in which he uses machine learning and statistical modeling to predict outcomes of esports matches. In this Q&A session, Stephan will be sharing his experiences on the job market and on making the transition to an industry environment.
Seminars in Fall 2021
10/01/2021
Speaker: Mathias Muia
Title: Markov Chains and Copulas
Abstract: Copulas are well known tools for understanding the dependence relation among random variables. They offer the choice of appropri- ate models for the dependence between random variables independent from selection of marginal distributions. Often in models it is neces- sary to understand the dependence structures between random vari- ables so as to be able draw wholesome conclusions from data. We are in particular interested in the dependence that arises in a Markov process in R. A copula allows us to describe the dependence between the variables of a random process.
09/17/2021
Speaker: Phil Kains
Title: Introduction to graph theory
Seminars in Spring 2021
03/25/2021
Speaker: Page Thorn
Title: The Fremlin Tensor Product on Subspaces of Continuous Functions
Abstract: The space of continuous functions on a compact Hausdorff space X, denoted C(X), is an excellent example of a Riesz space. Fremlin’s Riesz space tensor product, denoted by \overline{⨂}, enables us to combine two Riesz spaces to create a third, which is rarely accomplished by the algebraic tensor product. In this talk, we explore how various Riesz subspaces are affected in this process. In particular, we make statements of the type: “If A ⊆ C(X) and B ⊆ C(Y) are each of type P in their respective spaces, then A \overline{⨂} B is (or is not) of type P in C(X) \overline{⨂} C(Y).”
03/05/2021
Speaker: Derrick DeMars, Auburn University
Title: Ramsey Theory: An Introduction to Mixed Ramsey Problems
Abstract: We review basic definitions in graph theory that prepare us for concepts related to Ramsey Theory and Mixed Ramsey problems. We define the Mixed Ramsey Spectrum (from Axenovich and Choi) which is for graphs F,H, as the set of colors k, such that for some k-edge-coloring of K_n, there is neither a monochromatic copy of F nor a rainbow copy of H denoted MRS(K_n: F, H). Our result is for integers m and n, where 3 m n,
m {n+2-m, . . . ,(n+( (\binom{m-1}{2})+1)-m)} ⊆ MRS(K_n: C_m, C_n).
02/26/2021
Speaker: Andrew Pham
Title: Independent Bondage Number of Planar Graphs with Minimum Degree at Least 3
Abstract: Given a simple finite graph G, a vertex subset D ⊂ V(G) is said to be a dominating set of G if every vertex v ∈ V(G) – D is adjacent to a vertex in D. The independent domination number γ^i (G) is the minimum cardinality among all independent dominating sets of G, and the independent bondage number b_i (G) is the minimum cardinality among all edge sets E of G such that γ^i (G – E) > γ^i (G). In this talk, we make use of the discharging method and provide a constant upper bound for the independent bondage number of planar graphs with minimum degree at least 3.
Seminars in Fall 2020
10/22/2020
Speaker: Matthew Just
Title: Fermat and the Elite Prime
Abstract: The Fermat numbers are given by Fn = 22^n + 1 for n ≥ 0. It was conjectured by Fermat that all such numbers are prime, a conjecture that was proved for all n 5. Fermat was unable to determine whether F6 was prime, and indeed it is not. An efficient test now exists to determine whether a Fermat number is prime, and it is known that Fn is composite for 6 n 32. A sequence of primes related to this test, called elite primes, are discussed in this talk.
Seminars in Spring 2020
02/27/2020
Speaker: Qinghong Zhao
Title: Multicolor Ramsey numbers of Kipas in Gallai colorings
Abstract: A Gallai coloring of a complete graph is an edge-coloring such that no triangle has all its edges colored differently. A Gallai k-coloring is a Gallai coloring that uses k colors. Given a graph H and an integer k ≥ 1, the Gallai-Ramsey number GRk(H) is defined to be the minimum integer n such that every Gallai k-coloring of the edges of Kn contains a monochromatic copy of H. In this talk, I will present our recent results on Gallai-Ramsey numbers for some graphs with chromatic number three such as K^m for m ≥ 2, where K^m is a kipas with m+1 vertices obtained from the join of K1 and Pm, and a class of graphs with five vertices, denoted by H . We first study the general lower bound of such graphs and propose a conjecture for the exact value of GRk(K^m). Then we give a unified proof to determine the Gallai-Ramsey numbers for many graphs in H and obtain the exact value of GRk(K^4) for all k ≥ 1. Our outcomes not only indicate that the conjecture on GRk(K^m) is true for m = 4, but also imply several results on GRk(H) for some H in H which are proved individually in different papers.
02/20/2020
Speaker: Zachary Tripp
Title: Jensen Polynomials for the Riemann Xi Function
Abstract: In a recent paper by Griffin, Ono, Rolen, and Zagier, the authors discuss an equivalent formulation of the Riemann Hypothesis in terms of polynomials having real roots and give partial progress towards this conjecture. In this talk, which is intended for a general audience, we will explain basic properties of the Riemann zeta function, give the ideas behind this recent paper, and introduce the broader impact this paper has to other areas of number theory, such as partition theory. Moreover, we will discuss the extent to which the results of this paper can be made to be effective, which is based on joint work with Griffin, Ono, Rolen, Thorner, and Wagner.
Seminars in Fall 2019
12/03/2019
Speaker: Ayla Gafni
Title: Job Talk Demonstration
Abstract: The “job talk” is one of the most important aspects of your academic job search, but it’s also one of the most daunting to prepare. It is your chance to introduce yourself and your research to the entire department. A good job talk presents a clear and cohesive research program of past results and future projects, in a way that is understandable to all mathematicians. In this session, Dr. Gafni will demonstrate the job talk she used in interviews last year. After the talk, there will be an informal discussion about her experience on the academic job market.
11/08/2019
Speaker: Joe Lopez
Title: An introduction to Knot Theory
Abstract: When looking at DNA within a cell, how do we study the structure? One answer is to unknot the DNA. In order for cellular replication to occur, quickly knotted DNA must unknot itself. We can study and answer this question in Knot theory. A Knot is a smooth embedding f : S1 → R3. Knot theory is the study of loops in 3-dimensional space. Knot theory is a modern branch of mathematics and a subbranch of Topology. The largest objective of Knot Theory is classifying all Knots. In this talk, we will go over the following: What is a Knot? Classical Knot Invariants. Computing Knot Invariants. Conjectures in Knot Theory.
09/19/2019
Speaker: Carla Cotwright-Williams, Department of Defense, Women in Mathematics Society
Title: Secret Lives of Mathematicians
Abstract: Mathematics can be more than just a subject in school; it can be a career. The government is the number one single employer of mathematicians in the country. Many of those mathematicians end up at the National Security Agency, where they find careers in research, information assurance, and cryptanalysis. This talk will be an introduction to the roles of mathematicians at NSA, as well as basics of cryptography.
Seminars in Fall 2018
11/13/2018
Speaker: Meghann Moriah Gibson
Title: Algebraic Properties and Geometric Applications of Fibonacci Numbers
Abstract: The Fibonacci number sequence is famous for its connection to the Golden Ratio and its appearance within natural objects. There also exist Fibonacci-like sequences that share the same recursive definition without possessing its same connection to the Golden Ratio. We will explain a method for constructing such Fibonacci-like sequences, and subsequently give an example of the relationship between the Golden Ratio and the Fibonacci sequence using a sunflower.
09/23/2018
Speaker: Christopher Seaton, Rhodes College, Department of Mathematics and Computer Science
Title: Distinguishing between symplectic quotients by SU2 using invariant theory
Abstract: Let K be (closed) subgroup of the group of n×n unitary matrices, considered as a group of linear transformations from Cn to itself. Forgetting the complex structure on Cn, the underlying real vector space isomorphic to R2n is an example of a symplectic manifold. The action of K admits a moment map, which in this case is a collection of quadratic polynomials, and the symplectic quotient is denoted to be X = Z/K where Z is the set of points on which the moment map vanishes. While X is usually singular, it has several structures, including well-defined notions of what it means for a function X→R to be smooth or polynomial. The algebra R[X] of polynomial functions on X can be described using the real polynomial invariants associated to the group K. We will describe methods from invariant theory that have been used to study the properties of a symplectic quotient X via R[X]. We will in particular focus on the case K = SU2, the 2×2 unitary matrices with determinant 1, and discuss recent results regarding distinguishing between symplectic quotients.
09/20/2018
Speaker: Zhenchao Ge
Title: Subspaces in difference sets in vector spaces
Abstract: A common theme in additive combinatorics states that if A is a subset of positive density of a vector space Fpn, then the difference set A − A must contain a large subspace. A result of Sanders says that when p=2 and A has density at least 1/2 − c/√n, A − A must contain a subspace of co-dimension 1. We generalize this result to all p while giving a simpler and elementary proof. Our proof is based on an argument of Wirsing. This is joint work with Thai Hoang Le.
09/14/2018
Speaker: Brad Cole, Senior Mathematician for BEN Markets
Title: A Q&A with Brad Cole about his role at BEN Markets
Abstract: In 2011, local resident Jason Finch founded Binary Event Network, a start up, in New York City to combine public opinion, news and data to produce live odds on global event outcomes. Former graduate students Brad Cole and Adam Gray accepted positions with BEN Markets, and Mr. Cole is now Senior Mathematician on staff. Mr. Cole will discuss the day to day life of a sports quant, how he learned of the opportunity, and how recent changes to sports gaming laws may change the industry.
Seminars in Spring 2018
01/26/2018
Speaker: Irina Ilioaea, Georgia State University
Title: Finding Infeasible Cores of a Set of Polynomials using the Gröbner Basis Algorithm
Abstract: This talk investigates an algorithmic approach to identify a small unsatisfiable core of an ideal I in K[x1, . . . , xn], where K is a field and the ideal I is found to have an empty variety. The main aim of the talk will be to introduce the fundamental notions and to illustrate the concepts we use by examples. We identify certain conditions that are helpful in deciding whether or not a polynomial from the given generating set is a part of the unsat core. Our algorithm cannot guarantee a minimal unsat core; hence the talk discusses opportunities for refinement of the identified core.
Seminars in Fall 2017
11/03/2017
Speaker: Sumeet Kulkarni, Department of Physics
Title: Use of Randomised Techniques in Gravitational Wave Searches
Abstract: The recent joint discovery by LIGO-Virgo of Gravitational waves (GW) emitted from a pair of coalescing Neutron Stars, and the consequent observations of the system across the Electromagnetic spectrum by numerous observatories has kicked off the era of multi-messenger astronomy. To enable more such observations in the future, it is important to detect incoming gravitational wave signals as soon as possible to generate alerts for astronomers worldwide to follow-up. Gravitational wave detection involves cross-correlating the detector data with millions of modeled templates, each sampled at millions of templates. Given the magnitude of the databases, data reduction techniques provide extremely useful solutions to speeding up GW searches. In the past, singular value decomposition (SVD) has been applied to eliminate degeneracies that exist within a template bank. Weaddress some of its limitations by employing the technique of Random Projections, inspired by the Johnson-Lindenstrauss lemma (1984). Popular in other avenues of data science for dimensionality reduction, this method proves to be surprisingly helpful for reducing GW templates. I’ll present an overview of GW data analysis through this perspective and discuss the use of Randomised techniques in order to reduce the complexity of the search operations.
10/27/2017
Speaker: Zhenchao Ge
Title: Primes in Arithmetic Progression
Abstract: Euler proved the existence of infinitely many primes by showing that the sum of reciprocals of primes is divergent. By the same idea, Dirichlet proved that for every coprime a and q, the arithmetic progression a, a + q, a + 2q, … contains the infinitely many primes. In this talk, we sketch Euler and Dirichlet’s proof. Also, we introduce Dirichlet characters and character orthogonality.
10/13/2017
Speaker: Sashwat Tanay, Department of Physics
Title: Bivariate Power Series via a Perturbative Approach
Abstract: Power series are one of the most important inventions in mathematics and have been employed to solve a myriad of problems in the physical science. There are multiple ways to obtain a power series (Taylor’s formula, perturbative approach etc.) but the uniqueness theorem guarantees that all such methods yield the same answer. In this talk, the perturbative approach to obtain the power series (in two variables) is discussed. The perturbative approach is one where we first take into account only the leading order input information to obtain the leading order solution and gradually process the higher order terms of the input information to churn out the higher order terms of the solution as a power series. This happens order by order. This method is first demonstrated for the case of a much simpler univariate sine series because it captures the essence of the problem, which is then employed to the more complicated bivariate case. Finally, the physical context of this mathematical procedure is discussed briefly. The method finds application in simulating gravitational waves, which is crucial to their detection. The 2017 Nobel prize in physics was awarded to three key members of the LIGO collaboration (Profs. Weiss, Thorne and Barish) for their contributions to gravitational wave detections.
10/06/2017
Speaker: Khazhakanush Navoyan
Title: Connected Spaces
Abstract: Let X be a topological space. A separation of X is a pair of U, V of disjoint nonempty open subsets of X whose union is X. The space X is said to be connected if there does not exist a separation of X. Connectedness is obviously a topological property, since it is formulated entirely in terms of the collection of open sets of X. Said differently, if X is connected, so is any space homeomorphic to X. The definition of connectedness for a topological space is a quite natural one. One says that a space can be “separated” if it can be broken up into two “globs”- disjoint open sets. Otherwise, one says that it is connected.
09/08/2017
Speaker: Khazhakanush Navoyan
Title: On Spectrum and Resolvent Set, Part 1
Abstract: In this talk we define resolvent set and spectrum of a linear norm bounded operator T in a complex Banach space V. In particular, we deal with the case that V is a Banach lattice and the operator T is positive. We show that the spectrum is a closed and bounded set in the complex plane C and discuss the properties of spectral radius, which is the radius of the smallest circle having as its centre the origin and containing the spectrum..