A Proof Of Alons Second Eigenvalue Conjecture And Related Problems
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Author | : Joel Friedman |
Publisher | : American Mathematical Soc. |
Total Pages | : 114 |
Release | : 2008 |
Genre | : Mathematics |
ISBN | : 0821842803 |
A $d$-regular graph has largest or first (adjacency matrix) eigenvalue $\lambda_1=d$. Consider for an even $d\ge 4$, a random $d$-regular graph model formed from $d/2$ uniform, independent permutations on $\{1,\ldots,n\}$. The author shows that for any $\epsilon>0$ all eigenvalues aside from $\lambda_1=d$ are bounded by $2\sqrt{d-1}\;+\epsilon$ with probability $1-O(n^{-\tau})$, where $\tau=\lceil \bigl(\sqrt{d-1}\;+1\bigr)/2 \rceil-1$. He also shows that this probability is at most $1-c/n^{\tau'}$, for a constant $c$ and a $\tau'$ that is either $\tau$ or $\tau+1$ (``more often'' $\tau$ than $\tau+1$). He proves related theorems for other models of random graphs, including models with $d$ odd.
Author | : Irina D. Suprunenko |
Publisher | : American Mathematical Soc. |
Total Pages | : 168 |
Release | : 2009-06-05 |
Genre | : Mathematics |
ISBN | : 0821843699 |
The minimal polynomials of the images of unipotent elements in irreducible rational representations of the classical algebraic groups over fields of odd characteristic are found. These polynomials have the form $(t-1)^d$ and hence are completely determined by their degrees. In positive characteristic the degree of such polynomial cannot exceed the order of a relevant element. It occurs that for each unipotent element the degree of its minimal polynomial in an irreducible representation is equal to the order of this element provided the highest weight of the representation is large enough with respect to the ground field characteristic. On the other hand, classes of unipotent elements for which in every nontrivial representation the degree of the minimal polynomial is equal to the order of the element are indicated. In the general case the problem of computing the minimal polynomial of the image of a given element of order $p^s$ in a fixed irreducible representation of a classical group over a field of characteristic $p>2$ can be reduced to a similar problem for certain $s$ unipotent elements and a certain irreducible representation of some semisimple group over the field of complex numbers. For the latter problem an explicit algorithm is given. Results of explicit computations for groups of small ranks are contained in Tables I-XII. The article may be regarded as a contribution to the programme of extending the fundamental results of Hall and Higman (1956) on the minimal polynomials from $p$-solvable linear groups to semisimple groups.
Author | : Antonino Morassi |
Publisher | : American Mathematical Soc. |
Total Pages | : 74 |
Release | : 2009-06-05 |
Genre | : Mathematics |
ISBN | : 0821843257 |
The authors consider the inverse problem of determining a rigid inclusion inside an isotropic elastic body $\Omega$, from a single measurement of traction and displacement taken on the boundary of $\Omega$. For this severely ill-posed problem they prove uniqueness and a conditional stability estimate of log-log type.
Author | : Gelu Popescu |
Publisher | : American Mathematical Soc. |
Total Pages | : 105 |
Release | : 2009-06-05 |
Genre | : Mathematics |
ISBN | : 0821843966 |
This paper concerns unitary invariants for $n$-tuples $T:=(T_1,\ldots, T_n)$ of (not necessarily commuting) bounded linear operators on Hilbert spaces. The author introduces a notion of joint numerical radius and works out its basic properties. Multivariable versions of Berger's dilation theorem, Berger-Kato-Stampfli mapping theorem, and Schwarz's lemma from complex analysis are obtained. The author studies the joint (spatial) numerical range of $T$ in connection with several unitary invariants for $n$-tuples of operators such as: right joint spectrum, joint numerical radius, euclidean operator radius, and joint spectral radius. He also proves an analogue of Toeplitz-Hausdorff theorem on the convexity of the spatial numerical range of an operator on a Hilbert space, for the joint numerical range of operators in the noncommutative analytic Toeplitz algebra $F_n^\infty$.
Author | : Sergiu Aizicovici |
Publisher | : American Mathematical Soc. |
Total Pages | : 84 |
Release | : 2008 |
Genre | : Mathematics |
ISBN | : 0821841920 |
In this paper the authors examine the degree map of multivalued perturbations of nonlinear operators of monotone type and prove that at a local minimizer of the corresponding Euler functional, this degree equals one.
Author | : Yoshikata Kida |
Publisher | : American Mathematical Soc. |
Total Pages | : 206 |
Release | : 2008 |
Genre | : Mathematics |
ISBN | : 0821841963 |
The author obtains some classification result for the mapping class groups of compact orientable surfaces in terms of measure equivalence. In particular, the mapping class groups of different closed surfaces cannot be measure equivalent. Moreover, the author gives various examples of discrete groups which are not measure equivalent to the mapping class groups. In the course of the proof, the author investigates amenability in a measurable sense for the actions of the mapping class group on the boundary at infinity of the curve complex and on the Thurston boundary and, using this investigation, proves that the mapping class group of a compact orientable surface is exact.
Author | : Jonathan Brundan |
Publisher | : American Mathematical Soc. |
Total Pages | : 122 |
Release | : 2008 |
Genre | : Mathematics |
ISBN | : 0821842161 |
The authors study highest weight representations of shifted Yangians over an algebraically closed field of characteristic $0$. In particular, they classify the finite dimensional irreducible representations and explain how to compute their Gelfand-Tsetlin characters in terms of known characters of standard modules and certain Kazhdan-Lusztig polynomials. The authors' approach exploits the relationship between shifted Yangians and the finite W-algebras associated to nilpotent orbits in general linear Lie algebras.
Author | : Georgia Benkart |
Publisher | : American Mathematical Soc. |
Total Pages | : 164 |
Release | : 2009 |
Genre | : Mathematics |
ISBN | : 0821842269 |
"Volume 197, number 920 (second of 5 numbers)."
Author | : Yufei Zhao |
Publisher | : Cambridge University Press |
Total Pages | : 336 |
Release | : 2023-07-31 |
Genre | : Mathematics |
ISBN | : 1009310933 |
Using the dichotomy of structure and pseudorandomness as a central theme, this accessible text provides a modern introduction to extremal graph theory and additive combinatorics. Readers will explore central results in additive combinatorics-notably the cornerstone theorems of Roth, Szemerédi, Freiman, and Green-Tao-and will gain additional insights into these ideas through graph theoretic perspectives. Topics discussed include the Turán problem, Szemerédi's graph regularity method, pseudorandom graphs, graph limits, graph homomorphism inequalities, Fourier analysis in additive combinatorics, the structure of set addition, and the sum-product problem. Important combinatorial, graph theoretic, analytic, Fourier, algebraic, and geometric methods are highlighted. Students will appreciate the chapter summaries, many figures and exercises, and freely available lecture videos on MIT OpenCourseWare. Meant as an introduction for students and researchers studying combinatorics, theoretical computer science, analysis, probability, and number theory, the text assumes only basic familiarity with abstract algebra, analysis, and linear algebra.
Author | : Marius Junge |
Publisher | : American Mathematical Soc. |
Total Pages | : 168 |
Release | : 2010 |
Genre | : Mathematics |
ISBN | : 0821846558 |
Contains the proof of a noncommutative analogue of the inequality for sums of free random variables over a given von Neumann subalgebra.