Eigenvalues with Geometric Multiplicity One

2013 ◽  
pp. 13-31
Author(s):  
Yuli Eidelman ◽  
Israel Gohberg ◽  
Iulian Haimovici
Keyword(s):  
Geometric Multiplicity ◽  
Multiplicity One

scholarly journals Functional analysis method for the M/G/1 queueing model with single working vacation

2018 ◽  
Vol 16 (1) ◽  
pp. 767-791 ◽  
Author(s):  
Ehmet Kasim ◽  
Geni Gupur
Keyword(s):  
Adjoint Operator ◽  
Steady State Solution ◽  
Time Dependent ◽  
Queueing Model ◽  
Working Vacation ◽  
Geometric Multiplicity ◽  
Resolvent Set ◽  
Multiplicity One ◽  
Vacation Period ◽  
Time Dependent Solution

AbstractIn this paper, we study the asymptotic property of underlying operator corresponding to the M/G/1 queueing model with single working vacation, where both service times in a regular busy period and in a working vacation period are function. We obtain that all points on the imaginary axis except zero belong to the resolvent set of the operator and zero is an eigenvalue of both the operator and its adjoint operator with geometric multiplicity one. Therefore, we deduce that the time-dependent solution of the queueing model strongly converges to its steady-state solution. We also study the asymptotic behavior of the time-dependent queueing system’s indices for the model.

The k-Zero Bifurcation Theorem for m-Parameterized Systems

2018 ◽  
Vol 28 (08) ◽  
pp. 1850100
Author(s):  
Martin A. Carrillo ◽  
Fernando Verduzco ◽  
Francisco A. Carrillo
Keyword(s):  
Equilibrium Point ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Geometric Multiplicity ◽  
Parameterized Systems ◽  
Multiplicity One ◽  
The Family ◽  
Parameterized Family ◽  
Multiplicity Formula ◽  
Eigenvalue Zero

Given an [Formula: see text]-parameterized family of [Formula: see text]-dimensional vector fields, with an equilibrium point with linearization of eigenvalue zero with algebraic multiplicity [Formula: see text], with [Formula: see text], and geometric multiplicity one, our goal in this paper is to find sufficient conditions for the family of vector fields such that the dynamics on the [Formula: see text]-dimensional [Formula: see text]-parameterized center manifold around the equilibrium point becomes locally topologically equivalent to a given unfolding. Finally, the result is applied to the study of the Rössler system.

scholarly journals The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree

2019 ◽  
Vol 7 (1) ◽  
pp. 257-262
Author(s):  
Kenji Toyonaga
Keyword(s):  
Symmetric Matrix ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Geometric Multiplicity ◽  
Necessary And Sufficient ◽  
Multiplicity Of An Eigenvalue

Abstract Given a combinatorially symmetric matrix A whose graph is a tree T and its eigenvalues, edges in T can be classified in four categories, based upon the change in geometric multiplicity of a particular eigenvalue, when the edge is removed. We investigate a necessary and sufficient condition for each classification of edges. We have similar results as the case for real symmetric matrices whose graph is a tree. We show that a g-2-Parter edge, a g-Parter edge and a g-downer edge are located separately from each other in a tree, and there is a g-neutral edge between them. Furthermore, we show that the distance between a g-downer edge and a g-2-Parter edge or a g-Parter edge is at least 2 in a tree. Lastly we give a combinatorially symmetric matrix whose graph contains all types of edges.

On Lagrangian Catenoids

2007 ◽  
Vol 50 (3) ◽  
pp. 321-333 ◽  
Author(s):  
David E. Blair
Keyword(s):  
General Problem ◽  
Lagrangian Submanifolds ◽  
Minimal Submanifold ◽  
Conformally Flat ◽  
Totally Geodesic ◽  
Schouten Tensor ◽  
Multiplicity One

AbstractRecently I. Castro and F.Urbano introduced the Lagrangian catenoid. Topologically, it is ℝ × Sn–1 and its induced metric is conformally flat, but not cylindrical. Their result is that if a Lagrangian minimal submanifold in ℂn is foliated by round (n – 1)-spheres, it is congruent to a Lagrangian catenoid. Here we study the question of conformally flat, minimal, Lagrangian submanifolds in ℂn. The general problem is formidable, but we first show that such a submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an eigenvalue of multiplicity one. Then, restricting to the case of at most two eigenvalues, we show that the submanifold is either flat and totally geodesic or is homothetic to (a piece of) the Lagrangian catenoid.

scholarly journals All-multiplicity one-loop amplitudes in Born-Infeld electrodynamics from generalized unitarity

2020 ◽  
Vol 2020 (3) ◽  
Author(s):  
Henriette Elvang ◽  
Marios Hadjiantonis ◽  
Callum R. T. Jones ◽  
Shruti Paranjape
Keyword(s):  
Multiplicity One ◽  
Loop Amplitudes

scholarly journals Proof of de Smit’s conjecture: a freeness criterion

2017 ◽  
Vol 153 (11) ◽  
pp. 2310-2317
Author(s):  
Sylvain Brochard
Keyword(s):  
Complete Intersection ◽  
Local Rings ◽  
Embedding Dimension ◽  
Fermat's Last Theorem ◽  
Multiplicity One ◽  
Fermat’S Last Theorem ◽  
Invent Math

Let $A\rightarrow B$ be a morphism of Artin local rings with the same embedding dimension. We prove that any $A$-flat $B$-module is $B$-flat. This freeness criterion was conjectured by de Smit in 1997 and improves Diamond’s criterion [The Taylor–Wiles construction and multiplicity one, Invent. Math. 128 (1997), 379–391, Theorem 2.1]. We also prove that if there is a nonzero $A$-flat $B$-module, then $A\rightarrow B$ is flat and is a relative complete intersection. Then we explain how this result allows one to simplify Wiles’s proof of Fermat’s last theorem: we do not need the so-called ‘Taylor–Wiles systems’ any more.

scholarly journals On the Number of Partition Weights with Kostka Multiplicity One

10.37236/2574 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Zachary Gates ◽  
Brian Goldman ◽  
C. Ryan Vinroot
Keyword(s):  
Positive Integer ◽  
Ordered Set ◽  
Unitary Groups ◽  
Character Theory ◽  
Partition Functions ◽  
Young Tableaux ◽  
Multiplicity One ◽  
Restricted Partition ◽  
Finite Unitary Groups ◽  
Kostka Number

Given a positive integer $n$, and partitions $\lambda$ and $\mu$ of $n$, let $K_{\lambda \mu}$ denote the Kostka number, which is the number of semistandard Young tableaux of shape $\lambda$ and weight $\mu$.  Let $J(\lambda)$ denote the number of $\mu$ such that $K_{\lambda \mu} = 1$.  By applying a result of Berenshtein and Zelevinskii, we obtain a formula for $J(\lambda)$ in terms of restricted partition functions, which is recursive in the number of distinct part sizes of $\lambda$.  We use this to classify all partitions $\lambda$ such that $J(\lambda) = 1$ and all $\lambda$ such that $J(\lambda) = 2$.  We then consider signed tableaux, where a semistandard signed tableau of shape $\lambda$ has entries from the ordered set $\{0 < \bar{1} < 1 < \bar{2} < 2 < \cdots \}$, and such that $i$ and $\bar{i}$ contribute equally to the weight.  For a weight $(w_0, \mu)$ with $\mu$ a partition, the signed Kostka number $K^{\pm}_{\lambda,(w_0, \mu)}$ is defined as the number of semistandard signed tableaux of shape $\lambda$ and weight $(w_0, \mu)$, and $J^{\pm}(\lambda)$ is then defined to be the number of weights $(w_0, \mu)$ such that $K^{\pm}_{\lambda, (w_0, \mu)} = 1$.  Using different methods than in the unsigned case, we find that the only nonzero value which $J^{\pm}(\lambda)$ can take is $1$, and we find all sequences of partitions with this property.  We conclude with an application of these results on signed tableaux to the character theory of finite unitary groups.

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