scholarly journals MATRIX THEORY, HILBERT SCHEME AND INTEGRABLE SYSTEM

1998 ◽  
Vol 13 (34) ◽  
pp. 2731-2742 ◽  
Author(s):  
YUTAKA MATSUO
Keyword(s):  
Hilbert Scheme ◽  
Matrix Theory ◽  
Fock Space ◽  
Homology Class ◽  
Orthogonal Basis ◽  
Inner Product ◽  
Virasoro Symmetry ◽  
Hilbert Scheme Of Points ◽  
The Matrix ◽  
Boson Fock Space

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relates the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These bases can be related to the eigenfunctions of Calogero–Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model.

scholarly journals Donaldson–Thomas theory of 𝒜n×P1

2009 ◽  
Vol 145 (5) ◽  
pp. 1249-1276 ◽  
Author(s):  
Davesh Maulik ◽  
Alexei Oblomkov
Keyword(s):  
Hilbert Scheme ◽  
Fock Space ◽  
Complete Solution ◽  
Quantum Cohomology ◽  
Affine Algebra ◽  
Hilbert Scheme Of Points ◽  
Witten Theory

AbstractWe study the relative Donaldson–Thomas theory of 𝒜n×P1, where 𝒜n is the surface resolution of type An singularity. The action of divisor operators in the theory is expressed in terms of operators of the affine algebra $\glh $ on Fock space. Assuming a nondegeneracy conjecture, this gives a complete solution for the theory. The results complete the comparison of this theory with the Gromov–Witten theory of 𝒜n×P1 and the quantum cohomology of the Hilbert scheme of points on 𝒜n.

scholarly journals Partial Component Consensus of Discrete-Time Multiagent Systems

2019 ◽  
Vol 2019 ◽  
pp. 1-5
Author(s):  
Wenjun Hu ◽  
Gang Zhang ◽  
Zhongjun Ma ◽  
Binbin Wu
Keyword(s):  
Multiagent Systems ◽  
Discrete Time ◽  
Matrix Theory ◽  
Pinning Control ◽  
Directed Network ◽  
Strong Function ◽  
Partial Stability ◽  
Partial Component ◽  
The Matrix ◽  
Error System

The multiagent system has the advantages of simple structure, strong function, and cost saving, which has received wide attention from different fields. Consensus is the most basic problem in multiagent systems. In this paper, firstly, the problem of partial component consensus in the first-order linear discrete-time multiagent systems with the directed network topology is discussed. Via designing an appropriate pinning control protocol, the corresponding error system is analyzed by using the matrix theory and the partial stability theory. Secondly, a sufficient condition is given to realize partial component consensus in multiagent systems. Finally, the numerical simulations are given to illustrate the theoretical results.

scholarly journals Singular Bogoliubov transformations and inequivalent representations of canonical commutation relations

2019 ◽  
Vol 31 (08) ◽  
pp. 1950026 ◽  
Author(s):  
Asao Arai
Keyword(s):  
Fock Space ◽  
Sufficient Conditions ◽  
Bogoliubov Transformation ◽  
Fock Representation ◽  
Canonical Commutation Relations ◽  
Commutation Relations ◽  
Direct Sum ◽  
Boson Fock Space ◽  
Inequivalent Representations ◽  
Bogoliubov Transformations

We introduce a concept of singular Bogoliubov transformation on the abstract boson Fock space and construct a representation of canonical commutation relations (CCRs) which is inequivalent to any direct sum of the Fock representation. Sufficient conditions for the representation to be irreducible are formulated. Moreover, an example of such representations of CCRs is given.

scholarly journals The integral cohomology of the Hilbert scheme of points on a surface

2020 ◽  
Vol 8 ◽  
Author(s):  
Burt Totaro
Keyword(s):  
Natural Number ◽  
Hilbert Scheme ◽  
Obstruction Theory ◽  
Projective Surface ◽  
Hilbert Schemes ◽  
Complex Projective Surface ◽  
Smooth Complex ◽  
Hilbert Scheme Of Points ◽  
Complex Projective ◽  
Torsion Free

Abstract We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme $X^{[n]}$ has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas’s reduced obstruction theory for nested Hilbert schemes of surfaces.

scholarly journals All about the ⊥ with its applications in the linear statistical models

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Augustyn Markiewicz ◽  
Simo Puntanen
Keyword(s):  
Statistical Models ◽  
Inner Product ◽  
Real Matrix ◽  
Inner Products ◽  
Column Space ◽  
The Matrix ◽  
Linear Statistical Models ◽  
Extensive List

Abstract For an n x m real matrix A the matrix A⊥ is defined as a matrix spanning the orthocomplement of the column space of A, when the orthogonality is defined with respect to the standard inner product ⟨x, y⟩ = x'y. In this paper we collect together various properties of the ⊥ operation and its applications in linear statistical models. Results covering the more general inner products are also considered. We also provide a rather extensive list of references

scholarly journals LOCALIZATION OF THE NUMBER OF PHOTONS OF GROUND STATES IN NONRELATIVISTIC QED

2003 ◽  
Vol 15 (03) ◽  
pp. 271-312 ◽  
Author(s):  
FUMIO HIROSHIMA
Keyword(s):  
Radiation Field ◽  
Ground State ◽  
Fock Space ◽  
Adjoint Operator ◽  
Ground States ◽  
Electron System ◽  
Number Operator ◽  
Position Variable ◽  
Boson Fock Space ◽  
Quantized Radiation Field

One electron system minimally coupled to a quantized radiation field is considered. It is assumed that the quantized radiation field is massless, and no infrared cutoff is imposed. The Hamiltonian, H, of this system is defined as a self-adjoint operator acting on L2 (ℝ3) ⊗ ℱ ≅ L2 (ℝ3; ℱ), where ℱ is the Boson Fock space over L2 (ℝ3 × {1, 2}). It is shown that the ground state, ψg, of H belongs to [Formula: see text], where N denotes the number operator of ℱ. Moreover, it is shown that for almost every electron position variable x ∈ ℝ3 and for arbitrary k ≥ 0, ‖(1 ⊗ Nk/2) ψg (x)‖ℱ ≤ Dk e-δ|x|m+1 with some constants m ≥ 0, Dk > 0, and δ > 0 independent of k. In particular [Formula: see text] for 0 < β < δ/2 is obtained.

scholarly journals Toeplitz Operators with Lagrangian Invariant Symbols Acting on the Poly-Fock Space of ℂ n

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Jorge Luis Arroyo Neri ◽  
Armando Sánchez-Nungaray ◽  
Mauricio Hernández Marroquin ◽  
Raquiel R. López-Martínez
Keyword(s):  
Toeplitz Operators ◽  
Complex Space ◽  
Fock Space ◽  
Identity Matrix ◽  
The Matrix

We introduce the so-called extended Lagrangian symbols, and we prove that the C ∗ -algebra generated by Toeplitz operators with these kind of symbols acting on the homogeneously poly-Fock space of the complex space ℂ n is isomorphic and isometric to the C ∗ -algebra of matrix-valued functions on a certain compactification of ℝ n obtained by adding a sphere at the infinity; moreover, the matrix values at the infinity points are equal to some scalar multiples of the identity matrix.

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