scholarly journals The Minimum Numbers for Certain Positive Operators

2020 ◽  
Vol 12 (5) ◽  
pp. 15
Author(s):  
Ching-Yun Suen
Keyword(s):  
Hilbert Space ◽  
Lower Bounds ◽  
Positive Operators ◽  
Upper And Lower Bounds ◽  
Numerical Radius ◽  
Minimum Numbers

In this paper we give upper and lower bounds of the infimum of k  such that kI+2ReT⊗Sm  is positive, where Sm  is the m×m  matrix whose entries are all 0’s except on the superdiagonal where they are all 1’s and T∈BH  for some Hilbert space H. When T  is self-adjoint, we have the minimum of k. When m=3  and T∈B(H)  , we obtain the minimum of k  and an inequality Involving the numerical radius w(T) .

9.—Bivariational Bounds associated with Non-self-adjoint Linear Operators

1976 ◽  
Vol 75 (2) ◽  
pp. 109-118 ◽  
Author(s):  
Michael F. Barnsley ◽  
Peter D. Robinson
Keyword(s):  
Hilbert Space ◽  
Lower Bounds ◽  
Positive Constant ◽  
Real Hilbert Space ◽  
Linear Operators ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Inner Product ◽  
Upper And Lower Bounds ◽  
Fredholm Integral Equations

SYNOPSISLet A be a closed linear transformation from a real Hilbert space ℋ, with symmetric inner product 〈, 〉, into itself; and let f ∈ ℋ be given such that the problem Aø = f has a solution ø ∈ D(A), the domain of A. Then bivariational upper and lower bounds on 〈g, ø〉 for any g ∈ ℋ are exhibited when there exists a positive constant a such that 〈AΦ, AΦ⊖ ≧ a2〈Φ, Φ〉 for all Φ ∈ D(A). The applicability of the theory both to Fredholm integral equations and also to time-dependent diffusion equations is demonstrated.

scholarly journals Programmability of covariant quantum channels

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 488
Author(s):  
Martina Gschwendtner ◽  
Andreas Bluhm ◽  
Andreas Winter
Keyword(s):  
Hilbert Space ◽  
Lower Bounds ◽  
Group Action ◽  
Space Dimension ◽  
Upper And Lower Bounds ◽  
Quantum Channels ◽  
Covariant Quantum ◽  
Finite Dimensional ◽  
Channel Input ◽  
Quantum Processor

A programmable quantum processor uses the states of a program register to specify one element of a set of quantum channels which is applied to an input register. It is well-known that such a device is impossible with a finite-dimensional program register for any set that contains infinitely many unitary quantum channels (Nielsen and Chuang's No-Programming Theorem), meaning that a universal programmable quantum processor does not exist. The situation changes if the system has symmetries. Indeed, here we consider group-covariant channels. If the group acts irreducibly on the channel input, these channels can be implemented exactly by a programmable quantum processor with finite program dimension (via teleportation simulation, which uses the Choi-Jamiolkowski state of the channel as a program). Moreover, by leveraging the representation theory of the symmetry group action, we show how to remove redundancy in the program and prove that the resulting program register has minimum Hilbert space dimension. Furthermore, we provide upper and lower bounds on the program register dimension of a processor implementing all group-covariant channels approximately.

Complementary bivariational principles for linear problems involving non-self-adjoint operators

1980 ◽  
Vol 86 (1-2) ◽  
pp. 115-128 ◽  
Author(s):  
R. J. Cole
Keyword(s):  
Hilbert Space ◽  
Integral Equation ◽  
Linear Operator ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Fredholm Type ◽  
Linear Problems ◽  
Adjoint Operators

SynopsisBivariational principles are constructed that yield upper and lower bounds to the quantity 〈g0, f〉, where f is the solution of the equation f0−Tf = 0, g0 is a given function and T is a non-self-adjoint linear operator from a Hilbert space into itself. The theory is illustrated by an integral equation of Fredholm type.

Upper and lower bounds for solutions of linear operator problems with unilateral constraints

1977 ◽  
Vol 76 (2) ◽  
pp. 95-105 ◽  
Author(s):  
W. D. Collins
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Lower Bounds ◽  
Original Problem ◽  
Adjoint Operator ◽  
Decomposition Theorem ◽  
Positive Definite ◽  
Upper And Lower Bounds ◽  
Unilateral Constraints ◽  
Space Decomposition

SynopsisDual extremum principles characterising the solutions of problems for a positive-definite self-adjoint operator on a Hilbert space which involve unilateral constraints are formulated using a Hilbert space decomposition theorem due to Moreau. Various upper and lower bounds to these solutions are then obtained, these bounds involving the solutions to subsidiary problems with less restrictive conditions than the solution to the original problem.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Hadamard k-fractional inequalities of Fejér type for GA-s-convex mappings and applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hui Lei ◽  
Gou Hu ◽  
Zhi-Jie Cao ◽  
Ting-Song Du
Keyword(s):  
Lower Bounds ◽  
Hypergeometric Functions ◽  
Upper And Lower Bounds ◽  
Weighted Inequalities ◽  
Convex Mappings ◽  
Special Means

Abstract The main aim of this paper is to establish some Fejér-type inequalities involving hypergeometric functions in terms of GA-s-convexity. For this purpose, we construct a Hadamard k-fractional identity related to geometrically symmetric mappings. Moreover, we give the upper and lower bounds for the weighted inequalities via products of two different mappings. Some applications of the presented results to special means are also provided.

scholarly journals On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 512
Author(s):  
Maryam Baghipur ◽  
Modjtaba Ghorbani ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Distance Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Simple Connected Graph ◽  
Second Largest Eigenvalue ◽  
Reciprocal Distance Matrix

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

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