Efficiency increasing of the bidirectional teleportation protocol via weak and reversal measurements

In this suggested version of the bidirectional teleportation protocol, it is assumed that the used quantum channel passes through an amplitude damping channel. Therefore, some of its quantum correlations (entanglement) are lost and, consequently, its efficiency to implement this protocol decreases. The weak and the reversal measurements are used to recover the losses of these correlations, where the negativity, as a measure of entanglement is improved. In this context, we discussed the effect of the noisy strength on the fidelities of the bidirectional teleported states between the users. It is shown that, by applying the weak and the reversal measurements (WRM) on the initial quantum channel between the users, the fidelities of the teleported states are improved. Moreover, we showed that, the upper bounds of the teleported states depend on the initial states of the triggers and the strengths of WRM. It is worth noting that the WRM improves the quantum correlations of the shared channel and, hence, the fidelity of the teleported state if the initial fidelity of the teleported state is larger than 0.5

On the upper bounds for complexities of discrete functions

In this paper, we study two classes of complexity measures induced by new data structures (abstract reduction systems) for representing [Formula: see text]-valued functions (operations), namely subfunction and minor reductions. When assigning values to some variables in a function, the resulting functions are called subfunctions, and when identifying some variables, the resulting functions are called minors. The number of the distinct objects obtained under these reductions of a function [Formula: see text] is a well-defined measure of complexity denoted by [Formula: see text] and [Formula: see text], respectively. We examine the maximums of these complexities and construct functions which reach these upper bounds.

Some Upper Bounds on the First General Zagreb Index

The first general Zagreb index M γ G or zeroth-order general Randić index of a graph G is defined as M γ G = ∑ v ∈ V d v γ where γ is any nonzero real number, d v is the degree of the vertex v and γ = 2 gives the classical first Zagreb index. The researchers investigated some sharp upper and lower bounds on zeroth-order general Randić index (for γ < 0 ) in terms of connectivity, minimum degree, and independent number. In this paper, we put sharp upper bounds on the first general Zagreb index in terms of independent number, minimum degree, and connectivity for γ . Furthermore, extremal graphs are also investigated which attained the upper bounds.

MAD Dispersion Measure Makes Extremal Queue Analysis Simple

A notorious problem in queueing theory is to compute the worst possible performance of the GI/G/1 queue under mean-dispersion constraints for the interarrival- and service-time distributions. We address this extremal queue problem by measuring dispersion in terms of mean absolute deviation (MAD) instead of the more conventional variance, making available methods for distribution-free analysis. Combined with random walk theory, we obtain explicit expressions for the extremal interarrival- and service-time distributions and, hence, the best possible upper bounds for all moments of the waiting time. We also obtain tight lower bounds that, together with the upper bounds, provide robust performance intervals. We show that all bounds are computationally tractable and remain sharp also when the mean and MAD are not known precisely but are estimated based on available data instead. Summary of Contribution: Queueing theory is a classic OR topic with a central role for the GI/G/1 queue. Although this queueing system is conceptually simple, it is notoriously hard to determine the worst-case expected waiting time when only knowing the first two moments of the interarrival- and service-time distributions. In this setting, the exact form of the extremal distribution can only be determined numerically as the solution to a nonconvex nonlinear optimization problem. Our paper demonstrates that using mean absolute deviation (MAD) instead of variance alleviates the computational intractability of the extremal GI/G/1 queue problem, enabling us to state the worst-case distributions explicitly.

On Numerical Radius Bounds Involving Generalized Aluthge Transform

In this paper, we establish some upper bounds of the numerical radius of a bounded linear operator S defined on a complex Hilbert space with polar decomposition S = U ∣ S ∣ , involving generalized Aluthge transform. These bounds generalize some bounds of the numerical radius existing in the literature. Moreover, we consider particular cases of generalized Aluthge transform and give some examples where some upper bounds of numerical radius are computed and analyzed for certain operators.

Computability-theoretic complexity of effective Banach spaces

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

Computability-theoretic complexity of effective Banach spaces

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

Schur Complement-Based Infinity Norm Bounds for the Inverse of GDSDD Matrices

Based on the Schur complement, some upper bounds for the infinity norm of the inverse of generalized doubly strictly diagonally dominant matrices are obtained. In addition, it is shown that the new bound improves the previous bounds. Numerical examples are given to illustrate our results. By using the infinity norm bound, a lower bound for the smallest singular value is given.

Bounds for eigenvalues of the Dirichlet problem for the logarithmic Laplacian

We provide bounds for the sequence of eigenvalues { λ i ⁢ ( Ω ) } i {\{\lambda_{i}(\Omega)\}_{i}} of the Dirichlet problem L Δ ⁢ u = λ ⁢ u ⁢ in ⁢ Ω , u = 0 ⁢ in ⁢ ℝ N ∖ Ω , L_{\Delta}u=\lambda u\text{ in }\Omega,\quad u=0\text{ in }\mathbb{R}^{N}% \setminus\Omega, where L Δ {L_{\Delta}} is the logarithmic Laplacian operator with Fourier transform symbol 2 ⁢ ln ⁡ | ζ | {2\ln\lvert\zeta\rvert} . The logarithmic Laplacian operator is not positively defined if the volume of the domain is large enough. In this article, we obtain the upper and lower bounds for the sum of the first k eigenvalues by extending the Li–Yau method and Kröger’s method, respectively. Moreover, we show the limit of the quotient of the sum of the first k eigenvalues by k ⁢ ln ⁡ k {k\ln k} is independent of the volume of the domain. Finally, we discuss the lower and upper bounds of the k-th principle eigenvalue, and the asymptotic behavior of the limit of eigenvalues.