REFINEMENTS OF THE NORM OF TWO ORTHOGONAL PROJECTIONS

In this paper,some refinements of norm equalities and inequalities of combination of two orthogonal projections are established.We use certain norm inequalities for positive contraction operator to establish norm inequalities for combination of orthogonal projections on a Hilbert space.Furthermore,we give necessary and sufficient conditions under which the norm of the above combination of o`rthogonal projections attains its optimal value.

Association between operative position and postoperative nausea and vomiting in patients undergoing laparoscopic sleeve gastrectomy

BACKGROUND Bariatric surgery is one of the most effective ways to treat morbid obesity,and postoperative nausea and vomiting(PONV)is one of the common complications after bariatric surgery.At present,the mechanism of the high incidence of PONV after weight-loss surgery has not been clearly explained,and this study aims to investigate the effect of surgical position on PONV in patients undergoing bariatric surgery.AIM To explore the effect of the operative position during bariatric surgery on PONV.METHODS Data from obese patients,who underwent laparoscopic sleeve gastrectomy(LSG)in the authors’hospital between June 2020 and February 2022 were divided into 2 groups and retrospectively analyzed.Multivariable logistic regression analysis and the t-test were used to study the influence of operative position on PONV.RESULTS There were 15 cases of PONV in the supine split-leg group(incidence rate,50%)and 11 in the supine group(incidence rate,36.7%)(P=0.297).The mean operative duration in the supine split-leg group was 168.23±46.24 minutes and 140.60±32.256 minutes in the supine group(P<0.05).Multivariate analysis revealed that operative position was not an independent risk factor for PONV(odds ratio=1.192,95%confidence interval:0.376-3.778,P=0.766).CONCLUSION Operative position during LSG may affect PONV;however,the difference in the incidence of PONV was not statistically significant.Operative position should be carefully considered for obese patients before surgery.

Feynman Path Integral Using Operator Integration in Banach Space

Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.

Evaluation index system for positive operation of water conservancy projects

The conditions for the positive operation of water conservancy projects are described in this paper. A scientific and effective evaluation index system was established based on frequency analysis, theoretical analysis, and expert consultation. This evaluation index system can be divided into six first-level indices： the degree to which facilities are intact and functionality standards are reached, the status of operation and management funds, the rationality and degree of advancement of the management team structure, the adaptability and rationality of the water conservancy project management system, the degree of automatization and informationization of the management techniques, and the conduciveness of the exterior environment. The weights for evaluation indices were obtained through the analytic hierarchy process method with consideration of the difference between public welfare and profit-oriented water conservancy projects. This study provides a scientific method for evaluating the positive operation of water conservancy projects.

Reliability Analysis of a Cold Standby Repairable System with Repairman Extra Work

This paper deals with a cold standby repairman who can do extra work in idle time. The repairable system with two identical units and one authors are devoted to studying the unique existence and exponential stability of the system solution. C0-semigroup theory is used to prove the existence of a unique nonnegative time-dependent solution of the system. Then by using the theory of resolvent positive operator, the authors derive that dynamic solution of the system exponentially converges to its steady-state one which is the eigenfunction corresponding to eigenvalue 0 of the system operator. Some reliability indices of the system are discussed with a different method from traditional one. The authors also make a profit analysis to determine the optimal service time outside the system to maximize the system profit.

EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO A CLASS OF SEMILINEAR ELLIPTIC SYSTEMS

In this paper an existence and uniqueness theorem of positive solutions to a class of semilinear elliptic systems is proved. Also, a necessary condition for the existence of the positive solution is obtained. As the application of the main theorem, two examples are given.

Operator Matrix Forms of Positive Operators

If a 3-tuple (A:H_1→H_1,B:H_2→H_1,C:H_2→H_2) of operators on Hilbert spaces is given,we proved that the operator A:= on H=H_1⊕H_2 is≥0 if and only if A≥0,R(B) R(A1/2)and C≥B~* A^+ B, where A^+ is the generalized inverse of A. In general,A^+ is a closed operator,but since R(B) R(A1/2),B~* A^+ B is bounded yet.

THE RELATION BETWEEN SIGN-CHANGING SOLUTION AND POSITIVE-NEGATIVE SOLUTIONS FOR NONLINEAR OPERATOR EQUATIONS AND ITS APPLICATIONS

By fixed point index theory and a result obtained by Amann, existence of the solution for a class of nonlinear operator equations x = Ax is discussed. Under suitable conditions, a couple of positive and negative solutions are obtained. Finally, the abstract result is applied to nonlinear Sturm-Liouville boundary value problem, and at least four distinct solutions are obtained.

ASYMPTOTIC APPROXIMATION OF FUNCTIONS AND THEIR DERIVATIVES BY GENERALIZED BASKAKOV-SZAZS-DURRMEYER OPERATORS

We present the complete asymptotic expansion for a generalization of the Baskakov-Szasz-Durrmeyer operators and their derivatives.

On the Method of Multiplier-enlargement and Approximation of Unbounded Continuous Functions

By combining the classical appropriate functions “1, x, x 2” with the method of multiplier enlargement, this paper establishes a theorem to approximate any unbounded continuous functions with modified positive linear operators. As an example, Hermite Fejér interpolation polynomial operators are analysed and studied, and a general conclusion is obtained.

The Spectrum of Positive Operators on Hilbert Lattices

Some spectral characterizations of positive operators on Hilbert lattices are presented. The application of these results can yield some equivalent relations of an irreducible positive operator. Some related results for positive operators on Hilbert lattice are also obtained.

Remote interactions on two distributed quantum systems:nonlocal unambiguous quantum-state discrimination

Remote quantum-state discrimination is a critical step for the implementation of quantum communication network and distributed quantum computation. We present a protocol for remotely implementing the unambiguous discrimination between nonorthogonal states using quantum entanglements, local operations, and classical communications. This protocol consists of a remote generalized measurement described by a positive operator valued measurement （POVM）. We explicitly construct the required remote POVM. The remote POVM can be realized by performing a nonlocal controlled-rotation operation on two spatially separated qubits, one is an ancillary qubit and the other is the qubit which is encoded by two nonorthogonal states to be distinguished, and a conventional local Von Neumann orthogonal measurement on the ancilla. The particular pair of states that can be remotely and unambiguously distinguished is specified by the state of the ancilla. The probability of successful discrimination is not optimal for all admissible pairs. However, for some subset it can be very close to an optimal value in an ordinary local POVM.

Teleporting a quantum controlled-Not with one target/two targets gate using two partially entangled states

This paper considers the teleportation of quantum controlled-Not （CNOT） gate by using partially entangled states. Different from the known probability schemes, it presents a method for teleporting a CNOT gate with unit fidelity and unit probability by using two partially entangled pairs as quantum channel. The method is applicable to any two partially entangled pairs satisfying the condition that their smaller Schmidt coefficients μ and ν are （2μ ＋ 2ν - 2μν - 1） ≥ 0. In this scheme, the sender＇s local generalized measurement described by a positive operator valued measurement （POVM） lies at the heart. It constructs the required POVM. It also puts forward a scheme for teleporting a CNOT with two targets gate with unit fidelity by using same quantum channel. With assistance of local operations and classical communications, three spatially separated users are able to complete the teleportation of a CNOT with two targets gate with probability of （2μ ＋ 2ν - 1）. With a proper value of μ and ν, the probability could reach nearly 1.

AN ORDER PRESERVING INEQUALITY FOR THREE OPERATORS VIA FURUTA INEQUALITY

Furuta showed that if A≥B≥0,then for each r≥0,f（p）=（A^r/2 B^p A^r/2）^t＋r/p＋r is decreasing for p≥t≥0.Using this result,the following inequality（C^r/2（AB^2A）^δC^ r/2）^ p-1＋r/4δ＋r ≤C^p-1＋r is obtained for 0〈p ≤1,r≥1,1/4≤δ≤1 and three positive operators A, B, C satisfy（A^1/2BA^1/2）^p/2≤A^p,（B^1/2AB^1/2）^p/2≥B^p,（C^1/2AC^1/2）^p/2≤C^p,（A^1/2CA^1/2）^p/2≥A^p.

MEAN CONVERGENCE OF SOME POSITIVE OPERATORS

L'convergence with 0<p<infinity of the positive operators F.(da,f) and G.(da,f), introduced by Nevai P. [3], and Qn,8(da,f), introduced by Hermann T. and Vertesi P. [1] to f is an element of C[-1,1] is proved to hold for very general measures.

A Note on Generalized Aluthge Transformation for p-hyponormal and Log-hyponormal Operators

We shall give some results on generalized aluthge transformation for phyponormal and log-hyponormal operators. We shall also discuss the best possibility of these results.

ON THE TRANSPORT EQUATION WITH INTEGRAL BOUNDARY CONDITIONS AND ITS CONSERVAIVE LAW

The well-posedness for the time-dependent neutron transport equationwith integral boundary conditions is established in L^1 space. Some spectral propertiesof the transport operator are discussed, the dominant eigenvalue is proved existing,and furthermore, the conservative law of migrating particle numbers is established.

Dynamical time versus system time in quantum mechanics

Properties of an operator representing the dynamical time in the extended parameterization invariant formulation of quantum mechanics are studied. It is shown that this time operator is given by a positive operator measure analogously to the quantities that are known to represent various measurable time operators. The relation between the dynamical time of the extended formulation and the best known example of the system time operator, i.e., for the free one- dimensional particle, is obtained.

An application of Furuta inequality and its best possibility

This paper shows the best possibility for outer exponents of some inequalities under some conditions, and a counter example is obtained.

Multi-Attribute Selection Procedures Based on Regret and Rejoice for the Decision-Maker

Feelings influence human beings’decision-making;therefore,incorporation of feeling factors in decision-making is very important.Regret and rejoice are very important emotional feelings that can have a great impact on decision-making if they are considered together.While regret has received most of the attention in related research,rejoice has been less considered even though it can greatly influence people’s preferences in decision-making.Furthermore,systematically incorporating regret and rejoice in the multicriteria decision-making(MCDM)modeling frameworks for decision-making has received little research attention.In this paper,we introduce a new multiattribute selection procedure that incorporates both regret and rejoice to select the best choice.We utilize the positional advantage operator concept to develop regret and rejoice mathematical equations,and prove them.The proposed MCDM procedure that incorporates these two emotional factors offers a decision-maker the flexibility to trade off some benefits in order to gain a state of psychological satisfaction.More specifically,regret and rejoice are presentedmathematically to enable the decision-maker to determine the values of regret and rejoice,and then make the decision in which the rejoice value is higher than the regret value.To test the performance of this new procedure,we apply it to three numerical examples proposed in previous works.The results are matched with those obtained by other methods such as the regret model,VIKOR,PROMETHEE I,and PROMETHEE II,thereby proving the efficacy of the new procedure.