On the Comparison Theorems for the Volume of Tubes in Kahlerian Manifolds

In this note we give some comparison theorems for the volume of tubes around a complex submanifold in a Kahlerian monifold.

A General Converse Comparison Theorem for Backward Stochastic Differential Equation with Non-lipschitz Coefficient

In this article, we first introduce g-expectation via the solution of backward stochastic differential equation（BSDE in short） with non-Lipschitz coefficient, and give the properties of g-expectation, then we establish a general converse comparison theorem for backward stochastic differential equation with non-Lipschitz coefficient.

MULTI-DIMENSIONAL REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND THE COMPARISON THEOREM

In this article, we study the multi-dimensional reflected backward stochastic differential equations. The existence and uniqueness result of the solution for this kind of equation is proved by the fixed point argument where every element of the solution is forced to stay above the given stochastic process, i.e., multi-dimensional obstacle, respectively. We also give a kind of multi-dimensional comparison theorem for the reflected BSDE and then use it as the tool to prove an existence result for the multi-dimensional reflected BSDE where the coefficient is continuous and has linear growth.

A Comparison Theorem for Solution of the Fully Coupled Backward Stochastic Differential Equations

The comparison theorems of solutions for BSDEs in fully coupled forward-backward stochastic differential equations (FBSDEs) are studied in this paper, here in the fully coupled FBSDEs the forward SDEs are the same structure.

COMPARISON THEOREMS FOR MULTI-DIMENSIONAL GENERAL MEAN-FIELD BDSDES

In this paper we study multi-dimensional mean-field backward doubly stochastic differential equations(BDSDEs),that is,BDSDEs whose coefficients depend not only on the solution processes but also on their law.The first part of the paper is devoted to the comparison theorem for multi-dimensional mean-field BDSDEs with Lipschitz conditions.With the help of the comparison result for the Lipschitz case we prove the existence of a solution for multi-dimensional mean-field BDSDEs with an only continuous drift coefficient of linear growth,and we also extend the comparison theorem to such BDSDEs with a continuous coefficient.

ON LEVIN TYPE COMPARISON THEOREMS FOR CERTAIN SECOND ORDER DIFFERENTIAL EQUATIONS

In this paper we establish Levin type comparison theorems for certain second order differential equations. The results obtained here generalize and extend some of the earlier ones related to the Levin's comparison theorems.

A NOTE ON COMPARISON THEOREM OF TWO-STAGE ITERATIVE METHOD FOR HERMITIAN POSITIVE DEFINITE LINEAR SYSTEMS

We discuss two-stage iterative methods for the solution of linear systemAx = b, and give a new proof of the comparison theorems of two-stage iterative methodfor an Hermitian positive definite matrix. Meanwhile, we put forward two new versionsof well known comparison theorem and apply them to some examples.

Dilation,discrimination and Uhlmann's theorem of link products of quantum channels

We establish the Stinespring dilation theorem of the link product of quantum channels in two different ways,discuss the discrimination of quantum channels,and show that the distinguishability can be improved by self-linking each quantum channel n times as n grows.We also find that the maximum value of Uhlmann's theorem can be achieved for diagonal channels.

Efficacy of fexuprazan compared with rebamipide in Korean patients with gastritis:A matching-adjusted indirect comparison

BACKGROUND Gastritis is one of the most frequently diagnosed diseases requiring medical treatment in South Korea.Fexuprazan,a novel potassium-competitive acid blocker,has been approved for treating gastritis and erosive esophagitis.Meanwhile,rebamipide is the most commonly used mucoprotective agent for acute and chronic gastritis in real-world settings in South Korea.However,there have been no studies comparing the efficacy of these two drugs yet.AIM To compare the efficacy of fexuprazan with that of rebamipide for acute and chronic gastritis.METHODS This was a matching-adjusted indirect comparison.Individual patient data from a phase III study of fexuprazan(10 mg BID)were compared with cumulative data from two matching studies of rebamipide(100 mg TID).Erosion improvement and healing rates were compared between two weeks of fexurapan,two weeks of rebamipide,and four weeks of rebamipide.The two main outcome variables were presented as percentages,and the risk differences(RD)and 95%confidence intervals(CI)were calculated for the relative treatment effects.RESULTS In the primary analysis,the erosion improvement and healing rates after a twoweek treatment with fexuprazan were 64.5%and 53.2%,respectively,while a twoweek treatment with rebamipide resulted in erosion improvement and healing rates of 43.6%(RD:21.0%;95%CI:9.6-32.3;P<0.01)and 35.6%(RD:17.6%;95%CI:6.1-29.2;P=0.003),respectively.In the additional analysis,the erosion improvement and healing rates for the two-week fexuprazan treatment(64.2%and 51.2%,respectively)were similar to those obtained during a four-week treatment with rebamipide(60.6%;RD:3.6%;95%CI:-9.8,17.0;P=0.600 and 53.5%;RD:-2.3%;95%CI:-16.1,11.5;P=0.744,respectively).CONCLUSION The two-week fexuprazan treatment was superior to the two-week rebamipide treatment and similar to the fourweek rebamipide treatment for patients with gastritis.

New Asymptotic Results on Fermat-Wiles Theorem

We analyse the Diophantine equation of Fermat xp yp = zp with p > 2 a prime, x, y, z positive nonzero integers. We consider the hypothetical solution (a, b, c) of previous equation. We use Fermat main divisors, Diophantine remainders of (a, b, c), an asymptotic approach based on Balzano Weierstrass Analysis Theorem as tools. We construct convergent infinite sequences and establish asymptotic results including the following surprising one. If z y = 1 then there exists a tight bound N such that, for all prime exponents p > N , we have xp yp zp.

Multi-party semi-quantum private comparison protocol of size relation based on two-dimensional Bell states

Currently,all quantum private comparison protocols based on two-dimensional quantum states can only compare equality,via using high-dimensional quantum states that it is possible to compare the size relation in existing work.In addition,it is difficult to manipulate high-dimensional quantum states under the existing conditions of quantum information processing,leading to low practicality and engineering feasibility of protocols for comparing size relation.Considering this situation,we propose an innovative protocol.The proposed protocol can make size comparison by exploiting more manageable two-dimensional Bell states,which significantly enhances its feasibility with current quantum technologies.Simultaneously,the proposed protocol enables multiple participants to compare their privacies with the semi-quantum model.The communication process of the protocol is simulated on the IBM Quantum Experience platform to verify its effectiveness.Security analysis shows that the proposed protocol can withstand common attacks while preserving the privacies of all participants.Thus,the devised protocol may provide an important reference for implementation of quantum private size comparison protocols.

SOME LIOUVILLE-TYPE THEOREMS FOR THE STATIONARY 3D MAGNETO-MICROPOLAR FLUIDS

In this paper,we prove some Liouville-type theorems for the stationary magnetomicropolar fluids under suitable conditions in three space dimensions.We first prove that the solutions are trivial under the assumption of certain growth conditions for the mean oscillations of the potentials.And then we show similar results assuming that the solutions are contained in L^(p)(R^(3))with p∈[2,9/2).Finally,we show the same result for lower values of p∈[1,9/4)with the further assumption that the solutions vanish at infinity.

AAK Theorem and a Design of Multidimensional BIBO Stable Filters

Rational approximation theory occupies a significant place in signal processing and systems theory. This research paper proposes an optimal design of BIBO stable multidimensional Infinite Impulse Response filters with a realizable (rational) transfer function thanks to the Adamjan, Arov and Krein (AAK) theorem. It is well known that the one dimensional AAK results give the best approximation of a polynomial as a rational function in the Hankel semi norm. We suppose that the Hankel matrix associated to the transfer function has a finite rank.

Diophantine equations and Fermat's last theorem for multivariate(skew-)polynomials

Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely.The aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.

Some convergence theorems of fuzzy concave integral on fuzzyσ-algebra

In this paper,we consider the extension of the concave integral from classical crispσ-algebra to fuzzyσ-algebra of fuzzy sets.Firstly,the concept of fuzzy concave integral on a fuzzy set is introduced.Secondly,some important properties of such integral are discussed.Finally,various kinds of convergence theorems of a sequence of fuzzy concave integrals are proved.

Accuracy comparison and improvement for state of health estimation of lithium-ion battery based on random partial recharges and feature engineering

State of health(SOH)estimation of e-mobilities operated in real and dynamic conditions is essential and challenging.Most of existing estimations are based on a fixed constant current charging and discharging aging profiles,which overlooked the fact that the charging and discharging profiles are random and not complete in real application.This work investigates the influence of feature engineering on the accuracy of different machine learning(ML)-based SOH estimations acting on different recharging sub-profiles where a realistic battery mission profile is considered.Fifteen features were extracted from the battery partial recharging profiles,considering different factors such as starting voltage values,charge amount,and charging sliding windows.Then,features were selected based on a feature selection pipeline consisting of filtering and supervised ML-based subset selection.Multiple linear regression(MLR),Gaussian process regression(GPR),and support vector regression(SVR)were applied to estimate SOH,and root mean square error(RMSE)was used to evaluate and compare the estimation performance.The results showed that the feature selection pipeline can improve SOH estimation accuracy by 55.05%,2.57%,and 2.82%for MLR,GPR and SVR respectively.It was demonstrated that the estimation based on partial charging profiles with lower starting voltage,large charge,and large sliding window size is more likely to achieve higher accuracy.This work hopes to give some insights into the supervised ML-based feature engineering acting on random partial recharges on SOH estimation performance and tries to fill the gap of effective SOH estimation between theoretical study and real dynamic application.

Efficacy comparison of multipoint and single point scanning panretinal laser photocoagulation in non-proliferative diabetic retinopathy treatment

BACKGROUND Non-proliferative diabetic retinopathy(NPDR)poses a significant challenge in diabetes management due to its microvascular changes in the retina.Laser photocoagulation,a conventional therapy,aims to mitigate the risk of progressing to proliferative diabetic retinopathy(PDR).AIM To compare the efficacy and safety of multi-spot vs single-spot scanning panretinal laser photocoagulation in NPDR patients.METHODS Forty-nine NPDR patients(86 eyes)treated between September 2020 and July 2022 were included.They were randomly allocated into single-spot(n=23,40 eyes)and multi-spot(n=26,46 eyes)groups.Treatment outcomes,including bestcorrected visual acuity(BCVA),central macular thickness(CMT),and mean threshold sensitivity,were assessed at predetermined intervals over 12 months.Adverse reactions were also recorded.RESULTS Energy levels did not significantly differ between groups(P>0.05),but the multi-spot group exhibited lower energy density(P<0.05).BCVA and CMT improvements were noted in the multi-spot group at one-month posttreatment(P<0.05).Adverse reaction incidence was similar between groups(P>0.05).CONCLUSION While energy intensity and safety were comparable between modalities,multi-spot scanning demonstrated lower energy density and showed superior short-term improvements in BCVA and CMT for NPDR patients,with reduced laser-induced damage.

Global Research Progress on Municipal Waste and Future Prospect Based on the Cross-national Comparisons

Due to the acceleration of urbanization,the municipal waste(MW)problem has transformed into a global challenge for urb-an sustainability.To elucidate historical trends,current focal points,and future directions in MW research,we conducted a bibliometric analysis and employed knowledge graph visualization to scrutinize a total of 34212 articles,which were published between 1991 and 2021 in the Web of Science(WoS)core database.The results indicated that current major research themes encompass waste classifica-tion and recycling,waste management and public behavior,waste disposal methods and technologies,as well as environmental impact and evaluation.There has been a shift in the research focus from the environmental impacts of waste incineration to sustainable manage-ment related issues.A comparison of research from six typical countries revealed the differences in research priorities and techniques advantages.Scholars from the USA and Britain initiated MW research earlier than other countries and investigated management issues in depth,such as public behavior and willingness to pay.Meanwhile,Japanese,German,and Swedish scholars conducted extensive studies on advanced waste treatment technologies,such as disposal and recycling,risk assessment,and waste-to-energy techniques.Chinese scholars placed particular emphasis on end-of-pipe treatments and their associated environmental impacts.Hotspots and poten-tial future frontiers were identified by burst detection analysis.Keywords with high value of burst index(BI)worldwide are food waste and circular economy.Chinese scholars have put great efforts on waste environmental impact and its recycling technologies,while we’re expecting to further investigating vulnerable population.Furthermore,this study contributes to bridging the regional gap of scientific research among different countries and fostering international collaboration.

Small Modular Solutions to Fermat’s Last Theorem

The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.

Comparison and Selection of Water Restoration Techniques for Artificial Landscape Lakes in Nantong City

As an important part of urban infrastructure,urban water system is of great and far-reaching significance for ensuring urban flood control and waterlogging safety,protecting ecological environment and building livable homes.Taking the urban water system of Nantong as an example,Nantong Water Resources Bureau issued R evision of Nantong Urban Water System Planning in 2017,and put forward the construction of the"two circles,eight lakes and nine veins"water system layout,giving new vitality to the urban water system.In view of problems existing in newly excavated artificial landscape lakes,such as fragile water ecosystem,strong eutrophication trend,poor environmental sensory effect and unsatisfactory water landscape effect,it is urgent to study the in-situ water ecological restoration technique of"algae-controlling zooplankton+submerged plant community"to build a"grass-type clear water"ecosystem for artificial landscape lakes,so as to improve the water sensory index and self-purification ability and finally realize the double improvement of"sensory effect and water quality"of artificial landscape lakes.