Properties of Phase and Their Evolutions in Damped Odd and Even Coherent States

The properties of measured phase operators in damped odd and even coherent states have been studied. The fluctuations associated with measured phase and their squeezing in these states are investigated. The phase properties in damped superposition coherent states are considered too with the help of measured phase operators. These fluctuations and their squeezing are affected by damping and evolve with time elapsing.

Phase properties of odd and even circular states

Phase properties of the even and odd circular states are studied within the Hermitian phase formalism of Pegg and Barnett. Exact analytical formulas for the distribution function and the variance of the phase operator are obtained and used to examine whether or not the even and odd circular states exhibit photon-number squeezing and phase squeezing.

Continuous-variable quantum teleportation of even and odd coherent states through varied gain channels

This paper has investigated quantum teleportation of even and odd coherent states in terms of the EPR entanglement states for continuous variables. It discusses the relationship between the fidelity and the entanglement of EPR states, which is characterized by the degree of squeezing and the gain of classical channels. It shows that the quality of teleporting quantum states also depends on the characteristics of the states themselves. The properties of teleporting even and odd coherent states at different intensities are investigated. The difference of teleporting two such kinds of quantum states are analysed based on the quantum distance function.

Even and Odd Coherent States for Time-Dependent Harmonic Oscillator

The dynamical invariant for a general time-dependent harmonic oscillator is constructed by making use of two linearly independent solutions to the classical equation of motion. In terms of this dynamical invariant we define the time-dependent creation and annihilation operators and relevantly introduce even and odd coherent states for time dependent harmonic oscillator. The mathematical and quantum statistical properties of these states are discussed in detail. The harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach.

Wigner functions and tomograms of the photon-depleted even and odd coherent states

Using the coherent state representation of Wigner operator and the technique of integration within an ordered product （IWOP） of operators, this paper derives the Wigner functions for the photon-depleted even and odd coherent states （PDEOCSs）. Moreover, in terms of the Wigner functions with respect to the complex parameter a the nonclassical properties of the PDEOCSs are discussed. The results show that the nonclassicality for the state ｜β, m〉o （or ｜β,m〉e） is more pronounced when m is even （or odd）. According to the marginal distributions of the Wigner functions, the physical meaning of the Wigner functions is given. Further, the tomograms of the PDEOCSs are calculated with the aid of newly introduced intermediate coordinate-momentum representation in quantum optics.

Revolutionizing Treatment:AI-Driven Noninvasive Approaches for ODD and ADHD

Oppositional Defiant Disorder(ODD)and Attention Deficit/Hyperactivity Disorder(ADHD)are mental health conditions that have traditionally been managed through behavioral therapies and medication.However,the integration of Artificial Intelligence(AI)has brought about a revolutionary shift in treatment approaches.This article explores the role of AI-driven noninvasive treatments for ODD and ADHD.AI offers personalized treatment plans,predictive analytics,virtual therapeutic platforms,and continuous monitoring,enhancing the effectiveness and accessibility of interventions.Ethical considerations and the need for a balanced approach are discussed.As technology evolves,collaborative efforts between mental health professionals and technologists will shape the future of mental health care for individuals with ODD and ADHD.

Products of Odd Numbers or Prime Number Can Generate the Three Members’ Families of Fermat Last Theorem and the Theorem Is Valid for Summation of Squares of More Than Two Natural Numbers

Fermat’s last theorem, had the statement that there are no natural numbers A, B, and C such that An + Bn = Cn, in which n is a natural number greater than 2. We have shown that any product of two odd numbers can generate Fermat or Pythagoras triple (A, B, C) following n = 2 and also it is applicable A2 + B2 + C2 + D2 + so on =An2 where all are natural numbers.

Quasiprobability distributions of new kinds of even and odd nonlinear coherent states

Using a numerical computational method, quasiprobability distributions of new kinds of even and odd nonlinear coherent states （EONLCS） are investigated. The results show that the distributions of the new even nonlinear coherent states （NLCS） are distinct from those of the new odd NLCS and imply that the new EONLCS always exhibit some different nonclassical effects. Finally, with the aid of newly introduced intermediate coordinate-momentum representation in quantum optics, the tomograms of the new EONLCS are calculated. This is a new way of obtaining the tomogram function.

Finite-dimensional even and odd nonlinear pair coherent states and their some nonclassical properties

In this paper a new class of finite-dimensional even and odd nonlinear pair coherent states （EONLPCSs）, which can be realized via operating the superposed evolution operators D±（τ） on the state ｜q, 0）, is constructed, then their orthonormalized property, completeness relations and some nonclassical properties are discussed. It is shown that the finite-dimensional EONLPCSs possess normalization and completeness relations. Moreover, the finite-dimensional EONLPCSs exhibit remarkably different sub-Poissonian distributions and phase probability distributions for different values of parameters q, η and ξ.

Wigner functions and tomograms of the even and odd binomial states

Using the coherent state representation of Wigner operator and the technique of integration within an ordered product （IWOP） of operators, the Wigner functions of the even and odd binomial states （EOBSs） are obtained. The physical meaning of the Wigner functions for the EOBSs is given by means of their marginal distributions. Moreover, the tomograms of the EOBSs are calculated by virtue of intermediate coordinate-momentum representation in quantum optics.

Convergence Phenomenon with Fourier Series of tg(x2)and Alike

The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. .

Development and validation of prognostic nomogram based on log odds of positive lymph nodes for patients with gastric signet ring cell carcinoma

Objective: Our aims were to establish novel nomogram models, which directly targeted patients with signet ring cell carcinoma(SRC), for individualized prediction of overall survival(OS) rate and cancer-specific survival(CSS).Methods: We selected 1,365 SRC patients diagnosed from 2010 to 2015 from Surveillance, Epidemiology and End Results(SEER) database, and then randomly partitioned them into a training cohort and a validation cohort.Independent predicted indicators, which were identified by using univariate testing and multivariate analyses, were used to construct our prognostic nomogram models. Three methods, Harrell concordance index(C-index), receiver operating characteristics(ROC) curve and calibration curve, were used to assess the ability of discrimination and predictive accuracy. Integrated discrimination improvement(IDI), net reclassification improvement(NRI) and decision curve analysis(DCA) were used to assess clinical utility of our nomogram models.Results: Six independent predicted indicators, age, race, log odds of positive lymph nodes(LODDS), T stage, M stage and tumor size, were associated with OS rate. Nevertheless, only five independent predicted indicators were associated with CSS except race. The developed nomograms based on those independent predicted factors showed reliable discrimination. C-index of our nomogram for OS and CSS was 0.760 and 0.763, which were higher than American Joint Committee on Cancer(AJCC) 8 th edition tumor-node-metastasis(TNM) staging system(0.734 and 0.741, respectively). C-index of validation cohort for OS was 0.757 and for CSS was 0.773. The calibration curves also performed good consistency. IDI, NRI and DCA showed the nomograms for both OS and CSS had a comparable clinical utility than the TNM staging system.Conclusions: The novel nomogram models based on LODDS provided satisfying predictive ability of SRC both in OS and CSS than AJCC 8 th edition TNM staging system alone.

NUAT T-splines of odd bi-degree and local refinement

This paper presents a new kind of spline surfaces, named non-uniform algebraic- trigonometric T-spline surfaces （NUAT T-splines for short） of odd hi-degree. The NUAT T- spline surfaces are defined by applying the T-spline framework to the non-uniform algebraic- trigonometric B-spline surfaces （NUAT B-spline surfaces）. Based on the knot insertion algorithm of the NUAT B-splines, a local refinement algorithm for the NUAT T-splines is given. This algorithm guarantees that the resulting control grid is a T-mesh as the original one. Finally, we prove that, for any NUAT T-spline of odd hi-degree, the linear independence of its blending functions can be determined by computing the rank of the NUAT T-spline-to-NUAT B-spline transformation matrix.

Odd–even harmonic emission from asymmetric molecules:Identifying the mechanism

We study odd–even high-order harmonic generation（HHG） from oriented asymmetric molecules He H2＋numerically and analytically. The variational method is used to improve the analytical description of the ground-state wave function for the asymmetric system, with which the ground-state-continuum-state transition dipole is evaluated. The comparison between the odd–even HHG spectra and the improved dipoles allows us to identify and clarify the complex generation mechanism of odd–even harmonics from asymmetric molecules, providing deep insights into the relation between the odd–even HHG and the asymmetric molecular orbital.

Very Original Proofs of Two Famous Problems: “Are There Any Odd Perfect Numbers?” (Unsolved until to Date) and “Fermat’s Last Theorem: A New Proof of Theorem (Less than One and a Half Pages) and Its Generalization”

This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its generalization”. They are achieved with elementary mathematics. This is why these proofs can be easily understood by any mathematician or anyone who knows basic mathematics. Note that, in both problems, proof by contradiction was used as a method of proof. The first of the two problems to date has not been resolved. Its proof is completely original and was not based on the work of other researchers. On the contrary, it was based on a simple observation that all natural divisors of a positive integer appear in pairs. The aim of the first work is to solve one of the unsolved, for many years, problems of the mathematics which belong to the field of number theory. I believe that if the present proof is recognized by the mathematical community, it may signal a different way of solving unsolved problems. For the second problem, it is very important the fact that it is generalized to an arbitrarily large number of variables. This generalization is essentially a new theorem in the field of the number theory. To the classical problem, two solutions are given, which are presented in the chronological order in which they were achieved. Note that the second solution is very short and does not exceed one and a half pages. This leads me to believe that Fermat, as a great mathematician was not lying and that he had probably solved the problem, as he stated in his historic its letter, with a correspondingly brief solution. To win the bet on the question of whether Fermat was telling truth or lying, go immediately to the end of this article before the General Conclusions.

Amplitude-squared squeezing of the generalized odd-even coherent states of the anharmonic oscillator in a finite-dimensional Hilbert space

This paper discusses the properties of amplitude-squared squeezing of the generalized odd-even coherent states of anharmonic oscillator in finite-dimensional Hilbert space. It demonstrates that the generalized odd coherent states do exhibit strong amplitude-squared squeezing effects in comparison with the generalized even coherent states.

Advantage of log odds of positive lymph nodes in prognostic evaluation of patients with early-onset colon cancer

BACKGROUND Colon cancer(CC)is one of the most common cancers of the digestive tract,the third most common cancer worldwide,and the second most common cause of cancer-related deaths.Previous studies have demonstrated a higher risk of lymph node metastasis(LNM)in young patients with CC.It might be reasonable to treat patients with early-onset locally advanced CC with extended lymph node dissection.However,few studies have focused on early-onset CC(ECC)patients with LNM.At present,the methods of predicting and evaluating the prognosis of ECC patients with LNM are controversial.From the data of patients with CC obtained from the Surveillance,Epidemiology,and End Results(SEER)database,data of young patients with ECC(≤50 years old)was screened.Patients with unknown data were excluded from the study,while the remaining patients were included.The patients were randomly divided into a training group(train)and a testing group(test)in the ratio of 7:3,while building the model.The model was constructed by the training group and verified by the testing group.Using multiple Cox regression models to compare the prediction efficiency of LNM indicators,nomograms were built based on the best model selected for overall survival(OS)and cause-specific survival(CSS).In the two groups,the performance of the nomogram was evaluated by constructing a calibration plot,time-dependent area under the curve(AUC),and decision curve analysis.Finally,the patients were grouped based on the risk score predicted by the prognosis model,and the survival curve was constructed after comparing the survival status of the high and low-risk groups.RESULTS Records of 26922 ECC patients were screened from the SEER database.N classification,positive lymph nodes(PLN),lymph node ratio(LNR)and log odds of PLN(LODDS)were considered to be independent predictors of OS and CSS.In addition,independent risk factors for OS included gender,race,marital status,primary site,histology,grade,T,and M classification,while the independent prognostic factors for CSS included race,marital status,primary site,grade,T,and M classification.The prediction model including LODDS is composed of minimal Akaike information criterion,maximal concordance indexes,and AUCs.Factors including gender,race,marital status,primary site,histology,grade,T,M classification,and LODDS were integrated into the OS nomogram,while race,marital status,primary site,grade,T,M classification,and LODDS were included into the CSS nomogram.The nomogram representing both cohorts had been successfully verified in terms of prediction accuracy and clinical practicability.CONCLUSION LODDS is superior to N-stage,PLN,and LNR of ECC.The nomogram containing LODDS might be helpful in tumor evaluation and clinical decision-making,since it provides an appropriate prediction of ECC.

Gauss’ Problem, Negative Pell’s Equation and Odd Graphs

In this paper we present some results connected with still open problem of Gauss, negative Pell’s equation and some type graphs.In particular we prove in the Theorem 1 that all real quadratic fields K=Q( ) , generated by Fermat’s numbers with d=Fm+1=22m+1+1,m≥2, have not unique factorization. Theorem 2 give a connection of the Gauss problem with primitive Pythagorean triples. Moreover, in final part of our paper we indicate on some connections of the Gauss problem with odd graphs investigated by Cremona and Odoni in the papper [5].

Exposure to proton pump inhibitors and risk of diabetes:A systematic review and meta-analysis

BACKGROUND Exposure to proton pump inhibitors(PPIs)has been reported to have a potential role in the development of diabetes.AIM To determine the association between PPIs and diabetes.METHODS This meta-analysis is registered on PROSPERO(CRD42022352704).In August 2022,eligible studies were identified through a comprehensive literature search.In this study,odds ratios were combined with 95%confidence intervals using a random-effects model.The source of heterogeneity was assessed using sensitivity analysis and subgroup analysis.The publication bias was evaluated using Egger’s test and Begg’s test.RESULTS The meta-analysis included 9 studies with a total of 867185 participants.Results showed that the use of PPIs increased the risk of diabetes(odds ratio=1.23,95%confidence interval:1.05-1.43,n=9,I2=96.3%).Subgroup analysis showed that geographic location and study type had significant effects on the overall results.Both Egger’s and Begg’s tests showed no publication bias(P>0.05).Sensitivity analysis also confirmed the stability of the results.CONCLUSION The results of this study indicated that the use of PPIs was related to an increased risk of diabetes.However,more well-designed studies are needed to verify these results in the future.