Boundary Conditions for Infinite Conservation Laws

A continuous symmetry of a system of partial differential equations (PDEs) is a transformation that leaves invariant the solution manifold of the system, i.e, it maps (deforms) any solution of the system into a solution of the same system. This definition is topological in nature. However, in practice, the direct calculation of the continuous symmetries of a given system of PDEs restricts one to consider symmetries that are local transformations acting on its space of independent variables, dependent variables and their derivatives. Lie’s algorithm to determine Lie groups of point transformations (point symmetries) of differential equations was presented in Bluman & Anco (2002) [see also Ovsiannikov [(1962), (1982)]; Bluman & Cole (1974); Olver (1986); Bluman & Kumei (1989); Stephani (1989); Hydon (2000); Cantwell (2002)]. Point symmetries arise from solutions of linear systems of determining equations for components of infinitesimal generators for the independent and dependent variables of a given PDE system, where these components themselves depend only on the given PDE system’s independent and dependent variables. Point transformations acting on the space of the given independent and dependent variables of a given PDE system can be extended (prolonged) to point transformations acting on the space of the given independent variables, dependent variables, and their derivatives to any finite order.

The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schrödinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.

The two-dimensional cubic nonlinear Schrodinger equation is used to describe the propagation of an intense laser beam through a medium with Kerr nonlinearity. Coupled two-dimensional cubic nonlinear Schrodinger equations are used to describe the interaction of electromagnetic waves with different polarizations in nonlinear optics. In this chapter, we solve these equations by imposing a quadratic condition on the related argument functions and using their symmetry transformations. More complete families of exact solutions of this type are obtained, many of which are periodic, quasi-periodic, aperiodic, and singular solutions that may have practical significance. The Davey–Stewartson equations are used to describe the long time evolution of three-dimensional packets of surface waves. Assuming that the argument functions are quadratic in spatial variables, we find various exact solutions for the Davey–Stewartson equations.

The present study deals with male’s attitude towards contraceptive use behavior and fatalism in Peshawar city, Pakistan. Pakhtoon mostly inhabitsPeshawar city, which has its own culture, highly strict in terms of women emancipation and considered women freedom as a sin and culturallyasking for severe punishment even in some cases death for the offenders. A sample size of 613 married male respondents (15-49 years old) wasinterviewed through cluster sampling procedure from five randomly selected bazaars out of total fifteen. Likert and Semantic Differential Scales were used in devising independent and dependent variables while constructing interview schedule. At first stage univariate analysis and then bivariate analysis were carried out to reach into logical outcome. Both dependent variable (Contraceptive Use Behavior) and independent variable (Fatalism) were indexed and reliability analysis was carriedout through Cronbach’s alpha test. The reliability coefficients were found lying between .7 to .8. Both dependent and independent variables were indexed and cross-tabulated for conducting bi-variate analysis. Gamma, Chi-square and Fisher Exact Test was used to determine the strength, direction and level of association between dependent and independent variables. Majority of the respondents had a clear vision of family planning and found to be moderately consistent of contraceptive use behavior. Moreover, respondents negated that fate prescribes the final number of children, a childless woman is a handicapped person, a sonless womanis a sign of misfortune, more children ensure perpetuation of family; bring husband and wife closer and sign of prestige. Results based onindexation of variables showed a low intensity of fatalistic behavior amongst the respondents of the study area. Efforts are needed on part of the policy makers to properly portray the complexities involved in the bareness of a woman by recognizing more rights for women than just producing children. Moreover, Peshawar city needs to be declared as model to other parts of the country, which is still in the grip of fatalism.

Systematic methods are used to find all possible conservation laws of a given type for certain systems of partial differential equations, including some from fluid mechanics. The necessary and sufficient conditions for a vector to be divergence-free are found in the form of a system of first order, linear, homogeneous partial differential equations, usually overdetermined. Incompressible, inviscid fluid flow is treated in the unsteady two-dimensional and steady three-dimensional cases. A theorem about the degrees of freedom of partial differential equations, needed for finding conservation laws, is proven. Derivatives of the dependent variables are then included in the divergence-free vectors. Conservation laws for Laplace's equation are found with the aid of complex variables, used also to treat the two-dimensional steady flow case when first derivatives are included in the vectors. Conservation laws, depending on an arbitrary number of derivatives, are found for a general first order quasi-linear equation in two independent variables, using two differential operators, which are associated with the derivatives with respect to the independent variables. Linear totally hyperbolic systems are then treated using an obvious generalization of the above operators.

The spatio-temporal behaviors of possible solutions of driven and damped nonlinear Schrodinger and sine-Gordon equations are analyzed. In the first part, the nonlinear Schrodinger equation is considered also as the limiting case corresponding to weak driving and damping in the sine-Gordon equation, with driving frequency close to one. It is shown that in this limit a phase-locked breather with m/n = 1/1 exists and undergoes period-doubling bifurcations. The close analogy to the dynamics of a phase-locked Schrodinger soliton is demonstrated in detail. In the second part, for smaller driving frequencies, the existence and stability of subharmonically phase-locked breathers is discussed. The quasi-periodic route to chaos is re-investigated and special attention is given to the length-dependence of the results. The findings are compared with predictions of simpler models in nonlinear dynamics. The whole paper consists of analytical and numerical results. For the latter, improved diagnostic tools have been developed.

Traditional theoretical calculations, field measurements, and finite element methods sometimes fail to realize life cycle simulations of the temperature field and temperature effect of steel–concrete composite bridge deck systems. In this paper, a simulation method based on a back propagation–long short-term memory (BP-LSTM) network correlation model is proposed to predict the temperature field and temperature effect in a low-cost and highly efficient manner. A bridge was used as the engineering background according to a health monitoring system, and the finite element method based on the principle of heat transfer was used to expand the data sets. Data sets with structural, time, and environmental characteristics as the independent variables and temperature and the temperature effect as the dependent variables were formed. The correlation between the dependent and independent variables was verified using the maximal information coefficient. Thus, the BP-LSTM model was established, and the mean squared error loss function considering the time weight was designed. The data set was read in for training, verification, and testing, and a correlation model representing the relationship between the set’s independent and dependent variables was obtained with relatively high accuracy. Finally, combined with the bridge’s historical meteorological data, the established correlation model was used to simulate the bridge’s temperature field and temperature effect. The results indicated that the finite element calculation results of the structure’s temperature field based on the heat transfer principle were basically consistent with the measured results. The independent variables in the data set were non-linearly related to the dependent variables. The BP-LSTM’s prediction accuracy of the temperature field and temperature effect was above 98.8% and 94.5%, respectively, in good agreement with the target value. The variation law of the temperature field and temperature effect of the steel–concrete composite bridge deck system simulated by combining the historical meteorological statistics was in accordance with reality.

Overview of the Kadomtsev–Petviashvili-hierarchy reduction method for solitons

The covariation of independent and dependant variables in neurofeedback: a proposal framework to identify cognitive processes and brain activity variables.

A summary is presented of the principal types of completely integrable partial differential equations having soliton solutions. Each type is derived from an appropriate physical model of an electromagnetic wave problem, with the intention to show how known mathematical results apply to a coherent class of physical problems in electromagnetic waves. The non-linear Schrodinger (NS) equation appears when the induced non-linear dielectric polarization is expanded in a series of powers of the electric field, only the linear and third-order polarizations are retained, and the temporal spectrum of the wave is a narrow band far removed from any resonance of the medium. The sine-Gordon equation appears from a similar optical model of propagation in a dielectric consisting of identical 2-level atomic systems, but resonance occurs between the carrier frequency of the wave and the transition frequency of the atoms. The Boussinesq and Korteweg– de Vries equations appear at different levels of approximation to a potential wave on a transmission line having a non-linear capacitance such that the charge stored is a non-linear function of the line potential. In all cases the evolution variable is the propagation distance; the transverse variable is time, but in the case of the NS equation it may alternatively be a spatial coordinate, giving rise to the possibility of spatial solitons as well as temporal solitons for NS-type problems. Two examples are derived of non-integrable Hamiltonian systems having spatial solitary waves, namely the second-order cascade interaction and vector spatial solitary waves of the third-order interaction, and a brief survey of the analytical solutions for the plane waves and solitary waves of these two types is presented. Finally, the addition of a second spatial dimension to the non-linear transmission line problem leads to the Kadomtsev–Petviashvili equations, and a further approximation for weakly modulated travelling waves leads to the Davey–Stewartson equations. Both of these completely integrable systems support combined spatial–temporal solitons.

It is difficult to study the brain "at rest" with an approach generally pursued in science when external manipulation (independent variable) is used to obtain informative measurements (dependent variable) about the object of interest. External manipulation in its classic sense may suspend the resting state, and hence the object of interest will evade. Naturally, unless in a final and irreversible state, biological rest will always be an endogenously dynamic process. Combining two modalities, electroencephalography (EEG) and functional magnetic resonance imaging (fMRI), to simultaneously measure the brain's activity from two angles, one can be chosen to be interpreted as the independent variable and the other as the dependent variable, and without external manipulation the brain's spontaneous dynamics can be studied. The EEG, for example, observes endogenous modulations of vigilance and detects spontaneous events such as sleep spindles or epileptic discharges and can be used as the independent variable, i.e., to form a regressor to interrogate the fMRI data (dependent variable). The opposite is possible as well, and data fusion attempts try using all data both as dependent and independent variables at the same time. This review limits itself to an exemplary discussion of simultaneous EEG/fMRI studies in humans, and among a variety of proposed resting state networks only discusses a few, especially those for which non-resting state literature has proposed a functional meaning: the "default mode" network and an attentional network. It will be shown that one EEG feature can correlate with different fMRI activation maps and that a single resting state network may be associated with a variety of EEG patterns giving insight into the function of different resting states and the relationship between the two modalities in themselves.

Evidence for HIV transmission across key populations: a longitudinal analysis of HIV and AIDS rates among Black people who inject drugs and Black heterosexuals in 84 large U.S. metropolitan areas, 2008–2016

For people living with HIV/AIDS (PLWHA) positive social support networks (SSNs) can help improve quality of life, overall well-being, coping, and decrease mood disturbance, morbidity, mortality, sexual and substance associated...

This research mainly aimed at studying the extension needs of caraway farmers in El-Mesery Agricultural Cooperative-El Amereia - Alexandria Governorate. This necessitated the achievement of the following sub-objectives: (1) Identify some of the respondents 'characteristics, (2) Identify the cognitive level of the respondents, (3) Identify the related and influencing variables on the cognitive level of the respondents, (4) Identify the respondents' trends towards caraway cultivation, ( 5) Identifying the related and influencing variables in the respondents ’attitudes towards caraway cultivation, (6) Identifying the most important problems they face in cultivating and marketing the caraway crop. The personal interview questionnaire form was used to complete the field data from a random sample of 200 respondents, and the percentages, range, arithmetic mean, standard deviation, frequency tables and simple correlation coefficient of Pearson were used in the presentation, discussion and analysis of the results. statistical analysis.The study concluded many research results, the most prominent of which are the following: 1-With regard to the characteristics of the studied caraway farmers, the research results indicated that 66%Of the total respondents, their ages are less than 40-53 years, and that (32.5%) of the respondents are illiterate, and that the farmers studied have all previously cultivated medicinal and aromatic plants, and the percentage of those with moderate experience reached 42.5%, of the total respondents 2-The low cognitive level in caraway cultivation is 15%, the average cognitive level is 69%, and the high cognitive level is 16%, and the results showed a positive and significant correlation at the probability level 0.01 Between the cognitive level of the surveyed farmers and each of the following independent variables: full-time agricultural work, the number of years of experience cultivating medicinal and aromatic plants, media openness, agricultural information sources, membership of organizations, opinion leadership, readiness for change, living satisfaction, level of ambition and direction towards agricultural extension. While that correlation was positive and significant at the probability level 0.05 with both age and educational level, while that relationship was negative and significant at the probability level 0.05 with the variable of agricultural land tenure, It was not clear that there was a significant correlation relationship between the dependent variable and only two independent variables: the number of family members and community participation. The subsistence variable, devotion to agricultural work, and that these four independent variables collectively affect the dependent variable by 74.1% and that the organizational membership variable is more influential in the dependent variable as it helps explain 61% of the possible variation in the dependent variable. 3- The percentage of those with positive trends towards caraway cultivation was 25.5%, neutral trends 54.5%, and negative trends 20% of the total number of farmers surveyed, and it was found that there was a positive and significant correlation at the probability level 0.01 between the trends of the surveyed caraway farmers and each of the independent variables The following: exposure to the media, community participation, membership of organizations, opinion leadership, willingness to change, living satisfaction, and level of ambition. While that correlation was positive and significant at the probability level 0,05 with the variable of agricultural work, while that relationship was negative and significant at the probability level 0,01 with a variable Only one is the educational level, and the research results showed that there is no significant relationship between the dependent variable and each of the following independent variables: age, number of family members, agricultural land tenure, number of years of experience in growing medicinal and aromatic plants, and sources of agricultural information. The results of the multiple regression analysis indicated that There are only two independent variables that collectively affect the attitudes of caraway growers of the respondents towards cultivating medicinal and aromatic plants as a dependent variable, namely the level of ambition and community participation. It contributes to explaining 29.7% of the possible variance in the dependent variable 4- The research results indicated that the surveyed caraway growers face many problems related to the cultivation and marketing of the caraway crop, the most important of which are: the prevalence of nematodes and grass weeds 78%, the absence of agricultural extension 18.5%, the difficulty of obtaining fertilizers, 5.5%, and that the problems of marketing the caraway crop are The problem of low sales price 78,5%, lack of opportunities to contract with agencies to market the crop 50%, lack of sawmills 15%, absence of marketing guidance 13%, unavailability of packing packages 10%, and high transport costs 1.5% of the total number of farmers surveyed .

Using macroeconomic variables in the prediction of stock market indices:A theoretical and empirical assessment within BRICS and selected developed economies