Lax pair and first integrals for two of nonlinear coupled oscillators

The system of two nonlinear coupled oscillators is studied. As partial case this system of equation is reduced to the Duffing oscillator which has many applications for describing physical processes. It is well known that the inverse scattering transform is one of the most powerful methods for solving the Cauchy problems of partial differential equations. To solve the Cauchy problem for nonlinear differential equations we can use the Lax pair corresponding to this equation. The Lax pair for ordinary differential or systems or for system ordinary differential equations allows us to find the first integrals, which also allow us to solve the question of integrability for differential equations. In this report we present the Lax pair for the system of coupled oscillators. Using the Lax pair we get two first integrals for the system of equations. The considered system of equations can be also reduced to the fourth-order ordinary differential equation and the Lax pair can be used for the ordinary differential equation of fourth order. Some special cases of the system of equations are considered.Comment: 9 page

Experimental Tests of Discrete Strengthened Elements of Machine-Building Structures

Computer simulation and bench tests of components and full-scale structures of internal combustion engine are performed in order to evaluate discrete and continual strengthening technology. The contact pressure distributions, friction coefficients, wear, roughness and hardness of the contacting surfaces of the tested machine parts were determined. The numerical characteristics that determine the effectiveness of such combined strengthening method are established. Conceptual fundamentals of discrete continual strengthening have been developed. Positive effects in the “load – contact – friction – wear” chain were found due to the proposed strengthening method. The positive effect of the coordination of micro and macroscale processes and states of loaded parts, which are strengthened by the discrete and continuous method, is also established. It is confirmed that the entire set of tribo-mechanical characteristics is improved with such strengthening, in contrast to traditional methods, an application of which results in improvement in some characteristics at the cost of the others

Linear and Branched Glyco-Lipopeptide Vaccines Follow Distinct Cross-Presentation Pathways and Generate Different Magnitudes of Antitumor Immunity

Glyco-lipopeptides, a form of lipid-tailed glyco-peptide, are currently under intense investigation as B- and T-cell based vaccine immunotherapy for many cancers. However, the cellular and molecular mechanisms of glyco-lipopeptides (GLPs) immunogenicity and the position of the lipid moiety on immunogenicity and protective efficacy of GLPs remain to be determined.We have constructed two structural analogues of HER-2 glyco-lipopeptide (HER-GLP) by synthesizing a chimeric peptide made of one universal CD4(+) epitope (PADRE) and one HER-2 CD8(+) T-cell epitope (HER(420-429)). The C-terminal end of the resulting CD4-CD8 chimeric peptide was coupled to a tumor carbohydrate B-cell epitope, based on a regioselectively addressable functionalized templates (RAFT), made of four alpha-GalNAc molecules. The resulting HER glyco-peptide (HER-GP) was then linked to a palmitic acid moiety, attached either at the N-terminal end (linear HER-GLP-1) or in the middle between the CD4+ and CD8+ T cell epitopes (branched HER-GLP-2). We have investigated the uptake, processing and cross-presentation pathways of the two HER-GLP vaccine constructs, and assessed whether the position of linkage of the lipid moiety would affect the B- and T-cell immunogenicity and protective efficacy. Immunization of mice revealed that the linear HER-GLP-1 induced a stronger and longer lasting HER(420-429)-specific IFN-gamma producing CD8(+) T cell response, while the branched HER-GLP-2 induced a stronger tumor-specific IgG response. The linear HER-GLP-1 was taken up easily by dendritic cells (DCs), induced stronger DCs maturation and produced a potent TLR- 2-dependent T-cell activation. The linear and branched HER-GLP molecules appeared to follow two different cross-presentation pathways. While regression of established tumors was induced by both linear HER-GLP-1 and branched HER-GLP-2, the inhibition of tumor growth was significantly higher in HER-GLP-1 immunized mice (p<0.005).These findings have important implications for the development of effective GLP based immunotherapeutic strategies against cancers

Nonlinear differential equations with exact solutions expressed via the Weierstrass function

A new problem is studied, that is to find nonlinear differential equations with special solutions expressed via the Weierstrass function. A method is discussed to construct nonlinear ordinary differential equations with exact solutions. The main step of our method is the assumption that nonlinear differential equations have exact solutions which are general solution of the simplest integrable equation. We use the Weierstrass elliptic equation as building block to find a number of nonlinear differential equations with exact solutions. Nonlinear differential equations of the second, third and fourth order with special solutions-expressed via the Weierstrass function are given