Search

According to the new regulations on the ranking of Olympic and non-Olympic sports, there is a need to take into account the effectiveness of competitive activities under the new scoring system for participation in national and international competitions of weightlifters, coaches and national teams of Ukraine. The research done by experts indicates a fragmentary consideration of the effectiveness of competitive activities of weightlifters, coaches and various regions of Ukraine. Such studies are based mainly on the analysis of the results of competitive activities of weightlifters of the National Team of Ukraine. The goal is to determine the ratings of women weightlifters at national and international competitions, coaches of Ukraine and national teams of regions of Ukraine in 2021 by the effectiveness of competitive activities. To reach this goal, we used the following research methods: analysis, synthesis, induction, analysis of documentary materials, methods of mathematical statistics. The study revealed: the ranking of the best women-weightlifters in Ukraine (n = 20) according to the results of participation in national and international competitions in 2021; rating of the best coaches of Ukraine (n = 20) according to the results of participation of women-weightlifters in national and international competitions in 2021; ranking of regions of Ukraine by the total sum of points of participation of weightlifters in national and international competitions in 2021.

Monitoring the development of weightlifting in Ukraine requires taking into account the effectiveness of individual and team performance indicators in national and international competitions for women weightlifters of different ages. Research by experts indicates a fragmentary consideration of the effectiveness of competitive activities of women weightlifters in different regions of Ukraine and the National Team of Ukraine. The goal is to determine the effectiveness of team and individual indicators of competitive activity of weightlifters of all ages in different regions of Ukraine and the National Team of Ukraine at the national and international starts in 2021. To solve this goal, we used the following research methods: analogy, analysis, synthesis, induction, analysis of documentary materials, methods of mathematical statistics. As a result of the research the following were revealed: quantitative data of participants of women weightlifters of different age groups and teams of regions of Ukraine in national competitions in 2021; team championship of weightlifters of different age groups of different regions of Ukraine based on the results of participation in national competitions in 2021; quantitative data on the participation of Ukrainian weightlifters of different age groups in international competitions in 2021; personal and team medal count of women-weightlifters of Ukraine of different age groups according to the results of participation in international competitions in 2021.

Among the areas of improving the training system of athletes, the focus on finding new effective tools and methods of training is one of the most popular ways to improve athletic performance. Therefore, substantiation of the peculiarities of the use of occlusion in the training of highly qualified athletes in classical powerlifting is relevant. The aim is to identify the peculiarities of the use of occlusion in the training of highly qualified powerlifters. To solve this goal, we used the following research methods: analogy, analysis, synthesis, abstraction, induction, extrapolation, generalization of practical experience. The study revealed the features of the use of occlusion in the training of athletes in classical powerlifting. It is recommended to use occlusion devices in isolation and to include only one joint in the performance of exercises. Therefore, in the author's program occlusion means are used exclusively in bench press exercises and squats. In the process of building the training of athletes using the means of collusion, the intensity of the load should be 50-60% of the maximum, the number of repetitions should be 8-15 times and should decrease with each approach. The duration of rest between approaches in the training process using the means of occlusion should be 30 seconds. Means of occlusion should be removed only after the last approach in the exercise, and their implementation should be planned at the end of the main part of the training process.

December 25 marks the 75th anniversary of the famous Ukrainian specialist in mathematical physics, winner of the State Prize of Ukraine in Science and Technology (2001) and the M.M. Krylov Prize of the NAS of Ukraine (2010), Head of the Department of Mathematical Physics of the Institute of Mathematics of the NAS of Ukraine, Doctor of Physical and Mathematical Sciences (1987), Professor (2001), Corresponding Member of the NAS of Ukraine (2009) Anatoly G. Nikitin.

We classify zeroth-order conservation laws of systems from the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The classification is carried out up to equivalence generated by the equivalence group of this class. We find additional point equivalences between some of the listed cases of extensions of the space of zeroth-order conservation laws, which are inequivalent up to transformations from the equivalence group. Hamiltonian structures of systems of shallow water equations are used for relating the classification of zeroth-order conservation laws of these systems to the classification of their Lie symmetries. We also construct generating sets of such conservation laws under action of Lie symmetries.

Applying an original megaideal-based version of the algebraic method, we compute the point-symmetry pseudogroup of the dispersionless (potential symmetric) Nizhnik equation. This is the first example of this kind in the literature, where there is no need to use the direct method for completing the computation. The analogous studies are also carried out for the corresponding nonlinear Lax representation and the dispersionless counterpart of the symmetric Nizhnik system. We also first apply the megaideal-based version of the algebraic method to find the contact-symmetry (pseudo)group of a partial differential equation. It is shown that the contact-symmetry pseudogroup of the dispersionless Nizhnik equation coincides with the first prolongation of its point-symmetry pseudogroup. We check whether the subalgebras of the maximal Lie invariance algebra of the dispersionless Nizhnik equation that naturally arise in the course of the above computations define the diffeomorphisms stabilizing this algebra or its first prolongation. In addition, we construct all the third-order partial differential equations in three independent variables that admit the same Lie invariance algebra. We also find a set of geometric properties of the dispersionless Nizhnik equation that exhaustively defines it., Comment: 27 pages, extended version

We derive a nice representation for point symmetry transformations of the (1+1)-dimensional linear heat equation and properly interpret them. This allows us to prove that the pseudogroup of these transformations has exactly two connected components. That is, the heat equation admits a single independent discrete symmetry, which can be chosen to be alternating the sign of the dependent variable. We introduce the notion of pseudo-discrete elements of a Lie group and show that alternating the sign of the space variable, which was for a long time misinterpreted as a discrete symmetry of the heat equation, is in fact a pseudo-discrete element of its essential point symmetry group. The classification of subalgebras of the essential Lie invariance algebra of the heat equation is enhanced and the description of generalized symmetries of this equation is refined as well. We also consider the Burgers equation because of its relation to the heat equation and prove that it admits no discrete point symmetries. The developed approach to point-symmetry groups whose elements have components that are linear fractional in some variables can directly be extended to many other linear and nonlinear differential equations., Comment: 21 pages, published version, minor corrections, the application of an approach from arXiv:2205.13526 to the heat equation

We carry out the extended symmetry analysis of an ultraparabolic Fokker-Planck equation with three independent variables, which is also called the Kolmogorov equation and is singled out within the class of such Fokker-Planck equations by its remarkable symmetry properties. In particular, its essential Lie invariance algebra is eight-dimensional, which is the maximum dimension within the above class. We compute the complete point symmetry pseudogroup of the Fokker-Planck equation using the direct method, analyze its structure and single out its essential subgroup. After listing inequivalent one- and two-dimensional subalgebras of the essential and maximal Lie invariance algebras of this equation, we exhaustively classify its Lie reductions, carry out its peculiar generalized reductions and relate the latter reductions to generating solutions with iterative action of Lie-symmetry operators. As a result, we construct wide families of exact solutions of the Fokker-Planck equation, in particular, those parameterized by an arbitrary finite number of arbitrary solutions of the (1+1)-dimensional linear heat equation. We also establish the point similarity of the Fokker-Planck equation to the (1+2)-dimensional Kramers equations whose essential Lie invariance algebras are eight-dimensional, which allows us to find wide families of exact solutions of these Kramers equations in an easy way., Comment: 33 pages, 2 tables, corrected version

We revisit the entire framework of group classification of differential equations. After introducing the notion of weakly similar classes of differential equations, we develop the mapping method of group classification for such classes, which generalizes all the versions of this method that have been presented in the literature. The mapping method is applied to group classification of various classes of Kolmogorov equations and of Fokker-Planck equations in the case of space dimension one. The equivalence groupoids and the equivalence groups of these classes are computed. The group classification problems for these classes with respect to the corresponding equivalence groups are reduced to finding all inequivalent solutions of heat equations with inequivalent potentials admitting Lie-symmetry extensions. This reduction allows us to exhaustively solve the group classification problems for the classes of Kolmogorov and Fokker-Planck equations with time-independent coefficients., 42 pages, 2 tables, minor revision

Essentially generalizing Lie's results, we prove that the contact equivalence groupoid of a class of (1+1)-dimensional generalized nonlinear Klein-Gordon equations is the first-order prolongation of its point equivalence groupoid, and then we carry out the complete group classification of this class. Since it is normalized, the algebraic method of group classification is naturally applied here. Using the specific structure of the equivalence group of the class, we essentially employ the classical Lie theorem on realizations of Lie algebras by vector fields on the line. This approach allows us to enhance previous results on Lie symmetries of equations from the class and substantially simplify the proof. After finding a number of integer characteristics of cases of Lie-symmetry extensions that are invariant under action of the equivalence group of the class under study, we exhaustively describe successive Lie-symmetry extensions within this class., 31 pages, 1 figure, minor corrections

Viral hepatitis B course in HIV patients in Zambia is still bad studied. At present, the study of the influence of immunosuppression on the clinical and laboratory features of viral hepatitis B acquires a special relevance. The aim of research is to investigate clinical and laboratory features of viral hepatitis B course among HIV-patients in Zambia. Epidemiological, descriptive, serological (HBV, HIV markers), biochemical (cytolysis and cholestasis syndromes’ indicators), statistic method for determining the reliability using average errors (m), t-criterion of the probability difference of the Student) methods are used. The results show HBV/HIV-positive Zambian patients reliably differ from HBV-monoinfected ones. As well, HBV/HIV-positive patients have significantly higher indexes of asteno-vegetatic, dyspeptic syndromes, hepato- and splenomegaly, cytolysis indexes (alanine aminotransferase, lactate dehydrogenase), which are associated with increase of HIV-infection’s stage. Hepatocellular deficiency’s indicators don’t differ reliably in comparison groups. Therefore, HBV/HIV-infected patients have more severe clinical and laboratory course than HBV-monoinfected ones.

We propose the use of physics-informed neural networks for solving the shallow-water equations on the sphere in the meteorological context. Physics-informed neural networks are trained to satisfy the differential equations along with the prescribed initial and boundary data, and thus can be seen as an alternative approach to solving differential equations compared to traditional numerical approaches such as finite difference, finite volume or spectral methods. We discuss the training difficulties of physics-informed neural networks for the shallow-water equations on the sphere and propose a simple multi-model approach to tackle test cases of comparatively long time intervals. Here we train a sequence of neural networks instead of a single neural network for the entire integration interval. We also avoid the use of a boundary value loss by encoding the boundary conditions in a custom neural network layer. We illustrate the abilities of the method by solving the most prominent test cases proposed by Williamson et al. [J. Comput. Phys. 102 (1992), 211-224]., Comment: 24 pages, 9 figures, 1 tables, minor extensions

We describe the point and contact equivalence groupoids of an important class of two-dimensional quasilinear hyperbolic equations. In particular, we prove that this class is normalized in the usual sense with respect to point transformations, and its contact equivalence groupoid is generated by the first-order prolongation of its point equivalence groupoid, the contact vertex group of the wave equation and a family of contact admissible transformations between trivially Darboux-integrable equations., 8 pages, minor corrections

We introduce matrix coupled (local and non-local) dispersionless equations, construct GBDT (generalized Bäcklund-Darboux transformation) for these equations, derive wide classes of explicit multipole solutions, give explicit expressions for the corresponding Darboux and wave matrix valued functions and study their asymptotics in some interesting cases. We consider the scalar cases of coupled, complex coupled and non-local dispersionless equations as well.

We carry out the complete group classification of the class of (1+1)-dimensional linear Schrodinger equations with complex-valued potentials. After introducing the notion of uniformly semi-normaliz ...

The group classification problem for the class of (1+1)-dimensional linear $r$th order evolution equations is solved for arbitrary values of $r>2$. It is shown that a related maximally gauged class of homogeneous linear evolution equations is uniformly semi-normalized with respect to linear superposition of solutions and hence the complete group classification can be obtained using the algebraic method. We also compute exact solutions for equations from the class under consideration using Lie reduction and its specific generalizations for linear equations., Minor corrections, 24 pages, 1 table

We study the hydrodynamic-type system of differential equations modeling isothermal no-slip drift flux. Using the facts that the system is partially coupled and its subsystem reduces to the (1+1)-dimensional Klein--Gordon equation, we exhaustively describe generalized symmetries, cosymmetries and local conservation laws of this system. A generating set of local conservation laws under the action of generalized symmetries is proved to consist of two zeroth-order conservation laws. The subspace of translation-invariant conservation laws is singled out from the entire space of local conservation laws. We also find broad families of local recursion operators and a nonlocal recursion operator, and construct an infinite family of Hamiltonian structures involving an arbitrary function of a single argument. For each of the constructed Hamiltonian operators, we obtain the associated algebra of Hamiltonian symmetries., 36 pages, extended version, the proof on cosymmetries in presented with more details

We carry out the group classification of the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The equivalence group of this class is found by the algebraic method. Using algebraic techniques, we construct additional point equivalences between some of the listed cases of Lie-symmetry extensions, which are inequivalent up to transformations from the equivalence group., Comment: 26 pages, 1 figure, minor extension

The well-known solution theory for (systems of) linear ordinary differential equations undergoes significant changes when introducing an additional real parameter. Properties like the existence of fundamental sets of solutions or characterizations of such sets via nonvanishing Wronskians are sensitive to the topological properties of the underlying domain of the independent variable and the parameter. We give a complete characterization of the solvability of such parameter-dependent equations and systems in terms of topological properties of the domain. In addition, we also investigate this problem in the setting of Schwartz distributions., Comment: 26 pages, 7 figures, minor corrections

We carry out the extended symmetry analysis of a two-dimensional degenerate Burgers equation. Its complete point-symmetry group is found using the algebraic method, and all its generalized symmetries are proved equivalent to its Lie symmetries. We also prove that the space of conservation laws of this equation is infinite-dimensional and is naturally isomorphic to the solution space of the (1+1)-dimensional backward linear heat equation. Lie reductions of the two-dimensional degenerate Burgers equation are comprehensively studied in the optimal way and new Lie invariant solutions are constructed. We additionally consider solutions that also satisfy an analogous nondegenerate Burgers equation. In total, we construct four families of solutions of two-dimensional degenerate Burgers equation that are expressed in terms of arbitrary (nonzero) solutions of the (1+1)-dimensional linear heat equation. Various kinds of hidden symmetries and hidden conservation laws (local and potential ones) are discussed as well. As a by-product, we exhaustively describe generalized symmetries, cosymmetries and conservation laws of the transport equation, also called the inviscid Burgers equation, and construct new invariant solutions of the nonlinear diffusion and diffusion-convection equations with power nonlinearities of degree -1/2., Comment: revised version, 26 pp., application of techniques developed in arXiv:1709.02708 to a two-dimensional degenerate Burgers equation

We study admissible transformations and Lie symmetries for a class of variable-coefficient Burgers equations. We combine the advanced methods of splitting into normalized subclasses and of mappings between classes that are generated by families of point transformations parameterized by arbitrary elements of the original classes. A nontrivial differential constraint on the arbitrary elements of the class of variable-coefficient Burgers equations leads to its partition into two subclasses, which are related to normalized classes via families of point transformations parameterized by subclasses' arbitrary elements. One of the mapped classes is proved to be normalized in the extended generalized sense, and its effective extended generalized equivalence group is found. Using the mappings between classes and the algebraic method of group classification, we carry out the group classification of the initial class with respect to its equivalence groupoid., Comment: 22 pages, minor corrections

We study admissible and equivalence point transformations between generalized multidimensional nonlinear Schr\"odinger equations and classify Lie symmetries of such equations. We begin with a wide superclass of Schr\"odinger-type equations, which includes all the other classes considered in the paper. Showing that this superclass is not normalized, we partition it into two disjoint normalized subclasses, which are not related by point transformations. Further constraining the arbitrary elements of the superclass, we construct a hierarchy of normalized classes of Schr\"odinger-type equations. This gives us an appropriate normalized superclass for the non-normalized class of multidimensional nonlinear Schr\"odinger equations with potentials and modular nonlinearities and allows us to partition the latter class into three families of normalized subclasses. After a preliminary study of Lie symmetries of nonlinear Schr\"odinger equations with potentials and modular nonlinearities for an arbitrary space dimension, we exhaustively solve the group classification problem for such equations in space dimension two., Comment: 35 pages, minor corrections

The direct method based on the definition of conserved currents of a system of differential equations is applied to compute the space of conservation laws of the (1+1)-dimensional wave equation in the light-cone coordinates. Then Noether's theorem yields the space of variational symmetries of the corresponding functional. The results are also presented for the standard space-time form of the wave equation., 6 pages, minor corrections

The explicit formulation of the general inverse problem on conservation laws is presented for the first time. In this problem one aims to derive the general form of systems of differential equations that admit a prescribed set of conservation laws. The particular cases of the inverse problem on first integrals of ordinary differential equations and on conservation laws for evolution equations are studied. We also solve the inverse problem on conservation laws for differential equations admitting an infinite dimensional space of zeroth-order conservation-law characteristics. This particular case is further studied in the context of conservative first-order parameterization schemes for the two-dimensional incompressible Euler equations. We exhaustively classify conservative first-order parameterization schemes for the eddy-vorticity flux that lead to a class of closed, averaged Euler equations possessing generalized circulation, generalized momentum and energy conservation., 29 pages, extended version

We perform extended group analysis for a system of differential equations modeling an isothermal no-slip drift flux. The maximal Lie invariance algebra of this system is proved to be infinite-dimensional. We also find the complete point symmetry group of this system, including discrete symmetries, using the megaideal-based version of the algebraic method. Optimal lists of one- and two-dimensional subalgebras of the maximal Lie invariance algebra in question are constructed and employed for obtaining reductions of the system under study. Since this system contains a subsystem of two equations that involves only two of three dependent variables, we also perform group analysis of this subsystem. The latter can be linearized by a composition of a fiber-preserving point transformation with a two-dimensional hodograph transformation to the Klein-Gordon equation. We also employ both the linearization and the generalized hodograph method for constructing the general solution of the entire system under study. We find inter alia genuinely generalized symmetries for this system and present the connection between them and the Lie symmetries of the subsystem we mentioned earlier. Hydrodynamic conservation laws and their generalizations are also constructed., 29 pages, minor corrections

Diabetic foot syndrome (DFS) is a serious late complication of Diabetes mellitus (DM) and the unsolved problem of endocrine surgery. This pathology occurs in 80% of patients with diabetes and half of them leads to an extremity amputation. Combined drug and surgical treatment of patients with type II diabetes complications with pyonecrotic DFS stages and sacrococcygeal and perineum phlegmons was analysed. There were used the medical immune biological preparations (MIBP) based on cryopreserved placental blood which contribute to the normalization of immunological reactions, proliferation processes, stimulate a reparation, possess antioxidant, adaptogenic, anti-hypoxic, anabolic and anti-inflammatory effects. The patients were administered with "Cryocell-Cryocord", i.e. placental blood serum (108 patients), and in severe endointoxication they were additionally treated within "Cryocell-Hemocord", placental blood leukoconcentrate (37 patients). When applying MIBP the insulin dosage is reduced by 25-30%. A significant decrease in leukocyte index, the level of an average molecular weight peptides is observed. Cytoarchitecture of erythrocytes significantly improved. Potentiation of the preparations on the activity of metalloenzymes, hepatocyte cytolysis enzymes was found. The number of proteases inhibitors decreases with a parallel increase in the content of albumin, total protein amount, normalization of organ specific enzymes.The increase in total content of immunoglobulins, a sharp decline in the level of circulating immune complexes (antigen-antibody autoantigens) with a parallel increase in the antibodies of natural protection were revealed. In addition, under the influence of cord blood MIBP the wound cleansing and its healing starts faster, pyo-inflammatory process is restricted at the foot level, lower leg edema disappears. The data obtained can be interpreted as recovery of reparative processes in the wound due to effective correction of metabolic-homeostatic changes in the patient suffering from diabetes and its complications. Probl Cryobiol Cryomed 2014; 24(4):332-345.

The complete group classification problem for the class of (1+1)-dimensional $r$th order general variable-coefficient Burgers-Korteweg-de Vries equations is solved for arbitrary values of $r$ greater than or equal to two. We find the equivalence groupoids of this class and its various subclasses obtained by gauging equation coefficients with equivalence transformations. Showing that this class and certain gauged subclasses are normalized in the usual sense, we reduce the complete group classification problem for the entire class to that for the selected maximally gauged subclass, and it is the latter problem that is solved efficiently using the algebraic method of group classification. Similar studies are carried out for the two subclasses of equations with coefficients depending at most on the time or space variable, respectively. Applying an original technique, we classify Lie reductions of equations from the class under consideration with respect to its equivalence group. Studying of alternative gauges for equation coefficients with equivalence transformations allows us not only to justify the choice of the most appropriate gauge for the group classification but also to construct for the first time classes of differential equations with nontrivial generalized equivalence group such that equivalence-transformation components corresponding to equation variables locally depend on nonconstant arbitrary elements of the class. For the subclass of equations with coefficients depending at most on the time variable, which is normalized in the extended generalized sense, we explicitly construct its extended generalized equivalence group in a rigorous way. The new notion of effective generalized equivalence group is introduced., 40 pages, 1 table, minor corrections

Admissible point transformations between Burgers equations with linear damping and time-dependent coefficients are described and used in order to exhaustively classify Lie symmetries of these equations. Optimal systems of one- and two-dimensional subalgebras of the Lie invariance algebras obtained are constructed. The corresponding Lie reductions to ODEs and to algebraic equations are carried out. Exact solutions to particular equations are found. Some generalized Burgers equations are linearized to the heat equation by composing equivalence transformations with the Hopf-Cole transformation., 18 pages, 1 figure; the version accepted to Appl. Math. Comput

Using advanced classification techniques, we carry out the extended symmetry analysis of the class of generalized Burgers equations of the form $u_t+uu_x+f(t,x)u_{xx}=0$. This enhances all the previous results on symmetries of these equations and includes the description of admissible transformations, Lie symmetries, Lie and nonclassical reductions, conservation laws, potential admissible transformations and potential symmetries. The study is based on the fact that the class is normalized, and its equivalence group is finite-dimensional., Comment: 31 pages, 2 tables, minor corrections

Invariant discretization schemes are derived for the one- and two-dimensional shallow-water equations with periodic boundary conditions. While originally designed for constructing invariant finite difference schemes, we extend the usage of difference invariants to allow constructing of invariant finite volume methods as well. It is found that the classical invariant schemes converge to the Lagrangian formulation of the shallow-water equations. These schemes require to redistribute the grid points according to the physical fluid velocity, i.e., the mesh cannot remain fixed in the course of the numerical integration. Invariant Eulerian discretization schemes are proposed for the shallow-water equations in computational coordinates. Instead of using the fluid velocity as the grid velocity, an invariant moving mesh generator is invoked in order to determine the location of the grid points at the subsequent time level. The numerical conservation of energy, mass and momentum is evaluated for both the invariant and non-invariant schemes., Comment: 27 pages, 6 figures, minor corrections

The method of preliminary group classification is rigorously defined, enhanced and related to the theory of group classification of differential equations. Typical weaknesses in papers on this method are discussed and strategies to overcome them are presented. The preliminary group classification of the class of generalized diffusion equations of the form u_t=f(x,u)u_x^2+g(x,u)u_{xx} is carried out. This includes a justification for applying this method to the given class, the simultaneous computation of the equivalence algebra and equivalence group, as well as the classification of inequivalent appropriate subalgebras of the whole infinite-dimensional equivalence algebra. The extensions of the kernel algebra, which are induced by such subalgebras, are exhaustively described. These results improve those recently published in Commun. Nonlinear Sci. Numer. Simul., 22 pages, minor corrections

We study local conservation laws for evolution equations in two independent variables. In particular, we present normal forms for the equations admitting one or two low-order conservation laws. Examples include Harry Dym equation, Korteweg-de-Vries-type equations, and Schwarzian KdV equation. It is also shown that for linear evolution equations all their conservation laws are (modulo trivial conserved vectors) at most quadratic in the dependent variable and its derivatives., Comment: 16 pages

Better convenience and tolerability and sustained therapeutic concentrations might improve interferon (IFN) treatment for chronic hepatitis C virus (HCV) infection. In an open-label, randomized study, controlled release free (chemically unmodified) recombinant human IFN-α(2b) in poly(ether-ester) microspheres (CR-rhIFN-α(2b)), was injected at doses of 160, 320, 480 or 640 μg every 2 weeks for 12 weeks with concomitant weight-based oral ribavirin in 32 treatment-naive patients with chronic HCV genotype 1. Treatment was well tolerated, with 31 patients (97%) successfully completing the study. Full doses of CR-rhIFN-α(2b) were administered on 96% of scheduled occasions. Flu-like symptoms were generally mild and brief. Injection site reactions developed in 13 patients (41%), and neutropenia occurred in six of eight patients receiving 640 μg. In the 320, 480 and 640 μg groups, 62-75% of patients achieved a ≥2 log(10) HCV RNA reduction by 4 weeks and 88-100% by 12 weeks. For those groups, the pooled median time to ≥2 log(10) reduction was 11 days (95% confidence interval, 7-35 days). In those groups, viral reduction below the limit of detection was accomplished in 25% of patients by 4 weeks and in 62% by 12 weeks. The 160-μg dose was less potent. After CR-rhIFN-α(2b) injection, stable plateau levels of serum IFN-α(2b) were generally reached within 72 h. Treatment-emergent neutralizing antibodies to IFN-α(2b) were observed in one patient. No antibodies to host plant proteins were detected. CR-rhIFN-α(2b) with ribavirin cotherapy was well tolerated and displayed potent early antiviral activity in patients with chronic HCV genotype 1.

We present a simple and rigorous proof of the claim by Weimar-Woods [Rev. Math. Phys. 12 (2000) 1505-1529] that any diagonal contraction (e.g., a generalized In\"on\"u-Wigner contraction) is equivalent to a generalized In\"on\"u-Wigner contraction with integer parameter powers., Comment: 9 pages, extended version

A multi-dimensional simple wave formalism is employed to formulate a multi-dimensional quasi-simple wave for a weakly dissipative fluid. This is a natural but nontrivial generalization of the so-called unidirectional quasi-simple wave. The method is more close to multi-orders analysis which is different from the standard perturbation method. Detailed solving up to the second order is presented for a 2D sound simple wave. A new 2D Burgers equation is derived for the wave phase. It essentially differs from the known 2D generalizations of the Burgers equation, e.g., from the Zabolotskaya–Khokhlov equation. The derived equation is investigated in the framework of group analysis of differential equations. Multi-parameter families of its exact solutions are constructed. Simplest solutions are chosen for analysis of their physical relevance in the initial variables.

The invariants of solvable Lie algebras with nilradicals isomorphic to the algebra of strongly upper triangular matrices and diagonal nilindependent elements are studied exhaustively. Bases of the invariant sets of all such algebras are constructed by an original purely algebraic algorithm based on Cartan's method of moving frames., Comment: 21 pages, enhanced and extended version. Section 2 reviews the method of finding invariants of Lie algebras that was proposed in arXiv:math-ph/0602046 and arXiv:math-ph/0606045. The computation is based on developing a specific technique given in arXiv:0704.0937. Results generalize ones of arXiv:0705.2394 to the case of arbitrary relevant number of nilindependent elements

We carry out an extensive investigation of conservation laws and potential symmetries for the class of linear (1+1)-dimensional second-order parabolic equations. The group classification of this class is revised by employing admissible transformations, the notion of normalized classes of differential equations and the adjoint variational principle. All possible potential conservation laws are described completely. They are in fact exhausted by local conservation laws. For any equation from the above class the characteristic space of local conservation laws is isomorphic to the solution set of the adjoint equation. Effective criteria for the existence of potential symmetries are proposed. Their proofs involve a rather intricate interplay between different representations of potential systems, the notion of a potential equation associated with a tuple of characteristics, prolongation of the equivalence group to the whole potential frame and application of multiple dual Darboux transformations. Based on the tools developed, a preliminary analysis of generalized potential symmetries is carried out and then applied to substantiate our construction of potential systems. The simplest potential symmetries of the linear heat equation, which are associated with single conservation laws, are classified with respect to its point symmetry group. Equations possessing infinite series of potential symmetry algebras are studied in detail., Comment: 67 pages, minor corrections

The invariants of solvable triangular Lie algebras with one nilindependent diagonal element are studied exhaustively. Bases of the invariant sets of all such algebras are constructed using an original algebraic algorithm based on Cartan's method of moving frames and the special technique developed for triangular and related algebras in [J. Phys. A: Math. Theor. 40 (2007), 7557-7572]. The conjecture of Tremblay and Winternitz [J. Phys. A: Math. Gen. 34 (2001), 9085-9099] on the number and form of elements in the bases is completed and proved., Comment: 11 pages; enhanced version