the paper, we find sharp constants in a Kolmogorov-type inequality between the L-2r-norm of the derivative, the L-r-norm (quasinorm for r < 1) of the function, and the asymmetric L-infinity-norm (including the norm of the nonnegative part) of the second derivative for any 1/2 <= r < infinity on the real line and on the period.
the Ramsey number R(K-1,K-s, P-t) one means the least positive integer n such that, for every n-vertex graph G, the following condition holds: either G contains a vertex of degree at least s or the complement of G contains a simple t-path. In this paper, we find precise values of R(K-1,K-s, P-t) for certain values s > t.
Nonlinear inverse problems with a time-dependent parameter are considered both for equations solvable with respect to the Gerasimov–Caputo leading fractional derivative and for equations with a degenerate operator at the derivative. The global existence of a unique generalized solution to the nonlinear inverse problem is proven for a nondegenerate equation with a sectorial operator in the linear part. In the degenerate case, under the condition that the image of the nonlinear operator belongs to a subspace without degeneration, the local existence and uniqueness of the generalized and smooth solutions and the global existence and uniqueness of the generalized solution to the nonlinear inverse problem are obtained. Under the condition that the nonlinear operator is independent of the elements of the degeneration subspace, the local and global existence of a unique generalized solution to the inverse problem are also proven. The results thus obtained are illustrated by examples of coefficient inverse problems for degenerate systems of partial differential equations.
A criterion for the solvability of a holomorphic Cauchy problem for a certain class of matrix soliton equations is proved in terms of a scattering transform defined in a new way. For the equations under consideration, a method is indicated for constructing holomorphic solutions using the Zakharov-Shabat dressing method for the zero solution, as well as the possibility of extending the solutions with respect to the spatial variable to globally meromorphic functions.
We consider the class of nonlinear multivariate Urysohn type integral equations in the critical case. For certain Urysohn kernels, this class of equations has applications in various problems of physics and biology. We prove a constructive theorem on existence of a positive bounded continuous solution. We also show that the corresponding iterations converge geometrically to this solution. The solution thus constructed is shown to be unique in a certain conical segment. We also give examples of Urysohn kernels satisfying the conditions of our theorems.
Using an operational scheme, we explicitly construct a multiparameter family of solutions to a multidimensional hyperbolic equation with translation operators in the lowest derivatives acting in all coordinate directions is. We prove a theorem stating that the resulting solutions are classical under the condition that the real part of the symbol of the differential-difference operator.
In this note, we prove that the higher Bers maps induced by a kind of higher order Schwarzian derivatives on the Teichmüller spaces of circle diffeomorphisms with Hölder continuous derivative are holomorphic.
In this paper, a new block preconditioner based on the pre-2 order block splitting of the coefficient matrix is established to solve the three-by-three block saddle point problems. In theory, the new iteration method under the preconditioner is proved to be unconditionally convergent; then, the eigenvalues of the preconditioned matrix are shown to have good clustering properties; furthermore, the algorithm implementation and parameter selection of the preconditioner are also discussed in details. Finally, numerical experiments show the effectiveness of the new block preconditioner by comparing with other existing preconditioners.
There exist seven nonequivalent Lorentzian metrics on the nilpotent four-dimensional Lie group G_4 . We show that three of these metrics do not yield algebraic Ricci solitons, while two are confirmed to be algebraic Ricci solitons. Among the latter, one metric is Ricci-flat under a certain condition but not flat. The remaining two metrics admit an algebraic Ricci soliton structure only in specific cases, depending on particular parameter relations. In contrast, for the nilpotent Lie group H_3 ×ℝ , five Lorentzian metrics are algebraic Ricci solitons and one is flat.
A representation in the form of a multiplicative convolution is obtained for Hadamard type operators on spaces of holomorphic functions on a bounded convex complex domain which have polynomial growth near the boundary of the domain or are infinitely differentiable up to the boundary. It is proved that the spaces of Hadamard type operators under consideration endowed with the topology of uniform convergence on bounded sets are topologically isomorphic to the strong dual of the space of all C^∞ functions on the corresponding set of multipliers holomorphic in its interior.
Approximation properties of q -Cesáro sums of rational Fourier–Chebyshev integral operators with arbitrary fixed number of geometrically different poles are studied. For this method of rational approximation, an integral representation is obtained. The suprema of the q -Cesáro sums of the polynomial Fourier–Chebyshev series are evaluated on classes of γ -Hölder functions with γ∈ (0,1] on [-1,1] . Approximations of the function |x|^s , s ∈ (0,2) , on [-1,1] by the rational q -Cesáro sums are studied. An asymptotic expression of the majorant of uniform approximations depending on the parameters of an approximated function functions is derived, and the values of these parameters are given for which the majorant converges to zero most rapidly. For s ∈ (0,1] , the uniform rational approximation by our method are shown to have a greater decrease rate in comparison with their polynomial analogues.
In the paper, we find sharp constants in a Kolmogorov-type inequality between the L_2r -norm of the derivative, the L_r -norm (quasinorm for r<1 ) of the function, and the asymmetric L_∞ -norm (including the norm of the nonnegative part) of the second derivative for any 1/2 ≤ r< ∞ on the real line and on the period.
