Ranks of matrices with few distinct entries

It is consistent that P(ω ₁) is the union of less than \({2^{{\aleph _1}}}\) parts such that if A ₀,..., A _n?1, B ₀,..., B _m?1 are distinct elements of the same part, then |A ₀ ∩ · · · ∩ A _n?1 ∩ (ω ₁ ? B ₀) ∩ · · ·∩ (ω ₁ ? B _m?1)| = N₁.     

Serrin-type regularity criterion for the Navier-Stokes equations involving one velocity and one vorticity component

We consider the Cauchy problem for the three-dimensional Navier-Stokes equations, and provide an optimal regularity criterion in terms of u₃ and ω₃, which are the third components of the velocity and vorticity, respectively. This gives an affirmative answer to an open problem in the paper by P. Penel, M.Pokorný (2004).     

Quasi-Periodic Solutions for Non-Autonomous mKdV Equation

In this paper, we consider the non-autonomous modified Korteweg-de Vries (mKdV) equation

$${u_t} = {u_{xxx}} - 6f\left( {\omega t} \right){u^2}{u_x},x \in \mathbb{R}/2\pi \mathbb{Z}$$

, where f(ωt) is real analytic and quasi-periodic in t with frequency vector ω = (ω₁,ω₂, · · ·; ω _m ). Basing on an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field, we obtain the existence of Cantor families of smooth quasi-periodic solutions.

     

On real solutions of systems of equations

Systems of equations f ₁ = ··· = f _n?1 = 0 in ? ⁿ = {x} having the solution x = 0 are considered under the assumption that the quasi-homogeneous truncations of the smooth functions f ₁,..., f _n?1 are independent at x ≠ 0. It is shown that, for n ≠ 2 and n ≠ 4, such a system has a smooth solution which passes through x = 0 and has nonzero Maclaurin series.     

A test for identities satisfied in lattices of submodules

SupposeR is ring with 1, andH?(R) denotes the variety of modular lattices generated by the class of lattices of submodules of allR-modules. An algorithm using Mal'cev conditions is given for constructing integersm≧0 andn≧1 from any given lattice polynomial inclusion formulad∈e. The main result is thatd∈e is satisfied in every lattice inH?(R) if and only if there existsx inR such that (m·1)x=n·1 inR, where 0·1=0 andk·1=1+1...+1 (k times) fork≧1. For example, this “divisibility” condition holds form=2 andn=1 if and only if 1+1 is an invertible element ofR, and it holds form=0 andn=12 if and only if the characteristic ofR divides 12. This result leads to a complete classification of the lattice varietiesH?(R),R a ring with 1. A set of representative rings is constructed, such that for each ringR there is a unique representative ringS satisfyingH?(R)=H?(R). There is exactly one representative ring with characteristick for eachk≧1, and there are continuously many representative rings with characteristic zero. IfR has nonzero characteristic, then all free lattices inH?(R) have recursively solvable word problems. A necessary and sufficient condition onR is given for all free lattices inH?(R) to have recursively solvable word problems, ifR is a ring with characteristic zero. All lattice varieties of the formH?(R) are self-dual. A varietyH?(R) is a congruence variety, that is, it is generated by the class of congruence lattices of all members of some variety of algebras. A family of continuously many congruence varieties related to the varietiesH?(R) is constructed.     

Incompressible limit and stability of all-time solutions to 3-D full Navier-Stokes equations for perfect gases

This paper studies the incompressible limit and stability of global strong solutions to the threedimensional full compressible Navier-Stokes equations, where the initial data satisfy the "well-prepared" conditions and the velocity field and temperature enjoy the slip boundary condition and convective boundary condition, respectively. The uniform estimates with respect to both the Mach number ∈(0, ∈] and time t ∈ [0, ∞) are established by deriving a differential inequality with decay property, where ∈∈(0, 1] is a constant.As the Mach number vanishes, the global solution to full compressible Navier-Stokes equations converges to the one of isentropic incompressible Navier-Stokes equations in t ∈ [0, +∞). Moreover, we prove the exponentially asymptotic stability for the global solutions of both the compressible system and its limiting incompressible system.     

On oscillation of solutions of a nonautonomous quasilinear second-order equation

Sufficient conditions are obtained for the initial values of nontrivial oscillating (for t=ω) solutions of the nonautonomous quasilinear equation $$y'' \pm \lambda (t)y = F(t,y,y'),$$ wheret ∈ Δ=[a, ω[,-∞ <a < ω ≤+ ∞, λ(t) > 0, λ(t) ∈ C _Δ ⁽¹⁾ , |F((t,x,y))|≤L(t)(|x|+|y|)^1+α, L(t) ≥-0, α ∈ [0,+∞[, F: Δ × R² →R,F ∈C _Δ×R ²,R is the set of real numbers, and R² is the two-dimensional real Euclidean space.     

Asymptotic integration of symmetric second-order quasidifferential equations

This paper presents conditions on the coefficients of the equations $$\begin{array}{*{20}c} { - (p(f' - rf))' - \bar rp(f' - rf) + qf = 0,} \\ { - (P(f' - Rf))' - \bar RP(f' - Rf) + Qf = 0,} \\ \end{array}$$ where 1/p, 1/P, q, Q, r, R ∈ ? _loc ¹ (?₊), p, P, q, and Q are real-valued functions, while r and R are complex-valued functions, as well as on the fundamental system of solutions of the second equation, which ensure the asymptotic proximity of the solutions of these equations. The results obtained are applied to the study of the spectral properties of the differential operator generated by the expression $$ - y'' + \sum\limits_{k = 0}^{ + \infty } {h_k \delta (x - x_k )y, x_k \in \mathbb{R}_ + ,} h_k \in R,$$ , in the space ?²(?₊). In particular, we obtain conditions on h _k , x _k under which the limit-disk case is realized for this operator.     

Признак Дини-Липшица и сходимость рядов Фу рье по мультипликативны м системам

Fourier serieses with respect to Vilenkin multiplicative systems are studied. It is proved that a Dini-Lipschitz condition imposed on the group or group integral modulus of continuity is sufficient for the convergence of the series in the normC(·) orL(·), respectively. Necessity in the classesH _ω andH _ω ¹ of that condition is also proved. A boundedness criterion of the sequence of partial sums inH _ω andH _ω ¹ is also obtained.     

Some embedding theorems for generalized Nikol'skii classes from Lorentz spaces

Quasi-normed Lorentz spaces Λ_{ψ, q} of 2π-periodic functions with quasinorms $$\left\| f \right\|_{\psi ,q} = \left\{ {\int\limits_0^{2\pi } {\psi ^q (t)\left[ {\frac{1}{t}\int\limits_0^t {f * (x)} dx} \right]} ^q \frac{{dt}}{t}} \right\}^{{1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-\nulldelimiterspace} q}} $$ (0<q<∞,ω(t): [0,2π]→R is a continuous concave function with finite derivative everywhere on (0, 2gp)) and classes of functions $$H_{\psi ,q}^\omega \equiv \{ f(x):f(x) \in \Lambda _{\psi ,q} ;\mathop {\sup }\limits_{0 \leqq h \leqq \delta } \left\| {f(x + h) - f(x)} \right\|_{\psi ,q} = O\{ \omega (\delta )\} , \delta \to + 0\} $$ (ω(δ) — modulus of continuity) are studied. Precise embedding conditions of classes H _{ψ, q} ^ω into Lorentz spaces and into each other are obtained: $$\begin{array}{*{20}c} {H_{\psi ,q_1 }^\omega \subset \Lambda _{\psi ,q_2 } ;} & {H_{\psi ,q_1 }^\omega \subset {\rm H}_{\psi ,q_2 }^{\omega * } ,} & {0< q_2< q_1< \infty ,} \\ \end{array} $$ under conditions \(\mathop {\lim }\limits_{t \to \infty } \frac{{\psi (2t)}}{{\psi (t)}} > 1,\mathop {\overline {\lim } }\limits_{x \to \infty } \frac{{\psi (2t)}}{{\psi (t)}}< 2\) andω(δ)=O{ω(δ ²)},δ→+0, andω ^* (δ) is an arbitrary modulus of continuity.     

Global Unique Solvability of Inhomogeneous Navier-Stokes Equations with Bounded Density

In this paper, we prove the global existence and uniqueness of solution to d-dimensional (for d = 2, 3) incompressible inhomogeneous Navier-Stokes equations with initial density being bounded from above and below by some positive constants, and with initial velocity u₀ ∈ H^s( R ²) for s > 0 in 2-D, or u₀ ∈ H¹( R ³) satisfying ‖u₀‖_L² ‖?u₀‖_L² being sufficiently small in 3-D. This in particular improves the most recent well-posedness result in [10 Danchin , R. , Mucha , P.B. ( 2013 ). Incompressible flows with piecewise constant density . Arch. Rat. Mech. Anal. 207 : 991 – 1023 .[Crossref], [Web of Science ®] , [Google Scholar]], which requires the initial velocity u₀ ∈ H²( R ^d) for the local well-posedness result, and a smallness condition on the fluctuation of the initial density for the global well-posedness result.     

Quantization causes waves: Smooth finitely computable functions are affine

Every automaton (a letter-to-letter transducer) A whose both input and output alphabets are F _p = {0, 1,..., p - 1} produces a 1-Lipschitz map f _A from the space Z _p of p-adic integers to Z _p . The map fA can naturally be plotted in a unit real square I² ? R²: To an m-letter non-empty word v = γ _m-1γ _m-2... γ0 there corresponds a number 0.v ∈ R with base-p expansion 0.γ _m-1γ _m-2... γ0; so to every m-letter input word w = α _m-1α _m-2 ··· α0 of A and to the respective m-letter output word a(w) = β _m-1β _m-2 ··· β0 of A there corresponds a point (0.w; 0.a(w)) ∈ R². Denote P(A) a closure of the point set (0.w; 0.a(w)) where w ranges over all non-empty words.We prove that once some points of P(A) constitute a C ²-smooth curve in R², the curve is a segment of a straight line with a rational slope. Moreover, when identifying P(A) with a subset of a 2-dimensional torus T² ∈ R³, the smooth curves from P(A) constitute a collection of torus windings which can be ascribed to complex-valued functions ψ(x, t) = e ^i(Ax-2πBt) (x, t ∈ R), i.e., to matter waves. As automata are causal discrete systems, the main result may serve a mathematical reasoning why wave phenomena are inherent in quantum systems: This is just because of causality principle and discreteness of matter.     

Decay of the L∞-norm of solutions of Navier-Stokes equations in unbounded domains

We show that for nn? 4 the L_∞-norm of weak solutions of the Navier-Stokes equations on ?ⁿ with generalized energy inequality decays like $\parallel u(t, \cdot )\parallel _\infty = O(t^{ - ({{n + 1)} \mathord{\left/ {\vphantom {{n + 1)} 2}} \right. \kern-0em} 2}} ),if(1 + | \cdot |)|u(0, \cdot )| \in L_1 $ and $$\int_{\mathbb{R}^n } {u(0,x)} dx = 0$$ . The same holds for strong solutions in all dimensions, if additionally u(0, ·) ε L_p p >n.     

On some new nonlocal solvability theorems for various classes of nonlinear differential equations

We study nonlinear boundary value problems of the form $$ [\Psi u']' + F(x;u',u) = g, u(0) = u(1) = 0 $$ , where Φ is a coercive continuous operator from L _p to L _q , and $$ F(x;u'',u',u) = g, u(0) = u(1) = 0 $$ ; first- and second-order partial differential equations $$ \Phi (x_1 ,x_2 ;u'_1 ,u'_2 ,u) = 0, \sum\limits_{i = 1}^\infty {[\Psi _i (u'_{x_i } )]'_{x_i } + F(x; \ldots ,u'_{x_i } , \ldots ,u) = g_i } $$ ; and general equations F(x; ..., u″ _ii , ...., ...., u′ _i , ...; u) = g(x) of elliptic type. We consider the corresponding boundary value problems of parabolic and hyperbolic type. The proof is based on various a priori estimates obtained in the paper and a nonlocal implicit function theorem.     

On the deficiency index of the vector-valued Sturm–Liouville operator

Let R₊:= [0, +∞), and let the matrix functions P, Q, and R of order n, n ∈ N, defined on the semiaxis R₊ be such that P(x) is a nondegenerate matrix, P(x) and Q(x) are Hermitian matrices for x ∈ R₊ and the elements of the matrix functions P^?1, Q, and R are measurable on R₊ and summable on each of its closed finite subintervals. We study the operators generated in the space L_n²(R₊) by formal expressions of the form l[f] = ?(P(f' ? Rf))' ? R*P(f' ? Rf) + Qf and, as a particular case, operators generated by expressions of the form l[f] = ?(P₀f')' + i((Q₀f)' + Q₀f') + P'₁f, where everywhere the derivatives are understood in the sense of distributions and P₀, Q₀, and P₁ are Hermitianmatrix functions of order n with Lebesgue measurable elements such that P₀^?1 exists and ∥P0∥, ∥P₀^?1∥, ∥P₀^?1∥∥P₁∥², ∥P₀^?1∥∥Q₀∥² ∈ L_loc¹(R₊). Themain goal in this paper is to study of the deficiency index of the minimal operator L₀ generated by expression l[f] in L_n²(R₊) in terms of the matrix functions P, Q, and R (P₀, Q₀, and P₁). The obtained results are applied to differential operators generated by expressions of the form \(l[f] = - f'' + \sum\limits_{k = 1}^{ + \infty } {{H_k}} \delta \left( {x - {x_k}} \right)f\), where x_k, k = 1, 2,..., is an increasing sequence of positive numbers, with lim_k→+∞x_k = +∞, H_k is a number Hermitian matrix of order n, and δ(x) is the Dirac δ-function.     

An extended class of stabilizable uncertain systems

The system of equations \(\frac{{dx}}{{dt}} = A\left( \cdot \right)x + B\left( \cdot \right)u\), where A(·) ∈ ?^{n × n}, B(·) ∈ ?^{n × m}, S(·) ∈ R^{n × m}, is considered. The elements of the matrices A(·), B(·), S(·) are uniformly bounded and are functionals of an arbitrary nature. It is assumed that there exist k elements \({\alpha _{{i_i}{j_l}}}\left( \cdot \right)\left( {l \in \overline {1,k} } \right)\) of fixed sign above the main diagonal of the matrix A(·), and each of them is the only significant element in its row and column. The other elements above the main diagonal are sufficiently small. It is assumed that m = n ?k, and the elements β_ij(·) of the matrix B(·) possess the property \(\left| {{\beta _{{i_s}s}}\left( \cdot \right)} \right| = {\beta _0} > 0\;at\;{i_s}\; \in \;\overline {1,n} \backslash \left\{ {{i_1}, \ldots ,{i_k}} \right\}\). The other elements of the matrix B(·) are zero. The positive definite matrix H = {h_ij} of the following form is constructed. The main diagonal is occupied by the positive numbers h_ii = h_i, \({h_{{i_l}}}_{{j_l}}\, = \,{h_{{j_l}{i_l}}}\, = \, - 0.5\sqrt {{h_{{i_l}}}_{{j_l}}} \,\operatorname{sgn} \,{\alpha _{{i_l}}}_{{j_l}}\left( \cdot \right)\). The other elements of the matrix H are zero. The analysis of the derivative of the Lyapunov function V(x) = x*H^–1x yields h_i\(\left( {i \in \overline {1,n} } \right)\) and λ_i ≤ 0 \(\left( {i \in \overline {1,n} } \right)\) such that for S(·) = H^?1ΛB(·), Λ = diag(λ₁, ..., λ_n), the system of the considered equations becomes globally exponentially stable. The control is robust with respect to the elements of the matrix A(·).     

Asymptotic representations of solutions of two-term nonautonomous nth-order ordinary differential equations with exponential nonlinearity

We obtain necessary and sufficient conditions for the existence of a certain class of solutions of the differential equation $$ (|y^{(n - 1)} |^{\lambda - 1} y^{(n - 1)} )' = \alpha _0 p(t)e^{\sigma y} $$ , where α ₀ ∈ {?1, 1}, σ, λ ∈ R \ {0}, and p: [a, ω[→]0,+∞[(?∞ < a < ω ≤ + ∞) is a continuously differentiable function. We also establish asymptotic representations of such solutions.     

The maximal Fejér operator on the spacesH p (R×···×R)

It is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from thed-dimensional Hardy spaceH _p (R×···×R) toL _p (R ^d ) (1/2<p<∞) and is of weak type (H ₁ ^?i ,L ₁) (i=1,…,d), where the Hardy spaceH ₁ ^?i is defined by a hybrid maximal function. As a consequence, we obtain that the Fejér means of a functionf ∈H ₁ ^?i ?L(logL) ^d?1 converge a.e. to the function in question. Moreover, we prove that the Fejér means are uniformly bounded onH _p (R×···×R) whenever 1/2<p<∞. Thus, in casef ∈H _p (R×···×R) the Fejér means converge tof inH _p (R×···×R) norm. The same results are proved for the conjugate Fejér means, too.