L 2 existence theorems for the\bar \partial _b - Neumann problem on strongly pseudoconvex CR manifoldsproblem on strongly pseudoconvex CR manifolds

LetM be the boundary of a strongly pseudoconvex domain in \(\mathbb{C}^n \) ,n≥4 and ω be an open subset inM such that ?ω is the intersection ofM with a flat hypersurface. We establish theL ² existence theorems of the \(\bar \partial _b - Neumann\) problem on ω. In particular, we prove that the \(\bar \partial _b - Laplacian\) \(\square _b = \bar \partial _b \bar \partial _b^* + \bar \partial _b^* \bar \partial _b \) equipped with a pair of natural boundary conditions, the so-called \(\bar \partial _b - Neumann\) boundary conditions, has closed range when it acts on (0,q) forms, 1≤q≤n?3. Thus there exists a bounded inverse operator for \(\square _b \) , the \(\bar \partial _b - Neumann\) operatorN _b, and we have the following Hodge decomposition theorem on ω for \(\bar \partial _b \bar \partial _b^* N_b \alpha + \bar \partial _b^* \bar \partial _b N_b \alpha \) , for any (0,q) form α withL ²(ω) coefficients. The proof depends on theL ^p regularity of the tangential Cauchy-Riemann operators \(\bar \partial _b u = \alpha \) on ω?M under the compatibility condition \(\bar \partial _b \alpha = 0\) , where α is a (p, q) form on ω, where 1≤q≤n?2. The interior regularity ofN _b follows from the fact that \(\square _b \) is subelliptic in the interior of ω. The operatorN _b induces natural questions on the regularity up to the boundary ?ω. Near the characteristic point of the boundary, certain compatibility conditions will be present. In fact, one can show thatN _b is not a compact operator onL ²(ω).     

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.     

Gradient Estimates for Some Semi-Linear Hypoelliptic Equations

Given some smooth vector fields X ₁,X ₂,…,X _m on a compact manifold M, if they satisfy Hörmander’s condition, we establish global gradient estimates for the positive smooth solutions to the semi-linear hypoelliptic equations $$Lu+au\log u+bu=\partial_tu, \quad \mbox{on} \ M\times[0,\infty) $$ and $$Lu+au\log u+bu=0, \quad \mbox{on} \ M, $$ where a,b are constants, and $L=\sum_{i}X_{i}^{2}-X_{0}$ . We partially generalize the results of Cao and Yau (Math. Z. 211:485–504, 1992).     

A Sharp Remez Inequality for Trigonometric Polynomials

We obtain a sharp Remez inequality for the trigonometric polynomial T _n of degree n on [0,2π): $$\|T_n \|_{L_\infty([0,2\pi))} \le \biggl(1+2\tan^2 \frac{n\beta}{4m} \biggr) { \|T_n \|_{L_\infty ([0,2\pi) \setminus B )}}, $$ where $\frac{2\pi}{m}$ is the minimal period of T _n and $|B|=\beta<\frac {2\pi m}{n}$ is a measurable subset of $\mathbb {T}$ . In particular, this gives the asymptotics of the sharp constant in the Remez inequality: for a fixed n, $$\mathcal{C}(n, \beta)=1+ \frac{(n\beta)^2}{8} +O \bigl(\beta^4\bigr), \quad\beta \to0, $$ where $$\mathcal{C}(n,\beta):= \sup_{|B|=\beta}\sup_{T_n} \frac{ \|T_n \|_{L_\infty([0,2\pi ))}}{ \|T_n \|_{L_\infty ([0,2\pi) \setminus B )}}. $$ We also obtain sharp Nikol’skii’s inequalities for the Lorentz spaces and net spaces. Multidimensional variants of Remez and Nikol’skii’s inequalities are investigated.     

On the analytic continuation of Eisenstein series for Siegel's modular group of degreen

For \(M = \left( {\begin{array}{*{20}c} {A B} \\ {C D} \\ \end{array} } \right)\) ∈ Γ(n)=Sp(n?) andZ=Z+iY,Y > 0, set $$M\left\langle Z \right\rangle = (AZ + B)(CZ + D)^{ - 1} = X_M + iY_M ;M\{ Z\} = CZ + D.$$ Denote with Γ _n (n) the subgroup defined byC=0. Forr∈? and a complex variable ω form the Eisenstein series $$E(n,r,Z,\omega ) = \sum\limits_{M\varepsilon I'_n (n)\backslash \Gamma (n)} {(DetM\{ Z\} )^{ - 2r} (DetY_M )^{\omega - r} } .$$ It is proved thatE(n, r, Z, ω) can be meromorphically continued to the ω-plane and satisfies a functional equation. Forr=1, 2, [(n?1)/2], [(n+1)/2] the functionE(n, r, Z, ω) is holomorphic at ω-r. For 3≤r≤[(n?3)/2] the functionE(n, r, Z, ω) may have poles at ω=r. But the pole-order is for two smaller than known until now. This result says especially that the Eisenstein series has Hecke summation forr=1, 2, [(n?1)/2], [(n+1)/2].     

On the number of non-isomorphic subgraphs

Let $\mathcal{K}$ be the family of graphs on ω₁ without cliques or independent subsets of sizew ₁. We prove that

1. it is consistent with CH that everyGε $\mathcal{K}$ has 2^ω many pairwise non-isomorphic subgraphs,
2. the following proposition holds in L: (*)there is a Gε $\mathcal{K}$ such that for each partition (A, B) of ω₁ either G?G[A] orG?G[B],
3. the failure of (*) is consistent with ZFC.

     

Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Let M be a complete noncompact Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation $ \frac{\partial u}{\partial t} = \Delta _{f}u +au\,{\rm log}\, u + bu$ on ${M \times [0, + \infty)}Let M be a complete noncompact Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation

\frac?u?t = Dfu +au log u + bu \frac{\partial u}{\partial t} = \Delta _{f}u +au\,{\rm log}\, u + bu      10. On characterizations of sup-preserving functionals      Anna Mureńko  Jacek Tabor  Józef Tabor 《Acta Mathematica Hungarica》2009,122(1-2):161-172 Let (E, ≦) be a vector lattice and E + be the set of all nonnegative elements of E. We investigate M-functionals from E + into ?+, that is functions A: E + → ?+ such that $$ \Lambda (f \vee g) = \Lambda (f) \vee \Lambda (g),\Lambda (\alpha f) = \alpha \Lambda (f) $$ for α ≧ 0 and f, g ? E +. Let X be a set and Σ be an algebra of subsets of X. By an M-measure we understand the function μ: Σ → ?+ such that μ( $ \not 0 $ ) = 0 and $$ \mu (A \cup B) = \mu (A) \vee \mu (B)forA,B \in \Sigma ). $$ The main result of the paper is a Riesz type theorem. We prove that every M-functional on C(X, ?)+ can be expressed in terms of M-measure.      11. The asymptotic behavior of the spectral function for elliptic operators in an unbounded region      G. I. Bass 《Mathematical Notes》1969,5(2):149-152 We consider elliptic self-adjoint differential operators L of order 2m in a bounded region D? Rn. An asymptotic formula for the function N(λ) = \(N(\lambda ) = \sum\limits_{\lambda _n< \lambda } 1 \) the number of eigenvalues of the operator L less than A. is proved: $$N(\lambda ) = M_0 \lambda ^{n/2m} + o(\lambda ^{n/2m} )$$ whereλ → + ∞ and M0 is the following constant: $$M_0 = \frac{{V_D }}{{(2\pi )^n \Gamma (1 + n/2m)}}\int_{R_n } {e^{ - L(s)} ds} .$$       12. Evaluation of the limits of maximal means      O. P. Filatov 《Mathematical Notes》1996,59(5):547-553 It is proved that the limit $$\mathop {\lim }\limits_{\Delta \to \infty } \mathop {\sup }\limits_\gamma \tfrac{1}{\Delta }\int_0^\Delta {f(\gamma (t))dt} $$ , wheref: ? → ? is a locally integrable (in the sense of Lebesgue) function with zero mean and the supremum is taken over all solutions of the generalized differential equation γ ∈ [ω1, ω2], coincides with the limit $$\mathop {\lim }\limits_{T \to \infty } \mathop {\sup }\limits_{c \geqslant 0} \varphi _f (k,{\mathbf{ }}T,{\mathbf{ }}c)$$ , where $$\varphi _f = \frac{{(k - 1)\bar I_f (T,c)}}{{1 + (k - 1)\bar \lambda _f (T,c)}},k = \frac{{\omega _2 }}{{\omega _1 }}$$ . Here ¯λf = λf /T, ¯ If =If/T, and λf is the Lebesgue measure of the set $$\{ \gamma \in [\gamma _0 ,\gamma _0 + T]:f(\gamma ) \geqslant c\} = A_f ,I_f = \int_{A_f } {f(\gamma )d\gamma } $$ . It is established that this limit always exists for almost-periodic functionsf.      13. Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group      Martin Bauer  Martins Bruveris  Philipp Harms  Peter W. Michor 《Annals of Global Analysis and Geometry》2013,44(1):5-21 We study Sobolev-type metrics of fractional order s ≥ 0 on the group Diff c (M) of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M = S 1, the geodesic distance on Diff c (S 1) vanishes if and only if ${s\leq\frac12}$ . For other manifolds, we obtain a partial characterization: the geodesic distance on Diff c (M) vanishes for ${M=\mathbb{R}\times N, s < \frac12}$ and for ${M=S^1\times N, s\leq\frac12}$ , with N being a compact Riemannian manifold. On the other hand, the geodesic distance on Diff c (M) is positive for ${{\rm dim}(M)=1, s > \frac12}$ and dim(M) ≥ 2, s ≥ 1. For ${M=\mathbb{R}^n}$ , we discuss the geodesic equations for these metrics. For n = 1, we obtain some well-known PDEs of hydrodynamics: Burgers’ equation for s = 0, the modified Constantin–Lax–Majda equation for ${s=\frac12}$ , and the Camassa–Holm equation for s = 1.      14. The formal translation equation for iteration groups of type II      Harald Fripertinger  Ludwig Reich 《Aequationes Mathematicae》2010,79(1-2):111-156 We investigate the translation equation $$F(s+t, x) = F(s, F(t, x)),\quad \quad s,t\in{\mathbb{C}},\qquad\qquad\qquad\qquad({\rm T})$$ in ${\mathbb{C}\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right]}$ , the ring of formal power series over ${\mathbb{C}}$ . Here we restrict ourselves to iteration groups of type II, i.e. to solutions of (T) of the form ${F(s, x) \equiv x + c_k(s)x^k {\rm mod} x^{k + 1}}$ , where k ≥ 2 and c k ≠ 0 is necessarily an additive function. It is easy to prove that the coefficient functions c n (s) of $$F(s, x) = x + \sum_{n \ge q k}c_n(s)x^n$$ are polynomials in c k (s). It is possible to replace this additive function c k by an indeterminate. In this way we obtain a formal version of the translation equation in the ring ${(\mathbb{C}[y])\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right]}$ . We solve this equation in a completely algebraic way, by deriving formal differential equations or an Aczél–Jabotinsky type equation. This way it is possible to get the structure of the coefficients in great detail which are now polynomials. We prove the universal character (depending on certain parameters, the coefficients of the infinitesimal generator H of an iteration group of type II) of these polynomials. Rewriting the solutions G(y, x) of the formal translation equation in the form ${\sum_{n\geq 0}\phi_n(x)y^n}$ as elements of ${(\mathbb{C}\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right])\left[\kern-0.15em\left[{y}\right]\kern-0.15em\right]}$ , we obtain explicit formulas for ${\phi_n}$ in terms of the derivatives H (j)(x) of the generator ${H}$ and also a representation of ${G(y, x)}$ as a Lie–Gröbner series. Eventually, we deduce the canonical form (with respect to conjugation) of the infinitesimal generator ${H}$ as x k + hx 2k-1 and find expansions of the solutions ${G(y, x) = \sum_{r\geq 0} G_r(y, x)h^r}$ of the above mentioned differential equations in powers of the parameter h.      15. An extremal problem for algebraic polynomials in the symmetric discrete Gegenbauer-Sobolev space      B. P. Osilenker 《Mathematical Notes》2007,82(3-4):366-379 We study discrete Sobolev spaces with symmetric inner product $$\left\langle {f,g} \right\rangle _\alpha = \int_{ - 1}^1 {f g d\mu _\alpha } + M[f(1)g(1) + f( - 1)g( - 1)] + K[f'(1)g'(1) + f'( - 1)g'( - 1)]$$ , where M ≥ 0, k ≥ 0, and $$d\mu _\alpha (x) = \frac{{\Gamma (2\alpha + 2)}}{{2^{2\alpha + 1} \Gamma ^2 (\alpha + 1)}}(1 - x^2 )^\alpha dx, \alpha > - 1$$ , is the Gegenbauer probability measure. We obtain the solution of the following extremal problem: Calculate $$\mathop {\inf }\limits_{a_0 ,a_1 ,...,a_{N - r} } \left\{ {\langle P_N^{(r)} ,P_N^{(r)} \rangle _\alpha ,1 \leqslant r \leqslant N - 1, P_N^{(r)} (x) = \sum\limits_{j = N - r + 1}^N {a_j^0 x^j } + \sum\limits_{j = 0}^{N - r} {a_j x^j } } \right\}$$ , where the a j 0 , j = N ? r + 1, N ? r + 2, ..., N ? 1, N, a N 0 > 0, are fixed numbers, and find the extremal polynomial.      16. On the exact solitary wave solutions of a special class of Benjamin-Bona-Mahony equation      Reza Abazari 《Computational Mathematics and Mathematical Physics》2013,53(9):1371-1376 The general form of Benjamin-Bona-Mahony equation (BBM) is $u_t + au_x + bu_{xxt} + (g(u))_x = 0,a,b \in \mathbb{R},$ where ab ≠ 0 and g(u) is a C 2-smooth nonlinear function, has been proposed by Benjamin et al. in [1] and describes approximately the unidirectional propagation of long wave in certain nonlinear dispersive systems. In this payer, we consider a new class of Benjamin-Bona-Mahony equation (BBM) $u_t + au_x + bu_{xxt} + (pe^u + qe^{ - u} )_x = 0,a,b,p,q \in \mathbb{R},$ where ab ≠ 0, and qp ≠ 0, and we obtain new exact solutions for it by using the well-known (G′/G)-expansion method. New periodic and solitary wave solutions for these nonlinear equation are formally derived.      17. Quotients of Boolean algebras and regular subalgebras      B. Balcar  T. Pazák 《Archive for Mathematical Logic》2010,49(3):329-342       18. Chen’ s theorem in arithmetical progressions      Lu Minggao  Cai Yingchun 《中国科学A辑(英文版)》1999,42(6):561-569 LetN be a sufficiently large even integer and $$\begin{gathered} q \geqslant 1, (l_i ,q) = 1 (i = 1, 2), \hfill \\ l_1 + l_2 \equiv N(\bmod q). \hfill \\ \end{gathered} $$ . It is proved that the equation $$N = p + P_2 ,p \equiv l_1 (\bmod q), P_2 \equiv l_2 (\bmod q)$$ has infinitely many solutions for almost all $q \leqslant N^{\frac{1}{{37}}} $ , wherep is a prime andP 2 is an almost prime with at most two prime factors.      19. On orthogonal polynomials      P. Túrán 《Analysis Mathematica》1975,1(4):297-311 Пустьw(х)∈L[-1, +1] — неотрица тельная функция така я, что $$\frac{{\log ^ + \frac{1}{{w(x)}}}}{{\sqrt {1 - x^2 } }} \in L[ - 1, + 1]$$ и пусть {(р n (х)} — много члены, ортогональные и нормированные с весо мw(x). Мы доказываем следующие две теорем ы, являющиеся обобщен ием одного известного результа та Н. Винера. I. Для каждого δ, 0<δ<1, суще ствует числоB=B(δ, w) тако е, что если $$f_N (x) = \sum\limits_{j = 1}^N {a_j p_{v_j } (x)} $$ причем выполнено сле дующее условие лакун арности $$\begin{gathered} v_{j + 1} - v_j \geqq B(\delta ,w) (j = 1,2,...,N - 1), \hfill \\ v_1 \geqq B(\delta ,w) \hfill \\ \end{gathered} $$ , то для некоторого С(δ, w) и всехh и δ, для которых $$ - 1 \leqq h - \delta< h + \delta \leqq + 1$$ , имеет место неравенс тво $$\int\limits_{ - 1}^1 {|f_N (x)|^2 w(x)dx \leqq C(\delta ,w)} \int\limits_{h - \delta }^{h + \delta } {|f_N (x)|^2 w(x)dx} $$ каковы бы ни былиa j ,N и h. II. Если формальный ряд $$\sum\limits_{j = 1}^\infty {b_j p_{\mu _j } (x)} $$ удовлетворяет услов ию лакунарности μj+1-μj→∞ и суммируем, например, м етодом Абеля на произвольно малом отрезке [а, Ь] ?[0,1] к ф ункцииf(x) такой, что \(f(x)\sqrt {w(x)} \in L_2 [a,b]\) , то $$\sum\limits_j {|b_j |^2< \infty } $$ Теорема I — это первый ш аг в направлении проб лемы типа Мюнтца-Саса о замкнут ости подпоследовательно сти pvj(x)} последовател ьности {рn(х)} на отрезке [а, Ь] в метрике С[а, Ь] (см. теорему II стать и).      20. Local linear independence of the translates of a box spline      Rong-Qing Jia 《Constructive Approximation》1985,1(1):175-182 Let Ξ=(ξ i ) l n be a sequence of vectors inR m . The box splineM Ξ is defined as the distribution given by $$M_\Xi :\varphi \to \int_{[0,1]^n } \varphi \left( {\sum\limits_{i = 1}^n {\lambda (i)\xi _i } } \right)d\lambda ,\varphi \in C_c^\infty (R^m ).$$ . Suppose that Ξ contains a basis forR m . ThenM Ξ∈L ∞(R m ). Assume $$\Xi \subset V: = z^m .$$ . Consider the translatesM v :=M Ξ(·?v),v∈V. It is known that (M v ) V is linearly dependent unless (*) $$|\det Z| = 1forallbasesZ \subset \Xi$$ . This paper demonstrates that under condition (*), (M v ) V is locally linearly independent, i.e., $$\{ M_v ;\sup p M_v \cap A \ne \not 0\}$$ is linearly independent over any open setA.

On characterizations of sup-preserving functionals

Let (E, ≦) be a vector lattice and E ₊ be the set of all nonnegative elements of E. We investigate M-functionals from E ₊ into ?₊, that is functions A: E ₊ → ?₊ such that $$ \Lambda (f \vee g) = \Lambda (f) \vee \Lambda (g),\Lambda (\alpha f) = \alpha \Lambda (f) $$ for α ≧ 0 and f, g ? E ₊. Let X be a set and Σ be an algebra of subsets of X. By an M-measure we understand the function μ: Σ → ?₊ such that μ( $ \not 0 $ ) = 0 and $$ \mu (A \cup B) = \mu (A) \vee \mu (B)forA,B \in \Sigma ). $$ The main result of the paper is a Riesz type theorem. We prove that every M-functional on C(X, ?)₊ can be expressed in terms of M-measure.     

The asymptotic behavior of the spectral function for elliptic operators in an unbounded region

We consider elliptic self-adjoint differential operators L of order 2m in a bounded region D? R_n. An asymptotic formula for the function N(λ) = \(N(\lambda ) = \sum\limits_{\lambda _n< \lambda } 1 \) the number of eigenvalues of the operator L less than A. is proved: $$N(\lambda ) = M_0 \lambda ^{n/2m} + o(\lambda ^{n/2m} )$$ whereλ → + ∞ and M₀ is the following constant: $$M_0 = \frac{{V_D }}{{(2\pi )^n \Gamma (1 + n/2m)}}\int_{R_n } {e^{ - L(s)} ds} .$$      

Evaluation of the limits of maximal means

It is proved that the limit $$\mathop {\lim }\limits_{\Delta \to \infty } \mathop {\sup }\limits_\gamma \tfrac{1}{\Delta }\int_0^\Delta {f(\gamma (t))dt} $$ , wheref: ? → ? is a locally integrable (in the sense of Lebesgue) function with zero mean and the supremum is taken over all solutions of the generalized differential equation γ ∈ [ω1, ω2], coincides with the limit $$\mathop {\lim }\limits_{T \to \infty } \mathop {\sup }\limits_{c \geqslant 0} \varphi _f (k,{\mathbf{ }}T,{\mathbf{ }}c)$$ , where $$\varphi _f = \frac{{(k - 1)\bar I_f (T,c)}}{{1 + (k - 1)\bar \lambda _f (T,c)}},k = \frac{{\omega _2 }}{{\omega _1 }}$$ . Here ¯λf = λf /T, ¯ I_f =If/T, and λf is the Lebesgue measure of the set $$\{ \gamma \in [\gamma _0 ,\gamma _0 + T]:f(\gamma ) \geqslant c\} = A_f ,I_f = \int_{A_f } {f(\gamma )d\gamma } $$ . It is established that this limit always exists for almost-periodic functionsf.     

Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

We study Sobolev-type metrics of fractional order s ≥ 0 on the group Diff _c (M) of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M = S ¹, the geodesic distance on Diff _c (S ¹) vanishes if and only if ${s\leq\frac12}$ . For other manifolds, we obtain a partial characterization: the geodesic distance on Diff _c (M) vanishes for ${M=\mathbb{R}\times N, s < \frac12}$ and for ${M=S^1\times N, s\leq\frac12}$ , with N being a compact Riemannian manifold. On the other hand, the geodesic distance on Diff _c (M) is positive for ${{\rm dim}(M)=1, s > \frac12}$ and dim(M) ≥ 2, s ≥ 1. For ${M=\mathbb{R}^n}$ , we discuss the geodesic equations for these metrics. For n = 1, we obtain some well-known PDEs of hydrodynamics: Burgers’ equation for s = 0, the modified Constantin–Lax–Majda equation for ${s=\frac12}$ , and the Camassa–Holm equation for s = 1.     

The formal translation equation for iteration groups of type II

We investigate the translation equation $$F(s+t, x) = F(s, F(t, x)),\quad \quad s,t\in{\mathbb{C}},\qquad\qquad\qquad\qquad({\rm T})$$ in ${\mathbb{C}\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right]}$ , the ring of formal power series over ${\mathbb{C}}$ . Here we restrict ourselves to iteration groups of type II, i.e. to solutions of (T) of the form ${F(s, x) \equiv x + c_k(s)x^k {\rm mod} x^{k + 1}}$ , where k ≥ 2 and c _k ≠ 0 is necessarily an additive function. It is easy to prove that the coefficient functions c _n (s) of $$F(s, x) = x + \sum_{n \ge q k}c_n(s)x^n$$ are polynomials in c _k (s). It is possible to replace this additive function c _k by an indeterminate. In this way we obtain a formal version of the translation equation in the ring ${(\mathbb{C}[y])\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right]}$ . We solve this equation in a completely algebraic way, by deriving formal differential equations or an Aczél–Jabotinsky type equation. This way it is possible to get the structure of the coefficients in great detail which are now polynomials. We prove the universal character (depending on certain parameters, the coefficients of the infinitesimal generator H of an iteration group of type II) of these polynomials. Rewriting the solutions G(y, x) of the formal translation equation in the form ${\sum_{n\geq 0}\phi_n(x)y^n}$ as elements of ${(\mathbb{C}\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right])\left[\kern-0.15em\left[{y}\right]\kern-0.15em\right]}$ , we obtain explicit formulas for ${\phi_n}$ in terms of the derivatives H ^(j)(x) of the generator ${H}$ and also a representation of ${G(y, x)}$ as a Lie–Gröbner series. Eventually, we deduce the canonical form (with respect to conjugation) of the infinitesimal generator ${H}$ as x ^k + hx ^2k-1 and find expansions of the solutions ${G(y, x) = \sum_{r\geq 0} G_r(y, x)h^r}$ of the above mentioned differential equations in powers of the parameter h.     

An extremal problem for algebraic polynomials in the symmetric discrete Gegenbauer-Sobolev space

We study discrete Sobolev spaces with symmetric inner product $$\left\langle {f,g} \right\rangle _\alpha = \int_{ - 1}^1 {f g d\mu _\alpha } + M[f(1)g(1) + f( - 1)g( - 1)] + K[f'(1)g'(1) + f'( - 1)g'( - 1)]$$ , where M ≥ 0, k ≥ 0, and $$d\mu _\alpha (x) = \frac{{\Gamma (2\alpha + 2)}}{{2^{2\alpha + 1} \Gamma ^2 (\alpha + 1)}}(1 - x^2 )^\alpha dx, \alpha > - 1$$ , is the Gegenbauer probability measure. We obtain the solution of the following extremal problem: Calculate $$\mathop {\inf }\limits_{a_0 ,a_1 ,...,a_{N - r} } \left\{ {\langle P_N^{(r)} ,P_N^{(r)} \rangle _\alpha ,1 \leqslant r \leqslant N - 1, P_N^{(r)} (x) = \sum\limits_{j = N - r + 1}^N {a_j^0 x^j } + \sum\limits_{j = 0}^{N - r} {a_j x^j } } \right\}$$ , where the a _j ⁰ , j = N ? r + 1, N ? r + 2, ..., N ? 1, N, a _N ⁰ > 0, are fixed numbers, and find the extremal polynomial.     

On the exact solitary wave solutions of a special class of Benjamin-Bona-Mahony equation

The general form of Benjamin-Bona-Mahony equation (BBM) is $u_t + au_x + bu_{xxt} + (g(u))_x = 0,a,b \in \mathbb{R},$ where ab ≠ 0 and g(u) is a C ₂-smooth nonlinear function, has been proposed by Benjamin et al. in [1] and describes approximately the unidirectional propagation of long wave in certain nonlinear dispersive systems. In this payer, we consider a new class of Benjamin-Bona-Mahony equation (BBM) $u_t + au_x + bu_{xxt} + (pe^u + qe^{ - u} )_x = 0,a,b,p,q \in \mathbb{R},$ where ab ≠ 0, and qp ≠ 0, and we obtain new exact solutions for it by using the well-known (G′/G)-expansion method. New periodic and solitary wave solutions for these nonlinear equation are formally derived.     

Chen’ s theorem in arithmetical progressions

LetN be a sufficiently large even integer and $$\begin{gathered} q \geqslant 1, (l_i ,q) = 1 (i = 1, 2), \hfill \\ l_1 + l_2 \equiv N(\bmod q). \hfill \\ \end{gathered} $$ . It is proved that the equation $$N = p + P_2 ,p \equiv l_1 (\bmod q), P_2 \equiv l_2 (\bmod q)$$ has infinitely many solutions for almost all $q \leqslant N^{\frac{1}{{37}}} $ , wherep is a prime andP ₂ is an almost prime with at most two prime factors.     

On orthogonal polynomials

Пустьw(х)∈L[-1, +1] — неотрица тельная функция така я, что $$\frac{{\log ^ + \frac{1}{{w(x)}}}}{{\sqrt {1 - x^2 } }} \in L[ - 1, + 1]$$ и пусть {(р _n (х)} — много члены, ортогональные и нормированные с весо мw(x). Мы доказываем следующие две теорем ы, являющиеся обобщен ием одного известного результа та Н. Винера. I. Для каждого δ, 0<δ<1, суще ствует числоB=B(δ, w) тако е, что если $$f_N (x) = \sum\limits_{j = 1}^N {a_j p_{v_j } (x)} $$ причем выполнено сле дующее условие лакун арности $$\begin{gathered} v_{j + 1} - v_j \geqq B(\delta ,w) (j = 1,2,...,N - 1), \hfill \\ v_1 \geqq B(\delta ,w) \hfill \\ \end{gathered} $$ , то для некоторого С(δ, w) и всехh и δ, для которых $$ - 1 \leqq h - \delta< h + \delta \leqq + 1$$ , имеет место неравенс тво $$\int\limits_{ - 1}^1 {|f_N (x)|^2 w(x)dx \leqq C(\delta ,w)} \int\limits_{h - \delta }^{h + \delta } {|f_N (x)|^2 w(x)dx} $$ каковы бы ни былиa _j ,N и h. II. Если формальный ряд $$\sum\limits_{j = 1}^\infty {b_j p_{\mu _j } (x)} $$ удовлетворяет услов ию лакунарности μ_j+1-μ_j→∞ и суммируем, например, м етодом Абеля на произвольно малом отрезке [а, Ь] ?[0,1] к ф ункцииf(x) такой, что \(f(x)\sqrt {w(x)} \in L_2 [a,b]\) , то $$\sum\limits_j {|b_j |^2< \infty } $$ Теорема I — это первый ш аг в направлении проб лемы типа Мюнтца-Саса о замкнут ости подпоследовательно сти pv_j(x)} последовател ьности {р_n(х)} на отрезке [а, Ь] в метрике С[а, Ь] (см. теорему II стать и).     

Local linear independence of the translates of a box spline

Let Ξ=(ξ _i ) _l ⁿ be a sequence of vectors inR ^m . The box splineM _Ξ is defined as the distribution given by $$M_\Xi :\varphi \to \int_{[0,1]^n } \varphi \left( {\sum\limits_{i = 1}^n {\lambda (i)\xi _i } } \right)d\lambda ,\varphi \in C_c^\infty (R^m ).$$ . Suppose that Ξ contains a basis forR ^m . ThenM _Ξ∈L _∞(R ^m ). Assume $$\Xi \subset V: = z^m .$$ . Consider the translatesM _v :=M _Ξ(·?v),v∈V. It is known that (M _v ) _V is linearly dependent unless (*) $$|\det Z| = 1forallbasesZ \subset \Xi$$ . This paper demonstrates that under condition (*), (M _v ) _V is locally linearly independent, i.e., $$\{ M_v ;\sup p M_v \cap A \ne \not 0\}$$ is linearly independent over any open setA.