Long time small solutions to nonlinear parabolic equations

A sharp result on global small solutions to the Cauchy problem $$u_t = \Delta u + f\left( {u,Du,D^2 u,u_t } \right)\left( {t > 0} \right),u\left( 0 \right) = u_0 $$ In Rⁿ is obtained under the the assumption thatf is C^1+r forr>2/n and ‖u ₀‖C²(R ⁿ ) +‖u ₀‖W ₁ ² (R ⁿ ) is small. This implies that the assumption thatf is smooth and ‖u ₀ ‖W ₁ ^k (R ⁿ )+‖u ₀‖W ₂ ^k (R ⁿ ) is small fork large enough, made in earlier work, is unnecessary.     

On the Davenport constant and on the structure of extremal zero-sum free sequences

Let $G = C_{n_1 } \oplus \cdots \oplus C_{n_r }$ with 1 < n ₁ | ?? | n _r be a finite abelian group, d*(G) = n ₁ +??+n _r ?r, and let d(G) denote the maximal length of a zerosum free sequence over G. Then d(G) ?? d*(G), and the standing conjecture is that equality holds for G = C _n ^r . We show that equality does not hold for C ₂ ?? C _2n ^r , where n ?? 3 is odd and r ?? 4. This gives new information on the structure of extremal zero-sum free sequences over C _2n ^r .     

The computation and perturbation analysis for weighted group inverse of rectangular matrices

Cen (Math. Numer. Sin. 29(1):39–48, 2007) has defined a weighted group inverse of rectangular matrices. Let A∈C ^m×n ,W∈C ^n×m , if X∈C ^m×n satisfies the system of matrix equations $$(W_{1})\quad AWXWA=A,\quad\quad (W_{2})\quad XWAWX=X,\quad\quad (W_{3})\quad AWX=XWA$$ X is called the weighted group inverse of A with W, and denoted by A _W ^# . In this paper, we will study the algebra perturbation and analytical perturbation of this kind weighted group inverse A _W ^# . Under some conditions, we give a decomposition of B _W ^# ?A _W ^# . As a results, norm estimate of ‖B _W ^# ?A _W ^# ‖ is presented (where B=A+E). An expression of algebra of perturbation is also obtained. In order to compute this weighted group inverse with ease, we give a new representation of this inverse base on Gauss-elimination, then we can calculate this weighted inverse by Gauss-elimination. In the end, we use a numerical example to show our results.     

Comparison Inequalities for One Sided Normal Probabilities

Several sharp upper and lower bounds for the ratio of two normal probabilities $\mathbb{P}\Biggl(\,\bigcap_{i=1}^{n}\bigl\{\xi^{(1)}_i\leq \mu_i\bigr\}\Biggr)\Big/\mathbb{P}\Biggl(\,\bigcap_{i=1}^{n}\bigl\{\xi^{(0)}_i\leq \mu_i\bigr\}\Biggr)$ are given in this paper for various cases, where (ξ ₁ ⁽⁰⁾ ,ξ ₂ ⁽⁰⁾ ,…,ξ _n ⁽⁰⁾ ) and (ξ ₁ ⁽¹⁾ ,ξ ₂ ⁽¹⁾ , …,ξ _n ⁽¹⁾ ) are standard normal random variables with covariance matrices R ⁰=(r _ij ⁰ ) and R ¹=(r _ij ¹ ), respectively.     

On the chromatic numbers of spheres in ℝ n

Let χ(S _r ^n?1 )) be the minimum number of colours needed to colour the points of a sphere S _r ^n?1 of radius $r \geqslant \tfrac{1} {2}$ in ? ⁿ so that any two points at the distance 1 apart receive different colours. In 1981 P. Erd?s conjectured that χ(S _r ^n?1 )→∞ for all $r \geqslant \tfrac{1} {2}$ . This conjecture was proved in 1983 by L. Lovász who showed in [11] that χ(S _r ^n?1 ) ≥ n. In the same paper, Lovász claimed that if $r < \sqrt {\frac{n} {{2n + 2}}}$ , then χ(S _r ^n?1 ) ≤ n+1, and he conjectured that χ(S _r ^n?1 ) grows exponentially, provided $r \geqslant \sqrt {\frac{n} {{2n + 2}}}$ . In this paper, we show that Lovász’ claim is wrong and his conjecture is true: actually we prove that the quantity χ(S _r ^n?1 ) grows exponentially for any $r > \tfrac{1} {2}$ .     

Finite separable field extensions with prescribed extensions of valuations. II

Let A¹,...,A^k be pairwise independent valuation rings of K. Prescribing extensions Δ _i ^j . of the value group Γ^j and extensions \(\mathfrak{L}_i^j\) of the residue field \(H^j\) of A^j (i=1,...,r^j) such that \(\sum\limits_{i = 1}^{r^j } {(\Delta _i^j :\Gamma ^j )} \cdot [\mathfrak{L}_i^j :H^j ] = n\) , we provide sufficient conditions for the existence of a separable field extension L of K of degree n with exactly r^j pairwise independent valuation rings B _i ^j lying over A^j, which have Δ _i ^j as value groups and \(\mathfrak{L}_i^j\) as residue fields.     

Приближение алгебра ическими многочлена ми классов функций, являющихся д робными интегралами от суммируемых функц ий

Approximation in the mean (E _n(f)₁) by algebraic polynomials of order ≦n is studied in the paper, for classesW ₁ ^r of functionsf, which can be represented as $$f(x) = \frac{1}{{\Gamma (r)}}\int\limits_{ - 1}^1 {(x - t)_ + ^{^{r - 1} } } \varphi (t)dt,$$ where??L ₁-1, 1], ∥?∥₁≧1, (x-t) ₊ ^r1 =[max(0, x-t)]^r1, Г (r) stands for Euler's gamma-function. It is proved that for all realr≧1 and positive integersn≧[r]?1 the relation sup E_n(f)₁:f?W₁ ^r=∥(S_n)_r∥_t8, is valid, where $$(s_\Lambda )_{_r } (t) = \frac{1}{{\Gamma (r)}}\int\limits_{ - 1}^1 {(x - t)_ + ^{r - 1} } $$ sgn sin (n+2) arc cosx dx.     

On the uniqueness of symmetric bases in finite dimensional Banach spaces

Suppose{e _i} _i=1 ⁿ and{f _i} _i=1 ⁿ are symmetric bases of the Banach spacesE andF. Letd(E,F)≦C andd(E,l _n ² )≧n' for somer>0. Then there is a constantC _r=C_r(C)>0 such that for alla _i∈Ri=1,...,n $$C_r^{ - 1} \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\| \leqq \left\| {\sum\limits_{i = 1}^n {a_i f_i } } \right\| \leqq C_r \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\|$$ We also give a partial uniqueness of unconditional bases under more restrictive conditions.     

n-Widths of Sobolev spaces inL p

LetW _p ^(r) ={f:f∈C ^r?1[0, 1],f ^(r?1) abs.cont., ∥f ^(r)∥ _p <∞}, and setB _p ^(r) ={f:f∈W _p ^(r) ,∥f ^(r)∥ _p ≤1}. We find the exact Kolmogorov, Gel'fand, linear, and Bernsteinn-widths ofB _p ^(r) inL ^p for allp∈(1, ∞). For the Kolmogorovn-width we show that forn≥r there exists an optimal subspace of splines of degreer?1 withn?r fixed simple knots depending onp.     

Følner sequences and Hilbert’s Irreducibility Theorem over Q

Letf(X; T ₁, ...,T _n) be an irreducible polynomial overQ. LetB be the set ofb teZ ⁿ such thatf(X;b) is of lesser degree or reducible overQ. Let ?={F _j}{F _j } _j?1 ^∞ be a Følner sequence inZ ⁿ — that is, a sequence of finite nonempty subsetsF _j ?Z ⁿ such that for eachvteZ ⁿ , $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap (F_j + \upsilon )} \right|}}{{\left| {F_j } \right|}} = 1$ Suppose ? satisfies the extra condition that forW a properQ-subvariety ofP ⁿ ?A ⁿ and ?>0, there is a neighborhoodU ofW(R) in the real topology such that $\mathop {lim sup}\limits_{j \to \infty } \frac{{\left| {F_j \cap U} \right|}}{{\left| {F_j } \right|}}< \varepsilon $ whereZ ⁿ is identified withA ⁿ (Z). We prove $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap B} \right|}}{{\left| {F_j } \right|}} = 0$ .     

The error term in Nevanlinna’s inequality

An upper bound is given for the error termS(r, |a _j |,f) in Nevanlinna’s inequality. For given positive increasing functions p and $ with ∫ ₁ ^∞ dr/p(r) = ∫ ₁ ^∞ dr/r ?(r) = ∞, setP(r) = ∫ ₁ ^r dt/p,Ψ(r) = ∫ ₁ ^r dt/t ?(t) We prove that $$S(r, \left\{ {a_j } \right\}, f) \leqslant \log \frac{{T(r, f)\phi (T(r, f))}}{{p(r)}} + O(1)$$ holds, with a small exceptional set of r, for any finite set of points |a _j | in the extended plane and any meromorphic function f such thatΨ(T(r, f)) =O(P(r)). This improves the known results of A. Hinkkanen and Y. F. Wang. The sharpness of the estimate is also considered.     

Some generating functions of modified Gegenbauer polynomials

L Weisner's group theoretic method has been introduced in the study of special function. In this paper we obtain two differential operators, one of which simultaneously raises the index and lowers the parameter of modified Gegenbauer polynomialsC _n ^v+n (x) by unity and the other acts onC _n ^v+n(x) in the reversed way by suitable interpretation to the indexn and the parameterv ofC _n ^v+n(x) . We have also found out the extended form of the groups generated by the operatorsA _ij(i,j=1,2). We have also derived some novel generating functions ofC _n ^v+n (x) from which several special generating functions can be easily derived.     

On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spaces

We study the asymptotic behavior of the eigenvalues the Sturm-Liouville operator Ly = ?y″ + q(x)y with potentials from the Sobolev space W ₂ ^θ?1 , θ ≥ 0, including the nonclassical case θ ∈ [0, 1) in which the potential is a distribution. The results are obtained in new terms. Let s _2k (q) = λ _k ^1/2 (q) ? k, s _2k?1(q) = μ _k ^1/2 (q) ? k ? 1/2, where {λ _k } ₁ ^∞ and {μ _k } ₁ ^∞ are the sequences of eigenvalues of the operator L generated by the Dirichlet and Dirichlet-Neumann boundary conditions, respectively,. We construct special Hilbert spaces t ₂ ^θ such that the mapping F:W ₂ ^θ?1 →t ₂ ^θ defined by the equality F(q) = {s _n } ₁ ^∞ is well defined for all θ ≥ 0. The main result is as follows: for θ > 0, the mapping F is weakly nonlinear, i.e., can be expressed as F(q) = Uq + Φ(q), where U is the isomorphism of the spaces W ₂ ^θ?1 and t ₂ ^θ , and Φ(q) is a compact mapping. Moreover, we prove the estimate ∥Ф(q)∥_τ ≤ C∥q∥_θ?1, where the exact value of τ = τ(θ) > θ ? 1 is given and the constant C depends only on the radius of the ball ∥q∥_θ? ≤ R, but is independent of the function q varying in this ball.     

Presentations of alternating semigroups

One presentation of the alternating groupA _n hasn?2 generatorss ₁,…,s_n?2 and relationss ₁ ³ =s _i ² =(s_1?1s_i)³=(s_js_k)²=1, wherei>1 and |j?k|>1. Against this backdrop, a presentation of the alternating semigroupA _n ^c )A _n is introduced: It hasn?1 generatorss ₁,…,S _n?2,e, theA _n-relations (above), and relationse ²=e, (es ₁)⁴, (es _j)²=(es _j)⁴,es _i=s _i s ₁ ^-1 es ₁, wherej>1 andi≥1.     

Description of traces of functions in the Sobolev space with a Muckenhoupt weight

We characterize the trace of the Sobolev space W _p ^l (? ⁿ , γ) with 1 < p < ∞ and weight γ ∈ A _p ^loc (? ⁿ ) on a d-dimensional plane for 1 ≤ d < n. It turns out that for a function φ to be the trace of a function f ∈ W _p ^l (? ⁿ , γ), it is necessary and sufficient that φ belongs to a new Besov space of variable smoothness, $\overline B _p^l \left( {\mathbb{R}^d ,\left\{ {\gamma _{k,m} } \right\}} \right)$ , constructed in this paper. The space $\overline B _p^l \left( {\mathbb{R}^d ,\left\{ {\gamma _{k,m} } \right\}} \right)$ is compared with some earlier known Besov spaces of variable smoothness.     

Rows and diagonals of the Walsh array for entire functions with smooth Maclaurin series coefficients

Let \(f(z): = \sum\nolimits_{j = 0}^\infty {a_j z^J } \) be entire, witha _j≠0,j large enough, \(\lim _{J \to \infty } a_{j + 1} /a_J = 0\) , and, for someq∈C, \(q_j : = a_{j - 1} a_{j + 1} /a_j^2 \to q\) asj→∞. LetE _mn(f; r) denote the error in best rational approximation off in the uniform norm on |z‖≤r, by rational functions of type (m, n). We study the behavior ofE _mn(f; r) asm and/orn→∞. For example, whenq above is not a root of unity, or whenq is a root of unity, butq _m has a certain asymptotic expansion asm→∞, then we show that, for each fixed positive integern, ,m→∞. In particular, this applies to the Mittag-Leffler functions \(f(z): = \sum\nolimits_{j = 0}^\infty {z^j /\Gamma (1 + j/\lambda )} \) and to \(f(z): = \sum\nolimits_{j = 0}^\infty {z^j /(j!)^{I/\lambda } } \) , λ>0. When |q‖<1, we also handle the diagonal case, showing, for example, that ,n→∞. Under mild additional conditions, we show that we can replace 1+0(1) ⁿ by 1+0(1). In all cases we show that the poles of the best approximants approach ∞ asm→∞.     

Sharpening of the estimates for relative widths of classes of differentiable functions

We improve the earlier obtained upper estimates for the least value of the coefficient M for which the Kolmogorov widths d _n (W _C ^r , C) of the function class W _C ^r are equal to the relative widths K _n (W _C ^r , MW _C ^j , C) of the class W _C ^r with respect to the class MW _C ^j , j < r.     

Separate asymptotics of two series of eigenvalues for a single elliptic boundary-value problem

The spectral problem in a bounded domain Ω?Rⁿ is considered for the equation Δu= λu in Ω, ?u=λ?υ/?ν on the boundary of Ω (ν the interior normal to the boundary, Δ, the Laplace operator). It is proved that for the operator generated by this problem, the spectrum is discrete and consists of two series of eigenvalues {λ _j ⁰ } _j=1 ^∞ and {λ _j ^∞ } _j=1 ^∞ , converging respectively to 0 and +∞. It is also established that $$N^0 (\lambda ) = \sum\nolimits_{\operatorname{Re} \lambda _j^0 \geqslant 1/\lambda } {1 \approx const} \lambda ^{n - 1} , N^\infty (\lambda ) \equiv \sum\nolimits_{\operatorname{Re} \lambda _j^\infty \leqslant \lambda } {1 \approx const} \lambda ^{n/1} .$$ The constants are explicitly calculated.     

Necessary conditions for the weak generalized localization of Fourier series with “lacunary sequence of partial sums”

It has been established that, on the subsets $ \mathbb{T}^N = [ - \pi ,\pi ]^N $ describing a cross W composed of N-dimensional blocks, $ W_{x_s x_t } = \Omega _{x_s x_t } \times [ - \pi ,\pi ]^{N - 2} (\Omega _{x_s x_t } $ is an open subset of [?π, π]²) in the classes $ L_p (\mathbb{T}^N ),p > 1 $ , a weak generalized localization holds, for N ≥ 3, almost everywhere for multiple trigonometric Fourier series when to the rectangular partial sums $ S_n (x;f)(x \in \mathbb{T}^N ,f \in L_p ) $ of these series corresponds the number n = (n ₁,…, n _N ) ∈ ? ₊ ^N , some components n _j of which are elements of lacunary sequences. In the present paper, we prove a number of statements showing that the structural and geometric characteristics of such subsets are sharp in the sense of the numbers (generating W) of the N-dimensional blocks $ W_{x_s x_t } $ as well as of the structure and geometry of $ W_{x_s x_t } $ . In particular, it is proved that it is impossible to take an arbitrary measurable two-dimensional set or an open three-dimensional set as the base of the block.     

Multi-neighboring grids schemes for solving PDE eigen-problems

Instead of most existing postprocessing schemes,a new preprocessing approach,called multineighboring grids(MNG),is proposed for solving PDE eigen-problems on an existing grid G(Δ).The linear or multi-linear element,based on box-splines,are taken as the frst stage Kh1Uh=λh1Mh1Uh.In this paper,the j-th stage neighboring-grid scheme is defned asKh jUh=λh j Mh jUh,where Kh j:=Mh j 1Kh1and Mh jUh is to be found as a better mass distribution over the j-th stage neighboring-gridG(Δ),and Kh jcan be seen as an expansion of Kh1on the j-th neighboring-grid with respect to the(j 1)-th mass distribution Mh j 1.It is shown that for an ODE model eigen-problem,the j-th stage scheme with 2j-th order B-spline basis can reach2j-th order accuracy and even(2j+2)-th order accuracy by perturbing the mass matrix.The argument can be extended to high dimensions with separable variable cases.For Laplace eigen-problems with some 2-D and 3-D structured uniform grids,some 2j-th order schemes are presented for j 3.