Complete hypersurfaces with constant scalar curvature in spheres

To a given immersion ${i:M^n\to \mathbb S^{n+1}}$ with constant scalar curvature R, we associate the supremum of the squared norm of the second fundamental form sup |A|². We prove the existence of a constant C _n (R) depending on R and n so that R ≥ 1 and sup |A|² = C _n (R) imply that the hypersurface is a H(r)-torus ${\mathbb S^1(\sqrt{1-r^2})\times\mathbb S^{n-1} (r)}$ . For R > (n ? 2)/n we use rotation hypersurfaces to show that for each value C > C _n (R) there is a complete hypersurface in ${\mathbb S^{n+1}}$ with constant scalar curvature R and sup |A|² = C, answering questions raised by Q. M. Cheng.     

Singular oscillatory integrals on {\mathbb{R}^n}

Let ${\mathcal{P}_{d,n}}Let P_d,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRⁿ{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? P_d,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that

supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      3. Semiconcavity estimates for viscous Hamilton–Jacobi equations      Thomas Strömberg 《Archiv der Mathematik》2010,94(6):579-589 We present sharp Hessian estimates of the form D2 Se(t,x) ￡ g(t)I{D^2 S^\varepsilon(t,x)\leq g(t)I} for the solution of the viscous Hamilton–Jacobi equation llSet+\frac12|DSe|2+V(x)-eDSe = 0    in  QT=(0,T]× \mathbb Rn,                                  Se(0,x) = S0(x)   in \mathbb Rn.\begin{array}{ll}S^\varepsilon_t+\frac{1}{2}|DS^\varepsilon|^2+V(x)-\varepsilon\Delta S^\varepsilon = 0\quad {\rm in} \, Q_T=(0,T]\times\, {\mathbb {R}^n}, \\ \qquad \qquad \qquad \qquad \quad \, S^\varepsilon(0,x) = S_0(x)\quad{\rm in}\, {\mathbb {R}^n}.\end{array}      4. Projective Freeness of Algebras of Real Symmetric Functions      Amol Sasane 《Complex Analysis and Operator Theory》2012,6(2):465-475 Let \mathbb Dn:={z=(z1,?, zn) ? \mathbb Cn:|zj| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]n{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cn{\mathbb {C}^n}. Consider the ring Cr([`(\mathbbD)]n;\mathbb C) = {f:[`(\mathbbD)]n? \mathbb C:f   is   continuous   and  f(z)=[`(f([`(z)]))]   (z ? [`(\mathbbD)]n)}C_{\rm r}(\overline{\mathbb{D}}^n;\mathbb {C}) =\left\{f: \overline{\mathbb{D}}^n\rightarrow \mathbb {C}:f \,\, {\rm is \,\, continuous \,\, and}\,\, f(z)=\overline{f(\overline{z})} \;(z\in \overline{\mathbb{D}}^n)\right\}      5. The structure of {\phi}-stable minimal hypersurfaces in manifolds of nonnegative P-scalar curvature      Pak Tung Ho 《Mathematische Annalen》2010,348(2):319-332 Suppose (N n , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as ${P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}Suppose (N n , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as P(g)=Rm￥(g)=R(g)-2Dglog f-|?glog f|g2{P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}, where dm=f dvol(g){dm=\phi\,dvol(g)} and R(g) is the scalar curvature of (N n , g). In this paper, under a technical assumption on f{\phi}, we prove that f{\phi}-stable minimal oriented hypersurface in the three-dimensional manifold with nonnegative P-scalar curvature must be conformally equivalent to either the complex plane \mathbbC{\mathbb{C}} or the cylinder \mathbbR×\mathbbS1{\mathbb{R}\times\mathbb{S}^1}.      6. On the regular growth of Dirichlet series absolutely convergent in a half-plane      Yu. V. Stets’  M. M. Sheremeta 《Ukrainian Mathematical Journal》2011,63(5):797-814 For the Dirichlet series F(s) = ?n = 1￥ anexp{ sln } F(s) = \sum\nolimits_{n = 1}^\infty {{a_n}\exp \left\{ {s{\lambda_n}} \right\}} with abscissa of absolute convergence σ a =0, we establish conditions for (λ n ) and (a n ) under which lnM( s, F ) = TR( 1 + o(1) )exp{ rR / | s| } \ln M\left( {\sigma, F} \right) = {T_R}\left( {1 + o(1)} \right)\exp \left\{ {{{{{\varrho_R}}} \left/ {{\left| \sigma \right|}} \right.}} \right\} for σ ↑ 0, where M( s, F ) = sup{ | F( s+ it ) |:t ? \mathbbR } M\left( {\sigma, F} \right) = \sup \left\{ {\left| {F\left( {\sigma + it} \right)} \right|:t \in \mathbb{R}} \right\} and T R and ϱ R are positive constants.      7. Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions      V. F. Babenko  N.V. Parfinovych  S. A. Pichugov 《Ukrainian Mathematical Journal》2010,62(3):343-357 Let C( \mathbbRm ) C\left( {{\mathbb{R}^m}} \right) be the space of bounded and continuous functions x:\mathbbRm ? \mathbbR x:{\mathbb{R}^m} \to \mathbb{R} equipped with the norm || x ||C = || x ||C( \mathbbRm ): = sup{ | x(t) |:t ? \mathbbRm } \left\| x \right\|C = {\left\| x \right\|_{C\left( {{\mathbb{R}^m}} \right)}}: = \sup \left\{ {\left| {x(t)} \right|:t \in {\mathbb{R}^m}} \right\}      8. Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b      Pascual Lucas  H. Fabián Ramírez-Ospina 《Geometriae Dedicata》2011,153(1):151-175 We study hypersurfaces in the Lorentz-Minkowski space \mathbbLn+1{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L k ψ = Aψ + b, where L k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLn+1{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSn1(r){\mathbb{S}^n_1(r)} or \mathbbHn(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbSm1(r)×\mathbbRn-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbHm(-r)×\mathbbRn-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbLm×\mathbbSn-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRn+1{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).      9. Blow up of subcritical quantities at the first singular time of the mean curvature flow      Nam Q. Le 《Geometriae Dedicata》2011,151(1):361-371 Consider a family of smooth immersions F(·,t) : Mn? \mathbbRn+1{F(\cdot,t)\,:\,{M^n\to \mathbb{R}^{n+1}}} of closed hypersurfaces in \mathbbRn+1{\mathbb{R}^{n+1}} moving by the mean curvature flow \frac?F(p,t)?t = -H(p,t)·n(p,t){\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)}, for t ? [0,T){t\in [0,T)}. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example ò0tòMs\frac|A|n + 2 log (2 + |A|) dmds,{\int_{0}^{t}\int_{M_{s}}\frac{{\vert{\it A}\vert}^{n + 2}}{ log (2 + {\vert{\it A}\vert})}} d\mu ds, blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.      10. Spectral decomposition and Gelfand’s theorem      A. Driouich  O. El-Mennaoui  M. Jazar 《Semigroup Forum》2010,80(3):391-404 We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\}We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set s(A)={ilk;k ? \mathbb\mathbbZ*}\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\} is discrete and satisfies ?\frac1|lk|ldkn < ￥\sum \frac{1}{|\lambda_{k}|^{\ell}\delta_{k}^{n}}<\infty , where ℓ is a nonnegative integer and dk=min(\frac|lk+1-lk|2,\frac|lk-1-lk|2)\delta _{k}=\min(\frac{|\lambda_{k+1}-\lambda _{k}|}{2},\frac{|\lambda _{k-1}-\lambda _{k}|}{2}) . In this case, Theorem 3, we show by using Gelfand’s Theorem that there exists a family of projectors (Pk)k ? \mathbb\mathbbZ*(P_{k})_{k\in\mathbb{\mathbb{Z}}^{*}} such that, for any x∈D(A n+ℓ ), the decomposition ∑P k x=x holds.      11. Gaussian mean curvature flow      Alexander A. Borisenko  Vicente Miquel 《Journal of Evolution Equations》2010,10(2):413-423 In the Euclidean Space \mathbb Rn+1{\mathbb {R}^{n+1}} with a density ee\frac12 n m2 |x|2, (e = ±1){e^{\varepsilon \frac12 n \mu^2 |x|^2},} {(\varepsilon =\pm1}), we consider the flow of a hypersurface driven by its mean curvature associated to this density. We give a detailed account of the evolution of a convex hypersurface under this flow. In particular, when e = -1{ \varepsilon=-1} (Gaussian density), the hypersurface can expand to infinity or contract to a convex hypersurface (not necessarily a sphere) depending on the relation between the bound of its principal curvatures and μ.      12. Interpolating Thin-Shell and Sharp Large-Deviation Estimates for Lsotropic Log-Concave Measures      Olivier Gu��don  Emanuel Milman 《Geometric And Functional Analysis》2011,21(5):1043-1068 Given an isotropic random vector X with log-concave density in Euclidean space \mathbbRn{\mathbb{R}^n} , we study the concentration properties of |X| on all scales, both above and below its expectation. We show in particular that l \mathbbP( | |X| - ?n | 3 t?n ) ￡ C  exp ( -cn1/2 min(t3, t) )   "t 3 0, \begin{array}{l} \mathbb{P}\left ( \left | |X| - \sqrt{n} \right | \geq t\sqrt{n} \right ) \leq C \, {\rm exp} \left ( -cn^{1/2} {\rm min}(t^{3}, t) \right) \; \forall t \geq 0, \end{array}      13. A Hyperplane Inequality for Measures of Convex Bodies in ? n , n≤4      Alexander Koldobsky 《Discrete and Computational Geometry》2012,47(3):538-547 Let 2≤n≤4. We show that for an arbitrary measure μ with even continuous density in ℝ n and any origin-symmetric convex body K in ℝ n , m(K) ￡ \fracnn-1\frac|B2n|\fracn-1n|B2n-1|maxx ? Sn-1 m(K?x^)\operatornameVoln(K)1/n,\mu(K) \le\frac{n}{n-1}\frac{|B_2^n|^{\frac{n-1}{n}}}{|B_2^{n-1}|}\max_{\xi\in S^{n-1}} \mu\bigl(K\cap\xi^\bot\bigr)\operatorname{Vol}_n(K)^{1/n},      14. Maximum Hitting of a Set by Compressed Intersecting Families      Peter Borg 《Graphs and Combinatorics》2011,27(6):785-797 For a family A{\mathcal{A}} and a set Z, denote {A ? A \colon A ?Z 1 ?}{\{A \in \mathcal{A} \colon A \cap Z \neq \emptyset\}} by A(Z){\mathcal{A}(Z)}. For positive integers n and r, let Sn,r{\mathcal{S}_{n,r}} be the trivial compressed intersecting family {A ? (c[n]r ) \colon 1 ? A}{\{A \in \big(\begin{subarray}{c}[n]\\r \end{subarray}\big) \colon 1 \in A\}}, where [n] : = {1, ?, n}{[n] := \{1, \ldots, n\}} and (c[n]r ) : = {A ì [n] \colon |A| = r}{\big(\begin{subarray}{c}[n]\\r \end{subarray}\big) := \{A \subset [n] \colon |A| = r\}}. The following problem is considered: For r ≤ n/2, which sets Z í [n]{Z \subseteq [n]} have the property that |A(Z)| ￡ |Sn,r(Z)|{|\mathcal{A}(Z)| \leq |\mathcal{S}_{n,r}(Z)|} for any compressed intersecting family A ì (c[n]r ){\mathcal{A}\subset \big(\begin{subarray}{c}[n]\\r \end{subarray}\big)}? (The answer for the case 1 ? Z{1 \in Z} is given by the Erdős–Ko–Rado Theorem.) We give a complete answer for the case |Z| ≥ r and a partial answer for the much harder case |Z| < r. This paper is motivated by the observation that certain interesting results in extremal set theory can be proved by answering the question above for particular sets Z. Using our result for the special case when Z is the r-segment {2, ?, r+1}{\{2, \ldots, r+1\}}, we obtain new short proofs of two well-known Hilton–Milner theorems. At the other extreme end, by establishing that |A(Z)| ￡ |Sn,r(Z)|{|\mathcal{A}(Z)| \leq |\mathcal{S}_{n,r}(Z)|} when Z is a final segment, we provide a new short proof of a Holroyd–Talbot extension of the Erdős-Ko-Rado Theorem.      15. Dimension Free Estimates for Riesz Transforms Associated to the Harmonic Oscillator on \mathbb{R}^{n}      Françoise Lust-Piquard 《Potential Analysis》2006,24(1):47-62 Let L=?Δ+|ξ|2 be the harmonic oscillator on $\mathbb{R}^{n}Let L=−Δ+|ξ|2 be the harmonic oscillator on \mathbbRn\mathbb{R}^{n} , with the associated Riesz transforms R2j−1=(∂/∂ξj)L−1/2,R2j=ξjL−1/2. We give a shorter proof of a recent result of Harboure, de Rosa, Segovia, Torrea: For 1<p<∞ and a dimension free constant Cp, ||(?k=12n|Rk(f)|2)1/2||Lp(\mathbbRn,dx)\leqslant Cp||f||Lp(\mathbbRn,dx).\bigg\Vert \bigg(\sum_{k=1}^{2n}\vert R_{k}(f)\vert ^{2}\bigg)^{{1}/{2}}\bigg\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}\leqslant C_{p}\Vert f\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}.      16. Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem      V. V. Lebedev 《Functional Analysis and Its Applications》2012,46(2):121-132 We consider the space A(\mathbbT)A(\mathbb{T}) of all continuous functions f on the circle \mathbbT\mathbb{T} such that the sequence of Fourier coefficients [^(f)] = { [^(f)]( k ), k ? \mathbbZ }\hat f = \left\{ {\hat f\left( k \right), k \in \mathbb{Z}} \right\} belongs to l 1(ℤ). The norm on A(\mathbbT)A(\mathbb{T}) is defined by || f ||A(\mathbbT) = || [^(f)] ||l1 (\mathbbZ)\left\| f \right\|_{A(\mathbb{T})} = \left\| {\hat f} \right\|_{l^1 (\mathbb{Z})}. According to the well-known Beurling-Helson theorem, if f:\mathbbT ? \mathbbT\phi :\mathbb{T} \to \mathbb{T} is a continuous mapping such that || einf ||A(\mathbbT) = O(1)\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = O(1), n ∈ ℤ then φ is linear. It was conjectured by Kahane that the same conclusion about φ is true under the assumption that || einf ||A(\mathbbT) = o( log| n | )\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\log \left| n \right|} \right). We show that if $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right)$\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right), then φ is linear.      17. On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality      E. A. Sevost’yanov 《Ukrainian Mathematical Journal》2011,63(1):84-97 For open discrete mappings f:D\{ b } ? \mathbbR3 f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3} of a domain D ì \mathbbR3 D \subset {\mathbb{R}^3} satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point b ? \mathbbR3 b \in {\mathbb{R}^3} , we prove the following statement: Let a point y 0 belong to [`(\mathbbR3)] \f( D\{ b } ) \overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right) and let the inner dilatation K I (x, f) and outer dilatation K O (x, f) of the mapping f at the point x satisfy certain conditions. Let B f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y 0, the set V ∩ f(B f ) cannot be contained in a set A such that g(A) = I, where I = { t ? \mathbbR:| t | < 1 } I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\} and g:U ? \mathbbRn g:U \to {\mathbb{R}^n} is a quasiconformal mapping of a domain U ì \mathbbRn U \subset {\mathbb{R}^n} such that A ⊂ U.      18. Localization of a Corner-Point Gibbs Phenomenon for Fourier Series in Two Dimensions      Gilbert Helmberg 《Journal of Fourier Analysis and Applications》2002,8(1):29-42 Let VV be a convex neighborhood of the origin contained in the square Q = { (x,y) ? \mathbbR2 : |x| ￡ p, |y| ￡ p} Q = \{ (x,y) \in \mathbb{R}^2 : |x| \leq \pi, |y| \leq \pi \} . Let AA be a 'sector' of VV bounded by two convex or concave curves intersecting at the origin. Let ff be an integrable function on QQ, smooth on AA and on V  AV \ A but having a jump discontinuity at the origin, whose coordinate sections (i.e., the restrictions of ff to x = const x = \textit{const} , resp.\ y = const y = \textit{const} ) have uniformly bounded variation. Under essentially these conditions the partial sums Sn,n S_{n,n} of the Fourier series of ff display for n ? ￥ n \rightarrow \infty at the origin a corner point Gibbs phenomenon with an overshoot of up to 37,4% of half the jump size. This Gibbs phenomenon manifests itself in the pointwise convergence of Sn,n ( \fracxn, \fracyn ; f)S_{n,n} ( \frac{x}{n}, \frac{y}{n} ; f) as n ? ￥ n \rightarrow \infty for all (x,y) ? \mathbbR2 (x,y) \in \mathbb{R}^2 to a non-constant limiting function only depending on the slopes of the boundary curves of AA at the origin and on the jump of f (x,y) f (x,y) as (x,y) (x,y) approaches (0,0)(0,0) within AA resp.V  A V \ A.      19. A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations      Xiaochun Chen  Sadek Gala  Zhengguang Guo 《Acta Appl Math》2011,113(2):207-213 In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to L\frac21-r( 0,T;[(X)\dot]r( \mathbbR3) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique.      20. Uniform estimates for order statistics and Orlicz functions      Yehoram Gordon  Alexander E. Litvak  Carsten Schütt  Elisabeth Werner 《Positivity》2012,16(1):1-28 We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ 1, … , ξ n and a vector of scalars x = (x 1, … , x n ), and 1 ≤ k ≤ n, we provide estimates for \mathbb E   k-min1 ￡ i ￡ n |xixi|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and \mathbb E k-max1 ￡ i ￡ n|xixi|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||yx||M{\|y_x\|_M} of the vector y x  = (1/x 1, … , 1/x n ). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ 1|, G(t) = \mathbb P ({ |x1| ￡ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ 1 is the standard N(0, 1) Gaussian random variable, then G(t) = ?{\tfrac2p}ò0t e-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds }  and M(s)=?{\tfrac2p}ò0se-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Semiconcavity estimates for viscous Hamilton–Jacobi equations

We present sharp Hessian estimates of the form D² S^e(t,x) ￡ g(t)I{D^2 S^\varepsilon(t,x)\leq g(t)I} for the solution of the viscous Hamilton–Jacobi equation

Projective Freeness of Algebras of Real Symmetric Functions

Let \mathbb Dⁿ:={z=(z₁,?, z_n) ? \mathbb Cⁿ:|z_j| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]ⁿ{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cⁿ{\mathbb {C}^n}. Consider the ring

The structure of {\phi}-stable minimal hypersurfaces in manifolds of nonnegative P-scalar curvature

Suppose (N ⁿ , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as ${P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}Suppose (N ⁿ , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as P(g)=R^m_￥(g)=R(g)-2D_glog f-|?_glog f|_g²{P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}, where dm=f dvol(g){dm=\phi\,dvol(g)} and R(g) is the scalar curvature of (N ⁿ , g). In this paper, under a technical assumption on f{\phi}, we prove that f{\phi}-stable minimal oriented hypersurface in the three-dimensional manifold with nonnegative P-scalar curvature must be conformally equivalent to either the complex plane \mathbbC{\mathbb{C}} or the cylinder \mathbbR×\mathbbS¹{\mathbb{R}\times\mathbb{S}^1}.     

On the regular growth of Dirichlet series absolutely convergent in a half-plane

For the Dirichlet series F(s) = ?_{n = 1}^￥ a_nexp{ sl_n } F(s) = \sum\nolimits_{n = 1}^\infty {{a_n}\exp \left\{ {s{\lambda_n}} \right\}} with abscissa of absolute convergence σ _a =0, we establish conditions for (λ _n ) and (a _n ) under which lnM( s, F ) = T_R( 1 + o(1) )exp{ r_R

Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions

Let C( \mathbbR^m ) C\left( {{\mathbb{R}^m}} \right) be the space of bounded and continuous functions x:\mathbbR^m ? \mathbbR x:{\mathbb{R}^m} \to \mathbb{R} equipped with the norm

Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b

We study hypersurfaces in the Lorentz-Minkowski space \mathbbLⁿ⁺¹{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L _k ψ = Aψ + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR^(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLⁿ⁺¹{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSⁿ₁(r){\mathbb{S}^n_1(r)} or \mathbbHⁿ(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbS^m₁(r)×\mathbbR^n-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbH^m(-r)×\mathbbR^n-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbL^m×\mathbbS^n-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).     

Blow up of subcritical quantities at the first singular time of the mean curvature flow

Consider a family of smooth immersions F(·,t) : Mⁿ? \mathbbRⁿ⁺¹{F(\cdot,t)\,:\,{M^n\to \mathbb{R}^{n+1}}} of closed hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} moving by the mean curvature flow \frac?F(p,t)?t = -H(p,t)·n(p,t){\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)}, for t ? [0,T){t\in [0,T)}. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example ò₀^tò_Ms\frac|A|^{n + 2} log (2 + |A|) dmds,{\int_{0}^{t}\int_{M_{s}}\frac{{\vert{\it A}\vert}^{n + 2}}{ log (2 + {\vert{\it A}\vert})}} d\mu ds, blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.     

Spectral decomposition and Gelfand’s theorem

We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\}We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set s(A)={il_k;k ? \mathbb\mathbbZ^*}\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\} is discrete and satisfies ?\frac1|l_k|^ld_kⁿ < ￥\sum \frac{1}{|\lambda_{k}|^{\ell}\delta_{k}^{n}}<\infty , where ℓ is a nonnegative integer and d_k=min(\frac|l_k+1-l_k|2,\frac|l_k-1-l_k|2)\delta _{k}=\min(\frac{|\lambda_{k+1}-\lambda _{k}|}{2},\frac{|\lambda _{k-1}-\lambda _{k}|}{2}) . In this case, Theorem 3, we show by using Gelfand’s Theorem that there exists a family of projectors (P_k)_{k ? \mathbb\mathbbZ^*}(P_{k})_{k\in\mathbb{\mathbb{Z}}^{*}} such that, for any x∈D(A ^n+ℓ ), the decomposition ∑P _k x=x holds.     

Gaussian mean curvature flow

In the Euclidean Space \mathbb Rⁿ⁺¹{\mathbb {R}^{n+1}} with a density e^{e\frac12 n m2 |x|2}, (e = ±1){e^{\varepsilon \frac12 n \mu^2 |x|^2},} {(\varepsilon =\pm1}), we consider the flow of a hypersurface driven by its mean curvature associated to this density. We give a detailed account of the evolution of a convex hypersurface under this flow. In particular, when e = -1{ \varepsilon=-1} (Gaussian density), the hypersurface can expand to infinity or contract to a convex hypersurface (not necessarily a sphere) depending on the relation between the bound of its principal curvatures and μ.     

Interpolating Thin-Shell and Sharp Large-Deviation Estimates for Lsotropic Log-Concave Measures

Given an isotropic random vector X with log-concave density in Euclidean space \mathbbRⁿ{\mathbb{R}^n} , we study the concentration properties of |X| on all scales, both above and below its expectation. We show in particular that

A Hyperplane Inequality for Measures of Convex Bodies in ? n , n≤4

Let 2≤n≤4. We show that for an arbitrary measure μ with even continuous density in ℝ ⁿ and any origin-symmetric convex body K in ℝ ⁿ ,

Maximum Hitting of a Set by Compressed Intersecting Families

For a family A{\mathcal{A}} and a set Z, denote {A ? A \colon A ?Z 1 ?}{\{A \in \mathcal{A} \colon A \cap Z \neq \emptyset\}} by A(Z){\mathcal{A}(Z)}. For positive integers n and r, let S_n,r{\mathcal{S}_{n,r}} be the trivial compressed intersecting family {A ? (c[n]r ) \colon 1 ? A}{\{A \in \big(\begin{subarray}{c}[n]\\r \end{subarray}\big) \colon 1 \in A\}}, where [n] : = {1, ?, n}{[n] := \{1, \ldots, n\}} and (c[n]r ) : = {A ì [n] \colon |A| = r}{\big(\begin{subarray}{c}[n]\\r \end{subarray}\big) := \{A \subset [n] \colon |A| = r\}}. The following problem is considered: For r ≤ n/2, which sets Z í [n]{Z \subseteq [n]} have the property that |A(Z)| ￡ |S_n,r(Z)|{|\mathcal{A}(Z)| \leq |\mathcal{S}_{n,r}(Z)|} for any compressed intersecting family A ì (c[n]r ){\mathcal{A}\subset \big(\begin{subarray}{c}[n]\\r \end{subarray}\big)}? (The answer for the case 1 ? Z{1 \in Z} is given by the Erdős–Ko–Rado Theorem.) We give a complete answer for the case |Z| ≥ r and a partial answer for the much harder case |Z| < r. This paper is motivated by the observation that certain interesting results in extremal set theory can be proved by answering the question above for particular sets Z. Using our result for the special case when Z is the r-segment {2, ?, r+1}{\{2, \ldots, r+1\}}, we obtain new short proofs of two well-known Hilton–Milner theorems. At the other extreme end, by establishing that |A(Z)| ￡ |S_n,r(Z)|{|\mathcal{A}(Z)| \leq |\mathcal{S}_{n,r}(Z)|} when Z is a final segment, we provide a new short proof of a Holroyd–Talbot extension of the Erdős-Ko-Rado Theorem.     

Dimension Free Estimates for Riesz Transforms Associated to the Harmonic Oscillator on \mathbb{R}^{n}

Let L=?Δ+|ξ|² be the harmonic oscillator on $\mathbb{R}^{n}Let L=−Δ+|ξ|² be the harmonic oscillator on \mathbbRⁿ\mathbb{R}^{n} , with the associated Riesz transforms R_2j−1=(∂/∂ξ_j)L^−1/2,R_2j=ξ_jL^−1/2. We give a shorter proof of a recent result of Harboure, de Rosa, Segovia, Torrea: For 1<p<∞ and a dimension free constant C_p,

||(?k=12n|Rk(f)|2)1/2||Lp(\mathbbRn,dx)\leqslant Cp||f||Lp(\mathbbRn,dx).\bigg\Vert \bigg(\sum_{k=1}^{2n}\vert R_{k}(f)\vert ^{2}\bigg)^{{1}/{2}}\bigg\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}\leqslant C_{p}\Vert f\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}.      16. Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem      V. V. Lebedev 《Functional Analysis and Its Applications》2012,46(2):121-132 We consider the space A(\mathbbT)A(\mathbb{T}) of all continuous functions f on the circle \mathbbT\mathbb{T} such that the sequence of Fourier coefficients [^(f)] = { [^(f)]( k ), k ? \mathbbZ }\hat f = \left\{ {\hat f\left( k \right), k \in \mathbb{Z}} \right\} belongs to l 1(ℤ). The norm on A(\mathbbT)A(\mathbb{T}) is defined by || f ||A(\mathbbT) = || [^(f)] ||l1 (\mathbbZ)\left\| f \right\|_{A(\mathbb{T})} = \left\| {\hat f} \right\|_{l^1 (\mathbb{Z})}. According to the well-known Beurling-Helson theorem, if f:\mathbbT ? \mathbbT\phi :\mathbb{T} \to \mathbb{T} is a continuous mapping such that || einf ||A(\mathbbT) = O(1)\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = O(1), n ∈ ℤ then φ is linear. It was conjectured by Kahane that the same conclusion about φ is true under the assumption that || einf ||A(\mathbbT) = o( log| n | )\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\log \left| n \right|} \right). We show that if $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right)$\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right), then φ is linear.      17. On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality      E. A. Sevost’yanov 《Ukrainian Mathematical Journal》2011,63(1):84-97 For open discrete mappings f:D\{ b } ? \mathbbR3 f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3} of a domain D ì \mathbbR3 D \subset {\mathbb{R}^3} satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point b ? \mathbbR3 b \in {\mathbb{R}^3} , we prove the following statement: Let a point y 0 belong to [`(\mathbbR3)] \f( D\{ b } ) \overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right) and let the inner dilatation K I (x, f) and outer dilatation K O (x, f) of the mapping f at the point x satisfy certain conditions. Let B f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y 0, the set V ∩ f(B f ) cannot be contained in a set A such that g(A) = I, where I = { t ? \mathbbR:| t | < 1 } I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\} and g:U ? \mathbbRn g:U \to {\mathbb{R}^n} is a quasiconformal mapping of a domain U ì \mathbbRn U \subset {\mathbb{R}^n} such that A ⊂ U.      18. Localization of a Corner-Point Gibbs Phenomenon for Fourier Series in Two Dimensions      Gilbert Helmberg 《Journal of Fourier Analysis and Applications》2002,8(1):29-42 Let VV be a convex neighborhood of the origin contained in the square Q = { (x,y) ? \mathbbR2 : |x| ￡ p, |y| ￡ p} Q = \{ (x,y) \in \mathbb{R}^2 : |x| \leq \pi, |y| \leq \pi \} . Let AA be a 'sector' of VV bounded by two convex or concave curves intersecting at the origin. Let ff be an integrable function on QQ, smooth on AA and on V  AV \ A but having a jump discontinuity at the origin, whose coordinate sections (i.e., the restrictions of ff to x = const x = \textit{const} , resp.\ y = const y = \textit{const} ) have uniformly bounded variation. Under essentially these conditions the partial sums Sn,n S_{n,n} of the Fourier series of ff display for n ? ￥ n \rightarrow \infty at the origin a corner point Gibbs phenomenon with an overshoot of up to 37,4% of half the jump size. This Gibbs phenomenon manifests itself in the pointwise convergence of Sn,n ( \fracxn, \fracyn ; f)S_{n,n} ( \frac{x}{n}, \frac{y}{n} ; f) as n ? ￥ n \rightarrow \infty for all (x,y) ? \mathbbR2 (x,y) \in \mathbb{R}^2 to a non-constant limiting function only depending on the slopes of the boundary curves of AA at the origin and on the jump of f (x,y) f (x,y) as (x,y) (x,y) approaches (0,0)(0,0) within AA resp.V  A V \ A.      19. A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations      Xiaochun Chen  Sadek Gala  Zhengguang Guo 《Acta Appl Math》2011,113(2):207-213 In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to L\frac21-r( 0,T;[(X)\dot]r( \mathbbR3) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique.      20. Uniform estimates for order statistics and Orlicz functions      Yehoram Gordon  Alexander E. Litvak  Carsten Schütt  Elisabeth Werner 《Positivity》2012,16(1):1-28 We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ 1, … , ξ n and a vector of scalars x = (x 1, … , x n ), and 1 ≤ k ≤ n, we provide estimates for \mathbb E   k-min1 ￡ i ￡ n |xixi|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and \mathbb E k-max1 ￡ i ￡ n|xixi|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||yx||M{\|y_x\|_M} of the vector y x  = (1/x 1, … , 1/x n ). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ 1|, G(t) = \mathbb P ({ |x1| ￡ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ 1 is the standard N(0, 1) Gaussian random variable, then G(t) = ?{\tfrac2p}ò0t e-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds }  and M(s)=?{\tfrac2p}ò0se-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence.

Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem

We consider the space A(\mathbbT)A(\mathbb{T}) of all continuous functions f on the circle \mathbbT\mathbb{T} such that the sequence of Fourier coefficients [^(f)] = { [^(f)]( k ), k ? \mathbbZ }\hat f = \left\{ {\hat f\left( k \right), k \in \mathbb{Z}} \right\} belongs to l ¹(ℤ). The norm on A(\mathbbT)A(\mathbb{T}) is defined by || f ||_A(\mathbbT) = || [^(f)] ||_{l¹ (\mathbbZ)}\left\| f \right\|_{A(\mathbb{T})} = \left\| {\hat f} \right\|_{l^1 (\mathbb{Z})}. According to the well-known Beurling-Helson theorem, if f:\mathbbT ? \mathbbT\phi :\mathbb{T} \to \mathbb{T} is a continuous mapping such that || e^inf ||_A(\mathbbT) = O(1)\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = O(1), n ∈ ℤ then φ is linear. It was conjectured by Kahane that the same conclusion about φ is true under the assumption that || e^inf ||_A(\mathbbT) = o( log| n | )\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\log \left| n \right|} \right). We show that if $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right)$\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right), then φ is linear.     

On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality

For open discrete mappings f:D\{ b } ? \mathbbR³ f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3} of a domain D ì \mathbbR³ D \subset {\mathbb{R}^3} satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point b ? \mathbbR³ b \in {\mathbb{R}^3} , we prove the following statement: Let a point y ₀ belong to [`(\mathbbR³)] \f( D\{ b } ) \overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right) and let the inner dilatation K _I (x, f) and outer dilatation K _O (x, f) of the mapping f at the point x satisfy certain conditions. Let B _f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y ₀, the set V ∩ f(B _f ) cannot be contained in a set A such that g(A) = I, where I = { t ? \mathbbR:| t | < 1 } I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\} and g:U ? \mathbbRⁿ g:U \to {\mathbb{R}^n} is a quasiconformal mapping of a domain U ì \mathbbRⁿ U \subset {\mathbb{R}^n} such that A ⊂ U.     

Localization of a Corner-Point Gibbs Phenomenon for Fourier Series in Two Dimensions

Let VV be a convex neighborhood of the origin contained in the square Q = { (x,y) ? \mathbbR² : |x| ￡ p, |y| ￡ p} Q = \{ (x,y) \in \mathbb{R}^2 : |x| \leq \pi, |y| \leq \pi \} . Let AA be a 'sector' of VV bounded by two convex or concave curves intersecting at the origin. Let ff be an integrable function on QQ, smooth on AA and on V  AV \ A but having a jump discontinuity at the origin, whose coordinate sections (i.e., the restrictions of ff to x = const x = \textit{const} , resp.\ y = const y = \textit{const} ) have uniformly bounded variation. Under essentially these conditions the partial sums S_n,n S_{n,n} of the Fourier series of ff display for n ? ￥ n \rightarrow \infty at the origin a corner point Gibbs phenomenon with an overshoot of up to 37,4% of half the jump size. This Gibbs phenomenon manifests itself in the pointwise convergence of S_n,n ( \fracxn, \fracyn ; f)S_{n,n} ( \frac{x}{n}, \frac{y}{n} ; f) as n ? ￥ n \rightarrow \infty for all (x,y) ? \mathbbR² (x,y) \in \mathbb{R}^2 to a non-constant limiting function only depending on the slopes of the boundary curves of AA at the origin and on the jump of f (x,y) f (x,y) as (x,y) (x,y) approaches (0,0)(0,0) within AA resp.V  A V \ A.     

A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations

In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to L^\frac21-r( 0,T;[(X)\dot]_r( \mathbbR³) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique.     

Uniform estimates for order statistics and Orlicz functions

We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ ₁, … , ξ _n and a vector of scalars x = (x ₁, … , x _n ), and 1 ≤ k ≤ n, we provide estimates for \mathbb E   k-min_{1 ￡ i ￡ n} |x_ix_i|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and \mathbb E k-max_{1 ￡ i ￡ n}|x_ix_i|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||y_x||_M{\|y_x\|_M} of the vector y _x  = (1/x ₁, … , 1/x _n ). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ ₁|, G(t) = \mathbb P ({ |x₁| ￡ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ ₁ is the standard N(0, 1) Gaussian random variable, then G(t) = ?{\tfrac2p}ò₀^t e^-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds }  and M(s)=?{\tfrac2p}ò₀^se^-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence.