Two families of approximations for the gamma function

In this paper, we establish two families of approximations for the gamma function: $$ \begin{array}{lll} {\varGamma}(x+1)&=\sqrt{2\pi x}{\left({\frac{x+a}{{\mathrm{e}}}}\right)}^x {\left({\frac{x+a}{x-a}}\right)}^{-\frac{x}{2}+\frac{1}{4}} {\left({\frac{x+b}{x-b}}\right)}^{\sum\limits_{k=0}^m\frac{{\beta}_k}{x^{2k}}+O{{\left(\frac{1}{x^{2m+2}}\right)}}},\\ {\varGamma}(x+1)&=\sqrt{2\pi x}\cdot(x+a)^{\frac{x}{2}+\frac{1}{4}}(x-a)^{\frac{x}{2}-\frac{1}{4}} {\left({\frac{x-1}{x+1}}\right)}^{\frac{x^2}{2}}\\ &\quad\times {\left({\frac{x-c}{x+c}}\right)}^{\sum\limits_{k=0}^m\frac{{\gamma}_k}{x^{2k}}+O{\left({\frac{1}{x^{2m+2}}}\right)}}, \end{array}$$ where the constants ${\beta }_k$ and ${\gamma }_k$ can be determined by recurrences, and $a$ , $b$ , $c$ are parameters. Numerical comparison shows that our results are more accurate than Stieltjes, Luschny and Nemes’ formulae, which, to our knowledge, are better than other approximations in the literature.     

Pointwise estimates for Lagrange interpolation polynomials

Let f ∈ C[?1, 1]. Let the approximation rate of Lagrange interpolation polynomial of f based on the nodes $ \left\{ {\cos \frac{{2k - 1}} {{2n}}\pi } \right\} \cup \{ - 1,1\} $ be Δ _{n + 2}(f, x). In this paper we study the estimate of Δ _{n + 2}(f,x), that keeps the interpolation property. As a result we prove that $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left| {T_n (x)} \right|\ln (n + 1) + \omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}\left| {T_n (x)} \right|} \right)} \right\}, $$ where T _n (x) = cos (n arccos x) is the Chebeyshev polynomial of first kind. Also, if f ∈ C ^r [?1, 1] with r ≧ 1, then $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\frac{{\sqrt {1 - x^2 } }} {{n^r }}\left| {T_n (x)} \right|\omega \left( {f^{(r)} ,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left( {\left( {\sqrt {1 - x^2 } + \frac{1} {n}} \right)^{r - 1} \ln (n + 1) + 1} \right)} \right\}. $$      

On the best constants in the Khintchine inequality for Steinhaus variables

Let S _j : (Ω, P) → S ¹ ? ? be an i.i.d. sequence of Steinhaus random variables, i.e. variables which are uniformly distributed on the circle S ¹. We determine the best constants a _p in the Khintchine-type inequality $${a_p}{\left\| x \right\|_2} \leqslant {\left( {{\text{E}}{{\left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|}^p}} \right)^{1/p}} \leqslant {\left\| x \right\|_2};{\text{ }}x = ({x_j})_{j = 1}^n \in {{\Bbb C}^n}$$ for 0 < p < 1, verifying a conjecture of U. Haagerup that $${a_p} = \min \left( {\Gamma {{\left( {\frac{p}{2} + 1} \right)}^{1/p}},\sqrt 2 {{\left( {{{\Gamma \left( {\frac{{p + 1}}{2}} \right)} \mathord{\left/ {\vphantom {{\Gamma \left( {\frac{{p + 1}}{2}} \right)} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right)}^{1/p}}} \right)$$ . Both expressions are equal for p = p ₀ }~ 0.4756. For p ≥ 1 the best constants a _p have been known for some time. The result implies for a norm 1 sequence x ∈ ? ⁿ , ‖x‖₂ = 1, that $${\text{E}}\ln \left| {\frac{{{S_1} + {S_2}}}{{\sqrt 2 }}} \right| \leqslant {\text{E}}\ln \left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|$$ , answering a question of A. Baernstein and R. Culverhouse.     

Symplectic mean curvature flow in CP 2

Let Σ be an immersed symplectic surface in CP ² with constant holomorphic sectional curvature k > 0. Suppose Σ evolves along the mean curvature flow in CP ². In this paper, we show that the symplectic mean curvature flow exists for long time and converges to a holomorphic curve if the initial surface satisfies ${|A|^2 \leq \lambda|H|^2 + \frac{2\lambda-1}{\lambda}k}$ and ${\cos\alpha\geq\sqrt{\frac{7\lambda-3}{3\lambda}}\left(\frac{1}{2} < \lambda\leq\frac{2}{3}\right) {\rm or} |A|^2\leq \frac{2}{3}|H|^2+\frac{4}{5}k\cos\alpha\, {\rm and} \cos\alpha\geq 1-\varepsilon}$ , for some ${\varepsilon}$ .     

Two results on arithmetical functions

Generalizing two results of Rieger [8] and Selberg [10] we give asymptotic formulas for sums of type $${\matrix {\sum \limits_{n\leq x}\cr n\equiv l({\rm mod}k)\cr f_{\kappa}(n)\equiv s_{\kappa}({\rm mod}p_{\kappa})\cr (\kappa=1,\dots,r)\cr}}\qquad \chi(n)\qquad {\rm and} {\matrix {\sum \limits_{n\leq x}\cr n\equiv l({\rm mod}k)\cr f_{\kappa}(n)\equiv s_{\kappa}({\rm mod}p_{\kappa})\cr (\kappa=1,\dots,r)\cr}}\qquad \chi(n),$$ where χ is a suitable multiplicative function, f₁,…, f _r are “small” additive, prime-independent arithmetical functions and k, l are coprime. The proofs are based on an analytic method which consists of considering the Dirichlet series generated by $ \chi(n)z_{1}^{f_{1}(n)}\cdot... \cdot z_{r}^{f_{r}(n)},z_{1}\dots z_{r} $ complex.     

Some remarks on the moments of \left| {\varsigma \left( {\tfrac{1} {2} + it} \right)} \right| in short intervals

Some new results on power moments of the integral $$ J_k (t,G) = \frac{1} {{\sqrt {\pi G} }}\int_{ - \infty }^\infty { \left| {\varsigma \left( {\tfrac{1} {2} + it + iu} \right)} \right|^{2k} } e^{ - (u/G)^2 } du $$ (t ? T, T ^? ≦ G ? T, κ ∈ N) are obtained when κ = 1. These results can be used to derive bounds for moments of $ \left| {\varsigma \left( {\tfrac{1} {2} + it} \right)} \right| $ .     

Reelle Abstandsräume und hyperbolische Geometrie

Ein reeller Abstandsraum ist eine Menge S ≠ ø zusammen mit einer Abbildung d: S × S → ?. Für x,y ∈ S hei\t d(x,y) der Abstand von x und y. Für beliebige reelle Abstandsräume definieren wir Begriffe wie Gerade, sphärischer Teilraum, konvexe Teilmenge, Winkelma\e usf. derart, da\ diese Begriffe im Falle $$={\rm R}^{n}\ {\rm und}\ d(x,y)=\sqrt{x^{2}+y^{2}- 2xy}$$ klassische Objekte gleichen Namens ergeben. Sodann wenden wir unseren Begriffsapparat auf $$ S={\rm R}^{n}\ {\rm und}\ d(x,y)=\sqrt{1+x^{2}}\sqrt{1+y^{2}}- xy $$ an. Dabei entsteht dann die n-dimensionale hyperbolische Geometrie mit dem ?ⁿ als Punktmenge und geeigneten Geraden, Strecken usf. als dortige Grundobjekte.     

An explicit formula for |FM(1+1+n)|

We show that the number of elements in FM(1+1+n), the modular lattice freely generated by two single elements and an n-element chain, is 1 $$\frac{1}{{6\sqrt 2 }}\sum\limits_{k = 0}^{n + 1} {\left[ {2\left( {\begin{array}{*{20}c} {2k} \\ k \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {2k} \\ {k - 2} \\ \end{array} } \right)} \right]} \left( {\lambda _1^{n - k + 2} - \lambda _2^{n - k + 2} } \right) - 2$$ , where \(\lambda _{1,2} = {{\left( {4 \pm 3\sqrt 2 } \right)} \mathord{\left/ {\vphantom {{\left( {4 \pm 3\sqrt 2 } \right)} 2}} \right. \kern-0em} 2}\) .     

K-loops and quasidirect products in 2-dlmensional Linear Groups over a Pythagorean Field

For a group G = (G, ·), we define the (internal) quasidirect product f · U = F × U of a certain K-loop (F,+) with F ? G and a suitable subgroup il of G (cf. (3.1)). Let K be a commutative pythagorean field and let L = K(i) be the quadratic extension of K with i2 = ～-1. Then the future cone H:= A ∈ GL(2,L) ¦ A = A*, det A ∈ K⁺, Tr A ∈ K⁺ is a K-loop with respect to the binary operation $A?ggsquaredplus B:=sqrt{AB^{2}A},{? where}sqrt{A}=({? Tr}A+2sqrt {{? det}A})^{1?er 2}(sqrt {? det}AE+A)$} (cf. (2.4)), and the (internal) quasidirect product $H^{}</Emphasis>{\mathop \times\limits_{Q}}Q_{1}$ of the K-loop (H},+) and the group Q₁:= {X ∈ GL(2,L) ¦ X*X = E) is a subgroup of GL(2,L) (cf. (3.2)). Moreover, S L(2,1) = $H^{1+}{\mathop \times\limits_{Q}}Q^{1}$ , where H¹+ = SL(2,L)∩ H ≤} (H},+), Q¹ = S L(2, L) ∩ Q₁ (cf. (3.4)), and if K is euclidean, then (cf. (3.6)).     

Constructions of Large Graphs on Surfaces

We consider the degree/diameter problem for graphs embedded in a surface, namely, given a surface Σ and integers Δ and k, determine the maximum order N(Δ,k,Σ) of a graph embeddable in Σ with maximum degree Δ and diameter k. We introduce a number of constructions which produce many new largest known planar and toroidal graphs. We record all these graphs in the available tables of largest known graphs. Given a surface Σ of Euler genus g and an odd diameter k, the current best asymptotic lower bound for N(Δ,k,Σ) is given by $$\sqrt{\frac{3}{8}}g \Delta^{\lfloor k/2 \rfloor}.$$ Our constructions produce new graphs of order $$\left\{\begin{array}{ll}6 \Delta^{\lfloor k/2 \rfloor} \qquad \qquad \qquad \qquad {\rm if \Sigma\;is\;the\;Klein\;bottle} \\ \left(\frac{7}{2} + \sqrt{6g + \frac{1}{4}}\right) \Delta^{\lfloor k/2 \rfloor} \quad {\rm otherwise},\end{array}\right.$$ thus improving the former value.     

On the normalizing multiplier of the generalized Jackson kernel

We consider the question of evaluating the normalizing multiplier $$\gamma _{n,k} = \frac{1}{\pi }\int_{ - \pi }^\pi {\left( {\frac{{sin\tfrac{{nt}}{2}}}{{sin\tfrac{t}{2}}}} \right)^{2k} dt} $$ for the generalized Jackson kernel J _n,k (t). We obtain the explicit formula $$\gamma _{n,k} = 2\sum\limits_{p = 0}^{\left[ {k - \tfrac{k}{n}} \right]} {( - 1)\left( {\begin{array}{*{20}c} {2k} \\ p \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {k(n + 1) - np - 1} \\ {k(n - 1) - np} \\ \end{array} } \right)} $$ and the representation $$\gamma _{n,k} = \sqrt {\frac{{24}}{\pi }} \cdot \frac{{(n - 1)^{2k - 1} }}{{\sqrt {2k - 1} }}\left[ {1\frac{1}{8} \cdot \frac{1}{{2k - 1}} + \omega (n,k)} \right],$$ , where $$\left| {\omega (n,k)} \right| < \frac{4}{{(2k - 1)\sqrt {ln(2k - 1)} }} + \sqrt {12\pi } \cdot \frac{{k^{\tfrac{3}{2}} }}{{n - 1}}\left( {1 + \frac{1}{{n - 1}}} \right)^{2k - 2} .$$ .     

Nearly Quartic Mappings in β-Homogeneous F-Spaces

In this paper, we investigate the Hyers–Ulam stability of the following quartic equation $$\begin{array}{ll} {\sum\limits^{n}_{k=2}}\left({\sum\limits^{k}_{i_{1}=2}}{\sum\limits^{k+1}_{i_{2}=i_{1}+1}} \ldots {\sum\limits^{n}_{i_{n-k+1}=i_{n-k}+1}}\right)\\ \quad\times f \left({\sum\limits^{n}_{i=1,i \neq i_{1},\ldots,i_{n-k+1}}} x_{i}-{\sum\limits^{n-k+1}_{r=1}}x_{i_{r}}\right) + f \left({\sum\limits^{n}_{i=1}}x_{i}\right)\\ \quad-2^{n-2}{\sum\limits^{}_{1 \leq{i} \leq{j} \leq{n}}}(f(x_{i} + x_{j}){+f(x_{i} - x_{j})){+2^{n-5}(n - 2){\sum\limits^{n}_{i=1}}f(2x_{i})}} = \theta \end{array} $$ $({n \in \mathbb{N}, n \geq 3})$ in β-homogeneous F-spaces.     

Representation of Integers by Positive Quaternary Quadratic Forms

Let $f(x,y,x,w) = x^2 + y^2 + z^2 + Dw^2$ , where $D >1$ is an integer such that $D \ne d^2$ and ${{\sqrt n } \mathord{\left/ {\vphantom {{\sqrt n } {\sqrt D = n^\theta , 0 < \theta < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} \right. \kern-0em} {\sqrt D = n^\theta , 0 < \theta < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}$ . Let $rf(n)$ be the number of representations of n by f. It is proved that $r_f (n) = \pi ^2 \frac{n}{{\sqrt D }}\sigma _f (n) + O\left( {\frac{{n^{1 + \varepsilon - c(\theta )} }}{{\sqrt D }}} \right),$ where $\sigma _f (n)$ is the singular series, $c(\theta ) >0$ , and ε is an arbitrarily small positive constant. Bibliography: 14 titles.     

A sharp equivalence between H ∞ functional calculus and square function estimates

Let (T _t ) _{t?≥ 0} be a bounded analytic semigroup on L ^p (Ω), with 1?<?p?<?∞. Let ?A denote its infinitesimal generator. It is known that if A and A ^* both satisfy square function estimates ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{\frac{1}{2}} T_t(x)\vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^p} \lesssim \|x\|_{L^p}}$ and ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{*\frac{1}{2}} T_t^*(y) \vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^{p^\prime}} \lesssim \|y\|_{L^{p^\prime}}}$ for ${x\in L^p(\Omega)}$ and ${y\in L^{p^\prime}(\Omega)}$ , then A admits a bounded ${H^{\infty}(\Sigma_\theta)}$ functional calculus for any ${\theta>\frac{\pi}{2}}$ . We show that this actually holds true for some ${\theta<\frac{\pi}{2}}$ .     

Non-resonant boundary value problems with singular {\phi} -Laplacian operators

In this paper, using Leray–Schauder degree arguments, critical point theory for lower semicontinuous functionals and the method of lower and upper solutions, we give existence results for periodic problems involving the relativistic operator ${u \mapsto \left(\frac{u^\prime}{\sqrt{1-u^\prime 2}}\right)^\prime+r(t)u}$ with ${\int_0^Tr dt\neq 0}$ . In particular we show that in this case we have non-resonance, that is periodic problem $$\left(\frac{u^\prime}{\sqrt{1-u^\prime 2}}\right)^\prime+r(t)u=e(t),\quad u(0)-u(T)=0=u^\prime(0)-u^\prime(T),$$ has at least one solution for any continuous function ${e : [0, T] \to \mathbb {R}}$ . Then, we consider Brillouin and Mathieu-Duffing type equations for which ${r(t) \equiv b_1 + b_2 {\rm cos} t {\rm and} b_1, b_2 \in \mathbb{R}}$ .     

Some properties of elliptic Riesz means at critical index on Hp(TH)

Suppose f∈H^p(Tⁿ), 0 _r ^δ , δ=n/p?(n+1)/2. In this paper we eastablish the following inequality $$\mathop {\sup }\limits_{R > 1} \left\{ {\frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta } \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqslant C_{R,p} \left\| f \right\|_{H^p (T^R )} $$ It implies that $$\mathop {\lim }\limits_{R \to \infty } \frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta - f} \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} = 0$$ Moreover we obtain the same conclusion when p=1 and n=1.     

A generalized Cole-Hopf transformation for a two-dimensional burgers equation with a variable coefficient

The direct method is applied to the two dimensional Burgers equation with a variable coefficient (u _t + uu _x ? u _xx ) _x + s(t)u _yy = 0 is transformed into the Riccati equation $H' - \tfrac{1} {2}H^2 + \left( {\tfrac{\rho } {2} - 1} \right)H = 0$ via the ansatz $u\left( {x,y,t} \right) = \tfrac{1} {{\sqrt t }}H(\rho ) + \tfrac{y} {{2\sqrt t }}\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y$ , provided that s(t) = t ^?3/2. Further, a generalized Cole-Hopf transformations $u\left( {x,y,t} \right) = \tfrac{y} {{2\sqrt t }} - \tfrac{2} {{\sqrt t }}\tfrac{{U_\rho (\rho ,r)}} {{U(\rho ,r)}}$ , $\rho \left( {x,y,t} \right) = \tfrac{x} {{\sqrt t }} - y$ , r(t) = log t is derived to linearize (u _t + uu _x ? u _xx ) _x + t ^?3/2 u _yy to the parabolic equation $U_r = U_{\rho \rho } + \left( {\tfrac{\rho } {2} - 1} \right)U_\rho$ .     

The maximal variation of a bounded martingale

Let \(\chi _0^n = \left\{ {X_t } \right\}_0^n \) be a martingale such that 0≦X_i≦1;i=0, …,n. For 0≦p≦1 denote by ? _p ⁿ the set of all such martingales satisfying alsoE(X₀)=p. Thevariation of a martingale χ ₀ ⁿ is denoted byV ₀ ⁿ and defined by \(V(\chi _0^n ) = E\left( {\sum {_{l = 0}^{n - 1} } \left| {X_{l + 1} - X_l } \right|} \right)\) . It is proved that $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\mathop {Sup}\limits_{x_0^n \in \mathcal{M}_p^n } \left[ {\frac{1}{{\sqrt n }}V(\chi _0^n )} \right]} \right\} = \phi (p)$$ , where ?(p) is the well known normal density evaluated at itsp-quantile, i.e. $$\phi (p) = \frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi _p^2 ) where \int_{ - \alpha }^{x_p } {\frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi ^2 )} dx = p$$ . A sequence of martingales χ ₀ ⁿ ,n=1,2, … is constructed so as to satisfy \(\lim _{n \to \infty } (1/\sqrt n )V(\chi _0^n ) = \phi (p)\) .     

Rational approximations of real numbers

For anyx ∈ r put $$c(x) = \overline {\mathop {\lim }\limits_{t \to \infty } } \mathop {\min }\limits_{(p,q\mathop {) \in Z}\limits_{q \leqslant t} \times N} t\left| {qx - p} \right|.$$ . Let [x₀; x₁,..., x_n, ...] be an expansion of x into a continued fraction and let \(M = \{ x \in J,\overline {\mathop {\lim }\limits_{n \to \infty } } x_n< \infty \}\) .Forx∈M put D(x)=c(x)/(1?c(x)). The structure of the set \(\mathfrak{D} = \{ D(x),x \in M\}\) is studied. It is shown that $$\mathfrak{D} \cap (3 + \sqrt 3 ,(5 + 3\sqrt 3 )/2) = \{ D(x^{(n,3} )\} _{n = 0}^\infty \nearrow (5 + 3\sqrt 3 )/2,$$ where \(x^{(n,3)} = [\overline {3;(1,2)_n ,1} ].\) This yields for \(\mu = \inf \{ z,\mathfrak{D} \supset (z, + \infty )\}\) (“origin of the ray”) the following lower bound: μ?(5+3√3)/2=5.0n>(5 + 3/3)/2=5.098.... Suppose a∈n. Put \(M(a) = \{ x \in M,\overline {\mathop {\lim }\limits_{n \to \infty } } x_n = a\}\) , \(\mathfrak{D}(a) = \{ D(x),x \in M(a)\}\) . The smallest limit point of \(\mathfrak{D}(a)(a \geqslant 2)\) is found. The structure of (a) is studied completely up to the smallest limit point and elucidated to the right of it.