A conjecture of Segre on diophantine approximation

In 1945,B. Segre proved the following classical theorem: Every irrational ξ has an infinity of rational approximationsp/q such that (0) $$\frac{{ - 1}}{{q^2 \sqrt {1 + 4\tau } }}< \frac{p}{q} - \xi< \frac{\tau }{{q^2 \sqrt {1 + 4\tau } }},$$ where τ is any given non-negative real number. Segre conjectured that when τ≠0 and τ^?1 is not an integer, inequalities (0) can be improved by replacing \(\sqrt {1 + 4\tau } \) and \(\sqrt {1 + 4\tau } /\tau \) with larger numbers. In this paper we prove that these two numbers can be replaced with the larger numbers \(\sqrt {1 + 4\tau } + 0.2\tau ^2 \{ \tau ^{ - 1} \} (1 - \{ \tau ^{ - 1} \} )\) and \(\sqrt {1 + 4\tau } /\tau + 0.2\tau ^2 \{ \tau ^{ - 1} \} (1 - \{ \tau ^{ - 1} \} )\) respectively, where {τ^?1} is the fractional part of τ^?1.     

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 \(\lambda _{1,2} = {{\left( {4 \pm 3\sqrt 2 } \right)} \mathord{\left/ {\vphantom {{\left( {4 \pm 3\sqrt 2 } \right)} 2}} \right. \kern-0em} 2}\) .     

An interstice relationship for flowers with four petals

Given three mutually tangent circles with bends (related to the reciprocal of the radius) a, b and c respectively, an important quantity associated with the triple is the value ${\langle a,b,c \rangle:=ab+ac+bc}$ . In this note we show in the case when a central circle with bend b ₀ is “surrounded” by four circles, i.e., a flower with four petals, with bends b ₁, b ₂, b ₃,b ₄ that either $$\sqrt{\langle b_{0},b_{1},b_{2} \rangle}+\sqrt{\langle b_{0},b_{3},b_{4} \rangle}=\sqrt{\langle b_{0},b_{2},b_{3} \rangle}+\sqrt{\langle b_{0},b_{4},b_{1} \rangle}$$ or $$\sqrt{\langle b_{0},b_{1},b_{2} \rangle}=\sqrt{\langle b_{0},b_{2},b_{3} \rangle}+\sqrt{\langle b_{0},b_{3},b_{4} \rangle}+\sqrt{\langle b_{0},b_{4},b_{1} \rangle}$$ (where ${\langle b_{0},b_{1},b_{2} \rangle}$ is chosen to be maximal). As an application we give a sufficient condition for the alternating sum of the ${\sqrt{\langle a,b,c\rangle}}$ of a packing in standard position to be 0. (A packing is in standard position when we have two circles with bend 0, i.e., parallel lines, and the remaining circles are packed in between.)     

The Generalization of the Decomposition of Functions by Energy Operators

This work starts with the introduction of a family of differential energy operators. Energy operators $({\varPsi}_{R}^{+}, {\varPsi}_{R}^{-})$ were defined together with a method to decompose the wave equation in a previous work. Here the energy operators are defined following the order of their derivatives $(\varPsi^{-}_{k}, \varPsi^{+}_{k}, k=\{0,\pm 1,\pm 2,\ldots\})$ . The main part of the work demonstrates for any smooth real-valued function f in the Schwartz space $(\mathbf{S}^{-}(\mathbb{R}))$ , the successive derivatives of the n-th power of f ( $n \in \mathbb{Z}$ and n≠0) can be decomposed using only $\varPsi^{+}_{k}$ (Lemma); or if f in a subset of $\mathbf{S}^{-}(\mathbb{R})$ , called $\mathbf{s}^{-}(\mathbb{R})$ , $\varPsi^{+}_{k}$ and $\varPsi^{-}_{k}$ ( $k\in \mathbb{Z}$ ) decompose in a unique way the successive derivatives of the n-th power of f (Theorem). Some properties of the Kernel and the Image of the energy operators are given along with the development. Finally, the paper ends with the application to the energy function.     

Asymptotic properties for the loglog laws under positive association

Let {X _n : n ?? 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $S_n = \sum\limits_{k = 1}^n {X_k }$ , $Mn = \mathop {\max }\limits_{k \leqslant n} \left| {S_k } \right|$ , n ?? 1. Suppose that $0 < \sigma ^2 = EX_1^2 + 2\sum\limits_{k = 2}^\infty {EX_1 X_k < \infty }$ . In this paper, we prove that if E|X ₁|^2+?? < for some ?? ?? (0, 1], and $\sum\limits_{j = n + 1}^\infty {Cov\left( {X_1 ,X_j } \right) = O\left( {n^{ - \alpha } } \right)}$ for some ?? > 1, then for any b > ?1/2 $$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2b + 1} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^{b - 1/2} }} {{n^{3/2} \log n}}} E\left\{ {M_n - \sigma \varepsilon \sqrt {2n\log \log n} } \right\}_ + = \frac{{2^{ - 1/2 - b} E\left| N \right|^{2(b + 1)} }} {{(b + 1)(2b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2(b + 1)} }}}$$ and $$\mathop {\lim }\limits_{\varepsilon \nearrow \infty } \varepsilon ^{ - 2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^b }} {{n^{3/2} \log n}}E\left\{ {\sigma \varepsilon \sqrt {\frac{{\pi ^2 n}} {{8\log \log n}}} - M_n } \right\}} _ + = \frac{{\Gamma (b + 1/2)}} {{\sqrt 2 (b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2b + 2} }}} ,$$ where x ₊ = max{x, 0}, N is a standard normal random variable, and ??(·) is a Gamma function.     

Regularized sums of half-integer powers of a Sturm-Liouville operator

We investigate the question of the regularized sums of part of the eigenvalues z_n (lying along a direction) of a Sturm-Liouville operator. The first regularized sum is $$\sum\nolimits_{n = 1}^\infty {(z_n - n - \frac{{c_1 }}{n} + \frac{2}{\pi } \cdot z_n arctg \frac{1}{{z_n }} - \frac{2}{\pi }) = \frac{{B_2 }}{2} - c_1 \cdot \gamma + \int_1^\infty {\left[ {R(z) - \frac{{l_0 }}{{\sqrt z }} - \frac{{l_1 }}{z} - \frac{{l_2 }}{{z\sqrt z }}} \right]} } \sqrt z dz,$$ where the z_n are eigenvalues lying along the positive semi-axis, z _n ² =λ_n, $$l_0 = \frac{\pi }{2}, l_1 = - \frac{1}{2}, l_2 = - \frac{1}{4}\int_0^\pi {q(x) dx,} c_1 = - \frac{2}{\pi }l_2 ,$$ , B₂ is a Bernoulli number, γ is Euler's constant, and \(R(z)\) is the trace of the resolvent of a Sturm-Liouville operator.     

Completeness of the Grid Translates of Functions

If $f\in L^{p}(\mathbb{R}^{d})$ is a bounded real valued continuous function which has a unique maximum or a unique minimum at a point $x_{0}\in \mathbb{R}^{d}$ and if the inverse image of the neighborhoods of f(x ₀) shrinks regularly to x ₀, then $\mathrm{ span }\{f^{m}(x-2^{-m}\varSigma_{i=1}^{d} j_{i} e_{i})\mid m\in\mathbb{N}, j_{i}\in\mathbb{Z}\}$ is a dense subset of $L^{p}(\mathbb{R}^{d}), 1\le p<\infty$ where f ^m (x)=f(x) ^m and {e _i } is the natural basis of $\mathbb{R}^{d}$ . The result extends to all homogeneous groups, Riemannian symmetric spaces of noncompact type, Damek-Ricci spaces etc.     

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 \(\lim _{n \to \infty } (1/\sqrt n )V(\chi _0^n ) = \phi (p)\) .     

Curvature estimates for submanifolds with prescribed Gauss image and mean curvature

We study that the n-graph defined by a smooth map ${f:\Omega\subset\mathbb R^{n}\to \mathbb R^{m}, m\ge 2,}$ in ${\mathbb R^{m+n}}$ of the prescribed mean curvature and the Gauss image. Under the condition $$\Delta_f=\left[\text{det}\left(\delta_{ij}+\sum_\alpha{\frac {\partial {f^\alpha}}{\partial {x^i}}}{\frac {\partial {f^\alpha}}{\partial {x^j}}}\right)\right]^{\frac{1}{2}} < 2,$$ we derive the interior curvature estimates $$\sup_{D_R(x)}|B|^2\le{\frac{C}{R^2}}$$ when 2 ≤ n ≤ 5 with constant C depending on the given geometric data. If there is no dimension limitation we obtain $$\sup_{D_R(x)}|B|^2\le CR^{-a}\sup_{D_{2R}(x)}(2-\Delta_f)^{-\left({\frac{3}{2}}+{\frac{1}{s}}\right)},\quad s=\min(m, n)$$ with a < 1. If the image under the Gauss map is contained in a geodesic ball of the radius ${{\frac{\sqrt{2}}{4}}\pi}$ in G _n,m we also derive corresponding estimates.     

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 $$\left| {\omega (n,k)} \right| < \frac{4}{{(2k - 1)\sqrt {ln(2k - 1)} }} + \sqrt {12\pi } \cdot \frac{{k^{\tfrac{3}{2}} }}{{n - 1}}\left( {1 + \frac{1}{{n - 1}}} \right)^{2k - 2} .$$ .     

Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix

Let {X _k,i ; i ≥ 1, k ≥ 1} be a double array of nondegenerate i.i.d. random variables and let {p _n ; n ≥ 1} be a sequence of positive integers such that n/p _n is bounded away from 0 and ∞. In this paper we give the necessary and sufficient conditions for the asymptotic distribution of the largest entry ${L_{n}={\rm max}_{1\leq i < j\leq p_{n}}|\hat{\rho}^{(n)}_{i,j}|}$ of the sample correlation matrix ${{\bf {\Gamma}}_{n}=(\hat{\rho}_{i,j}^{(n)})_{1\leq i,j\leq p_{n}}}$ where ${\hat{\rho}^{(n)}_{i,j}}$ denotes the Pearson correlation coefficient between (X _1,i , ..., X _n,i )′ and (X _1,j ,...,X _n,j )′. Write ${F(x)= \mathbb{P}(|X_{1,1}|\leq x), x\geq0}$ , ${W_{c,n}={\rm max}_{1\leq i < j\leq p_{n}}|\sum_{k=1}^{n}(X_{k,i}-c)(X_{k,j}-c)|}$ , and ${W_{n}=W_{0,n},n\geq1,c\in(-\infty,\infty)}$ . Under the assumption that ${\mathbb{E}|X_{1,1}|^{2+\delta} < \infty}$ for some δ > 0, we show that the following six statements are equivalent: $$ {\bf (i)} \quad \lim_{n \to \infty} n^{2}\int\limits_{(n \log n)^{1/4}}^{\infty}\left( F^{n-1}(x) - F^{n-1}\left(\frac{\sqrt{n \log n}}{x}\right) \right) dF(x) = 0,$$ $$ {\bf (ii)}\quad n \mathbb{P}\left ( \max_{1 \leq i < j \leq n}|X_{1,i}X_{1,j} | \geq \sqrt{n \log n}\right ) \to 0 \quad{\rm as}\,n \to \infty,$$ $$ {\bf (iii)}\quad \frac{W_{\mu, n}}{\sqrt {n \log n}}\stackrel{\mathbb{P}}{\rightarrow} 2\sigma^{2},$$ $$ {\bf (iv)}\quad \left ( \frac{n}{\log n}\right )^{1/2} L_{n} \stackrel{\mathbb{P}}{\rightarrow} 2,$$ $$ {\bf (v)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (\frac{W_{\mu, n}^{2}}{n \sigma^{4}} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8\pi}} e^{-t/2}\right \}, - \infty < t < \infty,$$ $$ {\bf (vi)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (n L_{n}^{2} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8 \pi}} e^{-t/2}\right \}, - \infty < t < \infty$$ where ${\mu=\mathbb{E}X_{1,1}, \sigma^{2}=\mathbb{E}(X_{1,1} - \mu)^{2}}$ , and a _n  = 4 log p _n ? log log p _n . The equivalences between (i), (ii), (iii), and (v) assume that only ${\mathbb{E}X_{1,1}^{2} < \infty}$ . Weak laws of large numbers for W _n and L _n , n ≥  1, are also established and these are of the form ${W_{n}/n^{\alpha}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(\alpha > 1/2)$ and ${n^{1-\alpha}L_{n}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(1/2 < \alpha \leq 1)$ , respectively. The current work thus provides weak limit analogues of the strong limit theorems of Li and Rosalsky as well as a necessary and sufficient condition for the asymptotic distribution of L _n obtained by Jiang. Some open problems are also posed.     

An inverse problem of the three-dimensional wave equation for a general annular vibrating membrane with piecewise smooth boundary conditions

This paper deals with the very interesting problem about the influence of piecewise smooth boundary conditions on the distribution of the eigenvalues of the negative Laplacian inR ³. The asymptotic expansion of the trace of the wave operator $\widehat\mu (t) = \sum\limits_{\upsilon = 1}^\infty {\exp \left( { - it\mu _\upsilon ^{1/2} } \right)} $ for small ?t? and $i = \sqrt { - 1} $ , where $\{ \mu _\nu \} _{\nu = 1}^\infty $ are the eigenvalues of the negative Laplacian $ - \nabla ^2 = - \sum\limits_{k = 1}^3 {\left( {\frac{\partial }{{\partial x^k }}} \right)} ^2 $ in the (x ¹,x ²,x ³), is studied for an annular vibrating membrane Ω inR ³ together with its smooth inner boundary surfaceS ₁ and its smooth outer boundary surfaceS ₂. In the present paper, a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth componentsS ^* _i(i=1, …,m) ofS ₁ and on the piecewise smooth componentsS ^* _i(i=m+1, …,n) ofS ₂ such that $S_1 = \bigcup\limits_{i = 1}^m {S_i^* } $ and $S_2 = \bigcup\limits_{i = m + 1}^n {S_i^* } $ are considered. The basic problem is to extract information on the geometry of the annular vibrating membrane ω from complete knowledge of its eigenvalues by analyzing the asymptotic expansions of the spectral function $\widehat\mu (t)$ for small ?t?.     

First moment of Rankin–Selberg central L-values and subconvexity in the level aspect

Let 1≤N<M with N and M coprime and square-free. Through classical analytic methods we estimate the first moment of central L-values $L(\frac{1}{2},f\times g)$ where $f\in\mathcal{S}^{*}_{k}(N)$ runs over primitive holomorphic forms of level N and trivial nebentypus and g is a given form of level M. As a result, we recover the bound $L(\frac{1}{2},f\times g) \ll_{\varepsilon}(N + \sqrt{M}) N^{\varepsilon}M^{\varepsilon}$ when g is dihedral. The first moment method also applies to the special derivative $L'(\frac{1}{2},f\times g)$ under the assumption that it is non-negative for all $f\in\mathcal{S}^{*}_{k}(N)$ .     

Strong approximation by rectangular partial sums of double orthogonal series

Пусть {? _ik(x):i, k=1, 2,...} — орто нормированная систе ма в пространстве с полож ительной мерой и {a _ik} — последов ательность действит ельных чисел, для которой $$\mathop \sum \limits_{\iota = 1}^\infty \mathop \sum \limits_{\kappa = 1}^\infty a_{ik}^2 \kappa ^2 (i,k)< \infty ,$$ где {x(i, K)} — определенна я неубывающая последовательность положительных чисел. Тогда суммаf(x) двойног о ортогонального ряд а \(\mathop \sum \limits_{\iota = 1}^\infty \mathop \sum \limits_{\kappa = 1}^\infty a_{ik} \varphi _{ik} (x)\) существует в смысле с ходимости в метрикеL ² и сходимос ти почти всюду. Изучае тся порядок так называем ой сильной аппроксимац ииf(x) (при коэффициентн ых условиях) прямоуголь ными частными суммами \(s_{mn} (x) = \mathop \sum \limits_{\iota = 1}^\infty \mathop \sum \limits_{\kappa = 1}^\infty a_{ik} \varphi _{ik} (x)\) . Основной ре зультат состоит в сле дующем. Если {λ_j(m):m=1, 2,...} — неубывающи е последовательност и положительньк чисел, стремящиеся к ∞ и такие, что \(\mathop {\lim \sup }\limits_{m \to \infty } \lambda _j (2m)/\lambda _j (m)< \sqrt 2 \) дляj=1,2, и если $$\mathop \sum \limits_{\iota = 1}^\infty \mathop \sum \limits_{\kappa = 1}^\infty a_{ik}^2 \left[ {\log log (i + 3)} \right]^2 \left[ {\log log (k + 3)} \right]^2 (\lambda _1^2 (i) + \lambda _2^2 (k))< \infty ,$$ TO ПОЧТИ ВСЮДУ $$\left\{ {\frac{1}{{mn}}\mathop \sum \limits_{i = 1}^m \mathop \sum \limits_{\kappa = 1}^m \left[ {s_{ik} (x) - f(x)} \right]^2 } \right\}^{1/2} = o_x (\lambda _1^{ - 1} (m) + \lambda _2^{ - 1} (n))$$ при min (m, n) → ∞.     

A priori estimates for Donaldson’s equation over compact Hermitian manifolds

In this paper we prove a priori estimates for Donaldson equation’s $$\begin{aligned} \omega \wedge (\chi +\sqrt{-1}\partial \bar{\partial }\varphi )^{n-1} =e^{F}(\chi +\sqrt{-1}\partial \bar{\partial }\varphi )^{n}, \end{aligned}$$ over a compact complex manifold \(X\) of complex dimension \(n\) , where \(\omega \) and \(\chi \) are arbitrary Hermitian metrics. Our estimates answer a question of Tosatti-Weinkove (Asian J. Math. 14:19–40, 2010).     

Saddle solutions for bistable symmetric semilinear elliptic equations

This paper concerns the existence and asymptotic characterization of saddle solutions in ${\mathbb {R}^{3}}$ for semilinear elliptic equations of the form $$-\Delta u + W'(u) = 0,\quad (x, y, z) \in {\mathbb {R}^{3}} \qquad\qquad\qquad (0.1)$$ where ${W \in \mathcal{C}^{3}(\mathbb {R})}$ is a double well symmetric potential, i.e. it satisfies W(?s) =  W(s) for ${s \in \mathbb {R},W(s) > 0}$ for ${s \in (-1,1)}$ , ${W(\pm 1) = 0}$ and ${W''(\pm 1) > 0}$ . Denoted with ${\theta_{2}}$ the saddle planar solution of (0.1), we show the existence of a unique solution ${\theta_{3} \in {\mathcal{C}^{2}}(\mathbb {R}^{3})}$ which is odd with respect to each variable, symmetric with respect to the diagonal planes, verifies ${0 < \theta_{3}(x,y,z) < 1}$ for x, y, z >  0 and ${\theta_{3}(x, y, z) \to_{z \to + \infty} \theta_{2}(x, y)}$ uniformly with respect to ${(x, y) \in \mathbb {R}^{2}}$ .     

Phase transitions of iterated Higman-style well-partial-orderings

We elaborate Weiermann-style phase transitions for well-partial-orderings (wpo) determined by iterated finite sequences under Higman-Friedman style embedding with Gordeev’s symmetric gap condition. For every d-times iterated wpo ${\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}\right)}$ in question, d >? 1, we fix a natural extension of Peano Arithmetic, ${T \supseteq \sf{PA}}$ , that proves the corresponding second-order sentence ${\sf{WPO}\left({\rm S}{\textsc{eq}}^{d}, \trianglelefteq _{d}\right) }$ . Having this we consider the following parametrized first-order slow well-partial-ordering sentence ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}, r\right):}$ $$\left( \forall K > 0 \right) \left( \exists M > 0\right) \left( \forall x_{0},\ldots ,x_{M}\in {\rm S}\text{\textsc{eq}}^{d}\right)$$ $$\left( \left( \forall i\leq M\right) \left( \left| x_{i}\right| < K + r \left\lceil \log _{d} \left( i+1\right) \right\rceil \right)\rightarrow \left( \exists i < j \leq M \right) \left(x_{i} \trianglelefteq _{d} x_{j}\right) \right)$$ for a natural additive Seq ^d -norm |·| and r ranging over EFA-provably computable positive reals, where EFA is an abbreviation for IΔ ₀?+?exp. We show that the following basic phase transition clauses hold with respect to ${T = \Pi_{1}^{0}\sf{CA}_{ < \varphi ^{_{\left( d-1\right) }} \left(0\right) }}$ and the threshold point1.

1. If r <? 1 then ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d},r \right) }$ is provable in T.

1. If ${r > 1}$ then ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d},r \right) }$ is not provable in T.

Moreover, by the well-known proof theoretic equivalences we can just as well replace T by PA or ACA ₀ and ${\Delta _{1}^{1}\sf{CA}}$ , if d =? 2 and d =? 3, respectively.In the limit case d → ∞ we replaceEFA-provably computable reals r by EFA-provably computable functions ${f: \mathbb{N} \rightarrow \mathbb{R}_{+}}$ and prove analogous theorems. (In the sequel we denote by ${\mathbb{R}_{+}}$ the set of EFA-provably computable positive reals). In the basic case T?=? PA we strengthen the basic phase transition result by adding the following static threshold clause

1. ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1\right)}$ is still provable in T = PA (actually in EFA).

Furthermore we prove the following dynamic threshold clauses which, loosely speaking are obtained by replacing the static threshold t by slowly growing functions 1 _α given by ${1_{\alpha }\left( i\right)\,{:=}\,1+\frac{1}{H_{\alpha }^{-1}\left(i\right) }, H_{\alpha}}$ being the familiar fast growing Hardy function and ${H_{\alpha }^{-1}\left( i\right)\,{:=}\,\rm min \left\{ j \mid H_{\alpha } \left ( j\right) \geq i \right\}}$ the corresponding slowly growing inversion.

1. If ${\alpha < \varepsilon _{0}}$ , then ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1_{\alpha}\right)}$ is provable in T = PA.

1. ${\sf{SWP}\left( {\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2},1_{\varepsilon _{0}}\right)}$ is not provable in T = PA.

We conjecture that this pattern is characteristic for all ${T\supseteq \sf{PA}}$ under consideration and their proof-theoretical ordinals o (T ), instead of ${\varepsilon _{0}}$ .     

Sections of the Taylor expansion of exp(g(z)) whereg(z) is entire and admissible in the sense of Hayman

The author considers $$f(z) = \exp (g(z)) = \sum\limits_{j = 0}^\infty {a_j z^j ,}$$ , whereg(z) is a real, entire, transcendental function admissible in the sense of W. K. Hayman [(1956): Reine Angew. Math.,196:67-95]. The aim of the paper is to study, asm→+∞, the distribution of the zeros of the partial sums $$s_m (z) = \sum\limits_{j = 0}^m {a_j z^j .}$$ The results are stated in terms of Hayman's auxiliary functions Ifr>0 is large enough, botha(r) andb(r) are positive,a(r) is strictly increasing, and $$a(r) \to + \infty ,b(r) \to + \infty (r \to + \infty ).$$ Define the sequence (R _m ) (m>m ₀) by the relationsa(R _m )=m. From the following proposition, typical of those stated in the paper, it is easy to deduce accurate information regarding those zeros ofs _m (z) that lie near the positive axis: Letζ be an auxiliary complex variable; then asm→+∞, and forR=R _m , the functions $$\left\{ {1 + \zeta \left( {\frac{2}{{b(R)}}} \right)^{1/2} } \right\}^{ - m} \{ f(R)\} ^{ - 1} s_m \left( {R\left( {1 + \zeta \left( {\frac{2}{{b(R)}}} \right)^{1/2} } \right)} \right)$$ tend to $$\frac{1}{2}e^{\zeta ^2 } \left( {1 - \frac{2}{{\sqrt \pi }}\int_0^\zeta {e^{ - \sigma ^2 } d\sigma } } \right)$$ uniformly on every compact subset of theζ-plane. There are similar, equally precise, results covering those zeros ofs _m (z) that lie near any rayte ^i?(0<t<+∞,?≠0).     

The Concentration Function of Additive Functions with Special Weight

Suppose that $${g\left( n \right)}$$ is an additive real-valued function, W(N) = 4+ $$\mathop {\min }\limits_\lambda $$ ( λ₂ + $$\sum\limits_{p < N} {\frac{1}{2}} $$ min (1, ( g(p) - λlog p)²), E(N) = 4+1 $$\sum\limits_{\mathop {p < N,}\limits_{g(p) \ne 0} } {\frac{1}{p}.} $$ In this paper, we prove the existence of constants C₁, C₂ such that the following inequalities hold: $\mathop {\sup }\limits_a \geqslant \left| {\left\{ {n, m, k: m, k \in \mathbb{Z},n \in \mathbb{N},n + m^2 + k^2 } \right.} \right. = \left. {\left. {N,{\text{ }}g(n) \in [a,a + 1)} \right\}} \right| \leqslant \frac{{C_1 N}}{{\sqrt {W\left( N \right)} }},$ $\mathop {\sup }\limits_a \geqslant \left| {\left\{ {n, m, k: m, k \in \mathbb{Z},n \in \mathbb{N},n + m^2 + k^2 } \right.} \right. = \left. {\left. {N,{\text{ }}g(n) = a} \right\}} \right| \leqslant \frac{{C_2 N}}{{\sqrt {E\left( N \right)} }},$ . The obtained estimates are order-sharp.     

Some new estimates of the Fourier transform in \mathbb{L}_2 (ℝ)

Given a function $\mathbb{L}_2 $ (?), its Fourier transform $g(x) = \hat f(x) = F[f](x) = \frac{1} {{\sqrt {2\pi } }}\int\limits_{ - \infty }^{ + \infty } {f(x)e^{ - ixt} dt} ,f(t) = F^{ - 1} [g](t) = \frac{1} {{\sqrt {2\pi } }}\int\limits_{ - \infty }^{ + \infty } {g(x)e^{ - ixt} dx} $ and the inverse Fourier transform are considered in the space f ε $\mathbb{L}_2 $ (?). New estimates are presented for the integral $\int\limits_{|t| \geqslant N} {|g(t)|^2 dt} = \int\limits_{|t| \geqslant N} {|\hat f(t)|^2 dt} ,N \geqslant 1,$ in the vase of f ε $\mathbb{L}_2 $ (?) characterized by the generalized modulus of continuity of the kth order constructed with the help of the Steklov function. Some other estimates associated with this integral are proved.