On Bohr’s inequality

It is established that H. Bohr’s inequality \(\sum\nolimits_{k = 0}^\infty {\left| {{{f^{\left( k \right)} \left( 0 \right)} \mathord{\left/ {\vphantom {{f^{\left( k \right)} \left( 0 \right)} {\left( {2^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} k!} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} k!} \right)}}} \right| \leqslant \sqrt 2 \left\| f \right\|_\infty }\) is sharp on the class H _∞.     

Comparison of Exact Constants in Inequalities for Derivatives of Functions Defined on the Real Axis and a Circle

We investigate the relationship between the constants K(R) and K(T), where is the exact constant in the Kolmogorov inequality, R is the real axis, T is a unit circle,

Strong convergence in nonparametric estimation of regression functions

The nonparametric regression problem has the objective of estimating conditional expectation. Consider the model $$Y = R(X) + Z$$ , where the random variableZ has mean zero and is independent ofX. The regression functionR(x) is the conditional expectation ofY givenX = x. For an estimator of the form $$R_n (x) = \sum\limits_{i = 1}^n {Y_i K{{\left[ {{{\left( {x - X_i } \right)} \mathord{\left/ {\vphantom {{\left( {x - X_i } \right)} {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }}} \right]} \mathord{\left/ {\vphantom {{\left[ {{{\left( {x - X_i } \right)} \mathord{\left/ {\vphantom {{\left( {x - X_i } \right)} {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }}} \right]} {\sum\limits_{i = 1}^n {K\left[ {{{\left( {x - X_i } \right)} \mathord{\left/ {\vphantom {{\left( {x - X_i } \right)} {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }}} \right]} }}} \right. \kern-\nulldelimiterspace} {\sum\limits_{i = 1}^n {K\left[ {{{\left( {x - X_i } \right)} \mathord{\left/ {\vphantom {{\left( {x - X_i } \right)} {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }}} \right]} }}} $$ , we obtain the rate of strong uniform convergence $$\mathop {\sup }\limits_{x\varepsilon C} \left| {R_n (x) - R(x)} \right|\mathop {w \cdot p \cdot 1}\limits_ = o({{n^{{1 \mathord{\left/ {\vphantom {1 {(2 + d)}}} \right. \kern-\nulldelimiterspace} {(2 + d)}}} } \mathord{\left/ {\vphantom {{n^{{1 \mathord{\left/ {\vphantom {1 {(2 + d)}}} \right. \kern-\nulldelimiterspace} {(2 + d)}}} } {\beta _n \log n}}} \right. \kern-\nulldelimiterspace} {\beta _n \log n}}),\beta _n \to \infty $$ . HereX is ad-dimensional variable andC is a suitable subset ofR ^d .     

The prime-counting function and its analytic approximations

The paper describes a systematic computational study of the prime counting function π(x) and three of its analytic approximations: the logarithmic integral \({\text{li}}{\left( x \right)}: = {\int_0^x {\frac{{dt}}{{\log \,t}}} }\), \({\text{li}}{\left( x \right)} - \frac{1}{2}{\text{li}}{\left( {{\sqrt x }} \right)}\), and \(R{\left( x \right)}: = {\sum\nolimits_{k = 1}^\infty {{\mu {\left( k \right)}{\text{li}}{\left( {x^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} } \right)}} \mathord{\left/ {\vphantom {{\mu {\left( k \right)}{\text{li}}{\left( {x^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} } \right)}} k}} \right. \kern-\nulldelimiterspace} k} }\), where μ is the Möbius function. The results show that π(x)x) for 2≤x≤10¹⁴, and also seem to support several conjectures on the maximal and average errors of the three approximations, most importantly \({\left| {\pi {\left( x \right)} - {\text{li}}{\left( x \right)}} \right|} < x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}\) and \( - \frac{2}{5}x^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} < {\int_2^x {{\left( {\pi {\left( u \right)} - {\text{li}}{\left( u \right)}} \right)}du < 0} }\) for all x>2. The paper concludes with a short discussion of prospects for further computational progress.     

Su un problema di tipo nuovo relativo alle equazioni paraboliche d'ordine superiore al secondo

Sunto Si studia il problema della determinazione di una soluzione dell'equazione a_k(x)∂^ku/∂x^k=f(x, y) entro la semistriscia a≤x≤b, y≥0, che assuma assegnati valori per y=0 e per x=a, x₁, x₂, b (a<x₁<x₂<b). Analogamente si studia il problema della determinazione di una soluzione dell' equazione a_k(x)∂^ku/∂x^k+b(x)∂u/∂y=f(x,y), entro la medesima semistriscia, cha assuma assegnati valori per y=0 e per x=a, x₁, x₂, b e la cui ∂/∂y assuma assegnati valori per y=0. A Giovanni Sansone nel suo 70^mo compleanno.     

Eine Bemerkung über die Werte der Funktion σ (n)

By means of the Hoheisel—Montgomery prime number theorem it is shown that for every α≥1 the inequality $$|(\sigma (n)/n) - \alpha | \leqslant {1 \mathord{\left/ {\vphantom {1 {n^{({2 \mathord{\left/ {\vphantom {2 5}} \right. \kern-\nulldelimiterspace} 5}) - \varepsilon } }}} \right. \kern-\nulldelimiterspace} {n^{({2 \mathord{\left/ {\vphantom {2 5}} \right. \kern-\nulldelimiterspace} 5}) - \varepsilon } }}(\varepsilon > 0,\sigma (n) = \sum\limits_{d/n} d )$$ has infinitely many solutionsn∈N. It is highly probable that the exponent 2/5 can be replaced by 1.     

Some integral inequalities for the polar derivative of a polynomial

If P(z) is a polynomial of degree n which does not vanish in |z| 1,then it is recently proved by Rather [Jour.Ineq.Pure and Appl.Math.,9 (2008),Issue 4,Art.103] that for every γ 0 and every real or complex number α with |α|≥ 1,{∫02π |D α P(e iθ)| γ dθ}1/γ≤ n(|α| + 1)C γ{∫02π|P(eiθ)| γ dθ}1/γ,C γ ={1/2π∫0 2π|1+eiβ|γdβ}-1/γ,where D α P(z) denotes the polar derivative of P(z) with respect to α.In this paper we prove a result which not only provides a refinement of the above inequality but also gives a result of Aziz and Dawood [J.Approx.Theory,54 (1988),306-313] as a special case.     

Weighted monotonicity inequalities for traces on operator algebras

We study inequalities of the form $$ \tau (w(A)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} f(A)w(A)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ) \leqslant \tau (w(A)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} f(B)w(A)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ),A \leqslant B $$ where τ is a trace on a von Neumann algebra or a C*-algebra, A and B are self-adjoint elements of the algebra in question, f and w are real-valued functions, and the “weight” function w is nonnegative.     

Precise inequalities for norms of functions,third partial,second mixed,or directional derivatives

For functions f which are bounded throughout the plane R₂ together with the partial derivatives f(^3,0) f(_0,3), inequalities $$\left\| {f^{(1,1)} } \right\| \leqslant \sqrt[3]{3}\left\| f \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left\| {f^{(3,0)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left\| {f^{(0,3)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} ,\left\| {f_e^{(2)} } \right\| \leqslant \sqrt[3]{3}\left\| f \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left( {\left\| {f^{(3,0)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left| {e_1 } \right| + \left\| {f^{(0,3)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left| {e_2 } \right|} \right)^2 ,$$ are established, where ∥?∥denotes the upper bound on R² of the absolute values of the corresponding function, andf _fe ⁽²⁾ is the second derivative in the direction of the unit vector e=(e₁, e₂). Functions are exhibited for which these inequalities become equalities.     

Strong isochronism of a center and a focus for systems with homogeneous nonlinearities

We suggest a new approach to studying the isochronism of the system

${{dx} \mathord{\left/ {\vphantom {{dx} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = - y + p_n (x,y),{{dy} \mathord{\left/ {\vphantom {{dy} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = x + q_n (x,y),$

where p _n and q _n are homogeneous polynomials of degree n. This approach is based on the normal form

${{dX} \mathord{\left/ {\vphantom {{dX} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = - Y + XS(X,Y),{{dY} \mathord{\left/ {\vphantom {{dY} {dt}}} \right. \kern-\nulldelimiterspace} {dt}} = X + YS(X,Y)$

and its analog in polar coordinates. We prove a theorem on sufficient conditions for the strong isochronism of a center and a focus for the reduced system and obtain examples of centers with strong isochronism of degrees n = 4, 5. The present paper is the first to give examples of foci with strong isochronism for the system in question.

     

Multiple rational trigonometric sums and multiple integrals

We obtain an estimate of the modulus of a complete multiple rational trigonometric sum: $$\left| {\sum {_{x_{1, \ldots ,} x_r = 1^{\exp \left( {{{2\pi if\left( {x_{1, \ldots ,} x_r } \right)} \mathord{\left/ {\vphantom {{2\pi if\left( {x_{1, \ldots ,} x_r } \right)} q}} \right. \kern-\nulldelimiterspace} q}} \right)} }^q } } \right| \ll q^{{{r - 1} \mathord{\left/ {\vphantom {{r - 1} {n + \varepsilon }}} \right. \kern-\nulldelimiterspace} {n + \varepsilon }}} ,$$ where $$\begin{gathered} f\left( {x_{1, \ldots ,} x_r } \right) = \sum {_{0 \leqslant t_1 , \ldots ,t_r \leqslant n^a t_1 , \ldots ,t_r x_1^{t_1 } \ldots x_r^{t_r } ,} } \hfill \\ a_{0, \ldots ,0} = 0,\left( {a_{0, \ldots ,0,1} , \ldots ,a_{n, \ldots ,n,} q} \right) = 1 \hfill \\ \end{gathered} $$ , and an estimate of the modulus of a multiple trigonometric integral.     

On relatively aspherical presentations and their central extensions

Under the condition of asphericity of a quotient group , mutual commutants of the form in hyperbolic groups G are investigated together with the structure of central subgroups in central extensions of . In particular, quotients of the form G/[g ^m , G] are considered, where g is an element of infinite order from a hyperbolic group G and m is sufficiently large (depending on g). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 115–125, 2005.     

XXZ Spin Chain with the Asymmetry Parameter Δ = −1/2: Evaluation of the Simplest Correlators

We consider a finite XXZ spin chain with periodic boundary conditions and an odd number of sites. It appears that for the special value of the asymmetry parameter = –1/2, the ground state of this system described by the Hamiltonian has the energy E ₀ = –3N/2. Although the ground state is antiferromagnetic, we can find the corresponding solution of the Bethe equations. Specifically, we can explicitly construct a trigonometric polynomial Q(u) of degree n = (N–1)/2, whose zeros are the parameters of the Bethe wave function for the ground state of the system. As is known, this polynomial satisfies the Baxter T–Q equation. This equation also has a second independent solution corresponding to the same eigenvalue of the transfer matrix T. We use this solution to find the derivative of the ground-state energy of the XXZ chain with respect to the crossing parameter . This derivative is directly related to one of the spin–spin correlators, which appears to be . In turn, this correlator gives the average number of spin strings for the ground state of the chain, . All these simple formulas fail if the number N of chain sites is even.     

Asymptotic formulas for the hyperġeometric function2 F 1 of matrix argument,useful in multivariate analysis

Summary LetS _i have the Wishart distributionW _p(∑_i,n_i) fori=1,2. An asymptotic expansion of the distribution of for large n=n₁+n₂ is derived, when∑ ₁∑ ₂ ⁻¹ =I+n^−1/2θ, based on an asymptotic solution of the system of partial differential equations for the hypergeometric function₂ F ₁, obtained recently by Muirhead [2]. Another asymptotic formula is also applied to the distributions of −2 log λ and −log|S ₂(S ₁+S ₂)⁻¹| under fixed∑ ₁∑ ₂ ⁻¹ , which gives the earlier results by Nagao [4]. Some useful asymptotic formulas for₁ F ₁ were investigated by Sugiura [7].     

Remarks on Mean Convergence (Boundedness) of Partial Sums of Trigonometric Series

Fairly general conditions on the coefficients of even and odd trigonometric Fourier series under which L-convergence (boundedness) of partial sums of the series is equivalent to the relation are given.     

Estimate of a multiple sum involving the legendre symbol

Supposef(x₁,..., x_n) is a polynomial of even degree d having coefficients in the finite field k=[q] and satisfying certain natural conditions, and let χ be the quadratic character of k. Then $$\left| {\sum {x_1 , \ldots ,} x_n \in k\chi (f(x_1 , \ldots ,x_n ))} \right| \leqslant Cq^{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} $$ where the constant C depends only on d and n.     

On a conjecture of Zaremba

LetN _C (x) be the number of integersm≤x such that there is an integera with 1≤a<m, (a, m)=1 and all partial quotients in the continued fraction expansion ofa/m are at mostC. We prove for allx≥1 that $$N_c (x) > {1 \mathord{\left/ {\vphantom {1 {\sqrt {2C} x^{{1 \mathord{\left/ {\vphantom {1 {2(1 - 1/C^2 )}}} \right. \kern-\nulldelimiterspace} {2(1 - 1/C^2 )}}} }}} \right. \kern-\nulldelimiterspace} {\sqrt {2C} x^{{1 \mathord{\left/ {\vphantom {1 {2(1 - 1/C^2 )}}} \right. \kern-\nulldelimiterspace} {2(1 - 1/C^2 )}}} }}$$ .     

Hiding Cliques for Cryptographic Security

We demonstrate how a well studied combinatorial optimizationproblem may be used as a new cryptographic primitive. The problemin question is that of finding a "large" clique in a randomgraph. While the largest clique in a random graph with nvertices and edge probability p is very likely tobe of size about , it is widely conjecturedthat no polynomial-time algorithm exists which finds a cliqueof size with significantprobability for any constant > 0. We presenta very simple method of exploiting this conjecture by hidinglarge cliques in random graphs. In particular, we show that ifthe conjecture is true, then when a large clique—of size,say, is randomlyinserted (hidden) in a random graph, finding a clique ofsize remains hard.Our analysis also covers the case of high edge probabilitieswhich allows us to insert cliques of size up to . Our result suggests several cryptographicapplications, such as a simple one-way function.     

Optimal recovery methods for the integral on classes of differentiable functions

Найдены методы восст ановления интеграла по информации $$I\left( f \right) = \left\{ {f^{(j)} \left( {x_i } \right)\left( {j = 0, ..., \gamma _i - 1; i = 1, ..., n; 1 \leqq \gamma _i \leqq r; \gamma _i + ... + \gamma _n \leqq N} \right.} \right\},$$ оптимальные на класс ахW _p ^r ,r=1,2,...; 1≦p≦∞. Это позволило, в частност и, получить наилучшие для классаW _p ^r квадратурные форму лы вида $$\mathop \smallint \limits_0^1 f\left( x \right)dx = \mathop \Sigma \limits_{i = 1}^n \mathop \Sigma \limits_{j = 1}^{\gamma _i - 1} a_{ij} f^{(j)} \left( {x_i } \right) + \mathop \Sigma \limits_{j = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} b_j f^{(2j - 1)} \left( 0 \right) + \mathop \Sigma \limits_{k = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} c_k f^{(2k - 1)} \left( 1 \right) + R\left( f \right)$$ И $$\mathop \smallint \limits_0^1 f\left( x \right)dx = af\left( 0 \right) + \mathop \Sigma \limits_{i = 1}^n \mathop \Sigma \limits_{j = 0}^{\gamma _i - 1} a_{ij} f^{(j)} \left( {x_i } \right) + bf\left( 1 \right) + \mathop \Sigma \limits_{j = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} b_j f^{(2j - 1)} \left( 0 \right) + \mathop \Sigma \limits_{k = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} c_k f^{(2k - 1)} \left( 1 \right) + R\left( f \right).$$      

A sharp form of an embedding into multiple exponential spaces

Let Θ be a bounded open set in ℝ ⁿ , n ⩾ 2. In a well-known paper Indiana Univ. Math. J., 20, 1077–1092 (1971) Moser found the smallest value of K such that