A note on the diophantine equation x2 + 1 = dy4

In this paper we prove that the Diophantine equation as in the title has at most one integer solution if $ \in > 5 \times 10^7 $ where $ \in = u + \upsilon \sqrt d $ is the least positive solution of Pell’s equation $x^2 - dy^2 = - 1$      

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.     

Optimal Set of the Modulus of Continuity in the Sharp Jackson Inequality in the Space $$L_2 $$

To a function $f \in L_2 [ - \pi ,\pi ]$ and a compact set $Q \subset [ - \pi ,\pi ]$ we assign the supremum $\omega (f,Q) = \sup _{t \in Q} ||f( \cdot + t) - f( \cdot )||_{L_2 [ - \pi ,\pi ]} $ , which is an analog of the modulus of continuity. We denote by $K(n,Q)$ the least constant in Jackson's inequality between the best approximation of the function f by trigonometric polynomials of degree $n - 1$ in the space $L_2 [ - \pi ,\pi ]$ and the modulus of continuity $\omega (f,Q)$ . It follows from results due to Chernykh that $K(n,Q) \geqslant 1/\sqrt 2 $ and $K(n,[0,\pi /\pi ]) = 1/\sqrt 2 $ . On the strength of a result of Yudin, we show that if the measure of the set Q is less than $\pi /n$ , then $K(n,Q) >1/\sqrt 2 $ .     

Estimates of the levy constant for\sqrt p and class number one criterion for ℚ(\sqrt p )

Let p ≡ = 3 (mod 4) be a prime, let ?( $\sqrt p $ ) be the length of the period of the expansion of $\sqrt p $ into a continued fraction, and let h(4p) be the class number of the field ?( $\sqrt p $ ). Our main result is as follows. For p > 91, h(4p)=1 if and only if ?( $\sqrt p $ ) > 0.56 $\sqrt p $ L_4p(1), where L_4p(1) is the corresponding Dirichlet series. The proof is based on studying linear relations between convergents of the expansion of $\sqrt p $ into a continued fraction. Bibliography: 13 titles.     

A Criterion for Weak Generalized Localization in the Class L_1 for Multiple Trigonometric Series from the Viewpoint of Isometric Transformations

In this paper, we study the problem of the variation (if any) of the sets of convergence and divergence everywhere or almost everywhere of a multiple Fourier series (integral) of a function $f \in L_p $ , $p \geqslant 1$ , $f(x) = 0$ , on a set of positive measure $\mathfrak{A} \subset \mathbb{T}^N = [ - \pi ,\pi )^N $ , $N \geqslant 2$ , depending on the rotation of the coordinate system, i.e., depending on the element $\tau \in \mathcal{F}$ , where $\mathcal{F}$ is the rotation group about the origin in $\mathbb{R}^N $ . This problem has been reduced to the study of the change in the geometry of the sets $\tau ^{ - 1} (\mathfrak{A}) \cap \mathbb{T}^N $ (where $\tau ^{ - 1} \in \mathcal{F}$ satisfies $\tau ^{ - 1} \cdot \tau = 1$ ) and $\mathbb{T}^N \backslash {\text{supp}}(f \circ \tau )$ depending on the “rotation,” i.e., on $\tau \in \mathcal{F}$ . In the present paper, we consider two settings of this problem (depending on the sense in which the Fourier series of the function $f \circ \tau $ is understood) and give (for both cases) possible solutions of the problem in the class $L_1 (\mathbb{T}^N )$ , $N \geqslant 2$ .     

A Characterization of Projective Spaces by a Set of Planes

This note deals with the following question: How many planes of a linear space (P, $\mathfrak{L}$ ) must be known as projective planes to ensure that (P, $\mathfrak{L}$ ) is a projective space? The following answer is given: If for any subset M of a linear space (P, $\mathfrak{L}$ ) the restriction (M, $\mathfrak{L}$ )(M)) is locally complete, and if for every plane E of (M, $\mathfrak{L}$ (M)) the plane $\bar E$ generated by E is a projective plane, then (P, $\mathfrak{L}$ ) is a projective space (cf. 5.6). Or more generally: If for any subset M of P the restriction (M, $\mathfrak{L}$ (M)) is locally complete, and if for any two distinct coplanar lines G1, G2 ∈ $\mathfrak{L}$ (M) the lines $\bar G_1 ,\bar G_2 \varepsilon \mathfrak{L}$ generated by G1, G2 have a nonempty intersection and $\overline {G_1 \cup {\text{ }}G_2 }$ satisfies the exchange condition, then (P, $\mathfrak{L}$ ) is a generalized projective space.     

Integrability of the Majorants of Fourier Series and Divergence of the Fourier Series of Functions with Restrictions on the Integral Modulus of Continuity

We construct an example of a function from the class $H_1^{\omega ^ * } $ , where $\omega ^ * (t) = \sqrt {\log \log (t^{ - 1} )/\log (t^{ - 1} )} $ , $0 < t \leqslant t_0 $ , whose trigonometric Fourier series is divergent almost everywhere. We obtain sharp integrability conditions for the majorants of the partial sums of trigonometric Fourier series in terms of whether the functions in question belong to the classes $H_1^\omega $ .     

Finitely Generated Submodules in the Module of Entire Functions Determined by Restrictions on the Indicator Function

We study finitely generated submodules in the module $P$ of entire functions bounded by a system of $\rho $ -trigonometrically convex weights majorized by a given $\rho $ -trigonometrically convex function. Sufficient conditions for the ampleness of a finitely generated submodule in terms of the relative position of the zeros of its generators are obtained. Using these conditions, we prove that each ample submodule in $P$ is generated by two (possibly, coinciding) functions.     

Regular Semigroups of Endomorphisms of von Neumann Factors

We study a class of $E_0$ -semigroups of endomorphisms of a von Neumann factor $\mathcal{M}$ possessing the following property: an $e_0$ -semigroup of endomorphisms of $\mathcal{B}\left( \mathcal{H} \right)$ , where $\mathcal{H}$ is the standard representation space for $\mathcal{M}$ , and a product system of Hilbert spaces can be associated with each of these $E_0$ -semigroups.     

The index of centralizers of elements in classical Lie algebras

The index of a finite-dimensional Lie algebra $\mathfrak{g}$ is the minimum of dimensions of the stabilizers $\mathfrak{g}_\alpha $ over all covectors $\alpha \in \mathfrak{g}^ * $ . Let $\mathfrak{g}$ be a reductive Lie algebra over a field $\mathbb{K}$ of characteristic ≠ = 2. Élashvili conjectured that the index of $\mathfrak{g}_\alpha $ is always equal to the index, or, which is the same, the rank of $\mathfrak{g}$ . In this article, Élashvili’s conjecture is proved for classical Lie algebras. Furthermore, it is shown that if $\mathfrak{g} = \mathfrak{g}\mathfrak{l}_n $ or $\mathfrak{g} = \mathfrak{s}\mathfrak{p}_{2n} $ and $e \in \mathfrak{g}$ is a nilpotent element, then the coadjoint action of $\mathfrak{g}_e $ has a generic stabilizer. For $\mathfrak{g}$ , we give examples of nilpotent elements $e \in \mathfrak{g}$ such that the coadjoint action of $\mathfrak{g}_e $ does not have a generic stabilizer.     

On Milnor's Invariants of 4 -Component Links

We study the behavior of Milnor's µ-invariants of three- and four-component links with respect to the discriminant determined by $\Delta $ -moves of links. We introduce a new type of $\Delta $ -move, balanced $\Delta $ -moves, or, briefly, $B\Delta $ -moves. Since each four-component link is equivalent to a standard link under a sequence of balanced $\Delta $ -moves, $\Delta $ -moves that involve at most two components, and Reidemeister moves, we manage to define axiomatically µ-invariants of length $3$ for arbitrary semibounding links.     

Periodic Abelian Groups with $$UA$$ -Rings of Endomorphisms

A ring $R$ is said to be a unique addition ring (a $UA$ -ring) if its multiplicative semigroup $(R,{\text{ }} \cdot )$ can uniquely be endowed with a binary operation $ +$ in such a way that $(R,{\text{ }} \cdot ,{\text{ }} + )$ becomes a ring. An Abelian group is said to be an ${\text{End - }}UA$ -group if the endomorphism ring of the group is a $UA$ -ring. In the paper we study conditions under which an Abelian group is an ${\text{End - }}UA$ -group.     

Lower Bounds for the Riemann Zeta Function on the Critical Line

We establish a relation between the lower bound for the maximum of the modulus of $\zeta (1/2 + iT + s)$ in the disk $|s| \leqslant H$ and the lower bound for the maximum of the modulus of $\zeta (1/2 + iT + it)$ on the closed interval $|t| \leqslant H$ for $0 < H(T) \leqslant {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}$ . We prove a theorem on the lower bound for the maximum of the modulus of $0 < H(T) \leqslant {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}$ on the closed interval $|t| \leqslant H$ for $40 \leqslant H(T) \leqslant \log \log T$ .     

Regions of Values for the Systems \{ f(z_0 ),f'(z_0 ),c_2 \} and \{ f(r),f'(r),f(z_0 )\} in the Class of Typically Real Functions

Let T be the class of functions $f(z)$ having the following properties: these functions are regular and typically real in the disk $\left| z \right| < 1$ and have the expansions $f(z) = z + c_2 z^2 + c_3 z^3 + ....$ . We give algebraic and geometric characterizations of regions of values for the functionals in the class T mentioned in the title. In the same class of functions, we find regions of values for $f'(z_0 )$ with fixed $c_2$ and $f(z_0 )$ and for $f(z_0 )$ with fixed $f(r)$ and $f'(r)$ . Bibliography: 8 titles.     

The densest packing of equal circles into a parallel strip

What is the densest packing of points in an infinite strip of widthw, where any two of the points must be separated by distance at least I? This question was raised by Fejes-Tóth a number of years ago. The answer is trivial for \(w \leqslant \sqrt 3 /2\) and, surprisingly, it is not difficult to prove [M2] for \(w = n\sqrt 3 /2\) , wheren is a positive integer, that the regular triangular lattice gives the optimal packing. Kertész [K] solved the case \(w< \sqrt 2 \) . Here we fill the first gap, i.e., the maximal density is determined for \(\sqrt 3 /2< w \leqslant \sqrt 3 \) .     

Calderón constants of finite-dimensional couples

The Calderón constant æ( $\bar X$ ) is a numerical invariant of finite-dimensional Banach couple $\bar X = (X_0 ,X_1 )$ measuring its interpolation property with respect to linear operators acting in $\bar X$ . In the paper we prove the duality relation æ( $\bar X$ )≈ æ( $\bar X$ ^*)and calculate the asymptotic behavior of æ( $\bar X$ ) as dim $\bar X \to \infty $ for a few “classical” Banach couples.     

Converse Quadrature sum Inequalities for Freud Weights. ii

Let $W: = \exp \left( { - Q} \right)$ , where $Q$ is of smooth polynomial growth at $\infty$ , for example $Q\left( x \right) = \left| x \right|^\beta ,\beta >1$ . We call $W^2 $ a Freud weight. Let $\left\{ {x_{j{\kern 1pt} n} } \right\}_{j = 1}^n $ and $\left\{ {\lambda _{j{\kern 1pt} n} } \right\}_{j = 1}^n $ denote respectively the zeros of the $n$ th orthonormal polynomial $p_n$ for $W^2 $ and the Christoffel numbers of order $n$ . We establish converse quadrature sum inequalities associated with W, such as $$\left\| {\left( {PW} \right)\left( x \right)\left( {1 + \left| x \right|} \right)^r } \right\|_{L_p \left( R \right)} $$ with $C$ independent of $n$ and polynomials P of degree $ < n$ , and suitable restrictions on $r$ , $R$ . We concentrate on the case ${ \geqq 4}$ , as the case ${p < 4}$ was handled earlier. We are able to treat a general class of Freud weights, whereas our earlier treatment dealt essentially with $\left( { - \left| x \right|^\beta } \right),\beta = 2,4,6,....$ Some applications to Lagrange interpolation are presented.