How quadratic are the natural numbers?

Abstract. For natural numbers n we inspect all factorizations n = ab of n with a ￡ ba \le b in \Bbb N\Bbb N and denote by n=a_n b_nn=a_n b_n the most quadratic one, i.e. such that b_n - a_nb_n - a_n is minimal. Then the quotient k(n) : = a_n/b_n\kappa (n) := a_n/b_n is a measure for the quadraticity of n. The best general estimate for k(n)\kappa (n) is of course very poor: 1/n ￡ k(n) ￡ 11/n \le \kappa (n)\le 1. But a Theorem of Hall and Tenenbaum [1, p. 29], implies(logn)^-d-e ￡ k(n) ￡ (logn)^-d(\log n)^{-\delta -\varepsilon } \le \kappa (n) \le (\log n)^{-\delta } on average, with d = 1 - (1+log₂  2)/log2=0,08607 ?\delta = 1 - (1+\log _2 \,2)/\log 2=0,08607 \ldots and for every e > 0\varepsilon >0. Hence the natural numbers are fairly quadratic.¶k(n)\kappa (n) characterizes a specific optimal factorization of n. A quadraticity measure, which is more global with respect to the prime factorization of n, is k^*(n): = ?_{1 ￡ a ￡ b, ab=n} a/b\kappa ^*(n):= \textstyle\sum\limits \limits _{1\le a \le b, ab=n} a/b. We show k^*(n) ~ \frac 12\kappa ^*(n) \sim \frac {1}{2} on average, and k^*(n)=W(2^{\frac 12(1-e) log n/log ₂n})\kappa ^*(n)=\Omega (2^{\frac {1}{2}(1-\varepsilon ) {\log}\, n/{\log} _2n})for every e > 0\varepsilon>0.     

Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold

We show that for every closed Riemannian manifold X there exists a positive number¶ $ \varepsilon_0 > 0 $ \varepsilon_0 > 0 such that for all 0< e\leqq e₀ \varepsilon \leqq \varepsilon_0 there exists some¶ $ \delta > 0 $ \delta > 0 such that for every metric space Y with Gromov-Hausdorff distance to X less than¶ d \delta the geometric e \varepsilon -complex |Y_e| |Y_\varepsilon| is homotopy equivalent to X.¶ In particular, this gives a positive answer to a question of Hausmann [4].     

On groups acting freely on a tree

Suppose we are given a group G\mit\Gamma and a tree X on which G\mit\Gamma acts. Let d be the distance in the tree. Then we are interested in the asymptotic behavior of the numbers a_d: = # {w ? vertX : w=gv, g ? G , d(v₀,w)=d }a_d:= \# \{w\in {\rm {vert}}X : w=\gamma {v}, \gamma \in {\mit\Gamma} , d({v}_0,w)=d \} if d? ￥d\rightarrow \infty , where v, v_o are some fixed vertices in X.¶ In this paper we consider the case where G\mit\Gamma is a finitely generated group acting freely on a tree X. The growth function ?a_d x^d\textstyle\sum\limits a_d x^d is a rational function [3], which we describe explicitely. From this we get estimates for the radius of convergence of the series. For the cases where G\mit\Gamma is generated by one or two elements, we look a little bit closer at the denominator of this rational function. At the end we give one concrete example.     

Arithmetic progressions in sumsets

We prove several results concerning arithmetic progressions in sets of integers. Suppose, for example, that a \alpha and b \beta are positive reals, that N is a large prime and that C,D í \Bbb Z/N\Bbb Z C,D \subseteq {\Bbb Z}/N{\Bbb Z} have sizes gN \gamma N and dN \delta N respectively. Then the sumset C + D contains an AP of length at least e^{c ?{log} N} e^{c \sqrt{\rm log} N} , where c > 0 depends only on g \gamma and d \delta . In deriving these results we introduce the concept of hereditary non-uniformity (HNU) for subsets of \Bbb Z/N\Bbb Z {\Bbb Z}/N{\Bbb Z} , and prove a structural result for sets with this property.     

White noise eliminates instability

Let x₁,..., x_n be points in the d-dimensional Euclidean space E^d with || x_i-x_j|| ￡ 1\| x_{i}-x_{j}\| \le 1 for all 1 \leqq i,j \leqq n1 \leqq i,j \leqq n, where || .||\| .\| denotes the Euclidean norm. We ask for the maximum M(d,n) of \mathop?_i, j=1ⁿ|| x_i-x_j|| ²\textstyle\mathop\sum\limits _{i,\,j=1}^{n}\| x_{i}-x_{j}\| ^{2} (see [4]). This paper deals with the case d = 2. We calculate M(2, n) and show that the value M(2, n) is attained if and only if the points are distributed as evenly as possible among the vertices of a regular triangle of edge-length 1. Moreover we give an upper bound for the value \mathop?_i, j=1ⁿ|| x_i-x_j|| \textstyle\mathop\sum\limits _{i,\,j=1}^{n}\| x_{i}-x_{j}\| , where the points x₁,...,x_n are chosen under the same constraints as above.     

Equality of two variable weighted means: reduction to differential equations

Summary. Let F, Y \Phi, \Psi be strictly monotonic continuous functions, F,G be positive functions on an interval I and let n ? \Bbb N \{1} n \in {\Bbb N} \setminus \{1\} . The functional equation¶¶F^-1 ([(?_i=1ⁿF(x_i)F(x_i))/(?_i=1ⁿ F(x_i)]) Y^-1 ([(?_i=1ⁿY(x_i)G(x_i))/(?_i=1ⁿ G(x_i))])  (x₁,?,x_n ? I) \Phi^{-1}\,\left({\sum\limits_{i=1}^{n}\Phi(x_{i})F(x_{i})\over\sum\limits_{i=1}^{n} F(x_{i}}\right) \Psi^{-1}\,\left({\sum\limits_{i=1}^{n}\Psi(x_{i})G(x_{i})\over\sum\limits_{i=1}^{n} G(x_{i})}\right)\,\,(x_{1},\ldots,x_{n} \in I) ¶was solved by Bajraktarevi' [3] for a fixed n 3 3 n\ge 3 . Assuming that the functions involved are twice differentiable he proved that the above functional equation holds if and only if¶¶Y(x) = [(aF(x) + b)/(cF(x) + d)],       G(x) = kF(x)(cF(x) + d) \Psi(x) = {a\Phi(x)\,+\,b\over c\Phi(x)\,+\,d},\qquad G(x) = kF(x)(c\Phi(x) + d) ¶where a,b,c,d,k are arbitrary constants with k(c²+d²)(ad-bc) 1 0 k(c^2+d^2)(ad-bc)\ne 0 . Supposing the functional equation for all n = 2,3,... n = 2,3,\dots Aczél and Daróczy [2] obtained the same result without differentiability conditions.¶The case of fixed n = 2 is, as in many similar problems, much more difficult and allows considerably more solutions. Here we assume only that the same functional equation is satisfied for n = 2 and solve it under the supposition that the functions involved are six times differentiable. Our main tool is the deduction of a sixth order differential equation for the function j = F°Y^-1 \varphi = \Phi\circ\Psi^{-1} . We get 32 new families of solutions.     

Zyklische kubische monogene Erweiterungen der ganzalgebraischen Zahlen eines quadratischen Körpers

D'après [6] et [7] l'anneau des entiers du corps quadratique Q(?d), d \not = -3{\bf Q}(\sqrt {d}), d \not = -3, possède une extension cyclique cubique monogène (de discriminant 1) si, et seulement si, l'équation diophantienne¶¶ 4m³ = y²d + 274m^3 = y^2d + 27 a une solution avec d \not o 21d \not \equiv 21 (mod 36) et m \not o 3m \not \equiv 3 (mod 9).¶¶ On démontre ici que pour qu'une telle extension existe il faut que 3 divise h (d) et, lorsque d o 1d \equiv 1 (mod 8), d'où (2) = \frak p₁\frak p₂(2) = \frak p_1\frak p_2 où \frak p₁\frak p_1 et \frak p₂\frak p_2 sont deux idéaux premiers distincts de A_d, que la classe [\frak p₁][\frak p_1] de \frak p₁\frak p_1 dans le groupe de classes de Q(?d){\bf Q}(\sqrt {d}) ne soit pas un cube. Pour |d||d| < 100'000 cela élimine 68,37 % des valeurs restantes, les valeurs éliminées passent ainsi de 90 à 97 %.¶ De plus d ne doit pas être de la forme pq ou -3 pq pour lesquels le symbole d'Aigner T(p *q)T(p \star q) vale -1. L'article comporte aussi deux corrections, des résultats complétant [6] et [7], parus dans une thèse, et d'autres (en particulier l'indépendance des critères et des résultats numériques) parus ailleurs.     

On the derivatives of X-ray functions

We show that for every n \geqq 4, 0 \leqq k \leqq n - 3, p ? (0, 3] n \geqq 4, 0 \leqq k \leqq n - 3, p \in (0, 3] and every origin-symmetric convex body K in \mathbbRⁿ \mathbb{R}^n , the function ||x ||^-k₂ ||x ||^-n+k+p_K \parallel x \parallel^{-k}_{2} \parallel x \parallel^{-n+k+p}_{K} represents a positive definite distribution on \mathbbRⁿ \mathbb{R}^n , where ||·||₂ \parallel \cdot \parallel_{2} is the Euclidean norm and ||·||_K \parallel \cdot \parallel_{K} is the Minkowski functional of K. We apply this fact to prove a result of Busemann-Petty type that the inequalities for the derivatives of order (n - 4) at zero of X-ray functions of two convex bodies imply the inequalities for the volume of average m-dimensional sections of these bodies for all 3 \leqq m \leqq n 3 \leqq m \leqq n . We also prove a sharp lower estimate for the maximal derivative of X-ray functions of the order (n - 4) at zero.     

Résonances dans des Petits Domaines près d‚une Énergie Critique

Résumé. On travaille dans le cadre de l‘analyse semi-classique. Considérons p(x, hD_x) p(x, hD_{x}) , une perturbation de -h²D -h^{2}\Delta qui est analytique à l‘infini. On suppose que dans la surface d‘énergie E₀ > 0, les points critiques du symbole p(x, x) p(x, \xi) forment une sous-variété C \mathcal C et que p est non dégénéré dans l‘espace normal à C \mathcal C .¶En utilisant les résultats de [6] et [18], on obtient une majoration du nombre de résonances dans des disques de rayon d \delta centrés en E proche de E₀, où d \delta satisfait Ch < d \delta < 1/C pour une constante C > 0. En généralisant la formule de trace de Sjöstrand qui exprime la trace d‘une différence d‘opérateurs en fonction des résonances, on trouve une minoration du nombre de résonances proches de E₀.     

Uncorrelatedness and correlatedness of powers of random variables

Let x₁, ?, x_n \xi_1, \ldots, \xi_n be random variables and U be a subset of the Cartesian product \mathbbZ₊ⁿ, \mathbbZ₊ \mathbb{Z}_+^n, \mathbb{Z}_+ being the set of all non-negative integers. The random variables are said to be strictly U-uncorrelated if¶¶E(x₁^j₁ ?x_n^j_n) = E(x₁^j₁) ?E(x_n^j_n) ? (j₁, ... ,j_n) ? U. \textbf {E}\big(\xi_1^{j_1} \cdots \xi_n^{j_n}\big) = \textbf {E}\big(\xi_1^{j_1}\big) \cdots \textbf {E}\big(\xi_n^{j_n}\big) \iff (j_1, \dots ,j_n) \in U. ¶It is proved that for an arbitrary subset U \subseteqq \mathbbZ₊ⁿ U \subseteqq \mathbb{Z}_+^n containing all points with 0 or 1 non-zero coordinates there exists a collection of n strictly U-uncorrelated random variables.     

Inequalities of Hlawka type for matrices

A generalized Hlawka's inequality says that for any n (\geqq 2) (\geqq 2) complex numbers¶ x₁, x₂, ..., x_n,¶¶ ?_i=1ⁿ|x_i - ?_j=1ⁿx_j| \leqq ?_i=1ⁿ|x_i| + (n - 2)|?_j=1ⁿx_j|. \sum_{i=1}^n\Bigg|x_i - \sum_{j=1}^{n}x_j\Bigg| \leqq \sum_{i=1}^{n}|x_i| + (n - 2)\Bigg|\sum_{j=1}^{n}x_j\Bigg|. ¶¶ We generalize this inequality to the trace norm and the trace of an n x n matrix A as¶¶ ||A - Tr A ||₁ \leqq ||A||₁ + (n - 2)| Tr A|. ||A - {\rm Tr} A ||_1\ \leqq ||A||_1 + (n - 2)| {\rm Tr} A|. ¶¶ We consider also the related inequalities for p-norms (1 \leqq p \leqq ￥) (1 \leqq p \leqq \infty) on matrices.     

Polar factorization of maps on Riemannian manifolds

Let (M,g) be a connected compact manifold, C³ smooth and without boundary, equipped with a Riemannian distance d(x,y). If s : M ? M s : M \to M is merely Borel and never maps positive volume into zero volume, we show s = t °u s = t \circ u factors uniquely a.e. into the composition of a map t(x) = exp_x[-?y(x)] t(x) = {\rm exp}_x[-\nabla\psi(x)] and a volume-preserving map u : M ? M u : M \to M , where y: M ? \bold R \psi : M \to {\bold R} satisfies the additional property that (y^c)^c = y (\psi^c)^c = \psi with y^c(y) :=inf{c(x,y) - y(x) | x ? M} \psi^c(y) :={\rm inf}\{c(x,y) - \psi(x)\,\vert\,x \in M\} and c(x,y) = d²(x,y)/2. Like the factorization it generalizes from Euclidean space, this non-linear decomposition can be linearized around the identity to yield the Hodge decomposition of vector fields.¶The results are obtained by solving a Riemannian version of the Monge--Kantorovich problem, which means minimizing the expected value of the cost c(x,y) for transporting one distribution f 3 0 f \ge 0 of mass in L¹(M) onto another. Parallel results for other strictly convex cost functions c(x,y) 3 0 c(x,y) \ge 0 of the Riemannian distance on non-compact manifolds are briefly discussed.     

Gorenstein injective dimension and Tor-depth of modules

The main aim of this paper is to obtain a dual result to the now well known Auslander-Bridger formula for G-dimension. We will show that if R is a complete Cohen-Macaulay ring with residue field k, and M is a non-injective h-divisible Ext-finite R-module of finite Gorenstein injective dimension such that for each i 3 1i \geq 1 Extⁱ (E,M) = 0 for all indecomposable injective R-modules E 1 E(k)E \neq E(k), then the depth of the ring is equal to the sum of the Gorenstein injective dimension and Tor-depth of M. As a consequence, we get that this formula holds over a d-dimensional Gorenstein local ring for every nonzero cosyzygy of a finitely generated R-module and thus in particular each such n^th cosyzygy has its Tor-depth equal to the depth of the ring whenever n 3 dn \geq d.     

Quadratic matrix polynomials with Hamiltonian spectrum and oscillatory damped systems

We consider quadratic matrix polynomials of the form L(l) = l²A + elB + CL(\lambda) = \lambda^{2}A + \epsilon\lambda B + C, where e\epsilon is a real parameter, A is positive definite and B and C are symmetric. The main results of the paper are the characterization of the class of symmetric matrices B for which the spectrum of the polynomial is symmetric with respect to the imaginary axis and solutions of the corresponding differential equation oscillate in time. We also extend the results in [2] to allow us to study the asymptotic behaviour of the eigenvalues for large e\epsilon.     

Solvability of systems of partial differential equations for functions defined on nonconvex sets

We consider systems of partial differential equations with constant coefficients of the form ( R(D_x, D_y)f = 0, P(D_x)f = g), f,g ? C^￥(W),\big ( R(D_x, D_y)f = 0, P(D_x)f = {g}\big ), f,g \in {C}^{\infty}(\Omega),, where R (and P) are operators in (n + 1) variables (and in n variables, respectively), g satisfies the compatibility condition R(D_x, D_y)g = 0  and  W ì \Bbb Rⁿ⁺¹R(D_x, D_y){g} = 0 \ {\rm and} \ \Omega \subset {\Bbb R}^{n+1} is open. Let R be elliptic. We show that the solvability of such systems for certain nonconvex sets W\Omega implies that any localization at ￥\infty of the principle part P_m of P is hyperbolic. In contrast to this result such systems can always be solved on convex open sets W\Omega by the fundamental principle of Ehrenpreis-Palamodov.     

Solutions of the Dirac-Fock Equations without Projector

.In this paper we prove the existence of infinitely many solutions of the Dirac-Fock equations with N electrons turning around a nucleus of atomic charge Z, satisfying N < Z + 1 and amax(Z, N)\alpha\max(Z, N)<[2/([2/(P)]+[(P)/2])]{2\over {2\over \Pi}+{\Pi\over 2}}, where a ? [1/137]\alpha \approx {1\over 137} is the fundamental constant of the electromagnetic interaction. This work is an improvement of an article of Esteban-Séré, where the same result was proved under more restrictive assumptions on N.     

A rigidity result for highest order local cohomology modules

Let M be a finitely generated faithful module over a noetherian ring R of dimension d < ￥ \infty and let \mathfrak a \subseteqq R {\mathfrak a} \subseteqq R be an ideal. We describe the (finite) set Supp_R(H_{\mathfrak a}^d (M)) = Ass_R(H_{\mathfrak a}^d (M)) \textrm{Supp}_R(H_{\mathfrak a}^d (M)) = \textrm{Ass}_R(H_{\mathfrak a}^d (M)) of primes associated to the highest local cohomology module H_{\mathfrak a}^d (M) H_{\mathfrak a}^d (M) in terms of the local formal behaviour of \mathfrak a {\mathfrak a} . If R is integral and of finite type over a field, Supp_R(H_{\mathfrak a}^d (M)) \textrm{Supp}_R(H_{\mathfrak a}^d (M)) is the set of those closed points of X = Spec(R) whose fibre under the normalization morphism n: X￠? X \nu : X' \rightarrow X contains points which are isolated in n^-1(Spec(R/\mathfrak a)) \nu^{-1}(\textrm{Spec}(R/{\mathfrak a})) .     

Sharp bounds for the Bernoulli numbers

We determine the best possible real constants a\alpha and b\beta such that the inequalities [(2(2n)!)/((2p)²ⁿ)] [1/(1-2^a-2n)] \leqq |B_2n| \leqq [(2(2n)!)/((2p)²ⁿ)] [1/(1-2^b-2n)]{2(2n)! \over(2\pi)^{2n}} {1 \over 1-2^{\alpha -2n}} \leqq |B_{2n}| \leqq {2(2n)! \over (2\pi )^{2n}}\, {1 \over 1-2^{\beta -2n}}hold for all integers n\geqq 1n\geqq 1. Here, B₂, B₄, B₆,... are Bernoulli numbers.     

A linear functional equation with a singularity at a fixed point

Summary. Local solutions of the functional equation¶¶z^k f( z) = ?_k=1ⁿG_k( z) f( s_kz ) +g( z) z{^\kappa} \phi \left( z\right) =\sum_{k=1}^nG_k\left( z\right) \phi \left( s_kz \right) +g\left( z\right) ¶with k > 0 \kappa > 0 and | s_k| \gt 1 \left| s_k\right| \gt 1 are considered. We prove that the equation is solvable if and only if a certain system of k \kappa conditions on G_k (k = 1, 2, ... , n) and g is fulfilled.     

Integration of the lifting formulas and the cyclic homology of the algebras of differential operators

We integrate the Lifting cocycles Y_2n+1, Y_2n+3, Y_2n+5,? ([Sh1,2]) \Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,([\rm Sh1,2]) on the Lie algebra Dif_n of holomorphic differential operators on an n-dimensional complex vector space to the cocycles on the Lie algebra of holomorphic differential operators on a holomorphic line bundle l \lambda on an n-dimensional complex manifold M in the sense of Gelfand--Fuks cohomology [GF] (more precisely, we integrate the cocycles on the sheaves of the Lie algebras of finite matrices over the corresponding associative algebras). The main result is the following explicit form of the Feigin--Tsygan theorem [FT1]:¶¶ H^·_Lie(\frak g\frak l^fin_￥(Dif_n);\Bbb C) = ù^·(Y_2n+1, Y_2n+3, Y_2n+5,? ) H^\bullet_{\rm Lie}({\frak g}{\frak l}^{\rm fin}_\infty({\rm Dif}_n);{\Bbb C}) = \wedge^\bullet(\Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,) .