On a question of Erdös and Graham

We prove that for any $ \varepsilon > 0 $ \varepsilon > 0 there is k (e) k (\varepsilon) such that for any prime p and any integer c there exist k \leqq k(e) k \leqq k(\varepsilon) pairwise distinct integers x_i with 1 \leqq x_i \leqq p^e, i = 1, ?, k 1 \leqq x_{i} \leqq p^{\varepsilon}, i = 1, \ldots, k , and such that¶¶?_i=1^k [1/(x_i)] o c    (mod p). \sum\limits_{i=1}^k {{1}\over{x_i}} \equiv c\quad (\mathrm{mod}\, p). ¶¶ This gives a positive answer to a question of Erdös and Graham.     

Packing circuits in matroids

The purpose of this paper is to characterize all matroids M that satisfy the following minimax relation: for any nonnegative integral weight function w defined on E(M),

Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Let M be a complete noncompact Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation $ \frac{\partial u}{\partial t} = \Delta _{f}u +au\,{\rm log}\, u + bu$ on ${M \times [0, + \infty)}Let M be a complete noncompact Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation

\frac?u?t = Dfu +au log u + bu \frac{\partial u}{\partial t} = \Delta _{f}u +au\,{\rm log}\, u + bu      5. The minimal positive integer represented by a positive definite quadratic form      X. Wang 《Archiv der Mathematik》2003,80(3):245-254 In this paper we shall give an upper bound on the size of the gap between the constant term and the next nonzero Fourier coefficient of a holomorphic modular form of given weight for the group G0(2) \Gamma_{0}(2) . We derive an upper bound for the minimal positive integer represented by an even positive definite quadratic form of level two. In our paper we prove two conjectures given in [1]. In particular, we can prove the following result: let Q \mathcal{Q} be an even positive definite quadratic form of level two in v v variables, with v o 4(mod 8) v \equiv 4(\textrm{mod}\, 8) , then Q \mathcal{Q} represents a positive integer 2n ￡ 3+v/4 2n \leq 3+v/4 .      6. The mod 4 behaviour of total Lie algebra cohomology      G. Cairns  G. Kim 《Archiv der Mathematik》2001,77(2):177-180 We show that if L is a unimodular Lie algebra over a field of characteristic 1 2\ne 2, then the dimension s\sigma(L) of the total cohomology of L is a multiple of 4 when dim(L)\not o 3\dim(L)\not\equiv 3 (mod 4). However, contrary to a claim by Deninger and Singhof, we give an example of a rational nilpotent algebra L of dimension 15 with s(L)\not o 0\sigma(L)\not\equiv 0 (mod 4). Over fields of characteristic 2, we completely classify those algebras L with s(L)\not o 0\sigma(L)\not\equiv 0 (mod 4).      7. Product of nilpotent subgroups      M.J. Iranzo  J. Medina  F. Pérez-Monasor 《Archiv der Mathematik》2000,74(4):246-252 We will say that a subgroup X of G satisfies property C in G if CG(X?Xg)\leqq X?Xg{\rm C}_{G}(X\cap X^{{g}})\leqq X\cap X^{{g}} for all g ? G{g}\in G. We obtain that if X is a nilpotent subgroup satisfying property C in G, then XF(G) is nilpotent. As consequence it follows that if N\triangleleft GN\triangleleft G is nilpotent and X is a nilpotent subgroup of G then CG(N?X)\leqq XC_G(N\cap X)\leqq X implies that NX is nilpotent.¶We investigate the relationship between the maximal nilpotent subgroups satisfying property C and the nilpotent injectors in a finite group.      8. Wilson's functional equation on P3-groups      I. Corovei 《Aequationes Mathematicae》2001,61(3):212-220 Summary. Consider Wilson's functional equation¶¶f(xy) + f(xy-1) = 2f(f)g(y) f(xy) + f(xy^{-1}) = 2f(f)g(y) , for f,g : G ? K f,g : G \to K ¶where G is a group and K a field with char K 1 2 {\rm char}\, K\ne 2 .¶Aczél, Chung and Ng in 1989 have solved Wilson's equation, assuming that the function g satisfies Kannappan's condition g(xyz) = g(xzy) and f(xy) = f(yx) for all x,y,z ? G x,y,z\in G .¶In the present paper we obtain the general solution of Wilson's equation when G is a P3-group and we show that there exist solutions different of those obtained by Aczél, Chung and Ng.¶A group G is said to be a P3-group if the commutator subgroup G' of G, generated by all commutators [x,y] := x-1y-1xy, has the order one or two.      9. The Łojasiewicz exponent of an analytic function at an isolated zero      J. Gwoździewicz 《Commentarii Mathematici Helvetici》1999,74(3):364-375 Let f be a real analytic function defined in a neighborhood of 0 ? \Bbb Rn 0 \in {\Bbb R}^n such that f-1(0)={0} f^{-1}(0)=\{0\} . We describe the smallest possible exponents !, #, / for which we have the following estimates: |f(x)| 3 c|x|a |f(x)|\geq c|x|^{\alpha} , |grad f(x)| 3 c|x|b |{\rm grad}\,f(x)|\geq c|x|^{\beta} , |grad f(x)| 3 c|f(x)|q |{\rm grad}\,f(x)|\geq c|f(x)|^{\theta} for x near zero with c > 0 c > 0 . We prove that a = b+1 \alpha=\beta+1, q = b/a\theta=\beta/\alpha . Moreover b = N+a/b \beta=N+a/b where $ 0 h a < b h N^{n-1} $ 0 h a < b h N^{n-1} . If f is a polynomial then |f(x)| 3 c|x|(degf-1)n+1 |f(x)|\geq c|x|^{(\deg f-1)^n+1} in a small neighborhood of zero.      10. On minimal p-degrees in 2-transitive permutation groups      Höchsmann  Jens 《Archiv der Mathematik》1999,72(6):405-417 Suppose G is a transitive permutation group on a finite set W\mit\Omega of n points and let p be a prime divisor of |G||G|. The smallest number of points moved by a non-identity p-element is called the minimal p-degree of G and is denoted mp (G). ¶ In the article the minimal p-degrees of various 2-transitive permutation groups are calculated. Using the classification of finite 2-transitive permutation groups these results yield the main theorem, that mp(G) 3 [(p-1)/(p+1)] ·|W|m_{p}(G) \geq {{p-1} \over {p+1}} \cdot |\mit\Omega | holds, if Alt(W) \nleqq G {\rm Alt}(\mit\Omega ) \nleqq G .¶Also all groups G (and prime divisors p of |G||G|) for which mp(G) ￡ [(p-1)/(p)] ·|W|m_{p}(G)\le {{p-1}\over{p}} \cdot |\mit\Omega | are identified.      11. On groups acting freely on a tree      Ulrich Tipp 《Archiv der Mathematik》1999,72(4):261-269 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 ad: = # {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, vo 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 ?ad xd\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.      12. Invariants of free Lie algebras      A. I. Papistas 《Archiv der Mathematik》2001,76(5):338-342 Let k be a principal ideal domain with identity and characteristic zero. For a positive integer n, with n \geqq 2n \geqq 2, let H(n) be the group of all n x n matrices having determinant ±1\pm 1. Further, we write SL(n) for the special linear group. Let L be a free Lie algebra (over k) of finite rank n. We prove that the algebra of invariants LB(n) of B(n), with B(n) ? { H(n), SL(n)}B(n) \in \{ H(n), {\rm SL}(n)\} , is not a finitely generated free Lie algebra. Let us assume that k is a field of characteristic zero and let áSem(n) ?\langle {\rm Sem}(n) \rangle be the Lie subalgebra of L generated by the semi-invariants (or Lie invariants) Sem(n). We prove that áSem(n) ?\langle {\rm Sem}(n) \rangle is not a finitely generated free Lie algebra which gives a positive answer to a question posed by M. Burrow [4].      13. On Lipschitz selections of affine-set valued mappings      P. Shvartsman 《Geometric And Functional Analysis》2001,11(4):840-868 We prove a Helly-type theorem for the family of all k-dimensional affine subsets of a Hilbert space H. The result is formulated in terms of Lipschitz selections of set-valued mappings from a metric space (M,r) ({\cal M},\rho) into this family.¶Let F be such a mapping satisfying the following condition: for every subset M￠ ì M {\cal M'} \subset {\cal M} consisting of at most 2k+1 points, the restriction F|M￠ F|_{\cal M'} of F to M￠ {\cal M'} has a selection fM￠ (i.e. fM￠(x) ? F(x) for all x  ? M￠) f_{\cal M'}\,({\rm i.e.}\,f_{\cal M'}(x) \in F(x)\,{\rm for\,all}\,x\,\in {\cal M'}) satisfying the Lipschitz condition ||fM￠(x) - fM￠(y)||  ￡ r(x,y ), x,y ? M￠ \parallel f_{\cal M'}(x) - f_{\cal M'}(y)\parallel\,\le \rho(x,y ),\,x,y \in {\cal M'} . Then F has a Lipschitz selection f : M ? H f : {\cal M} \to H such that ||f(x) - f(y) ||  ￡ gr(x,y ), x,y ? M \parallel f(x) - f(y) \parallel\,\le \gamma \rho (x,y ),\,x,y \in {\cal M} where g = g(k) \gamma = \gamma(k) is a constant depending only on k. (The upper bound of the number of points in M￠ {\cal M'} , 2k+1, is sharp.)¶The proof is based on a geometrical construction which allows us to reduce the problem to an extension property of Lipschitz mappings defined on subsets of metric trees.      14. Orbits of permutation groups on the power set      S. Dolfi 《Archiv der Mathematik》2000,75(5):321-327 Let G be a permutation group on a finite set W\Omega . If G does not involve An for n \geqq 5 n \geqq 5 , then there exist two disjoint subsets of W\Omega such that no Sylow subgroup of G stabilizes both and four disjoint subsets of W\Omega whose stabilizers in G intersect trivially.      15. Quadratic matrix polynomials with Hamiltonian spectrum and oscillatory damped systems      P. Freitas 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1999,50(1):64-81 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.      16. Arithmetic progressions in sumsets      B. Green 《Geometric And Functional Analysis》2002,12(3):584-597 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 ec ?{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.      17. On the singularities of residual intertwining operators      W. Müller 《Geometric And Functional Analysis》2000,10(5):1118-1170 Let G be a reductive algebraic group defined over \Bbb Q {\Bbb Q} . Let P, P' be parabolic subgroups of G, defined over \Bbb Q {\Bbb Q} , and let _boxclose_boxclose, a_P') t \in W({\frak a}_{P}, {\frak a}_{P'}) . In this paper we study the intertwining operator MP￠|P(t,l), l ? \frak a*P,\Bbb C M_{P' \vert P}(t,\lambda),\,\lambda \in {\frak a}^*_{P,{\Bbb C}} , acting in corresponding spaces of automorphic forms. One of the main results states that each matrix coefficient of MP￠|P(t,l) M_{P' \vert P}(t,\lambda) is a meromorphic function of order ￡ n + 1 \le n + 1 , where n = dim G. Using this result, we further investigate the rank one intertwining operators, in particular, we study the distribution of their poles.      18. Solutions of the Dirac-Fock Equations without Projector      Eric Paturel 《Annales Henri Poincare》2000,1(6):1123-1157 .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.      19. Kazhdan's property (T), L^2-spectrum and isoperimetric inequalities for locally symmetric spaces      E. Leuzinger 《Commentarii Mathematici Helvetici》2003,78(1):116-133 Let V = G\G/KV =\Gamma\backslash G/K be a Riemannian locally symmetric space of nonpositive sectional curvature and such that the isometry group G of its universal covering space has Kazhdan's property (T). We establish strong dichotomies between the finite and infinite volume case. In particular, we characterize lattices (or, equivalently, arithmetic groups) among discrete subgroups G ì G\Gamma\subset G in various ways (e.g., in terms of critical exponents, the bottom of the spectrum of the Laplacian and the behaviour of the Brownian motion on V).      20. Finding the Best Face on a Voronoi Polyhedron – The Strong Dodecahedral Conjecture Revisited      Károly Bezdek  Endre Daróczy-Kiss 《Monatshefte für Mathematik》2005,74(5):191-206 In this paper we prove the following theorem. The surface area density of a unit ball in any face cone of a Voronoi cell in an arbitrary packing of unit balls of Euclidean 3-space is at most [(-9p+ 30 arccos([(?3)/2]sin ([(p)/5]) ))/(5 tan([(p)/5]))]=0.77836?,{-9\pi + 30\,{\rm arccos}\left({\sqrt{3}\over 2}{\rm sin}\,\left({\pi\over 5}\right) \right)\over 5\, {\rm tan}\left({\pi\over 5}\right)}=0.77836\ldots, and so the surface area of any Voronoi cell in a packing with unit balls in Euclidean 3-space is at least [(20p· tan ( [(p)/5]))/(-9p+ 30 arccos([(?3)/2]sin ([(p)/5]) ))]=16.1445? .{20\pi\cdot\,{\rm tan}\,\left( {\pi \over 5}\right) \over -9\pi + 30\,{\rm arccos}\left({\sqrt{3}\over 2}{\rm sin}\,\left({\pi\over 5}\right) \right)}=16.1445\ldots\ . This result and the ideas of its proof support the Strong Dodecahedral Conjecture according to which the surface area of any Voronoi cell in a packing with unit balls in Euclidean 3-space is at least as large as 16.6508..., the surface area of a regular dodecahedron of inradius 1.

The minimal positive integer represented by a positive definite quadratic form

In this paper we shall give an upper bound on the size of the gap between the constant term and the next nonzero Fourier coefficient of a holomorphic modular form of given weight for the group G₀(2) \Gamma_{0}(2) . We derive an upper bound for the minimal positive integer represented by an even positive definite quadratic form of level two. In our paper we prove two conjectures given in [1]. In particular, we can prove the following result: let Q \mathcal{Q} be an even positive definite quadratic form of level two in v v variables, with v o 4(mod 8) v \equiv 4(\textrm{mod}\, 8) , then Q \mathcal{Q} represents a positive integer 2n ￡ 3+v/4 2n \leq 3+v/4 .     

The mod 4 behaviour of total Lie algebra cohomology

We show that if L is a unimodular Lie algebra over a field of characteristic 1 2\ne 2, then the dimension s\sigma(L) of the total cohomology of L is a multiple of 4 when dim(L)\not o 3\dim(L)\not\equiv 3 (mod 4). However, contrary to a claim by Deninger and Singhof, we give an example of a rational nilpotent algebra L of dimension 15 with s(L)\not o 0\sigma(L)\not\equiv 0 (mod 4). Over fields of characteristic 2, we completely classify those algebras L with s(L)\not o 0\sigma(L)\not\equiv 0 (mod 4).     

Product of nilpotent subgroups

We will say that a subgroup X of G satisfies property C in G if C_G(X?X^g)\leqq X?X^g{\rm C}_{G}(X\cap X^{{g}})\leqq X\cap X^{{g}} for all g ? G{g}\in G. We obtain that if X is a nilpotent subgroup satisfying property C in G, then XF(G) is nilpotent. As consequence it follows that if N\triangleleft GN\triangleleft G is nilpotent and X is a nilpotent subgroup of G then C_G(N?X)\leqq XC_G(N\cap X)\leqq X implies that NX is nilpotent.¶We investigate the relationship between the maximal nilpotent subgroups satisfying property C and the nilpotent injectors in a finite group.     

Wilson's functional equation on P3-groups

Summary. Consider Wilson's functional equation¶¶f(xy) + f(xy^-1) = 2f(f)g(y) f(xy) + f(xy^{-1}) = 2f(f)g(y) , for f,g : G ? K f,g : G \to K ¶where G is a group and K a field with char K 1 2 {\rm char}\, K\ne 2 .¶Aczél, Chung and Ng in 1989 have solved Wilson's equation, assuming that the function g satisfies Kannappan's condition g(xyz) = g(xzy) and f(xy) = f(yx) for all x,y,z ? G x,y,z\in G .¶In the present paper we obtain the general solution of Wilson's equation when G is a P₃-group and we show that there exist solutions different of those obtained by Aczél, Chung and Ng.¶A group G is said to be a P₃-group if the commutator subgroup G' of G, generated by all commutators [x,y] := x^-1y^-1xy, has the order one or two.     

The Łojasiewicz exponent of an analytic function at an isolated zero

Let f be a real analytic function defined in a neighborhood of 0 ? \Bbb Rⁿ 0 \in {\Bbb R}^n such that f^-1(0)={0} f^{-1}(0)=\{0\} . We describe the smallest possible exponents !, #, / for which we have the following estimates: |f(x)| 3 c|x|^a |f(x)|\geq c|x|^{\alpha} , |grad f(x)| 3 c|x|^b |{\rm grad}\,f(x)|\geq c|x|^{\beta} , |grad f(x)| 3 c|f(x)|^q |{\rm grad}\,f(x)|\geq c|f(x)|^{\theta} for x near zero with c > 0 c > 0 . We prove that a = b+1 \alpha=\beta+1, q = b/a\theta=\beta/\alpha . Moreover b = N+a/b \beta=N+a/b where $ 0 h a < b h N^{n-1} $ 0 h a < b h N^{n-1} . If f is a polynomial then |f(x)| 3 c|x|^(degf-1)n+1 |f(x)|\geq c|x|^{(\deg f-1)^n+1} in a small neighborhood of zero.     

On minimal p-degrees in 2-transitive permutation groups

Suppose G is a transitive permutation group on a finite set W\mit\Omega of n points and let p be a prime divisor of |G||G|. The smallest number of points moved by a non-identity p-element is called the minimal p-degree of G and is denoted m_p (G). ¶ In the article the minimal p-degrees of various 2-transitive permutation groups are calculated. Using the classification of finite 2-transitive permutation groups these results yield the main theorem, that m_p(G) 3 [(p-1)/(p+1)] ·|W|m_{p}(G) \geq {{p-1} \over {p+1}} \cdot |\mit\Omega | holds, if Alt(W) \nleqq G {\rm Alt}(\mit\Omega ) \nleqq G .¶Also all groups G (and prime divisors p of |G||G|) for which m_p(G) ￡ [(p-1)/(p)] ·|W|m_{p}(G)\le {{p-1}\over{p}} \cdot |\mit\Omega | are identified.     

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.     

Invariants of free Lie algebras

Let k be a principal ideal domain with identity and characteristic zero. For a positive integer n, with n \geqq 2n \geqq 2, let H(n) be the group of all n x n matrices having determinant ±1\pm 1. Further, we write SL(n) for the special linear group. Let L be a free Lie algebra (over k) of finite rank n. We prove that the algebra of invariants L^B(n) of B(n), with B(n) ? { H(n), SL(n)}B(n) \in \{ H(n), {\rm SL}(n)\} , is not a finitely generated free Lie algebra. Let us assume that k is a field of characteristic zero and let áSem(n) ?\langle {\rm Sem}(n) \rangle be the Lie subalgebra of L generated by the semi-invariants (or Lie invariants) Sem(n). We prove that áSem(n) ?\langle {\rm Sem}(n) \rangle is not a finitely generated free Lie algebra which gives a positive answer to a question posed by M. Burrow [4].     

On Lipschitz selections of affine-set valued mappings

We prove a Helly-type theorem for the family of all k-dimensional affine subsets of a Hilbert space H. The result is formulated in terms of Lipschitz selections of set-valued mappings from a metric space (M,r) ({\cal M},\rho) into this family.¶Let F be such a mapping satisfying the following condition: for every subset M￠ ì M {\cal M'} \subset {\cal M} consisting of at most 2^k+1 points, the restriction F|_M￠ F|_{\cal M'} of F to M￠ {\cal M'} has a selection f_M￠ (i.e. f_M￠(x) ? F(x) for all x  ? M￠) f_{\cal M'}\,({\rm i.e.}\,f_{\cal M'}(x) \in F(x)\,{\rm for\,all}\,x\,\in {\cal M'}) satisfying the Lipschitz condition ||f_M￠(x) - f_M￠(y)||  ￡ r(x,y ), x,y ? M￠ \parallel f_{\cal M'}(x) - f_{\cal M'}(y)\parallel\,\le \rho(x,y ),\,x,y \in {\cal M'} . Then F has a Lipschitz selection f : M ? H f : {\cal M} \to H such that ||f(x) - f(y) ||  ￡ gr(x,y ), x,y ? M \parallel f(x) - f(y) \parallel\,\le \gamma \rho (x,y ),\,x,y \in {\cal M} where g = g(k) \gamma = \gamma(k) is a constant depending only on k. (The upper bound of the number of points in M￠ {\cal M'} , 2^k+1, is sharp.)¶The proof is based on a geometrical construction which allows us to reduce the problem to an extension property of Lipschitz mappings defined on subsets of metric trees.     

Orbits of permutation groups on the power set

Let G be a permutation group on a finite set W\Omega . If G does not involve A_n for n \geqq 5 n \geqq 5 , then there exist two disjoint subsets of W\Omega such that no Sylow subgroup of G stabilizes both and four disjoint subsets of W\Omega whose stabilizers in G intersect trivially.     

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.     

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.     

On the singularities of residual intertwining operators

Let G be a reductive algebraic group defined over \Bbb Q {\Bbb Q} . Let P, P' be parabolic subgroups of G, defined over \Bbb Q {\Bbb Q} , and let _boxclose_boxclose, a_P') t \in W({\frak a}_{P}, {\frak a}_{P'}) . In this paper we study the intertwining operator M_P￠|P(t,l), l ? \frak a^*_{P,\Bbb C} M_{P' \vert P}(t,\lambda),\,\lambda \in {\frak a}^*_{P,{\Bbb C}} , acting in corresponding spaces of automorphic forms. One of the main results states that each matrix coefficient of M_P￠|P(t,l) M_{P' \vert P}(t,\lambda) is a meromorphic function of order ￡ n + 1 \le n + 1 , where n = dim G. Using this result, we further investigate the rank one intertwining operators, in particular, we study the distribution of their poles.     

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.     

Kazhdan's property (T), L^2-spectrum and isoperimetric inequalities for locally symmetric spaces

Let V = G\G/KV =\Gamma\backslash G/K be a Riemannian locally symmetric space of nonpositive sectional curvature and such that the isometry group G of its universal covering space has Kazhdan's property (T). We establish strong dichotomies between the finite and infinite volume case. In particular, we characterize lattices (or, equivalently, arithmetic groups) among discrete subgroups G ì G\Gamma\subset G in various ways (e.g., in terms of critical exponents, the bottom of the spectrum of the Laplacian and the behaviour of the Brownian motion on V).     

Finding the Best Face on a Voronoi Polyhedron – The Strong Dodecahedral Conjecture Revisited

In this paper we prove the following theorem. The surface area density of a unit ball in any face cone of a Voronoi cell in an arbitrary packing of unit balls of Euclidean 3-space is at most [(-9p+ 30 arccos([(?3)/2]sin ([(p)/5]) ))/(5 tan([(p)/5]))]=0.77836?,{-9\pi + 30\,{\rm arccos}\left({\sqrt{3}\over 2}{\rm sin}\,\left({\pi\over 5}\right) \right)\over 5\, {\rm tan}\left({\pi\over 5}\right)}=0.77836\ldots, and so the surface area of any Voronoi cell in a packing with unit balls in Euclidean 3-space is at least [(20p· tan ( [(p)/5]))/(-9p+ 30 arccos([(?3)/2]sin ([(p)/5]) ))]=16.1445? .{20\pi\cdot\,{\rm tan}\,\left( {\pi \over 5}\right) \over -9\pi + 30\,{\rm arccos}\left({\sqrt{3}\over 2}{\rm sin}\,\left({\pi\over 5}\right) \right)}=16.1445\ldots\ . This result and the ideas of its proof support the Strong Dodecahedral Conjecture according to which the surface area of any Voronoi cell in a packing with unit balls in Euclidean 3-space is at least as large as 16.6508..., the surface area of a regular dodecahedron of inradius 1.