Covering and packing in \Bbb Zn{\Bbb Z}^n and \Bbb Rn{\Bbb R}^n, (I)

Given a finite subset A{\cal A} of an additive group \Bbb G{\Bbb G} such as \Bbb Zⁿ{\Bbb Z}^n or \Bbb Rⁿ{\Bbb R}^n , we are interested in efficient covering of \Bbb G{\Bbb G} by translates of A{\cal A} , and efficient packing of translates of A{\cal A} in \Bbb G{\Bbb G} . A set S ì \Bbb G{\cal S} \subset {\Bbb G} provides a covering if the translates A + s{\cal A} + s with s ? Ss \in {\cal S} cover \Bbb G{\Bbb G} (i.e., their union is \Bbb G{\Bbb G} ), and the covering will be efficient if S{\cal S} has small density in \Bbb G{\Bbb G} . On the other hand, a set S ì \Bbb G{\cal S} \subset {\Bbb G} will provide a packing if the translated sets A + s{\cal A} + s with s ? Ss \in {\cal S} are mutually disjoint, and the packing is efficient if S{\cal S} has large density. In the present part (I) we will derive some facts on these concepts when \Bbb G = \Bbb Zⁿ{\Bbb G} = {\Bbb Z}^n , and give estimates for the minimal covering densities and maximal packing densities of finite sets A ì \Bbb Zⁿ{\cal A} \subset {\Bbb Z}^n . In part (II) we will again deal with \Bbb G = \Bbb Zⁿ{\Bbb G} = {\Bbb Z}^n , and study the behaviour of such densities under linear transformations. In part (III) we will turn to \Bbb G = \Bbb Rⁿ{\Bbb G} = {\Bbb R}^n .     

Embedding metric spaces into normed spaces and estimates of metric capacity

Let M^d{\cal M}^d be an arbitrary real normed space of finite dimension d ≥ 2. We define the metric capacity of M^d{\cal M}^d as the maximal m ? \Bbb Nm \in {\Bbb N} such that every m-point metric space is isometric to some subset of M^d{\cal M}^d (with metric induced by M^d{\cal M}^d ). We obtain that the metric capacity of M^d{\cal M}^d lies in the range from 3 to ë\frac32d û+1\left\lfloor\frac{3}{2}d\right\rfloor+1 , where the lower bound is sharp for all d, and the upper bound is shown to be sharp for d ∈ {2, 3}. Thus, the unknown sharp upper bound is asymptotically linear, since it lies in the range from d + 2 to ë\frac32d û+1\left\lfloor\frac{3}{2}d\right\rfloor+1 .     

A rigidity theorem for the pair ${\cal q}{\Bbb C} P^n$ (complex hyperquadric, complex projective space)

Given a compact Kähler manifold M of real dimension 2n, let P be either a compact complex hypersurface of M or a compact totally real submanifold of dimension n. Let q\cal q (resp. \Bbb R Pⁿ{\Bbb R} P^n) be the complex hyperquadric (resp. the totally geodesic real projective space) in the complex projective space \Bbb C Pⁿ{\Bbb C} P^n of constant holomorphic sectional curvature 4l \lambda . We prove that if the Ricci and some (n-1)-Ricci curvatures of M (and, when P is complex, the mean absolute curvature of P) are bounded from below by some special constants and volume (P) / volume (M) ￡\leq volume (q\cal q)/ volume (\Bbb C Pⁿ)({\Bbb C} P^n) (resp. ￡\leq volume (\Bbb R Pⁿ)({\Bbb R} P^n) / volume (\Bbb C Pⁿ)({\Bbb C} P^n)), then there is a holomorphic isometry between M and \Bbb C Pⁿ{\Bbb C} P^n taking P isometrically onto q\cal q (resp. \Bbb R Pⁿ{\Bbb R} P^n). We also classify the Kähler manifolds with boundary which are tubes of radius r around totally real and totally geodesic submanifolds of half dimension, have the holomorphic sectional and some (n-1)-Ricci curvatures bounded from below by those of the tube \Bbb R Pⁿ_r{\Bbb R} P^n_r of radius r around \Bbb R Pⁿ{\Bbb R} P^n in \Bbb C Pⁿ{\Bbb C} P^n and have the first Dirichlet eigenvalue not lower than that of \Bbb R Pⁿ_r{\Bbb R} P^n_r.     

Conjugates of Rational Equivariant Holomorphic Maps of Symmetric Domains

Let t: D ?D￠\tau: {\cal D} \rightarrow{\cal D}^\prime be an equivariant holomorphic map of symmetric domains associated to a homomorphism r: \Bbb G ?\Bbb G￠{\bf\rho}: {\Bbb G} \rightarrow{\Bbb G}^\prime of semisimple algebraic groups defined over \Bbb Q{\Bbb Q} . If G ì \Bbb G (\Bbb Q)\Gamma\subset {\Bbb G} ({\Bbb Q}) and G￠ ì \Bbb G￠(\Bbb Q)\Gamma^\prime \subset {\Bbb G}^\prime ({\Bbb Q}) are torsion-free arithmetic subgroups with r (G) ì G￠{\bf\rho} (\Gamma) \subset \Gamma^\prime , the map G\D ?G￠\D￠\Gamma\backslash {\cal D} \rightarrow\Gamma^\prime \backslash {\cal D}^\prime of arithmetic varieties and the rationality of D{\cal D} and D￠{\cal D}^\prime as well as the commensurability groups of s ? Aut (\Bbb C)\sigma \in {\rm Aut} ({\Bbb C}) determines a conjugate equivariant holomorphic map t^s: D^s ?D^￠s\tau^\sigma: {\cal D}^\sigma \rightarrow{\cal D}^{\prime\sigma} of f^s: (G\D)^s ?(G￠\D￠)^s\phi^\sigma: (\Gamma\backslash {\cal D})^\sigma \rightarrow(\Gamma^\prime \backslash {\cal D}^\prime)^\sigma of . We prove that is rational if is rational.     

Successive Minima and Best Simultaneous Diophantine Approximations

We study the problem of best approximations of a vector a ? \Bbb Rⁿ\alpha\in{\Bbb R}^n by rational vectors of a lattice L ì \Bbb Rⁿ\Lambda\subset{\Bbb R}^n whose common denominator is bounded. To this end we introduce successive minima for a periodic lattice structure and extend some classical results from geometry of numbers to this structure. This leads to bounds for the best approximation problem which generalize and improve former results.     

Analytic Fourier integral operators, Monge-Ampère equation and holomorphic factorization

We will show that the factorization condition for the Fourier integral operators I_r ^m (X,Y;L )I_\rho ^\mu (X,Y;\it\Lambda ) leads to a parametrized parabolic Monge-Ampère equation. For an analytic operator, the fibration by the kernels of the Hessian of phase function is shown to be analytic in a number of cases, by considering a more general continuation problem for the level sets of a holomorphic mapping. The results are applied to obtain L^p-continuity for translation invariant operators in \Bbb Rⁿ{\Bbb R}^n with n ￡ 4n\leq 4 and for arbitrary \Bbb Rⁿ{\Bbb R}^n with dp_X×Y|_L ￡ n+2d\pi _{X\times Y}|_\Lambda \leq n+2.     

A Blichfeldt-type inequality for the surface area

In 1921, Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that #(K?\Bbb Zⁿ) ￡ n! vol(K)+n\#(K\cap{\Bbb Z}^n)\le n! {\rm vol}(K)+n , whenever K ì \Bbb RⁿK\subset{\Bbb R}^n is a convex body containing n + 1 affinely independent integral points. Here we prove an analogous inequality with respect to the surface area F(K), namely #(K?\Bbb Zⁿ) < vol(K) + ((?n+1)/2) (n-1)! F(K)\#(K\cap{\Bbb Z}^n) < {\rm vol}(K) + ((\sqrt{n}+1)/2) (n-1)! {\rm F}(K) . The proof is based on a slight improvement of Blichfeldt’s bound in the case when K is a non-lattice translate of a lattice polytope, i.e., K = t + P, where t ? \Bbb Rⁿ\\Bbb Zⁿt\in{\Bbb R}^n\setminus{\Bbb Z}^n and P is an n-dimensional polytope with integral vertices. Then we have #((t+P)?\Bbb Zⁿ) ￡ n! vol(P)\#((t+P)\cap{\Bbb Z}^n)\le n! {\rm vol}(P) . Moreover, in the 3-dimensional case we prove a stronger inequality, namely #(K?\Bbb Zⁿ) < vol(K) + 2 F(K)\#(K\cap{\Bbb Z}^n)< {\rm vol}(K) + 2 {\rm F}(K) .     

Constant angle surfaces in \Bbb S2×\Bbb R {\Bbb S}^2\times {\Bbb R}

In this article we study surfaces in \Bbb S²×\Bbb R {\Bbb S}^2\times {\Bbb R} for which the unit normal makes a constant angle with the \Bbb R {\Bbb R} -direction. We give a complete classification for surfaces satisfying this simple geometric condition.     

Quantization of Lie bialgebras and shuffle algebras of Lie algebras

To any field \Bbb K \Bbb K of characteristic zero, we associate a set (\mathbbK) (\mathbb{K}) and a group G₀(\Bbb K) {\cal G}_0(\Bbb K) . Elements of (\mathbbK) (\mathbb{K}) are equivalence classes of families of Lie polynomials subject to associativity relations. Elements of G₀(\Bbb K) {\cal G}_0(\Bbb K) are universal automorphisms of the adjoint representations of Lie bialgebras over \Bbb K \Bbb K . We construct a bijection between (\mathbbK)×G₀(\Bbb K) (\mathbb{K})\times{\cal G}_0(\Bbb K) and the set of quantization functors of Lie bialgebras over \Bbb K \Bbb K . This construction involves the following steps.? 1) To each element v \varpi of (\mathbbK) (\mathbb{K}) , we associate a functor \frak a?\operatornameSh^v(\frak a) \frak a\mapsto\operatorname{Sh}^\varpi(\frak a) from the category of Lie algebras to that of Hopf algebras; \operatornameSh^v(\frak a) \operatorname{Sh}^\varpi(\frak a) contains U\frak a U\frak a .? 2) When \frak a \frak a and \frak b \frak b are Lie algebras, and r_{\frak a\frak b} ? \frak a?\frak b r_{\frak a\frak b} \in\frak a\otimes\frak b , we construct an element ?^v (r_{\frak a\frak b}) {\cal R}^{\varpi} (r_{\frak a\frak b}) of \operatornameSh^v(\frak a)?\operatornameSh^v(\frak b) \operatorname{Sh}^\varpi(\frak a)\otimes\operatorname{Sh}^\varpi(\frak b) satisfying quasitriangularity identities; in particular, ?^v(r_{\frak a\frak b}) {\cal R}^\varpi(r_{\frak a\frak b}) defines a Hopf algebra morphism from \operatornameSh^v(\frak a)^* \operatorname{Sh}^\varpi(\frak a)^* to \operatornameSh^v(\frak b) \operatorname{Sh}^\varpi(\frak b) .? 3) When \frak a = \frak b \frak a = \frak b and r_\frak a ? \frak a?\frak a r_\frak a\in\frak a\otimes\frak a is a solution of CYBE, we construct a series r^v(r_\frak a) \rho^\varpi(r_\frak a) such that ?^v(r^v(r_\frak a)) {\cal R}^\varpi(\rho^\varpi(r_\frak a)) is a solution of QYBE. The expression of r^v(r_\frak a) \rho^\varpi(r_\frak a) in terms of r_\frak a r_\frak a involves Lie polynomials, and we show that this expression is unique at a universal level. This step relies on vanishing statements for cohomologies arising from universal algebras for the solutions of CYBE.? 4) We define the quantization of a Lie bialgebra \frak g \frak g as the image of the morphism defined by ?^v(r^v(r)) {\cal R}^\varpi(\rho^\varpi(r)) , where r ? \mathfrakg ?\mathfrakg^* r \in \mathfrak{g} \otimes \mathfrak{g}^* .<\P>     

Idempotent multipliers for LP (\Bbb R)(\Bbb R)

The algebra B_p(\Bbb R){\cal B}_p({\Bbb R}), p ? (1,￥)\{2}p\in (1,\infty )\setminus \{2\}, consisting of all measurable sets in \Bbb R{\Bbb R} whose characteristic function is a Fourier p-multiplier, forms an algebra of sets containing many interesting and non-trivial elements (e.g. all intervals and their finite unions, certain periodic sets, arbitrary countable unions of dyadic intervals, etc.). However, B_p(\Bbb R){\cal B}_p({\Bbb R}) fails to be a s\sigma -algebra. It has been shown by V. Lebedev and A. Olevskii [4] that if E ? B_p(\Bbb R)E\in {\cal B}_p({\Bbb R}), then E must coincide a.e. with an open set, a remarkable topological constraint on E. In this note we show if $2 < p < \infty $2 < p < \infty , then there exists E ? B_p(\Bbb R)E\in {\cal B}_p({\Bbb R}) which is not in B_q(\Bbb R){\cal B}_q({\Bbb R}) for any q > pq>p.     

Lagrangian barriers and symplectic embeddings

We prove that every symplectic Kähler manifold (M;W) (M;\Omega) with integral [W] [\Omega] decomposes into a disjoint union (M,W) = (E,w₀) \coprod D (M,\Omega) = (E,\omega_0) \coprod \Delta , where (E,w₀) (E,\omega_0) is a disc bundle endowed with a standard symplectic form w₀ \omega_0 and D \Delta is an isotropic CW-complex. We perform explicit computations of this decomposition on several examples.¶As an application we establish the following symplectic intersection phenomenon: There exist symplectically irremovable intersections between contractible domains and Lagrangian submanifolds. For example, we prove that every symplectic embedding j:B²ⁿ(l) ? \Bbb CPⁿ \varphi:B^{2n}(\lambda) \to {\Bbb C}P^n of a ball of radius l² 3 1/2 \lambda^2 \ge 1/2 must intersect the standard Lagrangian real projective space \Bbb RPⁿ ì \Bbb CPⁿ {\Bbb R}P^n \subset {\Bbb C}P^n .     

Geometric progressions in sumsets over finite fields

Given two sets A, B í \Bbb F_q^d{\cal A}, {\cal B}\subseteq {\Bbb F}_q^d , the set of d dimensional vectors over the finite field \Bbb F_q{\Bbb F}_q with q elements, we show that the sumset A+B = {a+b | a ? A, b ? B}{\cal A}+{\cal B} = \{{\bf a}+{\bf b}\ \vert\ {\bf a} \in {\cal A}, {\bf b} \in {\cal B}\} contains a geometric progression of length k of the form vΛ ^j , where j = 0,…, k − 1, with a nonzero vector v ? \Bbb F_q^d{\bf v} \in {\Bbb F}_q^d and a nonsingular d × d matrix Λ whenever # A # B 3 20 q^2d-2/k\# {\cal A} \# {\cal B} \ge 20 q^{2d-2/k} . We also consider some modifications of this problem including the question of the existence of elements of sumsets on algebraic varieties.     

Some blow-up results for a generalized Ginzburg-Landau equation

We investigate the blow-up of the solution to a complex Ginzburg-Landau like equation in u coupled with a Poisson equation in f\phi defined on the whole space \Bbb Rⁿ, n = 1{\Bbb R}^n, n = 1 or 2.     

Blocks with cyclic defect of Hecke orders of Coxeter groups

Let B\cal B be a p-block of cyclic defect of a Hecke order over the complete ring \Bbb Z[q] _{áq-1,p ?}\Bbb {Z}[q] _{\langle q-1,p \rangle}; i.e. modulo áq-1 ?\langle q-1 \rangle it is a p-block B of cyclic defect of the underlying Coxeter group G. Then B\cal B is a tree order over \Bbb Z[q]_{áq-1, p ?}\Bbb {Z}[q]_{\langle q-1, p \rangle } to the Brauer tree of B. Moreover, in case B\cal B is the principal block of the Hecke order of the symmetric group S(p) on p elements, then B\cal B can be described explicitly. In this case a complete set of non-isomorphic indecomposable Cohen-Macaulay B\cal B-modules is given.     

Surfaces with affine Gauss-Kronecker curvature zero

We classify surfaces with affine Gauss-Kronecker curvature zero in locally and globally. If x:M ? A³x:M \rightarrow A^3 is Euclidean complete and its affine conormal surface U:M ? \Bbb R³U:M \rightarrow \Bbb R^3 is complete, thenx(M) is an elliptic paraboloid or unimodular affine equivalent to a surface determined by equation x²₁-x²₂+C² x^2_1-x^2_2+C^2\,cos h²x₃=0\,\hbox {h}^2x_3=0 where x₂ \leqq Cx_2 \leqq C and C is a positive constant.     

Complex hyperbolic hyperplane complements

We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n − 1)-submanifold S. The main result is that the fundamental group of M\ S{M{\setminus} S} is relatively hyperbolic, relative to fundamental groups of the ends of M\ S{M{\setminus} S} , and M\ S{M{\setminus} S} admits a complete finite volume A-regular Riemannian metric of negative sectional curvature. It follows that for n > 1 the fundamental group of M\ S{M{\setminus} S} satisfies Mostow-type Rigidity, has solvable word and conjugacy problems, has finite asymptotic dimension and rapid decay property, satisfies Borel and Baum-Connes conjectures, is co-Hopf and residually hyperbolic, has no nontrivial subgroups with property (T), and has finite outer automorphism group. Furthermore, if M is compact, then the fundamental group of M\ S{M{\setminus} S} is biautomatic and satisfies Strong Tits Alternative.     

Jordan Triple Product Homomorphisms

Nondegenerate mappings that preserve Jordan triple product on M_n(\Bbb F){\cal M}_n({\Bbb F}) are characterized. Here, n ≥ 3 and \Bbb F{\Bbb F} is an arbitrary field.     

Compact Surfaces with Constant Gaussian Curvature in Product Spaces

We prove that the only compact surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} (resp. positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}}) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant angle along Γ, are the pieces of a rotational complete surface. We also obtain some area estimates for surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} and positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}} whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds are attained, the surface is again a piece of a rotational complete surface.     

The behaviour of microstructures with small shears of the austenite-martensite interface in martensitic phase transformations

Let W ì \BbbR²\Omega \subset \Bbb{R}^2 denote a bounded domain whose boundary ?W\partial \Omega is Lipschitz and contains a segment G₀\Gamma_0 representing the austenite-twinned martensite interface. We prove inf_{u ? W(W)} ò_W j(?u(x,y))dxdy=0\displaystyle{\inf_{{u\in \cal W}(\Omega)} \int_\Omega \varphi(\nabla u(x,y))dxdy=0}