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.     

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.     

Quermassintegrals of a random polytope in a convex body

Let K be a convex body in \mathbbRⁿ \mathbb{R}^n with volume |K| = 1 |K| = 1 . We choose N 3 n+1 N \geq n+1 points x₁,?, x_N x_1,\ldots, x_N independently and uniformly from K, and write C(x₁,?, x_N) C(x_1,\ldots, x_N) for their convex hull. Let f : \mathbbR⁺ ? \mathbbR⁺ f : \mathbb{R^+} \rightarrow \mathbb{R^+} be a continuous strictly increasing function and 0 ￡ i ￡ n-1 0 \leq i \leq n-1 . Then, the quantity¶¶E (K, N, f °W_i) = ò_K ?ò_K f[W_i(C(x₁, ?, x_N))]dx_N ?dx₁ E (K, N, f \circ W_{i}) = \int\limits_{K} \ldots \int\limits_{K} f[W_{i}(C(x_1, \ldots, x_N))]dx_{N} \ldots dx_1 ¶¶is minimal if K is a ball (W_i is the i-th quermassintegral of a compact convex set). If f is convex and strictly increasing and 1 ￡ i ￡ n-1 1 \leq i \leq n-1 , then the ball is the only extremal body. These two facts generalize a result of H. Groemer on moments of the volume of C(x₁,?, x_N) C(x_1,\ldots, x_N) .     

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.     

Helly-type theorems for appropriate colorings of visibility sets

Summary. For positive integers q and n, think of P as the vertex set of a (qn + r)-gon, 0 ￡ r ￡ q - 1 0 \leq r \leq q - 1 . For 1 ￡ i ￡ qn + r 1 \leq i \leq qn + r , define V(i) to be a set of q consecutive points of P, starting at p(i), and let S be a subset of {V(i) : 1 ￡ i ￡ qn + r } \lbrace V(i) : 1 \leq i \leq qn + r \rbrace . A q-coloring of P = P(q) such that each member of S contains all q colors is called appropriate for S, and when 1 ￡ j ￡ q 1 \leq j \leq q , the definition may be extended to suitable subsets P(j) of P. If for every 1 ￡ j ￡ q 1 \leq j \leq q and every corresponding P(j), P(j) has a j-coloring appropriate for S, then we say P = P(q) has all colorings appropriate for S. With this terminology, the following Helly-type result is established: Set P = P(q) has all colorings appropriate for S if and only if for every (2n + 1)-member subset T of S, P has all colorings appropriate for T. The number 2n + 1 is best possible for every r 3 1 r \geq 1 . Intermediate results for q-colorings are obtained as well.     

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.     

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.     

On the iterates of Euler's function

Asymptotic representations are given for the three sums ?_{n ￡ x} j(n)/j(j(n))\textstyle\sum\limits \limits _{n\le x} \varphi (n)/\varphi \bigl (\varphi (n)\bigr ), ?_{n ￡ x} j(n)/j(j(n))\textstyle\sum\limits \limits _{n\le x} \varphi (n)/\varphi \bigl (\varphi (n)\bigr ), ?_{n ￡ x} log j(n)/j(j(n)) ;  j\textstyle\sum\limits \limits _{n\le x}\ \log \, \varphi (n)/\varphi \bigl (\varphi (n)\bigr )\ ; \ \varphi is Euler's function.     

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.     

On conjugacy of some systems of functions

Summary. We investigate the bounded solutions j:[0,1]? X \varphi:[0,1]\to X of the system of functional equations¶¶j(f_k(x))=F_k(j(x)),    k=0,?,n-1,x ? [0,1] \varphi(f_k(x))=F_k(\varphi(x)),\;\;k=0,\ldots,n-1,x\in[0,1] ,(*)¶where X is a complete metric space, f₀,?,f_n-1:[0,1]?[0,1] f_0,\ldots,f_{n-1}:[0,1]\to[0,1] and F₀,...,F_n-1:X? X F_0,...,F_{n-1}:X\to X are continuous functions fulfilling the boundary conditions f₀(0) = 0, f_n-1(1) = 1, f_k+1(0) = f_k(1), F₀(a) = a,F_n-1(b) = b,F_k+1(a) = F_k(b), k = 0,?,n-2 f_{0}(0) = 0, f_{n-1}(1) = 1, f_{k+1}(0) = f_{k}(1), F_{0}(a) = a,F_{n-1}(b) = b,F_{k+1}(a) = F_{k}(b),\,k = 0,\ldots,n-2 , for some a,b ? X a,b\in X . We give assumptions on the functions f_k and F_k which imply the existence, uniqueness and continuity of bounded solutions of the system (*). In the case X = \Bbb C X= \Bbb C we consider some particular systems (*) of which the solutions determine some peculiar curves generating some fractals. If X is a closed interval we give a collection of conditions which imply respectively the existence of homeomorphic solutions, singular solutions and a.e. nondifferentiable solutions of (*).     

Moser's second one-dimensional inequality for a class of integral operators

Suppose that $1 < p < \infty $1 < p < \infty , q=p/(p-1)q=p/(p-1), and for non-negative f ? L^p(-￥ ,￥)f\in L^p(-\infty\! ,\infty ) and any real x we let F(x)-F(0)=ò₀^xf(t) dtF(x)-F(0)=\int _0^xf(t)\ dt; suppose in addition that ò_-￥^￥ F(t)exp(-|t|) dt=0\int\limits _{-\infty }^\infty F(t)\exp (-|t|)\ dt=0. Moser's second one-dimensional inequality states that there is a constant C_pC_p, such that ò_-￥^￥ exp[a |F(x)|^q-|x|]  dx ￡ C_p\int\limits _{-\infty }^\infty \exp [a |F(x)|^q-|x|] \ dx\le C_p for each f with ||f||_p ￡ 1||f||_p\le 1 and every a ￡ 1a\le 1. Moreover the value a = 1 is sharp. We replace the operation connecting f with F by a more general integral operation; specifically we consider non-negative kernels K(t,x) with the property that xK(t,x) is homogeneous of degree 0 in t, x. We state an analogue of the inequality above for this situation, discuss some applications and consider the sharpness of the constant which replaces a.     

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.     

On the mean square of the error term for the asymmetric two-dimensional divisor problems(II)

Let Δ(a, b; x) denote the error term of the asymmetric two-dimensional divisor problem. In this paper we shall study the relation between the discrete mean value ?_{n ￡ T} D²(a,b;n){\sum_{n\leq T} \Delta^2(a,b;n)} and the continuous mean value ò₁^TD²(a,b;x)dx{\int_1^T\Delta^2(a,b;x)dx} .     

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\,) .     

Maximum IPP Codes of Length 3

Let Q be an alphabet with q elements. For any code C over Q of length n and for any two codewords a = (a ₁, . . . , a _n ) and b = (b ₁, . . . , b _n ) in C, let ${D({\bf a, b}) = \{(x_1, . . . , x_n) \in {Q^n} : {x_i} \in \{a_i, b_i\}\,{\rm for}\,1 \leq i \leq n\}}Let Q be an alphabet with q elements. For any code C over Q of length n and for any two codewords a = (a ₁, . . . , a _n ) and b = (b ₁, . . . , b _n ) in C, let D(a, b) = {(x₁, . . . , x_n) ? Qⁿ : x_i ? {a_i, b_i} for 1 ￡ i ￡ n}{D({\bf a, b}) = \{(x_1, . . . , x_n) \in {Q^n} : {x_i} \in \{a_i, b_i\}\,{\rm for}\,1 \leq i \leq n\}}. Let C^* = è_{a, b ? C}D(a, b){C^* = {{\bigcup}_{\rm {a,\,b}\in{C}}}D({\bf a, b})}. The code C is said to have the identifiable parent property (IPP) if, for any x ? C^*{{\rm {\bf x}} \in C^*}, ?_{x ? D(a, b)}{a, b} 1 ?{{\bigcap}_{{\rm x}{\in}D({\rm a,\,b})}\{{\bf a, b}\}\neq \emptyset} . Codes with the IPP were introduced by Hollmann et al [J. Combin. Theory Ser. A 82 (1998) 21–133]. Let F(n, q) = max{|C|: C is a q-ary code of length n with the IPP}.T? and Safavi-Naini [SIAM J. Discrete Math. 17 (2004) 548–570] showed that 3q + 6 - 6 é?{q+1}ù ￡ F(3, q) ￡ 3q + 6 - é6 ?{q+1}ù{3q + 6 - 6 \lceil\sqrt{q+1}\rceil \leq F(3, q) \leq 3q + 6 - \lceil 6 \sqrt{q+1}\rceil}, and determined F (3, q) precisely when q ≤ 48 or when q can be expressed as r ² + 2r or r ² + 3r +2 for r ≥ 2. In this paper, we establish a precise formula of F(3, q) for q ≥ 24. Moreover, we construct IPP codes of size F(3, q) for q ≥ 24 and show that, for any such code C and any x ? C^*{{\rm {\bf x}} \in C^*}, one can find, in constant time, a ? C{{\rm {\bf a}} \in C} such that if x ? D (c, d){{\rm {\bf x}} \in D ({\bf c, d})} then a ? {c, d}{{\rm {\bf a}} \in \{{\rm {\bf c, d}}\}}.     

Homogenization for strongly anisotropic nonlinear elliptic equations

In this paper we present homogenization results for elliptic degenerate differential equations describing strongly anisotropic media. More precisely, we study the limit as e? 0 \epsilon \to 0 of the following Dirichlet problems with rapidly oscillating periodic coefficients:¶¶ . \cases {{ -div(\alpha(\frac{x}{\epsilon}}, \nabla u) A(\frac{x}{\epsilon}) \nabla u) = f(x) \in L^{\infty}(\Omega) \atop u = 0 su \eth\Omega\ } ¶¶where, p > 1,     a: \Bbb Rⁿ ×\Bbb Rⁿ ? \Bbb R,     a(y,x) ? áA(y)x,x?^p/2-1, A ? M^{n ×n}(\Bbb R) p>1, \quad \alpha : \Bbb R^n \times \Bbb R^n \to \Bbb R, \quad \alpha(y,\xi) \approx \langle A(y)\xi,\xi \rangle ^{p/2-1}, A \in M^{n \times n}(\Bbb R) , A being a measurable periodic matrix such that A^t(x) = A(x) 3 0A^t(x) = A(x) \ge 0 almost everywhere.¶¶The anisotropy of the medium is described by the following structure hypothesis on the matrix A:¶¶l^2/p(x) |x|² ￡ áA(x)x,x? ￡ L ^2/p(x) |x|², \lambda^{2/p}(x) |\xi|^2 \leq \langle A(x)\xi,\xi \rangle \leq \Lambda ^{2/p}(x) |\xi|^2, ¶¶where the weight functions l \lambda and L \Lambda (satisfying suitable summability assumptions) can vanish or blow up, and can also be "moderately" different. The convergence to the homogenized problem is obtained by a classical compensated compactness argument, that had to be extended to two-weight Sobolev spaces.     

Essential norms of some singular integral operators

Let a\alpha and b\beta be bounded measurable functions on the unit circle T. The singular integral operator S_a, bS_{\alpha ,\,\beta } is defined by S_a, b f = aPf + bQf(f ? L² (T))S_{\alpha ,\,\beta } f = \alpha Pf + \beta Qf(f \in L^2 (T)) where P is an analytic projection and Q is a co-analytic projection. In the previous paper, the norm of S_a, bS_{\alpha ,\,\beta } was calculated in general, using a,b\alpha ,\beta and a[`(b)] + H<sup>￥</sup>\alpha \bar {\beta } + H^\infty where H<sup>￥</sup>H^\infty is a Hardy space in L<sup>￥</sup> (T).L^\infty (T). In this paper, the essential norm ||S<sub>a, b</sub> ||<sub>e</sub>\Vert S_{\alpha ,\,\beta } \Vert _e of S<sub>a, b</sub>S_{\alpha ,\,\beta } is calculated in general, using a[`(b)] + H^￥ + C\alpha \bar {\beta } + H^\infty + C where C is a set of all continuous functions on T. Hence if a[`(b)]\alpha \bar {\beta } is in H^￥ + CH^\infty + C then ||S_a, b ||_e = max(||a||_￥ , ||b||_￥ ).\Vert S_{\alpha ,\,\beta } \Vert _e = \max (\Vert \alpha \Vert _\infty , \Vert \beta \Vert _\infty ). This gives a known result when a, b\alpha , \beta are in C.     

Truncated Complex Moment Problems with a $ \widetilde{zz} $ Relation

We solve the truncated complex moment problem for measures supported on the variety K o \mathcal{K}\equiv { z ? \in C: z [(z)\tilde]\widetilde{z} = A+Bz+C [(z)\tilde]\widetilde{z} +Dz ² ,D 1 \neq 0}. Given a doubly indexed finite sequence of complex numbers g o g⁽²ⁿ⁾:g₀₀,g₀₁,g₁₀,?,g_0,2n,g_1,2n-1,?,g_2n-1,1,g_2n,0 \gamma\equiv\gamma^{(2n)}:\gamma_{00},\gamma_{01},\gamma_{10},\ldots,\gamma_{0,2n},\gamma_{1,2n-1},\ldots,\gamma_{2n-1,1},\gamma_{2n,0} , there exists a positive Borel measure m\mu supported in K \mathcal{K} such that g_ij=ò[`(z)]<sup>i</sup>z<sup>j</sup> dm (0 ￡ 1+j ￡ 2n) \gamma_{ij}=\int\overline{z}^{i}z^{j}\,d\mu\,(0\leq1+j\leq2n) if and only if the moment matrix M(n)( g\gamma ) is positive, recursively generated, with a column dependence relation Z [(Z)\tilde]\widetilde{Z} = A1+BZ +C [(Z)\tilde]\widetilde{Z} +DZ <sup>2</sup>, and card V(g) 3\mathcal{V}(\gamma)\geq rank M(n), where V(g)\mathcal{V}(\gamma) is the variety associated to g \gamma . The last condition may be replaced by the condition that there exists a complex number g<sub>n,n+1</sub> \gamma_{n,n+1} satisfying g<sub>n+1,n</sub> o [`(g)]_n,n+1=Ag_n,n-1+Bg_n,n+Cg_n+1,n-1+Dg_n,n+1 \gamma_{n+1,n}\equiv\overline{\gamma}_{n,n+1}=A\gamma_{n,n-1}+B\gamma_{n,n}+C\gamma_{n+1,n-1}+D\gamma_{n,n+1} . We combine these results with a recent theorem of J. Stochel to solve the full complex moment problem for K \mathcal{K} , and we illustrate the connection between the truncated and full moment problems for other varieties as well, including the variety z ^k = p(z, [(Z)\tilde] \widetilde{Z} ), deg p < k.     

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.     

The Theorem of the k-1 Happy Divorces

Given a binary relation R between the elements of two sets X and Y and a natural number k, it is shown that there exist k injective maps f₁, f₂,...,f_k: X \hookrightarrow Y X \hookrightarrow Y with # {f₁(x), f₂(x),...,f_k(x)}=k    and    (x,f₁(x)), (x, f₂(x)),...,(x, f_k(x)) ? R \# \{f_1(x), f_2(x),...,f_k(x)\}=k \quad{\rm and}\quad (x,f_1(x)), (x, f_2(x)),...,(x, f_k(x)) \in R for all x ? X x \in X if and only if the inequality k ·# A ￡ ?_{y ? Y} min(k, #{a ? A | (a,y) ? R}) k \cdot \# A \leq \sum_{y \in Y} min(k, \#\{a \in A \mid (a,y) \in R\}) holds for every finite subset A of X, provided {y ? Y | (x,y) ? R} \{y \in Y \mid (x,y) \in R\} is finite for all x ? X x \in X .¶Clearly, as suggested by this paper's title, this implies that, in the context of the celebrated Marriage Theorem, the elements x in X can (simultaneously) marry, get divorced, and remarry again a partner from their favourite list as recorded by R, for altogether k times whenever (a) the list of favoured partners is finite for every x ? X x \in X and (b) the above inequalities all hold.¶In the course of the argument, a straightforward common generalization of Bernstein's Theorem and the Marriage Theorem will also be presented while applications regarding (i) bases in infinite dimensional vector spaces and (ii) incidence relations in finite geometry (inspired by Conway's double sum proof of the de Bruijn-Erdös Theorem) will conclude the paper.