On holomorphic isometric embeddings of the unit n-ball into products of two unit m-balls

We study holomorphic isometric embeddings of the complex unit n-ball into products of two complex unit m-balls with respect to their Bergman metrics up to normalization constants (the isometric constant). There are two trivial holomorphic isometric embeddings for m ?? n, given by F ₁(z)?=?(0, I _n;m (z)) with the isometric constant equal to (m?+?1)/(n?+?1) and F ₂(z)?=?(I _n;m (z), I _n;m (z)) with the isometric constant equal to 2(m?+?1)/(n?+?1). Here ${I_{n;m}:\mathbb{C}^n \longrightarrow \mathbb{C}^m}$ is the canonical embedding. We prove that when m < 2n, these are the only holomorphic isometric embeddings up to unitary transformations.     

Linear maps preserving M–P inverses of matrices between matrix spaces over fields

Suppose F is a field of characteristic not 2. Let n and m be two arbitrary positive integers with n≥2. We denote by M _n (F) and S _n (F) the space of n×n full matrices and the space of n×n symmetric matrices over F, respectively. All linear maps from S _n (F) to M _m (F) preserving M–P inverses of matrices are characterized first, and thereby all linear maps from S _n (F) (M _n (F)) to S _m (F) (M _m (F)) preserving M–P inverses of matrices are characterized, respectively.     

Difference Sets in {\mathbb{Z}}_4^m and {\mathbb{F}}_2^{2m}

Let R=GR(4,m) be the Galois ring of cardinality 4^m and let T be the Teichmüller system of R. For every map λ of T into { -1,+1} and for every permutation Π of T, we define a map φ _λ ^Π of Rinto { -1,+1} as follows: if x∈R and if x=a+2b is the 2-adic representation of x with x∈T and b∈T, then φ _λ ^Π (x)=λ(a)+2Tr(Π(a)b), where Tr is the trace function of R . For i=1 or i=-1, define D _i as the set of x in R such thatφ _λ ^Π =i. We prove the following results: 1) D _i is a Hadamard difference set of (R,+). 2) If φ is the Gray map of R into ${\mathbb{F}}_2^{2m}$ , then (D _i) is a difference set of ${\mathbb{F}}_2^{2m}$ . 3) The set of D _i and the set of φ(D _i) obtained for all maps λ and Π, both are one-to-one image of the set of binary Maiorana-McFarland difference sets in a simple way. We also prove that special multiplicative subgroups of R are difference sets of kind D _i in the additive group of R. Examples are given by means of morphisms and norm in R.     

Additive operators preserving rank-additivity on symmetry matrix spaces

We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. LetS _n(F) be the space of alln×n symmetry matrices over a fieldF with 2,3 ∈F ^*, thenT is an additive injective operator preserving rank-additivity onS _n(F) if and only if there exists an invertible matrixU∈M _n(F) and an injective field homomorphism ? ofF to itself such thatT(X)=cUX ^?U^T, ?X=(x_ij)∈S_n(F) wherec∈F ^*,X ^?=(?(x _ij)). As applications, we determine the additive operators preserving minus-order onS _n(F) over the fieldF.     

Følner sequences and Hilbert’s Irreducibility Theorem over Q

Letf(X; T ₁, ...,T _n) be an irreducible polynomial overQ. LetB be the set ofb teZ ⁿ such thatf(X;b) is of lesser degree or reducible overQ. Let ?={F _j}{F _j } _j?1 ^∞ be a Følner sequence inZ ⁿ — that is, a sequence of finite nonempty subsetsF _j ?Z ⁿ such that for eachvteZ ⁿ , $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap (F_j + \upsilon )} \right|}}{{\left| {F_j } \right|}} = 1$ Suppose ? satisfies the extra condition that forW a properQ-subvariety ofP ⁿ ?A ⁿ and ?>0, there is a neighborhoodU ofW(R) in the real topology such that $\mathop {lim sup}\limits_{j \to \infty } \frac{{\left| {F_j \cap U} \right|}}{{\left| {F_j } \right|}}< \varepsilon $ whereZ ⁿ is identified withA ⁿ (Z). We prove $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap B} \right|}}{{\left| {F_j } \right|}} = 0$ .     

On diversity and stability of unit bases for the Euclidean metric

We prove the existence of a family Ω(n) of 2 ^c (where c is the cardinality of the continuum) subgraphs of the unit distance graph (E ⁿ , 1) of the Euclidean space E ⁿ , n ≥ 2, such that (a) for each graph G ? Ω(n), any homomorphism of G to (E ⁿ , 1) is an isometry of E ⁿ ; moreover, for each subgraph G ₀ of the graph G obtained from G by deleting less than c vertices, less than c stars, and less than c edges (we call such a subgraph reduced), any homomorphism of G ₀ to (E ⁿ , 1) is an isometry (of the set of the vertices of G ₀); (b) each graph G ? Ω(n) cannot be homomorphically mapped to any other graph of the family Ω(n), and the same is true for each reduced subgraph of G.     

Contraction mappings in interval analysis

Under certain conditions, the contraction mapping fixed point theorem guarantees the convergence of the iterationx _i+1=f(x _i ) toward a fixed point of the functionf:R ⁿ →R ⁿ. When an interval extensionF off is used in a similar iteration scheme to obtain a sequence of interval vectors these conditions need not provide convergence to a degenerate interval vector representing the fixed point, even if the width of the initial interval vector is chosen arbitrarily small. We give a sufficient condition on the extensionF in order that the convergence is guaranteed. The centered form of Moore satisfies this condition.     

A sharp weightedL 2-estimate for the solution to the time-dependent Schrödinger equation

For Ξ∈R ⁿ ,t∈R andf∈S(R ⁿ ) define $\left( {S^2 f} \right)\left( t \right)\left( \xi \right) = \exp \left( {it\left| \xi \right|^2 } \right)\hat f\left( \xi \right)$ . We determine the optimal regularitys ₀ such that $\int_{R^n } {\left\| {(S^2 f)[x]} \right\|_{L^2 (R)}^2 \frac{{dx}}{{(1 + |x|)^b }} \leqslant C\left\| f \right\|_{H^s (R^n )}^2 ,s > s_0 } ,$ holds whereC is independent off∈S(R ⁿ ) or we show that such optimal regularity does not exist. This problem has been treated earlier, e.g. by Ben-Artzi and Klainerman [2], Kato and Yajima [4], Simon [6], Vega [9] and Wang [11]. Our theorems can be generalized to the case where the exp(it|ξ|²) is replaced by exp(it|ξ|^a),a≠2. The proof uses Parseval's formula onR, orthogonality arguments arising from decomposingL ²(R ⁿ ) using spherical harmonics and a uniform estimate for Bessel functions. Homogeneity arguments are used to show that results are sharp with respect to regularity.     

The maximal Fejér operator on the spacesH p (R×···×R)

It is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from thed-dimensional Hardy spaceH _p (R×···×R) toL _p (R ^d ) (1/2<p<∞) and is of weak type (H ₁ ^?i ,L ₁) (i=1,…,d), where the Hardy spaceH ₁ ^?i is defined by a hybrid maximal function. As a consequence, we obtain that the Fejér means of a functionf ∈H ₁ ^?i ?L(logL) ^d?1 converge a.e. to the function in question. Moreover, we prove that the Fejér means are uniformly bounded onH _p (R×···×R) whenever 1/2<p<∞. Thus, in casef ∈H _p (R×···×R) the Fejér means converge tof inH _p (R×···×R) norm. The same results are proved for the conjugate Fejér means, too.     

Minimal polynomials for compact sets of the complex plane

LetW _N(z)=a_Nz^N+... be a complex polynomial and letT _n be the classical Chebyshev polynomial. In this article it is shown that the polynomials (2a_N)^?n+1T_n(W_N), n ∈N, are minimal polynomials on all equipotential lines for {z∈C:|W _N(z)|≤1 Λ ImW _N(z)=0}     

Right inverses for linear, constant coefficient partial differential operators on distributions over open half spaces

Results of Hörmander on evolution operators together with a characterization of the present authors [Ann. Inst. Fourier, Grenoble 40, 619–655 (1990)] are used to prove the following: Let P ∈ ?[z₁,...,z _n ] and denote by P _m its principal part. If P ? P_m is dominated by P _m then the following assertions for the partial differential operators P(D) and P _m(D) are equivalent for N ∈ S ^n?1:

1. P(D) and/or P_m D)admit a continuous linear right inverse on C ^∞(H ₊(N)).
2. P(D) admits a continuous linear right inverse on C ^∞ (? ⁿ ) and a fundamental solution E ∈ C ^∞(?ⁿ) satisfying Supp $E \subset \overline {H - (N)} $

where H ₊(N) := {x ∈ ? ⁿ :±(x,N) τ; 0}.     

Periodic boundary value problem of functional differential equations with perturbation

By coincidence degree, the existence of solution to the periodic boundary value problem of functional differential equations with perturbation     

Congruences of multipartition functions modulo powers of primes

Let p _r (n) denote the number of r-component multipartitions of n, and let S _γ,λ be the space spanned by η(24z) ^γ ?(24z), where η(z) is the Dedekind’s eta function and ?(z) is a holomorphic modular form in \(M_{\lambda}(\mathrm{SL}_{2}(\mathbb{Z}))\) . In this paper, we show that the generating function of \(p_{r}(\frac{m^{k} n +r}{24})\) with respect to n is congruent to a function in the space S _γ,λ modulo m ^k . As special cases, this relation leads to many well known congruences including the Ramanujan congruences of p(n) modulo 5,7,11 and Gandhi’s congruences of p ₂(n) modulo 5 and p ₈(n) modulo 11. Furthermore, using the invariance property of S _γ,λ under the Hecke operator \(T_{\ell^{2}}\) , we obtain two classes of congruences pertaining to the m ^k -adic property of p _r (n).     

Ombres Convexité, régularité et sous-harmonicité

Letf:V→R be a function defined on a subsetV ofR ⁿ ×R ^d let?:x→inf{f(x t);t such that(x t)∈V} denote theshadow off and letΦ={(x t)∈V; f(x t)=?(x)} This paper deals with the characterization of some properties of ? in terms of the infinitesimal behavior off near points ζ∈Φ proving in particular a conjecture of J M Trépreau concerning the cased=1 Characterizations of this type are provided for the convexity the subharmonicity or theC ^{1 1} regularity of ? in the interior ofI={x∈ R ⁿ;εR ^d (x t)∈V} and in theC ^{1 1} case an expression forD ²? is given To some extent an answer is given to the following question: which convex function ?:I→R I interval ?R (resp which function √:I→R of classC ^{1 1}) is the shadow of aC ² functionf:I×R→R?     

Strong capacity-estimates for “fractional” norms

It is proved that for all fractionall the integral \(\int\limits_0^\infty {(p,\ell ) - cap(M_t )} dt^p\) is majorized by the P-th power norm of the functionu in the space ? _p ^l (Rⁿ) (here M_t={x∶¦u(x)¦?t} and (p,l)-cap(e) is the (p,l)-capacity of the compactum e?Rⁿ). Similar results are obtained for the spaces W _p ^l (Rⁿ) and the spaces of M. Riesz and Bessel potentials. One considers consequences regarding imbedding theorems of “fractional” spaces in ?_q(dμ), whereμ is a nonnegative measure in Rⁿ. One considers specially the case p=1.     

Weighted restriction theorems for Bessel potentials

Пустьk-мерное евклид ово пространствоR ^k рассматривается как подмножествоR ⁿ . Зафиксируемр, 1<р<∞ иα >(n?k)/p, α≠п. Как обычно, бесселев потенциалJαf обобщенной функции Шварцаf наR ⁿ определяется с помощ ью ее преобразования Фурь е \((\widehat{G_\alpha f})(\xi ) = (2\pi )^{ - n/2} [1 + |\xi |^2 ]^{\alpha /2} f(\xi ), \xi \in R^n .B\) , ξ∈R ⁿ . В работе характ еризуются положител ьные весовые функцииw(x ₁,...,x _k ), которые при продолжении наR ⁿ с помощью равенстваw(x ₁,...,x _k ,...,x _n )=w(x ₁, ...,x _k ) обладают с ледующим свойством: существует числос>0, не зависящее отf, такое, что $$\begin{gathered} \int\limits_{R^k } {|(G_\alpha f)(x_1 ,...,x_k ,0,...,0)w(x_1 ,...,x_k )|^p dx_1 ...dx_k \leqq } \hfill \\ \leqq C\int\limits_{R^n } {|f(x_1 ,...,x_n )w(x_1 ,...,x_n )|^p dx_1 ...dx_n } \hfill \\ \end{gathered} $$      

On rings whose number of centralizers of ideals is finite

LetR be a ring. For the setF of all nonzero ideals ofR, we introduce an equivalence relation inF as follows: For idealsI andJ, I～J if and only ifV _R (I)=V _R(J), whereV _R() is the centralizer inR. LetI _R=F/～. Then we can see thatn(I _R), the cardinality ofI _R, is 1 if and only ifR is either a prime ring or a commutative ring (Theorem 1.1). An idealI ofR is said to be a commutator ideal ifI is generated by{st?ts; s∈S, t∈T} for subsetS andT ofR, andR is said to be a ring with (N) if any commutator ideal contains no nonzero nilpotent ideals. Then we have the following main theorem: LetR be a ring with (N). Thenn(I _R) is finite if and only ifR is isomorphic to an irredundant subdirect sum ofS⊕Z whereS is a finite direct sum of non commutative prime rings andZ is a commutative ring (Theorem 2.1). Finally, we show that the existence of a ringR such thatn(I _R)=m for any given natural numberm.     

Local linear independence of the translates of a box spline

Let Ξ=(ξ _i ) _l ⁿ be a sequence of vectors inR ^m . The box splineM _Ξ is defined as the distribution given by $$M_\Xi :\varphi \to \int_{[0,1]^n } \varphi \left( {\sum\limits_{i = 1}^n {\lambda (i)\xi _i } } \right)d\lambda ,\varphi \in C_c^\infty (R^m ).$$ . Suppose that Ξ contains a basis forR ^m . ThenM _Ξ∈L _∞(R ^m ). Assume $$\Xi \subset V: = z^m .$$ . Consider the translatesM _v :=M _Ξ(·?v),v∈V. It is known that (M _v ) _V is linearly dependent unless (*) $$|\det Z| = 1forallbasesZ \subset \Xi$$ . This paper demonstrates that under condition (*), (M _v ) _V is locally linearly independent, i.e., $$\{ M_v ;\sup p M_v \cap A \ne \not 0\}$$ is linearly independent over any open setA.     

CAT(0) 4-manifolds possessing a single tame point are Euclidean

The concepts of geometric and topological tame point are introduced for a space of nonpositive curvature. These concepts are applied to the characterization problem forCAT(0) 4-manifolds. It is shown that everyCAT(0)M ⁴ having a single (geometric or topological) tame point is homeomorphic toR ⁴. Davis and Januszkiewicz have recently constructedCAT(0)n-manifolds,M ⁿ withn ≥ 5 such that the set of tame points form a dense open subset ofM ⁿ , butM ⁿ ≠R ⁿ .