The Maximum Principle for Holomorphic Operator Functions

We show that if an operator-valued analytic function f of a complex variable attains its maximum modulus at z ₀, then the coefficients of the nonconstant terms in the power series expansion about z ₀ cannot be invertible, provided a complex uniform convexity condition holds. One application is that the norm of the resolvent of an operator on a complex uniformly convex space cannot have a local maximum.     

On Unimodality of Hilbert Functions of Gorenstein Artin Algebras of Embedding Dimension Four

We prove that the Hilbert functions of Gorenstein Artin algebras R/I of embedding dimension four are unimodal provided I has a minimal generator in degree less than five. It is still an open question as to whether all Gorenstein Hilbert functions in codimension four are SI-sequences. It is not even known if they are all unimodal. In this article, we prove that Hilbert functions of all Gorenstein Artin algebras starting with (1, 4, 10, 20, h ₄,…), where h ₄ = 34 are unimodal. Combining this with previously known results, we obtain that all Gorenstein Hilbert functions (1, 4, h ₂, h ₃, h ₄,…4, 1) are unimodal if h ₄ ≤ 34.     

Permanence and Positive Periodic Solutions for Kolmogorov Competing Species Systems

We prove the existence of periodic solutions in a compact attractor of (R⁺)ⁿ for the Kolmogorov system x′_i = x_if_i(t, x₁, …, x_n), i = l, …, n in the competitive case. Extension to differential delay equations are con- sidered too. Applications are given to Lotka-Volterra systems with periodic coefficients.     

Intersection patterns of convex sets

LetK ₁,…K_n be convex sets inR ^d. For 0≦i denote byf _ithe number of subsetsS of {1,2,…,n} of cardinalityi+1 that satisfy ∩{K _i∶i∈S}≠Ø. We prove:Theorem.If f _d+r=0 for somer r>=0, then {fx161-1} This inequality was conjectured by Katchalski and Perles. Equality holds, e.g., ifK ₁=…=K_r=R^d andK _r+1,…,K_n aren?r hyperplanes in general position inR ^d. The proof uses multilinear techniques (exterior algebra). Applications to convexity and to extremal set theory are given.     

Linearization of holomorphic germs with quasi-Brjuno fixed points

Let f be a germ of holomorphic diffeomorphism of \mathbb Cⁿ{\mathbb {C}^{n}} fixing the origin O, with d f _O diagonalizable. We prove that, under certain arithmetic conditions on the eigenvalues of d f _O and some restrictions on the resonances, f is locally holomorphically linearizable if and only if there exists a particular f -invariant complex manifold. Most of the classical linearization results can be obtained as corollaries of our result.     

Simultaneous quasidiagonalization of complex matrices

Let

be the complex algebra generated by a pair of n × n Hermitian matrices A, B. A recent result of Watters states that A, B are simultaneously unitarily quasidiagonalizable [i.e., A and B are simultaneously unitarily similar to direct sums C₁⊕…⊕C_t,D₁⊕…⊕D_t for some t, where C_i, D_i are k_i × k_i and k_i?2(1?i?t)] if and only if [p(A, B), A]² and [p(A, B), B]² belong to the center of

for all polynomials p(x, y) in the noncommuting variables x, y. In this paper, we obtain a finite set of conditions which works. In particular we show that if A, B are positive semidefinite, then A, B are simultaneously quasidiagonalizable if (and only if) [A, B]², [A², B]² and [A, B²]² commute with A, B.     

Rekurrente zahlentheoretische Funktionen in mehreren Variablen

Some results of C. Methfessel (Multiplicative and additive recurrent sequences, Arch. Math. 63, 321–328 (1994)) on recurrent multiplicative arithmetical functions are generalized to the case of several variables. As ?[X ₁,…, X _r ] for r ≧ 2 is no principal ideal domain it is necessary to prove some elementary algebraic facts on the calculation of recurrent functions in several variables. The main tool is a vector space duality between ?[X ₁,…, X _r ] and the space of all functions f : ?^r → ?. This gives a correspondence between spaces of recurrent functions and ideals in ?[X ₁,…, X _r ]. The practical calculation can be done using Groebner bases. It is proved that a multiplicative function g in r variables which satisfies enough recurrences is of the form g (n ₁…, n _r )=n ₁ ^l … n _r ^rl h (n ₁ …, n _r ), with h multiplicative and periodic in each variable.     

Nonparametric estimation of mixed partial derivatives of a multivariate density

On the basis of a random sample of size n on an m-dimensional random vector X, this note proposes a class of estimators f_n^(p) of f^(p), where f is a density of X w.r.t. a σ-finite measure dominated by the Lebesgue measure on R^m, p = (p₁,…,p_m), p_j ≥ 0, fixed integers, and for x = (x₁,…,x_m) in R^m, f^(p)(x) = ?^p₁+…+p_m f(x)/(?^p₁x₁ … ?^p_mx_m). Asymptotic unbiasedness as well as both almost sure and mean square consistencies of f_n^(p) are examined. Further, a necessary and sufficient condition for uniform asymptotic unbisedness or for uniform mean square consistency of f_n^(p) is given. Finally, applications of estimators of this note to certain statistical problems are pointed out.     

Maximal function and Riesz potential on p-adic linear spaces

For maximal function and Riesz potential on p-adic linear space ? _p ⁿ we give sufficient conditions of its boundedness in generalized Morrey spaces. For radial weights of special kind this condition for Riesz potential is sharp. Also we prove that if Riesz potential I ^α(f) exists at point b, then b is L ^q Lebesgue point for some q.     

On the Brauer group of a two-dimensional local field

The two-dimensional local field K = F _q((u))((t)), char K = p, and its Brauer group Br(K) are considered. It is proved that, if L = K(x) is the field extension for which we have x ^p ? x = ut ^?p =: h, then the condition that (y, f | h]_K = 0 for any y ε K is equivalent to the condition f ε Im(Nm(L*)).     

重调和Hardy空间上的Toeplitz 算子

与调和Bergman 空间相对应, 本文研究重调和Hardy 空间h²(D²) 上的Toeplitz 算子. 本文发现, h²(D²) 上的Toeplitz 算子与经典的Hardy 空间、Bergman 空间及调和Bergman 空间上的Toeplitz算子的性质都有很大的差异. 例如解析Toeplitz 算子可以不是半可换及可交换的. 即使半可换, 其中任何一个符号可以不为常数; 即使可交换, 两个符号的非平凡线性组合也不一定是常数. 本文得到了h²(D²) 上两个解析Toeplitz 算子半可换和可换的充分必要条件.     

Inverse Hölder inequalities in one and several dimensions

We study certain functionals and obtain an inverse Hölder inequality for n functions f₁^a1,…,f_n^an (f_k concave, 1 dimension).We also prove a multidimensional inverse Hölder inequality for n functions f₁,…,f_n, where $\text{?}{}^2\text{f}{}_{\mathrm{k}}\text{?x}{}_{\mathrm{i}}{}^2\text{? 0, i = 1,…, d, k = 1,…, n}$.Finally we give an inverse Minkowski inequality for concave functions.     

Structural stability of C1 diffeomorphisms

In this paper we prove that if f is a C¹ diffeomorphism that satisfies Axiom A and the strong transversality condition then it is structurally stable. J. Robbin proved this theorem for C² diffeomorphisms. In addition to reducing the amount of differentiability necessary to prove the theorem, we also give a new proof combining the d_f metric of Robbin with the stable and unstable manifold proof of D. Anosov. We also prove structural stability in the neighborhood of a single hyperbolic basic set (independent of its being part of a diffeomorphism that satisfies Axiom A and the strong transversality condition). These proofs are adapted to prove the structural stability of C¹ flows in another paper.     

On the probability that given polynomials have a specified highest common factor

The polynomial functions f₁, f₂,…, f_m are found to have highest common factor h for a set of values of the variables x₁, x₂,…,x_m whose asymptotic density is

$\text{1}\text{h}{}^{\mathrm{n}}\text{∑}\text{d∣h}\text{μ(d)Π}{}^{\mathrm{m}}{}_{\mathrm{l\; =\; 1}}\text{?(f}{}_1\text{, dh)}\text{d}{}^{\mathrm{m}}\text{Π}\text{p∣h}\text{1?}\text{Π}{}^{\mathrm{m}}{}_{\mathrm{l\; =\; 1}}\text{?(f}{}_1\text{, p)}\text{p}{}^{\mathrm{m}}$

For the special case f₁(x) = f₂(x) = … = f_m(x) = x and h = 1 the above formula reduces to $\text{Π}{}_{\mathrm{?}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{m}}\text{) =}\text{1}\text{ζ}\text{(m)}$, the density if m-tuples with highest common factor 1. Necessary and sufficient conditions on the polynomials f₁, f₂,…, f_m for the asymptotic density to be zero are found. In particular it is shown that either the polynomials may never have highest common factor h or else h is the highest common factor infinitely often and in fact with positive density.     

CR-Warped Products in Complex Projective Spaces with Compact Holomorphic Factor

A submanifold of a Kaehler manifold is called a CR-warped product if it is the warped product N_T ×_fN of a complex submanifold N_T and a totally real submanifold N. There exist many CR-warped products N_T ×_fN in CP^h+p, h = dim_CN_T and p = dim_RN (see [5, 6]). In contrast, we prove in this article that the situation is quite different if the holomorphic factor N_T is compact. For such CR-wraped products in CP^m (4), we prove the following: (1) The complex dimension m of the ambient space is at least h + p + hp. (2) If m = h + p + hp, then N_T is CP^h(4). We also obtain two geometric inequalities for CR-warped products in CP^m with compact N_T.     

On norm form equations

In his paper [Ann. of Math.96 (1972)] Schmidt applied his results on the n-dimensional Roth theorem to study the n-dimensional Thue equation f(x₁,…, x_n) = c, and he proved, when f is of norm type, a necessary and sufficient criterion for the Thue equation to have only finitely many solutions. Here we shall study equations f(x₁,…, x_n) = p₁^z1 … p_t^zt and prove the p-adic analog of Schmidt's theorems.     

A Characteristic Property of Injective Holomorphic Germs

Let X ₀ be the germ at 0 of a complex variety and let f: X₀? \Bbb Cⁿ₀f:\ X_0\rightarrow {\Bbb C}^n_0 be a holomorphic germ. We say that f is pseudoimmersive if for any g: \Bbb R₀? X₀g:\ {\Bbb R}_0\rightarrow X_0 such that f °g ? C^￥ f \circ g \in C^{\infty} , we have g ? C^￥g\in C^{\infty} . We prove that f is pseudoimmersive if and only if it is injective. Some results about the real case are also considered.     

Some optimization problems with applications to canonical correlations and sphericity tests

Optimization problems are connected with maximization of three functions, namely, geometric mean, arithmetic mean and harmonic mean of the eigenvalues of (X′ΣX)^?1ΣY(Y′ΣY)^?1Y′ΣX, where Σ is positive definite, X and Y are p × r and p × s matrices of ranks r and s (≥r), respectively, and X′Y = 0. Some interpretations of these functions are given. It is shown that the maximum values of these functions are obtained at the same point given by X = (h₁ + ?₁h_p, …, h_r + ?_rh_p?r+1) and $\text{Y = (h}{}_1\text{? ?}{}_1\text{h}{}_{\mathrm{p}}\text{, …, h}{}_{\mathrm{r}}\text{? ?}{}_{\mathrm{r}}\text{h}{}_{\mathrm{p?r+1}}\text{,}\text{Y}{}_{\mathrm{r+1}}\text{, …,}\text{Y}{}_{\mathrm{s}}\text{)}$, where h₁, …, h_p are the eigenvectors of Σ corresponding to the eigenvalues λ₁ ≥ λ₂ ≥ … ≥ λ_p > 0, ?_j = +1 or ?1 for j = 1,2,…, r and $\text{Y}{}_{\mathrm{r+1}}\text{, …,}\text{Y}{}_{\mathrm{s}}$, are linear functions of h_r+1,…, h_p?r. These results are extended to intermediate stationary values. They are utilized in obtaining the inequalities for canonical correlations θ₁,…,θ_r and they are given by expressions (3.8)–(3.10). Further, some new union-intersection test procedures for testing the sphericity hypothesis are given through test statistics (3.11)–(3.13).     

Direct and converse results for operators of Baskakov-Durrmeyer type

We consder the n-th so-called operators of Baskakov-Durrmeyer type, which result from the classical Baskakov-type operators with weights p_nk, if the discrete values f(k/n) are replaced by the integral terms (n-c∫ ₀ ^∞ p _n k(t)f(t)dt. The main differences between these operators and their classical and Kantorovicvariants respectively is that they commute. We prove direct and converse theorems also for linear combinations of the operators and results of Zamansky-Sunouchi type. As an important tool for measuring the smootheness of a function we use the Ditzian-Totik modulus of smoothness and its equivalence to appropriate K-functionals. This paper is part of the author's dissertation.