A general induction theorem for rearrangements of n-tuples

In this paper, a general induction theorem for rearrangements of n-tuples in Rⁿ is proved, showing that a certain proposition regarding a pair of n-tuples related by the strong spectral order 〈 is true for any integer n ? 2 if and only if it is true for the case n = 2. With this theorem, a whole series of well-known theorems is derived as particular cases, and some new results are also obtained.     

On the theorem and constants of H. Whitney

Whitney's famous theorem shows that the error of approximation to a functionf by algebraic polynomials of degree <n can be estimated by thenth order modulus of smoothness off. We show that the constants in this theorem can be taken independent ofn.     

Lagrangian submanifolds in complex projective space CP n

We first prove a basic theorem with respect to the moving frame along a Lagrangian immersion into the complex projective space CP ⁿ . Applying this theorem, we study the rigidity problem of Lagrangian submanifolds in CP ⁿ .     

Embedding Tn-like continua in Euclidean space

Many authors have been concerned with embedding ∏-like continua in Rⁿ where ∏ is some collection of polyhedra or manifolds. A similar concern has been embedding ∏-like continua in Rⁿ up to shape. In this paper we prove two main theorems. Theorem: If n ? 2 and X is Tⁿ-like, then X embeds in R²ⁿ. This result was conjectured by McCord for the case H¹(X) finitely generated and proved by McCord for the case that H¹(X) = 0 using a theorem of Isbell. The second theorem is a shape embedding theorem. Theorem: If X is Tⁿ-like, then X embeds in Rⁿ⁺² up to shape. This theorem is proved by showing that an n-dimensional compact connected abelian topological group embeds in Rⁿ⁺². Any Tⁿ-like continuum is shape equivalent to a k-dimensional compact connected abelian topological group for some 0 ? k ? n.     

Mapping fibrations

We present a fibration theorem for mappings from C ⁿ to C ^p , withn <p that resembles the Milnor fibration theorem for isolated complete intersection singularities which is due to H. Hamm.     

On the enumeration of finite Abelian and solvable groups

Let a(n) be the number of nonisomorphic abelian groups of order n. We obtain a short interval result for the local density of a(n). More generally, we get short interval version of results of Ivi? on the local density of prime independent multiplicative functions. Also we prove a short interval version of the theorem of Erdös and Szekeres on the summatory function of a(n) and the theorem of Greenberg and Newman on the enumeration of a certain type of finite solvable groups.     

Perturbation thèorems for the generalized eigenvalue problem

Given a matrix pair Z = (A, B), the perturbation of its eigenvalues (α, β) is studied. Considering two pairs Z, W as points of the Grassman manifold G_{n, 2n} and its eigenvalues as points in G_{1, 2}, the projective complex plane, the distance of the spectra, measured in the chordal metric in G_{1, 2}, is bounded by some distance of the matrix pairs in G_{n, 2n}. Analogs of the Bauer-Fike theorem, Henrici's theorem, and the Hoffman-Weilandt theorem are obtained, from which the “classical” results can be derived.     

Spaces of rational maps and the Stone-Weierstrass theorem

It is shown that Segal's theorem on the spaces of rational maps from CP¹ to CPⁿ can be extended to the spaces of continuous rational maps from CP^m to CPⁿ for any m?n. The tools are the Stone-Weierstrass theorem and Vassiliev's machinery of simplicial resolutions.     

Analogues of Miyachi, cowling-price and Morgan theorems for compact extensions of ℝ n

Let K be a compact subgroup of automorphisms of ? ⁿ . We formulate and prove an analogue of Miyachi’s theorem for the semi-direct product K ? ? ⁿ . This allows us to solve the sharpness problems in the theorem of Cowling-Price and in the L ^p ? L ^q analogue of Morgan theorem for any compact extension of ? ⁿ . These upshots are proved using the representations theory and the Plancherel formula for the group Fourier transform.     

Sufficient conditions for open manifolds to be diffeomorphic to Euclidean spaces

Let M be a complete non-compact connected Riemannian n-dimensional manifold. We first prove that, for any fixed point p∈M, the radial Ricci curvature of M at p is bounded from below by the radial curvature function of some non-compact n-dimensional model. Moreover, we then prove, without the pointed Gromov-Hausdorff convergence theory, that, if model volume growth is sufficiently close to 1, then M is diffeomorphic to Euclidean n-dimensional space. Hence, our main theorem has various advantages of the Cheeger-Colding diffeomorphism theorem via the Euclidean volume growth. Our main theorem also contains a result of do Carmo and Changyu as a special case.     

A variation of Menger's theorem for long paths

In this paper we prove a Mengerian theorem for long paths, namely, that if in order to cut every uv-path of length at least n (n ≥ 2), in a diagraph D, we need to remove at least h points, then there exist {$\text{h}\text{(3n ? 5)}$} interior disjoint uv-paths in D of length at least n. Some variations and applications of this theorem are given as well.     

Another Rigidity Theorem for Affine Immersions

The theorem of Beez-Killing in Euclidean differential geometry states as follows [KN, p.46]. Let f: M ⁿ → Rⁿ⁺¹ be an isometric immersion of an n-dimensional Riemannian manifold into a Euclidean (n + l)-space. If the rank of the second fundamental form of f is greater than 2 at every point, then any isometric immersion of M ⁿ into R ^{n +} 1 is congruent to f. A generalization of this classical theorem to affine differential geometry has been given in [O] (see Theorem 1.5). We shall give in this paper another version of rigidity theorem for affine immersions.     

Generalized boundary-value problems

If each of n people defines a (suitable) measure on a compact convex cake I, then there exists a division of I into n connected parts, and an assignment of these n parts to the n people, in such a way that the piece of cake assigned to each person is at least as large (in his own measure) as that assigned to anyone else. (The proof uses Brouwer's fixed-point theorem and Hall's theorem on systems of distinct representatives.) Slight progress is made towards finding algorithms for this problem.     

Notes About the Carathéodory Number

In this paper we give sufficient conditions for a compactum in ? ⁿ to have Carathéodory number less than n+1, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Carathéodory theorem and give a Tverberg-type theorem for families of convex compacta.     

Differentiable sphere theorems for submanifolds of positive k-th ricci curvature

We investigate the differentiable pinching problem for compact immersed submanifolds of positive k-th Ricci curvature, and prove that if M ⁿ is simply connected and the k-th Ricci curvature of M ⁿ is bounded below by a quantity involving the mean curvature of M ⁿ and the curvature of the ambient manifold, then M ⁿ is diffeomorphic to the standard sphere ${\mathbb{S}^n}$ . For the case where the ambient manifold is a space form with nonnegative constant curvature, we prove a differentiable sphere theorem without the assumption that the submanifold M ⁿ is simply connected. Motivated by a geometric rigidity theorem due to S. T. Yau and U. Simon, we prove a topological rigidity theorem for submanifolds in a space form.     

Multipliers for ¦N,p n;δ¦ k summability of infinite series

In this paper a theorem on ¦N,p _n;δ¦_ksummability factors, which generalizes a theorem of Bor [3] on ¦N,p _n¦_ksummability factors, has been proved.     

On the basis property of root elements of ordinary differential operators in the space L 2,n (a, b)

We study a spectral problem for a system of linear ordinary differential operators in the vector function space L _2,n (a, b) with parameter-dependent boundary conditions. We prove a theorem stating that the system of root functions of the problem is a basis with parentheses in L _2,n (a, b). Corollaries of the theorem are considered.     

A symmetrical plane partition correspondence

MacMahon conjectured the form of the generating function for symmetrical plane partitions, and as a special case deduced the following theorem. The set of partitions of a number n whose part magnitude and number of parts are both no greater than m is equinumerous with the set of symmetrical plane partitions of 2n whose part magnitude does not exceed 2 and whose largest axis does not exceed m. This theorem, together with a companion theorem for the symmetrical plane partitions of odd numbers, are proved by establishing 1-1 correspondences between the sets of partitions.     

Principal points of a multivariate mixture distribution

A set of n-principal points of a distribution is defined as a set of n points that optimally represent the distribution in terms of mean squared distance. It provides an optimal n-point-approximation of the distribution. However, it is in general difficult to find a set of principal points of a multivariate distribution. Tarpey et al. [T. Tarpey, L. Li, B. Flury, Principal points and self-consistent points of elliptical distributions, Ann. Statist. 23 (1995) 103-112] established a theorem which states that any set of n-principal points of an elliptically symmetric distribution is in the linear subspace spanned by some principal eigenvectors of the covariance matrix. This theorem, called a “principal subspace theorem”, is a strong tool for the calculation of principal points. In practice, we often come across distributions consisting of several subgroups. Hence it is of interest to know whether the principal subspace theorem remains valid even under such complex distributions. In this paper, we define a multivariate location mixture model. A theorem is established that clarifies a linear subspace in which n-principal points exist.     

Contributions to zero-sum problems

A prototype of zero-sum theorems, the well-known theorem of Erd?s, Ginzburg and Ziv says that for any positive integer n, any sequence a₁,a₂,…,a_2n-1 of 2n-1 integers has a subsequence of n elements whose sum is 0 modulo n. Appropriate generalizations of the question, especially that for (Z/pZ)^d, generated a lot of research and still have challenging open questions. Here we propose a new generalization of the Erd?s-Ginzburg-Ziv theorem and prove it in some basic cases.