On a theorem about projectivities

The main result of this paper is a theorem about projectivities in then-dimensional complex projective spaceP ⁿ (n 2). Puttingn = 2, we showed in [3] that the theorem of Desargues inP ⁿ is a special case of this theorem. And not only the theorem of Desargues but also the converse of the theorem of Pascal, the theorem of Pappus-Pascal, the theorem of Miquel, the Newton line, the Brocard points and a lot of lesser known results in the projective, the affine and the Euchdean plane were obtained from this theorem as special cases without any further proof. Many extensions of classical theorems in the projective, affine and Euclidean plane to higher dimensions can be found in the literature and probably some of these are special cases of this theorem inP ⁿ. We only give a few examples and leave it as an open problem which other special cases could be found.     

Linear Systems of Cubics Singular at General Points of Projective Space

We present an elementary proof that given a general collection of d points in Pⁿ the linear system of cubics singular on each point has the expected codimension except when n=4 and d=7. In that case the cubic is unique. This, together with previous work of the author, gives a proof of the Alexander–Hirschowitz interpolation theorem.     

A combinatorial proof of the Borsuk-Ulam antipodal point theorem

We give a proof of Tucker’s Combinatorial Lemma that proves the fundamental nonexistence theorem: There exists no continuous map fromB ⁿ toS ^{n − 1} that maps antipodal points of∂B ⁿ to antipodal points ofS ^{n − 1}.     

Restricted chord projection and affine inequalities

In this note, we give two applications of the critical point theory of distance functions to Riemannian geometry. First, we present a new proof of the theorem: if a complete open nonnegatively curved n-manifold M has a volume growth of degree n, then M is diffeomorphic to R ⁿ. Second, we prove a sphere theorem about the mutually -critical points.Partially supported by a NSF grant.     

A short proof of the generalized Bonnet theorem

In three‐dimensional Euclidean space E³, the Bonnet theorem says that a curve on a ruled surface in three‐dimensional Euclidean space, having two of the following properties, has also a third one, namely, it can be a geodesic, that it can be the striction line, and that it cuts the generators under constant angle. In this work, in n dimensional Euclidean space Eⁿ, a short proof of the theorem generalized for (k + 1) dimensional ruled surfaces by Hagen in 4 is given. Copyright © 2013 John Wiley & Sons, Ltd.     

From Tanaka's formula to Ito's formula: The fundamental theorem of stochastic calculus

In this article we give a new proof of Ito's formula inR ⁿ starting from the one-dimensional Tanaka formula. The proof is algebraic and does not use any limiting procedure. It uses the integration by parts formula, Fubini's theorem for stochastic integrals and essential properties of local times.     

Dense Families of Selections and Finite-Dimensional Spaces

A characterization of n-dimensional spaces via continuous selections avoiding Z _n -sets is given, and a selection theorem for strongly countable-dimensional spaces is established. We apply these results to prove a generalized Ostrand's theorem, and to obtain a new alternative proof of the Hurewicz formula. It is also shown that our selection theorem yields an easy proof of a Michael's result.     

On the second differentiability of convex surfaces

Properties of pointwise second differentiability of real-valued convex functions in ⁿ are studied. Some proofs of the Busemann-Feller-Aleksandrov theorem are reviewed and a new proof of this theorem is presented.     

On the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domains

We present a uniqueness theorem for time-periodic solutions to the Navier–Stokes equations in unbounded domains. Thus far, results on the uniqueness of time-periodic solutions to the Navier–Stokes equations in unbounded domain, roughly speaking, have only found that a small time-periodic L ⁿ -solution is unique within the class of solutions which have sufficiently small L ^∞(L ⁿ )-norm. In this paper, we show that a small time-periodic L ⁿ -solution is unique within the class of all time-periodic L ⁿ -solutions, which contains large solutions. We also consider the uniqueness of solutions in weak-L ⁿ space. The proof of the present uniqueness theorem is based on the method of dual equations.      

Another Proof of Kasami's Theorem

We give a short direct proof for a famous theorem published by Kasami in 1971. In terms of Walsh analysis it states that for d = 2^2k - 2^k + 1 the Walsh spectrum of the Boolean function Tr(x ^d ) on GF(2 ⁿ ) consists precisely of the three values 0, ±2^(n+s)/2 if s = gcd(k, n) = gcd(2k, n).     

A proof of Devadze’s theorem on generators of the semigroup of Boolean matrices

In 1968 Devadze described, without a proof, minimal sets of generators of the semigroup of n×n Boolean matrices. We provide a proof of Devadze’s theorem.     

Stationary solutions to the drift–diffusion model in the whole spaces

We study the stationary problem in the whole space ?ⁿ for the drift–diffusion model arising in semiconductor device simulation and plasma physics. We prove the existence and uniqueness of stationary solutions in the weighted L^p spaces. The proof is based on a fixed point theorem of the Leray–Schauder type. Copyright © 2008 John Wiley & Sons, Ltd.     

On the Faith-Utumi theorem

In this paper we shall give a new proof of the well-known theorem of Faith-Utumi^[1]. Using our method we can show that every right order ofK _n is a prime right Goldie ring, whereK _n is the n×n-matrix ring over division ring K. Specially,D _n is a prime right Goldie ring, ifD is a right order ofK.The Project Supported by the National Natural Science Foundation of China.     

On the construction and topological invariance of the Pontryagin classes

We use sheaves and algebraic L-theory to construct the rational Pontryagin classes of fiber bundles with fiber \mathbbRⁿ{\mathbb{R}^n} . This amounts to an alternative proof of Novikov’s theorem on the topological invariance of the rational Pontryagin classes of vector bundles. Transversality arguments and torus tricks are avoided.     

Note on the heights of random recursive trees and random m-ary search trees

A process of growing a random recursive tree T_n is studied. The sequence {T_n} is shown to be a sequence of “snapshots” of a Crump–Mode branching process. This connection and a theorem by Kingman are used to show quickly that the height of T_n is asymptotic, with probability one, to c log n. In particular, c = e = 2.718 … for the uniform recursive tree, and c = (2γ)^?1, where γe^1+γ = 1, for the ordered recursive tree. An analogous reduction provides a short proof of Devroye's limit law for the height of a random m-ary search tree. We show finally a close connection between another Devroye's result, on the height of a random union-find tree, and our theorem on the height of the uniform recursive tree. © 1994 John Wiley & Sons, Inc.     

The gradient of certain harmonic functions on manifolds of almost nonnegative Ricci curvature

In this paper we prove that when the Ricci curvature of a Riemannian manifoldM ⁿ is almost nonnegative, and a ballB _L (p)⊂M ⁿ is close in Gromov-Hausdorff distance to a Euclidean ball, then the gradient of the harmonic functionb defined in [ChCo1] does not vanish. In particular, these functions can serve as harmonic coordinates on balls sufficiently close to an Euclidean ball. The proof, is based on a monotonicity theorem that generalizes monotonicity of the frequency for harmonic functions onR ⁿ .     

Herbrand's theorem revisited

We present recent results on the deepening connection between proof theory and formal language theory. To each first-order proof with prenex cuts of complexity at most Π_n/Σ_n, we associate a typed (non-deterministic) tree grammar of order n (equivalently, an order n recursion scheme) that abstracts the computation of Herbrand sets obtained through Gentzen-style cut elimination. Apart from offering a means to compute Herbrand expansions directly from proofs with cuts, these grammars provide a structural counterpart to Herbrand's theorem that opens the door to tackling a number of questions in proof theory such as proof equivalence, proof compression and proof complexity. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new proof of the three squares theorem

A theorem of Fein, Gordon, and Smith on the representation of ?1 as a sum of two squares is shown to yield a new proof of the three squares theorem. A positive integer k can be represented as a sum of three integer squares if and only if k ≠ 4^an with n ≡ 7 (mod 8) and a ≥ 0. This proof depends of the Brauer group and class field theory, not on ternary quadratic forms.