Witt’s Theorem for Noncommutative Conic Curves

Let k be a field. We extend the main result in Nyman (J. Algebra 434, 90–114, 2015) to show that all homogeneous noncommutative curves of genus zero over k are noncommutative \(\mathbb {P}^{1}\)-bundles over a (possibly) noncommutative base. Using this result, we compute complete isomorphism invariants of homogeneous noncommutative curves of genus zero, allowing us to generalize a theorem of Witt.     

Cartan’s Second Main Theorem and Mason’s Theorem for Jackson Difference Operator

Let f : C → Pⁿ be a holomorphic curve of order zero. The authors establish a Jackson difference analogue of Cartan’s second main theorem for the Jackson q-Casorati determinant and introduce a truncated second main theorem of Jackson difference operator for holomorphic curves. In addition, a Jackson difference Mason’s theorem is proved by using a Jackson difference radical of a polynomial. Furthermore, they extend the Mason’s theorem for m + 1 polynomials. Some examples are constructed to show that their results are accurate.     

Three-dimensional Pythagorean Theorem for Elliptic Space

Let M be an elliptic space with constant curvature K=1/R~2,and consider inM a right-angled tetrahedron,i.e.,a tetrahedron with three mutually perpendi-cular edges at a certain vertex.Denote the tetrahedron by VABC and assumethat each of the edges VA,VB and VC is perpendicular to others.The areas oftriangles VBC,VCA,VAB and ABC are denoted by S_1,S_2,S_3 and S,respective-ly.We obtain the following     

An Extension of Polyak’s Theorem in a Hilbert Space

Let H be an infinite-dimensional real Hilbert space equipped with the scalar product (⋅,⋅) _H . Let us consider three linear bounded operators,

Engel’s Theorem for Generalized Lie Algebras

We prove analogues of the classical Engel’s Theorem for Lie algebras in the category of comodules over a cotriangular Hopf algebra, generalizing the known result for Lie coloralgebras.     

Morley’s trisector Theorem for isosceles tetrahedron

We extend Morley’s trisector theorem in the plane to an isosceles tetrahedron in three-dimensional space. We will show that the Morley tetrahedron of an isosceles tetrahedron is also isosceles tetrahedron. Furthermore, by the formula for distance in barycentric coordinate, we introduce and prove a general theorem on an isosceles tetrahedron.

     

Generalized Bochner’s Theorem for radial function

A radial function Φ(x) can be expressed by its generator ?(·) through Φ(x)=?(‖x‖). The positive de finite of the function Φ plays an important role in the radial basis interpolation. We can naturally use Bochner’s Theorem to check if Φ is positive de finite. This requires however a n-dimensional Fourier transformation and it is not very easy to calculate. Furthermore in a lot of cases we will use ? for spaces of various dimensions too, then for every fixed n we need do the Fourier transformation once to check if the function is positive definite in the n-dimensional space. The completely monotone function, which is discussed in [4], is positive definite for arbitrary space dimensions. With this technique we can very easily characterize the positive definite of a radial function through its generator. Unfortunately there is only a very small subset of radial function which is completely monotone. Thus this criterion excluded a lot of interesting functions such as compactly supported radial function, which are very use ful in application. Can we find some conditions (as the completely monotone function) only for the 1-dimensional Fourier transform of the generator ? to characterize a radial function Φ, which is positive definite in n-dimensional (fixed n) space? In this paper we defined a kind of incompletely monotone function of order α, for α=0, 1/2, 1, 3/2, 2, … (we denote the function class by ICM), in this sence a normal positvie function is in ICM₀; a positive monotone decreasing function is in ICM₁ and a positive monotone decreasing and convex function is in ICM₂. Based on this definition we get a generalized Bochner’s Theorem for radial function: If1-dimensional Fourier transform of the generator of a radial function can be written as $F{}_1\varphi (t) = \tilde F(\frac{{t^2 }}{2})$ , then corresponding radial function Φ(x) is positive definite as a n-variate function iff $\tilde F$ is an incompletely monotone function of order α=(n-1)/2 (or simply $\tilde F \in ICM_{\frac{{n - 1}}{2}} $ .     

De Rham’s Theorem for Orlicz Cohomology

Siberian Mathematical Journal - We prove that the de Rham $ L^{phi} $ -cohomology of a Riemannian manifold $ M $ admitting a convenient triangulation $ X $ is isomorphic to the...     

Weyl’s Theorem for Algebraically Paranormal Operators

Let T be an algebraically paranormal operator acting on Hilbert space. We prove : (i) Weyls theorem holds for f(T) for every f $\in$ H((T)); (ii) a-Browders theorem holds for f(S) for every S $\prec$ T and f $\in$ H((S)); (iii) the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T.     

Wall Rational Functions and Khrushchev’s Formula for Orthogonal Rational Functions

The Nevalinna–Pick algorithm yields a continued fraction expansion of every Schur function, whose approximants are identified. These approximants are quotients of rational functions which can be understood as the rational analogs of the Wall polynomials. The properties of these Wall rational functions and the corresponding approximants permit us to obtain a Khrushchev’s formula for orthogonal rational functions. An introduction to the convergence of the Wall approximants in the indeterminate case is presented. This work was partially realized during two stays of the second author at the Norwegian University of Science and Technology (NTNU) financed respectively by Secretaría de Estado de Universidades e Investigación from the Ministry of Education and Science of Spain and by the Department of Mathematical Sciences of NTNU. The work of the second author was also partially supported by the Spanish grants from the Ministry of Education and Science, project code MTM2005-08648-C02-01, and the Ministry of Science and Innovation, project code MTM2008-06689-C02-01, and by Project E-64 of Diputación General de Aragón (Spain).     

Perron’s Method and Wiener’s Theorem for a Nonlocal Equation

We study the Dirichlet problem for non-homogeneous equations involving the fractional p-Laplacian. We apply Perron’s method and prove Wiener’s resolutivity theorem.     

On Weyl’s Theorem for Functions of Operators

Let H be a complex separable infinite dimensional Hilbert space. In this paper, a variant of the Weyl spectrum is discussed. Using the new spectrum, we characterize the necessary and sufficient conditions for both T and f(T) satisfying Weyl's theorem, where f ∈ Hol(σ(T)) and Hol(σ(T)) is defined by the set of all functions f which are analytic on a neighbourhood of σ(T) and are not constant on any component of σ(T). Also we consider the perturbations of Weyl's theorem for f(T).     

Weyl’s Theorem for Algebraically Quasi-class A Operators

If T or T* is an algebraically quasi-class A operator acting on an infinite dimensional separable Hilbert space then we prove that Weyl’s theorem holds for f(T) for every f ∈ H(σ(T)), where H(σ(T)) denotes the set of all analytic functions in an open neighborhood of σ(T). Moreover, if T* is algebraically quasi-class A then a-Weyl’s theorem holds for f(T). Also, if T or T* is an algebraically quasi-class A operator then we establish that the spectral mapping theorems for the Weyl spectrum and the essential approximate point spectrum of T for every f ∈ H(σ(T)), respectively. This research was supported by the Kyung Hee University Research Fund in 2007 (KHU- 20071605).     

Ramanujan’s Master Theorem

S. Ramanujan introduced a technique, known as Ramanujan??s Master Theorem, which provides an explicit expression for the Mellin transform of a function in terms of the analytic continuation of its Taylor coefficients. The history and proof of this result are reviewed, and a variety of applications is presented. Finally, a multi-dimensional extension of Ramanujan??s Master Theorem is discussed.     

The Milnor–Totaro Theorem for Space Polygons

If denotes the curvature and the torsion of a closed, generic, and oriented polygonal space curve X in , then we show that _X (² + ²) ds = _X ds + _X | | ds > 4 if is positive. We also show that _X (² + ²) ds 2n if no four consecutive vertices lie in a plane and X has linking number n with a straight line. These extend theorems of Milnor and Totaro.     

Heat Kernel and Hardy’s Theorem for Jacobi Transform

In this paper, the authors obtain sharp upper and lower bounds for the heat kernel associated with Jacobi transform, and get some analogues of Hardy’s Theorem for Jacobi transform by using the sharp estimate for the heat kernel.     

On Wiener’s Theorem for functions periodic at infinity

We consider the functions periodic at infinity with values in a complex Banach space. The notions are introduced of the canonical and generalized Fourier series of a function periodic at infinity. We prove an analog of Wiener’s Theorem on absolutely convergent Fourier series for functions periodic at infinity whose Fourier series are summable with weight. The two criteria are given: for the function periodic at infinity to be the sum of a purely periodic function and a function vanishing at infinity and for a function to be periodic at infinity. The results of the article base on substantially use on spectral theory of isometric representations.