Mathematics

Dual numbers and their higher order version are important tools for numerical computations, and in particular for finite difference calculus. Based upon the relevant algebraic rules and matrix realizations of dual numbers, we will present a novel point of view, embedding dual numbers within a formalism reminiscent of operational umbral calculus.

The so-called holographic principle, originally addressed to the high energy physics, suggests more generally that the information inseparatly bounded with a physical carrier (measured by its entropy) scales as the event horizon surface---a two-dimensional object. In this paper we present an idea of representing classical information in formalism of pure braid groups, characterized by an exceptionally rich structure for two-dimensional spaces. This leads to some interesting properties, e.g. information geometrization, multi-character alphabets energetic efficiency for information coding characteristic or a group structure decoding scheme. We also proved that proposed pure braid group approach meets all the conditions for storing and processing of information.

Let A=[ a ij ]∈ O 3 (R) . We give several different proofs of the fact that the vector V:= [ 1 a 23 + a 32 1 a 13 + a 31 1 a 12 + a 21 ] T , if it exists, is an eigenvector of A corresponding to the eigenvalue 1 .

We present an eight-dimensional even sub-algebra of the 2^4 = 16-dimensional associative Clifford algebra Cl(4,0) and show that its eight-dimensional elements denoted as X and Y respect the norm relation ||XY|| = ||X|| ||Y||, thus forming an octonion-like but associative normed division algebra, where the norms are calculated using the fundamental geometric product instead of the usual scalar product so that the underlying coefficient algebra resembles that of split complex numbers instead of reals. The corresponding 7-sphere has a topology that differs from that of octonionic 7-sphere.

Ausgehend von einer konkreten technischen Fragestellung diskutieren wir in dieser Notiz die Anwendung verschiedener Glättungsverfahren auf Datensätze mit vorgegebener Struktur. Wir stellen die Verfahren im Detail vor und besprechen die Vorund Nachteile. Insbesondere stellen wir hier die symmetrisierte exponentielle Glättung vor, die ein sehr gutes Glättungsverhalten mit einem hohen Maß~an Symmetrieerhaltung kombiniert. (Based on a specific technical question, we discuss the application of various smoothing methods to data sets with a particular structure. We present the methods in detail and discuss their advantages and disadvantages. In particular, we present the symmetrized exponential smoothing that combines very good smoothing behavior with a high degree of symmetry-conservation)

``Real Normed Algebras Revisited,'' the last paper of the late Gadi Moran, attempts to reconstruct the discovery of the complex numbers, the quaternions and the octonions, as well as proofs of their properties, using only what was known to 19th century mathematicians. In his research, Gadi had discovered simple and elegant proofs of the above-mentioned classical results using only basic properties of the geometry of Euclidean spaces and tools from high school geometry. His reconstructions underline an interesting connection between Euclidean geometry and these algebras, and avoid the advanced machinery used in previous derivations of these results. The goal of this article is to present Gadi's derivations in a way that is accessible to a wide audience of readers.

This note offers an elementary proof of the Siegel-Walfisz theorem for primes in arithmetic progressions.

For n positive numbers ( a k , 1≤k≤n ), enhanced inequalities about the arithmetic mean ( A n ≡ ∑ k a k n ) and the geometric mean ( G n ≡ Π k a k − − − − √ n ) are found if some numbers are known, namely, G n A n ≤( n− ∑ m k=1 r k n−m ) 1− m n ( Π m k=1 r k ) 1 n , if we know a k = A n r k ( 1≤k≤m≤n ) for instance, and G n A n ≤ 1 (1− m n ) Π m k=1 r −1 n−m k + 1 n ∑ m k=1 r k , if we know a k = G n r k ( 1≤k≤m≤n ) for instance. These bounds are better than those derived from S.~H.~Tung's work [1].

In this article, we introduce equivariant formal deformation theory of associative algebra morphisms. We introduce an equivariant deformation cohomology of associative algebra morphisms and using this we study the equivariant formal deformation theory of associative algebra morphisms.

The linear Navier-Stokes equations in three dimensions are given by: u it (x,t)−ρ△ u i (x,t)− p x i (x,t)= w i (x,t) , divu(x,t)=0,i=1,2,3 with initial conditions: u | (t=0)⋃∂Ω =0 . The Green function to the Dirichlet problem u | (t=0)⋃∂Ω =0 of the equation u it (x,t)−ρ△ u i (x,t)= f i (x,t) present as: G(x,t;ξ,τ)=Z(x,t;ξ,τ)+V(x,t;ξ,τ). Where Z(x,t;ξ,τ)= 1 8 π 3/2 (t−τ ) 3/2 ⋅ e − ( x 1 − ξ 1 ) 2 +( x 2 − ξ 2 ) 2 +( x 3 − ξ 3 ) 2 4(t−τ) is the fundamental solution to this equation and V(x,t;ξ,τ) is the smooth function of variables (x,t;ξ,τ) . The construction of the function G(x,t;ξ,τ) is resulted in the book [1 p.106]. By the Green function we present the Navier-Stokes equation as: u i (x,t)= ∫ t 0 ∫ Ω (Z(x,t;ξ,τ)+V(x,t;ξ,τ)) dp(ξ,τ) dξ dξdτ+ ∫ t 0 ∫ Ω G(x,t;ξ,τ) w i (ξ,τ)dξdτ . But divu(x,t)= ∑ 3 1 d u i (x,t) d x i =0. Using these equations and the following properties of the fundamental function: Z(x,t;ξ,τ) : dZ(x,t;ξ,τ) d x i =− dZ(x,t;ξ,τ) d ξ i , for the definition of the unknown pressure p(x,t) we shall receive the integral equation. From this integral equation we define the explicit expression of the pressure: p(x,t)=− d dt △ −1 ∗ ∫ t 0 ∫ Ω ∑ 3 1 dG(x,t;ξ,τ) d x i w i (ξ,τ)dξdτ+ρ⋅ ∫ t 0 ∫ Ω ∑ 3 1 dG(x,t;ξ,τ) d x i w i (ξ,τ)dξdτ. By this formula the following estimate: ∫ t 0 ∑ 3 1 ∥ ∥ ∂p(x,τ) ∂ x i ∥ ∥ 2 L 2 (Ω) dτ<c⋅ ∫ t 0 ∑ 3 1 ∥ w i (x,τ) ∥ 2 L 2 (Ω) dτ holds.