Hermitian K-theory and 2-regularity for totally real number fields

An algebraic description of basic discrete symmetries (space reversal P, time reversal T, and their combination PT) is studied. Discrete subgroups of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. In accordance with a division ring structure, a complete classification of automorphism groups is established for the Clifford algebras over the field of real numbers. The correspondence between eight double coverings (Dąbrowski groups) of the orthogonal group and eight types of the real Clifford algebras is defined with the use of isomorphisms between the automorphism groups and finite groups. Over the field of complex numbers there is a correspondence between two nonisomorphic double coverings of the complex orthogonal group and two types of complex Clifford algebras. It is shown that these correspondences associate with a well-known Atiyah–Bott–Shapiro periodicity. Generalized Brauer–Wall groups are introduced on the extended sets of the Clifford algebras. The structure of the inequality between the two Clifford–Lipschitz groups with mutually opposite signatures is elucidated. The physically important case of the two different double coverings of the Lorentz groups is considered in details.

The probability of spanning a classical space by two non-degenerate subspaces of complementary dimensions

Orthonormal bases play an important role in the geometric study of vector spaces. For inner product spaces over real or complex number fields, we can apply Gram-Schmidt algorithm to construct an orthonormal subset from a linearly independent subset. However, on sesquilinear spaces over finite fields, Gram-Schmidt algorithm fails to produce an orthonormal subset because of the presence of non-zero, self-orthogonal vectors. In fact, there is a subspace that does not contain an orthonormal basis. In this paper, we study sesquilinear spaces over finite fields and show that a non-zero subspace has an orthonormal basis if and only if it is non-degenerate. An Extended Gram-Schmidt Process (EG-SP) is then discussed to construct an orthogonal subset from a linearly independent subset having equal generated subspaces. An advantage of the proposed EG-SP is that the obtained orthogonal subset is orthonormal when the generated subspace is non-degenerate. In addition, we can also extend an orthonormal subset of a sesquilinear space to an orthonormal basis.

Let {Km}m≥4 be the family of non-normal totally real cubic number fields as- sociated with the Q-irreducible cubic polynomials Pm(x) = x 3 − mx 2 − (m + 1)x − 1, m ≥ 4. We determine all these Km's with class numbers hm ≤ 3: there are 14 such Km's. Assuming the Generalized Riemann hypothesis for all the real quadratic number fields, we also prove that the exponents em of the ideal class groups of these Km go to infinity with m and we determine all these Km's with ideal class groups of exponents em ≤ 3: there are 6 such Km with ideal class groups of exponent 2, and 6 such Km with ideal class groups of exponent 3.

For every integer k there exists a bound B=B(k) such that if the characteristic polynomial of gin textrm{SL}_n(q) is the product of le k pairwise distinct monic irreducible polynomials over mathbb {F}_q, then every element x of textrm{SL}_n(q) of support at least B is the product of two conjugates of g. We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions (p, q), in the special case that n=p is prime, if g has order frac{q^p-1}{q-1}, then every non-scalar element x in textrm{SL}_p(q) is the product of two conjugates of g. The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups.

In this paper, we give the complete structures of the equivalence canonical form of four matrices over an arbitrary division ring. As applications, we derive some practical necessary and sufficient conditions for the solvability to some systems of generalized Sylvester matrix equations using the ranks of their coefficient matrices. The results of this paper are new and available over the real number field, the complex number field, and the quaternion algebra.

A system of quaternary coupled Sylvester-type real quaternion matrix equations

Adaptive digital baseband predistortion is a highly effective and simple linearization technique which compensates for the nonlinear distortion of PAs. This paper presents an improved adaptive technique using discrete Newton's method efficiently applied to digital baseband lookup table (LUT) predisortion. We simplify and convert the hardship of adaptive updating LUT to the roots finding problem for a system of two element real equations on mathematics. And we deduce the improved adaptive iterative formula used for updating LUT based on discrete Newton's method. The iterative formula of the proposed method is in real number field, but secant method previously published is in complex number field. So the proposed method reduces the number of real multiplications and is implemented with ease by hardware. Furthermore, computer simulation results verify the LUT baseband predistortion using discrete Newton's method could rectify nonlinear distortion and improve system performance. Also, the simulation results reveal the proposed method reaches to the stable statement in fewer iteration times than secant method.

Let F be a field and let n, m ≥ 1 be integers. Then M m,n (F)denote the set of all m × n matrices over F, and M n (F) = M n,n (F). When F is not specified, or is the real numbers field or the complex number field, we omit the field and simply write M m,n and M n instead. For a matrix A = (a ij ) ∈ M m, n , and for 1 ≤i ≤ m and 1 ≤ j ≤ n, the symbol (A) ij denotes the (i, j)-entry of A, whereas A ij denote a block submatrix of A. If v = (x 1, x 2,⋯, x n ) T denotes an n-dimensional vector, then (v) i denotes the ith component x i of v.

As an important branch in the field of chaos, chaos synchronization (CHS) has attracted great attentions since it was firstly proposed. In this review article, numerous types of continuous time CHS are presented for the past 29 years. Previous CHS are divided into two major parts: one is CHS in real number field, and the other is CHS in complex number field. Some classic types of CHS, controllers and corresponding stability theories are preliminarily reviewed. Besides, a new type of chaos synchronization is proposed, which is called N-systems combination function projective synchronization (NCOFPS). Interestingly and importantly, most of these proposed types of CHS belong to NCOFPS. Then, time delay function projective synchronization (TDFPS) is also proposed, which is one of the types of NCOFPS and never studied in existed papers. And TDFPS controller, corresponding mathematical proof are obtained and simulation experiments illustrate the effectiveness of investigated controller. This paper provides a generalized view to catch the development of CHS in different fields. Finally, we give the conclusions and prospects of CHS.

Simultaneous asymptotic diophantine approximations to a basic of a real cubic number field

CHAPTER TEN - ROOTS OF POLYNOMIALS

A characterization of number fields by p-closed extensions

An extension of the numerical range over finite fields

In this article, we give an explicit elementary proof of a version of resolution of singularities in characteristic zero. Local means that the centres of blowing up are chosen locally, so that a finite number of finite sequences of blowings-up may be required to cover a neighbourhood of a given point. (Hence we use the term rather than local desingularization, although uniformization in our sense is considerably stronger than the original idea of Zariski [16].) In the last two decades, several mathematicians (notably Abhyankar [1], Hironaka and Spivakovsky [12], and Youssin [14, 15]) have proposed simpler or more explicit versions of the inductive procedure in Hironaka's proof of his great theorem [8]. These approaches would seem to lead to some form of uniformization, though full details of none of them have yet appeared. The idea of seeking an explicit procedure to determine the centres of blowing up is that a sufficiently good choice should globalize automatically. (Our method, for example, gives global resolution of singularities of surfaces in any codimension.) Throughout this paper, K denotes either the field of real numbers R or the field of complex numbers C. We will work in the category of analytic spaces over K, although our techniques apply as well to algebraic spaces over a field