A note on 2-dimensional arithmetic symbols on $Gl(n,\Sigma_S)$

In this article, we introduce the idea of Lie regular elements and study 2 × 2 Lie regular matrices. It is shown that the linear groups GL(2, ℤ2 n ), GL(2, ℤ p n ), and GL(2, ℤ2p ) (where p is an odd prime) can be genrated by Lie regular matrices. Presentations of linear groups GL(2, ℤ4), GL(2, ℤ6), GL(2, ℤ8), and GL(2, ℤ10) are also given.

Locally finite Sylp*-groups

Reconstruction of 2-D porous media using Karhunen–Lóeve expansion

10 - THE RESOLUTION OF SOME SINGULARITIES

The realization of a BRST cohomology of the 4D topological gauge-affine gravity is established in terms of a superconnection formalism. The identification of fields in the quantized theory occurs directly as is usual in terms of superconnection and its supercurvature components with the double covering of the general affine group GA¯(4,R). Then, by means of an appropriate decomposition of the metalinear double-covering group SL¯(5,R) with respect to the general linear double-covering group GL¯(4,R), one can easily obtain the enlargements of the fields while remaining consistent with the BRST algebra. This leads to the descent equations, allowing us to build the observables of the theory by means of the BRST algebra constructed using a sa¯(5,R) algebra-valued superconnection. In particular, we discuss the construction of topological invariants with torsion.

For an extension L/K of discrete valuation fields, let $$e_{L/K}$$ , $${\mathfrak {O}}_{L}$$ denote the ramification index and valuation ring of L/K respectively. Let K be a complete discrete valuation field and $$L_1/K$$ , $$L_2/K$$ be finite linearly disjoint extensions over K. We show that if $${\mathfrak {O}}_{L_1L_2} ={\mathfrak {O}}_{L_1}{\mathfrak {O}}_{L_2}$$ or $$\mathrm {gcd}(e_{L_1/K}, e_{L_2/K}) =1$$ , and one of the residue fields $$l_1/k,$$ $$l_2/k$$ is separable, then $$e_{L_1L_2/L_1} =e_{L_2/K}.$$ The analogous results for the residue degrees are also true.

An explicit construction is described for computing representatives in each isogeny class of one-dimensional formal groups over the ring of integers of a complete discrete valuation field of characteristic 0 with residue field of characteris- tic p. The logarithms of representatives are written out explicitly, and the number of nonisomorphic representatives of the form described in each isogeny class is com- puted. This result extends and generalizes the result obtained by Laffaile in the case of an algebraically closed residue field. The homomorphisms between the repre- sentatives constructed are described completely. The results obtained are applied to computation of the Newton polygon and the of the logarithm for an arbitrary one-dimensional formal group. Moreover, the valuations and the residues of the torsion elements of the formal module are calculated. A certain valuation of logarithms of formal groups is introduced and the equivalence of two definitions of the valuation is proved. One of these definitions is in terms of the valuations of the coefficients, and the other is in terms of the valuations of the roots of the logarithm (i.e., of the torsion elements of the formal module). This valuation only depends on the isomorphism class of a formal group, is nonpositive, and equals zero if and only if the formal group in question is isomorphic to one of the representatives considered. The classification results of M. V. Bondarko and S. V. Vostokov on formal groups are employed, including invariant Cartier-Dieudonnmodules and the fractional part invariant for the logarithm of a formal group.

For a prime p, a cyclic-by-p group G and a G-extension L|K of complete discrete valuation fields of characteristic p with algebraically closed residue field, the local lifting problem asks whether the extension L|K lifts to characteristic zero. In this thesis, we characterize D_4-extensions of fields of characteristic two, determine the ramification breaks of (suitable) D_4-extensions of complete discrete valuation fields of characteristic two, and solve the local lifting problem in the affirmative for every D_4-extension of complete discrete valuation fields of characteristic two with algebraically closed residue field; that is, we show that D_4 is a local Oort group for the prime 2. Furthermore, we characterize Q_8-extensions of fields of characteristic two, determine the ramification breaks of (suitable) Q_8-extensions of complete discrete valuation fields of characteristic two, and, by solving the local lifting problem in the negative for a family of Q_8-extensions of complete discrete valuation fields of characteristic two with algebraically closed residue field, show that neither Q_8 nor SL_2(Z/3Z) is an almost local Oort group for the prime 2.

Let E and G be free modules of rank e and g, respectively, over a commutative noetherian ring R. The identity map on E*⊗ G induces the Koszul complex and its dual Let H𝒩(m,n,p) be the homology of the top complex at Sym m E*⊗ Sym n G⊗ ∧ p (E*⊗ G) and Hℳ(m,n,p) the homology of the bottom complex at D m E⊗ D n G*⊗ ∧ p (E⊗ G*). It is known that H𝒩(m,n,p)≅ Hℳ(m′, n′, p′), provided m + m′ = g − 1, n + n′ = e − 1, p + p′ = (e − 1)(g − 1), and 1 − e ≤ m − n ≤ g − 1. In this article, we exhibit a complex 𝕐 and explicit quasi-isomorphisms from 𝕐 to two complexes, as described above, for the appropriate choice of parameters, which give rise to this isomorphism. Our quasi-isomorphisms may be formed over the ring of integers; they can be passed to an arbitrary ring or field by base change. All of our work is equivariant under the action of the group GL(E) × GL(G); that is, everything we do is independent of the choice of basis. Knowledge of the homology of the top complex is equivalent to knowledge of the modules in the resolution of the Segre module Segre(e,g, ℓ), for ℓ = m − n. The modules {Segre (e,g, ℓ)|ℓ ∈ ℤ} are a set of representatives of the divisor class group of the determinantal ring defined by the 2 × 2 minors of an e × g matrix of indeterminates. If R is the ring of integers, then the homology H𝒩(m,n,p) is not always a free abelian group. In other words, if R is a field, then the dimension of H𝒩(m,n,p) depends on the characteristic of R. The module H𝒩(m,n,p) is known when R is a field of characteristic zero; however, this module is not yet known over arbitrary fields. The modules in the minimal resolution of the universal ring for finite length modules of projective dimension two are equal to modules of the form H𝒩(m,n,p).

Let L be a free Lie algebra of finite rank r over an arbitrary field K of characteristic 0, and let L n denote the homogeneous component of degree n in L. Viewed as a module for the general linear group GL(r,K), L n is known to be semisimple with the isomorphism types of the simple summands indexed by partitions of n with at most r parts. Klyachko proved in 1974 that, for n > 6, almost all such partitions are needed here, the exceptions being the partition with just one part, and the partition in which all parts are equal to 1. This paper presents a combinatorial proof based on the Littlewood-Richardson rule. This proof also yields that if the composition multiplicity of a simple summand in L n is greater than 1, then it is at least \(\frac{n}{6}-1\).

The problem of roots existence for different classes of matrix such as simple and semisimple matrices from SL2(F), SL2(Z) and GL2(F) are solved. For this purpose, we first introduced the concept of an extended special linear group ESL2(F), which is generalisation of the matrix group SL2(F), where F is arbitrary perfect field. The group of unimodular matrices and extended symplectic group ESp2(R) are generalised by us, their structures are found. Our criterion oriented on a general class of matrix depending of the form of minimal and characteristic polynomials, moreover a proposed criterion holds in GL2(F), where F is an arbitrary field. The method of matrix factorisation is outlined. We show that ESL2(F) is a set of all square matrix roots from SL2(F) except of that established in our root existence criterion.

A generalization of a theorem of Ore

Let A_K be an abelian variety over a discrete valuation field K. Let A be the Neron model of A_K over the ring of integers O_K of K and A_k its special fibre. We study the set of rational points of the group of components \phi_A of A_k. In particular, we look at the cases where A_K is a Jacobian or where A_K is an abelian variety having semi-stable reduction. We also consider algebraic tori over K instead of abelian varieties.

Let $F$ be a field, $\ell$ a prime and $D$ a central division $F$-algebra of $\ell$-power degree. By the Rost kernel of $D$ we mean the subgroup of $F^*$ consisting of elements $\lambda$ such that the cohomology class $(D)\cup (\lambda)\in H^3(F,\,\mathbb{Q}_{\ell}/\mathbb{Z}_{\ell}(2))$ vanishes. In 1985, Suslin conjectured that the Rost kernel is generated by $i$-th powers of reduced norms from $D^{\otimes i},\,\forall i\ge 1$. Despite of known counterexamples, we prove some new cases of Suslin's conjecture. We assume $F$ is a henselian discrete valuation field with residue field $k$ of characteristic different from $\ell$. When $D$ has period $\ell$, we show that Suslin's conjecture holds if either $k$ is a $2$-local field or the cohomological $\ell$-dimension $\mathrm{cd}_{\ell}(k)$ of $k$ is $\le 2$. When the period is arbitrary, we prove the same result when $k$ itself is a henselian discrete valuation field with $\mathrm{cd}_{\ell}(k)\le 2$. In the case $\ell=\text{char}(k)$ an analog is obtained for tamely ramified algebras. We conjecture that Suslin's conjecture holds for all fields of cohomological dimension 3.

Improving mechanical performance in material extrusion parts via optimized toolpath planning