On the cohomology of tori over local fields with perfect residue field

If T is an algebraic torus defined over a discretely valued field K with perfect residue field k, we relate the K-cohomology of T to the k-cohomology of certain objects associated to T. When k has cohomological dimension ≤ 1, our results have a particularly simple form and yield, more generally, isomorphisms between Borovoi’s abelian K-cohomology of a reductive group G over K and the k-cohomology of a certain quotient of the algebraic fundamental group of G.     

Frobenius morphisms and stable module categories of repetitive algebras

Let k be the algebraic closure of a finite field F_q and A be a finite dimensional k-algebra with a Frobenius morphism F.In the present paper we establish a relation between the stable module category of the repetitive algebra  of A and that of the repetitive algebra of the fixed-point algebra A~F.As an application,it is shown that the derived category of A~F is equivalent to the subcategory of F-stable objects in the derived category of A when A has a finite global dimension.     

Homological Epimorphisms, Compactly Generated t-Structures and Gorenstein-Projective Modules

The aim of this paper is two-fold. Given a recollement (T′, T, T″, i*, i_*, i^!, j_!, j*, j_*), where T′, T, T″ are triangulated categories with small coproducts and T is compactly generated. First, the authors show that the BBD-induction of compactly generated t-structures is compactly generated when i_* preserves compact objects. As a con-sequence, given a ladder (T′, T, T″, T, T′) of height 2, then the certain BBD-induction of compactly generated t-structures is compactly generated. The authors apply them to the recollements induced by homological ring epimorphisms. This is the first part of their work. Given a recollement (D(B-Mod),D(A-Mod),D(C-Mod), i*, i_*, i^!, j_!, j*, j_*) induced by a homological ring epimorphism, the last aim of this work is to show that if A is Gorenstein, _A B has finite projective dimension and j_! restricts to D ^b (C-mod), then this recollement induces an unbounded ladder (B-Gproj,A-Gproj, C-Gproj) of stable categories of finitely generated Gorenstein-projective modules. Some examples are described.     

A method for analysing operational complexity in supply chains

This paper proposes a method for analysing the operational complexity in supply chains by using an entropic measure based on information theory. The proposed approach estimates the operational complexity at each stage of the supply chain and analyses the changes between stages. In this paper a stage is identified by the exchange of data and/or material. Through analysis the method identifies the stages where the operational complexity is both generated and propagated (exported, imported, generated or absorbed). Central to the method is the identification of a reference point within the supply chain. This is where the operational complexity is at a local minimum along the data transfer stages. Such a point can be thought of as a ‘sink’ for turbulence generated in the supply chain. Where it exists, it has the merit of stabilising the supply chain by attenuating uncertainty. However, the location of the reference point is also a matter of choice. If the preferred location is other than the current one, this is a trigger for management action. The analysis can help decide appropriate remedial action. More generally, the approach can assist logistics management by highlighting problem areas. An industrial application is presented to demonstrate the applicability of the method.     

Factorization of the reaction-diffusion equation,the wave equation,and other equations

We investigate equations of the form D _t u = Δu + ξ? _u for an unknown function u(t, x), t ∈ ?, x ∈ X, where D _t u = a ₀(u, t) + Σ _k=1 ^r a _k (t, u)? _t ^k u, Δ is the Laplace-Beltrami operator on a Riemannian manifold X, and ξ is a smooth vector field on X. More exactly, we study morphisms from this equation within the category PDE of partial differential equations, which was introduced by the author earlier. We restrict ourselves to morphisms of a special form—the so-called geometric morphisms, which are given by maps of X to other smooth manifolds (of the same or smaller dimension). It is shown that a map f: X → Y defines a morphism from the equation D _t u = Δu + ξ? _u if and only if, for some vector field Ξ and a metric on Y, the equality (Δ + ξ?)f*v = f*(Δ + Ξ?)v holds for any smooth function v: Y → ?. In this case, the quotient equation is D _t v = Δv + Ξ?v for an unknown function v(t, y), y ∈ Y. It is also shown that, if a map f: X → Y is a locally trivial bundle, then f defines a morphism from the equation D _t u = Δu if and only if fibers of f are parallel and, for any path γ on Y, the expansion factor of a fiber translated along the horizontal lift γ to X depends on γ only.     

Span of a DL-algebra

Unless otherswise specified, all objects are defined over a field k of characteristic 0. Let K be a field. The unessentialness of an extension of the algebra Der K by means of a splittable semisimple Lie algebra is established. Let D _K be the category of differential Lie algebras (DL-algebras) (g;K). In this paper for an extension L/K the functor η:D _K → D _L , defining the tensor product L ? _K of vector spaces and the homomorphism of Lie algebras, is constructed. If the extension L/K is algebraic, then η is unique. The results will be required for strengthening the progress on Gelfand–Kirillov problem and weakened conjecture [1, 2].     

Injective objects of monomorphism categories

For an acyclic quiver Q and a finite-dimensional algebra A, we give a unified form of the indecomposable injective objects in the monomorphism category Mon(Q,A) and prove that Mon(Q,A) has enough injective objects.     

The <Emphasis Type="Italic">H</Emphasis>-Closed Monoreflections,Implicit Operations,and Countable Composition,in Archimedean Lattice-Ordered Groups with Weak Unit

In the category of the title, called W, we completely describe the monoreflections \(\mathcal {R}\) which are H-closed (closed under homomorphic image) by means of epimorphic extensions S of the free object on ω generators, F(ω), within the Baire functions on \(\mathbb {R}^{\omega }\), \(B(\mathbb {R}^{\omega })\); label the inclusion \(e_{S} : F(\omega ) \rightarrow S\). Then (a) inj e _S (the class of objects injective for e _S ) is such an \(\mathcal {R}\), with e _S a reflection map iff S is closed under countable composition with itself (called ccc), (b) each such \(\mathcal {R}\) is inj e _S for a unique S with ccc, and (c) if S has ccc, then A∈inj e _S iff A is closed under countable composition with S. We think of (c) as expressing: A is closed under the implicit operations of W represented by S (and these are of at most countable arity). In particular, the family of H-closed monoreflections is a set, whereas the family of all monoreflections is consistently a proper class. There is a categorical framework to the proofs, valid in any sufficiently complete category with free objects and epicomplete monoreflection β which is H-closed and of bounded arity; in W the β is of countable arity, and \(\beta F(\omega ) = B(\mathbb {R}^{\omega })\). The paper continues our earlier work along similar lines.     

Maslov index in semi-Riemannian submersions

We study focal points and Maslov index of a horizontal geodesic γ : I → M in the total space of a semi-Riemannian submersion π : M → B by determining an explicit relation with the corresponding objects along the projected geodesic \({\pi\circ\gamma:I\to B}\) in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary spacetime which is orthogonal to a timelike Killing vector field.     

Symbol algebras and cyclicity of algebras after a scalar extension

For a field F and a family of central simple F-algebras we prove that there exists a regular field extension E/F preserving indices of F-algebras such that all the algebras from the family are cyclic after scalar extension by E. Let \( \mathcal{A} \) be a central simple algebra over a field F of degree n with a primitive nth root of unity ρ _n . We construct a quasi-affine F-variety Symb(\( \mathcal{A} \)) such that for a field extension L/F Symb(\( \mathcal{A} \)) has an L-rational point if and only if \( \mathcal{A}{ \otimes_F}L \) is a symbol algebra. Let \( \mathcal{A} \) be a central simple algebra over a field F of degree n and K/F be a cyclic field extension of degree n. We construct a quasi-affine F-variety C(\( \mathcal{A} \) ,K) such that, for a field extension L/F with the property [KL : L] = [K : F], the variety C(\( \mathcal{A} \) ,K) has an L-rational point if and only if KL is a subfield of \( \mathcal{A}{ \otimes_F}L \).     

On Schur Superfunctors

We introduce super-analogues of the Schur functors defined by Akin, Buchsbaum and Weyman. These Schur superfunctors may be viewed as characteristic-free analogues of finite dimensional irreducible polynomial representations of the Lie superalgebra ????(m|n) studied by Berele and Regev. Our construction realizes Schur superfunctors as objects of a certain category of strict polynomial superfunctors. We show that Schur superfunctors are indecomposable objects of this category. In characteristic zero, these correspond to the set of all simple supermodules for the Schur superalgebra, S(m|n, d), for any m, n, d ? 0. We also provide decompositions of Schur bisuperfunctors in terms of tensor products of skew Schur superfunctors.     

On collineation groups of finite projective spaces containing a Singer cycle

By a result of Kantor, any subgroup of GL(n, q) containing a Singer cycle normalizes a field extension subgroup. This result has as a consequence a projective analogue, and this paper gives the details of this deduction, showing that any subgroup of PΓL(n, q) containing a projective Singer cycle normalizes the image of a field extension subgroup GL(n/s, q^s) under the canonical homomorphism GL(n, q) → PGL(n, q), for some divisor s of n, and so is contained in the image of ΓL(n/s, q^s) under the canonical homomorphism ΓL(n, q) → PΓL(n, q). The actions of field extension subgroups on V (n, q) are also investigated. In particular, we prove that any field extension subgroup GL(n/s, q^s) of GL(n, q) has a unique orbit on s-dimensional subspaces of V (n, q) of length coprime to q. This orbit is a Desarguesian s-partition of V (n, q).     

Cleft extensions of Koszul twisted Calabi–Yau algebras

Let H be a twisted Calabi–Yau (CY) Hopf algebra and σ a 2-cocycle on H. Let A be an N-Koszul twisted CY algebra such that A is a graded H^σ- module algebra. We show that the cleft extension A#_σH is also a twisted CY algebra. This result has two consequences. Firstly, the smash product of an N-Koszul twisted CY algebra with a twisted CY Hopf algebra is still a twisted CY algebra. Secondly, the cleft objects of a twisted CY Hopf algebra are all twisted CY algebras. As an application of this property, we determine which cleft objects of U(D, λ), a class of pointed Hopf algebras introduced by Andruskiewitsch and Schneider, are Calabi–Yau algebras.     

Inner and outer approximation of convex sets using alignment

We show that there exists, for each closed bounded convex set C in the Euclidean plane with nonempty interior, a quadrangle Q having the following two properties. Its sides support C at the vertices of a rectangle r and at least three of the vertices of Q lie on the boundary of a rectangle R that is a dilation of r with ratio 2. We will prove that this implies that quadrangle Q is contained in rectangle R and that, consequently, the inner approximation r of C has an area of at least half the area of the outer approximation Q of C. The proof makes use of alignment or Schüttelung, an operation on convex sets.     

Seifert manifolds and (1, 1)-knots

The aim of this paper is to investigate the relations between Seifert manifolds and (1, 1)-knots. In particular, we prove that each orientable Seifert manifold with invariants

$\{ Oo,0| - 1;\underbrace {(p,q),...,(p,q)}_{n times},(l,l - 1)\} $

has the fundamental group cyclically presented by G _n ((x ₁ ^q ...x _n ^{q l} x _n ^?p ) and, moreover, it is the n-fold strongly-cyclic covering of the lens space L(|nlq ? p|, q) which is branched over the (1, 1)-knot K(q, q(nl ? 2), p ? 2q, p ? q) if p ≥ 2q and over the (1, 1)-knot K(p? q, 2q ? p, q(nl ? 2), p ? q) if p< 2q.

     

Polyboxes,Cube Tilings and Rigidity

A non-empty subset A of X=X ₁×???×X _d is a (proper) box if A=A ₁×???×A _d and A _i ?X _i for each i. Suppose that for each pair of boxes A, B and each i, one can only know which of the three states takes place: A _i =B _i , A _i =X _i ?B _i , A _i ?{B _i ,X _i ?B _i }. Let F and G be two systems of disjoint boxes. Can one decide whether ∪F=∪G? In general, the answer is ‘no’, but as is shown in the paper, it is ‘yes’ if both systems consist of pairwise dichotomous boxes. (Boxes A, B are dichotomous if there is i such that A _i =X _i ?B _i .) Several criteria that enable to compare such systems are collected. The paper includes also rigidity results, which say what assumptions have to be imposed on F to ensure that ∪F=∪G implies F=G. As an application, the rigidity conjecture for 2-extremal cube tilings of Lagarias and Shor is verified.     

New class of hybrid BDF methods for the computation of numerical solutions of IVPs

A new class of hybrid BDF-like methods is presented for solving systems of ordinary differential equations (ODEs) by using the second derivative of the solution in the stage equation of class 2 + 1hybrid BDF-like methods to improve the order and stability regions of these methods. An off-step point, together with two step points, has been used in the first derivative of the solution, and the stability domains of the new methods have been obtained by showing that these methods are A-stable for order p, p =?3,4,5,6,7and A(α)-stable for order p, 8 ≤ p ≤?14. The numerical results are also given for four test problems by using variable and fixed step-size implementations.     

Homology of braid groups,the Burau representation,and points on superelliptic curves over finite fields

The (reduced) Burau representation V _n of the braid group B _n is obtained from the action of B _n on the homology of an infinite cyclic cover of the disc with n punctures. In this paper, we calculate H _*(B _n ; V _n ). Our topological calculation has the following arithmetic interpretation (which also has different algebraic proofs): the expected number of points on a random superelliptic curve of a fixed genus over F _q is q.