Galois lines for the Artin–Schreier–Mumford curve

The arrangement of all Galois lines for the Artin–Schreier–Mumford curve in the projective 3-space is described. Surprisingly, there exist infinitely many Galois lines intersecting this curve.     

On the Cartan Map for Crossed Products and Hopf–Galois Extensions

We study certain aspects of the algebraic K-theory of Hopf–Galois extensions. We show that the Cartan map from K-theory to G-theory of such an extension is a rational isomorphism, provided the ring of coinvariants is regular, the Hopf algebra is finite dimensional and its Cartan map is injective in degree zero. This covers the case of a crossed product of a regular ring with a finite group and has an application to the study of Iwasawa modules.     

A Higher Atiyah–Patodi–Singer Index Theorem for the Signature Operator on Galois Coverings

Let (N, g) be a closed Riemannianmanifold of dimension 2m – 1 and let Ñ N be a Galois covering of N. We assumethat is of polynomial growth with respect to a word metric and that _Ñ is L ²-invertible in degree m. By employing spectral sections with asymmetry property with respect to the -Hodge operator, we define the higher eta invariant associatedwith the signature operator on Ñ, thus extending previous work of Lott. If ₁(M) M is the universal cover of a compact orientable even-dimensionalmanifold with boundary (M = N)then, under the above invertibility assumption on , andalways employing symmetric spectral sections, we define acanonical Atiyah–Patodi–Singer index class, in K ₀(C ^* _r ()), for the signature operator of . Using the higherAPS index theory developed in [6], we express the Chern character ofthis index class in terms of a local integral and of the higher etainvariant defined above, thus establishing a higher APS index theoremfor the signature operator on Galois coverings. We expect the notion ofa symmetric spectral section for the signature operator to have widerimplications in higher index theory for signatures operators.     

On the compatibility relation for the Föppl–von Kármán plate equations

By using a coordinate-free approach we propose a new derivation of the compatibility equation for the Föppl–von Kármán nonlinear plate theory.     

Canonical Extensions and Kripke–Galois Semantics for Non-distributive Logics

This article presents an approach to the semantics of non-distributive propositional logics that is based on a lattice representation (and duality) theorem that delivers a canonical extension of the lattice. Our approach supports both a plain Kripke-style semantics and, by restriction, a general frame semantics. Unlike the framework of generalized Kripke frames (RS-frames), the semantic approach presented in this article is suitable for modeling applied logics (such as temporal, or dynamic), as it respects the intended interpretation of the logical operators. This is made possible by restricting admissible interpretations.     

Note on the Hochschild cohomology of Hopf–Galois extension

We generalize [4, Theorem 4.3] to the case of Hopf–Galois extension, by introducing the cotensor product of a comodule algebra and its opposite algebra, and then give some applications.     

Estimation of the Bezout number for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function.In this paper.a coniecture on trianguation is confirmed The relation between the piecewise linear algebraiccurve and four-color conjecture is also presented.By Morgan-Scott triangulation, we will show the instabilityof Bezout number of piecewise algebraic curves. By using the combinatorial optimization method,an upper     

Integrals,Quantum Galois Extensions,and the Affineness Criterion for Quantum Yetter–Drinfel'd Modules

In this paper we shall generalize the notion of an integral on a Hopf algebra introduced by Sweedler, by defining the more general concept of an integral of a threetuple (H, A, C), where H is a Hopf algebra coacting on an algebra A and acting on a coalgebra C. We prove that there exists a total integral γ: C → Hom(C, A) of (H, A, C) if and only if any representation of (H, A, C) is injective in a functorial way, as a corepresentation of C. In particular, the quantum integrals associated to Yetter–Drinfel'd modules are defined. Let now A be an H-bicomodule algebra, ^H$\text{Y}$$\text{D}$_A the category of quantum Yetter–Drinfel'd modules, and B = {a ∈ A|∑S^− 1(a_〈1〉)a_{〈 − 1〉} ⊗ a_〈0〉 = 1_H ⊗ a}, the subalgebra of coinvariants of the Verma structure A ∈ ^H$\text{Y}$$\text{D}$_A. We shall prove the following affineness criterion: if there exists γ: H → Hom(H, A) a total quantum integral and the canonical map β: A ⊗ _B A → H ⊗ A, β(a ⊗ _B b) = ∑S^− 1(b_〈1〉)b_{〈 − 1〉} ⊗ ab_〈0〉 is surjective (i.e., A/B is a quantum homogeneous space), then the induction functor – ⊗ _B A: $\text{M}$_B → ^H$\text{Y}$$\text{D}$_A is an equivalence of categories. The affineness criteria proven by Cline, Parshall, and Scott, and independently by Oberst (for affine algebraic groups schemes) and Schneider (in the noncommutative case), are recovered as special cases.     

Even Galois representations and the Fontaine–Mazur conjecture

We prove, under mild hypotheses, that there are no irreducible two-dimensional ordinary even Galois representations of Gal([`(Q)]/Q)\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q}) with distinct Hodge–Tate weights. This is in accordance with the Fontaine–Mazur conjecture. If K/Q is an imaginary quadratic field, we also prove (again, under certain hypotheses) that Gal([`(Q)]/K)\mathrm{Gal}(\overline{\mathbf{Q}}/K) does not admit irreducible two-dimensional ordinary Galois representations of non-parallel weight.     

A mixed variational principle for the Föppl–von Kármán equations

A mixed variational principle is proposed for deducing the Föppl–von Kármán equations governing the large deflections of thin elastic plates or shallow shells. Proper boundary conditions are found for the case of applied in-plane tractions and displacements, and simple mechanical interpretations are achieved. Numerical implementation is carried out, along with examples and comparisons with the classical formulation in terms of displacements.     

Application of the trigonal curve to the Blaszak–Marciniak lattice hierarchy

We develop a method for constructing algebro-geometric solutions of the Blaszak–Marciniak (BM) lattice hierarchy based on the theory of trigonal curves. We first derive the BM lattice hierarchy associated with a discrete (3×3)-matrix spectral problem using Lenard recurrence relations. Using the characteristic polynomial of the Lax matrix for the BM lattice hierarchy, we introduce a trigonal curve with two infinite points, which we use to establish the associated Dubrovin-type equations. We then study the asymptotic properties of the algebraic function carrying the data of the divisor and the Baker–Akhiezer function near the two infinite points on the trigonal curve. We finally obtain algebro-geometric solutions of the entire BM lattice hierarchy in terms of the Riemann theta function.     

The curve selection lemma and the Morse–Sard theorem

We use an inequality due to Bochnak and Lojasiewicz, which follows from the Curve Selection Lemma of real algebraic geometry in order to prove that, given a C ^r function , we have

On the spectral curve for functional-difference Schrödinger equation

We suggest a method for constructing a set of finite-gap solutions for a functional-difference deformation of the Schr?dinger equation v(x)f(x +2h)+ f(x)= λf(x + h). It is shown that the edges of gaps of the corresponding spectral curve depend on x. Examples are given. Bibliography: 7 titles.     

Spectral sequence and finitely presented dimension for weak Gopf–Galois extensions

Let H be a weak Hopf algebra, A a right weak H-comodule algebra, and B the subalgebra of the H-coinvariant elements of A. Let A/B be a right weak H-Galois extension. In this paper, a spectral sequence for Ext which yields an estimate for the global dimension of A in terms of the corresponding data for H and B is constructed. Next, the relationship between the finitely presented dimensions of A and its subalgebra B are given. Further, the case in which A is an n-Gorenstein algebra is studied.     

The application of trigonal curve to the Mikhailov–Shabat–Sokolov flows

Resorting to the characteristic polynomial of Lax matrix for the Mikhailov–Shabat–Sokolov hierarchy associated with a \({3 \times 3}\) matrix spectral problem, we introduce a trigonal curve, from which we deduce the associated Baker–Akhiezer function, meromorphic functions and Dubrovin-type equations. The straightening out of the Mikhailov–Shabat–Sokolov flows is exactly given through the Abel map. On the basis of these results and the theory of trigonal curve, we obtain the explicit theta function representations of the Baker–Akhiezer function, the meromorphic functions, and in particular, that of solutions for the entire Mikhailov–Shabat–Sokolov hierarchy.