In the moduli space of smooth and complex irreducible projective curves of genus g, let be the locus of curves that do not satisfy the Gieseker-Petri theorem. Let be the subvariety of GPg formed by curves C of genus g with a pencil g1d=(V, L∈G1d(C) free of base points for which the Petri map μV:V⊗H0(C,KC⊗L−1)→H0(C,KC) is not injective. For g≥8, we construct in this work a family of irreducible plane curves of genus g with moduli

Let f(x, y, z, t) satisfy the equation

For certain purposes, particularly in connection with the propagation of a boundary of fusion etc., it is of interest to discover solutions of (1) which permit the equation:

to be solved explicitly in the form:

This suggests the examination of solutions of the type

where

and f, ϕ, ψ are functions to be determined. To save repetition, Roman capitals denote arbitrary constants throughout.

This paper considers the long-time behavior of solutions for the Cauchy problem of a class of second-order nonlinear differential equations: -x''+ f(t,x,x')x'+g(x)=h(t). Under appropriate conditions it is shown that the solutions of the problem possess some dichotomy properties.

Let G be a finite group acting freely on a finitistic space X having cohomology type (0, b) (for example, $\mathbb S$n × $\mathbb S$2n is a space of type (0, 1) and the one-point union $\mathbb S$n ∨ $\mathbb S$2n ∨ $\mathbb S$3n is a space of type (0, 0)). It is known that a finite group G that contains ℤp ⊕ ℤp ⊕ ℤp, p a prime, cannot act freely on $\mathbb S$n × $\mathbb S$2n. In this paper, we show that if a finite group G acts freely on a space of type (0, 1), where n is odd, then G cannot contain ℤp ⊕ ℤp, p an odd prime. For spaces of cohomology type (0, 0), we show that every p-subgroup of G is either cyclic or a generalized quaternion group. Moreover, for n even, it is shown that ℤ2 is the only group that can act freely on X.

Let G be a compact, simply-connected Lie group. The cohomology of the loop space ΏG has been described by Bott, both in terms of a cell decomposition [1] and certain homogeneous spaces called generating varieties [2]. It is possible to view ΏG as an infinite dimensional “Grassmannian” associated to an appropriate infinite dimensional group, cf. [3], [7]. From this point of view the above cell-decomposition of Bott arises from a Bruhat decomposition of the associated group. We choose a generator H ∈ H2(ΏG, ℤ) and call it the hyperplane class. For a finite-dimensional Grassmannian the highest power of H carries geometric information about the variety, namely, its degree. An analogous question for ΏG is: What is the largest integer Nk = Nk(G) which divides Hk ∈ H2k(ΏG, ℤ)?

Of course, if G = SU(2) = S3, one knows Nk = h!. In general, the deviation of Nk from k! measures the failure of H to generate a divided polynomial algebra in H*(ΏG, ℤ).

We determine the extent to which the collection of Γ-Euler–Satake characteristics classify closed 2-orbifolds. In particular, we show that the closed, connected, effective, orientable 2-orbifolds are classified by the Γ-Euler–Satake characteristics corresponding to free or free abelian Γ and are not classified by those corresponding to any finite set of finitely generated discrete groups. These results demonstrate that the Γ-Euler–Satake characteristics corresponding to free abelian Γ constitute new invariants of orbifolds. Similarly, we show that such a classification is neither possible for non-orientable 2-orbifolds nor for non-effective 2-orbifolds using any collection of groups Γ.

In 2006, F. Luca and I. E. Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) proved that there are only finitely many pairs (n, m) of positive integers which satisfy the Diophantine equation |τ(n!)|=m!, where τ is the Ramanujan function. In this paper, we follow the same approach of Luca and Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) to determine all solutions of the above equation. The proof of our main theorem uses linear forms in two logarithms and arithmetic properties of the Ramanujan function.

We show that Artin–Schelter regularity of a $\mathbb{Z}$-graded algebra can be examined by its associated $\mathbb{Z}$r-graded algebra. We prove that there is exactly one class of four-dimensional Artin–Schelter regular algebras with two generators of degree one in the Jordan type. This class is strongly noetherian, Auslander regular, and Cohen–Macaulay. Their automorphisms and point modules are described.

In this paper we consider for a non-unital ring R, the category of firm R-modules for a non-unital ring R, i.e. the modules M such that the canonical morphism μM : R ⊗RM → M given by r ⊗ m ↦ rm is an isomorphism. This category is a natural generalization of the usual category of unitary modules for a ring with identity and shares many properties with it. The only difference is that monomorphisms are not always kernels. It has been proved recently that this category is not Abelian in general by providing an example of a monomorphism that is not a kernel in a particular case. In this paper we study the lattices of monomorphisms and kernels, proving that the lattice of monomorphisms is a modular lattice and that the category of firm modules is Abelian if and only if the composition of two kernels is a kernel.

The centre of the maximal p-subgroup of (pkD2p) is described as an elementary abelian p-group, where p is a prime.

In this paper, we consider the following fractional Schrödinger–Poisson system with singularity

\begin{equation*} \left \{\begin{array}{lcl} ({-}\Delta)^s u+V(x)u+\lambda \phi u = f(x)u^{-\gamma}, &&\quad x\in\mathbb{R}^3,\\ ({-}\Delta)^t \phi = u^2, &&\quad x\in\mathbb{R}^3,\\ u>0,&&\quad x\in\mathbb{R}^3, \end{array}\right. \end{equation*}

where 0 < γ < 1, λ > 0 and 0 < s ≤ t < 1 with 4s + 2t > 3. Under certain assumptions on V and f, we show the existence, uniqueness, and monotonicity of positive solution uλ using the variational method. We also give a convergence property of uλ as λ → 0, when λ is regarded as a positive parameter.

It is shown that the countable saturated discrete linear ordering has the small index property, but that the countable 1-transitive linear orders which contain a convex subset isomorphic to ${\Bbb Z}^2$ do not. Similar results are also proved in the coloured case.

The equation of the title in positive integers x, y, z where D is a given integer has been considered for some 300 years [4, pp 634–639]. As observed by V. A. Lebesgue, and probably known to Euler, if x, y, z is one non-trivial solution i.e., one with xy(x2 – y2) ≠0, another is given by . It then follows that there are infinitely many such with (x, y) = 1. The question that remains is to determine for which values of D such solutions exist.

In this paper we calculate the localisation at the prime 3 of the integral cohomology ring of the Mathieu group M24, together with its mod-3 cohomology ring. The main results are

Theorem 1. The ring H*(M24, Z)(3)is the commutative graded Z(3)-algebra with generators

and relations v2 = 0 and βθ = 0. The Chern classes of the Todd representation in GL11F2

We construct a compactly generated, totally disconnected, locally compact group whose Hecke algebra with respect to any compact open subgroup does not have a $C^*$-enveloping algebra.

The purpose of this paper is to study an optimal control problem associated with the thermally coupled Navier-Stokes equations. An existence result for this problem is obtained. The most important result of this paper is the proof of the existence and regularity of a solution of the adjoint system. By defining several functions, this system (which is not a divergence free one) is replaced by a divergence free system.

In [1] it was shown that for a compact normal operator on a Hilbert space the numerical range was the convex hull of the point spectrum. Here it is shown that the same holds for a semi-normal operator whose point spectrum satisfies a density condition (Theorem 1). In Theorem 2 a similar condition is shown to imply that the numerical range of a semi-normal operator is closed. Some examples are given to indicate that the condition in Theorem 1 cannot be relaxed too much.

Let K be a compact Hausdorff space, and let C(K) be the corresponding Banach space of continuous functions on K. It is well-known that every 1-summing operator S:C(K)→l2 is also nuclear, and therefore factors S = S1S2, with S1:l2→l2 a Hilbert–Schmidt operator and S1:C(K)→l2 a bounded operator. It is easily seen that this latter property is preserved when C(K) is replaced by any quotient, and that a Banach space X enjoys this property if and only if its second dual, X**, does. This led A. Pełczyński [15] to ask if the second dual of a Banach space X must be isomorphic to a quotient of a C(K)-space if X has the property that every 1-summing operator X-→l2 factors through a Hilbert-Schmidt operator. In this paper, we shall first of all reformulate the question in an appropriate manner and then show that counter-examples are available among super-reflexive Tsirelson-like spaces as well as among quasi-reflexive Banach spaces.

In this brief note, we evaluate certain quartic and octic Gauss sums with the use of theorems on fourth and eighth power difference sets. We recall that a subset H of a finite (additive) abelian group G is said to be a difference set of G [5, p. 64] if for some fixed natural number λ, every nonzero element of G can be written as a difference of two elements of H fin exactly λ ways.

Let R be a commutative ring with an identity element, E a (unitary) R-module, and x1, x2, …, xs elements of R. In these circumstances it is possible to form the Koszul complex† K(x1, x2, …, xs|E) of E with respect to x1, x2,…, xs and to investigate the implications, for E and xl, x2, …, xs, if certain of the homology modules of this complex vanish. This was first undertaken by M. Auslander and D. A. Buchsbaum [1]. Among the many results they obtain, the following [1, Proposition 2.8, p. 632] is of particular interest in connection with the present paper:

If R is Noetherian, E is finitely generated, and x1x2,…, xs belong to the Jacobson radical of R, then the statements

(a) x1, x2,…, xsis an R-sequence on E,

(b) HpK(x1, x2,…,xs⃒E) = O for all p > 0,

(c) H1K(x1, x2,…., xs,⃒E) = 0,

are all equivalent.