In this paper we prove the existence of at least three classical solutions for the problem

$$ \left\{ \begin{aligned} \amp-(|u'|^{p-2}u')'=\lambda f(t,u)h(u'), \\ \ampu(a)=u(b)=0, \end{aligned} \right. $$

when $\lambda$ lies in an explicitly determined open interval.

Our main tool is a very recent three-critical-points theorem stated in a paper by D. Averna and G. Bonanno (Topolog. Meth. Nonlin. Analysis22 (2003), 93–103).

AMS 2000 Mathematics subject classification: Primary 34B15

Soit $G$ un groupe réductif $p$-adique connexe. Nous effectuons une décomposition spectrale sur $G$ à partir de la formule d’inversion de Fourier utilisée dans ‘Une formule de Plancherel pour l’algèbre de Hecke d’un groupe réductif $p$-adique’, V. Heiermann, Commun. Math. Helv.76 (2001), 388–415. Nous en déduisons essentiellement qu’une représentation cuspidale d’un sous-groupe de Levi M appartient au support cuspidal d’une représentation de carré intégrable de $G$ si et seulement si c’est un pôle de la fonction $\mu$ de Harish-Chandra d’ordre égal au rang parabolique de M. Ces pôles sont d’ordre maximal. Plus précisément, nous montrons que cette condition est nécessaire et que sa suffisance équivaut à une propriété combinatoire de la fonction $\mu$ de Harish-Chandra qui s’avère être une conséquence d’un résultat de E. Opdam. En outre, nous obtenons des identités entre des combinaisons linéaires de coefficients matriciels. Ces identités contiennent des informations sur le degré formel des représentations de carré intégrable ainsi que sur leur position dans la représentation induite.

Let $G$ be a reductive connected $p$-adic group. With help of the Fourier inversion formula used in ‘Une formule de Plancherel pour l’algèbre de Hecke d’un groupe réductif $p$-adique’, V. Heiermann, Commun. Math. Helv.76 (2001), 388–415, we give a spectral decomposition on $G$. In particular, we deduce from it essentially that a cuspidal representation of a Levi subgroup $M$ is in the cuspidal support of a square-integrable representation of $G$ if and only if it is a pole of Harish-Chandra’s $\mu$-function of order equal to the parabolic rank of $M$. These poles are of maximal order. In more explicit terms, we show that this condition is necessary and that its sufficiency is equivalent to a combinatorical property of Harish-Chandra’s $\mu$-function, which appears to be a consequence of a result of E. Opdam. We get also identities between some linear combinations of matrix coefficients. These identities contain information on the formal degree of square-integrable representations and on their position in the induced representation.

AMS 2000 Mathematics subject classification: Primary 22E50. Secondary 22E35; 11F70; 11F72; 11F85

For $q$ a non-negative integer, we introduce the internal $q$-homology of crossed modules and we obtain in the case $q=0$ the homology of crossed modules. In the particular case of considering a group as a crossed module we obtain that its internal $q$-homology is the homology of the group with coefficients in the ring of the integers modulo $q$.

The second internal $q$-homology of crossed modules coincides with the invariant introduced by Grandjeán and López, that is, the kernel of the universal $q$-central extension. Finally, we relate the internal $q$-homology of a crossed module to the homology of its classifying space with coefficients in the ring of the integers modulo $q$.

AMS 2000 Mathematics subject classification: Primary 18G50; 20J05. Secondary 18G30

For a smooth variety $X$ over a perfect field $k$ in characteristic $p$, an equivalence of category is established between the category of crystals on $X/W(k)$ and the category of $p$-adically nilpotent, integrable de Rham–Witt connections.

AMS 2000 Mathematics subject classification: Primary 14F30; 14F40

We obtain inverse factorial-series solutions of second-order linear difference equations with a singularity of rank one at infinity. It is shown that the Borel plane of these series is relatively simple, and that in certain cases the asymptotic expansions incorporate simple resurgence properties. Two examples are included. The second example is the large $a$ asymptotics of the hypergeometric function ${}_2F_1(a,b;c;x)$.

AMS 2000 Mathematics subject classification: Primary 34E05; 39A11. Secondary 33C05

This paper is a further contribution to the extensive study by a number of authors of the subalgebra lattice of a Lie algebra. We give some necessary and some sufficient conditions for a subalgebra to be upper modular. For algebraically closed fields of any characteristic these enable us to determine the structure of Lie algebras having abelian upper-modular subalgebras which are not ideals. We then study the structure of solvable Lie algebras having an abelian upper-modular subalgebra which is not an ideal and which has trivial intersection with the derived algebra; in particular, the structure is determined for algebras over the real field. Next we classify non-solvable Lie algebras over fields of characteristic zero having an upper-modular atom which is not an ideal. Finally, it is shown that every Lie algebra over a field of characteristic different from two and three in which every atom is upper modular is either quasi-abelian or a $\mu$-algebra.

AMS 2000 Mathematics subject classification: Primary 17B05; 17B50; 17B30; 17B20

We formulate and prove a free quantum analogue of the first fundamental theorems of invariant theory. More precisely, the polynomial function algebras on matrices are replaced by free algebras, while the universal cosovereign Hopf algebras play the role of the general linear group.

AMS 2000 Mathematics subject classification: Primary 16W30

We construct some series of polyhedral schemata which represent orientable closed connected 3-manifolds. We show that these manifolds have spines corresponding to certain balanced presentations of their fundamental groups. Then we study some covering properties of such manifolds and prove that many of them are cyclic branched coverings of lens spaces. Our theorems contain a number of published results from various sources as particular cases.

AMS 2000 Mathematics subject classification: Primary 57M12; 57M50; 57M60

Let $R$ be a semiprime ring with $U_{\trm{s}}$ its maximal symmetric ring of quotients and let $\rho_1$ and $\rho_2$ be two right ideals of $R$. We show that $\ell_R(\rho_1)=\ell_R(\rho_2)$ if and only if $\rho_1$ and $\rho_2$ satisfy the same differential identities with coefficients in $U_{\trm{s}}$, where $\ell_R(\rho_i)$ denotes the left annihilator of $\rho_i$ in $R$. This gives a generalization of several previous results in this area.

AMS 2000 Mathematics subject classification: Primary 6R50. Secondary 16N60

Using a connected sum construction, we prove the existence of complete non-compact embedded constant-mean-curvature-1 surfaces in hyperbolic 3-space with finitely many ends and non-trivial topology. These surfaces are obtained by desingularizing finitely many mutually tangent horospheres.

AMS 2000 Mathematics subject classification: Primary 53A10; 53A35

In this paper we compute the non-abelian tensor square for the free 2-Engel group of rank $n>3$. The non-abelian tensor square for this group is a direct product of a free abelian group and a nilpotent group of class 2 whose derived subgroup has exponent 3. We also compute the non-abelian tensor square for one of the group’s finite homomorphic images, namely, the Burnside group of rank $n$ and exponent 3.

AMS 2000 Mathematics subject classification: Primary 20F05; 20F45. Secondary 20F18

In this paper we study the eigenvalues associated with a positive eigenfunction of a quasilinear elliptic problem with an operator that is not necessarily bounded. For that, we use the bifurcation theory and obtain the existence of positive solutions for a range of values of the bifurcation parameter.

AMS 2000 Mathematics subject classification: Primary 35J60; 35J25. Secondary 35D05

In this work, the behaviour of solutions for the Dirichlet problem of the non-local equation

$$ u_t=\varDelta(\kappa(u))+\frac{\lambda f(u)}{(\int_{\varOmega}f(u)\,\mathrm{d}x)^p},\quad \varOmega\subset\mathbb{R}^N,\quad N=1,2, $$

is studied, mainly for the case where $f(s)=\mathrm{e}^{\kappa(s)}$. More precisely, the interplay of exponent $p$ of the non-local term and spatial dimension $N$ is investigated with regard to the existence and non-existence of solutions of the associated steady-state problem as well as the global existence and finite-time blow-up of the time-dependent solutions $u(x,t)$. The asymptotic stability of the steady-state solutions is also studied.

AMS 2000 Mathematics subject classification: Primary 35K60. Secondary 35B40

In this work we present a generalization of an exact sequence of normal bordism groups given in a paper by H. A. Salomonsen (Math. Scand.32 (1973), 87–111). This is applied to prove that if $h:M^n\to X^{n+k}$, $5\leq n\lt2k$, is a continuous map between two manifolds and $g:M^n\to BO$ is the classifying map of the stable normal bundle of $h$ such that $(h,g)_*:H_i(M,\mathbb{Z}_2)\to H_i(X\times BO,\mathbb{Z}_2)$ is an isomorphism for $i\lt n-k$ and an epimorphism for $i=n-k$, then $h$ bordant to an immersion implies that $h$ is homotopic to an immersion. The second remark complements the result of C. Biasi, D. L. Gonçalves and A. K. M. Libardi (Topology Applic.116 (2001), 293–303) and it concerns conditions for which there exist immersions in the metastable dimension range. Some applications and examples for the main results are also given.

AMS 2000 Mathematics subject classification: Primary 57R42. Secondary 55Q10; 55P60

We consider the eigenvalue equation for the largest eigenvalue of certain kinds of non-compact linear operators given as the sum of a multiplication and a kernel operator. It is shown that, under moderate conditions, such operators can be approximated arbitrarily well by operators of finite rank, which constitutes a discretization procedure. For this purpose, two standard methods of approximation theory, the Nyström and the Galerkin method, are generalized. The operators considered describe models for mutation and selection of an infinitely large population of individuals that are labelled by real numbers, commonly called continuum-of-alleles models.

AMS 2000 Mathematics subject classification: Primary 47A58; 45C05. Secondary 47B34