Positive solutions are obtained for the boundary value problem

$$\left\{ _{u\left( 0 \right)\,=\,u\left( 1 \right)\,=\,0}^{-{{\left( {{\left| {{u}'} \right|}^{p-2}}{u}' \right)}^{\prime }}\,=\,\lambda f\left( t,\,u \right),\,t\,\in \,\left( 0,\,1 \right),\,p\,>\,1} \right.$$

Here $f(t,u)\ge -M$, ($M$ is a positive constant) for $(t,u)\in [0,1]\times (0,\infty )$. We will show the existence of two positive solutions by using degree theory together with the upper–lower solution method.

This is a note on the classical Waring's problem for algebraic forms. Fix integers $(n,d,r,s)$, and let $\Lambda $ be a general $r$-dimensional subspace of degree $d$ homogeneous polynomials in $n+1$ variables. Let $\mathcal{A}$ denote the variety of $s$-sided polar polyhedra of $\Lambda $. We carry out a case-by-case study of the structure of $\mathcal{A}$ for several specific values of $(n,d,r,s)$. In the first batch of examples, $\mathcal{A}$ is shown to be a rational variety. In the second batch, $\mathcal{A}$ is a finite set of which we calculate the cardinality.

We extend the extension theorems to weighted Sobolev spaces $L_{w,k}^{p}\left( \mathcal{D} \right)$ on $(\varepsilon ,\delta )$ domains with doubling weight $w$ that satisfies a Poincaré inequality and such that ${{w}^{-1/p}}$ is locally ${{L}^{{{p}'}}}$. We also make use of the main theorem to improve weighted Sobolev interpolation inequalities.

In this paper we primarily consider two natural subgroups of the autohomeomorphism group of the real line $\mathbb{R}$, endowed with the compact-open topology. First, we prove that the subgroup of homeomorphisms that map the set of rational numbers $\mathbb{Q}$ onto itself is homeomorphic to the infinite power of $\mathbb{Q}$ with the product topology. Secondly, the group consisting of homeomorphisms that map the pseudoboundary onto itself is shown to be homeomorphic to the hyperspace of nonempty compact subsets of $\mathbb{Q}$ with the Vietoris topology. We obtain similar results for the Cantor set but we also prove that these results do not extend to ${{\mathbb{R}}^{n}}$ for $n\ge 2$, by linking the groups in question with Erdős space.

We consider Hausdorff and quasi-Hausdorff matrices as operators on classical spaces of analytic functions such as the Hardy and the Bergman spaces, the Dirichlet space, the Bloch spaces and $\text{BMOA}$. When the generating sequence of the matrix is the moment sequence of a measure $\mu$, we find the conditions on $\mu$ which are equivalent to the boundedness of the matrix on the various spaces.

We prove, for a field $K$ which is cyclic of odd prime power degree over the rationals, that the annihilator of the quotient of the units of $K$ by a suitable large subgroup (constructed from circular units) annihilates what we call the non-genus part of the class group. This leads to stronger annihilation results for the whole class group than a routine application of the Rubin–Thaine method would produce, since the part of the class group determined by genus theory has an obvious large annihilator which is not detected by that method; this is our reason for concentrating on the non-genus part. The present work builds on and strengthens previous work of the authors; the proofs are more conceptual now, and we are also able to construct an example which demonstrates that our results cannot be easily sharpened further.

In the Euclidean plane ${{\mathbb{R}}^{2}}$, we define the Minkowski difference $\mathcal{K}-\mathcal{L}$ of two arbitrary convex bodies $\mathcal{K},\mathcal{L}$ as a rectifiable closed curve ${{\mathcal{H}}_{h}}\subset {{\mathbb{R}}^{2}}$ that is determined by the difference $h={{h}_{K}}-{{h}_{\mathcal{L}}}$ of their support functions. This curve ${{\mathcal{H}}_{h}}$ is called the hedgehog with support function $h$. More generally, the object of hedgehog theory is to study the Brunn–Minkowski theory in the vector space of Minkowski differences of arbitrary convex bodies of Euclidean space ${{\mathbb{R}}^{n+1}}$, defined as (possibly singular and self-intersecting) hypersurfaces of ${{\mathbb{R}}^{n+1}}$. Hedgehog theory is useful for: (i) studying convex bodies by splitting them into a sum in order to reveal their structure; (ii) converting analytical problems into geometrical ones by considering certain real functions as support functions. The purpose of this paper is to give a detailed study of plane hedgehogs, which constitute the basis of the theory. In particular: (i) we study their length measures and solve the extension of the Christoffel–Minkowski problem to plane hedgehogs; (ii) we characterize support functions of plane convex bodies among support functions of plane hedgehogs and support functions of plane hedgehogs among continuous functions; (iii) we study the mixed area of hedgehogs in ${{\mathbb{R}}^{2}}$ and give an extension of the classical Minkowski inequality (and thus of the isoperimetric inequality) to hedgehogs.

In this paper we generalise the concept of a Steinberg cross section to non-connected affine Kac–Moody groups. This Steinberg cross section is a section to the restriction of the adjoint quotient map to a given exterior connected component of the affine Kac–Moody group. (The adjoint quotient is only defined on a certain submonoid of the entire group, however, the intersection of this submonoid with each connected component is non-void.) The image of the Steinberg cross section consists of a “twisted Coxeter cell”, a transversal slice to a twisted Coxeter element. A crucial point in the proof of the main result is that the image of the cross section can be endowed with a ${{\mathbb{C}}^{*}}$-action.

The equality of the centralizer and twisted centralizer is proved based on a case-by-case analysis when the unipotent radical of a maximal parabolic subgroup is abelian. Then this result is used to determine the poles of intertwining operators.