The study of the conformal group in $R^{p,q}$ usually involves the conformal compactification of $R^{p,q}$. This allows the transformations to be represented by linear transformations in $R^{p+1,q+1}$. So, for example, the conformal group of Minkowski space, $R^{1,3}$ leads to its isomorphism with $SO(2,4)$. This embedding into a higher dimensional space comes at the expense of the geometric properties of the transformations. This is particularly a problem in $R^{1,3}$ where we might well prefer to keep the geometric nature of the various types of transformations in sight. In this note, we show that this linearization procedure can be achieved with no loss of geometric insight, if, instead of using this compactification, we let the conformal transformations act on two copies of the associated Clifford algebra. Although we are mostly concerned with the conformal group of Minkowski space (where the geometry is clearest), generalization to the general case is straightforward.

In this paper, we prove that there are infinitely many positive integers $N$ such that the Diophantine equation $(x^2+y)(x+y^2)=N(x-y)^3$ has no nontrivial integer solution $(x,y)$.

We consider the generalized Wirtinger inequality \[ \left( \int_{0}^{T} a |u|^q \right)^{1/q} \le C \biggm(\int_{0}^{T} a^{1-p} |u'|^{p}\biggm)^{1/p}, \] with $p,q>1$, $T>0$, $a\in L^1[0,T]$, $a\ge0$, $a\not\equiv0$ and where $u$ is a $T$-periodic function satisfying the constraint \[ \int_{0}^{T} a |u|^{q-2}u =0. \] We provide the best constant $C>0$ as well as all extremals. Furthermore, we characterize the natural functional space where the inequality is defined.

In this paper we give some sufficient conditions, which do ensure the existence and uniqueness of a compact almost automorphic solution to some hyperbolic differential equations. As an illustration, we study the existence and uniqueness of almost automorphic solutions to a one-dimensional heat equation with small delays.

We study the following problem: $B_0$ and $B_1$ each has a sparse input vector $V_0$ and $V_1$; for each $j$ we need to decide whether $B_0[j]+B_1[j]>t$. We give a privacy-preserving algorithm, in which $B_0$ and $B_1$ do not need to reveal any information about their input vectors to each other, except the output of algorithm. Our algorithm is highly efficient.

It is well known that the functor $\Omega: {\bf Sp} \rightarrow {\bf Loc}$ does not preserve meets of subspaces. More generally, the meet of a family of spatial sublocales of a locale is in general not spatial. In this paper, we will give a characterization for those topological spaces for which the functor $\Omega$ preserves meets of subspaces. As a corollary, we give some characterizations for the meet of some spatial sublocales of a locale to be spatial.

The aim of the present paper is to investigate several interesting properties of a linear operator $L^{p}_{q,s}(\alpha_{i})$ associated with the Fox-Wright psi function.

We prove that from among simple multiveblen configurations only combinatorial Grassmannians can be embedded into a Desarguesian projective space. The class of regular multiveblen configurations which are projectively embeddable is determined.

Let $\mathcal{A}$ be the class of normalized analytic functions in the unit disc and let $P_{\gamma}(\alpha, \beta)$ be the class of all functions $f \in \mathcal{A}$ satisfying the condition \[ \exists \ \eta \in \mathbb{R}, \quad \Re \left \{ e^{i \eta}\left[(1-\gamma)\left(\frac{f(z)}{z}\right)^{\alpha} + \gamma \frac{zf'(z)}{f(z)}\left(\frac{f(z)}{z}\right)^{\alpha} - \beta \right] \right \} 0 .\] We consider the integral transform \[ V_{\lambda, \alpha}(f)(z)=\left\{\int_{0}^{1}\lambda(t) \left(\frac{f(tz)}{t} \right)^{\alpha} dt\right\}^{\frac{1}{\alpha}},\] where $\lambda(t)$ is a real-valued nonnegative weight function normalized by\linebreak $\int_{0}^{1}\lambda(t) dt=1$. In this paper we find conditions on the parameters $\alpha, \beta, \gamma, \mu $ such that $V_{\lambda, \alpha}(f)$ maps $P_{\gamma}(\alpha, \beta)$ into the class of starlike functions of order $\mu$. We also provide a number of applications for various choices of $\lambda(t)$. Our results generalize known results on this topic.

In this paper we obtain height estimates concerning to a compact spacelike hypersurface $\Sigma^n$ immersed with constant mean curvature $H$ in the Steady State space $\mathcal H^{n+1}$, when its boundary is contained into some hyperplane of this spacetime. As a first application of these results, when $\Sigma^n$ has spherical boundary, we establish relations between its height and the radius of its boundary. Moreover, under a certain restriction on the Gauss map of $\Sigma^n$, we obtain a sharp estimate for $H$. Finally, we also apply our estimates to describe the end of a complete spacelike hypersurface and to get theorems of characterization concerning to spacelike hyperplanes in $\mathcal H^{n+1}$.

As in the case of the associahedron and cyclohedron, the permutohedron can also be defined as an appropriate compactification of a configuration space of points on an interval or on a circle. The construction of the compactification endows the permutohedron with a projection to the cyclohedron, and the cyclohedron with a projection to the associahedron. We show that the preimages of any point via these projections might not be homeomorphic to (a cell decomposition of) a disk, but are still contractible. We briefly explain an application of this result to the study of knot spaces from the point of view of the Goodwillie-Weiss manifold calculus.

We show that the use of Brouwer fixed point theorem in Mironescu-Panaitopol's approach to the existence of a triangle with prescribed interior bisector lengths can be replaced by that of Banach fixed point theorem followed by an elementary limiting argument.

We characterize the boundedness and compactness of the weighted composition operators acting between some Fock-type spaces. Motivated by some recent ideas by S. Steviić we also calculate the operator norm of some of these operators. Also we determine the form of the symbol which induces a bounded or compact composition operator between some Fock-type spaces.

In this paper we characterize sequences of orthogonal polynomials on the unit circle whose Carathéodory function satisfies a Riccati differential equation with polynomial coefficients, in terms of matrix Sylvester differential equations. For the particular case of semi-classical orthogonal polynomials on the unit circle, it is derived a characterization in terms of first order linear differential systems.

We study diameter preserving linear bijections from ${\cal C}(X, V)$ onto ${\cal C}(Y, {\cal C}_0(L))$ where $X, Y$ are compact Hausdorff spaces, $L$ is a locally compact Hausdorff space and $V$ is a Banach space. In the case when $X$ and $Y$ are infinite and ${\cal C}_0(L)^*$ has the Bade property we prove that there is a diameter preserving linear bijection from ${\cal C}(X, V)$ onto ${\cal C}(Y, {\cal C}_0(L))$ if and only if $X$ is homeomorphic to $Y$ and $V$ is linearly isometric to ${\cal C}_0(L)$. Similar results are obtained in the case when $X$ and $Y$ are not compact but locally compact spaces.