We establish sufficient conditions on the shape of a set $A$ included in the space $\mathcal{L}_{s}^{n}\left( X,Y \right)$ of the $n$-linear symmetric mappings between Banach spaces $X$ and $Y$ , to ensure the existence of a ${{C}^{n}}$-smooth mapping $f:X\to Y$, with bounded support, and such that ${{f}^{\left( n \right)}}\left( X \right)=A$, provided that $X$ admits a ${{C}^{n}}$-smooth bump with bounded $n$-th derivative and dens $\text{dens }X=\text{dens }{{\mathcal{L}}^{n}}\left( X,Y \right)$. For instance, when $X$ is infinite-dimensional, every bounded connected and open set $U$ containing the origin is the range of the $n$-th derivative of such amapping. The same holds true for the closure of $U$, provided that every point in the boundary of $U$ is the end point of a path within $U$. In the finite-dimensional case, more restrictive conditions are required. We also study the Fréchet smooth case for mappings from ${{\mathbb{R}}^{n}}$ to a separable infinite-dimensional Banach space and the Gâteaux smooth case for mappings defined on a separable infinite-dimensional Banach space and with values in a separable Banach space.

We give a new proof of former results by $\text{G}$. Zampieri and the author on extension of holomorphic functions from one side $\Omega$ of a real hypersurface $M$ of ${{\mathbb{C}}^{n}}$ in the presence of an analytic disc tangent to $M$, attached to $\bar{\Omega }$ but not to $M$. Our method enables us to weaken the regularity assumptions both for the hypersurface and the disc.

Let $G$ be a compact group.

Let $\sigma$ be a continuous involution of $G$. In this paper, we are concerned by the following functional equation

$$\int\limits_{G}{f\left( xty{{t}^{-1}} \right)dt}\,+\int\limits_{G}{f\left( xt\sigma \left( y \right){{t}^{-1}} \right)dt}=2g\left( x \right)h\left( y \right),\,\,\,x,y\in G,$$

where $f,g,h:G\mapsto \mathbb{C}$, to be determined, are complex continuous functions on $G$ such that $f$ is central. This equation generalizes d’Alembert's and Wilson's functional equations. We show that the solutions are expressed by means of characters of irreducible, continuous and unitary representations of the group $G$.

We prove that a Minkowski plane is Euclidean if and only if Busemann's or Glogovskij's definitions of angular bisectors coincide with a bisector defined by an angular measure in the sense of Brass. In addition, bisectors defined by the area measure coincide with bisectors defined by the circumference (arc length) measure if and only if the unit circle is an equiframed curve.

Let $\mathcal{F}$ be an orthonormal basis for weight 2 cusp forms of level $N$. We show that various weighted averages of special values $L\left( f\otimes \text{ }\!\!\chi\!\!\text{ ,1} \right)$ over $f\in \mathcal{F}$ are equal to $4\text{ }\!\!\pi\!\!\text{ }c+O\left( {{N}^{-1+\in }} \right)$, where $c$ is an explicit nonzero constant. A previous result of Duke gives an error term of $O\left( {{N}^{-1/2}}\log N \right)$.

We study some global aspects of differential complex 2-forms and 3-forms on complex manifolds. We compute the cohomology classes represented by the sets of points on a manifold where such a form degenerates in various senses, together with other similar cohomological obstructions. Based on these results and a formula for projective representations, we calculate the degree of the projectivization of certain orbits of the representation ${{\Lambda }^{k}}{{\mathbb{C}}^{n}}$

We study Hamiltonian actions of compact groups in the presence of compatible involutions. We show that the Lagrangian fixed point set on the symplectically reduced space is isomorphic to the disjoint union of the involutively reduced spaces corresponding to involutions on the group strongly inner to the given one. Our techniques imply that the solution to the eigenvalues of a sum problem for a given real form can be reduced to the quasi-split real form in the same inner class. We also consider invariant quotients with respect to the corresponding real form of the complexified group.

Let $m={{p}^{e}}$ be a power of a prime number $p$. We say that a number field $F$ satisfies the property $\left( {{H}^{'}}_{m} \right)$ when for any $a\in {{F}^{\times }}$, the cyclic extension $F\left( {{\zeta }_{m}},{{a}^{1/m}} \right)/F\left( {{\zeta }_{m}} \right)$ has a normal $p$-integral basis. We prove that $F$ satisfies $\left( {{H}^{'}}_{m} \right)$ if and only if the natural homomorphism $C{{l}^{'}}_{F}\to C{{l}^{'}}_{K}$ is trivial. Here $K=F\left( {{\zeta }_{m}} \right)$, and $C{{l}^{'}}_{F}$ denotes the ideal class group of $F$ with respect to the $p$-integer ring of $F$.

Assume that $\Omega$ is a Hartogs domain in ${{\mathbb{C}}^{1+n}}$, defined as $\Omega =\left\{ \left( z,w \right)\,\in \,{{\mathbb{C}}^{1+n}}\,:\left| z \right|\,<\,\mu \left( w \right),w\,\in H \right\}$, where $H$ is an open set in ${{\mathbb{C}}^{n}}$ and $\mu$ is a continuous function with positive values in $H$ such that –ln $\mu$ is a strongly plurisubharmonic function in $H$. Let ${{\Omega }_{w}}=\Omega \cap \left( \mathbb{C}\times \left\{ w \right\} \right)$. For a given set $E$ contained in $H$ of the type ${{G}_{\delta }}$ we construct a holomorphic function $f\in \mathbb{O}\left( \Omega \right)$ such that

$$E=\left\{ w\in {{\mathbb{C}}^{n}}:\int\limits_{{{\Omega }_{w}}}{{{\left| f\left( \cdot ,w \right) \right|}^{2}}d{{\mathfrak{L}}^{2}}=\infty } \right\}.$$

Let ${{U}_{q}}{{\left( \mathfrak{s}{{\Iota }_{n+1}} \right)}^{+}}$ be the positive part of the quantized enveloping algebra ${{U}_{q}}\left( \mathfrak{s}{{\Iota }_{n+1}} \right)$. Using results of Alev–Dumas and Caldero related to the center of ${{U}_{q}}{{\left( \mathfrak{s}{{\Iota }_{n+1}} \right)}^{+}}$, we show that this algebra is free over its center. This is reminiscent of Kostant's separation of variables for the enveloping algebra $U\left( \mathfrak{g} \right)$ of a complex semisimple Lie algebra g, and also of an analogous result of Joseph–Letzter for the quantum algebra ${{\overset{\scriptscriptstyle\smile}{U}}_{q}}\left( \mathfrak{g} \right)$. Of greater importance to its representation theory is the fact that ${{U}_{q}}{{\left( \mathfrak{s}{{\Iota }_{n+1}} \right)}^{+}}$ is free over a larger polynomial subalgebra $N$ in $n$ variables. Induction from $N$ to ${{U}_{q}}{{\left( \mathfrak{s}{{\Iota }_{n+1}} \right)}^{+}}$ provides infinite-dimensional modules with good properties, including a grading that is inherited by submodules.

We show that the injective Kobayashi–Royden differential metric, as defined by Hahn, is upper semicontinous.

For a given irrational number $\theta$, we define Toeplitz operators with symbols in the irrational rotation algebra ${{\mathcal{A}}_{\theta }}$, and we show that the ${{C}^{*}}$ algebra $\mathfrak{J}\left( {{\mathcal{A}}_{\theta }} \right)$ generated by these Toeplitz operators is an extension of ${{\mathcal{A}}_{\theta }}$ by the algebra of compact operators. We then use these extensions to explicitly exhibit generators of the group $K{{K}^{1}}\left( {{\mathcal{A}}_{\theta }},\mathbb{C} \right)$. We also prove an index theorem for $\mathfrak{J}\left( {{\mathcal{A}}_{\theta }} \right)$ that generalizes the standard index theorem for Toeplitz operators on the circle.

Let $f:X\to Y$ be a $\sigma$-perfect $k$-dimensional surjective map of metrizable spaces such that dim $Y\le m$. It is shown that for every positive integer $p$ with $p\le m+k+1$ there exists a dense ${{G}_{\delta }}-\text{subset}\,\mathcal{H}\left( k,m,p \right)$ of $C\left( X,{{\mathbb{I}}^{k+p}} \right)$ with the source limitation topology such that each fiber of $f\Delta g$, $g\in \mathcal{H}\left( k,m,p \right)$, contains at most $\max \left\{ k+m-p+2,1 \right\}$ points. This result provides a proof the following conjectures of S. Bogatyi, V. Fedorchuk and J. van Mill. Let $f:X\to Y$ be a $k$-dimensional map between compact metric spaces with dim $Y\le m$. Then: (1) there exists a map $h:X\to {{\mathbb{I}}^{m+2k}}$ such that $f\Delta h:\,X\to Y\times {{\mathbb{I}}^{m+2k}}$ is 2-to-one provided $k\ge 1$; (2) there exists a map $h:X\to {{\mathbb{I}}^{m+k+1}}$ such that $f\Delta h:X\to Y\times {{\mathbb{I}}^{m+k+1}}$ is $\left( k+1 \right)$-to-one.

The Abhyankar–Sathaye Embedded Hyperplane Problem asks whether any hypersurface of ${{\mathbb{C}}^{n}}$ isomorphic to ${{\mathbb{C}}^{n-1}}$ is rectifiable, i.e., equivalent to a linear hyperplane up to an automorphism of ${{\mathbb{C}}^{n}}$. Generalizing the approach adopted by Kaliman, Vénéreau, and Zaidenberg, which consists in using almost nothing but the acyclicity of ${{\mathbb{C}}^{n-1}}$, we solve this problem for hypersurfaces given by polynomials of $\mathbb{C}\left[ x,y,{{z}_{1}},...,{{z}_{k}} \right]$ as in the title.