Let $T_\phi $ be a Toeplitz operator on the one variable Hardy space $H^2$. We show that if $T_\phi $ has a nontrivial invariant subspace in the set of invariant subspaces of $T_z$ then $\phi $ belongs to $H^\infty $. In fact, we also study such a problem for the several variables Hardy space $H^2$.

In this paper, we investigate the modified Hyers-Ulam stability of a mixed $n$-dimensional quadratic functional equation in Banach spaces and also Banach modules over a Banach algebra and a $C^*-$algebra. Finally, we study the stability using the alternative fixed point of the functional equation in Banach spaces: \begin{equation*} _{n-2}C_{m-2} f(\sum^n_{j=1} x_j) + _{n-2}C_{m-1}\sum^n_{i=1} f(x_i) =\sum_{1\leq i_1 < \cdots < i_m \leq n} f(x_{i_1}+\cdots+x_{i_m}), \end{equation*} for all $x_j (j=1,\cdots,n)$ where $n\geq 3$ is an integer number and $2\leq m \leq n-1.$

In this work, we deal with the fractal dimension D of a chaotic attractor which is generated by a bidimensional endomorphism (the Hogg-Huberman model).Using a modified box-counting method, we study the numerical behavior of D with respect to the number n of points of the considered set.One establishes an important relation D=D(n) which is valid for other dynamical systems.

In the present paper, we obtain some inclusion relations and convolution properties for certain subclasses of analytic functions in the unit disk which are defined by the generalized Sălăgean operator. Relevant connections of the results presented here with those obtained in earlier works are pointed out.

This paper deals with Moufang-Klingenberg planes $\boldsymbol{M}(\mathcal{A}) $ defined over a local\ alternative ring $\mathcal{A}$\ of dual numbers. The definition of cross-ratio is extended to $\boldsymbol{M}(\mathcal{A})$. Also, some properties of cross-ratios and 6-figures that arewell-known for Desarguesian planes are investigated in $\boldsymbol{M}(\mathcal{A})$; so we obtain relations between algebraic properties of $\mathcal{A}$ and geometric properties of $\boldsymbol{M}(\mathcal{A})$. In particular, we show that pairwise non-neighbour four points of the line $g$ are in harmonic position if and only if they are harmonic, and that $\mu $ is Menelaus or Ceva 6-figure if and only if $r\left( \mu \right) =-1$ or $r\left( \mu \right) =1, $ respectively.

We consider spacelike graphs $\Gamma_f$ of simple products $(M\times N, g\times -h)$ where $(M,g)$ and $(N,h)$ are Riemannian manifolds and $f:M\rightarrow N$ is a smooth map. Under the condition of the Cheeger constant of $M$ to be zero and some condition on the second fundamental form at infinity, we conclude that if $\Gamma_f\subset M\times N$ has parallel mean curvature $H$ then $H=0$. This holds trivially if $M$ is closed. If $M$ is the $m$-hyperbolic space then for any constant $c$, we describe an explicit foliation of $ {\mathbb H}^m\times \mathbb R$ by hypersurfaces with constant mean curvature $c$.

Following the works of van Asch and van der Blij, we show that the horn angles introduced by Euclide can be measured by superreal numbers as Tall defined them. We deduce from this the possibility to estimate, on the one hand, the ratio of the measures of a horn angle and of the mixed angle formed by a spiral of Archimede and, on the other hand, the ratio of the measures of two horn angles.

En nous appuyant sur des travaux de van Asch et van der Blij, nous montrons que les angles corniculaires introduits par Euclide peuvent se voir attribuer une mesure numérique donnée par un nombre superréel infiniment petit au sens de Tall. Nous en déduisons la possibilité d'estimer d'une part le rapport entre les mesures d'un angle corniculaire et d'un angle mixtiligne formé par une spirale d'Archimède, et d'autre part le rapport des mesures de deux angles corniculaires.

Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis, which has proven to be an appropriate framework for developing a higher dimensional continuous wavelet transform theory. In this setting a very specific construction of the wavelets has been established, encompassing all dimensions at once as opposed to the usual tensorial approaches, and being based on generalizations to higher dimension of classical orthogonal polynomials on the real line, such as the radial Clifford--Hermite polynomials, leading to Clifford--Hermite wavelets. More recently, Hermitian Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the orthogonal case. In this new setting a Clifford--Hermite continuous wavelet transform has already been introduced in earlier work, its norm preserving character however being expressed in terms of suitably adapted scalar valued inner products on the respective $L_2$--spaces of signals and of transforms involved. In this contribution we present an alternative Hermitian Clifford--Hermite wavelet theory with Clifford algebra valued inner products, based on an orthogonal decomposition of the space of square integral functions, which is obtained by introducing a new Hilbert transform in the Hermitian setting.

In this work we study bifurcation of forms of equilibrium of a thin circular elastic plate lying on an elastic base under the action of a compressive force. The forms of equilibrium may be found as solutions of the von Kármán equations with two real positive parameters defined on the unit disk in $\mathbb R^2$ centered at the origin. These equations are equivalent to an operator equation $F(x,p)=0$ in the real Hölder spaces with a nonlinear $S^{1}$-equivariant Fredholm map of index $0$. For the existence of bifurcation at a point $(0,p)$ it is necessary that $\dim\operatorname{Ker}F_{x}^{\prime}(0,p)>0$. The space $\operatorname{Ker}F_{x}^{\prime}(0,p)$ can be at most four-dimensional. We apply the Crandall-Rabinowitz theorem to prove that if $\dim\operatorname{Ker}F_{x}^{\prime}(0,p)=3$ then there is bifurcation of radial solutions at $(0,p)$. What is more, using the Lyapunov-Schmidt finite-dimensional reduction we investigate the number of branches of radial bifurcation at $(0,p)$.

We address the computation of the line that minimizes the sum of the least squared distances with respect to a given set of $n$ points in 3-space. This problem has a well known satisfying solution by means of PCA. We offer an alternative interpretation for this optimal line as the center of the screw motion that minimizes the sum of squared velocities in the given points. The numerical translation of this viewpoint is a generalized eigenproblem, where the total residue of the optimal line appears as the smallest generalized eigenvalue.

In this paper the stability of the quadratic equation is considered on arbitrary groups. Since the quadratic equation is stable on Abelian groups, this paper examines the stability of the quadratic equation on noncommutative groups. It is shown that the quadratic equation is stable on $n$-Abelian groups when $n$ is a positive integer. The stability of the quadratic equation is also established on the noncommutative group $T(2, K)$, where $K$ is an arbitrary commutative field. It is proved that every group can be embedded into a group in which the quadratic equation is stable.

This is a brief introduction to Fredholm theory for Hilbert space operators organized into ten sections. The classical partition of the spectrum into point, residual, and continuous spectra is reviewed in Section 1. Fredholm operators are introduced in Section 2, and Fredholm index in Section 3. The essential spectrum is considered in Section 4, the spectral picture is presented in Section 5, and Riesz points are discussed in Section 6. Weyl spectrum is the subject of Section 7 and, after bringing some basic results on ascent and descent in Section 8, Browder spectrum is investigated in Section 9. Finally, Weyl and Browder theorems close this expository paper in Section 10.