A bounded linear operator $T\in\mathbf{L}(\mathbb{X})$ acting on a Banach space satisfies property $(aw)$, a variant of Weyl's theorem, if the complement in the spectrum $\sigma(T)$ of the Weyl spectrum $\sigma_w(T)$ is the set of all isolated points of the approximate-point spectrum which are eigenvalues of finite multiplicity. In this article we consider the preservation of property $(aw)$ under a finite rank perturbation commuting with $T$, whenever $T$ is polaroid, or $T$ has analytical core $K(T-\lambda_0 I)=\{0\}$ for some $\lambda_0\in \mathbb{C}$. The preservation of property $(aw)$ is also studied under commuting nilpotent or under injective quasi-nilpotent perturbations or under Riesz perturbations. The theory is exemplified in the case of some special classes of operators.

In 2008, Ferrara Dentice and Marino provided a characterization theorem for Veronesean caps in $\mathsf{PG}(N,\mathbb{K})$, with $\mathbb{K}$ a skewfield. This result extends the theorem for the finite case proved by J.A. Thas and Van Maldeghem in 2004. However, although the statement of this theorem is correct, the proof given by Ferrara Dentice and Marino is incomplete, as they borrow some lemmas from the paper of J.A. Thas and Van Maldeghem, which are proved using counting arguments and hence require a different approach in the infinite case. In this paper we use the Veblen-Young theorem to fill these gaps. Moreover, we then use this classification of Veronesean caps to provide a further general geometric characterization.

In this paper, we investigate the growth of solutions of the linear differential equation \begin{multline*} f^{(k)}+\left( A_{k-1}(z)e^{P_{k-1}(z)}+B_{k-1}\left( z\right) \right) f^{(k-1)}+\cdots +\\\left( A_{1}(z)e^{P_{1}(z)}+B_{1}\left( z\right) \right) f^{\prime } +\left( A_{0}(z)e^{P_{0}(z)}+B_{0}\left( z\right) \right) f=0, \end{multline*} where $k\geq 2$\ is an integer, $P_{j}(z)$ $(j=0,1,\cdots ,k-1)$\ are nonconstant polynomials\ and $A_{j}(z)$ $\left( \not\equiv 0\right) ,$ $ B_{j}\left( z\right) $\ $\left( \not\equiv 0\right) $ $(j=0,1,\cdots ,k-1)$\ are meromorphic functions. Under some conditions, we determine the hyper-order of these solutions.

We study new general integral operators of $p$-valent functions by giving sufficient conditions of $p$-valently starlikeness, $p$-valently close-to-convexness, uniformly $p$-valent close-to-convexness and strongly starlikeness of order $\tau \,\,(0<\tau\leq 1)$ in $U$ (open unit disk). We end our investigation with an example from literature and some other references.

In this paper we prove an existence result for some class of variational boundary value problems for quasilinear elliptic equations in the Musielak-Orlicz spaces $W^mL_\varphi(\Omega)$, under the assumption that the conjugate function of $\varphi$ satisfies the $\Delta_2$ condition. An imbedding theorem has also been provided without assuming this condition.

Let $\eta$ be a Real bundle, in the sense of Atiyah, over a space $X$. This is a complex vector bundle together with an involution which is compatible with complex conjugation. We use the fact that $BU$ has a canonical structure of a conjugation space, as defined by Hausmann, Holm, and Puppe, to construct equivariant Chern classes in certain equivariant cohomology groups of $X$ with twisted integer coefficients. We show that these classes determine the (non-equivariant) Chern classes of $\eta$, forgetting the involution on $X$, and the Stiefel-Whitney classes of the real bundle of fixed points.

We prove some infinitesimal analogs of classical results of Menger, Schoenberg and Blumenthal giving the existence conditions for isometric embeddings of metric spaces in the finite-dimensional Euclidean spaces.

This paper is devoted to studying the growth of solutions of equations of type $f(z+n)+\sum_{j=0}^{n-1}\{P_{j}(e^{z})+Q_{j}(e^{-z})\}f(z+j)=0$ and $f(z+n)+ \sum_{j=0}^{n-1}\{P_{j}(e^{A(z)})+Q_{j}(e^{-A(z)})\}f(z+j)=0$, where $P_{j}(z)$ and $Q_{j}(z)$ are polynomials in $z$ and $A(z)$ is a transcendental entire function. We prove three theorems of such type, which improve some results in [6,7].

The paper studies bounded or unbounded operators which can act as analysis operators or synthesis operators of various signal processing including generalized frames, semi-frames, discrete frames, Fourier transforms, etc. The paper is concluded by a short discussion of the controllability of the behavior of the processed signals.

We extend the family of Meixner-Pollaczek polynomials $\{P_n^{(\lambda)}(\cdot;\phi)\}_{n=0}^{\infty}$, classically defined for $\lambda>0$ and $0<\phi<\pi$, to arbitrary complex values of the parameter $\lambda$, in such a way that both polynomial systems (the classical and the new {\it generalized} ones) share the same three term recurrence relation. The values $\lambda_N=(1-N)/2$, with $N$ a positive integer, are the only ones for which no orthogonality condition can be deduced from Favard's theorem. In this paper we introduce a non-standard discrete-continuous inner product with respect to which the generalized Meixner-Pollaczek polynomials $\{P_n^{(\lambda_N)}(\cdot;\phi)\}_{n=0}^{\infty}$ become orthogonal.

In this paper we study Dupin hypersurfaces in $\mathbb R^4$ parametrized by lines of curvature, with three distinct principal curvatures and $m_{jik}= 0$. We characterize locally a generic family of such hypersurfaces in terms of the principal curvatures and three vector valued functions of one variable, which are invariant under inversions and homotheties.

Some new existence theorems are obtained for periodic solutions of nonautonomous second order Hamiltonian systems with $(q,p)$-Laplacian by using the least action principle and the minimax methods.

In the present paper, by using a new defined -$\left\vert C,\alpha ,\sigma;\alpha_{n}\right\vert _{k}$ summability method and some classes of pairs of sequences, we generalize a result of Bor [5] dealing with $\varphi-\left\vert C,\alpha,\sigma;\beta\right\vert _{k}\ $summability factors.

Let $N$ be the counting function of a Beurling generalized number system and let $\pi$ be the counting function of its primes. We show that the $L^{1}$-condition $$ \int_{1}^{\infty}\left|\frac{N(x)-ax}{x}\right|\frac{\mathrm{d}x}{x}<\infty $$ and the asymptotic behavior $$N(x)=ax+O\left(\frac{x}{\log x}\right)\: ,$$ for some $a>0$, suffice for a Chebyshev upper estimate $$ \frac{\pi(x)\log x}{x}\leq B<\infty\: .$$