Subclasses of $p$-valent starlike and convex functions in the unit disk in the complex plane are investigated. Every $p$-valent convex function in a subclass is shown to belong to its corresponding subclass of starlike functions. A necessary and sufficient condition for functions to belong to these classes is obtained. Subordination properties, and sharp distortion, growth, covering and rotation estimates are obtained for these classes. Convolution results with prestarlike functions are also derived.

Our goal in this paper is to introduce some new sequences of some meromorphic function spaces, which will be called $b_q$ and $q_{K}$-sequences. Our study is motivated by the theories of normal, $Q^{\#}_K$ and meromorphic Besov functions. For a non-normal function $f$ the sequences of points $\{a_n\}$ and $\{b_n\}$ for which $$\lim_{n\rightarrow \infty}(1-|a_n|^2)f^{\#}(a_n)=+\infty\,\,\,\mbox{and} $$ $$ \lim_{n\rightarrow\infty}\iint_\Delta \bigl(f^{\#}(z)\bigr)^q (1-|z|^2)^{q-2}(1-|\varphi_{a_n}(z)|^2)^s dA(z)=+\infty\;$$ or $$ \lim_{n\rightarrow\infty}\iint_\Delta \bigl(f^{\#}(z)\bigr)^2 K(z,a_n)dA(z)=+\infty\;$$ are considered and compared with each other. Finally, non-normal meromorphic functions are described in terms of the distribution of the values of these meromorphic functions.

In this studies, we discuss the following Rayleigh equation with two delays: $$ x''(t)+f(t,x'(t))+g_{1}(t,x(t-\tau_{1}))+g_{2}(t,x(t-\tau_{2}))=e(t). $$ By using Mawhin's continuation theorem and some new techniques, some criteria to guarantee the existence and uniqueness of periodic solutions of this equation is given. Our results are new and complement the known results in the literature.

The aim of this paper is broadly two fold. Firstly, we define a new class of implicit function unifying a multitude of strict contractive conditions and utilize the same to prove a general common fixed point theorem for two pairs of weak compatible mappings satisfying common property $(E.A)$ when underlying space is not necessarily compact. Secondly, we show that common property $(E.A)$ relaxes the required containment of ranges of the involved mappings in common fixed point considerations up to two pairs of mappings.

The aim of this paper is to study some properties of secantoptics, defined in [8]. We show that any evolutoid of a given oval $C$ is a hedgehog and that any secantoptic of an oval $C$, is an isoptic of a pair of required evolutoids. We prove some Crofton-type formulas for secantoptics and give a necessary and sufficient condition for a secantoptic to be convex.

Invoking the Clifford-Hermite Wavelets from Clifford analysis, we use the covariances of affine groups to construct a kind of functional calculi for several non-commuting bounded operators. Functional calculi are the intertwining transforms between the representations of affine groups in the space $L^2(\mathbb R^m)$ and in the space of bounded operators. It turns out that the Weyl calculus is the value of this new functional calculus at the identity of affine groups. Our approach is inspired by the mathematical ideas contained in the paper ``V. V. Kisil. Wavelets in Banach spaces. Acta Appl. Math. 1999, {\bf 59}(1): 79-109".

In the note we give an uniform bound for the multiple point Seshadri constants on $\mathbb{P}^1\times\mathbb{P}^1$, improving the bound from [3].

For a given class $A$ and a set $D$ the sets $\bigcap_{f\in A}f(D)$ and $\bigcup_{f\in A}f(D)$ are called the Koebe set and the covering set for $A$ over $D$, respectively. These sets are found for the class $H$ of close-to-star functions $f$ of the form $f(z)=\frac{z}{1-z^2}p(z)$, where $Re p(z)>0, p(0)=1$. Analogous results concerning some other subclasses of close-to-star functions are established too.

Let $B$ be a Banach space and $T$ a bounded linear operator on $B.$ A celebrated theorem of Ansari says that whenever $T$ is hypercyclic so is any power $T^n$. We provide a very natural proof of this theorem by building on an approach by Bourdon. We also explore an extension to a non linear setting of a theorem of León-Saavedra and Müller which says that for $\lambda \in \mathbb C$ and $|\lambda|=1$ the operator $\lambda T$ is hypercyclic whenever $T$ is.

Let $D$ be the open unit disk in $\mathbb C$. In this article, we construct dense subspaces of $H^p(D)$, $1\leq p\leq \infty$, with certain barrelledness properties, such that their nonzero elements cannot be extended holomorphically outside $D$.

Harrison's criterion characterizes the isomorphy of the Witt rings of two fields in terms of properties of these fields. In this article, we discuss about the existence of such characterizations for the isomorphism of Witt groups of hermitian forms over certain algebras with involution. In the cases where we consider the Witt group of a quadratic extension with its non-trivial automorphism or the Witt group of a quaternion division algebra with its canonical involution, such criteria are proved. In the framework of global fields, these criteria are reformulated in terms of properties involving certain real places of the considered fields.

For a locally compact semigroup $\frak S$, we study a fixed point property in terms of left Banach $\frak S$-modules; we also use this property to give a characterization for inner amenability of $\frak S$.

We characterize those contact spaces which are isomorphic to some geometry of plane sections of a quadratic set of rank greater than 4 by a new incidence proposition combined with some richness conditions.

Let $(R,{\frak{m}},k)$ be a commutative Noetherian local ring. It is well-known that $R$ is regular if and only if the flat dimension of $k$ is finite. In this paper, we show that $R$ is Gorenstein if and only if the Gorenstein flat dimension of $k$ is finite. Also, we will show that if $R$ is a Cohen-Macaulay ring and $M$ is a Tor-finite $R$-module of finite Gorenstein flat dimension, then the depth of the ring is equal to the sum of the Gorenstein flat dimension and Ext-depth of $M$. As a consequence, we get that this formula holds for every syzygy of a finitely generated $R$-module over a Gorenstein local ring.

We study the class field theory of curves defined over two dimensional local fields. The approach used here is a combination of the work of Kato-Saito and Yoshida where the base field is one dimensional.