Abstract:
We hope to be able (in the future) to carefully analyze this structure and to tie the Jacobian Conjecture in dimension two to certain Zeta functions, thereby invoking a powerful arithmetic machinery to handle the two dimensional Jacobian Conjecture. Let us denote by ${\rm et}(\mathbb{C}^2)$ the semigroup of two dimensional Keller mappings. We would like to prove something like the following: a) That there exists an infinite index set $I$, and a family of mappings indexed by $I$, $\{F_i\,|\,i\in I\} \subset {\rm et}(\mathbb{C}^2)$ such that $$ {\rm et}(\mathbb{C}^2)={\rm Aut}(\mathbb{C}^2)\cup\bigcup_{i\in I} R_{F_i}({\rm et}(\mathbb{C}^2)), $$ where if $i\ne j$ then $R_{F_i}({\rm et}(\mathbb{C}^2))\cap R_{F_j}({\rm et}(\mathbb{C}^2))=\emptyset$. b) That the parallel representation to the representation described in (a) above holds true, this time with respect to the left composition operators $L_{G_j}$. These two claims will be the basis for a fractal structure on ${\rm et}(\mathbb{C}^2)$ because the pieces $R_{F_i}({\rm et}(\mathbb{C}^2))$ are similar to each other in the sense that they are homeomorphic, and we further have the scaling property of self-similarity, namely $R_F({\rm et}(\mathbb{C}^2))$ is homeomorphic to its proper subspace $R_{G\circ F}({\rm et}(\mathbb{C}^2))$ that is homeomorphic to its proper subspace $R_{H\circ G\circ F}({\rm et}(\mathbb{C}^2))$ etc . This is the right place to remark that the purpose of the current paper is to start and develop the parallel theory for entire functions in one complex variable. Results in this setting will hint that there are hopes to accomplish the above objective.

Abstract:
This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient in the linear parabolic equation with mixed boundary conditions. The aim of this paper is to investigate the distinguishability of the input-output mappings via semigroup theory. In this paper, we show that if the null space of the semigroup consists of only zero function, then the input-output mappings have the distinguishability property. It is shown that the types of the boundary conditions and the region on which the problem is defined, play an important role on the distinguishability property of these mappings. Moreover, under the light of measured output data (boundary observations) the values k(0) and k(1) of the unknown diffusion coefficient k(x) at x=0 and x=1, respectively, can be determined explicitly. In addition to these, the values k'(0) and k'(1) of the unknown coefficient k(x) at x=0 and x=1 , respectively, are also determined via the input data. Furthermore, it is shown that measured output data f(t) and h(t) can be determined analytically, by an integral representation. Hence the input-output mappings are given explicitly in terms of the semigroup.

Abstract:
In this paper we classify the maximal subsemigroups of the \emph{full transformation semigroup} $\Omega^\Omega$, which consists of all mappings on the infinite set $\Omega$, containing certain subgroups of the symmetric group $\sym(\Omega)$ on $\Omega$. In 1965 Gavrilov showed that there are five maximal subsemigroups of $\Omega^\Omega$ containing $\sym(\Omega)$ when $\Omega$ is countable and in 2005 Pinsker extended Gavrilov's result to sets of arbitrary cardinality. We classify the maximal subsemigroups of $\Omega^\Omega$ on a set $\Omega$ of arbitrary infinite cardinality containing one of the following subgroups of $\sym(\Omega)$: the pointwise stabiliser of a non-empty finite subset of $\Omega$, the stabiliser of an ultrafilter on $\Omega$, or the stabiliser of a partition of $\Omega$ into finitely many subsets of equal cardinality. If $G$ is any of these subgroups, then we deduce a characterisation of the mappings $f,g\in \Omega^\Omega$ such that the semigroup generated by $G\cup \{f,g\}$ equals $\Omega^\Omega$.

Abstract:
Let $S_{1}=\left\{F_t\right\}_{t\geq 0}$ and $S_{2}=\left\{G_t\right\}_{t\geq 0}$ be two continuous semigroups of holomorphic self-mappings of the unit disk $\Delta=\{z:|z|<1\}$ generated by $f$ and $g$, respectively. We present conditions on the behavior of $f$ (or $g$) in a neighborhood of a fixed point of $S_{1}$ (or $S_{2}$), under which the commutativity of two elements, say, $F_1$ and $G_1$ of the semigroups implies that the semigroups commute, i.e., $F_{t}\circ G_{s}=G_{s}\circ F_{t}$ for all $s,t\geq 0$. As an auxiliary result, we show that the existence of the (angular or unrestricted) $n$-th derivative of the generator $f$ of a semigroup $\left\{F_t\right\}_{t\geq 0}$ at a boundary null point of $f$ implies that the corresponding derivatives of $F_{t}$, $t\geq 0$, also exist, and we obtain formulae connecting them for $n=2,3$.

Abstract:
In this paper, the existence of a fixed point for TF -contractive mappings on complete metric spaces and cone metric spaces is proved, where T : X → X is a one to one and closed graph function and F : P → P is non-decreasing and right continuous, with F -1(0) = -0} and F(tn ) → 0 implies tn → 0. Our results, extend previous results given by Meir and Keeler (J. Math. Anal. Appl. 28, 326-329, 1969), Branciari (Int. J. Math. sci. 29, 531-536, 2002), Suzuki (J. Math. Math. Sci. 2007), Rezapour et al. (J. Math. Anal. Appl. 345, 719-724, 2010), Moradi et al. (Iran. J. Math. Sci. Inf. 5, 25-32, 2010) and Khojasteh et al. (Fixed Point Theory Appl. 2010). MSC(2000): 47H10; 54H25; 28B05.

Abstract:
Let C be a nonempty closed convex subset of a uniformly convex Banach space E with a Fr chet differentiable norm, G a right reversible semitopological semigroup, and ° ’ ={S(t):t ￠ G} a continuous representation of G as mappings of asymptotically nonexpansive type of C into itself. The weak convergence of an almost-orbit {u(t):t ￠ G} of ° ’ ={S(t):t ￠ G} on C is established. Furthermore, it is shown that if P is the metric projection of E onto set F(S) of all common fixed points of ° ’ ={S(t):t ￠ G}, then the strong limit of the net {Pu(t):t ￠ G} exists.

Abstract:
Let $(W,\Pi)$ be a Riemann domain over a complex manifold $M$ and $w_0$ be a point in $W$. Let $\mathbb D$ be the unit disk in $\mathbb C$ and $\mathbb T=\bd\mathbb D$. Consider the space ${\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ of continuous mappings $f$ of $\mathbb T$ into $W$ such that $f(1)=w_0$ and $\Pi\circ f$ extends to a holomorphic on $\mathbb D$ mapping $\hat f$. Mappings $f_0,f_1\in{\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ are called {\it $h$-homotopic} if there is a continuous mapping $f_t$ of $[0,1]$ into $\rS_{1,w_0}({\bar{\mathbb D}},W,M)$. Clearly, the $h$-homotopy is an equivalence relation and the equivalence class of $f\in{\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ will be denoted by $[f]$ and the set of all equivalence classes by $\eta_1(W,M,w_0)$. There is a natural mapping $\iota_1:\,\eta_1(W,M,w_0)\to\pi_1(W,w_0)$ generated by assigning to $f\in{\mathcal S}_{1,w_0}({\bar{\mathbb D}},W,M)$ its restriction to $\mathbb T$. We introduce on $\eta_1(W,M,w_0)$ a binary operation $\star$ which induces on $\eta_1(W,M,w_0)$ a structure of a semigroup with unity. Moreover, $\iota_1([f_1]\star[f_2])=\iota_1([f_1])\cdot\iota_1([f_2])$, where $\cdot$ is the standard operation on $\pi_1(W,w_0)$. Then we establish standard properties of $\eta_1(W,M,w_0)$ and provide some examples. In particular, we completely describe $\eta_1(W,M,w_0)$ when $W$ is a finitely connected domain in $M=\mathbb C$ and $\Pi$ is an identity. In particular, we show for a general domain $W\subset\mahbb C$ that $[f_1]=[f_2]$ if and only if $\iota_1([f_1])=\iota_1([f_2])$.

Abstract:
In this paper, using Kronecker's theorem, we discuss the set of common fixed points of an n-parameter continuous semigroup of mappings. We also discuss convergence theorems to a common fixed point of an n-parameter nonexpansive semigroup.

Abstract:
Let G be a semitopological semigroup, C a nonempty subset of a real Hilbert space H, and ￠ ‘={Tt:t ￠ G} a representation of G as asymptotically nonexpansive type mappings of C into itself. Let L(x)={z ￠ H:infs ￠ Gsupt ￠ G ￠ € –Tts ￠ € ‰x ￠ ’z ￠ € –=inft ￠ G ￠ € –Tt ￠ € ‰x ￠ ’z ￠ € –} for each x ￠ C and L( ￠ ‘)= ￠ x ￠ C ￠ € ‰L(x). In this paper, we prove that ￠ s ￠ Gconv ˉ{Tts ￠ € ‰x:t ￠ G} ￠ L( ￠ ‘) is nonempty for each x ￠ C if and only if there exists a unique nonexpansive retraction P of C into L( ￠ ‘) such that PTs=P for all s ￠ G and P(x) ￠ conv ˉ{Ts ￠ € ‰x:s ￠ G} for every x ￠ C. Moreover, we prove the ergodic convergence theorem for a semitopological semigroup of non-Lipschitzian mappings without convexity.

Abstract:
Let $\mathcal{S}$ be a finite commutative semigroup. The Davenport constant of $\mathcal{S}$, denoted $D(\mathcal{S})$, is defined to be the least positive integer $d$ such that every sequence $T$ of elements in $\mathcal{S}$ of length at least $d$ contains a subsequence $T'$ with the sum of all terms from $T'$ equaling the sum of all terms from $T$. Let $\F_p[x]$ be a polynomial ring in one variable over the prime field $\F_p$, and let $f(x)\in \F_p[x]$. In this paper, we made a study of the Davenport constant of the multiplicative semigroup of the quotient ring $\frac{\F_p[x]}{\langle f(x)\rangle}$. Among other results, we mainly prove that, for any prime $p>2$ and any polynomial $f(x)\in \F_p[x]$ which can be factorized into several pairwise non-associted irreducible polynomials in $\F_p[x]$, then $$D(\mathcal{S}_{f(x)}^p)=D(U(\mathcal{S}_{f(x)}^p)),$$ where $\mathcal{S}_{f(x)}^p$ denotes the multiplicative semigroup of the quotient ring $\frac{\F_p[x]}{\langle f(x)\rangle}$ and $U(\mathcal{S}_{f(x)}^p)$ denotes the group of units of the semigroup $\mathcal{S}_{f(x)}^p$.