Identities on the k-ary Lyndon words related to a family of zeta functions

The main aim of this paper is to investigate and introduce relations between the numbers of k-ary Lyndon words and unified zeta-type functions which was defined by Ozden et al [15, p. 2785]. Finally, we give some identities on generating functions for the numbers of k-ary Lyndon words and some special numbers and polynomials such as the Apostol-Bernoulli numbers and polynomials, Frobenius-Euler numbers, Euler numbers and Bernoulli numbers.Comment: 9 page

A Survey on Deep Learning-based Architectures for Semantic Segmentation on 2D images

Semantic segmentation is the pixel-wise labelling of an image. Since the problem is defined at the pixel level, determining image class labels only is not acceptable, but localising them at the original image pixel resolution is necessary. Boosted by the extraordinary ability of convolutional neural networks (CNN) in creating semantic, high level and hierarchical image features; excessive numbers of deep learning-based 2D semantic segmentation approaches have been proposed within the last decade. In this survey, we mainly focus on the recent scientific developments in semantic segmentation, specifically on deep learning-based methods using 2D images. We started with an analysis of the public image sets and leaderboards for 2D semantic segmantation, with an overview of the techniques employed in performance evaluation. In examining the evolution of the field, we chronologically categorised the approaches into three main periods, namely pre-and early deep learning era, the fully convolutional era, and the post-FCN era. We technically analysed the solutions put forward in terms of solving the fundamental problems of the field, such as fine-grained localisation and scale invariance. Before drawing our conclusions, we present a table of methods from all mentioned eras, with a brief summary of each approach that explains their contribution to the field. We conclude the survey by discussing the current challenges of the field and to what extent they have been solved.Comment: Updated with new studie

Combinatorial identities associated with new families of the numbers and polynomials and their approximation values

Recently, the numbers $Y_{n}(\lambda )$ and the polynomials $Y_{n}(x,\lambda)$ have been introduced by the second author [22]. The purpose of this paper is to construct higher-order of these numbers and polynomials with their generating functions. By using these generating functions with their functional equations and derivative equations, we derive various identities and relations including two recurrence relations, Vandermonde type convolution formula, combinatorial sums, the Bernstein basis functions, and also some well known families of special numbers and their interpolation functions such as the Apostol--Bernoulli numbers, the Apostol--Euler numbers, the Stirling numbers of the first kind, and the zeta type function. Finally, by using Stirling's approximation for factorials, we investigate some approximation values of the special case of the numbers $Y_{n}\left( \lambda \right)$.Comment: 17 page

A class of 3-dimensional contact metric manifolds

We classify the contact metric 3-manifolds that satisfy ||grad{\lambda}||=1 and \nabla_{{\xi}}{\tau}=2a{\tau}{\phi}.Comment: 12 pages, submitte