By using the geometric constraints on the control polygon of a Pythagorean hodograph (PH) quartic curve, we propose a sufficient condition for this curve to have monotone curvature and provide the detailed proof. Based on the results, we discuss the construction of spiral PH quartic curves between two given points and formulate the transition curve of a $G^2$ contact between two circles with one circle inside another circle. In particular, we deduce an attainable range of the distance between the centers of the two circles and summarize the algorithm for implementation. Compared with the construction of a PH quintic curve, the complexity of the solution of the equation for obtaining the transition curves is reduced.

In this paper we prove a global attractivity result for the unique positive equilibrium
point of a difference equation, which improves and generalizes some known ones in the existing literature. Especially, our results completely solve an open problem and some conjectures proposed in [1, 2, 3, 4].

In statistical parameter estimation problems, how well the parameters are estimated
largely depends on the sampling design used. In the current paper, a modification of ranked set
sampling (RSS) called moving extremes RSS (MERSS) is considered for the estimation of the
scale and shape parameters for the log-logistic distribution. Several traditional estimators and
ad hoc estimators will be studied under MERSS. The estimators under MERSS are compared
to the corresponding ones under SRS. The simulation results show that the estimators under
MERSS are significantly more efficient than the ones under SRS.

In this paper, the approximate solutions for two different type of two-dimensional
nonlinear integral equations: two-dimensional nonlinear Volterra-Fredholm integral equations
and the nonlinear mixed Volterra-Fredholm integral equations are obtained using the Laguerre
wavelet method. To do this, these two-dimensional nonlinear integral equations are transformed
into a system of nonlinear algebraic equations in matrix form. By solving these systems, unknown coefficients are obtained. Also, some theorems are proved for convergence analysis.
Some numerical examples are presented and results are compared with the analytical solution
to demonstrate the validity and applicability of the proposed method.

Tracy-Widom distribution was first discovered in the study of largest eigenvalues
of high dimensional Gaussian unitary ensembles (GUE), and since then it has appeared in a
number of apparently distinct research fields. It is believed that Tracy-Widom distribution
have a universal feature like classic normal distribution. Airy2 process is defined through finite
dimensional distributions with Tracy-Widom distribution as its marginal distributions. In this
introductory survey, we will briefly review some basic notions, intuitive background and fundamental properties concerning Tracy-Widom distribution and Airy2 process. For sake of reading,
the paper starts with some simple and well-known facts about normal distributions, Gaussian
processes and their sample path properties.

K-mer can be used for the description of biological sequences and k-mer distribution
is a tool for solving sequences analysis problems in bioinformatics. We can use k-mer vector as
a representation method of the k-mer distribution of the biological sequence. Problems, such as
similarity calculations or sequence assembly, can be described in the k-mer vector space. It helps
us to identify new features of an old sequence-based problem in bioinformatics and develop new
algorithms using the concepts and methods from linear space theory. In this study, we defined
the k-mer vector space for the generalized biological sequences. The meaning of corresponding
vector operations is explained in the biological context. We presented the vector/matrix form of
several widely seen sequence-based problems, including read quantification, sequence assembly,
and pattern detection problem. Its advantages and disadvantages are discussed. Also, we
implement a tool for the sequence assembly problem based on the concepts of k-mer vector
methods. It shows the practicability and convenience of this algorithm design strategy.