This is a review of the theory of compact surfaces with constant mean curvature with special attention on surfaces with non empty boundary. We analyze the relation between the geometry of the boundary curve with the one of the surface that spans. This will allow to give a mathematical support to describe the geometry of liquid drops and soap bubbles.

The p-Willmore energy $\mathcal{W}^p$, which extends the venerable Willmore energy by accommodating different powers of the mean curvature in its integrand, is a relevant geometric functional that bears both similarities and differences to its namesake. To elucidate this, some recent results in this area are surveyed. In particular, the first and second variations of $\mathcal{W}^p$ are given, and a flux formula is presented which reveals a connection between its critical points and the minimal surfaces. Finally, a model for the p-Willmore flow of graphs is presented, and this connection is visualized through computer implementation.

This paper presents the method of building surveillance, as the each build is separated to sides. Each side is segmented in viewing areas. Each viewing area is a field of view of a camera, mounted on a drone. The separate segments are connected sequentially and this gives the way for surveillance the this side, Connected the all paths for all sides will give the building surveillance. The proposed method is used to generate the coverage path of the whole buildings.

Conditions to guarantee that a point symmetry ${\bf X}$ of an $n^{th}$-order differential equation $q^{(n)}-\omega=0$ is simultaneously a point symmetry of its derived equation $q^{(n+1)}-\dot{\omega}=0$ are analyzed, and the possible types of vector fields established. It is further shown that only the simple Lie algebra $\mathfrak{sl}(2,\mathbb{R})$ for a very specific type of realization in the plane can be inherited by a derived equation.

The present paper aims to introduce a new algebra based on the knots and their ability to change dramatically the physical state, when we act on them with the braiding operator. The frame of all this ideas is the topological quantum computing. Nowadays, the quest for a reliable quantum computer technology refers, among others quantum phenomena, to the Quantum Hall Effect. Setting suitable quantum experiments (for instance a super lattice - two dimensional electron system that appears in the hetero structures of two semiconductors) involving exceptionally strong magnetic fields and very low temperatures, the QHE puts into evidence new quasi particles, evolving in 2+1 dimensions space. Such a topological quantum computer will successfully fight against the undesirable interaction between the quantum system and the environment (de-coherence). The knotted quantum world-lines of quasi-particles (anyons) provide the quantum computing physical support. The necessary algebra (which will deal with) is the algebra of the creation and annihilation of the knot's braids when the corresponding braiding operators act.

Motivated by the classical Euler elastic curves and the Bernoulli's lemniscate, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. The aim of this talk is the analysis of the authors' contribution to the above problem when the curvature is prescribed in terms of the distance to a line.

We present a new method for the parameterization of the characteristic melody of birdsongs that makes use of Gaussian Mixture Models (GMMs) in order to interpolate the individual segments of the characteristic curve computed for each acoustic event. Initially, the proposed method implements morphological processing and automatic segmentation of the birdsong to separate acoustic events based on the audio spectrogram. This is followed by an estimation of the dominant frequency component, smoothing of the dominant energy contour and reduction of the image to black and white. The characteristic curve obtained for each acoustic event is then interpolated with a six-component GMM. The outcome of this process is a parametric model which describes segments of a varying length and complexity with a predefined number of parameters. These parameters can serve well in birdsong modeling or species recognition applications. We illustrate the applicability of the proposed approach on publicly available field recordings of species Myrmotherula multostriata, which had been recorded in their natural habitats.

In the present work, we investigate the behavior of the Locally Recurrent Probabilistic Neural Network (LRPNN) with different activation functions in the recurrent layer neurons. Specifically, we evaluate the performance of the modified activation function proposed here, which belongs to the family of Rectified Linear Units (ReLU), and compare it with other ReLU-based functions, the traditional sigmoid activation function, as well as with the Swish and E-Swish activation functions. Furthermore, we investigate the efficiency of a training procedure which simultaneously adjusts the spread factor sigma and the weights in the recurrent layer of the LRPNN. This training helps for coping with practical tasks, such as the recognition of Parkinson condition from speech signals, which operate under limited amount of training data.

We propose a unified perspective of quantization using a categorical approach. From a fixed Poisson algebra, we define quantization categories as subcategories of the $R$-module category equipped with the structure of classical limits. The generalized quantization categories have a huge structure including matrix regularization, strict deformation quantization, prequantization, Poisson enveloping algebra and so on.

In this paper, we obtain necessary and sufficient conditions, which ensure that all pairs of the solution sets of Stampacchia variational-like inequality, Minty variational-like inequality and global minimization problem coincide.

Some recent results on variational principles of constrained supinf problems of the form $\sup_{x \in X} \inf_{y \in Kx} \{f(x,y)\}$ are presented. Conditions that ensure validity of the presented results in case of Stackelberg problem are provided.

In this paper we review some recent results concerning the variational deduction of a Canham-Helfrich model for biomembranes obtained starting from a mesoscopic model which implements the amphiphilic behavior of the lipid molecules and the head-tail connection. The two-dimensional analysis is complete while in the three-dimensional case we have partial results and open problems.

In this paper we will introduce a newly found knowledge above the existence and the uniqueness of isoperimetric extremals of rotation on two-dimensional (pseudo-) Riemannian manifolds and on surfaces on Euclidean space. We will obtain the fundamental equations of the existence of isoperimetric extremals of rotation on spheres. They are only and the only planar sections of the sphere.

The present paper is a review of the research in the area of representations of the rotational motions in the three-dimensional Euclidian space. The study starts with the topics of Euler angles and Rodrigues' formula, and passes through the investigations in quaternion, spinor and vector kinematics. The authors present the interconnections between the different parameterizations of SO(3) group and stress on their merits and negative characteristics, and applications. A special attention is paid on the vector-parameterization of the rotation group, and how its nice properties are used in different mechanical applications.

This paper elaborates on a composition of two set-valued mappings in partial metric spaces. We establish several fixed point theorems, which generalize and complement some already known results. Especially, even in a partial metric space, our main result is an extension of the fixed point theorems of Abdessalem Benterki.

We study invariant Willmore-like tori in total spaces of Killing submersions. In particular, using a relation with elastic curves with potentials in the base surfaces, we analyze Willmore tori in total spaces of Killing submersions. Finally, we apply our findings to construct foliations of these total spaces by constant mean curvature Willmore tori.

Recently Yoon, Ru and Mioduchowski have introduced a model based on the classical Timoshenko beam theory, describing the propagation of transverse waves in double-wall carbon nanotubes regarded as a system of two separate nested tubes interacting via van der Waals forces. In the present work, we study the group properties of a system of equations generalizing the governing equations of the aforementioned model in which some of the coefficients are assumed to depend on the spatial variable. The full group consisting of all local point one parameter smooth automorphisms admitted by the regarded system is established. Next, the subgroup of those of them that leave invariant the functional whose Fréchet gradient (Euler-Lagrange equations) is exactly the regarded system of equations is obtained.

Finally, all conservation laws bijective to the set of the divergence symmetries of the foregoing functional are determined.

We explore an interesting three parametric family of essentially non-circular torus-like thin shells of revolution deforming under symmetrical loading without bending. It was found that these non-bending torus-like shells split naturally into two classes of shells lying either outside or inside of a given right circular cylinder. For each of these classes we obtain uniform parameterizations in terms of elliptic integrals. Explicit formulas for the aspect ratio of the non-bending toroids measuring their non-circularity are presented as well.

We present three different two-dimensional nonlocal integrable cmKdV equations obtained by Ablowitz-Musslimani type of reductions which are respectively T-symmetric, S-symmetric, and ST-symmetric nonlocal cmKdV equations. Ablowiz - Musslimani type of reduction arises from remarkably simple symmetry reductions of general AKNS scattering problems where the nonlocality appears in both space and time or time alone. Moreover, we obtain exact solutions by Darboux transformation.

Image segmentation technology is widely used in image processing, object recognition, face recognition, and more. This article discusses segmentation of a color image for the purpose of recognizing water areas. After recognizing the water areas, their boundaries are outlined and the area of the contours is calculated.

The article addresses the specific characteristic features of mutually intersecting regular pyramids with a common base. Advanced is a new theorem that proves to be highly effective when the objects are placed in a particular position.

This paper considers a finite-horizon linear quadratic (LQ) optimal control problem for a class of stochastic discrete-time, linear systems of fractional order which are generated by the operator involved in the definition of the fractional-order derivative of Grünwald-Letnikov type. This subject is new for discrete-time, linear, fractional-order systems (DTLFSs) with infinite Markovian jumps. We use an equivalent linear expanded-state model of the DTLFS with jumps and an equivalent quadratic cost functional to reduce the original optimal control problem to a similar one for discrete-time, linear, integer-order systems with Markovian jumps. The obtained optimal control problem is then solved by applying a dynamic programming technique.

The work deals with a family of nonlinear differential equations of Lane–Emden–Fowler type. The original Lane–Emden equation was used to model the thermal behavior of spherical clouds of gas within the framework of classical thermodynamics. Slightly modified, it describes phase transitions in critical thermodynamic systems, cylindrical equilibrium shapes of lipid membranes and many other physical processes and phenomena. The aim of the present work is to obtain approximate analytical solutions of the regarded equations. The problem is formulated in terms of nonlinear Volterra integral equations of the second kind. The solutions are sought by He's homotopy perturbation technique, as series expansion in the independent variable, and by Picard's method of successive approximations.

In this paper, we study one of generalized Heisenberg ferromagnet equations with self-consistent sources, namely, the so-called Myrzakulov-CIV equation with self-consistent sources (M-CIVESCS). The Lax representation of the M-CIVESCS is presented. We have shown that the M-CIVESCS and the CH equation with self-consistent sources (CHESCS) is geometrically equivalent to each other. The gauge equivalence between these equations is proved. Soliton (peakon) and pseudo-spherical surfaces induced by these equations are considered. The one peakon solution of the M-CIVESCS is presented.

In this paper, we study the (1+1)-dimensional Schrödinger-Maxwell-Bloch equation (NLS-MBE) which is integrable by the Inverse Scattering Method. Its Lax pair is presented. We apply methods of the theory of integrable systems to the geometry of surfaces immersed in Euclidean spaces. Using Sym-Tafel formula we construct the first and second fundamental forms, the Gaussian (total) curvature, mean curvature and Christoffel symbols for the NLS-MBE.

Using a star product of one variable, we construct a deformation of the Kummer function, called a star Kummer function. We show the convergence of the power series of the star Kummer function and then we obtain that the star Kummer function is an entire function of the variable and also with respect to the defomation parameter.

Classical and quantum Hamiltonian systems in two-dimensional Euclidean plane and allowing separation of variables in polar coordinates are investigated. The additional integral of motion is assumed to be a polynomial of degree $N \geq 3$ in momenta. After analyzing the particular cases of $N = 3, 4$ and $5$, a general description will be given. This leads to a classification of superintegrable potentials into two major categories. For the exotic potentials, the existence of an infinite family of superintegrable potentials in terms of the sixth Painlevé transcendent $P_6$ is conjectured and will be demonstrated for the first few cases.