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The boundary layer equations for a plane unsteady flow are considered, for which the existence theorems with monotone initial data and with analytic initial data are obtained by Oleinik and Sammartino-Caflish, respectively. Here, without these assumptions, we construct some special smooth solutions.

The compressible Euler equations in R N with Coriolis force is the fundamental mathematical model for the atmospheric dynamics. We use the vector and matrix technique to the N -dimensional Euler equations, to construct the following Cartesian vector form solutions u = b ( t ) + A ( t ) x . One advantage of this approach is that the Euler equations with Coriolis force can be solved both theoretically and algebraically, which can be accomplished by constructing appropriate matrices A ( t ) , and vector b ( t ) . We can obtain the new exact solutions using the required matrices A ( t ) and vector b ( t ) . As exact solutions for the compressible Euler equations are rare, the new exact solutions are valuable for the verification of the corresponding computational methods.

In this letter, the stability problem of a class of planar positive nonlinear systems is investigated. Based on homogeneous nonlinear system theory and positive system theory, a necessary and sufficient condition for stability is established. To prove the necessity, a new method called special solution method is provided, and this method can be efficiently used to analyze instability. Three examples are provided to demonstrate the efficiency of the obtained results.

Landau solutions are special solutions to the stationary incompressible Navier-Stokes equations in the three dimensional space excluding the origin. They are self-similar and axisymmetric with no swirl. In fact, any self-similar smooth solution must be a Landau solution. In the effort of extending this result to the one for the solution class with the pointwise scale-invariant bound $|u(x)|\leq C_0|x|^{-1}$ for some $C_0>0$, we consider axisymmetric discretely self-similar solutions, and investigate the existence of such solution curve emanating from some Landau solution. We prove that the inclusion of the swirl component does not enhance the bifurcation and present numerical evidence of no bifurcation.

We derive the fractional version of one-phase one-dimensional Stefan model. We assume that the diffusive flux is given by the time-fractional Riemann-Liouville derivative, i.e. we impose the memory effect in the examined model. Furthermore, we find a special solution to this problem.

Contact lens-assisted corneal cross-linking (CACXL) was introduced by Jacob et al. in 2012 for treating thin keratoconic corneas using riboflavin soaked soft contact lens to artificially increase the functional corneal thickness. It is advantageous over other thin corneal cross-linking techniques as it works independent of swelling properties of the cornea, is an epi-off technique and does not require additional time, additional expensive equipments or special solutions. The only additional requirement as compared to all other techniques is a UV barrier-free soft contact lens (SoflensTM, B&L) which is easily available and inexpensive. Advantages include simplicity, easy adaptability, early visual rehabilitation, good visual outcomes, safety, and efficacy. Progression rates are acceptable and the need for re-treatment has been low. CACXL can help regularize corneal shape and may be used in isolation or synergistically with Intracorneal ring segments (ICRS) or Corneal allogenic intrastromal ring segments (CAIRS). It gives about 70% stiffening as compared to standard Dresden protocol CXL in less ideal porcine eye studies. Murine eye models that closely mimic thin corneas and show greater cross-linking effect as compared to porcine eyes may be a better model for evaluation of CACXL, however further studies are needed. Care should be taken in selecting the right kind of contact lens. Proper technique should be followed, especially by confirming thinnest functional pachymetry to be above 400 microns intra-operatively before application of UV-A. The sub-contact lens riboflavin film should be avoided as also an excessively thick supra-contact lens riboflavin film and too many re-applications.

In this paper we consider a class of chemotaxis models with two arbitrary constitutive functions g(u) and f(v). After having performed a complete symmetry group classification with respect to them the reduced systems are derived. By considering g(u) of the logistic form wide classes of exact solutions are found.

We study special solutions of Galactic Tidal models and the existence, stability and bifurcations of relative equilibria. Precisely, we found four equilibria and rectilinear, as well as circular so...

Reductions of the KP-Whitham system, namely the (2+1)-dimensional hydrodynamic system of five equations that describes the slow modulations of periodic solutions of the Kadomtsev-Petviashvili (KP) equation, are studied. Specifically, the soliton and harmonic wave limits of the KP-Whitham system are considered, which give rise in each case to a four-component (2+1)-dimensional hydrodynamic system. It is shown that a suitable change of dependent variables splits the resulting four-component systems into two parts: (i) a decoupled, independent two-component system comprised of the dispersionless KP equation, (ii) an auxiliary, two-component system coupled to the mean flow equations, which describes either the evolution of a linear wave or a soliton propagating on top of the mean flow. The integrability of both four-component systems is then demonstrated by applying the Haantjes tensor test as well as the method of hydrodynamic reductions. Various exact reductions of these systems are then presented that correspond to concrete physical scenarios., 17 pages

We propose a max-plus equation which includes Conway's Game of Life (GoL) as a special case. There are some special solutions to the equation which include and unify those to GoL. Moreover, the multi-value extension of GoL is derived from the equation and the behavior of solutions is discussed.

Self-similar reductions of the Sawada-Kotera and Kupershmidt equations are studied. Results of Painleve’s test for these equations are given. Lax pairs for solving the Cauchy problems to these nonlinear ordinary differential equations are found. Special solutions of the Sawada-Kotera and Kupershmidt equations expressed via the first Painleve equation are presented. Exact solutions of the Sawada-Kotera and Kupershmidt equations by means of general solution for the first member of K2 hierarchy are given. Special polynomials for expressions of rational solutions for the equations considered are introduced. The differential-difference equations for finding special polynomials corresponding to the Sawada-Kotera and Kupershmidt equations are found. Nonlinear differential equations of sixth order for special polynomials associated with the Sawada-Kotera and Kupershmidt equations are obtained. Lax pairs for nonlinear differential equations with special polynomials are presented. Rational solutions of the self-similar reductions for the Sawada-Kotera and Kupershmidt equations are given.

Waste of the metalworking industry is analyzed from the point of view of their further processing. There is a lack of effective methods of processing and disposal of grinding sludge and, as a consequence, a negative impact on the environment. Existing methods of grinding sludge processing are analyzed. The expediency of applying the method of metallurgical remelting under the condition of special preparation of metal sludges of grinding machines is proved. It is established that briquettes for their further processing should have a certain density and purity. It is suggested to wash the sludge from the LCPM (lubricant-cooling process media) in special solutions using washing machines, briquettes and dry before using them in metallurgical remelting.In order to determine the technological parameters of the process of briquetting sludge grinding machines and drying of briquettes, the effect of pressing pressure, binder concentration and concentration of liquid (water) on the density of briquette was experimentally investigated. Appropriate mathematical dependencies and graphs of surfaces were obtained. It was found that the most significant factor influencing the density of the briquette is the pressure of the press, less significant – the concentration of liquid and binder. In the case of increasing water concentration, the density of the briquette decreases. It is recommended to prevent the briquette from spilling out after drying, to add a binder element of 3–5 % by weight of dry sludge. To dissolve the binder, water is added, the optimal concentration of which is 6–8 %. Technological pressing parameters are recommended.

Software product lining is the act of providing different but related software products under the same brand, known as a software product line (SPL). As engineering, management and validation of SPLs is far from trivial, special solutions for software product line engineering (SPLE) have a continuous momentum in both academic and industry. In general, it is hard to judge when to reasonably favor SPLE over alternative solutions that are more common in the industry. In this paper, we illustrate how we as Elinvar manage variability within our WealthTech Platform as a Service (PaaS) at different granularity levels, and discuss methods for SPLE in this context. More in detail, we share our techniques and concepts to address configuration management, and show how we manage a single microservice SPL including inter-service communication. Finally, we provide insights into platform solutions by means of packages for our clients. We end with a discussion on SPLE techniques in context of service SPLs and our packaging strategy. We conclude that while we are good to go with industry-standard approaches for microservice SPLs, the variability modeling and analysis advantages within SPLE is promising for our packaging strategy.

ملخص تستعرض هذه الدراسة تقنية تصوير الفاصل الزمني بتقديمها أحد أهم الشرکات العالمية التي تقدم هذا النوع من التصوير باستخدام الکاميرات عالية الوضوح والدقة منذ العام 2006 حتى الآن ألا وهي شرکة EarthCam Inc, . وقد قام الباحث بهذا الأمر في ضوء أحد المشروعات السعودية التي تم استخدام هذا نظام هذه الشرکة بها وهو مشروع برج المملکة في مدينة جده. وقد اتبع الباحث منهج البحوث الوصفية حيث يستخدم هذا المنهج لوصف خصائص الظاهرة قيد الدراسة، والباحث هنا لم يجب على أسئلة حول "کيف / متى / لماذا" حدثت هذه الخصائص بل يعالج مسألة "ماذا" أو ما هي خصائص هذا البرنامج ومميزاته والسبب في ذلک هو أنه غالبا ما تسبق البحوث الوصفية البحوث التفسيرية. کما استخدمت هذه الدراسة اسلوب دراسة الحالة وتتمثل هنا في نظام Control 8 المستخدم من قبل شرکة EarthCam الامريکية حيث قام الباحث بتحليل متعمق قدر الامکان في خصائص هذا النظام ومميزاته. وفي جانب النتائج المستخلصة فإن هذه الدراسة تتبعت نتائج بحوث البرامج الحاسوبية والتي من ضمن أنواعها "الحلول الخاصة"، حيث قام الباحث بالتوصل لبعض نتائج وحلول إبداعية خاصة بمجال توثيق ومراقبة الإنشاءات العمرانية المترتبة على استخدام مثل هذه الکاميرات والتقنيات الحديثة الخاصة بالوسائط المتعددة. Abstract This study reviews the Time-lapse technology provided by one of the world's leading manufacturers of high definition and precision cameras from 2006 to date, EarthCam Inc. The researcher did this in the light of one of the Saudi projects that have been used this technology in “Kingdom Tower” project in Jeddah city. The researcher has followed the descriptive research approach. This method is used to describe the characteristics of the phenomenon under study, and the researcher here did not answer questions about "how / when / why" these characteristics occurred, but rather the question of what or what the characteristics of this program and its advantages. The study also used the case study method, which is the control 8 system which is invented by EarthCam USA, where the researcher analyzed as thoroughly as possible the characteristics of this system and its features. On the results side, this study followed the results of the research of the software programs, which are among the types of "special solutions". The researcher reached some creative results and solutions for the field of documentation and control of the construction of the use of such cameras and modern technologies for multimedia.

The inverse mean curvature flow (IMCF) has been extensively studied not only as a type of geometric flows, but also for its applications to geometric inequalities. The focus is primarily on homothetic solitons for the IMCF in this paper, which are special solutions deformed only homothetically under the flow. We completely classify the profile curves of the higher dimensional rotationally and birotationally symmetric homothetic solitons using the phase-plane analysis. As a characterization of the round cylinder, we obtain that any helicoidal homothetic soliton of IMCF must be round cylinders. In addition, we prove that any surface foliated by circles to be the homothetic soliton of IMCF, which is a cyclic homothetic soliton, is either a surface of revolution or a piece of a round sphere.