Special Cone Research Articles

Integer Carathéodory results with bounded multiplicity

AbstractThe integer Carathéodory rank of a pointed rational cone C is the smallest number k such that every integer vector contained in C is an integral non-negative combination of at most k Hilbert basis elements. We investigate the integer Carathéodory rank of simplicial cones with respect to their multiplicity, i.e., the determinant of the integral generators of the cone. One of the main results states that simplicial cones with multiplicity bounded by five have the integral Carathéodory property, that is, the integer Carathéodory rank equals the dimension. Furthermore, we give a novel upper bound on the integer Carathéodory rank that depends on the dimension and the multiplicity. This bound improves upon the best known upper bound on the integer Carathéodory rank if the dimension exceeds the multiplicity. At last, we present special cones that have the integral Carathéodory property such as certain dual cones of Gorenstein cones.

The motion of a particle on the surface of a general cone

We use the formalism of Lagrange to find the equations of motion of a particle on the inner surface of a “general cone”. The equations of motion are challenging to solve, but we can evaluate them numerically with different software, to obtain the particle’s trajectory on the surface as a function of parameters such as angular momentum Lθ, cone shape and initial conditions, and then we find the total free-fall time of the particle. The results show a special cone in which the free fall time has a minimum for a fixed angular momentum and fall height. Differences in the free-fall times and the particle’s trajectory also analyzed for a two-coordinate (r, z) and three-coordinate (r, θ, z) system. This work shows the importance of learning to use software (Wolfram Mathematica, Python, POV-Ray) to help with some complex theoretical problems. Finally, the results can easily be generalized for other more complex surfaces.

Study on the Multi-Point Boundary Value Problem for Second-Order Nonlinear Impulsive Integro-Differential Equation

In the field of biological control, there are a large number of systems that gradually evolve over a certain period of time. However, due to some natural or human intervention behavior, the system state will be subjected to some relatively short time interference, so that the system state changes in an instant. This sudden change of state makes the system not simply described by continuous or discrete dynamical systems, but by means of impulse dynamical systems. In this paper, the multi-point boundary value problem of a class of second-order nonlinear impulsive integral differential equations is studied on the basis of impulsive differential equation theory. The main results of this kind of equation are obtained by using the fixed point theorem of strict set contraction operator. Under certain assumptions, the existence of the solution of the equation is proved by constructing the operator on the special cone. Finally, combined with the practical application, the theory was applied to the stability prediction of biological ecosystem, and the correctness of the conclusion was verified.

Massively parallel computation of tropical varieties, their positive part, and tropical Grassmannians

We present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of symmetries using the workflow management system GPI-Space and the computer algebra system Singular. We determine the tropical Grassmannian TGr0(3,8). Our implementation works efficiently on up to 840 cores, computing the 14763 orbits of maximal cones under the canonical S8-action in about 20 minutes.Relying on our result, we show that the Gröbner structure of TGr0(3,8) refines the 16-dimensional skeleton of the coarsest fan structure of the Dressian Dr(3,8), except for 23 orbits of special cones, for which we construct explicit obstructions to the realizability of their tropical linear spaces. Moreover, we propose algorithms for identifying maximal-dimensional cones which belong to positive tropicalizations of algebraic varieties. We compute the positive Grassmannian TGr+(3,8) and compare it to the cluster complex of the classical Grassmannian Gr(3,8).

Existence of positive solutions for third-order semipositone boundary value problems on time scales

In this paper, we consider the existence of positive solutions for a semipositone third-order nonlinear ordinary differential equation on time scales. In suitable growth conditions, by considering the properties on time scales and establishing a special cone, some new results on the existence of positive solutions are established when the nonlinearity is semipositone.

Circumcentric directions of cones

Generalized circumcenters have been recently introduced and employed to speed up classical projection-type methods for solving feasibility problems. In this note, circumcenters are enforced in a new setting; they are proven to provide inward directions to sets given by convex inequalities. In particular, we show that circumcentric directions of finitely generated cones belong to the interior of their polars. We also derive a measure of interiorness of the circumcentric direction, which then provides a special cone of search directions, all being feasible to the convex region under consideration.

CONCERNING THE DEVELOPMENT OF MODERN TEST EQUIPMENT FOR THE EVALUATION OF QUALITY INDICATORS OF FIRE-FIGHTING LAYFLAT DELIVERY HOSES FOR FIRE AND RESCUE VEHICLES

Pressure fire hoses are one of the main types of fire-fighting equipment, on the serviceable condition of which depends the operational activities of fire and rescue units and successful firefighting. At present, Ukraine has entered into force with (01.08.2021) the national standard, which contains modern European requirements for assessing the quality of flat-hose fire hoses for fire and rescue vehicles. This standard was developed by the Institute of Public Administration and Research in Civil Defense in the framework of research work "Fire hoses – test methods" and adopted by order of SE "UkrNDNC" from 23.03.2021 № 107. In order to assist in solving this problem, the Institute of Public Administration and Research in Civil Defense put into operation, tested and verified the installation for testing fire pressure hoses, which will check the quality of pressure flat hose for fire and rescue vehicles in accordance with modern European requirements. The installation is intended for tests of pressure fire hoses carried out on the following quality indicators: working, test and burst pressure; abrasion resistance; resistance to hot objects; resistance to open flame. The unit is operated at air temperature from 10 ° С to 25 ° С and relative humidity from 15% to 90%. The installation is a metal table, which consists of two levels (upper and lower). The main part of the tests is to be performed at the upper level with the help of three removable modules. The lower level provides one stationary module on which tests for working, test and burst pressure are carried out. The upper level of the installation is designed to test pressure fire hoses for the following quality indicators: resistance to surface abrasion, resistance to point abrasion, resistance to hot objects and resistance to open flames. For versatility and convenience of work on installation on the top level of a table the possibility of change of modules depending on their need is provided. The lower level of the installation is intended for carrying out tests of pressure fire hoses on quality indicators – working and test pressure, bursting pressure. At the lower level of the metal table is a rectangular metal box, which has a closing lid. During the tests, the sample of the test sleeve is automatically filled with water, which drains after the test. To fix the samples of fire hoses of different diameters (25, 32, 38, 51, 66, 77, 100, 125 and 150) mm, the installation includes special cones and plates.

Multiple solutions for singular semipositone boundary value problems of fourth-order differential systems with parameters

The aim of this paper is to establish some results about the existence of multiple solutions for the following singular semipositone boundary value problem of fourth-order differential systems with parameters: {u(4)(t)+β1u″(t)−α1u(t)=f1(t,u(t),v(t)),0<t<1;v(4)(t)+β2v″(t)−α2v(t)=f2(t,u(t),v(t)),0<t<1;u(0)=u(1)=u″(0)=u″(1)=0;v(0)=v(1)=v″(0)=v″(1)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} u^{(4)}(t)+\\beta _{1}u''(t)-\\alpha _{1}u(t)=f_{1}(t,u(t),v(t)),\\quad 0< t< 1; \\\\ v^{(4)}(t)+\\beta _{2}v''(t)-\\alpha _{2}v(t)=f_{2}(t,u(t),v(t)),\\quad 0< t< 1; \\\\ u(0)=u(1)=u''(0)=u''(1)=0; \\\\ v(0)=v(1)=v''(0)=v''(1)=0, \\end{cases} $$\\end{document} where f_{1},f_{2}in C[(0,1)times mathbb{R}^{+}_{0}times mathbb{R}, mathbb{R}], mathbb{R}_{0}^{+}=(0,+infty ). By constructing a special cone and applying fixed point index theory, some new existence results of multiple solutions for the considered system are obtained under some suitable assumptions. Finally, an example is worked out to illustrate the main results.

On the theory of constitutive equations for composites with different resistance in compression and tension

The paper proposes a new approach to the construction of nonlinear constitutive equations for the description of elastic deformation of fibrous composites with different resistance in compression and tension. The framework of the approach is the generalized rheological method of constructing constitutive equations of the uniaxial stress–strain state. Transition to the tensor equations uses the hypothesis of existence of a special cone of all possible compression states in the space of strains. The cone structure is governed by the reinforcement pattern of a composite. When imposed with strains in the cone, the composite behaves as a linearly elastic medium, while beyond the cone, the composite is nonlinear, with higher stiffness conditioned by tension of high-modulus reinforcement fibers. It is highly critical that the constitutive equations obtained using this approach belong to the theory of hyperelasticity. The elastic potentials of stresses and strains are obtained for these equations, which guarantees implementation of fundamental principles of equilibrium thermodynamics. Application of the developed general theory is exemplified by construction of the constitutive equations for the plane stress state of a thin unidirectional fiber reinforced composite plate and for a multilayer plate with different-direction reinforcement of layers. Using the finite element method in combination with the initial stress method, the stress and strain patterns are analyzed in rectangular one-, two- and three-layer carbon fiber reinforcement polymer plates with a circular cut under the action of external stresses at the boundary.

Numerical Simulations of Novel Conning Designs for Future Super-Large Wind Turbines

In order to develop super-large wind turbines, new concepts, such as downwind load-alignment, are required. Additionally, segmented blade concepts are under investigation. As a simple example, the coned rotor needs be investigated. In this paper, different conning configurations, including special cones with three segments, are simulated and analyzed based on the DTU-10 MW reference rotor. It was found that the different force distributions of upwind and downwind coned configurations agreed well with the distributions of angle of attack, which were affected by the blade tip position and the cone angle. With the upstream coning of the blade tip, the blade sections suffered from stronger axial induction and a lower angle of attack. The downstream coning of the blade tip led to reverse variations. The cone angle determined the velocity and force projecting process from the axial to the normal direction, which also influenced the angle of attack and force, provided that correct inflow velocity decomposition occurred.

Multiple Solutions for a Class of Nonlinear Fourth-Order Boundary Value Problems

This paper is concerned with multiple solutions for a class of nonlinear fourth-order boundary value problems with parameters. By constructing a special cone and applying fixed point index theory, the multiple solutions for the considered systems are obtained under some suitable assumptions. The main feature of obtained solutions (u(t),v(t)) is that the solution u(t) is positive, and the other solution v(t) may change sign. Finally, two examples with continuous function f1 being positive and f2 being semipositone are worked out to illustrate the main results.

Microscopic characteristics of impinging spray sliced by a cone structure under increased injection pressures

Wall-impingement is a serious problem for engine research owing to the possible source of hydrocarbons and soot emission. Therefore, impinging spray becomes the hotpot for investigation, especially for the microscopic characteristics. Droplet behaviors are significantly important to evaluate the atomization for predicting the impingement phenomena, such as stick, spread or splash. Many studies on it for both gasoline and diesel engines were published. However, these researches failed to capture the droplet behaviors clearly in the dense region owing to the overlap of the droplets. Only periphery or far field of the spray was tested. Therefore, a special cone structure named “spray slicer” was designed to cut the spray before observation with the consideration of good observation and less effect on droplet velocity. The experiment was conducted in a constant volume chamber under increased injection pressures between 10 and 30 MPa. Heptane was used as the surrogate fuel injected by a mini-sac injector with a single hole. Microscopic behaviors were recorded by particle image analysis (PIA) method to obtain the droplets size, velocity and minimum distance (Md). Results showed that microscopic behaviors can be observed clearly at different locations. Sauter mean diameter (SMD) and mean velocity decrease with spray development, and the smallest values occur at the periphery. Md increases with spray development owing to better dispersion. Moreover, high injection pressure decreases the droplet size but increases the velocity and Md at a certain degree.

Current perspectives in the management of keratoconus with contact lenses.

Our aim is to review current and significant articles on contact lens wear in keratoconus patients. A comprehensive literature search of PubMed was performed for the following topics on contact lens wear in keratoconus patients: (1) characteristics of contact lens wearers, (2) safety and efficacy, (3) complications, (4) fitting techniques, (5) contact lens wear after procedures/surgeries, (6) patient satisfaction. A total of 104 studies were finally selected and reviewed. Gas permeable (GP) lens wear provided significantly better vision than glasses. Special cone design lenses had better patient comfort levels though there was no difference in best corrected visual acuity among the GP lenses. Soft lenses showed good efficacy for the treatment of mild-to-moderate keratoconus with newer designs improving visual performance such as customised hydrogel and novel pinhole lenses. Scleral and hybrid lenses provide good visual acuity and comfort for keratoconic patients previously intolerant to RGP lenses. RGP lens wear post-cross linking (CXL) is relatively well-tolerated. Contact lenses may still be required post intrastromal corneal ring procedures and post keratoplasty. Scheimpflug imaging and anterior segment optical coherent tomography (ASOCT) are useful in contact lens fitting. Computerized contact lens fitting techniques could reduce the chair time of lens fitting as well as improve comfort and visual performance. Contact lenses play an important role in the visual rehabilitation of keratoconus patients. New contact lens designs and materials have significantly expanded the available fitting options for keratoconus patients. Imaging technology can be utilized to guide contact lens fitting.

A special cone construction and its connections to structured tensors and their spectra

In this work we construct a cone comprised of a group of tensors hypermatrices satisfying a special condition, and we study its relations to structured tensors such as M-tensors and H-tensors. We also investigate its applications to spectra of certain Z-tensors. We obtain an inequality for the spectral radius of certain tensors when the order m is odd.

Rapidly Separating Earthworm from Vermicompost Using Two-Step Technology

The vermicomposting has become a widely accepted approach to dispose organic solid waste such as livestock manure, plant straw and municipal sludge. With the enlargement of vermicomposting scale, it is necessary to separate earthworm from vermicompost as soon as possible before marketing. However, nowadays, separating earthworm is a bottle-neck in vermicomposting industry because it is often done manually, which is inefficient and expensive. The objective of this study is to develop a rapid and highly efficient two-step technology of separating earthworm from large-scale vermicompost. During the separation process, small granule vermicompost (diameter size < 6 mm) were separated with rolling screen and the earthworm were thoroughly separated from vermicompost through a special cone structure equipment called conical separator. Three experiments were conducted to evaluate the separation efficiency and obtain the optimal parameters for Earthworm-Vermicompost Rolling Separator (EVRS). The results showed that the optimal parameters for the EVRS were as follows: screen mesh size was 6 mm, rolling speed was 33.05 r min−1, inclined angle of rolling screen was 6.40°, screening length of rolling screen was 1290 mm and the generatrix angle of conical separator was 28°. The optimal separation effect could be achieved for EVRS at 45–55% vermicompost moisture content. Ten kg earthworm and vermicompost mixture was treated within 41.55 s, consuming electric power in 0.0025 kWh. EVRS can separate 84.38 ± 3.20% of mature earthworm and earthworm can be separated with survival rate of 96.25 ± 2.50%. In this study, the developed two-step technology and separator is a rapid, less energy consumption, highly efficient and applicable, which is expected to lighten the labor burden and shorten the recycling time of vermicomposting product.

Positive Solutions of Fourth-Order Problem Subject to Nonlocal Boundary Conditions

In this paper, we study the fourth-order problem with the second derivative in nonlinearity under nonlocal boundary value conditions involving Stieltjes integrals. Some inequality conditions on nonlinearity and the spectral radius conditions of linear operators are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on special cone. The conditions allow that the nonlinearity has superlinear or sublinear growth. Two examples are provided to support the main results under mixed boundary conditions involving multi-point with sign-changing coefficients and integral with sign-changing kernel.

The c-map on groups

We study the projective special Kähler condition on groups, providing an intrinsic definition of homogeneous projective special Kähler that includes the previously known examples. We give intrinsic defining equations that may be used without resorting to computations in the special cone, and emphasise certain associated integrability equations. The definition is shown to have the property that the image of such structures under the c-map is necessarily a left-invariant quaternionic Kähler structure on a Lie group.

Positive solutions of beam equations under nonlocal boundary value conditions

In this article, we study the fourth-order problem with the first and second derivatives in nonlinearity under nonlocal boundary value conditions{u(4)(t)=h(t)f(t,u(t),u′(t),u″(t)),t∈(0,1),u(0)=u(1)=β1[u],u″(0)+β2[u]=0,u″(1)+β3[u]=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& \\left \\{ \\textstyle\\begin{array}{l}u^{(4)}(t)=h(t)f(t,u(t),u'(t),u''(t)),\\quad t\\in(0,1),\\\\ u(0)=u(1)=\\beta_{1}[u],\\qquad u''(0)+\\beta_{2}[u]=0,\\qquad u''(1)+\\beta_{3}[u]=0, \\end{array}\\displaystyle \\right . \\end{aligned}$$ \\end{document} where f: [0,1]timesmathbb{R}_{+}timesmathbb{R}timesmathbb{R}_{-}to mathbb{R}_{+} is continuous, hin L^{1}(0,1) and beta_{i}[u] is Stieltjes integral (i=1,2,3). This equation describes the deflection of an elastic beam. Some inequality conditions on nonlinearity f are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on a special cone in C^{2}[0,1]. Two examples are provided to support the main results under mixed boundary conditions involving multi-point with sign-changing coefficients and integral with sign-changing kernel.

Dynamics and steady-state analysis of an unstirred chemostat model with internal storage and toxin mortality

This study proposes and analyzes a reaction–diffusion system describing the competition of two species for a single limiting nutrient that is stored internally in an unstirred chemostat, in which each species also produces a toxin that increases the mortality of its competitors. The possibility of coexistence and bistability for the model system is studied by the theory of uniform persistence and topological degree theory in cones, respectively. More precisely, the sharp a priori estimates for nonnegative solutions of the system are first established, which assure that all of nonnegative solutions belong to a special cone. Then it turns out that coexistence and bistability can be determined by the sign of the principal eigenvalues associated with specific nonlinear eigenvalue problems in the special positive cones. The local stability of two semi-trivial steady states cannot be studied via the technique of linearization since a singularity arises from the linearization around those steady states. Instead, we introduce a 1-homogeneous operator to rigorously investigate their local stability.

3D INNOVATION TECHNOLOGIES IN EDUCATION

Introduction: Studying the discipline of stereometry is difficult for students who do not have spatial imagination. To overcome this problem, software applications are being created in addition to traditional training methods to improve the understanding of 3D geometry.&#x0D; Objectives: The subject of this article is the presentation of the functionality and tools of a created software application in teaching stereometry, with the help of which students will construct, observe, measure and study 3D geometric shapes, as well as create dynamic images.&#x0D; Methods: The methods: stereoscopy, polygon triangulation, cleaning/extrusion, and level of detail were used to create 3D geometric shapes via the software application designed for the secondary education of the Bulgarian students in stereometry&#x0D; Results: The software application created the following objects in 3D space: regular prism (three to six sides), general three-sided prism, special four-sided prism, regular pyramid (three to six sides), cube, sphere, cone and cylinder. The parameters of any geometric object can be adjusted, including the height and length of the sides, the slope of the sides, the number of sides, etc. Each object can be gradually opened or closed from its main network to the geometric object. Each parameter correction is interactive and visible immediately. Geometric shapes can be rotated in space, zoomed in to or out of, and viewed as a solid object or transparent net, as well as have the ability to change colors.&#x0D; Conclusion: With the software application created, new opportunities for teaching and learning in the subject of stereometry are provided, which helps students to more easily understand spatial geometry and the ability of teachers to explain abstract problems in 3D geometry.