geoTangle : Interactive Design of Geodesic Tangle Patterns on Surfaces

Tangles are complex patterns, which are often used to decorate the surface of real-world artisanal objects. They consist of arrangements of simple shapes organized into nested hierarchies, obtained by recursively splitting regions to add progressively finer details. In this article, we show that 3D digital shapes can be decorated with tangles by working interactively in the intrinsic metric of the surface. Our tangles are generated by the recursive application of only four operators, which are derived from tracing the isolines or the integral curves of geodesics fields generated from selected seeds on the surface. Based on this formulation, we present an interactive application that lets designers model complex recursive patterns directly on the object surface without relying on parametrization. We reach interactive speed on meshes of a few million triangles by relying on an efficient approximate graph-based geodesic solver.

Variational principles for conformal geodesics

Conformal geodesics are solutions to a system of third-order equations, which makes a Lagrangian formulation problematic. We show how enlarging the class of allowed variations leads to a variational formulation for this system with a third-order conformally invariant Lagrangian. We also discuss the conformally invariant system of fourth-order ODEs arising from this Lagrangian and show that some of its integral curves are spirals.

The Design of Individual Orthopedic Insoles for the Patients with Diabetic Foot Using Integral Curves to Describe the Plantar Over-Pressure Areas

Identification of over-pressure areas in the plantar side of the foot in patients with diabetic foot and reduction of plantar pressure play a major role in clinical practice. The use of individual orthopedic insoles is essential to reduce the over-pressure. The aim of the present study is to mark the over-pressure areas of the plantar part of the foot on a pedogram and describe them with high accuracy using a mathematical research method. The locally over-pressured areas with calluses formed due to repeated injuries were identified on the patients’ pedograms. The geometric shapes of the over-pressure areas were described by means of the integral curves of the solutions to Dirichlet singular boundary differential equations. Based on the mathematical algorithm describing those curves, the computer programs were developed. The individual orthopedic insoles were produced on a computer numerical control milling machine considering the locally over-pressured areas. The ethylene vinyl acetate polymers of different degrees of hardness were used to produce the individual orthopedic insoles. For the over-pressure areas, a soft material with a hardness of 20 Shore A was used, which reduces the pressure on the plantar side of the foot and increases the contact area. A relatively hard material with a hardness of 40 Shore A was used as the main frame, which imparts the stability of shape to the insole and increases its wear life. The individual orthopedic insoles produced by means of such technology effectively reduce the pressure on the plantar side of the foot and protect the foot from mechanical damage, which is important for the treatment of the diabetic foot.

Investigation of an Organogel by Micro-Differential Scanning Calorimetry: Quantitative Relationship between the Shapes of the Thermograms and the Phase Diagram

The phase diagrams of organogels are necessary for applications and fundamental aspects, for instance to understand their thermodynamics. Differential scanning calorimetry is one of the techniques implemented to map these diagrams. The thermograms of organogels upon heating show broad endotherms, increasing gradually to a maximum, at a temperature Tmax, and decreasing back to the baseline, sometimes 10 °C above. This broadening can lead to uncertainty in determining the molar enthalpies and the melting temperatures Tm of the gels. Herein, we have measured the thermograms of the 12-hydroxystearic acid/nitrobenzene gels for weight fractions ranging from 0.0015 to 0.04. Compared with transition temperatures measured by other techniques, the inflection points of the thermograms provide a measurement of Tm with less bias than Tmax. The phase diagram explains why the molar melting enthalpies derived from the thermograms for samples of low concentration are lower than expected. The shapes of the heat flows below the peak correlate quantitatively with the diagrams: after suitable correction and normalization, the integral curves superimpose with the phase diagram in their ascending branch and reach a plateau when the gel is fully melted. The shape of the thermograms upon cooling is also qualitatively explained within the frame of the diagrams.

Extended Hamilton–Jacobi Theory, Symmetries and Integrability by Quadratures

In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–Jacobi equation (HJE) related to X (and a given fibration on M). We do that along each open subset U⊆M, such that πU has a manifold structure and πU:U→πU, the restriction to U of the canonical projection π:M→M/G, is a surjective submersion. If XU is not vertical with respect to πU, we show that such complete solutions solve the reconstruction equations related to XU and G, i.e., the equations that enable us to write the integral curves of XU in terms of those of its projection on πU. On the other hand, if XU is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of XU up to quadratures. To do that, we give, for some elements ξ of the Lie algebra g of G, an explicit expression up to quadratures of the exponential curve expξt, different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of expξt is valid for all ξ inside an open dense subset of g.

Some Results on Ricci Almost Solitons

We find three necessary and sufficient conditions for an n-dimensional compact Ricci almost soliton (M,g,w,σ) to be a trivial Ricci soliton under the assumption that the soliton vector field w is a geodesic vector field (a vector field with integral curves geodesics). The first result uses condition r2≤nσr on a nonzero scalar curvature r; the second result uses the condition that the soliton vector field w is an eigen vector of the Ricci operator with constant eigenvalue λ satisfying n2λ2≥r2; the third result uses a suitable lower bound on the Ricci curvature S(w,w). Finally, we show that an n-dimensional connected Ricci almost soliton (M,g,w,σ) with soliton vector field w is a geodesic vector field with a trivial Ricci soliton, if and only if, nσ−r is a constant along integral curves of w and the Ricci curvature S(w,w) has a suitable lower bound.

Analysis of Definite Integral Material Topics for Improve Student Learning Using Apriori Algorithm

Definite Integral is one of the most important subjects in calculus. The use of integrals that must be studied is calculating the area and drawing curves based on the equation of functions. However, there are still many students have difficult to understand integral material, especially definite integral. Most students have difficult to understand Integral learning because they do not understand the basic and material that needs to be mastered. The purpose of this study is to find a pattern of relationship to the understanding the topic of integral material about calculation and drawing of integral curves using apriori algorithm. Apriori algorithms can be used to determine learning patterns and linkages between definite integral material. Apriori algorithms can be used to determine learning patterns and linkages between definite integral material. The results of this study indicate that understanding of the material is the topic of calculation with 1 functional equation and 2 functional equations, and the depiction of integral curves at X and Y coordinates with a confidence value of 96% and basic integral material such as understanding basic integral techniques, definite integral formulas, calculations and curve depiction on cartesian diagram coordinates X and Y with a confidence value of 76%.

DATA RECOVERY OF ANNUAL RIVER RUNOFF IN THE SYRDARYA RIVER BASIN

The data recovery of the annual runoff was carried out and correlation dependences were obtained, which were used to calculate the runoff rate for each of the selected rivers in the Syrdariya river basin. Differential integral curves were constructed from the runoff data using the variability index. When restoring the missing data on the annual runoff, the river-analogue method was applied. The actual series of observations are given for a longterm period. The base period was chosen from 1960 to 2015. Quantitative estimates of the effectiveness of bringing the average values to a multi-year period are also provided.

RETROSPECTIVE ANALYSIS OF WATERNESS CHANGE IN THE DON RIVER

Water resources are one of the most important problems of our time. Population growth, industrial and agricultural development all over the world entail an ever-increasing demand for clean fresh water. These circumstances induce hydrologists to actively and thoroughly study the problem of studying water resources, changing their quantitative and qualitative characteristics in time and space, and the peculiarities of changes in the water regime of river runoff under conditions of climatic changes. In this work, we performed a retrospective analysis and assessment of changes in the water content of the Upper Don basin over a long 126-year period of hydrometric observational data (1881/1882–2006/2007). To study the change in the water content of the Upper Don River, we used the difference integral curves of the annual and seasonal (spring flood, summerautumn and winter low-water periods) runoff. Regularities of long-term cyclical fluctuations in the water content of the annual and seasonal runoff of the Upper Don are obtained. A retrospective analysis of long-term data of hydrometric observations made it possible to distinguish long phases of high-water, medium-water-content and low-water years on the differential integral curves of river runoff. Each phase, which is long in terms of water content, contains groups of years of shorter duration, for example, 2–3-year and 4–5-year phases of increased and decreased water content in a river. The analysis of the differential integral curves of the annual and seasonal runoff made it possible to establish that the long-term fluctuations in the annual runoff of the Upper Don are rhythmic, in contrast to the runoff of the spring flood, summer-autumn and winter low-water periods, which are characterized by a monotonic regime.