A study of a family of monomial ideals

In this paper, we study a family of rational monomial parametrizations. We investigate a few structural properties related to the corresponding monomial ideal [Formula: see text] generated by the parametrization. We first find the implicit equation of the closure of the image of the parametrization. Then we provide a minimal graded free resolution of the monomial ideal [Formula: see text], and describe the minimal graded free resolution of the symmetric algebra of [Formula: see text]. Finally, we provide a method to compute the defining equations of the Rees algebra of [Formula: see text] using three moving planes that follow the parametrization.

On nonlinear problems of parabolic type with implicit constitutive equations involving flux

We study systems of nonlinear partial differential equations of parabolic type, in which the elliptic operator is replaced by the first-order divergence operator acting on a flux function, which is related to the spatial gradient of the unknown through an additional implicit equation. This setting, broad enough in terms of applications, significantly expands the paradigm of nonlinear parabolic problems. Formulating four conditions concerning the form of the implicit equation, we first show that these conditions describe a maximal monotone [Formula: see text]-coercive graph. We then establish the global-in-time and large-data existence of a (weak) solution and its uniqueness. To this end, we adopt and significantly generalize Minty’s method of monotone mappings. A unified theory, containing several novel tools, is developed in a way to be tractable from the point of view of numerical approximations.

On the Implicit Equation of Conics and Quadrics Offsets

A new determinantal representation for the implicit equation of offsets to conics and quadrics is derived. It is simple, free of extraneous components and provides a very compact expanded form, these representations being very useful when dealing with geometric queries about offsets such as point positioning or solving intersection purposes. It is based on several classical results in “A Treatise on the Analytic Geometry of Three Dimensions” by G. Salmon for offsets to non-degenerate conics and central quadrics.

Deformation and Failure Mechanism of Surrounding Rock in Mining-Influenced Roadway and the Control Technology

Aiming at the problem of surrounding rock deformation and failure of mining roadway and its control, a mechanical model of the circular roadway under the mining environment is established, and the implicit equation of the plastic zone boundary is derived. By analyzing the morphologic evolution law of the surrounding rock plastic zone in the mining roadway, the key factors affecting the morphologic change of the plastic zone are obtained, that is, the magnitude and direction of principal stress. The influence law of the magnitude and direction of principal stress on the plastic zone of the mining roadway is analyzed by using numerical simulation software, and the deformation and failure mechanism of surrounding rock of the mining roadway is revealed. The results showed that the size and morphology of the plastic zone were closely related to the confining pressure ratio (η). Taking the boundary of η valuing 1, the larger or smaller η value was, the more serious the deformation and failure of surrounding rock would be; the morphology of the plastic zone changed with the deflection of the principal stress, with the location of the maximum plastic zone influenced by the principal stress direction. For the surrounding rock control in the mining-influenced roadway, it is advised to take the following methods: firstly, it is necessary to consider how to reduce or remove the influence of mining on surrounding rock, improve the stress environment of surrounding rock, and reduce the failure depth of the plastic zone, so as to better maintain the roadway. Secondly, in view of the deformation and failure characteristics of the mining roadway, the fractional support method of “yielding first and then resisting” should be adopted, which applies the cable supplement support after mining instead of the one-off high-strength support during roadway excavation, so as to control the malignant expansion of the surrounding rock plastic zone and prevent roof falling accidents.

Friction Factor Equations Accuracy for Single and Two-Phase Flows

Estimate pressure drop throughout petroleum production and transport system has an important role to properly sizing the various parameters involved in those complex facilities. One of the most challenging variables used to calculate the pressure drop is the friction factor, also known as Darcy–Weisbach’s friction factor. In this context, Colebrook’ s equation is recognized by many engineers and scientists as the most accurate equation to estimate it. However, due to its computational cost, since it is an implicit equation, several explicit equations have been developed over the decades to accurately estimate friction factor in a straightforward way. This paper aims to investigate accuracy of 46 of those explicit equations and Colebrook implicit equation against 2397 experimental points from single-phase and two-phase flows, with Reynolds number between 3000 and 735000 and relative roughness between 0 and 1.40 × 10−3. Applying three different statistical metrics, we concluded that the best explicit equation, proposed by Achour et al. (2002), presented better accuracy to estimate friction factor than Colebrook’s equation. On the other hand, we also showed that equations developed by Wood (1966), Rao and Kumar (2007) and Brkić (2016) must be used in specifics conditions which were developed, otherwise can produce highly inaccurate results. The remaining equations presented good accuracy and can be applied, however, presented similar or lower accuracy than Colebrook’s equation.

Modeling technology of curved surface development for puffer fish

The drag reduction mechanism of puffer epidermis was closely related to its real geometry. In order to solve the modeling problem of epidermal spines on the puffer surface, a modeling method for the expansion of puffer shape was proposed. The three-dimensional scanning and non-uniform rational B-spline surface modeling technology was used to reconstruct the puffer model. According to the curvature characteristics, the surface mathematical equations including exponential, logarithmic, and sinusoidal functions were established based on the multinomial function. The surface was generated by a mathematical equation, and the surface was divided into several non-uniform rational B-spline patches according to curvature. After discretization, the point cloud Gaussian curvature and average value were calculated based on the implicit equation of moving least square surface, and whether the surface is approximately extensible or not was judged. Finally, the puffer surface was divided into 46 curved patches. In this article, the surface expansion algorithm gave priority to ensure the area unchanged, and four feature surfaces were selected according to the epidermal spines arrangement of the puffer surface. The results showed that the technique can simply and efficiently unfold the curved surface of the puffer fish, thus the mapping relationship between the epidermal spines on the surface and the plane was determined, which established a foundation for the accurate arrangement and modeling of the epidermal spines on the surface.

Non-relativistic Energy Spectrum of the Deng-Fan Oscillator via the WKB Approximation Method

The energy spectrum of the radial Schrodinger equation with the molecular Deng Fan potential has been obtained through the WKB approximation scheme. The radial WKB solution yields a transcendental or an implicit equation. The energy eigenvalues for non-physical and real molecular interacting systems are presented. In comparison with the numerical eigenvalues obtained with MATHEMATICA 3.0 package, the WKB approximation method produces improved results over the results obtained with other analytical methods in the literature.

Practical thermal model for a radiant tubes annealing furnace

Modeling of vertical radiant tube annealing furnaces has proven to be one of the best tools to improve the performance of a galvanizing line. However, there is a lack of a practical model able to consider the temperature and status of the radiant tubes, which are key elements in the capacity of the furnace. The model proposed divides the furnace in several segments and compares the radiated heat, that is exchanged between the radiant tubes and the strip, and the heat required to increase the temperature of the mass flow on the strip. This comparison is represented as an implicit equation where the strip‘s temperature is obtained by iteration. The model is validated calculating the final temperature of more than five hundred coils divided in four different steel families. The 90% of the calculated temperatures are within a 2% deviation range compared to the measured temperatures. This model combines good accuracy in the results with low computational times, allowing the simulation of hundreds of coils in a few minutes.