scholarly journals Lerner’s inequality associated to a critical radius function and applications

2013 ◽  
Vol 407 (1) ◽  
pp. 35-55 ◽  
Author(s):  
B. Bongioanni ◽  
A. Cabral ◽  
E. Harboure
Keyword(s):  
Critical Radius ◽  
Radius Function

scholarly journals Weighted Weak Local Hardy Spaces Associated with Schrödinger Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Hua Zhu
Keyword(s):  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Critical Radius ◽  
Schrödinger Operators ◽  
Riesz Transforms ◽  
Schrodinger Operators ◽  
Local Hardy Spaces ◽  
Radius Function

We characterize the weighted weak local Hardy spacesWhρp(ω)related to the critical radius functionρand weightsω∈A∞ρ,∞(Rn)which locally behave as Muckenhoupt’s weights and actually include them, by the atomic decomposition. As an application, we show that localized Riesz transforms are bounded on the weighted weak local Hardy spaces.

Monocrystalline Fiber Growth Technique: New Critical Radius Considerations

2021 ◽  
pp. 126199
Author(s):  
Jorge-Enrique Rueda-P ◽  
J.E.F.S. Rodrigues ◽  
Antonio Carlos Hernandes
Keyword(s):  
Critical Radius ◽  
Growth Technique ◽  
Fiber Growth

An Investigation on the Profile of Microlens by the Thermal Reflow Process Due to Surface Tension and Gravity

2008 ◽  
Author(s):  
Jhy-Cherng Tsai ◽  
Yong-Sung Hsu
Keyword(s):  
Surface Tension ◽  
Critical Radius ◽  
Critical Size ◽  
Liquid Droplet ◽  
Bond Number ◽  
Diameter Ratio ◽  
Experimental Result ◽  
Critical Number ◽  
Reflow Process ◽  
Thermal Reflow

Microlens and its mold fabricated by thermal reflow using photoresist have been widely used for forming patterns in different scales. When the photoresist solidifies from melting condition, for example by the reflow process, its profile is formed based on the balance between surface tension and gravity. This research is aimed to investigate the influence of surface tension and gravity on the profile of microlens in thermal reflow process. Theoretical analysis based on the interaction between surface tension and gravity of liquid droplet is first investigated. The result showed that the height to diameter ratio (h/D), or the sag ratio, of the liquid droplet is affected by the Bond number (Bo), a number defined as the ratio of gravity to surface tension. The sag ratio is not sensitive to Bo when Bo is small but the ratio decreases as Bo increases if Bo is over the critical number. Based on the analysis, the critical number for the AZ4620 photoresist on a silicon substrate is 1, corresponding to the critical radius of droplet R = 2,500μm. When the size of the droplet is less then the critical size, the profile is mainly controlled by the surface tension and thus the sag ratio is about the same regardless the size. The profile, in contrast, is highly affected by the gravity if the size of the droplet is larger then the critical size. The sag ratio decreases exponentially with respect to Bo in this case. Experiments are also designed and conducted to verify the analysis. Experimental result showed that the sag ratio of the photoresist reduces to 0.065 from 0.095 when Bo increases from 0.0048 to 0.192. The results showed that the trend is consistent to the theoretical model.

scholarly journals Gödel and Gödel-type solutions in the Palatini f(R,T) gravity theory

2020 ◽  
pp. 2150014
Author(s):  
J. S. Gonçalves ◽  
A. F. Santos
Keyword(s):  
Trace Formula ◽  
Arbitrary Function ◽  
Critical Radius ◽  
Energy Momentum Tensor ◽  
Gravitational Theory ◽  
Momentum Tensor ◽  
Gravity Theory ◽  
Ricci Scalar ◽  
Independent Variables ◽  
Hilbert Action

The Palatini [Formula: see text] gravity theory is considered. The standard Einstein–Hilbert action is replaced by an arbitrary function of the Ricci scalar [Formula: see text] and of the trace [Formula: see text] of the energy-momentum tensor. In the Palatini approach, the Ricci scalar is a function of the metric and the connection. These two quantities, metric and connection, are taken as independent variables. Then, it is examined whether Palatini [Formula: see text] gravity theory allows solutions in which lead to violation of causality. The Gödel and Gödel-type spacetimes are considered. In addition, a critical radius, which permits to examine limits for violation of causality, is calculated. It is shown that, for different matter contents, noncausal solutions can be avoided in this Palatini gravitational theory.

scholarly journals Modelling of Metallic Particle Binders for Increased Part Density in Binder Jet Printed Components

2019 ◽  
Author(s):  
Joseph Roberts ◽  
Peter Green ◽  
Kate Black ◽  
Christopher Sutcliffe
Keyword(s):  
Particle Size ◽  
Critical Radius ◽  
Detailed Knowledge ◽  
Particle Sizes ◽  
Powder Bed ◽  
Beneficial Effects ◽  
Drop Tests ◽  
Maximum Particle ◽  
Packing Efficiency ◽  
Selection Of

Binder jet printed components typically have low overall density in the green state and high shrinkage and deformation after heat treatment. It has previously been demonstrated that, by including nanoparticles of the same material in the binder, these properties can be improved as the nanoparticles can fill the interstices and pore throats between the bed particles. The beneficial effects from using these additive binder particles can be improved by maximising the binder particle size, enabling the space within the powder bed to be filled with a higher packing efficiency. The selection of maximum particle size for a binder requires detailed knowledge of the pores and pore throats between the powder bed particles. In this paper, a raindrop model is developed to determine the critical radius at which binder particles can pass between pores and penetrate the bed. The model is validated against helium pycnometry measurements and binder particle drop tests. It is found that the critical radius can be predicted, with acceptable accuracy, using a linear function of the mean and standard deviation of the particle radii. Percolation theory concepts have been employed in order to generalise the results for powder beds that have different mean particle sizes and size distributions. The results of this work can be employed to inform the selection of particle sizes required for binder formulations, to optimise density and reduce shrinkage in printed binder jet components.

Critical radius and dipole polarizability for a confined system

2006 ◽  
Vol 15 (6) ◽  
pp. 1172-1176 ◽  
Author(s):  
Xu Tian ◽  
Cao Zhuang-Qi ◽  
Ou Yong-Cheng ◽  
Shen Qi-Shun ◽  
Zhu Guo-Long
Keyword(s):  
Critical Radius ◽  
Dipole Polarizability ◽  
Confined System

scholarly journals The Numerical Simulation of the Crack Elastoplastic Extension Based on the Extended Finite Element Method

2013 ◽  
Vol 2013 ◽  
pp. 1-13
Author(s):  
Xia Xiaozhou ◽  
Zhang Qing ◽  
Wang Hong ◽  
Jiang Qun
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Basis Function ◽  
Extended Finite Element Method ◽  
Crack Tip ◽  
Function Form ◽  
Extended Finite Element ◽  
Polar Radius ◽  
Radius Function ◽  
Element Method

In the frame of the extended finite element method, the exponent disconnected function is introduced to reflect the discontinuous characteristic of crack and the crack tip enrichment function which is made of triangular basis function, and the linear polar radius function is adopted to describe the displacement field distribution of elastoplastic crack tip. Where, the linear polar radius function form is chosen to decrease the singularity characteristic induced by the plastic yield zone of crack tip, and the triangle basis function form is adopted to describe the displacement distribution character with the polar angle of crack tip. Based on the displacement model containing the above enrichment displacement function, the increment iterative form of elastoplastic extended finite element method is deduced by virtual work principle. For nonuniform hardening material such as concrete, in order to avoid the nonsymmetry characteristic of stiffness matrix induced by the non-associate flowing of plastic strain, the plastic flowing rule containing cross item based on the least energy dissipation principle is adopted. Finally, some numerical examples show that the elastoplastic X-FEM constructed in this paper is of validity.

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