Variation of Calderón–Zygmund operators with matrix weight

Let [Formula: see text], [Formula: see text] and [Formula: see text] be a matrix [Formula: see text] weight. In this paper, we introduce a version of variation [Formula: see text] for matrix Calderón–Zygmund operators with modulus of continuity satisfying the Dini condition. We then obtain the [Formula: see text]-boundedness of [Formula: see text] with norm [Formula: see text] by first proving a sparse domination of the variation of the scalar Calderón–Zygmund operator, and then providing a convex body sparse domination of the variation of the matrix Calderón–Zygmund operator. The key step here is a weak type estimate of a local grand maximal truncated operator with respect to the scalar Calderón–Zygmund operator.

Functional Geominimal Surface Area and Its Related Affine Isoperimetric Inequality

The first variation of the total mass of log-concave functions was studied by Colesanti and Fragalà, which includes the Lp mixed volume of convex bodies. Using Colesanti and Fragalà’s first variation formula, we define the geominimal surface area for log-concave functions, and its related affine isoperimetric inequality is also established.

f-Minimal Lagrangian Submanifolds in Kähler Manifolds with Real Holomorphy Potentials

The aim of this paper is to study variational properties for $f$-minimal Lagrangian submanifolds in Kähler manifolds with real holomorphy potentials. Examples of submanifolds of this kind including minimal Lagrangians and soliton solutions for Lagrangian mean curvature flow (LMCF). We derive 2nd variation formula for $f$-minimal Lagrangians as a generalization of Chen and Oh’s formula for minimal Lagrangians. As a corollary, we obtain stability of expanding and translating solitons for LMCF. We also define calibrated submanifolds with respect to $f$-volume in gradient steady Kähler–Ricci solitons as generalizations of special Lagrangians and translating solitons for LMCF and show that these submanifolds are necessarily noncompact. As a special case, we study the exact deformation vector fields on Lagrangian translators. Finally we discuss some generalizations and related problems.

3D sharp-boundary inversion of potential-field data with an adjustable exponential stabilizing functional

Potential-field sharp-boundary inversion will allow us to identify the sharp petrophysical deposit boundaries inside the host rocks. With the purpose to find a more simple and convenient way to achieve a sharp-boundary and well-focused image for 3D focusing inversion, we analyze and discuss the influence of the focusing parameter in several commonly used stabilizing functionals, such as minimum support (MS), minimum gradient support (MGS), modified total variation ([Formula: see text]TV) stabilizing functional, etc. Then, we evaluate an adjustable exponential stabilizing (AES) functional, the focus of which is adjustable by the base and the exponent of the exponential functional, and we apply it to 3D density and magnetic susceptibility inversion for synthetic and field data. Compared with MS, MGS, [Formula: see text]TV, and other stabilizers, the proposed stabilizer can generate well-focused images and provide sharp boundaries. We also determine that sharp-boundary images produced by the proposed AES stabilizer have a weak dependence on the focusing parameter. Furthermore, we can obtain a stable and focused result by controlling the focusing parameter in the exponent of AES adaptively increasing with the iterations. The model tests and inversion of field data verify the flexibility and stability of this adjustable stabilizer.

The study of the entropy generation in a thin film flow with variable fluid properties past over a stretching sheet

This research inspects the liquid film flow of the nanofluid in a permeable medium with the consequence of thermal radiation over a stretching sheet. The viscidness and thermal conduction of the nanofluid varies with temperature in such a manner that the thermal conductivity considered in direct relation while the viscosity considered inversely proportional to the temperature field. The invariable magnetic field applies vertically to the flow field in the existence of entropy generation. For the above-mentioned nanofluid study, Buongiorno’s model is used. The leading equations are changed into a set of third- and second-order nonlinear coupled differential equations. These nonlinear ordinary differential equations are solved using the optimal approach of homotopy analysis method. The physical appearance of the modelled parameters based on the liquid film thickness is mainly focused. Furthermore, the influence of embedded parameters like variable viscosity parameter [Formula: see text] Prandtl number [Formula: see text] Schmidt number [Formula: see text] Brinkman number [Formula: see text] Brownian motion constraint [Formula: see text] thermophoresis constraint [Formula: see text] magnetic parameter [Formula: see text] thermal radiation parameter [Formula: see text] Reynolds number [Formula: see text] diffusion coefficient [Formula: see text] non-dimension temperature variation [Formula: see text] and non-dimension concentration variation [Formula: see text] is observed on the velocity pitch, temperature gradient and concentration sketch. The consequence of parameters due to entropy generation and Bejan number has also been observed in this work. The important physically quantities of skin friction coefficient, the local Nusselt number and Sherwood number have also been studied. Residual error and optimal values have been calculated for the range of each physical parameter. The present work is compared with the published work and the comparison has been shown physically and numerically.

Fabrication of Millimeters-Sized Poly(Divinylbenzene) Foam Shells from Controllable Double Emulsion in Microfluidic Device

A geometrically confined dripping was employed to enable precise control over the dimension and structure of millimeters-sized double-emulsion precursors of poly(divinylbenzene) foam shells in a new kind of double Y-shaped compound channels. Due to the 3D axial-symmetric microfluidic device, a more stable and robust flow field was maintained to obtain a continuous and regular emulsification. Various factors were systematically investigated for the precise size control of dripping in confined channel geometry, such as outlet channel size, fluid properties and flow rates. It was seen that phase properties and synergistic effects of main factors played key roles in determining droplet size. Thus, we used the optimized microfluidic approach to fabricate predetermined size foams to satisfy inertial fusion energy experiments, ranging from 4 to 4.6[Formula: see text]mm in diameter with a 50–300[Formula: see text][Formula: see text]m wall thickness and a coefficient of variation [Formula: see text]%. The results presented in this work provided a practical guideline for creating size-desired polymersome from comparable double emulsions.

Willmore-Like Tori in Killing Submersions

The first variation formula and Euler-Lagrange equations for Willmore-like surfaces in Riemannian 3-spaces with potential are computed and, then, applied to the study of invariant Willmore-like tori with invariant potential in the total space of a Killing submersion. A connection with generalized elastica in the base surface of the Killing submersion is found, which is exploited to analyze Willmore tori in Killing submersions and to construct foliations of Killing submersions made up of Willmore tori with constant mean curvature.