Young Formula Research Articles

The Equivariant Inverse Kazhdan--Lusztig Polynomials of Uniform Matroids

Motivated by the concepts of the inverse Kazhdan--Lusztig polynomial and the equivariant Kazhdan--Lusztig polynomial, Proudfoot defined the equivariant inverse Kazhdan--Lusztig polynomial for a matroid. In this paper, we show that the equivariant inverse Kazhdan--Lusztig polynomial of a matroid is very useful for determining its equivariant Kazhdan--Lusztig polynomials, and we determine the equivariant inverse Kazhdan--Lusztig polynomials for Boolean matroids and uniform matroids. As an application, we give a new proof of Gedeon, Proudfoot, and Young's formula for the equivariant Kazhdan--Lusztig polynomials of uniform matroids. Inspired by Lee, Nasr, and Radcliffe's combinatorial interpretation for the ordinary Kazhdan--Lusztig polynomials of uniform matroids, we further present a new formula for the corresponding equivariant Kazhdan--Lusztig polynomials.

The Kazhdan-Lusztig polynomials of uniform matroids

The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias et al. (2016) [4]. Let Um,d denote the uniform matroid of rank d on a set of m+d elements. Gedeon et al. (2017) [7] pointed out that they can derive an explicit formula of the Kazhdan-Lusztig polynomials of Um,d using equivariant Kazhdan-Lusztig polynomials. In this paper we give an alternative explicit formula, which allows us to prove the real-rootedness of the Kazhdan-Lusztig polynomials of Um,d for 2≤m≤15 and all d's. The case m=1 was previously proved by Gedeon et al. (2017) [8]. We further determine the Z-polynomials of all Um,d's and prove the real-rootedness of the Z-polynomials of Um,d for 2≤m≤15 and all d's. Our formula also enables us to give an alternative proof of Gedeon, Proudfoot, and Young's formula for the Kazhdan-Lusztig polynomials of Um,d's without using the equivariant Kazhdan-Lusztig polynomials.

Effects of mold surface conditions on flow length in injection molding process

AbstractIn this study, to clarify the influence of a mold on thin‐wall molding, the effect of different mold surface conditions on the flow length and mobility (i.e., ease with, which melted plastics can be filled into the mold) in an injection molding process was investigated. Three different coatings were used for the mold surface. Several degrees of roughness were also selected for the mold surface. The results were evaluated by comparing flow length with interfacial tensions, which were derived from Young's formula. Although the interfacial tension exhibited different values, the influence on flow length was generally found to be small. On the other hand, in the mold that gives surface roughness, though the change of interfacial tension was small compared with coatings, the flow length increased linearly with the surface roughness when the roughness exceeded a certain level roughness. POLYM. ENG. SCI., 2011. © 2011 Society of Plastics Engineers

Aerodynamics Design and Genetic Algorithms for Optimization of Airship Bodies

A special and effective aerodynamics calculation method has been applied for the flow field around a body of revolution to find the drag coefficient for a wide range of Reynolds numbers. The body profile is described by a first order continuous axial singularity distribution. The solution of the direct problem then gives the radius and inviscid velocity distribution. Viscous effects are considered by means of an integral boundary layer procedure, and for determination of the transition location the forced transition criterion is applied. By avoiding those profiles, which result in the separation of the boundary layer, the drag can be calculated at the end of the body by using Young's formula. In this study, a powerful optimization procedure known as a Genetic Algorithms (GA) is used for the first time in the shape optimization of airship hulls. GA represents a particular artificial intelligence technique for large spaces, striking a remarkable balance between exploration and exploitation of search space. This method could reach to minimum objective function through a better path, and also could minimize the drag coefficient faster for different Reynolds number regimes. It was found that GA is a powerful method for such multi-dimensional, multi-modal and nonlinear objective function.

Pointwise dimensions for Poincaré recurrences associated with maps and special flows

We introduce pointwise dimensions and spectra associated with Poincaré recurrences. These quantities are then calculated for any ergodic measure of positive entropy on a weakly specified subshift. We show that they satisfy a relation comparable to Young's formula for the Hausdorff dimension of measures invariant under surface diffeomorphisms. A key-result in establishing these formula is to prove that the Poincaré recurrence for a 'typical' cylinder is asymptotically its length. Examples are provided which show that this is not true for some systems with zero entropy. Similar results are obtained for special flows and we get a formula relating spectra for measures of the base to the ones of the flow.