Kruskal--Katona-Type Problems via Entropy Method

In this paper, we investigate several extremal combinatorics problems that ask for the maximum number of copies of a fixed subgraph given the number of edges. We call this type of problems Kruskal--Katona-type problems. Most of the problems that will be discussed in this paper are related to the joints problem. There are two main results in this paper. First, we prove that, in a $3$-colored graph with $R$ red, $G$ green, $B$ blue edges, the number of rainbow triangles is at most $\sqrt{2RGB}$, which is sharp. Second, we give a generalization of the Kruskal--Katona theorem that implies many other previous generalizations. Both arguments use the entropy method, and the main innovation lies in a more clever argument that improves bounds given by Shearer's inequality.Comment: 18 page

Tight Bound and Structural Theorem for Joints

A joint of a set of lines $\mathcal{L}$ in $\mathbb{F}^d$ is a point that is contained in $d$ lines with linearly independent directions. The joints problem asks for the maximum number of joints that are formed by $L$ lines. Guth and Katz showed that the number of joints is at most $O(L^{3/2})$ in $\mathbb{R}^3$ using polynomial method. This upper bound is met by the construction given by taking the joints and the lines to be all the $d$-wise intersections and all the $(d-1)$-wise intersections of $M$ hyperplanes in general position. Furthermore, this construction is conjectured to be optimal. In this paper, we verify the conjecture and show that this is the only optimal construction by using a more sophisticated polynomial method argument. This is the first tight bound and structural theorem obtained using this method. We also give a new definition of multiplicity that strengthens the main result of a previous work by Tidor, Zhao and the second author. Lastly, we include some discussion on the constants for the joints of varieties problem.Comment: 39 page

Effect of Auricular Acupressure on Peri- and Early Postmenopausal Women with Anxiety: A Double-Blinded, Randomized, and Controlled Pilot Study

We tested effects of auricular acupressure on peri- and early postmenopausal women with anxiety (PPWA). Fifty PPWA were randomly assigned to the auricular acupressure group (AG) or the sham group (SG). After 3 meals and before sleep every day for 4 weeks, the AG received auricular acupressure on the bilateral ear shenmen and subcortex points for 3 min per point on alternating ears. The SG received sham auricular acupressure. The Alprazolam was reduced from 0.5 mg/day at baseline to 0.3 mg/day 4 weeks after auricular acupressure (4 W) in the AG (P < .05) whereas maintained at 0.5 mg/day in the SG (P > .05). The Zolpidem was reduced from 3.0 mg/day at baseline to 1.5 mg/day at 4 W (P < .05) whereas was reduced from 2.4 mg/day to 1.9 mg/day at 4 W in the SG (P > .05), thus, significant tapering medication, suggesting auricular acupressure is helpful to PPWA

Foam Properties and Detergent Abilities of the Saponins from Camellia oleifera

The defatted seed meal of Camellia oleifera has been used as a natural detergent and its extract is commercially utilized as a foam-stabilizing and emulsifying agent. The goal of this study was to investigate the foam properties and detergent ability of the saponins from the defatted seed meal of C. oleifera. The crude saponin content in the defatted seed meal of C. oleifera was 8.34 and the total saponins content in the crude saponins extract was 39.5% (w/w). The foaming power of the 0.5 crude saponins extract solution from defatted seed meal of C. oleifera was 37.1 of 0.5 SLS solution and 51.3% to that of 0.5% Tween 80 solution. The R5 value of 86.0% represents good foam stability of the crude saponins extracted from the defatted seed meal of the plant. With the reduction of water surface tension from 72 mN/m to 50.0 mN/m, the 0.5% crude saponins extract solution has wetting ability. The sebum-removal experiment indicated that the crude saponins extract has moderate detergency. The detergent abilities of the saponins from C. oleifera and Sapindus mukorossi were also compared

Nonlinear dynamic responses of shell structures using vector form intrinsic finite element method

In this paper, in order to compute nonlinear dynamic responses of shell structures, formulations of the internal forces of the shell element in vector form intrinsic finite element (VFIFE) method are developed. This novel shell element is named by VFIFE-DKT element. These elements are to compute internal forces from the deformations and the motion of the shell structures. The VFIFE method is a particle-based method. They have three key VFIFE processes such as the point value description, path element and convected material frame. Thus, the shell structure is represented by finite particles. Each particle is subjected to the external forces and internal forces. The particle satisfies the Newton’s Law. A fictitious reversed rigid body motion is used to remove the rigid body motion from the deformations of the element. The internal forces of the element in deformation coordinates satify the equilibrium equations. Through the numerical examples of the benchmark structures undergo extermly-large displacements, rotation and motion, the proposed procedures using the novel element demonstrates its accuracy and efficiency

Splint therapy for disc displacement with reduction of the temporomandibular joint. Part I: Modified mandibular splint therapy

AbstractThe aims of this preliminary study were to present a modified mandibular splint together with a treatment regimen and to evaluate their effects on the treatment of reciprocal joint sounds of the temporomandibular joint (TMJ). The study participants were recruited from 312 consecutive patients in the temporomandibular disorder clinic of a medical center in Taiwan from January 2003 to December 2003. From among these, 59 cases with typical reciprocal clicking were selected for this study. All participants were treated with a modified mandibular splint and then followed up for 6 months. Successful treatment was defined as leading to the disappearance of the joint sounds of TMJ, as described by patients. Based on clinical evaluation, the overall success rate was 71.2% (42/59) with minimal temporary complications. Patients with clicking at less than 3.5cm of interincisal opening had a success rate of 92.5%, which was higher than the success rate of patients with clicking at a mouth opening of 3.5cm or more. This study showed that a modified mandibular splint can be used to treat reciprocal clicking of the TMJ effectively and encouraged us to conduct further study on the efficacy of this splint to treat disc displacement with reduction of TMJ using magnetic resonance imaging examination