A unique pair of triangles

A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area. In the proof, we determine the set of rational points on a certain hyperelliptic curve by a standard but sophisticated argument which is based on the 2-descent on its Jacobian variety and Coleman's theory of $p$-adic abelian integrals.Comment: 5 pages, to appear in Journal of Number Theory, Some modifications are added to the article published onlin

A generalization of formulas for the discriminants of quasi-orthogonal polynomials with applications to hypergeometric polynomials

Let $K$ be a field. In this article, we derive a formula for the discriminant of a sequence $\{r_{A,n}+c r_{A,n-1}\}$ of polynomials. Here, $c \in K$ and $\{r_{A,n} \}$ is a sequence of polynomials satisfying a certain recurrence relation that is considered by Ulas or Turaj. There are several works calculating the discriminants of given polynomials. For example, Kaneko--Niiho and Mahlburg--Ono independently proved the formula for the discriminants of certain hypergeometric polynomials that are related to $j$-invariants of supersingular elliptic curves. Sawa--Uchida proved the formula for the discriminants of quasi-Jacobi polynomials. In this article, we present a uniform way to prove a vast generalization of the above formulas. In the proof, we use the formulas for the resultants Res$(r_{A,n},r_{A,n-1})$ by Ulas and Turaj that are generalizations of Schur's classical formula for the resultants.Comment: 17 page

Quadrature formulas for Bessel polynomials

A quadrature formula is a formula computing a definite integration by evaluation at finite points. The existence of certain quadrature formulas for orthogonal polynomials is related to interesting problems such as Waring's problem in number theory and spherical designs in algebraic combinatorics. Sawa and Uchida proved the existence and the non-existence of certain rational quadrature formulas for the weight functions of certain classical orthogonal polynomials. Classical orthogonal polynomials belong to the Askey-scheme, which is a hierarchy of hypergeometric orthogonal polynomials. Thus, it is natural to extend the work of Sawa and Uchida to other polynomials in the Askey-scheme. In this article, we extend the work of Sawa and Uchida to the weight function of the Bessel polynomials. In the proofs, we use the Riesz--Shohat theorem and Newton polygons. It is also of number theoretic interest that proofs of some results are reduced to determining the sets of rational points on elliptic curves.Comment: 14 page

Analyzing the Development of a Remote Debate Program Using Video Annotation through a Systems Approach

This study aims to apply a systems approach to analyze the development of an education system for remote asynchronous debates. Although several studies have applied a systems approach to various objects, they attempted to improve projects using the approach by intention. However, in our study, the use of a systems approach was an unintentional part of the process, with stakeholders not being explicitly aware of the concepts behind systems thinking. In the midst of the process, a variety of systems thinking methods were partially adopted, making the project more systemic; however, the project was not oriented to follow specific problem-solving techniques. An educational program and information system for education support with sufficient systemic properties was created with such a systems approach. We propose that working on a problem with significant underlying systemic properties may help one to naturally adopt a systems approach

Zn-induced wipeout effect on Cu NQR spectra in La2−x_{2-x}Srx_xCu1−y_{1-y}Zny_yO4_4

We report a systematic study of Zn-substitution effect on Cu NQR spectrum for high $T_c$ superconductors La$_{2-x}$Sr$_x$Cu$_{1-y}$Zn$_y$O$_4$ from carrier-underdoped to -overdoped regimes (polycrystalline samples, $x$ =0.10, 0.15, and 0.20). We observed no appreciable wipeout effect for the overdoped samples, a gradual and partial wipeout effect below about 80 K for the optimally doped ones, and very abrupt and full wipeout effect below about 40 K for the underdoped ones. The wipeout effect indicates a highly enhanced spectral weight of Cu spin fluctuations at a low frequency. We associate the wipeout effect with a Zn-induced local magnetism far above 40 K and with a localization effect below 40 K.Comment: 2 pages, 3 figures, accepted for publication in Physica C (LT23, Hiroshima 2002

IL-1β Suppresses the Formation of Osteoclasts by Increasing OPG Production via an Autocrine Mechanism Involving Celecoxib-Related Prostaglandins in Chondrocytes

Elevated interleukin (IL)-1 concentrations in synovial fluid have been implicated in joint bone and cartilage destruction. Previously, we showed that IL-1β stimulated the expression of prostaglandin (PG) receptor EP4 via increased PGE2 production. However, the effect of IL-1β on osteoclast formation via chondrocytes is unclear. Therefore, we examined the effect of IL-1β and/or celecoxib on the expression of macrophage colony-stimulating factor (M-CSF), receptor activator of NF-κB ligand (RANKL), and osteoprotegerin (OPG) in human chondrocytes, and the indirect effect of IL-1β on osteoclast-like cell formation using RAW264.7 cells. OPG and RANKL expression increased with IL-1β; whereas M-CSF expression decreased. Celecoxib blocked the stimulatory effect of IL-1β. Conditioned medium from IL-1β-treated chondrocytes decreased TRAP staining in RAW264.7 cells. These results suggest that IL-1β suppresses the formation of osteoclast-like cells via increased OPG production and decreased M-CSF production in chondrocytes, and OPG production may increase through an autocrine mechanism involving celecoxib-related PGs

Effectiveness of Dance/Movement Therapy Intervention for Children with Intellectual Disability at an Early Childhood Special Education Preschool

Children with intellectual disability (ID) often have deficits in gross motor skills and static and dynamic balance abilities, poor lower muscle strength, and an increased risk of serious falls. They also face difficulty in continuing physical activity programs due to cognitive impairment and easy loss of motivation. However, dance/movement therapy (DMT) has been found to help children with ID perform static and dynamic movements. This study aimed to assess the effectiveness of DMT group sessions for children with ID as part of an early childhood special education preschool program. The outcome measures involved employing a hand-held dynamometer to assess knee extensor muscles, the one-leg stand test for static balance, and the timed “up and go” test for dynamic balance, and administering the Child Behavior Checklist and Caregiver-Teacher Report Form for children’s adaptive functions and behavioral problems, as reported by parents or relatives and teachers respectively. Twenty-one children with ID aged 36 to 72 months participated in the study. Ten 60-min DMT group sessions were conducted as manualized intervention, once a week. The measurements were done before and after the 10 DMT group session, and then compared. The results showed statistically significant changes in both knee extensor muscles, the standing time for both legs in the one-leg stand test, attention problems and affective problems in the Checklist, and total score, internalizing problems (including emotionally reactive and somatic complaints), externalizing problems (including attention problems and aggressive behavior), affective problems, anxiety problems, and attention deficit/hyperactivity problems in the Report Form. This study found that the DMT group sessions as part of an early childhood special education preschool program for children with ID aged 36 to 72 months helped improve their knee extensor muscles and static balance while reducing maladaptive behaviors, enabling them to enjoy the sessions for the full study period