Pancharatnam geometric phase originating from successive partial projections

The spin of a polarized neutron beam subjected to a partial projection in another direction, traces a geodesic arc in the 2-sphere ray space. We delineate the geometric phase resulting from two successive partial projections on a general quantal state and derive the direction and strength of the third partial projection that would close the geodesic triangle. The constraint for the three successive partial projections to be identically equivalent to a net spin rotation regardless of the initial state, is derived.      

Characterizations by Automorphism Groups of Some Rank 3 Buildings — II: A Half Strongly-Transitive Locally Finite Triangle Building is a Bruhat—Tits Building

We complete the proof of the fact that every locally finite triangle building with a half strongly-transitive automorphism group G (e.g., this happens when is defined via a (B, N)-pair in G) is a Bruhat—Tits building associated with a classical linear group over a locally finite local skewfield.     

On permutation representations of polyhedral groups

We answer affirmatively the following question of Derek Holt: Given integers , can one, in a simple manner, find a finite set and permutations such that has order , has order and has order ? The method of proof enables us to prove more general results (Theorems 2 and 3).

     

Packing similar triangles into a triangle

There exists a triangle T and a number \frac{1}{2}$$ " align="middle" border="0"> such that any sequence of triangles similar to T with total area not greater than times the area of T can be packed into T.     

On smallest triangles

Pick n points independently at random in ?², according to a prescribed probability measure μ, and let Δ ≤ Δ ≤ … be the areas of the () triangles thus formed, in nondecreasing order. If μ is absolutely continuous with respect to Lebesgue measure, then, under weak conditions, the set {n³Δ : i ≥ 1} converges as n → ∞ to a Poisson process with a constant intensity κ(μ). This result, and related conclusions, are proved using standard arguments of Poisson approximation, and may be extended to functionals more general than the area of a triangle. It is proved in addition that if μ is the uniform probability measure on the region S, then κ(μ) ≤ 2/|S|, where |S| denotes the area of S. Equality holds in that κ(μ) = 2/|S| if S is convex, and essentially only then. This work generalizes and extends considerably the conclusions of a recent paper of Jiang, Li, and Vitányi. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 206–223, 2003     

Application of binomial coefficients in representing central difference solution to a class of PDE arising in chemistry

The use of differential equations for modeling chemical systems and solving by numerical approaches (e.g. finite difference methods) are prevalent in chemistry-related problems. As an extension to the direct use of Pascal’s Triangle to obtain the forward and backward difference equations to partial differentials by Lim [Mathematical Medley 31 (2004) 2], this paper proposes the use of binomial coefficient to generate central difference equations to odd-ordered partial differentials in a single-step operation. All finite difference equations to partial differentials shown herein display finite series of palindromic coefficients with alternating signs     

图的结合数猜想的新结果

［１］中Ｗｏｏｄａｌ猜想：若图Ｇ的结合数ｂｉｎｄ（Ｇ）≥３２,则图Ｇ包含三角形,本文证明：若ｂｉｎｄ（Ｇ）≥７＋√６９１０,则图Ｇ包含三角形,从而进一步改进了［２］的结果     

任意三角形平板刚体的转动惯量与惯量主轴

用一种简单的方法计算了均质的任意三角形平板刚体的转动惯量,并由此结果给出一般三角形的惯量主轴.     

等腰直角三角形上混合边界条件的本征函数

假设等腰直角三角形上满足混合边界条件的本征函数是8个平面波函数的线性叠加,由边界条件,确定了8个待定系数.