A novel approach to an old symmetry problem is developed. A proof is given for the following symmetry problem, studied earlier: if ∆u = 1 in D ⊂ R3, u = 0 on S, the boundary of D, and uN = const on S, then S is a sphere. It is assumed that S is a Lipschitz surface homeomorphic to a sphere. This result has been proved in different ways by various authors. Our proof is based on a simple new idea....

We study the symmetry of divisor sums functions σ−s(n) def = ∑ d|n d −s (for σ = Re(s) > 0) in almost all short intervals; by elementary methods (based on the Large Sieve) we give an exact asymptotic estimate for the mean-square (over N < x ≤ 2N) of their “symmetry sum” ∑ |n−x|≤h sgn(n− x)σ−s(n) (here sgn(0) = 0 and sgn(t) def = t/|t|, for t 6= 0).

abstract different definitions have been proposed for soil quality. various soil properties have been suggetsted as soil quality indices by researchers. this research was conducted to study the relationship between slope of retention curve at its inflection point (s index) and other soil physical quality indices, because the reserches between s index and soil physical properties are little. thi...

The prevalence of three symmetries, reverse, complement, and inverse (or reverse-complement), in about 800 complete genomic sequences extant in GenBank is studied. Each symmetry is measured by an index, χ, where χ=1 indicates lack of symmetry and χ=0, perfect symmetry. All random sequences have χ∼1. For reverse and complement symmetries it is found that χ∼1 in all complete sequences. In sharp c...

in this study, species richness, evenness and vegetation biodiversity were investigated in stabilized and non-stabilized cut edge areas of chachkam district forest roads in sari forest of iran. 18 micro plots 1×1m (36 micro plots) were sampled to carry out this research on two cut edge areas. number of trees, shrubs and herbaceous were recorded in each micro-plot. significant difference was inv...

We study a wide class of free-fermion systems on a hypercubic lattice in spatial dimensions d ≥ 1. When the system is insulating, the topological properties, e.g., the integral quantization of the topological invariant, are protected by certain symmetries of the Hamiltonian against disorder. This generic feature is characterized by a generalized index theorem which is a noncommutative analogue ...

Quantitative measurement of limb loading is important in orthopedic and neurological rehabilitation. In current practice, mathematical models such as Symmetry index (SI), Symmetry ratio (SR), and Symmetry angle (SA) are used to quantify limb loading asymmetry. Literatures have identified certain limitations with the above mathematical models. Hence this study presents two new mathematical model...

the estimate of entropy (sample entropy), has been introduced by vasicek (1976), for the first time. in this paper, we provide an estimate of entropy of order statistics, that is the extention of the entropy estimate. then we present an application of the entropy estimate of order statistics as a test statistic for symmetry of distribution versus skewness. the proposed test has been compared wi...