In this paper we provide exact formula for the commute times of random walks on spherically symmetric random trees. Using this formula we sharpen some of the results presented in Al-Awadhi et al to the form of equalities rather than inequalities.

Worldwide energy consumption is generally related with fossil fuels that increase CO2 and other greenhouse gases (GHG) emissions. The transport sector represents a significant part of energy demand. As its share comes mostly from petroleum products, known for their highly polluting effects, there is the need to quantify energy use by transport. This assessment supports the planning and implemen...

Sphingolipid imbalance can take out an eye Y onamine et al. report that the relative amounts of two sphingolipids control the breakdown of light-sensing proteins and determine whether eye cells survive. Sphingolipids aren’t just building blocks of membranes. Some of the molecules, such as sphingosine 1 phosphate (S1P) and dihydrosphingosine 1 phosphate (DHS1P), take part in signaling pathways t...

A standard result in quantum mechanics is this: if two observables are commuting then they have a classical joint distribution in every state. A converse is demonstrated here: If a classical joint distribution for the pair agrees with standard quantum facts, then the observables must commute. This has consequences for some historical and recent quantum nonlocal models: they are analytically dis...

It it shown that geometric morphisms between elementary toposes can be represented as adjunctions between the corresponding categories of locales. These adjunctions are characterised as those that preserve the order enrichment, commute with the double power locale monad and whose right adjoints preserve finite coproduct. They are also characterised as those adjunctions that preserve the order e...

We introduce and study k-commuting permutations. One of our main results is a characterization of permutations that k-commute with a given permutation. Using this characterization, we obtain formulas for the number of permutations that k-commute with a permutation [Formula: see text], for some cycle types of [Formula: see text]. Our enumerative results are related with integer sequences in "The...

We give a simple condition on a matrix for which if the exponential matrix is diagonal, lower or upper triangular, then so is . It is also shown that for diagonalizable and any matrix , and commute if and only if and commute. These results are useful in problems in which knowledge about has to be extracted from structural information about its exponential, such as in large-scale sampled-data sy...

From the classical differential equation of Jacobi fields, one naturally defines the Jacobi operator of a Riemannian manifold with respect to any tangent vector. A straightforward computation shows that any real, complex and quaternionic space forms satisfy that any two Jacobi operators commute. In this way, we classify the real hypersurfaces in quaternionic projective spaces all of whose tange...

Let F be an algebraically closed field of characteristic 0 and f(x) a polynomial of degree strictly greater than one in F [x]. We show that the number of degree k polynomials with coefficients in F that commute with f (under composition) is either zero or equal to the number of degree one polynomials with coefficients in F that commute with f . As a corollary, we obtain a theorem of E. A. Bertr...

This behavior has been seen in some notable cases. Kirillov [3] shows that elementary symmetric polynomials in noncommuting variables commute (and, in some cases, all Schur functions) when elementary symmetric polynomials of degree at most three commute when restricted to at most three of the variables. Generalizing this, Blasiak and Fomin [1] give a wider theory for rules of three of generatin...