Joint Distribution of Rises and Falls

The marginal distributions of the number of rises and the number of falls have been used successfully in various areas of statistics, especially in non-parametric statistical inference. Carlitz (1972, Duke Math. J. 39, 268–269) showed that the generating function of the joint distribution for the numbers of rises and falls satisfies certain complex combinatorial equations, and pointed out that he had been unable to derive the explicit formula for the joint distribution from these equations. After more than two decades, this latter problem remains unsolved. In this article, the joint distribution is obtained via the probabilistic method of finite Markov chain imbedding for random permutations. A numerical example is provided to illustrate the theoretical results and the corresponding computational procedures.     

Asymptotic convergence of solutions for Laplace reaction–diffusion equations

We study the initial–boundary value problem for a Laplace reaction–diffusion equation. After constructing local solutions by using the theory of abstract degenerate evolution equations of parabolic type, we show asymptotic convergence of bounded global solutions if they exist under the assumption that the reaction function is analytic in neighborhoods of their $\omega$-limit sets. Reduction of degenerate evolution equation to multivalued evolution equation enables us to use the theory of the infinite-dimensional Łojasiewicz–Simon gradient inequality.     

Semiclassical analysis for highly degenerate potentials

This paper characterizes the semi-classical limit of the fundamental energy,

and ground state of the Schrödinger operator in a bounded domain , in the highly degenerate case when and consists of two components, say and . The main result establishes that

and that approximates in the ground state of in if

     

The role of aromatic groups in the Tobey-Simon additivity rule for vinylic protons

The Tobey-Simon (additivity) rule for aromatic groups which was devised about 40 years ago has been found to need revision. The rule shows an aromatic group attached to a C==C double bond as causing a downfield chemical shift of a cis-related vinylic proton and a small upfield shift of a trans-related proton. A search of data in the recent literature has shown that this rule should apply mainly to monosubstituted phenyl groups and some polynuclear aromatics. In contrast with them, 2,6-disubstituted phenyl, 2,4,6-trisubstituted phenyl and 9-anthracenyl groups cause cis-related vinylic protons to resonate upfield of comparable trans-related protons. Further, the current rule for o-substituted phenyl groups has been found to be inaccurate. In writing a rule for aromatic groups, therefore, greater attention needs to be given to the diversity of effects that these groups have on chemical shifts of vinylic protons.     

The Norton–Simon hypothesis and the onset of non-genetic resistance to chemotherapy induced by stochastic fluctuations

By studying a simple but realistic biophysical model of tumor growth in the presence of a constant continuous chemotherapy, we show that if an extended Norton–Simon hypothesis holds, the system may have multiple equilibria. Thus, the stochastic bounded fluctuations that affect both the tumor carrying capacity and/or the drug pharmacodynamics (and/or the drug pharmacokinetics) may cause the transition from a small equilibrium to a far larger one, not compatible with the life of the host. In particular, we mainly investigated the effects of fluctuations that involve parameters nonlinearly affecting the deterministic model. We propose to frame the above phenomena as a new and non-genetic kind of resistance to chemotherapy.     

A deuterium isotope effect on the lifetime of the second triplet state of naphthalene

From a Stern-Volmer kinetic analysis of the naphthalene-T₂ sensitized reaction of endodicyclopentadiene with benzene as the intermediate excitation carrier, it was concluded that deuterium substitution increases the T₂-lifetime of naphthalene. An isotope effect of 1.4 ± 0.3 was observed for per-deuteronaphthalene.     

Sufficient conditions for permutation equivalence to a WHS-matrix

Square matrices with positive leading principal minors, called WHS-matrices (weak Hawkins–Simon), are considered in economics. Some sufficient conditions for a matrix to be a WHS-matrix after suitable row and/or column permutations have recently appeared in the literature. New and unified proofs and generalizations of some results to rectangular matrices are given. In particular, it is shown that if left multiplication of a rectangular matrix A by some nonnegative matrix is upper triangular with positive diagonal, then some row pemutation of A is a WHS-matrix. For a nonsingular A with either the first nonzero entry of each of its rows positive or the last nonzero entry of each column of A ^?1 positive, again some row permutation of A is a WHS-matrix. In addition, any rectangular full rank semipositive matrix is shown to be permutation equivalent to a WHS-matrix.     

Restricted words by adjacencies

A recurrence, a determinant formula, and generating functions are presented for enumerating words with restricted letters by adjacencies. The main theorem leads to refinements (with up to two additional parameters) of known results on compositions, polyominoes, and permutations. Among the examples considered are (1) the introduction of the ascent variation on compositions, (2) the enumeration of directed vertically convex polyominoes by upper descents, area, perimeter, relative height, and column number, (3) a tri-variate extension of MacMahon's determinant formula for permutations with prescribed descent set, and (4) a combinatorial setting for an entire sequence of bibasic Bessel functions.