Chromatic roots are dense in the whole complex plane

I show that the zeros of the chromatic polynomials P-G(q) for the generalized theta graphs Theta((s.p)) are taken together, dense in the whole complex plane with the possible exception of the disc \q - l\ < l. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z(G)(q,upsilon) outside the disc \q + upsilon\ < \upsilon\. An immediate corollary is that the chromatic roots of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha-Kahane-Weiss theorem oil the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof

An elementary proof of Takagi's theorem on the differential composition of polynomials

I give a short and completely elementary proof of Takagi's 1921 theorem on the zeros of a composite polynomial f(d/dz)g(z)

How to generalize (and not to generalize) the Chu-Vandermonde identity

We consider two different interpretations of the Chu--Vandermonde identity: as an identity for polynomials, and as an identity for infinite matrices. Each interpretation leads to a class of possible generalizations, and in both cases we obtain a complete characterization of the solutions

The multivariate Tutte polynomial (alias Potts model) for graphs and matroids

The multivariate Tutte polynomial (known to physicists as the Potts-model partition function) can be defined on an arbitrary finite graph G, or more generally on an arbitrary matroid M, and encodes much important combinatorial information about the graph (indeed, in the matroid case it encodes the full structure of the matroid). It contains as a special case the familiar two-variable Tutte polynomial -- and therefore also its one-variable specializations such as the chromatic polynomial, the flow polynomial and the reliability polynomial -- but is considerably more flexible. I begin by giving an introduction to all these problems, stressing the advantages of working with the multivariate version. I then discuss some questions concerning the complex zeros of the multivariate Tutte polynomial, along with their physical interpretations in statistical mechanics (in connection with the Yang--Lee approach to phase transitions) and electrical circuit theory. Along the way I mention numerous open problems. This survey is intended to be understandable to mathematicians with no prior knowledge of physics

Wall's continued-fraction characterization of Hausdorff moment sequences: A conceptual proof

I give an elementary proof of Wall's continued-fraction characterization of Hausdorff moment sequences

Linear Bound in Terms of Maxmaxflow for the Chromatic Roots of Series-Parallel Graphs

We prove that the (real or complex) chromatic roots of a series-parallel graph with maxmaxflow $\Lambda$ lie in the disc $|q-1| < (\Lambda-1)/\log 2$. More generally, the same bound holds for the (real or complex) roots of the multivariate Tutte polynomial when the edge weights lie in the “real antiferromagnetic regime” $-1 \le v_e \le 0$. For each $\Lambda \geq 3$, we exhibit a family of graphs, namely, the “leaf-joined trees”, with maxmaxflow $\Lambda$ and chromatic roots accumulating densely on the circle $|q-1|=\Lambda -1$, thereby showing that our result is within a factor $1/\log 2 \approx 1.442695$ of being sharp

Lattice paths and branched continued fractions II. Multivariate Lah polynomials and Lah symmetric functions

We introduce the generic Lah polynomials Ln,k(ϕ), which enumerate unordered forests of increasing ordered trees with a weight ϕi for each vertex with i children. We show that, if the weight sequence ϕ is Toeplitz-totally positive, then the triangular array of generic Lah polynomials is totally positive and the sequence of row-generating polynomials Ln(ϕ,y) is coefficientwise Hankel-totally positive. Upon specialization we obtain results for the Lah symmetric functions and multivariate Lah polynomials of positive and negative type. The multivariate Lah polynomials of positive type are also given by a branched continued fraction. Our proofs use mainly the method of production matrices; the production matrix is obtained by a bijection from ordered forests of increasing ordered trees to labeled partial Łukasiewicz paths. We also give a second proof of the continued fraction using the Euler–Gauss recurrence method

The Complex Dynamics of Wishful Thinking The Critical Positivity Ratio

We examine critically the claims made by Fredrickson and Losada (2005) concerning the construct known as the “positivity ratio.” We find no theoretical or empirical justification for the use of differential equations drawn from fluid dynamics, a subfield of physics, to describe changes in human emotions over time; furthermore, we demonstrate that the purported application of these equations contains numerous fundamental conceptual and mathematical errors. The lack of relevance of these equations and their incorrect application lead us to conclude that Fredrickson and Losada’s claim to have demonstrated the existence of a critical minimum positivity ratio of 2.9013 is entirely unfounded. More generally, we urge future researchers to exercise caution in the use of advanced mathematical tools, such as nonlinear dynamics, and in particular to verify that the elementary conditions for their valid application have been met

Spanning forests and OSP(N|2M) -invariant σ-models

The present paper is part of our ongoing work on OSP(N|2M) supersymmetric σ-models, their relation with the Potts model at q = 0 and spanning forests, and the rigorous analytic continuation of the partition function as an entire function of N - 2M, a feature first predicted by Parisi and Sourlas in the 1970s. Here we accomplish two main steps. First, we analyze in detail the role of the Ising variables that arise when the constraint in the OSP(1|2) model is solved, and we point out two situations in which the Ising and forest variables decouple. Second, we establish the analytic continuation for the OSP(N|2M) model in some special cases: when the underlying graph is a forest, and for the Nienhuis action on a cubic graph. We also make progress in understanding the series-parallel graphs

Positive Psychology and Romantic Scientism

Replies to the comments of Nickerson (see record 2014-36500-010), Guastello (see record 2014-36500-011), Musau (see record 2014-36500-013), Hämäläinen et al. (see record 2014-36500-014), and Lefebvre and Schwartz (see record 2014-36500-015) on the authors article (see record 2013-24609-001). Fredrickson and Losada’s (2005) article was the subject of over 350 scholarly citations before our critique (Brown et al., 2013) appeared, and its principal “conclusions” have been featured in many lectures and public presentations by senior members of the positive psychology research community, although its deficiencies ought to have been visible to anyone with a modest grasp of mathematics and a little curiosity. Unfortunately— because human behavior is, after all, complex and difficult to understand—we have no way of knowing whether the fact that it took so long for these deficiencies to be recognized was due to an unwarranted degree of optimism about the reliability of the peer-review process, a reluctance to make waves in the face of powerful interests, a general lack of critical thinking within positive psychology, or some other factor. We hope that our revelation of the problems with the critical positivity ratio ultimately demonstrates the success of science as a self-correcting endeavor; however, we would have greatly preferred it if our work had not been necessary in the first place. (PsycINFO Database Record (c) 2016 APA, all rights reserved