We define an infinite sequence of generalizations, parametrized by an integer m≥1, of the Stieltjes--Rogers and Thron--Rogers polynomials; they arise as the power-series expansions of some branched continued fractions, and as the generating polynomials for m-Dyck and m-Schröder paths with height-dependent weights. We prove that all of these sequences of polynomials are coefficientwise Hankel-totally positive, jointly in all the (infinitely many) indeterminates. We then apply this theory to prove the coefficientwise Hankel-total positivity for combinatorially interesting sequences of polynomials.

One of the world's most creative mathematicians offers a new way to look at math--focusing on questions, not answers
Where do we learn math: From rules in a textbook? From logic and deduction? Not really, according to mathematician Eugenia Cheng: we learn it from human curiosity--most importantly, from asking questions. This may come as a surprise to those who think that math is about finding the one right answer, or those who were told that the "dumb" question they asked just proved they were bad at math. But Cheng shows why people who ask questions like "Why does 1 + 1 = 2?" are at the very heart of the search for mathematical truth.
Is Math Real? is a much-needed repudiation of the rigid ways we're taught to do math, and a celebration of the true, curious spirit of the discipline. Written with intelligence and passion, Is Math Real? brings us math as we've never seen it before, revealing how profound insights can emerge from seemingly unlikely sources.  

In this two part work we prove that for every finitely generated subgroup Γ<Out(Fn), either Γ is virtually abelian or H2b(Γ;R) contains an embedding of ℓ1. The method uses actions on hyperbolic spaces, for purposes of constructing quasimorphisms. Here in Part I, after presenting the general theory, we focus on the case of infinite lamination subgroups Γ - those for which the set of all attracting laminations of all elements of Γ is infinite - using actions on free splitting complexes of free groups.

In this paper, we study the dynamics of fluids in porous media governed by Darcy's law: the Muskat problem. We consider the setting of two immiscible fluids of different densities and viscosities under the influence of gravity in which one fluid is completely surrounded by the other.

Part 4 is devoted to hydrodynamic stability theory which aims at predicting the conditions under which the laminar state of a flow turns into a turbulent state. The phenomenon of laminar-turbulent transition remains one of the main challenges of modern physics. The resolution of this problem is important not only from a theoretical viewpoint but also for practical applications. For instance, in the flow past a passenger aircraft wing, the laminar-turbulent transition causes a fivefold increase in the viscous drag.

Let p be a prime. In this paper we investigate finite K{2,p}-groups G which

have a subgroup H ≤ G such that K ≤ H = NG(K) ≤ Aut(K) for K a simple

group of Lie type in characteristic p, and |G : H| is coprime to p. If G is of local

characteristic p, then G is called almost of Lie type in characteristic p. Here G is

of local characteristic p means that for all nontrivial p-subgroups P of G, and Q

the largest normal p-subgroup in NG(P) we have the containment CG(Q) ≤ Q.

We determine details of the structure of groups which are almost of Lie type in

characteristic p. In particular, in the case that the rank of K is at least 3 we prove

that G = H. If H has rank 2 and K is not PSL3(p) we determine all the examples

where G = H. We further investigate the situation above in which G is of parabolic

characteristic p. This is a weaker assumption than local characteristic p. In this

case, especially when p ∈ {2, 3}, many more examples appear.

We develop a unified theory of Eulerian spaces by combining the combinatorial theory of infinite, locally finite Eulerian graphs as introduced by Diestel and Kühn with the topological theory of Eulerian continua defined as irreducible images of the circle, as proposed by Bula, Nikiel and Tymchatyn.
First, we clarify the notion of an Eulerian space and establish that all competing definitions in the literature are in fact equivalent. Next, responding to an unsolved problem of Treybig and Ward from 1981, we formulate a combinatorial conjecture for characterising the Eulerian spaces, in a manner that naturally extends the characterisation for finite Eulerian graphs. Finally, we present far-reaching results in support of our conjecture which together subsume and extend all known results about the Eulerianity of infinite graphs and continua to date. In particular, we characterise all one-dimensional Eulerian spaces.

This monograph resolves - in a dense class of cases - several open problems concerning geodesics in i.i.d. first-passage percolation on Zd. Our primary interest is in the empirical measures of edge-weights observed along geodesics from 0 to nξ, where ξ is a fixed unit vector. For various dense families of edge-weight distributions, we prove that these empirical measures converge weakly to a deterministic limit as n→∞, answering a question of Hoffman.

I present a proof of Kirchberg's classification theorem: two separable, nuclear, O∞-stable C∗-algebras are stably isomorphic if and only if they are ideal-related KK-equivalent. In particular, this provides a more elementary proof of the Kirchberg--Phillips theorem which is isolated in the paper to increase readability of this important special case.

Émile Borel, one the early developers of measure theory and probability, was among the first to show the importance of the calculus of probability as a tool for the experimental sciences. A prolific and gifted researcher, his scientific works, so wast in number and scope, earned him international recognition. In addition, at the origin of the foundation of the Intitut Henri Poincaré in Paris and longtime its director, he also served as member of the French Parliament, minister of the Navy, president of the League of Nations Union, and president of the French Academy of Sciences