How many matrices can be spectrally balanced simultaneously?

We prove that any ? positive definite d × d matrices, M₁,...,M_?, of full rank, can be simultaneously spectrally balanced in the following sense: for any k < d such that ? ≤ \(\ell \leqslant \left\lfloor {\frac{{d - 1}}{{k - 1}}} \right\rfloor \), there exists a matrix A satisfying \(\frac{{{\lambda _1}\left( {{A^T}{M_i}A} \right)}}{{Tr\left( {{A^T}{M_i}A} \right)}} < \frac{1}{k}\) 1/k for all i, where λ₁(M) denotes the largest eigenvalue of a matrix M. This answers a question posed by Peres, Popov and Sousi ([PPS13]) and completes the picture described in that paper regarding sufficient conditions for transience of self-interacting random walks. Furthermore, in some cases we give quantitative bounds on the transience of such walks.     

How many atoms can be defined by boxes?

We study the functionb(n, d), the maximal number of atoms defined byn d-dimensional boxes, i.e. parallelopipeds in thed-dimensional Euclidean space with sides parallel to the coordinate axes. We characterize extremal interval families definingb(n, 1)=2n-1 atoms and we show thatb(n, 2)=2n ²-6n+7. We prove that for everyd, exists and . Moreover, we obtainb*(3)=8/9.     

How many quasiplatonic surfaces?

We show that the number of isomorphism classes of quasiplatonic Riemann surfaces of genus ≦ g has a growth of type The number of non-isomorphic regular dessins of genus ≦ g has the same growth type. Received: 8 February 2005; revised: 2 May 2005     

How many consumer levels can survive in a chemotactic food chain?

We investigate the effect and the impact of predator-prey interactions, diffusivity and chemotaxis on the ability of survival of multiple consumer levels in a predator-prey microbial food chain. We aim at answering the question of how many consumer levels can survive from a dynamical system point of view. To solve this standing issue on food-chain length, first we construct a chemotactic food chain model. A priori bounds of the steady state populations are obtained. Then under certain sufficient conditions combining the effect of conversion efficiency, diffusivity and chemotaxis parameters, we derive the co-survival of all consumer levels, thus obtaining the food chain length of our model. Numerical simulations not only confirm our theoretical results, but also demonstrate the impact of conversion efficiency, diffusivity and chemotaxis behavior on the survival and stability of various consumer levels.      

How many T‐tessellations on k lines? Existence of associated Gibbs measures on bounded convex domains

The paper bounds the number of tessellations with T‐shaped vertices on a fixed set of k lines: tessellations are efficiently encoded, and algorithms retrieve them, proving injectivity. This yields existence of a completely random T‐tessellation, as defined by Kiêu et al. (Spat Stat 6 (2013) 118–138), and of its Gibbsian modifications. The combinatorial bound is sharp, but likely pessimistic in typical cases. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 561–587, 2015     

How many different cascades on a surface can have coinciding hyperbolic attractors?

It is shown that the number of essentially nonconjugate (i.e., not being iterations of topologically conjugate) diffeomorphisms of a surface having homeomorphic one-dimensional hyperbolic attractors can be arbitrarily large, provided that the genus of the surface is large enough. A lower bound for this number depending on the surface genus is given. The corresponding result for pseudo-Anosov homeomorphisms is stated.     

How many graphs are unions of k‐cliques?

We study the number of n‐vertex graphs that can be written as the edge‐union of k‐vertex cliques. We obtain reasonably tight estimates for in the cases (i) k = n ? o(n) and (ii) k = o(n) but . We also show that exhibits a phase transition around . We leave open several potentially interesting cases, and raise some other questions of a similar nature. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 87–107, 2006     

How many lottery tickets to buy?

We investigate the optimal number of tickets an expected-utility-maximizing individual who participate in a lottery will buy. We show that the expected utility is not always unimodal in the number of tickets. We also show that a risk-averse individual will not buy more than one ticket. (S)he will be willing to own a share in several tickets only if that share is sufficiently small.     

How to find many counterfeit coins?

We propose an algorithm for findingm defective coins, that uses at most + 15m weighings on a balance scale, wheren is the number of all coins.     

How many diffusions exist on the Vicsek snowflake?

We consider Kusuoka's construction of Dirichlet forms on the Vicsek snowflake. a nested fractal. His method is generalized and all irreducible, local Dirichlet forms which can be constructed in this way are characterized. We end up with a one-parameter family of different possible forms. This proves that Brownian motion on fractals is not unique if the isometry group of the fractal is too small.     

How many points have the same Engel and Sylvester expansions?

In this paper, we prove that the set of points in (0,1] with the same Engel and Sylvester expansions is of Hausdorff dimension .