The Cohn-Elkies linear program for sphere packing, which was used to solve the 8 and 24 dimensional cases, is conjectured to not be sharp in any other dimension d > 2. By mapping feasible points of this infinite-dimensional linear program into a finite-dimensional problem via discrete reduction, we provide a general method to obtain dual bounds on the Cohn-Elkies linear program. This reduces the number of variables to be finite, enabling computer optimization techniques to be applied. Using this method, we prove that the Cohn-Elkies bound cannot come close to the best packing densities known in dimensions 3 ≤ d ≤ 13 except for the solved case d = 8. In particular, our dual bounds show the Cohn-Elkies bound is unable to solve the 3 and 4 dimensional sphere packing problems.

Inversion sequences of length $n$ are integer sequences $e_1,\ldots ,e_n$ with $0\le e_i<i$ for all $i$, which are in bijection with the permutations of length $n$. In this paper, we classify all Wilf equivalence classes of pattern-avoiding inversion sequences of length-4 patterns except for one case (whether 3012 $\equiv$ 3201) and enumerate some of the length-4 pattern-avoiding inversion sequences that are in the OEIS.

The Delsarte linear program is used to bound the size of codes given their block length n and minimal distance d by taking a linear relaxation from codes to quasicodes. We study for which values of (n, d) this linear program has a unique optimum: while we show that it does not always have a unique optimum, we prove that it does if $$d>n/2$$ d > n / 2 or if $$d \le 2$$ d ≤ 2 . Introducing the Krawtchouk decomposition of a quasicode, we prove there exist optima to the (n, 2e) and $$(n-1,2e-1)$$ ( n - 1 , 2 e - 1 ) linear programs that have essentially identical Krawtchouk decompositions, revealing a parity phenomenon among the Delsarte linear programs. We generalize the notion of extending and puncturing codes to quasicodes, from which we see that this parity relationship is given by extending/puncturing. We further characterize these pairs of optima, in particular demonstrating that they exhibit a symmetry property, effectively halving the number of decision variables.

We introduce a deterministic analogue of Markov chains that we call the hunger game. Like rotor‐routing, the hunger game deterministically mimics the behavior of both recurrent Markov chains and absorbing Markov chains. In the case of recurrent Markov chains with finitely many states, hunger game simulation concentrates around the stationary distribution with discrepancy falling off like Nprefix−1$$ {N}^{-1} $$, where N$$ N $$ is the number of simulation steps; in the case of absorbing Markov chains with finitely many states, hunger game simulation also exhibits concentration for hitting measures and expected hitting times with discrepancy falling off like Nprefix−1$$ {N}^{-1} $$ rather than Nprefix−1false/2$$ {N}^{-1/2} $$. When transition probabilities in a finite Markov chain are rational, the game is eventually periodic; the period seems to be the same for all initial configurations and the basin of attraction appears to tile the configuration space (the set of hunger vectors) by translation, but we have not proved this.

No abstract