A two variable Artin conjecture

Let a and b be non-zero rational numbers that are multiplicatively independent. We study the natural density of the set of primes p for which the subgroup of the multiplicative group of the finite field with p elements generated by (a\mod p) contains (b\mod p). It is shown that, under assumption of the generalized Riemann hypothesis (GRH), this density exists and equals a positive rational multiple of the universal constant S=\prod_{p prime}(1-p/(p^3-1)). An explicit value of the density is given under mild conditions on a and b. This extends and corrects earlier work of P.J. Stephens (1976). Our result, in combination with earlier work of the second author, allows us to deduce that any second order linear recurrence with reducible characteristic polynomial having integer elements, has a positive density of prime divisors (under GRH)

Statistical Physics Analysis of Maximum a Posteriori Estimation for Multi-channel Hidden Markov Models

The performance of Maximum a posteriori (MAP) estimation is studied analytically for binary symmetric multi-channel Hidden Markov processes. We reduce the estimation problem to a 1D Ising spin model and define order parameters that correspond to different characteristics of the MAP-estimated sequence. The solution to the MAP estimation problem has different operational regimes separated by first order phase transitions. The transition points for $L$-channel system with identical noise levels, are uniquely determined by $L$ being odd or even, irrespective of the actual number of channels. We demonstrate that for lower noise intensities, the number of solutions is uniquely determined for odd $L$, whereas for even $L$ there are exponentially many solutions. We also develop a semi analytical approach to calculate the estimation error without resorting to brute force simulations. Finally, we examine the tradeoff between a system with single low-noise channel and one with multiple noisy channels.Comment: The paper has been submitted to Journal of Statistical Physics with submission number JOSS-S-12-0039

Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II

We deliver here second new $\textit{H(x)}-binomials'$ recurrence formula, were $H(x)-binomials'$ array is appointed by $Ward-Horadam$ sequence of functions which in predominantly considered cases where chosen to be polynomials . Secondly, we supply a review of selected related combinatorial interpretations of generalized binomial coefficients. We then propose also a kind of transfer of interpretation of $p,q-binomial$ coefficients onto $q-binomial$ coefficients interpretations thus bringing us back to $Gy{\"{o}}rgy P\'olya$ and Donald Ervin Knuth relevant investigation decades ago.Comment: 57 pages, 8 figure

Provably Good Solutions to the Knapsack Problem via Neural Networks of Bounded Size

The development of a satisfying and rigorous mathematical understanding of the performance of neural networks is a major challenge in artificial intelligence. Against this background, we study the expressive power of neural networks through the example of the classical NP-hard Knapsack Problem. Our main contribution is a class of recurrent neural networks (RNNs) with rectified linear units that are iteratively applied to each item of a Knapsack instance and thereby compute optimal or provably good solution values. We show that an RNN of depth four and width depending quadratically on the profit of an optimum Knapsack solution is sufficient to find optimum Knapsack solutions. We also prove the following tradeoff between the size of an RNN and the quality of the computed Knapsack solution: for Knapsack instances consisting of $n$ items, an RNN of depth five and width $w$ computes a solution of value at least $1-\mathcal{O}(n^2/\sqrt{w})$ times the optimum solution value. Our results build upon a classical dynamic programming formulation of the Knapsack Problem as well as a careful rounding of profit values that are also at the core of the well-known fully polynomial-time approximation scheme for the Knapsack Problem. A carefully conducted computational study qualitatively supports our theoretical size bounds. Finally, we point out that our results can be generalized to many other combinatorial optimization problems that admit dynamic programming solution methods, such as various Shortest Path Problems, the Longest Common Subsequence Problem, and the Traveling Salesperson Problem.Comment: A short version of this paper appears in the proceedings of AAAI 202

Iterated Monodromy Groups of Quadratic Polynomials, I

We describe the iterated monodromy groups associated with post-critically finite quadratic polynomials, and explicit their connection to the `kneading sequence' of the polynomial. We then give recursive presentations by generators and relations for these groups, and study some of their properties, like torsion and `branchness'.Comment: 18 pages, 3 EPS figure