Universal Peculiar Linear Mean Relationships in All Polynomials

In any cubic polynomial, the average of the slopes at the $3$ roots is the negation of the slope at the average of the roots. In any quartic, the average of the slopes at the $4$ roots is twice the negation of the slope at the average of the roots. We generalize such situations and present a procedure for determining all such relationships for polynomials of any degree. E.g., in any septic $f$, letting $\overline{f}_n$ denote the mean $f$ value over all zeroes of the derivative $f^{\left(n\right)}$, it holds that $37$ $\overline{f}_1-150$ $\overline{f}_3+200\,\overline{f}_4-135\,\overline{f}_5+48\,\overline{f}_6=0$; and in any quartic it holds that $5$ $\overline{f}_1-6$ $\overline{f}_2+1\,\overline{f}_3=0$. Having calculated such relationships in all dimensions up to 40, in all even dimensions there is a single relationship, in all odd dimensions there is a two-dimensional family of relationships. We come upon connections to Tchebyshev, Bernoulli, \& Euler polynomials, and Stirling numbers.Comment: 27 pages; 1 figur

Multivariate Ap\'ery numbers and supercongruences of rational functions

One of the many remarkable properties of the Ap\'ery numbers $A (n)$, introduced in Ap\'ery's proof of the irrationality of $\zeta (3)$, is that they satisfy the two-term supercongruences \begin{equation*} A (p^r m) \equiv A (p^{r - 1} m) \pmod{p^{3 r}} \end{equation*} for primes $p \geq 5$. Similar congruences are conjectured to hold for all Ap\'ery-like sequences. We provide a fresh perspective on the supercongruences satisfied by the Ap\'ery numbers by showing that they extend to all Taylor coefficients $A (n_1, n_2, n_3, n_4)$ of the rational function \begin{equation*} \frac{1}{(1 - x_1 - x_2) (1 - x_3 - x_4) - x_1 x_2 x_3 x_4} . \end{equation*} The Ap\'ery numbers are the diagonal coefficients of this function, which is simpler than previously known rational functions with this property. Our main result offers analogous results for an infinite family of sequences, indexed by partitions $\lambda$, which also includes the Franel and Yang--Zudilin numbers as well as the Ap\'ery numbers corresponding to $\zeta (2)$. Using the example of the Almkvist--Zudilin numbers, we further indicate evidence of multivariate supercongruences for other Ap\'ery-like sequences.Comment: 19 page

Binomial coefficients involving infinite powers of primes

If p is a prime and n a positive integer, let v(n) denote the exponent of p in n, and u(n)=n/p^{v(n)} the unit part of n. If k is a positive integer not divisible by p, we show that the p-adic limit of (-1)^{pke} u((kp^e)!) as e goes to infinity is a well-defined p-adic integer, which we call z_k. In terms of these, we give a formula for the p-adic limit of binom{a p^e +c, b p^e +d) as e goes to infinity, which we call binom(a p^\infty +c, b p^\infty +d). Here a \ge b are positive integers, and c and d are integers.Comment: 5 page

Inverse of multivector: Beyond p+q=5 threshold

The algorithm of finding inverse multivector (MV) numerically and symbolically is of paramount importance in the applied Clifford geometric algebra (GA) $Cl_{p,q}$. The first general MV inversion algorithm was based on matrix representation of MV. The complexity of calculations and size of the answer in a symbolic form grow exponentially with the GA dimension $n=p+q$. The breakthrough occurred when D. Lundholm and then P. Dadbeh found compact inverse formulas up to dimension $n\le5$. The formulas were constructed in a form of Clifford product of initial MV and its carefully chosen grade-negation counterparts. In this report we show that the grade-negation self-product method can be extended beyond $n=5$ threshold if, in addition, properly constructed linear combinations of such MV products are used. In particular, we present compact explicit MV inverse formulas for algebras of vector space dimension $n=6$ and show that they embrace all lower dimensional cases as well. For readers convenience, we have also given various MV formulas in a form of grade negations when $n\le5$.Comment: 17 pages, new important conjecture was adde

Some identities involving polynomial coefficients

By polynomial (or extended binomial) coefficients, we mean the coefficients in the expansion of integral powers, positive and negative, of the polynomial $1+t +\cdots +t^{m}$; $m\geq 1$ being a fixed integer. We will establish several identities and summation formul\ae\ parallel to those of the usual binomial coefficients.Comment: 13 page

Relaxation and Metastability in the RandomWalkSAT search procedure

An analysis of the average properties of a local search resolution procedure for the satisfaction of random Boolean constraints is presented. Depending on the ratio alpha of constraints per variable, resolution takes a time T_res growing linearly (T_res \sim tau(alpha) N, alpha < alpha_d) or exponentially (T_res \sim exp(N zeta(alpha)), alpha > alpha_d) with the size N of the instance. The relaxation time tau(alpha) in the linear phase is calculated through a systematic expansion scheme based on a quantum formulation of the evolution operator. For alpha > alpha_d, the system is trapped in some metastable state, and resolution occurs from escape from this state through crossing of a large barrier. An annealed calculation of the height zeta(alpha) of this barrier is proposed. The polynomial/exponentiel cross-over alpha_d is not related to the onset of clustering among solutions.Comment: 23 pages, 11 figures. A mistake in sec. IV.B has been correcte

Macdonald polynomials and symmetric functions

The ring of symmetric functions $\Lambda$, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the symmetric group. One may define a coproduct on $\Lambda$ by the plethystic addition on alphabets. In this way the ring of symmetric functions becomes a Hopf algebra. The Littlewood--Richardson numbers may be viewed as the structure constants for the co-product in the Schur basis. In the first part of this thesis we show that by using a generalization of the classical umbral calculus of Gian-Carlo Rota, one may deform the basis of Schur functions to find many other bases for which the Littlewood--Richardson numbers as coproduct structure constants. The Macdonald polynomials are a somewhat mysterious qt-deformation of the Schur functions. The second part of this thesis contains a proof a generating function identity for the Macdonald polynomials which was originally conjectured by Kawanaka

Gaussian binomial coefficients with negative arguments

Loeb showed that a natural extension of the usual binomial coefficient to negative (integer) entries continues to satisfy many of the fundamental properties. In particular, he gave a uniform binomial theorem as well as a combinatorial interpretation in terms of choosing subsets of sets with a negative number of elements. We show that all of this can be extended to the case of Gaussian binomial coefficients. Moreover, we demonstrate that several of the well-known arithmetic properties of binomial coefficients also hold in the case of negative entries. In particular, we show that Lucas' Theorem on binomial coefficients modulo $p$ not only extends naturally to the case of negative entries, but even to the Gaussian case.Comment: 21 page

Exact Results for Amplitude Spectra of Fitness Landscapes

Starting from fitness correlation functions, we calculate exact expressions for the amplitude spectra of fitness landscapes as defined by P.F. Stadler [J. Math. Chem. 20, 1 (1996)] for common landscape models, including Kauffman's NK-model, rough Mount Fuji landscapes and general linear superpositions of such landscapes. We further show that correlations decaying exponentially with the Hamming distance yield exponentially decaying spectra similar to those reported recently for a model of molecular signal transduction. Finally, we compare our results for the model systems to the spectra of various experimentally measured fitness landscapes. We claim that our analytical results should be helpful when trying to interpret empirical data and guide the search for improved fitness landscape models.Comment: 13 pages, 5 figures; revised and final versio

Beating the random assignment on constraint satisfaction problems of bounded degree

We show that for any odd $k$ and any instance of the Max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a $\frac{1}{2} + \Omega(1/\sqrt{D})$ fraction of constraints, where $D$ is a bound on the number of constraints that each variable occurs in. This improves both qualitatively and quantitatively on the recent work of Farhi, Goldstone, and Gutmann (2014), which gave a \emph{quantum} algorithm to find an assignment satisfying a $\frac{1}{2} + \Omega(D^{-3/4})$ fraction of the equations. For arbitrary constraint satisfaction problems, we give a similar result for "triangle-free" instances; i.e., an efficient algorithm that finds an assignment satisfying at least a $\mu + \Omega(1/\sqrt{D})$ fraction of constraints, where $\mu$ is the fraction that would be satisfied by a uniformly random assignment.Comment: 14 pages, 1 figur