The Polchinski version of the exact renormalisation group equations is applied to multicritical fixed points, which are present for dimensions between two and four, for scalar theories using both the local potential approximation and its extension, the derivative expansion. The results are compared with the epsilon expansion by showing that the non linear differential equations may be linearised at each multicritical point and the epsilon expansion treated as a perturbative expansion. The results for critical exponents are compared with corresponding epsilon expansion results from standard perturbation theory. The results provide a test for the validity of the local potential approximation and also the derivative expansion. An alternative truncation of the exact RG equation leads to equations which are similar to those found in the derivative expansion but which gives correct results for critical exponents to order $epsilon$ and also for the field anomalous dimension to order $epsilon^2$. An exact marginal operator for the full RG equations is also constructed.
Let ${mathcal D}(n)$ be the maximal determinant for $n times n$ ${pm 1}$-matrices, and $mathcal R(n) = {mathcal D}(n)/n^{n/2}$ be the ratio of ${mathcal D}(n)$ to the Hadamard upper bound. Using the probabilistic method, we prove new lower bounds on ${mathcal D}(n)$ and $mathcal R(n)$ in terms of $d = n-h$, where $h$ is the order of a Hadamard matrix and $h$ is maximal subject to $h le n$. For example, $mathcal R(n) > (pi e/2)^{-d/2}$ if $1 le d le 3$, and $mathcal R(n) > (pi e/2)^{-d/2}(1 - d^2(pi/(2h))^{1/2})$ if $d > 3$. By a recent result of Livinskyi, $d^2/h^{1/2} to 0$ as $n to infty$, so the second bound is close to $(pi e/2)^{-d/2}$ for large $n$. Previous lower bounds tended to zero as $n to infty$ with $d$ fixed, except in the cases $d in {0,1}$. For $d ge 2$, our bounds are better for all sufficiently large $n$. If the Hadamard conjecture is true, then $d le 3$, so the first bound above shows that $mathcal R(n)$ is bounded below by a positive constant $(pi e/2)^{-3/2} > 0.1133$.
We consider several families of binomial sum identities whose definition involves the absolute value function. In particular, we consider centered double sums of the form [S_{alpha,beta}(n) := sum_{k,;ell}binom{2n}{n+k}binom{2n}{n+ell} |k^alpha-ell^alpha|^beta,] obtaining new results in the cases $alpha = 1, 2$. We show that there is a close connection between these double sums in the case $alpha=1$ and the single centered binomial sums considered by Tuenter.
We present here the first observationally based determination of the rate of occurrence of circumbinary planets. This is derived from the publicly available Kepler data, using an automated search algorithm and debiasing process to produce occurrence rates implied by the seven systems already known. These rates depend critically on the planetary inclination distribution: if circumbinary planets are preferentially coplanar with their host binaries, as has been suggested, then the rate of occurrence of planets with $R_p>6R_oplus$ orbiting with $P_p<300$ d is $10.0 ^{+18}_{-6.5}$% (95% confidence limits), higher than but consistent with single star rates. If on the other hand the underlying planetary inclination distribution is isotropic, then this occurrence rate rises dramatically, to give a lower limit of 47%. This implies that formation and subsequent dynamical evolution in circumbinary disks must either lead to largely coplanar planets, or proceed with significantly greater ease than in circumstellar disks. As a result of this investigation we also show that giant planets (${>}10R_oplus$) are significantly less common in circumbinary orbits than their smaller siblings, and confirm that the proposed shortfall of circumbinary planets orbiting the shorter period binaries in the Kepler sample is a real effect.
Let $D(n)$ be the maximal determinant for $n times n$ ${pm 1}$-matrices, and ${mathcal R}(n) = D(n)/n^{n/2}$ be the ratio of $D(n)$ to the Hadamard upper bound. We give several new lower bounds on ${mathcal R}(n)$ in terms of $d$, where $n = h+d$, $h$ is the order of a Hadamard matrix, and $h$ is maximal subject to $h le n$. A relatively simple bound is [{mathcal R}(n) ge left(frac{2}{pi e}right)^{d/2} left(1 - d^2left(frac{pi}{2h}right)^{1/2}right) ;text{ for all }; n ge 1.] An asymptotically sharper bound is [{mathcal R}(n) ge left(frac{2}{pi e}right)^{d/2} expleft(dleft(frac{pi}{2h}right)^{1/2} + ; Oleft(frac{d^{5/3}}{h^{2/3}}right)right).] We also show that [{mathcal R}(n) ge left(frac{2}{pi e}right)^{d/2}] if $n ge n_0$ and $n_0$ is sufficiently large, the threshold $n_0$ being independent of $d$, or for all $nge 1$ if $0 le d le 3$ (which would follow from the Hadamard conjecture). The proofs depend on the probabilistic method, and generalise previous results that were restricted to the cases $d=0$ and $d=1$.
We give upper and lower bounds on the determinant of a perturbation of the identity matrix or, more generally, a perturbation of a nonsingular diagonal matrix. The matrices considered are, in general, diagonally dominant. The lower bounds are best possible, and in several cases they are stronger than well-known bounds due to Ostrowski and other authors. If $A = I-E$ is an $n times n$ matrix and the elements of $E$ are bounded in absolute value by $varepsilon le 1/n$, then a lower bound of Ostrowski (1938) is $det(A) ge 1-nvarepsilon$. We show that if, in addition, the diagonal elements of $E$ are zero, then a best-possible lower bound is [det(A) ge (1-(n-1)varepsilon),(1+varepsilon)^{n-1}.] Corresponding upper bounds are respectively [det(A) le (1 + 2varepsilon + nvarepsilon^2)^{n/2}] and [det(A) le (1 + (n-1)varepsilon^2)^{n/2}.] The first upper bound is stronger than Ostrowskis bound (for $varepsilon < 1/n$) $det(A) le (1 - nvarepsilon)^{-1}$. The second upper bound generalises Hadamards inequality, which is the case $varepsilon = 1$. A necessary and sufficient condition for our upper bounds to be best possible for matrices of order $n$ and all positive $varepsilon$ is the existence of a skew-Hadamard matrix of order $n$.
