System for indicating fuel-efficient aircraft altitude

A method and apparatus are provided for indicating the altitude at which an aircraft should fly so the W/d ratio (weight of the aircraft divided by the density of air) more closely approaches the optimum W/d for the aircraft. A passive microwave radiometer on the aircraft is directed at different angles with respect to the horizon to determine the air temperature, and therefore the density of the air, at different altitudes. The weight of the aircraft is known. The altitude of the aircraft is changed to fly the aircraft at an altitude at which is W/d ratio more closely approaches the optimum W/d ratio for that aircraft

Some New Bounds For Cover-Free Families Through Biclique Cover

An $(r,w;d)$ cover-free family $(CFF)$ is a family of subsets of a finite set such that the intersection of any $r$ members of the family contains at least $d$ elements that are not in the union of any other $w$ members. The minimum number of elements for which there exists an $(r,w;d)-CFF$ with $t$ blocks is denoted by $N((r,w;d),t)$. In this paper, we show that the value of $N((r,w;d),t)$ is equal to the $d$-biclique covering number of the bipartite graph $I_t(r,w)$ whose vertices are all $w$- and $r$-subsets of a $t$-element set, where a $w$-subset is adjacent to an $r$-subset if their intersection is empty. Next, we introduce some new bounds for $N((r,w;d),t)$. For instance, we show that for $r\geq w$ and $r\geq 2$ $$N((r,w;1),t) \geq c{{r+w\choose w+1}+{r+w-1 \choose w+1}+ 3 {r+w-4 \choose w-2} \over \log r} \log (t-w+1),$$ where $c$ is a constant satisfies the well-known bound $N((r,1;1),t)\geq c\frac{r^2}{\log r}\log t$. Also, we determine the exact value of $N((r,w;d),t)$ for some values of $d$. Finally, we show that $N((1,1;d),4d-1)=4d-1$ whenever there exists a Hadamard matrix of order 4d

Rotations and Tangent Processes on Wiener Space

The paper considers (a) Representations of measure preserving transformations (``rotations'') on Wiener space, and (b) The stochastic calculus of variations induced by parameterized rotations \{T_\theta w, 0 \le \theta \le \eps\}: ``Directional derivatives'' $(dF(T_\theta w)/d \theta)_{\theta=0}$, ``vector fields'' or ``tangent processes'' $(dT_\theta w /d\theta)_{\theta=0}$ and flows of rotations.Comment: 29 page

Ethical intuitionism and the linguistic analogy

It is a central tenet of ethical intuitionism as defended by W. D. Ross and others that moral theory should reflect the convictions of mature moral agents. Hence, intuitionism is plausible to the extent that it corresponds to our well-considered moral judgments. After arguing for this claim, I discuss whether intuitionists offer an empirically adequate account of our moral obligations. I do this by applying recent empirical research by John Mikhail that is based on the idea of a universal moral grammar to a number of claims implicit in W. D. Ross’s normative theory. I argue that the results at least partly vindicate intuitionism

Quantum Diffusion and Delocalization for Band Matrices with General Distribution

We consider Hermitian and symmetric random band matrices $H$ in $d \geq 1$ dimensions. The matrix elements $H_{xy}$, indexed by $x,y \in \Lambda \subset \Z^d$, are independent and their variances satisfy \sigma_{xy}^2:=\E \abs{H_{xy}}^2 = W^{-d} f((x - y)/W) for some probability density $f$. We assume that the law of each matrix element $H_{xy}$ is symmetric and exhibits subexponential decay. We prove that the time evolution of a quantum particle subject to the Hamiltonian $H$ is diffusive on time scales $t\ll W^{d/3}$. We also show that the localization length of the eigenvectors of $H$ is larger than a factor $W^{d/6}$ times the band width $W$. All results are uniform in the size \abs{\Lambda} of the matrix. This extends our recent result \cite{erdosknowles} to general band matrices. As another consequence of our proof we show that, for a larger class of random matrices satisfying $\sum_x\sigma_{xy}^2=1$ for all $y$, the largest eigenvalue of $H$ is bounded with high probability by $2 + M^{-2/3 + \epsilon}$ for any $\epsilon > 0$, where M \deq 1 / (\max_{x,y} \sigma_{xy}^2).Comment: Corrected typos and some inaccuracies in appendix

Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One

We present a method to obtain infinitely many examples of pairs $(W,D)$ consisting of a matrix weight $W$ in one variable and a symmetric second-order differential operator $D$. The method is based on a uniform construction of matrix valued polynomials starting from compact Gelfand pairs $(G,K)$ of rank one and a suitable irreducible $K$-representation. The heart of the construction is the existence of a suitable base change $\Psi_{0}$. We analyze the base change and derive several properties. The most important one is that $\Psi_{0}$ satisfies a first-order differential equation which enables us to compute the radial part of the Casimir operator of the group $G$ as soon as we have an explicit expression for $\Psi_{0}$. The weight $W$ is also determined by $\Psi_{0}$. We provide an algorithm to calculate $\Psi_{0}$ explicitly. For the pair $(\mathrm{USp}(2n),\mathrm{USp}(2n-2)\times\mathrm{USp}(2))$ we have implemented the algorithm in GAP so that individual pairs $(W,D)$ can be calculated explicitly. Finally we classify the Gelfand pairs $(G,K)$ and the $K$-representations that yield pairs $(W,D)$ of size $2\times2$ and we provide explicit expressions for most of these cases

Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model

We consider Hermitian and symmetric random band matrices $H$ in $d \geq 1$ dimensions. The matrix elements $H_{xy}$, indexed by $x,y \in \Lambda \subset \Z^d$, are independent, uniformly distributed random variables if \abs{x-y} is less than the band width $W$, and zero otherwise. We prove that the time evolution of a quantum particle subject to the Hamiltonian $H$ is diffusive on time scales $t\ll W^{d/3}$. We also show that the localization length of an arbitrarily large majority of the eigenvectors is larger than a factor $W^{d/6}$ times the band width. All results are uniform in the size \abs{\Lambda} of the matrix.Comment: Minor corrections, Sections 4 and 11 update