arXiv subjects
Publications and source records attributed to Dan Levy.
Let p be any prime, and $p^(\nu_p(n!))$ the maximal power of $p$ dividing $n!$. It is proved that there exists a positive integer $n_0$, which depends only on $p$, such that $q^(\nu_q(n!)) < p^(\nu_p(n!))$ for all $n \ge n_0$ and all primes $q > p$. For twin primes $p$ and $q = p + 2$ it is proved that the minimal $n_0$ satisfying $q^(\nu_q(n!)) < p^(\nu_p(n!))$ for all $n \ge n_0$ is given by $n_0 = (p^2+p)/2$.
We report on the first simultaneous measurement of high-order harmonics, relativistic electrons and low divergence proton beams generated from plasma mirrors driven at kHz repetition rate by relativistic-intensity milliJoule-energy femtosecond laser pulses. This setup enables detailed parametric studies of the particle and radiation spatio-spectral beam properties for a wide range of controlled interaction conditions, such as pulse duration and plasma density scale length. This versatile setup should aid in further understanding the collective laser absorption mechanisms at play during the laser-plasma interaction and in optimizing the secondary beam properties for potential applications.
Proton beams with up to 100 pC bunch charge, 0.48 MeV cut-off energy and divergence as low as a $3^{\circ}$ were generated from solid targets at kHz repetition rate by a few-mJ femtosecond laser under controlled plasma conditions. The beam spatial profile was measured using a small aperture scanning time-of-flight detector. Detailed parametric studies were performed by varying the surface plasma scale length from 8 to 80 nm and the laser pulse duration from 4 fs to 1.5 ps. Numerical simulations are in good agreement with observations and, together with an in-depth theoretical analysis of the acceleration mechanism, indicate that high repetition rate femtosecond laser technology could be used to produce few-MeV protons beams for applications.
We study numerically the mechanisms of proton acceleration in gas-foil targets driven by an ultraintense femtosecond laser pulse. The target consists of a near-critical-density hydrogen gas layer of a few tens of microns attached to a solid carbon foil with a contaminant thin proton layer at its back side. Two-dimensional particle-in-cell simulations show that, at optimal gas density, the maximum energy of the contaminant protons is increased by a factor of $\sim 4$ compared to a single foil target. This improvement originates from the near-complete laser absorption into relativistic electrons in the gas. Several energetic electron populations are identified, and their respective effect on the proton acceleration is quantified by computing the electrostatic fields that they generate at the protons' positions. While each of those electron groups is found to contribute substantially to the overall accelerating field, the dominant one is the relativistic thermal bulk that results from the nonlinear wakefield excited in the gas, as analyzed recently by Debayle et al. [New J. Phys. 19, 123013 (2017)]. Our analysis also reveals the important role of the neighboring ions in the acceleration of the fastest protons, and the onset of multidimensional effects caused by the time-increasing curvature of the proton layer.
Laser-plasma proton acceleration was investigated in the Target Normal Sheath Acceleration (TNSA) regime using a novel gas-foil target. The target is designed for reaching higher laser intensity at the foil plane owing to relativistic self-focusing and self compression of the pulse in the gas layer. Numerical 3D particle-in-cell (PIC) simulations were used to study pulse propagation in the gas, showing a nearly seven-fold increase in peak intensity. In the experiment, maximum proton energies showed high dependence on the energy transmission of the laser through the gas and a lesser dependence on the size and shape of the pulse. At high gas densities, laser energy depletion and pulse distortion suppressed proton energies. At low densities, self-focusing was observed and comparable or higher proton energies were measured with the gas.
We consider factorizations $G=XY$ where $G$ is a general group, $X$ and $Y$ are normal subsets of $G$ and any $g\in G$ has a unique representation $g=xy$ with $x\in X$ and $y\in Y$. This definition coincides with the customary and extensively studied definition of a direct product decomposition by subsets of a finite abelian group. Our main result states that a group $G$ has such a factorization if and only if $G$ is a central product of $\left\langle X\right\rangle $ and $\left\langle Y\right\rangle $ and the central subgroup $\left\langle X\right\rangle \cap \left\langle Y\right\rangle $ satisfies certain abelian factorization conditions. We analyze some special cases and give examples. In particular, simple groups have no non-trivial set-direct factorization.
A partition of a group is a dioid partition if the following three conditions are met: The setwise product of any two parts is a union of parts, there is a part that multiplies as an identity element, and the inverse of a part is a part. This kind of a group partition was first introduced by Tamaschke in 1968. We show that a dioid partition defines a dioid structure over the group, analogously to the way a Schur ring over a group is defined. After proving fundamental properties of dioid partitions, we focus on three part dioid partitions of cyclic groups of prime order. We provide classification results for their isomorphism types as well as for the partitions themselves.
We present constructions of symmetric complete sum-free sets in general finite cyclic groups. It is shown that the relative sizes of the sets are dense in $[0,\frac{1}{3}]$, answering a question of Cameron, and that the number of those contained in the cyclic group of order $n$ is exponential in $n$. For primes $p$, we provide a full characterization of the symmetric complete sum-free subsets of $\mathbb{Z}_p$ of size at least $(\frac{1}{3}-c) \cdot p$, where $c>0$ is a universal constant.
We prove that there exists a universal constant $c$ such that any finite primitive permutation group of degree $n$ with a non-trivial point stabilizer is a product of no more than $c\log n$ point stabilizers.
We present a global optimization algorithm for clustering data given the ratio of likelihoods that each pair of data points is in the same cluster or in different clusters. To define a clustering solution in terms of pairwise relationships, a necessary and sufficient condition is that belonging to the same cluster satisfies transitivity. We define a global objective function based on pairwise likelihood ratios and a transitivity constraint over all triples, assigning an equal prior probability to all clustering solutions. We maximize the objective function by implementing max-sum message passing on the corresponding factor graph to arrive at an O(N^3) algorithm. Lastly, we demonstrate an application inspired by mutational sequencing for decoding random binary words transmitted through a noisy channel.
We consider factorizations of a finite group $G$ into conjugate subgroups, $G=A^{x_{1}}\cdots A^{x_{k}}$ for $A\leq G$ and $x_{1},\ldots ,x_{k}\in G$, where $A$ is nilpotent or solvable. First we exploit the split $BN$-pair structure of finite simple groups of Lie type to give a unified self-contained proof that every such group is a product of four or three unipotent Sylow subgroups. Then we derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group. Finally, using conjugate factorizations of a general finite solvable group by any of its Carter subgroups, we obtain an upper bound on the minimal length of a nilpotent conjugate factorization of a general finite group.
Let $G$ be a finite non-solvable group. We prove that there exists a proper subgroup $A$ of $G$ such that $G$ is the product of three conjugates of $A$, thus replacing an earlier upper bound of $36$ with the smallest possible value. The proof relies on an equivalent formulation in terms of double cosets, and uses the following theorem which is of independent interest and wider scope: Any group $G$ with a $BN$-pair and a finite Weyl group $W$ satisfies $G=\left( Bn_{0}B\right) ^{2}=BB^{n_{0}}B$ where $n_{0}$ is any preimage of the longest element of $W$. The proof of the last theorem is formulated in the dioid consisting of all unions of double cosets of $B$ in $G$. Other results on minimal length product covers of a group by conjugates of a proper subgroup are given.
For every non-nilpotent finite group $G$, there exists at least one proper subgroup $M$ such that $G$ is the setwise product of a finite number of conjugates of $M$. We define $γ_{\text{cp}}\left( G\right) $ to be the smallest number $k$ such that $G$ is a product, in some order, of $k$ pairwise conjugated proper subgroups of $G$. We prove that if $G$ is non-solvable then $γ_{\text{cp}}\left( G\right) \leq36$ while if $G$ is solvable then $γ_{\text{cp}}\left( G\right) $ can attain any integer value bigger than $2$, while, on the other hand, $γ_{\text{cp}}\left( G\right) \leq4\log_{2}\left\vert G\right\vert $.
Guralnick, Kunyavskii, Plotkin and Shalev have shown that the solvable radical of a finite group $G$ can be characterized as the set of all $x\in G$ such that $ $ is solvable for all $y\in G$. We prove two generalizations of this result. Firstly, it is enough to check the solvability of $ $ for every $p$-element $y\in G$ for every odd prime $p$. Secondly, if $x$ has odd order, then it is enough to check the solvability of $ $ for every 2-element $y\in G$.
Genomic copy number variation (CNV) is a large source of variation between organisms, and its consequences include phenotypic differences and genetic disorders. CNVs are commonly detected by hybridizing genomic DNA to microarrays of nucleic acid probes. System noise caused by operational and probe performance variability complicates the interpretation of these data. To minimize the distortion of genetic signal by system noise, we have explored the latter in an archive of hybridizations in which no genetic signal is expected. This archive is obtained by comparative genomic hybridization (CGH) of a sample in one channel to the same sample in the other channel, or 'self-self' data. These self-self hybridizations trap a variety of system noise inherent in sample-reference (test) data. Through singular value decomposition (SVD) of self-self data, we have determined the principal components of system noise. Assuming simple linear models of noise generation, the linear correction of test data with self-self data -or 'system normalization'- reduces local and long-range correlations and improves signal-to-noise metrics, yet does not introduce detectable spurious signal. Using this method, 90% of hybridizations displayed improved signal-to-noise ratios with an average increase of 7.0%, due mainly to a reduced median average deviation (MAD). In addition, we have found that principal component loadings correlate with specific probe variables including array coordinates, base composition, and proximity to the 5' ends of genes. The correlation of the principal component loadings with the test data depends on operational variables, such as the temporal order of processing and the localization of individual samples within 96-well plates.
The neighbor-joining algorithm is a popular phylogenetics method for constructing trees from dissimilarity maps. The neighbor-net algorithm is an extension of the neighbor-joining algorithm and is used for constructing split networks. We begin by describing the output of neighbor-net in terms of the tessellation of $\bar{\MM}_{0}^n(\mathbb{R})$ by associahedra. This highlights the fact that neighbor-net outputs a tree in addition to a circular ordering and we explain when the neighbor-net tree is the neighbor-joining tree. A key observation is that the tree constructed in existing implementations of neighbor-net is not a neighbor-joining tree. Next, we show that neighbor-net is a greedy algorithm for finding circular split systems of minimal balanced length. This leads to an interpretation of neighbor-net as a greedy algorithm for the traveling salesman problem. The algorithm is optimal for Kalmanson matrices, from which it follows that neighbor-net is consistent and has optimal radius 1/2. We also provide a statistical interpretation for the balanced length for a circular split system as the length based on weighted least squares estimates of the splits. We conclude with applications of these results and demonstrate the implications of our theorems for a recently published comparison of Papuan and Austronesian languages.
We show that the neighbor-joining algorithm is a robust quartet method for constructing trees from distances. This leads to a new performance guarantee that contains Atteson's optimal radius bound as a special case and explains many cases where neighbor-joining is successful even when Atteson's criterion is not satisfied. We also provide a proof for Atteson's conjecture on the optimal edge radius of the neighbor-joining algorithm. The strong performance guarantees we provide also hold for the quadratic time fast neighbor-joining algorithm, thus providing a theoretical basis for inferring very large phylogenies with neighbor-joining.
The Neighbor-Joining algorithm is a recursive procedure for reconstructing trees that is based on a transformation of pairwise distances between leaves. We present a generalization of the neighbor-joining transformation, which uses estimates of phylogenetic diversity rather than pairwise distances in the tree. This leads to an improved neighbor-joining algorithm whose total running time is still polynomial in the number of taxa. On simulated data, the method outperforms other distance-based methods. We have implemented neighbor-joining for subtree weights in a program called MJOIN which is freely available under the Gnu Public License at http://bio.math.berkeley.edu/mjoin/ .