Search results for "combinatoric"
showing 10 items of 1776 documents
Random Tensor Theory: Extending Random Matrix Theory to Mixtures of Random Product States
2012
We consider a problem in random matrix theory that is inspired by quantum information theory: determining the largest eigenvalue of a sum of p random product states in $${(\mathbb {C}^d)^{\otimes k}}$$ , where k and p/d k are fixed while d → ∞. When k = 1, the Marcenko-Pastur law determines (up to small corrections) not only the largest eigenvalue ( $${(1+\sqrt{p/d^k})^2}$$ ) but the smallest eigenvalue $${(\min(0,1-\sqrt{p/d^k})^2)}$$ and the spectral density in between. We use the method of moments to show that for k > 1 the largest eigenvalue is still approximately $${(1+\sqrt{p/d^k})^2}$$ and the spectral density approaches that of the Marcenko-Pastur law, generalizing the random matrix…
Adjacency matrices of random digraphs: singularity and anti-concentration
2017
Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We show that $M$ is invertible with probability at least $1-C\ln^{3} d/\sqrt{d}$ for $C\leq d\leq cn/\ln^2 n$, where $c, C$ are positive absolute constants. To this end, we establish a few properties of $d$-regular directed graphs. One of them, a Littlewood-Offord type anti-concentration property, is of independent interest. Let $J$ be a subset of vertices of $G$ with $|J|\approx n/d$. Let $\delta_i$ be the indicator of the event that the vertex $i$ is connected to $J$ and define $\delta = (\delta_1, …
Restricted compositions and permutations: from old to new Gray codes
2011
Any Gray code for a set of combinatorial objects defines a total order relation on this set: x is less than y if and only if y occurs after x in the Gray code list. Let @? denote the order relation induced by the classical Gray code for the product set (the natural extension of the Binary Reflected Gray Code to k-ary tuples). The restriction of @? to the set of compositions and bounded compositions gives known Gray codes for those sets. Here we show that @? restricted to the set of bounded compositions of an interval yields still a Gray code. An n-composition of an interval is an n-tuple of integers whose sum lies between two integers; and the set of bounded n-compositions of an interval si…
Statistics-preserving bijections between classical and cyclic permutations
2012
Recently, Elizalde (2011) [2] has presented a bijection between the set C"n"+"1 of cyclic permutations on {1,2,...,n+1} and the set of permutations on {1,2,...,n} that preserves the descent set of the first n entries and the set of weak excedances. In this paper, we construct a bijection from C"n"+"1 to S"n that preserves the weak excedance set and that transfers quasi-fixed points into fixed points and left-to-right maxima into themselves. This induces a bijection from the set D"n of derangements to the set C"n"+"1^q of cycles without quasi-fixed points that preserves the weak excedance set. Moreover, we exhibit a kind of discrete continuity between C"n"+"1 and S"n that preserves at each s…
The Matrix Effect and Application of the Multi-Parameter Optimization Method for X-Ray Spectrometric Determination of the Quantitative Composition of…
2018
Determining the quantitative composition of clay samples with X-ray fluorescent spectrometry is complicated because of the matrix effect, in which any element can increase or decrease the analytical signals of other elements. In order to predict the properties of clays, it is essential to know their precise chemical composition. Therefore, using the standard addition method was determined calibration and empirical influence coefficients, as well as the true composition of the elements. Farther, these coefficients were used to correct the matrix effect and develop a multi-parameter optimization method. It was determined that in clay samples, consisting of Si, Al, Fe, K, Mg, Ca, Na and Ti oxi…
A Generalized Semiempirical Approach to the Modeling of the Optical Band Gap of Ternary Al-(Ga, Nb, Ta, W) Oxides Containing Different Alumina Polymo…
2021
A generalization of the modeling equation of optical band gap values for ternary oxides, as a function of cationic ratio composition, is carried out based on the semiempirical correlation between the differences in the electronegativity of oxygen and the average cationic electronegativity proposed some years ago. In this work, a novel approach is suggested to account for the differences in the band gap values of the different polymorphs of binary oxides as well as for ternary oxides existing in different crystalline structures. A preliminary test on the validity of the proposed modeling equations has been carried out by using the numerous experimental data pertaining to alumina and gallia p…
Activity-composition relations for the calculation of partial melting equilibria in metabasic rocks
2020
A set of thermodynamic models is presented that, for the first time, allows partial melting equilibria to be calculated for metabasic rocks. The models consist of new activity–composition relations combined with end-member thermodynamic properties from the Holland & Powell dataset, version 6. They allow for forward modelling in the system Na (Formula presented.) O–CaO–K (Formula presented.) O–FeO–MgO–Al (Formula presented.) O (Formula presented.) –SiO (Formula presented.) –H (Formula presented.) O–TiO (Formula presented.) –Fe (Formula presented.) O (Formula presented.). In particular, new activity–composition relations are presented for silicate melt of broadly trondhjemitic–tonalitic compo…
Stage–Discharge Relationship for an Upstream Inclined Grid with Transversal Bars
2016
AbstractCheck dams with grids upgrading upstream are often used in mountain rivers, where intense sediment transport and steep slopes occur. In some cases, sloping grids are used in the construction of debris flow breakers. In this paper, the outflow process of an upstream-inclined grid with transversal bars is studied by using the dimensional analysis and the incomplete self-similarity theory. Next, the theoretical analysis shows that a power equation can be used for establishing the stage-discharge equation. The coefficient of the power equation depends on both the slope angle and the void ratio, whereas the exponent depends only on the slope angle. Finally, this deduced stage–discharge r…
Variational Approximations for Generalized Linear Latent Variable Models
2017
Generalized linear latent variable models (GLLVMs) are a powerful class of models for understanding the relationships among multiple, correlated responses. Estimation, however, presents a major challenge, as the marginal likelihood does not possess a closed form for nonnormal responses. We propose a variational approximation (VA) method for estimating GLLVMs. For the common cases of binary, ordinal, and overdispersed count data, we derive fully closed-form approximations to the marginal log-likelihood function in each case. Compared to other methods such as the expectation-maximization algorithm, estimation using VA is fast and straightforward to implement. Predictions of the latent variabl…
A segmentation algorithm for noisy images
2005
International audience; This paper presents a segmentation algorithm for gray-level images and addresses issues related to its performance on noisy images. It formulates an image segmentation problem as a partition of a weighted image neighborhood hypergraph. To overcome the computational difficulty of directly solving this problem, a multilevel hypergraph partitioning has been used. To evaluate the algorithm, we have studied how noise affects the performance of the algorithm. The alpha-stable noise is considered and its effects on the algorithm are studied. Key words : graph, hypergraph, neighborhood hypergraph, multilevel hypergraph partitioning, image segmentation and noise removal.