scholarly journals Bounded discrete walks

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
C. Banderier ◽  
P. Nicodème
Keyword(s):  
Generating Functions ◽  
High Speed ◽  
Symmetric Functions ◽  
Kernel Method ◽  
Discrete Distribution ◽  
Lattice Paths ◽  
Taylor Coefficients ◽  
Brownian Bridges ◽  
Finite Set ◽  
Speed Up

International audience This article tackles the enumeration and asymptotics of directed lattice paths (that are isomorphic to unidimensional paths) of bounded height (walks below one wall, or between two walls, for $\textit{any}$ finite set of jumps). Thus, for any lattice paths, we give the generating functions of bridges ("discrete'' Brownian bridges) and reflected bridges ("discrete'' reflected Brownian bridges) of a given height. It is a new success of the "kernel method'' that the generating functions of such walks have some nice expressions as symmetric functions in terms of the roots of the kernel. These formulae also lead to fast algorithms for computing the $n$-th Taylor coefficients of the corresponding generating functions. For a large class of walks, we give the discrete distribution of the height of bridges, and show the convergence to a Rayleigh limit law. For the family of walks consisting of a $-1$ jump and many positive jumps, we give more precise bounds for the speed of convergence. We end our article with a heuristic application to bioinformatics that has a high speed-up relative to previous work.

scholarly journals Analytic Combinatorics of Lattice Paths: Enumeration and Asymptotics for the Area

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Cyril Banderier ◽  
Bernhard Gittenberger
Keyword(s):  
Symmetric Functions ◽  
Kernel Method ◽  
Brownian Bridge ◽  
Lattice Paths ◽  
Reflected Brownian Motion ◽  
Analytic Combinatorics ◽  
Average Area ◽  
First Case ◽  
Finite Set ◽  
International Audience

International audience This paper tackles the enumeration and asymptotics of the area below directed lattice paths (walks on $\mathbb{N}$ with a finite set of jumps). It is a nice surprise (obtained via the "kernel method'') that the generating functions of the moments of the area are algebraic functions, expressible as symmetric functions in terms of the roots of the kernel. For a large class of walks, we give full asymptotics for the average area of excursions ("discrete'' reflected Brownian bridge) and meanders ("discrete'' reflected Brownian motion). We show that drift is not playing any role in the first case. We also generalise previous works related to the number of points below a path and to the area between a path and a line of rational slope.

scholarly journals Slicings of Parallelogram Polyominoes: Catalan, Schröder, Baxter, and Other Sequences

10.37236/7375 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Nicholas R. Beaton ◽  
Mathilde Bouvel ◽  
Veronica Guerrini ◽  
Simone Rinaldi
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Kernel Method ◽  
Lattice Paths ◽  
Schröder Numbers ◽  
Special Cases ◽  
Generating Tree

We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Baxter slicings: the $m$-skinny slicings and the $m$-row-restricted slicings, for $m \in \mathbb{N}$. Using functional equations and the kernel method, their generating functions are computed in some special cases, and we conjecture that they are algebraic for any $m$.

scholarly journals Asymptotics of lattice walks via analytic combinatorics in several variables

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Stephen Melczer ◽  
Mark C. Wilson
Keyword(s):  
Generating Functions ◽  
Kernel Method ◽  
Rational Functions ◽  
Two Dimensional ◽  
Combinatorial Properties ◽  
Analytic Combinatorics ◽  
Several Variables ◽  
Algebra Approach ◽  
Finite Set ◽  
International Audience

International audience We consider the enumeration of walks on the two-dimensional non-negative integer lattice with steps defined by a finite set S ⊆ {±1, 0}2 . Up to isomorphism there are 79 unique two-dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Pade ́-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.

Scanning x-ray fluorescence microscopy and principal component analysis

1992 ◽  
Vol 50 (2) ◽  
pp. 1752-1753
Author(s):  
Brian Cross
Keyword(s):  
Principal Component Analysis ◽  
Fluorescence Microscopy ◽  
Spatial Resolution ◽  
High Speed ◽  
Image Acquisition ◽  
Principal Component ◽  
Fluorescence Microscope ◽  
Acquisition Rate ◽  
X Ray ◽  
Speed Up

A relatively new entry, in the field of microscopy, is the Scanning X-Ray Fluorescence Microscope (SXRFM). Using this type of instrument (e.g. Kevex Omicron X-ray Microprobe), one can obtain multiple elemental x-ray images, from the analysis of materials which show heterogeneity. The SXRFM obtains images by collimating an x-ray beam (e.g. 100 μm diameter), and then scanning the sample with a high-speed x-y stage. To speed up the image acquisition, data is acquired "on-the-fly" by slew-scanning the stage along the x-axis, like a TV or SEM scan. To reduce the overhead from "fly-back," the images can be acquired by bi-directional scanning of the x-axis. This results in very little overhead with the re-positioning of the sample stage. The image acquisition rate is dominated by the x-ray acquisition rate. Therefore, the total x-ray image acquisition rate, using the SXRFM, is very comparable to an SEM. Although the x-ray spatial resolution of the SXRFM is worse than an SEM (say 100 vs. 2 μm), there are several other advantages.

SYMMETRIC FUNCTIONS OF BINARY PRODUCTS OF TRIBONACCI LUCAS NUMBERS AND ORTHOGONAL POLYNOMIALS

2021 ◽  
Vol 21 (2) ◽  
pp. 461-478
Author(s):  
HIND MERZOUK ◽  
ALI BOUSSAYOUD ◽  
MOURAD CHELGHAM
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Functions ◽  
Symmetric Functions ◽  
Lucas Numbers

In this paper, we will recover the new generating functions of some products of Tribonacci Lucas numbers and orthogonal polynomials. The technic used her is based on the theory of the so called symmetric functions.

Development of the LBM non-isothermal flows with arbitrarily large Mach number

2021 ◽  
pp. 62-71
Author(s):  
Елизавета Вячеславовна Зипунова ◽  
Анастасия Юрьевна Перепёлкина ◽  
Андрей Владимирович Закиров
Keyword(s):  
Mach Number ◽  
High Speed ◽  
Classical Method ◽  
Discrete Distribution ◽  
Explicit Calculation ◽  
Time Layer ◽  
Boltzmann Equations ◽  
On Demand ◽  
Small Flow ◽  
Lattice Boltzmann Equations

При решении задач динамики жидкостей и газов в области малых скоростей потока и при изотермических условиях с успехом применяется метод решеточных уравнений Больцмана (LBM). Для решения дискретного уравнения Больцмана может быть использован новый метод Particles-on-Demand (PonD), в котором в каждой точке сетки дискретизация функции распределения в пространстве скоростей центрирована относительно текущей скорости потока. В отличие от классического LBM, метод PonD применим не только для задач с малыми скоростями потока и при изотермических условиях. В данной работе реализован метод PonD D1Q5 с итерационным расчетом скорости переноса и явным расчетом первых трех моментов, включая скорости переноса. Показано, что рассмотренная модификация метода PonD хоть и накладывает ограничения на параметры, позволяет проводить расчеты в большем диапазоне допустимых скоростей. The purpose of the paper is to demonstrate applicability of the Particle on Demand (PonD) D1Q5 method with the explicit calculation of the first three moments to problem with high speed of the flow. The standard LBM is applicable for small flow velocities. Thus to overcome this limitation we use PonD. In this work, we use conservative version of PonD - the D1Q5 method with the explicit calculation of the first three moments. Methodology. The Pond over LBM was applied to the Riemann problem in order to demonstrate the advantage of the method. In this work, we choose the case when contact discontinuities could propagate at variable speed. Findings. If the interpolation pattern is fixed relative to the point at which there is a current update of the discrete distribution function, then the transfer step can be written explicitly, thus the scheme is conservative. On the other hand, this imposes additional restrictions on the temperature and the flow rate. But even if the PonD scheme is limited to a fixed interpolation pattern, it is possible to simulate flows with larger values of the Mach number than in the case when the classical method of lattice Boltzmann equations is used. Originality/value. In the described particular case of the PonD method, it is possible to avoid iterations by calculating the temperature and velocity values directly at a new time layer. In this work, we have investigated the properties and the range of applicability (admissible values of temperature and velocity) of such modification of PonD.

Experimental Study on the Dynamic Deformation of Simply Supported Beam in Chinese High-speed Railway

2018 ◽  
Author(s):  
Gonglian Dai ◽  
Meng Wang ◽  
Tianliang Zhao ◽  
Wenshuo Liu
Keyword(s):  
High Speed ◽  
Structure Design ◽  
Dynamic Coefficient ◽  
Bridge Structure ◽  
High Speed Railway ◽  
Simply Supported Beam ◽  
Vertical Stiffness ◽  
Simply Supported ◽  
Speed Up ◽  
Design Speed

<p>At present, Chinese high-speed railway operating mileage has exceeded 20 thousand km, and the proportion of the bridge is nearly 50%. Moreover, high-speed railway design speed is constantly improving. Therefore, controlling the deformation of the bridge structure strictly is particularly important to train speed-up as well as to ensure the smoothness of the line. This paper, based on the field test, shows the vertical and transverse absolute displacements of bridge structure by field collection. What’s more, resonance speed and dynamic coefficient of bridge were studied. The results show that: the horizontal and vertical stiffness of the bridge can meet the requirements of <b>Chinese “high-speed railway design specification” (HRDS)</b>, and the structure design can be optimized. However, the dynamic coefficient may be greater than the specification suggested value. And the simply supported beam with CRTSII ballastless track has second-order vertical resonance velocity 306km/h and third-order transverse resonance velocity 312km/h by test results, which are all coincide with the theoretical resonance velocity.</p>

scholarly journals AC_ICAP: A Flexible High Speed ICAP Controller

2015 ◽  
Vol 2015 ◽  
pp. 1-15 ◽  
Author(s):  
Luis Andres Cardona ◽  
Carles Ferrer
Keyword(s):  
High Speed ◽  
Access Port ◽  
Field Programmable ◽  
Programmable Gate Arrays ◽  
Speed Up ◽  
Run Time ◽  
Ip Cores ◽  
Similar Solutions ◽  
Internal Configuration ◽  
Run Time Reconfiguration

The Internal Configuration Access Port (ICAP) is the core component of any dynamic partial reconfigurable system implemented in Xilinx SRAM-based Field Programmable Gate Arrays (FPGAs). We developed a new high speed ICAP controller, named AC_ICAP, completely implemented in hardware. In addition to similar solutions to accelerate the management of partial bitstreams and frames, AC_ICAP also supports run-time reconfiguration of LUTs without requiring precomputed partial bitstreams. This last characteristic was possible by performing reverse engineering on the bitstream. Besides, we adapted this hardware-based solution to provide IP cores accessible from the MicroBlaze processor. To this end, the controller was extended and three versions were implemented to evaluate its performance when connected to Peripheral Local Bus (PLB), Fast Simplex Link (FSL), and AXI interfaces of the processor. In consequence, the controller can exploit the flexibility that the processor offers but taking advantage of the hardware speed-up. It was implemented in both Virtex-5 and Kintex7 FPGAs. Results of reconfiguration time showed that run-time reconfiguration of single LUTs in Virtex-5 devices was performed in less than 5 μs which implies a speed-up of more than 380x compared to the Xilinx XPS_HWICAP controller.

scholarly journals Combinatorial Expansions in $K$-Theoretic Bases

10.37236/2320 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Jason Bandlow ◽  
Jennifer Morse
Keyword(s):  
Generating Functions ◽  
Symmetric Functions ◽  
Schur Functions ◽  
Geometric Information ◽  
Combinatorial Description ◽  
Littlewood Polynomials ◽  
Stanley Symmetric Functions ◽  
K Theory ◽  
Combinatorial Representation

We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape.  Included in this class are the Hall-Littlewood polynomials, $k$-Schur functions, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur functions represent the cohomology of the Grassmannian variety of $GL_n$, Grothendieck functions $\{G_\lambda\}$ represent the $K$-theory of the same space.  In this paper, we give a combinatorial description of the coefficients when any element of $\mathcal C$ is expanded in the $G$-basis or the basis dual to $\{G_\lambda\}$.

scholarly journals The Expected Characteristic and Permanental Polynomials of the Random Gram Matrix

10.37236/3709 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Jacob G. Martin ◽  
E. Rodney Canfield
Keyword(s):  
Random Matrix ◽  
Generating Functions ◽  
Numerical Algorithms ◽  
Singular Values ◽  
Gram Matrix ◽  
Efficient Computation ◽  
Underlying Distribution ◽  
Characteristic Roots ◽  
Speed Up ◽  
Theoretical Results

A $t \times n$ random matrix $A$ can be formed by sampling $n$ independent random column vectors, each containing $t$ components. The random Gram matrix of size $n$, $G_{n}=A^{T}A$, contains the dot products between all pairs of column vectors in the randomly generated matrix $A$, and has characteristic roots coinciding with the singular values of $A$. Furthermore, the sequences $\det{(G_{i})}$ and $\text{perm}(G_{i})$ (for $i = 0, 1, \dots, n$) are factors that comprise the expected coefficients of the characteristic and permanental polynomials of $G_{n}$. We prove theorems that relate the generating functions and recursions for the traces of matrix powers, expected characteristic coefficients, expected determinants $E(\det{(G_{n})})$, and expected permanents $E(\text{perm}(G_{n}))$ in terms of each other. Using the derived recursions, we exhibit the efficient computation of the expected determinant and expected permanent of a random Gram matrix $G_{n}$, formed according to any underlying distribution. These theoretical results may be used both to speed up numerical algorithms and to investigate the numerical properties of the expected characteristic and permanental coefficients of any matrix comprised of independently sampled columns.

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