scholarly journals A Formal Approach to the Description and Manipulation of Structured Mathematical Objects

2005 ◽  
Vol Volume 3, Special Issue... ◽  
Author(s):  
Bernard Fotsing Talla ◽  
Georges-Edouard Kouamou
Keyword(s):  
Square Root ◽  
Two Dimensional ◽  
Attribute Grammars ◽  
Formal Approach ◽  
Mathematical Symbols ◽  
Nous Montrons ◽  
The Matrix ◽  
Mathematical Formulas ◽  
International Audience ◽  
Structured Objects

International audience We present in this paper a formal approach of description, posting and handling of the mathematical structured objects; based on the formalism of attribute grammars. We are interested particularly in the problem of two-dimensional and bidirectional posting of certain expressions and mathematical formulas. Indeed, in more of the two-dimensional character that presents certain mathematical symbols like the square root or the matrix, we also note the problem of posting rightto-left of an Arab text in a context planned for a posting left-to-right of an Indo-European text, or a bidirectional posting mixing the two modes. After a study of some solutions suggested in the literature, we show how the method of attribute grammars adapts easily to these types of problem. Nous présentons dans ce papier une approche formelle de description, d'affichage et de manipulation des objets structurés mathématiques ; basée sur le formalisme des grammaires attribuées. Nous nous intéressons particulièrement au problème d'affichage bidimensionnel et bidirectionnel de certaines expressions et formules mathématiques. En effet, en plus du caractère bidimensionnel que présentent certains symboles comme la racine carrée ou la matrice, on note le problème d'affichage de droite à gauche d'un texte arabe dans un contexte prévu pour un affichage de gauche à droite d'un texte indo-européen, ou encore un affichage bidirectionnel mélangeant les deux modes. Après une étude de quelques méthodes proposées dans la littérature, nous montrons comment la méthode des grammaires attribuées s'adapte facilement à ces types de problèmes.

scholarly journals Interactive layout and handling of mathematical formulas in structured documents

2002 ◽  
Vol Volume 1, 2002 ◽  
Author(s):  
Hanane Naciri ◽  
Laurence Rideau
Keyword(s):  
User Interfaces ◽  
Graphical User Interfaces ◽  
Two Dimensional ◽  
Mathematical Proofs ◽  
Nous Montrons ◽  
Mathematical Formulas ◽  
International Audience ◽  
Structured Documents

International audience Tools dedicated to mathematics need to display formulas and to interact with them. In this paper, we present a summary of existing tools, then we describe FIGUE, an incremental two dimensional layout engine, developed at INRIA, to get a specialized toolbox for building customized editors and graphical user interfaces. Finally we give an exemple of interface using FIGUE to develop mathematical proofs on computer. Afficher des formules mathématiques et interagir avec ces formules sont des atouts primordiaux pour les outils informatiques dédiés aux mathématiques. Dans cet article, nous faisons un bilan des outils existants puis nous décrivons FIGUE, moteur d'affichage interactif incrémental et bidimensionnel, développé à l'INRIA, pour obtenir une bibliothèque dédiée au développement d'éditeurs de documents structurés et d'interfaces graphiques. Enfin nous montrons un exemple d'utilisation de FIGUE, dans le cadre du développement de preuves mathématiques sur ordinateur.

scholarly journals Tunable characteristics of low-frequency bandgaps in two-dimensional multivibrator phononic crystal plates under prestrain

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Hai-Fei Zhu ◽  
Xiao-Wei Sun ◽  
Ting Song ◽  
Xiao-Dong Wen ◽  
Xi-Xuan Liu ◽  
...  
Keyword(s):  
Phononic Crystal ◽  
Real Life ◽  
Low Frequency ◽  
Acoustic Vibration ◽  
Crystal Plate ◽  
Frequency Noise ◽  
Two Dimensional ◽  
The Matrix ◽  
Noise Frequency ◽  
Realtime Control

AbstractIn view of the influence of variability of low-frequency noise frequency on noise prevention in real life, we present a novel two-dimensional tunable phononic crystal plate which is consisted of lead columns deposited in a silicone rubber plate with periodic holes and calculate its bandgap characteristics by finite element method. The low-frequency bandgap mechanism of the designed model is discussed simultaneously. Accordingly, the influence of geometric parameters of the phononic crystal plate on the bandgap characteristics is analyzed and the bandgap adjustability under prestretch strain is further studied. Results show that the new designed phononic crystal plate has lower bandgap starting frequency and wider bandwidth than the traditional single-sided structure, which is due to the coupling between the resonance mode of the scatterer and the long traveling wave in the matrix with the introduction of periodic holes. Applying prestretch strain to the matrix can realize active realtime control of low-frequency bandgap under slight deformation and broaden the low-frequency bandgap, which can be explained as the multiple bands tend to be flattened due to the localization degree of unit cell vibration increases with the rise of prestrain. The presented structure improves the realtime adjustability of sound isolation and vibration reduction frequency for phononic crystal in complex acoustic vibration environments.

scholarly journals Divergent ⇒ complex amplitudes in two dimensional string theory

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Ashoke Sen
Keyword(s):  
Scattering Amplitudes ◽  
String Theory ◽  
Matrix Model ◽  
Two Dimensional ◽  
Full Matrix ◽  
Instanton Contribution ◽  
The Matrix ◽  
Theory Result ◽  
The One ◽  
Remarkable Agreement

Abstract In a recent paper, Balthazar, Rodriguez and Yin found remarkable agreement between the one instanton contribution to the scattering amplitudes of two dimensional string theory and those in the matrix model to the first subleading order. The comparison was carried out numerically by analytically continuing the external energies to imaginary values, since for real energies the string theory result diverges. We use insights from string field theory to give finite expressions for the string theory amplitudes for real energies. We also show analytically that the imaginary parts of the string theory amplitudes computed this way reproduce the full matrix model results for general scattering amplitudes involving multiple closed strings.

scholarly journals A new fourth-order explicit group method in the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Muhammad Asim Khan ◽  
Norhashidah Hj. Mohd Ali ◽  
Nur Nadiah Abd Hamid
Keyword(s):  
Finite Difference ◽  
Stokes Problem ◽  
Stability And Convergence ◽  
Second Grade ◽  
Group Method ◽  
Two Dimensional ◽  
Second Grade Fluid ◽  
The Matrix ◽  
Grade Fluid ◽  
Generalized Second Grade Fluid

Abstract In this article, a new explicit group iterative scheme is developed for the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid. The proposed scheme is based on the high-order compact Crank–Nicolson finite difference method. The resulting scheme consists of three-level finite difference approximations. The stability and convergence of the proposed method are studied using the matrix energy method. Finally, some numerical examples are provided to show the accuracy of the proposed method.

scholarly journals Lymphocyte locomotion and attachment on two-dimensional surfaces and in three-dimensional matrices

1982 ◽  
Vol 92 (3) ◽  
pp. 747-752 ◽  
Author(s):  
WS Haston ◽  
JM Shields ◽  
PC Wilkinson
Keyword(s):  
Three Dimensional ◽  
Two Dimensional ◽  
Collagen Gels ◽  
Peripheral Lymph Node ◽  
Cellulose Esters ◽  
Peripheral Lymph ◽  
The Matrix ◽  
Identical Material ◽  
Variable Morphology ◽  
Collagen Lattices

The adhesion and locomotion of mouse peripheral lymph node lymphocytes on 2-D protein- coated substrata and in 3-D matrices were compared. Lymphocytes did not adhere to, or migrate on, 2-D substrata suck as serum- or fibronectin-coated glass. They did attach to and migrate in hydrated 3-D collagen lattices. When the collagen was dehydrated to form a 2-D surface, lymphocyte attachment to it was reduced. We propose that lymphocytes, which are poorly adhesive, are able to attach to and migrate in 3-D matrices by a nonadhesive mechanism such as the extension and expansion of pseudopodia through gaps in the matrix, which could provide purchase for movement in the absence of discrete intermolecular adhesions. This was supported by studies using serum-coated micropore filters, since lymphocytes attached to and migrated into filters with pore sizes large enough (3 or 8 mum) to allow pseudopod penetration but did not attach to filters made of an identical material (cellulose esters) but of narrow pore size (0.22 or 0.45 mum). Cinematographic studies of lymphocyte locomotion in collagen gels were also consistent with the above hypothesis, since lymphocytes showed a more variable morphology than is typically seen on plane surfaces, with formation of many small pseudopodia expanded to give a marked constriction between the cell and the pseudopod. These extensions often remained fixed with respect to the environment as the lymphocyte moved away from or past them. This suggests that the pseudopodia were inserted into gaps in the gel matrix and acted as anchorage points for locomotion.

scholarly journals Evaluation of Matrix Square Root Operations for UKF within a UAV GPS/INS Sensor Fusion Application

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Matthew Rhudy ◽  
Yu Gu ◽  
Jason Gross ◽  
Marcello R. Napolitano
Keyword(s):  
Sensor Fusion ◽  
Unscented Kalman Filter ◽  
Attitude Estimation ◽  
Square Root ◽  
Fusion Algorithm ◽  
Matrix Square Root ◽  
Global Positioning ◽  
The Matrix ◽  
Aerial Vehicle ◽  
Diagonalization Method

Using an Unscented Kalman Filter (UKF) as the nonlinear estimator within a Global Positioning System/Inertial Navigation System (GPS/INS) sensor fusion algorithm for attitude estimation, various methods of calculating the matrix square root were discussed and compared. Specifically, the diagonalization method, Schur method, Cholesky method, and five different iterative methods were compared. Additionally, a different method of handling the matrix square root requirement, the square-root UKF (SR-UKF), was evaluated. The different matrix square root calculations were compared based on computational requirements and the sensor fusion attitude estimation performance, which was evaluated using flight data from an Unmanned Aerial Vehicle (UAV). The roll and pitch angle estimates were compared with independently measured values from a high quality mechanical vertical gyroscope. This manuscript represents the first comprehensive analysis of the matrix square root calculations in the context of UKF. From this analysis, it was determined that the best overall matrix square root calculation for UKF applications in terms of performance and execution time is the Cholesky method.

Finite Element Study of Oxygen Diffusion in the Intervertebral Disc

2000 ◽  
Author(s):  
E. Sélard ◽  
A. Shirazi-Adl ◽  
J. P. G. Urban
Keyword(s):  
Intervertebral Disc ◽  
Blood Supply ◽  
Oxygen Diffusion ◽  
Consumption Rate ◽  
The Body ◽  
Cellular Viability ◽  
Two Dimensional ◽  
Concentration Gradients ◽  
Exchange Area ◽  
The Matrix

Abstract The intervertebral disc consists of a water-rich extra-cellular matrix which is synthesized and maintained by its cells. The disc is the largest avascular tissue in the body with its cells lying as much as 8mm away from the blood supply. Nutrients, essential for maintaining cellular viability, diffuse through the matrix from blood supply under a concentration gradient arising from cellular demand. The oxygen concentration gradients in the intervertebral disc are investigated to examine the effects of exchange area and disc thickness on oxygen flux in the disc. The concentration gradients are computed using the two-dimensional Poisson’s equation and measured values for oxygen consumption rate and oxygen diffusion.

Service quality function deployment by the C-shaped QFD 3D matrix

2018 ◽  
Vol 25 (9) ◽  
pp. 3386-3405 ◽  
Author(s):  
Maryam Hassani ◽  
Arash Shahin ◽  
Manouchehr Kheradmandnia
Keyword(s):  
Quality Function Deployment ◽  
Design Methodology ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Customer Requirements ◽  
Content Type ◽  
Process Characteristics ◽  
The Matrix ◽  
3D Matrix ◽  
Simultaneous Comparison

Purpose The purpose of this paper is to examine the application of C-shaped QFD 3D Matrix in comparing process characteristics (PC), performance aspects (PA) and customer requirements, simultaneously and to prioritize the first two sets, respectively. Design/methodology/approach A three dimensional matrix has been developed with three sets of PC, PA and customers’ requirements and C-shaped matrix has been applied for simultaneous comparison of the dimensions and prioritization of the subsets of PC and PA. The proposed approach has been examined in a post bank. Findings Findings confirm the possibility of simultaneous comparison and prioritization of the three sets of dimensions of this study in post bank services. In addition, “growth and learning” and “bilateral relationship with suppliers” had the first priorities among PA and PC, respectively. Research limitations/implications While the proposed approach has many advantages, filling the matrixes is time-consuming. Since illustrating the 3D matrix was not possible, the matrix was separated into five two-dimensional matrixes. Originality/value Compared to the studied literature, the proposed approach is practically new in the post bank services.

SHARP POINCARÉ–SOBOLEV INEQUALITIES AND THE SHORTEST LENGTH OF SIMPLE CLOSED GEODESICS ON A TOPOLOGICAL TWO SPHERE

2004 ◽  
Vol 06 (05) ◽  
pp. 781-792 ◽  
Author(s):  
MEIJUN ZHU
Keyword(s):  
Riemannian Manifold ◽  
Sobolev Inequalities ◽  
Square Root ◽  
Two Dimensional ◽  
Closed Geodesics ◽  
Sharp Constants ◽  
Simple Closed Geodesics

We show that the sharp constants of Poincaré–Sobolev inequalities for any smooth two dimensional Riemannian manifold are less than or equal to [Formula: see text]. For a smooth topological two sphere M2, the sharp constants are [Formula: see text] if and only if M2 is isometric to two sphere S2 with the standard metric. In the same spirit, we show that for certain special smooth topological sphere the ratio between the shortest length of simple closed geodesics and the square root of its area is less than or equals to [Formula: see text].

scholarly journals Macdonald polynomials at $t=q^k$

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Jean-Gabriel Luque
Keyword(s):  
Macdonald Polynomials ◽  
Nous Montrons ◽  
International Audience

International audience We investigate the homogeneous symmetric Macdonald polynomials $P_{\lambda} (\mathbb{X} ;q,t)$ for the specialization $t=q^k$. We show an identity relying the polynomials $P_{\lambda} (\mathbb{X} ;q,q^k)$ and $P_{\lambda} (\frac{1-q}{1-q^k}\mathbb{X} ;q,q^k)$. As a consequence, we describe an operator whose eigenvalues characterize the polynomials $P_{\lambda} (\mathbb{X} ;q,q^k)$. Nous nous intéressons aux propriétés des polynômes de Macdonald symétriques $P_{\lambda} (\mathbb{X} ;q,t)$ pour la spécialisation $t=q^k$. En particulier nous montrons une égalité reliant les polynômes $P_{\lambda} (\mathbb{X} ;q,q^k)$ et $P_{\lambda} (\frac{1-q}{1-q^k}\mathbb{X} ;q,q^k)$. Nous en déduisons la description d'un opérateur dont les valeurs propres caractérisent les polynômes $P_{\lambda} (\mathbb{X} ;q,q^k)$.

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