scholarly journals The Primitive Derivation and Discrete Integrals

2021 ◽  
Author(s):  
Daisuke Suyama ◽  
◽  
Masahiko Yoshinaga ◽  
Keyword(s):  
Integral Formula ◽  
Root Systems ◽  
Type A ◽  
Integral Formulas ◽  
Discrete Analogues

he modules of logarithmic derivations for the (extended) Catalan and Shi arrangements associated with root systems are known to be free. However, except for a few cases, explicit bases for such modules are not known. In this paper, we construct explicit bases for type A root systems. Our construction is based on Bandlow-Musiker's integral formula for a basis of the space of quasiinvariants. The integral formula can be considered as an expression for the inverse of the primitive derivation introduced by K. Saito. We prove that the discrete analogues of the integral formulas provide bases for Catalan and Shi arrangements.

scholarly journals Integral formulas with weight factors

1992 ◽  
Vol 52 (1) ◽  
pp. 26-39
Author(s):  
Telemachos Hatziafratis
Keyword(s):  
Complete Intersection ◽  
Integral Formula ◽  
Integral Formulas ◽  
Weight Factors ◽  
Type Integral

AbstractA Bochner-Martinelli-Koppelman type integral formula with weight factors is derived on complete intersection submanifolds of domains of Cn.

SOME INTEGRAL FORMULAS FOR THE SOLUTIONS OF THE sl2 dKZ EQUATION WITH LEVEL −4

1994 ◽  
Vol 09 (32) ◽  
pp. 5673-5687 ◽  
Author(s):  
ATSUSHI NAKAYASHIKI
Keyword(s):  
Correlation Function ◽  
Integral Formula ◽  
Time Integration ◽  
Direct Proof ◽  
Vertex Operators ◽  
Type I ◽  
Xxz Model ◽  
Integral Formulas ◽  
Xxx Model

A direct proof is given for the fact that the integral formula for the XXX limit of the trace of the type I q-vertex operators satisfies the deformed Knizhnik-Zamolodchikov (dKZ) equation with level −4. We have also carried out one-time integration by taking the residue at infinity. As a corollary of these we can construct a family of the integral formulas for solutions to the dKZ equation. Another corollary is the integral formula for the correlation function of the inhomogeneous XXX model, whose number of integrals is less than that of the previously obtained correlator for the XXZ model.

Parameters Analysis of Line Segment Fabrication on Near-Field Photolithography

2012 ◽  
Vol 579 ◽  
pp. 330-337
Author(s):  
Ching Been Yang ◽  
Hsiu Lu Chiang ◽  
Jaw Ren Lin
Keyword(s):  
Energy Density ◽  
Integral Formula ◽  
Near Field ◽  
Relative Concentration ◽  
Work Piece ◽  
Integral Formulas ◽  
Group A ◽  
Parameters Analysis ◽  
Group B ◽  
Exposure Energy

This paper proposes the integral formula of exposure energy density during the movement of work piece to investigate the exposure energy distribution on the photoresist surface. The photoresist was divided into finite nodes to combine the integral formulas of exposure energy density to calculate the relative concentration variation of the photoactive compound (PAC) at each finite node of photoresist interior layer. This paper further combines with Mack’s development model and calculates the average full-width at half maximum (FWHM) of near field photolithography. This study conducted sensitivity analysis to determine how adjusting groups of control parameters influences FWHM and working depth (Hmax). Group A, in which the probe aperture was the adjusted parameter, had the most influence, followed by Group B, in which exposure energy/μm was the adjusted parameter, and Group C, in which development time was the adjusted parameter.

scholarly journals The modified integral method for the determination of gravity disturbance near the Earth’s surface

2019 ◽  
Vol 9 (1) ◽  
pp. 65-70
Author(s):  
D. Zhao ◽  
S. Li ◽  
Q. Wang ◽  
Z. Gong
Keyword(s):  
Integral Formula ◽  
Radial Component ◽  
Target Point ◽  
Surface Point ◽  
Gravity Disturbance ◽  
Integral Formulas ◽  
Modified Formula ◽  
Stokes Kernel ◽  
Better Than

Abstract For the calculation of gravity disturbance in the Earth’s external gravity field, the Stokes-Pizzetti integral is a commonly used method. However, when the target point approaches the Earth’s surface, such problems as singularity and discontinuity arise due to the Stokes kernel structure itself. To settle the problems, firstly the reason for singularity and discontinuity was discussed, and then modification was made to the integral formula, by which the singularity at the surface point is eliminated. Finally the non-singular integral formulas for the calculation of disturbing gravity were derived. In numerical experiments, an area in China was selected to test the modified formula. Numerical results show that the modified formula performs much better than classical Stokes-Pizzetti integral formula when dealing with the calculation of the radial component of gravity disturbance near the Earth’s surface.

scholarly journals ON GENERAL EULERIAN INTEGRAL FORMULAS AND FRACTIONAL INTEGRATION

1999 ◽  
Vol 30 (2) ◽  
pp. 155-164
Author(s):  
K. C. GUPTA ◽  
S. P. GOYAL ◽  
R. K. LADDHA
Keyword(s):  
Integral Operator ◽  
Infinite Series ◽  
Fractional Integral ◽  
Integral Formula ◽  
Polynomial System ◽  
Fractional Integration ◽  
Compact Form ◽  
Integral Formulas ◽  
Special Cases ◽  
Type Integral

In the present work, we evaluate a unified Eulerian type integral whose integrand involves the product of a polynomial system and the multivariable H-function having general arguments. Our integral formula encompasses a very large number of integrals and provides interesting unifieation and extensions of several known (e.g., [1], [3], [4], [5], [9], [11], etc.) and new results. Since the integral has been given in a compact form free from infinite series, it is likely to prove useful in applications. Three special cases of the main integral (which are also sufficiently general in nature and are of interest in themselves) have also been given. Finally, the main integral formula has been expressed as a fractional integral operator to make it more useful in applications.

scholarly journals A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 62 ◽  
Author(s):  
Maxie D. Schmidt
Keyword(s):  
Generating Functions ◽  
Hadamard Product ◽  
Integral Formula ◽  
Short Note ◽  
Integral Representations ◽  
Integral Transformations ◽  
Integral Formulas ◽  
Exponential Generating Function ◽  
The Laplace Transform ◽  
Integration Techniques

The purpose of this note is to provide an expository introduction to some more curious integral formulas and transformations involving generating functions. We seek to generalize these results and integral representations which effectively provide a mechanism for converting between a sequence’s ordinary and exponential generating function (OGF and EGF, respectively) and vice versa. The Laplace transform provides an integral formula for the EGF-to-OGF transformation, where the reverse OGF-to-EGF operation requires more careful integration techniques. We prove two variants of the OGF-to-EGF transformation integrals from the Hankel loop contour for the reciprocal gamma function and from Fourier series expansions of integral representations for the Hadamard product of two generating functions, respectively. We also suggest several generalizations of these integral formulas and provide new examples along the way.

scholarly journals Some results on Ricci-Bourguignon solitons and almost solitons

2020 ◽  
pp. 1-14
Author(s):  
Shubham Dwivedi
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Integral Formula ◽  
Constant Scalar Curvature ◽  
Ricci Solitons ◽  
Euclidean Sphere ◽  
Integral Formulas

Abstract We prove some results for the solitons of the Ricci–Bourguignon flow, generalizing the corresponding results for Ricci solitons. Taking motivation from Ricci almost solitons, we then introduce the notion of Ricci–Bourguignon almost solitons and prove some results about them that generalize previous results for Ricci almost solitons. We also derive integral formulas for compact gradient Ricci–Bourguignon solitons and compact gradient Ricci–Bourguignon almost solitons. Finally, using the integral formula, we show that a compact gradient Ricci–Bourguignon almost soliton is isometric to a Euclidean sphere if it has constant scalar curvature or its associated vector field is conformal.

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