Generalized Hermite Reduction, Creative Telescoping and Definite Integration of D-Finite Functions

The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative A-polynomial of twist knots. Our multivariable creative telescoping method allows us to compute linear recursions for sums of the form [Formula: see text] given a recursion relation for [Formula: see text] and the hypergeometric kernel c(n, k). As an application of our method, we explicitly compute the non-commutative A-polynomial for twist knots with -15 and 15 crossings. The non-commutative A-polynomial of a knot encodes the monic, linear, minimal order q-difference equation satisfied by the sequence of colored Jones polynomials of the knot. Its specialization to q = 1 is conjectured to be the better-known A-polynomial of a knot, which encodes important information about the geometry and topology of the knot complement. Unlike the case of the Jones polynomial, which is easily computable for knots with 50 crossings, the A-polynomial is harder to compute and already unknown for some knots with 12 crossings.

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Stationary random vibration of a viscoelastic Timoshenko cantilever beam under diverse random processes

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1177/0954406219885960 ◽

2019 ◽

Vol 234 (4) ◽

pp. 849-861

Author(s):

Qingzhao Zhou ◽

David He ◽

Yaping Zhao

Keyword(s):

White Noise ◽

Cantilever Beam ◽

Time Correlation ◽

Random Excitation ◽

Transverse Displacement ◽

Mean Square ◽

Concentrated Force ◽

The Mean ◽

Definite Integration ◽

Band Limited

In this paper, the stochastic properties of a uniform Timoshenko cantilever beam are investigated systematically. Based on the external viscous damping and Kelvin–Voigt viscoelastic damping, the partial differential equations of the Timoshenko beam subjected to random excitation are derived. The applied load is the concentrated force, and the excitation related to includes the ideal white noise, the band-limited white noise, and the exponential noise. Expressions are obtained for the space–time correlation functions and the space–frequency power spectral density functions of the transverse displacement response. The evident improvement is that the infinite integral and the definite integration in the mean square responses are worked out by means of the residue integral method and the integration by partial fraction, and the exact solutions of the mean square response are obtained in the form of an infinite series finally. This improvement provides a basis for both the mode truncation and the modal cross-spectral densities whether which can be ignored. Providing the numerical example, the numerical results obtained show the effectiveness of the theoretical analysis.

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Adaptive synchronization of uncertain chaotic systems with definite integration scaling function

Optik ◽

10.1016/j.ijleo.2015.05.049 ◽

2015 ◽

Vol 126 (19) ◽

pp. 1999-2002 ◽

Cited By ~ 2

Author(s):

Weixun Gao ◽

Jun Li ◽

Yuhua Xu

Keyword(s):

Chaotic Systems ◽

Scaling Function ◽

Adaptive Synchronization ◽

Uncertain Chaotic Systems ◽

Definite Integration

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Hermite reduction and creative telescoping for hyperexponential functions

Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation - ISSAC '13 ◽

10.1145/2465506.2465946 ◽

2013 ◽

Cited By ~ 22

Author(s):

Alin Bostan ◽

Shaoshi Chen ◽

Frédéric Chyzak ◽

Ziming Li ◽

Guoce Xin

Keyword(s):

Creative Telescoping

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The Effect of Friction at the Strip Footings’ Underside on the Expansion of Plastic Zones in Subgrade

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.1008-1009.1227 ◽

2014 ◽

Vol 1008-1009 ◽

pp. 1227-1233

Author(s):

Nian Chun Xu ◽

Wen Jing Xia ◽

Tong Qing Wu

Keyword(s):

Critical Load ◽

Bearing Capacity ◽

Ultimate Bearing Capacity ◽

Horizontal Direction ◽

Strength Theory ◽

Subgrade Soil ◽

Plastic Zones ◽

Definite Integration ◽

Strip Footings ◽

Two Sides

There exists horizontal friction besides vertically pressure at the foundations' underside. Considering the effect of friction on the expansion of plastic zones in subgrade is needed to accurately evaluate subgrade’s safety. A strip footing is chose as the research object. Assuming the distribution of friction at the strip footing’s underside is two symmetrical triangles. With the help of Flamant formula and via definite integration, the formulas of stress in subgrade induced by the friction are got. Setting the Coulomb-Mohr strength theory as the yielding criterion for the subgrade soil, through the comparison among the different friction angles in expansion characteristics of plastic zones, the research object is achieved. Two major conclusions as following: (1) the friction makes the plastic zones appear in advance, the initial critical load get smaller with the friction get bigger; (2) the plastic zones get broader in horizontal direction under the action of the friction, this makes the plastic zones in two sides beneath the footing run-through later and so enhances the subgrade’s ultimate bearing capacity.

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