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# Star time on main sequence

## Homework Statement

A B0 star has a luminosity of ~ 16000 L(solar) and a mass of ~ 16 M(solar). Can you estimate the time on the main sequence for this star simply from considering the rest mass energy of the core of the star (~10% of the total, and fusion burns with an efficiency of about 0.7%) and its luminosity?

## Homework Equations

I know that for main sequence stars, L is proportional to M^3

## The Attempt at a Solution

I have no idea where to go with this question, i've trawled through so many webpages and textbooks without success. Any ideas?

Dick


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# The boy and the bird

Once there was a tree, and a bird landed on it and started pulling the leaves off. Eventually a boy came along and ate the bird, and the tree grew back its leaves.

What happened?

Hint:

This happens on a regular basis

• Nice riddle! :D – Feeds Sep 2 '18 at 16:41

Might it be

'The changing of seasons' in which the bird would be autumn or abscission (The process in which a tree loses its leaves). The boy would then be the change to a next season (getting rid of the bird). In which the tree starts to get his leaves back. And of course the changing of seasons occurs on a regular basis

• Yes! Well done on getting the answer – Rohit Jose Sep 3 '18 at 8:45
• @RohitJose Nice riddle had me wondering for some time, until this morning I had a discussion with a friend of mine between summer and winter hours and then it struck me :p. – mrdeadsven Sep 3 '18 at 8:50

No laughing, but:

The boy's name was Robin. So:

Robin ate the robin who was robbin' pieces of the Picea Rubens, and the Picea Rubens re-budded.

• What... I have no idea what you said but thank you for taking the effort of saying it! – Rohit Jose Sep 2 '18 at 17:31

Answered before the hint was included.

This mimics the story of

The Selfish Giant by Oscar Wilde (one of my favourite tales!).

Once there was a tree,

The Giant's beautiful garden.

and a bird landed on it

The Giant surrounds his garden with a high wall and a notice-board (the bird; "bird" sounds like "board") reading TRESPASSERS WILL BE PROSECUTED.

and started pulling the leaves off

There suddenly came an eternal winter in the Giant's garden for which all the leaves fell off and the trees bore no golden fruit.

Eventually a boy came along

The Giant.

and ate the bird

The Giant took a great axe ("axe" sounds like "ate") and cut down the high wall.

and the tree grew back its leaves.

The Giant puts a crying boy up into a tree, wherefore it blossoms and the Garden stays beautiful again, forever and ever.

Both stories are split into three parts:

1. The tree with the bird. (Expulsion from the Garden)

1. The bird pulling off the leaves. (Dances of Eternal Winter)

1. The leaves grew back and the bird is gone. (The Return of Spring)

• I read that story! It's very nice and an interesting interpretation of my riddle, but again, not the correct answer. – Rohit Jose Sep 2 '18 at 17:29
• @RohitJose I thought I had it for a sec... eh, oh well. I need to go to bed. I want to see if by, say, tomorrow morning, someone has answered it correctly; I really want to know what the answer is :D – Feeds Sep 2 '18 at 17:34
• Well, if anything were to solve a riddle, it would be stack exchange puzzling, right? – Rohit Jose Sep 2 '18 at 17:43
• @RohitJose and there you 'ave it, folks! It's our new motto! :P – Feeds Sep 2 '18 at 17:46

Time elapsed.
Time will heal each and every thing happened to you, no matter how horrible it may be.

• That's very poetic of you, but not the right answer – Rohit Jose Sep 2 '18 at 16:27

format: plain


text: ER 26 26 VE 18 18 SP 4 4 Supporting Materials Periodic Table Fundamental Constants Drag each item to the appropriate bin. The Lewis dot structures of NO2– highlight a difference in the bond order of the two N-O bonds. It is the simplest carboxylate anion. There are a total of 18 valence electrons in the HCO 2 - Lewis structure . A formate (compound) is a salt or ester of formic acid. Solution for The formate ion is HCOO-. A Lewis structure of OCl- ion is drawn below. The other atoms are bonded only to the carbon atom. (e) How many electrons are in the π system of the ion? Include All Lone Pairs Of Electrons. Compared to the more common ethyl esters, formate esters are less commonly used commercially because they are less stable. (Assign lone pairs, radical electrons, and atomic charges where appropriate.) [1], Werner Reutemann and Heinz Kieczka "Formic Acid" in. Which statement best describes the bonding character of this ion? They follow the duet rule (2 electrons). Production. Methyl formate, also called methyl methanoate, is the methyl ester of formic acid. ... difference between the number of valence electrons in a free atom and the number of electrons assigned to the atom in a Lewis structure. (a) Draw a Lewis structure for the formate ion. The formal charge of an atom in a molecule is the hypothetical charge the atom would have if we could redistribute the electrons in the bonds evenly between the atoms. It is the simplest carboxylate anion. Formate (IUPAC name: methanoate) is the anion derived from formic acid. Because carbon is the least electronegative element, we place it in the central position: HCO2-is also called Formate Ion. In the formate ion, $\mathrm{HCO}_{2}^{-}$ , the carbon atom is the central atom with the other three atoms attached to it. Draw the Lewis structure of the formate ion HCO 2 Assign lone pairs radical from CH 101 at North Carolina State University (b) What hybridization is exhibited by the C atom? What is the Lewis structure for hco2? The most important formate ester is methyl formate, which is produced as an intermediate en route to formic acid. 1. Aliphatic Compounds" in Ullmann's Encyclopedia of Industrial Chemistry, 2015, Wiley-VCH, Weinheim. For example, hydrated nickel formate decarboxylates at about 200 °C to give finely powdered nickel metal: Such fine powders are useful as hydrogenation catalysts. In many cases, following the steps for writing Lewis structures may lead to more than one possible molecular structure—different multiple bond and lone-pair electron placements or different arrangements of atoms, for instance. (e) 007 James Bonds. Decide which is the central atom in the structure. The exception, of course, being the hydrogen's. Draw the Lewis structure of the formate ion, HCO 2 −. The arrows show how you can think of the electronsshifting as one resonance structure changes to another. BTo explore the relationship between bond order and bond length. How many possible resonance structures exist for the formate ion, HCO2−? It is the product of deprotonation of formic acid. It is a precursor to many other compounds of commercial interest. With HCO2-you'll need to form a double bond between one of the Oxygen atoms and the Carbon atom to fill the octets and still use … - Draw the best Lewis structure for the… They follow the duet rule (2 electrons). (d) One single bond and two equivalent 1 1/2 bonds. the first lewis structure of formate has a central carbon atom. //-->, Lewis Dot of the Formate Ion (methanoate). You cannot draw a Lewis structure that would suggest all three bonds are the same length. In the nitrite ion, the bond lengths of both nitrogen-oxygen bonds are equal. Johannes Panten and Horst Surburg "Flavors and Fragrances, 2. Draw a trial structure by putting electron pairs around every atom until each gets an octet. (d) How many electrons are in the $\pi$ system of the ion? Draw a skeleton structure in which the other atoms are single-bonded to the central atom: 3. Tell me about the atomic charges, dipole moment, bond lengths, angles, bond orders, molecular orbital energies, or total energy. The resonance hybrid of this polyatomic ion, obtained from its different resonance structures, can be used to explain the equal bond lengths, as illustrated below.The resonance hybrid of NO2– suggests that each oxygen atom holds a partial charge of magnitude -½. The following is the general form for resonance ina structure of this type. The b… That will normally be the least electronegative atom (C). Many form spontaneously when alcohols dissolve in formic acid. The exception, of course, being the hydrogen's. (d) Which of the atoms in the ion have p π orbitals? Tell me about the best Lewis structure. I figured out the Lewis structure and the true form, but I'm unsure as to why it is more stable compared to other anions. Its formula is represented in various equivalent ways: HCOO− or CHOO− or HCO2−. However, carbonate ion is an exception to the rule. It has a role as a human metabolite and a Saccharomyces cerevisiae metabolite. See the answer. 70 More Lewis Dot Structures. [3] Ethyl formate is found in some confectionaries.[1]. Unlike O 3, though, the actual structure of CO 3 2− is an average of three resonance structures. A single Lewis Structure of the formate ion (HCO 2-) is shown below. (c) Are there multiple equivalent resonance structures for the ion? Formate esters often are fragrant or have distinctive odors. Carbon (C) is the least electronegative atom and goes at the center of the HCO 2 - Lewis structure . https://en.wikipedia.org/w/index.php?title=Formate&oldid=891999456, Creative Commons Attribution-ShareAlike License, This page was last edited on 11 April 2019, at 14:50. (c) Are there multiple equivalent resonance structures for the ion? The H and the two O atoms are bonded to the central C atom, (a) Draw the best Lewis structure (s) for this ion. A formate (compound) is a salt or ester of formic acid. (d) How many electrons are in the $\pi$ system of the ion? Thus, we calculate formal charge as follows: Comparing the three formal charges, we can def… Such salts are prone to decarboxylation. 2. CH3O- ? 2. (b) What hybridization is exhibited by the C atom? Carbon is the central atom. Since all the atoms are in either period 1 or 2, this molecule will adhere to the octet rule. Consider the resonance structures of formate. The … (c) Are there multiple equivalent resonance structures for the ion? Answer and Explanation: The formate ion or methanoate, HCO − 2 2 − is composed of one atom each of H and C and two atoms O, with the ion itself having a charge of -1. (b) One double bond and two delocalized single bonds. Resonance structures have the same number of electrons, but may have a di erent number of shared pairs or the atoms in the structure may have di erent formal charges. Formate salts have the formula M(O2CH)(H2O)x. a hydrogen atom and two osygen atoms are bonded to the carbon atom. Structure, properties, spectra, suppliers and links for: Formate ion, Formate. 2. Hydrolysis of methyl formate gives formic acid and regenerates methanol: Formic acid is used for many applications in industry. It is the product of deprotonation of formic acid. O. H-C-H. Two electrons are being shared between the carbon atom and the oxygen atom. A number of possible resonance structures for this ion are shown. Which of these structures are valid and which are invalid? Atomic Charges and Dipole Moment C1 charge= 0.728 O2 charge=-0.780 O3 charge=-0.779 H4 charge=-0.168 with a dipole moment of 1.33721 Debye Bond Lengths: 1. View Answer The ion, PO4^(3-), possesses more than one resonance structure. The formate ion, \rm {HCO_2}^-, is formed when formic acid dissolves in water. In fact, carbonate ion is a symmetric, trigonal planar molecule. (No Response) Calculate the electrons required (ER), valence electrons (VE), and shared pairs (SP). The simplest example of an ester, it is a colorless liquid with an ethereal odour, high vapor pressure, and low surface tension. The ion charge is -1. (c) Three equivalent 1 1/3 bonds. Formate esters have the formula ROC(O)H (alternative way of writing formula RO2CH). This problem has been solved! Instead, you can use resonance structures to understand this fact.


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Browse Questions

# Which of the following diseases is due to an alergic reaction?

$\begin{array}{1 1}(A)\;\text{Goitre}\\(B)\;\text{Enteric fever}\\(C)\;\text{Skin cancer}\\(D)\;\text{Hay fever}\end{array}$

Hay fever
Hence (D) is the correct answer.


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## Shen, Jie

Compute Distance To:
 Author ID: shen.jie Published as: Shen, Jie; Shen, J. External Links: MGP
 Documents Indexed: 265 Publications since 1988, including 2 Books 5 Contributions as Editor Biographic References: 1 Publication Co-Authors: 108 Co-Authors with 192 Joint Publications 4,313 Co-Co-Authors
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### Co-Authors

 27 single-authored 22 Wang, Lilian 21 Yang, Xiaofeng 17 Liu, Chun 10 Cheng, Qing 10 Guo, Ben-Yu 8 Sheng, Changtao 8 Xu, Chuanju 8 Xu, Jie 8 Yu, Haijun 7 Chen, Feng 7 Feng, James J. 7 Guermond, Jean-Luc 7 Li, Xiaoli 7 López, Juan Manuel 6 Huang, Fukeng 6 Marques, Francisco 6 Yue, Pengtao 5 An, Jing 5 Mao, Zhiping 5 Wang, Qi 5 Wang, Shouhong 5 Wang, Yingwei 4 Bao, Weizhu 4 Chen, Lizhen 4 Ma, Lina 4 Nicholls, David P. 4 Tang, Tao 4 Temam, Roger Meyer 4 Wang, Zhongqing 4 Zhuang, Qingqu 3 Cai, Yongyong 3 Forest, Mark Gregory 3 Gu, Yiqi 3 He, Ying 3 Li, Huiyuan 3 Wise, Steven M. 3 Yang, Jiang 2 Alici, Haydar 2 Azaïez, Mejdi 2 Cao, Duo 2 Chen, Hongtao 2 Choi, Heejun 2 Fang, Qirong 2 Hu, Jingwei 2 Hu, Lina 2 Kwan, Yuen-Yick 2 Li, Peijun 2 Lu, Zhiming 2 Luo, Lishi 2 Mao, Jingjing 2 Michaux, Bertrand 2 Minev, Peter Dimitrov 2 Park, Jinhae 2 Pyo, Jae-Hong 2 Rakotoson, Jean-Michel 2 Sengul, Mustafa Taylan 2 Wang, Cheng 2 Wang, Langdi 2 Wang, Ruodu 2 Wang, Xiaoming 2 Wu, Jiahong 2 Wu, Ke 2 Xia, Jianlin 2 Xiu, Dongbin 2 Xu, Chenglong 2 Yang, Zhiguo 2 Yuan, Juanming 2 Zhang, Xiangxiong 2 Zhang, Yanrong 2 Zhao, Jia 1 Antoine, Xavier 1 Brahim, M. Mint 1 Brauner, Claude-Michel 1 Cai, Jiaxiang 1 Chandy, Abhilash J. 1 Chen, Falai 1 Chen, Jia 1 Chen, Wenbin 1 Chen, Zhi 1 Cheng, Ting 1 Cheng, Yongping 1 Colbert-Kelly, Sean 1 Cui, Zhenlu 1 Dai, C. M. 1 Delafrouz, Pourya 1 Demkowicz, Leszek F. 1 Deng, Beichuan 1 Dijkstra, Henk A. 1 Du, Qiang 1 Fang, Yuqiang 1 Fang, Zhiwei 1 Feng, Xinlong 1 Fisher, Thomas J. 1 Frankel, Steven H. 1 Gallagher, Colin M. 1 Gebremariam, Tesfay 1 Goubet, Olivier 1 Gresho, Philip M. 1 Guo, Benqi 1 Guo, Zhichang ...and 72 more Co-Authors
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### Serials

 40 Journal of Computational Physics 27 SIAM Journal on Numerical Analysis 23 Journal of Scientific Computing 21 SIAM Journal on Scientific Computing 10 Mathematics of Computation 9 Applied Numerical Mathematics 9 Communications in Computational Physics 7 Computer Methods in Applied Mechanics and Engineering 7 Numerische Mathematik 5 Journal of Fluid Mechanics 5 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 5 Advances in Computational Mathematics 5 Discrete and Continuous Dynamical Systems 5 Communications in Mathematical Sciences 5 Numerical Mathematics: Theory, Methods and Applications 4 Journal of Computational and Applied Mathematics 4 Discrete and Continuous Dynamical Systems. Series B 4 International Journal of Numerical Analysis and Modeling 3 Mathematical Methods in the Applied Sciences 3 Physica D 3 Mathematical Problems in Engineering 3 Journal of Mathematical Study 3 East Asian Journal on Applied Mathematics 2 IMA Journal of Numerical Analysis 2 International Journal for Numerical Methods in Fluids 2 Chinese Annals of Mathematics. Series B 2 Journal of Computational Mathematics 2 RAIRO. Modélisation Mathématique et Analyse Numérique 2 Applied Mathematics Letters 2 Machine Learning 2 Journal of Non-Newtonian Fluid Mechanics 2 Physics of Fluids 2 Journal of Liaoning Normal University. Natural Science Edition 2 Scientia Sinica. Mathematica 1 Applicable Analysis 1 Computers and Fluids 1 Computers & Mathematics with Applications 1 Computer Physics Communications 1 International Journal of Theoretical Physics 1 Physics Letters. A 1 Wave Motion 1 Fluid Dynamics Research 1 Indiana University Mathematics Journal 1 International Journal for Numerical Methods in Engineering 1 Numerical Functional Analysis and Optimization 1 Acta Applicandae Mathematicae 1 Journal of Statistical Computation and Simulation 1 SIAM Journal on Applied Mathematics 1 SIAM Review 1 Stochastic Processes and their Applications 1 Matemática Aplicada e Computacional 1 Advances in Differential Equations 1 Finance and Stochastics 1 Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 1 IEEE Transactions on Image Processing 1 Foundations of Computational Mathematics 1 Journal of Machine Learning Research (JMLR) 1 Multiscale Modeling & Simulation 1 Mathematical Biosciences and Engineering 1 Springer Series in Computational Mathematics 1 Computational & Mathematical Methods in Medicine 1 Kinetic and Related Models 1 Science China. Mathematics 1 Mathematics Monograph Series 1 Communications on Applied Mathematics and Computation
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### Fields

 205 Numerical analysis (65-XX) 128 Partial differential equations (35-XX) 103 Fluid mechanics (76-XX) 14 Approximations and expansions (41-XX) 13 Statistical mechanics, structure of matter (82-XX) 10 Optics, electromagnetic theory (78-XX) 8 Ordinary differential equations (34-XX) 7 Special functions (33-XX) 7 Dynamical systems and ergodic theory (37-XX) 7 Computer science (68-XX) 6 Real functions (26-XX) 6 Probability theory and stochastic processes (60-XX) 6 Quantum theory (81-XX) 5 Geophysics (86-XX) 5 Biology and other natural sciences (92-XX) 4 General and overarching topics; collections (00-XX) 4 Classical thermodynamics, heat transfer (80-XX) 4 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 3 Statistics (62-XX) 3 Mechanics of deformable solids (74-XX) 3 Operations research, mathematical programming (90-XX) 2 Harmonic analysis on Euclidean spaces (42-XX) 2 Systems theory; control (93-XX) 1 History and biography (01-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Potential theory (31-XX) 1 Integral equations (45-XX) 1 Operator theory (47-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Convex and discrete geometry (52-XX) 1 Information and communication theory, circuits (94-XX)

### Citations contained in zbMATH Open

230 Publications have been cited 7,965 times in 3,625 Documents Cited by Year
An overview of projection methods for incompressible flows. Zbl 1122.76072
Guermond, J. L.; Minev, P.; Shen, Jie
2006
Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Zbl 1201.65184
Shen, Jie; Yang, Xiaofeng
2010
Spectral methods. Algorithms, analysis and applications. Zbl 1227.65117
Shen, Jie; Tang, Tao; Wang, Li-Lian
2011
The scalar auxiliary variable (SAV) approach for gradient flows. Zbl 1380.65181
Shen, Jie; Xu, Jie; Yang, Jiang
2018
Efficient spectral-Galerkin method. I: Direct solvers of second- and fourth-order equations using Legendre polynomials. Zbl 0811.65097
Shen, Jie
1994
A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Zbl 1092.76069
Liu, Chun; Shen, Jie
2003
A diffuse-interface method for simulating two-phase flows of complex fluids. Zbl 1130.76437
Yue, Pengtao; Feng, James J.; Liu, Chun; Shen, Jie
2004
Spectral and high-order methods with applications. Zbl 1234.65005
Shen, Jie; Tang, Tao
2006
Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: Application to thin film epitaxy. Zbl 1247.65088
Shen, Jie; Wang, Cheng; Wang, Xiaoming; Wise, Steven M.
2012
A new class of efficient and robust energy stable schemes for gradient flows. Zbl 1422.65080
Shen, Jie; Xu, Jie; Yang, Jiang
2019
A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities. Zbl 1410.76464
Shen, Jie; Yang, Xiaofeng
2010
Generalized Jacobi functions and their applications to fractional differential equations. Zbl 1335.65066
Chen, Sheng; Shen, Jie; Wang, Li-Lian
2016
On error estimates of projection methods for Navier-Stokes equations: First-order schemes. Zbl 0741.76051
Shen, Jie
1992
Applications of semi-implicit Fourier-spectral method to phase field equations. Zbl 1017.65533
Chen, L. Q.; Shen, Jie
1998
Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows. Zbl 1327.65178
Shen, Jie; Yang, Xiaofeng
2015
Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows. Zbl 1403.65047
Shen, Jie; Xu, Jie
2018
Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method. Zbl 1158.76319
Yang, Xiaofeng; Feng, James J.; Liu, Chun; Shen, Jie
2006
Efficient spectral-Galerkin method. II: Direct solvers of second- and fourth-order equations using Chebyshev polynomials. Zbl 0840.65113
Shen, Jie
1995
On the error estimates for the rotational pressure-correction projection methods. Zbl 1093.76050
Guermond, J. L.; Shen, Jie
2004
On error estimates of the projection methods for the Navier-Stokes equations: Second-order schemes. Zbl 0855.76049
Shen, Jie
1996
Numerical approximations for a three-component Cahn-Hilliard phase-field model based on the invariant energy quadratization method. Zbl 1393.80003
Yang, Xiaofeng; Zhao, Jia; Wang, Qi; Shen, Jie
2017
Velocity-correction projection methods for incompressible flows. Zbl 1130.76395
Guermond, J. L.; Shen, Jie
2003
Stable and efficient spectral methods in unbounded domains using Laguerre functions. Zbl 0979.65105
Shen, Jie
2000
Long time stability and convergence for fully discrete nonlinear Galerkin methods. Zbl 0684.65095
Shen, Jie
1990
Optimal spectral-Galerkin methods using generalized Jacobi polynomials. Zbl 1102.76047
Guo, Benyu; Shen, Jie; Wang, Li-Lian
2006
On error estimates of the penalty method for unsteady Navier-Stokes equations. Zbl 0822.35008
Shen, Jie
1995
Some recent advances on spectral methods for unbounded domains. Zbl 1364.65265
Shen, Jie; Wang, Li-Lian
2009
Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval. Zbl 0969.65094
Guo, Ben-Yu; Shen, Jie
2000
Energy stable schemes for Cahn-Hilliard phase-field model of two-phase incompressible flows. Zbl 1400.65049
Shen, Jie; Yang, Xiaofeng
2010
A new class of truly consistent splitting schemes for incompressible flows. Zbl 1032.76529
Guermond, J. L.; Shen, Jie
2003
A time-stepping scheme involving constant coefficient matrices for phase-field simulations of two-phase incompressible flows with large density ratios. Zbl 1277.76118
Dong, S.; Shen, J.
2012
A fourth-order time-splitting Laguerre-Hermite pseudospectral method for Bose-Einstein condensates. Zbl 1084.35083
Bao, Weizhu; Shen, Jie
2005
A new dual-Petrov-Galerkin method for third and higher odd-order differential equations: Application to the KdV equation. Zbl 1053.65085
Shen, Jie
2003
Efficient energy stable numerical schemes for a phase field moving contact line model. Zbl 1351.76184
Shen, Jie; Yang, Xiaofeng; Yu, Haijun
2015
On the maximum principle preserving schemes for the generalized Allen-Cahn equation. Zbl 1361.65059
Shen, Jie; Tang, Tao; Yang, Jiang
2016
Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations. Zbl 1382.65222
Mao, Zhiping; Chen, Sheng; Shen, Jie
2016
Hopf bifurcation of the unsteady regularized driven cavity flow. Zbl 0725.76059
Shen, Jie
1991
Efficient spectral-Galerkin methods. III: Polar and cylindrical geometries. Zbl 0890.65117
Shen, Jie
1997
A rational approximation and its applications to differential equations on the half line. Zbl 0984.65104
Guo, Benyu; Shen, Jie; Wang, Zhongqing
2000
Efficient spectral-Galerkin methods for fractional partial differential equations with variable coefficients. Zbl 1352.65395
Mao, Zhiping; Shen, Jie
2016
Decoupled energy stable schemes for a phase-field model of two-phase incompressible flows with variable density. Zbl 1326.76064
Liu, Chun; Shen, Jie; Yang, Xiaofeng
2015
Generalized Jacobi polynomials/functions and their applications. Zbl 1171.33006
Guo, Ben-Yu; Shen, Jie; Wang, Li-Lian
2009
A decoupled energy stable scheme for a hydrodynamic phase-field model of mixtures of nematic liquid crystals and viscous fluids. Zbl 1349.76019
Zhao, Jia; Yang, Xiaofeng; Shen, Jie; Wang, Qi
2016
An efficient, energy stable scheme for the Cahn-Hilliard-Brinkman system. Zbl 1373.76161
Collins, Craig; Shen, Jie; Wise, Steven M.
2013
Multiple scalar auxiliary variable (MSAV) approach and its application to the phase-field vesicle membrane model. Zbl 1403.65045
Cheng, Qing; Shen, Jie
2018
Efficient, adaptive energy stable schemes for the incompressible Cahn-Hilliard Navier-Stokes phase-field models. Zbl 1352.65229
Chen, Ying; Shen, Jie
2016
Gauge-Uzawa methods for incompressible flows with variable density. Zbl 1109.76037
Pyo, Jae-Hong; Shen, Jie
2007
Fourierization of the Legendre-Galerkin method and a new space-time spectral method. Zbl 1118.65111
Shen, Jie; Wang, Li-Lian
2007
Decoupled energy stable schemes for phase-field models of two-phase complex fluids. Zbl 1288.76057
Shen, Jie; Yang, Xiaofeng
2014
Error analysis of pressure-correction schemes for the time-dependent Stokes equations with open boundary conditions. Zbl 1083.76044
Guermond, J. L.; Minev, P.; Shen, J.
2005
An efficient moving mesh spectral method for the phase-field model of two-phase flows. Zbl 1159.76032
Shen, Jie; Yang, Xiaofeng
2009
Spectral and pseudospectral approximations using Hermite functions: Application to the Dirac equation. Zbl 1032.33004
Guo, Ben-yu; Shen, Jie; Xu, Cheng-long
2003
Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn-Hilliard systems. Zbl 1388.65117
Chen, Feng; Shen, Jie
2013
Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval. Zbl 1001.65129
Guo, Benyu; Shen, Jie; Wang, Zhongqing
2002
Efficient spectral sparse grid methods and applications to high-dimensional elliptic problems. Zbl 1233.65094
Shen, Jie; Yu, Haijun
2010
Efficient stochastic Galerkin methods for random diffusion equations. Zbl 1161.65008
Xiu, Dongbin; Shen, Jie
2009
Energy stability and convergence of SAV block-centered finite difference method for gradient flows. Zbl 1422.65165
Li, Xiaoli; Shen, Jie; Rui, Hongxing
2019
On error estimates of some higher order penalty-projection methods for Navier-Stokes equations. Zbl 0782.76025
Shen, Jie
1992
An efficient spectral-projection method for the Navier-Stokes equations in cylindrical geometries. I: Axisymmetric cases. Zbl 0908.76068
Lopez, J. M.; Shen, Jie
1998
Diffuse-interface simulations of drop coalescence and retraction in viscoelastic fluids. Zbl 1195.76120
Yue, Pengtao; Feng, James J.; Liu, Chun; Shen, Jie
2005
Highly efficient and accurate numerical schemes for the epitaxial thin film growth models by using the SAV approach. Zbl 1444.76020
Cheng, Qing; Shen, Jie; Yang, Xiaofeng
2019
An energetic variational formulation with phase field methods for interfacial dynamics of complex fluids: advantages and challenges. Zbl 1181.76019
Feng, James J.; Liu, Chun; Shen, Jie; Yue, Pengtao
2005
A generalized-Laguerre-Fourier-Hermite pseudospectral method for computing the dynamics of rotating Bose-Einstein condensates. Zbl 1205.82096
Bao, Weizhu; Li, Hailiang; Shen, Jie
2009
Pressure boundary condition for the time-dependent incompressible Navier-Stokes equations. Zbl 1155.76322
Sani, R. L.; Shen, J.; Pironneau, O.; Gresho, P. M.
2006
Remarks on the pressure error estimates for the projection methods. Zbl 0802.76060
Shen, Jie
1994
Spectral approximation of the Helmholtz equation with high wave numbers. Zbl 1091.65119
Shen, Jie; Wang, Li-Lian
2005
On spectral approximations using modified Legendre rational functions: Application to the Korteweg-de Vries equation on the half line. Zbl 0992.65111
Guo, Ben-Yu; Shen, Jie
2001
Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains. Zbl 1154.65082
Shen, Jie; Wang, Li-Lian
2007
Mass and volume conservation in phase field models for binary fluids. Zbl 1373.76317
Shen, Jie; Yang, Xiaofeng; Wang, Qi
2013
A spectral-element method for transmission eigenvalue problems. Zbl 1292.65119
An, Jing; Shen, Jie
2013
A new Lagrange multiplier approach for gradient flows. Zbl 1442.65211
Cheng, Qing; Liu, Chun; Shen, Jie
2020
Efficient and accurate numerical schemes for a hydro-dynamically coupled phase field diblock copolymer model. Zbl 1380.65203
Cheng, Qing; Yang, Xiaofeng; Shen, Jie
2017
Viscoelastic effects on drop deformation in steady shear. Zbl 1082.76007
Yue, Pengtao; Feng, James J.; Liu, Chun; Shen, Jie
2005
On numerical approximations of forward-backward stochastic differential equations. Zbl 1177.60064
Ma, Jin; Shen, Jie; Zhao, Yanhong
2008
Sparse spectral approximations of high-dimensional problems based on hyperbolic cross. Zbl 1215.65179
Shen, Jie; Wang, Li-Lian
2010
On error estimates of some higher order penalty-projection methods for Navier-Stokes equations. Zbl 0739.76017
Shen, Jie
1992
A stable high-order method for two-dimensional bounded-obstacle scattering. Zbl 1130.78007
Nicholls, David P.; Shen, Jie
2006
Modeling and simulations of drop pinch-off from liquid crystal filaments and the leaky liquid crystal faucet immersed in viscous fluids. Zbl 1286.65112
Yang, Xiaofeng; Forest, M. Gregory; Li, Huiyuan; Liu, Chun; Shen, Jie; Wang, Qi; Chen, Falai
2013
Stabilized semi-implicit spectral deferred correction methods for Allen-Cahn and Cahn-Hilliard equations. Zbl 1335.65078
Liu, Fei; Shen, Jie
2015
Generalized Laguerre approximation and its applications to exterior problems. Zbl 1073.65130
Guo, Benyu; Shen, Jie; Xu, Chenglong
2005
A triangular spectral element method using fully tensorial rational basis functions. Zbl 1197.65187
Shen, Jie; Wang, Li-Lian; Li, Huiyuan
2009
A stable, high-order method for three-dimensional, bounded-obstacle, acoustic scattering. Zbl 1123.76049
Fang, Qirong; Nicholls, David P.; Shen, Jie
2007
An efficient spectral-projection method for the Navier-Stokes equations in cylindrical geometries. II: Three-dimensional cases. Zbl 1130.76392
Lopez, J. M.; Marques, F.; Shen, Jie
2002
An efficient spectral method for acoustic scattering from rough surfaces. Zbl 1164.65303
Xiu, Dongbin; Shen, Jie
2007
Dynamics of defect motion in nematic liquid crystal flow: modeling and numerical simulation. Zbl 1164.76422
Liu, Chun; Shen, Jie; Yang, Xiaofeng
2007
Mode interactions in an enclosed swirling flow: A double Hopf bifurcation between azimuthal wavenumbers 0 and 2. Zbl 1016.76029
Marques, F.; Lopez, J. M.; Shen, J.
2002
Two classes of linearly implicit local energy-preserving approach for general multi-symplectic Hamiltonian PDEs. Zbl 1453.65437
Cai, Jiaxiang; Shen, Jie
2020
Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle. Zbl 1197.65176
Li, Huiyuan; Shen, Jie
2010
Hermite spectral methods for fractional PDEs in unbounded domains. Zbl 1373.65075
Mao, Zhiping; Shen, Jie
2017
Instability and mode interactions in a differentially driven rotating cylinder. Zbl 1042.76026
Lopez, J. M.; Hart, J. E.; Marques, F.; Kittelman, S.; Shen, J.
2002
Efficient energy stable schemes for isotropic and strongly anisotropic Cahn-Hilliard systems with the Willmore regularization. Zbl 1395.65014
Chen, Ying; Lowengrub, John; Shen, Jie; Wang, Cheng; Wise, Steven
2018
A highly efficient and accurate new scalar auxiliary variable approach for gradient flows. Zbl 1451.65210
Huang, Fukeng; Shen, Jie; Yang, Zhiguo
2020
The IEQ and SAV approaches and their extensions for a class of highly nonlinear gradient flow systems. Zbl 1484.65196
Shen, Jie; Yang, Xiaofeng
2020
Spectral element method with geometric mesh for two-sided fractional differential equations. Zbl 1397.65305
Mao, Zhiping; Shen, Jie
2018
Legendre and Chebyshev dual-Petrov-Galerkin methods for hyperbolic equations. Zbl 1173.65342
Shen, Jie; Wang, Li-Lian
2007
On the dual Petrov-Galerkin formulation of the KdV equation on a finite interval. Zbl 1157.35093
Goubet, Olivier; Shen, Jie
2007
On liquid crystal flows with free-slip boundary conditions. Zbl 1014.76005
Liu, Chun; Shen, Jie
2001
An unconditionally stable rotational velocity-correction scheme for incompressible flows. Zbl 1425.76058
Dong, S.; Shen, J.
2010
Generalized Jacobi spectral-Galerkin method for nonlinear Volterra integral equations with weakly singular kernels. Zbl 1349.65721
Shen, Jie; Sheng, Changtao; Wang, Zhongqing
2015
Error estimates for time discretizations of Cahn-Hilliard and Allen-Cahn phase-field models for two-phase incompressible flows. Zbl 1382.35184
Cai, Yongyong; Choi, Heejun; Shen, Jie
2017
New SAV-pressure correction methods for the Navier-Stokes equations: stability and error Analysis. Zbl 1479.35625
Li, Xiaoli; Shen, Jie; Liu, Zhengguang
2022
A new Lagrange multiplier approach for constructing structure preserving schemes. I: Positivity preserving. Zbl 07487691
Cheng, Qing; Shen, Jie
2022
Log orthogonal functions: approximation properties and applications. Zbl 1481.65037
Chen, Sheng; Shen, Jie
2022
Bound/positivity preserving and unconditionally stable schemes for a class of fourth order nonlinear equations. Zbl 07525157
Huang, Fukeng; Shen, Jie; Wu, Ke
2022
A new class of implicit-explicit BDF$$k$$ SAV schemes for general dissipative systems and their error analysis. Zbl 07526020
Huang, Fukeng; Shen, Jie
2022
A new class of higher-order decoupled schemes for the incompressible Navier-Stokes equations and applications to rotating dynamics. Zbl 07527723
Wu, Ke; Huang, Fukeng; Shen, Jie
2022
A generalized SAV approach with relaxation for dissipative systems. Zbl 07540356
Zhang, Yanrong; Shen, Jie
2022
Discrete maximum principle of a high order finite difference scheme for a generalized Allen-Cahn equation. Zbl 1489.65124
Shen, Jie; Zhang, Xiangxiong
2022
Generalized SAV approaches for gradient systems. Zbl 1475.65066
Cheng, Qing; Liu, Chun; Shen, Jie
2021
Bound/positivity preserving and energy stable scalar auxiliary variable schemes for dissipative systems: applications to Keller-Segel and Poisson-Nernst-Planck equations. Zbl 07364339
Huang, Fukeng; Shen, Jie
2021
Unconditionally positivity preserving and energy dissipative schemes for Poisson-Nernst-Planck equations. Zbl 1491.65086
Shen, Jie; Xu, Jie
2021
Stability and error analysis of a class of high-order IMEX schemes for Navier-Stokes equations with periodic boundary conditions. Zbl 07453432
Huang, Fukeng; Shen, Jie
2021
Scalar auxiliary variable/Lagrange multiplier based pseudospectral schemes for the dynamics of nonlinear Schrödinger/Gross-Pitaevskii equations. Zbl 07505911
Antoine, Xavier; Shen, Jie; Tang, Qinglin
2021
Efficient and accurate SAV schemes for the generalized Zakharov systems. Zbl 1465.35362
Shen, Jie; Zheng, Nan
2021
Computing interface with quasiperiodicity. Zbl 07508468
Cao, Duo; Shen, Jie; Xu, Jie
2021
A bound-preserving high order scheme for variable density incompressible Navier-Stokes equations. Zbl 07508502
Li, Maojun; Cheng, Yongping; Shen, Jie; Zhang, Xiangxiong
2021
Second-order SAV schemes for the nonlinear Schrödinger equation and their error analysis. Zbl 1491.65111
Deng, Beichuan; Shen, Jie; Zhuang, Qingqu
2021
A new Lagrange multiplier approach for gradient flows. Zbl 1442.65211
Cheng, Qing; Liu, Chun; Shen, Jie
2020
Two classes of linearly implicit local energy-preserving approach for general multi-symplectic Hamiltonian PDEs. Zbl 1453.65437
Cai, Jiaxiang; Shen, Jie
2020
A highly efficient and accurate new scalar auxiliary variable approach for gradient flows. Zbl 1451.65210
Huang, Fukeng; Shen, Jie; Yang, Zhiguo
2020
The IEQ and SAV approaches and their extensions for a class of highly nonlinear gradient flow systems. Zbl 1484.65196
Shen, Jie; Yang, Xiaofeng
2020
Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation. Zbl 1442.65294
Li, Xiaoli; Shen, Jie
2020
Error analysis of the SAV-MAC scheme for the Navier-Stokes equations. Zbl 1452.65167
Li, Xiaoli; Shen, Jie
2020
Fast Fourier-like mapped Chebyshev spectral-Galerkin methods for PDEs with integral fractional Laplacian in unbounded domains. Zbl 1450.65154
Sheng, Changtao; Shen, Jie; Tang, Tao; Wang, Li-Lian; Yuan, Huifang
2020
Optimal error estimates for the scalar auxiliary variable finite-element schemes for gradient flows. Zbl 1440.65192
Chen, Hongtao; Mao, Jingjing; Shen, Jie
2020
On a SAV-MAC scheme for the Cahn-Hilliard-Navier-Stokes phase-field model and its error analysis for the corresponding Cahn-Hilliard-Stokes case. Zbl 1471.65106
Li, Xiaoli; Shen, Jie
2020
A spectrally accurate approximation to subdiffusion equations using the log orthogonal functions. Zbl 1434.65291
Chen, Sheng; Shen, Jie; Zhang, Zhimin; Zhou, Zhi
2020
Global constraints preserving scalar auxiliary variable schemes for gradient flows. Zbl 1456.65087
Cheng, Qing; Shen, Jie
2020
Unconditionally bound preserving and energy dissipative schemes for a class of Keller-Segel equations. Zbl 1440.65101
Shen, Jie; Xu, Jie
2020
Bound preserving and energy dissipative schemes for porous medium equation. Zbl 1436.65101
Gu, Yiqi; Shen, Jie
2020
Error estimate of Gauge-Uzawa methods for incompressible flows with variable density. Zbl 1427.76124
Chen, Hongtao; Mao, Jingjing; Shen, Jie
2020
Accurate and efficient spectral methods for elliptic PDEs in complex domains. Zbl 1440.65160
Gu, Yiqi; Shen, Jie
2020
An efficient and accurate numerical method for the spectral fractional Laplacian equation. Zbl 1433.65235
Chen, Sheng; Shen, Jie
2020
A Petrov-Galerkin spectral method for the inelastic Boltzmann equation using mapped Chebyshev functions. Zbl 1446.65128
Hu, Jingwei; Shen, Jie; Wang, Yingwei
2020
A total fractional-order variation model for image super-resolution and its SAV algorithm. Zbl 1439.65100
Yao, Wenjuan; Shen, Jie; Guo, Zhichang; Sun, Jiebao; Wu, Boying
2020
A new class of efficient and robust energy stable schemes for gradient flows. Zbl 1422.65080
Shen, Jie; Xu, Jie; Yang, Jiang
2019
Energy stability and convergence of SAV block-centered finite difference method for gradient flows. Zbl 1422.65165
Li, Xiaoli; Shen, Jie; Rui, Hongxing
2019
Highly efficient and accurate numerical schemes for the epitaxial thin film growth models by using the SAV approach. Zbl 1444.76020
Cheng, Qing; Shen, Jie; Yang, Xiaofeng
2019
Efficient SAV approach for imaginary time gradient flows with applications to one- and multi-component Bose-Einstein condensates. Zbl 1452.65116
Zhuang, Qingqu; Shen, Jie
2019
An efficient space-time method for time fractional diffusion equation. Zbl 1466.65214
Shen, Jie; Sheng, Chang-Tao
2019
Stability and convergence analysis of rotational velocity correction methods for the Navier-Stokes equations. Zbl 1435.65119
Chen, Feng; Shen, Jie
2019
Efficient and accurate structure preserving schemes for complex nonlinear systems. Zbl 1446.65213
Shen, Jie
2019
Distributional compatibility for change of measures. Zbl 1420.60027
Shen, Jie; Shen, Yi; Wang, Bin; Wang, Ruodu
2019
Fast structured Jacobi-Jacobi transforms. Zbl 07040627
Shen, Jie; Wang, Yingwei; Xia, Jianlin
2019
Fundamental gaps of the fractional Schrödinger operator. Zbl 1429.35195
Bao, Weizhu; Ruan, Xinran; Shen, Jie; Sheng, Changtao
2019
The scalar auxiliary variable (SAV) approach for gradient flows. Zbl 1380.65181
Shen, Jie; Xu, Jie; Yang, Jiang
2018
Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows. Zbl 1403.65047
Shen, Jie; Xu, Jie
2018
Multiple scalar auxiliary variable (MSAV) approach and its application to the phase-field vesicle membrane model. Zbl 1403.65045
Cheng, Qing; Shen, Jie
2018
Efficient energy stable schemes for isotropic and strongly anisotropic Cahn-Hilliard systems with the Willmore regularization. Zbl 1395.65014
Chen, Ying; Lowengrub, John; Shen, Jie; Wang, Cheng; Wise, Steven
2018
Spectral element method with geometric mesh for two-sided fractional differential equations. Zbl 1397.65305
Mao, Zhiping; Shen, Jie
2018
Error estimates for a fully discretized scheme to a Cahn-Hilliard phase-field model for two-phase incompressible flows. Zbl 1390.35225
Cai, Yongyong; Shen, Jie
2018
Laguerre functions and their applications to tempered fractional differential equations on infinite intervals. Zbl 1398.65314
Chen, Sheng; Shen, Jie; Wang, Li-Lian
2018
Enriched spectral methods and applications to problems with weakly singular solutions. Zbl 1406.65123
Chen, Sheng; Shen, Jie
2018
A space-time Petrov-Galerkin spectral method for time fractional diffusion equation. Zbl 1438.65308
Sheng, Changtao; Shen, Jie
2018
A tight bound of hard thresholding. Zbl 1473.62287
Shen, Jie; Li, Ping
2018
Wavenumber explicit analysis for time-harmonic Maxwell equations in a spherical shell and spectral approximations. Zbl 1477.78001
Ma, Lina; Shen, Jie; Wang, Li-Lian; Yang, Zhiguo
2018
Highly efficient and accurate spectral approximation of the angular Mathieu equation for any parameter values $$q$$. Zbl 1424.65119
Alici, Haydar; Shen, Jie
2018
Stabilized predictor-corrector schemes for gradient flows with strong anisotropic free energy. Zbl 1475.82027
Shen, Jie; Xu, Jie
2018
Approximations on $$\mathrm{SO}(3)$$ by Wigner D-matrix and applications. Zbl 1398.65321
Shen, Jie; Xu, Jie; Zhang, Pingwen
2018
Preconditioning techniques in Chebyshev collocation method for elliptic equations. Zbl 1408.35025
Fang, Zhi-Wei; Shen, Jie; Sun, Hai-Wei
2018
Numerical approximations for a three-component Cahn-Hilliard phase-field model based on the invariant energy quadratization method. Zbl 1393.80003
Yang, Xiaofeng; Zhao, Jia; Wang, Qi; Shen, Jie
2017
Efficient and accurate numerical schemes for a hydro-dynamically coupled phase field diblock copolymer model. Zbl 1380.65203
Cheng, Qing; Yang, Xiaofeng; Shen, Jie
2017
Hermite spectral methods for fractional PDEs in unbounded domains. Zbl 1373.65075
Mao, Zhiping; Shen, Jie
2017
Error estimates for time discretizations of Cahn-Hilliard and Allen-Cahn phase-field models for two-phase incompressible flows. Zbl 1382.35184
Cai, Yongyong; Choi, Heejun; Shen, Jie
2017
A hybrid spectral element method for fractional two-point boundary value problems. Zbl 1413.65307
Sheng, Changtao; Shen, Jie
2017
Unconditionally stable Gauge-Uzawa finite element schemes for incompressible natural convection problems with variable density. Zbl 1380.76132
Wu, Jilian; Shen, Jie; Feng, Xinlong
2017
A stable scheme and its convergence analysis for a 2D dynamic $$Q$$-tensor model of nematic liquid crystals. Zbl 1368.76048
Cai, Yongyong; Shen, Jie; Xu, Xiang
2017
A nodal sparse grid spectral element method for multi-dimensional elliptic partial differential equations. Zbl 1429.65287
Rong, Zhijian; Shen, Jie; Yu, Haijun
2017
Online optimization for max-norm regularization. Zbl 1454.68132
Shen, Jie; Xu, Huan; Li, Ping
2017
Highly accurate pseudospectral approximations of the prolate spheroidal wave equation for any bandwidth parameter and zonal wavenumber. Zbl 1416.65224
Alıcı, H.; Shen, Jie
2017
Numerical analysis and simulation for a generalized planar Ginzburg-Landau equation in a circular geometry. Zbl 1367.65142
Colbert-Kelly, Sean; McFadden, Geoffrey B.; Phillips, Daniel; Shen, Jie
2017
Generalized Jacobi functions and their applications to fractional differential equations. Zbl 1335.65066
Chen, Sheng; Shen, Jie; Wang, Li-Lian
2016
On the maximum principle preserving schemes for the generalized Allen-Cahn equation. Zbl 1361.65059
Shen, Jie; Tang, Tao; Yang, Jiang
2016
Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations. Zbl 1382.65222
Mao, Zhiping; Chen, Sheng; Shen, Jie
2016
Efficient spectral-Galerkin methods for fractional partial differential equations with variable coefficients. Zbl 1352.65395
Mao, Zhiping; Shen, Jie
2016
A decoupled energy stable scheme for a hydrodynamic phase-field model of mixtures of nematic liquid crystals and viscous fluids. Zbl 1349.76019
Zhao, Jia; Yang, Xiaofeng; Shen, Jie; Wang, Qi
2016
Efficient, adaptive energy stable schemes for the incompressible Cahn-Hilliard Navier-Stokes phase-field models. Zbl 1352.65229
Chen, Ying; Shen, Jie
2016
Müntz-Galerkin methods and applications to mixed Dirichlet-Neumann boundary value problems. Zbl 1416.65482
Shen, Jie; Wang, Yingwei
2016
Efficient splitting schemes for magneto-hydrodynamic equations. Zbl 1388.76224
Choi, Heejun; Shen, Jie
2016
Fast structured direct spectral methods for differential equations with variable coefficients. I. The one-dimensional case. Zbl 1330.65124
Shen, Jie; Wang, Yingwei; Xia, Jianlin
2016
Two-phase Stefan problem with smoothed enthalpy. Zbl 1349.35433
Azaïez, M.; Jelassi, F.; Brahim, M. Mint; Shen, J.
2016
Efficient spectral-Galerkin method and analysis for elliptic PDEs with non-local boundary conditions. Zbl 1375.65159
Hu, Lina; Ma, Lina; Shen, Jie
2016
Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows. Zbl 1327.65178
Shen, Jie; Yang, Xiaofeng
2015
Efficient energy stable numerical schemes for a phase field moving contact line model. Zbl 1351.76184
Shen, Jie; Yang, Xiaofeng; Yu, Haijun
2015
Decoupled energy stable schemes for a phase-field model of two-phase incompressible flows with variable density. Zbl 1326.76064
Liu, Chun; Shen, Jie; Yang, Xiaofeng
2015
Stabilized semi-implicit spectral deferred correction methods for Allen-Cahn and Cahn-Hilliard equations. Zbl 1335.65078
Liu, Fei; Shen, Jie
2015
Generalized Jacobi spectral-Galerkin method for nonlinear Volterra integral equations with weakly singular kernels. Zbl 1349.65721
Shen, Jie; Sheng, Changtao; Wang, Zhongqing
2015
A pressure correction scheme for generalized form of energy-stable open boundary conditions for incompressible flows. Zbl 1349.76198
Dong, S.; Shen, J.
2015
Dynamic transitions of quasi-geostrophic channel flow. Zbl 1329.35232
Dijkstra, Henk; Sengul, Taylan; Shen, Jie; Wang, Shouhong
2015
Pattern formations of 2D Rayleigh-Bénard convection with no-slip boundary conditions for the velocity at the critical length scales. Zbl 1333.76036
Sengul, Taylan; Shen, Jie; Wang, Shouhong
2015
Accurate solution and approximations of the linearized BGK equation for steady Couette flow. Zbl 1410.76412
Li, Wei; Luo, Li-Shi; Shen, Jie
2015
Spectral approximation to a transmission eigenvalue problem and its applications to an inverse problem. Zbl 1443.65306
An, Jing; Shen, Jie
2015
Spectral approximation of time-harmonic Maxwell equations in three-dimensional exterior domains. Zbl 1329.65286
Ma, Lina; Shen, Jie; Wang, Li-Lian
2015
Modeling and simulation of fingering pattern formation in a combustion model. Zbl 1318.80002
Hu, Lina; Brauner, Claude-Michel; Shen, Jie; Sivashinsky, Gregory I.
2015
A semi-implicit spectral deferred correction method for two water wave models with nonlocal viscous term and numerical study of their decay rates. Zbl 1488.65513
Mao, Zhiping; Shen, Jie
2015
Decoupled energy stable schemes for phase-field models of two-phase complex fluids. Zbl 1288.76057
Shen, Jie; Yang, Xiaofeng
2014
Approximations by orthonormal mapped Chebyshev functions for higher-dimensional problems in unbounded domains. Zbl 1293.65161
Shen, Jie; Wang, Li-Lian; Yu, Haijun
2014
A new spectral method for numerical solution of the unbounded rough surface scattering problem. Zbl 1349.65646
He, Ying; Li, Peijun; Shen, Jie
2014
Unconditionally stable pressure-correction schemes for a linear fluid-structure interaction problem. Zbl 1324.76040
He, Ying; Shen, Jie
2014
Efficient spectral methods for transmission eigenvalues and estimation of the index of refraction. Zbl 1313.78022
An, Jing; Shen, Jie
2014
...and 130 more Documents
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### Cited by 4,211 Authors

 114 Shen, Jie 88 Yang, Xiaofeng 57 Guo, Ben-Yu 52 He, Yinnian 50 Wang, Cheng 44 Wang, Zhongqing 43 Wang, Lilian 41 Kim, Junseok 36 Wang, Qi 35 Feng, Xinlong 35 Wang, Yushun 33 Wise, Steven M. 31 Liu, Chun 30 Chen, Yanping 30 Dehghan Takht Fooladi, Mehdi 30 Zhao, Jia 29 Li, Huiyuan 27 Mei, Liquan 27 Xu, Chuanju 27 Zhang, Zhimin 26 Karniadakis, George Em 24 Chen, Wenbin 24 Gong, Yuezheng 24 Huang, Yunqing 24 Wang, Hong 23 Bhrawy, Ali Hassan 23 Doha, Eid H. 22 Nicholls, David P. 22 Tang, Tao 21 Du, Qiang 21 Guermond, Jean-Luc 21 Wang, Tianjun 21 Wang, Xiaoming 20 Bao, Weizhu 20 Han, Daozhi 20 Lee, Hyun Geun 20 Li, Xiaoli 20 Qiao, Zhonghua 19 Ju, Lili 18 Chen, Chuanjun 18 Miranville, Alain M. 18 Sun, Shuyu 18 Zaky, Mahmoud A. 17 Cai, Wenjun 17 Hou, Yanren 17 Li, Yibao 17 López, Juan Manuel 17 Ma, Heping 17 Parand, Kourosh 17 Yang, Junxiang 16 Cai, Yongyong 16 Chen, Lizhen 16 Dong, Suchuan 16 He, Xiaoming 16 Kou, Jisheng 16 Li, Jian 16 Wu, Boying 15 Li, Buyang 15 Lin, Ping 15 Liu, Wen-Jie 15 Marques, Francisco 15 Minev, Peter Dimitrov 15 Sherwin, Spencer J. 15 Yang, Jiang 14 Guillén González, Francisco M. 14 Jiang, Chaolong 14 Lowengrub, John Samuel 14 Mao, Zhiping 14 Wang, Quan 14 Yi, Lijun 13 An, Jing 13 Auteri, Franco 13 Codina, Ramon 13 Feng, James J. 13 Jia, Hongen 13 Li, Peijun 13 Yu, Haijun 13 Zhang, Luming 13 Zheng, Xiangcheng 12 Abd-Elhameed, Waleed Mohamed 12 An, Rong 12 Li, Xiao 12 Liu, Fawang 12 Liu, Zhengguang 12 Lu, Shujuan 12 Tachim Medjo, Theodore 12 Wang, Kun 12 Yue, Pengtao 12 Zhang, Zhengru 12 Zhou, Jianwei 11 Azaïez, Mejdi 11 Cheng, Qing 11 Grasselli, Maurizio 11 Lai, Mingchih 11 Liu, Hailiang 11 Si, Zhiyong 11 Yang, Yidu 11 Yu, Xuhong 10 Cheng, Kelong 10 Hafez, Ramy Mahmoud ...and 4,111 more Authors
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### Cited in 306 Serials

 509 Journal of Computational Physics 301 Journal of Scientific Computing 180 Applied Numerical Mathematics 154 Journal of Computational and Applied Mathematics 141 Computer Methods in Applied Mechanics and Engineering 123 Computers & Mathematics with Applications 111 Computers and Fluids 100 Applied Mathematics and Computation 97 SIAM Journal on Scientific Computing 89 Journal of Fluid Mechanics 73 Communications in Computational Physics 70 SIAM Journal on Numerical Analysis 66 Mathematics of Computation 60 Numerical Algorithms 55 International Journal of Computer Mathematics 54 Advances in Computational Mathematics 45 Numerische Mathematik 45 Applied Mathematics Letters 38 Mathematical Methods in the Applied Sciences 35 Numerical Methods for Partial Differential Equations 35 Physics of Fluids 35 Discrete and Continuous Dynamical Systems. Series B 34 Computational and Applied Mathematics 32 Communications in Nonlinear Science and Numerical Simulation 30 Journal of Mathematical Analysis and Applications 29 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 27 Advances in Applied Mathematics and Mechanics 26 International Journal for Numerical Methods in Fluids 26 Applied Mathematical Modelling 23 Mathematics and Computers in Simulation 20 Advances in Difference Equations 20 Science China. Mathematics 19 Physica D 19 Mathematical Problems in Engineering 18 Computer Physics Communications 17 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 17 Communications on Applied Mathematics and Computation 16 East Asian Journal on Applied Mathematics 15 Journal of Differential Equations 14 Abstract and Applied Analysis 13 Calcolo 13 SIAM Journal on Applied Mathematics 13 Computational Methods in Applied Mathematics 13 International Journal of Numerical Analysis and Modeling 13 Discrete and Continuous Dynamical Systems. Series S 11 BIT 11 Engineering Analysis with Boundary Elements 11 European Journal of Mechanics. B. Fluids 10 Applicable Analysis 10 Numerical Functional Analysis and Optimization 10 International Journal of Computational Fluid Dynamics 10 Nonlinear Dynamics 10 Journal of Mathematical Fluid Mechanics 10 Journal of Applied Mathematics and Computing 10 Frontiers of Mathematics in China 9 Archive for Rational Mechanics and Analysis 9 Chaos, Solitons and Fractals 9 International Journal for Numerical Methods in Engineering 9 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 9 Applied Mathematics and Mechanics. (English Edition) 9 Journal of Non-Newtonian Fluid Mechanics 9 International Journal of Numerical Methods for Heat & Fluid Flow 9 Discrete and Continuous Dynamical Systems 9 Journal of Applied Mathematics 9 Comptes Rendus. Mathématique. Académie des Sciences, Paris 9 Communications on Pure and Applied Analysis 8 Chinese Annals of Mathematics. Series B 8 Fractional Calculus & Applied Analysis 8 Nonlinear Analysis. Real World Applications 8 International Journal of Applied and Computational Mathematics 7 ZAMP. Zeitschrift für angewandte Mathematik und Physik 7 Computational Mechanics 7 SIAM Journal on Mathematical Analysis 7 Mathematical Modelling and Analysis 7 International Journal of Computational Methods 7 S$$\vec{\text{e}}$$MA Journal 7 AIMS Mathematics 6 Acta Mechanica 6 Journal of Inequalities and Applications 6 International Journal of Nonlinear Sciences and Numerical Simulation 6 Multiscale Modeling & Simulation 6 Boundary Value Problems 6 SIAM/ASA Journal on Uncertainty Quantification 6 Journal of Function Spaces 5 Journal of Engineering Mathematics 5 Journal of Statistical Physics 5 Mathematical and Computer Modelling 5 SIAM Review 5 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 5 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 4 International Journal of Engineering Science 4 Physica A 4 Bulletin of Mathematical Biology 4 Theoretical and Computational Fluid Dynamics 4 Journal of Approximation Theory 4 Acta Applicandae Mathematicae 4 Bulletin of the Iranian Mathematical Society 4 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 4 RAIRO. Modélisation Mathématique et Analyse Numérique 4 Calculus of Variations and Partial Differential Equations ...and 206 more Serials
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# Solution of following ODE

1. Feb 15, 2016

### Atr cheema

• OP warned about not using the homework template
Let’s consider Uc to be transformed form of h(x,t) by applying Fourier transform
Then solution of Eq 1 by integrating factor is as in Eq 2
And by applying on Eq 2 inverse fourier transform & some simplification gives us final solution as Eq 3
But what if f(t) in Eq 1 is equal to Eq 4
Putting value of f(t) from Eq 4 in Eq 1 gives us Eq 5
Similarly by integrating factor solution of Eq 5 comes out to be as in Eq 6
Applying fourier inverse on Eq 6 gives final solution as Eq 7
The problem is in final equation (Eq 7)I have h(x,tow) on right hand side which is actually same what I want to find out and is on left side of eq 7. How can I modify solution of Eq 5 that h(x,tow) in Eq 6 does not remain on right hand side?

https://www.physicsforums.com/attachments/equations-png.95912/ [Broken]

Last edited by a moderator: May 7, 2017
2. Feb 15, 2016

### HallsofIvy

Staff Emeritus
When I click on that link, I get a message that says "you do not have permission to view this page".


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# (Function|Operator) - Associative Property

## Definition

An operator or function op is associative if the following holds:

$$(a \text{ op } b) \text{ op } c == a \text{ op } (b \text{ op } c)$$

## Parallel

The importance of this to parallel evaluation can be seen if we expand this to four terms:

$$a \text{ op } b \text{ op } c \text{ op } d == (a \text{ op } b) \text{ op } (c \text{ op } d)$$

So we can evaluate <wrap box>(a op b)</note> in parallel with <wrap box>(c op d)</note>, and then invoke op on the results.

## Linear Algebra

Associativity of function composition: $h \circ (g \circ f) = (h \circ g) \circ f$

 $(h \circ (g \circ f))(x)$ = $h(g \circ f)(x)$ $h(g(f(x)))$ $(h \circ g)(f(x))$ $((h \circ g) \circ f)(x)$

## Example

Examples of associative operations include:

$$(x + y) + z = x + (y + z)$$


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# Quest:Hillshire quests

WARNING: This page contains Quest spoilers. ~ Solutions ~ Table ~ List ~ All ~ Category ~

Highlight to view quest spoilers.

## Hillshire quests

Walking inside Hillshire requires a pass (see Hillshire Tourist) or bypassing the troll at the entry of town.

#### "Hillshire Tourist", in which you visited scenic Hillshire. (Quest level 2)

Go to the campsite on the road northwest from Pekan Ford, and search the ashes. Take the tarnished ring from the circle of stones and go to Hillshire by going northwest then northeast at the junction outside the tourism office. Give the tarnished ring to the bridge troll and he will give you a pass to enter Hillshire.

#### "Occult Apprentice", in which you became more knowledgeable in magical control. (Quest level 7)

REQUIRED SKILLS/BONUSES NEEDED (Below scores CONFIRMED. Minimum bonus may be less.)
*Magic.methods.physical.evoking (feather) - 79 bonus
*Magic.methods.elemental.fire (book) - 121 bonus
*Magic.methods.mental.channeling (spider Ice cube)- 183 bonus (may be up to 20 points lower)
*Magic.spell.misc (spider Ice cube)- between 134 and 138 bonus
*Magic.methods.mental.animating (bottle of alcohol) - 181 bonus

REQUIRED ITEMS
*Feather - available from corpses of birds. Also available in a pillow south from main quest room.
*Book - available in various shops.
*Spider Ice Cube - available in Krazzander's, Ankh-Morpork.
*Bottle of Alcohol - must be a bottle, not a glass.  Beware that most bars and restaurants don't sell alcohol in bottles.

The quest is in northwestern Hillshire, in the Wizards house. Go upstairs and then north (if the door is not open, say friend). You then meet a wizard who asks if you'd like to be an apprentice. Say yes. He will tell you to retrieve the above items, in potentially varying order. Read the scroll he tells you to read for each item, then perform the following actions in line with the appropriate item.

Feather - Drop feather, then lift feather.
Book - Drop book, then force book to combust.
Cube - Drop the cube and cast Flizzlofrob's Morphogenic Resonator at cube.
Alcohol - Drop bottle, then induce weather from bottle.


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Spelling principal and lens

Spelling principal

The most important person in the opera helps you to spell principal by reminding you that:

Everyone wants to be the principal singer's PAL.

Principal ends in PAL.

Spelling lens

It's also worth pointing out how to spell lens,

The plural of lens is lenses, but one lens is not spelt "lense".

To remember that "lense" is incorrect, try pronouncing the E.

LENS – EE

We don't want to be calling a single lens a lens-ee, although it would be cute.


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Deepak Scored 45->99%ile with Bounce Back Crack Course. You can do it too!

# Define degree of a polynomial.

Question:

Define degree of a polynomial.

Solution:

The exponent of the highest degree term in a polynomial is known as its degree.

In other words, the highest power of $x$ in a polynomial $f(x)$ is called the degree of the polynomial $f(x)$.

For Example: $g(x)=2 x^{2}+3 x+4$ is a polynomial in the variable $x$ of degree 2 .


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## Algebra and Trigonometry 10th Edition

$$-114.592^{\circ}$$
Using the fact that there are 180 degrees per $\pi$ radians, we find: $$-2\times \frac{180^{\circ}}{\pi\ radians} =-114.592^{\circ}$$


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6

# Questions

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sep 2 11 at 23:38 Mark Sapir 28k257120

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### Example of the completion of a noetherian domain at a prime that is not a domain

may 13 at 7:30 Sándor Kovács 23.8k25373

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jun 10 10 at 18:49 Arturo Magidin 2,72111225

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feb 15 10 at 19:38 Pete L. Clark 40k4103227

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### Varieties of groups with infinite relatively free group of rank 2 finite, infinite in rank 3

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Find a particular nesting of directory names, e.g. .../project/dir1/dir2

I'm having some concerns managing the find command ; I do not recognize the syntax for it quite possibly and also I require to utilize it in a manuscript to find a details directory framework that can be situated anywhere.

Primarily, I have a framework like "project/dir1/dir2" that I recognize will certainly constantly be specifically those 3 directory sites, with the very same names, because order. What might not coincide is just how deep this framework is nested ; as an example you can have:

$HOME/project/dir1/dir2 or $HOME/workspace/project/dir1/dir2

Basically I require a basic means to situate the project/dir1/dir2 framework despite just how much deeply hidden it is.

This will certainly constantly get on Ubuntu. I thought of making use of situate, yet after that I would certainly need to assure that the situate db is constantly approximately day.

5
2022-07-25 20:44:10
Source Share


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# How can escape velocity be independent of the direction of projection of a body?

I read that the escape velocity of a body is independent of the direction of projection.

For example, I could throw a ball at $$11.2$$ km/s velocity horizontally, and it would still leave earth.

I am unable to visualise or understand this. Could someone explain?

Edit: I think I should put my exact problem down more clearly.

At point A: I release the ball. It has horizontal velocity of 11.2 km/s.

From A to B: Gravity acts on the ball. One component slows down the horizontal velocity and one component constantly attracts the ball to earth.

At B: the ball is trying to leave earth with less than escape velocity.

What I realised: B might not be exactly on earth, the ball might have already covered some height.

So right now, I would just like someone to confirm if what I realised is right. (Or correct me.)

Thank you.

• Have you taken into consideration that the ground will eventually curve away from the horizontal trajectory, and at such a speed(if constant) the particle will escape. Dec 26, 2015 at 6:04
• It's simply a matter of energy conservation. If the kinetic energy exceeds the (negative) potential energy at the launch point, the body will escape. Dec 26, 2015 at 7:34
• Please see the edit, and thank you both for getting me thinking. Dec 26, 2015 at 9:03
• Dec 26, 2015 at 11:12

Because it is a energy balance between a potential $$V(r)$$ and the kinetic energy of the object $$T = 1/2mv^2$$, so the work to do provide in order to escape: $$T > V$$ does not depend on the path followed by the object (it only depends on the module of it’s velocity).

• I am not sure your answer fully clarifies the issue. Why is the condition that the kinetic energy be larger than the potential energy enough to ensure that the projectile ‘escapes’? If this had been clear to the OP, he may not have posed the question in the first place. Aug 8, 2021 at 5:11
• Think of it as jumping a fence. In order to jump a fence, one merely has to jump in the air high enough (the threshold being the height of the fence). The result does not depend on the angle at which one runs towards the fence. Aug 8, 2021 at 10:26
• With a fence, it depends on the vertical component of the velocity. It won’t be of any use running very fast toward the fence if your jump speed is low; you will just hit the fence very fast. So it does not seem to be the modulus of the velocity that is crucial. Aug 8, 2021 at 11:00
• It is just a metaphor of a case where a single parameter matters (height) and another don’t (angle). I was trying to give a graphical metaphor for a potential barrier. Aug 8, 2021 at 11:05
• Ok, I get that. And I am sure the analogy is there. However, I think the source of his confusion (as is mine) is the fact that some escape trajectories actually intersect the planet, which seems to contradict the idea of ‘escape’. Aug 8, 2021 at 11:14

Ignoring atmospheric friction and buoyancy, if you threw the ball at a and it arrived at b, the velocity you threw the ball at would have to be about 8000 m\s, and the ball's velocity at b would not have changed.

If you threw the ball at escape velocity it would follow a parabolic path, with the vertex of the parabola at the throwing position. The distance of the ball from the center of the Earth would be constantly increasing, and the velocity would be constantly decreasing, without ever reaching zero.

Of course, the escape velocity is all about energy, and a very special 'thing' about $$1/r^2$$ forces is that energy of an orbit does not depend on the eccentricity. Hence, if you are at a $$r_0$$ with energy $$V(r_0)<0$$, you just need:

$$T(\vec v) = T(|v|) =\frac 1 2 mv^2 \ge -V(r_0)$$

to escape.

In principle, the independence of $$E$$ from $$\epsilon$$ can be related to the conservation of the Laplace-Runge-Lenz vector:

$${\bf A} = {\bf p}\times {\bf L} -mk{\bf {\hat r}}$$

but at this moment, I am not having a "Eureka" moment as to how that applies here.


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# PSP Bibliography

 Notice: Clicking on the title will open a new window with all details of the bibliographic entry. Clicking on the DOI link will open a new window with the original bibliographic entry from the publisher. Clicking on a single author will show all publications by the selected author. Clicking on a single keyword, will show all publications by the selected keyword.

## Found 21 entries in the Bibliography.

### Showing entries from 1 through 21

 2022 On the utility of flux rope models for CME magnetic structure below 30 R$_\ensuremath\odot$ We present a comprehensive analysis of the three-dimensional magnetic flux rope structure generated during the Lynch et al. (2019, ApJ 880:97) magnetohydrodynamic (MHD) simulation of a global-scale, 360 \textdegree -wide streamer blowout coronal mass ejection (CME) eruption. We create both fixed and moving synthetic spacecraft to generate time series of the MHD variables through different regions of the flux rope CME. Our moving spacecraft trajectories are derived from the spatial coordinates of Parker Solar Probe s past enc ... Lynch, Benjamin; Al-Haddad, Nada; Yu, Wenyuan; Palmerio, Erika; Lugaz, No\; Published by: Advances in Space Research      Published on: sep YEAR: 2022     DOI: 10.1016/j.asr.2022.05.004 Higher-order Turbulence Statistics in the Sub-Alfv\ enic Solar Wind Observed by Parker Solar Probe Parker Solar Probe has been the first spacecraft to enter the deep corona below the Alfv\ en critical point. Here we examine the higher-order statistical properties of magnetic-field fluctuations in the sub-Alfv\ enic solar wind and compare the results with the neighboring super-Alfv\ enic region. The intermittency and multifractal properties are analyzed by inspecting the probability density functions, the scale- dependent kurtosis, and fractal spectrum of magnetic-field fluctuations. It is found that the magnetic-field flu ... Zhang, J.; Huang, S.~Y.; Yuan, Z.~G.; Jiang, K.; Xu, S.~B.; Bandyopadhyay, R.; Wei, Y.~Y.; Xiong, Q.~Y.; Wang, Z.; Yu, L.; Lin, R.~T.; Published by: \apj      Published on: oct YEAR: 2022     DOI: 10.3847/1538-4357/ac8c34 Higher-order Turbulence Statistics in the Sub-Alfv\ enic Solar Wind Observed by Parker Solar Probe Parker Solar Probe has been the first spacecraft to enter the deep corona below the Alfv\ en critical point. Here we examine the higher-order statistical properties of magnetic-field fluctuations in the sub-Alfv\ enic solar wind and compare the results with the neighboring super-Alfv\ enic region. The intermittency and multifractal properties are analyzed by inspecting the probability density functions, the scale- dependent kurtosis, and fractal spectrum of magnetic-field fluctuations. It is found that the magnetic-field flu ... Zhang, J.; Huang, S.~Y.; Yuan, Z.~G.; Jiang, K.; Xu, S.~B.; Bandyopadhyay, R.; Wei, Y.~Y.; Xiong, Q.~Y.; Wang, Z.; Yu, L.; Lin, R.~T.; Published by: \apj      Published on: oct YEAR: 2022     DOI: 10.3847/1538-4357/ac8c34 Multi-scale image preprocessing and feature tracking for remote CME characterization Coronal Mass Ejections (CMEs) influence the interplanetary environment over vast distances in the solar system by injecting huge clouds of fast solar plasma and energetic particles (SEPs). A number of fundamental questions remain about how SEPs are produced, but current understanding points to CME-driven shocks and compressions in the solar corona. At the same time, unprecedented remote and in situ (Parker Solar Probe, Solar Orbiter) solar observations are becoming available to constrain existing theories. Here we present a ... Stepanyuk, Oleg; Kozarev, Kamen; Nedal, Mohamed; Published by: Journal of Space Weather and Space Climate      Published on: may YEAR: 2022     DOI: 10.1051/swsc/2022020 Categorizing MHD Discontinuities in the Inner Heliosphere by Utilizing the PSP Mission The interplanetary discontinuities (IDs) have been widely observed in astrophysical and space plasmas, while their characteristics and evolutions within 0.3 AU are still unclear due to the limitation of spacecraft orbits in previous missions. Here, we report three ID events, including a rotational discontinuity (RD), a tangential discontinuity (TD), and a suspected contact discontinuity (CD), detected by the Parker Solar Probe in a previously unexplored region of the heliosphere as close to the Sun as 0.13 AU. By the combina ... Liu, Y.~Y.; Fu, H.~S.; Cao, J.~B.; Yu, Y.; Liu, C.~M.; Wang, Z.; Guo, Z.~Z.; He, R.~J.; Published by: Journal of Geophysical Research (Space Physics)      Published on: mar YEAR: 2022     DOI: 10.1029/2021JA029983 Parker Data Used Three-Dimensional Anisotropy and Scaling Properties of Solar Wind Turbulence at Kinetic Scales in the Inner Heliosphere: Parker Solar Probe Observations We utilize the data from the Parker Solar Probe mission at its first perihelion to investigate the three-dimensional (3D) anisotropies and scalings of solar wind turbulence for the total, perpendicular, and parallel magnetic-field fluctuations at kinetic scales in the inner heliosphere. By calculating the five-point second-order structure functions, we find that the three characteristic lengths of turbulence eddies for the total and the perpendicular magnetic-field fluctuations in the local reference frame $(\hatL_\perp ,\ha ... Zhang, J.; Huang, S.~Y.; He, J.~S.; Wang, T.~Y.; Yuan, Z.~G.; Deng, X.~H.; Jiang, K.; Wei, Y.~Y.; Xu, S.~B.; Xiong, Q.~Y.; Lin, R.~T.; Yu, L.; Published by: \apjl Published on: jan YEAR: 2022 DOI: 10.3847/2041-8213/ac4027 Three-Dimensional Anisotropy and Scaling Properties of Solar Wind Turbulence at Kinetic Scales in the Inner Heliosphere: Parker Solar Probe Observations We utilize the data from the Parker Solar Probe mission at its first perihelion to investigate the three-dimensional (3D) anisotropies and scalings of solar wind turbulence for the total, perpendicular, and parallel magnetic-field fluctuations at kinetic scales in the inner heliosphere. By calculating the five-point second-order structure functions, we find that the three characteristic lengths of turbulence eddies for the total and the perpendicular magnetic-field fluctuations in the local reference frame$(\hatL_\perp ,\ha ... Zhang, J.; Huang, S.~Y.; He, J.~S.; Wang, T.~Y.; Yuan, Z.~G.; Deng, X.~H.; Jiang, K.; Wei, Y.~Y.; Xu, S.~B.; Xiong, Q.~Y.; Lin, R.~T.; Yu, L.; Published by: \apjl      Published on: jan YEAR: 2022     DOI: 10.3847/2041-8213/ac4027 Double-power-law Feature of Energetic Particles Accelerated at Coronal Shocks Recent observations have shown that in many large solar energetic particle (SEP) events the event-integrated differential spectra resemble double power laws. We perform numerical modeling of particle acceleration at coronal shocks propagating through a streamer-like magnetic field by solving the Parker transport equation, including protons and heavier ions. We find that for all ion species the energy spectra integrated over the simulation domain can be described by a double power law, and the break energy depends on the ion ... Yu, Feiyu; Kong, Xiangliang; Guo, Fan; Liu, Wenlong; Jiang, Zelong; Chen, Yao; Giacalone, Joe; Published by: \apjl      Published on: feb YEAR: 2022     DOI: 10.3847/2041-8213/ac4cb3 Small-Scale Magnetic Holes in the Solar Wind Observed by Parker Solar Probe The small-scale magnetic hole (SSMH), characterized by magnetic field depression, is a structure with the size in the order of proton gyro-radius. SSMHs near the Earth or other planets have been widely observed in recent years. However, SSMHs in the solar wind near the Sun are rarely investigated due to mission constraints. In the present study, SSMHs in the pristine solar wind within a wide heliocentric distance range are analyzed based on the Parker Solar Probe (PSP) Mission measurements. A total of 2,416 SSMHs are success ... Yu, L.; Huang, S.~Y.; Yuan, Z.~G.; Jiang, K.; Wei, Y.~Y.; Zhang, J.; Xu, S.~B.; Xiong, Q.~Y.; Wang, Z.; Lin, R.~T.; Li, Y.~J.; Wang, C.~M.; Song, G.~J.; Published by: Journal of Geophysical Research (Space Physics)      Published on: aug YEAR: 2022     DOI: 10.1029/2022JA030505 Small-Scale Magnetic Holes in the Solar Wind Observed by Parker Solar Probe The small-scale magnetic hole (SSMH), characterized by magnetic field depression, is a structure with the size in the order of proton gyro-radius. SSMHs near the Earth or other planets have been widely observed in recent years. However, SSMHs in the solar wind near the Sun are rarely investigated due to mission constraints. In the present study, SSMHs in the pristine solar wind within a wide heliocentric distance range are analyzed based on the Parker Solar Probe (PSP) Mission measurements. A total of 2,416 SSMHs are success ... Yu, L.; Huang, S.~Y.; Yuan, Z.~G.; Jiang, K.; Wei, Y.~Y.; Zhang, J.; Xu, S.~B.; Xiong, Q.~Y.; Wang, Z.; Lin, R.~T.; Li, Y.~J.; Wang, C.~M.; Song, G.~J.; Published by: Journal of Geophysical Research (Space Physics)      Published on: aug YEAR: 2022     DOI: 10.1029/2022JA030505 Solar Chromospheric Network as a Source for Solar Wind Switchbacks Recent studies suggest that the magnetic switchbacks (SBs) detected by the Parker Solar Probe carry information on the scales of solar supergranulation (large scale) and granulation (medium scale). We test this claim using high-resolution H\ensuremath\alpha images obtained with the visible spectropolarimeters of the Goode Solar Telescope in Big Bear Solar Observatory. As possible solar sources, we count all the spicule-like features standing along the chromospheric networks near the coronal hole boundary visible in the H\ens ... Lee, Jeongwoo; Yurchyshyn, Vasyl; Wang, Haimin; Yang, Xu; Cao, Wenda; Oliveros, Juan; Published by: \apjl      Published on: aug YEAR: 2022     DOI: 10.3847/2041-8213/ac86bf Anisotropy of Magnetic Field Spectra at Kinetic Scales of Solar Wind Turbulence as Revealed by the Parker Solar Probe in the Inner Heliosphere Using the Parker Solar Probe data taken in the inner heliosphere, we investigate the power and spatial anisotropy of magnetic field spectra at kinetic scales (i.e., around sub-ion scales) in solar wind turbulence in the inner heliosphere. We find that strong anisotropy of the magnetic spectra occurs at kinetic scales with the strongest power in the perpendicular direction with respect to the local magnetic field (forming an angle \ensuremath\theta $_B$ with the mean flow velocity). The spectral index of the magnetic spectra ... Huang, S.~Y.; Xu, S.~B.; Zhang, J.; Sahraoui, F.; es, Andr\; He, J.~S.; Yuan, Z.~G.; Deng, X.~H.; Jiang, K.; Wei, Y.~Y.; Xiong, Q.~Y.; Wang, Z.; Yu, L.; Lin, R.~T.; Published by: \apjl      Published on: apr YEAR: 2022     DOI: 10.3847/2041-8213/ac5f02 Anisotropy of Magnetic Field Spectra at Kinetic Scales of Solar Wind Turbulence as Revealed by the Parker Solar Probe in the Inner Heliosphere Using the Parker Solar Probe data taken in the inner heliosphere, we investigate the power and spatial anisotropy of magnetic field spectra at kinetic scales (i.e., around sub-ion scales) in solar wind turbulence in the inner heliosphere. We find that strong anisotropy of the magnetic spectra occurs at kinetic scales with the strongest power in the perpendicular direction with respect to the local magnetic field (forming an angle \ensuremath\theta $_B$ with the mean flow velocity). The spectral index of the magnetic spectra ... Huang, S.~Y.; Xu, S.~B.; Zhang, J.; Sahraoui, F.; es, Andr\; He, J.~S.; Yuan, Z.~G.; Deng, X.~H.; Jiang, K.; Wei, Y.~Y.; Xiong, Q.~Y.; Wang, Z.; Yu, L.; Lin, R.~T.; Published by: \apjl      Published on: apr YEAR: 2022     DOI: 10.3847/2041-8213/ac5f02 2021 Predicting the Magnetic Fields of a Stealth CME Detected by Parker Solar Probe at 0.5 au Stealth coronal mass ejections (CMEs) are eruptions from the Sun that are not associated with appreciable low-coronal signatures. Because they often cannot be linked to a well-defined source region on the Sun, analysis of their initial magnetic configuration and eruption dynamics is particularly problematic. In this article, we address this issue by undertaking the first attempt at predicting the magnetic fields of a stealth CME that erupted in 2020 June from the Earth-facing Sun. We estimate its source region with the aid o ... Palmerio, Erika; Kay, Christina; Al-Haddad, Nada; Lynch, Benjamin; Yu, Wenyuan; Stevens, Michael; Pal, Sanchita; Lee, Christina; Published by: \apj      Published on: oct YEAR: 2021     DOI: 10.3847/1538-4357/ac25f4 The Ion Transition Range of Solar Wind Turbulence in the Inner Heliosphere: Parker Solar Probe Observations The scaling of the turbulent spectra provides a key measurement that allows us to discriminate between different theoretical predictions of turbulence. In the solar wind, this has driven a large number of studies dedicated to this issue using in situ data from various orbiting spacecraft. While a semblance of consensus exists regarding the scaling in the magnetohydrodynamic (MHD) and dispersive ranges, the precise scaling in the transition range and the actual physical mechanisms that control it remain open questions. Using ... Huang, S; Sahraoui, F.; Andrés, N.; Hadid, L.; Yuan, Z.; He, J.; Zhao, J.; Galtier, S.; Zhang, J.; Deng, X.; Jiang, K.; Yu, L.; Xu, S.; Xiong, Q; Wei, Y; de Wit, Dudok; Bale, S.; Kasper, J.; Published by: The Astrophysical Journal      Published on: 03/2021 YEAR: 2021     DOI: 10.3847/2041-8213/abdaaf The Ion Transition Range of Solar Wind Turbulence in the Inner Heliosphere: Parker Solar Probe Observations The scaling of the turbulent spectra provides a key measurement that allows us to discriminate between different theoretical predictions of turbulence. In the solar wind, this has driven a large number of studies dedicated to this issue using in situ data from various orbiting spacecraft. While a semblance of consensus exists regarding the scaling in the magnetohydrodynamic (MHD) and dispersive ranges, the precise scaling in the transition range and the actual physical mechanisms that control it remain open questions. Using ... Huang, S; Sahraoui, F.; Andrés, N.; Hadid, L.; Yuan, Z.; He, J.; Zhao, J.; Galtier, S.; Zhang, J.; Deng, X.; Jiang, K.; Yu, L.; Xu, S.; Xiong, Q; Wei, Y; de Wit, Dudok; Bale, S.; Kasper, J.; Published by: The Astrophysical Journal      Published on: 03/2021 YEAR: 2021     DOI: 10.3847/2041-8213/abdaaf Characteristics of Magnetic Holes in the Solar Wind Revealed by Parker Solar Probe Yu, L.; Huang, S.~Y.; Yuan, Z.~G.; Jiang, K.; Xiong, Q.~Y.; Xu, S.~B.; Wei, Y.~Y.; Zhang, J.; Zhang, Z.~H.; Published by: \apj      Published on: 02/2021 YEAR: 2021     DOI: 10.3847/1538-4357/abb9a8 Characteristics of Magnetic Holes in the Solar Wind Revealed by Parker Solar Probe Yu, L.; Huang, S.~Y.; Yuan, Z.~G.; Jiang, K.; Xiong, Q.~Y.; Xu, S.~B.; Wei, Y.~Y.; Zhang, J.; Zhang, Z.~H.; Published by: \apj      Published on: 02/2021 YEAR: 2021     DOI: 10.3847/1538-4357/abb9a8 2020 Kinetic Scale Slow Solar Wind Turbulence in the Inner Heliosphere: Coexistence of Kinetic Alfv\ en Waves and Alfv\ en Ion Cyclotron Waves The nature of the plasma wave modes around the ion kinetic scales in highly Alfv\ enic slow solar wind turbulence is investigated using data from the NASA\textquoterights Parker Solar Probe taken in the inner heliosphere, at 0.18 au from the Sun. The joint distribution of the normalized reduced magnetic helicity σm (θRB, τ) is obtained, where θRB is the angle between the local mean magnetic field and the radial direction and τ is the temporal scale. Two populations around ion scales a ... Huang, S; Zhang, J.; Sahraoui, F.; He, J.; Yuan, Z.; es, Andr\; Hadid, L.; Deng, X.; Jiang, K.; Yu, L.; Xiong, Q; Wei, Y; Xu, S.; Bale, S.; Kasper, J.; Published by: The Astrophysical Journal      Published on: 07/2020 YEAR: 2020     DOI: 10.3847/2041-8213/ab9abb Kinetic Scale Slow Solar Wind Turbulence in the Inner Heliosphere: Coexistence of Kinetic Alfv\ en Waves and Alfv\ en Ion Cyclotron Waves The nature of the plasma wave modes around the ion kinetic scales in highly Alfv\ enic slow solar wind turbulence is investigated using data from the NASA\textquoterights Parker Solar Probe taken in the inner heliosphere, at 0.18 au from the Sun. The joint distribution of the normalized reduced magnetic helicity σm (θRB, τ) is obtained, where θRB is the angle between the local mean magnetic field and the radial direction and τ is the temporal scale. Two populations around ion scales a ... Huang, S; Zhang, J.; Sahraoui, F.; He, J.; Yuan, Z.; es, Andr\; Hadid, L.; Deng, X.; Jiang, K.; Yu, L.; Xiong, Q; Wei, Y; Xu, S.; Bale, S.; Kasper, J.; Published by: The Astrophysical Journal      Published on: 07/2020 YEAR: 2020     DOI: 10.3847/2041-8213/ab9abb 2015 ION-SCALE TURBULENCE IN THE INNER HELIOSPHERE: RADIAL DEPENDENCE The evolution of the ion-scale plasma turbulence in the inner heliosphere is studied by associating the plasma parameters for hybrid-code turbulence simulations to the radial distance from the Sun via a Solar wind model based mapping procedure. Using a mapping based on a one-dimensional solar wind expansion model, the resulting ion-kinetic scale turbulence is related to the solar wind distance from the Sun. For this purpose the mapping is carried out for various values of ion beta that correspond to the heliocentric dista ... Comişel, H.; Motschmann, U.; üchner, J.; Narita, Y.; Nariyuki, Y.; Published by: The Astrophysical Journal      Published on: 10/2015 YEAR: 2015     DOI: 10.1088/0004-637X/812/2/175

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## Chebyshev’s Inequality

Let $X$ be a random variable with mean $\mu$ and finite variance $\sigma^2$. Then for any real constant $k>0$, $$\begin{equation*} P[|X-\mu| \geq k\sigma] \leq \frac{1}{k^2}\quad \text{ and }\quad P[|X-\mu|< k\sigma] \geq 1-\frac{1}{k^2} \end{equation*}$$


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# Meeting Details

Title: Metrics with strongly positive curvature on Flag manifolds Geometry Working Seminar Ricardo Mendes, U. Muenster (Joint work with Renato G. Bettiol) This work concerns a curvature condition for Riemannian manifolds called strongly positive curvature''. It lies strictly between positive sectional curvature and positive definite curvature operator, and was introduced by J. Thorpe in the 1970s. We identify the moduli space of homogeneous metrics satisfying this condition on the manifolds W6, W12 and W24 of complete K-flags in K^3, where K is the algebra of complex numbers, quaternions and octonions, respectively. In particular, this finishes the classification of manifolds admitting a homogeneous metric with strongly positive curvature initiated in our previous work. It also suggests a general deformation conjecture.


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# Details

Model used in Get Your Vitamin C! Robust Fact Verification with Contrastive Evidence (Schuster et al., NAACL 21).

For more details see: https://github.com/TalSchuster/VitaminC

When using this model, please cite the paper.

# BibTeX entry and citation info

@inproceedings{schuster-etal-2021-get,
title = "Get Your Vitamin {C}! Robust Fact Verification with Contrastive Evidence",
author = "Schuster, Tal  and
Barzilay, Regina",
booktitle = "Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies",
month = jun,
year = "2021",
publisher = "Association for Computational Linguistics",
url = "https://aclanthology.org/2021.naacl-main.52",
doi = "10.18653/v1/2021.naacl-main.52",
pages = "624--643",
abstract = "Typical fact verification models use retrieved written evidence to verify claims. Evidence sources, however, often change over time as more information is gathered and revised. In order to adapt, models must be sensitive to subtle differences in supporting evidence. We present VitaminC, a benchmark infused with challenging cases that require fact verification models to discern and adjust to slight factual changes. We collect over 100,000 Wikipedia revisions that modify an underlying fact, and leverage these revisions, together with additional synthetically constructed ones, to create a total of over 400,000 claim-evidence pairs. Unlike previous resources, the examples in VitaminC are contrastive, i.e., they contain evidence pairs that are nearly identical in language and content, with the exception that one supports a given claim while the other does not. We show that training using this design increases robustness{---}improving accuracy by 10{\%} on adversarial fact verification and 6{\%} on adversarial natural language inference (NLI). Moreover, the structure of VitaminC leads us to define additional tasks for fact-checking resources: tagging relevant words in the evidence for verifying the claim, identifying factual revisions, and providing automatic edits via factually consistent text generation.",
}
`


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Taylor theorem for function of several variables

Where one could find the proof of the following version of Taylor theorem for functions of several variables?

Assume that $f$ is a function of class $C^{n+k}$ defined in a neighbourhood $W$ of zero in $\mathbb{R}^m$ with values in $\mathbb{R}.$ Then there exist a functions $S_\alpha$, where $\alpha$ is multiindex with length $n$, defined in $W$ with values in $\mathbb{R}$ such that

1. $S_\alpha$ is of clas $C^k$ in $W$, $S_\alpha(0)=0$, $D^i S_\alpha(0)=\left(\frac{i!}{(n+i)!}\right) D^{n+i}f(0)$ for every multiindex $i$ with $0 \leq |i| \leq k$.

2. $S_\alpha$ is of class $C^{n+k}$ in $W\setminus \{0\}$.

3. $\lim_{x\rightarrow 0} \|x\|^{|i|} D^{i+k}S_\alpha(x)=0$ for each multiindex $i$ with $1\leq |i|\leq n$ and for each $\alpha$ (with $| \alpha|=n$).

4. $f(x)=\sum_{|\alpha|\leq n}\left(\frac{D^\alpha f(0)}{\alpha !}\right)x^\alpha+\sum_{|\alpha| =n} S_\alpha(x) x^\alpha$ for $x \in W$.

For $m=1$ representation 4. follows from Taylor's formula with integral's remainder: $$f(x)=f(0)+\frac{f'(0)}{1!}x+...+\frac{f^{(n-1)}(0)}{(n-1)!}x^{n-1}+\frac{f^{(n)}(0)}{n!}x^n+S_n(x) x^n,$$ where $$S_n(x):=\frac{1}{(n-1)!}\int_0^1 (f^{(n)}(sx)-f^{(n)}(0)) (1-s)^{n-1}ds.$$ By differentiation under integral we obtain statement 1. The 2. is obvious, because $S_n(x)=\frac{f(x)-P_n(x)}{x^n}$ for $x \neq 0$, where $P_n$ is a polynomial function.
By substitution $y=sx$ formula for $S_n$ can be written in the following form $$S_n(x):=\frac{1}{(n-1)!} \frac{1}{x^n} \int_0^x (f^{(n)}(y)-f^{(n)}(0)) (x-y)^{n-1}dy.$$ By this and Leibniz formula on higher-order derivatives of multiplication follows 3.
For arbitrary $m$ 4. follows from the case $m=1$ applied to the function $g(t):=f(tx)$ for $t \in [0,1].$ Here $$S_\alpha(x)=\frac{1}{(n-1)!}\int_0^1 (D^\alpha f(sx)-D^\alpha f(0))(1-s)^{n-1}ds$$ (where $\alpha$ is multiindex with length $n$.)
The proof of statement 1. goes as the case $m=1$. But I'm most interested how to obtain the 2 and 3.


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#### DOCUMENTA MATHEMATICA, Vol. 5 (2000), 351-364

Andrew Baker

$I_n$-Local Johnson-Wilson Spectra and their Hopf Algebroids

We consider a generalization $\mathcal{E}(n)$ of the Johnson-Wilson spectrum $E(n)$ for which $\mathcal{E}(n)_*$ is a local ring with maximal ideal $I_n$. We prove that the spectra $E(n)$, $\mathcal{E}(n)$ and $\widehat{E(n)}$ are Bousfield equivalent. We also show that the Hopf algebroid $\mathcal{E}(n)_*\mathcal{E}(n)$ is a free $\mathcal{E}(n)_*$-module, generalizing a result of Adams and Clarke for $KU_*KU$.

1991 Mathematics Subject Classification: Primary 55N20 55N22

Keywords and Phrases: Johnson-Wilson spectrum, Hopf algebroid, localization, free module

Full text: dvi.gz 23 k, dvi 62 k, ps.gz 141 k, pdf 193 k.


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# H. Cochran, Inc., is considering a new three-year expansion project that requires an initial fixed asset...

##### REN webassign.net/web/Student/Assignment-Responses/submit?dep=24043711&tags=autosavequestion 382 Evaluate the function for the given values. g(x) = -3[X + 2] +...
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##### Question Two [Total 50 marks] Suppose the market of carpets is competitive. The demand for and...
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##### Find the class width for the frequency table below. Select one: A. 2 B. 1.5 C....
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# (Hyper)volume of the demi(hyper)cube

The demihypercube is formed by taking the convex hull of the set of points in $$\{0, 1\}^n$$ with an even number of $$1$$s. It can be thought of as taking a hypercube, and cutting off "every other" vertex.

### Examples

For example, the $$3$$-demihypercube (which is also a regular tetrahedron) can be described with vertices $$(0,0,0)$$, $$(0,1,1)$$, $$(1,0,1)$$, and $$(1,1,0)$$.

The $$4$$-demicube is the $$16$$-cell.

### Question

What is the hypervolume of the demihypercube under this construction?

(I couldn't find this information on Wikipedia or elsewhere, and I'm curious about it in part because it relates to my question "https://math.stackexchange.com/q/3813522/121988".)

• hal-upec-upem.archives-ouvertes.fr/hal-00793781/document – Jean Marie Sep 5 at 1:16
• @JeanMarie, thanks! If you want to put this as a comment with the answer and a link to this, I'll delete my answer and accept yours. – Peter Kagey Sep 5 at 5:48
• Let us keep it like this. It has been a pleasure for me to discover this new geometrical "object". – Jean Marie Sep 5 at 15:18

This might be a naive approach.

The demi-cube is the cube but, as you said, with $$2^n/2=2^{n-1}$$ of its vertices "cut off". How much do we cut off each time? We cut off a simplex spanned by the standard basis vectors $$e_1,...,e_n$$, the volume of which is $$1/n!$$.

So the demi-cube (constructed from the cube of edge-length 1) should have volume

$$V_n = 1-\frac{2^{n-1}}{n!}.$$

It is not quite what you got in your answer. But maybe the difference is just about the scale of the initial cube.

• This is because I had misunderstood the paper, and I quoted from the wrong section. Your answer is lovely, and I especially like the illustrations! – Peter Kagey Sep 7 at 19:07

As Jean Marie commented, the formula appears at the bottom of page 26 in the the paper "The volume product of convex bodies with many hyperplane symmetries" by Franck Barthe and Matthieu Fradelizi:

The demihypercube is obtained from the cube by cutting off $$2^{n-1}$$ simplicies whose vertices are a vertex $$x \in \{-1; 1\}^n$$ of the cube such that $$\prod x_i = -1$$ and its $$n$$ neighbours. They have a right angle at $$x$$ and the length of the edges through this vertex are $$2$$ thus $$|DH_n| = 2^n - 2^{n-1} \times \frac{2^n}{n!} = 2^n\left(1 - \frac{2^{n-1}}{n!}\right).$$

Note, the authors define the demihypercube on the convex hull of points with coordinates of $$\{-1, 1\}$$, so to get the volume of the construction in the question, divide by a factor of $$2^n$$, yielding $$1 - \frac{2^{n-1}}{n!}.$$

Edit: the original quote that I had taken from the paper gave the volume product, not the volume as I had asked for in the question. This answer has been corrected to reflect this.

• What is the scale of this formula? When I plug in $n=3$, I obtain $8/3>1$, which is apparently not the demi-cube contained in the unit cube of volume 1. – M. Winter Sep 7 at 9:52
• @M.Winter, I had quoted the volume product formula from the paper instead of the volume formula. The answer has now been updated to reflect this. – Peter Kagey Sep 7 at 19:08