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Get Full Access to College Physics - 1 Edition - Chapter 5 - Problem 4cq
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# Solved: When you push a piece of chalk across a

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## Solution for problem 4CQ Chapter 5

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Problem 4CQ

When you push a piece of chalk across a chalkboard, it sometimes screeches because it rapidly alternates between slipping and sticking to the board. Describe this process in more detail, in particular explaining how it is related to the fact that kinetic friction is less than static friction. (The same slip-grab process occurs when tires screech on pavement.)

Step-by-Step Solution:

Step 1 of 2

When the chalk is pushed along the board, it encounters kinetic friction. However the chalk also encounters static friction at each interval.

Step 2 of 2

##### ISBN: 9781938168000

Since the solution to 4CQ from 5 chapter was answered, more than 540 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 4CQ from chapter: 5 was answered by , our top Physics solution expert on 07/07/17, 04:39PM. This textbook survival guide was created for the textbook: College Physics , edition: 1. This full solution covers the following key subjects: Friction, process, particular, chalk, chalkboard. This expansive textbook survival guide covers 34 chapters, and 3125 solutions. College Physics was written by and is associated to the ISBN: 9781938168000. The answer to “?When you push a piece of chalk across a chalkboard, it sometimes screeches because it rapidly alternates between slipping and sticking to the board. Describe this process in more detail, in particular explaining how it is related to the fact that kinetic friction is less than static friction. (The same slip-grab process occurs when tires screech on pavement.)” is broken down into a number of easy to follow steps, and 58 words.

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# How to take into consideration gravity when moving AI

I'm making a 2D game. There is currently a helicopter flying around, controlled by the player. It's controlled using the arrow keys: UP, LEFT and RIGHT.

It's speed along the y axis is dy, and speed along the x axis is dx.

It's physics is as follows:

Whenever UP isn't pressed, dy accelerates in constant acceleration, indefinitely towards down. (Gravity). dx stays in it's current value.

When UP is pressed, dy accelerates in constant acceleration from whatever it currently is, up to 4 (upward, until it reaches speed 4). dx stays in it's current value.

When LEFT is pressed, dx accelerates in constant acceleration, from whatever it currently is, up to -4.

When RIGHT is pressed, dx accelerates in constant acceleration, from whatever it currently is, up to 4.

(When LEFT or RIGHT are pressed and UP is not pressed simultaneously, as I said: dy increasingly becomes smaller and smaller, because gravity is affecting the helicopter)

All of this makes the helicopter often follow arches in the air, rather than straight lines.

This creates physics that seem quite realistic.

My question is:

The opponent helicopter, an AI, should move using the same physics system.

Lets say the AI wants to get from where it currently is, to point B.

If there was no gravity and no gradual acceleration in the game, it would be easy. I would simply draw a vector from the AI's position to point B, and make the AI follow it.

But since there is gravity and gradual acceleration, the AI can never move in a straight line (almost). what would be the best way to make the AI go to point B, as time-efficiently as possible?

How can I take the gravity into consideration when moving the AI toward a specific destination?

(If it's easier to explain, please consider point B to be on the same level on the y-axis as the AI, and not in diagonal to it.)

Thanks

• Can you disable physics on the AI whilst it's moving? If so, then you could disable it while it's moving towards the point, and enable when it reached it. – Zhafur Jan 31 '14 at 9:10
• @Zhafur This will make the movement of the AI seem unrealistic, or at least difference from how the movement of the player seems. I want the movement of the AI to look the same as the movement of the player. – user3150201 Jan 31 '14 at 9:12

tl;dr:

TimeToStop.x = CurrentSpeed.x / Accelaration.x;

if (TimeToStop.x * CurrentSpeed.x >= 1.99 * DistanceFromTarget.x)
slow_down_x(); // CurrentSpeed += Acc.x * direction;
else
speed_up_towards_target_x(); // CurrentSpeed += Acc.x * direction;

Same with y. Remember to bound speed so it's between zero and the max speed. If the enemy has a very slow speed at some point and it"s trying to stop, it may begin to move in the opposite direction. Do not allow it. Stop it if it's slowing down and its speed is under 1 * Acc.

Long version: If there are no obstacles, movement on y-axis is completely irrelevant to (and does not affect) movement on x-axis. So the question you describe can be broken into two separate questions.

1. Moving there on x-Axis
2. And moving there on y-Axis.

CS.x & CS.y are our current speed on the x and y axises.

TS.x & TS.y Is the time it would take you stop only vertically or horizontally considering your current speed in the relevant axis.

D.x & D.y are the distance on each axis.

tl;dr:you continue to accelerate (if possible [unless you reached top speed]) on x-axis until you reach a spot where the following condition is true:

if (TS.x * CS.x >= 1.99 * D.x) hit_the_breaks_on_x();

The same with y.

One approach I've used for a similar problem was to first draw a vector to the target, then compare that to the current velocity direction.

So, take the signed angle between the velocity vector and the target vector and apply a thrust relative to the size of the vector.

So if the angle a between the red velocity vector and the green target vector is negative then apply positive thrust, otherwise apply no thrust and let gravity sort it out.

Note that you'll have to treat the four quadrants where the target may be differently (as if the target had been in the upper left corner a positive angle would have required applying thrust).

Also, make sure to apply the amount of thrust relative to the deviation from the target velocity.

Technically this isn't taking gravity into account, it's compensating for the current state and it means the AI helicopter may not fly in a straight line to the target but will wobble a bit. This may or may not work for your game.

• Thanks. Several questions about this: 1- "...apply a thrust relative to the size of the vector". Did you mean, "size of the angle"? 2- By 'signed angle', you mean positive or negative - correct? If so, than an angle is positive if it's above the brown line, and negative if it's below the brown line. Right? 3- If I understand correctly, what you mean is: Every frame, get the angle between the current velocity vector and the current vector to the target. Then add/subtract from dx and dy accordingly. Is this correct? – user3150201 Jan 31 '14 at 11:36
• @user3150201 Yes on the angle question. Yes on the signed/unsigned question, atan2 can give you the signed angle between two vectors. It looks like you understood my point, apologies if I was unclear. – bornander Jan 31 '14 at 13:08

You could simply recalculate the movement each X frames. Assuming you don't put too many frames between, but enough so it doesn't affect performance, the gravity on a helicopter shouldn't be strong enough to really see the trajectory changing. Plus, I think a player could possibly move the same way, adapting to the gravity while moving and not having it planned.


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# Configuration of settings¶

The current settings of chemcoord can be seen with cc.settings. This is a dictionary that can be changed in place. If it is necessary to change these settings permamently there is the possibility to write a configuration file of the current settings, that is read automatically while importing the module. The configuration file is in the INI format and can be changed with any text editor.

The possible settings and their defaults are:

['defaults']

['atomic_radius_data'] = 'atomic_radius_cc'
Determines which atomic radius is used for calculating if atoms are bonded
['use_lookup_internally'] = True
Look into get_bonds() for an explanation
['viewer'] = 'gv.exe'
Which one is the default viewer used in chemcoord.Cartesian.view() and chemcoord.xyz_functions.view().
 write_configuration_file([filepath, overwrite]) Create a configuration file. read_configuration_file([filepath]) Read the configuration file.


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## Souplet, Philippe

Compute Distance To:
 Author ID: souplet.philippe Published as: Souplet, Philippe; Souplet, Ph.
 Documents Indexed: 121 Publications since 1991, including 2 Books Co-Authors: 45 Co-Authors with 79 Joint Publications 1,089 Co-Co-Authors
all top 5

### Co-Authors

 42 single-authored 14 Quittner, Pavol 9 Weissler, Fred B. 5 Tayachi, Slim 4 Guo, Jong-Shenq 4 Mizoguchi, Noriko 4 Winkler, Michael 4 Zhang, Qi-Shuhuason 3 Fila, Marek 3 Laurençot, Philippe 3 Matos, Júlia 3 Porretta, Alessio 3 Sirakov, Boyan Slavchev 2 Attouchi, Amal 2 Ben-Artzi, Matania 2 Esteve, Carlos 2 Filippucci, Roberta 2 Martel, Yvan 2 Montaru, Alexandre 2 Phan Quoc Hung 2 Poláčik, Peter 2 Pucci, Patrizia 1 Ammar, Kaouther 1 Arrieta, José M. 1 Balabane, Mikhael 1 Bartier, Jean-Philippe 1 Bedjaoui, Nabil 1 Benachour, Said 1 Dickstein, Flávio 1 Dupaigne, Louis 1 Ghidouche, Hamid 1 Haraux, Alain 1 Jazar, Mustapha 1 Jleli, Mohamed Boussairi 1 Kavallaris, Nikos I. 1 Ling, Chia-Tung 1 Mahmoudi, Nejib 1 Manasevich, Raul F. 1 Phan, Quoc Hung 1 Rodríguez-Bernal, Aníbal 1 Rossi, Julio Daniel 1 Samet, Bessem 1 Schmitt, Didier 1 Tarzia, Domingo Alberto 1 Vazquez, Juan Luis 1 Yuxiang, Li
all top 5

### Serials

 6 Journal of Differential Equations 6 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 5 SIAM Journal on Mathematical Analysis 5 Discrete and Continuous Dynamical Systems 4 Proceedings of the American Mathematical Society 4 Comptes Rendus de l’Académie des Sciences. Série I 3 Communications in Mathematical Physics 3 Advances in Mathematics 3 Indiana University Mathematics Journal 3 Mathematische Annalen 3 Differential and Integral Equations 3 Communications in Partial Differential Equations 3 Journal de Mathématiques Pures et Appliquées. Neuvième Série 3 Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 3 Nonlinear Analysis. Theory, Methods & Applications 2 Archive for Rational Mechanics and Analysis 2 Journal d’Analyse Mathématique 2 Journal of Mathematical Analysis and Applications 2 Mathematical Methods in the Applied Sciences 2 Nonlinearity 2 ZAMP. Zeitschrift für angewandte Mathematik und Physik 2 Colloquium Mathematicum 2 Duke Mathematical Journal 2 Journal of Functional Analysis 2 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 2 Asymptotic Analysis 2 IMRN. International Mathematics Research Notices 2 Calculus of Variations and Partial Differential Equations 2 Advances in Differential Equations 2 Journal of Evolution Equations 2 Comptes Rendus. Mathématique. Académie des Sciences, Paris 2 Birkhäuser Advanced Texts. Basler Lehrbücher 1 Houston Journal of Mathematics 1 Journal of Mathematical Biology 1 Mathematical Biosciences 1 Annali di Matematica Pura ed Applicata. Serie Quarta 1 Bulletin of the London Mathematical Society 1 Journal of the Mathematical Society of Japan 1 Portugaliae Mathematica 1 Quarterly of Applied Mathematics 1 Transactions of the American Mathematical Society 1 European Journal of Applied Mathematics 1 Acta Mathematica Universitatis Comenianae. New Series 1 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 1 Journal of Dynamics and Differential Equations 1 Electronic Journal of Differential Equations (EJDE) 1 NoDEA. Nonlinear Differential Equations and Applications 1 The Journal of Fourier Analysis and Applications 1 Interfaces and Free Boundaries 1 Journal of the European Mathematical Society (JEMS) 1 Portugaliae Mathematica. Nova Série 1 Advanced Nonlinear Studies 1 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V 1 Networks and Heterogeneous Media 1 Discrete and Continuous Dynamical Systems. Series S 1 SN Partial Differential Equations and Applications
all top 5

### Fields

 114 Partial differential equations (35-XX) 8 Ordinary differential equations (34-XX) 6 Biology and other natural sciences (92-XX) 4 Functional analysis (46-XX) 4 Operator theory (47-XX) 3 Integral equations (45-XX) 3 Mechanics of deformable solids (74-XX) 2 Global analysis, analysis on manifolds (58-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Statistical mechanics, structure of matter (82-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Potential theory (31-XX) 1 Dynamical systems and ergodic theory (37-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Numerical analysis (65-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX)

### Citations contained in zbMATH Open

113 Publications have been cited 2,830 times in 1,698 Documents Cited by Year
Superlinear parabolic problems. Blow-up, global existence and steady states. Zbl 1128.35003
Quittner, Pavol; Souplet, Philippe
2007
Singularity and decay estimates in superlinear problems via Liouville-type theorems. I: Elliptic equations and systems. Zbl 1146.35038
Poláčik, Peter; Quittner, Pavol; Souplet, Philippe
2007
Blow-up in nonlocal reaction-diffusion equations. Zbl 0909.35073
Souplet, Philippe
1998
Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source. Zbl 0923.35077
Souplet, Philippe
1999
The proof of the Lane-Emden conjecture in four space dimensions. Zbl 1171.35035
Souplet, Philippe
2009
Nondegeneracy of blow-up points for the parabolic Keller-Segel system. Zbl 1302.35075
Mizoguchi, Noriko; Souplet, Philippe
2014
Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Zbl 1109.58025
Souplet, Philippe; Zhang, Qi S.
2006
Singularity and decay estimates in superlinear problems via Liouville-type theorems. II: Parabolic equations. Zbl 1122.35051
Poláčik, Peter; Quittner, Pavol; Souplet, Philippe
2007
Liouville-type theorems and bounds of solutions of Hardy-Hénon equations. Zbl 1233.35093
Phan Quoc Hung; Souplet, Philippe
2012
The local theory for viscous Hamilton–Jacobi equations in Lebesgue spaces. Zbl 1046.35046
Ben-Artzi, Matania; Souplet, Philippe; Weissler, Fred B.
2002
Finite time blow-up for a nonlinear parabolic equation with a gradient term and applications. Zbl 0858.35067
Souplet, Philippe
1996
Optimal condition for non-simultaneous blow-up in a reaction-diffusion system. Zbl 1059.35049
Souplet, Philippe; Tayachi, Slim
2004
Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions. Zbl 1015.35016
Souplet, Philippe
2002
A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces. Zbl 1113.35062
Quittner, P.; Souplet, Ph.
2004
Superlinear parabolic problems. Blow-up, global existence and steady states. 2nd revised and updated edition. Zbl 1423.35004
Quittner, Pavol; Souplet, Philippe
2019
Self-similar subsolutions and blowup for nonlinear parabolic equations. Zbl 0892.35011
Souplet, Philippe; Weissler, Fred B.
1997
Coexistence of simultaneous and nonsimultaneous blow-up in a semilinear parabolic system. Zbl 1212.35219
Rossi, Julio D.; Souplet, Philippe
2005
Recent results and open problems on parabolic equations with gradient nonlinearities. Zbl 0982.35054
Souplet, Philippe
2001
Global solutions of inhomogeneous Hamilton-Jacobi equations. Zbl 1149.35050
Souplet, Philippe; Zhang, Qi S.
2006
Linear and nonlinear heat equations in $$L^q_\delta$$ spaces and universal bounds for global solutions. Zbl 0993.35023
Fila, Marek; Souplet, Philippe; Weissler, Fred B.
2001
Decay of global solutions, stability and blow up for a reaction-diffusion problem with free boundary. Zbl 0959.35087
Ghidouche, Hamid; Souplet, Philippe; Tarzia, Domingo
2001
Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem. Zbl 1006.35103
Fila, Marek; Souplet, Philippe
2001
Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains. Zbl 1218.35052
Yuxiang, Li; Souplet, Philippe
2010
Symmetry of components for semilinear elliptic systems. Zbl 1255.35111
Quittner, Pavol; Souplet, Philippe
2012
The blow-up rate for semilinear parabolic problems on general domains. Zbl 0993.35046
Fila, Marek; Souplet, Philippe
2001
Uniform blow up profile and boundary behaviour for a non-local reaction-diffusion equation with critical damping. Zbl 1058.35121
Souplet, Philippe
2004
Grow-up rate and refined asymptotics for a two-dimensional Patlak-Keller-Segel model in a disk. Zbl 1387.35354
Kavallaris, Nikos I.; Souplet, Philippe
2009
Optimal Liouville-type theorems for noncooperative elliptic Schrödinger systems and applications. Zbl 1254.35077
Quittner, Pavol; Souplet, Philippe
2012
Critical blowup exponents for a system of reaction-diffusion equations with absorption. Zbl 0996.35029
Bedjaoui, Nabil; Souplet, Philippe
2002
Optimal uniqueness condition for the antiperiodic solutions of some nonlinear parabolic equations. Zbl 0892.35078
Souplet, Philippe
1998
Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state. Zbl 1029.35106
Souplet, Philippe; Weissler, Fred B.
2003
Monotonicity of solutions and blow-up for semilinear parabolic equations with nonlinear memory. Zbl 1099.35049
Souplet, Philippe
2004
Global existence of solutions for a chemotaxis-type system arising in crime modelling. Zbl 1284.35445
Manásevich, Raúl; Phan Quoc Hung; Souplet, Philippe
2013
Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena. Zbl 1072.35098
Rodriguez-Bernal, Anibal; Arrieta, José M.; Souplet, Philippe
2004
Extinction and non-extinction for viscous Hamilton-Jacobi equations in $$\mathbb{R}^N$$. Zbl 1046.35053
Benachour, S.; Laurençot, Ph.; Schmitt, D.; Souplet, Ph.
2002
On the growth of mass for a viscous Hamilton-Jacobi equation. Zbl 1031.35054
Laurençot, Philippe; Souplet, Philippe
2003
Exact self-similar blow-up of solutions of a semilinear parabolic equation with a nonlinear gradient term. Zbl 0990.35061
Souplet, Philippe; Tayachi, Slim; Weissler, Fred B.
1996
Poincaré’s inequality and global solutions of a nonlinear parabolic equation. Zbl 0924.35065
Souplet, Philippe; Weissler, Fred B.
1999
Single-point blow-up for a semilinear parabolic system. Zbl 1171.35059
Souplet, Philippe
2009
Initial blow-up rates and universal bounds for nonlinear heat equations. Zbl 1044.35027
Quittner, Pavol; Souplet, Philippe; Winkler, Michael
2004
Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem. Zbl 1116.35070
Souplet, Philippe; Vázquez, Juan Luis
2006
Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up. Zbl 1063.35087
Guo, Jong-Shenq; Souplet, Philippe
2005
Optimal regularity conditions for elliptic problems via $$L^p_\delta$$-spaces. Zbl 1130.35057
Souplet, Philippe
2005
Uniqueness and nonuniqueness results for the antiperiodic solutions of some second-order nonlinear evolution equations. Zbl 0855.34075
Souplet, Philippe
1996
Geometry of unbounded domains, Poincaré inequalities and stability in semilinear parabolic equations. Zbl 0926.35064
Souplet, Philippe
1999
Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions $${n\geq 3}$$. Zbl 1411.35140
Souplet, Philippe; Winkler, Michael
2019
Global existence for reaction-diffusion systems with dissipation of mass and quadratic growth. Zbl 07003366
Souplet, Philippe
2018
Transversality of stable and Nehari manifolds for a semilinear heat equation. Zbl 1233.35045
Dickstein, Flavio; Mizoguchi, Noriko; Souplet, Philippe; Weissler, Fred
2011
Admissible $$L_p$$ norms for local existence and for continuation in semilinear parabolic systems are not the same. Zbl 1006.35048
Quittner, Pavol; Souplet, Philippe
2001
Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities. Zbl 0984.35077
Souplet, Philippe; Tayachi, Slim
2001
Proportionality of components, Liouville theorems and a priori estimates for noncooperative elliptic systems. Zbl 1298.35060
Montaru, Alexandre; Sirakov, Boyan; Souplet, Philippe
2014
Universal blow-up rates for a semilinear heat equation and applications. Zbl 1028.35065
Matos, Júlia; Souplet, Philippe
2003
No touchdown at zero points of the permittivity profile for the MEMS problem. Zbl 1332.35160
Guo, Jong-Shenq; Souplet, Philippe
2015
Stability and continuous dependence of solutions of one-phase Stefan problems for semilinear parabolic equations. Zbl 1007.35040
Souplet, Philippe
2002
On nonexistence and nonuniqueness of solutions of the Cauchy problem for a semilinear parabolic equation. (Sur la non-existence et la non-unicité des solutions du problème de Cauchy pour une équation parabolique semi-linéaire.) Zbl 0938.35068
Ben-Artzi, Matania; Souplet, Philippe; Weissler, Fred B.
1999
Analysis of the loss of boundary conditions for the diffusive Hamilton-Jacobi equation. Zbl 1391.35078
Porretta, Alessio; Souplet, Philippe
2017
Morrey spaces and classification of global solutions for a supercritical semilinear heat equation in $$\mathbb{R}^n$$. Zbl 1372.35157
Souplet, Philippe
2017
Improved conditions for single-point blow-up in reaction-diffusion systems. Zbl 1320.35097
Mahmoudi, Nejib; Souplet, Philippe; Tayachi, Slim
2015
Stability for semilinear parabolic equations with decaying potentials in $$\mathbb{R}^n$$ and dynamical approach to the existence of ground states. Zbl 1017.35033
Souplet, Philippe; Zhang, Qi S.
2002
Decay of heat semigroups in $$L^\infty$$ and applications to nonlinear parabolic problems in unbounded domains. Zbl 0954.35032
Souplet, Philippe
2000
The profile of boundary gradient blowup for the diffusive Hamilton-Jacobi equation. Zbl 1405.35097
Porretta, Alessio; Souplet, Philippe
2017
Nonexistence of global solutions to some differential inequalities of the second order and applications. Zbl 0843.34017
Souplet, Philippe
1995
Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source. Zbl 1195.35077
Ammar, Kaouther; Souplet, Philippe
2010
Parabolic Liouville-type theorems via their elliptic counterparts. Zbl 1306.35066
Quittner, Pavol; Souplet, Philippe
2011
Excluding blowup at zero points of the potential by means of Liouville-type theorems. Zbl 1394.35229
Guo, Jong-Shenq; Souplet, Philippe
2018
On the asymptotics of global solutions for a semilinear heat equation in unbounded domains. (Sur l’asymptotique des solutions globales pour une équation de la chaleur semi-linéaire dans des domaines non bornés.) Zbl 0860.35051
Souplet, Philippe
1996
Blow-up and regularization rates, loss and recovery of boundary conditions for the superquadratic viscous Hamilton-Jacobi equation. Zbl 1437.35112
Porretta, Alessio; Souplet, Philippe
2020
Finite time blow up for a nonlinear heat equation. (Résultats d’explosion en temps fini pour une équation de la chaleur non linéaire.) Zbl 0843.35044
Souplet, Philippe
1995
Small time boundary behavior of solutions of parabolic equations with noncompatible data. Zbl 0959.35079
Martel, Yvan; Souplet, Philippe
2000
Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices. Zbl 1263.35100
Souplet, Philippe
2012
Non-self-similar dead-core rate for the fast diffusion equation with strong absorption. Zbl 1194.35059
Guo, Jong-Shenq; Ling, Chia-Tung; Souplet, Philippe
2010
A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient. Zbl 1318.35018
Souplet, Philippe
2015
Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with $$p$$-Laplacian diffusion. Zbl 1353.35070
Attouchi, Amal; Souplet, Philippe
2017
A Liouville-type theorem for the 3-dimensional parabolic Gross-Pitaevskii and related systems. Zbl 1362.35063
Phan, Quoc Hung; Souplet, Philippe
2016
Single-point blow-up for parabolic systems with exponential nonlinearities and unequal diffusivities. Zbl 1383.35098
Souplet, Philippe; Tayachi, Slim
2016
Existence of exceptional growing-up solutions for a class of nonlinear second order ordinary differential equations. Zbl 0849.34024
Souplet, Philippe
1995
The influence of gradient perturbations on blow-up asymptotics in semilinear parabolic problems: a survey. Zbl 1094.35060
Souplet, Philippe
2005
Bounds of global solutions of parabolic problems with nonlinear boundary conditions. Zbl 1080.35039
Quittner, Pavol; Souplet, Philippe
2003
A survey on $$L^p_\delta$$ spaces and their applications to nonlinear elliptic and parabolic problems. Zbl 1080.35026
Souplet, Philippe
2004
An optimal Liouville-type theorem for radial entire solutions of the porous medium equation with source. Zbl 1172.35038
Souplet, Philippe
2009
Critical exponents, special large-time behavior and oscillatory blow-up in nonlinear ODE’s. Zbl 1015.34038
Souplet, Philippe
1998
Oscillatory blow-up in nonlinear second order ODE’s: The critical case. Zbl 1048.34074
Balabane, Mikhaël; Jazar, Mustapha; Souplet, Philippe
2003
A priori and universal estimates for global solutions of superlinear degenerate parabolic equations. Zbl 1072.35099
Souplet, Philippe
2002
A Liouville-type theorem in a half-space and its applications to the gradient blow-up behavior for superquadratic diffusive Hamilton-Jacobi equations. Zbl 1436.35010
Filippucci, Roberta; Pucci, Patrizia; Souplet, Philippe
2020
Blow-up rate of solutions of parabolic problems with nonlinear boundary conditions. Zbl 1364.35175
Quittner, Pavol; Souplet, Philippe
2012
A note on diffusion-induced blow-up. Zbl 1145.35068
Souplet, Philippe
2007
Self-similar subsolutions and blow-up for nonlinear parabolic equations. Zbl 0893.35056
Souplet, Philippe; Weissler, Fred B.
1997
Étude des solutions globales de certaines équations différentielles ordinaires du second ordre non linéaires. (Study of the global solutions of certain nonlinear second order ordinary differential equations). Zbl 0738.34003
Souplet, Philippe
1991
A priori estimates of global solutions of superlinear parabolic problems without variational structure. Zbl 1029.35049
Quittner, Pavol; Souplet, Philippe
2003
An example of uniformly recurrent function which is not almost periodic. Zbl 1051.42008
Haraux, Alain; Souplet, Philippe
2004
Optimal condition for blow-up of the critical $$L^q$$ norm for the semilinear heat equation. Zbl 1420.35139
Mizoguchi, Noriko; Souplet, Philippe
2019
Global existence from single-component $$L_{p}$$ estimates in a semilinear reaction-diffusion system. Zbl 0993.35048
Quittner, Pavol; Souplet, Philippe
2002
Optimal growth rates for a viscous Hamilton-Jacobi equation. Zbl 1090.35036
Laurençot, Philippe; Souplet, Philippe
2005
The influence of space dimension on the large-time behavior in a reaction-diffusion system modeling diallelic selection. Zbl 1303.92086
Souplet, Philippe; Winkler, Michael
2011
Discontinuous critical Fujita exponents for the heat equation with combined nonlinearities. Zbl 1439.35199
Jleli, Mohamed; Samet, Bessem; Souplet, Philippe
2020
A simplified approach to the refined blowup behavior for the nonlinear heat equation. Zbl 1411.35176
Souplet, Philippe
2019
Small time boundary estimates for solutions of the heat equation with non-compatible data. (Estimation optimales en temps petit et près de la frontière pour les solutions de l’équation de la chaleur avec données non compatibles.) Zbl 0990.35051
Martel, Yvan; Souplet, Philippe
1998
An optimal uniqueness condition for the antiperiodic solutions of parabolic evolution equations. (Une condition optimale d’unicité pour les solutions antipériodiques d’équations d’évolution paraboliques.) Zbl 0809.35036
Souplet, Philippe
1994
Gradient bounds for solutions of semilinear parabolic equations without Bernstein’s quadratic condition. Zbl 1073.35045
Bartier, Jean-Philippe; Souplet, Philippe
2004
Instantaneous smoothing estimates for the Hermite semigroup in uniformly local spaces and related nonlinear equations. Zbl 1052.35090
Matos, Júlia; Souplet, Philippe
2004
The Vázquez maximum principle and the Landis conjecture for elliptic PDE with unbounded coefficients. Zbl 1471.35138
Sirakov, Boyan; Souplet, Philippe
2021
Blow-up and regularization rates, loss and recovery of boundary conditions for the superquadratic viscous Hamilton-Jacobi equation. Zbl 1437.35112
Porretta, Alessio; Souplet, Philippe
2020
A Liouville-type theorem in a half-space and its applications to the gradient blow-up behavior for superquadratic diffusive Hamilton-Jacobi equations. Zbl 1436.35010
Filippucci, Roberta; Pucci, Patrizia; Souplet, Philippe
2020
Discontinuous critical Fujita exponents for the heat equation with combined nonlinearities. Zbl 1439.35199
Jleli, Mohamed; Samet, Bessem; Souplet, Philippe
2020
A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient. Zbl 1440.35106
Filippucci, Roberta; Pucci, Patrizia; Souplet, Philippe
2020
Gradient blow-up rates and sharp gradient estimates for diffusive Hamilton-Jacobi equations. Zbl 1448.35051
Attouchi, Amal; Souplet, Philippe
2020
Superlinear parabolic problems. Blow-up, global existence and steady states. 2nd revised and updated edition. Zbl 1423.35004
Quittner, Pavol; Souplet, Philippe
2019
Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions $${n\geq 3}$$. Zbl 1411.35140
Souplet, Philippe; Winkler, Michael
2019
Optimal condition for blow-up of the critical $$L^q$$ norm for the semilinear heat equation. Zbl 1420.35139
Mizoguchi, Noriko; Souplet, Philippe
2019
A simplified approach to the refined blowup behavior for the nonlinear heat equation. Zbl 1411.35176
Souplet, Philippe
2019
No touchdown at points of small permittivity and nontrivial touchdown sets for the MEMS problem. Zbl 1444.35104
Esteve, Carlos; Souplet, Philippe
2019
Corrigendum to: “Nondegeneracy of blow-up points for the parabolic Keller-Segel system”. Zbl 1415.35062
Mizoguchi, N.; Souplet, Ph.
2019
Global existence for reaction-diffusion systems with dissipation of mass and quadratic growth. Zbl 07003366
Souplet, Philippe
2018
Excluding blowup at zero points of the potential by means of Liouville-type theorems. Zbl 1394.35229
Guo, Jong-Shenq; Souplet, Philippe
2018
Quantitative touchdown localization for the MEMS problem with variable dielectric permittivity. Zbl 1397.35127
Esteve, Carlos; Souplet, Philippe
2018
Analysis of the loss of boundary conditions for the diffusive Hamilton-Jacobi equation. Zbl 1391.35078
Porretta, Alessio; Souplet, Philippe
2017
Morrey spaces and classification of global solutions for a supercritical semilinear heat equation in $$\mathbb{R}^n$$. Zbl 1372.35157
Souplet, Philippe
2017
The profile of boundary gradient blowup for the diffusive Hamilton-Jacobi equation. Zbl 1405.35097
Porretta, Alessio; Souplet, Philippe
2017
Single point gradient blow-up on the boundary for a Hamilton-Jacobi equation with $$p$$-Laplacian diffusion. Zbl 1353.35070
Attouchi, Amal; Souplet, Philippe
2017
A Liouville-type theorem for the 3-dimensional parabolic Gross-Pitaevskii and related systems. Zbl 1362.35063
Phan, Quoc Hung; Souplet, Philippe
2016
Single-point blow-up for parabolic systems with exponential nonlinearities and unequal diffusivities. Zbl 1383.35098
Souplet, Philippe; Tayachi, Slim
2016
No touchdown at zero points of the permittivity profile for the MEMS problem. Zbl 1332.35160
Guo, Jong-Shenq; Souplet, Philippe
2015
Improved conditions for single-point blow-up in reaction-diffusion systems. Zbl 1320.35097
Mahmoudi, Nejib; Souplet, Philippe; Tayachi, Slim
2015
A priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient. Zbl 1318.35018
Souplet, Philippe
2015
Nondegeneracy of blow-up points for the parabolic Keller-Segel system. Zbl 1302.35075
Mizoguchi, Noriko; Souplet, Philippe
2014
Proportionality of components, Liouville theorems and a priori estimates for noncooperative elliptic systems. Zbl 1298.35060
Montaru, Alexandre; Sirakov, Boyan; Souplet, Philippe
2014
Symmetry of components and Liouville theorems for noncooperative elliptic systems on the half-space. (Symétrie des composantes et théorèmes de Liouville pour des systèmes elliptiques non coopératifs dans le demi-espace.) Zbl 1300.35032
Montaru, Alexandre; Souplet, Philippe
2014
Global existence of solutions for a chemotaxis-type system arising in crime modelling. Zbl 1284.35445
Manásevich, Raúl; Phan Quoc Hung; Souplet, Philippe
2013
Liouville-type theorems and bounds of solutions of Hardy-Hénon equations. Zbl 1233.35093
Phan Quoc Hung; Souplet, Philippe
2012
Symmetry of components for semilinear elliptic systems. Zbl 1255.35111
Quittner, Pavol; Souplet, Philippe
2012
Optimal Liouville-type theorems for noncooperative elliptic Schrödinger systems and applications. Zbl 1254.35077
Quittner, Pavol; Souplet, Philippe
2012
Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices. Zbl 1263.35100
Souplet, Philippe
2012
Blow-up rate of solutions of parabolic problems with nonlinear boundary conditions. Zbl 1364.35175
Quittner, Pavol; Souplet, Philippe
2012
Classification of large-time behaviors in a reaction-diffusion system modeling diallelic selection. Zbl 1319.92032
Souplet, Philippe; Winkler, Michael
2012
Transversality of stable and Nehari manifolds for a semilinear heat equation. Zbl 1233.35045
Dickstein, Flavio; Mizoguchi, Noriko; Souplet, Philippe; Weissler, Fred
2011
Parabolic Liouville-type theorems via their elliptic counterparts. Zbl 1306.35066
Quittner, Pavol; Souplet, Philippe
2011
The influence of space dimension on the large-time behavior in a reaction-diffusion system modeling diallelic selection. Zbl 1303.92086
Souplet, Philippe; Winkler, Michael
2011
Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains. Zbl 1218.35052
Yuxiang, Li; Souplet, Philippe
2010
Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source. Zbl 1195.35077
Ammar, Kaouther; Souplet, Philippe
2010
Non-self-similar dead-core rate for the fast diffusion equation with strong absorption. Zbl 1194.35059
Guo, Jong-Shenq; Ling, Chia-Tung; Souplet, Philippe
2010
The proof of the Lane-Emden conjecture in four space dimensions. Zbl 1171.35035
Souplet, Philippe
2009
Grow-up rate and refined asymptotics for a two-dimensional Patlak-Keller-Segel model in a disk. Zbl 1387.35354
Kavallaris, Nikos I.; Souplet, Philippe
2009
Single-point blow-up for a semilinear parabolic system. Zbl 1171.35059
Souplet, Philippe
2009
An optimal Liouville-type theorem for radial entire solutions of the porous medium equation with source. Zbl 1172.35038
Souplet, Philippe
2009
Superlinear parabolic problems. Blow-up, global existence and steady states. Zbl 1128.35003
Quittner, Pavol; Souplet, Philippe
2007
Singularity and decay estimates in superlinear problems via Liouville-type theorems. I: Elliptic equations and systems. Zbl 1146.35038
Poláčik, Peter; Quittner, Pavol; Souplet, Philippe
2007
Singularity and decay estimates in superlinear problems via Liouville-type theorems. II: Parabolic equations. Zbl 1122.35051
Poláčik, Peter; Quittner, Pavol; Souplet, Philippe
2007
A note on diffusion-induced blow-up. Zbl 1145.35068
Souplet, Philippe
2007
A remark on the large time behavior of solutions of viscous Hamilton-Jacobi equations. Zbl 1134.35019
Souplet, Philippe
2007
Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Zbl 1109.58025
Souplet, Philippe; Zhang, Qi S.
2006
Global solutions of inhomogeneous Hamilton-Jacobi equations. Zbl 1149.35050
Souplet, Philippe; Zhang, Qi S.
2006
Stabilization towards a singular steady state with gradient blow-up for a diffusion-convection problem. Zbl 1116.35070
Souplet, Philippe; Vázquez, Juan Luis
2006
Coexistence of simultaneous and nonsimultaneous blow-up in a semilinear parabolic system. Zbl 1212.35219
Rossi, Julio D.; Souplet, Philippe
2005
Fast rate of formation of dead-core for the heat equation with strong absorption and applications to fast blow-up. Zbl 1063.35087
Guo, Jong-Shenq; Souplet, Philippe
2005
Optimal regularity conditions for elliptic problems via $$L^p_\delta$$-spaces. Zbl 1130.35057
Souplet, Philippe
2005
The influence of gradient perturbations on blow-up asymptotics in semilinear parabolic problems: a survey. Zbl 1094.35060
Souplet, Philippe
2005
Optimal growth rates for a viscous Hamilton-Jacobi equation. Zbl 1090.35036
Laurençot, Philippe; Souplet, Philippe
2005
Optimal condition for non-simultaneous blow-up in a reaction-diffusion system. Zbl 1059.35049
Souplet, Philippe; Tayachi, Slim
2004
A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces. Zbl 1113.35062
Quittner, P.; Souplet, Ph.
2004
Uniform blow up profile and boundary behaviour for a non-local reaction-diffusion equation with critical damping. Zbl 1058.35121
Souplet, Philippe
2004
Monotonicity of solutions and blow-up for semilinear parabolic equations with nonlinear memory. Zbl 1099.35049
Souplet, Philippe
2004
Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena. Zbl 1072.35098
Rodriguez-Bernal, Anibal; Arrieta, José M.; Souplet, Philippe
2004
Initial blow-up rates and universal bounds for nonlinear heat equations. Zbl 1044.35027
Quittner, Pavol; Souplet, Philippe; Winkler, Michael
2004
A survey on $$L^p_\delta$$ spaces and their applications to nonlinear elliptic and parabolic problems. Zbl 1080.35026
Souplet, Philippe
2004
An example of uniformly recurrent function which is not almost periodic. Zbl 1051.42008
Haraux, Alain; Souplet, Philippe
2004
Gradient bounds for solutions of semilinear parabolic equations without Bernstein’s quadratic condition. Zbl 1073.35045
Bartier, Jean-Philippe; Souplet, Philippe
2004
Instantaneous smoothing estimates for the Hermite semigroup in uniformly local spaces and related nonlinear equations. Zbl 1052.35090
Matos, Júlia; Souplet, Philippe
2004
Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state. Zbl 1029.35106
Souplet, Philippe; Weissler, Fred B.
2003
On the growth of mass for a viscous Hamilton-Jacobi equation. Zbl 1031.35054
Laurençot, Philippe; Souplet, Philippe
2003
Universal blow-up rates for a semilinear heat equation and applications. Zbl 1028.35065
Matos, Júlia; Souplet, Philippe
2003
Bounds of global solutions of parabolic problems with nonlinear boundary conditions. Zbl 1080.35039
Quittner, Pavol; Souplet, Philippe
2003
Oscillatory blow-up in nonlinear second order ODE’s: The critical case. Zbl 1048.34074
Balabane, Mikhaël; Jazar, Mustapha; Souplet, Philippe
2003
A priori estimates of global solutions of superlinear parabolic problems without variational structure. Zbl 1029.35049
Quittner, Pavol; Souplet, Philippe
2003
The local theory for viscous Hamilton–Jacobi equations in Lebesgue spaces. Zbl 1046.35046
Ben-Artzi, Matania; Souplet, Philippe; Weissler, Fred B.
2002
Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions. Zbl 1015.35016
Souplet, Philippe
2002
Critical blowup exponents for a system of reaction-diffusion equations with absorption. Zbl 0996.35029
Bedjaoui, Nabil; Souplet, Philippe
2002
Extinction and non-extinction for viscous Hamilton-Jacobi equations in $$\mathbb{R}^N$$. Zbl 1046.35053
Benachour, S.; Laurençot, Ph.; Schmitt, D.; Souplet, Ph.
2002
Stability and continuous dependence of solutions of one-phase Stefan problems for semilinear parabolic equations. Zbl 1007.35040
Souplet, Philippe
2002
Stability for semilinear parabolic equations with decaying potentials in $$\mathbb{R}^n$$ and dynamical approach to the existence of ground states. Zbl 1017.35033
Souplet, Philippe; Zhang, Qi S.
2002
A priori and universal estimates for global solutions of superlinear degenerate parabolic equations. Zbl 1072.35099
Souplet, Philippe
2002
Global existence from single-component $$L_{p}$$ estimates in a semilinear reaction-diffusion system. Zbl 0993.35048
Quittner, Pavol; Souplet, Philippe
2002
Recent results and open problems on parabolic equations with gradient nonlinearities. Zbl 0982.35054
Souplet, Philippe
2001
Linear and nonlinear heat equations in $$L^q_\delta$$ spaces and universal bounds for global solutions. Zbl 0993.35023
Fila, Marek; Souplet, Philippe; Weissler, Fred B.
2001
Decay of global solutions, stability and blow up for a reaction-diffusion problem with free boundary. Zbl 0959.35087
Ghidouche, Hamid; Souplet, Philippe; Tarzia, Domingo
2001
Existence of global solutions with slow decay and unbounded free boundary for a superlinear Stefan problem. Zbl 1006.35103
Fila, Marek; Souplet, Philippe
2001
The blow-up rate for semilinear parabolic problems on general domains. Zbl 0993.35046
Fila, Marek; Souplet, Philippe
2001
Admissible $$L_p$$ norms for local existence and for continuation in semilinear parabolic systems are not the same. Zbl 1006.35048
Quittner, Pavol; Souplet, Philippe
2001
Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities. Zbl 0984.35077
Souplet, Philippe; Tayachi, Slim
2001
Existence of ground states for semilinear elliptic equations with decaying mass: A parabolic approach. Zbl 0992.35029
Souplet, Philippe; Zhang, Qi S.
2001
Decay of heat semigroups in $$L^\infty$$ and applications to nonlinear parabolic problems in unbounded domains. Zbl 0954.35032
Souplet, Philippe
2000
Small time boundary behavior of solutions of parabolic equations with noncompatible data. Zbl 0959.35079
Martel, Yvan; Souplet, Philippe
2000
Blow-up behavior in nonlocal vs. local heat equations. Zbl 0954.35091
Souplet, Philippe
2000
Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source. Zbl 0923.35077
Souplet, Philippe
1999
Poincaré’s inequality and global solutions of a nonlinear parabolic equation. Zbl 0924.35065
Souplet, Philippe; Weissler, Fred B.
1999
Geometry of unbounded domains, Poincaré inequalities and stability in semilinear parabolic equations. Zbl 0926.35064
Souplet, Philippe
1999
On nonexistence and nonuniqueness of solutions of the Cauchy problem for a semilinear parabolic equation. (Sur la non-existence et la non-unicité des solutions du problème de Cauchy pour une équation parabolique semi-linéaire.) Zbl 0938.35068
Ben-Artzi, Matania; Souplet, Philippe; Weissler, Fred B.
1999
Sharp stability estimates for quasi-autonomous evolution equations of hyperbolic type. Zbl 1031.34059
Souplet, Philippe
1999
Blow-up in nonlocal reaction-diffusion equations. Zbl 0909.35073
Souplet, Philippe
1998
Optimal uniqueness condition for the antiperiodic solutions of some nonlinear parabolic equations. Zbl 0892.35078
Souplet, Philippe
1998
Critical exponents, special large-time behavior and oscillatory blow-up in nonlinear ODE’s. Zbl 1015.34038
Souplet, Philippe
1998
...and 13 more Documents
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### Cited by 1,578 Authors

 53 Souplet, Philippe 50 Winkler, Michael 47 Mu, Chunlai 31 Liu, Bingchen 31 Zheng, Sining 26 Li, Fengjie 22 Ishige, Kazuhiro 20 Wei, Juncheng 19 Fang, Zhongbo 18 Quittner, Pavol 17 Laurençot, Philippe 17 Zhang, Zhengce 16 Ding, Juntang 15 Fujishima, Yohei 15 Xiang, Zhaoyin 15 Xie, Chunhong 14 Lankeit, Johannes 14 Li, Yuxiang 14 Weissler, Fred B. 14 Zhou, Jun 13 Iagar, Razvan Gabriel 13 Kawakami, Tatsuki 13 Yanagida, Eiji 12 Quaas, Alexander 12 Wang, Liangchen 11 Fila, Marek 11 Guo, Zongming 11 Lin, Zhigui 11 Liu, Qilin 11 Mizoguchi, Noriko 11 Phan, Quoc Hung 11 Zheng, Pan 11 Zou, Wenming 10 Duong, Anh Tuan 10 Harrabi, Abdellaziz 10 Villavert, John 10 Wang, Mingxin 10 Wang, Wei 10 Wang, Yulan 10 Yang, Zuodong 10 Zaag, Hatem 9 Del Pino, Manuel A. 9 García-Melián, Jorge 9 Karch, Grzegorz 9 Poláčik, Peter 9 Tang, Bao Quoc 9 Tayachi, Slim 9 Wan, FangShu 8 Deng, Weibing 8 Du, Lili 8 Fellner, Klemens 8 Gao, Wenjie 8 Ghergu, Marius 8 Guo, Jong-Shenq 8 Lei, Yutian 8 Liu, Dengming 8 Musso, Monica 8 Parshad, Rana D. 8 Sierżȩga, Mikołaj 8 Song, Xianfa 8 Taliaferro, Steven D. 8 Weth, Tobias 8 Yin, Jingxue 7 Biler, Piotr 7 Chen, Caisheng 7 Chen, Youpeng 7 Chen, Yuqing 7 Harada, Junichi 7 Jin, Hai-Yang 7 Kong, Linghua 7 Liu, Baiyu 7 Loayza, Miguel 7 Nguyễn Thạc Dũng 7 Punzo, Fabio 7 Sánchez, Ariel 7 Stinner, Christian 7 Tao, Youshan 7 Wang, Xiaoliu 7 Yang, Hui 7 Yu, Xiaohui 6 Abdellaoui, Boumediene 6 Bidaut-Veron, Marie-Françoise 6 Dai, Wei 6 Ferreira, Raúl 6 Filippucci, Roberta 6 Fujie, Kentarou 6 Hu, Xuegang 6 Kirane, Mokhtar 6 Li, Congming 6 Li, Huiling 6 Morgan, Jeff J. 6 O’Regan, Donal 6 Rahal, Belgacem 6 Reichel, Wolfgang 6 Robinson, James Cooper 6 Rodríguez-Bernal, Aníbal 6 Shang, Haifeng 6 Takahashi, Jin 6 Vazquez, Juan Luis 6 Vernier-Piro, Stella ...and 1,478 more Authors
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### Cited in 230 Serials

 153 Journal of Differential Equations 147 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 138 Journal of Mathematical Analysis and Applications 47 Discrete and Continuous Dynamical Systems 45 Boundary Value Problems 38 Calculus of Variations and Partial Differential Equations 37 ZAMP. Zeitschrift für angewandte Mathematik und Physik 36 Journal of Functional Analysis 35 Applied Mathematics Letters 35 Nonlinear Analysis. Real World Applications 35 Communications on Pure and Applied Analysis 31 Computers & Mathematics with Applications 31 Journal de Mathématiques Pures et Appliquées. Neuvième Série 30 Applicable Analysis 27 Proceedings of the American Mathematical Society 26 Mathematische Annalen 26 Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 25 Journal of Evolution Equations 23 Mathematical Methods in the Applied Sciences 23 Discrete and Continuous Dynamical Systems. Series B 21 Applied Mathematics and Computation 20 Advanced Nonlinear Studies 18 NoDEA. Nonlinear Differential Equations and Applications 14 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 14 Science China. Mathematics 13 Nonlinearity 13 Advances in Mathematics 13 Archiv der Mathematik 13 Communications in Partial Differential Equations 12 Journal of Mathematical Physics 12 SIAM Journal on Mathematical Analysis 12 Communications in Contemporary Mathematics 11 Transactions of the American Mathematical Society 11 Journal of Dynamics and Differential Equations 10 Journal of Inequalities and Applications 10 Comptes Rendus. Mathématique. Académie des Sciences, Paris 10 Advances in Nonlinear Analysis 9 Annali di Matematica Pura ed Applicata. Serie Quarta 9 Abstract and Applied Analysis 9 Acta Mathematica Sinica. English Series 9 Complex Variables and Elliptic Equations 9 SN Partial Differential Equations and Applications 8 Journal of Computational and Applied Mathematics 7 Journal d’Analyse Mathématique 7 Acta Applicandae Mathematicae 7 Mediterranean Journal of Mathematics 7 Discrete and Continuous Dynamical Systems. Series S 6 Communications in Mathematical Physics 6 Monatshefte für Mathematik 6 European Journal of Applied Mathematics 6 Applied Mathematics. Series B (English Edition) 6 Journal of Applied Mathematics and Computing 6 Journal of Elliptic and Parabolic Equations 5 Archive for Rational Mechanics and Analysis 5 Journal of Mathematical Biology 5 Quarterly of Applied Mathematics 5 Mathematical and Computer Modelling 5 The Journal of Geometric Analysis 5 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 5 SIAM Journal on Applied Mathematics 5 Stochastic Processes and their Applications 5 Potential Analysis 5 Milan Journal of Mathematics 5 Advances in Difference Equations 5 Journal of Function Spaces 5 AIMS Mathematics 4 Mathematical Notes 4 Duke Mathematical Journal 4 Journal of the Mathematical Society of Japan 4 Manuscripta Mathematica 4 Results in Mathematics 4 Tohoku Mathematical Journal. Second Series 4 Applied Mathematics and Mechanics. (English Edition) 4 Bulletin of the Iranian Mathematical Society 4 Japan Journal of Industrial and Applied Mathematics 4 Electronic Journal of Differential Equations (EJDE) 4 European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations 4 Revista Matemática Complutense 4 Open Mathematics 4 Electronic Research Archive 3 Israel Journal of Mathematics 3 Rocky Mountain Journal of Mathematics 3 Chaos, Solitons and Fractals 3 International Journal of Mathematics and Mathematical Sciences 3 Journal of the Korean Mathematical Society 3 Mathematische Nachrichten 3 Pacific Journal of Mathematics 3 Chinese Annals of Mathematics. Series B 3 Physica D 3 Differential Geometry and its Applications 3 Bulletin des Sciences Mathématiques 3 Taiwanese Journal of Mathematics 3 Journal of Mathematical Fluid Mechanics 3 Communications in Nonlinear Science and Numerical Simulation 3 Annales Henri Poincaré 3 Differential Equations 3 Journal of Applied Mathematics 3 Journal of Fixed Point Theory and Applications 3 Frontiers of Mathematics in China 3 International Journal of Differential Equations ...and 130 more Serials
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text:
# Week 2 – Geometric Sequences

This week in pre-calculus 11, we learned geometric sequences and geometric series. I learned how to find $S_n$ with just the first term and r. I did find it a little complicated but when I used the formula $S_n=\frac{a(r{^n}-1)}{r-1}$ I could figure it out and it made sense.

For example (Page 49, #5a, Pre-calculus worktext):

$S_n=\frac{a(r{^n}-1)}{r-1}$

$S_8=\frac{1(0.3{^7}-1)}{0.3-1}$

$S_8=\frac{-0.99}{-0.7}$

$S_n=1.41$


text:
# The Gravitational Force of Rubbish

Imagine, for just a moment, that you were one a group of scientists that had proven the most important, the most profound, the most utterly amazing scientific discovery of all time. Where would you publish it?

Maybe Nature? Science? Or maybe you’d prefer to go open-access, and go with PLOS ONE? Or more mainstream, and send a press release to the NYT?

Well, in the case of today’s crackpots, they bypassed all of those boring journals. They couldn’t be bothered with a pompous rag like the Times. No, they went for the really serious press: America Now with Leeza Gibbons.

What did they go to this amazing media outlet to announce? The most amazing scientific discovery of all time: gravity is an illusion! There’s no gravity. In fact, not just is there no gravity, but all of that quantum physics stuff? It’s utter rubbish. You don’t need any of that complicated stuff! No – you need only one thing: the solar wind.

A new theory on the forces that control planetary orbit refutes the 400-year old assumptions currently held by the scientific community. Scientific and engineering experts Gerhard and Kevin Neumaier have established a relationship between solar winds and a quantized order in both the position and velocity of the solar system’s planets, and movement at an atomic level, with both governed by the same set of physics.

The observations made bring into question the Big Bang Theory, the concept of black holes, gravitational waves and gravitons. The Neumaiers’ paper, More Than Gravity, is available for review at MoreThanGravity.com

Pretty damned impressive, huh? So let’s follow their instructions, and go over to their website.

Ever since humankind discovered that the Earth and the planets revolved around the Sun, there was a question about what force was responsible for this. Since the days of Newton, science has held onto the notion that an invisible force, which we have never been able to detect, controls planetary motion. There are complicated theories about black holes that have never been seen, densities of planets that have never been measured, and subatomic particles that have never been detected.

However, it is simpler than all of that and right in front of us. The Sun and the solar wind are the most powerful forces in our solar system. They are physically moving the planets. In fact, the solar wind spins outward in a spiral at over a million miles per hour that controls the velocity and distances that planets revolve around the Sun. The Sun via the solar wind quantizes the orbits of the planets – their position and speed.

The solar wind also leads to the natural log and other phenomenon from the very large scale down to the atomic level. This is clearly a different idea than the current view that has been held for over 400 years. We have been working on this for close 50 years and thanks to satellite explorations of space have data that just was not available when theories long ago were developed. We think that we have many of the pieces but there are certainly many more to be found. We set this up as a web site, rather as some authoritative book so that there would be plenty of opportunity for dialog. The name for this web site, www.MorethanGravity.com was chosen because we believe there is far more to this subject than is commonly understood. Whether you are a scientific expert in your field or just have a general interest in how our solar system works, we appreciate your comments.

See, it’s all about the solar wind. There’s no such thing as gravity – that’s just nonsense. The sun produces the solar wind, which does absolutely everything. The wind comes out of the sun, and spirals out from the sun. That spiral motion has eddies in it an quantized intervals, and that’s where the planets are. Amazing, huh?

Remember my mantra: the worst math is no math. This is a beautiful demonstration
of that.

Of course… why does the solar wind move in a spiral? Everything we know says that in the absence of a force, things move in a straight line. It can’t be spiraling because of gravity, because there is no gravity. So why does it spiral? Our brilliant authors don’t bother to say. What makes it spiral, instead of just move straight? Mathematically, spiral motion is very complicated. It requires a centripetal force which is smaller than the force that would produce an orbit. Where’s that force in this framework? There isn’t any. They just say that that’s how the solar wind works, period. There are many possible spirals, with different radial velocities – which one does the solar wind follow according to this, and why? Again, no answer from the authors.

Or… why is the sun producing the solar wind at all? According to those old, stupid theories that this work of brilliance supercedes, the sun produces a solar wind because it’s fusing hydrogen atoms into helium. That’s happening because gravity is causing the atoms of the sun to be compressed together until they fuse. Without gravity, why is fusion happening at all? And given that it’s happening, why does the sun not just explode into a supernova? We know, from direct observation, that the energy produced by fusion creates an outward force. But gravity can’t be holding the sun together – so why is the sun there at all? Still, no answers.

They do, eventually, do some math. One of the big “results” of this hypothesis is about the “quantization” of the orbits of planets around the sun. They were able to develop a simple equation which predicts the locations where planets could exist in their “solar wind” system.

Let’s start with the distance between the planets and the Sun. We guessed that if the solar system was like an atom, that planetary distance would be quantized. This is to say that we thought that the planets would have definite positions and that they would be either in the position or it would be empty. In a mathematical sense, this would be represented by a numerical integer ordering (0,1,2,3,…). If the first planet, Mercury was in the 0 orbital, how would the rest of the planets line up? Amazingly well we found.

If we predict the distance from the surface of the Sun to each planet in this quantized approach, the results are astounding. If D equals the mean distance to the surface of the Sun, and d0 as the distance to Mercury, we can describe the relationship that orders the planets mathematically as:

$D=d_0 S^n$

Each planetary position can be predicted from this equation in a simple calculation as we increase the integer (or planet number) n. S is the solar factor, which equals 1.387. The solar factor is found in the differential rotation of the Sun and the profile of the solar wind which we will discuss later.

Similar to the quantized orbits that exist within an atom, the planetary bodies are either there or not. Mercury is in the zero orbital. The next orbital is missing a planet. The second, third, and fourth orbitals are occupied by Venus, Earth, and Mars respectively. The fifth orbital is missing. The sixth orbital is filled with Ceres. Ceres is described as either the largest of all asteroids or a minor planet (with a diameter a little less than half that of Pluto), depending on who describes it. Ceres was discovered in 1801 as astronomers searched for the missing planets that the Titius-Bode Law predicted would exist.

So. What they found was an exponential equation which products very approximate versions of the size of first 8 planets’ orbits, as well as a couple of missing ones.

This is, in its way, interesting. Not because they found anything, but rather because they think that this is somehow profound.

We’ve got 8 data points (or 9, counting the asteroid belt). More precisely, we have 9 ranges, because all of the orbits are elliptical,but the authors of this junk are producing a single number for the size of the orbits, and they can declare success if their number falls anywherewithin the range from perihelion to aphelion in each of the orbits.

It would be shocking if there weren’t any number of simple equations that described exactly the 9 data points of the planet’s orbits.

But they couldn’t even make that work directly. They only manage to get a partial hit – getting an equation that hits the right points, but which also generates a bunch of misses. There’s nothing remotely impressive about that.

From there, they move on to the strawmen. For example, they claim that their “solar wind” hypothesis explains why the planets all orbit in the same direction on the same plane. According to them, if orbits were really gravitational, then planets would orbit in random directions on random planes around the sun. But their theory is better than gravity, because it says why the planets are in the same plane, and why they’re all orbiting in the same direction.

The thing is, this is a really stupid argument. Why are the planets in the same plane, orbiting in the same direction? Because the solar system was formed out of a rotating gas cloud. There’s a really good, solid, well-supported explanation of why the planets exist, and why they orbit the sun the way they do. Gravity doesn’t explain all of it, but gravity is a key piece of it.

What they don’t seem to understand is how amazingly powerful the theory of gravity is as a predictive tool. We’ve sent probes to the outer edges of the solar system. To do that, we didn’t just aim a rocket towards Jupiter and fire it off. We’ve done things like the Cassini probe, where we launched a rocket towards Venus. It used the gravitational field of Venus twice to accelerate it with a double-slingshot maneuver, and send it back towards earth, using the earth’s gravity to slingshot it again, to give it the speed it needed to get to Jupiter.

This wasn’t a simple thing to do. It required an extremely deep understanding of gravity, with extremely accurate predictions of exactly how gravity behaves.

How do our brilliant authors answer this? By handwaving. The extend of their response is:

Gravitational theory works for things like space travel because it empirically measures the force of a planet, rather than predicting it.

That’s a pathetic handwave, and it’s not even close to true. The gravitational slingshot is a perfect answer to it. A slingshot doesn’t just use some “empirically measured” force of a planet. It’s a very precise prediction of what the forces will be at different distances, how that force will vary, and what effects that force will have.

They do a whole lot more handwaving of very much the same order. Pure rubbish.

## 19 thoughts on “The Gravitational Force of Rubbish”

1. dete

Do they ever try to explain why people stick to the earth on all sides?

Or how a force which is originating from the sun doesn’t push things away from the sun, at least slowly?

Or comets? Comets trash their argument not only with their solar-wind-induced tails (which extend directly out from the sun, not around the sun in a circle) and their highly elliptical orbits.

And if physicists are so very wrong, how come they are willing to trust their measurements of the solar wind?

2. Rob Ryan

The scary part isn’t that these people have crackpot theories (after all, Kevin Neumaier is in Mensa and the “Triple Nine Society,” IQ above the 99.9 percentile so he MUST be correct) but that they have an outlet for it and many people (it’s Leeza Gibbons for Heaven’s sake!) will lap it up. I suppose that there’s always been an audience for the lunatic fringe but it seems to be growing as the filtering function previously served by published media has weakened.

3. Deen

So is the orbit of our moon explained by an earth-wind, and those of Jupitrer by a jovian wind?

4. Kevin

Apparently the Neumaiers run a company with a market cap of 55M. Unless this page is a hoax someone set up to make them look bad?

I could imagine a scenario in which someone got annoyed if Kevin Neumaier was an ass who went around bragging about Mensa and such, so made a website to make him look stupid. Perhaps a disgruntled employee who disliked family politics or heard “I’m in Mensa so shut up and listen” one time too many, so sought out to ruin their reputation.

Or maybe they are just bizarre.

1. MarkCC Post author

You can never be sure on the internet, but all signs seem to indicate that this is legit – that they really do believe this, and that the site is completely serious.

The fact that they’re successful businessmen isn’t particularly surprising. As my dad used to say: Everyone is, roughly, equally intelligent. If you’re a genius in one field, you’re an idiot in all others.

Very smart people often fall into the trap of believing that success at one thing means that they’re universally exceptional, and that they’ll be successful at whatever they try. This seems particularly true for people who are obsessed about their rankings according to personality profiles or intelligence tests.

So they look at something like the theory of gravity, which they know next to nothing about, and conclude that since they’re in the top 0.1 percent of human intelligence according to their test, they must know more about science than people who spend their entire lives studying science.

(This is a pet peeve of mine. I don’t know my IQ. I don’t *want* to know my IQ. But I grew up dealing with two kinds of people who saw my personal records at school.

On one hand, there were the people who saw my IQ, and wanted to tell me all about the things I *should* be doing because I’m so smart. I should join mensa, I should take this special writing class, I should go to a different school, etc.

On the other hand, there were people who saw that I’m learning disabled, with a perceptual impairment, and wanted to tell me that I *shouldn’t* be doing things like taking advanced math classes, because I was over-stressing myself.

I hated both kinds.)

1. Kevin Neumaier

Interesting discussions – but straying considerably from the points we are making. To bring them back a little, we don’t say that gravity doesn’t exist. We say that the equation for gravity is incorrect and the solar wind is another force that is working in our solar system – serving to provide motion, direction of orbit and positioning of the planets. In a good math/bad math sense we say that math shouldn’t replace scientific observations and when it does, math doesn’t give logical results.

So why doesn’t a space probe prove that the equation for gravity is correct?
Space missions work. In part, they do prove that the end result of the calculation, the force on the spacecraft is correct.

There are only two measurable terms that are in play with any spacecraft – these are the distance to the planet and the surface gravity of the planet. So, we contend that the net result of the equation for gravity is correct but the terms themselves are implying something that is not happening.

The equation for gravity is F= Gm1m2/d^2

Let’s assign m1 to be our spacecraft and m2 to be the mass of the planet.
The terms Gm2 are always observed as one term, the surface gravity of the planet. The only way that the mass of the planet was determined was through using Newton’s equation for gravity with the assumption that G is a universal constant.

The separation of one observation (the acceleration due to gravity) into two terms (Gravitational constant and mass) is both why the equation works for space missions and the equation may also not be correct.

To clarify, in addition to the gravitational force of the Sun, there are other forces that order and move the planets. One of the other forces is the solar wind.

We don’t explicitly prove the equation for gravity to be wrong, but strongly question using two terms for one observation. Theories that use gravity as the primary and only force for planetary motion and the universe’s workings are ones that we think are incorrect.

“Why are the planets in the same plane, orbiting in the same direction? Because the solar system was formed out of a rotating gas cloud. There’s a really good, solid, well-supported explanation of why the planets exist, and why they orbit the sun the way they do.”

The argument doesn’t have support of a lot of observational data. There are no accounts of what happened 4.6 Billion years ago. To create a past that is based on what is now in place is a very difficult thing to do and be correct. There is maybe 100 years of data on how our solar system has been working. To project backward 4.6 Billion years is a stretch. When I do the math, 100 years/4,600,000,000 years there is no observational data for 99.999998% of the history that is being said to have occurred.

We point out that the solar wind today is providing the force and direction to move the planets along their orbits. This is a force in addition to the attractive force of the Sun.

1. MarkCC Post author

So… Your defense is, basically, two points.

First, you’re arguing that the results of the equation for gravity is absolutely perfectly precise to the limits of our ability to measure, but that that’s coincidental, because in reality, the force is partially gravity, and partially something else caused by the solar wind. But you can’t quantify that in any way.

Second, you’re pulling out the old “you weren’t there, so you don’t know” thing.

In response to the first – there’s a whole lot more to our understanding of gravity than you seem to understand. Trying doing some research on experimental gravitational physics. It’s a fascinating field of work. And using it, there are tons of ways of actually testing gravity. We can measure differences in the gravitational field at different elevations. We can measure the gravitational attraction between different objects, not just between objects and the earth. We can measure the time dilation effects of gravity. In orbital physics, we can observe the differential gravitational forces caused by different bodies. It’s a very rich field, with tons of data. It’s a lot more than a circular “we figure out the gravitational force using the mass of the earth, and we figure out the mass of the earth using the gravitational force”.

In response to the second, we don’t have direct observational evidence, for obvious reasons. But we have plenty of evidence. The fact that you either don’t know about or don’t understand that evidence is irrelevant.

“There are only two measurable terms that are in play with any spacecraft – these are the distance to the planet and the surface gravity of the planet.”

Absolutely not, photon pressure is also taken into account see the Pioneer Anomaly.

“The equation for gravity is F= Gm1m2/d^2”

Actually there is a more accurate theory of gravity that is used… I can’t remember the guys name but he had a big bushy mustache and wild hair, and his theory of gravitation has been tested to a spectacular degree using binary systems consisting of a pulsar and white dwarf in tight orbits around one another.

“The argument doesn’t have support of a lot of observational data. There are no accounts of what happened 4.6 Billion years ago.”

We have plenty of observational data, we can actually see the proto-planetary disks forming around other stars.

5. Bruno Saboia

There is something very simple that you forgot to mention, Mark. Space bodies with sufficient mass are round. Why? Because gravity would “crush” a very high mountain or peak, “smoothening” its surface. That’s why every sufficiently large body in space is round-like.

6. kaleberg

This makes Some Sophomore’s Law – There is no gravity. The Earth sucks. – sound as profound and solid as Newton or Einstein.

7. shafiqifs

This confirms the alternative theory of gravity. The very space-time concept, on which theories of relativity are founded, has been mathematically, theoretically & experimentally proved as baseless and openly challenged on the basis of published scientific articles. Since the very space-time concept has been proved as baseless the question of curvature of space-time being correct does not arise. Gravity has been shown to be an electromagnetic force as foreseen by Maxwell due to the curl/vortices of aether (the electric dipoles) in the published article ‘Revised Foundation of Theory of Everything: Non-living Things & Living Things’ (www.indjst.org; Sep 2010) Revised version of this article is available on vixra & General Science Journal in my profile. Standing open challenge could be seen at http://www.worldsci.org/php/index.php?tab0=Abstracts&tab1=Display&id=6476&tab=2 and http://gsjournal.net/Science-Journals/Essays/View/4018.


text:
## Class files

When it comes to bibliography management in L a T e X the program natbib is an alternative used in several journals. The program is not actively developed, but is very stable and widely used. This article explains how to use natbib to format and cite bibliographic sources.

Note: If you are starting from scratch it's recommended to use biblatex since that package provides localization in several languages, it's actively developed and makes bibliography management easier and more flexible.

## Introduction

A minimal working example is presented below:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}

\usepackage{natbib}
\bibliographystyle{unsrtnat}

\title{Bibliography management: \texttt{natbib} package}
\author{Share\LaTeX}
\date { }

\begin{document}

\maketitle

This document is an example of \texttt{natbib} package using in bibliography
management. Three items are cited: \textit{The \LaTeX\ Companion} book
\cite{latexcompanion}, the Einstein journal paper \cite{einstein}, and the
Donald Knuth's website \cite{knuthwebsite}. The \LaTeX\ related items are
\cite{latexcompanion,knuthwebsite}.

\medskip

\bibliography{sample}

\end{document}

In this example there are four basic commands to manage the bibliography:

 usepackage{natbib}
Imports the package natbib .
 \bibliographystyle{unsrtnat}
 \cite{labelcompanion}
Prints a reference to the citation entry, what is printed depends on the citation style . The word inside the braces corresponds to a particular entry in the bibliography file .
 bibliography{sample}
Imports the file sample.bib that contains bibliography sources. See the bibliography file section.

## Basic usage

A simple working example was shown at the introduction, there are more bibliography-related commands available.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}

\usepackage[square,numbers]{natbib}
\bibliographystyle{abbrvnat}

\title{Bibliography management: \texttt{natbib} package}
\author{Share\LaTeX}
\date { }

\begin{document}

\maketitle

This document is an example of \texttt{natbib} package using in bibliography
management. Three items are cited: \textit{The \LaTeX\ Companion} book \cite{latexcompanion}, the Einstein journal paper \citet{einstein}, and the
Donald Knuth's website \cite{knuthwebsite}. The \LaTeX\ related items are
\cite{latexcompanion,knuthwebsite}.

\medskip

\bibliography{sample}

\end{document}

There are a few changes in this example:

• The options  square  and  numbers  in  \usepackage[square,numbers]{natbib}  enable squared brackets and numeric citations respectively. See the reference guide for a list of package options
• The command  \citet  adds the name of the author to the citation mark, regardless of the citation style .

## The bibliography file

The bibliography files must have the standard bibtex syntax and the extension .bib. They contain a list of bibliography sources and several fields with information about each entry.

@article{einstein,
author =       "Albert Einstein",
title =        "{Zur Elektrodynamik bewegter K{\"o}rper}. ({German})
[{On} the electrodynamics of moving bodies]",
journal =      "Annalen der Physik",
volume =       "322",
number =       "10",
pages =        "891--921",
year =         "1905",
DOI =          "http://dx.doi.org/10.1002/andp.19053221004"
}

@book{latexcompanion,
author    = "Michel Goossens and Frank Mittelbach and Alexander Samarin",
title     = "The \LaTeX\ Companion",
year      = "1993",
}

@misc{knuthwebsite,
author    = "Donald Knuth",
title     = "Knuth: Computers and Typesetting",
url       = "http://www-cs-faculty.stanford.edu/\~{}uno/abcde.html"

This file contains records in a special format, for instance, the first bibliographic reference is defined by:

 @article{...}
This is the first line of a record entry,  @article  tells BibTeX that the information stored here is about an article. The information about this entry is enclosed within braces. Besides the entry types shown in the example (  article  ,  book  and  misc  there are a lot more, see the reference guide .
 einstein
The label  einstein  is assigned to this entry, is a unique identifier that can be used to refer this article within the document.
 author = "Albert Einstein",
This is the first field in the bibliography entry, indicates that the author of this article is Albert Einstein. Several comma-separated fields can be added using the same syntax  key = value  , for instance: title, pages, year, URL, etc. See the reference guide for a list of possible fields.

The information in this file can later be printed and referenced within a L a T e X document, as shown in the previous sections, with the command  \bibliography{sample  . Not all the information in the .bib file will be displayed, it depends on the bibliography style set in the document.

If you want the bibliography to be included in the table of contents, importing the package tocbibind in the preamble will do the trick:

\documentclass[a4paper,10pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}

\usepackage[nottoc]{tocbibind}

\begin{document}

\tableofcontents

\section{First Section}
This document ...

\bibliographystyle{unsrt}
\bibliography{sample}

\end{document}

\usepackage[nottoc]{tocbibind}

## Reference guide

natbib package options

•  round  for round parentheses
•  square  uses square brackets
•  curly  curly braces
•  angle  angle braces or chevrons
•  semicolon  separates multiple citations with semicolons
•  colon  same as  semicolon
•  comma  separate multiple citations with commas
•  authoryear  for author-year citations
•  numbers  for numerical citations
•  super  superscripts for numerical citations, as in Nature
•  sort  orders multiple citations according to the list of references
•  sort&compress  same as  sort  but multiple numerical citations are compressed if possible
•  compress  compress without sorting
•  longnamefirst  the full name of the author will appear in the first citation of any reference
•  sectionbib  To be used with the package chapterbib to add the bibliography to the table of contents as a unnumbered section instead of an unnumbered chapter
•  nonamebreak  prevents hyphenation of author names
•  elide  to omit common elements of merged references

Standard entry types

 article
Article from a magazine or journal
 book
A published book
 booklet
A work that is printed but have no publisher or sponsoring institution
 conference
An article in a conference proceedings
 inbook
A part of a book (section, chapter and so on)
 incollection
A part of a book having its own title
 inproceedings
An article in a conference proceedings
 manual
Technical documentation
 masterthesis
A Master's thesis
 misc
Something that doesn't fit in any other type
 phdthesis
A PhD thesis
 proceedings
The same as  conference
 techreport
 unpublished


text:
# Affine subspace: Wikis

Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.

# Encyclopedia

(Redirected to Affine space article)

In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. It can be thought of informally as a vector space where one has forgotten which point is the origin. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point that serves as an origin.

## Informal descriptions

The following characterization may be easier to understand than a precise definition: an affine space is what is left of a vector space after you've forgotten which point is the origin (or, in the words of mathematical physicist John Baez, "An affine space is a vector space that's forgotten its origin"). Imagine that Smith knows that a certain point is the true origin, and Jones believes that another point — call it p — is the origin. Two vectors, a and b, are to be added. Jones draws an arrow from p to a and another arrow from p to b, and completes the parallelogram to find what Jones thinks is a + b, but Smith knows that it is actually p + (ap) + (bp). Similarly, Jones and Smith may evaluate any linear combination of a and b, or of any finite set of vectors, and will generally get different answers. However — and note this well:

If the sum of the coefficients in a linear combination is 1, then Smith and Jones will agree on the answer!

Here is the punch line: Smith knows the "linear structure", but both Smith and Jones know the "affine structure"—i.e. the values of affine combinations, defined as linear combinations in which the sum of the coefficients is 1. An underlying set with an affine structure is an affine space.

## Precise definition

An affine space can most easily be defined in terms of a vector space over a field (or division ring), as here, but can also be defined intrinsically by axioms, without reference to an auxiliary vector space or field.

An affine space is a set with a faithful freely transitive vector space action, i.e. a torsor (or principal homogeneous space) for the vector space.

Equivalently an affine space is a set A, together with a vector space V, and a map

$V \times A\to A$, written as (v, a) → v + a.

The map has the properties that:

1. for every a in A the map $V \to A\colon v \mapsto v + a\,$ is a bijection, and
2. for every v, w in V and a in A we have $(v+w)+a = v+(w+a).\,$

We can define subtraction of points of an affine space as follows:

ab is the unique vector in V such that (ab) + b = a.

By choosing an origin a we can thus identify A with V, hence change A into a vector space.

Conversely, any vector space V is an affine space over itself.

If O, a and b are points in A and $\ell$ is a scalar, then

$\ell a+(1-\ell)b = O+\ell(a-O)+(1-\ell)(b-O)\,$

is independent of O. Instead of arbitrary linear combinations, only such affine combinations of points have meaning.

## Examples

• Any coset of a subspace V of a vector space is an affine space over V.
• If A is a matrix and b lies in its column space, the set of solutions of the equation Ax = b is an affine space over the subspace of solutions of Ax = 0.
• The solutions of an inhomogeneous linear differential equation form an affine space over the solutions of the corresponding homogeneous linear equation.
• More generally, if $T\colon V\to W$ is a linear mapping and y lies in its image, the set of solutions $x\in V$ to the equation T(x) = y is a coset of the kernel of T, and is therefore an affine space over Ker(T).

## Affine subspaces

An affine subspace (sometimes called a linear manifold) of a vector space V is a subset closed under affine combinations of vectors in the space. For example, the set

$A=\Bigl\{\sum^N_i \alpha_i \mathbf{v}_i \Big| \sum^N_i\alpha_i=1\Bigr\}$

is an affine space, where {vi}i is a family of vectors in V. To see that this is indeed an affine space, observe that this set carries a transitive action of the vector subspace W of V

$W=\Bigl\{\sum^N_i \beta_i\mathbf{v}_i \Big| \sum^N_i \beta_i=0\Bigr\}.$

This affine subspace can be equivalently described as the coset of the W-action

$S=\mathbf{p}+W,\,$

where p is any element of A. A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations. A linear subspace is an affine subspace containing the origin, or, equivalently, a subspace that is closed under linear combinations.

For example, in R3, the origin, lines and planes through the origin and the whole space are linear subspaces, while points, lines and planes in general as well as the whole space are the affine subspaces.

## Affine combinations and affine dependence

An affine combination is a linear combination in which the sum of the coefficients is 1. Just as members of a set of vectors are linearly independent if none is a linear combination of the others, so also they are affinely independent if none is an affine combination of the others. The set of linear combinations of a set of vectors is their "linear span" and is always a linear subspace; the set of all affine combinations is their "affine span" and is always an affine subspace. For example, the affine span of a set of two points is the line that contains both; the affine span of a set of three non-collinear points is the plane that contains all three. Vectors

v1, v2, ..., vn

are linearly dependent if there exist scalars a1, a2, … ,an, not all zero, for which

a1v 1 + a2v2 + … + anvn = 0

(1)

Similarly they are affinely dependent if the same is true and also

$\sum_{i=1}^n a_i = 0.$

Equation (1) is an affine relation among the vectors v1, v2, …, vn.

## Axioms

Affine space is usually studied as analytic geometry using coordinates, or equivalently vector spaces. It can also be studied as synthetic geometry by writing down axioms, though this approach is much less common. There are several different systems of axioms for affine space.

Coxeter (1969, p.192) axiomatizes affine geometry (over the reals) as ordered geometry together with an affine form of Desargues's theorem and an axiom stating that in a plane there is at most one line through a given point not meeting a given line.

Affine planes satisfy the following axioms (Cameron 1991, chapter 2): (in which two lines are called parallel if they are equal or disjoint):

• Any two distinct points lie on a unique line.
• Given a point and line there is a unique line which contains the point and is parallel to the line
• There exist three non-collinear points.

As well as affine planes over fields (or division rings), there are also many non-Desarguesian planes satisfying these axioms. An affine plane can be obtained from any projective plane by removing a line and all the points on it, and conversely any affine plane can be used to construct a projective plane by adding a "line at infinity" whose points correspond to equivalence classes of parallel lines.

(Cameron 1991, chapter 3) gives axioms for higher dimensional affine spaces.