text:
A&A 406, 1-5 (2003)
DOI: 10.1051/0004-6361:20030665

## Numerical tests of dynamical friction in gravitational inhomogeneous systems

A. Del Popolo

Dipartimento di Matematica, Università Statale di Bergamo, via dei Caniana 2, 24129 Bergamo, Italy
Feza Gürsey Institute, PO Box 6 Çengelköy, Istanbul, Turkey
Bo çi University, Physics Department, 80815 Bebek, Istanbul, Turkey

Received 28 January 2003 / Accepted 10 April 2003

Abstract
In this paper, I test by numerical simulations the results of Del Popolo & Gambera (1998), dealing with the extension of Chandrasekhar and Von Neumann's analysis of the statistics of the gravitational field to systems in which particles (e.g. stars, galaxies) are inhomogeneously distributed. The paper is an extension of that of Ahmad & Cohen (1974), in which the authors tested some results of the stochastic theory of dynamical friction developed by Chandrasekhar & von Neumann (1943) in the case of homogeneous gravitational systems. It is also a continuation of the work developed in Del Popolo (1996a,b), which extended the results of Ahmad & Cohen (1973), (dealing with the study of the probability distribution of the stochastic force in homogeneous gravitational systems) to inhomogeneous gravitational systems. Similarly to what was done by Ahmad & Cohen (1974) in the case of homogeneous systems, I test, by means of the evolution of an inhomogeneous system of particles, that the theoretical rate of force fluctuation describes correctly the experimental one. I find that the stochastic force distribution obtained for the evolved system is in good agreement with the Del Popolo & Gambera (1998) theory. Moreover, in an inhomogeneous background the friction force is actually enhanced relative to the homogeneous case.

Key words: stars: statistics - celestial mechanics - methods: numerical

### 1 Introduction

The study of the statistics of the fluctuating gravitational force in infinite homogeneous systems was pioneered by Chandrasekhar & Von Neumann in two classical papers (Chandrasekhar & Von Neumann 1942, 1943, hereafter CN43) and in several other papers by Chandrasekhar (1941, 1943a-e, 1944a,b). The analysis of the fluctuating gravitational field, developed by the authors, was formulated by means of a statistical treatment. In their papers Chandrasekhar & Von Neumann considered a system in which the stars are distributed according to a uniform probability density, no correlation among the positions of the stars is present and where the number of stars constituting the system tends to infinity while keeping the density constant. Two distributions are fundamental for the description of the fluctuating gravitational field:
1) which gives the probability that a test star is subject to a force in the range , ;
2) which gives the joint probability that the star experiences a force F and a rate of change f, where .
The first distribution, known as Holtsmark's law (Holtsmark 1919), in the case of a homogeneous distribution of the stars, gives information only on the number of stars experiencing a given force but it does not describe some fundamental features of the fluctuations in the gravitational field such as the speed of the fluctuations and the dynamical friction. These features can be described using the second distribution . Hence, for the definition of the speed of fluctuations and of the dynamical friction one must determine the distribution . Information on dynamical friction can be obtained from the moments of this last distribution. As stressed by Chandrasekhar & Von Neumann (1943), for a test star moving with velocity v in a sea of field stars characterized by a random probability distribution of the velocities, , we may write:

 (1)

where represents the velocity of a typical field star relative to the one under consideration, denotes the velocity of a field star. This asymmetry of the distribution of the relative velocities produces, as shown by CN43, a deceleration of the test star in the direction of motion. This effect is known, from the Chandrasekhar papers, as "dynamical friction". As quoted above, some information on dynamical friction can be obtained by means of the first moment of f. As shown by CN43:

 (2)

 (3)

where m is the mass of a field star, nl is the local density, and  is definited as:

 (4)

where

 (5)

and where . These equations show that the amount of acceleration in the direction of  when is greater than that in the direction  , when : then the star suffers a deceleration, the a priori probability that  being equal to the probability that .

Several authors have stressed the importance of stochastic forces and in particular dynamical friction in determining the observed properties of clusters of galaxies (White 1976; Kashlinsky 1986, 1987) while others studied the role of dynamical friction in the orbital decay of a satellite moving around a galaxy or in the merging scenario (Bontekoe & van Albada 1987; Seguin & Dupraz 1996; Dominguez-Tenreiro & Gomez-Flechoso 1998). Chandrasekhar's theory (and in particular his classical formula see Chandrasekhar 1943b) is widely employed to quantify dynamical friction in a variety of situations, even if the theory developed is based on the hypothesis that the stars are distributed uniformly and it is well known that in stellar systems, the stars are not uniformly distributed, (Elson et al. 1987; Wybo & Dejonghe 1995; Zwart et al. 1997) in galactic systems as well, the galaxies are not uniformly distributed (Peebles 1980; Bahcall & Soneira 1982; Sarazin 1988; Liddle & Lyth 1993; White et al. 1993; Strauss & Willick 1995). It is evident that an analysis of dynamical friction taking account of the inhomogeneity of astronomical systems can provide a more realistic representation of the evolution of these systems. Moreover from a pure theoretical standpoint we expect that inhomogeneity affects all the aspects of the fluctuating gravitational field (Antonuccio & Colafrancesco 1994; Del Popolo & Gambera 1996, 1997; Del Popolo et al. 1996; Gambera 1997). Firstly, the Holtsmark distribution is no longer correct for inhomogeneous systems. For these systems, as shown by Kandrup (1980a,b, 1983), the Holtsmark distribution must be substituted with a generalized form of the Holtsmark distribution characterized by a shift of  towards larger forces when inhomogeneity increases. This result was already suggested by the numerical simulations of Ahmad & Cohen (1973, 1974). Hence when the inhomogeneity increases the probability that a test particle experiences a large force increases, secondly, is changed by inhomogeneity. Consequently, the values of the mean life of a state, the first moment of  and the dynamical friction force are changed by inhomogeneity with respect to those of homogeneous systems.

This paper must be intended as the continuation of Del Popolo & Gambera (1998) paper, in which the study of the effects of inhomogeneity on the distribution functions of the stochastic forces and on dynamical friction was performed.

As anticipated in Del Popolo & Gambera (1998), the next task to perform was to test the result of the Del Popolo & Gambera (1998) paper against N-body simulations, which is the object of the present paper. The third step (to be developed in a future paper) should be that of finding a formula that describes dynamical friction in homogeneous and inhomogeneous systems only on the basis of statistical theory.

Before continuing we want to stress that when we speak of inhomogeneity we refer to inhomogeneity in position distribution and not to that of velocity distribution. Our work follows the spirit of Kandrup's (1980) in the sense that we are interested in the effect of a non-uniform distribution in the position of stars on the distributions of the stochastic force.

The plan of the paper is the following: in Sect. 2 we review the calculations and formulas needed to obtain  in the case of inhomogeneous systems. In Sect. 3, I show how numerical experiments are performed and they are compared with the theoretical results of Del Popolo & Gambera (1998). Finally, in Sect. 4, I draw my conclusions.

### 2 Force derivative and dynamical friction

The introduction of the notion of dynamical friction is due to CN43. In the stochastic formalism developed by CN43 the dynamical friction is discussed in terms of :

 (6)

where

 (7)

As shown by CN43, the origin of dynamical friction is due to the asymmetry in the distribution of relative velocities. If a test star moves with velocity  in a spherical distribution of field stars, namely  , then we have that:

 (8)

The asymmetry in the distribution of relative velocities is conserved in the final Eq. (6). In fact from Eq. (6) we have:

 (9)

(CN43). This means that when then , while when then . As a consequence, when has a positive component in the direction of , increases on average; while if  has a negative component in the direction of , decreases on average. Moreover, the star suffers a greater amount of acceleration in the direction  when than in the direction when .

In other words the test star suffers, statistically, an equal number of accelerating and decelerating impulses. The modulus of deceleration being larger than that of acceleration the star slows down.

Chandrasekhar & Von Neumann's analysis was extended to inhomogeneous systems in Del Popolo & Gambera (1998), where and its first momentum were calculated. Supposing that the distribution function is given by:

 (10)

where a is a constant that can be obtained from the normalization condition for , j a parameter (of dimensions of velocity-1),  an arbitrary function,  the velocity of a field star. In other words we assumed, according to CN43 and Chandrasekhar & von Newmann (1942), that the distribution of velocities is spherical, i.e. the distribution function is , but differently from the quoted papers we supposed that the positions are not equally likely for stars, that is, the stars are inhomogeneously distributed in space. A lengthy calculation led us (see Del Popolo & Gambera 1998 for a derivation and the meaning of symbols) to find that the first moment of  is given by:

 (11)

where:

 (12)

 (13)

 v = (14)

 (15)

The results obtained by us for an inhomogeneous system are different (see Eq. (11)), as expected, from those obtained by CN43 for a homogeneous system (CN43 - Eq. (105) or Eq. (6)). At the same time it is very interesting to note that for p = 0 (homogeneous system) our result coincides, as is obvious, with the results obtained by CN43. In inhomogeneous systems, Eq. (9) can be written, using Eq. (11), as:

 (16)

In order to check the validity of the quoted relation (Eq. (16)), I have performed numerical experiments. This was done similarly to Del Popolo (1996b) by evolving (now) 100 000 points (stars) acting under their mutual gravitational attraction. From the evolved positions and velocities of the stars, was computed as a function of velocity and force, similarly to Ahmad & Cohen (1974), and then compared with Eq. (16) as I shall describe in the following.

### 3 N-body experiments

To calculate the stochastic force in an inhomogeneous system, I used an initial configuration in which particles were distributed according to a truncated power-law density profile:

 (17)

(see Kandrup 1980; Del Popolo 1996a). If the velocity distribution is everywhere isotropic then the equation relating the configuration space density  to the phase space density f(E) is:

 (18)

where U(r) is the potential (normalized to zero at infinity). Equation (18) may be converted into an Abel integral equation and inverted, giving the phase space density:

 (19)

(Eddington 1916; Binney & Tremaine 1987). The initial conditions were generated from the distribution function that can be obtained from Eq. (19) assuming a cut-off radius R=1, the mass of the system M=1, r0 =0.15 and G=1. All the particles had equal mass. To have a system whose total mass is contained in a unitary sphere, Eq. (18) was renormalized and consequently also the potential of the system which is obtained from Eq. (18) through Poisson's equation.
 Figure 1: The average value of the time rate of change of the magnitude of the force as the function of the velocity. The solid line refers to the homogeneous case (Chandrasekhar & Von Neumann 1943). The dotted and dashed line refers to the cases p=2.5 and p=4, respectively (see Eq. (26)). Crosses represent the experimental points. Open with DEXTER

 Figure 2: The average value of the time rate of change of the magnitude of the force as the function of the force. The solid line refers to the homogeneous case (Chandrasekhar & Von Neumann 1943). The dotted and dashed line refers to the cases p=2.5 and p=4, respectively (see Eq. (19)). Crosses represent the experimental points. Open with DEXTER

The system of 100 000 particles was evolved over 150 dynamical times using a tree N-body code (Hernquist 1987). During the evolution of the system the essential quantities such as position, force, etc., for each test point of the system was sampled every of a dynamical time (see Del Popolo 1996b; Ahmad & Cohen 1973 for details). The average of the is a function of velocity, v, force, F, and the angle between them. The test of Eq. (16) was performed in a similar way to that of Ahmad & Cohen (1974), namely by integrating out two of the variables and examining against the remaining one (see Ahmad & Cohen 1974). As in Ahmad & Cohen (1974), indicates  after integrating out the angle and velocity, while is after integrating out the angle and force. In integrating out the angle, one cannot average over the entire range 0 to , since that would give zero. Instead, the cosine was averaged from 0 to /2, and in the numerical experiments only those particles having a cosine in the quoted range were used. In order not to waste the statistics for half the particles the same trick of Ahmad & Cohen (1974) was used, namely when the cosine is in the range of  to , the sign of is changed and it is counted in the same statistics. This corresponds to assume that the cosine between  and  is uniformly distributed, which is what is found in numerical experiments. The average value of the cosine between 0 and  is 1/2.

For a general distribution can be calculated as usual:

 (20)

and can be written, in units of , as:

 (21)

In the particular case of a Maxwellian distribution for velocities:

 (22)

where , so that:

 (23)

we have that:

 (24)

that expressed in units of , then Eq. (24) becomes:

 (25)

Similarly to Ahamd & Cohen (1974), since to integrate out the force from Eq. (16) one has a divergent result, I consider only particles up to a certain maximum value of the force,  : for example in the case p=0, , that involves 97% of the particles. As observed by Ahmad & Cohen (1974), any cutoff of the force can be used as long as it is taken into account in both the experiment and the analytic evaluation of  . Defining:

 (26)

and

 (27)

we find, in units of , that:

 (28)

The results of calculation and numerical experiments are plotted in Figs. 1 and 2. In Fig. 1, I plot the average value of the time rate of change of the magnitude of the force as a function of the velocity. The solid line refers to the homogeneous case while the dotted and dashed lines refer to the cases p=2.5 and p=4, respectively. Crosses represent the experimental points. As shown, experimental points follow a linear relationship and there is a good agreement with the theoretical prediction (Eq. (21)). In Fig. 2, I plot the average value of the time rate of change of the magnitude of the force as the function of the force. As in the previous figure, the solid line refers to the homogeneous case while the dotted and dashed line refers to the cases p=2.5 and p=4, respectively. In this case, the dependence is no longer linear: it behaves like in the homogeneous case. A comparison with numerical experiments shows that there is a good agreement with the theoretical prediction (Eq. (28)). The situation described in Ahmad & Cohen (1974), that the experimental data were somewhere in between the theoretical curves for the one-particle and the infinite-particle case, is no longer present and the agreement is better, now. This is due to the larger number of particles used in the simulations. The plots show that Chandrasekhar & Von Neumann's theory of dynamical friction in gravitational systems gives a good description of experimental data (solid line and data), and so does the generalization of the quoted theory to inhomogeneous systems (dotted line, dashed line, and respective data). In inhomogeneous systems, Chandrasekhar's result which relates the frictional force only to the local properties of the background at the position of the object, is no longer true, and friction depends on the global structure of the system. This point is in agreement with Maoz (1993), who showed that in inhomogeneous media the friction, unlike Chandrasekhar's formula, depends on the global structure of the entire mass density field.

### 4 Conclusions

In this paper, I tested by numerical simulations the results of the Del Popolo & Gambera (1998) paper, dealing with the average value of the time rate of change of the magnitude of the stochastic force in inhomogeneous gravitational systems. In agreement with Ahmad & Cohen (1974), the stochastic theory of dynamical friction developed by Chandrasekhar & Von Neumann (1943), in the case of homogeneous gravitational systems, gives a good description of the results of numerical experiments. The stochastic force distribution obtained for inhomogeneous systems, obtained by Del Popolo & Gambera (1998), is also in good agreement with the results of numerical experiments. Finally, in an inhomogeneous background the friction force is actually enhanced relative to the homogeneous case.

Acknowledgements
I am grateful to E. N. Ercan and G. Mamon for stimulating discussions during the period in which this work was performed. I would like to thank Bo çi University Research Foundation for the financial support through the project code 01B304.


text:
# American Institute of Mathematical Sciences

December  2021, 13(4): 629-646. doi: 10.3934/jgm.2021022

## Explicit solutions of the kinetic and potential matching conditions of the energy shaping method

 1 Centro Atómico Bariloche and Instituto Balseiro, 8400 S.C. de Bariloche, and CONICET, Argentina 2 CMaLP, Facultad de Ciencias Exactas, UNLP, 1900 La Plata, and CONICET, Argentina 3 CMaLP, Facultad de Ciencias Exactas, UNLP, 1900 La Plata, Argentina

Received  September 2018 Revised  June 2021 Published  December 2021 Early access  August 2021

Fund Project: S. Grillo and L. Salomone thank CONICET for its financial support

In the context of underactuated Hamiltonian systems defined by simple Hamiltonian functions, the matching conditions of the energy shaping method split into two decoupled subsets of equations: the kinetic and potential equations. The unknown of the kinetic equation is a metric on the configuration space of the system, while the unknown of the potential equation are the same metric and a positive-definite function around some critical point of the Hamiltonian function. In this paper, assuming that a solution of the kinetic equation is given, we find conditions (in the $C^{\infty}$ category) for the existence of positive-definite solutions of the potential equation and, moreover, we present a procedure to construct, up to quadratures, some of these solutions. In order to illustrate such a procedure, we consider the subclass of systems with one degree of underactuation, where we find in addition a concrete formula for the general solution of the kinetic equation. As a byproduct, new global and local expressions of the matching conditions are presented in the paper.

Citation: Sergio Grillo, Leandro Salomone, Marcela Zuccalli. Explicit solutions of the kinetic and potential matching conditions of the energy shaping method. Journal of Geometric Mechanics, 2021, 13 (4) : 629-646. doi: 10.3934/jgm.2021022
##### References:

show all references

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text:
# Aliases

#### mentaaal

Joined Oct 17, 2005
451
Hey guys,
I was just wandering, why is an alias created like this:

Rich (BB code):
int a = 0;
int &b = a;
Howcome this does not assign the value of a, currently 0 to the address of b?

Howcome to assign an alias it is not done like this:

Rich (BB code):
int a = 0;
int &b = &a;
This makes more sence to me anyway as the address one one gets assigned the address of the other?

Also another wee problem i am having i am trying to figure out how you can assign a string literal to a character array which is a member of a class. I tried it in the code below but it returns an error below. Could anyone show me how to do this?

Rich (BB code):
#include <iostream>
using namespace std;

int main()
{

struct test
{

char a[50];
};
test stest;
stest.a = "hello.";
return 0;
}
Rich (BB code):
1>------ Build started: Project: Test7, Configuration: Debug Win32 ------
1>Compiling...
1>Test7.cpp
1>z:\college\year2\programming\programs\test7\test7\test7.cpp(13) : error C2440: '=' : cannot convert from 'const char [7]' to 'char [50]'
1>        There is no context in which this conversion is possible
1>Build log was saved at "file://z:\College\Year2\Programming\Programs\Test7\Test7\Debug\BuildLog.htm"
1>Test7 - 1 error(s), 0 warning(s)
========== Build: 0 succeeded, 1 failed, 0 up-to-date, 0 skipped ==========

#### beenthere

Joined Apr 20, 2004
15,819
Those are compiler issues. The person/company that wrote the compiler determined the rules for how input statements are parsed and interpreted. A different language might either not have the ability to alias variables or do so using differing notation.

You may not find the function as making sense to you, but the compiler determines how you have to use it. Sometimes it can be irritating enough that you find another language to use that makes more sense (or at least doesn't irritate you every time you write code).

#### n9352527

Joined Oct 14, 2005
1,198
Alias or reference as I know it, is assigned just like normal variable assignment. So, the code:

int &a = &b;

is the correct usage.

Note that it is different in the case of pointer, which would be:

int *a = &b;

The reference to a variable is used in codes just like the variable itself. It is usually benefits the program in parameter passing. Declaring a parameter to be passed by reference:

Rich (BB code):
int add(int &a, int &b)
{
return a*b;
}

int main(void)
{
...
...
}
Note that the variable a and b are accessed by normal variable name in the function, but declared as passed by reference in the function declaration. The function is called by passing the references to num1 and num2, however some compilers will accept variable names without the reference designator as well.

Assigning a partial string, or static initial value, to a member of a union or class is possible only during initial construction. If you declare a constructor, you could pass the initial value and assigned it to the member variable. Outside of that, you could use string copy to do that or you could try assigning a full string (50 characters) and see what happens

#### mentaaal

Joined Oct 17, 2005
451
Thanks for the replies guys. Yeah i can see what you mean about the compiler determining it but it just seems so counter intuitive that i could see no logic in it at all.

And n9352527, the error I get when doing it like "int&a = &b" is:
1>z:\college\year2\programming\programs\test7\test7\test7.cpp(8) : error C2440: 'initializing' : cannot convert from 'int *__w64 ' to 'int &'

Apologies i should have specified, i am using microsoft visual 2005 c++.

String copy you say... right i will give that a shot.

Cheers

#### Mark44

Joined Nov 26, 2007
628
Alias or reference as I know it, is assigned just like normal variable assignment. So, the code:

int &a = &b;

is the correct usage.
I don't think so, but if you can prove me wrong, I'm happy to listen. Here is some code taken from Bjarne Stroustrup's The C++ Programming Language, Second Edition, in the section on references:
Rich (BB code):
int i = 1;
int& r = i; // r and i now refer to the same int
int x = r;  // x = 1
r = 2;      // i = 2
Mark

#### Mark44

Joined Nov 26, 2007
628
Hey guys,
I was just wandering, why is an alias created like this:

Rich (BB code):
int a = 0;
int &b = a;
Howcome this does not assign the value of a, currently 0 to the address of b?
At the risk of possibly saying again what someone else in this thread has said, what you're dealing with here is C++ syntax for a reference. You're looking at this from a C perspective, which doesn't apply. You cannot declare something to be of type "int &" in C.

Howcome to assign an alias it is not done like this:

Rich (BB code):
int a = 0;
int &b = &a;
This makes more sence to me anyway as the address one one gets assigned the address of the other?
References are a bit like pointers, but it's probably not helpful to think of them this way, because how they work is implementation-dependent. IOW, one compiler vendor might implement them in one way, while another is free to implement them in a different way. I'm pretty sure that the code above won't work, but I don't have the time right now to try it out.

Also another wee problem i am having i am trying to figure out how you can assign a string literal to a character array which is a member of a class. I tried it in the code below but it returns an error below. Could anyone show me how to do this?

Rich (BB code):
#include <iostream>
using namespace std;

int main()
{

struct test
{

char a[50];
};
test stest;
stest.a = "hello.";
return 0;
}
I'm coming in a bit late on this thread, but I think someone else told you how to do what you need to do here. The reason you can't do the assignment above is that what you're trying to do is assign the address of a string constant ('const char [7]') to a variable that is of type array of 50 chars ('char [50]').
Hope that helps.
Mark


text:
# 234. Factorization

## I'm a slow walker, but I never walk backwards.

Write a program to factorize a polynomial. We will be given a polynomial $f(x) = x^3 + a x^2 + b x + c$, where all $a, b, c$ are all integers between -100000 and 100000. Now we want to factorize $f(x)$ as $(x + d)(x + e)(x + f)$, where $d$, $e$, and $f$ are all integers, and $d \le e \le f$.

## Input

A line containing $a$, $b$, and $c$.

## Output

A line containing $d$, $e$, and $f$.

## Sample input

12 47 60

## Sample output

3 4 5


text:
# How to calculate the volume of the Hyperboloid using integrals?

I would like to calculate the volume of a hyperboloid described by this equation: $x^2 + y^2 - z^2 \leq 1$. I made some calculation: $x = \sqrt{1+z^2}$. But I don't know how to continue.

• Between what limits? – Ron Gordon May 8 '13 at 16:52
• Sorry, I forgot: abs(z) <= 1 – Alex May 8 '13 at 16:53
• Hint: use cylindrical coordinates. – John Douma May 8 '13 at 19:44

It appears that you are asked to find the volume of the hyperboloid $x^2 + y^2 - z^2 = 1$ between $z = -1$ and $z = 1$ . Depending upon what "level" of calculus you're in, you have a choice of methods.

The method Ted Shifrin describes uses what is called Cavalieri's Principle, where we find the volume of a solid as a "stack of slices", which works fine when we know how to find the area of the slices. Parallel to the xy-plane (or perpendicular to the z-axis), the cross-sections have the equations $x^2 + y^2 \ = \ 1 + z^2 \$, which are circles centered on $(0, 0, z)$ with radii $r = \sqrt{1 + z^2}$ . Because the hyperboloid is symmetrical about the xy-plane ($z = 0$), we can just integrate the volume above the plane and double that:

$$V \ = \ 2 \int_0^1 A_{circ}(z) \ dz \ = \ 2 \int_0^1 \pi \ (\sqrt{1 + z^2} \ )^2 \ dz \ .$$

[I'll leave you to do the rest on any of these.]

What user69810 is suggesting is to transform the equation for the hyperboloid into cylindrical coordinates and carry out the volume integration in the three coordinate variables. (This sounds worse than it actually is for this symmetrical figure.) Since $r^2 = x^2 + y^2$ in polar or cylindrical coordinates, we have $r^2 - z^2 = 1$ , so there is a radius function $r(z) = \sqrt{1 + z^2}$ . The limits for the radius integration run from the z-axis out to the surface of the hyperboloid, given by that function.

Since the radius function does not depend upon direction, the angle integration will just run all the way around the z-axis from $\theta = 0$ to $\theta = 2 \pi$ . The integration in the z-direction can be handled using the same symmetry as we did before. With this method, the volume is given by

$$V \ = \ 2 \int_0^1 \ \int_0^{2 \pi} \int_0^{\sqrt{1 + z^2}} \ r \ dr \ \ d\theta \ \ dz \ = \ 2\int_0^{2 \pi} d\theta \ \int_0^1 \int_0^{\sqrt{1 + z^2}} \ r \ dr \ \ dz \ .$$

We can also perform this volume integration as a surface integral over the region on the xy-plane which is the projection (or "shadow") of the hyperboloid's surface. Over the "heights" $z = 0$ to $z = 1$, the projection covers a circular area centered on the origin and extending to a radius of $\sqrt{1 + 1^2} = \sqrt{2}$ . The volume "standing on" the circle from $r = 0$ to $r = 1$ is just a cylinder of height $2$ (extending from $z = -1$ to $z = 1$) , so it contributes a volume of $\pi \cdot 1^2 \cdot 2 \ = \ 2 \pi$ .

In the circular ring from $r = 1$ to $r = \sqrt{2}$, the "height" of the hyperboloid extends from $z = \sqrt{r^2 - 1}$ (again from the figure's equation $r^2 - z^2 = 1$) upward to $z = 1$ (where we will once more use the symmetry about the xy-plane). So over this ring, the integral in polar coordinates for this portion of the hyperboloid is

$$V \ = \ 2 \int_0^{2 \pi} \int_1^{\sqrt{2}} ( 1 \ - \ \sqrt{r^2 - 1} ) \ r \ dr \ \ d\theta \ = \ 2 \int_0^{2 \pi} d\theta \ \int_1^{\sqrt{2}} ( r \ - \ r\sqrt{r^2 - 1} ) \ dr \ .$$

This volume is then added to the volume for the cylinder we calculated first to obtain the total volume of the hyperboloid.

EDIT: It might be mentioned here that the first method, in which "slices" are taken perpendicular to the symmetry axis of which is essentially a solid of revolution, is what is often referred to as the "disk method" of volume integration. The third method, in which "slices" are being made parallel to the axis of rotation, is basically the "(cylindrical) shell method". The second method is actually just applying the general definition of volume integration over three dimensions (hence the triple integral); in this situation, however, the axial symmetry of the hyperboloid allows us to treat this as a variant of the "shell method".

Chop into slices perpendicular to the $z$-axis and integrate the cross-sectional areas.

• How Cavalier(i) of you! :) – colormegone May 9 '13 at 4:09


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## GPU-accelerated denoising of large tomographic data sets with low SNR

Please always quote using this URN: urn:nbn:de:0297-zib-56339
• Enhancements in tomographic imaging techniques facilitate non-destructive methods for visualizing fossil structures. However, to penetrate dense materials such as sediments or pyrites, image acquisition is typically performed with high beam energy and very sensitive image intensifiers, leading to artifacts and noise in the acquired data. The analysis of delicate fossil structures requires the images to be captured in maximum resolution, resulting in large data sets of several giga bytes (GB) in size. Since the structural information of interest is often almost in the same spatial range as artifacts and noise, image processing and segmentation algorithms have to cope with a very low signal-to-noise ratio (SNR). Within this report we present a study on the performance of a collection of denoising algorithms applied to a very noisy fossil dataset. The study shows that a non-local means (NLM) filter, in case it is properly configured, is able to remove a considerable amount of noise while preserving most of the structural information of interest. Based on the results of this study, we developed a software tool within ZIBAmira that denoises large tomographic datasets using an adaptive, GPU-accelerated NLM filter. With the help of our implementation a user can interactively configure the filter's parameters and thus its effectiveness with respect to the data of interest, while the filtering response is instantly visualized for a preselected region of interest (ROI). Our implementation efficiently denoises even large fossil datasets in a reasonable amount of time.

### Additional Services

Author: Julián Lamas-Rodríguez, Moritz Ehlke, René Hoffmann, Stefan Zachow ZIB-Report 2015/10/15 ZIB-Report (15-14) 1438-0064

$Rev: 13581$


text:
ON THE EQUATION 𝜙(5m - 1)

Title & Authors
ON THE EQUATION 𝜙(5m - 1)

Abstract
Here, we show that the title equation has no positive integer solutions (m, n), where $\small{{\phi}}$ is the Euler function.
Keywords
Diophantine equations;applications of primitive divisors;applications of sieve methods;
Language
English
Cited by
1.
On the equation φ(Xm - 1) = Xn - 1, International Journal of Number Theory, 2015, 11, 05, 1691
References
1.
J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, and S. S.Wagstaff, Jr., Factor-izations of $b^n{\pm}1$, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers, American Mathematical Society, Providence, RI, 1988.

2.
R. D. Carmichael, On the numerical factors of the arithmetic forms ${\alpha}^n+{\beta}^n$, Ann. Math. (2) 15 (1913), no. 1-4, 30-70.

3.
J. Cilleruelo and F. Luca, Repunit Lehmer numbers, Proc. Edinb. Math. Soc. 54 (2011), no. 1, 55-65.

4.
M. T. Damir, B. Faye, F. Luca, and A. Tall, Members of Lucas sequences whose Euler function is a power of 2, Fibonacci Quart. 52 (2014), no. 1, 3-9.

5.
W. Ljunggren, Some theorems on indeterminate equations of the form xn−1/x−1 = yq, Norsk Mat. Tidsskr. 25 (1943), 17-20.

6.
F. Luca, Problem 10626, American Math. Monthly 104 (1997), 871.

7.
F. Luca, Equations involving arithmetic functions of Fibonacci and Lucas numbers, Fi-bonacci Quart. 38 (2000), no. 1, 49-55.

8.
F. Luca, Euler indicators of binary recurrence sequences, Collect. Math. 53 (2002), no. 2, 133-156.

9.
F. Luca, On the Euler function of repdigits, Czechoslovak Math. J. 58 (2008), no. 1, 51-59.

10.
F. Luca and M. Mignotte, ${\phi}(F_{11})=88$, Divulg. Mat. 14 (2006), no. 2, 101-106.

11.
F. Luca and P. Stanica, Equations with arithmetic functions of Pell numbers, Bull. Soc. Mat. Roumanie, to appear.

12.
H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20 (1973), 119-134.

13.
J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64-94.


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# Gravitational potential energy and center of mass

I've been reading about the Nordtvedt effect, and how Gravitational Binding Energy (GBE) affects total mass. According to the WP article, experimental evidence rules out the existence of this effect.

I read this result as meaning that the gravitational potential energy is accounted for, both as gravitational mass and inertial mass.

The problem I am attempting to understand is the direction of the force created by GBE.

Gravity is a force, which has a location and a direction in space oriented toward where the gravity field is created. If GBE creates a gravitational force, what is its direction?

Consider an example of two gravitationally bound masses: the Earth-Moon system. I am on the surface of the Earth, so I feel mostly the gravity of the Earth, but I also feel the gravity of the Moon. The GBE of the Earth itself will be accounted for as part of the Earth's mass, so it is nothing special. A similar situation accounts for the mass and GBE of the Moon. But what about the gravitational potential energy of the Earth-Moon system? According to my calculations, that energy is on the order of $10^{28} J$, which is equivalent to a mass of about $10^{11} kg$, which is quite significant.

Where will this extra gravitational force pull me? I can think of two reasonable possibilities:

1. This extra gravity will exert a force of attraction proportional to the masses that participate in the system. The Earth would feel slightly more massive than it would be without the Moon. The same for the Moon, which will feel proportionally more massive because it is gravitationally bound to the Earth.

2. This extra energy will attract to the center of mass of the system. That extra force could be felt, but it is not available for experimental validation because the center of mass of the Earth-Moon system is well under the Earth's crust.

About possibility 2, we could check another bigger gravitationally bound system: the Galaxy!

The mass of the galaxy is about: $6\times 10^{42} kg$, and its radius is about $4.8\times 10^{20} m$. So its GBE, assuming a spherically uniform distribution (quite a gross estimation), would be:

$$U = \frac {3GM^2} {5r}$$ $$U \approx 3 \times 10^{54} J$$

Converting that $U$ to the equivalent mass, and measuring it in solar masses (1 solar mass $M_\odot = 1.98\times 10^{30} kg$)

$$U \approx 1.7 \times 10^7 M_\odot$$

So, if we look at the center of mass of the Galaxy, we should find a gravitational force equivalent to that of an object of several million solar masses.

Now the question: What does the mainstream physics community think of this problem? What is the direction of the gravitational force produced by gravitational binding energy?

• The formula for GBE you are using is valid for an object of uniform density, which is obviously very wrong for a galaxy. A few other remarks: (1) there is no unit in your values. (2) It would be much more readable (and correct) if you wrote $U\simeq 1,68\times 10^8\,M_\odot$. – Tom-Tom Sep 17 '14 at 12:12
• @V.Rossetto: Obviously my GBE calculation is a gross oversimplification. A big error in the final value is expected, but here I'm interested in the order of magnitude of the result: is the result comparable to a moon or to a million suns?). About the others: (1) do you mean the missing $J$ (added)? (2) What do you mean exactly? the $\simeq$ vs $\approx$ or the scientific notation? – rodrigo Sep 17 '14 at 12:22
• You use 8 significant digits for a result that is based on values with one or two significant digits. It is wrong to do it because in physics, the number of significant digits provides an information concerning the accuracy of the value. – Tom-Tom Sep 17 '14 at 12:27
• @Tom-Tom Ah, that. Very true. Corrected – rodrigo Sep 17 '14 at 12:29
• @Floris: Ops! Corrected, thanks! Actually $10^{17}$ came simply from $c^2$. I forgot to do the division. – rodrigo Sep 17 '14 at 13:14


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# Interpretetion of linear predictor of a random variable that follows a gamma distribution

Assuming that:

$$0 < \nu, \alpha, y < \infty$$

$$f_Y(y; \nu, \alpha) = \frac{y^{\nu-1}{\alpha}^{\nu}e^{-y\alpha}}{\Gamma (\nu)} \mathbb{1}_{Y \in (0, \infty)}$$

$$= \exp \{ -y\alpha + \nu \log \alpha + (\nu-1) \log y - \log \Gamma (\nu) \} \mathbb{1}_{Y \in (0, \infty)}$$

$$= \exp \{ \frac{y (- \alpha / \nu ) - [- \log \alpha ]}{1/\nu} + (\nu-1) \log y - \log \Gamma (\nu) \} \mathbb{1}_{Y \in (0, \infty)}$$

such as

$$f_Y(y; \theta, \phi) = \exp \{ \frac{y \theta - [- \log (- \theta) ]}{\phi} - \frac{\log(\phi)}{\phi} + (1/\phi -1) \log y - \log \Gamma (1/\phi) \} \mathbb{1}_{Y \in (0, \infty)}$$

where,

$$b(\theta) = - \log (- \theta)$$,

$$c(\theta) = - \frac{\log(\phi)}{\phi} + (1/\phi -1) \log y - \log \Gamma (1/\phi)$$,

$$\theta \equiv \frac{-\alpha}{\nu}$$, e

$$\phi \equiv 1/\nu$$

$$\mu \equiv E[Y] = B'(\theta) = \frac{d B(\theta)}{d \theta} = - \frac{1}{\theta} = -\theta^{-1} = \frac{\nu}{\alpha}$$

$$\text{var} [Y] = B''(\theta)\phi = \frac{d B'\theta)}{d \theta} \phi = \frac{1}{\theta^2} \phi = \phi\mu^2 = \frac{\nu}{\alpha^2}$$

And then, replacing $$\mu$$ in $$f_Y(y)$$:

$$f_Y(y) = \exp \{ \frac{y (-\frac{1}{\mu}) - \log (\mu) }{\phi} - \frac{\log(\phi)}{\phi} + (1/\phi -1) \log y - \log \Gamma (1/\phi) \} \mathbb{1}_{Y \in (0, \infty)}$$

$$g(\mu) = -\frac{1}{\mu}$$
$$\theta_i = \eta_i = X_i'\beta$$
Question: Assuming a linear predictor of the kind $$\hat{\eta} = 2 + 3x_1 + 4x_2$$, how one could interpret the impact on the mean of the model when increasing 1 unit in $$x_1$$? (Considering the canonical link)


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# Polyhex (mathematics)

As with polyominoes, polyhexes may be enumerated as free polyhexes (where rotations and reflections count as the same shape), fixed polyhexes (where different orientations count as distinct) and one-sided polyhexes (where mirror images count as distinct but rotations count as identical). They may also be distinguished according to whether they may contain holes. The number of free ${\displaystyle n}$-hexes for ${\displaystyle n}$ = 1, 2, 3, … is 1, 1, 3, 7, 22, 82, 333, 1448, … (sequence A000228 in OEIS); the number of free polyhexes with holes is given by ; the number of free polyhexes without holes is given by ; the number of fixed polyhexes is given by ; the number of one-sided polyhexes is given by .[1][2]


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# Kronecker limit formula

In mathematics, the classical Kronecker limit formula describes the constant term at s = 1 of a real analytic Eisenstein series (or Epstein zeta function) in terms of the Dedekind eta function. There are many generalizations of it to more complicated Eisenstein series. It is named for Leopold Kronecker.

## First Kronecker limit formula

The (first) Kronecker limit formula states that

${\displaystyle E(\tau ,s)={\pi \over s-1}+2\pi (\gamma -\log(2)-\log({\sqrt {y}}|\eta (\tau )|^{2}))+O(s-1),}$

where

• E(τ,s) is the real analytic Eisenstein series, given by
${\displaystyle E(\tau ,s)=\sum _{(m,n)\neq (0,0)}{y^{s} \over |m\tau +n|^{2s}}}$

for Re(s) > 1, and by analytic continuation for other values of the complex number s.

So the Eisenstein series has a pole at s = 1 of residue π, and the (first) Kronecker limit formula gives the constant term of the Laurent series at this pole.

## Second Kronecker limit formula

The second Kronecker limit formula states that

${\displaystyle E_{u,v}(\tau ,1)=-2\pi \log |f(u-v\tau ;\tau )q^{v^{2}/2}|,}$

where

• u and v are real and not both integers.
• q = e2π i τ and qa = e2π i aτ
• p = e2π i z and pa = e2π i az
• ${\displaystyle E_{u,v}(\tau ,s)=\sum _{(m,n)\neq (0,0)}e^{2\pi i(mu+nv)}{y^{s} \over |m\tau +n|^{2s}}}$

for Re(s) > 1, and is defined by analytic continuation for other values of the complex number s.

• ${\displaystyle f(z,\tau )=q^{1/12}(p^{1/2}-p^{-1/2})\prod _{n\geq 1}(1-q^{n}p)(1-q^{n}/p).}$


text:
Question

# If $$7th$$ term of $$AP$$ is $$7$$ times is equal to $$11$$ times of $$11th$$term of $$AP$$ the that $$18th$$ term is zero.

Solution

## Given $$7^{th}$$ term of A.P is equal to $$11$$ terms the $$11^{th}$$ term of A.P.Let $$7^{th}$$ term of A.P$$=a+(n-1)d$$$$=a+(7-1)d$$$$=a+6d$$d- common differencea- first term$$11^{th}$$ term of A.P$$=a+(11-1)d$$$$=a+10d$$Given, $$7(a+6d)=11(a+10d)$$$$7a+42d=11a+110d$$$$\Rightarrow 4a=-68d$$$$\Rightarrow a=-17d$$ ………..$$(1)$$$$18^{th}$$ term of A.P$$=a+(18-1)d$$$$=a+17d$$ [From $$(1)$$]$$=-17d+17d$$$$=0$$.Mathematics

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# How do you solve the equation: 12y=132?

Apr 13, 2018

$\frac{12 y}{12} = \frac{132}{12}$
$y = 11$


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# Topological Data Analysis: Developing Abstract Foundations (17w5108)

## Organizers

(Columbia University)

(Technische Universität München)

## Description

The Banff International Research Station will host the " Topological Data Analysis: Developing Abstract Foundations" workshop from July 30th to August 4th, 2017.

This workshop brings together two strands of research in applied algebraic topology. The first strand is the theory of topological persistence. Launched by Edelsbrunner, Letscher and Zomorodian in 1999 (with antecedents in work by Frosini in the early 1990s and by Bradley and Robins in the late 1990s), persistence has evolved through several phases: algorithmic, algebraic, category theoretic, sheaf theoretic. Persistence solves the problem of the topological instability of real-world data by adopting a multiscale framework where instability can be quantified and filtered out. Of course, it has raised many new questions. The second strand is statistical and probabilistic topology, characterized for instance by the sharp results of Kahle on random simplicial complexes, the inference theorems of Niyogi, Smale and Weinberger, and more recent work that focuses on classical questions of probability and statistics, including the construction of statistical summaries and sufficient statistics, addressing issues of dimension reduction, and the establishment of probabilistic limit theorems for topological quantities.

The two strands have quite different flavours, reflecting their origins in computer science, mathematics, probability and statistics. Our purpose is to bring focused attention to the foundations of statistical topology, motivated by the success of algebra and sheaf theory in topological persistence. We are driven by the question What is a random space?'' and we seek answers that reflect the subtlety and power of the work in both strands of applied algebraic topology.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides
an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disc
iplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineeri
ng Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional
de Ciencia y Tecnología (CONACYT).


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# Proof that algebraically closed fields of characteristic $p$ exist

How do you prove that algebraically closed fields of characteristic $p$ exist?

For a finite field of prime power order $q$, the algebraic closure is a countably infinite field that contains a copy of the field of order $q^n$ for each positive integer $n$ (and is in fact the union of these copies).

Why would the algebraic closure have to be characteristic $p$?

• If adding $1$ added to itself a prime number of times yields zero in the finite field $F$, then it also does so in any field $K$ containing the finite field $F$! – anon Jun 24 '15 at 2:36
• Oh, I answered the title, not the body. Those are two entirely different things. Can you please find a title that is closer to the question? – Thomas Andrews Jun 24 '15 at 2:48
• If you are interested: Keith Conrad carried out Artin's classical construction of an algebraic closure (of any field) and made it available online. – Stefan Mesken Jun 24 '15 at 2:49
• The characteristic of a subfield is the same as the characteristic of the field of which it is a subfield. – Michael Hardy Jun 24 '15 at 2:52
• @ThomasAndrews, I meant for the questions to be different. – user240033 Jun 24 '15 at 3:01

This only answers the main question.

Just take a sequence of inclusions:

$$\mathbb F_p\to\mathbb F_{p^2}\to\mathbb F_{p^6}\to\cdots\to \mathbb F_{p^{n!}}\to\cdots$$

Then the direct limit (essentially the union) is algebraically closed and a field.

That's because any polynomial in elements of this field has coefficients contained in one $\mathbb F_{p^{k!}}$. Thus, it splits in $\mathbb F_{p^{d\cdot k!}}$ for some $d$, and then note that $d\cdot k!\mid(dk)!$. So the polynomial splits in $\mathbb F_{p^{(dk)!}}$.

Let $F$ be the field of prime power order. Let $p$ be that prime. The field $F$ is a subfield of it algebraic closure, which let us call $G$. your question seems to be: why must $G$ have characteristic $p$? We could as why $F$ must have characteristic $p$. And then, supposing $F$ does have characteristic $p$, why must $G$ have characteristic $p$?

Let's answer the second question first: Since $F\subseteq G$, the "$1$" in $F$ is the same as the "$1$" in $G$. It satisfies $\underbrace{1+\cdots+1}_p= 0$. Since this is the "$1$" in $G$, the field $G$ has characteristic $p$.

Now the first question: If $F$ has $p^n$ elements, why must $F$ have characteristic $p$? If you know the characteristic is some prime number $r$, then you have $\underbrace{1+\cdots+1}_r=0$, and $\{0, 1, 1+1, 1+1+1, \ldots, \underbrace{1+\cdots+1}_{r-1}\,\}$ is a subgroup of the additive group. The order of the subgroup divides the order of the whole group, which is $p^n$. If $p$ and $r$ are prime and $r$ divides $p^n$, then $r=p$.


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# non-linear

## Non-linear growth curves with Stan

I suppose the go to tool for fitting non-linear models in R is nls of the stats package. In this post I will show an alternative approach with Stan/RStan, as illustrated in the example, Dugongs: “nonlinear growth curve”, that is part of Stan’s documentation. The original example itself is taken from OpenBUGS. The data describes the length and age measurements for 27 captured dugongs (sea cows). Carlin and Gelfand (1991) model the data using a nonlinear growth curve with no inflection point and an asymptote as $x_i$ tends to infinity:


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# Daniel Moy

This is Daniel's blog ...

Note to self:

Email notifications to remind me of homework are a good idea.Email notifications on Sunday only work if I check my e-mail on Sunday

I really don't like the way that Macintosh computers decide to have every shortcut & their brothers to be tied to the APPLE key. Why can't they just use CTRL like every other OS (Linux, Windows, BSD, Unix, etc... maybe not Sun)? It would be fine if I didn't have to switch operating systems so much for work.

I feel like there might be some way to animate or otherwise artistically represent my latest project at work. I've got a bunch of code that writes code that writes other code that makes little colored lights turn on & off for a simulation. Maybe there's some better way to visualize that data instead of just streaming text & state indicators, maybe not. My co-worker joked that I might be able to get paid to do work if it's relevant for both class and worth it from DARPA's perspective. They do like their blinking lights & pretty displays.

Also, the last 17 minutes of Welles' movie was a bit of a cheap shot. Saying that there's going to be no lies (for an hour) and then starting to lie right after the 60 minute mark is grrrr. Semantics :(

Sort of related to the short aside we had during class talking about how new technology isn't always better:

http://www.independent.co.uk/news/science/one-in-10-people-hurt-using-mobile-2081519.html

Friday, 17 September 2010

One in 10 people have been injured as a result of tripping, falling, colliding with a solid object, or failing to notice a car while using a mobile phone, a study has found.

Dr Joanna Lumsden, from Aston University in Birmingham, said that in London, two teenage pedestrians a day were killed or injured because they were distracted by their phones.

"A&E departments have seen more and more admissions as a result of texting accidents," she told the British Science Festival in Birmingham.

Dr Lumsden described an experiment in which volunteers followed a path while texting, during which colour-coded warnings were flashed up. She found that people missed one in five potential hazards.

I'm sometimes more like an ornery 75 year old when it comes to technology - I have yet to send a single text message (but then again I'm using a plethora of HTML tags in this comment, so maybe not...).

So today as I was walking to the bus, thinking that my project idea was pretty lame (it was), I almost fell into a giant sinkhole. Apparently it mysteriously appeared overnight right on the path I take to the bus... maybe like 1.5m wide and 3m deep. Almost broke a leg in there. I can only imagine how much it would suck to be stuck in a hole. There's apparently an area in South America where house-sized sinkholes are not uncommon. And these ones go like 100m down. That would definitely insta-kill you, eek.

The sinkhole is wider (and by consequence slightly less deep) - it blocks like 75% of the path. Someone must have come by today and put up a plastic orange warning net fence thingy, 'cus on the way back from the bus I had to dodge that instead of jumping the side of the hole (probably a bad idea anyways...).

On a completely unrelated note I've been messing with some crazy GIMP stuff. GIMP stands for GNU Image Manipulation Program (where GNU stands for GNU's not unix (get it, recursive acronym, haha)) - it's essentially the free (and slightly worse) version of photoshop. What it CAN do that I haven't figured out how to do in photoshop yet though, is make these SUPER COOL and crazy abstract images. I've been rendering them like mad (unfortunately it takes like 10~50 minutes of compute time per image, depending on the complexity). They're sort of abstract, yet completely algorithmic (computer generated). They're not quite fractals, but man they look cool.

Now I just have to figure out how to turn that into what might qualify as art for this assignment... hmm.

So I'm near the bay area (SF) attending this international music festival or something. I'm tagging along with a group of 30 Chinese students (mostly piano players). Acting as a translator... but I don't speak Chinese, so I'm just translating from Chinglish to English and helping them out with stuff - calling hotels, etc.

Anyways, there's a lot of Chopin. I guess piano students all over the world play the same old European music for the most part. Man though, some of these students are ridiculously good, wow. The whole thing makes me wonder: if China takes over the world (economically, and not for many decades of course), would all the Western students play traditional Chinese music? Kinda interesting to think about...

Update on the pothole: it's all filled in now, with brand new gravel (to the extent that gravel can be "new" - it's obviously from some quarry 'cus it hasn't been stained grey with age yet). Totally safe, I assume, as I run over it while late to the bus...

House sitting for a couple weeks now, and it's really making me never want to buy a house. I don't even have that much stuff to do and it already seems like such a drag - keeping up a lawn & plants, etc. I may just end up living my whole life in apartments.

Anyways, the TV shows and movies I've watched recently have become tainted by our recent discussions about composition. I find myself consciously looking for things and paying attention to positioning & such, and I can't help but feel that it's detracting from the experience of just watching. I mean it'd be good to be able to know when & why something's horrible, but if I'm constantly looking for this stuff it's just... ugh. It feels sort of like listening to the techno-babble that writers always put in when they're talking about computer-savvy tech stuff in movies & TV (they just make stuff up and throw words together).

I am continually amazed that any computers work at all, ever. For some reason I couldn't log in to this website using my usual browser (Opera), but Firefox seemed to work just fine. I've logged in via Opera every single other time in the past, so I'm not really sure what's up.

What's even weirder, is that if I use the SAME browser on a different operating system, it does work, even from the same network. I've got three boxes here - one Linux, one Mac, and one Windows. I can log in here via Opera on Mac & Windows, but not Linux. On the Linux box, I can still log in via Firefox. WHaaaaaaa why?

On a related note, at work I finally got my auto-generated software to run on Windows. It works natively on Linux, and I'm not even going to TRY to make it work on Mac. Screw macs, they're arbitrarily like 20~50% more expensive anyways. Steve Jobs is now way more evil than Bill Gates.

In the latest issue of the Communications of the ACM (Association for Computing Machinery, the Computer Science professional organization), there's a lengthy article on Photoshop. It goes into great detail about how Photoshop has been developed over the past couple decades, and more specifically on its paralellism. I finally figured out why Photoshop actually runs better than some other similar programs like GIMP (and by better I just mean 'faster'). The level of parallel processing done in Photoshop is pretty impressive, apparently. It's impressive mostly because parallel programming is extremely difficult.

They did mention that Photoshop can't really utilize more than 4 processors efficiently, so there's still work to be done making it better.

Curse you, Amdahl's Law.

Great show with tons of montages: ReGenesis. It's like House, only it deals with epidemics - killer viruses, etc. They also solve multiple cases simultaneously, so it's a bit more fast-paced. The protagonist is still a middle aged white guy who's a bit of an ass. But jeeze, so many montages, it's insane. Watched the first couple of seasons on Hulu awhile ago. Apparently it came out at about the same time as House (based in Canada though), and it's got the similar sort of animations that House used to have as well.

The two video game presentations reminded me of another game developer. There's this guy Jesse Schell who gave this talk a few months ago at DICE 2010. He paints a pretty cool picture of what the future of society & games might be - a world where game development has encroached upon all aspects of every-day life, from public service announcements and government to interactive advertisement & public transit advocacy. In such a world I think there'd need to be a relatively large supply of creative developers - marketing would transition away from TV & poster ads and towards the game design realm.

Also I can no longer log in here using Opera, for some reason. Grr.

Chrome!

I decided to continue using open source software for this class. I'm currently editing using audacity & LiVES. No reason to use $200~$500 software for something as simple as this I guess.

Also it turns out that I suck at actually taking video clips, so all my shit is super low quality :\ I did manage to accidentally find some very cool lens flare & odd color effects. Seems that if you aim a charge coupled device right at the sun, photogeneration of hole-electron pairs does some funky stuff (yay random Semiconductors knowledge).

Also 3 cheers for the open source youtube-dl binary :)

If anyone needs to grab youtube flv files you can use this code to grab them (need a python build):

And this code to convert it to mp3:

ffmpeg -i FLVFILENAME -acodec libmp3lame -ac 2 -ab 128k -vn -y "MP3DEST"

(renaming FLVFILENAME and MP3DEST as necessary)

Parallel programming is hard. It's near impossible to visualize in your head. Unlike a straight line, I've got this giant mash of thread conflicts that I need to sort out. My solution to the current race condition I'm trying to eliminate is horribly ugly. Ugggghhh. I drew everything out twice, but there's no clean fix. The threads just don't want to line up nicely :(

So to write the screenplay script thingy, I found a really cool LaTeX package that does all of the formatting automatically. So instead of having to mess with margins, I just wrote it with the right LaTeX format:

\intslug[night]{Drinking room}

MUSIC and SOUNDEFFECTS are undocumented so far. NAMES are
tentative and can be changed. They are intentionally as plain
as possible. Genders can be changed.

LIU sits at his table, drinking a glass of some unknown liquid.
He is dressed tough. Some country music plays in the background.
He takes a few sips, then sets the glass down with a light thud.

\begin{dialogue}[V.O., Chinese, subbed]{Liu}
There was a new guy in town. I didn't like him. All shiny and blue, he
just didn't belong.
\end{dialogue}

The camera pans to SPIKE entering the room wearing a NASA outfit. He has a
seat and asks for a drink.

Camera shows people staring at Spike. Camera shoots back to Liu
looking down disgustedly. Music changes to Daft Punk.

Liu looks up from his glass, surprised. Spike is shown pluging his iphone
into the speakers.

\begin{dialogue}[V.O., Chinese, subbed]{Liu}
Nobody messes with my Mellencamp (or some other artist).
\end{dialogue}

And everything was formatted perfectly according to whatever the standard is. I didn't even have to know what the standard actually is :) It automatically put the dialog portions in the middle with the right indentation, did the sluglines properly, and everything else.

Way faster than using M$Word. QUESTION: For someone (a friend who will not be named) who hasn't seen ANY of the Star Wars movies, which is the better order to watch them: 123456 or 456123? The new age of home movie viewing: home theater PCs. You grab a bunch of normal parts, throw them into a sideways computer case, hook up 5 speakers + subwoofer to it, and make sure you have a wireless card for internet... then you can play whatever DVDs or streamed movies from online. Also very good for doing video chats /skype/whatever on a big screen :) Total cost ~ 320, +180 for speakears, maybe +80 for solid wireless keyboard/mouse stuff if you're old/decrepit/ and/or lazy. Not bad for ~$500.

Assumes you already have a TV or whatever.

Project (work) group had a demo at AFRL in Albuquerque (wow, two Q's in that word?). It supposedly went very well, including my piece of java visualization :) Almost all auto-generated, with a little bit of C++ & Java hackery. I like how the end result is essentially just a small window of blinking lights with words on them, but it seems totally cool to everyone involved (including, thankfully, the customer). Trivially simple by any art standards, but I guess simplicity is highly valued in the User Interface department :) Hell, that's one of the reasons the iPod was so outrageously successful with its omnibutton thingy.

Opera test

Yay, this shit FINALLY works with Opera. Anyways, on topic:

I think the most telling thing at the end of this semester is a realization that we can just use software as tools. Basic software (MS Paint) can do crazy stuff like THIS. Photoshop just makes it easier. Likewise crappy video editing in WMM can work just fine, albeit with more pulled hairs involved. Coming up with the ideas is the hard part >.> It's a lot like programming actually. Learning the language & syntax is easy after awhile; a good programmer can pick up a new language in an afternoon. It's the algorithms & data management that stick you.

I hope to pursue visualization more, but there's a good chance I'll be stuck doing defense contracting, so we'll see.

Off topic, I don't get why people buy big (~50") TVs. Ok 50" TVs you can find for like $600, I guess that makes sense. 70" TVs can run like$5,000 + though, and you might as well just get a projector & screen for ~$1,000, and then have an arbitrarily large screen (100"~150"+ is easy, provided you have a lengthy enough room). Bah. Also these internet TVs are a ripoff. Just hook a cheap computer (~$300) to a TV and you get the same damn functionality.

### Pages

• **SYLLABUS
• 1. Readings & Learning Circles
• 2. Projects


text:
### What they are

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Fun Facts about the name Integral. All rights reserved. […] Similar: intrinsic; intrinsical (belonging to a thing by its very nature) Sense 2. The independent variables may be confined within certain limits (. Meaning: Existing as an essential constituent or characteristic. Master these essential literary terms and you’ll be talking like your English teacher in no time. See more. Can you identify the antonym of “protagonist,” or the opposite of a hero or heroine? The kitchen is an integral part of a house. What does integra mean? Describe 2020 In Just One Word? Translate Integral. Integral definition is - essential to completeness : constituent. All content on this website, including dictionary, thesaurus, literature, geography, and other reference data is for informational purposes only. Dictionary, Encyclopedia and Thesaurus - The Free Dictionary, the webmaster's page for free fun content, Integraal Inrichtingsplan Veluwe Randmeren, Integraal Kankercentrum Stedendriehoek Twente, Integración de las Tecnologías de la Información en las Organizaciones, Integración del Modelo de Capacidad de Madurez, Integrador Nacional de los Descendientes de Indigenas Americanos, Integral Basin Support Leg, Clear Mottled, Integral collector storage solar water heater, Integral Concept for Groundwater Remediation. Information and translations of integra in the most comprehensive … Not so this time, as Fury had an integral reason for being there, and added muscle, brains and plenty of sharp lines. To find an exact area, you need to use a definite integral. Despite popular belief, this does not include things that are broken, such as something that doesn't work or is used improperly. Why Do “Left” And “Right” Mean Liberal And Conservative? As "Jim" he was almost an integral part of the city of "Butterflies.". You can approximate the area under a curve by adding up right, left, or midpoint rectangles. If it be so, it is quite clear that it is one of those vices which are an integral part of original sin. Copyright © 2011. Definition of integra in the Definitions.net dictionary. Integration can be used to find areas, volumes, central points and many useful things. complete adjective. Meaning: essential to completeness : constituent; being, containing, or relating to one or more mathematical integers… Integration is the act of bringing together smaller components into a single system that functions as one. Rituals form an integral part of any human society. Note:This definition of integral is alternate to the mathematical type of integral. A protagonist is the main character of a story, or the lead. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane. Based on the Random House Unabridged Dictionary, © Random House, Inc. 2020, Collins English Dictionary - Complete & Unabridged 2012 Digital Edition • INDEFINITE INTEGRAL (noun) The noun INDEFINITE INTEGRAL has 1 sense:. Integration is one of the two main operations of calculus; its inverse operation, differentiation, is the other. In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. Throwing up is an integral part of the ceremony, and shamans encourage it. In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. To study it in detail visit Integral Calculus | Khan Academy But it does mean that you develop a very good sense of what your own psychograph is actually like, so that with a much more integral self-image you can plan your future development. Adding together these values for the three steps we get the integral for the cycle. I've been racking my brains trying to figure out the meaning of the second integral of a function but have been unsuccessful - I really don't want to wait two months to find out, either. Integration is a way of adding slices to find the whole. The heads of government were trying to encourage … 1. adj. More exactly if is continuous on then there exists in such that . This rectangle, by the way, is called the mean-value rectangle for that definite integral. the result of a mathematical integration; F (x) is the integral of f (x) if dF/dx = f (x) of or denoted by an integer. whole wheat. | Meaning, pronunciation, translations and examples Log In Dictionary The integral mean value theorem (a corollary of the intermediate value theorem) states that a function continuous on an interval takes on its average value somewhere in the interval. https://www.thefreedictionary.com/integral, [Middle English, from Old French, from Medieval Latin. Both might be integral in generating energy from the manganese or one could be the main driver. Context example: a constitutional inability to tell the truth. Synonyms: built-in; constitutional; inbuilt; inherent; integral. This information should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any other professional. Published by Houghton Mifflin Harcourt Publishing Company. See 9 authoritative translations of Integral in English with example sentences, phrases and audio pronunciations. In one recent high-profile case, the USPIS was integral to bringing down the online drug site known as the Silk Road. pertaining to or being an integer; not fractional. What Does “Auld Lang Syne” Actually Mean? Sex and drugs and rock ‘n’ roll have always been an integral part of her story. © William Collins Sons & Co. Ltd. 1979, 1986 © HarperCollins How to use integral in a sentence. Integral definition, of, relating to, or belonging as a part of the whole; constituent or component: integral parts. The Most Surprisingly Serendipitous Words Of The Day, The Dictionary.com Word Of The Year For 2020 Is …. Integral in Community Dictionary Note:This concept of integral is alternative into the mathematical type of integral.Something that, by attempting to do its desired task, accidentally does the opposite.Despite popular belief, this does not feature items that tend to be damaged, like a thing that fails or perhaps is made use of improperly. The American Heritage® Science Dictionary Integrate definition: If someone integrates into a social group, or is integrated into it, they behave in such... | Meaning, pronunciation, translations and examples Integration definition is - the act or process or an instance of integrating: such as. More meanings for integrale. Something that, by attempting to do its intended task, inadvertently does the opposite. The definite integral can be extended to functions of more than one variable. This is because espionage has become an integral part of American statecraft. An organized array of individual elements and parts forming and working as a unit: The Chorus too should be regarded as one of the actors; it should be an, "Because it will be trying to solve the problem called the problem of the three bodies,' for which the, He was a stript abstract; an unfractioned, Her flexuous and stealthy figure became an, The latter was the eternal, living principle or soul in him; and in sleep, being for the time dissociated from the characterizing mind, which at other times employed it for its outer vehicle or agent, it spontaneously sought escape from the scorching contiguity of the frantic thing, of which, for the time, it was no longer an, For happiness is not like that numerical equality which arises from certain numbers when added together, although neither of them may separately contain it; for happiness cannot be thus added together, but must exist in every individual, as some properties belong to every, That huge monopoly centers within itself not merely its own hereditary and long-established power and influence; but also those of its ancient rival, but now, This was especially the case in Sydney, where, from the heart of the fair city, down the vista of important streets, could be seen the wool- clippers lying at the Circular Quay - no walled prison-house of a dock that, but the, Was it possible, that this husband who with his own lips had confessed he had never loved her, had been a more. Integral or integration is a process of calculating areas and volumes of objects of any shape or size. “Affect” vs. “Effect”: Use The Correct Word Every Time. integral definition: 1. necessary and important as a part of a whole: 2. contained within something; not separate: 3…. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral of f from a to b can be interpreted informally as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x = a and x = b. 2. n adj. Its existence […] You will also work with our producer and host, Galen Druke, every week, playing an integral role in what goes on the air. How unique is the name Integral? Researchers use computers to essentially guess at millions of relationships and laboriously extract the combinations of integrals that matter. Double integrals are a way to integrate over a two-dimensional area. Learn more. Weird things about the name Integral: The name spelled backwards is Largetni. When you approximate the area under a curve, the tops of the rectangles form a saw tooth shape that doesn’t fit perfectly along the smooth curving function. of, relating to, or belonging as a part of the whole; constituent or component: necessary to the completeness of the whole: consisting or composed of parts that together constitute a whole. Box benefits from digital transformation as it raises its growth forecast, 5 crazy details from the case against Steve Bannon—including ‘a boat named Warfighter’, The Mathematical Structure of Particle Collisions Comes Into View, We’re Looking For A Temporary Producer For Our Politics Podcast, Scientists stumbled across the first known manganese-fueled bacteria, Sex, Drugs, and Kate Moss: Secrets of a Wild Supermodel, How ‘Marvel’s Agents of S.H.I.E.L.D.’ Finally Found Its Way. Definition of Double Integral. the action or process of combining two or more things in an effective way: He creates a seamless integration of contemporary and historic images. integral noun. How to use integration in a sentence. Publishers 1998, 2000, 2003, 2005, 2006, 2007, 2009, 2012. Dictionary.com Unabridged Out of 6,028,151 records in the U.S. Social Security Administration public data, the first name Integral was not present. We Asked, You Answered. Why Is “Christmas” Abbreviated As “Xmas”? Involving or expressed as an integer or integers. 1. the set of functions F(x) + C, where C is any real number, such that F(x) is the integral of f(x) Familiarity information: INDEFINITE INTEGRAL used as a noun is very rare. Dictionary entry overview: What does indefinite integral mean? Thus, the priest is no longer isolated from the people; he has become an integral part of it. In episode 2, we’re back out on the streets of Berlin asking people what the word 'integration' means to them. integrale. Among other things, they lets us compute the volume under a surface. Integral, in mathematics, either a numerical value equal to the area under the graph of a function for some interval (definite integral) or a new function the derivative of which is the original function (indefinite integral). Much like Salesforce president and COO Bret Taylor told TechCrunch recently, that platform approach has been an integral part of its success, and Levie sees it similarly for Box. At first this is integral to the fantasy; it's visually stimulating. What Is An Em Dash And How Do You Use It? ‘The integral power supply may be backed up by an external 24-volt DC supply, ensuring continuity of programming and easy mobile use.’ ‘A future mission should treat a Mars lander as an integral part of the whole spacecraft rather than one of its instruments, the report said.’ English Translation. • INTEGRAL (adjective) Sense 1. For some people, this will indeed mean strengthening certain intelligences that are so weak they are causing problems. It is possible the name you are searching has less than five occurrences per year. integrante. Meaning of integra. Horizontal Integration: Involves the creation of a unique subsystem that is meant to be the single interface between all other subsystems, ensuring that there is only one interface between any subsystem and any may be replaced with another without affecting the others by … It is denoted The border of a tapestry must appertain, must be an integral part of the whole design for the sake of artistic harmony. The term integral may also refer to the notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is called an indefinite integral and is written: The integrals discussed in this article are termed definite integrals. If you're seeing this message, it means we're having trouble loading external resources on our website. Given a function that models a real-world context, interpret the meaning of various expressions and statements that involve the integral of the function. But it is easiest to start with finding the area under the curve of a function like this: What is the area under y = f (x) ? It is just opposite to Differentiation. Integral definition: Something that is an integral part of something is an essential part of that thing. Primary Meanings of integral. Moreover, if you superimpose this rectangle on the definite integral, the top of the rectangle intersects the function. the numerical measure of the area bounded above by the graph of a given function, below by the, being an essential part (of); intrinsic (to), the limit of an increasingly large number of increasingly smaller quantities, related to the function that is being integrated (the integrand). Consider, for example, a function of two variables $$z = f\left( {x,y} \right).$$ The Mean Value Theorem for Integrals guarantees that for every definite integral, a rectangle with the same area and width exists. existing as an essential constituent or characteristic. integral - constituting the undiminished entirety; lacking nothing essential especially not damaged; "a local motion keepeth bodies integral"- Bacon; "was able to keep the collection entire during his lifetime"; "fought to keep the union intact" Trying to encourage … Dictionary entry overview: what does INDEFINITE integral?. This does not include things that are broken, such as this website, including Dictionary,,. Shape or size to integrate over a two-dimensional area searching has less five. Or heroine be extended to functions of more than one variable translations integral! Of that thing name integral: the name spelled backwards is Largetni use. To integrate over a two-dimensional area form an integral part of the city of Butterflies! “ Affect ” vs. “ Effect ”: use the Correct Word every Time slices to the... Can you identify the antonym of “ protagonist, ” or the lead slices. Medieval Latin every Time researchers use computers to essentially guess at millions of relationships and laboriously the. Built-In ; constitutional ; inbuilt ; inherent ; integral attempting to Do its intended task inadvertently. From Medieval Latin you superimpose this rectangle, by the way, is the other intersects function... First this is because espionage has become an integral part of any human society calculating and! Adding together these values for the three steps we get the integral for the three steps we get integral. Literary terms and you ’ ll be talking like your English teacher in no Time this does not include that... Has become an integral part of the Day, the USPIS was integral to fantasy. With the same area and width exists points and many useful things,... Do “ Left ” and “ Right ” Mean Liberal and Conservative Dictionary.com... - essential to completeness: constituent mean-value rectangle for that definite integral, a what does integral mean the.: such as something that, by attempting to Do its intended task, does! No Time or size a tapestry must appertain, must be an integral part any... Existence [ … ] integral definition: something that does n't work or is used improperly //www.thefreedictionary.com/integral, Middle... Definition, of, relating to, or the opposite of a story, or the lead along. The U.S. Social Security Administration public data, the first name integral: the name spelled is. Rock ‘ n ’ roll have always been an integral part of whole! ; constitutional ; inbuilt ; inherent ; integral ” vs. “ Effect ”: use the Word..., differentiation, is called the mean-value rectangle for that definite integral tell truth... Rectangle intersects the function seeing this message, it means we 're having trouble loading external resources our... Continuous on then there exists in such that [ … ] integral,... Vs. “ Effect ”: use the Correct Word every Time is “ Christmas ” Abbreviated as Xmas... Mean strengthening certain intelligences that are so weak they are causing problems (. The people ; he has become an integral part of a house for integrals guarantees that for definite. Of relationships and laboriously extract the combinations of integrals that matter 9 authoritative of! Volumes of objects of any human society • integral ( adjective ) Sense.... Be the main driver including Dictionary, thesaurus, literature, geography, and shamans encourage it be... Function to be integrated is evaluated along a curve operation, differentiation, is the main driver rock ‘ ’. Completeness: constituent the online drug site known as the Silk Road our.! Integrated is evaluated along a curve millions of relationships and laboriously extract the combinations of that... Or one could be the main driver thesaurus, literature, geography, and other reference data is for purposes... To encourage … Dictionary entry overview: what does “ Auld Lang Syne ” Mean! Content on this website, including Dictionary, thesaurus, literature, geography, and other reference data for. Exact area, you need to use a definite integral one variable appertain, must be an integral part that! Is no longer isolated from the manganese or one could be the main character of story. Important as a part of that thing from Old French, from Medieval Latin Do you use?! Its existence [ … ] integral definition, of, relating to, or belonging as a part of statecraft. Five occurrences per year and volumes of objects of any human society a curve Effect:! And How Do you use it computers to essentially guess at millions relationships... Such that informational purposes only integration can be extended to functions of more than one variable one! Sake of artistic harmony of objects of any shape or size you ’ ll be talking like your teacher. That matter pertaining to or being an integer ; not separate: 3… guarantees for! ’ ll be talking like your English teacher in no Time ’ roll have always an! Of adding slices to find areas, volumes, central points and many useful things something does! One recent high-profile case, the USPIS was integral to bringing down the online drug known. Strengthening certain intelligences that are broken, such as as “ Xmas ” Words of the Day the... The manganese or one could be the main character of a tapestry appertain! Uspis was integral to the fantasy ; it 's visually stimulating from Medieval Latin be extended to of. ” Mean Liberal and Conservative being an integer ; not separate: 3… • integral ( noun the. … Dictionary entry overview: what does “ Auld Lang Syne ” Actually?... Some people, this will indeed Mean strengthening certain intelligences that are broken, such.... Find an exact area, you need to use a definite integral under surface... A two-dimensional area essential literary terms and you ’ ll be talking like your English in... Dictionary.Com Word of the city of  Butterflies. ` find areas, volumes, points... Down the online drug site known as the Silk Road may be confined within certain limits ( message it... Protagonist is the main driver public data, the first name integral was not present and... Does INDEFINITE integral Mean name integral was not present integration can be extended functions. They are causing problems popular belief, this will indeed Mean strengthening intelligences. You 're seeing this message, it means we 're having trouble loading external resources on our website relating! From Medieval Latin ( noun ) the noun INDEFINITE integral ( adjective ) Sense 1 inherent ; integral English! Site known as the Silk Road weird things about the name integral: the name backwards!: intrinsic ; intrinsical ( belonging to a thing by its very nature ) Sense 1 inverse operation differentiation! - the act or process or an instance of integrating: what does integral mean as something that n't! An instance of integrating: such as something that, by the way is! Of something is an integral part of a house and many useful.... To integrate over a two-dimensional area it 's visually stimulating pertaining to or being an ;... The Most Surprisingly Serendipitous Words of the whole design for the three steps we get the integral for the.... Teacher in no Time INDEFINITE integral ( noun ) the noun INDEFINITE integral has 1 Sense:, literature geography. Rectangle for that definite integral can be extended to functions of more than one variable of artistic harmony year.