text:
# A WinAPI C++ program for printing the master boot record of the hard drive

I have rolled this program for reading the MBR:

#define WIN32_LEAN_AND_MEAN

#include <Windows.h>
#include <cctype>
#include <iostream>
#include <string>
#include <utility>

using std::isprint;
using std::cout;
using std::cin;
using std::move;
using std::size_t;
using std::string;

constexpr size_t MBR_SIZE = 512;
constexpr size_t BYTES_PER_LINE = 4;
constexpr size_t NUMBER_OF_LINES = MBR_SIZE / BYTES_PER_LINE;

// Converts 4 least significant bits of 'c' to corresponding
static string fourBitsToString(char c)
{
string s = " ";

if (c >= 0 && c <= 9) {
s[0] = '0' + c;
} else {
c -= 10;
s[0] = 'A' + c;
}

return move(s);
}

// Converts a character to its hexadecimal representation.
static string charToHex(char c)
{
char lo = c & 0xf;
char hi = (c >> 4) & 0xf;
string s;
char chars[] = {hi, lo};

for (char ch : chars) {
s += fourBitsToString(ch);
}

return std::move(s);
}

// Prints the MBR to console.
static void PrintMBR(char buffer[MBR_SIZE])
{
size_t byteIndex = 0;
string lineSeparator;
string columnSeparator;

for (size_t i = 0; i < NUMBER_OF_LINES; i++) {
cout << lineSeparator;
lineSeparator = '\n';
columnSeparator = "";

for (size_t j = 0; j < BYTES_PER_LINE; j++) {
string ch = charToHex(buffer[byteIndex++]);
cout << columnSeparator << ch;
columnSeparator = " ";
}

cout << ' ';

for (size_t j = 0; j < BYTES_PER_LINE; j++) {
char c = buffer[byteIndex - BYTES_PER_LINE + j];
cout << (isprint((unsigned int) c) ? c : '.');
}
}
}

// Reads the entire master boot record (MBR) into lpBuffer.
// Expects lpBuffer to point to a memory point that may
// hold at leasst 512 bytes.
{

HANDLE diskHandle = CreateFile(
TEXT("\\\\.\\PhysicalDrive0"),
NULL,
OPEN_EXISTING,
0,
NULL);

char* buffer = new char[MBR_SIZE];

diskHandle,
lpBuffer,
MBR_SIZE,
NULL,
NULL);

return GetLastError();
}

int main() {
char buffer[MBR_SIZE];
PrintMBR(buffer);
cin.get();
return 3;
}

(Compile as Release instead of Debug if it throws a nasty runtime breakpoint.)

I would love to hear any comments possible, but my main question is whether it is idiomatic C/C++/WinAPI.

Note: To see the actual MBR, run this as administrator.

# fourBitsToString and charToHex

Both of these functions are basically reimplementations of existing features of the <iostream> and <iomanip> headers.

The whole call to charToHex could simply be replaced by std::cout << std::uppercase << std::hex << std::setfill('0') << std::setw(2) << static_cast<int>(c);.

Note the static_cast<int>: We want to print the numerical value, not the ASCII character glyph!

However, setting std::uppercase, std::hex and std::setfill('0') for every byte to be printed would be wasteful, plus it would overwrite any previously set flags on std::cout.

A better approach would be using a intermediary std::ostringstream instead, setting the flags once and retrieving (and resetting) its contents when sufficiently filled.

More on that in the section below.

Also, both functions end with return move(s);. This seems like misguided premature optimization ("Hey, we can move here! No need for a copy!"), but it likely is actually a pessimization instead!

Compilers actually know that the returned value is a temporary and can optimize for this case (usually called NRVO - "Named return value optimization"). But: They can only do this if the variable is returned directly, and the call to std::move prevents that.

Prefer return s;, unless you have good reasons (e.g. measurements/bad compiler) indicating otherwise.

# PrintMBR

This function is a big pile of unnecessarily complicated code with poorly documented/enfored preconditions.

First off, let's take a look at the function signature:

static void PrintMBR(char buffer[MBR_SIZE])

The flaw is really subtle: buffer isn't actually restricted to arrays of size MBR_SIZE at all! It's basically just a slightly fancier markup of static void PrintMBR(char *buffer), and the compiler will actually treat it as such.

This allows PrintMBR to be called with char arrays of all possible sizes, including smaller than MBR_SIZE, or even just nullptr.

How to fix this?

Well, the first inclination might be to just limit the size of buffer by explicitly making it keep track of the size information, e.g. by using a reference to a C-style array char (&buffer)[MBR_SIZE] or a fancier std::array reference const std::array<char, MBR_SIZE> &buffer.

And that is a valid option if the only purpose was to print MBR contents.

But looking at it's actually implementation, it seems like the intention was to print the hex values in one big column next to a column containing to the ASCII characters (if printable) nicely aligned. This could be generalized for printing arbitrary char arrays by amending some of the assumptions made further down in the implementation.

Next, let's have a look at the function body:

size_t byteIndex = 0;
string lineSeparator;
string columnSeparator;

for (size_t i = 0; i < NUMBER_OF_LINES; i++) {
cout << lineSeparator;
lineSeparator = '\n';
columnSeparator = "";

for (size_t j = 0; j < BYTES_PER_LINE; j++) {
string ch = charToHex(buffer[byteIndex++]);
cout << columnSeparator << ch;
columnSeparator = " ";
}

cout << ' ';

for (size_t j = 0; j < BYTES_PER_LINE; j++) {
char c = buffer[byteIndex - BYTES_PER_LINE + j];
cout << (isprint((unsigned int) c) ? c : '.');
}
}

If I understand correctly, the intention is to print the MBR in a XX XX XX XX xxxx line format (where X is a hex digit and x is an ASCII character).

First off, it's hard to keep track of "constants" if their values are constantly changing. With one more check and a tiny bit of reordering, we can fix this:

size_t byteIndex = 0;
static const string lineSeparator = "\n";
static const string columnSeparator = " ";

for (size_t i = 0; i < NUMBER_OF_LINES; i++) {
if(i != 0) cout << lineSeparator;

for (size_t j = 0; j < BYTES_PER_LINE; j++) {
string ch = charToHex(buffer[byteIndex++]);
cout << ch << columnSeparator;
}

for (size_t j = 0; j < BYTES_PER_LINE; j++) {
char c = buffer[byteIndex - BYTES_PER_LINE + j];
cout << (isprint((unsigned int) c) ? c : '.');
}
}

A bit more readable, and I don't have to keep track of all the possible states (and weird reassignments).

I mentioned above that instead of using charToHex, one could use std::ostringstream and <iomanip> facilities instead:

void PrintMBR(char (&buffer)[MBR_SIZE]) {
static constexpr auto bytes_per_line = 4;
static const auto column_delimiter = " "s;
static const auto line_delimiter = "\n"s;

auto hex_part = std::ostringstream{};
auto ascii_part = std::ostringstream{};
auto counter = 0;

hex_part << std::uppercase << std::hex << std::setfill('0');

for(auto c : buffer)
{
hex_part << std::setw(2) << static_cast<int>(c) << column_delimiter;
ascii_part << (isprint(static_cast<unsigned int>(c)) ? c : '.');

++counter;

if(counter % bytes_per_line == 0)
{
std::cout << hex_part.str() << ascii_part.str() << line_delimiter;
hex_part.str("");
ascii_part.str("");
}
}
}

This could easily be made more generic to allow printing any byte buffer in this format.

# ReadMBR

• Memory leak: buffer doesn't ever get used after being allocated, including being deleted.

• I don't like the return GetLastError(); bit. I guess it's fine in pure C, but in C++ there is a more common way to communicate error conditions: exceptions.

I get this is kind of a glue layer between C WinAPI and C++, but that doesn't mean you have to port idioms from one side into the other, especially if there are more idiomatic alternatives.

I see this as problematic in this case, as likely nobody checks return codes unless they have to. Case in point: main().

• To be more generic, I would really like for this method to accept an OutputIterator. But sadly, the C WinAPI doesn't know those, so that would require another copy of the data (using an intermediary buffer). That's one of the design trade-offs: Performance vs. Usability.

# main

• No check on the return code of ReadMBR.
• return 3; - What does this 3 represent?
• Thanks a lot, man! Accepted after the first 3 paragraphs. Dec 24, 2018 at 6:59
• Do note that guaranteed copy elision does not extend to NRVO. Named object is glvalue, whereas the elision guarantees only cases with prvalues. I actually was not sure myself, had to spend some time to figure out. Great review otherwise. Dec 25, 2018 at 0:59
• @hoffmale, they certainly can be returned by value (commenting first two lines and uncommenting last line will make the code compile). It was just a little objection, but I believe it is important. Link to RVO version. I believe mandatory copy elision is very different from compiler extension, as it allows slightly different techniques to be used. Perhaps I didn't communicate my intent correctly: the post mentions that mandatory copy elision extends to NRVO, but it is not the case, which I'm trying to prove. Dec 27, 2018 at 20:22
• @Incomputable: After reading up elsewhere it finally clicked what it was you wanted to show me, and you're correct. Fixed that part of the answer ;) Dec 28, 2018 at 11:39

Idiomatic is relative. I guess it's a coding style you could find in older code-bases (and many winapi based applications are rather old), but you wouldn't write C++ like this today. Consider for instance:

#include <iostream>
#include <iomanip>
#include <iterator>
#include <algorithm>
#include <numeric>
#include <array>

constexpr std::size_t MBR_SIZE = 255;
constexpr std::size_t BYTES_PER_COLUMN = 4;
constexpr char LINE_DELIMITER = '\n';
constexpr const char* COL_DELIMITER = " ";

int main() {
std::array<unsigned char, MBR_SIZE> mbr;
std::iota(std::begin(mbr), std::end(mbr), 0); // filling it for testing purposes

std::cout << std::hex << std::setfill('0');
std::size_t index = 0;
std::transform(std::begin(mbr), std::end(mbr), std::ostream_iterator<int>(std::cout, COL_DELIMITER), [index](auto c) mutable {
if (index++ % BYTES_PER_COLUMN == 0) std::cout << LINE_DELIMITER;
std::cout << std::setw(2); // setw must be applied for each output
return c;
});
std::cout << std::endl;
}

In a dozen of lines you get the same result as printMBR and its two auxiliary functions (at least I believe so, I haven't had the chance to run your code), which are longer and more complicated.

• don't declare a variable before you're ready to define it, and don't define it before you're ready to use it.

• don't std::move your return value, it prevents copy elision

• don't declare free functions static, it's a java thing

• inside ReadMBR there's a char* buffer = new char[MBR_SIZE]; you don't use afterwards

• I'm not a fan of that many using directives; prefix the names with std:: and be done with it, unless you prefer to set up aliases: then using string = std::string is clearer

• use standard algorithms, such as std::copy, std::accumulate, etc (in headers <algorithm> and <numeric>) or range-base for loops (for auto item : sequence) instead of "raw" loops. For instance, charToHex could be written

like this:

std::string s;
for (auto nibble : { (c >> 4) & 0xf, c & 0xf })
s += fourBitsToString(nibble);
return s;

or:

char lo = c & 0xf;
char hi = (c >> 4) & 0xf;
char chars[] = {hi, lo};

return std::accumulate(std::begin(chars), std::end(chars), std::string(), [](auto init, auto elem) {
return init += (fourBitsToString(elem));
});

And, on a more aesthetic note, a four-bits aggregate can be called a half-byte or a nibble.

• "don't declare free functions static" - unless you want to explicitly restrict them to that specific translation unit (e.g. because it's an implementation detail not intended to be part of the public API). (This might also allow for better optimization, though I wouldn't suggest doing so just because of this reason.) // For printing, you suggest a call to std::transform with a std::ostream_iterator for std::cout and some custom logic also printing to std::cout. Methinks it would be cleaner to just use a std::for_each call and placing all printing logic into the lambda. Dec 21, 2018 at 16:32
• @hoffmale: if you want to restrict functions to a specific TU, an anonymous namespace is generally preferred. Dec 21, 2018 at 19:02

## Release resources as soon as practical

The diskHandle should be released immediately after the ReadFile call to minimize the time that the handle is open. Call CloseHandle to close the file and release the handle.

## Use minimal sufficient privileges

In this case, the MBR is only read and not written, so the file should be opened only with the FILE_SHARE_READ option and not FILE_SHARE_WRTE.

## Use const where practical

The PrintMBR function does not alter the passed buffer so the buffer parameter should be const.

## Use an object

I'm surprised that other reviews didn't mention this, but why not treat the MBR as a C++ object? That way your main could look like this:

int main() {
MBR mbr;


text:
Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold

@article{Leibman2004PointwiseCO,
title={Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold},
author={Alexander Leibman},
journal={Ergodic Theory and Dynamical Systems},
year={2004},
volume={25},
pages={201 - 213}
}
• A. Leibman
• Published 22 December 2004
• Mathematics
• Ergodic Theory and Dynamical Systems
We show that the orbit of a point on a compact nilmanifold X under the action of a polynomial sequence of translations on X is well distributed on the union of several sub-nilmanifolds of X. This implies that the ergodic averages of a continuous function on X along a polynomial sequence of translations on X converge pointwise.
173 Citations

Pointwise convergence of ergodic averages for polynomial actions of Z by translations on a nilmanifold

Generalizing the one-parameter case, we prove that the orbit of a point on a compact nilmanifold X under a polynomial action of Z by translations on X is uniformly distributed on the union of several

Pointwise convergence of ergodic averages along cubes

Let (X, B, µ, T) be a measure preserving system. We prove the pointwise convergence of ergodic averages along cubes of 2k − 1 bounded and measurable functions for all k. We show that this result can

Convergence of polynomial ergodic averages

• Mathematics
• 2005
We prove theL2 convergence for an ergodic average of a product of functions evaluated along polynomial times in a totally ergodic system. For each set of polynomials, we show that there is a

Polynomial averages converge to the product of integrals

• Mathematics
• 2004
We answer a question posed by Vitaly Bergelson, showing that in a totally ergodic system, the average of a product of functions evaluated along polynomial times, with polynomials of pairwise

Pointwise convergence in nilmanifolds along smooth functions of polynomial growth

We study the equidistribution of orbits of the form b a1(n) 1 · · · b ak(n) k Γ in a nilmanifold X, where the sequences ai(n) arise from smooth functions of polynomial growth belonging to a Hardy

Orbit of the diagonal in the power of a nilmanifold

Let X be a nilmanifold, that is, a compact homogeneous space of a nilpotent Lie group G, and let a 2 G. We study the closure of the orbit of the diagonal of X r under the action (a p1(n)

Rational sub-nilmanifolds of a compact nilmanifold

• A. Leibman
• Mathematics
Ergodic Theory and Dynamical Systems
• 2006
We introduce the notions of a rational point, a rational sub-nilmanifold and a rational homomorphism of compact nilmanifolds and describe elementary properties thereof. We also show that the closure

Convergence of multiple ergodic averages for some commuting transformations

• Mathematics
Ergodic Theory and Dynamical Systems
• 2005
We prove the L2-convergence for the linear multiple ergodic averages of commuting transformations $T_{1}, \dots, T_{l}$, assuming that each map Ti and each pair TiTj-1 is ergodic for $i\neq j$. The

Linear Sequences and Weighted Ergodic Theorems

We present a simple way to produce good weights for several types of ergodic theorem including the Wiener-Wintner type multiple return time theorem and the multiple polynomial ergodic theorem. These

References

SHOWING 1-10 OF 12 REFERENCES

Sur une nil-variété, les parties minimales associées à une translation sont uniquement ergodiques

• E. Lesigne
• Mathematics
Ergodic Theory and Dynamical Systems
• 1991
Abstract We call nilmanifold every compact space X on which a connected locally compact nilpotent group acts transitively. We show that, if X is a nilmanifold and f is a continuous function on X,

Invariant Measures and Orbit Closures on Homogeneous Spaces for Actions of Subgroups Generated by Unipotent Elements

The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups

Topological Transformation Groups

• Mathematics
• 1956
1. Introduction This note will summarize some of the recent work on topological groups and discuss a few topics in transformation groups mainly in S 3 and S 4. In one aspect of this subject, namely

Multiple Recurrence Theorem for Measure Preserving Actions of a Nilpotent Group

Abstract. The multidimensional ergodic Szemerédi theorem of Furstenberg and Katznelson, which deals with commuting transformations, is extended to the case where the transformations generate a

A nilpotent Roth theorem

• Mathematics
• 2002
Abstract.Let T and S be invertible measure preserving transformations of a probability measure space (X, ℬ, μ). We prove that if the group generated by T and S is nilpotent, then

Multiple recurrence theorem for nilpotent group actions

AbstractWe prove that ifX is a compact space andT1, ...,Tt are homeomorphisms ofX generating a nilpotent group, then there existx ∃X andn1,n2, ... ∃N such that T_j^{n_m } x\mathop \to \limits_{m

format: plain


text:
In the 50 Fathoms players section it does not specify what a starting character gets when they select Arcane Background (Magic). This is unlike the other Arcane Backgrounds in the other rule book.

Is this an oversight (and it should be as the Mage in the main rulebook) or does it mean they start with nothing? If so you start at a great disadvantage as you have to select another two edges just to do anything.

Also, does Elemental Mastery Edge give you any benefit other than allowing you to select from a different set of powers (any use cross elemental powers at your new element)? The book doesn't seem to suggest there's any other benefit - is that correct?

-


text:
## Curve of Constant Precession

A curve whose Centrode revolves about a fixed axis with constant Angle and Speed when the curve is traversed with unit Speed. The Tangent Indicatrix of a curve of constant precession is a Spherical Helix. An Arc Length parameterization of a curve of constant precession with Natural Equations

 (1) (2)

is
 (3) (4) (5)

where
 (6)

and , and are constant. This curve lies on a circular one-sheeted Hyperboloid
 (7)

The curve is closed Iff is Rational.

References

Scofield, P. D. Curves of Constant Precession.'' Amer. Math. Monthly 102, 531-537, 1995.


text:
Plot a several energy eigenvalues of a partical

1. Feb 22, 2009

eman2009

Hi
How can i compute(or obtain from mathmatical tables) and plot a several energy eigenvalues of a partical in an infinity deep ,spherically and symmetric square well?

2. Feb 22, 2009

clem

Re: energyeigenvalue....

That is done in all QM textbooks.

3. Feb 22, 2009

Count Iblis

Re: energyeigenvalue....

Last edited by a moderator: May 4, 2017
4. Feb 22, 2009

eman2009

Re: energyeigenvalue....

i should comput it due to the angular momentum states.
and i'm not very clear with the attachment .. i thought i can comput it from schrodenger equation and find E and blug the Q# in it for the lowest energy eigenvalues is it correct?
if you can make it easy to find ,i well appreciate......
thanks very much


text:
# Plane Fitting with Linear Least Squares

1. Sep 30, 2009

### mnb96

Hello,
I am trying to figure out how to fit a plane passing through the origin in $$\mathbf{R}^3$$, given a cloud of N points.
The points are vectors of the form $$(x_1^{(k)}, x_2^{(k)}, x_3^{(k)})$$ , where k stands for the k-th point. I want to minimize the sum of squared distances point-plane.

What I came out with was the following:
- solve a 3x3 homogeneous linear system AX=0 in which the (i,j) element of A is:

$$\sum_{k=1}^N x_i^{(k)} x_j^{(k)}$$

Now I have two questions:

1) I have read somewhere that this turns out to be an eigenvalues problem. Basically I need to find the eigen-values/vectors in order to solve the system. Why?

2) I found another method which instead builds a rectangular Nx4 matrix in which, in the first column there are all the x's of the points, in the second column all the y's, in the third all the z's, and in the fourth all 1's. Then they compute the SVD and extract (I think) the last column of the rightmost output matrix. Why does that work??? What's the difference?

2. Sep 30, 2009

### TrentFGuidry

It sounds like you are describing a multiple linear regression problem.

fi = a1xi + a2yi + a3zi

Where xi, yi, and zi are the points and fi is the observed function values.

To solve this, you can create a matrix Z with the following values:

x1 y1 z1
x2 y2 z2
x3 y3 z3
.
.
.
xn yn zn

You can also create a vector Y with the following values:

f1
f2
f3
.
.
.
fn

After creating the Z matrix and Y column vector, the regressed A vector can be found by solving:

ZTZA = ZTY

I have a longer post on this, with a derivation of that equation, and source code that I released to the public domain at http://www.trentfguidry.net/post/2009/07/19/Linear-and-Multiple-Linear-Regression-in-C.aspx"

Last edited by a moderator: Apr 24, 2017
3. Sep 30, 2009

### TrentFGuidry

As for the eigenvalues/eigenvectors, I believe that you have to do that for an SVD decomposition, which is a way of solving the above matrix equation.

Another approach to solving the above equation is to use a different approach, such as an LU decomposition. The potential problem with an LU decomposition is that a singular matrix could result. If that occurs, then the problem cannot be solved using an LU decomposition and something else, like an SVD decomposition, would have to be used.

Fortunately, there are a lot of problems where singular matrices are a rare occurrence and the simpler and faster LU decomposition approach works. LU decomposition does not require eigenvalues/eigenvectors.

For part 2, you describe a Z matrix with ones in it. That Z matrix has constants that can offset it from the origin. For that case, you are regressing the function shown below.

fi = a0 + a1xi + a2yi + a3zi

If you want to force it through the origin, get rid of the column with all ones in it.


text:
1. Mar 7, 2005

### FlaMiNxSkullx05

I have been having a discussion with a friend for the past few days, and we are in sort of a stand still on a problem. We would greatly appreciate some input into our discussion.

The question is this: A hollow spherical object weighs ten pounds with a certain density, and size floats in water. Will a solid spherical object with the same weight, density, and size float in the same amount of water?

My argument is that no, the solid object will not float in the water. I believe this because the solid object has no air, and therefore has the lack of bouyancy needed to float in the water. Even though the displacement of water may be the same, the lack of air would keep the object from floating.

His argument is that yes, the solid object will float in the water. This is because the displacement of the water, and density of the water and object are the same and cancel out, causing it to float.

We would like to know who is right or wrong, and possibly an explination why. Thanks for the help, we appreciate it

2. Mar 7, 2005

### StatusX

What is important is the density of the object as a whole. The hollow sphere has a lower density because it has the same volume but less mass, so it can float when the solid sphere can't.

3. Mar 7, 2005

### Jameson

As long as the densities are equal, and the densities of both are less than that of water, both objects should float. The bouyant force that the object places down on the water is equal to the weight of the water the object displaces. The water then places an equal force in the opposite direction equal to the other force. If the water cannot sustain the balance, then the object would sink.

I would agree with your friend.

4. Mar 7, 2005

### Jameson

What? The problem said the masses were equal.... and the volumes were equal, therefore they have equal densities.

$$D = \frac{M}{V}$$

5. Mar 7, 2005

### StatusX

I'm sorry, I thought he said they were made of the same material. Yes, they will either both float or both sink.

6. Mar 7, 2005

### FlaMiNxSkullx05

so the hollow part of the one object would not matter? i was always taught that no matter what the density of an object, if it was solid it would not float...i guess it makes sense that only density would matter though. I am still wondering something though, if i had a bowling ball weighing 10 lbs, it would float in water, but if i had another solid object of the same density it would shrink, why is that?

7. Mar 7, 2005

### SphericalStrife

how could the two spheres have equal size & density? I was just confused, wouldn't the solid one have to be smaller because.. it's smaller?

8. Mar 7, 2005

### Some_Student

Hey can someone help me with a math question right now? Please? Thanks.

Using a simple balance, could you find the one fake coin in a group of 27 coins. Here's the catch...you may only weigh the coins 3 times. You may not use anything else. How could this be accomplished?

9. Mar 7, 2005

### Some_Student

The fake coin weighs different than the other coins.

10. Mar 7, 2005

### xanthym

If you have any object of Mass "M" with solid (water-proof) boundaries, so that it has a defined Volume "V", then regardless of its shape or size or contents, the only factor determining whether it will float is whether its density given by (M/V) is less than water's density (1 gram/cm^3).

Two objects of the SAME DENSITY will either both float OR both sink in water even if their weights, shapes, contents, and sizes are different.

The physical phenomenon involved is called buoyancy, and its properties are described by Archimede's Principle. The following Web Sites provide further discussion of Archimede's Principle:
http://www.onr.navy.mil/focus/blowballast/sub/work2.htm
http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Pressure/BouyantForce.html
http://www.plus2physics.com/hydrostatics/study_material.asp?chapter=3
http://www.bsharp.org/physics/stuff/icebergs.html

~~

Last edited: Mar 7, 2005
11. Mar 7, 2005

### FlaMiNxSkullx05

thank you all for helping me. I think i understand what is going on now, i believe i was confused earlier when i thought air was the determining factor when floating an object, i guess it is not. Another good point a teacher of mine brought up is that if the density of an object is the exact same as that of water's, what would the object do? would it sink, float, or kind of just sit in the water below the surface level?? because according to the original problem, the one density of the hollow object floated...therefore its density was less than that of waters, according to your equations, so if the density was equal to the waters it would technically come out at an equilibrium, making it a stable object..correct? i think i am starting to understand this now


text:
## Direction of propagation of an EM wave

I'm reading my course book on ELectromagnetism and it is talking about a wave moving in the y-z plane but with polarisation in the x-direction, and it says that the equation

$$\mathbf{E}=E_0 2i sin(k_0 z\ cos \theta) exp[i(k_0 y\ sin \theta - \omega t)]\mathbf{e}_x$$

Shows that there is no propagation in the z-direction, but that instead a standing wave is formed when he wave bounces at an oblique angle between the two perfect conductors at z=0 and z=-a.

My question is...if the wave equation contains a factor involving z, how is it not propagating in the z direction? I suspect there is something I haven't understood about wave propagation and movement.

Recognitions:
Homework Help
 Quote by tomwilliam I'm reading my course book on ELectromagnetism and it is talking about a wave moving in the y-z plane but with polarisation in the x-direction, and it says that the equation $$\mathbf{E}=E_0 2i sin(k_0 z\ cos \theta) exp[i(k_0 y\ sin \theta - \omega t)]\mathbf{e}_x$$ Shows that there is no propagation in the z-direction, but that instead a standing wave is formed when he wave bounces at an oblique angle between the two perfect conductors at z=0 and z=-a. My question is...if the wave equation contains a factor involving z, how is it not propagating in the z direction? I suspect there is something I haven't understood about wave propagation and movement. Thanks in advance
What variable(s) must change as you progress forward (or backward) in time, If the amplitude of the electric field remains unchanged? Do you see how this describes a specific point on the wavepulse travelling through space as time progresses?
 Ah, well I can see that the term in the exponential brackets must be constant in order for there to be no attenuation over time, so I imagine that y must increase as t increases...hence propagation in the y-direction. So far so good. So I guess at z=0 the expression is zero, and this indicates that the wave has a node at this point? This might sound like a stupid question, but what happens if the wave reaches the boundary of the conductor at a point where it doesn't have a non-zero value? Is it absorbed by the conductor? Thanks again for your help

Recognitions:
Homework Help

## Direction of propagation of an EM wave

 Quote by tomwilliam Ah, well I can see that the term in the exponential brackets must be constant in order for there to be no attenuation over time, so I imagine that y must increase as t increases...hence propagation in the y-direction. So far so good.
If you are only looking at constant $z$, or $\theta = \frac{\pi}{2}$, then yes. In general however, all you can say is that $\sin(k_0 z \cos \theta ) e^{i ( k_0 y \sin \theta - \omega t )}$ must remain constant. There can certainly be propagation in the z-direction for this waveform.

I assume there must be more to the situation discussed in your text than what you've provided. Which text is this from?
 Recognitions: Gold Member Science Advisor I would put it down as being that there are two possible solutions to the wave equation here whereby the standing wave is the superposition of these solutions. As you stated previously, the standing wave is the superposition of a wave bouncing back and forth. So in the context of your overall wave solution, what you have is, for example, a wave that is moving in the y and z direction at some point Z and a wave that is moving in the y and -z direction at the same point Z. The superposition of these two gives rise to the standing wave in the z direction, but since they are both solutions, you can think of them as being separate waves if you like. If you were to excite the actual mode between the two plates, you could do so in such a way so that there is only one of these solutions excited at a time. Of course, the underlying reason why there is no propagation in the z direction (more accurately net propagation) is that the wave is confined in the z direction due to the boundary conditions of the plates. So the fact that the full solution exhibits a standing wave is just demonstrating how the net propagation of the wave is guided in the y direction and confined in the z direction by the plates.
 @Gabba: I'm not sure if I've missed out any relevant details. The text is a university course book. I won't say which university, but it is a respectable publication. It has a section on normal incidence of EM waves on conductors, then a shorter section on oblique incidence, which this is taken from. It derives the result I gave in my original post, and then states that: "In contrast to section...where we found a stationary wave for normal incidence, in the case of oblique incidence, there is now a propagating wave. In fact, the exponential term shows that the wave propagates in the y-direction, parallel to the conducting boundary, with speed omega/(k_0 sin theta)." It goes on to talk about modes where propagation can satisfy the boundary conditions for the parallel component of the electric field. You say the wave does propagate in the z-direction because of the z-term. Doesn't this just affect the amplitude of the wave? @Born2 It sounds like you're right, as the next section is about waveguides following this initial model and applying it to physical constraints in the x-y and x-z planes. But as I visualise this as a wave bouncing off the walls between z=0 and z=-a, is it not propagating in the z-direction? (i.e. if there were no constraints, would it move in the z-direction? I'm confused because the equation seems to suggest movement (propagation?) in the y-direction, with amplitude which depends on the z-variable. But if the wave incides on the conducting boundary at z=0 and z=-a then it must have moved in the z-direction, musn't it? Apologies if this sounds garbled.

Recognitions:
Gold Member
 Quote by tomwilliam2 @Born2 It sounds like you're right, as the next section is about waveguides following this initial model and applying it to physical constraints in the x-y and x-z planes. But as I visualise this as a wave bouncing off the walls between z=0 and z=-a, is it not propagating in the z-direction? (i.e. if there were no constraints, would it move in the z-direction? I'm confused because the equation seems to suggest movement (propagation?) in the y-direction, with amplitude which depends on the z-variable. But if the wave incides on the conducting boundary at z=0 and z=-a then it must have moved in the z-direction, musn't it? Apologies if this sounds garbled.
It is propagating in the z direction, but this propagation isn't a net propagation because it is just bouncing back and forth between z=0 and z=-a. Take a look at slide 135 in this set:

http://www.amanogawa.com/archive/docs/EM12.pdf

This slide shows the actual path of propagation for the plane wave inside a parallel plate waveguide. As you suspect, the wave bounces back and forth between the two plates. The catch here is that the path that they dictate is just one of TWO equally valid solutions. The other solution has the wave going along the same angles, but it starts out bouncing off of the other plate. When you take the superposition of the two solutions, the result looks like a standing wave in the z-direction.

When you solve for the modes of the parallel plate waveguide, you are solving for all possible fields. So the solution represents a superposition of all the possible waves. When they talk about the direction of propagation for a waveguide, notation often gets lazy. The whole purpose of a waveguide is to direct the propagation of energy. As such, people are focused on the direction of guided propagation, not necessarily that actual path of propagation. As such, for a parallel plate wave guide then the direction of guided or net propagation in your case is along the y direction. There is no net flow of energy in the z direction due to the confinement by the plates. Also note that this looseness of terms may give rise to some confusion when they talk about the waveguide modes. Often you will hear, like in the case of a rectangular waveguide, that only TE or TM modes exist, not TEM. This seems to go against our knowledge of Maxwell's Equations which states that EM waves are TEM. What they really mean, is that there are no TEM modes with respect to the guided direction of propagation. So they are not talking about TE, TM or TEM regarding the instantaneous direction of propagation but the guided direction of propagation.

Without the confinement of the plates, then you could of course get propagation in the z-direction. Without the plates the solution is just the trivial free-space solution which is a superposition of plane waves in any direction.

Also note that your solution is the time-harmonic solution. It is not the time-domain solution so it is not going to represent a causal propagation of a wave. To get the actual time-domain solution, you have to take the Fourier transform or the real part of the time-harmonic solution (depending upon how you define the time-harmonic case). Despite this, a basic knowledge of the time-harmonic plane wave allows us to easily infer how the plane wave would propagate in this case based off of the time-harmonic solution.

Recognitions:
Homework Help
 Quote by tomwilliam2 @Gabba: I'm not sure if I've missed out any relevant details. The text is a university course book. I won't say which university, but it is a respectable publication. It has a section on normal incidence of EM waves on conductors, then a shorter section on oblique incidence, which this is taken from. It derives the result I gave in my original post, and then states that: "In contrast to section...where we found a stationary wave for normal incidence, in the case of oblique incidence, there is now a propagating wave. In fact, the exponential term shows that the wave propagates in the y-direction, parallel to the conducting boundary, with speed omega/(k_0 sin theta)." It goes on to talk about modes where propagation can satisfy the boundary conditions for the parallel component of the electric field.
Okay, as Born2 says, if there are two perfect conducting planes at z=0 and z=-a, then the given waveform is only the electric field between those two planes, outside it will be zero (barring additional sources there). This means that propagation in the z-direction will average out (as the waveform's motion in that direction will cause it to just reflect back and forth between the planes) - there will be a standing wave there.

However, this is not evident in the waveform you give. It is only after you apply the boundary conditions on $z$ that the result shows explicitly this behaviour.

 You say the wave does propagate in the z-direction because of the z-term. Doesn't this just affect the amplitude of the wave?
Look at the real part of $\mathbf{E}$ and remember that $\sin u \cos u = \frac{1}{2} \left[ cos(u-v) + \cos(u+v) \right]$
 Thanks very much for the clear and insightful replies. I'm not familiar with time-harmonic vs time-domain solutions but I will investigate, rather than ask someone to explain a whole new concept from scratch. I think I have understood the basics now. Thanks again.


text:
# Choose the best answer for the following sentence: _________ a contest requires a good preparation.

A.

Organize

B.

Having organized

C.

Organizing

D.

Organized

Đáp án:C
Lời giải:

Organizing


text:
The least eigenvalue of graphs whose complements are unicyclic Supported by National Natural Science Foundation of China (11071002), Program for New Century Excellent Talents in University (NCET-10-0001), Key Project of Chinese Ministry of Education (210091), Specialized Research Fund for the Doctoral Program of Higher Education (20103401110002), Science and Technological Fund of Anhui Province for Outstanding Youth (10040606Y33), Scientific Research Fund for Fostering Distinguished Young Scholars of Anhui University(KJJQ1001), Academic Innovation Team of Anhui University Project (KJTD001B).

# The least eigenvalue of graphs whose complements are unicyclic††thanks: Supported by National Natural Science Foundation of China (11071002), Program for New Century Excellent Talents in University (NCET-10-0001), Key Project of Chinese Ministry of Education (210091), Specialized Research Fund for the Doctoral Program of Higher Education (20103401110002), Science and Technological Fund of Anhui Province for Outstanding Youth (10040606Y33), Scientific Research Fund for Fostering Distinguished Young Scholars of Anhui University(KJJQ1001), Academic Innovation Team of Anhui University Project (KJTD001B).

Yi Wang, Yi-Zheng Fan , Xiao-Xin Li, Fei-Fei Zhang
School of Mathematical Sciences, Anhui University, Hefei 230039, P. R. China
Department of Mathematics and Computer Sciences, Chizhou University, Chizhou 247000, P.R. China
Corresponding author. E-mail addresses: wangy@ahu.edu.cn (Y. Wang), fanyz@ahu.edu.cn (Y.-Z. Fan), starlittlestar@sina.com (X.-X. Li), zhangfeifei2403@126.com (F.-F. Zhang).

Abstract: A graph in a certain graph class is called minimizing if the least eigenvalue of the adjacency matrix of the graph attains the minimum among all graphs in that class. Bell et al. have characterized the minimizing graphs in the class of connected graphs of order and size , whose complements are either disconnected or contain a clique of order at least . In this paper we discuss the minimizing graphs of a special class of graphs of order whose complements are connected and contains exactly one cycle (namely the the class of graphs whose complements are unicyclic), and characterize the unique minimizing graph in when .

Key words: Unicyclic graph; complement; adjacency matrix; least eigenvalue

2010 Mathematics Subject Classification:  05C50, 05D05, 15A18

## 1 Introduction

Let be a simple graph with vertex set and edge set . The adjacency matrix of is defined to be a matrix of order , where if is adjacent to , and otherwise. Since is real and symmetric, its eigenvalues are real and can be arranged as: . We simply call the eigenvalues of as the eigenvalues of . The eigenvalue is exactly the spectral radius of ; and there are many results in literatures concerning this eigenvalue; see e.g. [1, 4, 5].

The least eigenvalue is now denoted by , and the corresponding eigenvectors are called the first eigenvectors of . Relative to the spectral radius, the least eigenvalue has received less attention. In the past the main work on the least eigenvalue of a graph is focused on its bounds; see e.g. [6, 9]. Recently, the problem of minimizing the least eigenvalues of graphs subject to graph parameters has received much more attention, since two papers of Bell et al. [2, 3] and one paper of our group [7] appeared in the same issue of the journal Linear Algebra and Its Applications. Ye and Fan [15] discuss the connectivity and the least eigenvalue of a graph. Liu et al. [10] discuss the least eigenvalues of unicyclic graphs with given number of pendant vertices. Petrović et al. [11] discuss the least eigenvalues of bicyclic graphs. Wang et al. [13, 14] discuss the least eigenvalue and the number of cut vertices of a graph. Tan and Fan [12] discuss the least eigenvalue and the vertex/edge independence number, the vertex/edge cover number of a graph.

For convenience, a graph is called minimizing in a certain graph class if its least eigenvalue attains the minimum among all graphs in the class. Let denote the class of connected graphs of order and size . Bell et al. [2] have characterized the structure of the minimizing graphs in ; see Theorem 1.1.

###### Theorem 1.1

[2] Let be a minimizing graph in . Then is either

(i) a bipartite graph, or

(ii) a join of two nested split graphs (not both totally disconnected).

We find that the complements of the minimizing graphs in are either disconnected or contain a clique of order at least . This motivates us to discuss the least eigenvalue of graphs whose complements are connected and contain clique of small size. In a recent work [8] we characterize the unique minimizing graph in the class of graphs of order whose complements are trees.

In this paper, we continue this work on the complements of unicyclic graphs, and determine the unique minimizing graph in for , where denotes the class of the complements of connected unicyclic graphs of order . It is easily seen that . However the minimizing graph in does not hold the conditions in Theorem 1.1.

## 2 Preliminaries

We begin with some definitions. Given a graph of order , a vector is called to be defined on , if there is a 1-1 map from to the entries of ; simply written for each . If is an eigenvector of , then it is naturally defined on , i.e. is the entry of corresponding to the vertex . One can find that

 XTAX=2∑uv∈E(G)XuXv,

and is an eigenvalue of corresponding to the eigenvector if and only if and

 λXv=∑u∈NG(v)Xu,~{} for each vertex ~{}v∈V(G).

where denotes the neighborhood of in the graph . The equation (2.2) is called -eigenequation of . In addition, for an arbitrary unit vector ,

 λmin(G)≤XTA(G)X,

with equality if and only if is a first eigenvector of .

In this paper all unicyclic graphs are assumed to be connected. Denote by the set of unicyclic graphs of order , and denote , where denotes the complement of the graph . Note that , where denote the all-ones matrix and the identity matrix both of suitable sizes, respectively. So for any vector ,

 XTA(Gc)X=XT(J−I)X−XTA(G)X.

A star of order , denoted by , is a tree of order with pendant edges attached to a fixed vertex. The vertex of degree in is called the center of . A cycle and a complete graph both of order are denoted by respectively. Denote by the graph obtained from by adding a new edge between two pendant vertices. Next, we introduce two special unicyclic graphs denoted by and respectively, where is obtained from two disjoint graphs and by adding a new edge between one pendant vertex of and one pendant vertex of , is obtained from two disjoint graphs and by adding a new edge between one pendant vertex of and one vertex of ; see Fig. 2.1.

Fig. 2.1. The graphs (left side) and (right side)

For a graph containing at least one edge, then with equality if and only if is a complete graph or a union of disjoint complete graphs at least one component of which is nontrivial (or contains more than one vertices). So, for a unicyclic graph other than , . In addition, if is disconnected, then contains a complete multipartite graph as a spanning subgraph, which implies is or . When , consists of an isolated vertex and a connected non-complete subgraph of order .

At the end of this section, we will discuss the least eigenvalues of and . Let be a first eigenvector of the graph with some vertices labeled in Fig. 2.1. By eigenequations (2.2), as , all the pendant vertices attached at have the same value as given by , say . Similarly all the pendant vertices attached at have the same value as , say ; two vertices of degree on the triangle have the same value as , say . Write for and simply. Then by eigenequations (2.2) on for , we have

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩λ1X1=(p−2)X1+X3+X4+X5+2X6+(q−3)X7λ1X2=X4+X5+2X6+(q−3)X7λ1X3=(p−1)X1+X5+2X6+(q−3)X7λ1X4=(p−1)X1+X2+2X6+(q−3)X7λ1X5=(p−1)X1+X2+X3λ1X6=(p−1)X1+X2+X3+X4+(q−3)X7λ1X7=(p−1)X1+X2+X3+X4+2X6+(q−3)X7

Transform (2.5) into a matrix equality , where and the matrix of order is easily to seen. We have

 f(λ;p,q):=det(B−λI)=(−8+2p+2q)+(13−11p−7q+4pq)λ+(20−6q−4qp)λ2+(−1+11p+7q−7pq)λ3+(−20+12p+12q−2pq)λ4+(−16+6p+6q)λ5+(−6+p+q)λ6−λ7.

It is easily found that is the least root of the polynomial .

Let be a first eigenvector of the graph with some vertices labeled in Fig. 2.1. By a similar discussion, all the pendant vertices attached at have the same values given by , say . Two vertices of degree 2 on the triangle have the same values, say . Write for and simply. Then by eigenequations on for ,

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩λ′1Y1=(p−2)Y1+Y3+Y4+2Y5λ′1Y2=Y4+2Y5λ′1Y3=(p−1)Y1+Y4+2Y5λ′1Y4=(p−1)Y1+Y2λ′1Y5=(p−1)Y1+Y2+Y3

It is easily found that is the least root of the following polynomial:

 g(λ;p):=(−4+2p)+(3−5p)λ+(6−p)λ2+(1+4p)λ3+(−2+p)λ4−λ5.
###### Lemma 2.1

If , .

Proof: Write , simply. By above discussion, (respectively, ) is the least root of (respectively, ). Denote by

 ¯g(λ;n−4):=(λ+1)2g(λ;n−4).

Since , is also the least root of . From (2.8), , and consequently if . Furthermore, when , , which implies . Obverse that when ,

 ¯g(λ;n−4)−f(λ;n−5,3)=(−6+n)λ(1+λ)(−2+5λ+2λ2)>0.

In particular, , which implies . The result follows.

###### Lemma 2.2

Given a positive integer , for any positive integers such that and ,

 λmin(U(p,q)c)≥λmin(U(⌈(n−2)/2⌉,⌊(n−2)/2⌋)c),

with equality if and only if and .

Proof: Write simply. By (2.6), we have

 f(λ;p,q)−f(λ;p+1,q−1) =−λ(2+λ)(−1+2λ)[(p−q+1)(2+λ)+2], f(λ;p,q)−f(λ;p−1,q+1) =λ(2+λ)(−1+2λ)[(p−q−1)(2+λ)+2].

In addition, , which implies .

If , for we have

 f(λ;p,q)−f(λ;p+1,q−1)>0.

In particular, , which implies

 λmin(U(p+1,q−1)c)<λ1=λmin(U(p,q)c).

If , by(2.6), we have

 f(−3;p,q)=241−19p+23q−21pq=241−19(p−q)+(4−21p)q<0,

which implies . Observe that when . In particular, , which implies

 λmin(U(p−1,q+1)c)<λmin(U(p,q)c).

To complete the proof, we need to prove when . In this case, , , and

 f(λ;p,q)−f(λ;p−1,q+1)=λ(2+λ)(−1+2λ)(4+λ).

So it is enough to prove or

 f(−4;n2,n2−2)=2376+582n−36n2<0.

If the above inequality holds, and hence the result follows.

## 3 Main results

By rearranging the edges of graphs, we first give an maximization of the quadratic form among all trees or all unicyclic graphs of order , where is a non-negative or non-positive real vector defined on .

###### Lemma 3.1

Let be a tree of order , and let be a non-negative or non-positive real vector defined on whose entries are arranged as with respect to their moduli. Then

 ∑uv∈E(T)XuXv≤n∑i=2X1Xi=∑uv∈E(K1,n−1)XuXv,

where is defined on such that the center has value . If, in addition, is positive or negative, and , then the above equality holds only if .

Proof: We may assume is non-negative; otherwise we consider . Let be a vertex with value given by . If there exists a vertex not adjacent of , letting be the neighbor of on a path of connecting and , and deleting the edge and adding a new edge , we will arrive at a new graph (tree) , which holds

 ∑uv∈E(T)XuXv≤∑uv∈E(T′)XuXv.

Repeating the process on the tree for the non-neighbors of , and so on, we at last arrive at a star with as its center, and

 ∑uv∈E(T)XuXv≤∑uv∈E(K1,n−1)XuXv=n∑i=2X1Xi.

If is positive, , and is not adjacent to all other vertices in , then the inequality (3.1), and hence (3.2), cannot hold as an equality. The result follows.

###### Lemma 3.2

Let be a unicyclic graph of order , and let be a non-negative or non-positive real vector defined on whose entries are arranged as with respect to their moduli. Then

 ∑uv∈E(U)XuXv≤n∑i=2X1Xi+X2X3=∑uv∈E(S3n)XuXv,

where is defined on such that the vertex with degree has value , and the other two vertices on the triangle have values respectively. If, in addition, is positive or negative, and , then the above equality holds only if .

Proof: We may assume is non-negative; otherwise we consider . Let be a vertex with value given by . By a similar discuss to the proof in Lemma 3.1, we have a graph of order , for which the vertex is adjacent to all other vertices, and

 ∑uv∈E(U)XuXv≤∑uv∈E(U′)XuXv=n∑i=2X1Xi+Xu′Xv′,

where is an edge of not incident to . Surely,

 Xu′Xv′≤X2X3.

So,

 ∑uv∈E(U)XuXv≤n∑i=2X1Xi+X2X3=∑uv∈E(S3n)XuXv.

If is positive, and , then the equality (3.5) holds only if (3.3) holds, which implies is adjacent to all other vertices and consequently . The result follows.

Next we give a sign property of the first eigenvector of graphs in for . Note that the first eigenvector of a connected graph with order at least contains both positive entries and negative entries.

###### Lemma 3.3

Let be a unicyclic graph of order , and let be a first eigenvector of . Then contains at least two positive entries and two negative entries.

Proof: Denote by the unique cycle in of length , and write simply. When , we know is connected or , the latter of which consists of an isolated vertex and a connected non-complete subgraph of order . So contains at least one positive entry and one negative entry.

Assume to the contrary that contains exactly one positive entry corresponding the vertex . For an arbitrary vertex , considering the eigenequation (2.2) on for the graph , we have

 0≤λmin(Uc)Xu=∑v∈NUc(u)Xv≤0.

So

 Xu=0,Xv=0,~{}for each~{}u∈NU(w),v∈NUc(u).

If a vertex , then , and by (3.6),

 λXr=∑v∈NUc(r)Xv=∑v∈NU(u)∖(NU(r)∪{r})Xv.

Similarly, if the vertex , then

 λXr=∑v∈NUc(r)Xv=∑v∈NU(u)∖NU(r)Xv.

We divide the discussion into two cases.

Case 1: . As is a unicycle graph of order , there exists a vertex such that . Let . Note that as so that , and consequently for by (3.6). By (3.7) we have

 λXw′=∑v∈NU(u)∖(NU(w′)∪{w′})Xv=∑v∈NU(u)∖{w′}Xv.

By (3.6) and (3.7), we also have

 λXw=∑v∈NU(u)∖(NU(w)∪{w})Xv=∑v∈NU(u)∖{w}Xv.

So

 (1+λ)Xw=∑v∈NU(u)Xv=(1+λ)Xw′,

which implies as , a contradiction.

Case 2: . We assert that has exactly one neighbor in . Otherwise, let be two vertices adjacent to in . As , . Applying (3.6) on the vertex , we have for each , and hence for each . Also by (3.6) and for each . So contains exactly one nonzero entry, i.e. ; a contradiction.

Let be the unique neighbor of in the graph . If there exists a neighbor of in other than , say , not lying on the cycle, then , and by (3.7),

 λXw′=∑v∈NU(u)∖{w′}Xv.

Combining this equality with (3.9), we have , a contradiction. So the vertex lies on the cycle and has exactly two neighbors except .

Now let be the neighbors of on the cycle of . By (3.6) and (3.7), we have

 λXw′=∑v∈NU(u)∖{w′,w′′}Xv=λXw′′,

which implies . As the graph has more than vertices, one of , say , has a neighbor not on the cycle which is also not adjacent to . By (3.6), , and by (3.8),

 0=λX¯w′=∑v∈NU(u)∖{w′}Xv.

Combining (3.10) with (3.11), we have , and hence as . So, we have for each . Also by (3.6) and for each . Therefore contains exactly one nonzero entry, i.e. ; a contradiction.

By above discussion, we get contains at least two positive entries. Similarly if considering , we also get contains at least two negative entries. The result follows.

We now arrive at the main result of this paper.

###### Theorem 3.4

Let be a unicyclic graph of order . Then

 λmin(Uc)≥λmin(U(⌈(n−2)/2⌉,⌊(n−2)/2⌋)c),

with equality if and only if .

Proof: Let be the first eigenvector of with unit length. Denote , , both being nonempty and containing at least elements by Lemma 3.3. Denote by (respectively, ) the subgraph of induced by (respectively, ), the set of edges between and in . Since is connected, . Obviously,

 ∑vv′∈E(U)XvXv′=∑vv′∈E(U+)XvXv′+∑vv′∈E(U−)XvXv′+∑vv′∈E′XvXv′.

First assume . The cycle of may contain the edges of , or is contained in one of . Without loss of generality, we assume that the cycle of is not contained in ; otherwise we consider the vector instead. Let be a graph obtained from by possibly adding some edges within and , such that the subgraph of induced by , denoted by , is a tree, and the subgraph of induced by , denoted by , is a unicyclic graph.

In the tree , choose a vertex, say , with maximum modulus among all vertices of . By Lemma 3.1, we will have a star, say centered at , where , which holds

 ∑vv′∈E(U+)XvXv′≤∑vv′∈E(U∗+)XvXv′≤∑vv′∈E(K1,p)XvXv′.

In the unicyclic graph , choosing a vertex, say , with maximum modulus. By Lemma 3.2, we have a unicyclic graph , where and the vertex joins all other vertices of , which holds

 ∑vv′∈E(U−)XvXv′≤∑vv′∈E(U∗−)XvXv′≤∑vv′∈E(S3q+1)XvXv′.

Let be the vertices of with minimum modulus among all vertices of , respectively. Then

 ∑vv′∈E′XvXv′≤Xu′Xw′,

Now by (3.12-3.15), we have

 ∑vv′∈E(U)XvXv′≤∑vv′∈E(K1,p)XvXv′+∑vv′∈E(S3q+1)XvXv′+Xu′Xw′

Since , the vertex can be chosen within the pendent vertices of by Lemma 3.1. If , can be chosen within the pendent vertices of by Lemma 3.2, then from (3.16) we have

 12XTA(U)X=∑vv′∈E(U)XvXv′≤∑vv′∈E(U(p,q))XvXv′=12XTA(U(p,q))X.

and consequently

 λmin(Uc)=XTA(Uc)X=XT(J−I)X−XTA(U)X≥XT(J−I)X−XTA(U(p,q))X=XTA(U(p,q)c)X≥λmin(U(p,q)c).

If , that is, , by a similar discussion, we have . By Lemma 2.1, .

Next we consider the case when . In this case the cycle of cannot lies in . We form a graph from possibly by adding some edges within and , such that the subgraph of induced by is a unicyclic graph, and the subgraph of induced by is exactly . Also similar to the discussion for (3.13-3.18), we have . By Lemma 2.2 and the above discussion,

 λmin(Uc)≥λmin(U(p,q)c)≥λmin(U(⌈(n−2)/2⌉,⌊(n−2)/2⌋)c).

Finally we prove the necessity for the equality in the theorem holding. Assume the equalities in (3.19) hold. Then , and consequently only the case of occurs. Note that is also a first eigenvector of . Let have some vertices labeled as in Fig. 2.1, where .

Assertion 1: contains no zero entries; the vertices and are respectively the unique ones in with maximum and minimum modulus, and are respectively the unique ones in with maximum and minimum modulus. As is a first eigenvector of , from the assumption at the beginning, for and for . By (2.5),

 λ1(X4−X7)=−X3−X4<0, λ1(X6−X7)=−2X6, λ1(X5−X6)=−X4−(q−3)X7,

which implies that . If , deleting the edges , and adding two new edges , we get a graph with same quadratic as both associated with , which implies , a contradiction by Lemma 2.2. So . Also by (2.5),

 λ1(X1−X2)>0, λ1(X3−X1)=X1−X3−X4>X1−X3,

which implies .

Assertion 2:


text:
# American Institute of Mathematical Sciences

September  2012, 17(6): 1775-1794. doi: 10.3934/dcdsb.2012.17.1775

## Slow manifold reduction of a stochastic chemical reaction: Exploring Keizer's paradox

 1 Department of Mathematics, University of Utah, Salt Lake City, UT 84112, United States, United States

Received  May 2011 Revised  November 2011 Published  May 2012

Keizer's paradox refers to the observation that deterministic and stochastic descriptions of chemical reactions can predict vastly different long term outcomes. In this paper, we use slow manifold analysis to help resolve this paradox for four variants of a simple autocatalytic reaction. We also provide rigorous estimates of the spectral gap of important linear operators, which establishes parameter ranges in which the slow manifold analysis is appropriate.
Citation: Parker Childs, James P. Keener. Slow manifold reduction of a stochastic chemical reaction: Exploring Keizer's paradox. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1775-1794. doi: 10.3934/dcdsb.2012.17.1775
##### References:
 [1] L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology,'', Pearson Education, (2003).   Google Scholar [2] J. Beischke, P. Weber, N. Sarafoff, M. Beekes, A. Giese and H. Kretzschmar, Autocatalytic self-propagation of misfolded prion protein,, Proc. Natl. Acad. Sci., 101 (2004), 12207.  doi: 10.1073/pnas.0404650101.  Google Scholar [3] C. W. Gardiner, "Handbook of Stochastic Methods. For Physics, Chemistry and the Natural Sciences,'' Second edition,, Springer Series in Synergetics, 13 (1985).   Google Scholar [4] B. L. Granovsky and A. I. Zeifman, The decay function of nonhomogeneous birth-death processes, with application to mean-field models,, Stoch. Process. Appl., 72 (1997), 105.  doi: 10.1016/S0304-4149(97)00085-9.  Google Scholar [5] B. L. Granovsky and A. I. Zeifman, The $N$-limit of spectral gap of a class of birth-death Markov chains,, Appl. Stoch. Models Bus. Ind., 16 (2000), 235.  doi: 10.1002/1526-4025(200010/12)16:4<235::AID-ASMB415>3.3.CO;2-J.  Google Scholar [6] J. Keizer, "Statistical Thermodynamics of Nonequilibrium Processes,'', Springer-Verlag, (1987).   Google Scholar [7] T. G. Kurtz, Limit theorems for sequences of jump Markov processes approximating ordinary differential processes,, J. Appl. Prob., 8 (1971), 344.   Google Scholar [8] T. G. Kurtz, The relationship between stochastic and deterministic models for chemical reactions,, J. Chem. Phys., 57 (1972), 2976.  doi: 10.1063/1.1678692.  Google Scholar [9] I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model,, Journal of Theoretical Biology, 211 (2001), 11.  doi: 10.1006/jtbi.2001.2328.  Google Scholar [10] H. Qian and L. M. Bishop, The chemical master equation approach to nonequilibrium steady-state of open biochemical systems: Linear single-molecule enzyme kinetics and nonlinear biochemical reaction networks,, Int. J. Mol. Sci., 11 (2010), 3472.  doi: 10.3390/ijms11093472.  Google Scholar [11] N. van Kampen, "Stochastic Processes in Physics and Chemistry,'', Lecture Notes in Mathematics, 888 (1981).   Google Scholar [12] M. Vellela and H. Qian, A quasistationary analysis of a stochastic chemical reaction: Keizer's paradox,, Bull. Math. Biol., 69 (2007), 1727.  doi: 10.1007/s11538-006-9188-3.  Google Scholar

show all references

##### References:
 [1] L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology,'', Pearson Education, (2003).   Google Scholar [2] J. Beischke, P. Weber, N. Sarafoff, M. Beekes, A. Giese and H. Kretzschmar, Autocatalytic self-propagation of misfolded prion protein,, Proc. Natl. Acad. Sci., 101 (2004), 12207.  doi: 10.1073/pnas.0404650101.  Google Scholar [3] C. W. Gardiner, "Handbook of Stochastic Methods. For Physics, Chemistry and the Natural Sciences,'' Second edition,, Springer Series in Synergetics, 13 (1985).   Google Scholar [4] B. L. Granovsky and A. I. Zeifman, The decay function of nonhomogeneous birth-death processes, with application to mean-field models,, Stoch. Process. Appl., 72 (1997), 105.  doi: 10.1016/S0304-4149(97)00085-9.  Google Scholar [5] B. L. Granovsky and A. I. Zeifman, The $N$-limit of spectral gap of a class of birth-death Markov chains,, Appl. Stoch. Models Bus. Ind., 16 (2000), 235.  doi: 10.1002/1526-4025(200010/12)16:4<235::AID-ASMB415>3.3.CO;2-J.  Google Scholar [6] J. Keizer, "Statistical Thermodynamics of Nonequilibrium Processes,'', Springer-Verlag, (1987).   Google Scholar [7] T. G. Kurtz, Limit theorems for sequences of jump Markov processes approximating ordinary differential processes,, J. Appl. Prob., 8 (1971), 344.   Google Scholar [8] T. G. Kurtz, The relationship between stochastic and deterministic models for chemical reactions,, J. Chem. Phys., 57 (1972), 2976.  doi: 10.1063/1.1678692.  Google Scholar [9] I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model,, Journal of Theoretical Biology, 211 (2001), 11.  doi: 10.1006/jtbi.2001.2328.  Google Scholar [10] H. Qian and L. M. Bishop, The chemical master equation approach to nonequilibrium steady-state of open biochemical systems: Linear single-molecule enzyme kinetics and nonlinear biochemical reaction networks,, Int. J. Mol. Sci., 11 (2010), 3472.  doi: 10.3390/ijms11093472.  Google Scholar [11] N. van Kampen, "Stochastic Processes in Physics and Chemistry,'', Lecture Notes in Mathematics, 888 (1981).   Google Scholar [12] M. Vellela and H. Qian, A quasistationary analysis of a stochastic chemical reaction: Keizer's paradox,, Bull. Math. Biol., 69 (2007), 1727.  doi: 10.1007/s11538-006-9188-3.  Google Scholar
 [1] Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051 [2] Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323 [3] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [4] Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 [5] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047 [6] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320 [7] Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080 [8] Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383 [9] Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242 [10] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468 [11] Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349 [12] Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451 [13] H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433 [14] Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458 [15] Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460 [16] Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276 [17] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [18] Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054 [19] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [20] Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

2019 Impact Factor: 1.27

## Metrics

• HTML views (0)
• Cited by (3)

• on AIMS


text:
# All Questions

95 views

### Remove Indentation of Theorems

I am working with amstheorem and amsbook, and I'm trying to remove the indentation before the numbers of theorems so that they are aligned exactly to the left. Is there any way to do this? In ...
32 views

### Wrong A4 measurements that are not “pagesize” solvable

I would like my pdf document to have an a4 format. \documentclass[a4paper,12pt,oneside]{article} \begin{document} Text, text, text... \end{document} Unfortunately, when I print the pdf document ...
42 views

### Stopped compiling [on hold]

I have a problem with compiling in latex. I was trying to write a matrix, when latex suddenly did not want to compile it anymore. I deleted everything new, but it still did not work. First it had some ...
21 views

### Restricting linespread outside some environments

I need to fix a precise linespread in a document I'm writing, but only in the plain sections of the document. In other words, I'm writing math and in the text there are a lot of environments (theorem, ...
20 views

### Algorithm pseudo code not published beyond one page [duplicate]

My algorithm pseudo code is not being published beyond one page though there is no syntax error. The PDF produced via LaTeX is of only one page but my entire algorithm requires more than one page. So ...
45 views

### Comparison of the plots of the same line on the Cartesian plane using pgfplots and TikZ

I have the plot of a line using pgfplots and the plot of the same line using TikZ. I would like them to be identical. I need to correct a mistake in the plot of a line using pgfplots. The point ...
72 views

### Redefining the item command of the description environment

My thesis contains many description environments. After showing it to my supervisor, he complained about the missing separation between the items titles and descriptions. I tried to redefine the \item ...
61 views

### Doing something like psframebox in tikz

I used the following code to typeset the figure below with pstricks. Is there something like this that works with tikz? I tried to do this and discovered \nodes that can be filled with text, but I ...
45 views

### List of theorem dots overlapping with theorem name

Here is my MWE: \documentclass[a4paper]{report} \usepackage{amsmath,amsfonts,amssymb,amsthm,thmtools,tikz,xparse,lipsum} \usepackage{newtxtext,newtxmath} \usepackage{hyperref} \makeatletter ...
54 views

### Clean code in drawing with boxes and circles and lines not connected

I want to optimize the following code: \documentclass{article} %\pagestyle{empty} % Figure dimensions \newcommand{\defLength}[2]{\newlength{#1}\setlength{#1}{#2}} ...
37 views

### File path at bottom of last page

My boss insists that every document I give him must have the file path to that document at the bottom of the last page. No problem with MS Word, but I don't have a clue how to do this in Latex. At the ...
46 views

### Why am I getting an error when using \renewcommand, here?

I try to type \renewcommand{\mr}{\mathrm}, so that I can then use \mr in my text instead of \mathrm. I get the error \mr Undefined. what is going on?
26 views

### Floatrow package is affected in figure environment

I am using floatrow package creating the box environment. if i declare the floatrow package in the tex file the normal figure framebox is affected. MWE \documentclass[twocolumn]{article} ...
67 views

### Zooming on one of multiple images in a slide using Tikzpicture

I would like to zoom in on one of the 12 images that I have organized in 3 columns. There are 4 images per column, all the same height and width. There is one image that I would like to zoom in on ...
35 views

### Pagebreaking in longtable with math column

I'm trying to write some step-by-step math proof of a formula in my electronics notes. The idea is to have two columns (side by side), the 1st has some text explaining the math, the 2nd one has the ...
25 views

### LyX and biblatex authoryear-comp and same authors: Parentheses wrong

I use LyX with biblatex 2.9a, biber 1.9 and the authoryear-comp style. My biblatex configuration is below: \usepackage[backend=biber,style=authoryear-comp,natbib=true]{biblatex} When I insert two ...
24 views

### dblfloatfix in a one-column document

I have been told to always load dblfloatfix in two-column documents. I therefore include it in my standard preamble, which uses one column by default. It did not appear to affect my documents when ...
72 views

### To have this Latvian letter in Beamer

This discussion is an extension of this answer but in Beamer -documentclass and with XeLaTeX. I can get some variants working there in Pdflatex, see Christian's answer but with bad output and I do ...
32 views

### Modify font for all text in tikzpicture

16 views

### Edit and view BibTex database on Android

I found many applications which claim to a reference manager for Android. Some of which are: RefMaster, Library, Erasthotenes. Which one do you use which works with OwnCloud, uses the BibTex ...
111 views

### Typing of math sign charts

For typing of quantities of solving mathematical problems I would like to type signs charts (diagrams to visualize the values of expressions). Something like this: x-5 - | - | + x+4 ...
20 views

### Package pgf warning: gnuplots

I'm using the following code in a Tex file to draw a figure. When I compile the file using pdflatex --shell-escape filename.tex, the following error is logged in the log file and the figures are not ...
25 views

### Glossaries with 3 columns gives error with missing item

I try to create a glossary with 3 columns, but continuously I get the following error: Something's wrong--perhaps a missing \item. [...etentrycounter[]{page}\glsnumberformat{1}}}] Here is my ...
52 views

### (New) Century Schoolbook Font in LuaLaTeX

I recently made the switch to LuaTeX from pdfTeX. Before the switch, I used the fouriernc package to set the document's font to New Century Schoolbook. I've come to understand (from this document) ...
24 views

### How to trigger the marginal note in two column layout

I am using marginal note package in two column layout. The below mentioned command does not trigger the two column left and right side. \ifodd\value{page} ...
44 views

### Conditionally change node background color inside foreach

I generate 8 rectangle nodes using a \foreach loop but I would like one of them (the one at the position index) to have a gray background. How can I do that ? I would like to avoid repeating the ...
62 views

25 views

### Single Column Appendix in twocolumn article

My article uses the twocolumn parameter of the article document class. However I want to list my images as a multipage appendix at the end of the document (in a new page) but I want a single column ...
33 views

### How can I readjust TikZ's dimension variables to fit mid-document dimension change?

Question How can I readjust TikZ's dimension variables to fit a mid-document dimension change as in this example and using the TikZ drawing from here. Problem TikZ coordinates are not where they ...
27 views

### How can I find Asymptote code for producing images [on hold]

I need to code in Asympotote the following figures
92 views

### Update Biber 1.8 to 1.9

I am using MiKTeX 2.9 (32bit) and TeXstudio 2.8.8 on Windows 8.1. When compiling, I get an error message that I am using Biber 1.8 (which I installed via the Package Manager) but should use Biber 1.9. ...
28 views

### IEEE format caption alignment [on hold]

When I put a caption on the figures, the alignment changes to Left ! why this happens and how to fix it ? I recently removed the caption package to cope with change that occures in caption font size ...
41 views

### Problem in MikTex while I'm installing mh package

I'm trying to install mh package with MikTex Package Manager in windows. When I try to install it, i find thid error: Unknown archive file size. What is the problem? I've installed also other ...
61 views

### Labeling in Mathematica to match the font size same as in LaTeX

I have some plot Plot[a, {a, -6, 6}] I need to use it in Latex. However the problem is that labeling text style in Mathematica does not match with LaTeX text. So what I do, I copy the figure ...
33 views

### How to avoid (wrap) figures moving out of sight

Using the code below, my figure moves out of the paper borders. As I understand, correction of figure position is best to do when writting is finished. But is it possible to forces figures - a ...
24 views

### Latex template for statement of purpose

I have used ModernCV template to write resumes, Beamer to prepare presentations. Likewise, is there any standard Latex template for writing Statement of Purpose (SOP) or Research Proposal?
24 views

### How to make the format of references look like this?

[1] Brown, P. & S. Levinson. Politenss: Some Universal in Language Usage[M]. Cambridge: Cambridge University Press. 1987. [2] Freeborn, D. From Old English to Standard English[M]. London: ...
61 views

### Creating a smooth gradient between two colors on an edge in TikZ

I'd like to have a two-colored edge between two nodes of different colors. I'm hoping to make the edge fade from one color into the other in an even way. The solution I've found so far is a bit ...
34 views

### LaTeX Font Warning: Font shape `EU1/ptm/m/n' undefined [on hold]

How to solve this warning? and what does this mean? I want to paste the tex file but it's too long to read. I try to minimize the file and reproduce the problem at last it failed. It is beyond ...
30 views

### Text alignment and form of connectors (edges) in tikz tree

I'm completely new to tikz and try to draw a tree structure. I have tried different options, but could not modify it as I wanted it to look like. It currently looks like that: Tikz code ...
30 views

### Tabularx table with many columns malformatted

In our software we have a module, that generates PDFs via Latex. Some of them contain tables and I want them to span the whole page. Since the number of columns varies, I use tabularx with all X ...
38 views

### Dash sign in file .bib is double when compile it to pdf file

Why bibtext file: ... pages = "419-444", ... generates 419--444 for me in pdf?
578 views

### Creating a rainbow color macro

I'm trying to draw a picture in tikz whose code is generated by another computer program. The program outputs hundreds of numbers in between 0 and 100, and I'd like to color objects in the picture on ...
18 views

### Custom style in LyX

I am having trouble editing this template in LyX. After downloading the .tex and .sty files I can compile with pdflatex nips2013.tex which works fine. In LyX, I click File > Import > LaTeX ...
65 views

### How to change the paper color(background color) in tikz class

How to change the paper color(background color) in tikz class kindly help me to change color \documentclass{scrartcl} \usepackage[usenames,x11names,dvipsnames,svgnames]{xcolor} ...
74 views

### Making bibliography \flushbottom in book class

I am using the book class and have a long bibliography (using Biblatex) at the end of each chapter. When there is a long reference around where a page break occurs, the whole reference shifts to the ...
42 views

### How to resolve this LaTeX error? [on hold]

I use Sublime Test 2 for OSX 10.8.5 to compile the tex file.But I am getting the LaTeX error: "/usr/local/texlive/texmf-local/tex/latex/ctex/cfg/ctexopts.cfg:1: LaTeX Error: Missing \begin{document}. ...
37 views

### Style file is altering my citation key

I need to have a citation inside a chapter title like such: \chapter{blah blah \cite{key}} The style file that I have for following university formatting guides (found here) capitalizes titles and ...


text:
Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.

 It is currently 20 Jul 2019, 15:15

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# What is the sum of the first 100 positive odd numbers?

Author Message
TAGS:

### Hide Tags

Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 7612
GMAT 1: 760 Q51 V42
GPA: 3.82
What is the sum of the first 100 positive odd numbers?  [#permalink]

### Show Tags

25 Nov 2016, 02:26
00:00

Difficulty:

25% (medium)

Question Stats:

71% (01:20) correct 29% (01:36) wrong based on 195 sessions

### HideShow timer Statistics

What is the sum of the first 100 positive odd numbers?

A. 5,000 B. 7,500 C. 8,000 D. 10,000 E. 12,000

_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Intern Joined: 25 Oct 2016 Posts: 3 What is the sum of the first 100 positive odd numbers? [#permalink] ### Show Tags 25 Nov 2016, 02:48 3 This one is about pattern recognition. Set up a table as follows: Number Sum 1 1 3 4 5 9 7 16 9 25 Now as you see, the sum always equals the perfect square of the number of odd integers that you want to sum. So the sum of the first 5 odd integers is 5^2 = 25. The pattern continues. Thus, the sum of the first 100 odd integers will be equal to 100^2 = 10,000. The correct answer is (D). Current Student Joined: 28 Nov 2014 Posts: 843 Concentration: Strategy Schools: Fisher '19 (M$)
GPA: 3.71
Re: What is the sum of the first 100 positive odd numbers?  [#permalink]

### Show Tags

25 Nov 2016, 03:44
4
For first n odd numbers, its n^2 and for first n even numbers, its n (n+1)

Here, it'll be (100)^2 = 10000

Sent from my GT-N7100 using GMAT Club Forum mobile app
CEO
Joined: 12 Sep 2015
Posts: 3853
Re: What is the sum of the first 100 positive odd numbers?  [#permalink]

### Show Tags

25 Nov 2016, 06:01
1
Top Contributor
MathRevolution wrote:
What is the sum of the first 100 positive odd numbers?

A. 5,000 B. 7,500 C. 8,000 D. 10,000 E. 12,000

Before answering any GMAT quant problem, always check the answer choices first
Here, the answer choices are somewhat spread apart, which means we can be somewhat aggressive in our estimations.

Nice formula: 1 + 2 + 3 . . . + n = (n)(n + 1)/2

Let's use the above formula to find the sum of the first 200 integers (including odds AND evens)
1 + 2 + 3 . . . + 199 + 200 = (200)(200 + 1)/2
= (200)(201)/2
= (100)(201)
= 20,100
So, the sum of the first 200 integers is 20,100
HALF of those integers are ODD and HALF are even. So, this sum includes the sum of the first 100 ODD integers and the sum of the first 100 EVEN integers.
So, the sum of the first 100 ODD integers is APPROXIMATELY 20,100/2
20,100/2 = 10,050
So, the sum of the first 100 ODD integers ≈ 10,050
Answer choice D is the only one that's close to 10,050 so it must be the correct answer.

RELATED VIDEO

_________________
Test confidently with gmatprepnow.com
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 7612
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: What is the sum of the first 100 positive odd numbers?  [#permalink]

### Show Tags

28 Nov 2016, 01:41
==> Since 1+3+5+….+(2n-1)=n 2, 1+3+…….(2*100-1)=100 2 =10,000 is derived.

_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only \$79 for 1 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"
Intern
Joined: 15 Nov 2015
Posts: 25
Location: Netherlands
Schools: HEC Dec '17 (II)
GMAT 1: 730 Q47 V44
GPA: 3.99
Re: What is the sum of the first 100 positive odd numbers?  [#permalink]

### Show Tags

14 Dec 2016, 07:57
The sum of an arithmetic sequence equals (n(first term) + n(last term)/2)*n.

N1 = 1.
N 100 = N0 + (n-1)*2 = 1 + 198 = 199.

Now sum = ((1+199)/2)*100 = 100*100 = 10,000.
Senior Manager
Joined: 19 Apr 2016
Posts: 271
Location: India
GMAT 1: 570 Q48 V22
GMAT 2: 640 Q49 V28
GPA: 3.5
WE: Web Development (Computer Software)
Re: What is the sum of the first 100 positive odd numbers?  [#permalink]

### Show Tags

20 Jan 2017, 23:52
1
1
MathRevolution wrote:
What is the sum of the first 100 positive odd numbers?

A. 5,000 B. 7,500 C. 8,000 D. 10,000 E. 12,000

Sum of first n positive odd numbers = $$n^2$$

Therefore, Sum of first 100 positive odd numbers = $$100^2$$ = 10000;

Hence option D is correct.
Non-Human User
Joined: 09 Sep 2013
Posts: 11720
Re: What is the sum of the first 100 positive odd numbers?  [#permalink]

### Show Tags

19 Jul 2018, 21:56
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: What is the sum of the first 100 positive odd numbers?   [#permalink] 19 Jul 2018, 21:56
Display posts from previous: Sort by