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# Squares and Square Roots Class 8 Notes: Chapter 6

Squares and Square Roots Class 8 Notes available here is specially designed by math experts who have taken extra care to use simple language and style so that students can understand concepts easily and quickly. Chapter 6 notes are a great reference tool and will help students to quickly go through or revise all the important topics before the exams as well as score better marks.

## Perfect Squares

To know more about Perfect Square Formula, visit here.

### Properties of Square Numbers

Properties of square numbers are:

• If a number has 0, 1, 4, 5, 6 or 9 in the unit’s place, then it may or may not be a square number. If a number has 2, 3, 7 or 8 in its units place then it is not a square number.
• If a number has 1 or 9 in unit’s place, then it’s square ends in 1.
• If a square number ends in 6, the number whose square it is, will have either 4 or 6 in unit’s place.

To know more about Square Numbers, visit here.

### Finding square of a number with unit’s place 5

The square of a number N5 is equal to (N(N+1))×100+25, where N can have one or more than one digit.
For example: 152=(1×2)×100+25=200+25=225
2052=(20×21)×100+25=42000+25=42025

## Square Roots

### Square Root of a Number

Finding the number whose square is known is known as finding the square root. Finding square root is inverse operation of finding the square of a number.
For example:
12=1,  square root of 1 is 1.
22=4, square root of 4 is 2.
32=9, square root of 9 is 3.

To know more about Square Roots, visit here.

### Estimating the number of digits in the square root of a number

If a perfect square has n digits, then its square root will have $\frac{n}{2}$ digits if n is even and $\frac{(n+1)}{2}$ digits if n is odd.
For example: 100 has 3 digits, and its square root(10) has $\frac{(3+1)}{2}$ =2 digits.

## Estimating Square Roots

### Estimating the Square Root

Estimating the square root of 247:
Since: 100 < 247 < 400
i.e. 10<247<20
But it is not very close.
Also, 152=225<247 and 162=256>247
15<247<16.
256 is much closer to 247 than 225.
Therefore, 247 is approximately equal to 16.

## Introduction to Squares and Square Roots

### Introduction to Square Numbers

If a natural number m can be expressed as n2, where n is also a natural number, then m is a square number.
Example: 1, 4, 9, 16 and 25.

### Finding the Square of a Number

If n is a number, then its square is given as n×n=n2.
For example: Square of 5 is equal to  5×5=25

### Finding square of a number using identity

Squares of numbers having two or more digits can easily be found by writing the number as the sum of two numbers.
For example:
232=(20+3)=20(20+3)+3(20+3)
=202+20×3+20×3+32
=400+60+60+9
=529

To know more about Squares and Square Roots, visit here.

## Interesting Patterns

### Interesting Patterns

There exists interesting patterns in:

• Numbers between square numbers
• A sum of consecutive natural numbers
• Product of two consecutive even or odd natural numbers

Triangular numbers: It is a sequence of the numbers 1, 3, 6, 10, 15 etc. It is obtained by continued summation of the natural numbers. The dot pattern of a triangular number can be arranged as triangles.

If we add two consecutive triangular numbers, we get a square number.
Example: 1+3=4=22 and 3+6=9=32.
To know more about Number Patterns, visit here.

### Numbers between Square Numbers

There are 2n non-perfect square numbers between squares of the numbers n and (n + 1), where is any natural number.

Example:

• There are two non-perfect square numbers (2, 3) between12=1 and 22=4.
• There are four non-perfect square numbers (5, 6, 7, 8) between 22=4 and 32=9.

The sum of first n odd natural numbers is n2.
Example:
1+3=4=22
1+3+5=9=32

### Square of an odd number as a sum

Square of an odd number n can be expressed as sum of two consecutive positive integers $\frac{(n^{2}-1)}{2}$ and $\frac{(n^{2}+1)}{2}$.
For example: 32=9=4+5=$\frac{(3^{2}-1)}{2}$ + $\frac{(3^{2}+1)}{2}$ Similarly, 52=25=12+13=$\frac{(5^{2}-1)}{2}$ $\frac{(5^{2}+1)}{2}$

### Product of Two Consecutive Even or Odd Natural Numbers

The product of two even or odd natural number can be calculated as, (a+1)×(a1)=(a21), where a is a natural number, and a1, a+1, are the consecutive odd or even numbers.
For example:
11×13=(121)×(12+1)=1221=1441=143

### Random Interesting Patterns Followed by Square Numbers

Patterns in numbers like 1, 11, 111, … :
12=                        1
112=                   1 2 1
1112=               1 2 3 2 1
11112=          1 2 3 4 3 2 1
111112=     1 2 3 4 5 4 3 2 1
111111112=1 2 3 4 5 6 7 8 7 6 5 4 3 2 1
Patterns in numbers like 6, 67, 667, … :
72=49
672=4489
6672=444889
66672=44448889
666672=4444488889
6666672=444444888889

## Pythagorean Triplets

### Pythagorean Triplets

For any natural number m>1, we have (2m)2+(m21)2=(m2+1)2.
2m, (m21) and (m2+1) forms a Pythagorean triplet.
For m=2, 2m=4, m21=3 and m2+1=5.
So, 3, 4, 5 is the required Pythagorean triplet.

## Square Roots of Perfect Squares

### Finding square root through repeated subtraction

Every square number can be expressed as a sum of successive odd natural numbers starting from one.

The square root can be found through repeated subtraction. To find the square root of a number n:
Step 1: subtract successive odd numbers starting from one.
Step 2: stop when you get zero.

The number of successive odd numbers that are subtracted gives the square root of that number. Suppose we want to find the square root of 36.

• 361=35
• 353=32
• 325=27
• 277=20
• 209=11
• 1111=0

Here 6 odd numbers (1, 3, 5, 7, 9, 11) are subtracted to from 36 to get 0.
So, the square root of 36 is 6.

## Square Roots – Generalised

### Finding Square Root by Long Division Method

Steps involved in finding the square root of 484 by Long division method:
Step 1: Place a bar over every pair of numbers starting from the digit at units place. If the number of digits in it is odd, then the left-most single-digit too will have a bar.
Step 2: Take the largest number as divisor whose square is less than or equal to the number on the extreme left. Divide and write quotient.

Step 3: Bring down the number which is under the next bar to the right side of the remainder.

Step 4: Double the value of the quotient and enter it with a blank on the right side.

Step 5: Guess the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new
quotient the product is less than or equal to the dividend.

The remainder is 0, therefore, 484=22.
To know more about Finding the Square Root of a Number, visit here.

## Be More Curious!

### Finding Pythagorean Triplets for Any Given Number

If we are given any member of a Pythagorean triplet, then we can find the Pythagorean triplet by using general form 2m, m21, m2+1.
For example, If we want to find the Pythagorean triplet whose smallest number is 8.
Let, m21=8m=3
2m=6 and m2+1=10
The triplet is 6, 8 and 10.
But 8 is not the smallest number of this triplet.
So, we substitute 2m=8m=4
m21=15 and m2+1=17.
Therefore, 8, 15, 17 is the required triplet.
To know more about Pythagorean Triplets, visit here.


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 Chebyshevskii Sb.: Year: Volume: Issue: Page: Find

 Chebyshevskii Sb., 2015, Volume 16, Issue 3, Pages 246–275 (Mi cheb417)

Binary additive problem with numbers of special type

A. A. Zhukovaa, A. V. Shutovb

a Russian Academy of National Economy and Public Administration under the President of the Russian Federation (Vladimir Branch)

Abstract: In this paper we consider binary additive problem of the form $n_1 + n_2 = N$ with $n_1 \in \mathbb {N} (\alpha, I_1)$, $N_2 \in \mathbb{N} (\beta, I_2)$, where $\mathbb{N} (\alpha, I) = \{n \in \mathbb{N}: \{n \alpha \} \in I \}$. Main examples of such sets are the sets of natural numbers with specified ending of greedy expansion of the number by linear recurrence sequences associated with Pisot numbers. Besides that, the sets $\mathbb{N} (\alpha, I)$ are special cases of quasilattices. Previously additive problems on the sets of this type are considered only for the case $\alpha = \beta$. In this case was obtained asymptotic formulaes for the number of solutions of the additive problem with an arbitrary number of terms, and for number of solutions in analogues of ternary Goldbach problem, Hua-Loken problem, Waring problems, and Lagrange problem about the representation number of natural numbers as a sum of four squares. Wherein, Gritsenko and Motkina discovered that in the case of linear problems we have the following nontrivial effect: apprearence of a rather complicated function in the main term of the asymptotics for the number of solutions. For nonlinear problems corrsponding effect is missing and the form of the main term can be obtained by the density considerations.
In our problem, we show that the behavior of the main term of the asymptotic formula for the number of solutions significantly depends on the arithmetic of $\alpha$ and $\beta$. If $1$, $\alpha$ and $\beta$ are linearly independent over the ring of integers $\mathbb{Z}$, then the main term of the asymptotic has the "density" form, i.e. it is equal to $| I_1 || I_2 | N$. In the case of linear dependence of $1$, $\alpha$ and $\beta$ we have the Gritsenko-Motkina effect, i.e. the main term is $\rho (\{N \beta \}) N$, where $\rho$ is a rather complicated efficiently computable piecewise linear function of the fractional part $\{N \beta \}$. we obtain an algorithm for computation of the function $\rho$, and study basic properties of this function. In particular, we obtain sufficient conditions for its non-vanishing. Also we give a numerical example of the computation of this function for some concrete sets $\mathbb{N} (\alpha, I_1)$, $\mathbb {N} (\beta, I_2)$. In the final part of the paper we discuss some open problems in this area.
Bibliography: 23 titles.

Full text: PDF file (3377 kB)
References: PDF file   HTML file
UDC: 511.34

Citation: A. A. Zhukova, A. V. Shutov, “Binary additive problem with numbers of special type”, Chebyshevskii Sb., 16:3 (2015), 246–275

Citation in format AMSBIB
\Bibitem{ZhuShu15} \by A.~A.~Zhukova, A.~V.~Shutov \paper Binary additive problem with numbers of special type \jour Chebyshevskii Sb. \yr 2015 \vol 16 \issue 3 \pages 246--275 \mathnet{http://mi.mathnet.ru/cheb417} \elib{http://elibrary.ru/item.asp?id=24398936}

format: plain


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# Some Thoughts on a Real-Valued Function

## Problem

Spending so much time to work out the locale support for dates and valid HTML code in my second blog which was powered by Jekyll-Bootstrap, I couldn’t recall this fact.

Suppose that $\mathcal{D}$ is a subset of $\R^2$ that contains an $\varepsilon$-neighbourhood of a point $(x_0,y_0)$. If

1. $f: \mathcal{D} \to \R$ is has first-order partial derivatives in the $\varepsilon$-neighbourhood of $(x_0,y_0)$.
2. The first-order partial derivatives of $f$ are continuous at $(x_0,y_0)$.

Then we can write

$$\Delta f = \frac{\partial f}{\partial x} (x_0,y_0) \Delta x + \frac{\partial f}{\partial y} (x_0,y_0) \Delta y + \varepsilon_1 \Delta x + \varepsilon_2 \Delta y, \label{fact}$$

where $\Delta f := f(x_0+\Delta x,y_0+\Delta y)-f(x_0,y_0), \varepsilon_1$ and $\varepsilon_2 \to 0$ as $\Delta x$ and $\Delta y \to 0$.

## The First-Order Approximation For a Differentiable Function

At first, I thought that equation \eqref{fact} followed from formula \eqref{thm}.

$$\lim_{\vect{h} \to \zeros} \frac{f(\vect{x}+\vect{h})-[f(\vect{x})+\langle\nabla f(\vect{x}),\vect{h}\rangle]}{\norm{\vect{h}}} = 0 \label{thm}$$

I forgot that this statement assumed the continuous differentiability on an open set. However, we only know that the first-order partial derivatives are continuous at one point $(x_0,y_0)$.

## Solve an easier problem first

This question should be much easier and much more intuitive if the domain of the function $f$ in \eqref{fact} is one-dimensional. Then, by drawing a curve and sketching its tangent line at a point, one can intuitively realise that the geometric meaning of $\varepsilon$.

In fact, one constructs

$$\varepsilon (\Delta x) := \begin{cases} \frac{\Delta f}{\Delta x}-f'(x_0) &\text{if }\Delta x \ne 0,\\ 0 &\text{if } \Delta x = 0, \end{cases} \label{def1}$$

where $\Delta f := f(x_0+\Delta x)-f(x_0)$ in this section since $f$ is now one-dimensional.

Then one can make use of the differentiability of $f$ at $(x_0)$ to say that $\varepsilon \to 0$ as $\Delta x \to 0$.

To get the one-dimensional version of \eqref{fact}, we get rid of the denominator by multiplying both sides by $\Delta x$ in the case of $\Delta x \ne 0$.

\begin{aligned} \varepsilon \Delta x &= \Delta f - f' (x_0) \Delta x\\ \Delta f &= f' (x_0) \Delta x + \varepsilon \Delta x \end{aligned} \label{mult1}

Observe that equality \eqref{mult1} also holds when $\Delta x = 0$.

## Back to the problem

### Make use of the previous section

From \eqref{def1}, we observe that $\varepsilon$ is defined as the difference between a difference quotient between two points $x_0$ and $x_0+\Delta x$ and the derivative $f'(x_0)$. We can define $\varepsilon_1$ and $\varepsilon_2$ in a similar way.

\begin{align} &\begin{split} & \varepsilon_1(\Delta x,\Delta y) :=\\ &\begin{cases} \frac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x}-\frac{\partial f}{\partial x}(x_0,y_0) &{\small \text{if } \Delta x \ne 0},\\ 0 &\text{if } \Delta x = 0, \end{cases} \end{split} \label{def2a}\\ &\begin{split} & \varepsilon_2(\Delta x,\Delta y) :=\\ &\begin{cases} \frac{f(x_0+\Delta x,y_0+\Delta y) - f(x_0+\Delta x,y_0)}{\Delta y} - \frac{\partial f}{\partial y}(x_0+\Delta x,y_0) &\text{if } \Delta y \ne 0,\\ 0 &\text{if } \Delta y = 0, \end{cases} \end{split} \label{def2b} \end{align}

We then multiply \eqref{def2a} and \eqref{def2b} by $\Delta x$ and $\Delta y$ respectively, just like what we’ve done in \eqref{mult1}.

\begin{align} &\begin{split} \varepsilon_1(\Delta x,\Delta y) \Delta x =& f(x_0+\Delta x,y_0) - f(x_0,y_0)\\ &- \frac{\partial f}{\partial x}(x_0,y_0) \Delta x \end{split} \label{mult2a}\\ &\begin{split} \varepsilon_2(\Delta x,\Delta y) \Delta y =& f(x_0+\Delta x,y_0+\Delta y) - f(x_0+\Delta x,y_0)\\ &- \frac{\partial f}{\partial y}(x_0+\Delta x,y_0) \Delta y \end{split} \label{mult2b} \end{align}

The trick is to change the independent variables one-by-one since we can only make use of partial derivatives. After defining \eqref{def2a}, there’s no need to scratch our head for \eqref{def2b} to fit the pizzle. We may first try to use \eqref{def2a} to get \eqref{mult2a}. Then by observing the term $\Delta f$ in \eqref{fact}, we realize that we should add the term $f(x_0+\Delta x,y_0+\Delta y)$ and remove $f(x_0+\Delta x,y_0)$ so as to get $\Delta f$ in \eqref{fact}.

### Remaining problem

In \eqref{mult2b}, the partial derivative with respect to $y$ is taken at $(x_0+\Delta x,y_0)$, which shouldn’t appear in \eqref{fact}. Therefore, we need a way to get rid of the $\Delta x$ inside the bracket. That leads us to the one unused condition—the second given condition in \eqref{fact}.

By the continuity of the first-order partial derivatives at $(x_0,y_0)$,

$$\lim_{\Delta x \to 0} \frac{\partial f}{\partial y}(x_0+\Delta x,y_0) = \frac{\partial f}{\partial y}(x_0,y_0)\\ \lim_{\Delta x \to 0} \left ( \frac{\partial f}{\partial y}(x_0+\Delta x,y_0) - \frac{\partial f}{\partial y}(x_0,y_0) \right ) = 0 \label{cts1}$$

Thus, we define

$$\varepsilon_3(\Delta x) := \frac{\partial f}{\partial y}(x_0+\Delta x,y_0) - \frac{\partial f}{\partial y}(x_0,y_0) \label{cts2}$$

so that $\varepsilon_3 \to 0$ as $\Delta x \to 0$. With \eqref{cts2}, we can replace the first-order partial derivative with respect to $y$ at $(x_0+\Delta x,y_0)$ in \eqref{mult2b} by the one at $(x_0,y_0)$.

$$\begin{split} \varepsilon_2(\Delta x,\Delta y) \Delta y =& f(x_0+\Delta x,y_0+\Delta y) - f(x_0+\Delta x,y_0)\\ &- \left ( \frac{\partial f}{\partial y}(x_0,y_0) + \varepsilon_3(\Delta x) \right ) \Delta y \end{split} \label{b4repl}$$

Therefore, from \eqref{b4repl}, we see that it’s legitimate for us to rename $\varepsilon_2-\varepsilon_3$ as \varepsilon_2 to get an equation which looks more similar to \eqref{mult2a} than \eqref{mult2b} does.

\begin{gather} \begin{split} &\varepsilon_2(\Delta x,\Delta y) \Delta y\\ =& f(x_0+\Delta x,y_0+\Delta y) - f(x_0+\Delta x,y_0) - \frac{\partial f}{\partial y}(x_0,y_0) \Delta y \end{split} \label{mult2bb}\\ \begin{split} &\varepsilon_1(\Delta x,\Delta y) + \varepsilon_2(\Delta x,\Delta y)\\ =& f(x_0+\Delta x, y_0+\Delta y) - f(x_0,y_0)\\ &- \frac{\partial f}{\partial x}(x_0,y_0) \Delta x - \frac{\partial f}{\partial y}(x_0,y_0) \Delta y \quad \text{(\eqref{mult2a}+\eqref{mult2bb})} \end{split}\\ \begin{split} &f(x_0+\Delta x, y_0+\Delta y) - f(x_0,y_0)\\ =& \frac{\partial f}{\partial x}(x_0,y_0) \Delta x + \frac{\partial f}{\partial y}(x_0,y_0) \Delta y + \varepsilon_1(\Delta x,\Delta y) + \varepsilon_2(\Delta x,\Delta y) \end{split} \label{result} \end{gather}

The result \eqref{result} is what we desired in \eqref{fact}.

## Generalisation to $n$-dimension

By reusing the trick of changing the variables once at a time from \eqref{def2a} to \eqref{mult2b}, and a suitable renaming and rearrangment of terms, one can generalise the result in \eqref{fact} to a function $f:\mathcal{D} \to \R$ defined on a subset $\mathcal{D}$ of $\R^n$ containing a $\varepsilon$-neighbourhood of a point $\vect{x}_0 \in \R^n$.

$$\Delta f = \langle \nabla f(\vect{x_0}) + \vect{\varepsilon},\Delta \vect{x} \rangle, \label{generalisation}$$

where $\Delta f := f(\vect{x_0}+\Delta \vect{x})-f(\vect{x_0})$ and $\vect{\varepsilon} \to \zeros$ as $\Delta \vect{x} \to \zeros$.

As you can see in \eqref{generalisation}, writing the statement in its vector form is more concise than writing out each partial derivative in \eqref{fact}.


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# [SOLVED]Powers of polylogarithms

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
We should consider the general form

$$\displaystyle L^m_n (a) = \int^1_0 x^{a-1} \mathrm{Li}_{\, n}(x)^m \, dx$$

This is NOT a tutorial , any attempts or comments are always welcomed.

Last edited:

#### DreamWeaver

##### Well-known member
The case $$\displaystyle m=1$$ and $$\displaystyle a>0 \in \mathbb{R}$$ is fairly straightforward. Additionally, if we generalize your integral to

$$\displaystyle \text{Li}_{p,q}(m, n; a, z)=\int_0^zx^{a-1} (\text{Li}_p(x))^m (\text{Li}_q(x))^n\,dx$$

Then the special cases

$$\displaystyle \text{Li}_{p,q}(1, 1; 0, z)=\int_0^z \frac{ \text{Li}_p(x) \text{Li}_q(x) }{x}\,dx$$

require nothing more than repeated integration by parts, provided $$\displaystyle p+q \in 2\mathbb{N}+1$$, since

$$\displaystyle \frac{d}{dx} \text{Li}_p(x)=\frac{\text{Li}_{p-1}(x)}{x}$$

So, for example, we have:

$$\displaystyle (01) \quad \int_0^z \frac{ \text{Li}_1(x) \text{Li}_2(x) }{x}\,dx= \frac{1}{2} \text{Li}_2(z)^2$$

$$\displaystyle (02) \quad \int_0^z \frac{ \text{Li}_2(x) \text{Li}_3(x) }{x}\,dx= \frac{1}{2} \text{Li}_3(z)^2$$

$$\displaystyle (03) \quad \int_0^z \frac{ \text{Li}_3(x) \text{Li}_4(x) }{x}\,dx= \frac{1}{2} \text{Li}_4(z)^2$$

Integrating (02) by parts also gives:

$$\displaystyle (04) \quad \int_0^z \frac{ \text{Li}_1(x) \text{Li}_4(x) }{x}\,dx= \text{Li}_2(z) \text{Li}_4(z) - \frac{1}{2} \text{Li}_3(z)^2$$

Similarly, an integration by parts of (03) gives

$$\displaystyle (05) \quad \int_0^z \frac{ \text{Li}_2(x) \text{Li}_5(x) }{x}\,dx= \text{Li}_3(z) \text{Li}_5(z) - \frac{1}{2} \text{Li}_4(z)^2$$

as well as

$$\displaystyle (06) \quad \int_0^z \frac{ \text{Li}_1(x) \text{Li}_6(x) }{x}\,dx=$$

$$\displaystyle \text{Li}_2(z) \text{Li}_6(z)- \text{Li}_3(z) \text{Li}_5(z) + \frac{1}{2} \text{Li}_4(z)^2$$

Last edited:

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
Hey DW , thanks for your input (which I always wait for). I think the generalization you put is interesting to look at.

I was playing a little bit with these integrals and found interesting to look at the infinite sum

$$\displaystyle \mathscr{C}(\alpha , k) =\sum_{n\geq 1}\frac{1}{n^{\alpha}(n+k)}\,\,\, ; \,\,\,\,\mathscr{C}(1, k)=\frac{H_k}{k}$$

We can find a general formula to find the sum

\begin{align}
\mathscr{C}(\alpha , k) &=\sum_{n\geq 1}\frac{1}{k\, n^{\alpha-1}}\left( \frac{1}{n}-\frac{1}{n+k}\right)\\ &= \frac{1}{k}\zeta(\alpha)-\frac{1}{k}\mathscr{C}(\alpha-1 , k)\\ &= \frac{1}{k}\zeta(\alpha)-\frac{1}{k^2}\zeta(\alpha-1)+\frac{1}{k^2}\mathscr{C}(\alpha-2 , k)\\ &= \sum_{n=1}^{\alpha-1}(-1)^{n-1}\frac{\zeta(\alpha-n+1)}{k^n}+(-1)^{\alpha-1}\frac{H_k}{k^\alpha}
\end{align}

Hence we have the general formula

$$\displaystyle \mathscr{C}(\alpha , k) = \sum_{n=1}^{\alpha-1}(-1)^{n-1}\frac{\zeta(\alpha-n+1)}{k^n}+(-1)^{\alpha-1}\frac{H_k}{k^\alpha}$$

Dividing by $$\displaystyle \frac{1}{k^{\beta}}$$ and summing w.r.t to $k$ we have

$$\displaystyle \sum_{k\geq 1}\frac{\mathscr{C}(\alpha , k)}{k^{\beta}} = \sum_{n=1}^{\alpha-1}(-1)^{n-1}\zeta(\alpha-n+1)\zeta(\beta+n)+(-1)^{\alpha-1}\sum_{k\geq 1}\frac{H_k}{k^{\alpha+\beta}}$$

Now we use that

$$\displaystyle \sum_{n=1}^\infty \frac{H_n}{n^q}= \left(1+\frac{q}{2} \right)\zeta(q+1)-\frac{1}{2}\sum_{k=1}^{q-2}\zeta(k+1)\zeta(q-k)$$

Hence we have

$$\displaystyle \sum_{n=1}^\infty \frac{H_n}{n^{\alpha+\beta}}= \left(1+\frac{{\alpha+\beta}}{2} \right)\zeta({\alpha+\beta}+1)-\frac{1}{2}\sum_{k=1}^{{\alpha+\beta}-2}\zeta(k+1)\zeta({\alpha+\beta}-k)$$

And the generalization is the following formula

\begin{align}
\sum_{k\geq 1}\frac{\mathscr{C}(\alpha , k)}{k^{\beta}} &= \sum_{n=1}^{\alpha-1}(-1)^{n-1}\zeta(\alpha-n+1)\zeta(\beta+n) -\frac{1}{2}\sum_{n=1}^{{\alpha+\beta}-2}(-1)^{\alpha-1}\zeta(n+1)\zeta({\alpha+\beta}-n)\\ &+(-1)^{\alpha-1}\left(1+\frac{{\alpha+\beta}}{2} \right)\zeta({\alpha+\beta}+1)\end{align}

Interestingly we also have that

$$\displaystyle \sum_{k\geq 1}\frac{\mathscr{C}(\alpha , k)}{k^{\beta}}=\sum_{k\geq 1}\frac{\mathscr{C}(\beta, k)}{k^{\alpha}}$$

Generally we will use the symbol

$$\displaystyle \mathscr{H}(\alpha,\beta)=\sum_{k\geq 1}\frac{\mathscr{C}(\alpha , k)}{k^{\beta}}$$

So we have the symmetric property

$$\displaystyle \mathscr{H}(\alpha,\beta) = \mathscr{H}(\beta,\alpha)$$

$$\displaystyle \mathscr{H}(1,\beta)= \left(1+\frac{{\beta}}{2} \right)\zeta({\beta}+1)-\frac{1}{2}\sum_{k=1}^{{\beta}-2}\zeta(k+1)\zeta({\beta}-k)$$

Wasn't that fun , yes it was. I'll have to check my work because I might have some errors.

#### DreamWeaver

##### Well-known member
Hey DW , thanks for your input (which I always wait for). I think the generalization you put is interesting to look at.
Thanks Zaid! The feeling's mutual. Very much so.

Wasn't that fun , yes it was. I'll have to check my work because I might have some errors.

Certainly was...

Incidentally, I saw a paper some years ago on arXiv that dealt with Polylog Integrals of the exact same type as I posted above, but where $$\displaystyle p+q \in 2\mathbb{N}$$

Can't seem to find it now, but the author used Euler Sums to evaluate them (nudge, nudge, wink, wink... )

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
Incidentally, I saw a paper some years ago on arXiv that dealt with Polylog Integrals of the exact same type as I posted above, but where $$\displaystyle p+q \in 2\mathbb{N}$$

Can't seem to find it now, but the author used Euler Sums to evaluate them (nudge, nudge, wink, wink... )
Really , that would be interesting if you can find it. Generally, I can find a general formula for the integral

$$\displaystyle \int^1_0 \frac{\mathrm{Li}_n(x) \mathrm{Li}_m(x)}{x}\, dx$$

which is related to $$\displaystyle \mathscr{H}(\alpha ,\beta)$$. Will post that when I have time. Currently I read about the G-Barnes function which is really interesting to play with.

#### DreamWeaver

##### Well-known member
Really , that would be interesting if you can find it. Generally, I can find a general formula for the integral

$$\displaystyle \int^1_0 \frac{\mathrm{Li}_n(x) \mathrm{Li}_m(x)}{x}\, dx$$

which is related to $$\displaystyle \mathscr{H}(\alpha ,\beta)$$. Will post that when I have time. Currently I read about the G-Barnes function which is really interesting to play with.

I'll keep trying to find that old paper, and will let you know if I have any luck.

Likewise, I look forward to your general solution to

$$\displaystyle \int^1_0 \frac{\mathrm{Li}_n(x) \mathrm{Li}_m(x)}{x}\, dx$$

as and when you have time.

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
We find series and integral representations of $$\displaystyle \mathscr{H}(\alpha,\beta)$$

$$\displaystyle \mathscr{H}(p,q)=\sum_{k\geq 1}\frac{\mathscr{C}(p,k)}{k^{q}}= \sum_{k\geq 1}\sum_{n\geq 1}\frac{1}{k^{q}n^{p}(n+k)}$$

Also we have

$$\displaystyle \mathscr{H}(p,q)= \int^1_0 \frac{\mathrm{Li}_p(x)\,\, \mathrm{Li}_q(x)\, }{x}\, dx$$

So we have the following

\begin{align}
\mathscr{H}(p,q)=\int^1_0 \frac{\mathrm{Li}_p(x)\,\, \mathrm{Li}_q(x)\, }{x}\, dx&= \sum_{n=1}^{p-1}(-1)^{n-1}\zeta(p-n+1)\zeta(q+n) -\frac{1}{2}\sum_{n=1}^{{p+q}-2}(-1)^{p-1}\zeta(n+1)\zeta({p+q}-n)\\ &+(-1)^{p-1}\left(1+\frac{{p+q}}{2} \right)\zeta({p+q}+1)\end{align}

Now for the special case $$\displaystyle p=q$$ we have

\begin{align}
\mathscr{H}(q,q)=\int^1_0 \frac{ \mathrm{Li}_q(x) ^2 \, }{x}\, dx&= \sum_{n=1}^{q-1}(-1)^{n-1}\zeta(q-n+1)\zeta(q+n) -\frac{1}{2}\sum_{n=1}^{{2q}-2}(-1)^{q-1}\zeta(n+1)\zeta({2q}-n)\\ &+(-1)^{q-1}\left(1+q \right)\zeta(2q+1)\end{align}

We have for the special case $$\displaystyle q=2$$

$$\displaystyle \mathscr{H}(2,2)=\int^1_0 \frac{ \mathrm{Li}_2(x) ^2 \, }{x}\, dx=2\zeta(3)\zeta(2)-3\zeta(5)$$

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
An interesting form that I wasn't able to solve is the following

$$\displaystyle \int^1_0 \frac{\mathrm{Li}_{q-1}(x)\mathrm{Li}_{q}(x)\mathrm{Li}_{q+1}(x)}{x}\, dx \,\,\,\, q>1$$

For simplicity we can consider

$$\displaystyle \int^1_0 \frac{\mathrm{Li}_{1}(x)\mathrm{Li}_{2}(x)\mathrm{Li}_{3}(x)}{x}\, dx$$

#### DreamWeaver

##### Well-known member
Excellent work, Zaid!

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
Consider the following integral

$$\displaystyle \int^1_0 \frac{\mathrm{Li}_{q-1}(x)\mathrm{Li}_{q}(x)\mathrm{Li}_{q+1}(x)}{x}\, dx$$

Integrating by parts we obtain

$$\displaystyle \int^1_0 \frac{\mathrm{Li}_{q-1}(x)\mathrm{Li}_{q}(x)\mathrm{Li}_{q+1}(x)}{x}\, dx= \frac{\zeta(q+1)\zeta(q)^2}{2}-\frac{1}{2}\int^1_0 \frac{\mathrm{Li}_{q}(x)^3}{x}\, dx$$

Hence we have

$$\displaystyle \int^1_0\frac{\mathrm{Li}_{q}(x)^3}{x}\, dx = \zeta(q+1)\zeta(q)^2-2\int^1_0 \frac{\mathrm{Li}_{q-1}(x)\mathrm{Li}_{q}(x)\mathrm{Li}_{q+1}(x)}{x}\, dx$$

Integrating by parts again we obtain

$$\displaystyle \int^1_0 \frac{\mathrm{Li}_{q-1}(x)\mathrm{Li}_{q}(x)\mathrm{Li}_{q+1}(x)}{x}\, dx=\frac{\zeta(q-1)\zeta^2(q+1)}{2}-\frac{1}{2}\int^1_0 \frac{\mathrm{Li}_{q+1}(x)^2\mathrm{Li}_{q-2}(x)}{x}\, dx$$

So we have the following formula

$$\displaystyle \int^1_0\frac{\mathrm{Li}_{q}(x)^3-\mathrm{Li}_{q+1}(x)^2\mathrm{Li}_{q-2}(x)}{x}\, dx = \zeta(q+1)\zeta(q)^2-\zeta(q-1)\zeta^2(q+1)$$

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
Hey , I found a paper by PEDRO FREITAS that exactly dealt with the form I had in mind namely the author discusses

$$\displaystyle J(m,p,q)= \int^1_0 x^m \mathrm{Li}_p(x) \mathrm{Li}_q(x) \, dx$$

The author gives a general formula for $$\displaystyle J(-1,p,q) =\mathscr{H}(p,q)$$ which is just like the formula I gave proof for but he uses a different method. He discusses many special cases for $$\displaystyle m=-2$$.

Here is a link for the paper.

#### DreamWeaver

##### Well-known member
Thanks for the link, Zaid!

I read that paper years ago, and have been trying to locate it again more recently.

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
Thanks for the link, Zaid!

I read that paper years ago, and have been trying to locate it again more recently.
Yup , very interesting . He treats a generalization of my results. I am glad I found it .


text:
## New and efficient DCA based algorithms for minimum sum-of-squares clustering.(English)Zbl 1326.68225

Summary: The purpose of this paper is to develop new efficient approaches based on DC (Difference of Convex functions) programming and DCA (DC Algorithm) to perform clustering via minimum sum-of-squares Euclidean distance. We consider the two most widely used models for the so-called Minimum Sum-of-Squares Clustering (MSSC in short) that are a bilevel programming problem and a mixed integer program. Firstly, the mixed integer formulation of MSSC is carefully studied and is reformulated as a continuous optimization problem via a new result on exact penalty technique in DC programming. DCA is then investigated to the resulting problem. Secondly, we introduce a Gaussian kernel version of the bilevel programming formulation of MSSC, named GKMSSC. The GKMSSC problem is formulated as a DC program for which a simple and efficient DCA scheme is developed. A regularization technique is investigated for exploiting the nice effect of DC decomposition and a simple procedure for finding good starting points of DCA is developed. The proposed DCA schemes are original and very inexpensive because they amount to computing, at each iteration, the projection of points onto a simplex and/or onto a ball, and/or onto a box, which are all determined in the explicit form. Numerical results on real word datasets show the efficiency, the scalability of DCA and its great superiority with respect to k-means and kernel k-means, standard methods for clustering.

### MSC:

 68T05 Learning and adaptive systems in artificial intelligence 90C90 Applications of mathematical programming
Full Text:

### References:

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text:
Paul's Online Notes
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## Chapter 4 : Multiple Integrals

Here are a set of practice problems for the Multiple Integrals chapter of the Calculus III notes.

1. If you’d like a pdf document containing the solutions the download tab above contains links to pdf’s containing the solutions for the full book, chapter and section. At this time, I do not offer pdf’s for solutions to individual problems.
2. If you’d like to view the solutions on the web go to the problem set web page, click the solution link for any problem and it will take you to the solution to that problem.

Note that some sections will have more problems than others and some will have more or less of a variety of problems. Most sections should have a range of difficulty levels in the problems although this will vary from section to section.

Here is a list of all the sections for which practice problems have been written as well as a brief description of the material covered in the notes for that particular section.

Double Integrals – In this section we will formally define the double integral as well as giving a quick interpretation of the double integral.

Iterated Integrals – In this section we will show how Fubini’s Theorem can be used to evaluate double integrals where the region of integration is a rectangle.

Double Integrals over General Regions – In this section we will start evaluating double integrals over general regions, i.e. regions that aren’t rectangles. We will illustrate how a double integral of a function can be interpreted as the net volume of the solid between the surface given by the function and the $$xy$$-plane.

Double Integrals in Polar Coordinates – In this section we will look at converting integrals (including $$dA$$) in Cartesian coordinates into Polar coordinates. The regions of integration in these cases will be all or portions of disks or rings and so we will also need to convert the original Cartesian limits for these regions into Polar coordinates.

Triple Integrals – In this section we will define the triple integral. We will also illustrate quite a few examples of setting up the limits of integration from the three dimensional region of integration. Getting the limits of integration is often the difficult part of these problems.

Triple Integrals in Cylindrical Coordinates – In this section we will look at converting integrals (including $$dV$$) in Cartesian coordinates into Cylindrical coordinates. We will also be converting the original Cartesian limits for these regions into Cylindrical coordinates.

Triple Integrals in Spherical Coordinates – In this section we will look at converting integrals (including $$dV$$) in Cartesian coordinates into Spherical coordinates. We will also be converting the original Cartesian limits for these regions into Spherical coordinates.

Change of Variables – In previous sections we’ve converted Cartesian coordinates in Polar, Cylindrical and Spherical coordinates. In this section we will generalize this idea and discuss how we convert integrals in Cartesian coordinates into alternate coordinate systems. Included will be a derivation of the $$dV$$ conversion formula when converting to Spherical coordinates.

Surface Area – In this section we will show how a double integral can be used to determine the surface area of the portion of a surface that is over a region in two dimensional space.

Area and Volume Revisited – In this section we summarize the various area and volume formulas from this chapter.


text:
# Global Health Inequalities: Vaccines

December 2020 has been a pivotal month in the global fight against Coronavirus. Having pre-ordered enough COVID-19 shots to immunise their populaces a few times over, subject to governmental approvals, the US, UK, Canada and the European Union have already begun administering the first shots of Pfizer and BioNTech’s vaccine.

But in the world’s poorest countries, the wait for any doses at all could be much longer.

As governments across the world negotiate vaccine deals in this critical game of medical logistics – where supply is minimal but demand is immediate and universal – wealthy nations are accused of snatching up orders before they are even ready and in greater amounts than they need. According to research at Duke University’s Global Health Innovation Center, this race for deals could prevent poor countries from gaining access to enough vaccines for most of their populations until 2024. People’s Vaccine Alliance, a collaboration between several aid groups which include Amnesty International, has also stated that, given the current situation, an estimate of nearly 70 poor countries will be unable to inoculate 90% of their populations in 2021.

Looking into barriers that affect the global access to a vaccine, scientists and researchers have been able to find countless factors that allow the western world to put themselves at the front of the line. It is not just the obvious cost and availability of vaccines that puts the lower-income countries at a disadvantage, but also, these countries lack the basic infrastructure they need to transport, store and distribute these vaccines with the same capacity as other more developed countries.

### Manufacturing

Earlier in December, Pfizer became the first company in the world to have developed a COVID-19 vaccine authorised for emergency use in the West. Now, a little under a month later, more than 10.8 million doses have been supplied and administered in 29 countries. Expectedly enough, 83% of the total number of doses have been given out only in the US, UK and China alone, while the remaining 17% have been administered across Canada and Europe.

However, it takes time to manufacture doses.

The most effective vaccines use several different technologies, such as mRNA, recombinant protein and adenoviruses, and each of these has its own complex manufacturing process. This means that, although the leading vaccine developers and providers have been extensively scaling up production, we can safely conclude that worldwide demand will not be met anytime soon. Additionally, supply is further constrained given that all authorised vaccines require people to get two shots a few weeks apart. But the limitations of the production process do not end here; the last mile of the journey may be the most challenging. Many of the vaccines, including Pfizer’s and Moderna’s, have special storage requirements and conditions. For example, both developers must keep the doses in freezers at very low temperatures. This poses a major infrastructure issue for the developing world, especially where electricity may not be widespread.

Pfizer’s target vaccine supply goal for 2021 is at least 1.3 billion doses, while Moderna aims to produce between 500 million and 1 billion doses. While we can now see the light shining through at the end of the tunnel, we are nowhere near celebrating this triumph. According to researchers from Duke University, the current production capacity indicates that it could take at least three to four years to produce enough vaccines for the global population. Wealthier nations will most likely be able to issue multiple doses to inoculate their entire populations before the vaccines even become widespread in lower-income countries.

### Cost

One other obvious factor which limits the spread of the vaccine to poorer countries is the expensive cost. BioNTech announced that Pfizer has charged $19.50 per dose for the first 100 million doses that were supplied to the US. Given that one person requires two doses of the vaccine to be effectively immunized, the cost rises to$39 per person. Meanwhile, Moderna is currently charging from $25 to$37 per dose and $50 to$74 per person.

These numbers immediately eliminate any chances of achieving herd immunity amongst the poor communities. However, several drugmakers have announced that those extremely vulnerable segments of society will also be guaranteed access to the vaccines. Having promised to reserve 400 million doses of its vaccine for low- and middle-income countries, AstraZeneca will also be selling all vaccines at a cost between $3 and$5 per dose. In addition, Johnson & Johnson has announced a no-profit guarantee from its sales to the poorer nations as well, while China’s vaccine will be made a “global public good”.

### Covax

Since May, and way before being anywhere near developing safe and effective vaccines, wealthier countries have been placing pre-orders for potential ones. In an effort to prevent them from snatching up scarce doses, a scheme called Covax was launched in April by the World Health Organization (WHO), Gavi (the Vaccine Alliance) and the Coalition for Epidemic Preparedness Innovations (CEPI). The scheme allows countries access to an equal share of successful vaccine candidates, ensuring that doses are shared amongst all countries, regardless of their classification based on income and standard of living. Covax’s aim is to ensure that the poorer countries are provided with enough vaccines to cover at least 20% of their population.

So, Covax is quite literally a lifeline for the citizens of lower-income countries, who would otherwise be unable to get vaccinated. Further, as of January 1, evidence of any direct deals made between manufacturers and these poor countries is yet to be found. This suggests that they are relying entirely on the 20% population coverage from Covax.

Despite being a remarkable effort of global collaboration and solidarity, Covax is so extremely underfunded that even its founders are worried. Gavi has stated in a report issued in December that the likelihood of Covax failing its mission is “very high”. According to the report, the scheme has only managed to buy 200 million doses, which is just a 10th of the 2 billion it aimed for over the next year. Having originally signed advanced purchase orders for 700 million doses, Covax is currently unable to finance the purchase of the remaining 500 million, as it has fallen \$5 billion short of the money needed.

This is mainly because some countries, including the world’s largest economy, haven’t contributed. The US alone could eventually have access to 1.8 billion doses, which is estimated to be about a quarter of the world’s near-term supply. The number is not only shocking because this is more than their population of 328 million people could possibly ever use or need, but also since none of it would be shared with lower-income countries via Covax.

As for the many wealthy countries which have joined the scheme, such as the UK, Canada and the European Union, they have also made “side-deals” with manufacturing companies to guarantee their own supply. But this hoarding and stockpiling of vaccines does nothing but undermine the global efforts to ensure that everyone across the world is provided with a chance for protection from Coronavirus. Richer nations have simple human-rights commitments not only to abstain from activities that make the access of vaccines elsewhere challenging but also to aid countries that need this assistance the most.

### Transport and storage

Distributing the vaccines globally is proving to be yet another formidable hurdle. Cargo airline representatives have already said that distributing COVID-19 vaccines to everyone in the world could very easily take up to two years, making this one of the greatest challenges facing the transportation industry.

Some vaccines require significant investment to satisfy the conditions needed for transporting and distributing them. For example, Pfizer’s vaccine must be transported at -70 degrees Celsius through a system of deep-freeze airport warehouses and dry ice refrigerated vehicles. Not only do lower-income countries lack the finances to afford these specially-adapted vehicles, but they also lack the transport links and road networks needed to distribute the vaccines to all those in need.

Nevertheless, even if developing countries can push past these restrictions and vaccines eventually reach their target destinations, they will still have to be kept in cold-chain storage. Given that some of the most well-established US hospitals lack adequate facilities to store the doses, it is safe to conclude that lower-income countries will most likely not have access to the ultra-cold freezers and storage systems needed. Additionally, the vaccines can be thawed in regular fridges – and in this case, must be injected within five days. However, even then, several villages and informal settlements may not have a working fridge or electricity.

The various vaccines developed have different storage needs that are all proving to be unsuitable for the developing world. But still, some may be better than others. For example, AstraZeneca’s vaccine can be stored, transported and handled at normal fridge temperature for over six months. This means that it can be administered within existing healthcare settings and does not require investment in expensive cold-chain facilities.

### Cause for optimism

Despite the myriad of challenges facing lower-income countries today, mass global vaccination is still possible. There may always be a selfish reason motivating wealthier countries to ensure the vaccination of the whole world. Even if the US, the UK and other large economies achieve herd immunity, immunizing between 60 to 72% of their populations, the rest of the world would still serve as a threat to their safety. Hopefully, this will be enough incentive to provide aid, because like German President Frank-Walter Steinmeier said, “even those who conquer the virus within their own borders remain prisoners within these borders until it is conquered everywhere.”


text:
# Matt's arXiv selection: week ending 27 May 2007

From: Matthew Davis <mdavis_at_physics.uq.edu.au>
Date: Mon, 14 May 2007 09:54:15 +1000 (EST)

The following message was sent to the matts_arxiv list by Matthew Davis <mdavis_at_physics.uq.edu.au>

Hi all,

I think I've broken my previous record for the lateness of a week's mailing.
Thankfully by the time you finish reading this the following two weeks emails
will be sitting in your inbox ready to be enjoyed, and once again you will be
entirely up to date.

I've gone through so many papers in this scanning session that I can't even
remember the interesting ones. I'll have to revisit them again later. There are
28 new preprints and 14 replacements this week.

------------------------------------------------------------------------------
\\
arXiv:0704.2623
Date: Thu, 19 Apr 2007 22:11:20 GMT (413kb)

Title: Bragg spectroscopy of a superfluid Bose-Hubbard gas
Authors: X. Du, Shoupu Wan, Emek Yesilada, C. Ryu, D. J. Heinzen, Z. X. Liang,
and Biao Wu
Categories: cond-mat.other
\\
Bragg spectroscopy is used to measure excitations of a trapped,
quantum-degenerate gas of 87Rb atoms in a 3-dimensional optical lattice. The
measurements are carried out over a range of optical lattice depths in the
superfluid phase of the Bose-Hubbard model. For fixed wavevector, the resonant
frequency of the excitation is found to decrease with increasing lattice depth.
A numerical calculation of the resonant frequencies based on Bogoliubov theory
shows a less steep rate of decrease than the measurements.
\\ ( http://arxiv.org/abs/0704.2623 , 413kb)
------------------------------------------------------------------------------
\\
arXiv:0704.2660
Date: Fri, 20 Apr 2007 07:10:25 GMT (54kb)

Title: Competing superfluid and density-wave ground-states of fermionic
mixtures with mass imbalance in optical lattices
Authors: Tung-Lam Dao, Antoine Georges, Massimo Capone.
Categories: cond-mat.supr-con cond-mat.str-el
\\
We study the effect of mass imbalance on the phase diagram of a two-component
fermionic mixture with attractive interactions in optical lattices. Using
static and dynamical mean-field theories, we show that the pure superfluid
phase is stable for all couplings when the mass imbalance is smaller than a
limiting value. For larger imbalance, phase separation between a superfluid and
a charge-density wave takes place when the coupling exceeds a critical
strength. The harmonic trap induces a spatial segregation of the two phases,
with a rapid variation of the density at the boundary.
\\ ( http://arxiv.org/abs/0704.2660 , 54kb)
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\\
arXiv:0704.2721
Date: Fri, 20 Apr 2007 14:45:33 GMT (211kb)

Title: Half-vortices in polariton condensates
Authors: Yuri Rubo
Categories: cond-mat.other
\\
It is shown that vortices in linearly polarized polariton condensates in
planar semiconductor microcavities carry two winding numbers (k,m). These
numbers can be either integer or half-integer simultaneously. Four half-integer
vortices (1/2,1/2), (-1/2,-1/2), (1/2,-1/2), and (-1/2,1/2) are anisotropic,
possess the smallest energy, and define the Kosterlitz-Thouless transition
temperature. The condensate concentration remains finite within the core of
half-vortex and the polarization becomes fully circular in the core center.
\\ ( http://arxiv.org/abs/0704.2721 , 211kb)
------------------------------------------------------------------------------
\\
arXiv:0704.2648
Date: Fri, 20 Apr 2007 02:31:21 GMT (19kb)

Title: Prospects for making polar molecules with microwave fields
Authors: Svetlana Kotochigova
Categories: quant-ph
\\
We propose a new mechanism to produce ultracold polar molecules with
microwave fields. The proposed mechanism converts trapped ultracold atoms of
different species into vibrationally excited molecules by a single microwave
transition and entirely depends on the existence of a permanent dipole moment
in the molecules. As opposed to production of molecules by photoassociation or
magnetic-field Feshbach resonances our method does not rely on the structure
and lifetime of excited states or existence of Feshbach resonances. In
addition, we determine conditions for optimal creation of polar molecules in
vibrationally excited states of the ground-state potential by changing
frequency and intensity of the microwave field. We also explore the possibility
to produce vibrationally cold molecules by combining the microwave field with
an optical Raman transition or by applying a microwave field to Feshbach
molecules. The production mechanism is illustrated for two polar molecules: KRb
and RbCs.
\\ ( http://arxiv.org/abs/0704.2648 , 19kb)
------------------------------------------------------------------------------
\\
arXiv:0704.2757
Date: Fri, 20 Apr 2007 17:16:45 GMT (259kb)

Title: Polaron Physics in Optical Lattices
Authors: Martin Bruderer, Alexander Klein, Stephen R. Clark, and Dieter Jaksch
Categories: quant-ph
\\
We investigate the effects of a nearly-uniform Bose-Einstein condensate (BEC)
on the properties of immersed trapped impurity atoms. Using a weak-coupling
expansion in the BEC-impurity interaction strength we derive a model describing
polarons, i.e. impurities dressed by a coherent state of Bogoliubov phonons,
and apply it to ultra-cold bosonic atoms in an optical lattice. We show that
with increasing BEC temperature the transport properties of the impurities
change from coherent to diffusive. Furthermore, stable polaron clusters are
formed via a phonon-mediated off-site attraction.
\\ ( http://arxiv.org/abs/0704.2757 , 259kb)
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\\
arXiv:0704.2627
Date: Thu, 19 Apr 2007 22:31:10 GMT (228kb)

Title: Atom-wave diffraction between the Raman-Nath and the Bragg regime:
Effective Rabi frequency, losses, and phase shifts
Authors: Holger Mueller, Sheng-wey Chiow, and Steven Chu
Categories: physics.atom-ph
\\
We present an analytic theory of the diffraction of (matter) waves by a
lattice in the "quasi-Bragg" or "channelling" regime between the Bragg and the
short-interaction Raman-Nath regime. The Schroedinger equation is solved by
starting point, and re-inserting the result into the Schroedinger equation to
yield a second order correction. Closed expressions for arbitrary pulse shapes
and diffraction orders are obtained and the losses of the population to output
states otherwise forbidden by the Bragg condition are derived. We consider the
phase shift due to couplings of the desired output to these states that depends
on the interaction strength and duration and show how these can be kept
negligible by a choice of smooth (e.g., Gaussian) envelope functions even in
situations that substantially violate the adiabaticity condition. We also give
an efficient method for calculating the effective Rabi frequency (which is
related to the eigenvalues of Mathieu functions) in the quasi-Bragg regime.
\\ ( http://arxiv.org/abs/0704.2627 , 228kb)
------------------------------------------------------------------------------
\\
arXiv:0704.2856
Date: Sun, 22 Apr 2007 00:21:54 GMT (421kb,D)

Title: State preparation and dynamics of ultracold atoms in higher lattice
orbitals
Authors: Torben Mueller, Simon Foelling, Artur Widera, Immanuel Bloch
Categories: cond-mat.other cond-mat.str-el
\\
We report on the realization of a multi-orbital system with ultracold atoms
in the excited bands of a 3D optical lattice by selectively controlling the
band population along a given lattice direction. The lifetime of the atoms in
the excited band is found to be considerably longer (10-100 times) than the
characteristic time scale for inter-site tunneling, thus opening the path for
orbital selective many-body physics with ultracold atoms. Upon exciting the
atoms from an initial lowest band Mott insulating state to higher lying bands,
we observe the dynamical emergence of coherence in 1D (and 2D), compatible with
Bose-Einstein condensation to a non-zero momentum state.
\\ ( http://arxiv.org/abs/0704.2856 , 421kb)
------------------------------------------------------------------------------
\\
arXiv:0704.2957
Date: Mon, 23 Apr 2007 09:29:01 GMT (434kb)

Title: Superfluid-Insulator Transition of Strongly Interacting Fermi Gases in
Optical Lattices
Authors: Hui Zhai and Tin-Lun Ho
Categories: cond-mat.supr-con
\\
We study a quantum phase transition from fermionic suerpfluid (SF) to
band-insulator (BI) of fermions in optical lattices. The destruction of the
band insulator is driven by the energy gain in promoting fermions in valance
band to various conducting bands to form Cooper pairs. We have obtained the
phase boundary of this transition for systems with two fermions per site using
mean-field theory, which is accurate at fermion side of a Feshbach resonance,
and should give qualitative correct results near resonance. As one moves across
resonance to the molecule side, the SF-BI transition evolves into the SF-Mott
insulator transition of bosonic molecules. The physics becomes more intriguing
at higher fermion densities, and we shall present a qualitative global phase
diagram for SF-Insulator transition for the entire BCS-BEC family. Our results
are directly related to the recent MIT experiment on lattices fermions.
\\ ( http://arxiv.org/abs/0704.2957 , 434kb)
------------------------------------------------------------------------------
\\
arXiv:0704.3011
Date: Sun, 22 Apr 2007 23:06:35 GMT (4810kb)

Title: Many-Body Physics with Ultracold Gases
Authors: Immanuel Bloch, Jean Dalibard, Wilhelm Zwerger
Categories: cond-mat.other
Comments: Article for Review of Modern Physics, 76 pages
\\
many-body phenomena in dilute, ultracold gases. Its focus are effects beyond
standard weak-coupling descriptions, like the Mott-Hubbard-transition in
optical lattices, strongly interacting gases in one dimension or quasi
two-dimensional gases in fast rotation. Strong correlations in fermionic gases
are discussed in optical lattices or near Feshbach resonances in the BCS-BEC
crossover.
\\ ( http://arxiv.org/abs/0704.3011 , 4810kb)
------------------------------------------------------------------------------
\\
arXiv:0704.2932
Date: Mon, 23 Apr 2007 07:16:20 GMT (274kb)

Title: Controlling statistical properties of stored light
Authors: A. Raczynski, K. Slowik, J. Zaremba, S. Zielinska-Kaniasty
Categories: quant-ph
\\
Statistical properties of outgoing light pulses are studies after they have
been stored in a medium of atoms in the tripod configuration. A generalized
Hong-Ou-Mandel interference, storing of squeezed states and homodyne signal
analysis are discussed in the context of their dependence on the parameters of
the control fields used for light storage and release.
\\ ( http://arxiv.org/abs/0704.2932 , 274kb)
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\\
arXiv:0704.2935
Date: Mon, 23 Apr 2007 07:42:39 GMT (397kb)

Title: Formation of Ultracold Heteronuclear Dimers in Electric Fields
Authors: Rosario Gonz\'alez-F\'erez, Michael Mayle and Peter Schmelcher
Categories: quant-ph
\\
The formation of ultracold molecules via stimulated emission followed by a
radiative deexcitation cascade in the presence of a static electric field is
investigated. By analyzing the corresponding cross sections, we demonstrate the
possibility to populate the lowest rotational excitations via photoassociation.
narrow rotational state distributions in the vibrational ground state. External
fields might therefore represent an additional valuable tool towards the
ultimate goal of quantum state preparation of molecules.
\\ ( http://arxiv.org/abs/0704.2935 , 397kb)
------------------------------------------------------------------------------
\\
arXiv:0704.3047
Date: Mon, 23 Apr 2007 19:31:05 GMT (437kb)

Title: Atom-atom correlations in spontaneous four wave mixing of two colliding
Bose-Einstein Condensates
Authors: A. Perrin (LCFIO), H. Chang (LCFIO), V. Krachmalnicoff (LCFIO), M.
Schellekens (LCFIO), D. Boiron (LCFIO), A. Aspect (LCFIO), C. I. Westbrook
(LCFIO)
Categories: quant-ph
Proxy: ccsd hal-00142991
\\
We have used a position sensitive, single atom detector to observe atom
scattering during the collision of two Bose-Einstein condensates. The process
can also be thought of as spontaneous, degenerate four wave mixing of deBroglie
waves. The observed two-particle correlation function shows a clear peak
corresponding to the production of back to back atom pairs. The correlation
function also exhibits a peak corresponding to pairs of atoms emitted in the
same direction, the Hanbury Brown-Twiss effect. Our experiment is in the regime
of well separated atom pairs, opening the route to the study of quantum atom
optics effects with individual entangled pairs.
\\ ( http://arxiv.org/abs/0704.3047 , 437kb)
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\\
arXiv:0704.3010
Date: Mon, 23 Apr 2007 14:05:19 GMT (233kb)

Title: Ground-state magneto-optical resonances in Cesium vapour confined in an
extremely thin cell
Authors: C. Andreeva, A. Atvars, M. Auzinsh, K.Bluss, S. Cartaleva, L. Petrov,
and D. Slavov
Categories: physics.atom-ph
\\
Experimental and theoretical studies are presented related to the
ground-state magneto-optical resonance prepared in Cesium vapour confined in an
Extremely Thin Cell (ETC, with thickness equal to the wavelength of the
irradiating light). It is shown that the utilization of the ETC allows one to
examine the formation of a magneto-optical resonance on the individual
hyperfine transitions, thus distinguishing processes resulting in dark (reduced
absorption) or bright (enhanced absorption) resonance formation. We report on
an experimental evidence of the bright magneto-optical resonance sign reversal
in Cs atoms confined in the ETC. A theoretical model is proposed based on the
optical Bloch equations that involves the elastic interaction processes of
atoms in the ETC with its walls resulting in depolarization of the Cs excited
to the sign reversal of the bright resonance. Using the proposed model, the
magneto-optical resonance amplitude and width as a function of laser power are
calculated and compared with the experimental ones. The numerical results are
in good agreement with the experiment.
\\ ( http://arxiv.org/abs/0704.3010 , 233kb)
------------------------------------------------------------------------------
\\
arXiv:0704.3114
Date: Tue, 24 Apr 2007 18:32:09 GMT (606kb)

Title: Topological Winding and Unwinding in Metastable Bose-Einstein
Condensates
Authors: Rina Kanamoto, Lincoln D. Carr, Masahito Ueda
Categories: cond-mat.other
\\
Topological winding and unwinding in a quasi-one-dimensional metastable
Bose-Einstein condensate are shown to be manipulated by changing the strength
of interaction or the frequency of rotation. Exact diagonalization analysis
reveals that quasidegenerate states emerge spontaneously near the transition
point, allowing a smooth crossover between topologically distinct states. On a
mean-field level, the transition is accompanied by formation of grey solitons,
or density notches, which serve as an experimental signature of this
phenomenon.
\\ ( http://arxiv.org/abs/0704.3114 , 606kb)
------------------------------------------------------------------------------
\\
arXiv:0704.3116
Date: Tue, 24 Apr 2007 04:46:59 GMT (88kb)

Title: Combinatorics and Boson normal ordering: A gentle introduction
Authors: P. Blasiak (1), A. Horzela (1), K. A. Penson (2), A. I. Solomon (2),
G. H. E. Duchamp (3), ((1) Polish Academy of Sciences, Krakow, Poland, (2)
LPTMC, University of Paris VI, France, (3) Institut Galilee, University of
Paris 13, France)
Categories: quant-ph
\\
We discuss a general combinatorial framework for operator ordering problems
by applying it to the normal ordering of the powers and exponential of the
boson number operator. The solution of the problem is given in terms of Bell
and Stirling numbers enumerating partitions of a set. This framework reveals
several inherent relations between ordering problems and combinatorial objects,
and displays the analytical background to Wick's theorem. The methodology can
be straightforwardly generalized from the simple example given herein to a wide
class of operators.
\\ ( http://arxiv.org/abs/0704.3116 , 88kb)
------------------------------------------------------------------------------
\\
arXiv:0704.3137
Date: Tue, 24 Apr 2007 07:58:46 GMT (169kb)

Title: Scanning magnetoresistance microscopy of atom chips
Authors: M. Volk, S. Whitlock, B. V. Hall, A. I. Sidorov
Categories: physics.atom-ph
\\
Surface based geometries of microfabricated wires or patterned magnetic films
can be used to magnetically trap and manipulate ultracold neutral atoms or
Bose-Einstein condensates. We investigate the magnetic properties of such atom
chips using a scanning magnetoresistive (MR) microscope with high spatial
resolution and high field sensitivity. We show that MR sensors are ideally
suited to observe small variations of the magnetic field caused by
imperfections in the wires or magnetic materials which ultimately lead to
fragmentation of ultracold atom clouds. Measurements are also provided for the
magnetic field produced by a thin current-carrying wire with small geometric
modulations along the edge. Comparisons of our measurements with a full numeric
calculation of the current flow in the wire and the subsequent magnetic field
show excellent agreement. Our results highlight the use of scanning MR
microscopy as a convenient and powerful technique for precisely characterizing
the magnetic fields produced near the surface of atom chips.
\\ ( http://arxiv.org/abs/0704.3137 , 169kb)
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\\
arXiv:0704.3275
Date: Tue, 24 Apr 2007 21:20:57 GMT (187kb)

Title: Matter-wave interference in s-wave and p-wave Fermi condensates
Authors: Wei Zhang and C. A. R. Sa de Melo
Categories: cond-mat.str-el cond-mat.supr-con
Comments: extension of earlier work cond-mat/0603561. 8 pages, 5 figures
\\
We discuss the time evolution and matter-wave interference of Fermi
condensates on the BEC side of Feshbach resonances for $s$ and p-wave
superfluids, upon release from harmonic traps. In s-wave systems, where the
order parameter is a complex scalar, we find that the interference patterns
depend on the relative phase of the order parameters of the condensates. In
p-wave systems involving the mixture of two-hyperfine states, we show that the
interference pattern exhibits a polarization effect depending on the relative
orientation of the two vector order parameters. Lastly, we also point out that
p-wave Fermi condensates exhibit an anisotropic expansion, reflecting the
spatial anisotropy of the underlying interaction between fermions and the
orbital nature of the vector order parameter. Potential applications of our
results include systems of ultra-cold atoms that exhibit p-wave Feshbach
resonances such as $^6$Li or $^{40}$K.
\\ ( http://arxiv.org/abs/0704.3275 , 187kb)
------------------------------------------------------------------------------
\\
arXiv:0704.3366
Date: Wed, 25 Apr 2007 13:07:37 GMT (151kb)

Title: Dynamics of a strongly interacting Fermi gas: the radial quadrupole mode
Authors: A. Altmeyer (1), S. Riedl (1 and 2), M. J. Wright (1), C. Kohstall
(1), J. Hecker Denschlag (1) and R. Grimm (1 and 2) ((1) Inst. of
Experimental Physics and Center for Quantum Physics, Univ. Innsbruck,
Technikerstra{\ss}e 25, Austria, (2) Inst. for Quantum Optics and Quantum
Categories: cond-mat.other
\\
We report on measurements of an elementary surface mode in an ultracold,
strongly interacting Fermi gas of 6Li atoms. The radial quadrupole mode allows
us to probe hydrodynamic behavior in the BEC-BCS crossover without being
influenced by changes in the equation of state. We examine frequency and
damping of this mode, along with its expansion dynamics. In the unitarity limit
and on the BEC side of the resonance, the observed frequencies agree with
standard hydrodynamic theory. However, on the BCS side of the crossover, a
striking down shift of the oscillation frequency is observed in the
hydrodynamic regime as a precursor to an abrupt transition to collisionless
behavior; this indicates coupling of the oscillation to fermionic pairs.
\\ ( http://arxiv.org/abs/0704.3366 , 151kb)
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\\
arXiv:0704.3412
Date: Wed, 25 Apr 2007 17:15:25 GMT (214kb)

Title: The effect of dipole-dipole interactions between atoms in an active
medium
Authors: V.V.Berezovsky, L.I.Men'shikov, S.Oberg and C.D.Latham
Categories: cond-mat.str-el cond-mat.other
\\
Based on the results of numerical modeling, it is shown that dipole-dipole
interactions among atoms in the active medium influences strongly the character
of the associated superradiation. The main effect is to make the nuclear
subsystem behave chaotically. Its strength increases with the atom density, and
Near correlations between the atoms are established, causing a confinement
effect: a shielding of radiation in the active medium.
\\ ( http://arxiv.org/abs/0704.3412 , 214kb)
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\\
arXiv:0704.3430
Date: Wed, 25 Apr 2007 19:28:34 GMT (273kb)

Title: Prospects for precision measurements of atomic helium using direct
frequency comb spectroscopy
Authors: E. E. Eyler, D. E. Chieda, Matthew C. Stowe, Michael J. Thorpe, T. R.
Schibli, and Jun Ye
Categories: physics.atom-ph
\\
We analyze several possibilities for precisely measuring electronic
transitions in atomic helium by the direct use of phase-stabilized femtosecond
frequency combs. Because the comb is self-calibrating and can be shifted into
the ultraviolet spectral region via harmonic generation, it offers the prospect
of greatly improved accuracy for UV and far-UV transitions. To take advantage
of this accuracy an ultracold helium sample is needed. For measurements of the
triplet spectrum a magneto-optical trap (MOT) can be used to cool and trap
metastable $2 ^3S$ state atoms. We analyze schemes for measuring the two-photon
$2 ^3S \to 4 ^3S$ interval, and for resonant two-photon excitation to high
Rydberg states, $2 ^3S \to 3 ^3P \to n^3S,D$. We also analyze experiments on
the singlet-state spectrum. To accomplish this we propose schemes for producing
and trapping ultracold helium in the $1 ^1S$ or $2 ^1S$ state via
intercombination transition. A particularly intriguing scenario is the
possibility of direct singlet state spectroscopy, including a measurement of
the $1 ^1S \to 2 ^1S$ transition with extremely high accuracy by use of
two-photon excitation in a magic wavelength trap that operates identically for
both states. We predict a triple magic wavelength'' at 412 nm that could
facilitate numerous experiments on trapped helium atoms, because here the
polarizabilities of the $1 ^1S$, $2 ^1S$ and $2 ^3S$ states are all similar,
small, and positive.
\\ ( http://arxiv.org/abs/0704.3430 , 273kb)
------------------------------------------------------------------------------
\\
arXiv:0704.3273
Date: Tue, 24 Apr 2007 20:42:25 GMT (37kb)

Title: Forward and non-forward symplectic integrators in solving classical
dynamics problems
Authors: Siu A. Chin
Categories: physics.comp-ph physics.class-ph
Comments: 26 pages, 17 figures, to be published in a "Special Issue on
Splitting Methods for Differential Equation", Intl. J. of Comp. Math
\\
Forward time step integrators are splitting algorithms with only positive
splitting coefficients. When used in solving physical evolution equations,
these positive coefficients correspond to positive time steps. Forward
algorithms are essential for solving time-irreversible equations that cannot be
evolved using backward time steps. However, forward integrators are also better
in solving time-reversible equations of classical dynamics by tracking as
closely as possible the physical trajectory. This work compares in detail
various forward and non-forward fourth-order integrators using three, fourth,
five and six force evaluations. In the case of solving the 2D Kepler orbit, all
non-forward integrators are optimized by simply minimizing the size of their
backward time steps.
\\ ( http://arxiv.org/abs/0704.3273 , 37kb)
------------------------------------------------------------------------------
\\
arXiv:0704.3439
Date: Wed, 25 Apr 2007 21:49:31 GMT (36kb)

Title: Dilute two-component Fermi superfluids and the symmetric heavy-light
ansatz
Authors: Dean Lee (North Carolina State University)
Categories: cond-mat.supr-con
\\
The symmetric heavy-light ansatz is an approximate method for finding the
superfluid ground state of dilute fermions with two degenerate components.
While the Hamiltonian has an exact SU(2) symmetry, the heavy-light ansatz
breaks this symmetry by skewing the mass ratio of the two components. In the
limit where one component is infinitely heavy, the many-body problem can be
solved in terms of single-particle orbitals. The original SU(2) symmetry is
then reintroduced by requiring N-body density correlations for the two
components be equal for all N. In the simple approximation where only the
lowest orbitals are filled, the method is exact for the two-body system, exact
in the strong-coupling Bose gas limit, exact in the weak-coupling Fermi gas
limit, and in good agreement with recent lattice Monte Carlo results for the
ground state energy at unitarity.
\\ ( http://arxiv.org/abs/0704.3439 , 36kb)
------------------------------------------------------------------------------
\\
arXiv:0704.3440
Date: Wed, 25 Apr 2007 22:02:28 GMT (36kb)

Title: Dynamical models and the phase ordering kinetics of the s=1 spinor
condensate
Authors: Subroto Mukerjee Cenke Xu Joel E. Moore
Categories: cond-mat.stat-mech cond-mat.soft
Comments: 15 pages with 9 figures
\\
The $s=1$ spinor Bose condensate at zero temperature supports ferromagnetic
and polar phases that combine magnetic and superfluid ordering. We investigate
the formation of magnetic domains at finite temperature and magnetic field in
two dimensions in an optical trap. We study the general ground state phase
diagram of a spin-1 system and focus on a phase that has a magnetic Ising order
parameter and numerically determine the nature of the finite temperature
superfluid and magnetic phase transitions. We then study three different
dynamical models: model A, which has no conserved quantities, model F, which
has a conserved second sound mode and the Gross-Pitaevskii (GP) equation which
has a conserved density and magnetization. We find the dynamic critical
exponent to be the same for models A and F ($z=2$) but different for GP ($z \approx 3$). Externally imposed magnetization conservation in models A and F
yields the value $z \approx 3$, which demonstrates that the only conserved
density relevant to domain formation is the magnetization density.
\\ ( http://arxiv.org/abs/0704.3440 , 36kb)
------------------------------------------------------------------------------
\\
arXiv:0704.3483
Date: Thu, 26 Apr 2007 07:18:31 GMT (35kb)

Title: Collective oscillations of a Bose-Fermi mixture: Effect of unequal mass
of Bose and Fermi particles
Authors: Arup Banerjee
Categories: cond-mat.other cond-mat.mes-hall
\\
We investigate the effect of different mass of a Bose- and a Fermi-particle
on the collective oscillations of the degenerate boson-fermion mixtures. In
particular we consider the monopole and the quadrupole modes of the
oscillations and study their characters and the frequencies by using
variational-sum-rule approach. We find that for both the modes there exists a
critical value of the ratio of boson-fermion mass below and above which the
character and the frequency of the respective modes are significantly
different.
\\ ( http://arxiv.org/abs/0704.3483 , 35kb)
------------------------------------------------------------------------------
\\
arXiv:0704.3522
Date: Thu, 26 Apr 2007 12:07:32 GMT (69kb)

Title: Superradiance and multiple scattering of photons in atomic gases
Authors: A. Gero and E. Akkermans
Categories: cond-mat.mes-hall
\\
We study the influence of cooperative effects such as superradiance and
subradiance, on the scattering properties of dilute atomic gases. We show that
cooperative effects lead to an effective potential between pairs of atoms that
decays like $1/r$. In the case of superradiance, this potential is attractive
for close enough atoms and can be interpreted as a coherent mesoscopic effect.
We consider a model of multiple scattering of a photon among superradiant pairs
and calculate the elastic mean free path and the group velocity. We study first
the case of a scalar wave which allows to obtain and to understand basic
features of cooperative effects and multiple scattering. We then turn to the
general problem of a vector wave. In both cases, we obtain qualitatively
similar results and derive, for the case of a scalar wave, analytic expressions
of the elastic mean free path and of the group velocity for an arbitrary (near
resonance) detuning.
\\ ( http://arxiv.org/abs/0704.3522 , 69kb)
------------------------------------------------------------------------------
\\
arXiv:0704.3460
Date: Thu, 26 Apr 2007 01:05:34 GMT (909kb)

Title: Two-color atom guide and 1D optical lattice using evanescent fields of
high-order transverse modes
Authors: Jian Fu, Xiang Yin, and Limin Tong
Categories: quant-ph
Comments: 18 pages, 12 figures, 1 table
\\
We propose a two-color scheme of atom guide and 1D optical lattice using
evanescent light fields of different transverse modes. The optical waveguide
carries a red-detuned light and a blue-detuned light, with both modes far from
resonance. The atom guide and 1D optical lattice potentials can be transformed
to each other by using a Mach-Zehnder interferometer to accurately control mode
transformation. This might provide a new approach to realize flexible
transition between the guiding and trapping states of atoms.
\\ ( http://arxiv.org/abs/0704.3460 , 909kb)
------------------------------------------------------------------------------

The replacements:

------------------------------------------------------------------------------
\\
arXiv:0704.1642
replaced with revised version Fri, 20 Apr 2007 18:18:23 GMT (51kb)

Title: Collective excitations of hard-core Bosons at half filling on square and
triangular lattices: Development of roton minima and collapse of roton gap
Authors: Tyler Bryant and Rajiv R. P. Singh
Categories: cond-mat.str-el
\\ ( http://arxiv.org/abs/0704.1642 , 51kb)
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\\
arXiv:cond-mat/0510436
replaced with revised version Fri, 20 Apr 2007 09:10:22 GMT (38kb)

Title: A simple renormalization group approximation of the groundstate
properties of interacting bosonic systems
Authors: Roman Werpachowski and Jerzy Kijowski
Categories: cond-mat.stat-mech
Comments: RevTex 4, 12 figures (revised and updated); to appear in Physical
Review A
\\ ( http://arxiv.org/abs/cond-mat/0510436 , 38kb)
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\\
arXiv:cond-mat/0608491
replaced with revised version Thu, 19 Apr 2007 21:37:16 GMT (321kb)

Title: Supersolids in confined fermions on one-dimensional optical lattices
Authors: F. Karim Pour, M. Rigol, S. Wessel, A. Muramatsu
Categories: cond-mat.str-el cond-mat.supr-con
Comments: 4 pages, 5 figures, published version
Journal-ref: Phys. Rev. B 75, 161104(R) (2007)
DOI: 10.1103/PhysRevB.75.161104
\\ ( http://arxiv.org/abs/cond-mat/0608491 , 321kb)
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\\
arXiv:cond-mat/0608566
replaced with revised version Thu, 19 Apr 2007 20:08:29 GMT (499kb)

Title: Excitations from a Bose-Einstein condensate of magnons in coupled spin
Authors: V. O. Garlea, A. Zheludev, T. Masuda, L.-P. regnault, E. Ressouche, B.
Brenier, H. Manaka, J.-H. Chung, Y. Qiu, K. Habicht, K. Kiefer and M. Boehm
Categories: cond-mat.str-el cond-mat.supr-con
Journal-ref: Phys. Rev. Lett. 98, 167202 (2007)
\\ ( http://arxiv.org/abs/cond-mat/0608566 , 499kb)
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\\
arXiv:quant-ph/0612129
replaced with revised version Fri, 20 Apr 2007 16:51:44 GMT (66kb)

Title: Photon number states generated from a continuous-wave light source
Authors: Anne E. B. Nielsen and Klaus Molmer
Categories: quant-ph
Comments: 8 pages, 7 figures, v2: shortened version
Journal-ref: Phys. Rev. A 75, 043801 (2007)
\\ ( http://arxiv.org/abs/quant-ph/0612129 , 66kb)
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\\
arXiv:cond-mat/0512537
replaced with revised version Tue, 24 Apr 2007 16:05:20 GMT (29kb)

Title: Dynamical Aspects of Analogue Gravity: The Backreaction of Quantum
Fluctuations in Dilute Bose-Einstein Condensates
Authors: Uwe R. Fischer
Categories: cond-mat.other gr-qc hep-th
Comments: 21+epsilon pages; has appeared in Springer Lecture Notes in Physics
Journal-ref: Lect. Notes Phys. 718, 93-113 (2007)
DOI: 10.1007/3-540-70859-6_5
\\ ( http://arxiv.org/abs/cond-mat/0512537 , 29kb)
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\\
arXiv:cond-mat/0701280
replaced with revised version Tue, 24 Apr 2007 12:33:30 GMT (37kb)

Title: Phase diagram for interacting Bose gases
Authors: K. Morawetz, M. Maennel and M. Schreiber
Categories: cond-mat.stat-mech cond-mat.other
\\ ( http://arxiv.org/abs/cond-mat/0701280 , 37kb)
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\\
arXiv:quant-ph/0703155
replaced with revised version Tue, 24 Apr 2007 13:16:31 GMT (229kb)

Title: Trapping cold atoms near carbon nanotubes: thermal spin flips and
Casimir-Polder potential
Authors: Rachele Fermani, Stefan Scheel, Peter L. Knight
Categories: quant-ph cond-mat.other
\\ ( http://arxiv.org/abs/quant-ph/0703155 , 229kb)
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\\
arXiv:quant-ph/0609134
replaced with revised version Tue, 24 Apr 2007 19:20:24 GMT (248kb)

Title: Fast Quantum State Control of a Single Trapped Neutral Atom
Authors: Matthew P. A. Jones (LCFIO), Jerome Beugnon (LCFIO), Alpha Ga\"etan
(LCFIO), Junxiang Zhang (LCFIO), Gaetan Messin (LCFIO), Antoine Browaeys
(LCFIO), Philippe Grangier (LCFIO)
Categories: quant-ph
Journal-ref: Physical review A 75 (11/04/2007) 040301(R)
DOI: 10.1103/PhysRevA.75.040301
\\ ( http://arxiv.org/abs/quant-ph/0609134 , 248kb)
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arXiv:0704.2623
replaced with revised version Wed, 25 Apr 2007 14:23:35 GMT (413kb)

Title: Bragg spectroscopy of a superfluid Bose-Hubbard gas
Authors: X. Du, Shoupu Wan, Emek Yesilada, C. Ryu, D. J. Heinzen, Z. X. Liang,
and Biao Wu
Categories: cond-mat.other
\\ ( http://arxiv.org/abs/0704.2623 , 413kb)
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\\
arXiv:cond-mat/0510675
replaced with revised version Wed, 25 Apr 2007 17:45:39 GMT (119kb)

Title: Spin-correlation functions in ultracold paired atomic-fermion systems:
sum rules, self-consistent approximations, and mean fields
Authors: Zhenhua Yu and Gordon Baym
Categories: cond-mat.other
Journal-ref: Physical Review A 73, 063601 (2006)
\\ ( http://arxiv.org/abs/cond-mat/0510675 , 119kb)
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arXiv:cond-mat/0612538
replaced with revised version Wed, 25 Apr 2007 17:57:46 GMT (19kb)

Title: Quasi-Adiabatic Continuation in Gapped Spin and Fermion Systems:
Goldstone's Theorem and Flux Periodicity
Authors: M. B. Hastings
Categories: cond-mat.stat-mech
Comments: 19 pages, 2 figures, final version in press at JSTAT
\\ ( http://arxiv.org/abs/cond-mat/0612538 , 19kb)
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arXiv:physics/0611237
replaced with revised version Wed, 25 Apr 2007 15:47:22 GMT (177kb)

Title: Interaction-induced crossover versus finite-size condensation in a
weakly interacting trapped one-dimensional Bose gas
Authors: Isabelle Bouchoule (LCFIO), Karen V. Kheruntsyan (ARC Centre of
Excellence for Quantum-Atom Optics), Gora Shlyapnikov (LPTMS)
Categories: physics.atom-ph cond-mat.other cond-mat.stat-mech
Journal-ref: Physical Review A 75 (28/03/2007) 031606
\\ ( http://arxiv.org/abs/physics/0611237 , 177kb)
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arXiv:cond-mat/0610536
replaced with revised version Thu, 26 Apr 2007 05:19:03 GMT (77kb)

Title: Bose-Einstein condensation of magnons in magnets with predominant
ferromagnetic interaction
Authors: A. V. Syromyatnikov
Categories: cond-mat.str-el
Comments: 9 pages, 4 figure. The paper has been rewritten considerably. In
particular, discussion of crossovers in the critical temperature in
Journal-ref: Phys. Rev. B 75, 134421 (2007)
DOI: 10.1103/PhysRevB.75.134421
\\ ( http://arxiv.org/abs/cond-mat/0610536 , 77kb)
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Next one will be in a few minutes....
Matt.

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School of Physical Sciences,  email: mdavis_at_physics.uq.edu.au
University of Queensland,     ph   : +61 7 334 69824
Brisbane, QLD 4072,           fax  : +61 7 336 51242
Australia.                    http://www.physics.uq.edu.au/people/mdavis/
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text:
## Cutoff for the Ising model on the lattice

• Lubetzky E
• Sly A
• 21

• 15

Citations

#### Abstract

Introduced in 1963, Glauber dynamics is one of the most practiced and extensively studied methods for sampling the Ising model on lattices. It is well known that at high temperatures, the time it takes this chain to mix in $L^1$ on a system of size $n$ is $O(\log n)$. Whether in this regime there is cutoff, i.e. a sharp transition in the $L^1$-convergence to equilibrium, is a fundamental open problem: If so, as conjectured by Peres, it would imply that mixing occurs abruptly at $(c+o(1))\log n$ for some fixed $c>0$, thus providing a rigorous stopping rule for this MCMC sampler. However, obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple Markov chains. Already for the one-dimensional Ising model, showing cutoff is a longstanding open problem. We settle the above by establishing cutoff and its location at the high temperature regime of the Ising model on the lattice with periodic boundary conditions. Our results hold for any dimension and at any temperature where there is strong spatial mixing: For $\Z^2$ this carries all the way to the critical temperature. Specifically, for fixed $d\geq 1$, the continuous-time Glauber dynamics for the Ising model on $(\Z/n\Z)^d$ with periodic boundary conditions has cutoff at $(d/2\lambda_\infty)\log n$, where $\lambda_\infty$ is the spectral gap of the dynamics on the infinite-volume lattice. To our knowledge, this is the first time where cutoff is shown for a Markov chain where even understanding its stationary distribution is limited. The proof hinges on a new technique for translating $L^1$ to $L^2$ mixing which enables the application of log-Sobolev inequalities. The technique is general and carries to other monotone and anti-monotone spin-systems.

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# Engine compilation.. (or something to that effect)

## Recommended Posts

Hello, I have been busy programming a game engine over the past month. It can''t do that much at the moment, but it''s continually evolving. However, so far I have about 25+ files (header and source). I use VC++, and the only way I have been able to get the intellisense thing to work is to include ALL of those files into my project. This is really a pain, and I was wondering if there is an easier way to do it. I have lots of classes (i split up my classes one per file), so I would like to avoid adding all the class descriptions to one big file. I was wondering if I could create, say, a DLL or something, so that I could include all of my files and still get intellisense to work. I have organized my files neatly into folders so it is not that hard to add them to my project, but they still take up a lot of room in VC++ (and yes, I organize them into folders in vc++ too.) Anyone have any ideas? Thanks in advance! Fuzztrek ¬_¬

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Intellisense is broken. Get Visual Assist from Whole Tomato.

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hate to sound cheap..but is there another way? or is VS.NET better?

¬_¬

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Yes, vs.net is much better.

The real question is this: Is Intellisense worth $70-$80 (or whatever Visual Assist is), or is it worth the several hundred or more dollars for visual studio .net ?

Send one of your friends to get hired at Microsoft, and tell him to send you a discounted ($20) version of VC++.NET to you! #### Share this post ##### Link to post ##### Share on other sites quote: Original post by Nypyren Send one of your friends to get hired at Microsoft, and tell him to send you a discounted ($20) version of VC++.NET to you!

Nice friend you are!

And to the original post, yes, make DLL''s if its a windows only engine.

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Alright! thanks so much! I actually just got VS.NET yesterday :D

¬_¬

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VS.net''s Intellisense isn''t much better than what was in VC 6. Visual Assist is much more useful IMHO.

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