text:
## Section6.5Improper Integrals

###### Motivating Questions
• What are improper integrals and why are they important?

• What does it mean to say that an improper integral converges or diverges?

• What are some typical improper integrals that we can classify as convergent or divergent?

Another important application of the definite integral regards how the likelihood of certain events can be measured. For example, consider a company that manufactures incandescent light bulbs, and suppose that based on a large volume of test results, they have determined that the fraction of light bulbs that fail between times $t = a$ and $t = b$ of use (where $t$ is measured in months) is given by

\begin{equation*} \int_a^b 0.3 e^{-0.3t} \, dt\text{.} \end{equation*}

For example, the fraction of light bulbs that fail during their third month of use is given by

\begin{align*} \int_2^3 0.3e^{-0.3t} \, dt \amp = -e^{-0.3t} \bigg \vert_2^3\\ \amp = -e^{-0.9} + e^{-0.6}\\ \amp \approx 0.1422\text{.} \end{align*}

Thus about 14.22% of all lightbulbs fail between $t = 2$ and $t = 3\text{.}$ Clearly we could adjust the limits of integration to measure the fraction of light bulbs that fail during any time period of interest.

###### Preview Activity6.5.1

A company with a large customer base has a call center that receives thousands of calls a day. After studying the data that represents how long callers wait for assistance, they find that the function $p(t) = 0.25e^{-0.25t}$ models the time customers wait in the following way: the fraction of customers who wait between $t = a$ and $t = b$ minutes is given by

\begin{equation*} \int_a^b p(t) \, dt\text{.} \end{equation*}

Use this information to answer the following questions.

1. Determine the fraction of callers who wait between 5 and 10 minutes.

2. Determine the fraction of callers who wait between 10 and 20 minutes.

3. Next, let's study the fraction who wait up to a certain number of minutes:

1. What is the fraction of callers who wait between 0 and 5 minutes?

2. What is the fraction of callers who wait between 0 and 10 minutes?

3. Between 0 and 15 minutes? Between 0 and 20?

4. Let $F(b)$ represent the fraction of callers who wait between $0$ and $b$ minutes. Find a formula for $F(b)$ that involves a definite integral, and then use the First FTC to find a formula for $F(b)$ that does not involve a definite integral.

5. What is the value of the limit $\lim_{b \to \infty} F(b)\text{?}$ What is its meaning in the context of the problem?

### Subsection6.5.1Improper Integrals Involving Unbounded Intervals

In light of our example with light bulbs that fail, as well as with the problem involving customer wait time in Preview Activity 6.5.1, we see that it is natural to consider questions where we desire to integrate over an interval whose upper limit grows without bound. For example, if we are interested in the fraction of light bulbs that fail within the first $b$ months of use, we know that the expression

\begin{equation*} \int_0^b 0.3e^{-0.3t} \, dt \end{equation*}

measures this value. To think about the fraction of light bulbs that fail eventually, we understand that we wish to find

\begin{equation*} \lim_{b \to \infty} \int_0^b 0.3e^{-0.3t} \, dt\text{,} \end{equation*}

for which we will also use the notation

$$\int_0^\infty 0.3e^{-0.3t} \, dt\text{.}\label{E-ImpInt1}\tag{6.5.1}$$

Note particularly that we are studying the area of an unbounded region, as pictured in Figure 6.5.1.

Anytime we are interested in an integral for which the interval of integration is unbounded (that is, one for which at least one of the limits of integration involves $\infty$), we say that the integral is improper. For instance, the integrals

\begin{equation*} \int_1^{\infty} \frac{1}{x^2} \, dx, \ \ \int_{-\infty}^0 \frac{1}{1+x^2} \, dx, \ \ \text{and} \int_{-\infty}^{\infty} e^{-x^2} \, dx \end{equation*}

are all improper due to having limits of integration that involve $\infty\text{.}$ We investigate the value of any such integral by replacing the improper integral with a limit of proper integrals; for an improper integral such as $\int_0^\infty f(x) \, dx\text{,}$ we write

\begin{equation*} \int_0^\infty f(x) \, dx = \lim_{b \to \infty} \int_0^b f(x) \,dx\text{.} \end{equation*}

We can then attempt to evaluate $\int_0^b f(x) \,dx$ using the First FTC, after which we can evaluate the limit. An immediate and important question arises: is it even possible for the area of such an unbounded region to be finite? The following activity explores this issue and others in more detail.

###### Activity6.5.2

In this activity we explore the improper integrals $\int_1^{\infty} \frac{1}{x} \, dx$ and $\int_1^{\infty} \frac{1}{x^{3/2}} \, dx\text{.}$

1. First we investigate $\int_1^{\infty} \frac{1}{x} \, dx\text{.}$

1. Use the First FTC to determine the exact values of $\int_1^{10} \frac{1}{x} \, dx\text{,}$ $\int_1^{1000} \frac{1}{x} \, dx\text{,}$ and $\int_1^{100000} \frac{1}{x} \, dx\text{.}$ Then, use your calculator to compute a decimal approximation of each result.

2. Use the First FTC to evaluate the definite integral $\int_1^{b} \frac{1}{x} \, dx$ (which results in an expression that depends on $b$).

3. Now, use your work from (ii.) to evaluate the limit given by

\begin{equation*} \lim_{b \to \infty} \int_1^{b} \frac{1}{x} \, dx\text{.} \end{equation*}
2. Next, we investigate $\int_1^{\infty} \frac{1}{x^{3/2}} \, dx\text{.}$

1. Use the First FTC to determine the exact values of $\int_1^{10} \frac{1}{x^{3/2}} \, dx\text{,}$ $\int_1^{1000} \frac{1}{x^{3/2}} \, dx\text{,}$ and $\int_1^{100000} \frac{1}{x^{3/2}} \, dx\text{.}$ Then, use your calculator to compute a decimal approximation of each result.

2. Use the First FTC to evaluate the definite integral $\int_1^{b} \frac{1}{x^{3/2}} \, dx$ (which results in an expression that depends on $b$).

3. Now, use your work from (ii.) to evaluate the limit given by

\begin{equation*} \lim_{b \to \infty} \int_1^{b} \frac{1}{x^{3/2}} \, dx\text{.} \end{equation*}
3. Plot the functions $y = \frac{1}{x}$ and $y = \frac{1}{x^{3/2}}$ on the same coordinate axes for the values $x = 0 \ldots 10\text{.}$ How would you compare their behavior as $x$ increases without bound? What is similar? What is different?

4. How would you characterize the value of $\int_1^{\infty} \frac{1}{x} \, dx\text{?}$ of $\int_1^{\infty} \frac{1}{x^{3/2}} \, dx\text{?}$ What does this tell us about the respective areas bounded by these two curves for $x \ge 1\text{?}$

### Subsection6.5.2Convergence and Divergence

Our work so far has suggested that when we consider a nonnegative function $f$ on an interval $[1,\infty]\text{,}$ such as $f(x) = \frac{1}{x}$ or $f(x) = \frac{1}{x^{3/2}}\text{,}$ there are at least two possibilities for the value of $\lim_{b \to \infty} \int_1^b f(x) \, dx\text{:}$ the limit is finite or infinite. With these possibilities in mind, we introduce the following terminology.

If $f(x)$ is nonnegative for $x \ge a\text{,}$ then we say that the improper integral $\int_a^{\infty} f(x) \, dx$ converges provided that

\begin{equation*} \lim_{b \to \infty} \int_a^{b} f(x) \, dx \end{equation*}

exists and is finite. Otherwise, we say that $\int_a^{\infty} f(x) \, dx$ diverges.

We normally restrict our interest to improper integrals for which the integrand is nonnegative. Further, we note that our primary interest is in functions $f$ for which $\lim_{x \to \infty} f(x) = 0\text{,}$ for if the function $f$ does not approach $0$ as $x \to \infty\text{,}$ then it is impossible for $\int_a^{\infty} f(x) \, dx$ to converge.

###### Activity6.5.3

Determine whether each of the following improper integrals converges or diverges. For each integral that converges, find its exact value.

1. $\int_1^{\infty} \frac{1}{x^2} \, dx$

2. $\int_0^{\infty} e^{-x/4} \, dx$

3. $\int_2^{\infty} \frac{9}{(x+5)^{2/3}} \, dx$

4. $\int_4^{\infty} \frac{3}{(x+2)^{5/4}} \, dx$

5. $\int_0^{\infty} x e^{-x/4} \, dx$

6. $\int_1^{\infty} \frac{1}{x^p} \, dx\text{,}$ where $p$ is a positive real number

### Subsection6.5.3Improper Integrals Involving Unbounded Integrands

It is also possible for an integral to be improper due to the integrand being unbounded on the interval of integration. For example, if we consider

\begin{equation*} \int_0^1 \frac{1}{\sqrt{x}} \, dx\text{,} \end{equation*}

we see that because $f(x) = \frac{1}{\sqrt{x}}$ has a vertical asymptote at $x = 0\text{,}$ $f$ is not continuous on $[0,1]\text{,}$ and the integral is attempting to represent the area of the unbounded region shown at right in Figure 6.5.2.

Just as we did with improper integrals involving infinite limits, we address the problem of the integrand being unbounded by replacing such an improper integral with a limit of proper integrals. For example, to evaluate $\int_0^1 \frac{1}{\sqrt{x}} \, dx\text{,}$ we replace $0$ with $a$ and let $a$ approach 0 from the right. Thus,

\begin{equation*} \int_0^1 \frac{1}{\sqrt{x}} \, dx = \lim_{a \to 0^+} \int_a^1 \frac{1}{\sqrt{x}} \, dx\text{,} \end{equation*}

and then we evaluate the proper integral $\int_a^1 \frac{1}{\sqrt{x}} \, dx\text{,}$ followed by taking the limit. In the same way as with improper integrals involving unbounded regions, we will say that the improper integral converges provided that this limit exists, and diverges otherwise. In the present example, we observe that

\begin{align*} \int_0^1 \frac{1}{\sqrt{x}} \, dx \amp = \lim_{a \to 0^+} \int_a^1 \frac{1}{\sqrt{x}} \, dx\\ \amp = \lim_{a \to 0^+} 2\sqrt{x} \big\vert_a^1\\ \amp = \lim_{a \to 0^+} 2\sqrt{1} - 2\sqrt{a}\\ \amp = 2\text{,} \end{align*}

and therefore the improper integral $\int_0^1 \frac{1}{\sqrt{x}} \, dx$ converges (to the value 2).

We have to be particularly careful with unbounded integrands, for they may arise in ways that may not initially be obvious. Consider, for instance, the integral

\begin{equation*} \int_1^3 \frac{1}{(x-2)^2} \, dx\text{.} \end{equation*}

At first glance we might think that we can simply apply the Fundamental Theorem of Calculus by antidifferentiating $\frac{1}{(x-2)^2}$ to get $-\frac{1}{x-2}$ and then evaluate from $1$ to $3\text{.}$ Were we to do so, we would be erroneously applying the FTC because $f(x) = \frac{1}{(x-2)^2}$ fails to be continuous throughout the interval, as seen in Figure 6.5.3.

Such an incorrect application of the FTC leads to an impossible result ($-2$), which would itself suggest that something we did must be wrong. Indeed, we must address the vertical asymptote in $f(x) = \frac{1}{(x-2)^2}$ at $x = 2$ by writing

\begin{equation*} \int_1^3 \frac{1}{(x-2)^2} \, dx = \lim_{a \to 2^-} \int_1^a \frac{1}{(x-2)^2} \, dx + \lim_{b \to 2^+} \int_b^3 \frac{1}{(x-2)^2} \, dx \end{equation*}

and then evaluate two separate limits of proper integrals. For instance, doing so for the integral with $a$ approaching $2$ from the left, we find

\begin{align*} \int_1^2 \frac{1}{(x-2)^2} \, dx \amp = \lim_{a \to 2^-} \int_1^a \frac{1}{(x-2)^2} \, dx\\ \amp = \lim_{a \to 2^-} -\frac{1}{(x-2)} \bigg\vert_1^a\\ \amp = \lim_{a \to 2^-} -\frac{1}{(a-2)} + \frac{1}{1-2}\\ \amp =\infty\text{,} \end{align*}

since $\frac{1}{a-2} \to -\infty$ as $a$ approaches 2 from the left. Thus, the improper integral $\int_1^2 \frac{1}{(x-2)^2} \, dx$ diverges; similar work shows that $\int_2^3 \frac{1}{(x-2)^2} \, dx$ also diverges. From either of these two results, we can conclude that that the original integral, $\int_1^3 \frac{1}{(x-2)^2} \, dx$ diverges, too.

###### Activity6.5.4

For each of the following definite integrals, decide whether the integral is improper or not. If the integral is proper, evaluate it using the First FTC. If the integral is improper, determine whether or not the integral converges or diverges; if the integral converges, find its exact value.

1. $\int_0^1 \frac{1}{x^{1/3}} \, dx$

2. $\int_0^2 e^{-x} \, dx$

3. $\int_1^4 \frac{1}{\sqrt{4-x}} \, dx$

4. $\int_{-2}^2 \frac{1}{x^2} \, dx$

5. $\int_0^{\pi/2} \tan(x) \, dx$

6. $\int_0^1 \frac{1}{\sqrt{1-x^2}} \, dx$

### Subsection6.5.4Summary

• An integral $\int_a^b f(x) \, dx$ can be improper if at least one of $a$ or $b$ is $\pm \infty\text{,}$ making the interval unbounded, or if $f$ has a vertical asymptote at $x = c$ for some value of $c$ that satisfies $a \le c \le b\text{.}$ One reason that improper integrals are important is that certain probabilities can be represented by integrals that involve infinite limits.

• When we encounter an improper integral, we work to understand it by replacing the improper integral with a limit of proper integrals. For instance, we write

\begin{equation*} \int_a^\infty f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx\text{,} \end{equation*}

and then work to determine whether the limit exists and is finite. For any improper integral, if the resulting limit of proper integrals exists and is finite, we say the improper integral converges. Otherwise, the improper integral diverges.

• An important class of improper integrals is given by

\begin{equation*} \int_1^{\infty} \frac{1}{x^p} \, dx \end{equation*}

where $p$ is a positive real number. We can show that this improper integral converges whenever $p \gt 1\text{,}$ and diverges whenever $0 \lt p \le 1\text{.}$ A related class of improper integrals is $\int_0^1 \frac{1}{x^p} \, dx\text{,}$ which converges for $0 \lt p \lt 1\text{,}$ and diverges for $p \ge 1\text{.}$

### SubsectionExercises

###### 6

Determine, with justification, whether each of the following improper integrals converges or diverges.

1. $\int_e^{\infty} \frac{\ln(x)}{x} \, dx$

2. $\int_e^{\infty} \frac{1}{x\ln(x)} \, dx$

3. $\int_e^{\infty} \frac{1}{x(\ln(x))^2} \, dx$

4. $\int_e^{\infty} \frac{1}{x(\ln(x))^p} \, dx\text{,}$ where $p$ is a positive real number

5. $\int_0^1 \frac{\ln(x)}{x} \, dx$

6. $\int_0^1 \ln(x) \, dx$

###### 7

Sometimes we may encounter an improper integral for which we cannot easily evaluate the limit of the corresponding proper integrals. For instance, consider $\int_1^{\infty} \frac{1}{1+x^3} \, dx\text{.}$ While it is hard (or perhaps impossible) to find an antiderivative for $\frac{1}{1+x^3}\text{,}$ we can still determine whether or not the improper integral converges or diverges by comparison to a simpler one. Observe that for all $x \gt 0\text{,}$ $1 + x^3 \gt x^3\text{,}$ and therefore

\begin{equation*} \frac{1}{1+x^3} \lt \frac{1}{x^3}\text{.} \end{equation*}

It therefore follows that

\begin{equation*} \int_1^b \frac{1}{1+x^3} \, dx \lt \int_1^b \frac{1}{x^3} \, dx \end{equation*}

for every $b \gt 1\text{.}$ If we let $b \to \infty$ so as to consider the two improper integrals $\int_1^\infty \frac{1}{1+x^3} \, dx$ and $\int_1^\infty \frac{1}{x^3} \, dx\text{,}$ we know that the larger of the two improper integrals converges. And thus, since the smaller one lies below a convergent integral, it follows that the smaller one must converge, too. In particular, $\int_1^\infty \frac{1}{1+x^3} \, dx$ must converge, even though we never explicitly evaluated the corresponding limit of proper integrals. We use this idea and similar ones in the exercises that follow.

1. Explain why $x^2 + x + 1 \gt x^2$ for all $x \ge 1\text{,}$ and hence show that $\int_1^{\infty} \frac{1}{x^2 + x + 1} \, dx$ converges by comparison to $\int_1^{\infty} \frac{1}{x^2} \, dx\text{.}$

2. Observe that for each $x \gt 1\text{,}$ $\ln(x) \lt x\text{.}$ Explain why

\begin{equation*} \int_2^b \frac{1}{x} \, dx \lt \int_2^b \frac{1}{\ln(x)} \,dx \end{equation*}

for each $b \gt 2\text{.}$ Why must it be true that $\int_2^b \frac{1}{\ln(x)} \, dx$ diverges?

3. Explain why $\sqrt{\frac{x^4+1}{x^4}} \gt 1$ for all $x \gt 1\text{.}$ Then, determine whether or not the improper integral

\begin{equation*} \int_1^{\infty} \frac{1}{x} \cdot \sqrt{\frac{x^4+1}{x^4}} \, dx \end{equation*}

converges or diverges.


text:
How to suppress undesirable kerning rules in math inline cells?

Update: (11/29/2012) Looks like Mathematica 9 has the same behavior.

The following graphic shows how Mathematica kerns italic characters in an inline math formula cell (first row) where an \[InvisibleSpace] has been inserted between consecutive characters (which could represent either multiplication of terms or multi-indices depending on context) vs. in an ordinary, non-math text cell which was switched to italics using Ctrl-I (second row).

When the letters ijk for example appear together in a subscript as indices, the spacing between the i and the j appears too wide relative to the distance between the j and the k.

Original Q: Is it possible to turn off this special kerning in math formula cells?

Updated Q: (10/1/2012) A reasonable but nevertheless somewhat inconvenient method to work around this issue has been suggested below. Does anyone know the reason why the kerning rules for juxtaposing italic characters inside math formula cells are different from those for normal, non-math italic text?

For ease of copying:

Cell[TextData[{
"Text ",
Cell[BoxData[
FormBox[
RowBox[{
"a", "\[InvisibleSpace]", "b", "\[InvisibleSpace]", "c",
"\[InvisibleSpace]", "d", "\[InvisibleSpace]", "e",
"\[InvisibleSpace]", "f", "\[InvisibleSpace]", "g",
"\[InvisibleSpace]", "h", "\[InvisibleSpace]", "i",
"\[InvisibleSpace]", "j", "\[InvisibleSpace]", "k",
"\[InvisibleSpace]", "l", "\[InvisibleSpace]", "m",
"\[InvisibleSpace]", "n", "\[InvisibleSpace]", "o",
"\[InvisibleSpace]", "p", "\[InvisibleSpace]", "q",
"\[InvisibleSpace]", "r", "\[InvisibleSpace]", "s",
"\[InvisibleSpace]", "t", "\[InvisibleSpace]", "u",
"\[InvisibleSpace]", "v", "\[InvisibleSpace]", "w",
"\[InvisibleSpace]", "x", "\[InvisibleSpace]", "y",
" Text"
}], "Text"]

-
Can you specify exactly how you typed/created the two lines? Making more than one letter in an inline math cell switches the fontslant back to plain. Therefore, if I just type abc.. in an inline cell it is upright. When I use explicitly italic, than it looks equally spaced. I tested OSX here, which system do you have? –  halirutan Sep 18 '12 at 11:33
First row: Between each letter I inserted an [InvisibleSpace] to prevent the sequence of letters from switching from italic back to roman (which as you point out is the default behavior inside a math inline cell). Second row: typeset as italic text inside a text cell. Perhaps the real issue here has to do with how Mma varies the width of the [\InvisibleSpace] according to the letter forms that surround it. –  StackExchanger Sep 19 '12 at 2:36

You can redefine RowBox to automatically wrap specific characters in an AdjustmentBox which reduces the right-side spacing. This isn't perfect but might get you on the right track.

Unprotect[RowBox];
RowBox[chars_List  /; MemberQ[chars, "i" | "e" | "f" | "o"]] :=
RowBox[Replace[chars , {
"i" -> AdjustmentBox["i", BoxMargins -> {{0, -0.1}, {0, 0}}],
"e" -> AdjustmentBox["e", BoxMargins -> {{0, -0.15}, {0, 0}}],
"f" -> AdjustmentBox["f", BoxMargins -> {{0, -0.15}, {0, 0}}],
"o" -> AdjustmentBox["o", BoxMargins -> {{0, -0.1}, {0, 0}}]
}, {1}]];
Protect[RowBox];

Comparison:

This won't catch sub/superscripted characters, and it will affect all RowBoxes, not just the ones used for TraditionalForm. I could not figure out how to do the same trick, but only for FormBox[RowBox[chars_List], TraditionalForm]. Maybe someone else can figure that part out.

-
Your suggestion is a nice application of the AdjustmentBox, but raises the question now of why this kind of tweaking should even be necessary. Does anyone know the reason why the kerning rules for juxtaposing italic characters inside math formula cells are different from those for normal, non-math italic text? –  StackExchanger Sep 19 '12 at 19:30
I like your proposed solution (in lieu of a more direct fix) and will probably post my updates as a new question later. –  StackExchanger Dec 7 '12 at 6:00

format: plain


text:
In his paper Invariant Kahler metrics and projective embeddings of the flag manifold, Bull. Austal. Math. Soc. 49 (1994), K. Yang considers the flag manifold $$F_{1,2,3}(\mathbb{C}^3):=SU(3)/S(U(1)^3)$$ and determines the space of invariant Hermitian and inveriant Kahler metrics on it.
On the other hand, by applying Kodaira embedding theorem, he deduces proves that $F_{1,2,3}$ F_{1,2,3}(\mathbb{C}^3)$is projective algebraic and provides an explicit projective embedding of it. The computations are made quite explicitly in terms of the Maurer-Cartan form of$SU(3)$. So this could be one of the examples you are looking for. 1 In his paper Invariant Kahler metrics and projective embeddings of the flag manifold, Bull. Austal. Math. Soc. 49 (1994), K. Yang considers the flag manifold $$F_{1,2,3}(\mathbb{C}^3):=SU(3)/S(U(1)^3)$$ and determines the space of Hermitian and inveriant Kahler metrics on it. In particular, he shows that a Killing metric is not Kahler. Moreover, by applying Kodaira embedding theorem he deduces that$F_{1,2,3}$is projective algebraic. The computations are made quite explicitly in terms of the Maurer-Cartan form of$SU(3)\$.


text:
errors in running smartpca
2
2
Entering edit mode
4.6 years ago
ShirleyDai ▴ 40

Hi, I've got an error when running smartpca. Anyone has ideas what was going wrong? Thanks.

$perl smartpca.perl -i raw-GWA-data.hapmap3r2.pruned.bed -a raw-GWA-data.hapmap3r2.pruned.pedsnp -b raw-GWA-data.hapmap3r2.pruned.pedind -o raw-GWA-data.hapmap3r2.pruned.pca -p raw-GWA-data.hapmap3r2.pruned.plot -e raw-GWA-data.hapmap3r2.pruned.eval -l raw-GWA-data.hapmap3r2.pruned.log -k 2 -t 2 -w pca-populations.txt smartpca -p raw-GWA-data.hapmap3r2.pruned.pca.par >raw-GWA-data.hapmap3r2.pruned.log fatalx: snps out of order and packed format. Run convertf with pordercheck: NO Aborted (core dumped) ploteig -i raw-GWA-data.hapmap3r2.pruned.pca.evec -c 1:2 -p -x -y -o raw-GWA-data.hapmap3r2.pruned.plot.xtxt evec2pca.perl 2 raw-GWA-data.hapmap3r2.pruned.pca.evec raw-GWA-data.hapmap3r2.pruned.pedind raw-GWA-data.hapmap3r2.pruned.pca  plink smartpca eigensoft GWAS • 3.4k views ADD COMMENT 4 Entering edit mode 4.1 years ago Vincent Laufer ★ 1.1k This problem can be caused by a variety of issues, but in my experience all of them relate to the .map or .bim file. 1) you have chr 0 or chr 24-26 data in your data set. Solution: Fix with plink --exclude or plink --chr 1-23 2) you have multiple data points with a genetic distance listed as "0" found after SNPs with a higher genetic distance. Solution: remove using a command like awk '($3==0){print \$2}' mysnpdata.map > snpstoexclude.txt then remove with plink.

3) you have data between chr 1-23 and genetic distances are assigned appropriately, but the snps are out of order. Solution: Fix with plink --recode.

2
Entering edit mode
4.5 years ago
spsoisson ▴ 20

I was getting the same error. I had SNPs labeled as being on chromosome 0. I went back to plink and removed those using --chr 1-23. I ran smartpca again and it worked.

0
Entering edit mode

Thanks. I will try it.


text:
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Some statistical physics book use: $Z=\sum e^{-H}$ and $F=-lnZ$ as defination for partition function and free energy. I think they should be $Z=\sum e^{-\frac{H}{kT}}$ and $F=-kT lnZ$ Are they ...


text:
# HP Forums

Full Version: desolve y'=(x+y)^2
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hi all,
there is a way to solve with Prime this kind of differential equation, like y'=(x+y)^2 ?
I get [[]]...
I should get TAN(x+G_0) - x
I'm in CAS, real mode (no complex checked)

Thank you
Salvo
(05-01-2015 02:45 PM)salvomic Wrote: [ -> ]hi all,
there is a way to solve with Prime this kind of differential equation, like y'=(x+y)^2 ?
I get [[]]...
I should get TAN(x+G_0) - x
I'm in CAS, real mode (no complex checked)

Thank you
Salvo
No success here.
The 50g also fails on that one with the message "Can't find ODE type".
(05-01-2015 04:11 PM)Tugdual Wrote: [ -> ]No success here.
The 50g also fails on that one with the message "Can't find ODE type".

yes, in fact!
Also my 50g fails with that message...

thank you
Funny, yesterday I had a similar Problem to solve, nearly the same diffeq, clearly that didn't work. So I did the Substitution u=x+y by hand, after that I entered the new equation in desolve and it worked. Itried then mathexplorer on my mobile, it didn't solve either equation. I think that this little trick can easily be implemented to the CAS.
Hth Arno
(05-01-2015 04:41 PM)Arno K Wrote: [ -> ]Funny, yesterday I had a similar Problem to solve, nearly the same diffeq, clearly that didn't work. So I did the Substitution u=x+y by hand, after that I entered the new equation in desolve and it worked. Itried then mathexplorer on my mobile, it didn't solve either equation. I think that this little trick can easily be implemented to the CAS.
Hth Arno

yes, I hope indeed that the substitution could be really implementable to the CAS as well.
I've tried a lot of ODE and Prime handles them well, except few types (i.e the Riccati type); this is a common form and it would be important that the Prime (and CAS) could solve it...

Another problematic form is y'= f(y/x), almost in the tries I did...

Salvo
I didn't give up but couldn't conclude.
Tried Maxima, failed.
Tried Wolfram Alpha and it found something:
$$y(x)=\frac { 1 }{ { c }_{ 1 }.{ e }^{ 2.i.x }-\frac { i }{ 2 } } -x-i$$
I checked on the 50g and this seems good but I haven't been able to relate this with a tangent...
(05-01-2015 07:06 PM)Tugdual Wrote: [ -> ]I didn't give up but couldn't conclude.
Tried Maxima, failed.
Tried Wolfram Alpha and it found something:
$$y(x)=\frac { 1 }{ { c }_{ 1 }.{ e }^{ 2.i.x }-\frac { i }{ 2 } } -x-i$$
I checked on the 50g and this seems good but I haven't been able to relate this with a tangent...

yes, I noted too...
However, if in Wolfram you clic on "step-by-step" you can get the trigonomtric solution (see image attached)...
(05-01-2015 02:45 PM)salvomic Wrote: [ -> ]hi all,
there is a way to solve with Prime this kind of differential equation, like y'=(x+y)^2 ?
I get [[]]...
I should get TAN(x+G_0) - x
I'm in CAS, real mode (no complex checked)

Thank you
Salvo
Didn't understand how quote works...my CASIO Classpad 400 comes up with the intended answer that I quoted.
(05-01-2015 07:57 PM)lrdheat Wrote: [ -> ]Didn't understand how quote works...my CASIO Classpad 400 comes up with the intended answer that I quoted.

so, in Classpad 400 the result is
TAN(x+c) - x ?

thank you
(05-01-2015 08:06 PM)salvomic Wrote: [ -> ]
(05-01-2015 07:57 PM)lrdheat Wrote: [ -> ]Didn't understand how quote works...my CASIO Classpad 400 comes up with the intended answer that I quoted.

so, in Classpad 400 the result is
TAN(x+c) - x ?

thank you
Just checked on the emulator.
I'm impressed...
(05-01-2015 08:26 PM)Tugdual Wrote: [ -> ]Just checked on the emulator.
I'm impressed...
...

great!
Xcas can solve this equation. It is a Ricatti equation, you can solve it by giving a particular solution, otherwise the system rewrites it as a 2nd order equation.
(05-02-2015 05:30 AM)parisse Wrote: [ -> ]Xcas can solve this equation. It is a Ricatti equation, you can solve it by giving a particular solution, otherwise the system rewrites it as a 2nd order equation.

thank you, Parisse.
The Xcas now is more updated than the CAS on Prime...
So, we hope that in the next FW upgrade of Prime it will be ok...

the type of equation is generally y' = f(ax+by+c) and the general solution can be got substituting u (in our case u=x+y -> u'=1+y' and so on)
The classic Riccati ODE is y'+g(x)y+h(x)y^2=k(x)

I had issue with a Riccati equation like $y' + \frac{2x+1}{x}y - \frac{1}{x}y^{2} = x+2$
Prime gives [], but the solution is $x + \frac{1}{1+cx}$
Indeed, you can also solve it by this kind of substitution. Maybe I can add another special case in desolve.
(c_1*x^2+c_2*x+c_2)/(c_1*x+c_2)
with the latest Xcas. You can get your solution with c_2=1.
You are missing y=x with your general solution.
(05-02-2015 10:50 AM)parisse Wrote: [ -> ]Indeed, you can also solve it by this kind of substitution. Maybe I can add another special case in desolve.
(c_1*x^2+c_2*x+c_2)/(c_1*x+c_2)
with the latest Xcas. You can get your solution with c_2=1.
You are missing y=x with your general solution.

thank you!
I'll try, hoping in the future to have the XCas feature also in the Prime...

(I wonder if there is a regular version of XCas that run into Mac OS X (without Windows emulator): I need it...)
EDIT: found! http://www-fourier.ujf-grenoble.fr/~pari...all_en#osx
There are native versions of Giac/Xcas for the 3 major desktop OS: windows, linux and mac. Giac is also ported to pure javascript code (embeddable into a web application), android (Xcas pad UI) and all TI nspire up to OS 3.9 (khicas, run also on non-CAS models). For ios, there is no free port I know, pocketcas is using giac as math kernel.
(05-02-2015 12:43 PM)parisse Wrote: [ -> ]There are native versions of Giac/Xcas for the 3 major desktop OS: windows, linux and mac. Giac is also ported to pure javascript code (embeddable into a web application), android (Xcas pad UI) and all TI nspire up to OS 3.9 (khicas, run also on non-CAS models). For ios, there is no free port I know, pocketcas is using giac as math kernel.

well :-)
I hope to have also an iOS version too (for my iPhone), paid or not!

Installed just now the Mac OS X version (unstable).
Trying the ODE I get an error (see image): have I missed something in syntax?

EDIT:
http://www-fourier.ujf-grenoble.fr/~pari...ble.dmg.gz
http://www-fourier.ujf-grenoble.fr/~pari...sx6.dmg.gz


text:
# SNLSDP

SNLSDP version 0 -- a MATLAB software for sensor network localization It implemented an SDP based approach with regularization for solving sensor network localization problems. The algorithm first solves an SDP relaxation (with regularization) of the non-convex minimization problem (1), and use the SDP computed solution as the starting point for a gradient descent method with backtracking line search to solve the smooth unconstrained problem (2). This software package is designed for solving small size senor network localization problems with up to 200 sensors and a few thousands given distances. (Source: http://plato.asu.edu)


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# Converting hair particles to a mesh while keeping the animation

There's this game that I'm working on, using Unity as the engine and Blender to make the models, animations etc.

After going though a lot of trouble trying to make the hair look like feathers, I stuck trying to export my model as a .fbx to Unity.

Only the model and the animation work properly. The feathers don't scale with the model and the rotation is mess inside the engine. I read in many sites that Unity doesn't support the particle system of Blender, so I'm trying to apply the particle modifier, but then I lose the animation. The feathers are not connected to the mesh, is a scalp with the same animation of the body.

If anyone even understands the problem I'm trying to express with my low-level English, I would be grateful for any help.

• – Brenticus Feb 4 '18 at 3:47


text:
# When a quantity of monatomic ideal gas expands at a constant pressure of 4.00 x 10^{4} Pa, the...

## Question:

When a quantity of monatomic ideal gas expands at a constant pressure of {eq}\rm 4.00 \times 10^{4} \ Pa {/eq}, the volume of the gas increases from {eq}\rm 2.00 \times 10^{-3} \ m^3 {/eq} to {eq}\rm 8.44 \times 10^{-3} \ m^3 {/eq}. What is the change in the internal energy of the gas?

## Internal Energy of an Ideal Gas:

Internal energy is a thermodynamic parameter defined as a sum of kinetic energies of all molecules of the sample and total potential energy of all inter-molecular interactions. For an ideal gas, we can neglect the potential energy. The average kinetic energy of gas molecules could be estimated in the frameworks of the kinetic theory where it was demonstrated that the average kinetic energy of the molecules is related with gas temperature.

The change in the internal energy of the gas {eq}\Delta U=386\ \rm J {/eq}.

The internal energy of a mono-atomic ideal gas

{eq}U=3/2nRT {/eq}

where n is the number of moles, R is the gas constant, T is the temperature.

On the other hand the equation of state of an ideal gas

{eq}pV=nRT {/eq}

Combining the two equations we obtain

{eq}U=3/2pV {/eq}

The change of the internal energy {eq}\Delta U = 3/2p\Delta V = 3/2 \cdot \rm 4.00 \times 10^{4} (\rm 8.44 \times 10^{-3}-\rm 2.00 \times 10^{-3})= 386 \ J {/eq}


text:
# Arbital page summaries

Each Arbital page can have multiple summaries. These summaries show up in places where the user needs a quick overview of the concept the page is explaining: link popovers and in an embedded vote bar. Markdown may be used in all summaries. Some encouraged summary types are listed below, though pages may work best with a different set of summaries (for example, if the page is about an overloaded term it should have a summary for each definition).

### Brief

Brevity is the soul of wit.

This is a short summary for people who just want a gist of what the page is about. Keep it to a few sentences. If the normal summary is short already, this is unnecessary.

### Summary

Summary of the page goes here. *Markdown* is okay!

This is the summary shown by default. It should work for most readers. Keep it to one or two paragraphs.

If this summary is not specified, Arbital will use the first paragraph instead.

### Technical

Very technical summary. *Markdown* is okay!

Feel free to use domain specific jargon in this summary. Keep it to one or two paragraphs.

If the page has [3r voting enabled, and another page embed this one, this summary will be displayed. It should clearly state what is being voted on and how the correct answer will be decided.

Parents:


text:
## Encyclopedia > Ecuador

Article Content

# Ecuador

The Republic of Ecuador was one of three countries that emerged from the collapse of Gran Colombia[?] in 1830 (the others being Colombia and Venezuela). Between 1904 and 1942, Ecuador lost territories in a series of conflicts with its neighbors. A border war with Peru that flared in 1995 was resolved in 1999, although Peru still occupies and claims a large portion of territory to the east of Ecuador that was once Ecuadorean territory and which Ecuador still considers to belong to Ecuador.

The capital is Quito and the largest city is Guayaquil.

On 24 November 2002, Lucio Gutiérrez won the presidential elections. He became president on 15 January 2003.

This article contains information from the CIA World Factbook 2000 and the U.S. Department of State website.

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Exponential function ... ${d \over dx} a^{bx} = \ln(a) b a^{bx}.$ If a variable's growth or decay rate is proportional to its size, as is the case in unlimited population ...


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# Inequality proof

#### Townsend

Show that for each complex sequence $$c_1, c_2, ..., c_n$$ and for each integer $$1 \leq H < N$$ one has the inequality

$$| \sum_{n=1}^N c_n|^2 \leq \frac{4N}{H+1} ( \sum_{n=1}^N |c_n|^2 + \sum_{h=1}^H | \rho_N(h)|)$$

Any one.....matt grime perhaps?

note: if anyone actually wants to work this out let me know and I will fill in the missing parts...but don't ask me to do it.... :tongue2:

Last edited:
Related General Discussion News on Phys.org

#### honestrosewater

Gold Member
Yeah, that's good. The more fancy meaningless symbols, the better.

#### Townsend

honestrosewater said:
Yeah, that's good. The more fancy meaningless symbols, the better.
There is an important part missing but it is a true bound...not just meaningless...

#### honestrosewater

Gold Member
Okay, I'll take your word for it. It would be nice if someone were around to explain it to me.

#### Townsend

honestrosewater said:
Okay, I'll take your word for it. It would be nice if someone were around to explain it to me.
Yeah...it would take a really smart....creative....mathematician to do so...who would be able to do that I wonder??????

#### honestrosewater

Gold Member
Townsend said:
Yeah...it would take a really smart....creative....
... and patient. I've never even worked with complex numbers before. You can just treat them as ordered pairs of real numbers, right? I think I'd like that approach.

#### Johnny5

I know the solution, but I won't say to give other people a chance. It's not that hard.

#### wolram

Gold Member
Its packman eating flies, so the ansewer must be MxBxHs

### Physics Forums Values

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text:
Copper cable certification

10 Gb Certification Kit

In copper twisted pair wire networks, copper cable certification is achieved through a thorough series of tests in accordance with Telecommunications Industry Association (TIA) or International Organization for Standardization (ISO) standards. These tests are done using a certification-testing tool, which provide “Pass” or “Fail” information. While certification can be performed by the owner of the network, certification is primarily done by datacom contractors. It is this certification that allows the contractors to warranty their work.

Need for certification

Installers who need to prove to the network owner that the installation has been done correctly and meets TIA or ISO standards need to certify their work. Network owners who want to guarantee that the infrastructure is capable of handling a certain application (e.g. Voice over Internet) will use a tester to certify the network infrastructure. In some cases, these testers are used to pinpoint specific problems. Certification tests are vital if there is a discrepancy between the installer and network owner after an installation has been performed.

The Standards

The performance tests and their procedures have been defined in the ANSI/TIA/EIA-568-B.1 standard and the ISO/IEC 11801 standard. The TIA standard defines performance in categories (Cat 3, Cat 5e, Cat 6, Cat 6A) and the ISO defines classes (Class C, D, E, EA, F and FA). These standards define the procedure to certify that an installation meets performance criteria in a given category or class.

The significance of each category or class is the limit values of which the Pass/Fail and frequency ranges are measured: Cat 3 and Class C (no longer used) test and define communication with 16 MHz bandwidth, Cat 5e and Class D with 100 MHz bandwidth, Cat 6 and Class E up to 250 MHz, Cat6A and Class EA up to 500 MHz, Cat7 and Class F up to 600 MHZ and Cat 7A and Class FA with a frequency range through 1000 MHz.

The standards also define that data from each test result must be collected and stored in either print or electronic format for future inspection.

The Tests

Test Parameter TIA-568-B ISO 11801:2002
Wiremap Pass/Fail Pass/Fail
Propagation Delay Pass/Fail Pass/Fail
Delay Skew Pass/Fail Pass/Fail
Cable Length Pass/Fail Information only
Insertion Loss (IL) Pass/Fail Pass/Fail
Return Loss (RL) Pass/Fail (except Cat3) Pass/Fail
Near-End Crosstalk (NEXT) Pass/Fail Pass/Fail
Power Sum NEXT (PSNEXT) Pass/Fail Pass/Fail
Equal-Level Far-End Crosstalk (ELFEXT) Pass/Fail Pass/Fail
Power Sum ELFEXT (PSELFEXT) Pass/Fail Pass/Fail
Attenuation-to-Crosstalk Ratio (ACR) Information only Pass/Fail (except Class C)
Power sum ACR (PSACR) Information only Pass/Fail (except Class C)
DC Loop Resistance Pass/Fail

Wiremap

The wiremap test is used to identify physical errors of the installation; proper pin termination at each end, shorts between any two or more wires, continuity to the remote end, split pairs, crossed pairs, reversed pairs, and any other mis-wiring.

See TIA/EIA-568-B for wiring diagram via Mudit

Propagation Delay

The Propagation Delay test tests for the time it takes for the signal to be sent from one end and received by the other end.

Delay Skew

The Delay Skew test tests for the difference in propagation delay between the fastest and slowest set of wire pairs. An ideal skew is between 25 and 50 nanoseconds over a 100-meter cable. The lower this skew the better; less than 25 ns is excellent, but 45 to 50 ns is marginal. (100 meters in a cable electromagnetic wave passes behind 334ns)

Cable Length

The Cable Length test verifies that the copper cable from the transmitter to receiver does not exceed the maximum recommended distance of 100 meters in a 10BASE-T/100BASE-TX/1000BASE-T network.

Insertion Loss

Insertion loss, also referred to as attenuation, refers to the loss of signal strength at the far end of a line compared to the signal that was introduced into the line. This loss is due to the electrical impedance of the copper cable, the loss of energy through the cable insulation and the impedance caused by the connectors. Insertion loss is usually expressed in decibels dB with a minus sign. Insertion loss increases with distance and frequency. For every 3 dB of loss, signal power is reduced by a factor of ${\displaystyle 2}$ and signal amplitude is reduced by a factor of ${\displaystyle {\sqrt {2}}}$.

Return Loss

Return Loss is the measurement (in dB) of the amount of signal that is reflected back toward the transmitter. The reflection of the signal is caused by the variations of impedance in the connectors and cable and is usually attributed to a poorly terminated wire. The greater the variation in impedance, the greater the return loss reading. If 3 pairs of wire pass by a substantial amount, but the 4 pair barely passes, it usually is an indication of a bad crimp or bad connection at the RJ45 plug. Return loss is usually not significant in the loss of a signal, but rather signal jitter.

Near-End Crosstalk (NEXT)

Near-End Crosstalk (NEXT) is an error condition that describes the occurrence of a signal from one wire pair radiating to and interfering with the signal of another wire pair. It is the difference in amplitude (in dB) between a transmitted signal and the crosstalk received on other cable pairs at the same end of the cabling. Higher NEXT values correspond to better cabling performance. A higher value is desirable as it would indicate that the power transmitted is greater in magnitude than the power induced onto another wire pair given that the NEXT measurement is simply a difference calculation. NEXT must be measured from each pair to each other pair in twisted pair cabling and from each end of the connection. NEXT is measured 30 meters (about 98 feet) from the injector / generator. Higher near end crosstalk values correspond to higher overall circuit performance. Low NEXT values on a UTP LAN that will be using an older signaling standard (IEEE 802.3i and earlier) are particularly detrimental. It could be an indication of improper termination.

Power Sum NEXT (PSNEXT)

Power Sum NEXT (NEXT) is the sum of NEXT values from 3 wire pairs as they affect the other wire pair. The combined effect of NEXT can be very detrimental to the signal.

The Equal-Level Far-End Crosstalk (ELFEXT)

The Equal-Level Far-End Crosstalk (ELFEXT) test measures Far-End Crosstalk (FEXT). FEXT is very similar to NEXT, but happens at the receiver side of the connection. Due to attenuation on the line, the signal causing the crosstalk diminishes as it gets further away from the transmitter. Because of this, FEXT is usually less detrimental to a signal than NEXT, but still important nonetheless. Recently the designation was changed from ELFEXT to ACR-F (far end ACR).

Power Sum ELFEXT (PSELFEXT)

Power Sum ELFEXT (PSELFEXT) is the sum of FEXT values from 3 wire pairs as they affect the other wire pair, minus the insertion loss of the channel. Recently the designation was changed from PSELFEXT to PSACR-F (far end ACR).

Attenuation-to-Crosstalk ratio (ACR)

Attenuation-to-Crosstalk ratio (ACR) is the difference between the signal attenuation produced NEXT and is measured in decibels (dB). The ACR indicates how much stronger the attenuated signal is than the crosstalk at the destination (receiving) end of a communications circuit. The ACR figure must be at least several decibels for proper performance. If the ACR is not large enough, errors will be frequent. In many cases, even a small improvement in ACR can cause a dramatic reduction in the bit error rate. Sometimes it may be necessary to switch from un-shielded twisted pair (UTP) cable to shielded twisted pair (STP) in order to increase the ACR.

Power Sum ACR (PSACR)

Power Sum ACR (PSACR) done in the same way as ACR, but using the PSNEXT value in the calculation rather than NEXT.

DC Loop Resistance

DC Loop Resistance measures the total resistance through one wire pair looped at one end of the connection. This will increase with the length of the cable. DC resistance usually has less effect on a signal than insertion loss, but plays a major role if power over Ethernet is required. Also measured in ohms as the characteristic impedance of the cable, which is independent of the cable length.


text:
Lemma 10.135.8. Let $k$ be a field. Let $S$ be a finite type $k$-algebra. Let $\mathfrak q$ be a prime of $S$. The following are equivalent:

1. The local ring $S_{\mathfrak q}$ is a complete intersection ring (Definition 10.135.5).

2. There exists a $g \in S$, $g \not\in \mathfrak q$ such that $S_ g$ is a local complete intersection over $k$.

3. There exists a $g \in S$, $g \not\in \mathfrak q$ such that $S_ g$ is a global complete intersection over $k$.

4. For any presentation $S = k[x_1, \ldots , x_ n]/I$ with $\mathfrak q' \subset k[x_1, \ldots , x_ n]$ corresponding to $\mathfrak q$ any of the equivalent conditions (1) – (5) of Lemma 10.135.4 hold.

Proof. This is a combination of Lemmas 10.135.4 and 10.135.7 and the definitions. $\square$

There are also:

• 2 comment(s) on Section 10.135: Local complete intersections

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).


text:
# Flowing triangles

I have admired the work of the artist Bridget Riley for a long time. She is now in her eighties, but as it seems still very creative and productive. Some of her recent work combines simple triangles in fascinating compositions. The longer I look at them, the more patterns I recognise.

Yet, the actual painting can be explained easily, in a sense of a specification document to reproduce the pattern precisely. However, seeing the real print, as I had the chance at the London Art Fair last week, and a reproduction on the screen is incommensurable.

Having said that, I could not resist programming a figure that resembles the artwork labelled Bagatelle 2. Well, at least I can say that I learned more about grid [1], grid.path [2] and gridSVG [3] in R.

### References

[1] P. Murrell. R Graphics, Second Edition. CRC Press. 2011
[2] P. Murrell. What's in a Name? . The R Journal, 4(2):5–12, dec 2012.
[3] P. Murrell and S. Potter. gridSVG: Export grid graphics as SVG. R package 1.5-0. 2015

### Session Info

R version 3.2.3 (2015-12-10)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X 10.11.2 (El Capitan)

locale:
[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8

attached base packages:
[1] grid      stats     graphics  grDevices utils  datasets
[7] methods   base

other attached packages:
[1] gridSVG_1.5-0    data.table_1.9.6

loaded via a namespace (and not attached):
[1] tools_3.2.3   RJSONIO_1.3-0 chron_2.3-47  XML_3.98-1.3


text:
Spaces with a quasi triangle inequality - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T12:49:19Z http://mathoverflow.net/feeds/question/56118 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/56118/spaces-with-a-quasi-triangle-inequality Spaces with a quasi triangle inequality John H 2011-02-20T23:17:56Z 2011-02-21T09:17:31Z
How do you call a space with a function which is symmetric, non negative, positive definite and which satisfies a quasi-triangle inequality:

$d(x,z) \leq C( d(x,y)+d(y,z) )$

for all $x,y,z$ and some $C > 1$?

That is, it satisfies all the axioms of a metric space except for the triangle inequality, which is replaced by the one above.

Can anyone provide any reference on these spaces?

Thanks.
 http://mathoverflow.net/questions/56118/spaces-with-a-quasi-triangle-inequality/56119#56119 Answer by ex falso quodlibet for Spaces with a quasi triangle inequality ex falso quodlibet 2011-02-20T23:50:12Z 2011-02-21T02:04:41Z
Your construction is a special case of semimetric spaces with relaxed triangle inequality: http://en.wikipedia.org/wiki/Semimetric_space#Semimetrics. This type of metric is sometimes also called non-Archimedian metric. There is a classical paper of W.A.Wilson "On semi-metric spaces", Amer. J. Math. 53 (1931) 361–373, on the subject. Also, I have seen this type of construction mostly used in fixed-point theory, so this would be an additional keyword to look for.

EDIT: To answer your second question about whether Banach fixed-point theorem would be applicable to semimetric spaces: In general one needs $(X,d)$ to be bounded, otherwise there are counter-examples. Consider $X=\mathbb{N}$, $d(n,m):=\frac{|n-m|}{2^{\min(n,m)}}$ and $f(n):=n+1$. Then $(X,d)$ is $d$-Cauchy complete semimetric space (!), but $f$ has no fixed points, even though it is a contraction w.r.t. $d$ with contraction constant $1/2$. This example is taken from the paper "Nonlinear Contractions on Semimetric Spaces" by J. Jachymski, J. Matkowski, T. Swiatkowski, Journal of Applied Analysis Vol. 1, No. 2 (1995), pp. 125–134, where you can also find the proof of the Banach Fixed-Point Theorem for bounded semimetric spaces and some more related results.
 http://mathoverflow.net/questions/56118/spaces-with-a-quasi-triangle-inequality/56120#56120 Answer by Mark Sapir for Spaces with a quasi triangle inequality Mark Sapir 2011-02-20T23:58:07Z 2011-02-20T23:58:07Z
This is called "C-relaxed triangular inequality". See, for example, this  paper by Fagin and Stockmeyer.
 http://mathoverflow.net/questions/56118/spaces-with-a-quasi-triangle-inequality/56155#56155 Answer by Gideon Schechtman for Spaces with a quasi triangle inequality Gideon Schechtman 2011-02-21T09:12:06Z 2011-02-21T09:17:31Z
Here is a negative answer for your additional remark concerning Banach's fixed point theorem: Consider $d(x,y)=(\int_0^1|x-y|^p)^{1/p},$ $0<p<1$ which satisfies the quasi-triangle inequality. Look at the set of all (measurable real) functions on $[0,1]$ boundaed between 0 and 2 and of integral 1. Look at the Baker transformation on this set: first map $x$ to $y(t)=2x(2t)$, $0\le t\le 1/2$ then trancate at hight 2 and shift what remains ($(y-1)^{+}$) by $1/2$ to the right and $2$ down. I think I checked that it is a contraction (with constant $2^{p-1}$). This map is known not to have a fixed point, see the following paper of Dale Alspach: http://www.claremontmckenna.edu/math/moneill/Math%20138/papers138/Alspach.pdf


text:
# All Questions

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#### A 5 \mu W Standard Cell Memory-based Configurable Hyperdimensional Computing Accelerator for Always-on Smart Sensing

##### Manuel Eggimann, Abbas Rahimi, Luca Benini

Hyperdimensional computing (HDC) is a brain-inspired computing paradigm based on high-dimensional holistic representations of vectors. It recently gained attention for embedded smart sensing due to its inherent error-resiliency and suitability to highly parallel hardware implementations. In this work, we propose a programmable all-digital CMOS implementation of a fully autonomous HDC accelerator for always-on classification in energy-constrained sensor nodes. By using energy-efficient standard cell memory (SCM), the design is easily cross-technology mappable. It achieves extremely low power, 5 $\mu W$ in typical applications, and an energy-efficiency improvement over the state-of-the-art (SoA) digital architectures of up to 3$\times$ in post-layout simulations for always-on wearable tasks such as EMG gesture recognition. As part of the accelerator's architecture, we introduce novel hardware-friendly embodiments of common HDC-algorithmic primitives, which results in 3.3$\times$ technology scaled area reduction over the SoA, achieving the same accuracy levels in all examined targets. The proposed architecture also has a fully configurable datapath using microcode optimized for HDC stored on an integrated SCM based configuration memory, making the design "general-purpose" in terms of HDC algorithm flexibility. This flexibility allows usage of the accelerator across novel HDC tasks, for instance, a newly designed HDC applied to the task of ball bearing fault detection.

arrow_drop_up


text:
245 views
How many 3 letter words can be formed from word PRACTICES?

P R A C T I C E S

This word contains 8 distinct letters ( P, R, A, C, T, I, E, S), among them C is repeated twice (2).

The qs. is to make 3-letter words out of this letter.

So, we can form 3-letter words in this ways:

1.  All the letters of the word (3-letter word) will be Distinct.

OR

2. The word will be having a letter which is repeated Twice.

So, the 1st one:

There are 8 distinct letters, out of which we have to choose 3 letters and arrange them.

Hence,  8P3 = 8!/(8-3)! = 8!/5! = 336

2nd:

So, we're having only 1 letter which is repeated Twice (1C1) & we have to select another letter from remaining 7 letters (7C1) & then we have to arrange the word (3!/2!).

(here 1 thing may comes into mind that why 3!/2! ? : the 3 -letter word is containing 3 letters, out of which 1 letter is distinct and another 2 letter is identical(alphabet 'C'), so 1 letter is repeated twice in a 3 letter word, therefore 3!/2!)

Hence,    1C1 7C1 * 3!/2! = 1 * 7!/{1!*(7-1)!} * 3 = 7*3 = 21

Total ways = 1st one OR 2nd one

= 336 + 21

= 357

answered by (5.7k points) 7 22 138
selected
$PRA'CC'TIES$
No of words with 0 repetition $: 8_{c_3}*3!=336$
No of words with 1 repetition $: 7*3=21 \\ Total : \color{Blue}{357}$
answered by (870 points) 2 16

+1 vote


text:
The No.1 Website for Pro Audio
15th March 2009
#1
Lives for gear

DIY transformer box to add color

I heard an engineer talking about running tracks through a 1:1 transformer to add color to the sound. I think he said it was a Sowter that he took out of a broken Allen & Heath desk. After I did a search on google it seems that there might be a way of building a box with ins and outs, connected to the transformer that you can do this with, but I've found little more about it.

Can anyone tell me how I could wire something like this or give me schematics for it?

What might the results of doing something like this be sound-wise?
16th March 2009
#2
Lives for gear

Get yourself some 600:600 ohm transformers. You'll need to consult the datasheet from the manufacturer whose transformer you are using to know which colored wire is what. But you basically wire XLR connectors to the input and output of the transformer. Simple as that.

If it's a steel core transformer you will experience increasing low frequency distortion with level. If it's a nickel core then it will sound really clean and you probably won't notice much difference in sound.

16th March 2009
#3
Gear Nut

interested in this aswell, i got a clean preamp and lets say i want to add color to the sound im recording lets say vocals
so how does the connection goes?

Clean Preamp Output ---> DIY Colored Transfo Input

then

DIY Colored Transo Output ---> Interface (A/D, Soundcard, Mixer)

Correct? would it need powersupply or simply just wiring the xlr in and out adapters?

thanks
16th March 2009
#4
Lives for gear

Quote:
Originally Posted by tWoKp.
interested in this aswell, i got a clean preamp and lets say i want to add color to the sound im recording lets say vocals
so how does the connection goes?

Clean Preamp Output ---> DIY Colored Transfo Input

then

DIY Colored Transo Output ---> Interface (A/D, Soundcard, Mixer)

Correct? would it need powersupply or simply just wiring the xlr in and out adapters?

thanks
i think this could be a brilliant idea as well. could be incredibly easy to wire a few into a 1ru 'trafo color box' too. or at the least i've got a useless behr di-100 di box i could swap the di trafo out of and presto.

so hypothetically, if you were doing it with cinemags:
Line Input Transformer
Cinemag Audio Output Transformers

you'd take a CMLI-600/600C wired directly to the input jack, then hook that to a CMOL-2x600T2 wired to the output jack?

regarding 1:1 trafos, if they are 1:1, why is there any diff between input and output? or isn't there?

also, do 1:1's have less color/effect than bigger/smaller ratios?

and lastly, from an engineering standpoint, what's usually the point of 1:1 (600:600) trafos anyway? i understand they can be used as signal splitters. cinemags pdf's suggest that adding them to a signal chain will also cut hum. is this via their high cmrr?

thanks
16th March 2009
#5
Lives for gear
I think this is a great idea and cheap idea for all the ITB guys...
16th March 2009
#6
Gear Guru

This is a brilliant idea based on the hype of transformers on this very site.

But in practice, waste of time IMHO.

Keep in mind, I know nothing about making gear but…

This project would be the equivalent of putting a nice piece of analog gear into bypass.

And a lot of those great transformers are impossible to find. Pultec, LA2a, Neve 2254s.

So you'll be stuck with a high quality modern sounding transformer.

At about $65 a pop, and using both an input and an output transformer on each channel, we're talking about just over$2,000.00 for a 16 channel unit that Superman couldn't distinguish in a blind A/B test.

No?
16th March 2009
#7
Lives for gear

Quote:
Originally Posted by Kenny Gioia
This is a brilliant idea based on the hype of transformers on this very site.

But in practice, waste of time IMHO.

Keep in mind, I know nothing about making gear but…

This project would be the equivalent of putting a nice piece of analog gear into bypass.

And a lot of those great transformers are impossible to find. Pultec, LA2a, Neve 2254s.

So you'll be stuck with a high quality modern sounding transformer.

At about $65 a pop, and using both an input and an output transformer on each channel, we're talking about just over$2,000.00 for a 16 channel unit that Superman couldn't distinguish in a blind A/B test.

No?
that's the problem - getting good colored trafos these days...

regardless, i think the idea would be to just do 1 or 2 channels and throw them in after a bland pre-amp or other place where desired in a given OTB signal chain.

no, it wouldn't make sense to go otb just to hit a tranny due to d/a/d degradation, but if you're already out for something else it could be a handy option to chain in.

many high end summing buses get a large % of their sound from the trafos they do or do not use, so even with modern parts, transformers are not inconsequential in sound or character.

but ya, i guess no matter what you talk about there are always people who say it makes no difference. ie. no one i know can a/b a panasonic fm cap vs a xicon, but that doesn't mean they are equivalent.
16th March 2009
#8

Do you mean something like this?

Mixmasters Pro Audio

Cheers
16th March 2009
#9
Lives for gear
What about that 8 channel dohicky from ART? Probably for the price, the trannys might suck ( don't know for sure, haven't herd it....).But at least there's the casing/connectors, even if you need to swap out for better trannnys.. I think it's about $130.00'ish... 16th March 2009 #10 Gear Guru Quote: Originally Posted by audiovisceral many high end summing buses get a large % of their sound from the trafos they do or do not use... the sound comes from the entire output stage, which includes the trafo but is very much a sum total of the whole circuit around it. usually, the more toney boxes will also have some sort of custom op-amp(s) in there as well. gregory scott - ubk . 16th March 2009 #11 Lives for gear the benefit you get from transformers are huge. i wouldn´t care about the ADDA, because you´re getting so much more sound and won´t loose much by ADDA. Transformers can do things like this: saturation, adding punch, overtone enhancment, phase shift, resulting in a more 3D sound, by letting the high frequencies later through the transformer...you can´t do this with plug-ins this great and natural. if you look at an analog mix bus, it´s often that there is a a line to mic signal converter, which gives you a low ohm, low gain signal and this is followed by a preamp. You´ll hear the whole preamp, not just the transformer. But the transformer has a major sonic imprint. If you look at a "normal" preamp design, like API, its an opamp going into a transformer. The shematic is very simple. You wont hear the shematic, you´ll hear an opamp feeding a transformer, while both have sonic shapes and the transformer does more than the opamp. There are many different transformers, some are "clean", some give you more sound, like there are API and NEVE style trafos out there. 16th March 2009 #12 Lives for gear A word of caution: depending on the ratio of the trans and the load (Input) it's feeding just make sure your output circuit can handle the extra current... So if the output is an IC it may not be able to handle the extra load... If it's a 1:1 or 600 to 600 then it may not be an issue... If it's a 1:3 for example and feeding a 1176 (600 Ohm) then that will look like a 150 ohm load, or around that amount depending on trans... 16th March 2009 #13 Lives for gear Quote: Originally Posted by Kenny Gioia This is a brilliant idea based on the hype of transformers on this very site. But in practice, waste of time IMHO. Keep in mind, I know nothing about making gear but… This project would be the equivalent of putting a nice piece of analog gear into bypass. And a lot of those great transformers are impossible to find. Pultec, LA2a, Neve 2254s. So you'll be stuck with a high quality modern sounding transformer. At about$65 a pop, and using both an input and an output transformer on each channel, we're talking about just over $2,000.00 for a 16 channel unit that Superman couldn't distinguish in a blind A/B test. No? Quote: Originally Posted by UBK the sound comes from the entire output stage, which includes the trafo but is very much a sum total of the whole circuit around it. usually, the more toney boxes will also have some sort of custom op-amp(s) in there as well. I was curious too. I hung two Pearless 600:600 transformers off the back of my SSL compressor... no difference. I made two submixes in Protools, both going round-trip AD/DA, one having the transformers inserted. Flipped the phase and they null'd so far down it was silly. Sum is greater than the parts gang. This is why an FX386 doesn't sound the same as the same compressor in the 4000. A 1073 channel strip doesn't sound like a Neve console... No free lunch 16th March 2009 #14 Gear Guru Quote: Originally Posted by juniorhifikit No free lunch not only is the lunch not free, it tends to be quite pricey. there aren't many humans who are capable of designing a circuit such that what you put in invariably comes out sounding better; it takes a form of genius that is, in modern culture, generally scooped up by the military or large commercial interests. iow, those with the $$. take german mics and other broadcast gear post wwii; these were mostly ex-military engineers who were forbidden from working in any kind of military context anymore, so all these 'best and brightest' poured their talents into whatever other work was available. one of the fields that benefited from that was the state-funded broadcasting industry. the legacy of having that level of skill and technical/aesthetic understanding is still with us today, in the form of 10,000 tube mics. gregory scott - ubk . 16th March 2009 #15 Lives for gear Quote: Originally Posted by u b k not only is the lunch not free, it tends to be quite pricey. there aren't many humans who are capable of designing a circuit such that what you put in invariably comes out sounding better; it takes a form of genius that is, in modern culture, generally scooped up by the military or large commercial interests. iow, those with the$$$.

take german mics and other broadcast gear post wwii; these were mostly ex-military engineers who were forbidden from working in any kind of military context anymore, so all these 'best and brightest' poured their talents into whatever other work was available. one of the fields that benefited from that was the state-funded broadcasting industry.

the legacy of having that level of skill and technical/aesthetic understanding is still with us today, in the form of $10,000 tube mics. gregory scott - ubk . Fortunately, there are still some brilliant and genius designers with us today, but yeah, their stuff costs a small fortune. And it should. Nice job on the Fatso thingy by the way. Looks really cool. 8th August 2009 #16 Lives for gear Broadcast Professional Audio - Looks like the right solution. £75 inc postage from Aus. Handmade too. The passive summing mixer looks rigt up my street too. 8th August 2009 #17 Lives for gear Remember that a transformer is an inductor and is reacts differently with each type of circuit it is part of. The inductance changes according to the frequencies coursing through it as well. some type more than others. Some circuits more than others. I have a pair of DBX 161s that I have NOT modded into 160VUs. In place of the original circuit's balancing amp I have used an output transformer from a late '60s UA/UREI channel strip. The sound is quite different from either a 160VU or a 161. It is a damn good sound, but different. The output stage of the 161 reacts differently to the transformer than a 160VU's output circuit does. It also interfaces with the next piece of equipment in the chain in a different manner than a 160VU or 161 would. It is a valid sound, but idfferent from the original sound of the 160 or 161. As far as a 1073, 1081 or any circuit from a console sounding different than an actual console goes... If the device is is connected to the same components that would be between it and the recording device there would not be much difference. If the device is used as a stand-alone piece without everything that followed in a console it would sound potentially very different. This is why when I racked the two pairs of Sphere mic pres and 920/930 EQs I have re-created the component path from the actual console EXACTLY (remember I had the console for years prior to racking the pairs.) I have used the mic pres and EQs without the same output trannies and circuitry. They DO sound different. They are quite good sounding, but they sound different. I have also spoken to people who have stand alone Sphere EQs that do not have the BST circuit (fader amp) in use. These units also do not have transformers in place either. I can tell by talking to these people that they are not experiencing the true sound of the Sphere console. I was lucky that I was able to obtain a pair of original Reichenbach output transformers out of a Quad Eight. These were virtually identical to those that were used in the Sphere. Consoles are a "system" and if you only use one part of the "system" you generally can't obtain the sound of the "system." 25th October 2009 #18 Gear Maniac I think with transformers and gear "color" in general is that it is additive and really needs to be stacked to really hear the difference. But I am looking into building one of these to go between my d/a's to my analog console. Then I can go ITB for super clean, OTB w/o transformer for some color, or with transformers for that real old vintage console sound. I think to really hear the effect you are going to need to run everything through it. So just strapping it on the 2-bus probably won't make enough difference to matter. Maybe do a summing box with transformers in and out. I think for me I will probably build a 16-channel box. But to keep the price down, check out Edcor transformers. They are M6 Silicon Steel for that color. Everything I have used them for I have been happy with. Plus they are around$10 a pop! Can't beat that!
2nd December 2009
#19
Lives for gear

What about running your mic into the XLR side of a passive DI,coming out of the 1/4" jack into the 1/4" jack of a second passive and then from the xlr out to your Preamp??? I know it sounds weird, but would it work, or could I hurt something???
17th April 2010
#20
Here for the gear

Hi have made many transformer connections for the sake of color and saturation. I use 1:1 600/600 output Transformers . prefer vintage . iron core

leaving the recording industry and setting up a home studio with limited budget and sub-par equipment means you have to do mods and become a diy person real quick.......nothing is handed to you on a platter.......what people are taking about and trying to get a handle on in this forum is the truth......Transformers are wonderful devices that can transform the sterile sound of cheap mixers and preamps into something much more presentable.

I use transformers after the mixer or preamps before hitting the sound-card....I even use a transformer into the inserts of the mixer to give me transformer sound ......I may or may not use one on the mix buss . Just the transformer in the signal path has an affect ...

I use Nuendo which has an external affect plugin that allows the use of external affects in the software domain. which also gives an opportunity to dial in the amount of transformer sound .....

I use Tamura Tk1 on the mix buss sometimes.....
I use vintage Ferrodyne's (I believe TRIAD bought them out)
I also have small wind modern transformers Edcore WSM600/600 .in total no more than a few hundred dollars worth of transformers .

You will notice a smoothness like a compression to the sound . The lows sound bigger , The middles become sweeter , the sound seems to become wider (probable due to phase problems in older Transformers) It's all had to put into words . but there is an affect .

Some people try to tell me that transformer don't have an affect on sound well they do .

I Nuendo I recorded a track.....no transformers used. Copied the track and reversed the phase , which nulls both tracks completely. I hooked up the external transformer to the soundcard in and outs .Put the Vst external plugin on the original and you can only hear the affect the transformer is having on the sound. I is there and loads of it.
7th March 2012
#21
Here for the gear

i tried this tonight, wired up to an input TRS jack and output TRS jack an edcor PC10k:10k i had bought for a diy remamping box project that never panned out. i love it so much that im gonna buy another one so i can run things out as stereo pairs. it gets rid of the MUD below 20hz and tightens the bass a bit, maybe not idea for metal kick drum tracks. but VERY VERY nice an usable. i ran a song mono\left output from Reaper, out of my m audio fast track ultra 8r and back into a line input. it sounds great, adds some type of THD that i like, and i foresee cheap steel core line input transformers in my future
12th March 2012
#22
Registered User
I remember trialling a cheap passive DI box - probably Phonic or some **** like that. I quite liked what it did to a bass I was playing with, and on the strength of that I bought my first Radial JDI. I was initially disappointed that the JDI was so much cleaner ... not at all the effect I expected. I love my JDI's for what they are, but don't understimate what cheap iron can do when you want a more overt effect.

DI's can be connected back to back for close to 1:1 ... I haven't tried it, but I could probably abuse my JDI's this way and probably make them saturate ...

I have many passive DIs, Reamps, line-splitters, Hum Eliminators, Line Level Shifters, and transformer preamps ... they solve so many problems, and generally I quite like the softening effect they have.

Transformers definately have a time-based effect, and convolution of transformer impulses is one way to get a little of this effect (maybe couple that with a saturation plug). Delaying the higher frequencies tends to make the sound softer even though they are still there ... this is basically what an All Pass Filter does, so that's another option.
12th March 2012
#23
Registered User
Quote:
Originally Posted by gurlygurl
so I could use a three-way, line level crossover and run the mid and high through separate delays (HF longer natch) and play Hass Effect games?
Why, yes! Yes - you could do that.

Feck all to do with transformer sound, but Yes - you could do that ...

12th March 2012
#24
Here for the gear

For me I really like the low frequency harmonic distortion. Not over the top. But it adds something nice
16th October 2012
#25
Lives for gear

doesn't look like anybody found the "tranny farm" yet.

dubolds "Tranny" farm

go on, click it, it's not dirty....
16th October 2012
#26
Gear Maniac

Quote:
Get yourself some 600:600 ohm transformers. You'll need to consult the datasheet from the manufacturer whose transformer you are using to know which colored wire is what. But you basically wire XLR connectors to the input and output of the transformer. Simple as that.

If it's a steel core transformer you will experience increasing low frequency distortion with level. If it's a nickel core then it will sound really clean and you probably won't notice much difference in sound.

Just curious why you would use 600 ohm 1:1 tranformers? Why not 10k ohm if it's line level? Is 600 just a happy medium that wouldn't freak out if you put it after older gear vs. putting after the D to A outputs of your soundcard?

Looks like the Tranny farm action is to be inserted before the preamps they built it for. I wonder what the benefit of that would be sonically... putting a steel transformer in front of a preamp, after the microphone,,, in which case,, wouldn't you want to use a 150 ohm 1:1 transformer, if it's before the preamp? In that scenario, you'd probably have to hit it pretty hard (like with a kick or snare) before you started noticing any harmonic distortion.. I talked to someone recently who was explaining why it's the output transformer in well maintained Neve 1073 preamps from a certain era that give it the harmonic distortion and that the input transformer actually sounds very clean in contrast.. I'd really like to understand this better. I'd like to be able to wire up a couple transformers (but what impedance value) for use as an insert on my 2 mix.

Thanks.
16th October 2012
#27
Lives for gear

I'm going to scrounge up a bunch of old electrolytics and build a Sagging Cap Farm and sell it on EBay. Makes your gear sound vintage!
16th October 2012
#28
Lives for gear

Quote:
Originally Posted by jtienhaara
I'm going to scrounge up a bunch of old electrolytics and build a Sagging Cap Farm and sell it on EBay. Makes your gear sound vintage!
it'd be cheaper then the tranny farm i suppose.
16th October 2012
#29
Lives for gear

Quote:
Originally Posted by lerone
Broadcast Professional Audio - Looks like the right solution. £75 inc postage from Aus. Handmade too. The passive summing mixer looks rigt up my street too.
Here is a reveiw ofRob's products.

I totally agree with running thru transformers before your soundcard,

Depending on the quality of those you source the sound is smoother, richer.

I might get some more.
19th October 2012
#30
Gear Maniac

Definitely going to do this.

What would running a 1:1 transformer into another 1:1 transformer sound like? Too much like over-egging the pudding or more palatably saturated?

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text:
# Isothermal expansion

1. Mar 5, 2006

### Pixter

have a question.
the viral eqn of state for one mole of aragon is:

pV=RT(1+B/V +c/V^2)

T is known and so is B and C. but p is not known.
initial V is known and so is final V.

the question is "find out the work of a reversible isothermal expansion at this temperature"

Know i'm wondering is there something wrong with the eqn. is there suppose to be some subscripts for inital and final on the V? or am i suppose to figure out a way of rearranging or exchanging part of the eqn with other stuff?

can I exchange stuff that would normally describe an ideal gas?.. it doesn't say that it acts ideal and by the first eqn it doesn't seem as it does. so......
if the eqn is correct do i need to know V?

edit: oki if a move the v over to the left and then integrate.. do i get work or do i have to put something else in there.... W=integral of pdV

ps: sorry for posting in the wrong forum.. now the homework. but donät know how to delete it.. if someone could please move it or something.. thanx

Last edited: Mar 5, 2006