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# Post Hoc Power Calculations are Not Useful

It is well documented that post hoc power calculations are not useful (Goodman and Berlin 1994, Hoenig and Heisey 2001, Althouse 2020). Also known as observed power or retrospective power, post hoc power purports to estimate the power of a test given an observed effect size. The idea is to show that a “non-significant” hypothesis test failed to achieve significance because it wasn’t powerful enough. This allows researchers to entertain the notion that their hypothesized effect may actually exist, they just needed to use a bigger sample size. The problem with this idea is that post hoc power calculations are completely determined by the p-value. High p-values (ie, non-significance) will always have low power. Low p-values will always have high power. Nothing is learned from post hoc power calculations.

In this article we demonstrate this idea using simulations in R. To begin let’s simulate some data and perform a t test. We first draw 10 samples from two Normal distributions. One has a mean of 10 and the other a mean of 10.1. Therefore the “true” effect size is 0.1 (ie, 10.1 – 10 = 0.1). Notice both also have the same variance. The set.seed(11) function allows us to always generate the same “random” data for this example. This allows you to follow along in case you want to replicate the result. It may help to pretend that we ran an experiment and the x1 group is the control and the x2 group received some sort of treatment.

set.seed(11)
x1 <- rnorm(10, mean = 10, sd = 1)
x2 <- rnorm(10, mean = 10.1, sd = 1)

Next we run a t test using the t.test function. The t test for these data returns a very high p-value of 0.72. If this were a real analysis we would conclude that we’re unable to determine what effect, if any, exists. The 95% confidence interval on the difference in means ranges from -0.67 to 0.95. Every value in that range, including 0, is compatible with the data. The effect is almost certainly not exactly 0, but we can’t tell if the effect is positive or negative.

ttest <- t.test(x1, x2, var.equal = TRUE)
ttest
##
##  Two Sample t-test
##
## data:  x1 and x2
## t = 0.3608, df = 18, p-value = 0.7224
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.6707225  0.9488640
## sample estimates:
## mean of x mean of y
##  9.769937  9.630867

Now let’s do a so-called post hoc power calculation using the observed effect. This is the observed difference in means. We can calculate power with the power.t.test function. We need to give it delta (the effect, or difference in means), sd (the pooled standard deviation), n (sample size), and sig.level (the significance level or type I error tolerance). We get the difference in means by calling the diff function on the estimate element of the “ttest” object.

power.t.test(delta = diff(ttest$estimate), sd = sqrt((var(x1) + var(x2))/2), n = 10, sig.level = 0.05, type = "two.sample", alternative = "two.sided") ## ## Two-sample t test power calculation ## ## n = 10 ## delta = 0.1390708 ## sd = 0.8618848 ## sig.level = 0.05 ## power = 0.05283998 ## alternative = two.sided ## ## NOTE: n is number in *each* group  The post hoc power is calculated to be about 0.053, which is very low. Proponents of post hoc power might conclude that the experiment was under-powered, but as we stated in the opening, post hoc power is completely determined by the p-value. The experiment may very well be under-powered, but a post hoc power calculation doesn’t prove that. You will always get low post hoc power on a hypothesis test with a large p-value. The exercise is useless. Hoenig and Heisey (2001) demonstrate this mathematically. We’ll demonstrate it via simulation. The replicate function makes quick work of this. Simply pass our previous code (minus the set.seed function) as an expression surrounded by curly braces and specify the number of times to replicate the code. Below we “replicate” the code 2000 times. The p-value of the test and post hoc power (or “observed power”) are returned in a vector. The results are stored in an object called “sim_out”. sim_out <- replicate(n = 2000, expr = { x1 <- rnorm(10, mean = 10, sd = 1) x2 <- rnorm(10, mean = 10.1, sd = 1) ttest <- t.test(x1, x2, var.equal = TRUE) pwr <- power.t.test(delta = diff(ttest$estimate),
sd = sqrt((var(x1) + var(x2))/2),
sig.level = 0.05,
n = 10)
c(pvalue = ttest$p.value, obs_power = pwr$power)
})

Next we create a scatterplot. Since the “sim_out” object has a dimension of 2 x 2000, we need to transpose it for easier plotting. We can do that with the t function.

plot(t(sim_out),
xlim = c(0,1), ylim = c(0,1),
xlab = "p-value", ylab = "observed power",
main = "Two-sample t test simulation")

The resulting scatterplot reveals that high p-values always have low power and vice versa. The only time you’ll see high post hoc power is with a very small p-value. The point is you learn nothing new by calculating post hoc power.

We can demonstrate this with a chi-square test as well. Below we generate two groups labeled “A” and “B”, each with 40 members. Then we assign them probabilities of “success” of 0.5 and 0.7, respectively.

grp <- rep(c("A", "B"), each = 40)
p <- rep(c(0.5, 0.7), each = 40)

Next we use the rbinom function to randomly return a 0 or 1 depending on the probability we give it. A 1 corresponds to treatment response, a 0 indicates no response. According to how we generated probabilities, group B should have more subjects respond. However that doesn’t guarantee a chi-square test will detect that association as we see below.

set.seed(2)
resp <- rbinom(n = length(grp), size = 1, prob = p)
table(grp, resp)
##    resp
## grp  0  1
##   A 21 19
##   B 15 25

chisq.test(table(grp, resp))
##
##  Pearson's Chi-squared test with Yates' continuity correction
##
## data:  table(grp, resp)
## X-squared = 1.2626, df = 1, p-value = 0.2612

We got a p-value of about 0.26. What’s the post hoc power of this test? We can calculate this by saving the results of the chi-square test, extracting the proportions, and using the power.prop.test function. Notice we use the base R pipe operator |>, introduced in version 4.1, to pass the observed counts to the proportions function.

cout <- chisq.test(table(grp, resp))
obs <- cout$observed |> proportions(margin = 1) power.prop.test(n = length(grp)/2, p1 = obs["A","1"], p2 = obs["B","1"], sig.level = 0.05) ## ## Two-sample comparison of proportions power calculation ## ## n = 40 ## p1 = 0.475 ## p2 = 0.625 ## sig.level = 0.05 ## power = 0.2680786 ## alternative = two.sided ## ## NOTE: n is number in *each* group  It should come as no surprise that a p-value of 0.26 leads to a low “observed power”, in this case about 0.27. As we did with the t test, we can replicate this many times and plot the p-values and post hoc power calculations. sim_out <- replicate(n = 2000, expr = { resp <- rbinom(n = length(grp), size = 1, prob = p) chisqtest <- chisq.test(table(grp, resp)) obs <- chisqtest$observed |> proportions(margin = 1)
pwr <- power.prop.test(n = length(grp)/2,
p1 = obs["A","1"],
p2 = obs["B","1"],
sig.level = 0.05)
c(pvalue = chisqtest$p.value, obs_power = pwr$power)
})

plot(t(sim_out),
xlim = c(0,1), ylim = c(0,1),
xlab = "p-value", ylab = "observed power",
main = "Chi-square test simulation")

Again, nothing is gained performing post hoc power calculations. An insignificant result will always have low “power”. (The gaps in the scatterplot line are due to the discrete counts in the table.)

Power is something to think about before running an experiment, not after. Power is the probability of correctly rejecting a null hypothesis test when a hypothesized effect really exists. We pretend some meaningful effect is actually real and then determine the sample size we need to achieve a high level of power, such as 0.90.

In our first example, the real effect was 0.1 (with a standard deviation of 1). How large a sample size do we need to have a 90% chance of correctly rejecting the null hypothesis of no difference in the means at a significance level of 0.05? We can use the power.t.test function to answer this.

power.t.test(delta = 0.1, sd = 1, sig.level = 0.05, power = 0.90)
##
##      Two-sample t test power calculation
##
##               n = 2102.445
##           delta = 0.1
##              sd = 1
##       sig.level = 0.05
##           power = 0.9
##     alternative = two.sided
##
## NOTE: n is number in *each* group

It appears we need over 2100 subjects per group. Assuming our effect size estimates are realistic and meaningful, this provides us guidance on how many subjects we need to recruit. But it doesn’t guarantee a significant result. High power is a probability, not a certainty.

In addition to considering power before an experiment, we should focus less on p-values after the analysis and more on confidence intervals on the effect. There is a long tradition of making binary decisions of “some effect” or “no effect” based on a p-value falling below an arbitrary threshold. For example, get a p-value of 0.08 and declare there is “no effect” because the p-value is not less than 0.05. Or get a p-value of 0.000023 and declare a “highly significant effect” because the p-value is very small. In both cases we’re making decisions about the effect without actually investigating the effect.

We previously ran a chi-square test on simulated data. Let’s do that again. Below we use set.seed(8) to allow you to replicate our result.

grp <- rep(c("A", "B"), each = 40)
p <- rep(c(0.5, 0.7), each = 40)
set.seed(8)
resp <- rbinom(n = length(grp), size = 1, prob = p)
chisq.test(table(grp, resp))

##
##  Pearson's Chi-squared test with Yates' continuity correction
##
## data:  table(grp, resp)
## X-squared = 1.8569, df = 1, p-value = 0.173

The p-value of the chi-square test is 0.17. In a real analysis we might conclude “no effect” since the p-value is greater than 0.05. But that decision doesn’t even consider the estimated effect. One way to estimate the effect is to calculate the difference in proportions between those who responded in group A and those who responded in group B, and then calculate a 95% confidence interval on the difference in proportions. This can be done with the prop.test function. Before we do that we use the table and proportions function to determine the observed effect is 0.675 – 0.500 = 0.175.

table(grp, resp) |> proportions(margin = 1)
##    resp
## grp     0     1
##   A 0.500 0.500
##   B 0.325 0.675

To use the prop.test function, the table needs to have “successes” in the first column and “failures” in the second column, so we swap the order of the columns. In addition we swap the order of the rows so the estimated effect is B – A.

tab <- table(grp, resp)[c("B","A"),c("1","0")]
prop.test(x = tab)
##
##  2-sample test for equality of proportions with continuity correction
##
## data:  tab
## X-squared = 1.8569, df = 1, p-value = 0.173
## alternative hypothesis: two.sided
## 95 percent confidence interval:
##  -0.06231373  0.41231373
## sample estimates:
## prop 1 prop 2
##  0.675  0.500

Notice the p-value is exactly the sames as the chi-square test. (A chi-square test and 2-sample proportion test are the same thing statistically speaking.) More importantly notice the 95 percent confidence interval on the difference in proportions. Every proportion in that range is compatible with the data. Even though the p-value is “not significant”, plausible effects range from -0.06 to 0.4. To declare “no effect” is to decide that 0 is the most likely effect. But why is 0 more likely than 0.1 or 0.2? Instead of compressing this information into a single yes/no decision, we should report the confidence interval and let the reader see the range of plausible effect sizes.

References

• Althouse A. Post Hoc Power: Not Empowering, Just Misleading. J Surg Res. 2021 Mar; 259:A3-A6.
• Goodman SN, Berlin JA. The use of predicted confidence intervals when planning experiments and the misuse of power when interpreting results. Ann Intern Med. 1994; 121:200-206.
• Hoenig JM, Heisey DM. The abuse of power: the pervasive fallacy of power calculations for data analysis. Am Stat. 2001; 55:19-24.
• R Core Team (2021). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/.

For questions or clarifications regarding this article, contact the UVA Library StatLab: statlab@virginia.edu

View the entire collection of UVA Library StatLab articles.

Clay Ford
Statistical Research Consultant
University of Virginia Library
August 4, 2021


text:
# Properties

 Label 1350.2.e.k Level 1350 Weight 2 Character orbit 1350.e Analytic conductor 10.780 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1350 = 2 \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1350.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.7798042729$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 450) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + q^{8} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + q^{8} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{11} + ( -2 \beta_{1} + \beta_{2} ) q^{13} + ( -2 \beta_{1} + \beta_{2} ) q^{14} + ( -1 + \beta_{1} ) q^{16} -2 \beta_{3} q^{17} + ( 5 + \beta_{3} ) q^{19} -2 \beta_{2} q^{22} -\beta_{2} q^{23} + ( 2 - \beta_{3} ) q^{26} + ( 2 - \beta_{3} ) q^{28} + ( \beta_{2} - \beta_{3} ) q^{29} + ( -2 \beta_{1} - \beta_{2} ) q^{31} -\beta_{1} q^{32} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{34} + ( -4 - 3 \beta_{3} ) q^{37} + ( -5 + 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{38} + 9 \beta_{1} q^{41} + ( -5 + 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} + 2 \beta_{3} q^{44} + \beta_{3} q^{46} + ( -6 + 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{47} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{49} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{52} + ( 6 - \beta_{3} ) q^{53} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{56} -\beta_{2} q^{58} + ( -3 \beta_{1} - \beta_{2} ) q^{59} + ( -8 + 8 \beta_{1} ) q^{61} + ( 2 + \beta_{3} ) q^{62} + q^{64} + ( 7 \beta_{1} - 3 \beta_{2} ) q^{67} + 2 \beta_{2} q^{68} + ( -6 - 3 \beta_{3} ) q^{71} - q^{73} + ( 4 - 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{74} + ( -5 \beta_{1} - \beta_{2} ) q^{76} + ( 12 \beta_{1} - 4 \beta_{2} ) q^{77} + ( -2 + 2 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{79} -9 q^{82} + ( -3 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{83} + ( -5 \beta_{1} + \beta_{2} ) q^{86} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{88} + 9 q^{89} + ( 10 - 4 \beta_{3} ) q^{91} + ( \beta_{2} - \beta_{3} ) q^{92} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{94} + ( 1 - \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{97} + ( 3 - 4 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 2q^{4} - 4q^{7} + 4q^{8} + O(q^{10})$$ $$4q - 2q^{2} - 2q^{4} - 4q^{7} + 4q^{8} - 4q^{13} - 4q^{14} - 2q^{16} + 20q^{19} + 8q^{26} + 8q^{28} - 4q^{31} - 2q^{32} - 16q^{37} - 10q^{38} + 18q^{41} - 10q^{43} - 12q^{47} - 6q^{49} - 4q^{52} + 24q^{53} - 4q^{56} - 6q^{59} - 16q^{61} + 8q^{62} + 4q^{64} + 14q^{67} - 24q^{71} - 4q^{73} + 8q^{74} - 10q^{76} + 24q^{77} - 4q^{79} - 36q^{82} - 6q^{83} - 10q^{86} + 36q^{89} + 40q^{91} - 12q^{94} + 2q^{97} + 12q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{3} + 4 \beta_{2}$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1027$$ $$\chi(n)$$ $$-\beta_{1}$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 −1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −2.22474 + 3.85337i 1.00000 0 0
451.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 0.224745 0.389270i 1.00000 0 0
901.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −2.22474 3.85337i 1.00000 0 0
901.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 0.224745 + 0.389270i 1.00000 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.e.k 4
3.b odd 2 1 450.2.e.m yes 4
5.b even 2 1 1350.2.e.n 4
5.c odd 4 2 1350.2.j.g 8
9.c even 3 1 inner 1350.2.e.k 4
9.c even 3 1 4050.2.a.by 2
9.d odd 6 1 450.2.e.m yes 4
9.d odd 6 1 4050.2.a.br 2
15.d odd 2 1 450.2.e.l 4
15.e even 4 2 450.2.j.f 8
45.h odd 6 1 450.2.e.l 4
45.h odd 6 1 4050.2.a.bu 2
45.j even 6 1 1350.2.e.n 4
45.j even 6 1 4050.2.a.bl 2
45.k odd 12 2 1350.2.j.g 8
45.k odd 12 2 4050.2.c.w 4
45.l even 12 2 450.2.j.f 8
45.l even 12 2 4050.2.c.y 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.l 4 15.d odd 2 1
450.2.e.l 4 45.h odd 6 1
450.2.e.m yes 4 3.b odd 2 1
450.2.e.m yes 4 9.d odd 6 1
450.2.j.f 8 15.e even 4 2
450.2.j.f 8 45.l even 12 2
1350.2.e.k 4 1.a even 1 1 trivial
1350.2.e.k 4 9.c even 3 1 inner
1350.2.e.n 4 5.b even 2 1
1350.2.e.n 4 45.j even 6 1
1350.2.j.g 8 5.c odd 4 2
1350.2.j.g 8 45.k odd 12 2
4050.2.a.bl 2 45.j even 6 1
4050.2.a.br 2 9.d odd 6 1
4050.2.a.bu 2 45.h odd 6 1
4050.2.a.by 2 9.c even 3 1
4050.2.c.w 4 45.k odd 12 2
4050.2.c.y 4 45.l even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1350, [\chi])$$:

 $$T_{7}^{4} + 4 T_{7}^{3} + 18 T_{7}^{2} - 8 T_{7} + 4$$ $$T_{11}^{4} + 24 T_{11}^{2} + 576$$ $$T_{17}^{2} - 24$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + T^{2} )^{2}$$
$3$
$5$
$7$ $$1 + 4 T + 4 T^{2} - 8 T^{3} - 17 T^{4} - 56 T^{5} + 196 T^{6} + 1372 T^{7} + 2401 T^{8}$$
$11$ $$1 + 2 T^{2} - 117 T^{4} + 242 T^{6} + 14641 T^{8}$$
$13$ $$1 + 4 T - 8 T^{2} - 8 T^{3} + 199 T^{4} - 104 T^{5} - 1352 T^{6} + 8788 T^{7} + 28561 T^{8}$$
$17$ $$( 1 + 10 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 - 10 T + 57 T^{2} - 190 T^{3} + 361 T^{4} )^{2}$$
$23$ $$1 - 40 T^{2} + 1071 T^{4} - 21160 T^{6} + 279841 T^{8}$$
$29$ $$1 - 52 T^{2} + 1863 T^{4} - 43732 T^{6} + 707281 T^{8}$$
$31$ $$1 + 4 T - 44 T^{2} - 8 T^{3} + 2143 T^{4} - 248 T^{5} - 42284 T^{6} + 119164 T^{7} + 923521 T^{8}$$
$37$ $$( 1 + 8 T + 36 T^{2} + 296 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 + 10 T - 5 T^{2} + 190 T^{3} + 4876 T^{4} + 8170 T^{5} - 9245 T^{6} + 795070 T^{7} + 3418801 T^{8}$$
$47$ $$1 + 12 T + 38 T^{2} + 144 T^{3} + 2259 T^{4} + 6768 T^{5} + 83942 T^{6} + 1245876 T^{7} + 4879681 T^{8}$$
$53$ $$( 1 - 12 T + 136 T^{2} - 636 T^{3} + 2809 T^{4} )^{2}$$
$59$ $$1 + 6 T - 85 T^{2} + 18 T^{3} + 9036 T^{4} + 1062 T^{5} - 295885 T^{6} + 1232274 T^{7} + 12117361 T^{8}$$
$61$ $$( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 14 T + 67 T^{2} )^{2}( 1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4} )$$
$71$ $$( 1 + 12 T + 124 T^{2} + 852 T^{3} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + T + 73 T^{2} )^{4}$$
$79$ $$1 + 4 T + 70 T^{2} - 848 T^{3} - 4589 T^{4} - 66992 T^{5} + 436870 T^{6} + 1972156 T^{7} + 38950081 T^{8}$$
$83$ $$1 + 6 T - 133 T^{2} + 18 T^{3} + 18684 T^{4} + 1494 T^{5} - 916237 T^{6} + 3430722 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 - 9 T + 89 T^{2} )^{4}$$
$97$ $$1 - 2 T - 95 T^{2} + 190 T^{3} + 4 T^{4} + 18430 T^{5} - 893855 T^{6} - 1825346 T^{7} + 88529281 T^{8}$$

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text:
# Is it safe to use a 125PSI rated auxiliary air tank with a 150PSI air compressor?

I have a small, 5 gallon, pancake compressor which has a maximum 150PSI rating. Unfortunately, it doesn't have the air capacity to power the demands of an HVLP paint sprayer or a die grinder without the loud motor kicking on.

To increase the air capacity, I'm considering adding an auxiliary, portable air tank . The manual clearly says it is rated for 125PSI maximum, but the air relief valve has 140PSI Max rating punched on it. To keep it from blowing off, I'm planning on replacing it with a 150PSI one.

I also thought about derating the air compressor by setting it to 125PSI, but the pressure switch cannot be adjusted, at least not without replacing it.

I'm terrified of the tank exploding and covering me with shrapnel. At the same time, however, I have a hard time believing a mere 25PSI or 10PSI (depending on whether you believe the manual or the relief value) would cause the tank to explode. Is this a valid concern?

Don't do it!!

Just buy a pressure regulator to put between the two tanks that will decrease the pressure to below 125. You can get these for \$17 at home depot (https://www.homedepot.com/p/Husky-1-4-in-NPT-Regulator-with-Gauge-41155HOM/205331831) and maybe even cheaper elsewhere. I'm terrified of the tank exploding and covering me with shrapnel. This is a valid concern. You should treat pressure vessels as if they were bombs, because in some sense they are. Failures are "Not Good". I have a hard time believing a mere 25PSI or 10PSI (depending on whether you believe the manual or the relief value) would cause the tank to explode. Well, in some sense, you are right. If you take a tank rated to 125 psi and put 135 psi in it, it probably won't have an issue. But the magic word there is "probably". Let's quantify that. We can't answer the question "is it safe?" as if it was a yes or no answer. It's more like a probability that there will be in issue. why should we think in terms of probability? Well, what determines the strength of the tank? For starters, there is the thickness of the wall. There are always manufacturing variations. Some tanks will be a tiny (thousandths of an inch) thicker, and others thinner. So due to this, some will fail at a slightly higher pressure, and others at slightly lower pressure. There are many other sources of variation, but this easiest to think about, so let's stop here. Now I'm going to make up some numbers here, but it will illustrate the point. At the rate pressure, 125 psi, the probability of failure is very very small. Maybe not zero, but small. Let's just say 1 in 10 billion for sake of argument (i.e. every person in the world has an air tank, one of them explodes at the rated pressure). Now, if you go a little above the rated pressure, say 130 psi, now maybe 10 out of those 10 billion have an issue. It's still a pretty small number of tanks that have an issue, so you keep going. At 135 psi, now maybe 1000 out of the 10 billion (or 1 out of every 10 million) have an issue. At 140 psi, it's 1 out of every 100,000. At 145 psi, it's one out of every 10,000. and at 150 psi, it's 1 out of every 1,000. Again, I'm just making up the numbers, but you get the point. It's like a game of russian roulette. If you are just a tiny bit over in pressure, you're "probably" good. But the higher you go, the riskier it becomes. Most tanks won't have an issue at a little bit over the rated pressure, but how do you know if yours is the unlucky weakest one? Given that you can stay within the rated pressure with a$17 part, it's just not worth taking the risk

• Thanks. I won't need the \$17 part, since my compressor has a built in output pressure regulator. I was hoping for something like a pressure regulator but limits the incoming pressure to a fixed 125psi. It would prevent manually exceeding the 125 PSI. Unfortunately, I don't know what they are called. Jul 24, 2020 at 3:45
• @user148298 try to look for "pressure regulator valve" they should be built to stop the flow of medium as soon as the pressure on its output exceeds some set value. Jul 24, 2020 at 14:45

A 10 or 25 psig overpressure of a tank is well into a potential failure for the tank. The 140 psig safety valve is used because that accounts for the true pressure capability of the tank plus the tolerance in lift pressure for the safety valve. If you replace it with a 150 psig valve, you are risking the tank building up pressure past it's breaking point to reach the point when the 150 psig safety valve opens - the tank will vent via rupture before the valve has a chance to do anything.

• Wow! Your answer is a bit unsettling. It's terrifying a tank with such a tight tolerance can be sold for consumer use, especially when it is sold alongside compressors with higher PSI ratings. I would expect a 20% margin as these tanks are meant to be filled by other compressors. Jul 24, 2020 at 3:19
• These are commodity items, so they are designed to be as inexpensive as possible, so that means using the least amount of metal. The relief valve is there to prevent a catastrophic failure if a higher discharge pressure compressor is filling the tank, but if you change that relief valve, you are likely removing that layer of protection. Jul 24, 2020 at 20:17
• A typical maximum test pressure is 30% above maximum working pressure. At standard temperature. A consumer pressure vessel should be good for 130% at maximum rated temperature, which means it shouldn't kill you at 120% at room temperature, unless you use it more than once. Which is good, because killing consumers is generally a bad thing. If it does fail, it won't be on the first attempt: it will be after you've forgotten about it, and your kids are playing nearby. Pressure ratings are to protect you from failure, not from stupidity :~) Jul 25, 2020 at 5:07

If you have the Porter Cable compressor pictured, there's a regulator on the outlet. You could set that to 100 psi and fill your tank and be fine. As everyone has mentioned if you fill it to 150 psi you'll probably kill yourself or at least soil some underwear when one of the end caps explode.


text:
# Difference between impulse and magnitude of force

we learned about the impulse of a force , but just then a question came to my mind , whats the difference between the magnitude of a force and its impulse , or are they the same ... ??

jtbell
Mentor
The magnitude of a (vector) force ##\vec F## is ##|\vec F| = \sqrt{F_x^2 + F_y^2 + F_z^2}##. The impulse of the force when it acts for time ##\Delta t## is ##\vec F \Delta t##.

thanks !! :)

Orodruin
Staff Emeritus


text:
## Electronic Journal of Probability

### Geometric ergodicity of asymmetric volatility models with stochastic parameters

#### Abstract

In this paper, we consider a general family of asymmetric volatility models with stationary and ergodic coefficients. This family can nest several non-linear asymmetric GARCH models with stochastic parameters into its ambit. It also generalizes Markov-switching GARCH and GJR models. The geometric ergodicity of the proposed process is established. Sufficient conditions for stationarity and existence of moments have also been investigated. Geometric ergodicity of various volatility models with stochastic parameters has been discussed as special cases.

#### Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 90, 12 pp.

Dates
Accepted: 21 October 2013
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465064315

Digital Object Identifier
doi:10.1214/EJP.v18-1871

Mathematical Reviews number (MathSciNet)
MR3126573

Zentralblatt MATH identifier
1291.60145

Rights

#### Citation

Rohan, Neelabh; Ramanathan, T. V. Geometric ergodicity of asymmetric volatility models with stochastic parameters. Electron. J. Probab. 18 (2013), paper no. 90, 12 pp. doi:10.1214/EJP.v18-1871. https://projecteuclid.org/euclid.ejp/1465064315

#### References

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• Bauwens, Luc; Preminger, Arie; Rombouts, Jeroen V. K. Theory and inference for a Markov switching GARCH model. Econom. J. 13 (2010), no. 2, 218–244.
• Basrak, Bojan; Davis, Richard A.; Mikosch, Thomas. Regular variation of GARCH processes. Stochastic Process. Appl. 99 (2002), no. 1, 95–115.
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• P. and Spokoiny, V.: Varying coefficient GARCH models. In Handbook of Financial Time Series edited by T. G. Andersen, R. A. Davis, J. P. KreiÎ² and T. Mikosch. phSpringer-Verlag, Berlin Heidelberg, 2009, 169–183.
• Dahlhaus, Rainer; Subba Rao, Suhasini. Statistical inference for time-varying ARCH processes. Ann. Statist. 34 (2006), no. 3, 1075–1114.
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• Francq, Christian; Zakoïan, Jean-Michel. The $L^ 2$-structures of standard and switching-regime GARCH models. Stochastic Process. Appl. 115 (2005), no. 9, 1557–1582.
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• Fryzlewicz, Piotr; Subba Rao, Suhasini. Mixing properties of ARCH and time-varying ARCH processes. Bernoulli 17 (2011), no. 1, 320–346.
• Granger, Clive W. J. Non-linear models: Where do we go next-time varying parameter models? Stud. Nonlinear Dyn. Econom. 12 (2008), no. 3, Art. 1, 11 pp.
• Gray, S. F.: Modeling the conditional distribution of interest rates as a regime-switching process. Journal of Financial Economics 42, (1996), 27–62.
• Haas, M., Mittnik, S. and Paolella, M.: A new approach to Markov switching GARCH models. Journal of Financial Econometrics 2, (2004), 493–530.
• Hamilton, J. D. and Susmel, R.: Autoregressive conditional heteroscedasticity and changes in regime. Journal of Econometrics 64, (1994), 307–337.
• Hardy, G. H.; Littlewood, J. E.; Pólya, G. Inequalities. Reprint of the 1952 edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1988. xii+324 pp. ISBN: 0-521-35880-9
• He, Zhongfang; Maheu, John M. Real time detection of structural breaks in GARCH models. Comput. Statist. Data Anal. 54 (2010), no. 11, 2628–2640.
• Hentschel, L.: All in the family: Nesting symmetric and asymmetric GARCH models. Journal of Financial Economics 39, (1995), 71–104.
• Hwang, Soosung; Pereira, Pedro L. Valls. The effects of structural breaks in ARCH and GARCH parameters on persistence of GARCH models. Comm. Statist. Simulation Comput. 37 (2008), no. 3-5, 571–578.
• Klaassen, F.: Improving GARCH volatility forecasts with regime-switching GARCH. Empirical Economics 27, (2002), 363–394.
• Kokoszka, Piotr; Leipus, Remigijus. Change-point estimation in ARCH models. Bernoulli 6 (2000), no. 3, 513–539.
• Kristensen, Dennis. On stationarity and ergodicity of the bilinear model with applications to GARCH models. J. Time Series Anal. 30 (2009), no. 1, 125–144.
• Lamoureux, C. G. and Lastrapes W. D.: Persistence in variance, structural change, and the GARCH model. Journal of Business and Economic Statistics 8, (1990), 225–234.
• Meitz, Mika; Saikkonen, Pentti. Ergodicity, mixing, and existence of moments of a class of Markov models with applications to GARCH and ACD models. Econometric Theory 24 (2008), no. 5, 1291–1320.
• Meitz, Mika; Saikkonen, Pentti. Stability of nonlinear AR-GARCH models. J. Time Ser. Anal. 29 (2008), no. 3, 453–475.
• Meyn, S. P.; Tweedie, R. L. Markov chains and stochastic stability. Communications and Control Engineering Series. Springer-Verlag London, Ltd., London, 1993. xvi+ 548 pp. ISBN: 3-540-19832-6
• Mikosch, T. and Starica, C.: Non-stationarities in financial time series, the long-range dependence and the IGARCH effects. Review of Economics and Statistics 86, (2004), 378–390.
• Nelson, Daniel B. Stationarity and persistence in the GARCH$(1,1)$ model. Econometric Theory 6 (1990), no. 3, 318–334.
• Rohan, Neelabh; Ramanathan, T. V. Asymmetric volatility models with structural breaks. Comm. Statist. Simulation Comput. 41 (2012), no. 9, 1519–1543.
• Rohan, Neelabh; Ramanathan, T. V. Nonparametric estimation of a time-varying GARCH model. J. Nonparametr. Stat. 25 (2013), no. 1, 33–52.
• Saïdi, Youssef; Zakoïan, Jean-Michel. Stationarity and geometric ergodicity of a class of nonlinear ARCH models. Ann. Appl. Probab. 16 (2006), no. 4, 2256–2271.
• Stelzer, Robert. On Markov-switching ARMA processes-stationarity, existence of moments, and geometric ergodicity. Econometric Theory 25 (2009), no. 1, 43–62.
• Tjostheim, D.: Non-linear time series and Markov chains. Advances in Applied Probability 22, (1990), 587–611.


text:
## Show explicitly that two matrices are similar

Let $B = \begin{bmatrix} 0 & -4 & 85 \\ 1 & 4 & -30 \\ 0 & 0 & 3 \end{bmatrix}$ and let $C = \begin{bmatrix} 2 & 2 & 1 \\ 0 & 2 & -1 \\ 0 & 0 & 3 \end{bmatrix}$. Find matrices $P$ and $Q$ such that $P^{-1}BP$ and $Q^{-1}CQ$ are in Jordan Canonical Form. Give an explicit matrix $R$ such that $R^{-1}BR = C$.

We will follow the algorithm given on page 496 of D&F.

We begin by finding a sequence of ERCOs which transforms the matrix $xI-B$ into Smith Normal Form. Evidently the following sequence works.

1. $xR_2 + R_1 \rightarrow R_1$
2. $-C_1 \rightarrow C_1$
3. $(4-x)C_1 + C_2 \rightarrow C_2$
4. $-30C_1 + C_3 \rightarrow C_3$
5. $-30R_3 + R_1 \rightarrow R_1$
6. $-\frac{1}{5}(x-2)^2C_3 + C_2 \rightarrow C_2$
7. $\frac{1}{5}R_1 \rightarrow R_1$
8. $(3-x)R_1 + R_3 \rightarrow R_3$
9. $C_1 \leftrightarrow C_3$
10. $C_2 \leftrightarrow C_3$
11. $-5C_3 \rightarrow C_3$

The result is $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & (x-2)^2(x+3) \end{bmatrix}$.

In particular, the elementary divisors of $B$ are $(x-2)^2$ and $x-3$, so that the Jordan canonical form of $B$ is $\begin{bmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$.

We used four row operations, whose corresponding column operations are as follows.

1. $C_2 - BC_1 \rightarrow C_2$
2. $C_3 + 30C_1 \rightarrow C_3$
3. $C_1 + (B-3I)C_3 \rightarrow C_1$
4. $5C_1 \rightarrow C_1$

We now perform this sequence of column operations (in order) on the identity matrix. The result is $\begin{bmatrix} 0 & 0 & 30 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Multiplying the only nonzero column by each of $(B-2I)(B-3I)$, $B-3I$, and $B-2I$ to construct a matrix $P$, we have $P = \begin{bmatrix} 10 & -5 & 35 \\ -5 & 0 & -5 \\ 0 & 0 & 1 \end{bmatrix}$. Using Gauss-Jordan elimination on $[P|I]$, we can compute $P^{-1} = \frac{1}{5}\begin{bmatrix} 0 & -1 & -5 \\ -1 & -2 & 25 \\ 0 & 0 & 5 \end{bmatrix}$. We can easily verify that $P^{-1}BP$ is in Jordan Canonical Form. (WolframAlpha agrees.)

Next we consider $C$. Evidently $xI-C$ is put into Smith normal form by the following sequence of ERCOs.

1. $R_2 + R_1 \rightarrow R_1$
2. $-C_1 + C_2 \rightarrow C_2$
3. $-\frac{1}{2}C_2 \rightarrow C_2$
4. $(2-x)C_2 + C_1 \rightarrow C_1$
5. $\frac{1}{2}(x-2)R_1 + R_2 \rightarrow R_2$
6. $-\frac{1}{2}(x-2)^2C_3 + C_1 \rightarrow C_1$
7. $(3-x)R_2 + R_3 \rightarrow R_3$
8. $C_1 \leftrightarrow C_2$
9. $C_2 \leftrightarrow C_3$
10. $-2C_3 \rightarrow C_3$

The resulting matrix is $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & (x-2)^2(x-3) \end{bmatrix}$, so that the elementary divisors of $C$ are $(x-2)^2$ and $x-3$. At this point, we know that $B$ and $C$ are similar since they have the same elementary divisors.

We used three elementary row operations, whose corresponding column operations are as follows.

1. $C_1 - C_2 \rightarrow C_2$
2. $C_1 - \frac{1}{2}(C-2I)C_2 \rightarrow C_1$
3. $C_2 + (C-3I)C_3 \rightarrow C_2$

Performing these operations (in order) on the identity matrix yields $\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$. Multiplying the only nonzero column of this matrix by $(C-2I)^2$, $(C-2I)(C-3I)$, and $C-3I$ to construct a matrix $Q$, we have $Q = \begin{bmatrix} -2 & 1 & -1 \\ 0 & -1 & -1 \\ 0 & 0 & 1 \end{bmatrix}$. Evidently $Q^{-1} = \frac{1}{2} \begin{bmatrix} -1 & -1 & -2 \\ 0 & -2 & -2 \\ 0 & 0 & 2 \end{bmatrix}$, and $Q^{-1}CP$ is in Jordan Canonical Form. (WolframAlpha agrees.)

In particular, we have $P^{-1}BP = Q^{-1}CQ$, so that $(PQ^{-1})^{-1}B(PQ^{-1}) = C$.


text:
Django application with some addons regarding PostgreSQL trigram text comparison

## Project description

djorm-ext-pgtrgm
================

Django pgtrgm is a Django application with some addons regarding PostgreSQL trigram
(or trigraph) text comparison

A trigram is a group of three consecutive characters taken from a string. We can
measure the similarity of two strings by counting the number of trigrams they share.
This simple idea turns out to be very effective for measuring the similarity of words
in many natural languages.

For example, the set of trigrams in the string "cat" is " c", " ca", "cat", and "at ".

With this extension, we could use this feature inside Django ORM with a simple
QuerySet filter keyword

It is distributed under the terms of the [GNU Lesser General Public

PostgreSQL syntax
-----------------

text % text -> boolean Returns true if its arguments have a similarity that is greater
than the current similarity threshold set by set_limit.

[How to install and use pg_trgm extension in PostgreSQL 9.2](http://www.postgresql.org/docs/9.2/static/pgtrgm.html)

Requeriments
------------

- Django 1.5 (or higher). It's possible that works with other versions lower 1.5
- Postgres pg_trgm extension. In debian/ubuntu: sudo apt-get install postgres-contrib

Installation
------------

- In your settings.py

python

INSTALLED_APPS = (

'djorm_pgtrgm',
)

- You need to *register* the extension in your database. Run ./manage.py dbshell and then execute:

sql
CREATE EXTENSION pg_trgm;

- Optionally, you can create an index over a text column for the purpose of very fast similarity searches. For example, supose you will filter a lot over the field
description of the model myapp.Product:

sql

CREATE INDEX desctiption_trgm_idx ON myapp_product USING gist (description gist_trgm_ops);

Usage
-----------

Adds a new query set filter keyword to allow text searching.

python
MyModel.objects.filter(field_name__similar='whatever')

To ensure results ordered by similarity, you could do this:

python
MyModel.objects.filter(field_name__similar='whatever').
extra(select={'distance': "similarity(name, 'whatever')"}).
order_by('-distance'))

In a sake of brevity, you could use a provided SimilarManager that has a filter_o
method.

python

from djorm_pgtrgm import SimilarManager

class MyModel(models.Model):
objects = SimilarManager()

...

filter_o is a shortcut for the filter + extra + order_by in the snippet above.

python
MyModel.objects.filter_o(field_name__similar='whatever')

So, this will return every similar MyModel instance with a field_name *similar*
to 'whatever' and sorted by the distance ot each intance's field_name value to target 'whatever'. In addition, an extra field field_name_distance is added
to each item in the queryset.

Development
-----------

You can get the last version of djorm-ext-pgtrgm by doing a clone
of its repository:

git clone git://github.com/jleivaizq/djorm-ext-pgtrgm.git

0.0.1 (2013-10-23)
===================

* First version to djorm-ext-pgtrgm

## Project details

Uploaded source


text:
Subjects -> ENGINEERING (Total: 2844 journals)     - CHEMICAL ENGINEERING (259 journals)    - CIVIL ENGINEERING (255 journals)    - ELECTRICAL ENGINEERING (182 journals)    - ENGINEERING (1420 journals)    - ENGINEERING MECHANICS AND MATERIALS (454 journals)    - HYDRAULIC ENGINEERING (60 journals)    - INDUSTRIAL ENGINEERING (101 journals)    - MECHANICAL ENGINEERING (113 journals) ENGINEERING (1420 journals)                  1 2 3 4 5 6 7 8 | Last
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• Separation of circulating tumor cells from blood using dielectrophoretic
DLD manipulation

Abstract: Abstract Circulating Tumor Cells (CTCs) play a prominent role in early cancer detection. Emerging label-free techniques can be promising to CTC detection due to advantages in preserving cell integrity and minimal sample consumption. Deterministic Lateral Displacement (DLD) is a size-based label-free technique employing laminar flow for continuous sorting of suspended cells. However, separation based solely on size is challenging as the size distributions of CTCs tend to overlap with blood cells. Moreover, the rarity of CTCs in blood requires high throughput processing of samples for clinical utility. In this work, a dielectrophoretic DLD technique is presented to segregate CTCs from blood. This technique utilizes the cell size and dielectric properties as well as particle movement caused by polarization effect to accomplish continuous separation at high flow rates. A numerical model is developed and validated to investigate the effects of various parameters related to the fluid flow, micro-post array, and electric field. It is demonstrated that the dielectrophoretic DLD with specific post arrangement can continuously separate A549 lung CTCs from WBCs by applying a field frequency close to the crossover frequency of CTCs. The analysis further indicates that such a device can perform well despite uncertainties of CTC crossover frequencies. Additionally, efficient separation with minimum clogging can be achieved by setting the electric field perpendicular to fluid flow. The presented platform offers distinct advantages and can be potentially combined with techniques such as antibody-based immune-binding methods for rapid detection of CTCs.
PubDate: 2021-09-28

• Electrically controlled nicotine delivery through Carbon nanotube
membranes via electrochemical oxidation and nanofluidically enhanced
electroosmotic flow

Abstract: Abstract A promising tool for nicotine addiction treatment is a programmable nicotine delivery device coupled to smart phone-assisted behavioral therapies. Key metrics for such a device are delivery of adjustable nicotine doses tailored to individual needs, compact size and power efficiency. Reported here is a detailed optimization of carbon nanotube (CNT) membrane fabrication based on electrochemical oxidation, to improve its electrically driven performance for nicotine fluxes and switching ON (-1.5 V)-OFF (0 V) flux ratio. ON- state nicotine flux of ~ 6 µmoles/cm2/h at -1.5 V applied bias was achieved allowing ~ 6-folds decrease in the size of device (4 cm2) to attain flux equivalent to high dose nicotine gum (1.1 µmoles/cm2/h). Application of + 1.5 V bias in OFF state reduced diffusional background flux, giving an ON (-1.5 V)/OFF (+ 1.5 V) flux ratio of 68 that enabled device to deliver between the highest nicotine gum (1.1 µmoles/cm2/h) and lowest nicotine patch (0.08 µmoles/cm2/h) doses, as well as taper off nicotine doses for long term addiction treatment. The nicotine transport mechanism was studied as a function of pH and applied bias, using neutral tracer molecule, showing a mechanism of both electroosmosis and electrophoresis in the atomically smooth nanofluidic pores of CNTs. Optimal power consumption/flux efficiency of 111(µW/cm2)/µmoles/cm2/h was achieved allowing watch-battery lifetimes of 7–62 days for conventional treatment dosing regimens. Bluetooth-enabled, remotely controlled CNT membrane system has potential for treatments of nicotine, opioid and alcohol addictions that needs dose adjustment with precise temporal control.
PubDate: 2021-09-25

• CO2 laser fabrication of hydrogel-based open-channel microfluidic devices

Abstract: Abstract This study proposed a rapid and low-cost fabrication method for open-channel hydrogel-based microfluidic devices using CO2 laser ablation. The agarose hydrogel substrate was prepared with agarose gelation in DI water upon microwave heating, then a commercial CO2 laser system was used for the direct laser ablation of microchannels on the surface of agarose hydrogel substrate, the hydrophilic nature of the microchannels fabricated on hydrogel substrate enables the self-driven of the liquid inside the microchannels with capillary force. The profiles of the laser ablated microchannels on agarose hydrogel substrate with various laser power and scan speed were studied in detail. Due to the loss of water when exposed to the atmosphere, significant deformation of the fabricated microchannels was observed, and the profile change was recorded for 48 h for comparison. An easy-to-access storage method of hydrogel-based microfluidic device in DI water was also proposed in this study. Unlike compact silicon or polymer-based microfluidic devices, the hydrogel is formed by cross-linked polymer chains filled with water, for a better understanding of the diffusion of small molecules into the bulk hydrogel material during fluid propagation inside the microchannel, the Nile red fluorescent was added into the liquid and the diffusion across the hydrogel-based microchannels with time was measured and discussed in this study. Several open-channel agarose hydrogel-based microfluidic devices were fabricated in this study for the demonstration of the proposed fabrication method. The CO2 laser ablation approach for agarose hydrogel-based microfluidic devices has the advantages of rapid processing time, low-cost, highly biocompatible, and self-driven without pumps and could have wide application potentials in biological and medical fields.
PubDate: 2021-09-22

• Enhanced aptasensor performance for targeted HER2 breast cancer detection
by using screen-printed electrodes modified with Au nanoparticles

Abstract: Abstract The development of an Aptamer based biosensor for the selective detection of human epidermal growth factor receptor 2 (HER2) with high sensitivity and specificity was achieved. A screen-printed carbon electrode was used in the scope of this work. The HER2 Aptamer was immobilized via electrostatic adsorption on the surface of a screen-printed electrode, which was modified with Au Nanoparticles (~ 20 nm diameter) to support the Aptamer immobilization. The Aptasensor was extensively investigated using Cyclic voltammetry, Differential pulse voltammetry, Electrochemical impedance spectroscopy, Fourier transform infrared spectroscopy and Atomic force microscopy. The Aptasensor exhibits a fast response with a binding time of only 5 min and shows a log-linear response over a wide concentration range of 0.001—100 ng/mL. Moreover, it has high sensitivity and enhanced detection limit reaching 52.85 μA/ng/mL, and 0.001 ng/mL, respectively, with a relative standard deviation < 5%. The Aptasensor selectivity was studied by using different interfering substances, and the results demonstrate that the Aptasensor is efficient for the detection of HER2 with approximately 8% extent of the interference.
PubDate: 2021-09-21
DOI: 10.1007/s10544-021-00586-9

• A portable multi-sensor module for monitoring external ventricular drains

Abstract: Abstract External ventricular drains (EVDs) are used clinically to relieve excess fluid pressure in the brain. However, EVD outflow rate is highly variable and typical clinical flow tracking methods are manual and low resolution. To address this problem, we present an integrated multi-sensor module (IMSM) containing flow, temperature, and electrode/substrate integrity sensors to monitor the flow dynamics of cerebrospinal fluid (CSF) drainage through an EVD. The impedimetric sensors were microfabricated out of biocompatible polymer thin films, enabling seamless integration with the fluid drainage path due to their low profile. A custom measurement circuit enabled automated and portable sensor operation and data collection in the clinic. System performance was verified using real human CSF in a benchtop EVD model. Impedimetric flow sensors tracked flow rate through ambient temperature variation and biomimetic pulsatile flow, reducing error compared with previous work by a factor of 6.6. Detection of sensor breakdown using novel substrate and electrode integrity sensors was verified through soak testing and immersion in bovine serum albumin (BSA). Finally, the IMSM and measurement circuit were tested for 53 days with an RMS error of 61.4 μL/min.
PubDate: 2021-09-20
DOI: 10.1007/s10544-021-00579-8

• Label-free rapid isolation of saccharomyces cerevisiae with optically
induced dielectrophoresis-based automatic micromanipulation

Abstract: Abstract Saccharomyces cerevisiae is well-known in the baking and brewing industries and always used for the preparation of probiotics, especially its subtype, Saccharomyces boulardii, to prevent and treat various diarrhea and intestinal diseases. However, case reports on the side effects of a wide range of serious infections for the elderly, immunocompromised and critically ill patients after treatment with the S. cerevisiae have been increasing in recent years. The existing diagnose methods of the invasive S. cerevisiae infections in clinical, especially, the key step of the method—cell isolation, is time-consuming that always miss timey diagnose and early prevention. Here, we propose a new automatic micromanipulation method to label-free rapid isolation of S. cerevisiae based on the optically-induced dielectrophoresis (ODEP) technology, combining with image processing and recognition. S. cerevisiae is firstly identified by the image recognition method and then, automatically captured and moved to the target location by designing optical patterns. The results indicate the method can flexibly and automatically manipulate multiple S. cerevisiae cells simultaneously, such as, arranging S. cerevisiae cells, moving an array of the cells at any directions, aggregating the cells, and separating S. cerevisiae from the solution mixed with impurities. This work represents a step toward the use of automatic micromanipulation of ODEP technology to automatically and rapidly isolate S. cerevisiae for the detection of the invasive S. cerevisiae infections.
PubDate: 2021-09-18
DOI: 10.1007/s10544-021-00582-z

• Cellulose nanocrystals-based materials as hemostatic agents for wound
dressings: a review

Abstract: Abstract Wound dressings are devices used to stop bleeding and provide appropriate environmental conditions to accelerate wound healing. The effectiveness of wound dressing materials can be crucial to prevent deaths from excessive bleeding in surgeries and promote complete restoration of the injury. Some requirements for an ideal wound dressing are rapid hemostatic effect, high swelling capacity, antibacterial properties, biocompatibility, biodegradability, and mechanical strength. However, finding all these properties in a single material remains a challenge. In this context, nanocomposites have demonstrated an excellent capacity for this application because of their multifunctionality. One of the emerging materials used in nanocomposite manufacture is cellulose nanocrystals (CNCs), which are rod-like crystalline nanometric structures present on cellulose chains. These nanoparticles are attractive for wound healing applications because of their high aspect ratio, high mechanical properties, functionality and low density. Hence, this work aimed to present an overview of nanocomposites constituted by CNCs for wound healing applications. The review focuses on the most common materials used as matrices, the types of dressing, and their fabrication techniques. Novel wound dressings composites have improved hemostatic, swelling, and mechanical properties compared to other pure biopolymers while preserving their other biological properties. Films, nanofibers mats, sponges, and hydrogels have been prepared with CNCs nanocomposites, and in vitro and in vivo tests have proved their suitability for wound healing.
PubDate: 2021-09-07
DOI: 10.1007/s10544-021-00581-0

• Hand-drawn electrode based disposable paper chip for artificial sweat
analysis using impedance spectroscopy

Abstract: Abstract Low cost, disposable paper based electrical sensor to examine the analyte concentration in an extremely small volume of sample solution is essential for environmental and healthcare applications. For the development of paper based devices, sophisticated instruments are essential to pattern electrode on the top surface of the paper. In most cases, such fabricated device results in direct contact with the analyte solution on the surface of the electrode during electrical detection and leads to high electrical double layer capacitance. In this work, we have focused to reduce the double layer capacitance by fabricating hand drawn electrode paper sensor utilising the reverse side of the paper. This design acts as a sample storage and facilitate impedimetric sensing of ionic concentration of analyte solution using a few microlitre. Droplet formation at the bottom of the paper in the confined area is visually monitored to reduce sample wastage. The interaction between two different electrode materials (graphite and silver) on the paper substrate with the different volume and concentration of the electrolyte is analysed to improve the robustness and sensitivity of the measurement. Simultaneously, we observed a reduction in the electrical double layer effect on the low sample volumes. The proposed paper based sensor shows the enhanced impedance stability on silver electrode patterned paper chip than graphite electrode paper chip to detect the different ionic concentration of artificial sweat sample. Finally, it demonstrates that paper chip has great potential as a disposable diagnostics sensor in healthcare applications.
PubDate: 2021-09-01
DOI: 10.1007/s10544-021-00578-9

• Inertial microfluidics: Determining the effect of geometric key parameters
on capture efficiency along with a feasibility evaluation for bone marrow
cells sorting

Abstract: Abstract Despite great developments in inertial microfluidics, there is still a lack of knowledge to precisely define the particles’ behavior in the microchannels. In the present study, as a prerequisite to experimental studies, numerical simulations have been used to study the capture efficiency of target particles in the contraction–expansion microchannel, aiming to provide an estimation of the conditions at which the channel performs best. Fluid analysis based on Navier–Stokes equations is conducted using the finite element method to determine the streamlines and vortices. The highest capture efficiency for 10, 15, and 19-micron particles occurs when the center of the vortex is approximately in the middle of the wide section (at the flow rate of 0.35 ml/min). In addition to investigating the effect of particle diameter and input flow rate, the effect of channel geometry parameters (channel height and initial length of the channel) on particle trapping has also been studied. Also, to consider great interest in separating different-sized bioparticles from a sample, a three-stage platform has been designed to separate four types of bone marrow cells and evaluate the possibility of using contraction–expansion channels in this application.
PubDate: 2021-08-11
DOI: 10.1007/s10544-021-00577-w

• Dean migration of unfocused micron sized particles in low aspect ratio
spiral microchannels

Abstract: Abstract We present an analysis of the microfluidic Dean migration of 2.5 µm particles, which do not meet focus criterion, in tall and low aspect ratio microchannels. We demonstrate the use of such low aspect ratio and tall spirals (h > 50 µm) for isolating high concentration (> 106 particles or cells/mL) micron sized particles without an initial off-chip dilution step. We specifically show the need for a sheath fluid for isolation and systematically analyze the particle stream profile (i.e. thickness and distance from the channel wall) as a function of downstream channel length and curvature ratio, with changes in the fluid velocity and the flow rate ratio of particles to sheath fluid (FRR). We also show that the width of the particle stream can control the particle migration and that a threshold stream width and Dean drag is necessary to initiate the particle stream migration from the channel wall. We then propose a design guide based on the selection of optimum curvatures, flow velocities and the FRRs required for achieving a narrow particle stream through a particular outlet. Finally, we use the design guide to demonstrate the isolation of bacteria from bladder epithelial cells.
PubDate: 2021-07-26
DOI: 10.1007/s10544-021-00575-y

• Copper coating formed by micro-arc oxidation on pure Mg improved
antibacterial activity, osteogenesis, and angiogenesis in vivo and in
vitro

Abstract: Abstract Micro-arc oxidation (MAO) was used to improve the resistance of pure magnesium (Mg). Copper (Cu), a good antibacterial, angiogenic, and osteogenic element, was added by reaction in a Cu-containing electrolyte to improve the osteogenic and pro-angiogenic activities of Mg. The surface microstructures of the resulting MAO were evaluated by a scanning electron microscope (SEM) and energy-dispersive X-ray spectroscopy (EDS) mapping. The release of Cu ions was detected by ICP-OES. The antibacterial activity of films with different concentrations of Cu ions was assessed against Staphylococcus aureus (S. aureus). The osteogenesis of films was confirmed by cell morphology and proliferation, ALP activity, alizarin red staining, and osteogenic-related gene expression in the MC3T3-E1 cell line. The angiogenesis of the films was tested in human umbilical vein endothelial cells (HUVECs) by cell migration, tube formation, and VEGF quantification in vitro, and by a chicken embryo chorioallantoic membrane (CAM) assay in vivo. The results showed that the microporous structure was shaped by MAO, and the Cu group was denser and more uniform. The Cu coating showed effective antibacterial activity against S. aureus while also enhancing osteogenesis and angiogenesis in vitro. According to the CAM assay, the Cu group showed not only biocompatibility but also a significant angiogenic response, which was consistent with in vitro studies. The findings indicate that a Cu coating on Mg-MAO enhances osteogenesis and angiogenesis.
PubDate: 2021-07-24
DOI: 10.1007/s10544-021-00573-0

• Fabrication of novel-shaped microneedles to overcome the disadvantages of
solid microneedles for the transdermal delivery of insulin

Abstract: Abstract In this study, we fabricated two different microneedles (MNs) — semi-hollow and bird-bill — to overcome the limitations of solid and coated MNs, respectively. The two MN arrays were developed using a general injection molding process to obtain high-quality MNs with uniform shape. The semi-hollow and bird-bill MNs could penetrate the micropores of swine skin up to depths of 178.5 ± 27.6 µm and 232.1 ± 51.3 µm, respectively. When the semi-hollow MNs were used for the transdermal delivery of insulin in diabetic rats, it was observed that the blood glucose concentration (BGC) decreased remarkably within 30 min, and the desired effect of insulin was maintained for an additional 3 h after the removal of insulin from the skin surface. The bird-bill MN was able to load a coating gel at a maximum capacity of 3.20 ± 0.21 mg per MN array, and the BGC continued to decrease significantly after MN application for up to 2–6 h. In summary, we fabricated semi-hollow and bird-bill MN arrays using the injection molding method; these can be mass produced and are capable of effectively producing micro-holes in the stratum corneum. The two MN arrays could provide effective transdermal delivery of large-molecular-weight drugs such as insulin.
PubDate: 2021-07-21
DOI: 10.1007/s10544-021-00576-x

budesonide-Soluplus film

Abstract: Abstract Micro-reservoir based drug delivery systems have the potential to provide targeted drug release locally in the intestine, i.e. at the inflamed areas of the intestine of patients with inflammatory bowel disease (IBD). In this study, microcontainers with a diameter of 300 µm and a height of 100 µm, asymmetrical geometry and the possibility to provide unidirectional release, are fabricated in the biodegradable polymer poly-ɛ-caprolactone (PCL) using hot punching. As a first step towards local treatment of IBD, a novel method for loading of microcontainers with the corticosteroid budesonide is developed. For this purpose, a budesonide-Soluplus drug-polymer film is prepared by spin coating and loaded into the microcontainer reservoirs using hot punching. The processing parameters are optimized to achieve a complete loading of a large number of containers in a single step. A poly(lactic-co-glycolic acid) (PLGA) 50:50 lid is subsequently applied by spray coating. Solid-state characterization indicates that the drug is in an amorphous state in the drug-polymer films and the in vitro drug release profile showed a 68% release over 10 h. The results demonstrate that hot punching can be employed both as a production and loading method for PCL microcontainers with the perspective of local treatment of IBD.
PubDate: 2021-07-16
DOI: 10.1007/s10544-021-00572-1

• Non-enzymatic and rapid detection of glucose on PVA-CuO thin film using
ARDUINO UNO based capacitance measurement unit

Abstract: Abstract Glucose measurement is one of the essential health monitoring practices for maintaining blood sugar levels. Here, we have fabricated a highly specific capacitive nano-sensor for non-enzymatic glucose detection. Capacitance measurements were carried out on polyvinyl alcohol capped copper oxide (PVA-CuO) thin films on indium tin oxide (ITO) coated glass using ARDUINO UNO. The capacitance study shows a decrease in capacitance with an increase in glucose concentrations. The applicability in real samples was performed by studying the glucose in the presence of fetal bovine serum. Most commonly found interfering agents were used for interference studies, which confirmed the capacitive nano-sensor specificity. The system was further checked for repeatability up to six readings and reproducibility up to 5 chips. The shelf-life study showed stability for four weeks of a chip. These studies indicate that this capacitance-based measurement unit can be used for reliable, rapid, and non-enzymatic detection of glucose in real sample.
PubDate: 2021-07-14
DOI: 10.1007/s10544-021-00568-x

JournalTOCs
School of Mathematical and Computer Sciences
Heriot-Watt University
Edinburgh, EH14 4AS, UK
Email: journaltocs@hw.ac.uk
Tel: +00 44 (0)131 4513762


text:
## Files in this item

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## Description

 Title: Buffer Gas Cooled Molecule Source For Cpmmw Spectroscopy Author(s): Barnum, Timothy J. Contributor(s): Field, Robert W.; Klein, Ethan; Grimes, David; Zhou, Yan Subject(s): Chirped pulse Abstract: We have built a new molecular beam source that implements 20 K Neon buffer gas cooling for the study of the spectra of small molecules. In particular, laser ablation of BaF$_{2}$ pellets has been optimized to produce a molecular beam of BaF with a number density more than 100 times greater than what we have previously obtained from a typical Smalley-type photoablation supersonic beam source. Moreover, the forward beam velocity of ~150 m/s in our apparatus represents an approximate 10-fold reduction, improving spectroscopic resolution from 500 kHz to better than 50 kHz at 100 GHz in a chirped-pulse millimeter-wave experiment in which resolution is limited by Doppler broadening. Novel improvements in our buffer gas source and advantages for CPmmW spectroscopy studies will be discussed. We thank David Patterson, John Barry, John Doyle, and David DeMille for help in the design of our source. Issue Date: 2014-06-19 Publisher: International Symposium on Molecular Spectroscopy Citation Info: Barnum, T.J.; Field, R.W.; Klein, E.; Grimes, D.; Zhou, Y. BUFFER GAS COOLED MOLECULE SOURCE FOR CPMMW SPECTROSCOPY. Proceedings of the International Symposium on Molecular Spectroscopy, Urbana, IL, June 16-21, 2014. DOI: 10.15278/isms.2014.RE04 Genre: CONFERENCE PAPER/PRESENTATION Type: Text Language: English URI: http://hdl.handle.net/2142/51081 DOI: https://doi.org/10.15278/isms.2014.RE04 Rights Information: Copyright 2014 by the authors. Licensed under a Creative Commons Attribution 4.0 International License. http://creativecommons.org/licenses/by/4.0/ Date Available in IDEALS: 2014-09-172015-04-14


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Trio

Trio of workers earn 750 euros. Money divided by the work that each of them made.

First received twice as the second, the second received three times more than the third. How many got everyone from workers?

Correct result:

x1 =  450
x2 =  225
x3 =  75

Solution:

$\ \\ 750 = x_1 + x_2 + x_3 \ \\ 750 = 2 x_2 + x_2 + x_3 \ \\ 750 = 6 x_3 + 3 x_3 + x_3 \ \\ \ \\ 750 = 10x_3 \ \\ \ \\ x_3 = 75 \ \text{Eur} \ \\ x_2 = 3 x_3 = 225 \ Eur \ \\ x_1 = 2 x_2 = 450 \ Eur$

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# Homework Help: Deriving an expression for the energy required to separate charges

1. Feb 27, 2017

### Taylor_1989

1. The problem statement, all variables and given/known data
Hi guys, would just too make sure my derivation and insight to why is correct.
Question: a) only

2. Relevant equations

3. The attempt at a solution
$$dU=-wd_{ext}$$
$$dU=-F_{ext} \cdot dx$$
Now as the $F_{ext}$ is in the same direction and the direction vector { have not figured how to put direction vector in} then the equation becomes $dU=-Fdx$ this assume the direction is along the x axis
So now if I intergrate both sides $\int_{U(a)}^{U\infty}dU=-\int_{a}^{\infty}Fdx$

Subbing in a factoring columbs force law I get:

$$U(\infty)-U(a)=-kq_1q_2[-\frac{1}{x}]_a^\infty$$

Now U infity is zero because there is no force acting on it anymore and so I am left with$-U(a)$ on the LHS on the RHS I am left with $-kq_1q_2\frac{1}{a}$
thus the two negative cancel and I am left with the electric poteinal energy. Is this correct?

2. Feb 27, 2017


text:
help me

I think I may mistake myself. For my homework. Here
 12345678910111213141516171819202122232425262728 #include #include using namespace std; int main() { int n; int i; int is_prime = true; cout << "Enter a number and press Enter: "; cin >> n; i = 2; while (i <= sqrt(n)) { if (n % i == 0) is_prime = false; i++; } if (is_prime) cout << "Number is prime." << endl; else cout << "Number is not prime." << endl; system("pause"); return 0; }


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# Arch Linux – Conflicting Files and the Arch Wiki

Contents

It is no secret that my favorite distro for Linux after much trial and error, landed on Arch Linux.

I found I prefer the rolling release model vs major version upgrades and the AUR (Arch User Repository) is incredible for finding and installing packages. That being said, it’s biggest win is the Arch Wiki. I find however, that no matter how often that is repeated in the Arch circles, you still find forums full of solutions that the Arch Wiki covers better, or even conflict the Wiki.

I wrote this today, because I saw that example again in searching, where probably due to some previous mistake, when I went to update my system today, I was getting a failure for a pair of conflicting packages, which prevents updating as part of Arch giving you a chance to make sure you know what your doing…

A search for Arch Pacman + Conflicting Files returns all sorts of results, from various BBS Sites, Reddit, and Forums, with the Arch Wiki nestled tightly among them.

However, in a rather large number of the forums I took a quick browse at, were suggestions like: pacman -Syuf and pacman -Suf, among others, which most importantly are encouraging the f force flag. In some cases, other responses speak up right away much as I am, and suggest not taking that approach, as it’s a very strong-handed and dangerous approach to updating an Arch system.

Sure enough, in a bunch of cases (not a scientific sampling), there are follow-up posts of panic stricken users, who range from system won’t boot to pacman is completely broken, and other results. Where a quick search for this same problem to the Arch Wiki, leads you down the Pacman page to it’s troubleshooting section.

The core solution detailed:

This is happening because pacman has detected a file conflict, and by design, will not overwrite files for you. This is by design, not a flaw.

The problem is usually trivial to solve. A safe way is to first check if another package owns the file (pacman -Qo /path/to/file). If the file is owned by another package, file a bug report. If the file is not owned by another package, rename the file which ‘exists in filesystem’ and re-issue the update command. If all goes well, the file may then be removed.

Sure enough, in my case, I ran pacman -Qo /usr/lib/thefileinquestion, and it returned that no package owned that file. So following the wiki, I did a quick rename of that file (not delete, in case it does end up being needed!). I re-ran my update process, no conflicts, and everything appears to be working! Simple :)

In the case of Arch, when in doubt, start with the Arch Wiki, and then ask for elaborations on those solutions if they are not working, or not solving your problem, and you likely will save yourself a lot of headache!

I even went so far for a while, as to run one of my self-hosted servers on Vultr on Arch, given they allowed custom ISOs! It is not always the best solution for a server, depending on the applications running, but its worth evaluating, and again, it’s hard to fault the documentation power of the Wiki.


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# American Institute of Mathematical Sciences

July  2009, 3(3): 379-405. doi: 10.3934/jmd.2009.3.379

## Discontinuity-growth of interval-exchange maps

 1 Department of Mathematics and Statistics, University of Michigan - Dearborn, 4901 Evergreen Rd., Dearborn,MI 48128, United States

Received  November 2008 Revised  May 2009 Published  August 2009

For an interval-exchange map $f$, the number of discontinuities $d(f^n)$ either exhibits linear growth or is bounded independently of $n$. This dichotomy is used to prove that the group $\mathcal{E}$ of interval-exchanges does not contain distortion elements, giving examples of groups that do not act faithfully via interval-exchanges. As a further application of this dichotomy, a classification of centralizers in $\mathcal{E}$ is given. This classification is used to show that $\text{Aut}(\mathcal{E}) \cong \mathcal{E}$ $\mathbb{Z}$/$2 \mathbb{Z}$.
Citation: Christopher F. Novak. Discontinuity-growth of interval-exchange maps. Journal of Modern Dynamics, 2009, 3 (3) : 379-405. doi: 10.3934/jmd.2009.3.379
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