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# Walleye

Walleye
Scientific classification
Kingdom: Animalia
Phylum: Chordata
Class: Actinopterygii
Order: Perciformes
Family: Percidae
Genus: Sander
Species: S. vitreus
Binomial name
Sander vitreus
(Mitchill, 1818)
Subspecies

Walleye (Sander vitreus, formerly Stizostedion vitreum) is a freshwater perciform fish native to most of Canada and to the northern United States. It is a North American close relative of the European pikeperch. The walleye is sometimes called the yellow walleye to distinguish it from the blue walleye, which is a subspecies that can be found in the southern Ontario and Quebec.[1]

In some parts of its range, the walleye is known as the colored pike, yellow pike or pickerel (esp. in English-speaking Canada), although the fish is not related to other species of pikes which are members of the family Esocidae.

Walleyes show a fair amount of variation across watersheds. In general, fish within a watershed are quite similar and are genetically distinct from those of nearby watersheds. The species has been artificially propagated for over a century and has been planted on top of existing populations or introduced into waters naturally devoid of the species, sometimes reducing the overall genetic distinctiveness of populations.

## Meaning of the name

Walleye, Sander vitreus

The common name, "walleye", comes from the fact that the fish's eyes point outward, as if looking at the walls. In medicine, the term "walleyed" (as opposed to "cross-eyed")is used to refer to lateral strabismus, where the pupil deviates outward. This externally facing orientation of the eyes gives anglers an advantage in the dark because there is a certain eyeshine given off by the eye of the walleye in the dark, similar to that of lions and other night dwelling animals. This "eyeshine" is the result of a light-gathering layer in the eyes called the tapetum lucidum, which allows the fish to see well in low-light conditions. In fact, many anglers look for walleyes at night since this is when major feeding patterns occur. The fish's eyes also allow them to see well in turbid waters (stained or rough, breaking waters), which gives them an advantage over their prey. Thus, walleye anglers will commonly look for days and locations where there is a good "walleye chop" (i.e., rough water). This excellent vision also allows the fish to populate the deeper regions in a lake, and they can often be found in deeper water, particularly during the warmest part of the summer.[citation needed]

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## Physical description

Walleyes are largely olive and gold in colour (hence the French common name: doré — golden). The dorsal side of a walleye is olive, grading into a golden hue on the flanks. The olive/gold pattern is broken up by five darker saddles that extend to the upper sides. The colour shades to white on the belly. The mouth of a walleye is large and is armed with many sharp teeth. The first dorsal and anal fins are spinous, as is the operculum. Walleyes are distinguished from their close cousin the sauger by the white colouration on the lower lobe of the caudal fin which is absent on the sauger. In addition, the two dorsals and the caudal fin of the sauger are marked with distinctive rows of black dots which are absent from or indistinct on the same fins of walleyes.[2]

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## Length and weight

Weight vs. length of walleyes

Walleyes grow to about 80 cm (31 in) in length, and weigh up to about 9 kg (20 lb). The maximum recorded size for the fish is 107 cm (42 in) in length and 11.3 kilograms (25 lb) in weight. The growth rate depends partly on where in their range they occur, with southern populations often growing faster and larger. In general, females grow larger than males. Walleyes may live for decades; the maximum recorded age is 29 years. In heavily fished populations, however, few walleye older than five or six years of age are encountered. In North America, where they are heavily prized, their typical size when caught is on the order of 30 to 50 cm (12 to 20 in), substantially below their potential size.

As walleye grow longer, they increase in weight. The relationship between length and weight is not linear. The relationship between total length (L) and total weight (W) for nearly all species of fish can be expressed by an equation of the form

$W = cL^b \,$

Invariably, b is close to 3.0 for all species, and c is a constant that varies among species. For walleye, b = 3.180 and c = 0.000228 (with units in inches and pounds).[3]

The relationship described in this section suggests a 50 cm (20 in) walleye will weigh about 1.5 kg (3.3 lb), while a 60 cm (24 in) walleye will likely weigh about 2.5 kg (5.5 lb).

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## Reproduction

In most of the species' range, the majority of male walleyes mature at age three or four. Females normally mature about a year later. Adults migrate to tributary streams in late winter or early spring to lay eggs over gravel and rock, although there are open water reef or shoal spawning strains as well. Some populations are known to spawn on sand or on vegetation. Spawning occurs at water temperatures of 6 to 10 °C (43 to 50 °F). A large female can lay up to 500,000 eggs, and no care is given by the parents to the eggs or fry. The eggs are slightly adhesive and fall into spaces between rocks. The incubation period for the embryos is temperature-dependent, but generally lasts from 12 to 30 days. After hatching, the free-swimming embryo spends about a week absorbing the relatively small amount of yolk. Once the yolk has been fully absorbed, the young walleye begins to feed on invertebrates, such as fly larvæ and zooplankton. After 40 to 60 days, juvenile walleyes become piscivorous. Thenceforth, both juvenile and adult walleyes eat fish almost exclusively, frequently yellow perch or ciscoes, moving onto bars and shoals at night to feed. Walleye also feed heavily on crayfish, minnows, and leeches.

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## As food

The walleye is considered to be a quite palatable freshwater fish, and, consequently, is fished recreationally and commercially for food.[4] Because of its nocturnal feeding habits, it is most easily caught at night using live minnows or lures that mimic small fish. Most commercial fisheries for walleye are situated in the Canadian waters of the Great Lakes.

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## Fishing

Because walleyes are popular with anglers, fishing for walleyes is regulated by most natural resource agencies. Management may include the use of quotas and length limits to ensure that populations are not over-exploited. As one example, in the state of Michigan, walleye shorter than 15 in (38 cm) may not be legally kept, except in Lake St. Clair and the St. Clair River where fish as short as 13 in (33 cm) may be taken.

Since walleyes have excellent visual acuity under low illumination levels, they tend to feed more extensively at dawn and dusk, on cloudy or overcast days and under choppy conditions when light penetration into the water column is disrupted. Although anglers interpret this as light avoidance, it is merely an expression of the walleye's competitive advantage over its prey under those conditions. Similarly, in darkly stained or turbid waters, walleye tend to feed throughout the day. In the spring and fall walleye are located near the shallower areas due to the spawning grounds; and they are most often located in shallower areas during higher winds due to the murkier, higher oxygenated water at around six feet deep.[5] On calm spring days the walleye are more often located at the deep side of the shoreline drop-off and around shore slopes around or deeper than ten feet.[6]

"Walleye chop" is a term used by walleye anglers for rough water typically with winds of 10 to 25 km/h (6 to 16 mph), and is one of the indicators for good walleye fishing due to the walleye's increased feeding activity during such conditions. In addition to fishing the "Walleye chop", night fishing with live bait can be very effective.

On January 4, 2005, Father Mariusz Zajac from Carrot River, Saskatchewan caught an 18.30 lb (8.30 kg) walleye while ice fishing on Tobin Lake. This walleye stands as the world record for ice fishing and had a girth of 22.5 in (57 cm) and was 36.5 in (93 cm) in length.[7]

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## Cultural aspects

Large walleye statue at Mille Lacs in Garrison, Minnesota

The walleye is the state fish of Minnesota and South Dakota and the official fish of Saskatchewan.[8] It is very popular with Minnesota residents; more walleye is eaten in Minnesota than in any other jurisdiction of the United States. Both Garrison and Baudette, Minnesota, claim to be the "Walleye Capital of the World," each with a large statue of the fish.[9]

Winnipeg, Manitoba considers the walleye its most important local fish. Icelandic fishermen in Lake Winnipeg have traditionally supplied the Winnipeg market.

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## References

1. ^ http://www.lapresse.ca/le-soleil/200809/08/01-662528-le-dore-bleu-existe.php
2. ^ In-Fisherman – All About Walleye
3. ^ R. O. Anderson and R. M. Neumann, Length, Weight, and Associated Structural Indices, in Fisheries Techniques, second edition, B.E. Murphy and D.W. Willis, eds., American Fisheries Society, 1996.
4. ^ "Walleye, Sander vitreus". Department of Natural Resources (DNR), State of Michigan. Retrieved 15 March 2013.
5. ^ Joe Fellegy, Jr., Walleyes and Walleye Fishing (Dillon Press, 1974), 57, 58
6. ^ Fellegy, 60
7. ^ "World Record Walleye", Nipawin.com. Wapiti Valley CAP.
9. ^ "Walleyed War of the Walleye Capitals". RoadsideAmerica.com. Doug Kirby, Ken Smith, Mike Wilkins.
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• "Sander vitreus". Integrated Taxonomic Information System. Retrieved 19 March 2006.
• "Sander vitreus, Walleye". Fishbase. Retrieved 15 March 2013.
• Cena, Christopher J; George E. Morgan, Michael D. Malette, and Daniel D. Heath (2006). "Inbreeding, Outbreeding and Environmental Effects on Genetic Diversity in 46 Walleye (Sander Vitreus) Populations". Molecular Ecology 15 (2): 303–20.
• Grant, Gerold C; Paul Radomski, and Charles S. Anderson (2004). "Using Underwater Video to Directly Estimate Gear Selectivity: the Retention Probability for Walleye (Sander Vitreus) in Gill Nets". Canadian Journal of Fisheries and Aquatic Sciences 61 (2): 168–74.
• Huppert, Boyd (7 December 2004). "Walleye or Zander? What Are You Really Eating?". Kare 11. Retrieved 15 March 2013.
• Kaufman, Scott D.; John M. Gunn, George E. Morgan, and Patrice Couture (2006). "Muscle Enzymes Reveal Walleye (Sander Vitreus) Are Less Active When Larger Prey (cisco, Coregonus Artedi) Are Present". Canadian Journal of Fisheries and Aquatic Sciences 63 (5): 970–79.
• Simoneau, M (2005). "Fish Growth Rates Modulate Mercury Concentrations in Walleye (Sander Vitreus) from Eastern Canadian Lakes". Environmental Research 98 (1): 73–82.
• Suski, C. D.; S. J. Cooke, S. S. Killen, D. H. Wahl, and B. L. Tufts (2005). "Behaviour of Walleye, Sander Vitreus, and Largemouth Bass, Micropterus Salmoides, Exposed to Different Wave Intensities and Boat Operating Conditions during Livewell Confinement". Fisheries Management and Ecology 12 (1): 119–26.
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## Some theorems in fuzzy metric spaces.(English)Zbl 0870.54007

Summary: We prove the Ascoli-Arzela theorem for fuzzy metric spaces. We also prove that every fuzzy metric space has a basis that is countably locally finite. We use this result to conclude that every fuzzy metric space is metrizable.

### MSC:

 54A40 Fuzzy topology 54E35 Metric spaces, metrizability


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# A sextic counterexample to Euler's sum of powers conjecture

 Importance: Medium ✭✭
 Author(s): Euler, Leonhard P.
 Subject: Number Theory » Computational Number Theory
 Keywords:
 Recomm. for undergrads: yes
 Posted by: maxal on: August 5th, 2007
Problem   Find six positive integers such that or prove that such integers do not exist.

Euler's sum of powers conjecture states that for the Diophantine equation does not have solutions in positive integers as soon as For it corresponds to a particular case of Fermat Last Theorem and hence is true. For and , counterexamples to the Euler's sum of powers conjecture were found by N. Elkies in 1986 and L. J. Lander, T. R. Parkin in 1966 respectively. For , no counterexamples are currently known.

## Bibliography

* indicates original appearance(s) of problem.


text:
# Probabilistic abstraction for model checking: an approach based on property testing

## Abstract

The goal of model checking is to verify the correctness of a given program, on all its inputs. The main obstacle, in many cases, is the intractably large size of the program's transition system. Property testing is a randomized method to verify whether some fixed property holds on individual inputs, by looking at a small random part of that input. We join the strengths of both approaches by introducing a new notion of probabilistic abstraction, and by extending the framework of model checking to include the use of these abstractions. Our abstractions map transition systems associated with large graphs to small transition systems associated with small random subgraphs. This reduces the original transition system to a family of small, even constant-size, transition systems. We prove that with high probability, “sufficiently” incorrect programs will be rejected (-robustness). We also prove that under a certain condition (exactness), correct programs will never be rejected (soundness). Our work applies to programs for graph properties such as bipartiteness, ${\displaystyle k}$-colorabilityor any ${\displaystyle \exists \forall }$ first order graph properties. Our main contribution is to show how to apply the ideas of property testing to syntactic programs for such properties. We give a concrete example of an abstraction for a program for bipartiteness. Finally, we show that the relaxation of the test alone does not yield transition systems small enough to use the standard model checking method. More specifically, we prove, using methods from communication complexity, that the OBDD size remains exponential for approximate bipartiteness.

## Bibtex (lrde.bib)

```@Article{	  laplante.07.tocl,
author	= {Sophie Laplante and Richard Lassaigne and Fr\'ed\'eric
Magniez and Sylvain Peyronnet and Michel de Rougemont},
title		= {Probabilistic abstraction for model checking: an approach
based on property testing},
journal	= {ACM Transactions on Computational Logic},
year		= 2007,
month		= aug,
volume	= 8,
number	= 4,
abstract	= {The goal of model checking is to verify the correctness of
a given program, on all its inputs. The main obstacle, in
many cases, is the intractably large size of the program's
transition system. Property testing is a randomized method
to verify whether some fixed property holds on individual
inputs, by looking at a small random part of that input. We
join the strengths of both approaches by introducing a new
notion of probabilistic abstraction, and by extending the
framework of model checking to include the use of these
abstractions. Our abstractions map transition systems
associated with large graphs to small transition systems
associated with small random subgraphs. This reduces the
original transition system to a family of small, even
constant-size, transition systems. We prove that with high
probability, ``sufficiently'' incorrect programs will be
rejected (\$\eps\$-robustness). We also prove that under a
certain condition (exactness), correct programs will never
be rejected (soundness). Our work applies to programs for
graph properties such as bipartiteness, \$k\$-colorability,
or any \$\exists\forall\$ first order graph properties. Our
main contribution is to show how to apply the ideas of
property testing to syntactic programs for such properties.
We give a concrete example of an abstraction for a program
for bipartiteness. Finally, we show that the relaxation of
the test alone does not yield transition systems small
enough to use the standard model checking method. More
specifically, we prove, using methods from communication
complexity, that the OBDD size remains exponential for
approximate bipartiteness.}
}```


text:
RSA-CCA security of $m || H(m)$ as input to modular exponentiation

If we encrypt a message $m$ as $m || H(m)$ using textbook RSA encryption algorithm, $H$ is random oracle modeled. In this case the message can be recovered.

Is this encryption CCA secure? If so how can we prove it or not?

• It isn't even CPA secure.... – SEJPM May 4 '18 at 16:08

CCA security assumes CPA security, and $m || H(m)$ is not CPA secure. This is simple to show: say that $C = E(m || H(m))$ is known, then an adversary can simply guess $m'$ and encrypt that. Then $C' = C$ will show the adversary that $m'=m$.

To show a CCA attack you can use this attack. You can of course replace the value $2$ in that question with $y$ where $y = x || H(x)$ for any $x$. In that case you end up with $z \cdot m || H(m)$ after decryption. Calculating $m || H(m)$ will then just consist of division with $z$.

To avoid this it is of vital important that the signature is over data protected by randomized padding, and that this randomization is well distributed over the input to the RSA exponentiation. This is what OAEP does with the MGF(1) padding.

Alternatively a hybrid scheme such as RSA-KEM combined with a block cipher could be used.

Deterministic encryption cannot be fully CPA or CCA secure; some kind of randomness or at least uniqueness is required to be CPA / CCA secure to not leak information for repeated messages.

• Also there is a general result that a public key encryption operation must be randomized in order to be CPA secure... – SEJPM May 4 '18 at 17:53
• Anything missing from my answer, pinofer? – Maarten Bodewes May 9 '18 at 15:01


text:
2019
Том 71
№ 10

# Volume 59, № 5, 2007

Anniversaries (Ukrainian)

### Mark Grigorievich Krein (to the centenary of his birth)

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 579-587

Article (English)

### A generalization of an extended stochastic integral

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 588–617

We propose a generalization of an extended stochastic integral to the case of integration with respect to a broad class of random processes. In particular, we obtain conditions for the coincidence of the considered integral with the classical Itô stochastic integral.

Article (Russian)

### Passive impedance systems with losses of scattering channels

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 618–649

A new model of the passive impedance system with minimal losses of scattering channels and with bilaterally stable evolution semigroup is studied. In the case of discrete time, the passive linear stationary bilaterally stable impedance system $\Sigma$ is considered as a part of some minimal scattering-impedance lossless transmission system, that has a $(\tilde{J}_1, \tilde{J}_2)$-unitary system operator and a bilaterally $(J_1, J_2)$-inner (in certain weak sense) transmission function in the unit disk 22-block of which coincides with the impedance matrix of system $\Sigma$, belongs to the Caratheodory class, and has a pseudocontinuation. If the external space of the system $\Sigma$ is infinite-dimensional, then instead of the last mentioned property, we consider more complicated necessary and sufficient conditions on the impedance matrix of the system $\Sigma$. Different kinds of passive bilaterally stable impedance realizations with minimal losses of scattering channels (minimal, optimal, *-optimal, minimal and optimal, minimal and *-optimal) are studied.

Article (Ukrainian)

### Phragmén-Lindelöf theorem for solutions of elliptic differential equations in a banach space

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 650–657

For a second-order elliptic differential equation considered on a semiaxis in a Banach space, we show that if the order of growth of its solution at infinity is not higher than the exponential order, then this solution tends exponentially to zero at infinity.

Article (Russian)

### On the nature of the de Branges Hamiltonian

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 658–678

We prove the theorem announced by the author in 1995 in the paper "Criterion for discreteness of spectrum of singular canonical system" (Functional Analysis and Its Applications, Vol. 29, No. 3).

In developing the theory of Hilbert spaces of entire functions (we call them the Krein - de Branges spaces or, briefly, K-B spaces), L. de Branges arrived at some class of canonical equations of phase dimension 2. He proved that, for any given K-B space, there exists a canonical equation of the considered class such that it restores the chain of included K-B spaces. The Hamiltonians of such canonical equations are called the de Branges Hamiltonians. The following question arises: Under which conditions the Hamiltonian of some canonical equation should be a de Branges Hamiltonian. The basic theorem of the present paper together with Theorem 1 of the mentioned paper gives the answer to this question.

Article (Russian)

### Improved scales of spaces and elliptic boundary-value problems. III

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 679–701

We study elliptic boundary-value problems in improved scales of functional Hilbert spaces on smooth manifolds with boundary. The isotropic Hörmander-Volevich-Paneyakh spaces are elements of these scales. The local smoothness of a solution of an elliptic problem in an improved scale is investigated. We establish a sufficient condition under which this solution is classical. Elliptic boundary-value problems with parameter are also studied.

Article (English)

### On spectra of a certain class of quadratic operator pencils with one-dimensional linear part

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 702–716

We consider a class of quadratic operator pencils that occur in many problems of physics. The part of such a pencil linear with respect to the spectral parameter describes viscous friction in problems of small vibrations of strings and beams. Patterns in the location of eigenvalues of such pencils are established. If viscous friction (damping) is pointwise, then the operator in the linear part of the pencil is one-dimensional. For this case, rules in the location of purely imaginary eigenvalues are found.

Article (Ukrainian)

### On indecomposable and transitive systems of subspaces

Ukr. Mat. Zh. - 2007. - 59, № 5. - pp. 717–720

We prove that the indecomposability of a system of subspaces of a finite-dimensional Hilbert space implies the transitivity of this system under the condition of the linear coherence of the corresponding system of orthogonal projectors.


text:
In this post, we'll take a look at how to approximate integrals in Haskell and Python. These methods are probably not the most efficient, and there already are existing implementations like scipy.integrate in Python or Numeric.Tools.Integration in Haskell with a range of features and optimizations. The point of this post is to show a direct implementation of integration from definitions.

The Darboux Integral

From introductory calculus you should have a familiarity with the Riemann integral. An equivlent formulation is the Darboux integral. Look at the below diagram:

We have divided our interval $[a, b]$ into the partition $P = \{x_i\}$ where $x_0 < x_1 < \dots < x_n$.

For each of these subintervals we will choose two values of our function: \begin{align} M_i = \sup_{x\in[x_{i-1},x_{i}]} f(x)\\ m_i = \inf_{x\in[x_{i-1},x_{i}]} f(x) \end{align}

which represent the upper and lower bounds of the function. We then can construct rectangles to approximate our (signed) area under the curve: \begin{align} U_{f, P} &= \sum_{i=1}^n (x_{i}-x_{i-1}) M_i\\ L_{f, P} &= \sum_{i=1}^n (x_{i}-x_{i-1}) m_i \end{align} which are the upper and lower estimates of the integral given a particular partition, as represented in the above diagram.

From here, we have upper and lower integrals expressed in terms of the behavior of the upper and lower sums, but across all partitions: \begin{align} U_f &\equiv \overline{\int_{a}^{b}} f(x) \, \mathrm{d}x = \inf\{U_{f,P} \colon P \text{ is a partition of } [a,b]\}\\ L_f &\equiv \underline{\int_{a}^{b}} f(x) \, \mathrm{d}x = \sup\{L_{f,P} \colon P \text{ is a partition of } [a,b]\} \end{align}

and if both are equal, we say that $f$ is Darboux-integrable, denoted by the integral written without an underline or overline. A useful equivalent is that for all $\epsilon > 0$, there exists a partition $P_{\epsilon}$ such that:

$$U_{f, P} - L_{f, P} < \epsilon$$

Implementation in Python

For our numerical approximations, I will be making the large assumption of using a partition with subintervals of equal size. For most integrals we encounter, this is sufficient, but note the statement of the epsilon version of the Darboux integral. It is possible to construct a function where the upper and lower sums fail to converge for even spaced partitions, but there is another partition that does converge. However, as long as we're working with "nice" functions, this will not occur.

Here is my function:

import numpy as np

def darboux(f, a, b, epsilon, precision, increment=1, n=1):
record = []

while True:
upper_terms = []
lower_terms = []

partition = np.linspace(a, b, n+1)
delta = partition[1] - partition[0]

for i in range(len(partition)-1):
lower, upper = partition[i], partition[i+1]

options = np.linspace(lower, upper, precision)
values = list(map(f, options))

upper_terms.append(max(values)*delta)
lower_terms.append(min(values)*delta)

upper_sum = sum(upper_terms)
lower_sum = sum(lower_terms)

record.append((lower_sum, upper_sum, n))

e = upper_sum - lower_sum

if e < epsilon:
return record
else:
n+= increment

Let's walk through what is happening at each step. First, we have seven arguments:

- $f$, our function to be integrated
- $a$ and $b$, our interval over which we are integrating
- epsilon, how close we want our upper and lower sums in the approximation
- precision, how many points to evaluate in each subinterval to pick $M_i$ and $m_i$
- n, how many subintervals to begin with
- increment, how much to increase the number of subintervals each loop

And in the function we have the following steps:

Line 4: initialize a list to store the tuple (upper_sum, lower_sum, n)
Line 6-7: initialize lists to store our summation terms for each partition
Line 10: divide the interval $[a,b]$ into a partition of $n$ subintervals of equal size
Line 11: store the subinterval width (written $x_i - x_{i-1}$ above)
Line 13: begin loop through each subinterval
Line 14-16: select points in the subinterval (number determined by precision)
Line 17-20: evaluate $f$ and store the min/max multiplied by our interval width (rectangle area)
Line 22-23: sum the area of our rectangles and store the resulting upper/lower summations
Line 27-32: if the upper and lower bounds are within epsilon break, otherwise increment n

So for instance, if I call:

d = darboux(lambda x: x**2, a=0, b=1, epsilon=.1, precision=5)

for iteration in d:
print(iteration)

to approximate $\int_0^1 x^2 \, \mathrm{d}x = \frac{1}{3}$ the result is:

(0.0, 1.0, 1)
(0.125, 0.625, 2)
(0.18518518518518517, 0.5185185185185185, 3)
(0.21875, 0.46875, 4)
(0.24000000000000005, 0.44000000000000006, 5)
(0.2546296296296296, 0.4212962962962963, 6)
(0.26530612244897955, 0.4081632653061224, 7)
(0.2734375, 0.3984375, 8)
(0.279835390946502, 0.3909465020576131, 9)
(0.2850000000000001, 0.3850000000000001, 10)
(0.2892561983471075, 0.3801652892561984, 11)

I can make epsilon smaller for more precise results, but the speed is very dependent upon appropriately setting our increment and epsilon to match the scale of our integral/precision.

Consider the approximation of $\int_3^{4.5} (x^2+2x) \, \mathrm{d}x = 32.625$:

%%time
d = darboux(lambda x: (x**2+2*x),
a=3, b=4.5,
epsilon=.1,
precision=5,
increment=1)

print(d[-1])

(32.575070693947964, 32.67495387151806, 214)
Wall time: 595 ms

compared with:

%%time
d = darboux(lambda x: (x**2+2*x),
a=3, b=4.5,
epsilon=.001,
precision=5,
increment=500)

print(d[-1])

(32.62450293126423, 32.625497071071685, 21501)
Wall time: 11.6 s

Without adjusting the increment, the calculation would have taken much longer. Since I have stored each iteration, I can also graph the values and visualize the convergence of the upper and lower summations:

This problem also lends itself well to Haskell, with the added benefit that we don't have to manually allocate lists to store the values of our summations or use any loops. First I define two small helper functions, one to simulate np.linspace, and another a version of takeWhile that includes the boundary:

linspace :: Double -> Double -> Int -> [Double]
linspace a b n =
map (\ i -> a + (fromIntegral i) * inc) [(0::Int) .. (n - 1)]
where
inc = (b - a) / (fromIntegral (n - 1))

takeWhileInclusive :: (a -> Bool) -> [a] -> [a]
takeWhileInclusive _ [] = []
takeWhileInclusive p (x:xs) =
x : if p x then takeWhileInclusive p xs
else []

The actual function itself is nearly identical in concept:

darboux :: (Double -> Double) -> Double -> Double -> Double -> Int -> Int -> [(Double, Double, Int)]
darboux f a b epsilon precision increment = res
where
n = map (\x-> increment*x) [1..]

partitions = map (\n -> linspace a b (n+1)) n
delta = map (\xs -> (xs !! 1) - (xs !! 0)) partitions

pairs = map (\xs -> zip xs (tail xs)) partitions
options = (map.map) (\(start, end) -> linspace start end precision) pairs
values  = (map.map.map) (\x -> f x) options

upper = map sum $(map.map) maximum values lower = map sum$ (map.map) minimum values

upper_sum = zipWith (*) upper delta
lower_sum = zipWith (*) lower delta

s = zip3 lower_sum upper_sum n
res = takeWhileInclusive (\(low, high, n) -> (high - low) > epsilon) s

Again estimating $\int_3^{4.5} (x^2+2x) \, \mathrm{d}x = 32.625$ with epsilon = .001 and an increment of 500:

main = do
let test = darboux (\x -> x**2+2*x) 3 4.5 0.001 5 500
print (last test)

(32.62450290825767,32.62549709430418,21500)

The output is nearly identical, except for n being one different due to my indexing. What is noticeably different is our performance. Measure-Command in Powershell gives the following:

Days              : 0
Hours             : 0
Minutes           : 0
Seconds           : 4
Milliseconds      : 891
Ticks             : 48916287
TotalDays         : 5.66160729166667E-05
TotalHours        : 0.00135878575
TotalMinutes      : 0.081527145
TotalSeconds      : 4.8916287
TotalMilliseconds : 4891.6287

showing that the Haskell implementation was over twice as fast for this particular integral.


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ISSN 0439-755X
CN 11-1911/B

中国科学院心理研究所

• 研究报告 •

### 预期对注意的影响受制于被预期主体是目标还是分心物

1. 广州大学教育学院心理系/脑与认知科学中心, 广州 510006
• 收稿日期:2021-07-21 出版日期:2022-03-25 发布日期:2022-01-25
• 通讯作者: 傅世敏 E-mail:fusm@gzhu.edu.cn
• 基金资助:
国家自然科学基金资助项目(31970993)

### The effects of expectation on attention are dependent on whether expectation is on the target or on the distractor?

ZHOU Zinuan, CHEN Yanzhang, FU Shimin()

1. Department of Psychology and Center for Brain and Cognitive Sciences, School of Education, Guangzhou University, Guangzhou 510006, China
• Received:2021-07-21 Online:2022-03-25 Published:2022-01-25
• Contact: FU Shimin E-mail:fusm@gzhu.edu.cn

Abstract:

Both attention and expectation are important mechanisms to help the brain filter and process information in a complex environment. However, there is still no unified conclusion about whether they act interactively or additively on perception. Previous studies that used the spatial cueing paradigm to address this issue might have confounded attention and expectation. Recently, researchers have made efforts to define attention and expectation orthogonally. Using ERP and fMRI techniques, they explored the relationship between endogenous attention and probabilistic expectation in a way that defining attention by task relevance and defining expectation by probability. In the present study, we explored the relationship between exogenous attention and probabilistic expectation, by making clear orthogonal definitions for them. Moreover, previous studies mainly considered the relationship between target expectation and attention. Considering the possible difference between expectation on a target and expectation on a distractor, we also focus on the influence of expected subject in the relationship between expectation and attention.
In the present study, double peripheral cuing combined with visual search paradigm were used to explore the relationship between expectation and attention. We defined attention using peripheral cues without probabilistic information. Attended was defined when a peripheral cue was consistent with the location of the target, and unattended was defined when the peripheral cue was inconsistent with the location of the target. Expectations were defined by using spatial probabilities, and the participants were told beforehand where the stimulus would be more likely to occur through instruction. Four experiments were carried out in this study, and each experiment included 24 valid participants. Before the formal experiment, the participants were told which locations the target would appear more frequently (the distractor was randomly presented) or which locations the distractor will appear more frequently (the target was randomly presented). In a formal trial, the participants were presented with a 500 ms fixation point first, and then there were 50 ms double-cues without any validity probability. After the double-cues disappeared for 150 ms, a target array (consisted of a target and a distractor) appeared for 1000 ms. Participants were asked to find the target and to perform a discrimination task by pressing corresponding keys.
In experiment 1, the double-cues were presented randomly in the left or right visual field, such that attention was manipulated in left or right visual field. In contrast, the participants were expected the targets or distractors to occur more in the upper or lower visual field, that is, expectation was manipulated in the upper or lower visual filed, orthogonal to the manipulation of attention. In Experiment 1, we found that when the expected subject is different (on the target or on the distractor), the relationship between expectation and attention was different. When the target is the expected, expectation and attention influenced performance in an interactive way. However, when the distractor is the expected, they influenced performance in an additive way. In experiment 2, the double-cues were randomly presented in the upper or lower visual field, while the participants were expected the targets or distractors to occur more frequently in the left or right visual field. Contrary to our prediction, under this manipulation of attention, the main effect of attention was not significant. We suspected that this was due to the presence of attention cues across the left and right visual field. Therefore, we improved Experiment 3 by presenting the attention cues with diagonal locations centered on the fixation point, and manipulating the expectation on the orthogonal diagonal location to the attention cues. We replicated the basic findings of Experiment 1 in Experiment 3, that is, when the target was expected, expectation and attention interactively influenced performance. However, when the distractor was expected, expectation and attention influenced performance separately. Furthermore, in order to explore whether task difficulty affects the relationship between expectation and attention, we increased the number of stimuli in Experiment 4. Again, we found that when the target was expected, expectation and attention were interacted in Experiment 4. Moreover, by comparing with Experiment 1, we found that the relationship between expectation and attention did not change as a function of task difficulty.
Our results suggested that when the expected subject is different (on the target or on the distractor), the relationship between expectation and attention was different. Previous studies have suggested that in the process of visual search, the promotion of the target and the inhibition of the distractor are controlled by two different neural mechanisms. Therefore, the relationship between expectation and attention is different when target is the expected subject relative to when distractor is the expected subject.When the target is expected, the interaction between expectation and attention on performance can be explained by predictive coding theory. Under the expected condition, the prediction error is small and there is no need to update the prediction model in the brain, so the efficiency of attention is higher. Under unexpected conditions, the prediction model needs to be updated, which leads to relatively low efficiency of attention. When the distractor is expected, expectation and attention affect performance additively, which may be due to the low correlation between the distractor expectation and the target task, or the processing of the distractor and the target are completed by different neural mechanisms. In conclusion, we believe that when expectation is anchored to different subjects, the relationship between expectation and attention are different. Specifically, when expectation is anchored to the distractor, they influence performance independently, and when expectation is anchored to the target, they are influence performance interactively.


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Research Paper

Applied Science and Convergence Technology 2021; 30(6): 163-166

Published online November 30, 2021

https://doi.org/10.5757/ASCT.2021.30.6.163

Temperature-Dependence Study on the Hydrogen Transport Properties of Polymers Used for Hydrogen Infrastructure

Jin-Sub Janga , b , Chunjoong Kimb , and Nak-Kwan Chunga , ∗

aKorea Research Institute of Standards and Science (KRISS), Daejeon 34113, Republic of Korea
bDepartment of Materials Science & Engineering, Chungnam National University, Daejeon 34134, Republic of Korea

Correspondence to:E-mail: nk.chung@kriss.re.kr

Received: November 4, 2021; Revised: November 19, 2021; Accepted: November 23, 2021

Hydrogen infrastThis is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.ructure, Polymers, Hydrogen permeation, Activation energy

Polymeric materials used in hydrogen infrastructure degrade with temperature changes, thereby increasing the risk of hydrogen leakage. Therefore, it is essential to evaluate the hydrogen permeation characteristics at varying temperatures, particularly at low temperatures, for the safe use of hydrogen energy. This study aimed to measure hydrogen permeability at a temperature range of 243–313 K for exploring the hydrogen permeation characteristics according to the temperature of ethylene-propylene diene monomer (EPDM), fluorine-rubber (FKM), and nitrilebutadiene rubber (NBR) for hydrogen infrastructure. Results show that permeability and diffusivity rapidly decreased as the temperature decreased; however, there was no significant change in solubility with the variation in temperature. Activation energies of permeability, diffusivity, and solubility were calculated from the measurement results, among which the activation energy for permeability exhibited the largest value in the order of FKM, NBR, and EPDM.

Keywords: Hydrogen infrastructure, Polymers, Hydrogen permeation, Activation energy

Polymeric materials are widely used in high-pressure hydrogen storage containers and sealing parts owing to its low weight and excellent moldability [1]. These polymers are exposed at various environmental temperatures ranging from 228–358 K, which can induce the degradation of the polymer characteristics and blister fractures or ruptures [2, 3]. Therefore, it is important to evaluate the hydrogen transport properties of polymers at various temperatures for the safety of the hydrogen infrastructure.

To date, several studies have investigated the hydrogen permeation properties of polymers under different temperature conditions [47]. Amerongen et al. measured the permeability (P), the diffusivity (D), and the solubility (S) of various rubbers such as natural rubber, methyl rubber, and butyl rubber in the temperature range of 273–373 K and derived the activation energies (EP and ED) of P and D and the heat of sorption (ΔHS) from the measured data. Humpenoder et al. [6] studied the gas permeation of plastics in the temperature range of 200–293 K and calculated their EP and ED of P and D. However, few attempts have been made to measure the low-temperature transport properties of hydrogen in rubbers.

This study detailed the development of the hydrogen permeability testing apparatus equipped with a temperature control device, and measured P, D, and S of EPDM, NBR, and FKM used in hydrogen infrastructures in the temperature range of 243–313 K. Then, EP, ED, and ΔHS were obtained from the linear slope of an Arrhenius plots of the logarithm of the data against inverse temperature.

2.1. Materials

The material used for the test was purchased in the form of a 1 mm thick sheet from Kraiburg TPE GmbH (Germany), with the hardness values listed in Table I. The glass transition temperatures (Tg) were measured by differential scanning calorimetry (DMA 242E; ASTM1356), and the hardness values were provided by the supplier of the materials. For the measurements, the samples were cut into a circle of 35 mm in diameter and then washed with distilled water and ethanol, before finally drying in a vacuum chamber for two days to reduce its outgassing rate.

Table I. Hardness and glass transition temperatures of specimens.

PolymerHardness (Shore A)Tg (K)
EPDM95215
NBR75252
FKM90265

2.2. Method

Figure 1(a) shows a schematic of a device developed based on the differential-pressure method according to ISO 15105-1 [8]. The differential- pressure method follows Fick’s diffusion law and Henry’s gas solubility law, whilst the parameters P, D, and S can be calculated using this method. The experimental procedures were as follows: a round sample of 35 mm diameter was mounted on the holder, and the chamber was evacuated using a dry pump for 1–2 days until the outgassing rate was lowered to less than 1 × 10−7 Pa/s. After a sufficiently long time, the desired sample temperature was adjusted using a bath circulator. Then, the gate valve was closed and the rate of pressure rise was checked to ensure it did not rise above 1 × 10−7 Pa/s. If the outgassing rate was sufficiently low, the valve in the high-pressure chamber was opened to bring hydrogen gas to 0.1 MPa and the increase of the pressure in the low-pressure part was recorded and graphed. As shown in Fig. 1(b), the permeation curve of dp/dt slope is slowly changed and the measurement is continued until the gas permeation reaches a constant rate.

Figure 1. (a) Schematics of the hydrogen permeation measurement apparatus and (b) the representative hydrogen permeation curve through polymers.

From the graph, the slope (dp/dt) can be obtained and P can be calculated from Eq. (1) [8]:

where R, T, Pext, Vc, p, d, and A are the ideal gas constant [8.31 × 103 l·Pa/K·mol], a sample temperature [K], an external hydrogen pressure [Pa], a chamber volume [m3], a pressure in the measurement chamber, the thickness of a sample [m], and the area of a sample [m2], respectively.

Therefore, D can be deduced from the time lag value as follows [8].

Finally, S can be obtained from P and D as follows.

Figure 2 shows the schematic of the temperature-controlled sample holder and its photograph. Vacuum grease was used to ensure tight sealing between the sample and the holder. Filter paper was inserted between the sample and the metal-sintered filter to allow the permeated gases to be released into the measurement chamber, and its diameter is equal to the inner diameter of the O-ring to clearly define the effective permeation area of the sample. The sample holders were maintained at the set temperature using a circulating bath, and the bath fluid circulates into the cell from one to four. Additional thermal insulating jackets were used (not shown); later, it was confirmed the temperature difference between the set value of the bath and the cell was within 5 K, and the temperature was then set to the desired value.

Figure 2. (a) Schematics of the temperature-controlled cell and (b) its photo-graph. The bath fluid circulates into the cell in the order 1 to 4.

To establish traceability to national flow standards, the proposed system was calibrated with the helium leak standard and the capacitance diaphragm gauge was also calibrated with the low-vacuum primary standard in KRISS [9]. Figure 3 shows a rate-of-rise curve in the system due to helium flow from the leak standard [10], and the background pressure rise (dp′/dt) due to outgassing rate (q) of the system was 1.6 × 10-4 Pa/s before He injection.

Figure 3. The pressure-rise curve by a flow injection using He leak standard. The curve is fitted after the He injection (t = 0).

The pressure rises (dp/dt) became 8.2×10-3 Pa/s when He was injected. Finally, the outgassing rate (Q) was obtained using Eq. (4) with the volume of the system being V = 6.22 ×10-5 m3.

$Q=V·dpdt−dp′dt$

3. Results and discussion

Figure 4 shows the curves of hydrogen permeation of EPDM, NBR, and FKM at room temperature (296 K). The pressure increase rates of EPDM, NBR, and FKM were 1.8 ×10−2, 4.6×10-3, and 2.1×10-3 Pa/s, respectively, while the time-lags were measured as 2267, 10297, and 13032 s, respectively. According to Eqs. (1),(3), P, D, and S of the samples were calculated and are listed in Table II. Similar to previous studies, FKM exhibits the lowest P for hydrogen gas at room temperature. It was also determined that D of EPDM was much larger than those of NBR and FKM, whilst S was relatively similar. Therefore, it can be seen that the hydrogen transport property in rubber is more related to D than S [25].

Table II. Hydrogen transport properties of rubbers at 296 K..

PolymerPermeability [mol•m/(m2•s•MPa)]Diffusivity (m2/s)Solubility [mol/(m3⋅MPa)]
EPDM17.25.233.4
NBR3.80.942.8
FKM2.10.826.4

Figure 4. Hydrogen permeation curves of EPDM, NBR, and FKM. The curves are obtained after the H2 injection (t = 0), and the dotted lines describe the tangent line to obtain the pressure-rise rate.

To analyze the effects of temperature on the hydrogen transport properties of rubbers, this study measured the P, D, and S in a temperature range of 243–313 K. As shown in Table IIIV, the P and D of rubber rapidly decrease as the T decreases. Finally, although EPDM is highly permeable to hydrogen below 253 K, the permeation through NBR and FKM were not observed even after 72 h experiments. Although S exhibited extremely weak temperature dependency, those of NBR and FKM abruptly decreased below 263 and 273 K, respectively. Therefore, it was determined that the rubbery characteristics of NBR and FKM begin to transition to the glassy and rigid states from approximately 10 K above Tg [13].

Table III. Hydrogen transport properties of EPDM at various temperatures..

Temperature (K)Permeability 10-9 [mol•m/(m2•s•MPa)]Diffusivity 10-10(m2/s)Solubility [mol/(m3⋅MPa)]
29617.25.233.4
2837.92.630.2
2735.31.534.1
2632.80.736.0
2531.20.429.5
2430.60.237.2

Table IV. Hydrogen transport properties of NBR at various temperatures..

Temperature (K)Permeability 10-9 [mol•m/(m2•s•MPa)]Diffusivity 10-10(m2/s)Solubility [mol/(m3⋅MPa)]
3035.01.338.6
2963.80.942.8
2831.60.446.3
2730.80.238.4
2630.20.210.8
2530.0--

Table V. Hydrogen transport properties of FKM at various temperatures..

Temperature (K)Permeability [mol•m/(m2•s•MPa)]Diffusivity (m2/s)Solubility [mol/(m3⋅MPa)]
3136.42.827.3
3033.01.126.8
2961.91.116.5
2830.90.425.2
2730.40.214.4
2630.0--

To explore the temperature dependence of P, D, and S of the rubbers, the data was fitted with the Arrhenius equations for P, D and S [14]:

where P0, D0, and S0 are the pre-exponential factors, Ep, ED, and ΔHS are the activation energy of permeation and diffusion and the heat of sorption, T is the temperature, and R is the ideal gas constant. The parameters EP, ED, and ΔHS were determined from the slopes of the least-squares regression lines from the data, as shown in Fig. 5. The values in the open symbols were excluded from the calculation of EP, ED, and ΔHS for the rubbery states. The activation energies and the heat of sorption of the rubbers calculated with the filled symbols in Fig. 5 are listed in Table VI. The EP of FKM was the highest, followed by that of NBR and EPDM. Notably, ED exhibited a similar trend with EP whilst ΔHS was extremely low. The value of S was hardly affected by temperature, indicating an extremely small temperature dependence. Therefore, it can be seen that the permeation properties are mainly related to D [10, 11].

Table VI. Hydrogen transport properties of rubbers..

MaterialTemperature Range (K)EP(kJ/mol)ED(kJ/mol)ΔHS(kJ/mol)
EPDM253 - 29637.937.30.5
NBR273 - 30341.542.0-0.5
FKM283 - 31347.544.92.1

Figure 5. The Arrhenius plots of (a) permeability, (b) diffusivity, and (c) solubility of EPDM, NBR, and FKM. The filled symbols represent the data obtained above their Tg, that is, in the rubbery state and the open symbols represent the data obtained near or below their Tg, that is, in the glassy state.

This study measured P, D, and S of EPDM, NBR, and FKM in a temperature range between 253–313 K. As the temperature decreased, it was observed that P and D rapidly decreased, while in contrast, S exhibited less temperature dependence. The activation energies of permeation and diffusion as well as the heat of sorption of the rubbers were determined from the slope of the Arrhenius plot of each material. As a result, it was concluded that the activation energy of the hydrogen permeation is higher in FKM than in NBR, and EPDM has the lowest value among them.

This work was supported by the 2019~2021 KAIA/MOLIT Project (No. 19TLRP-C152334-01) leaded by Korea Research Institute of Standards and Science (KRISS), Republic of Korea, and Development of Reliability Measurement Technology for Hydrogen Refueling Station funded by KRISS (KRISS – 2021 – GP2021-0007).

Conflicts of Interest

The authors declare no conflicts of interest.

1. R. R. Barth, K. L. Simmons, and C. San Marchi, Sandia Report No. SAND2013-8904 (2013).
2. J. Yamabe and S. Nishimura, Int. J. Hydrog. Energy 34, 1977 (2009).
3. R. M. Barrer and E. K. Ridal, Trans. Faraday Soc. 35, 628 (1939).
4. G. J. Van Amerongen, C. Koningsberger, and G. Salomon, J. Polym. Sci. 6, 639 (1950).
5. G. J. Van Amerongen, J. Appl. Phys. 17, 972 (1946).
6. J. Humpenoder, Cryogenics 38, 143 (1998).
7. J. Jang and N. Chung, Proceedings of the ASME 2020 Pressure Vessels & Piping Conference of the Pressure Vessels and Piping Division, edited by ASME (Virtual, Online, August 3, 2020) Vol. 1, pp. V001T01A041.
8. ISO (the International Organization for Standardization) https://www.iso.org/standard/41677.html (accessed Nov. 24, 2021).
9. S. Hong, J. Kim, and Y. Shin, Meas. Sci. Technol. 19, 015102 (2008).
10. Y. Shin, S. Hong, I. Lim, and K. Chung, Key Eng. Mater. 270-273, 1674 (2004).
11. T. Komatsuka and K. Nagai, Polym. J. 41, 455 (2009).
12. M. H. Klopffer and B. Flaconneche, Oil Gas Sci. Technol. 56, 223 (2001).
13. H. Yasuda and T. Hirotsu, J. Appl. Polym. Sci. 21, 105 (1977).
14. S. C. George and S. Thomas, Prog. Polym. Sci. 26, 985 (2001).


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Edit Article

# How to Integrate by Partial Fractions

Community Q&A

When integrating functions involving polynomials in the denominator, partial fractions can be used to simplify integration. New students of calculus will find it handy to learn how to decompose functions into partial fractions not just for integration, but for more advanced studies as well, such as ordinary differential equations.

## Steps

1. 1
Check to make sure that the fraction you are trying to integrate is proper. A proper fraction has a larger power in the denominator than in the numerator. If the power of the numerator is larger than or equal to the power of the denominator, it is improper and must be divided using long division.
• In this example, the fraction is indeed improper because the power of the numerator, 3, is larger than the power of the denominator, 2. Therefore, long division must be used.
2. 2
After dividing your improper fraction with long division, your new fraction can be put in the form of quotient + (remainder / divisor). In this example, we must use partial fractions to integrate the second fraction.
3. 3
Factor the polynomials in the denominator.
4. 4
Separate the fraction that you wish to decompose in to multiple fractions. The number of fractions in decomposition should equal the number of factors of x. The numerators of these decomposed fractions should be represented with constants.
• If a factor of x in the denominator has a power higher than 1, then the constants in the numerator should reflect this higher power. For example, x^2 in the denominator should be represented with the constant Ax + B.
5. 5
Multiply both sides by the denominator of the original fraction in order to get rid of all denominators. Notice that right now, the right side is factored by coefficients.
6. 6
On the side of the decomposed fractions, multiply everything out.
7. 7
Factor the decomposed side by the x terms.
8. 8
Set the coefficients equal on both sides. Because both sides are equal, that means that the coefficients of the x terms are equal.
9. 9
Solve for all constants.
10. 10
Plug the constants in to the decomposed fractions.
11. 11
If you divided earlier because the fraction you wished to integrate was improper, don't forget to plug in the rest of the integrand.
12. 12

## Tips

• Do not forget a + C if evaluating an indefinite integral!

## Article Info

Featured Article

Categories: Featured Articles | Calculus

Thanks to all authors for creating a page that has been read 28,435 times.


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### Admissible Covers and Compactness Arguments for Ill-founded Models of Set Theory

This was a talk at the CUNY Set Theory Seminar.

The Barwise compactness theorem is a powerful tool, allowing one to prove many interesting results which cannot be gotten just from the ordinary compactness theorem. For example, it can be used to show that every countable transitive model of set theory has an end extension which is a model of $V = L$.

However, the Barwise compactness theorem only applies to transitive sets. What are we to do if we want to have compactness arguments for ill-founded models of set theory?

This is where Barwise’s notion of the admissible cover of a (possibly ill-founded) model of set theory comes in. In this talk, we will see how to construct admissible covers and how they can be used to extend compactness arguments to the ill-founded case.

This talk is a prequel of sorts to my next talk in this seminar. Some of the results discussed in this talk will play a crucial role in the arguments there.

(Image due to Barwise)


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David Coveney

# Gateshead/Newcastle Pics from Thinking Digital (part 1)

Just a few quick shots from last night.  I’d arrived late for the first afternoon thanks to a failed gearbox mount (thanks for that, VW Warrington) but at least did get there thanks to a courtesy car (also thanks, VW Warrington!) that got me here just a couple of hours late…

Anyway, to the pics.  I’ll add more later, I’m sure.


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anonymous 4 years ago evaluate the limit as x approaches pi/2 (xcosx)

1. Zarkon

replace x by $$\pi/2$$ and evaluate

2. anonymous

deja vu,that certain feeling you get when you think "i've been here before"

3. Zarkon

you should just post this every time you get that certain feeling http://www.youtube.com/watch?v=oMsDKzTNhKY


text:
# Chemistry - Why are solids and liquids not included in the equilibrium constant? What about in a reaction rate calculation?

It very much depends on what definition of the equilibrium constant you are looking. The most common usage of the same has quite a variety of possible setups, see goldbook:

Equilibrium Constant
Quantity characterizing the equilibrium of a chemical reaction and defined by an expression of the type $$K_x = \Pi_B x_B^{\nu_B},$$ where $\nu_B$ is the stoichiometric number of a reactant (negative) or product (positive) for the reaction and $x$ stands for a quantity which can be the equilibrium value either of pressure, fugacity, amount concentration, amount fraction, molality, relative activity or reciprocal absolute activity defining the pressure based, fugacity based, concentration based, amount fraction based, molality based, relative activity based or standard equilibrium constant (then denoted $K^\circ$ ), respectively.

The standard equilibrium constant is always unitless, as it is defined differently (goldbook)

Standard Equilibrium Constant $K$, $K^\circ$ (Synonym: thermodynamic equilibrium constant) Quantity defined by $$K^\circ = \exp\left\{-\frac{\Delta_rG^\circ}{RT}\right\}$$ where $\Delta_rG^\circ$ is the standard reaction Gibbs energy, $R$ the gas constant and $T$ the thermodynamic temperature. Some chemists prefer the name thermodynamic equilibrium constant and the symbol $K$.

It is worthwhile to note, that the first definition is always an approximation to the standard definition and in the standard definition all compounds regardless of their state are included in the equilibrium.

If you are looking at a reaction that takes place in gas phase, solid materials will play a constant role, since their partial pressure will solely depend on their vapour pressure and can therefore be regarded as constant. Therefore it can be seen as a part of the equilibrium constant. (The same applies to liquids while deriving for gas phase.) If you would increase the amount of solid in the system you would still not change it's concentration in gas phase.

For the reaction $$\ce{H2 (g) + I2 (s) <=> 2HI (g)}$$ you can form the standard equilibrium constant with activities/ fugacities $$K^\circ = \frac{a^2(\ce{HI})}{a(\ce{H2})\cdot{}a(\ce{I2})},$$ with $$a=\frac{f}{p^\circ}.$$ The activity for a pure solid is normally defined as one ($a(\ce{I2})=1$) and therefore $$K^\circ\approx K = \frac{a^2(\ce{HI})}{a(\ce{H2})}$$

For concentration dependent equilibrium constants, the following assumption is to be considered: $c(\ce{H2O})\approx55.6~\mathrm{mol/L}$ and is usually always much larger than that of any other component of the system in the range where the equilibrium approximation can be used. Also in most solution based reactions, the solute itself does not directly influence the reaction. Its concentration will therefore not change (significantly) and can be included in the equilibrium constant.

You are correct in assuming that the kinetics of a surface reaction will only depend of the actual area of the surface (and of course the reactants forming the products). But here again this area is most likely to be considered as constant and will result in a scalar for the reaction rate (in the area where an accurate description of the reaction is possible).

As of Greg's comment, He is of course right. In an equilibrium process the surface area will affect both directions of the reaction, hence it will not show up in the reaction rate at all. (The only effect it might have is, that the equilibrium state will be reached faster.)


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# Underfitting and overfitting illustrated

In this article, we define underfitting and overfitting and show some nice ways to vizualize them on polynomial regressions.

## In short

Underfitting and overfitting describe the ability of a machine-learning model to make good predictions on datasets it wasn’t trained on.

Underfitting happens when:

• The model is too rigid to learn the true relationship in the data.
• Both test error and train error are large.
• The error is dominated by the bias error.

Overfitting happens when:

• The model is not rigid enough and mistakes noise for signal.
• The train error is low but the test error is large.
• The error is dominated by the variance error.

• The left hand side graph shows the underfitting scenario. We can see that the model is so rigid that is doesn’t fit the general shape of the data. This is called the bias of the model.
• The graph in the middle displays a good fit. The model fits the general shape of the data but does not wiggle too much between data points. This is the good bias-variance equilibrium.
• The right hand side graph shows the overfitting scenario. The model wiggles too much. And the general shape of the data is obfuscated by this high variance.

## The signal and the noise

Let $\mathcal{X} \in \realvset{\inputdim}$ be a random vector. Given a deterministic function $\truemodel:\inputvec \mapsto \outputval \in \realset$, we say that the random variable $\model\paren{\mathcal{X}}$ is a signal.

Let $\epsilon \sim \gaussian(0, \sigma)$ a gaussian noise with $0$ mean. The random variable $\mathcal{Y} = \model\paren{\mathcal{X}} + \epsilon$ is our signal polluted by the noise:

The train dataset, $\trainset$ is made of $\ndataset$ independent observations of $(\mathcal{X}, \mathcal{Y})$:

This means that for each observation $\idataset \leq \ndataset$, we have:

Where:

• $\ninputvec{\idataset}$ is a realization of $\mathcal{X}$, whose value is known, and:
• $\epsilon_n$ is a realization of $\epsilon$ whose value is unknown.

## Learning the signal

The goal of machine-learning is to learn the signal function $\model$ from the inputs generated by $\mathcal{X}$ and the outputs generated by $\mathcal{Y}$. This task is made complex because of the unknown noise observations $\epsilon_n$.

To do so, we suppose a model $\model_{\text{ML}}$ for the function $\model$ and minimize the error between the observed outputs $\ioutputval{\idataset}$ and the predictions $\model_{\text{ML}}(\ninputvec{\idataset})$.

While our end goal is to approximate $\model\paren{\inputvec}$, we can only do so by comparing our predictions with $\model\paren{\inputvec} + \epsilon$. Overfitting happens when our model “learns” the specifics of the noise realizations in the train dataset.

To illustrate overfitting, we will use polynomial regression. As the degree of the fitted polynomial increases, the model has more freedom to fit complex signals. But also more freedom to fit the unwanted noise.

This is illustrated on the picture below, where:

• the red curve is the real signal $\model\paren{\mathcal{X}}$;
• the red points are observed values for this signal, polluted by the random noise $\epsilon$;
• the blue curve is the regression line learned by a polynomial regression.

## High variance

To better visualize the implications of overfitting on the regression curve, we can generate multiple train datasets, each with the same signal curve (in red), but with different random values for the noise $\epsilon$. For instance:

and:

where $\epsilon_{n, 1}$ and $\epsilon_{n, 2}$ are different values for the noise, drawn from the same $\gaussian(0, 1)$ distribution.

Let’s generate a lot of train datasets like those.

If we fit a polynomial regression to each train dataset thus generated and we graph all the regression lines (in blue) on the same plot, we can visualize the high variance induced by overfitting. See picture below.

Since we used a gaussian noise with $0$ mean, taking the mean of the regression curves should yield the signal. Indeed, as the number $\ndataset_s$ of train sets used increases, the mean of the noise observations converges to $0$:


text:
# Effects of differing production levels on absorption costing inc

Effects of differing production levels on absorption costing income. Metrics to minimize inventory buildups. Effects of differing production levels on absorption costing income: Metrics to minimize inventory buildups. University Press produces textbooks for college courses. They recently hired a new editor, Leslie White, to handle production and sales of books for an introduction to accounting course. Leslie’s compensation depends on the gross margin associated with sales of this book. Leslie needs to decide how many copies of the book to produce. The following information is available for the fall semester 2010:

The fixed cost allocation rate is based on expected sales and is therefore equal to
$120,000/10,000 books =$12 per book
Leslie has decided to produce either 10,000, 12,000, or 16,000 books.
1. Calculate expected gross margin if Leslie produces 10,000, 12,000, or 16,000 hooks. (Make sure you include the production-volume variance as part of cost of goods sold)
2. Calculate ending inventory in units and in dollars for each production level.
3. Managers who are paid a bonus that is a function of gross margin may be inspired to produce product in excess of demand to maximize their own bonus. The chapter suggested metrics to discourage managers from producing products in excess of demand. Do you think the following metrics will accomplish this objective? Show your work.
a. Incorporate a charge of 10% of the cost of the ending inventory as an expense for evaluating the manager.
b. Include non-financial measures when evaluating management and rewardingperformance.


text:
# 1. Initiation to Bayesian models

This vignette can be referred to by citing the package:

• Makowski, D., Ben-Shachar, M. S., & Lüdecke, D. (2019). bayestestR: Describing Effects and their Uncertainty, Existence and Significance within the Bayesian Framework. Journal of Open Source Software, 4(40), 1541. https://doi.org/10.21105/joss.01541

Now that you’ve read the Get started section, let’s dive in the subtleties of Bayesian modelling using R.

Once you’ve installed the necessary packages, we can load rstanarm (to fit the models), bayestestR (to compute useful indices), and insight (to access the parameters).

library(rstanarm)
library(bayestestR)
library(insight)

## Simple linear (regression) model

We will begin by conducting a simple linear regression to test the relationship between Petal.Length (our predictor, or independent, variable) and Sepal.Length (our response, or dependent, variable) from the iris dataset which is included by default in R.

### Fitting the model

Let’s start by fitting a frequentist version of the model, just to have a reference point:

model <- lm(Sepal.Length ~ Petal.Length, data = iris)
summary(model)
>
> Call:
> lm(formula = Sepal.Length ~ Petal.Length, data = iris)
>
> Residuals:
>     Min      1Q  Median      3Q     Max
> -1.2468 -0.2966 -0.0152  0.2768  1.0027
>
> Coefficients:
>              Estimate Std. Error t value Pr(>|t|)
> (Intercept)    4.3066     0.0784    54.9   <2e-16 ***
> Petal.Length   0.4089     0.0189    21.6   <2e-16 ***
> ---
> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
>
> Residual standard error: 0.41 on 148 degrees of freedom
> Multiple R-squared:  0.76,    Adjusted R-squared:  0.758
> F-statistic:  469 on 1 and 148 DF,  p-value: <2e-16

We can also zoom in on the parameters of interest to us:

insight::get_parameters(model)
>      Parameter Estimate
> 1  (Intercept)     4.31
> 2 Petal.Length     0.41

In this model, the linear relationship between Petal.Length and Sepal.Length is positive and significant ($$\beta = 0.41, t(148) = 21.6, p < .001$$). This means that for each one-unit increase in Petal.Length (the predictor), you can expect Sepal.Length (the response) to increase by 0.41. This effect can be visualized by plotting the predictor values on the x axis and the response values as y using the ggplot2 package:

library(ggplot2) # Load the package

# The ggplot function takes the data as argument, and then the variables
# related to aesthetic features such as the x and y axes.
ggplot(iris, aes(x = Petal.Length, y = Sepal.Length)) +
geom_point() + # This adds the points
geom_smooth(method = "lm") # This adds a regression line

Now let’s fit a Bayesian version of the model by using the stan_glm function in the rstanarm package:

model <- stan_glm(Sepal.Length ~ Petal.Length, data = iris)

You can see the sampling algorithm being run.

### Extracting the posterior

Once it is done, let us extract the parameters (i.e., coefficients) of the model.

posteriors <- insight::get_parameters(model)

head(posteriors) # Show the first 6 rows
>   (Intercept) Petal.Length
> 1         4.4         0.39
> 2         4.4         0.40
> 3         4.3         0.41
> 4         4.3         0.40
> 5         4.3         0.40
> 6         4.3         0.41

As we can see, the parameters take the form of a lengthy dataframe with two columns, corresponding to the intercept and the effect of Petal.Length. These columns contain the posterior distributions of these two parameters. In simple terms, the posterior distribution is a set of different plausible values for each parameter. Contrast this with the result we saw from the frequentist linear regression mode using lm, where the results had single values for each effect of the model, and not a distribution of values. This is one of the most important differences between these two frameworks.

Let’s look at the length of the posteriors.

nrow(posteriors) # Size (number of rows)
> [1] 4000

Why is the size 4000, and not more or less?

First of all, these observations (the rows) are usually referred to as posterior draws. The underlying idea is that the Bayesian sampling algorithm (e.g., Monte Carlo Markov Chains - MCMC) will draw from the hidden true posterior distribution. Thus, it is through these posterior draws that we can estimate the underlying true posterior distribution. Therefore, the more draws you have, the better your estimation of the posterior distribution. However, increased draws also means longer computation time.

If we look at the documentation (?sampling) for the rstanarm’s "sampling" algorithm used by default in the model above, we can see several parameters that influence the number of posterior draws. By default, there are 4 chains (you can see it as distinct sampling runs), that each create 2000 iter (draws). However, only half of these iterations are kept, as half are used for warm-up (the convergence of the algorithm). Thus, the total for posterior draws equals 4 chains * (2000 iterations - 1000 warm-up) = 4000.

We can change that, for instance:

model <- stan_glm(Sepal.Length ~ Petal.Length, data = iris, chains = 2, iter = 1000, warmup = 250)

nrow(insight::get_parameters(model)) # Size (number of rows)
[1] 1500

In this case, as would be expected, we have 2 chains * (1000 iterations - 250 warm-up) = 1500 posterior draws. But let’s keep our first model with the default setup (as it has more draws).

#### Visualizing the posterior distribution

Now that we’ve understood where these values come from, let’s look at them. We will start by visualizing the posterior distribution of our parameter of interest, the effect of Petal.Length.

ggplot(posteriors, aes(x = Petal.Length)) +
geom_density(fill = "orange")

This distribution represents the probability (the y axis) of different effects (the x axis). The central values are more probable than the extreme values. As you can see, this distribution ranges from about 0.35 to 0.50, with the bulk of it being at around 0.41.

Congrats! You’ve just described your first posterior distribution.

And this is the heart of Bayesian analysis. We don’t need p-values, t-values, or degrees of freedom. Everything we need is contained within this posterior distribution.

Our description above is consistent with the values obtained from the frequentist regression (which resulted in a $$\beta$$ of 0.41). This is reassuring! Indeed, in most cases, Bayesian analysis does not drastically differ from the frequentist results or their interpretation. Rather, it makes the results more interpretable and intuitive, and easier to understand and describe.

We can now go ahead and precisely characterize this posterior distribution.

### Describing the Posterior

Unfortunately, it is often not practical to report the whole posterior distributions as graphs. We need to find a concise way to summarize it. We recommend to describe the posterior distribution with 3 elements:

1. A point-estimate which is a one-value summary (similar to the $$beta$$ in frequentist regressions).
2. A credible interval representing the associated uncertainty.
3. Some indices of significance, giving information about the relative importance of this effect.

#### Point-estimate

What single value can best represent my posterior distribution?

Centrality indices, such as the mean, the median, or the mode are usually used as point-estimates. But what’s the difference between them?

Let’s answer this by first inspecting the mean:

mean(posteriors$Petal.Length) > [1] 0.41 This is close to the frequentist $$\beta$$. But, as we know, the mean is quite sensitive to outliers or extremes values. Maybe the median could be more robust? median(posteriors$Petal.Length)
> [1] 0.41

Well, this is very close to the mean (and identical when rounding the values). Maybe we could take the mode, that is, the peak of the posterior distribution? In the Bayesian framework, this value is called the Maximum A Posteriori (MAP). Let’s see:

map_estimate(posteriors$Petal.Length) > MAP Estimate: 0.41 They are all very close! Let’s visualize these values on the posterior distribution: ggplot(posteriors, aes(x = Petal.Length)) + geom_density(fill = "orange") + # The mean in blue geom_vline(xintercept = mean(posteriors$Petal.Length), color = "blue", size = 1) +
# The median in red
geom_vline(xintercept = median(posteriors$Petal.Length), color = "red", size = 1) + # The MAP in purple geom_vline(xintercept = map_estimate(posteriors$Petal.Length), color = "purple", size = 1)

Well, all these values give very similar results. Thus, we will choose the median, as this value has a direct meaning from a probabilistic perspective: there is 50% chance that the true effect is higher and 50% chance that the effect is lower (as it divides the distribution in two equal parts).

#### Uncertainty

Now that the have a point-estimate, we have to describe the uncertainty. We could compute the range:

range(posteriors$Petal.Length) > [1] 0.33 0.48 But does it make sense to include all these extreme values? Probably not. Thus, we will compute a credible interval. Long story short, it’s kind of similar to a frequentist confidence interval, but easier to interpret and easier to compute — and it makes more sense. We will compute this credible interval based on the Highest Density Interval (HDI). It will give us the range containing the 89% most probable effect values. Note that we will use 89% CIs instead of 95% CIs (as in the frequentist framework), as the 89% level gives more stable results (Kruschke, 2014) and reminds us about the arbitrariness of such conventions (McElreath, 2018). hdi(posteriors$Petal.Length, ci = 0.89)
> 89% HDI: [0.38, 0.44]

Nice, so we can conclude that the effect has 89% chance of falling within the [0.38, 0.44] range. We have just computed the two most important pieces of information for describing our effects.

#### Effect significance

However, in many scientific fields it not sufficient to simply describe the effects. Scientists also want to know if this effect has significance in practical or statistical terms, or in other words, whether the effect is important. For instance, is the effect different from 0? So how do we assess the significance of an effect. How can we do this?

Well, in this particular case, it is very eloquent: all possible effect values (i.e., the whole posterior distribution) are positive and over 0.35, which is already substantial evidence the effect is not zero.

But still, we want some objective decision criterion, to say if yes or no the effect is ‘significant’. One approach, similar to the frequentist framework, would be to see if the Credible Interval contains 0. If it is not the case, that would mean that our effect is ‘significant’.

But this index is not very fine-grained, no? Can we do better? Yes!

## A linear model with a categorical predictor

Imagine for a moment you are interested in how the weight of chickens varies depending on two different feed types. For this example, we will start by selecting from the chickwts dataset (available in base R) two feed types of interest for us (we do have peculiar interests): meat meals and sunflowers.

### Data preparation and model fitting

library(dplyr)

# We keep only rows for which feed is meatmeal or sunflower
data <- filter(chickwts, feed %in% c("meatmeal", "sunflower"))

Let’s run another Bayesian regression to predict the weight with the two types of feed type.

model <- stan_glm(weight ~ feed, data = data)

### Posterior description

posteriors <- insight::get_parameters(model)

ggplot(posteriors, aes(x = feedsunflower)) +
geom_density(fill = "red")

This represents the posterior distribution of the difference between meatmeal and sunflowers. It seems that the difference is positive (since the values are concentrated on the right side of 0). Eating sunflowers makes you more fat (at least, if you’re a chicken). But, by how much?

Let us compute the median and the CI:

median(posteriors$feedsunflower) > [1] 53 hdi(posteriors$feedsunflower)
> 95% HDI: [-0.12, 97.72]

It makes you fat by around 51 grams (the median). However, the uncertainty is quite high: there is 89% chance that the difference between the two feed types is between 14 and 91.

Is this effect different from 0?

### ROPE Percentage

Testing whether this distribution is different from 0 doesn’t make sense, as 0 is a single value (and the probability that any distribution is different from a single value is infinite).

However, one way to assess significance could be to define an area around 0, which will consider as practically equivalent to zero (i.e., absence of, or a negligible, effect). This is called the Region of Practical Equivalence (ROPE), and is one way of testing the significance of parameters.

How can we define this region?

Driing driiiing

The easystats team speaking. How can we help?

I am Prof. Sanders. An expert in chicks… I mean chickens. Just calling to let you know that based on my expert knowledge, an effect between -20 and 20 is negligible. Bye.

Well, that’s convenient. Now we know that we can define the ROPE as the [-20, 20] range. All effects within this range are considered as null (negligible). We can now compute the proportion of the 89% most probable values (the 89% CI) which are not null, i.e., which are outside this range.

rope(posteriors$feedsunflower, range = c(-20, 20), ci = 0.89) > # Proportion of samples inside the ROPE [-20.00, 20.00]: > > inside ROPE > ----------- > 5.28 % 5% of the 89% CI can be considered as null. Is that a lot? Based on our guidelines, yes, it is too much. Based on this particular definition of ROPE, we conclude that this effect is not significant (the probability of being negligible is too high). That said, to be honest, I have some doubts about this Prof. Sanders. I don’t really trust his definition of ROPE. Is there a more objective way of defining it? Yes! One of the practice is for instance to use the tenth (1/10 = 0.1) of the standard deviation (SD) of the response variable, which can be considered as a “negligible” effect size (Cohen, 1988). rope_value <- 0.1 * sd(data$weight)
rope_range <- c(-rope_value, rope_value)
rope_range
> [1] -6.2  6.2

Let’s redefine our ROPE as the region within the [-6.2, 6.2] range. Note that this can be directly obtained by the rope_range function :)

rope_value <- rope_range(model)
rope_value
> [1] -6.2  6.2

Let’s recompute the percentage in ROPE:

rope(posteriors$feedsunflower, range = rope_range, ci = 0.89) > # Proportion of samples inside the ROPE [-6.17, 6.17]: > > inside ROPE > ----------- > 0.00 % With this reasonable definition of ROPE, we observe that the 89% of the posterior distribution of the effect does not overlap with the ROPE. Thus, we can conclude that the effect is significant (in the sense of important enough to be noted). ### Probability of Direction (pd) Maybe we are not interested in whether the effect is non-negligible. Maybe we just want to know if this effect is positive or negative. In this case, we can simply compute the proportion of the posterior that is positive, no matter the “size” of the effect. n_positive <- posteriors %>% filter(feedsunflower > 0) %>% # select only positive values nrow() # Get length n_positive / nrow(posteriors) * 100 > [1] 98 We can conclude that the effect is positive with a probability of 98%. We call this index the Probability of Direction (pd). It can, in fact, be computed more easily with the following: p_direction(posteriors$feedsunflower)
> Probability of Direction: 0.98

Interestingly, it so happens that this index is usually highly correlated with the frequentist p-value. We could almost roughly infer the corresponding p-value with a simple transformation:

pd <- 97.82
onesided_p <- 1 - pd / 100
twosided_p <- onesided_p * 2
twosided_p
> [1] 0.044

If we ran our model in the frequentist framework, we should approximately observe an effect with a p-value of 0.04. Is that true?

#### Comparison to frequentist

summary(lm(weight ~ feed, data = data))
>
> Call:
> lm(formula = weight ~ feed, data = data)
>
> Residuals:
>     Min      1Q  Median      3Q     Max
> -123.91  -25.91   -6.92   32.09  103.09
>
> Coefficients:
>               Estimate Std. Error t value Pr(>|t|)
> (Intercept)      276.9       17.2   16.10  2.7e-13 ***
> feedsunflower     52.0       23.8    2.18     0.04 *
> ---
> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
>
> Residual standard error: 57 on 21 degrees of freedom
> Multiple R-squared:  0.185,   Adjusted R-squared:  0.146
> F-statistic: 4.77 on 1 and 21 DF,  p-value: 0.0405

The frequentist model tells us that the difference is positive and significant ($$\beta = 52, p = 0.04$$).

Although we arrived to a similar conclusion, the Bayesian framework allowed us to develop a more profound and intuitive understanding of our effect, and of the uncertainty of its estimation.

## All with one function

And yet, I agree, it was a bit tedious to extract and compute all the indices. But what if I told you that we can do all of this, and more, with only one function?

Behold, describe_posterior!

This function computes all of the adored mentioned indices, and can be run directly on the model:

describe_posterior(model, test = c("p_direction", "rope", "bayesfactor"))
> Summary of Posterior Distribution
>
> Parameter     | Median |           95% CI |     pd |          ROPE | % in ROPE |  Rhat |     ESS |     BF
> ---------------------------------------------------------------------------------------------------------
> (Intercept)   | 276.62 | [240.03, 310.68] |   100% | [-6.17, 6.17] |        0% | 1.002 | 2793.00 | > 1000
> feedsunflower |  52.73 | [ -0.12,  97.72] | 98.38% | [-6.17, 6.17] |     1.89% | 1.000 | 2860.00 |  0.806

Tada! There we have it! The median, the CI, the pd and the ROPE percentage!

Understanding and describing posterior distributions is just one aspect of Bayesian modelling. Are you ready for more?! Click here to see the next example.

## References

Cohen, J. (1988). Statistical power analysis for the social sciences.
Kruschke, J. (2014). Doing bayesian data analysis: A tutorial with r, JAGS, and stan. Academic Press.
McElreath, R. (2018). Statistical rethinking: A bayesian course with examples in r and stan. Chapman; Hall/CRC.


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What is the value of the two-digit positive integer n?

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What is the value of the two-digit positive integer n? [#permalink]

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What is the value of the two-digit positive integer n?

(1) When n is divided by 5, the remainder is equal to the tens digit of n.
(2) When n is divided by 9, the remainder is equal to the tens digit of n.
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Re: What is the value of the two-digit positive integer n? [#permalink]

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alexpavlos wrote:
What is the value of the two-digit positive integer n?

(1) When n is divided by 5, the remainder is equal to the tens digit of n.

(2) When n is divided by 9, the remainder is equal to the tens digit of n.

Any help on this one would be much appreciated! Thanks

Take the LCM of 9 and 5 which is 45. Since remainder is tens digit, add 4 to 45 = 49

Now, 49/5 = 9 + remainder ->4
49/9 = 5 + remainder ->4
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Re: What is the value of the two-digit positive integer n? [#permalink]

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alex1233 wrote:
What is the value of the two-digit positive integer n?

(1) When n is divided by 5, the remainder is equal to the tens digit of n.

(2) When n is divided by 9, the remainder is equal to the tens digit of n.

Any help on this one would be much appreciated! Thanks

Let's take each statement at a time.

(1) When n is divided by 5, the remainder is equal to the tens digit of n.
Think of a two digit number which is divisible by 5 - say 15. The remainder should be 1 so say n = 16.
Think of another number which is divisible by 5 - say 25. The remainder should be 2 so say n = 27
There will be more such numbers so we can see that this is certainly not sufficient.

(2) When n is divided by 9, the remainder is equal to the tens digit of n.
Think of a two digit number which is divisible by 9 - say 18. The remainder should be 1 so say n = 19.
Think of another number which is divisible by 9 - say 27. The remainder should be 2 so say n = 29
There will be more such numbers so we can see that this is certainly not sufficient.

What do we do when we consider both statements together?
We need to think of a number divisible by both 5 and 9, say 45 (their LCM). The remainder should be 4 so add 4 to 45 to get n = 49
Think of another number divisible by both which will be the next multiple of 45 i.e. 90. The remainder should be 9 but when we divide a number by 5, the remainder cannot be greater than 4. So n cannot be 99.
Hence, there is only one such two digit number i.e. n = 49.

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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Manager Joined: 19 Aug 2012 Posts: 58 Followers: 1 Kudos [?]: 42 [0], given: 9 Re: What is the value of the two-digit positive integer n? [#permalink] Show Tags 18 Mar 2013, 12:21 clearly statement 1 leads to many options and same to statement 2 now taking both the statements together. it should be a no. which is common multiple of both 5 and 9 and also is 2 digit no. which has 10th place digit as reminder... we have 9x5=45 so for reminder to be 4 the no. should be 49, which gives us reminder as 4. no other no. satisfies all the criteria mentioned in question. clearly option C is answer _________________ giving kudos is the best thing you can do for me.. Manager Joined: 20 Jun 2012 Posts: 105 Location: United States Concentration: Finance, Operations GMAT 1: 710 Q51 V25 Followers: 1 Kudos [?]: 35 [2] , given: 52 Re: What is the value of the two-digit positive integer n? [#permalink] Show Tags 27 Jun 2013, 06:38 2 This post received KUDOS alex1233 wrote: What is the value of the two-digit positive integer n? (1) When n is divided by 5, the remainder is equal to the tens digit of n. (2) When n is divided by 9, the remainder is equal to the tens digit of n. Any help on this one would be much appreciated! Thanks we really dont need any calculation in this question. This is quite conceptual. we know remainder of any number when divided by 5 can only be 1,2,3 or 4. Its given remainder equals to tens digit. we'll take the four remainders one by one. 1 >> we know tens digit should be 1 so number could only be 11 OR 16. 2 >> we know tens digit should be 2 so number could only be 22 OR 27 3 >> we know tens digit should be 3 so number could only be 33 OR 38. 4 >> we know tens digit should be 4 so number could only be 44 OR 49. we cant get an answer from st. 1. we can do same reasoning for st. 2 1 >> 10,19 2 >> 20,29 3 >> 30,39 4 >> 40,49 .... so on .. also no point going forward. we found a match from st. 1(and in st. 1 we wrote all the possible outcomes, so possibility of another such no. is zero) .. the no. is 49. hence C. _________________ Forget Kudos ... be an altruist Current Student Joined: 06 Sep 2013 Posts: 2035 Concentration: Finance GMAT 1: 770 Q0 V Followers: 43 Kudos [?]: 463 [0], given: 355 Re: What is the value of the two-digit positive integer n? [#permalink] Show Tags 15 Feb 2014, 16:30 VeritasPrepKarishma wrote: alex1233 wrote: What is the value of the two-digit positive integer n? (1) When n is divided by 5, the remainder is equal to the tens digit of n. (2) When n is divided by 9, the remainder is equal to the tens digit of n. Any help on this one would be much appreciated! Thanks Let's take each statement at a time. (1) When n is divided by 5, the remainder is equal to the tens digit of n. Think of a two digit number which is divisible by 5 - say 15. The remainder should be 1 so say n = 16. Think of another number which is divisible by 5 - say 25. The remainder should be 2 so say n = 27 There will be more such numbers so we can see that this is certainly not sufficient. (2) When n is divided by 9, the remainder is equal to the tens digit of n. Think of a two digit number which is divisible by 9 - say 18. The remainder should be 1 so say n = 19. Think of another number which is divisible by 9 - say 27. The remainder should be 2 so say n = 29 There will be more such numbers so we can see that this is certainly not sufficient. What do we do when we consider both statements together? We need to think of a number divisible by both 5 and 9, say 45 (their LCM). The remainder should be 4 so add 4 to 45 to get n = 49 Think of another number divisible by both which will be the next multiple of 45 i.e. 90. The remainder should be 9 but when we divide a number by 5, the remainder cannot be greater than 4. So n cannot be 99. Hence, there is only one such two digit number i.e. n = 49. Answer (C) Hi Karishma, I'm a bit stucked with both statements together. How do you know that the remainder has to be 4 and not 1,2 or 3? Could you please elaborate on this? Many thanks! Cheers J Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 6573 Location: Pune, India Followers: 1790 Kudos [?]: 10764 [0], given: 211 Re: What is the value of the two-digit positive integer n? [#permalink] Show Tags 16 Feb 2014, 20:55 Expert's post 1 This post was BOOKMARKED jlgdr wrote: VeritasPrepKarishma wrote: alex1233 wrote: What is the value of the two-digit positive integer n? (1) When n is divided by 5, the remainder is equal to the tens digit of n. (2) When n is divided by 9, the remainder is equal to the tens digit of n. Any help on this one would be much appreciated! Thanks Let's take each statement at a time. (1) When n is divided by 5, the remainder is equal to the tens digit of n. Think of a two digit number which is divisible by 5 - say 15. The remainder should be 1 so say n = 16. Think of another number which is divisible by 5 - say 25. The remainder should be 2 so say n = 27 There will be more such numbers so we can see that this is certainly not sufficient. (2) When n is divided by 9, the remainder is equal to the tens digit of n. Think of a two digit number which is divisible by 9 - say 18. The remainder should be 1 so say n = 19. Think of another number which is divisible by 9 - say 27. The remainder should be 2 so say n = 29 There will be more such numbers so we can see that this is certainly not sufficient. What do we do when we consider both statements together? We need to think of a number divisible by both 5 and 9, say 45 (their LCM). The remainder should be 4 so add 4 to 45 to get n = 49 Think of another number divisible by both which will be the next multiple of 45 i.e. 90. The remainder should be 9 but when we divide a number by 5, the remainder cannot be greater than 4. So n cannot be 99. Hence, there is only one such two digit number i.e. n = 49. Answer (C) Hi Karishma, I'm a bit stucked with both statements together. How do you know that the remainder has to be 4 and not 1,2 or 3? Could you please elaborate on this? Many thanks! Cheers J The tens digit of 45 is 4. 45 is the first positive two digit number which is divisible by both 5 and 9. So when n is divided by 5 or 9, the remainder should be 4 so n should be 49. The remainder will be 4 which is the tens digit of 49. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: What is the value of the two-digit positive integer n?   [#permalink] 16 Sep 2015, 09:54
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