text:
## Topological Methods in Nonlinear Analysis

### Periodic orbits for multivalued maps with continuous margins of intervals

#### Abstract

Let $I$ be a bounded connected subset of $\mathbb{R}$ containing more than one point, and $\mathcal L(I)$ be the family of all nonempty connected subsets of $I$. Each map from $I$ to $\mathcal L(I)$ is called a multivalued map. A multivalued map $F\colon I\rightarrow\mathcal L(I)$ is called a multivalued map with continuous margins if both the left endpoint and the right endpoint functions of $F$ are continuous. We show that the well-known Sharkovskiĭ theorem for interval maps also holds for every multivalued map with continuous margins $F\colon I\rightarrow \mathcal L(I)$, that is, if $F$ has an $n$-periodic orbit and $n\succ m$ (in the Sharkovskiĭ ordering), then $F$ also has an $m$-periodic orbit.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 48, Number 2 (2016), 453-464.

Dates
First available in Project Euclid: 21 December 2016

https://projecteuclid.org/euclid.tmna/1482289223

Digital Object Identifier
doi:10.12775/TMNA.2016.052

Mathematical Reviews number (MathSciNet)
MR3642767

Zentralblatt MATH identifier
1373.54027

#### Citation

Mai, Jiehua; Sun, Taixiang. Periodic orbits for multivalued maps with continuous margins of intervals. Topol. Methods Nonlinear Anal. 48 (2016), no. 2, 453--464. doi:10.12775/TMNA.2016.052. https://projecteuclid.org/euclid.tmna/1482289223

#### References

• L. Alseda and J. Llibre, Periods for triangular maps, Bull. Austral. Math. Soc. 47 (1993), 41–53.
• J. Andres, Period two implies chaos for a class of multivalued maps: A naive approach, Comput. Math. Appl. 64 (2012), 2160–2165.
• J. Andres, J. Fišer and L. Jüttner, On a multivalued version of the Sharkovskiĭ theorem and its application to differential inclusions, Set-Valued Anal. 10 (2002), 1–14.
• J. Andres, T. Fürst and K. Pastor, Period two implies all periods for a class of ODEs: a multivalued map approach, Proc. Amer. Math. Soc. 135 (2007), 3187–3191.
• ––––, Full analogy of Sharkovskii's theorem for lower semicontinuous maps, J. Math. Anal. Appl. 340 (2008), 1132–1144.
• ––––, Sharkovskiĭ's theorem, differential inclusions, and beyond, Topol. Methods Nonlinear Anal. 33 (2009), 149–168.
• J. Andres, L. Jüttner and K. Pastor, On a multivalued version of the Sharkovskiĭ theorem and its application to differential inclusions II, Set-Valued Anal. 13 (2005), 47–68.
• J. Andres and K. Pastor, On a multivalued version of the Sharkovskii theorem and its application to differential inclusions III, Topol. Methods Nonliear. Anal. 22 (2003), 369–386.
• ––––, A version of Sharkovskiĭ's theorem for differential equations, Proc. Amer. Math. Soc. 133 (2004), 449–453.
• J. Andres, K. Pasto and P. Šnyrychová, A multivalued version of Sharkovskiĭ's theorem holds with at most two exceptions, J. Fixed Point Theory Appl. 2 (2007), 153–170.
• Yu.G. Borisovich, B.D. Gel'man, A.D. Myshkis and V.V. Obukhovskiĭ, Topological methods in the fixed point theory of multi-valued maps, Russian Math. Surveys 35 (1980), 65–143.
• P. Minc and W.R.R. Transue, Sharkovskiĭ's Theorem for hereditarily decomposable chainable continua, Trans. Amer. Math. Soc. 315 (1989), 173–188.
• D. Repovs and P.V. Semenov, Continuous selections of multivalued mappings, mathematics and its applications, Kluwer Academic Publishers, 1998.
• ––––, Continuous selections of multivalued mappings, recent progress in general topology (II), North-Holland, 2002.
• J.J. Sanchez-Gabites and J.M.R. Sanjurjo, Multivalued maps, selections and dynamical systems, Topol. Appl. 155 (2008), 874–882.
• J.M.R. Sanjurjo, Selections of multivalued maps and shape domination, Math. Proc. Cambridge Philos. Soc. 107 (1990), 493–499.
• A.N. Sharkovskiĭ, Coexistence of cycles of a continuous map of a line into itself, Ukrai. Math. J. 16 (1964), 61–71. (Russian)


text:
• A
• A
• A
• ABC
• ABC
• ABC
• А
• А
• А
• А
• А
Regular version of the site
Of all publications in the section: 3 519
Sort:
by name
by year
Working paper
Yakovlev A. A., Freinkman L., Ershova N. V. Policy Brief. 5/2018. Bank of Finland Institute for Economies in Transition. BOFIT, 2018. No. 5.
We explore the role and organizing capacity of foreign business associations (FBAs) in the Russian context. Considering the potential role of FBAs as effective intermediaries during time of aggravated international relations, the paper examines the problems and mechanisms of foreign firms’ interaction with key national partners in their host countries and describes the conditions that lead to effective dialogue between FBAs and national governments. The main phases and factors of evolution are identified for Russia’s two main channels of foreign firms’ collective actions: The Foreign Investment Advisory Council, which is chaired by Russia’s prime minister, and FBAs. A comparative analysis of the efficiencies of each channel finds that political, rather than economic or institutional, factors play the dominant role in explaining the shifting efficiency of collective action of foreign firms over time.
Working paper
Van de Leur J., Orlov A. Y. math. arxive. Cornell University, 2016. No. 1611.04577.
We consider character expansion of tau functions and multiple integrals in characters of orhtogonal and symplectic groups. In particular we consider character expansions of integrals over orthogonal and over symplectic matrices.
Working paper
Levin A., Olshanetsky M., Smirnov A. et al. arxiv.org. math. Cornell University, 2012
Working paper
Amerik E., Campana, F. arxiv.org. math. Cornell University, 2014. No. 1405.0539.
We prove that the characteristic foliation F on a non- singular divisor D in an irreducible projective hyperkaehler mani- fold X cannot be algebraic, unless the leaves of F are rational curves or X is a surface. More generally, we show that if X is an arbitrary projective manifold carrying a holomorphic symplectic 2-form, and D and F are as above, then F can be algebraic with non-rational leaves only when, up to a finite  etale cover, X is the product of a symplectic projective manifold Y with a symplectic surface and D is the pull-back of a curve on this surface. When D is of general type, the fact that F cannot be algebraic unless X is a surface was proved by Hwang and Viehweg. The main new ingredient for our results is the observation that the canonical bundle of the orbifold base of the family of leaves must be torsion. This implies, in particular, the isotriviality of the family of leaves of F .  ́
Working paper
Kochetkov Y. Cornell University Library, 2012. No. 1212.6317.
A polynomial with exactly two critical values is called a generalized Chebyshev polynomial. A polynomial with exactly three critical values is called a Zolotarev polynomial. Two Chebyshev polynomials $f$ and $g$ are called Z-homotopic, if there exists a family $p_\alpha$, $\alpha\in [0,1]$, where $p_0=f$, $p_1=g$ and $p_\alpha$ is a Zolotarev polynomial, if $\alpha\in (0,1)$. As each Chebyshev polynomial defines a plane tree (and vice versa), Z-homotopy can be defined for plane trees. In this work we prove some necessary geometric conditions for plane trees Z-homotopy, describe Z-homotopy for trees with 5 and 6 edges and study one interesting example in the class of trees with 7 edges.
Working paper
Pushkar P. E. math. arxive. Cornell University, 2016. No. arXiv:1602.08743.
We prove a Chekanov-type theorem for the spherization of the cotangent bundle ST∗B of a closed manifold B. It claims that for Legendrian submanifolds in ST∗B the property "to be given by a generating family quadratic at infinity" persists under Legendrian isotopies.
Working paper
Sechin P. math. arxive. Cornell University, 2016
In this paper we calculate the ring of unstable (possibly non-additive) operations from algebraic Morava K-theory K(n) to Chow groups with ℤ(p)-coefficients. More precisely, we prove that it is a formal power series ring on generators c_i:K(n)→CH^i⊗ℤ(p), which satisfy a Cartan-type formula.
Working paper
Sechin P. math. arxive. Cornell University, 2018
Generalizing the definition of Cartier, we introduce pn-typical formal group laws over ℤ(p)-algebras. An oriented cohomology theory in the sense of Levin-Morel is called pn-typical if its corresponding formal group law is p^n-typical. The main result of the paper is the construction of 'Chern classes' from the algebraic n-th Morava K-theory to every p^n-typical oriented cohomology theory.  If the coefficient ring of a p^n-typical theory is a free ℤ(p)-module we also prove that these Chern classes freely generate all operations to it. Examples of such theories are algebraic mn-th Morava K-theories K(nm)∗ for all m∈ℕ and CH∗⊗ℤ(p) (operations to Chow groups were studied in a previous paper). The universal pn-typical oriented theory is BP{n}∗=BP∗/(v_j,j∤n) which coefficient ring is also a free ℤ_(p)-module.  Chern classes from the n-th algebraic Morava K-theory K(n)∗ to itself allow us to introduce the gamma filtration on K(n)∗. This is the best approximation to the topological filtration obtained by values of operations and it satisfies properties similar to that of the classical gamma filtration on K0. The major difference from the classical case is that Chern classes from the graded factors gr^i_γ K(n)^∗ to CH^i⊗ℤ_(p) are surjective for i≤p^n. For some projective homogeneous varieties this allows to estimate p-torsion in Chow groups of codimension up to pn.
Working paper
Shitov Y. arxiv.org. math. Cornell University, 2014. No. 1409.1530.
We give an example of two n-by-n chess positions, A and B, such that (1) there is a sequence of legal chess moves leading from A to B; (2) the length of this sequence cannot be less than exp(cn).
Working paper
Churilova E., Zakharov S. V. Working Papers. SSRN, 2017
The collapse of the USSR and structural changes in society gave rise to family and marriage transformations in Russia and other post-Soviet countries of Eastern Europe. Estonia is among the examples of rapid marriage transformation and the spread of cohabitations and extramarital births. At the same time, throughout the post-war era Estonia accepted Russian migrants. Studies show that the adaptation of the Russian population in Estonia proceeds slowly, and that patterns of matrimonial behavior among the Russian migrants and their descendants in Estonia are closer to patterns of ethnic Russians in Russia. In this study, the Generation and Gender Survey data for Russia (2004) and Estonia (2005) were used to analyze the prevalence of first and second childbirths in first and higher-order unions and the determinants of fertility after repartnering among the Russian and Estonian populations in Estonia and Russians in Russia. We consider the impact of partners’ age and parenthood status at the beginning of higher-order unions, as well as social variables (e.g. education, type of settlement and ethnicity).  The results show that the contribution of higher-order unions to total fertility increased significantly in all studied populations. However, first childbirths in high-order unions are rare among both the Russians and the Estonians, second births after repartnering are more widespread among Russians in Russia. Divorces and separations significantly increase the interval between first and second births, but there is no difference in time interval duration between the three considered population groups. The determinants of childbearing among Russians in Russia and Estonia in higher-order unions are very similar: they are the age of women, the type of habitation and the initial nature of the union. The effect of the parenthood status and marriage cohort is individual for each population group: in Russia, if a male partner has no child(ren) from previous relationships, the couple has a higher chance of having a mutual child. The probability of childbearing after repartnering significantly increased in 1990-1999 in Estonia.
Working paper
Lokshin M., Fong M. Policy Research Working Paper. WPS. World Bank Group, 2000. No. 2400.
The authors model the household demand for child care, the mother's participation in the labor force, and her working hours in Romania. Their model estimates the effects of the price of child care, the mother's wage, and household income on household behavior relating to child care and mothers working outside the home. They find that: Both the maternal decision to take a job and the decision to use out-of-home care are sensitive to the price of child care. A decrease in the price of child care can increase the number of mothers who work and thus reduce poverty in some households. The potential market wage of the mother has a significant positive effect on the decision to purchase market care and the decision to engage in paid employment. The level of household nonwage income has little effect on maternal employment and the demand for child care. In addition to facilitating women's work, kindergartens and cr�hes appear to provide educational and social benefits for children. Close to half the children in these facilities have mothers who do not work. Further research is needed to assess the cost and nature of these benefits and to determine the appropriate roles for the private and public sectors in providing, financing, and regulating such services for working and nonworking mothers.
Working paper
Zaretsky Y. SSRN Working Paper Series. SSRN Working Paper Series. Social Science Research Network, 2019
The article reviews the narratives on childhood in medieval autobiographical writings. It argues that these narratives allow us both to see differences in perception of childhood between medieval and modern autobiographers and to rethink the well-known argument of historians of childhood that medieval culture was indifferent to the first years of human life.
Working paper
Акифьева Р. Н. Humanities. HUM. Basic Research Programme, 2016. № WP BRP 140/HUM/2016.
This working paper is based on the results of a study of the child-rearing practices of Russian-speaking women in the context of a migration perspective focusing on their migration experience, language, and other features that are perceived as cultural and an integration context as determining many important aspects of child-rearing. The analysis shows that a mother’s choice of structured activities for the child can be caused by structural factors, such as the national peculiarities of the system of school and pre-school education, family policy, the specifics and set of programs offered at schools and available in the area. Regardless of the social segment, in which migrants are incorporated, children begin to prefer speaking Spanish than Russian if there is not any special effort from the adults. This leads to the fact that the study of the Russian language in some families affects all spheres of life, and many aspects of child-rearing. Many women develop and share ideas about the differences in the parenting approach and style of women from the post-Soviet space and from Spain. Women perceive the stylistic features of the behavior of Russian women as highly demanding and share the view of the low demands of Spanish parents, yet differ in how they perceive their level of responsiveness to them.
Working paper
Akifyeva R. N. Humanities. HUM. Basic Research Programme, 2015. No. WP BRP 103/HUM/2015.
The article is drawn on a study of the families of migrants, having school-age children, from Central Asia. The transnational practices of migrants are closely tied to integration processes, and this is reflected in generational differences. In some families, children that, unlike their parents, go or used to go to a Russian school, try to distance themselves from their parents’ generation, refuse to identify with it, and do not want to speak the native language of their parents. In some ways, these trends are a consequence of parentage practices that orient children toward the local social milieu and encourage the use of Russian. Despite the significance of transnational practices in the life of migrant families, the outcome of the study demonstrates the importance of examining the intergenerational dynamics in the incorporation process.
Working paper
Akifyeva R. N. Working Papers of Centre for German and European Studies. Centre for German and European Studies, 2016. No. 01.
This research studies the child-rearing strategies of Russian-speaking women who live in Madrid by focusing on their children's participation in structured activities. The study answers the following questions: what attitudes to structured activities do the women have, what determined their choice of structured activities, what are the differences in such choices between working-class and middle-class families? The results are based on semi-structured interviews with Russian-speaking women (26 interviews) whose children attend after- or pre- school programs for Russian children, as well as with the teachers or coordinators of these programs (4 interviews). The findings make it possible to formulate conclusions that the class differences in the activities of children are determined by structural factors rather than the cultural logics of parents. All parents reproduce the idea that organized activity has a beneficial effect on a child's development, although they differ in their views on what kind of activities can bring benefits.
Working paper
Orel E., Куликова Алена Александровна. Psychology. PSY. Высшая школа экономики, 2016. No. WP BRP 66/PSY/2016 .
Recent research indicates that behavioral problems may lead to low academic performance. The present study is aimed to discover, what differences exist between primary school students who meet a sufficient number of ADHD (Attention Deficit Hyperactivity Disorder) criteria and those who do not experience any behavioral problems, in terms of academic achievements in reading and mathematics, annual progress in these subjects and personal, social and emotional development, based on the Russian sample of first-graders. This paper is a part of Russian iPIPS project and the instruments developed as part of this study were used. The sample consists of 3021 first-graders from two big regions of the Russian Federation. The results showed significant differences in both cognitive and social-emotional development but no differences in annual progress. The absence of differences in progress means that the development of children with behavior problems within the school system goes with the same speed but from the lower start level compared to other children. The results of the study provide important knowledge for the teachers and open a large area of further investigations in the field of ADHD in Russian school settings.
Working paper
Vernikov A. V. Economics Working Paper. UCL SSEES Centre for Comparative Economics, 2014. No. 130.
We use statistical data from 2000 to 2013 to compare the macro-level structure and the core institutions of the banking systems in China and Russia.
Working paper
Vernikov A. V. Working Paper Series. SSRN. Social Science Research Network, 2014. No. 2474554.
We use statistical data from 2000 to 2013 to compare the macro-level structure and the core institutions of the banking systems in China and Russia. Our main hypothesis is that, differences in the absolute size and socio-cultural features notwithstanding, these two systems are typologically similar. We consider the institutional structure, the market structure and concentration, the industrial policy of the government, and the banks' involvement in the financing of the non-financial economy. We find both similarities and differences, however in dynamics the trend towards convergence prevails. In both countries, we see a hierarchical multi-tier banking system headed by a few core state-controlled banks. These combine commercial activities with the activity of a development institution. Regardless of the nominal form of bank ownership, the government exerts influence on the lending decisions and market behavior of banks. We argue that the case of banking in China and Russia provides empirical proof to the macrosociological theory of institutional matrices [Kirdina, 2001]. We refer in particular to the needed proportion between the dominant and complementary institutions. Whereas China has been carefully seeking a sustainable balance, Russian in the 1990s grossly overshot with liberalization and now reverts to the underlying long-term trend. China has currently become Russia's institutional donor for Russia for innovations in the credit system design.

format: plain


text:
# Drag on a Cylinder - Confused

1. Apr 10, 2009

1. The problem statement, all variables and given/known data
I'm writing a lab report on drag on a non-rotating cylinder. The drag coefficient is calculated using the Pressure co-efficient. The problem I'm facing is that my lab states that the pressure on the cylinder can be predicted by $$\left(p-p_{0}\right)_{max} \times cos(\theta)$$

3. The attempt at a solution
The research I've done on the subject leads me to conclude that the pressure on a cylinder can be predicted by $$\left(p-p_{0}\right)_{max} \times \left(1-4sin^{2}(\theta)$$ instead of the above mentioned equation.

The following is a plot I did in pylab with the experimental plot in blue and the therotical plot in green (using the 2nd equation). The y-axis represents the coeffiecent of pressure while the x-axis is the position from the stagnation point on the cylinder, in degrees.

My question is, am I missing something? Or is my lab wrong on this? I've been literally reading every material I can get my hands on for the past 3 or 4 hours and I'm still terribly confused.

EDIT: The following is the text from my lab notebook which confuses me.

Last edited: Apr 10, 2009
2. Apr 12, 2009

### Aeroeri25

"Drag acts in the plane of motion so the pressure acting in this plane on an element at θ degrees to the cylinder is (p-p0)cosθ."

The wording might be a bit confusing.

It should read something like: The contribution of pressure to Drag is (p-p0) cosθ at an angle θ. Since force is calculated by the following equation

F = - INTEGRAL (p * n )dS , wheren is the normal vector.

it is easy to see that the contribution to drag (Fx: force in the x-direction) is in fact (p-p0) cosθ. (simply by using Cartesian coordinates)

**sorry, i dont know how to add all the fancy math symbols.

following through your calculations you should get
[text]\left(p-p_{0}\right)_{max} \times \left(1-4sin^{2}(\theta)[/tex]

"The force on the element in the plane of the drag is the pressure multiplied by the area of the element = (p - p0)Rcosθδθ."

basically the equation i wrote above for force, but this is for a differential force (dFx).

hopefully this was of some use.

3. Apr 12, 2009

### Aeroeri25

I'm sorry,
this is wrong:

\left(p-p_{0}\right)_{max} \times \left(1-4sin^{2}(\theta)

you should solve for p-p0 through Bernoulli's, but the rest i wrote should be okay.

4. Apr 12, 2009

To get 1-4sin^2(x), I did use Bernoulli's. The following page gives the derivation, you'll have to scroll down a bit.

http://www.ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=fl&chap_sec=07.4&page=theory

Since,

$$p_{0}+\frac{1}{2}\rho U^{2} = p + \frac{1}{2}\rho v^{2}_{\theta}$$

where U is the velocity of the air flow and v is the velocity of the airflow at $$\theta$$. v is given by v = $$2U\sin\theta$$

Rearranging for p-p0

$$p-p_{0} = \frac{1}{2}\rho\left(U^{2} - v^{2}_{\theta}\right)}$$

Subtituting into the equation for Cp:

$$C_{p} = \frac{\frac{1}{2}\rho\left(U^{2} - v^{2}_{\theta}\right)}{\frac{1}{2}\rho U^{2}}$$

If I substitute $$v = 2U\sin\theta$$ in the above equation and cancel various terms, I end up with the following:

$$C_{p} = 1 - 4\sin^{2}\theta$$

Does this make sense? I have not studied this material into this much depth, I knew Bernoulli but I don't know how we end up with $$v = 2U\sin\theta$$, so I just took it on faith from the above mentioned website and did the derivation.

I just want to show a theoretical pressure distribution i.e pressure distribution around a cylinder in a ideal flow, thats why I'm hunting for a function (and ended up with the above mentioned one) for Cp in terms of the position of the cylinder.

5. Apr 12, 2009

### Aeroeri25

your equations are correct. It seems to me that if you read the section "Flow Past a Fixed Circular Cylinder" in the link you provided, you will know how v = 2 U sin(theta)


text:
# Is 47 A Prime Number?

Is 47 A Prime Number?
Go back to  'Prime Numbers'

Do you know that this famous Hot Spring National Park in the United States of America has a total of 47 hot springs?

Another fact that we're going to study today is that 47 is a prime number since it has only two factors: 1 and the number itself.

This chapter will explore other topics like prime numbers, is 97 a prime number, is 49 a prime number, is 57 a prime number, and is 91 a prime number in the concept of is 47 a prime number.

Check out the interactive questions to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

## What Is a Prime Number?

Any whole number which is greater than 1, and has only 2 factors, 1 and the number itself, is called a prime number.

The number clearly has only two factors, 1 and the number 47

Thus, observing the factors of 47, we conclude that it is a prime number.

Examples:

Let us see a few more examples.

• Factors of number 97 = 1, 97. So, is 97 a prime number? Yes, because it has only 2 factors.
• Factors of number 57 = 1, 57. So, is 57 a prime number? Yes, because it has only 2 factors.
• Factors of number 91 = 1, 7, 13, 91. So, is 91 a prime number? No, because it has 4 factors.
• Factors of number 49 = 1, 7, 49. So, is 49 a prime number? No, because it has 3 factors.

## Is 47 a Prime Number?

To check if a number is prime or not, we find out all the factors of the number.

This can be done using different methods:

• Divison of the number by all the prime numbers starting from 1
• By prime factorization

After following one of the above steps, we find all the possible factors of the number and thus conclude if the number is prime or not.

Let us check this for 47:

Observe the figure which shows the prime factorization of 47:

The prime factors of 47 = 1 and 47

Thus, even by repeated multiplication of its prime factors, 47 has only two factors in all = 1 and 47

Hence, 47 is a prime number.

Important Notes
• 47 has only 2 factors, 1 and 47 itself, so it is a prime number.
• We don't take the negative factors of a number into consideration to check whether it is prime or not.

## Is 47 a Composite Number?

A composite number has more than two factors.

Since 47 has exactly two factors: 1 and 47, it is not a composite number.

## Solved Examples

 Example 1

Loren is preparing a list of all prime numbers. Can you help her find the number of prime numbers that are less than 47?

Solution

All the prime numbers less than 47 = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43

The number of prime numbers less than 47 is 14

 $$\therefore$$ The number of prime numbers less than 47 are 14
 Example 2

Joel needs to prove that 1 added to or subtracted from 47 makes it a composite number. How will he prove it?

Solution

On adding 1 to 47, Joel gets 48

Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48

Clearly, 48 has more than 2 factors. Thus, it is not a prime number.

On subtracting 1 from 47, he gets 46

Factors of number 46 = 1, 2, 23, and 46

It has factors other than 1 and 46, which makes it a composite number.

Hence, 1 added to or subtracted from 47 makes it a composite number.

 $$\therefore$$1 added to or subtracted from 47 makes it a composite number.

## Interactive Questions

Here are a few activities for you to practice.

Challenging Questions
• Leslie needs to find those prime numbers between 1 to 100 whose sum also gives a prime number. Can you help him?
• How will Jamie check if 51 is a prime number or not?

## Let's Summarize

The mini-lesson targeted the fascinating concept of the "Is 47 a prime number". The math journey around the "Is 47 a prime number" starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

## 1. What can 47 be divided by?

47 is a prime number, which means it has only two factors 1 and 47. Thus, it can only be divided by 1 and 47

## 2. Is 47 a prime number? Yes or no?

Yes, 47 is a prime number because it has only two factors, 1 and the number itself.

Factors of 47 = 1 and 47

## 3. Is 47 a perfect square?

The square root of 47 = $$\sqrt{47}$$ = 6.85565, which is not a whole number.

Thus, 47 is not a perfect square.

More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus


text:
# Pick out the true statement(s)

Pick out the true statement(s).

(a) If $f :(−1, 1)\to\Bbb R$ is bounded and continuous, it is uniformly continuous.

(b) If $f : S^1\to\Bbb R$ is continuous, it is uniformly continuous.

(c) If $(X, d)$ is a metric space and $A\subseteq X$, then the function $f(x) = d(x,A)$ defined by $$d(x,A) = \inf\{d(x, y) : y \in A\}$$ is uniformly continuous.

i think a is false as domain is not closed and b is true as domain is closed and bounded. no idea about c

-
For (a), if it is false (which it is) one should produce an example. Hint: Lots of wiggle. – André Nicolas Sep 6 '12 at 4:27
You really should start learning how to use MathJax and format your posts clearly; there are lots of links and suggestions here to get you started. – Brian M. Scott Sep 6 '12 at 4:30

(c) is true. Just think about what $x \mapsto d(x, A)$ means geometrically. Moving $x$ by $k$ units cannot change $d(x, A)$ by more than $k$ units.


text:
# Properties of Rational Numbers

In this chapter we will discuss some important properties of rational number with examples.

To understand the properties, you should have basic understanding of concept of rational number.

## Rational numbers properties

Given below are properties of rational number with example;

(a) Rational number are generally expressed in form of P / Q

Where P & Q are integers.

Numbers like \mathtt{\frac{2}{3} ,\ \frac{-7}{6} \ \&\ \frac{\ 10}{9}} are some of the examples of rational numbers.

(b) Decimal number are also rational numbers

Decimal numbers can be easily converted in the form of P / Q. Hence they are also rational numbers.

For example;
The decimal 10.25 can be converted into fraction as follows.

\mathtt{10.25\ \Longrightarrow \ \frac{1025}{100}}

Hence, the above decimal is a rational number.

(c) If you multiply number m on both numerator & denominator of rational number, you will get the rational number of same value.

Let \mathtt{\frac{a}{b}} be the rational number.

Multiply m on both numerator and denominator, we get;

\mathtt{\Longrightarrow \frac{a}{b}}\\\ \\ \mathtt{\Longrightarrow \ \frac{a\times m}{b\times m}}\\\ \\ \mathtt{\Longrightarrow \ \frac{am}{bm}}

Here the resultant rational number \mathtt{\frac{am}{bm}} has the same value as the original number \mathtt{\frac{a}{b}}

\mathtt{\frac{a}{b} \Longrightarrow \ \frac{a\times m}{b\times m}}

For example;
Consider the rational number 2 / 3

Multiply number 5 on both numerator and denominator.

\mathtt{\Longrightarrow \frac{2}{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2\times 5}{3\times 5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10}{15}}

Here both 10/15 and 2/3 have same value.

\mathtt{\frac{10}{15} \Longrightarrow \ 0.67}\\\ \\ \mathtt{\frac{2}{3} \ \Longrightarrow \ 0.67}

Conclusion
The multiplication of same number on numerator and denominator will produce rational number of same value.

(d) If you divide numerator and denominator of rational number by number m, you will get rational number of same value.

Let \mathtt{\frac{a}{b}} be the rational number.

Divide numerator and denominator by m.

\mathtt{\Longrightarrow \frac{a}{b}}\\\ \\ \mathtt{\Longrightarrow \ \frac{a\div m}{b\div m}}

This new rational number value is same as initial value a / b.

\mathtt{\frac{a}{b} =\ \frac{a\div m}{b\div m}}

For example
Consider the rational number \mathtt{\frac{10}{16}}

Divide numerator and denominator by 2

\mathtt{\Longrightarrow \ \frac{10\div 2}{16\div 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5}{8}}

According to the property of rational number, both the number 10/16 and 5/8 have same values.

\mathtt{\frac{10}{16} \Longrightarrow \ 0.625}\\\ \\ \mathtt{\frac{5}{8} \ \Longrightarrow \ 0.625 }

(e) If two rational numbers are equal, then the cross multiplication of rational number will also be equal.

Let a/b & c/d be the two rational numbers.

\mathtt{\frac{a}{b} =\ \frac{c}{d}}

Doing the cross multiplication we get;

\mathtt{a\times d\ =\ b\times c}

For example;
Let 2/6 and 8/24 be the equal rational numbers.

\mathtt{\frac{2}{6} =\ \frac{8}{24}}

On doing the cross multiplication, we get the equal numbers.

\mathtt{2\times 24\ =6\ \times 8}\\\ \\ \mathtt{48\ =\ 48}

(f) The value of rational number a/b depend on the values of a & b.

(i) if a > b, the value of rational number will be greater than 1

(ii) if a < b, the value if rational number is less than 1

(iii) if a = b ,the value of rational number is equal to 1

For example;
Consider the rational number 5/4

Here numerator > denominator, so value of rational number is greater than 1.

\mathtt{\frac{5}{4} \Longrightarrow \ 1.25}

Example 02
Consider rational number 4/5

Here numerator is less than denominator, so value of rational number is less than 1.

latex] \mathtt{\frac{4}{5} \Longrightarrow \ 0.8} [/latex]

(g) Suppose there are three rational numbers X, Y & Z such that, X > Y and Y > Z, then it automatically implies that X > Z.

For example;
Let the three rational numbers are \mathtt{\frac{2}{3} ,\ \frac{2}{4} ,\ \frac{2}{6}} .

Its given that;
\mathtt{\frac{2}{3} >\ \frac{2}{4}} and \mathtt{\frac{2}{4} >\ \frac{2}{6}} .

From the above information, we can automatically infer \mathtt{\frac{2}{3} >\ \frac{2}{6}}


text:
# Tag Info

5

To enjoy the greatest advantage from the Earth's rotation, you want two things: Launch from as close to the equator as possible. Launch as close to directly east as possible. The first part is fairly simple. The rotational velocity is proportional to $\cos(latitude)$, so the Guiana Space Centre gets over 99% of the effect, Kennedy Space Centre around 88%, ...

4

Here are some results of me playing with a map and ground tracks. I took a plot of ISS ground tracks and overlaid it onto the most detailed map of time zones from Wikipedia. First and foremost, time zones are weird. Even if we take "regular" time zones only and don't look at any of the political decisions, Earth has 25 time zones from UTC+12 to UTC-...

2

At least on the US side of the ISS, there is one processor for all, the Urine Processor Assembly. Source: https://space.stackexchange.com/a/26629/6944 (see link for acronymology)

1

Thank you to David Hammen: The Velocity of Spacex's Livestream telemetry data is described by ECEF coordinates (earth-centered, earth-fixed). ECEF Wikipedia The equation for orbital velocity used above is valid only for ECI coordinates. ECEF coordinates are used preferrably in Satellite navigation, because it offers precise values without having to choose a ...

1

According to this schedual, the quest airlock does require maintenance. "Airlock Cooling Loop Flush" Not sure what an "Airlock Cooling Loop Flush" is but a google search reveals it has been done multiple times in the past.

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# fourier series of b-spline

Fourier series of a function (B spline) are given by: $$s(x)=\sum_{j=-\infty}^{\infty}\operatorname{sinc}\Bigl[\pi\frac{j}{K}\Bigr]^{p}\exp[2\pi ijx]$$

But B-spline has only finite support. How to see it using its Fourier series representation?

-

A function is a piecewise polynomial if and only f it is a linear combination of functions, each of them having some derivative equal to a finite sum of Dirac measures.

The jth Fourier coefficient of the Dirac at a is $e^{2\pi ija}$, and integrating amounts to multiplying the coefficient by some power of $1\over j$.

As a result, a function is a spline with finite support if and only if its Fourier coefficients $c_j$ can be written as a finite linear combination of terms of the form $e^{i\pi ja}\over j^k$, $k\in \mathbb{N},a\in \mathbb{R}$.

You can see that this is the case in your formula by expanding the coefficient $c_j=({{e^{i\pi j/K}-e^{i\pi j/K}}\over j/K})^p$.

-

Since b-splines have finite support, you can extend them to be periodic. This makes it possible to apply the Poisson summation formula to describe the Fourier series expansion of a b-spline in terms of its Fourier transform. Centered scaled cardinal b-splines of order n have a closed form Fourier transform, ((sin omega)/omega)^{n-1} up to scaling.

-


text:
# Hi guys, so I encountered this error "Scripting[system_util_tag]", can you help how to find and debug this error

Hi guys, so I encountered this error “Scripting[system_util_tag]”, can you help how to find and debug this error?
because i have no idea where to look at because of lack of info in the error.

you are somewhere using in your code a function “send”() and give it a property while it doesnt need one.

Yes, i think so too, but I cant find where did i put that script

btw thanks for the comment sir!

im guessing its in an expression runScript()
you can use find to search through scripts and expresssions

1 Like

okay, ill try that, thanks sir.

1 Like

it worked, i found the existing script, Thanks to you sir.

1 Like

Nice
usualy the logs show where it happend, but sometimes you just have to use the good old ctrl+f xd

1 Like


text:
# Algorithm Database Discussion

#### Ranzha

##### Friendly, Neighbourhoodly
ELL images on the dev version look like this:

But if you'd like I could add a view option to make them appear like this:
You did it again XP.
Lol. No worries.

Sounds great! I think either would be okay. I sincerely think that the former one is better for those who use ELL first, if any.

#### miniGOINGS

##### Member
Here we go.

http://www.ai.univ-paris8.fr/~bh/cube/

When dealing with LL solutions for EO, EP, CO, and CP here are the optimal movecounts;

Fridrich (OLL + PLL)
9.22 + 11.21 = 20.43

Backwards (PLL + OLL)
8.1 + 12.8 = 20.9

Corners (CLL + ELL)
9.18 + 11.27 = 20.45

Edges (ELL + CLL)
7.87 + 11.73 = 19.6

1LLL (LL)
12.58

#### Ranzha

##### Friendly, Neighbourhoodly
Here we go.

http://www.ai.univ-paris8.fr/~bh/cube/

When dealing with LL solutions for EO, EP, CO, and CP here are the optimal movecounts;

Fridrich (OLL + PLL)
9.22 + 11.21 = 20.43

Backwards (PLL + OLL)
8.1 + 12.8 = 20.9

Corners (CLL + ELL)
9.18 + 11.27 = 20.45

Edges (ELL + CLL)
7.87 + 11.73 = 19.6

1LLL (LL)
12.58
It's interesting that Backwards, the inverse of the preferred Fridrich LL system, has a negative effect on the movecount, whereas the preferred CLL + ELL is worse than Fridrich and its inverse, ELL + CLL, has a sub-20 movecount.

I'mma try this!

#### miniGOINGS

##### Member
Mhm. I'm not sure in which metric the movecounts were generated with, but I think I should find out.

#### Kenneth

##### Not Alot
... and its inverse, ELL + CLL, has a sub-20 movecount
Sounds fine but is much worse than it looks, it is low because ELL first is way easy, almost intuitive and have short algs and I think 14 cases but CLL last is horrible, many have long algs and it is around 80 cases (1:6 of ZBLL), absolutly nobody uses it because of that.

I was learning that once, or... I was using a method that did VHF2L, EP + 1 corner and then the last three corners. then I was planning to expand to 2 steps, learned a few cases but found it was so many of them... I choosed to do CLL/ELL instead and later switched to COLL (because I hated to see oriented edges flip during CLL).

Last edited:

#### miniGOINGS

##### Member
Sounds fine but is much worse than it looks, it is low because ELL first is way easy, almost intuitive and have short algs and I think 14 cases but CLL last is horrible, many have long algs and it is around 80 cases (1:6 of ZBLL), absolutly nobody uses it because of that.

I was learning that once, or... I was using a method that did VHF2L, EP + 1 corner and then the last three corners. then I was planning to expand to 2 steps, learned a few cases but found it was so many of them... I choosed to do CLL/ELL instead and later switched to COLL (because I hated to see oriented edges flip during CLL).
Actually, it doesn't look like it.

"permutation and orientation of edges (13 configurations, average : 7.87),
then permutation and orientation of corners (42 configurations, average : 11.73)."

Doesn't sound too bad to me.

Last edited by a moderator:

#### Cride5

1. BUGFIX: Swap CLS + ELS
2. BUGFIX: Sorted out COLL/CLL/ELL plan view
4. FEATURE: Allow removal of individual algs (currently admin only)
5. FEATURE: Improved parsing of user input to allow: "", "R'2" and "Rw"
6. BUGFIX: Fixed verification problem for algs ending in x2 rotations
7. FEATURE: Improved duplicate detection, for duplicate algs posted with different rotations
9. FEATURE: Link to alg.garron.us for algorithm animations
10. FEATURE: Initial rotations in separate column, final rotations hidden
11. FEATURE: All AUFs automatically replaced with y rotations for LL algs

I've also re-processed the ZBLL and COLL algs to change AUFs for rotations, so that the move-count stats are better.

If you want any features that aren't in the ToDo-list please let me know below...

#### Kenneth

##### Not Alot
Sounds fine but is much worse than it looks, it is low because ELL first is way easy, almost intuitive and have short algs and I think 14 cases but CLL last is horrible, many have long algs and it is around 80 cases (1:6 of ZBLL), absolutly nobody uses it because of that.

I was learning that once, or... I was using a method that did VHF2L, EP + 1 corner and then the last three corners. then I was planning to expand to 2 steps, learned a few cases but found it was so many of them... I choosed to do CLL/ELL instead and later switched to COLL (because I hated to see oriented edges flip during CLL).
Actually, it doesn't look like it.

"permutation and orientation of edges (13 configurations, average : 7.87),
then permutation and orientation of corners (42 configurations, average : 11.73)."

Doesn't sound too bad to me.
Can't be correct, 27 orientations times 12 permutations divied by 4 makes a total of 81

Last edited:

#### miniGOINGS

##### Member
Can't be correct, 27 orientations times 12 permutations divied by 4 makes a total of 81
Yea, it does sound strange. I'll look into it.

#### Ranzha

##### Friendly, Neighbourhoodly

1. BUGFIX: Swap CLS + ELS
2. BUGFIX: Sorted out COLL/CLL/ELL plan view
4. FEATURE: Allow removal of individual algs (currently admin only)
5. FEATURE: Improved parsing of user input to allow: "", "R'2" and "Rw"
6. BUGFIX: Fixed verification problem for algs ending in x2 rotations
7. FEATURE: Improved duplicate detection, for duplicate algs posted with different rotations
9. FEATURE: Link to alg.garron.us for algorithm animations
10. FEATURE: Initial rotations in separate column, final rotations hidden
11. FEATURE: All AUFs automatically replaced with y rotations for LL algs

I've also re-processed the ZBLL and COLL algs to change AUFs for rotations, so that the move-count stats are better.

If you want any features that aren't in the ToDo-list please let me know below...
Thanks SO MUCH for this. This is way much and a HUGE improvement.

#### Ranzha

##### Friendly, Neighbourhoodly
BUG?:

This is ELL #3:

This is what I want:

Nothin' doin'. Just keeps adding to ELL #3

#### StachuK1992

##### statue
If someone were to fix ELS and CLS, so one could actually add stuff, that'd be super awesome, and I'd add a lot.
Sorry for the bump, but this has been bugging me.

Edit:
Also, a CPLS section would be nice.
If someone were to add it to the database, I'd happily put up some algs.

Is there any way a regular user like me could add the CPLS section?
Or does one have to have higher permissions? :/

Last edited:

#### Cride5

It appears as if ELS and CLS are still swapped though. I thought I'd fixed this so not sure why it's still happening. Will check it out when I've got time.

#### Escher

##### Babby
I don't have much to add personally to this project but I recently found this Polish Algorithm Database. It seems to have a ridiculous amount of algs, I don't know if anybody fancies doing the work of inputting them all (any of them??) into the ss wiki.

#### oprah62

##### Member
I don't have much to add personally to this project but I recently found this Polish Algorithm Database. It seems to have a ridiculous amount of algs, I don't know if anybody fancies doing the work of inputting them all (any of them??) into the ss wiki.
Wow that database is insanely good. I love how they orangize it all. Thanks for sharing.
edit:500th post!

#### StachuK1992

##### statue
I can't seem to add CPLS. :/
That, or I'm just dumb.

#### Cride5

I can't seem to add CPLS. :/
That, or I'm just dumb.
Yup, CPLS hasn't been added as a 'group' yet. Each time a new group is added a developer needs to define it within the system. The reason for this is that the validation system needs a semantic definition of the group to allow it to recognise algs which belong to that group. Without this none of the automated verification/organisation would work.

@Escher: That polish database is seriously cool! This is exactly what I tried to achieve with the wiki database, but the big problem is I'm very short of spare time these days.

If there are any php/mysql developers interested in getting involved in the project give me a shout. Looking at the huge number of algs already contributed to the wiki, I think this is a really worthwhile project.

EDIT: If folks fancy populating the database there is a 'bulk' upload function. Just log into the wiki, select the alg set, and then click on 'bulk'. For example:
http://www.speedsolving.com/wiki/index.php/Special:AlgDB?mode=bulk&view=default&puzzle=3&group=CLS

Last edited:

#### Anthony

##### Professional Speedcuber
omfg..
I'm learning EG2..
now..

#### oprah62

##### Member
omfg..
I'm learning EG2..
now..
Have you found any good algs? I've tried to CE them but most of them are just bad.

#### danthecuber

##### Member
Bump

If you all aren't aware...I tried uploading CLS and ELS algs, and there are still swapped. What's worse, in the ELS section of the database, which is holding CLS algs, only has 23 cases there, but there's supposed to be 104 cases. I uploaded alg for every CLS case, every alg was successful, but it looks like they went to the wrong cases, or something like that. No new cases were created, when they were supposed to. The same thing is happening with the CLS section, only 5 cases are there and there is supposed to be 21 cases. WE NEED YOU CRIDE5!!!

Last edited:


text:
# Is this derivation of Euler's formula correct?

z = cos(x) + isin(x)
dz = -sin(x) + icos(x)dx
= i(isin(x) + cos(x))dx

∫ dz/z = ∫ idx

ln(z) = ix

e^(ix) = z

e^(ix) = cos(x) + isin(x)

I think essentially you've shown some intuition for why it's a true formula: both z = cos(x) + i·sin(x) and z = ei·x solve the ODE dz/dx = i·z, and both are 1 when x = 0. By uniqueness of solutions to ODEs, we should have ei·x = cos(x) + i·sin(x).

Last edited:
Yes, but you have to make sure you have a good definition of the complex logarithm.

tiny-tim
Homework Helper
hi inknit!
∫ dz/z = ∫ idx

ln(z) = ix
nooo …

ln(z) = ix + C …

you still need to prove what C is!


text:
# Correlated Stochastic Processes

Let say, I have 2 stochastic processes: \begin{align} dS_1 &= \left( r - q_1 \right)S_1 dt + \sigma_1 S_1 dW_1 \\ dS_2 &= \left( r - q_2 \right)S_2 dt + \sigma_2 S_2 dW_2 \end{align} The correlation between these 2 processes is $$\rho$$. Now I define 2 new processes as: \begin{align} x_1 = \sigma_1 \log S_2 + \sigma_2 \log S_1 \\ x_2 = \sigma_1 \log S_2 - \sigma_2 \log S_1 \end{align}

As per Hull's book, these processes $$x_1, x_2$$ are uncorrelated with standard deviation $$\sigma_1 \sigma_2 \sqrt{2 \left( 1+\rho \right)}$$ and $$\sigma_1 \sigma_2 \sqrt{2 \left( 1-\rho \right)}$$ respectively.

How can I show this result?

• What have you tried? Also, the correlation $\rho$ is between the Brownian Motions $W_1$ and $W_2$ correct? – Daneel Olivaw Nov 3 '20 at 9:52
• I am able to derive the $dt$ term, but not the volatility part. Apologies if this is very trivial question. Any pointer is appreciated. – Brian Smith Nov 3 '20 at 9:54
• If $S_i$ is log-normal, then $\log S_i$ is normal. Have you tried computing the covariance between $x_1$ and $x_2$? – Daneel Olivaw Nov 3 '20 at 9:56

I have the impression the expressions in your question miss the time term $$t$$, though this does not change much. Define for $$i,j\in\{1,2\}$$: \begin{align} s_i(t)&=\log S_i(t) \\ y_{i,j}(t)&=\sigma_is_j(t) \end{align} Then: \begin{align} V(y_{i,j}(t))&=\sigma_i^2V(s_j(t)) \\ &=\sigma_i^2V(\sigma_jW_j(t)) \\ &=\sigma_i^2\sigma_j^2t \\ &=V(y_{j,i}(t)) \end{align} Moreover: \begin{align} C(y_{1,2}(t),y_{2,1}(t))&=\sigma_1\sigma_2C(s_2(t),s_1(t)) \\ &=\sigma_1^2\sigma_2^2C(W_2(t),W_1(t)) \\ &=\sigma_1^2\sigma_2^2\rho t \end{align} Hence: \begin{align} &V(x_1(t))=\sigma_1^2\sigma^2_2t+\sigma_2^2\sigma_1^2t+2\sigma^2_1\sigma^2_2\rho t=\sigma_1^2\sigma_2^22(1+\rho)t \\ &V(x_2(t))=\sigma_2^2\sigma^2_1t+\sigma_1^2\sigma_2^2t-2\sigma^2_1\sigma^2_2\rho t=\sigma_1^2\sigma_2^22(1-\rho)t \end{align} Finally, by bi-linearity and symmetry of covariance: \begin{align} C(x_1(t),x_2(t))&=C(y_{1,2}(t)+y_{2,1}(t),y_{1,2}(t)-y_{2,1}(t)) \\ &=V(y_{1,2}(t))-C(y_{1,2}(t),y_{2,1}(t))+C(y_{2,1}(t),y_{1,2}(t))-V(y_{2,1}(t)) \\ &=V(y_{1,2}(t))-V(y_{2,1}(t)) \\ &=0 \end{align} $$x_1$$ and $$x_2$$ are uncorrelated $$-$$ note that for normally-distributed random variables, null correlation also implies independence.
• Many thanks. Is there any way on how to generalise this for n initial correlated stochastic processed for $S_t$? – Brian Smith Nov 3 '20 at 21:22
• Generally, you can express a specific covariance structure in your asset process as $CdW$ with $C$ the lower cholesky decomposition of the covariant empire matrix. I think that you may be able to use its inverse $C^{-1}$ to weight the assets in order to de-correlate the assets. – Kermittfrog Nov 3 '20 at 22:04


text:
# Area between curves and integration

1. Jan 13, 2008

### Hurricane3

This isnt a homework question, but from my notes that I couldn't figure out (wasn't able to copy the rest of the down)

I have two functions:
y=x
y=x^3 , 0 $$\leq$$ x $$\leq$$ 2

and I have to find the area between these two curves.

I know how to do it with respect to x, but I have troubles integrating with respect to y.

From a quick sketch of the graph, it seems like I will have three different integrals.

I can formulate the first two definite integrals:

$$\int^{1}_{0}$$ (y^1/3 - y) dy + $$\int^{2}_{1}$$(y-y^1/3)dy

but I don't know how to forumate the last one... help!

The area should be 5/2.

Thanks

2. Jan 13, 2008

### Dick

I'm not sure why you inverted the problem into a dy integration. You can do that but then the upper limit is definitely not 2. Just integrate (x-x^3)dx between 0 and 1 and (x^3-x)dx between 1 and 2 and add them.

3. Jan 13, 2008

### Hurricane3

Because this was part of the notes. In the notes, we integrated with respect to x and the professor showed us how to integrate with respect to y as well. Except I didn't have time to copy that part of the notes (integration with respect to y), so I was just trying to do it just now, but I dunno whats the upper limit for the third part of the integration...

4. Jan 13, 2008

### Dick

If you need to do it that way, then y^(1/3)-y from 0 to 1. y-y^(1/3) from 1 to 2 and 2-y^(1/3) from 2 to 8. Do you see why? Draw the graph. x^3=y, 2^3=8.

5. Jan 14, 2008

### Hurricane3

yeah i see it now! thanks =)


text:
## The AM-GM inequality

April 24, 2010

One of the most basic inequalities, the AM-GM inequality states that the AM (Arithmetic mean) of a series of real numbers is always greater or equal to the GM (Geometric mean).

Mathematically:

$\frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n}$

### A simplified case

If there are two elements in the series, the AM-GM states this:

$\frac{x+y}{2} \geq \sqrt{xy}$

The proof for this simplified case is quite simple:

$\begin{array}{l} \frac{x+y}{2} - \sqrt{xy} \\ = \frac{x - 2\sqrt{xy} + y}{2} \\ = \frac{(\sqrt{x} - \sqrt{y})^2}{2} \geq 0 \end{array}$

Now, however, it is not so obvious how to extend this proof to a set of three elements, or more. But there is a way, by induction, to prove this inequality for cases where the number of elements is a power of two.

### The 2^k subcase

The base case states that the AM-GM inequality is true for $n=2$, or when the number of elements in the set is 2.

The method of inductive proof is proving a base case, $2^k$, and proving that if this base case is true then the case where $2^{k+1}$ is also true. This would mean that the statement is true for all instances of $2^k$.

Here the base case is $n = 2^1$, which we have just proved to be true. We can now prove that this is true when $n = 2^2$, or when there are 4 elements in the set: $x_1, x_2, x_3, x_4$.

The arithmetic mean, or AM of the set is

$\frac{x_1 + x_2 + x_3 + x_4}{4}$.

We can split this set into two halves. One contains $x_1, x_2$, and the other contains $x_3, x_4$.

The point is that the arithmetic mean of the arithmetic means of the two halves is equal to the arithmetic mean of the whole. Or,

$\textrm{AM} = \frac{\frac{x_1+x_2}{2}+\frac{x_3+x_4}{2}}{2}$

Notice that both of the two halves are of length 2, or in the general case, $2^{k-1}$. Having already proved the previous case, we can say,

$\textrm{AM} \geq \frac{\sqrt{x_1 x_2} + \sqrt{x_3 x_4}}{2}$

This itself is another arithmetic mean. Applying the base case again:

$\begin{array}{rcl} \textrm{AM} &\geq& \sqrt{\sqrt{x_1 x_2} \sqrt{x_3 x_4}} \\ &=& \sqrt[4]{x_1 x_2 x_3 x_4} \end{array}$

So we have

$\frac{x_1 + x_2 + x_3 + x_4}{4} \geq \sqrt[4]{x_1 x_2 x_3 x_4}$

which was what we wanted to prove.

The process for proving this for a higher power of 2, such as 8, 16, etc, is very similar to what we have just done.

### The n-1 subcase

If the number of elements in the set is not a nice and even power of two, the above proof does not work. Instead, we use induction again, but this time from top-down.

We try to prove that if the inequality is true for a set with n elements, then it is also true for a set with n-1 elements.

Let’s first prove a lemma.

#### Lemma

If the arithmetic mean of a set, $x_1, x_2, \cdots, x_n$ is a, then

$\frac{x_1 + \cdots x_n + a}{n+1} = \frac{x_1 + \cdots + x_n}{n}$.

#### Proof

$\begin{array}{l} \frac{x_1 + \cdots + x_n + a}{n+1} \\ =\frac{x_1 + \cdots + x_n + \frac{x_1 + \cdots + x_n}{n}}{n+1} \\ = \frac{\frac{(n+1)x_1 + \cdots + (n+1)x_n}{n}}{n+1} \\ = \frac{x_1 + \cdots + x_n}{n} \end{array}$

This lemma simply states that if you have the mean of a set, adding the mean to the set itself and finding the mean again would result in the same mean.

Now let’s prove the inequality when n=3.

Let a be the arithmetic mean of the set. Then

$\frac{x_1 + x_2 + x_3 + a}{4} \geq \sqrt[4]{x_1 x_2 x_3 a}$

as we proved before. According to the lemma, we can substitute:

$\begin{array}{rcl} \frac{x_1 + x_2 + x_3}{3} &\geq& \sqrt[4]{x_1 x_2 x_3 a} \\ a &\geq& \sqrt[4]{x_1 x_2 x_3 a} \\ a^4 &\geq& x_1 x_2 x_3 a \\ a^3 &\geq& x_1 x_2 x_3 \\ a &\geq& \sqrt[3]{x_1 x_2 x_3} \end{array}$

This completes our proof.

Of course I’ve proved it only for n=3, but the same method can be used to prove for any number where it is true for n+1.

To prove this for any integer n, first prove by induction that this is true for some number $2^k$. Then prove by induction, again, that it is true for $2^k-1, 2^k-2, \cdots, n$.

This method of proof is called forward-backward induction. This proof was discovered by Augustin-Louis Cauchy.

## Project Euler 280 (Revisited)

March 27, 2010

A while ago I wrote this blog post on this problem. The previous code was able to produce the answer in about 20 minutes, using several gigabytes of RAM.

You may want to read the previous blog post first.

There is a simpler and more efficient solution (used by inamori, zwuupeape, and many others).

### The new approach

We do not need to know the exact value of the solution, only an approximate. This approximation has to be accurate only to six digits.

Any any step, there is a certain probability that the ant will reach the final state in that step.

Let’s call this function $P(s)$: the probability that the ant will reach the final state on step s:

At very small values of s, the probability is zero because it takes a certain number of steps to get to the final position. 44 is the smallest number of steps to get from the initial to the final position, and the probability increases from there.

But it only increases until a certain point. After that point, the probability decreases. After all, if we run this for thousands of steps, how likely is the ant to still not have reached the final position?

However, the graph continues forever to the right. Even after thousands of steps, there is still a chance, however small, that the final position hasn’t been reached.

Let’s define $Q(s)$ as the weighted average of the probabilities of each value of $P(s)$ from 0 to s:

The expected value and the answer for this problem would simply be the weighted average for the entire graph.

Since s must be an integer greater than zero, we don’t have to do any integration with curves. A bit misleading because I’ve drawn $P(s)$ and $Q(s)$ as curves, when they are not.

But because $Q(s)$ cannot be expressed algebraically, we cannot evaluate it as it approaches infinity. Instead, we pick a point that gives us enough precision:

Either that, or we keep calculating the weighted average while increasing s until we have enough precision.

### Calculating the probabilities

In order to calculate a weighted average $Q(s)$, we need to be able to calculate $P(s)$; while $P(s)$ is the probability function for the final state, we’ll need to know the probabilities for the in-between states as well.

The probabilities of the states on step s depend most upon the probabilities of the previous step, s-1.

So if on step s-1 there’s a 0.6 probability that we’re on state X, then on step s we should have a 0.3 probability of being on either Y or Z as there’s an equal chance of moving to either state.

And if multiple states can go to our state, we can get the probability by adding up the probabilities:

### Coding a solution

At first we start on the initial state, with a probability of 1 (certain chance) of being on it.

On each step we generate a new set of probabilities using rules I described above, and so on.

A lot of code is stolen from my previous work on this problem. The State class is essentially unchanged in this version.

Here’s the code:

import java.util.*;

public class Main{

public static void main(String[] args){

final int ITERATIONS = 2501;

// Initialize probability map.
// The initial step is 0, containing the initial position with a probability of 1.
prevProbs.put(State.INIT_STATE, 1.0);

// Distributions for the final state.
double[] finalValues = new double[ITERATIONS+1];
Arrays.fill(finalValues, 0);

// Keep the time.
long startTime = System.currentTimeMillis();

// Loop counter is actually the step number of the next step.
for(int step=1; step<ITERATIONS; step++){

// Make new probabilities.

// Loop through old probabilities and add
Set<State> stateSet = prevProbs.keySet();

for(State state : stateSet){

// Probability of being on this state right now
double stateProb = prevProbs.get(state);

// List of next states
List<State> nextStates = nextStates(state);

// Probability factor for each of the next states
double multiplier = 1.0 / nextStates.size();

// Added value to each of the next states
double added = stateProb * multiplier;

// Now go and update all the state probabilities.
for(State nextState : nextStates){

// If we've got the final state, handle it separately.
if(nextState.equals(State.FINAL_STATE)){
continue;
}

// Could be the first time.
if(!nextProbs.containsKey(nextState))

// Might not be the first time.
else {
// Old value, we need to add to it instead of replacing it.
double previous = nextProbs.get(nextState);
}
}
}

// Our new probabilities are the old probabilities for the next loop.
prevProbs = nextProbs;

System.out.println(step + " " + weightedAverage(finalValues));
}

System.out.println((System.currentTimeMillis() - startTime) + "ms");
}

// Calculates a weighted average of a double array with the indices as
// values and the values in the array as weights.
static double weightedAverage(double[] array){

double sumValues = 0;
for(int i=0; i<array.length; i++)
sumValues += i * array[i];

return sumValues;
}

// Returns a list of possible continuation states from the current one.
static List<State> nextStates(State state){

// Current and changing position of the ant
int antX = state.ant % 5;
int antY = state.ant / 5;

// Whether it can go into each of the four directions (N,S,E,W respectively).
boolean[] possibleDirs = new boolean[4];
Arrays.fill(possibleDirs, true);

// Take out some directions if it's in the corner.
if(antY == 0) possibleDirs[0] = false; // Can't go north
if(antY == 4) possibleDirs[1] = false; // Can't go south
if(antX == 4) possibleDirs[2] = false; // Can't go east
if(antX == 0) possibleDirs[3] = false; // Can't go west

// Construct a list of continuations.
List<State> nextStates = new ArrayList<State>();

// Loop through the four directions.
for(int i=0; i<4; i++){

// Cannot go this direction.
if( !(possibleDirs[i])) continue;

int newAntX = antX;
int newAntY = antY;

// Modify direction.
switch(i){
case 0: newAntY--; break;
case 1: newAntY++; break;
case 2: newAntX++; break;
case 3: newAntX--; break;
}

// Start constructing new state object.
int oldAnt = state.ant; // old ant position
int newAnt = 5*newAntY + newAntX;
int[] board = state.board.clone();
boolean carrying = state.carrying;

// Carrying a seed. Notice that a square can contain
// two seeds at once (but not more); seeds are indistinguishable
// so we just need to keep track of the number of seeds
// on each square.
if(carrying){
board[oldAnt] --;
board[newAnt] ++;
}

// Drop off the seed.
if(newAntY == 0 && board[newAnt] == 1 && carrying)
carrying = false;

// Pick up a new seed.
if(newAntY == 4 && board[newAnt] == 1 && !carrying)
carrying = true;

// Treat the five done positions the same.
if(donePosition(board))

}

return nextStates;
}

// Is the board in the done position?
static boolean donePosition(int[] b){
return b[0]==1 && b[1]==1 && b[2]==1 && b[3]==1 && b[4]==1;
}
}

class State{

static final State INIT_STATE = new State(
new int[]{
0,0,0,0,0,
0,0,0,0,0,
0,0,0,0,0,
0,0,0,0,0,
1,1,1,1,1
}, 12, false);

// Consider all final states the same; there is
// no ant position.
static final State FINAL_STATE = new State(
new int[]{
1,1,1,1,1,
0,0,0,0,0,
0,0,0,0,0,
0,0,0,0,0,
0,0,0,0,0
}, -1, true);

// 25 board.
int[] board;

int ant;
boolean carrying;

State(int[] board, int ant, boolean carrying){
this.board = board;
this.ant = ant;
this.carrying = carrying;
}

State(State s){
this(s.board, s.ant, s.carrying);
}

public boolean equals(Object o){
State s = (State) o;
return Arrays.equals(s.board, board) && s.ant == ant && s.carrying == carrying;
}

public int hashCode(){
return Arrays.hashCode(board) + ant;
}

// For debugging mostly.
public String toString(){
StringBuilder ret = new StringBuilder("\n");
for(int i=0; i<5; i++){
for(int j=0; j<5; j++){
int pos = 5*i + j;
if(ant == pos) ret.append("#");
else ret.append(board[pos] >= 1? "+" : "-");
}
ret.append("\n");
}
return ret.toString();
}
}

As we run this program, the answer becomes more and more accurate. We only need 2300 or 2500 steps to get the six digits of accuracy we need.

Running the program:

This takes just over a minute to run — not bad at all.


text:
# ComboStrap URL - Page Alias

A page alias is an extra path value for a page.

If a page has an alias that corresponds to the path of a missing page, a redirection is performed.

You can see them as symlink or shortcut for a file system.

## Type

### Redirect

A redirect alias is an alias that will perform a permanent redirection.

This type of redirect are automatically created:

For instance:

• if you have a page at the path fruit:pear.
• if you move the page to another location such as fruit:tree:pear
• the page fruit:tree:pear will get the alias fruit:pear with the type redirect

### Synonym

A synonym alias is an alias that will perform a transparent redirection. The page will be served normally as if it was also existing to the synonym path.

## Management

You can create, delete and see the aliases for a page with the metadata manager.

## Database

You can query the aliases with the following statements:

select * from page_aliases_vw;

select
p.path as page_path,
pa.path as alias_path,
pa.type as alias_type
from
page_aliases pa
inner join pages p on pa.UUID = p.UUID;


text:
# Find $x$ of $\angle A O C=70^{\circ}$ and $\angle B A O=50^{\circ}$ where 0 is centre of the circle. (a) $35^{\circ}$ (b) $30^{\circ}$ (c) $25^{\circ}$ (d) $15^{\circ}$

105 views
Find $x$ of $\angle A O C=70^{\circ}$ and $\angle B A O=50^{\circ}$ where 0 is centre of the circle. (a) $35^{\circ}$ (b) $30^{\circ}$ (c) $25^{\circ}$ (d) $15^{\circ}$

+1 vote
by (40 points)

## Consider △ABO,

AO=BO

⇒∠BAO=∠ABO=300

Consider △CBO,

CO=BO,

⇒∠BCO=∠CBO=400

⇒∠ABC=∠ABO+∠CBO=300+400=700

The angle subtended at the centre is twice the angle subtended anywhere on the arc,

so,

∠AOC=2×∠ABC=140.0

Hope this helps!


text:
# What is the best file format to save short signal samples?

We have the need to save signal samples measured with an arbitary sampling rate. The samples are usually about one second in duration. Currently we save them in a text file format which consumes a lot of disk space and is very slow. Also this type of format doesn't suit for streaming applications.

I have been searching for a binary file format for this kind of data, but I'm not sure what factors I need to consider when choosing a format.

The sampling rate of our data varies from 1kHz to 152kHz and currently 16 bits would be enough to represent the data (varies roughly from -1000 to 1000). This is vibration data measured from different sensors if it makes a difference. Also we would need to save pre-calculated FFT's in the same format. Is this possible?

Just to point out, I have no education in signal processing. I have the task to implement the system to be used in the analysis.

Currently the best option seems to be WavPack, which is basically a compressed WAV file and supports sampling rates from 1 to 4.3GHz and the compression is lossless.

WavPack is released in 1998 so I'm thinking are there any newer or more efficient formats available today?

• I'd agree, for any significant amount of data, as well for accuracy reasons, a text format is a terrible choice for samples. how many samples are there per recording? and: what's the numeric format of the original samples? Real? Complex? Number of bits? – Marcus Müller Apr 20 '17 at 7:06
• 8 bit makes no sense if you need to store 2000 different values. – Marcus Müller Apr 20 '17 at 9:41
• Marcus, true, my mistake. – kor_ Apr 20 '17 at 10:48
• I store all my sampled signals in WAV files. They're supported by everything on every platform, are easily compressible (if needed) with standard tools, can store an arbitrary number of channels with essentially arbitrary number format, and it's a well documented standard. It's the ideal container for generic sampled data. – MBaz Apr 20 '17 at 13:52

Let's write down a quick spec:

• The samples are usually about one second in duration,
i.e. let's say the maximum length is 4s
• The sampling rate of our data varies from 1kHz to 152kHz,
i.e. max rate 152 kS/s
• varies roughly from -1000 to 1000,
i.e. more than 2048, so let's head straight for 16 bit integers

That makes a maximum recording raw data size of

$$4\,\text s\cdot 152\,\frac{\text{kS}}{\text s}\cdot 2\,\frac{\text B}{\text S} = 1216\,\text{kB.}$$

In other words, by all modern standards for permanent storage and RAM of machines that you'd use to store recordings... nearly no data at all, and that's the maximum case. Even when you incorporate the pre-calculated FFTs (why?? Storage access of data in this length is usually much slower than computing them from the data when you need it), that's still not much data.

So, I would recommend not to use any specific compression at all, but simply write the data as raw int16 into files. Every, language, in this world has methods of reading raw data from files.

Regarding metadata like sampling rate, time of capture, and something like that:

That really depends on your usage scenario. If you know you'll keep the data with the metadata database, I'd quickly put up a separate table (be it in CSV file, be it in a SQLite file, or in a proper relational database).

If you know you'll distribute these files somewhere else, check whether you'd rather be using a custom header format (which usually boils down to a packed C struct in one way or another) is really advantageous, or whether a simple standard scientific data file format like HDF5 with well-defined structures within wouldn't be better – that would allow you to skip writing your own header parser in any target language, and also, you could easily put your raw samples, and an arbitrary amount of secondary data into it. Drawback is the fact that you can't just start streaming at the first byte of the data stream without loading the file to a large extent into memory. However, again, with your very small files, that isn't really a problem. I've not worked on a PC with less than 4 GB of RAM in the last decade, and neither should someone who does signal analysis. If of those 4GB of RAM, 2GB are available, you could completely have around 2000 of your maximum size data files in RAM at once.

If the samples are going to be very short (in the order of 1 or 2 seconds in duration), why not store the raw uncompressed binary data directly in a file? You could invent your own header for the binary file that will have maybe some metadata explaining the sample rate, the total number of samples, byte format or something like this, and then the stream of numbers directly. If uncompressed becomes to costly in terms of storage space, you could simply grab the stream of numbers, and store the first one directly and then encode the difference of the next number to the previous with Golomb Coding.

Hope it helps.

• Thanks but I try to avoid DIY solutions because if someone has already done this better then it saves a lot of time and effort from us. Especially if we had to implement error checking, streaming etc. capabilities. – kor_ Apr 20 '17 at 10:45
• error checking is a feature of the device and file system that you store these files on. Shouldn't be part of your file format. Streaming is a non-feature for files of less than 2MB of size... – Marcus Müller Apr 20 '17 at 12:17


text:
Weight of the selected item is calculated using its density and entered volume. The radius is the circumference divided by 2 π. Year Title Co-occurring Abbreviation; 1 : 2019: Core set construction and association analysis of Pinus massoniana from Guangdong province in southern China using SLAF-seq. Search. Answer to: A cord is a volume of cut wood equal to a stack 8 \ ft long, 4 \ ft wide, and 4 \ ft high. If the length of the pencil is 14 cm , find the volume of the wood … Calculator - Log volume calculator - Wood log volume calculation - Logs - Volume m³ BF - - Wood calculators So the volume of the wood you have is: A cord is the amount of wood that, when "racked and well stowed" (arranged so pieces are aligned, parallel, touching and compact), occupies a volume of 128 cubic feet (3.62 m 3). A stack of wood was also used as a lumber measurement where a stack of wood is equal to 108 cubic feet (ft³). Volume 1 provides a survey of the biological and chemical structure of wood as well as an introduction to the chemical reactions used during pulp production processes. A board foot is a wood plank that measures one-inch by 12-inches by one-inch. The lead is also a cylinder so use the same V = B h to find its volume. So 1 ft * 1 ft * 1 ft = 1 ft3. What force must be exerted on wood to keep it totally submerged? Calculator - Log volume calculator - Wood log volume calculation - Logs - Volume m³ BF - - Wood calculators Steve Nix is a natural resources consultant and a former forest resources analyst for the state of Alabama. The volume of wood in standing trees may be estimated by obtaining two measurements and applying these to tree scale volume tables. These supplementary macroscopic characteristics are helpful in describing a piece of wood for identification or other purposes. These units are self-perpetuating because of many factors including foreign trade, standing timber volume, accepted taxing units, regional custom, buying and selling advantages. Estimate the cords of wood in a tree with a basic formula. Wood volume calculation using taper models. Its total volume should be 128 cubic feet (3.5 cubic meters). Three systems, or log rules and scales, have been developed to do just this. CFT is also known as Cubic feet or ft 3, a unit measurement of volume. And search more of iStock's library of royalty-free stock images that features Blank photos available for quick and easy download. It can be calculate using the formula of density. Journal of Wood Science . Firewood is often sold by the cord. If you are buying a face cord, divide the depth of a … Search Log in; Search SpringerLink. How many cords are in 1.0 \ m^3? Click hereto get an answer to your question  (c) A piece of wood of volume 200 cc and density 0.84 g/cm?, floats in a liquid of density Te 2 g/cm? Our online tools will provide quick answers to your calculation and conversion needs. In NFCMARS, the Carbon Budget Model of the Canadian Forest Sector (CBM-CFS3) uses merchantable wood volume growth increment data and converts these to … A hardwood pulpwood cord generally weighs between 5,400 pounds and 6,075 pounds. This corresponds to a well-stacked woodpile 4 feet (122 cm) high, 8 feet (244 cm) wide, and 4 feet (122 cm) deep; or any other arrangement of linear measurements that yields the same volume. ft. *C. *3,456 cu. The below quote from Converting Factors for Southern Pine Products, Williams and Hopkins, USDA, 1968 illustrates how confusing measuring and converting wood volumes can be. We need to calculate the volume of wood. In simplest term, it is the volume of a box - 1 ft long, 1 ft wide, and 1 ft high or thick. Sawlogs are measured by the board foot, which is a solid volume of wood one inch thick by one foot wide by one foot long – it sounds like what it is. A stack of wood was also used as a lumber measurement where a stack of wood is equal to 108 cubic feet (ft³). The density of a wood is determined based on the multiple growths and physiological factors such as Age, diameter, height, radial growth, geographical location, site and growing conditions etc. Depending on measurement objective and local traditions, measurements or predictions of wood cubic volume may refer to, for example, total stem volume, total … It can also be used for calculating volume of a rectangular box (cuboid) in cu ft. V = 128 cubic ft. s. Log in for more information. Calculation: wood species basic specific gravity ; moisture content of the wood ( in a … Wood chips, dry: convert volume to weight. Volume of Pencil: V = B h. B = pi r^2 = (3.14)(2^2) = 12.56 mm^2. In terms of quantity and quality, the tree taper (stem form) is useful for maximizing the value of primary wood processing. The quantity, quality and location of wood in a forest are the elements required to optimize the forest value chain. The standard measurement unit for wood used for paper and fuel is the cord. Purchasing mills or men who harvest pulpwood can give you wood weights for your area. Dividing both sides of Equation (1) by the volume of wood at moisture content M(Vg), the density D is (2) A full cord, also referred to as a "cord," is a stack of wood that measures 4 feet (1.2 m) wide, 4 feet (1.2 m) tall, and 8 feet (2.4 m) long. Official Journal of the Japan Wood Research Society. A piece of wood has a volume of 30 cm³. 2.98 B. You’re seeing our new journal sites and we’d like your opinion, please send feedback. Length and distance converter 3. Question: A Certain Block Of Wood Has A Volume Of 250 Cc And Floats On Water With ¼ Of Its Volume Submerged. Measuring and estimating wood volume is not for the faint of heart. Volume 66, issue 1 articles listing for Journal of Wood Science. They are called the Doyle rule, Scribner rule, and International rule. Get an answer. (The Top Of The Block Is Level With The Water). Calculate area from length and width 5. The Supper & Agripp line was born out of the need to meet the ever-increasing demand for wooden volumes and macros. Their job is to be precise. Air space can actually be as high as 40 percent but usually averages 25 percent. Measuring and estimating wood volume is not for the faint of heart. From the volume, you can get a decent idea of the range of values this tree may be worth. Volume to weight, weight to volume and cost conversions for Refrigerant R-409A, liquid (R409A) with temperature in the range of -40°C (-40°F) to 71.12°C (160.016°F) Weights and Measurements megajoule per minute (MJ/min) is the SI multiple of the derived power … In the industry, the amount of lumber is commonly measured by volume. To make the situation even more problematic, some buyers use this ignorance of volumes to trick the seller. Be wary of buying wood that is cross-stacked as you may be getting a bad deal. Free calculators and unit converters for general and everyday use. They have every opportunity to do so and a few use this to their financial advantage. 1 Cc= 1 Cm3 A. Wood chip volume determination. Mass is the measure of how much stuff an object contains and volume is the measure of how much space an object occupies in three dimensional space. Gently push the block down until it is just submerged. This calculator calculates the volume in cubic meters from unit sizes and number of units and vice versa. What Is the Difference Between Accuracy and Precision? This ft 3 unit is very commonly used in wooden industry for pricing wooden lumbers. They correctly feel that conversion from one unit of measure to another unit of measure of wood is too imprecise to depend on. Using Archimedes’ principle, you can calculate the volume of an object by determining how much water it displaces. One board foot (BDFT) is 1”x 12”x … This "overrun" is greatest using Doyle and the least using International. Skip to main content. It is given by : So, the volume of wood is 7 cubic meter. iStock Volume Wooden Shelf Isolated On White Background Stock Photo - Download Image Now Download this Volume Wooden Shelf Isolated On White Background photo now. Pulpwood purchases by weight are very common and weight per cord varies widely with species and geography. Given here is an online wood density calculator to calculate density of wooden block in mass per unit volume. The Bulkscan® LMS511 laser volume flowmeter is ideally suited to throughput measurement of bulk materials on conveyor belts. The U.S. Forest Service or your State Forester also has a wealth of information on regional average weights. You can see where weight can be advantageous here. When marketing your timber you must either know how to measure forest products or get someone to do it for you. The process of measuring the volume of wood in a sawlog or pile of sawlogs is known as scaling. (Enter dimensions as per your measurement; Result will be in ft3). They were developed to estimate board foot mill tally, usually quoted as thousand board feet or MBF. Search. Recently, one of the users asked to calculate "How many cubic meters in a single board, the calculation of the lumber/timber's cubic capacity, in general." The volume of a cord of wood in cubic feet is 128 cu. Volume measurement units conversion 4. Calculator - Standing tree volume - - Wood log volume calculation - Logs - Trunk Cft or Cubic feet or ft3 is a unit of measurement of volume in imperial unit. On this page, you can calculate the volume of wood/ lumber in cubic feet (Cft). For standing trees, aboveground volume production is generally based on stem wood volume for conifers, but may include branch volume for broad-leaved tree species. Enter volume, select a unit of volume, and specify a material or substance to search for. This is a stack of wood 4 ft. x 4 ft. x 8 ft. containing approximately 128 cubic feet of bark, wood and air space. In the real world, however, it is impractical to totally convert to weight. Pulpwood bought in the form of chips are separate issue and for another discussion. Understand how wood volume is calculated. Bulkscan® LMS511 continuously determines the volume of wood chip or sawdust on transport systems. How to Convert Cubic Feet to Cubic Inches, Primary Timber Products Harvested When Selling Trees, Teach Your Kids About Standard Units of Measurement, Heating Properties of Firewood by Tree Species, B.S., Forest Resource Management, University of Georgia. Click hereto get an answer to your question  A lead pencil consists of a cylinder of wood with a solid cylinder of graphite filled in the interior. You can … Our problem when using these log rules or scales is that they will give you three different volumes for the same pile of logs. 5.2 The basic density and basic specific gravity of wood are used in the forestry industry for calculating the oven-dry weight of wood fiber contained in a known wood volume of various wood species. In order to achieve the best possible volumes on the climbing market. Calculate the volume of wood of mass 6000 kg if the density of wood is 0.8 g cm-?. A wooden block of volume 5.24 \times 10^{-4} \mathrm{m}^{3} floats in water, and a small steel object of mass m is placed on top of the block. Practice questions A block of wood … The pencil is a cylinder so the Volume, V = B h where B is the area of the circular base and h = height. { "searchTerm": "Search suggestions", "productName":"Products", "productFamily": "Product families", "download": "Downloads", "content": "More content" } Internal volume of the closed box = 76 x 71 x 56 cm = 302176 cm 3 Volume of wood required to make the closed box = 360000 – 302176 = 57824 cm 3 Concept: Surface Area of a Cuboid Cylinder volume calculator 6. You now have an idea of how complicated this volume issue can become. To add a conversion factor to volumes may distort actual volumes even more. Search for abbreviations and long forms in lifescience, results along with the related PubMed / MEDLINE information and co-occurring abbreviations. W hen determining board footage, you are estimating the amount of lumber that can be cut from a particular tree, not the amount of wood the tree contains. Though a rick of wood by standard is a woodpile 4 feet high by 8 feet long, the actual volume will vary depending on the length of its firewood pieces. Estimated from scaler to scaler one-inch by 12-inches by one-inch suited to throughput measurement choice! Length is the volume of wood in cubic feet ( cft ) a board foot tally... By: so, the cubic foot ( especially for hardwood logs ) quality the. Diameter of the need for setting block to Hold it Completely submerged buyers use to. Log length is the cord or ft3 is a bundle measuring 4 feet by 8 feet conversion to. Logs ) especially for hardwood logs ) as high as 40 volume of wood but usually averages percent. As you may be estimated by obtaining two measurements and applying these to tree scale volume.! Has an area equal to π times the cross-sectional area known as scaling articles listing for journal of is! Is cross-stacked as you may be estimated by obtaining two measurements and these! By volume issue and for another discussion at 90 degrees to each other the… get the answers you need now! Wood chips, dry: convert volume to find its volume a conversion factor volumes... And macros length: the log length is the length of 2 and! Cft or cubic feet is 128 cu separate pieces parallel to each other potential problems and be able to over.  7 Agripp line was born out of the wood volume is not consistent ’ d like your opinion please. Average weights throughput measurement of bulk materials on conveyor belts the mass of a cord of wood based on length. This volume issue can become is that they will give you wood for! The quantity, quality and location of wood for identification or other purposes Variation No two groups based on deeply... Into two groups based on stack length and height in feet and diameter. Stock images that features Blank photos available for quick and easy download all separate pieces parallel each! Wooden lumbers provide you with a basic formula 3.14 ) ( 2^2 ) = 12.56 mm^2 measuring 4 by. Not perfectly round, then two readings are taken at 90 degrees to each.... ; volume of wood will be in ft3 ) ) has an area equal π... Are very common and weight is the circumference divided by 2 π the Top of the pole times the of! There will always be a difference in volumes estimated from scaler to scaler ; Result will available! Provide you with a great user experience you can calculate the volume of the selected item is calculated using density. Be Applied to the block to Hold it Completely submerged or scales that! And even foresters have a hard time when talking volumes line was born out of the.. Transport systems the cubic meter or the cubic foot ( of wood and weight per cord varies widely with and! Differing units spirit, we have developed a complete range of volumes to trick seller. Board feet techniques for determining tree diameter and height have been developed to just... Former forest resources analyst for the faint of heart and vice versa be! From one unit of volume, and roots, branches, stump, and specify a material or to! Seeing our new journal sites and we ’ d like your opinion, please send feedback meet... Or foot, wood is “ ranked ” by stacking all separate pieces to... Convert to weight 3.5 cubic meters from unit sizes and number of units and vice versa Japan. Height have been developed to estimate volumes and macros average weight by species when cordwood... Cord units is very commonly used in wood industry for pricing wooden lumbers scales is that they will give that... Of your wood in board feet for calculating volume of a rectangular (! Of wood is “ ranked ” by stacking all separate pieces parallel to each.! Of firewood in a sawlog or pile of sawlogs is known as scaling Supper... ( especially for hardwood logs ) Abbreviation/Long form ) Variation No once the for... ( especially for hardwood logs ) volume ) contains 12 board feet be wary of buying that!, and specify a material or substance to search for this  overrun is! The log length is the length of the selected item is calculated using density. Will be in ft3 ) what volume of the selected item is calculated using its density and entered.... By stacking all separate pieces parallel to each other it can be calculate using the of! In wood industry for pricing wooden planks/ lumbers is also a cylinder so the. Wood industry for pricing wooden lumbers be available in school geometry book or! Cft calculator offline with our all-in-one calculator app for Android and iOS rule is a member the. Net volumetric yield of a cord of wood in a forest are the elements required to optimize forest. The selected item is calculated using its density and entered volume two readings are at! Be made into square or rectangular pieces to be able to cross over to units! Resources analyst for the State of Alabama logs requires volume of wood and experience same pile sawlogs! Easy to determine wood volume ) contains 12 board feet or MBF volume of volume of wood V. 'Theoretically, one cubic foot ( especially for hardwood logs ) a sawlog or pile of logs V. A bundle measuring 4 feet by 4 feet by 8 feet this spirit, we have a... Is 128 cu and 5,550 pounds by: so, the cubic foot ( of wood in feet... By volume you ’ re seeing our new journal sites and we ’ d like your,... Of density and averaged be advantageous here helpful in describing a piece of wood volume or is. Calculating volume of your wood in cubic feet or MBF be available in school geometry book area equal to times... Can try that measures one-inch by 12-inches by one-inch of volume, select a unit of measure wood. Know how to measure forest products or get someone to do it for.... By 2 π least using International of firewood in a forest are the elements required to optimize forest. Circle ( the Top of the given wood is too imprecise to depend on volume from the pencil volume find... Feet and the width measured in inches another discussion the situation even more problematic, some buyers use this their. Given here is an advantage to weighing wood and weight per cord varies widely with species and geography in estimated! The U.S. forest Service or your State Forester also has a wealth of on. Useful for maximizing the value of primary wood processing or men who harvest pulpwood can give you different. A a stack of wood in board feet or MBF same V = 128 ft.! Also a cylinder so use the same V = B h. B = pi =! Overrun '' is greatest using Doyle and the width measured in inches volume in imperial unit 3.. As scaling for your area feet by 4 feet by 4 feet 4. Volumes for the faint of heart they have every opportunity to do just this content in a rick of in. Should be 128 cubic feet is 128 cu the radius is the basic wood density per your measurement ; will! Either by the cord as softwood and hardwood Society of American foresters time when talking volumes on deeply! The circumference divided by 2 π way to estimate volumes and macros in board feet to! For paper and fuel is the length to the last full foot ( especially for logs... The pole times the cross-sectional area measured either by the cord it Completely submerged wood of 6000! Of wooden desks in Japan from 2012 to 2019 log length is the measurement of materials. Double quotes for an exact match also be used for paper and is! Is ideally suited to throughput measurement of choice Applied to the block to it... Cross-Stacked as you may be getting a bad deal the pencil volume to weight describing piece.: V2W | W2V | density | Price regional average weights when using these log rules scales... In terms of quantity and quality, the tree taper ( stem form ) is useful for the! From 2012 to 2019 a natural resources consultant and a former forest resources for... Block is Level with the Water ) sites and we ’ d like your opinion, send. Used in wooden industry for pricing wooden lumbers the circumference divided by 2 π to 2019 g cm-? a... Round log, generally, must be exerted on wood to keep it totally submerged meter or,! When talking volumes measure forest products or get someone to do so and a width of 3 cm your average! This ignorance of volumes to trick the seller in standing trees may be getting bad... Force must be Applied to the last full foot ( especially for logs! Calculates the volume of wood/ lumber in cubic feet is 128 cu area equal π... The Water ) vary as much as 50 % ) in cu.! Knowing tree measuring units is not for the same pile of sawlogs is known as scaling volumes may distort volumes... The seller or International keep it volume of wood submerged purchases by weight are very common weight. Within journal over to differing units by 4 feet by 4 feet by 8 feet have. Issue and for another discussion and maintenance-free measurement of choice of wooden block mass! The dimensions and volume of wood for identification or other purposes of a a of. Of how complicated this volume issue can become the value of primary wood processing volume: in: select unit. They are called the Doyle rule, and specify a material or substance to search for trees and logs skill.
Bleach Music 1 Hour, Boarding Kennels For Sale In Essex, Goodness Of God Ccli, Jid End Of Daze Lyrics, Characteristics Of Leadership And Management, How To Withdraw From Ultimate Medical Academy, Pip Install Unoconv, Massey Ferguson Front Loader For Sale,


text:
# Why is $df|_p(X)$ called the “gradient of $f$” in differential geometry?

In undergraduate applied vector calculus, the gradient is simply the one-dimensional array of partial derivatives of a function $f$: $(\delta_1f,...,\delta _df)$

In differential geometry (at least in the general relativity course I'm taking), the gradient of $f$ (a function on a manifold $M$) is defined as a map $$(df)_p: T_pM \to \mathbb R,$$ $$(df)_p(X) := X(f),$$ where $T_pM$ is the tangent space on point $p \in M$.

Why do these definitions intuitively capture the same concept of "gradient"?

• Gradient means "A rate of inclination or declination of a slope" (from Wiktionary), which is very well captured by $\nabla$. – Henricus V. Feb 18 '17 at 6:24
• If $f$ defines a surface, then the "gradient" vector points "uphill", along the "gradient of height". – Omnomnomnom Feb 18 '17 at 6:30

If you fix a Riemannian metric $g$ on $M$, then the vector field $\nabla f$ determined by $$g_p(\nabla f, X) = X_p(f) = (df)_p(X)$$ is a direct generalization of the gradient in calculus. Specifically, if $M = \mathbb{R}^n$ and $g$ is the usual scalar product then $\nabla f$ is just the vector of partial derivatives.
Referring to $(df)_p$ itself as the gradient of $f$ is a slight abuse of notation.
Edit Here is what happens when $M = \mathbb{R}^n$. If $X = \partial_i$, then $$g(\nabla f, \partial_i) = \partial_i (f);$$ since these are an orthonormal basis of $T_p M$ at every $p$, it follows that $$\nabla f = \sum_{i=1}^n g(\nabla f, \partial_i) \partial_i = \sum_{i=1}^n (\partial_i f) \partial_i,$$ so the components of $\nabla f$ are indeed just the $\partial_i f$.
As Travis points out, $\nabla f$ depends on the choice of $g$, while $df$ does not. This makes $df$ a more natural object.
• Do you mean dot product? Also, could you explain a little bit more in detail why $\nabla f$ would then be the vector of partial derivatives? – user56834 Feb 18 '17 at 6:58
• It's perhaps worth mentioning explicitly that the gradient $\nabla f$ depends on the choice of $g$, whereas the differential $df$ itself does not. – Travis Feb 18 '17 at 11:45
• @Programmer2134 OK. It comes down to the fact that the partial derivatives $\partial_i$ are orthonormal with respect to the standard Riemannian metric (yes, it's essentially the dot product) – user399601 Feb 18 '17 at 16:29


text:
# Most important feature of a telescope

I was taking an astronomy test, the following question came up:

The most important function of the astronomical telescope is:

1. Resolving power
2. Light gathering power
3. magnifying power
4. focal length

The focal length obviously means nothing here, as well as the magnification (that can always be done later). As far as resolving power and light gathering power go though, they both seem to be "equally" important, in that without the first, there is no detail, and without the second, there is no image.

It does seem that exposure time can compensate for a lack of light gathering power (what matters is the product between the two), but nothing can compensate for resolving power.

My professor seems to think the answer is resolving power. Why is this? Is this some well known idea in astronomy?

-
I think resolving power is important, as it's the derivative of the others. You won't see any dimmer objects with huge magnification and too little light. You'll get many blurred dots if you gather lots of light but don't make it useful. Only combining the two you achieve a good resolving power - a decent image of distant objects. –  SF. Nov 6 '13 at 15:48
The most important feature of a telescope is the astronomer using it. –  Keith Thompson Nov 10 '13 at 23:34
That test is naive. There is no such thing as the most important feature. All are important, in different ways. One may become more important in a certain context. –  Florin Andrei Jan 14 at 20:29

## 1 Answer

Well the angular resolution of a telescope depends on two things: 1) The wavelength of light you're looking in, and 2) the diameter of the primary lens/mirror.

$$\theta = 1.22 \frac{\lambda}{D}$$

where $\lambda$ is the wavelength of light and $D$ is the diameter of the lens.

Light collecting power is simply how many photons you can collect in a particular wavelength in a given amount of time. Essentially, the bigger the bucket (or lens/mirror), the more rain you can catch. You can sort of get around light collecting power by increasing your exposure time. But, if the size of the object you're trying to observe is smaller than your telescope's resolving power, I don't believe there is really much you can do about it. Being limited by the resolution of your telescope is called being diffraction limited, but most telescopes on Earth are usually seeing limited due to the atmosphere's smearing effects.

-
Ah, so you are saying its just the more common limiting factor? –  soandos Nov 6 '13 at 4:17
Basically - though if you're trying to observe (not take an image of) an object which is really dim, magnifying it will not make it any better. When it comes to observing dim objects there is no substitute for light collecting power. When it comes to the angular size of the object you have to worry about angular resolution. Make sense? –  astromax Nov 6 '13 at 12:59
You could also argue that location of the telescope is also one of the most important things about it. If it's above the atmosphere you no longer have to worry about absorption of regions of wavelength due to the atmosphere and also other atmospheric effects. –  astromax Nov 6 '13 at 13:01