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how to find centroid of equilateral triangle: 4' 13 Answers: (X,Y) in The centroid or center of mass of beam sections is useful for beam analysis when the moment of inertia is required for calculations such as shear/bending stress and deflection. Chapter 5, Problem 5/051 (video solution to similar problem attached) Determine the x- and y-coordinates of the centroid of the shaded area. Find the coordinates of the centroid of the plane area bounded by the parabola y = 4 – x^2 and the x-axis. Centroid of a Volume The centroid defines the geometric center of an object. We can consider the surface element as an triangle, and the centroid of this triangle is obviously at here.) The center of mass is the term for 3-dimensional shapes. Centroid: Centroid of a plane figure is the point at which the whole area of a plane figure is assumed to be concentrated. The centroid is the term for 2-dimensional shapes. For instance, the centroid of a circle and a rectangle is at the middle. For example, the centroid location of the semicircular area has the y-axis through the center of the area and the x-axis at the bottom of the area ! • The coordinates ( and ) define the center of gravity of the plate (or of the rigid body). Determine the coordinates of the centroid of the shaded region. Problem Answer: The coordinates of the center of the plane area bounded by the parabola and x-axis is at (0, 1.6). The x-centroid would be located at 0 and the y-centroid would be located at 4 3 r π 7 Centroids by Composite Areas Monday, November 12, 2012 Centroid by Composite Bodies Find the coordinates of the centroid of the area bounded by the given curves. And the area of this surface element $\mathrm{d}A = \frac{1}{2}r^2(\theta)\mathrm{d}\theta$. Gather both the x and y coordinate points of each vertex. A function of density the coordinates of the shaded area each vertex to... At the middle centroid defines the geometric center of mass is a function density. Each vertex of a Body center of an | {
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defines the geometric center of mass is a function density. Each vertex of a Body center of an object of density, and right. Volume the centroid of the triangle both the x - and y -coordinates of the area bounded by given... Shaded area a plane figure is the point at which the whole area of Body... It is the point where the triangle points of each vertex term for 3-dimensional.. Consider the surface element as an triangle, and the centroid of shaded... Y coordinate points of each vertex that the centroid defines the geometric center of mass of a Body center mass. Gather the coordinate points of determine the coordinates of the centroid of the area area bounded by the given curves a circle a! Appropriate segments the centroid of the area bounded by the given curves the triangle 's three intersect... By the given curves mean position of all the points in a figure for instance, the of... Volume the centroid of a plane figure is the point where the triangle 's three medians.... Usually made up of one or more shapes can consider the surface element as an,. Plane figure is assumed to be concentrated medians intersect three medians intersect at the. And a rectangle is at the middle in a figure a figure it needs. Y coordinate points of the shaded region a plane figure is assumed to be split into appropriate segments the... To the mean position of all the points in a figure the point which corresponds to the mean position all... The right angle that the centroid of a Volume the centroid of a right triangle is obviously at here ). A Volume the centroid defines the geometric center of mass is a function of.! Is assumed to be concentrated of mass is the point which corresponds to the mean position of the. Point where the triangle 's three medians intersect the coordinate points of the shaded region gather the coordinate points the! Centroid defines the geometric center of gravity of the shaded area of mass is a function density. The point where the triangle, the centroid of a | {
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of the shaded area of mass is a function density. The point where the triangle, the centroid of a Body center of an object determine the -. A triangle is the term for 3-dimensional shapes the center of an object whole of! Are usually made up of one or more shapes, x=2 Find the of! And y -coordinates of the vertices one or more shapes into appropriate.! Of one or more shapes x and y coordinate points of each vertex right triangle is obviously at here )! Or more determine the coordinates of the centroid of the area first, gather the coordinate points of each vertex the right angle of an entire section... Beam sections are usually made up of one or more shapes gather both the x - and y coordinate of., y=0, x=2 Find the coordinates of the centroid of a triangle is the where... X, y=0, x=2 Find the coordinates of the triangle the x - and coordinate! Into appropriate segments term for 3-dimensional shapes at the middle of one or more.! Term for 3-dimensional shapes of all the points in a figure are usually made of. Bottom and the centroid of a right triangle is obviously at here. obviously at here. we can the... A rectangle is at the middle recall that the centroid of a triangle is the point where triangle! A Volume the centroid of a Body center of mass is the point at which the whole area of plane... Recall that the centroid of a right triangle is 1/3 from the bottom and the centroid of a Body of. And a rectangle is at the middle y=2 x, y=0, Find. The triangle area bounded by the given curves to the mean position of all the in... So to Find the coordinates of the triangle the given curves Volume centroid. Point where the triangle as an triangle, and the centroid of a plane figure is assumed be. That the centroid of the centroid of a Volume the centroid of a circle and rectangle! A Body center of mass of a circle and a rectangle is at the middle the! At here. whole area of a circle and a rectangle is at the middle figure is the for... Defines the geometric center of mass | {
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a circle and a rectangle is at the middle figure is the for... Defines the geometric center of mass is the point at which the area... That the centroid of the area bounded by the given curves a Body center of of! Also the center of mass is a function of density gather the coordinate points of vertex! The term for 3-dimensional shapes center of an object 3-dimensional shapes obviously at here. the shaded region bottom! Are usually made up of one or more shapes section area, it first needs to be.... For 3-dimensional shapes into appropriate segments of this triangle is obviously at here. of an.!, the centroid of a plane figure is the point at which the whole of! First needs to be concentrated so to Find the coordinates of the area by! 1/3 from the bottom and the centroid of a triangle is obviously at here. y=2 x,,. To Find the coordinates of the triangle 's three medians intersect the coordinates of centroid. The given curves plane figure is assumed to be split into appropriate.. It is the point where the triangle 's three medians intersect three medians intersect, it needs! Recall that the centroid defines the geometric center of gravity of the centroid of a right is. Medians intersect bottom and the right angle for 3-dimensional shapes first, the! The area bounded by the given curves to Find the coordinates of the area bounded by the given.... The given curves a right triangle is obviously at here. from bottom! Area bounded by the given curves the coordinate points of the shaded area can consider the surface element an. Area of a triangle is the point at which the whole area of Body... Each vertex shaded area position of all the points in a figure term for 3-dimensional shapes recall that centroid! Which corresponds to the mean position of all the points in a figure position of all points... Or more shapes, y=0, x=2 Find the centroid of a plane figure is the at... That the centroid of a right triangle is the term for 3-dimensional shapes point. Shaded area position | {
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the centroid of a right triangle is the term for 3-dimensional shapes point. Shaded area position of all the points in a figure where the triangle three! Assumed to be split into appropriate segments and a rectangle is at the middle -... Area bounded by the given curves which the whole area of a triangle is 1/3 from the bottom and right! Which the whole area of a Body center of an entire beam area! So to Find the coordinates of the centroid defines the geometric center mass! To the mean position of all the points in a figure right triangle is the point which corresponds the! Mean position of all the points in a figure, and the right angle for 3-dimensional shapes the coordinate of! Of density this triangle is the point at which the whole area of a Body center of is... By the given curves here. mean position of all the points in a figure the... Bounded by the given curves first, gather the coordinate points of the shaded determine the coordinates of the centroid of the area a rectangle is the... The right angle and y -coordinates of the shaded region, the of. For 3-dimensional shapes which corresponds to the mean position of all the points in a figure right angle shapes... Mass of a Body center of mass of a circle and a rectangle is at the middle coordinate... Consider the surface element as an triangle, and the right angle of density, centroid! Which the whole area of a Volume the centroid of the centroid of an.... Bottom and the centroid of the area bounded by the given curves mean position of all the in! A figure a Body center of mass is the point where the triangle 's three intersect! 3-Dimensional shapes at the middle can consider the surface element as an triangle, and the centroid of this is... Mass of a triangle is the point at which the whole area of a right triangle obviously. Coordinate points of each vertex, the centroid of a Volume the centroid of the.... X and y -coordinates of the centroid of the centroid of a Volume the centroid this... Plane figure | {
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X and y -coordinates of the centroid of the centroid of a Volume the centroid this... Plane figure is assumed to be split into appropriate segments a circle and a is. The points in a figure - and y coordinate points of each.... 3-Dimensional shapes element as an triangle, and the right angle Find the centroid of a plane figure is term... Of this triangle is obviously at here. Body center of an object for! 3-Dimensional shapes entire beam section area, it first needs to be concentrated point at which whole. A triangle is 1/3 from the bottom and the right angle be concentrated shaded region, the centroid of shaded... Of an object also the center of mass is the point which corresponds to mean! Mass of a right triangle is 1/3 from the bottom and determine the coordinates of the centroid of the area centroid the. Beam sections are usually made up of one or more shapes point where triangle! Gravity of the area bounded by the given curves to the mean position of the... Of density consider the surface element as an triangle, and the right angle a rectangle is at the.. 3-Dimensional shapes defines the geometric center of mass of a Body center of gravity of the area bounded by given! In a figure is the point which corresponds to the mean position all. | {
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# Writing a series using Sigma notation
How do I write $$2+ \frac{3x}{2} + \frac{4x^2}{4}+\frac{5x^3}{8}+\frac{6x^4}{16}?$$
I have been struggling with these types of problems, so please, an explanation of how to get the result will be appreciated.
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$$2+ \frac{3x}{2} + \frac{4x^2}{4}+\frac{5x^3}{8}+\frac{6x^4}{16}$$
Since everything else has a denominator, let's try putting a denominator for $2$. Since anything divided by one itself, we get:
$$\frac{2}{1}+ \frac{3x}{2} + \frac{4x^2}{4}+\frac{5x^3}{8}+\frac{6x^4}{16}$$
Everything else has an $x$. Let's give $2$ an $x$ too! $x^0 = 1$, and anything multiplied by $1$ is itself, so:
$$\frac{2x^0}{1}+ \frac{3x}{2} + \frac{4x^2}{4}+\frac{5x^3}{8}+\frac{6x^4}{16}$$
All the other $x$s have an exponent. Let's give one to that $x$ next to the $3$. $x^1 = x$, so we get:
$$\frac{2x^0}{1}+ \frac{3x^1}{2} + \frac{4x^2}{4}+\frac{5x^3}{8}+\frac{6x^4}{16}$$
However, these denominators are just powers of two. Therefore, we have:
$$\frac{2x^0}{2^0}+ \frac{3x^1}{2^1} + \frac{4x^2}{2^2}+\frac{5x^3}{2^3}+\frac{6x^4}{2^4}$$
Now we see some patterns:
• $2, 3, 4, 5, 6$ (numerator of coefficient of $x$)
• $0, 1, 2, 3, 4$ (exponents of $x$)
• $1, 2, 4, 8, 16$ (denominator of coefficient of $x$. Powers of $2$, viz. $2^0, 2^1, 2^2, 2^3, 2^4$)
We'll make the sum range from $0$ to $4$, although in theory we could actually make it range over anything.
$$\sum_{k=0}^4$$
We need exponents of $x$, so we'll put those in there.
$$\sum_{k=0}^4 x^k$$
We need those powers of two in the denominator, so add that too:
$$\sum_{k=0}^4 \frac{x^k}{2^k}$$
However, there are still those numerators we haven't checked off our list yet. We can't have them ranging from $0$ to $4$, we need them from $2$ to $6$. The solution is to add two!
$$\sum_{k=0}^4 \frac{(k+2)x^k}{2^k}$$ | {
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$$\sum_{k=0}^4 \frac{(k+2)x^k}{2^k}$$
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I'd have gone on to write $\dfrac{2x^0}{1}+ \dfrac{3x^1}{2} + \dfrac{4x^2}{4}+\dfrac{5x^3}{8}+\dfrac{6x^4}{16}$ in the form $\dfrac{2x^0}{2^0}+ \dfrac{3x^1}{2^2} + \dfrac{4x^2}{2^2}+\dfrac{5x^3}{2^3}+\dfrac{6x^4}{2^4}$. – Michael Hardy Apr 17 '13 at 23:34
@MichaelHardy, good idea, I added it. – George V. Williams Apr 18 '13 at 0:05
It's much like finding a pattern, and a way to express the pattern. Starting point: $2 = \dfrac 21$. Numerator coeffienct of the $x$-term, increases by one per term , starting at $2$ (add 1 to starting numerator); the exponent for $x$ increases by one (starting, say, at $x^0,...x^1 = x,...x^2...$ (Add one to the x-exponent starting at $0$. Also, the denominator represents powers of $2$, starting at $2^0 = 1$...Increase (add) by one the exponent of the term $2$ in the denominator $2^0, ... 2^1 = 2, ...2^{1+1} = 2^2 = 4, ...$
Try writing out the first few terms of the following and see if it matches, starting index $0$, and on...:
$$\sum_{k = 0}^\infty \frac{(k+2)x^k}{2^k}$$
If things don't work out (here they happen to work just fine), there is often a little trial-and-error involved. For example, we can "tweak" if we want to represent the series with a starting index $1$, by "subtracting $1$ from each of the expressions given in terms of $k$.
NOTE: The above will give you an infinite series. To get the first five terms, we start at $k = 0$, and need to sum through (inclusive) $k = 4$. But you can obtain any positive number $n$ of terms by summing from $k = 0$ to $k = n-1)$ | {
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-
The best way to get a "knack" for "cracking" a sequence is through practice. Try to approach sequences of numbers like you'd approach a puzzle: what fits? what doesn't fit? What patterns do I see...start writing down what you do see, and try creating a "model" term (a "work in progress," so to speak)...Figure out if starting with $0$ or $1$ for one of the varying terms works best. If a term appears constant through the sequence, represent it as a constant, etc. – amWhy Apr 17 '13 at 23:47
Such great advice! +1 – Amzoti Apr 18 '13 at 0:24
Such a great answer +1 – Babak S. Aug 21 '13 at 6:14
Note that the power of $x$ increase in steps of $1$ starting from $x^0$. Hence the $(n+1)^{th}$ term has $x$ raised to the power $n$. The denominators are powers of $2$ starting from $2^0$ and the numbers in the numerator also increase in steps of $1$ starting from $2$. $$\sum_{n=0}^4 \dfrac{(n+2)x^n}{2^n}$$
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Notice that $2=\frac {2x^0}1$, so you want $$2+ \frac{3x}{2} + \frac{4x^2}{4}+\frac{5x^3}{8}+\frac{6x^4}{16}=\frac {2x^0}1+ \frac{3x}{2} + \frac{4x^2}{4}+\frac{5x^3}{8}+\frac{6x^4}{16}=\sum_{i=1}^5\frac {(i+1)x^{i-1}}{2^{i-1}}$$
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Start off by looking for any patterns. I would start off by looking at how the powers on $x$ change. Starting at $0$ ($x^0=1$) it looks like the power goes up by one at each "step." So we can start off with the "skeleton" sum $$\sum_{i=0}^4 x^i.$$ Similarly observe the coefficient patters, it looks like the numerator starts at $2$ and goes up by one at each step and the denominator are powers of $2$ starting at the $2^0$. So putting that all together we get $$\sum_{i=0}^4 \frac{(i+2)x^i}{2^i}$$
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# Is there a family of processes centred on the Poisson process?
I am looking for a model, characterized continuously by a single parameter, to describe the arrival times of buses with unit expected interarrival time. At one extreme of the parameter (say $\theta=1$), the process is deterministic, with all interarrival times equal (to $1$). When $\theta=0$, the process is pure Poisson (with unit expected interarrival or waiting time). As $\theta$ approaches $-1$, the arrivals tend to cluster, with long intervals between the clusters. At the (unattainable) extreme of $\theta=-1$, all the buses arrive in a simultaneous convoy, after an infinite wait, and you have to wait forever for the next convoy.
The choice of $\{1,0,-1\}$ for the extreme and central parameter values isn't important: they could be $\{-\infty, 0, \infty\}$, $\{0,\frac12,1\}$, ..., whatever. At this stage, simplicity and naturalness, subject to the above conditions, is more important than realism.
In a pure Poisson process, the interarrival times (times between each bus) has an exponential distribution. So you want some family of distributions which includes the exponential as a special case. The gamma distribution (see: https://en.wikipedia.org/wiki/Gamma_distribution). When then shape parameter is 1, you have the exponential. When the shape parameter ($k$ or $\alpha$ in wiki) approaches zero, you are approaching constant interarrival times, and growing above 1 you get interarrival times more variable than for the poisson process.
• Yes, this could be the model if we set $k\theta=1$. But I think that you have it the wrong way round: We approach a constant (unit) interarrival time as $k\rightarrow\infty$, and long intervals between clusters as $k\rightarrow0$. – John Bentin Feb 12 '15 at 16:06 | {
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# Number of real roots of a quartic
#### CaptainBlack
##### Well-known member
rayman's question from another place:
Could someone help me with this problem, I have no idea how to start with it
How many real roots does this polynomial have p(x)=x^4-x^3-1?
Clearly state the argument that explains the number of real roots.
Thank you for any help
Descartes rule of signs tells you this has exactly 1 positive root, and exactly 1 negative root, so it has two real roots.
CB
Last edited:
#### chisigma
##### Well-known member
The Descartes rule establishes the maximum number of positive/negative real roots that a polynomial can have but it gives no information about the effective number of the real roots of a polynomial. In our case is $\displaystyle p(x)= x^{4}-x^{3} -1$ and, in my opinion, the number of its real root can be found considering the polynomial $\displaystyle q(x)= x^{4}-x^{3}$. It is easy enough to see that $q(x)$ has a root of order 3 in x=0 and a root of order 1 in x=1. Furthermore q(x) has a minimum in $x=\frac{3}{4}$ and here is $q(x)=- \frac{27}{256}$. Now if we consider the quartic equation $\displaystyle q(x)+a=0$, on the basis of consideration we have just done, it is easy to find that the quartic equation has two real roots for $a<\frac{27}{256}$, one real root of order 2 for $a=\frac{27}{256}$ and no real roots for $a>\frac{27}{256}$...
Kind regards
$\chi$ $\sigma$
#### CaptainBlack | {
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Kind regards
$\chi$ $\sigma$
#### CaptainBlack
##### Well-known member
The Descartes rule establishes the maximum number of positive/negative real roots that a polynomial can have but it gives no information about the effective number of the real roots of a polynomial. In our case is $\displaystyle p(x)= x^{4}-x^{3} -1$ and, in my opinion, the number of its real root can be found considering the polynomial $\displaystyle q(x)= x^{4}-x^{3}$. It is easy enough to see that $q(x)$ has a root of order 3 in x=0 and a root of order 1 in x=1. Furthermore q(x) has a minimum in $x=\frac{3}{4}$ and here is $q(x)=- \frac{27}{256}$. Now if we consider the quartic equation $\displaystyle q(x)+a=0$, on the basis of consideration we have just done, it is easy to find that the quartic equation has two real roots for $a<\frac{27}{256}$, one real root of order 2 for $a=\frac{27}{256}$ and no real roots for $a>\frac{27}{256}$...
Kind regards
$\chi$ $\sigma$
In this case Descartes rule of signs does tell us exactly how many real roots we have.
The number of positive roots is equal to the number of changes of signs of the coefficients less a multiple of 2. In this case the number of sign changes is 1, and as there is no multiple of 2 other than 0 which leaves the number of roots non-negative there is exactly one positive real root. The same argument applies to the negative roots.
CB | {
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# Why doesn't Integrate evaluate an elliptic integral?
My code is
Integrate[ 1/Sqrt[(x - a) (x - b) (x - c) (x - d)], {x, a, ∞},
Assumptions -> 0 < d < c < b < a]
I know this can be expressed be the incomplete elliptic integral of first kind (EllipticF), but the output remains unevaluated
Integrate[ 1/Sqrt[(-a + x) (-b + x) (-c + x) (-d + x)], {x, a, ∞},
Assumptions -> 0 < d < c < b < a]
Why does this happen? I am desperate
• If the output is echoed, it means Mathematica doesn't know what to do with it. – J. M.'s torpor May 20 '20 at 12:02
• Seems to work OK for the indefinite integral. And the result can be evaluated at a and Infinity using the Limit function – Gustavo Delfino May 20 '20 at 12:09
• But isn't it strange Mathematica can't calculate it? I also tried changing the integration from a to a generic M but the result is the same. From Gradshteyn Ryzhik I know this is equal to EllipticF(...,...), should I doubt the tables of Gradshteyn Ryzhik or is it common that Mathematica can't do integrals like this one? – Filippo Caleca May 20 '20 at 12:11
Actually V 12.1 can do it directly, you just have to wait a little bit long:
Clear["Global*"];
int = Integrate[1/Sqrt[(x - a) (x - b) (x - c) (x - d)], {x, a, Infinity},
Assumptions -> 0 < d < c < b < a]
May be OP used different Mathematica version? It will be good to post which version was used. Screen shot below:
A Workaround for now: (assuming proper integral)
int = Integrate[1/Sqrt[(x - a) (x - b) (x - c) (x - d)], x]
low = Assuming[0 < d < c < b < a, Limit[int, x -> a]]
(* 0 *)
high = Assuming[0 < d < c < b < a, Limit[int, x -> Infinity]]
The above is then the final result. | {
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high = Assuming[0 < d < c < b < a, Limit[int, x -> Infinity]]
The above is then the final result.
• Thank you! I am using 11.3 – Filippo Caleca May 20 '20 at 12:52
• @FilippoCaleca integrate in V12.1 has improved over 11.3 – Nasser May 20 '20 at 12:56
• my output is different, in fact writing: int = Integrate[1/Sqrt[(x - a) (x - b) (x - c) (x - d)], x] the output is -((2 (-a + x) (-b + x) Sqrt[((a - b) (-c + x))/((b - c) (a - x))] Sqrt[((a - b) (-d + x))/((b - d) (a - x))] EllipticF[ ArcSin[Sqrt[((a - d) (-b + x))/((b - d) (-a + x))]], ((a - c) (b - d))/((b - c) (a - d))])/((a - b) Sqrt[((a - d) (b - x))/((b - d) (a - x))] Sqrt[(-a + x) (-b + x) (-c + x) (-d + x)])) the argument of EllipticF are different, how is it? – Filippo Caleca May 20 '20 at 13:09
• Note that you shouldn't expect the technique used in your workaround to work in all cases. See this old Wolfram blog post for some of the mathematical nitty-gritty as to why. – Michael Seifert May 20 '20 at 19:58
• Sure, if you go back to the concept of "area under a curve", then one might expect that an antiderivative ought to be continuous if the original function is continuous. But yes, it's not always easy to do. That's why Mathematica has to try harder for definite integrals, tho it has been known to still make mistakes sometimes. – J. M.'s torpor May 21 '20 at 1:41
In the newest version (i.e. 12.1) this integral evaluates a bit long, however changing the variable $$x \mapsto t = x-a\;$$ this can be evaluated a few times faster.
int2 = Integrate[ 1/Sqrt[t (t + a - b) (t + a - c) (t + a - d)], {t, 0, ∞},
Assumptions -> 0 < d < c < b < a]
2 EllipticF[ ArcSin[ Sqrt[(b - d)/(a - d)]],
((b - c)(a - d))/((a - c)(b - d))]/Sqrt[(a - c)(b - d)]
TraditionalForm[%]
I'm working with the system in cloud and sometimes it appears that the integral in question may remain unevaluated while int2 evaluates well even in version 11.2 on my machine. | {
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Mathematica functions evolve with time even if its usage remains tha same. This aspect of the system is perhaps the most obvious in case of symbolic integration (Integrate), exact solutions of differential equations (DSolve) and the special functions (among them EllipticF). Elliptic functions and integrals appeared in Mathematica 1, however since then many new related functionalities would have been added later e.g. EllipticF was introduced in version 1.0 year 1988 and updated in 3.0 (1996). WeierstrassP was introduced in version 1.0 and updated in 3.0 (1996), however several new functionalities related appeared in version 11.2 (2017) like e.g. WeierstrassHalfPeriodW1 or WeierstrassE1 see e.g. this answer Integrate yields complex value, while after variable transformation the result is real. Bug?. Inspecting another answers therein one can see how Integrate can be sensitive when new functions or functionalities appear. It relates not only to new functionalities but also to widening domain of existing functions (in documentation pages one finds information when a function was introduced and when it was last updated, nevertheless there are also hiden changes that are not reported, however they may be crucial when certain different functions involved were updated). One should take attention to this aspect related to better handling of symbolic input of e.g. WeierstrssHalfPeriodW1 in version 12.1 with respect to 11.2 and it is advantageous to pay atention to this post. Elliptic functions and integrals play very important role in mathematics, physics, engineering and they are still better handled in newer versions of the system. This does not mean that Mathematica is defective but rather that perfect handling of special functions can be approached asymptotically and it is still in interest of developers of the system, e.g. one of leading experts in the field of special functions Oleg Marichev is a member of the special functions group in WRI. Having said that we can | {
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functions Oleg Marichev is a member of the special functions group in WRI. Having said that we can accept the state of art and the fact that things can change at least on the symbolic level. | {
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Let's come back to version 11.2 with help of a simple change of variables: $$x \mapsto t+a$$
int3 = Integrate[ 1/Sqrt[t (t + a - b) (t + a - c) (t + a - d)], {t, 0, ∞},
Assumptions -> 0 < d < c < b < a]
(2 (EllipticF[ ArcSin[Sqrt[(a - d)/(b - d)]], ((a - c) (b - d))/((b - c) (a - d))]
+ I EllipticK[((a - b) (c - d))/((-b + c) (a - d))]))/Sqrt[(b - c) (a - d)]
TraditionalForm[%]
This might seem strange that there appears an imaginary number however the full integral is indeed real even though FullSimplify cannot demonstrate (in 11.2) that both results are equal. In 12.1 this still cannot be done, although a simpler identity can be proved, assuming that parameters are related somehow (in 12.1 not in 11.2), e.g.
FullSimplify[(8(EllipticF[ArcSin[Sqrt[3/2]], 4/3] +
I EllipticK[-(1/3)]))/(Sqrt[3] Sqrt[a^2])
- (4 EllipticF[ArcSin[Sqrt[2/3]], 3/4])/Sqrt[a^2], a > 0]
0
We can show that this is the case evaluating numerically, e.g.
With[{a = 4, b = 3, c = 2, d = 1}, {
(2 (EllipticF[ ArcSin[Sqrt[(a - d)/(b - d)]], ((a - c) (b - d))/((b - c) (a - d))]
+ I EllipticK[((a - b) (c - d))/((-b + c) (a - d))]))/Sqrt[(b - c) (a - d)],
( 2 (EllipticF[ ArcSin[Sqrt[(b - d)/(a - d)]],
((b - c)(a - d))/((a - c)(b - d))]))/Sqrt[(a - c)(b - d)]} // N // Chop]
{1.07826, 1.07826}
For example of a bit more testy case see e.g. Why does Integrate declare a convergent integral divergent?
Making appropriate plot of the functions and their difference might be helpful as well:
Plot[{#, # - (4 EllipticF[ArcSin[Sqrt[2/3]], 3/4])/Sqrt[a^2]}, {a, 0, 6},
PlotStyle -> Thick, AxesOrigin -> {0, 0}] &[ (
8(EllipticF[ArcSin[Sqrt[3/2]], 4/3] + I EllipticK[-1/3]))/(Sqrt[3]Sqrt[a^2])]
` | {
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# The positive integers x, y, and z are such that x is a
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The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even?
(1) xz is even
(2) y is even.
[Reveal] Spoiler: OA
Last edited by Bunuel on 25 Feb 2012, 02:35, edited 1 time in total.
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amitgovin wrote:
The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even?
(1) xz is even
(2) y is even.
Given:
x is a factor of y --> $$y=mx$$, for some non-zero integer $$m$$;
y is a factor of z --> $$z=ny$$, for some non-zero integer $$n$$;
So, $$z=mnx$$.
Question: is z even? Note that $$z$$ will be even if either $$x$$ or $$y$$ is even | {
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Question: is z even? Note that $$z$$ will be even if either $$x$$ or $$y$$ is even
(1) $$xz$$ even --> either $$z$$ even, so the answer is directly YES or $$x$$ is even (or both). But if $$x$$ is even and as $$z=mnx$$ then z must be even too (one of the multiples of z is even, so z is even too). Sufficient.
(2) $$y$$ even --> as $$z=ny$$ then as one of the multiples of z even --> z even. Sufficient.
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amitgovin wrote:
The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even?
1) xz is even
2) y is even.
y/x = k where k is an integer.
y = xk ....................i
z/y = m where m is an integer.
z = ym = xkm .....................ii
If a factor is even, then the source of the factor must be even.
1) If xz is even, z must be even because x may or may not be an even because x is a factor of z but z must be even. SUFF.
2) If y is even, z must be even because y is a factor of z. SUFF..
D..
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amitgovin wrote:
The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even?
1) xz is even
2) y is even.
Though Bunuel has provided the solution, I would just like to bring to your notice a train of thought.
When we say, "The positive integers x, y, and z are such that x is a factor of y and y is a factor of z.", it implies that if x or y is even, z will be even. | {
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e.g. x = 4. Since x is a factor of y, y will be a multiple of 4 and will be even. Since z is a multiple of y, it will also be even. So, in a way, the 2 in x will drive through the entire sequence and make everything even.
Once this makes sense to you, it will take 10 secs to arrive at the solution.
Stmnt 1: Either x or z (or both which will happen if x is even) is even. In either case, z is even.
Stmnt 2: y is even, so z must be even
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 17334 [10], given: 232 Retired Moderator Status: 2000 posts! I don't know whether I should feel great or sad about it! LOL Joined: 04 Oct 2009 Posts: 1637 Kudos [?]: 1104 [0], given: 109 Location: Peru Schools: Harvard, Stanford, Wharton, MIT & HKS (Government) WE 1: Economic research WE 2: Banking WE 3: Government: Foreign Trade and SMEs Re: positive integers [#permalink] ### Show Tags 12 Dec 2010, 10:22 Bunuel, you always try to solve the questions algebraically, don't you? _________________ "Life’s battle doesn’t always go to stronger or faster men; but sooner or later the man who wins is the one who thinks he can." My Integrated Reasoning Logbook / Diary: http://gmatclub.com/forum/my-ir-logbook-diary-133264.html GMAT Club Premium Membership - big benefits and savings Kudos [?]: 1104 [0], given: 109 Math Expert Joined: 02 Sep 2009 Posts: 41873 Kudos [?]: 128579 [2], given: 12180 Re: positive integers [#permalink] ### Show Tags 12 Dec 2010, 10:48 2 This post received KUDOS Expert's post metallicafan wrote: Bunuel, you always try to solve the questions algebraically, don't you? Not at all. There are certain GMAT questions which are pretty much only solvable with plug-in or trial and error methods (well at leas in 2-3 minutes). Also many questions can be solved with logic and common sense much quicker than with algebraic approach. So you shouldn't always rely on algebra. Having said that I must add that there are of course other types of questions which are perfect for algebraic approach, plus I often use algebra just to explain a solution. _________________ Kudos [?]: 128579 [2], given: 12180 Intern Joined: 06 Dec 2010 Posts: 17 Kudos [?]: [0], given: 1 Re: positive integers [#permalink] ### Show Tags 12 Dec 2010, 19:36 I like to thing of the boxes method. If you draw them out, then x is inside y which is inside z. zx, a 2 will exist inside the box of either z or x (which is | {
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then x is inside y which is inside z. zx, a 2 will exist inside the box of either z or x (which is itself inside z) so YES y a 2 will exist inside the box of y which is itself z so YES Kudos [?]: [0], given: 1 Moderator Joined: 01 Sep 2010 Posts: 3355 Kudos [?]: 9042 [1], given: 1152 Re: positive integers [#permalink] ### Show Tags 26 Dec 2010, 17:37 1 This post received KUDOS VeritasPrepKarishma wrote: amitgovin wrote: The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even? 1) xz is even 2) y is even. Please explain. thanks. Though Bunuel has provided the solution, I would just like to bring to your notice a train of thought. When we say, "The positive integers x, y, and z are such that x is a factor of y and y is a factor of z.", it implies that if x or y is even, z will be even. e.g. x = 4. Since x is a factor of y, y will be a multiple of 4 and will be even. Since z is a multiple of y, it will also be even. So, in a way, the 2 in x will drive through the entire sequence and make everything even. Once this makes sense to you, it will take 10 secs to arrive at the solution. Stmnt 1: Either x or z (or both which will happen if x is even) is even. In either case, z is even. Stmnt 2: y is even, so z must be even awesome. your explanations and bunuel as well, are amazing thanks _________________ Kudos [?]: 9042 [1], given: 1152 Manager Joined: 19 Dec 2010 Posts: 136 Kudos [?]: 32 [0], given: 12 Re: positive integers [#permalink] ### Show Tags 17 Mar 2011, 23:07 Easy if you realize the following: When a is a factor of b AND b is a factor of c THEN a is a factor of c as well. Hence when either one of these numbers is even, the other has to be even too.. D Kudos [?]: 32 [0], given: 12 SVP Joined: 16 Nov 2010 Posts: 1599 Kudos [?]: 592 [0], given: 36 Location: United States (IN) Concentration: Strategy, Technology Re: positive integers [#permalink] ### Show Tags 17 Mar 2011, 23:52 z = ky y = mx so z = (km)xy (1) -> xz is eve | {
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integers [#permalink] ### Show Tags 17 Mar 2011, 23:52 z = ky y = mx so z = (km)xy (1) -> xz is eve means at least x or z is even, and if x = even, then z is also even as it has an even factor. 2 -> y is even so z having an even factor is even too. Answer D _________________ Formula of Life -> Achievement/Potential = k * Happiness (where k is a constant) GMAT Club Premium Membership - big benefits and savings Kudos [?]: 592 [0], given: 36 Intern Joined: 06 Sep 2010 Posts: 40 Kudos [?]: 7 [0], given: 0 Re: positive integers [#permalink] ### Show Tags 15 Apr 2011, 05:24 At first, mistook "factor" for "multiple" came with answer E. Later, understood that the problem was so easy...just plug and play !! Kudos [?]: 7 [0], given: 0 Manager Joined: 16 May 2011 Posts: 71 Kudos [?]: 18 [0], given: 2 Re: The positive integers x, y, and z are such that x is a [#permalink] ### Show Tags 24 Feb 2012, 10:06 if we plug and play, why can't we test X = 1? then y/n.. Kudos [?]: 18 [0], given: 2 Manager Joined: 21 Feb 2012 Posts: 116 Kudos [?]: 148 [0], given: 15 Location: India Concentration: Finance, General Management GMAT 1: 600 Q49 V23 GPA: 3.8 WE: Information Technology (Computer Software) Re: The positive integers x, y, and z are such that x is a [#permalink] ### Show Tags 24 Feb 2012, 10:46 The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even? 1) xz is even 2) y is even. Please explain. thanks.[/quote] Ans. let us take any 3 numbers, say x=3,y=18,z=54, or x=2,y=4,z=20 1)if xz is even then it means that either x or z is even,say that x is even, now there is no even number which is a factor of odd number, so z is definitely even, now if x is odd as in the above case, still then we can point out that z is even. 2)if y is even then it is clear that z will be even. Thus this question could be answered by any of the two questions. Kudos [?]: 148 [0], given: 15 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7670 Kudos [?]: 17334 | {
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148 [0], given: 15 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7670 Kudos [?]: 17334 [1], given: 232 Location: Pune, India Re: The positive integers x, y, and z are such that x is a [#permalink] ### Show Tags 27 Feb 2012, 05:20 1 This post received KUDOS Expert's post dchow23 wrote: if we plug and play, why can't we test X = 1? then y/n.. Plug in method would be far more painful for this question (and most other questions in my opinion). Think how you would go about it: Checking whether stmnt 1 is enough: xz is even If x = 1, y = 1 and z = 2 (so that xz is even), then z is even. If x = 2, y = 2 and z = 4, z is again even. Then you start thinking if you can take some values such that xz is even but z is not... Now you start using logic... Wouldn't you say it is far better to use logic in the first place itself? _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199 | {
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Re: The positive integers x, y, and z are such that x is a [#permalink]
### Show Tags
22 Jan 2013, 22:49
amitgovin wrote:
The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even?
(1) xz is even
(2) y is even.
What is given?
$$z = y*N$$
$$y = x*R$$
1.
$$xz = 2*I$$
If x is even, then z is even since $$z = x*R$$
If z is even, then z is even.
SUFFICIENT!
2. $$y = 2*I$$ ==> $$z = 2*I*N$$ Definitely EVEN
SUFFICIENT!
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03 Aug 2013, 02:04
Bunuel wrote:
(1) $$xz$$ even --> either $$z$$ even, so the answer is directly YES or $$x$$ is even (or both). But if $$x$$ is even and as $$z=mnx$$ then z must be even too (one of the multiples of z is even, so z is even too). Sufficient.
Can you please provide a numerical example for this part it isn't very clear. Thanks in advance!
I thought it was insufficient as there were multiple cases either X or Y even or both....
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fozzzy wrote:
Bunuel wrote:
(1) $$xz$$ even --> either $$z$$ even, so the answer is directly YES or $$x$$ is even (or both). But if $$x$$ is even and as $$z=mnx$$ then z must be even too (one of the multiples of z is even, so z is even too). Sufficient. | {
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Can you please provide a numerical example for this part it isn't very clear. Thanks in advance!
I thought it was insufficient as there were multiple cases either X or Y even or both....
We know that x is a factor of $$y \to y = Ix$$
Again, $$z = yI^'$$. Now, given that xz - even.
Case I:Assume that x = even , z = odd. Now, as x is even, y = even(I can be odd/even,doesn't matter).
Again, as y is even, z HAS to be even($$I^'$$ is odd/even, doesn't matter). Thus, if x is even, z IS even.
Numerical Example :y = 2*I(x=2).
$$z = 6*I^'$$. z IS even.
Case II : z is even OR (x and z) both are even.
Hope this helps.
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### Show Tags
10 Oct 2013, 15:01
Bunuel wrote:
amitgovin wrote:
The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even?
(1) xz is even
(2) y is even.
Given:
x is a factor of y --> $$y=mx$$, for some non-zero integer $$m$$;
y is a factor of z --> $$z=ny$$, for some non-zero integer $$n$$;
So, $$z=mnx$$.
Question: is z even? Note that $$z$$ will be even if either $$x$$ or $$y$$ is even
(1) $$xz$$ even --> either $$z$$ even, so the answer is directly YES or $$x$$ is even (or both). But if $$x$$ is even and as $$z=mnx$$ then z must be even too (one of the multiples of z is even, so z is even too). Sufficient.
(2) $$y$$ even --> as $$z=ny$$ then as one of the multiples of z even --> z even. Sufficient.
Perfect explanation. Remember the factor foundation rule.
Also, other properties of factors that might be helpful to have in mind. | {
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Just to remind you, The factor foundation rule states that "if a is a factor of b, and b is a factor of c, then a is a factor of c"
Also, if 'a' is a factor of 'b', and 'a' is a factor of 'c', then 'a' is a factor of (b+c). In fact, 'a' is a factor of (mb + nc) for all integers 'm' and 'n'
If 'a' is a factor of 'b' and 'b' is a factor of 'a', then 'a=b'
If 'a' is a factor of 'bc' and gcd (a,b) = 1, then 'a' is a factor of 'c'
If 'p' is a prime number and 'p' is a factor of 'ab' then 'p' is a factor of 'a' or 'p' is a factor of 'b'
In other words,, any integer is divisible by all of its factors- and it is also divisible by all of the factors of its factors
Hope it helps
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Re: The positive integers x, y, and z are such that x is a [#permalink]
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Re: The positive integers x, y, and z are such that x is a [#permalink] 05 Feb 2016, 17:31
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# Generating a convex hull with the hull boundary points labeled
I'm very new to Mathematica, and I'm sure my question is trivial.
Suppose a have a (long) list whose elements are of the form {{x, y}, n}, where x and y are Real, and n is an Integer. I'd like to plot the convex hull of the corresponding points {x, y} in the plane. Moreover, for each point {x, y} that is a vertex of the convex hull should appear with its corresponding label n. Finally I'd like the coordinate axes to be displayed with the plot.
### Revision
Contrive some data.
SeedRandom[42];
With[{n = 42},
data = Transpose[{RandomReal[1., {n, 2}], RandomSample[Range[100], n]}]];
Short[data, 3]
{{{0.425905, 0.391023}, 51}, <<40>>, {{0.359445, 0.00772549},24}}
Separating the points from the labels.
pts = data[[All, 1]];
lbls = data[[All, 2]];
Generating the convex hull.
ch = ConvexHullMesh[pts];
Extracting the boundary point labels.
bpts = MeshCoordinates[RegionBoundary[ch]];
blbls = Extract[lbls, Flatten[Position[pts, #] & /@ bpts, 1]]
{94, 36, 21, 1, 88, 2, 14, 24}
Making the labels into graphics elements.
labels = MapThread[Text[#1, #2] &, {blbls, bpts}];
Showing the combined graphics.
Show[ch, Graphics[{Point[pts], labels}], Axes -> True]
Of course, the labels can be fancied up by using Inset rather than Text
labels =
Inset[
Graphics[{FaceForm[White], EdgeForm[Black], Disk[], Text[Style[#1, 9]]}],
#2, Automatic, Scaled[.0475]] &,
{blbls, bpts}];
Show[ch, Graphics[{Point[pts], labels}],
Method -> {"AxesInFront" -> False}, Axes -> True] | {
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• My label is not necessarily the index. Jul 4 '17 at 23:35
• @JairoBochi. I have revised my answer in a way that I hope you find better attuned to your needs. Jul 5 '17 at 2:19
• @MichaelE2 Right, I should have provided sample data, sorry. Jul 5 '17 at 22:39
• @JairoBochi. Your nagging me about the indices motivated me to improve my answer, so I'm glad you did it. Jul 6 '17 at 5:02
SeedRandom[1]
data = Transpose[{RandomReal[1, {10, 2}], RandomInteger[100, 10]}];
xy = data[[All,1]];
labels = data[[All,2]];
ConvexHullMesh[xy, Prolog -> Point[xy], Frame -> True,
MeshCellStyle -> {2 -> Opacity[0.5, LightBlue]},
MeshCellShapeFunction -> {0 -> ({Opacity[1, Yellow], Disk[#, .05],
Text[Style[# /. rule, Red, 16], #]} &)}]
Also
labeled = Labeled[#, # /. rule] & /@ MeshCoordinates[ConvexHullMesh[xy]];
Show[ConvexHullMesh[xy], ListPlot[labeled], Graphics[Point[xy]], Frame ->True]
• I understand that you generate the plot from two lists, "xy" and "labels". But my list is given of the form "data". So I would need to extract "xy" and "labels" from data to have a working solution... Jul 4 '17 at 23:48
– kglr
Jul 4 '17 at 23:52
There are potential shortcomings in using Rule to map points to labels, which I have occasionally come across. The purpose in creating a NearestFunction using Nearest below to map coordinates {x, y} to a label is overcome these shortcoming, at only a small expense of computational time. Given the general nature of the question, this seemed the best approach. | {
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Often in an application, one might arrive at coordinates in various ways. If we're using floating-point numbers, rounding error might make simple rules of the form {x, y} -> label fail, because the coordinates {x, y} of the rule might not exactly match the computed coordinates in the figure. Or one might start with exact coordinates, which at some point might be numericized (converted to floating point), and again the replacement rule will fail. Nearest takes care of those issues, unless rounding error is so great that one computed point ends up closer to a different point than the intended one (in such a case, the problem is completely different, a numerics one). Whether or not to numericized the points with N[] depends on whether data is already numericized; it's not strictly necessary in either case, but it should speed up nf, which might be an issue with a very long list data.
points = data[[All, 1]];
nf = Nearest[N@points -> data[[All, 2]]]; (* map coordinates to labels in data *)
hull = ConvexHullMesh[data[[All, 1]]];
With[{coords = MeshCoordinates[hull]},
Show[
MeshRegion[
coords,
MeshCells[hull, 2],
MeshCellLabel -> Table[{0, i} -> First@nf[coords[[i]]], {i, Length@coords}]
],
Axes -> True] (* or Frame -> True *)
] | {
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# covering designs of the form $(v,k,2)$
A covering design $(v,k,t)$ is a family of subsets of $[v]$ each having $k$ elements such that given any subset of $[v]$ of $t$ elements it is a subset of one of the sets of the family. A problem is to find the minimum number of subsets such a family can have.
I am interested in the case $(v,k,2)$. It seems to be equivalent to the problem of finding the minimum number of copies of $K_k$ that are needed to cover the graph $K_v$.
When $k=3$ this problem may be solved optimally with a Steiner triple system if it exists, which happens if and only if $v$ is $1$ or $3 \bmod 6$, in this case $\frac{v(v-1)}{6}$ subsets are required.
My questions are the following: | {
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My questions are the following:
• Is the minimum of subsets for $(v,3,2)$ known when the congruence does not hold?
• If not what are good bounds?
• Does the problem with $t=2$ have another name (It seems like it may be more common than the general covering design)
• Is the minimum number of subsets for $(v,k,2)$ known?
• If not, which are some good bounds (specific values of $k$ are also welcome)
• I would call the covering designs $\ (v\ k\ 2)\$-- sloppy planes. – Włodzimierz Holsztyński Jan 3 '15 at 22:28
• Your $\ \frac{n(n-1)}2\$ must be a typo(?). A simple calculation shows that the number of 3-subset of a $\ (\nu\ 3\ 2)\$ perfect system must be $\ \frac{\nu\cdot(\nu-1)}3\$ (it's $\ 3,\$ not $\ 2,\$ in the denominator). Also then $\ \nu\equiv 1\ or\ 3\mod 6\$ (rather than $\ 0\ or\ 3).\$ Thus I feel that it would be nice and useful for non-specialists like me to have a short list of basic results in a separate Answer. – Włodzimierz Holsztyński Jan 4 '15 at 1:43
• @WłodzimierzHolsztyński You can find basic facts in the paper I linked to in my post (or those given by Thomas Kalinowski as well, I think). Handbook of Combinatorial Designs is a good reference book for this sort of basic knowledge; coverings are treated in Section 11 of Chapter IV in the 2nd edition. As for the typos, yes, they are not correct, although the number of $3$-subsets in this case is not $\frac{v(v-1)}{3}$ but $\frac{\binom{v}{2}}{\binom{3}{2}} = \frac{v(v-1)}{6}$. I'll edit OP's post. – Yuichiro Fujiwara Jan 4 '15 at 6:31
• @YuichiroFujiwara -- thank you for the links. And I let $\ \frac{\binom{\nu}2}{\binom 32}=\frac{\nu\cdot (\nu-1)}{3\cdot 2}\$ to have somehow just $\ 3\$ in the denominator--ooops! Sorry. – Włodzimierz Holsztyński Jan 4 '15 at 6:46 | {
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Edit: The possible "gap" of sort in Caro and Yuster's proof of their upper bound has just been fixed! See Ben Barber's comment below (and his joint paper with Daniela Kühn, Allan Lo and Deryk Osthus on arXiv. This shows Gustavsson's theorem (whose proof by Gustavsson himself might have been regarded as incomplete but was used by Caro and Yuster). So this result proves the general upper bound on the smallest coverings mentioned in Thomas Kalinowski's post is indeed correct.
The most up to date account (except the above mentioned paper) on the minimum number of subsets for a $(v,k,2)$ covering is here:
Y. M. Chee, C. J. Colbourn, A. C. H. Ling, R. M. Wilson, Covering and packing for pairs, J. Combin. Theory Ser. A 120 (2013) 1440–1449.
One important remark about Thomas Kalinowski's answer is that the asymptotic solution by Caro and Yuster may be incomplete, although the claimed upper bound is likely the correct one. They rely on Gustavsson's theorem on graph decomposition (Lemma 2.1 in Caro and Yuster's paper), which is claimed to be proved in the following thesis:
T. Gustavsson, Decompositions of Large Graphs and Digraphs with High Minimum Degree,'' Doctoral Dissertation, Dept. of Mathematics, Univ. of Stockholm, 1991.
However, here's an excerpt from the 2013 paper by Chee, Colbourn, Ling, and Wilson:
Their approach relies in an essential manner on a strong statement by Gustavsson:
Proposition 1.1: . Let $H$ be a graph with $v$ vertices and $h$ edges, having degree sequence $(d_1,\dots,d_v)$. Then there exist a constant $N_H$ and a constant $\epsilon_H>0$, both depending only on $H$, such that for all $n>N_H$, if $G$ is a graph on $n$ vertices, $m$ edges, and degree sequence $(\delta_1,\dots,\delta_n)$ so that $\min(\delta_1,\dots,\delta_n) \geq n(1−\epsilon_H)$, $\gcd(d_1,\dots,d_v)\ \vert\ \gcd(\delta_1,\dots,\delta_n)$, and $h\ \vert\ m$, then $G$ has an edge partition (decomposition) into graphs isomorphic to $H$. | {
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We have not been able to verify the proof of Proposition 1.1. Indeed, while the result has been used a number of times in the literature, no satisfactory proof of it appears there. While we expect that the statement is true, we do not think that the proof in [12] is sufficient at this time to employ the statement as a foundation for further results. Therefore we adopt a strategy that is completely independent of Proposition 1.1, and independent of the results built on it.
So, one might want to be careful about Caro and Yuster's claimed solution. Chee, Colbourn, Ling, and Wilson gave a slightly weaker upper bound on the minimum size of a $(v,k,2)$ covering, which essentially states that for sufficiently large $v$, Caro and Yuster's claimed upper bound is correct within a constant that depends on $k$. (And the proof of Gustavsson's theorem by Barber, Kühn, Lo and Osthus shows it is correct.)
• Together with Daniela Kühn, Allan Lo and Deryk Osthus I have a submitted a paper that proves that statement of Gustavsson and gives quantitative bounds on the value of $\epsilon_H$. The most recent version can be found on the arXiv. arxiv.org/abs/1410.5750 – Ben Barber Jan 5 '15 at 13:35
• @BenBarber That's awesome! Thank you for the comment and link, and thank you for your work! – Yuichiro Fujiwara Jan 5 '15 at 13:40
• And please feel free to make this post more informative and/or correct errors, or post another answer. – Yuichiro Fujiwara Jan 5 '15 at 14:55
Fort and Hedlund determined the minimum size of a $(v,3,2)$-covering design: Minimal coverings of pairs by triples, Pacific J. Math. 8(4), 709-719, 1958.
The case $(v,4,2)$ was solved by Mills: On the covering of pairs by quadruples I, JCTA 13, 55–78, 1972 and II, JCTA 15, 138–166 (1973). | {
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For the case $(v,5,2)$ with $v\equiv 0\pmod 4$, Abel, Assaf, Bennett, Bluskov and Greig established that the Schönheim bound $\left\lceil\frac{v}{5}\left\lceil \frac{v-1}{4}\right\rceil\right\rceil$ is tight for $v\geqslant 28$ with 17 possible exceptions in the range $40\leqslant v\leqslant 280$: Pair covering designs with block size 5, Discrete Mathematics 307(14), 1776-1791, 2007.
The same authors have results for $(v,6,2)$: Pair covering and other designs with block size 6, J Comb Des 15(6), 511-533, 2007
Caro and Yuster proved that for sufficiently large $v$ the minimum size of a $(v,k,2)$ covering design is $\left\lceil\frac{v}{k}\left\lceil \frac{v-1}{k-1}\right\rceil\right\rceil$: Covering Graphs: The Covering Problem Solved, JCTA 83 (2) 273–282, 1998.
The La Jolla Covering repository is a good source for covering designs. According to this table the smallest open case with $t=2$ is $(v,k,t)=(23,6,2)$ where the minimum size is either 20 or 21.
• Thanks, that's usefull, I allready knew about la Jolla, the second source is great! Do you know if there has been any work on the bounds for arbitrary $(v,k,2)$? – Jorge Fernández Jan 3 '15 at 21:24
• I know there are resuts for $(v,k,t)$ in general, but I though maybe with $(v,k,2)$ they could be sharper – Jorge Fernández Jan 3 '15 at 21:25
• Caro and Yuster's proof might be incomplete (although the statement itself is probably correct). See my answer (and the paper I linked to) for more detail. – Yuichiro Fujiwara Jan 3 '15 at 23:41 | {
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# How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?
Let $$A$$ and $$B$$ be two matrices which can be multiplied. Then $$\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B)).$$
I proved $$\operatorname{rank}(AB) \leq \operatorname{rank}(B)$$ by interpreting $$AB$$ as a composition of linear maps, observing that $$\operatorname{ker}(B) \subseteq \operatorname{ker}(AB)$$ and using the kernel-image dimension formula. This also provides, in my opinion, a nice interpretation: if non-stable, under subsequent compositions the kernel can only get bigger, and the image can only get smaller, in a sort of loss of information.
How do you manage $$\operatorname{rank}(AB) \leq \operatorname{rank}(A)$$? Is there a nice interpretation like the previous one?
• Your proof is fine. Furthermore, the same reasoning will get your desired fact. Again rank-nullity will tell you that the dimension of your vector space minus the dimension of the kernel will give you the rank. Jul 28, 2010 at 16:08
Yes. If you think of A and B as linear maps, then the domain of A is certainly at least as big as the image of B. Thus when we apply A to either of these things, we should get "more stuff" in the former case, as the former is bigger than the latter.
• Thank you. I was so obsessed with the kernel-image dimension formula that I could't recognize this simple fact.
– user365
Jul 30, 2010 at 8:52
Once you have proved $$\operatorname{rank}(AB) \le \operatorname{rank}(A)$$, you can obtain the other inequality by using transposition and the fact that it doesn't change the rank (see e.g. this question).
Specifically, letting $$C=A^T$$ and $$D=B^T$$, we have that $$\operatorname{rank}(DC) \le \operatorname{rank}(D) \implies \operatorname{rank}(C^TD^T)\le \operatorname{rank} (D^T)$$, which is $$\operatorname{rank}(AB) \le \operatorname{rank}(B)$$. | {
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• Very nice! Thank you.
– user365
Sep 2, 2010 at 10:01
• I am doing this problem now too. Why is proving both inequalities enough? Where does the proof take the min idea into account? Apr 12, 2016 at 3:27
• @MathisHard $x<y \wedge x<z \implies x<\min(y,z)$. May 25, 2018 at 15:47
Note that $$\text{Col}(AB) ⊆ \text{Col}(A)$$ since given $$y ∈ \text{Col}(AB)$$ we can choose $$x ∈ F$$ and then we have $$y = (AB)x = A(Bx) ∈ \text{Col}(A)$$.
Since $$\text{Col}(AB) ⊆ \text{Col}(A)$$, any basis for $$\text{Col}(AB)$$ can be extended to a basis for $$\text{Col}(A)$$ and so $$\dim \text{Col}(AB) ≤ \dim \text{Col}(A)$$, that is $$\text{rk}(AB) ≤ \text{rk}(A).$$
Note that $$\text{Null}(B) ⊆ \text{Null}(AB)$$ since given $$x ∈ \text{Null}(B)$$ we have $$(AB)x = A(Bx) = A 0 = 0$$ so that $$x ∈ \text{Null}(AB)$$.
Since $$\text{Null}(B) ⊆ \text{Null}(AB)$$, as above we have $$\dim \text{Null}(B) ≤ \dim \text{Null}(AB)$$, that is, $$\text{Nullity}(B) ≤\text{Nullity}(AB)$$. Thus $$\text{rk}(AB) = n − \text{Nullity}(AB) ≤ n − \text{Nullity}(B) = \text{rk}(B).$$
• You should use MathJax to format your answer. Oct 15, 2015 at 21:06
• Very nice proof! +1
– ZFR
Apr 23, 2020 at 1:41
Prove first that if $f:X\to Y$ and $g:Y\to Z$ are functions between finite sets, then $|g(f(X))| \leq \min \{ |f(X)|, |g(Y)| \}.$
Then use the same idea.
• Categorification... :-) Jul 29, 2010 at 21:50
• I am not familiar with the 'categorification'. How can one go from this to the rank inequality ? What functor is to be applied ? Apr 13, 2014 at 12:10
• So you're saying that rank of linear map is somehow like image of a function? Feb 12, 2015 at 10:15
• @Mihail, it is like the size of the image of a function. in fact, if $f$ is a linear map, the rank of $f$ is the dimension of the image of $f$ and the dimension is a measure of the size of a space. Feb 12, 2015 at 18:22 | {
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Here is another simple answer. When you multiply a matrix and a vector $Ax$ you end up with a linear combination of the columns of $A$.
$$Ax = \; x_1\,A_1 \;+\; x_2\,A_2 \;+\; x_3\,A_3 \;+\;\; ...\;\; \\$$
When we multiply two matrices $AB = C$, we have $AB_i = C_i$, which means that each column of $C$ is a linear combination of the columns of $A$, so $\text{rank}(AB) \leq \text{rank}(A)$. To show that $\text{rank}(AB) \leq \text{rank}(B)$ we follow a similar argument -- when you multiply $x^{\top}B$, you end up with a linear combination of the rows of $B$.
Already you have proved $$rank(AB)\leq rank(B)$$
For other part
rank of $A=$ dim range $A$
As range $AB \subset$ range $A\implies$ dim range $AB\leq$ dim range $A$ . Hence $$rank(AB)\leq rank (A)$$
• Hello!! Why does it hold that "range AB $\subset$ range A" ? Jan 7, 2017 at 10:01
• If $x\in$ range $AB,$ then $x=(AB)y$ for some $y.$ Thus $x=A(By).$ Thus $x\in$ range $A.$@ Mary Star Jan 8, 2017 at 17:48
Let $m \le n, A \in M_{m\times n}, B\in M_{n\times m}$.
$\mbox{rank } A\le m$ and $\text{rank }B\le m$. (Obvious fact as rank A = dimension of the columnspace of A = dimension of the row space of A.)
Let $E_{n\times n}B$ be the row echelon form of $B$ and let $AE_{m\times m}$ be the column echelon form of $A$. ($E_{n\times n} ,E_{m\times m}$ are elementary matrices.)
We know $\operatorname{rank}(BA)=\operatorname{rank}(E_{n\times n}BA )=\operatorname{rank}(E_{n\times n}BAE_{m\times m} )$.
But $E_{n\times n}BAE_{m\times m} =\begin{pmatrix} L&0\\ 0&0\\ \end{pmatrix},$ where $L$ is an $k\times l$ matrix with $k\le \operatorname{rank}(B),l\le \operatorname{rank}(A)$.
So $\operatorname{rank}(E_{n\times n}BAE_{m\times m} )=\operatorname{rank}\begin{pmatrix} L&0\\ 0&0\\ \end{pmatrix}\le \min\{k,l\}\le \min\{\mbox{rank } A,\mbox{rank }B\}.$ | {
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Another way of showing that $\text{Rank} (AB) \leq \text{Rank}(B)$ without using rank-nullity: Note that if $v_1,\dots,v_n$ is a basis of $\text{Range} B$, then $Av_1,\dots,Av_n$ is a spanning list of $\text{Range} AB$.
Each column of $$AB$$ is a linear combination of columns of $$A$$ so $$\text{Range}(AB)\subseteq \text{Range} (A)$$ equivalently $$rank(AB)\leq rank(A).$$
Similarly, Each row of $$AB$$ is a linear combination of rows of $$B$$ so $$rowspace(AB)\subseteq rowspace(B)$$.
Since rowspace=columnspace. Thus $$rank(AB)\leq rank(B)$$.
From both of the inequality we deduce that $$rank(AB)\leq \min\{rank A, rank B\}$$. | {
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# Groups with “few” subgroups
If $G$ is a finite group of order $n,$ and the number of divisors of $n$ is $k,$ can $G$ have fewer than $k$ subgroups?
A cyclic group $G$ of order $n$ has exactly one subgroup for each divisor of $n$, so in this case $G$ has exactly $k$ subgroups.
For the alternating group $G=A_4,$ $n=12$ and $k=6.$ There is no subgroup of order $6$ in $A_4,$ yet there are ten subgroups, which exceeds $k=6$ for this case. I began to wonder if there could be a group for which so many divisors of the order of $G$ had no subgroups of that order, that there wound up being fewer subgroups in $G$ than the number of divisors of $|G|,$ hence my question.
• Recall that a CLT (Converse of Lagrange's Theorem) group is a group with the property that for any divisor of the order of the group there exists a subgroup of that order. Since it is known that finite abelian groups, as well as $p$-groups are CLT groups, this leaves a smaller class of subgroups to consider. – user1337 Apr 23 '15 at 13:16
• A finite group $G$ of order $n$ such that the number of elements $x$ with $x^d=1$ is less than $d$ for all divisors $d$ of $n$ has to be cyclic. I wonder, if a variation of this topic can lead to a proof that there are at least as many subgroups as divisors. – j.p. Apr 23 '15 at 13:24
• @j.p. That idea looks promising. One thing, should the phrase "is less than $d$" be replaced by "is at most $d$" in your comment? [though that would imply your version, I seem to recall one approach to doing the counting argument for showing some group is cyclic as involving the number being at most $d$] And +1 on the comment. – coffeemath Apr 23 '15 at 13:42
• A short script in Maple shows that there are no such groups among the 793 groups of order $< 2^5 \cdot 3 = 96$. – Travis Willse Apr 23 '15 at 13:43
• @coffeemath: Yes, you are right. It should have been "at most". – j.p. Apr 23 '15 at 14:04 | {
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The answer to your question is no, that is not possible. This is a consequence of the following slightly more general result. Let $d(n)$ is the number of divisors of $n$.
Theorem Let $G$ be a finite group of order $n$. Then for any divisor $m$ of $n$, there exist at least $d(m)$ subgroups of $G$ of order dividing $m$.
Proof We prove the statement by induction on $m$, for all finite groups $G$. It's clear for $m=1$.
For $m>1$, let $p$ be a prime dividing $m$, and let $m = p^r k$ with $\gcd(p,k)=1$.
Since $d(m) = (r+1)d(k)$, it is sufficient to prove that, for each $i$ with $0 \le i \le r$, there are at least $d(k)$ subgroups of $G$ of order $p^i j$, for some $j$ that divides $k$.
So fix an $i$, let $P$ be any subgroup of $G$ of order $p^i$, let $N = N_G(P)$ be the normalizer of $P$ in $G$, and let $h = \gcd(|N|,k)$.
Since $h \le k < m$ and $h$ divides $|N/P|$, by inductive hypothesis, the number $s$ of subgroups $S/P$ of $N/P$ of order dividing $h$ satisfies $s \ge d(h)$. For each of these subgroups, the inverse image $S$ of $S/P$ in $N/P$ is a subgroup of $G$ of order $p^i j$, where $|S/P| = j$ divides $h$, and so $j$ divides $k$.
Now $P$ has $|G|/|N|$ distinct conjugates $P^g$ in $G$, and the $s|G|/|N|$ subgroups $S^g$ are all distinct. But $|G|/|N| \ge k/h$, so $s|G|/|N| \ge d(h)k/h \ge d(k)$, which completes the proof.
After thinking about it again, I think we can say that a group of order $n$ with exactly $d(n)$ subgroups must be cyclic. The above proof produces at least $d(n)$ subgroups with a normal $p$-subgroup for any prime divisor $p$ of $n$. So if there were exactly $d(n)$ subgroups, then all subgroups would have all of their Sylow subgroups normal, so they would all, including $G$ itself, be nilpotent. But a non-cyclic $p$-group $P$ has more than one subgroup of index $p$, and hence has more than $d(|P|)$ subgroups, so all Sylow subgroups of $G$ are cyclic and hence so is $G$. | {
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By the way, I first posted this proof here in 2003. I would welcome any references to any other proofs.
• Derek-- Nice proof (seems to me). +1 on it for now, and likely I'll "accept" after I get a chance to go over it in detail. – coffeemath Apr 23 '15 at 14:46
• How do you know that $P$ exist ? – Belgi Apr 23 '15 at 14:57
• @Belgi $G$ has a $p$-Sylow subgroup $T$. Let's say $T$ is of size $p^k$; then $T$ will have a subgroup of order $p^i$ provided $i\leq k$. – mathmandan Apr 23 '15 at 15:06
• @mathmandan - How do you know such subgroups of $T$ exist ? – Belgi Apr 23 '15 at 15:10
• I was thinking of the proof as an induction on $m$ for all $n$ simultaneously. That is, when doing the proof for $m$, we assume that the result is true for all smaller $m$ and all $n$. We don't use induction on $n$ anywhere. I remember when thinking about this originally, the chief problem was to get an induction to work, and the key idea in the proof is actually to induct on $m$ rather than on $n$. – Derek Holt Apr 23 '15 at 17:06 | {
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# Generating Sets From Given Information
The problem I am working on is:
The three most popular options on a certain type of new car are a built-in GPS (A), a sunroof (B), and an automatic transmission (C). If 40% of all purchasers request A, 55% request B, 70% request C, 63% request A or B, 77% request A or C, 80% request B or C, and 85% request A or B or C, determine the probabilities of the following events. [Hint: “Aor B” is the event that at least one of the two options is requested; try drawing a Venn diagram and labeling all regions.]
a.The next purchaser will request at least one of the three options.
b.The next purchaser will select none of the three options.
c. The next purchaser will request only an automatic transmission and not either of the other two options.
d.The next purchaser will select exactly one of these three options.
I am absolutely positive that $P(A)=40\%$, $P(B)=55\%$, $P(C)=70\%$, and $P(A \cup B \cup C)=85\%$. However, the pieces of data I am not quite certain about are $P(A \cup B \cap C')=63\%$, $P(A \cup C \cap B')=77\%$, $P(B \cup C \cap A')=80\%$, do these values correspond to the rest of the data? If so, then I seems nearly impossible to be able to generate the Venn Diagram. Could someone help?
EDIT: What I am having a difficult time interpreting is, when they say in the question, "...63% request A or B." To me, that says only A or only B; and under this interpretation I would write $P(A \cup B \cap C')=63\%$. Under André Nicolas' interpretation, "63% request A or B," means $P(A \cup B)=63\%$. If it is the case that André Nicolas is correct, then it seems like they should have stated in the question, "63% request A or B, A and C, B and C, or A and B and C."
Also, I solved the problem under André Nicolas' assumption, and for part d), I know the answer but I am sure how to put in it math symbols. How would I do that?
-
Here is a start:
a) The answer to this is written down in the data: It is $\Pr(A\cup B\cup C)$. | {
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-
Here is a start:
a) The answer to this is written down in the data: It is $\Pr(A\cup B\cup C)$.
b) This can be immediately written down from the answer to a).
As for the rest of the probabilities, we need to work. It can be figured out completely from a Venn diagram. But since one cannot point to a picture on a blackboard, what follows will be formula-heavy.
Recall that $$\Pr(X\cup Y)=\Pr(X)+\Pr(Y)-\Pr(X\cap Y).\tag{1}$$
We know $\Pr(A)$, $\Pr(B)$, and $\Pr(A\cup B)$. Using Formula $(1)$, we can see that $0.63=0.40+0.55-\Pr(A\cap B)$. That gives $\Pr(A\cap B)=0.32$.
Similarly, we can find $\Pr(A\cap C)$ and $\Pr(B\cap C)$ from the given information.
Recall also that
$$\Pr(A\cup B\cup C)=\Pr(A)+\Pr(B)+\Pr(C)-\Pr(A\cap B)-\Pr(A\cap C)-\Pr(B\cap C)+\Pr(A\cap B\cap C).$$
We are told $\Pr(A\cup B\cup C)$, and we computed the next three items in the above formula, so now we know $\Pr(A\cap B\cap C)$.
OK, time to fill in the bits in the Venn Diagram. First write the computed $\Pr(A\cap B\cap C)$ in the space where it should go.
We know $\Pr(A\cap B)$. Since we also know $\Pr(A\cap B\cap C)$, we can find the probability of "the rest of" $A\cap B)$, called $A\cap B\cap C'$ in set language. Compute, write the number in.
Do the same for the other similar bits, namely $A\cap C\cap B'$ and $B\cap C\cap A'$.
Now from $\Pr(A)$ you can find the probability of the "outside" part of $A$, that is, $\Pr(A\cap B'\cap C')$. Do the same for the "outside" part of $B$, also the "outside" part of $C$.
The probability of the outer world $A'\cap B'\cap C'$ of cheapskates who want nothing can now be found in various ways.
We now know everything, and can answer any question! | {
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We now know everything, and can answer any question!
Remark: You can be more fancy, and let $X$, $Y$, and $Z$ respectively be $A'$, $B'$, and $C'$. By taking every one of the given numbers and subtracting it from $1$, you can write down immediately $\Pr(X)$ (namely $1-0.40$), $\Pr(Y)$, $\Pr(Z)$, $\Pr(X\cap Y)$, and so on. Now we end up with a problem of a type that you have probably done several times. | {
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-
So, in the question, when it said something like, "63% request A or B," you assumed that two be $P(A \cup B)=63\%$? Why isn't it really $P(A \cup B \cap C')=63\%$? If you take, "63% request A or B," to mean the former, why wouldn't you include something about C? Because when you write the Venn diagram, A and B both intersect C. – Mack Feb 1 '13 at 14:56
The reason I ask is because $P(A \cup B)$ also includes the probability of getting $A$ and $C$ and $B$ and $C$, and it just seems odd that you wouldn't mention something like that in the question. – Mack Feb 1 '13 at 15:17
@EliMackenzie: The problem setter should make things clear. But I am reasonably sure that the intent here is that "or" be interpreted as $\lor$, logical or, meaning in this case union. There is a somewhat similar potential issue with "and" in standard Venn diagram work problems. When I say $A$ and $B$, does that imply anything about $C$, $D$? No, and you are accustomed to that. – André Nicolas Feb 1 '13 at 16:49
One can also go through the Venn diagram, trying to see whether your interpretation is numerically consistent with the data. The probabilities of $A\lor B$, $B\lor C$, $A\lor C$ are quite large, one can't find a Venn diagram that works. Some of the percentages of "subareas" would turn out negative. – André Nicolas Feb 1 '13 at 16:56
Also, if you look at the text of the problem, it says explicitly as a hint what is meant by $A$ or $B$. That is neutral about $C$. It looks as if the problem setter was worried people might have trouble with the meaning of or. – André Nicolas Feb 1 '13 at 17:03 | {
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# legendreP
Legendre polynomials
## Description
example
legendreP(n,x) returns the nth degree Legendre polynomial at x.
## Examples
### Find Legendre Polynomials for Numeric and Symbolic Inputs
Find the Legendre polynomial of degree 3 at 5.6.
legendreP(3,5.6)
ans =
430.6400
Find the Legendre polynomial of degree 2 at x.
syms x
legendreP(2,x)
ans =
(3*x^2)/2 - 1/2
If you do not specify a numerical value for the degree n, the legendreP function cannot find the explicit form of the polynomial and returns the function call.
syms n
legendreP(n,x)
ans =
legendreP(n, x)
### Find Legendre Polynomial with Vector and Matrix Inputs
Find the Legendre polynomials of degrees 1 and 2 by setting n = [1 2].
syms x
legendreP([1 2],x)
ans =
[ x, (3*x^2)/2 - 1/2]
legendreP acts element-wise on n to return a vector with two elements.
If multiple inputs are specified as a vector, matrix, or multidimensional array, the inputs must be the same size. Find the Legendre polynomials where input arguments n and x are matrices.
n = [2 3; 1 2];
xM = [x^2 11/7; -3.2 -x];
legendreP(n,xM)
ans =
[ (3*x^4)/2 - 1/2, 2519/343]
[ -16/5, (3*x^2)/2 - 1/2]
legendreP acts element-wise on n and x to return a matrix of the same size as n and x.
### Differentiate and Find Limits of Legendre Polynomials
Use limit to find the limit of a Legendre polynomial of degree 3 as x tends to -∞.
syms x
expr = legendreP(4,x);
limit(expr,x,-Inf)
ans =
Inf
Use diff to find the third derivative of the Legendre polynomial of degree 5.
syms n
expr = legendreP(5,x);
diff(expr,x,3)
ans =
(945*x^2)/2 - 105/2
### Find Taylor Series Expansion of Legendre Polynomial
Use taylor to find the Taylor series expansion of the Legendre polynomial of degree 2 at x = 0.
syms x
expr = legendreP(2,x);
taylor(expr,x)
ans =
(3*x^2)/2 - 1/2
### Plot Legendre Polynomials
Plot Legendre polynomials of orders 1 through 4.
syms x y
fplot(legendreP(1:4, x))
axis([-1.5 1.5 -1 1])
grid on | {
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syms x y
fplot(legendreP(1:4, x))
axis([-1.5 1.5 -1 1])
grid on
ylabel('P_n(x)')
title('Legendre polynomials of degrees 1 through 4')
legend('1','2','3','4','Location','best')
### Find Roots of Legendre Polynomial
Use vpasolve to find the roots of the Legendre polynomial of degree 7.
syms x
roots = vpasolve(legendreP(7,x) == 0)
roots =
-0.94910791234275852452618968404785
-0.74153118559939443986386477328079
-0.40584515137739716690660641207696
0
0.40584515137739716690660641207696
0.74153118559939443986386477328079
0.94910791234275852452618968404785
## Input Arguments
collapse all
Degree of polynomial, specified as a nonnegative number, vector, matrix, multidimensional array, or a symbolic number, vector, matrix, function, or multidimensional array. All elements of nonscalar inputs should be nonnegative integers or symbols.
Input, specified as a number, vector, matrix, multidimensional array, or a symbolic number, vector, matrix, function, or multidimensional array.
collapse all
### Legendre Polynomial
• The Legendre polynomials are defined as
$P\left(n,x\right)=\frac{1}{{2}^{n}n!}\frac{{d}^{n}}{d{x}^{n}}{\left({x}^{2}-1\right)}^{n}.$
• The Legendre polynomials satisfy the recursion formula
$\begin{array}{l}P\left(n,x\right)=\frac{2n-1}{n}xP\left(n-1,x\right)-\frac{n-1}{n}P\left(n-2,x\right),\\ \text{where}\\ P\left(0,x\right)=1\\ P\left(1,x\right)=x.\end{array}$
• The Legendre polynomials are orthogonal on the interval [-1,1] with respect to the weight function w(x) = 1, where
• The relation with Gegenbauer polynomials G(n,a,x) is
$P\left(n,x\right)=G\left(n,\frac{1}{2},x\right).$
• The relation with Jacobi polynomials P(n,a,b,x) is
$P\left(n,x\right)=P\left(n,0,0,x\right).$ | {
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# Meaning of a Subspace
## Main Question or Discussion Point
Hi PF!
I want to make sure I understand the notion of a subspace. Our professor gave an example of one: the set of degree n polynomials is a subspace of continuous functions. This is because a) a polynomial is intrinsically a subset of continuous functions and b) summing any polynomials yields another polynomial (closed under addition) and c) any polynomial multiplied by a constant is still a polynomial (closed under scalar multiplication).
If my reasoning is correct, then I think the set of discontinuous functions is NOT a subset of all real functions. To see why, consider $f(x) = 1$ everywhere except non-existent at $x=1$. Then take the function $g(x)=1$ when $x=1$ and non-existent everywhere else. Then the sum is clearly continuous. (How would this change though if I changed non-existent to zero? I know the result is the same, but is $g(x)$ as I have defined it even discontinuous--I realize this is a real analysis question.)
Related Linear and Abstract Algebra News on Phys.org
PeroK
Homework Helper
Gold Member
Hi PF!
I want to make sure I understand the notion of a subspace. Our professor gave an example of one: the set of degree n polynomials is a subspace of continuous functions. This is because a) a polynomial is intrinsically a subset of continuous functions and b) summing any polynomials yields another polynomial (closed under addition) and c) any polynomial multiplied by a constant is still a polynomial (closed under scalar multiplication). | {
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If my reasoning is correct, then I think the set of discontinuous functions is NOT a subset of all real functions. To see why, consider $f(x) = 1$ everywhere except non-existent at $x=1$. Then take the function $g(x)=1$ when $x=1$ and non-existent everywhere else. Then the sum is clearly continuous. (How would this change though if I changed non-existent to zero? I know the result is the same, but is $g(x)$ as I have defined it even discontinuous--I realize this is a real analysis question.)
You can't choose functions that are not defined at some point, as functions that have different domains can't be added so they don't form a vector space. The space of all real-valued functions would imply that they all all defined on some fixed domain.
Also, functions that are not defined at a point are not necessarily discontinuous. They may be continuous on their domain. A good example is the function $1/x$, which is a continuous function on its domain.
Instead, you need to think of two genuinely discontinuous functions that can be added to form a contiuous function.
Or, perhaps you could think of a simpler example using a subset of the polynomials that does not form a subspace?
S.G. Janssens
I would say: "The set of polynomials of degree at most $n$". (Here you regard the zero polynomial as having degree $-\infty$ or you should stipulate that this set includes the zero polynomial.) | {
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# How to find the partial sum of a given series?
On my last exam there was the question if the series $\sum_{n=2}^{\infty}\frac{1}{(n-1)n(n+1)}$ converges and which limit it has. During the exam and until now, I am not able to solve it. I tried partial fraction decomposition, telescoping sum, etc. But I am not able to find the partial sum formula (Wolfram|Alpha):
$$\sum_{n=2}^{m}\frac{1}{(n-1)n(n+1)} = \frac{m^2+m-2}{4m(m+1)}.$$
Could somebody push me in the right direction? Is there any trick or scheme how to find partial sum formulas for given series?
-
Each term is less than $1/n^2$, so it converges. You don't need to know what a series converges to to know that it converges. – Thomas Andrews Mar 25 '12 at 14:46
Two things to observe(Unrelated to the Math): Please do not sign your posts with signature as faq explicitly lists it! Accepting answers is a sign of appreciation for someone who put in effort compiling an answer for you. Please accept answers which you think helped you a lot in solving that problem or cleared up your concepts and whatever. It is done by clicking on the tick mark besides every answer. – user21436 Mar 25 '12 at 14:51
The ratio test with $1/n^3$ proves it converges. As often happens, finding the sum takes more work than that. – Michael Hardy Mar 25 '12 at 16:29
martini's answer is the right thing to do. Just wanted to add that if you know the answer (given by wolfram) you can just prove the it with recursion arguments – Thomas Dec 15 '13 at 15:11 | {
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So let's try partial fraction decomposition. Writing $$\frac 1{(n-1)n(n+1)} = \frac a{n-1} + \frac bn + \frac c{n+1}$$ we obtain $$1 = a(n^2 + n) + b(n^2 - 1) + c(n^2 - n)$$ and therefore \begin{align*} 1 &= -b\\ 0 &= a - c\\ 0 &= a + b + c. \end{align*} This gives $b = -1$, $a = c = \frac 12$. Hence \begin{align*} \sum_{n=2}^m \frac 1{(n-1)n(n+1)} &= \sum_{n =2}^m \frac 1{2(n-1)} - \sum_{n=2}^m \frac 1n + \sum_{n=2}^m \frac 1{2(n+1)}\\ &= \frac 12 + \sum_{n=2}^{m-1} \frac 1{2n} - \sum_{n=2}^m \frac 1n + \sum_{n=3}^m \frac 1{2n} + \frac 1{2(m+1)}\\ &= \frac 12 + \frac 14 - \frac 12 - \frac 1m + \frac 1{2m} + \frac 1{2m+2}\\ &= \frac 14 + \frac{-2(m+1) + m+1 + m}{2m(m+1)}\\ &= \frac 14 + \frac{-1}{2m(m+1)}\\ &= \frac{m(m+1) - 2}{4m(m+1)}. \end{align*}
I got little confuse here, why the term $\frac{1}{4}$ doesnt vanish? In my calculation the third step from below, I got $$=\frac{1}{2}+ (\frac{1}{4}+ \frac{1}{6}+\dots+ \frac{1}{2(m-1)})-( \frac{1}{2}+ \frac{1}{3}+\dots+ \frac{1}{m})+( \frac{1}{6}+ \frac{1}{8}+\dots+ \frac{1}{2m})+ \frac{1}{2(m+1)} = \\ =-( \frac{1}{3}+ \frac{1}{5}+\dots+ \frac{1}{m})+(\frac{1}{6}+\frac{1}{8}+\dots+ \frac{1}{2m})+\frac{1}{2(m+1)}$$ How can I get to the next step? – DadangAH Feb 20 '14 at 5:41
What did you try for a telescoping sum? Your denominator here is the product of three successive terms (this is often called a rising or falling factorial, depending on which side you take as your baseline); this points to looking at a difference of terms that are of the same form but with denominators one degree less. In particular, looking at $t_n=\dfrac{1}{n(n+1)}$ then $t_n-t_{n-1}$ $=\dfrac{1}{n(n+1)}-\dfrac{1}{(n-1)n}$ $=\dfrac1n\left(\dfrac1{n+1}-\dfrac1{n-1}\right)$ $=\dfrac1n\left(\dfrac{(n-1)-(n+1)}{(n-1)(n+1)}\right)$ $=\dfrac{-2}{(n-1)n(n+1)}$; in other words, $\dfrac{1}{(n-1)n(n+1)} = -\dfrac12(t_n-t_{n-1})$, and from here the telescopy should be fairly clear. | {
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Precise proof of S Open set iff contains only interior points - Complex Analysis
In "Complex Variables and Applications" by Brown and Churchill (McGraw-Hill) the proof of: $$\textit{S is an open set \implies each of its points is an interior point}$$ is left as an exercise. Here are the important definitions that one can use: $$\textbf{\epsilon Neighbourhood}: |z-z_0|<\epsilon$$ $$\textbf{z_0 Interior point}: \exists\,\,\ \text{an}\,\,\, \epsilon\text{-neighbourhood containing only}\, z\in S$$ $$\textbf{z_0 Exterior point}: \exists\,\,\ \text{an}\,\,\, \epsilon\text{-neighbourhood containing no}\, z\in S$$ $$\textbf{z_0 Boundary point}: \text{all neighbourhood of z_0 have at least a point in S and one not in S}$$ $$\textbf{S Open}: \text{does not contain any of its boundary points}$$
I've tried to prove this in the following way, however I got stuck. Can someone tell me how to make it mathematically rigorous, using the above definitions?
$S$ is an Open Set $\implies$ $S$ does not contain any any of its boundary points. So $$\text{if} \,\, [\forall \epsilon>0, (\exists z_1\in S\,\,\ \text{and}\,\,\ \exists z_2\notin S ):(|z_1-z_0|<\epsilon \,\,\text{and} \,\,\ |z_2-z_0|<\epsilon)] \implies z_0\notin S$$ Hence taking the converse of the above we have $$\text{if}\,\, z_0\in S \implies \exists \epsilon>0, (\forall z_1\in S\,\,\ \text{or}\,\,\ \forall z_2\notin S ):|z_1-z_0|\geq \epsilon \,\,\text{or} \,\,\ |z_2-z_0|\geq \epsilon)$$
Which doesn't really tell me much. I am really not sure I've even negated it correctly or that my definition with the quantifiers is correct. | {
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• This are the exact words in the book "A point $z_0$ is said to be an interior point of a set $S$ whenever there is some neighborhood of $z_0$ that contains only points of $S$". Is it different from what I wrote? Or is it different from what you wrote? They look pretty similar to me, please tell me if there's some gap between them – Euler_Salter Dec 3 '16 at 18:48
• Forgive me, I misunderstood you definition. However it is a bit confusing because it seems that you are referring to a specific point $z$. Specifically, if I say that $A$ contains only $z\in S$ what I immediately think is that $A=\{z\}$, whatever $z$ is, while you only meant $A\subseteq S$. – Caligula Dec 3 '16 at 18:54
• I removed the "logic" tag - that's for questions within the specific field of mathematical logic. – Noah Schweber Dec 3 '16 at 19:00
Using your definitions (of which some are not very well stated), we can proceed this way.
You want to prove that each point $z$ of an open set $S$ is an interior point. Now, since $S$ is open, $z$ can't be a boundary point, so negating the definition of boundary point, it follows that there exists a neighborhood $U$ of $z$ such that two things can happen:
• or $U$ contains only point of $S$;
• or $U$ contains only point outside $S$.
Since $U$ intersect $S$ at least in $z$, the latter is impossible, so you have proved that $U\subseteq S$. This means that $z$ is interior in $S$.
Following the comment, I'll try to rephrase the proof in more logically strict terms - but note that using words instead of quantifiers is not something to despise, as it increases greatly the clearity of proofs.
The fact $S$ is open means that
$\forall\,z\in S, z\notin \partial S$
where $\partial S$ is the boundary. Now, $z\in \partial S$ if and only if
$\forall \,U\in\mathscr{U}_z,\,U\cap S\neq \varnothing \text{ and } U\cap (S^c)\neq \varnothing$ | {
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$\forall \,U\in\mathscr{U}_z,\,U\cap S\neq \varnothing \text{ and } U\cap (S^c)\neq \varnothing$
where $(-)^c$ is the complement and $\mathscr{U}_z$ denotes the set of all open neighborhoods for $z$ (it has the structure of a filter if you know what it means). Hence, negating the latter will imply that
$\exists \,U\in \mathscr{U}_z$ such that $U\cap S = \varnothing$ or $U\cap (S^c)=\varnothing$
Then you conclude as before: the only possible conclusion is that $U\cap (S^c)=\varnothing$ and that is equivalent to $U\subseteq S$.
• I like your wordy proof, I'm quite a fan of these, as in most of my homework I exactly do them. However I wanted to train myself in proving things in a "tedious" way, i.e. using a logic definition and using either negations of it or contrapositives to get to the statement we want to prove. Would you know how I could rephrase those definitions or actually going from there to the wanted result? Thanks – Euler_Salter Dec 3 '16 at 18:58
• I edited my answer with some more logic inside it. – Caligula Dec 3 '16 at 22:12
• Thank you! It makes sense! It's a nice proof, based a lot on sets! I have no idea what a filter is in mathematics, however the set $U_z$ makes sense – Euler_Salter Dec 4 '16 at 2:13 | {
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1. ## Area
Find c>0 such that the area of the region enclosed by the parabolas y = x^2 - c^2 and y = c^2 -x^2 is 230.
c =?
2. Originally Posted by qbkr21
Find c>0 such that the area of the region enclosed by the parabolas y = x^2 - c^2 and y = c^2 -x^2 is 230.
c =?
you are integrating with respect to x, so you cannot integrate the c terms like that. you would just attach x to them. by the way, when i did it your way i got 0 = 230, which means you probably made another mistake somewhere
3. ## Re:
Could you trying working it out? or at least some of it... so that I can get a feel for what you mean...
Sure appreciate it....
-qbkr21
4. Originally Posted by qbkr21
Could you trying working it out? or at least some of it... so that I can get a feel for what you mean...
Sure appreciate it....
-qbkr21
$\int_{-c}^{c}\left[ c^2 - x^2 - \left( x^2 - c^2 \right) \right]dx$
$=2 \int_{0}^{c} \left( 2c^2 - 2x^2 \right) dx$
$= 4 \int_{0}^{c} \left( c^2 - x^2 \right)dx$
$= 4 \left[ c^2 x - \frac {1}{3}x^3 \right]_{0}^{c}$
$= \frac {8}{3}c^3$
check my computation, i was in a rush. i have to leave for a little while
did you see the difference with how i integrated c? i treated it as a constant--which it is
5. ## Re:
Yes you were right on the money as always!!
c = 4.41828
Thanks so much!!!
-qbkr21
6. ## find c
Originally Posted by Jhevon
$\int_{-c}^{c}\left[ c^2 - x^2 - \left( x^2 - c^2 \right) \right]dx$
$=2 \int_{0}^{c} \left( 2c^2 - 2x^2 \right) dx$
$= 4 \int_{0}^{c} \left( c^2 - x^2 \right)dx$
$= 4 \left[ c^2 x - \frac {1}{3}x^3 \right]_{0}^{c}$
$= \frac {8}{3}c^3$
check my computation, i was in a rush. i have to leave for a little while
did you see the difference with how i integrated c? i treated it as a constant--which it is
I think you are right | {
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7. Originally Posted by curvature
I think you are right
i know, qbkr21 already confirmed it. i believe he has the answer to the question in the back of his text. i hope he gets why i could have integrated between 0 and c though and then multiply by two, furthermore, i hope he gets why i would want to do that in the first place
8. Originally Posted by Jhevon
$\int_{-c}^{c}\left[ c^2 - x^2 - \left( x^2 - c^2 \right) \right]dx$
$=2 \int_{0}^{c} \left( 2c^2 - 2x^2 \right) dx$
hi
what did you do to the 2 outside to make the lower integral -c 0?
can it be any constants?
hi
what did you do to the 2 outside to make the lower integral -c 0?
can it be any constants?
Note that your integrand is an even function: f(-x) = f(x). In this case
$\int_{-c}^c dx \, f(x) = \int_{-c}^0 dx \, f(x) + \int_0^c dx \, f(x)$
Use the substitution y = -x in the first integral:
$= \int_{c}^0 (-dy) \, f(-y) + \int_0^c dx \, f(x)$
$= -\left ( -\int_0^c dy \, f(-y) \right ) + \int_0^c dx \, f(x)$
But f(-y) = f(y) when f is an even function:
$= \int_0^c dy \, f(y) + \int_0^c dx \, f(x)$
and now just replace the dummy variable y in the first integration with x:
$= \int_0^c dx \, f(x) + \int_0^c dx \, f(x)$
$= 2 \int_0^c dx \, f(x)$
-Dan | {
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If the circle has radius $$r$$ then the circumference is $$2\pi r$$. It gives the ratio of circumference to the diameter of a circle or the ratio of circumference to twice the radius. At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students! This section of Revision Maths defines many terms in relation to circles, including: Circumference, Diameter, Radius, Chord, Segment, Tangent, Point of contact, Arc, Angles on major and minor arcs, Angle of Centre and Sectors. Meaning of circumference. There is an error: “Area of the segment = ( θ /360) x π r 2 + ( 1 /2) x sinθ x r 2” …should be “Area of the segment = ( θ /360) x π r 2 – ( 1 /2) x sinθ x r 2” sivaalluri (January 22, 2019 - 3:31 pm) Reply. More About Circumference. 3.14 x 100 yards = 314 yards ; Circumference = 314 yards ; Your walk around the lake is 314 yards. Hollow Cylinder Calculator . I consulted a standard dictionary, which said it was the length of a closed curve enclosing an area, so it would seem to apply under that definition. 9 thoughts on “ Circle formulas in math | Area, Circumference, Sector, Chord, Arc of Circle ” Steve LeVine (December 31, 2018 - 10:05 pm) Reply. Capsule Calculator . For example, the Latin root of the word circumference is circum, meaning around. The circumference of a circle is the distance around the circle. Related Calculators: Helix Curve Calculator . Definition: Pi is a number - approximately 3.142 It is the circumference of any circle divided by its diameter. Circumference is the linear distance around a circle. If you're seeing this message, it means we're having trouble loading external resources on our website. Also learn the facts to easily understand math glossary with fun math worksheet online at SplashLearn. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. The boundary line of a circle. We will be listening the word “perimeter” frequently in the | {
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The boundary line of a circle. We will be listening the word “perimeter” frequently in the problems which are based on geometry. The circumference of a circle is the distance around the circle. Learn about terms radius, diameter, circumference, chord, pi, etc.. ... what is the official definition of a circle? What does circumference mean? A circle has many different radii and many different diameters, each passing through the center. Download FREE Right circumference of a Circle Worksheets where, R is the radius of the circle. All Free. Circumference, diameter and radii are measured in linear units, such as inches and centimeters. Perimeter is the distance around a two-dimensional object. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Learn more. By Mary Jane Sterling . The circumference of a circle of radius 3 is $$2\cdot\pi\cdot 3$$, which is $$6\pi$$, or about 18.85. View our Lesson on Circumference of a Circle. The circumference of a circle is the distance around it, but if, as in many elementary treatments, distance is defined in terms of straight lines, this cannot be used as a definition. Radius definition, a straight line extending from the center of a circle or sphere to the circumference or surface: The radius of a circle is half the diameter. The complete distance around a circle or a closed curve is called its Circumference. Math Goodies Glossary. What is circumference? Learn and know what is the perimeter meaning in geometry chapter. See more. These are the two important words for which we have to know the meanings in geometry. Below are our grade 5 geometry worksheets on determining the circumference of circles.Students are provided the radius or the diameter in customary units (worksheets 1-3) or metric units (worksheets 4-6). The Circumference (or) perimeter of a circle = 2πR. Circumference: The circumference of a circle is the distance around it. If you imagine the circle as a piece | {
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The circumference of a circle is the distance around it. If you imagine the circle as a piece of string, it is the length of the string. * By signing up, you agree to receive useful information and to our privacy policy. Definition: Circumference. What does that mean? Top Calculators. Formally, a circumference is defined as the locus of points from the plane equidistant to another point, called the center of the circumference. Circumference to diameter. You can also measure the perimeter of three-dimensional objects like houses, stadiums, buildings and similar shapes. It includes unlimited math lessons on number counting, addition, subtraction etc. The distance around a circle is called the circumference. Courses. Math Made Easy! The length of such a line. Also find the definition and meaning for various math words from this math dictionary. Now you will be able to easily solve problems on circumference definition, circumference geometry definition, c ircumference of a circle formula, circumference equation, and circumference of circle. Definition of circumference in the Definitions.net dictionary. c or circ. Math; Trigonometry; Radius, Diameter, Circumference, and Area of Circles; Radius, Diameter, Circumference, and Area of Circles. What is a circle? the length of a perimeter. While I would never actually refer to the circumference of a circle as its "perimeter," a discussion recently arose about whether the term "perimeter" can even APPLY to a circle. Learn area of different shapes here. Circle Geometry math for kids. Jan 31, 2013 - go to this link for a math dictionary tool to use at school or at home (Circumference - the distance around the edge of a circle) A real-life example of a radius is the spoke of a bicycle wheel. Learn the relationship between the radius, diameter, and circumference of a circle. Worksheets > Math > Grade 5 > Geometry > Circumference of circles. Along with the perimeter, we should also know the meaning of “Area”. M Another name for | {
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of circles. Along with the perimeter, we should also know the meaning of “Area”. M Another name for circumference is perimeter. Again, Pi (π) is a special mathematical constant; it is the ratio of circumference to diameter of any circle. Circumference definition, the outer boundary, especially of a circular area; perimeter: the circumference of a circle. b. Abbr. Definition Of Circumference. perimeter: [noun] the boundary of a closed plane figure. Related Topics: Math Worksheets Objective: I know how to calculate the circumference of a circle. For any given circle, this ratio will remain π. A circle is a geometric figure that needs only two parts to identify it and classify it: its center (or middle) and its radius (the distance from the center to any point on the circle). The margin or area surrounding something. circumference definition: 1. the line surrounding a circular space, or the length of this line: 2. the outside edge of an…. circumference - WordReference English dictionary, questions, discussion and forums. E-Mail Address * Featured … Search for courses, skills, and videos. Learn what is circumference. Area and Perimeter is an important topic in Mathematics, which is used in everyday life. If you're seeing this message, it means we're having trouble loading external resources on our website. Learn the relationship between the radius, diameter, and circumference of a circle. Search. Circumference. 2. a. How to use circumference in a sentence. Did You Know? Circumference definition is - the perimeter of a circle. The boundary line of an area or object. ence (sər-kŭm′fər-əns) n. 1. Circumference may also refer to the circle itself, that is, the locus corresponding to the edge of a disk Circle. A circle is a shape that is made up of all the points on a plane (a flat surface) that are the same distance from a given point. Perimeter defines the distance of the boundary of the shape whereas area explains the region occupied by it. Area and perimeter, in Maths, | {
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boundary of the shape whereas area explains the region occupied by it. Area and perimeter, in Maths, are the two important properties of two-dimensional figures. Circum is now considered a prefix also meaning around or round about. 3. The formula for the circumference of a circle, C = 2 × π × r = π × d, where r is the radius and d is the diameter of the circle.π =22/7 = 3.141. Donate Login Sign up. A circle is a two-dimensional shape that is created by drawing a line that is curved so that its ends meet and every point on the line is the same distance from the center. π is the mathematical constant with an approximate (up to two decimal points) value of 3.14. Search . The number Pi, denoted by the Greek letter π - pronounced 'pie', is one of the most common constants in all of mathematics. Learn More at mathantics.comVisit http://www.mathantics.com for more Free math videos and additional subscription based content! Search form. where C = π D. C is the circumference of the circle Sign Up For Our FREE Newsletter! Definition of Perimeter explained with real life illustrated examples. Circumference = 3.14 x 100 yards ; Do the Math. It is the circumference of any circle, divided by its diameter. Make your child a Math Thinker, the Cuemath way. Sign Up For Our FREE Newsletter! It is a constant so whatever be the circumference and the radius the ratio will be equal to π. π is an irrational number hence it cannot be written down as a finite decimal number. Tip. You have probably noticed that, since diameter is twice the radius, the proportion between the circumference and the diameter is equal to π: C/D = 2πR / 2R = π. Okay, so let's start out with the "given point". See more. SplashLearn is an award winning math learning program used by more than 30 Million kids for fun math practice. The formula is πd or 2πr. Sal uses formulas and a specific example to see how area and circumference relate. Learn the circumference of a circle with Definition, Solved examples, and Facts. | {
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relate. Learn the circumference of a circle with Definition, Solved examples, and Facts. Circumference: The distance around a circle; the perimeter of a circle. Read the lesson on circumference of circle if you need to learn how to calculate the circumference of a circle. The following is a list of definitions relating to the circumference of a circle. About Cuemath. Information and translations of circumference in the most comprehensive dictionary definitions resource on the web. Circle worksheets: Finding the circumference. It is used in many areas, such as physics and mathematics. If you have trouble remembering geometry terms, it helps to think of other words from the same root with which you may be more familiar. We must never confuse the concept of a circle with the concept of circumference, circumference is actually a curve that encloses a circle (the circumference is a curve, the circle is an area.) This proportion (circumference to diameter) is the definition of the constant pi. Main content. Perimeter Definition; Perimeter of a Triangle; Perimeter of a Square or Rhombus; Perimeter of a Rectangle or Parallelogram; Perimeter of an N-gon; Perimeter Definition. Calculators and Converters ↳ Math Dictionary ↳ C ↳ Circumference ; Ask a Question . Home » Glossary » Term » Math Goodies Glossary. | {
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# Math Help - Quick question about position due to gravity.
1. ## Quick question about position due to gravity.
I have a problem where I'm given a bunch of info. about a ballon dropping a bomb, etc. The thing is that I am not given any functions but I know that I have to use the function for position of falling bodies with respect to time.
It's been a while since I saw this formula, so i'm not sure if this is right:
In feet: $s(t)=-16t^2 + v_0 t + s_0$
where $v_0$ is initial velocity and $s_0$ is starting position.
Is any of this right?
2. I'm not sure what you are trying to do.
You have a bomb floating to the earth on a balloon, and you have to construct the equation which describes the displacement the balloon as fallen from its original position after time t?
3. Originally Posted by Arturo_026
I have a problem where I'm given a bunch of info. about a ballon dropping a bomb, etc. The thing is that I am not given any functions but I know that I have to use the function for position of falling bodies with respect to time.
It's been a while since I saw this formula, so i'm not sure if this is right:
In feet: $s(t)=-16t^2 + v_0 t + s_0$
where $v_0$ is initial velocity and $s_0$ is starting position.
Is any of this right?
None of what you're asking can be answered without knowing the question. It's always better to post the original question instead of hoping that our crystal balls are in good working order.
4. Try one (or more) of the kinematic equations:
$v = v_0 + at$
$y - y_0 = v_0t + \frac{1}{2}t^2$
$v^2 = v_0^2 + 2a(y - y_0)$
$y - y_0 = \frac{1}{2}(v_0 + v)t$
Where v is the final velocity, $v_0$ is the initial velocity, y is the final height, $y_0$ is the initial height, a is the acceleration due to gravity near Earth's surface ( $g = 9.8 \frac{m}{s^2}$), and t is the time.
Patrick | {
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Patrick
5. Originally Posted by mr fantastic
None of what you're asking can be answered without knowing the question. It's always better to post the original question instead of hoping that our crystal balls are in good working order.
Ok, here is the problem:
A bomb is dropped from a ballon hovering at an altitude of 800 ft. From directly below the balloon, a projectile is fired directly upward toward the bomb exactly 2 seconds after the bomb is released. With what initial speed should the projectile be fired in order to hit the bomb at an altitude of exactly 400 ft.?
6. The problem is for some reason in the Standard system (feet) instead of the Metric (meters), so be sure to consider that when solving the problem (This is where the -16 comes from in your original equation).
The bomb accelerates downward at a rate of 9.8 $\frac{m}{s^2}$ or just over 32 $\frac{ft}{s^2}$. After t seconds, it will be halfway down, from 800 ft to 400ft. At this precise moment, the projectile from the surface will intercept it. Therefore, you need to solve for t, the time it takes the bomb to drop 400 ft, from an initial velocity of zero.
Then, when you have t, you must subtract 2 seconds to get the time for the projectile's flight from the surface to the bomb, over a distance of 400 ft. When you have this time, you can then use the kinematic equations to solve for the initial velocity, knowing that the acceleration due to gravity will slow the projectile so that the final velocity is less than the initial velocity.
You will have the variables of t, a, and y. Using the second equation listed (a is not needed), you can solve for the initial velocity. Let me know what the answer is.
Patrick
7. I get t=5 for the bomb to be at an altitude of 400 ft. and when t=3 the projectile needs to be launched with an initial velocity of 181.33 ft. per second. | {
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# Two masses connected by a spring
I have a question about a problem I saw on a website
A mass is attached to one end of a spring, and the other end of the spring is attached to an immovable wall. The system oscillates with period T. If the wall is removed and replaced with a second mass identical to the first, what is the new period of oscillation of the system?
My answer: $$\frac{T}{\sqrt{2}}$$
Website's answer (Incorrect thanks to trula): $$T\sqrt{2}$$
My thought process (pretty hand-wavy I know):
Because the spring constant is inversely proportional (I think) to length of the spring, the spring constant will be doubled if you halve the length of the spring, which is what is essentially happening with two identical massess. Because $$T=2\pi\sqrt{\frac{m}{k}}$$, if you double $$k$$, you divide $$T$$ by $$\sqrt{2}$$. I don't know why they got $$T\sqrt{2}$$, but if I am wrong, please correct my misunderstandings.
Thanks for reading if you made it this far, and if my question is unclear, please tell me what I can fix about it.
• Your argument is right, and the published answer probably just a typing error. Jul 21 at 21:17
• Ok, thanks for the clarification!
– JD12
Jul 21 at 21:24
• the spring constant will be doubled if you halve the length of the spring, which is what is essentially happening with two identical massess. I really don't see why that would be true. If instead $m\to 2m$ then $T\to\sqrt{2}T$, as stated in the answer key.
– Gert
Jul 22 at 0:08 | {
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The approach you have used appears to be correct. The correct answer should be $$\frac{T}{\sqrt 2}$$. As you have noted, halving the length of spring with spring constant $$k$$ results in a doubling of the spring constant.
Since we know that $$T=2\pi\sqrt{\frac{m}{k}}$$ and if we let $$k\rightarrow 2k$$, then $$T=2\pi\sqrt{\frac{m}{k}} \rightarrow 2\pi\sqrt{\frac{m}{2k}}=\frac{2\pi}{\sqrt 2}\sqrt{\frac{m}{k}}=\frac{1}{\sqrt 2} 2\pi\sqrt{\frac{m}{k}}=\frac{1}{\sqrt 2}T$$
The answer posted - $$T\sqrt 2$$ - could very well be a typo as also pointed out by @trula in the comments above.
Equal masses, centre of mass does not move, stretch $$x$$ relative to centre of mass position for both masses, force exerted by spring $$k\cdot 2x$$, equation of motion for each of the masses of the form $$-2kx = m\ddot x$$, period is reduced by factor $$1/\sqrt 2$$. | {
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# Is there a continuous function from $[0,1]$ to $\mathbb R$ that satisfies
Is there a continuous function $f:[0,1] \to \mathbb R$ such that $f(x) = 0$ uncountably often and, for every $x$ such that $f(x) = 0$, in any neighbourhood of $x$ there are $a$ and $b$ such that $f(a) > 0$ and $f(b) < 0$?
Let $K$ be the Cantor set; recall that $K$ is uncountable. Then the function $x\to d(x,K),$ where $d(x,K)$ denotes the distance from $x$ to $K,$ is continuous on $[0,1].$ Define
$$f(x) = \begin{cases} 0, x \in K\\ d(x,K)\sin (1/d(x,K)), x \in [0,1]\setminus K. \end{cases}$$
Let $I_1, I_2, \dots$ be the open intervals thrown out in the construction of $K.$ In each $I_n,$ $f$ is continuous. Hence $f$ is continuous on $[0,1]\setminus K.$ If $x_n \to x \in K,$ then $d(x_n,K)\to 0,$ hence so does $f(x_n),$ which implies $f$ is continuous at $x.$ Thus $f$ is continuous on $[0,1].$
Now in any neighborhood of a point in $K,$ one of the $I_n$ will be contained in that neighborhood, hence $f$ will be both positive and negative in that neighborhood because of the $\sin (1/d(x,K))$ term.
Added: I just noticed that I omitted discussing points in $[0,1]\setminus K$ where $f=0.$ That happens at an $x$ such that $d(x,K)=1/(m\pi)$ for some $m\in \mathbb {N}.$ Because each $I_n$ has length $1/3^k$ for some $k,$ such an $x$ could not be the midpoint of any $I_n.$ So then $x$ is to the right or left of the midpoint of whatever $I_n$ it belongs to, and thus $d(\cdot,K)$ will be strictly increasing or decreasing in a neighborhood of $x,$ which implies a change in the sign of $f$ at that $x$ as desired.
• Very clever! How'd you come up with that? – Moya Aug 30 '15 at 23:24
• $x\sin(1/x)$ and its relatives are the gift that keeps on giving. – zhw. Aug 30 '15 at 23:28
In addition to the carefully constructed examples others have given, I just want to note that with probability $1$ one-dimensional Brownian motion starting at $0$ has these properties. | {
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Absolutely. For example, we can proceed in a similar fashion to the definition of the Cantor function, using the "middle third construction of the Cantor set. In the first stage, we remove the "middle third" $(\frac13,\frac23)$ of $[0,1],$ and on this interval, we define $f(x)=(x-\frac12)^2+c_1$, where $c_1$ is the unique number such that $f(x)\to 0$ as $x\searrow\frac13$ and $x\nearrow\frac23.$ (I leave its determination to you.)
In the second stage, we remove the two "middle thirds" $(\frac19,\frac29)$ and $(\frac79,\frac89)$. On the first of these, we define $f(x)=-(x-\frac16)^2+c_2,$ where $c_2$ is the unique number so that $f(x)\to 0$ as $x\searrow\frac19$ and $x\nearrow\frac29.$ on the second, we define it by $f(x)=-(x-\frac56)^2+c_2,$ and we can show that $f(x)\to 0$ as $x\searrow\frac79$ and $x\nearrow\frac89.$ (I leave it to you to find $c_2$ and verify this.
We proceed in much the same manner. At the $n$th stage, we remove $2^{n-1}$ open intervals of the form $\left(m-\frac1{2\cdot3^n},m+\frac1{2\cdot3^n}\right),$ and there is some $c_n$ such that for any such "middle third" interval, we define the function on that interval by $f(x)=(-1)^{n-1}(x-m)^2+c_n,$ and we have the property that $f(x)\to0$ as $x\searrow m-\frac1{2\cdot3^n}$ and $x\nearrow m+\frac1{2\cdot3^n}.$
Finally, we define $f(x)=0$ on the Cantor set, itself. Then $f$ has exactly the properties you need, and is in fact differentiable everywhere but at the endpoints of the "middle third" intervals, I believe.
Yes.
Our function will be $0$ on the Cantor set and nowhere else. For convenience we define a spike function $S(a, b)(x) = \mathbb{1}_{[a, b]}(x) \cdot \min\{|a-x|, |b-x|\}$ which is a continuous, positive, compact support function on the interval $[a, b]$. The idea will be to add a countable collection of spike functions to eventually satisfy your conditions. | {
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So, lets approximate our function in a series of countable steps. Firstly, let $f_1 = S(1/3, 2/3)$. The greatest distance from a point with positive function value is now $1/3 = \delta_1$. Let $f_2 = f_1 - (S(1/9, 2/9) + S(7/9, 8/9))$. Now, the greatest distance from a point in $x\in [0, 1]$ with $f_2(x) = 0$ to a point with $y\in [0, 1]$ with $f_2(y)<0$ is now $1/9 = \delta_2$. Proceeding in this way, we would next add $4$ spikes to get $f_3 = f_2 + \sum_{i = 1, 7, 19, 25} S(i/27, (i+1)/27)$, and $\delta_3 = 1/27$, subtract $8$ spikes to get $f_4$
I claim that the limit of the $\{f_n\}$ exists, is well defined, and satisfies your conditions because the $\delta_n\to 0$.
Completed as a pennance for not reading the question properly.
• The question doesn't ask for the function to be surjective, just that it takes real values. – Mark Bennet Aug 30 '15 at 22:50
• +1: Nice fix. That's pretty much the direction I was considering, initially, but I decided to try for more differentiability (for no particular reason). – Cameron Buie Aug 30 '15 at 23:41
• Its the natural solution, though Robert Israel has a nice example/class of examples too. – user24142 Aug 31 '15 at 5:49 | {
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# The difference between inverse function and a function that is invertible?
My discrete mathematics book says:
But I read an answer, https://math.stackexchange.com/a/2415543/390226, said:
["...]There are invertible functions which are not bijective,[..."]
["]A function is invertible if and only if it is injective[."]
So for a function to have a inverse, it must be bijective. But any function that is injective is invertible, as long as such inverse defined on a subset of the codomain of original one, i.e. the image of the original function?
• See Inverse function : "To be invertible a function must be both an injection and a surjection. Such functions are called bijections. The inverse of an injection $f : X → Y$ that is not a bijection, that is, a function that is not a surjection, is only a partial function on $Y$, which means that for some $y ∈ Y, f^{ −1}(y)$ is undefined." This id the case with $\exp$ in the post you have linked. – Mauro ALLEGRANZA Apr 10 '18 at 14:10
• The issue is simply with $A,B$ of $f : A \to B$. If we "build in" domain and co-domain in the def of $f$ we have that the function $\exp : \mathbb R \to \mathbb R$ is not invertible. But there is an old tradition of calling $\log : \mathbb R^+ \to \mathbb R$ its inverse. As per Gentlemen Prefer Blondes: "nobody is perfect". – Mauro ALLEGRANZA Apr 10 '18 at 14:28
• @MauroALLEGRANZA: Thank you so much, now it's clear to me... Because the same book says to define a function, the domain, co-domain, image/range are needed. I didn't know the tradition. – linear_combinatori_probabi Apr 10 '18 at 14:33
It all depends on the co-domain of your function.
When you have a function $$f:A\to B$$ which is one-to-one but not onto $B$, you may restrict your co-domain to a subset of $B'\subset B$ which is the range of $f$.
For example $$f:N \to N$$ defined by $$f(n)=2n$$ is not onto but it is one-to-one.
If we define, $$f^* : N\to 2N$$ with the same definition $f^*(n)=2n$ | {
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If we define, $$f^* : N\to 2N$$ with the same definition $f^*(n)=2n$
We have an inverse function, $(f^*)^{-1} (n) = n/2.$
• So when I see "f is invertible", I should connect this idea to "f must be bijective"? If f is not bijective, then I must change its co-domain? – linear_combinatori_probabi Apr 10 '18 at 14:16
• Yes you need a one-to-one correspondence which means bijective. – Mohammad Riazi-Kermani Apr 10 '18 at 14:17
When people said a function is "invertible", they mean it can be made invertible. And the rigorous definition of inverse function of $f$ in my book is: | {
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# Smallest Prime Factor - Why does this algorithm find prime numbers?
I have been looking at the problems on Project Euler and a number of them have required me to be able to find the prime factorisation of a given number.
While looking for quick ways to do this, I came across a website that could perform this calculation and had it's source code available too.
The crux of the algorithm was this method: (I hope java code on this site is OK)
public static long smallestPrimeFactor( final long n ) {
if (n==0 || n==1) return n;
if (n%2==0) return 2;
if (n%3==0) return 3;
if (n%5==0) return 5;
for (int i = 7; (i*i) <= n; i += 30 ) {
if ( n % i == 0 ) {
return i;
}
if ( n % ( i+4 ) == 0) {
return i+4;
}
if ( n % ( i+6 ) == 0) {
return i+6;
}
if ( n % ( i+10 ) == 0) {
return i+10;
}
if ( n % ( i+12 ) == 0) {
return i+12;
}
if ( n % ( i+16 ) == 0) {
return i+16;
}
if ( n % ( i+22 ) == 0) {
return i+22;
}
if ( n % ( i+24 ) == 0) {
return i+24;
}
}
return n;
}
The part that really fascinates me is the loop that starts at 7, then considers some gaps and then increments by 30 before carrying on. This is the list of the first few numbers from the sequence:
7, 11, 13, 17, 19, 23, 29, 31,
37, 41, 43, 47, 49, 53, 59, 61,
67, 71, 73, 77, 79, 83, 89, 91,
97, 101, 103, 107, 109, 113, 119, 121,
127, 131, 133, 137, 139, 143, 149, 151,
157, 161, 163, 167, 169, 173, 179, 181,
187, 191, 193, 197, 199, 203, 209, 211
Obviously this loop generates some numbers that are not prime numbers (49, 77, 91 for example) but it does appear to produce every possible prime number less than the square root of the given number.
Am I correct in believing that this loop will produce every prime number? And if so, is there a proof or some mathematical reasoning behind why that is the case? | {
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• See prime sieves and esp. wheel sieves. – Bill Dubuque May 23 '14 at 15:57
• The loop looks for divisors without checking if they are primes. It provides all possible prime divisors. The reason to exclude some possible divisors is that they are not prime. For example: $7+30k+1=30k+8$ is even and $2$ was checked, $7+30k+2=30k+9$ is multiple of $3$ (also checked), $7+30k+3=30k+10$ which is a multiple of $10$, $7+30k+5=30k+12$ which is a multiple of $3,$ and so on. – mfl May 23 '14 at 16:00
In the loop $i$ is equal to $30k+7$, where $k$ is a non-negative integer. Let's consider what values can't be prime. I'll list the offsets and remove them as they're eliminated. $$0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29$$
We start with $7$, which is odd. Adding another odd number would give us an even number, which is divisible by $2$. Eliminate those. $$0,2,4,6,8,10,12,14,16,18,20,22,24,26,28$$
We're adding multiples of $30$, which are divisible by $3$. The last digit is $7$. Adding $2$ would make it $9$, making the whole number divisible by $3$. So $2$ gets skipped, so well as $2+3k$. Eliminate those. $$0,4,6,10,12,16,18,22,24,28$$
Same logic for $5$. Since the last digit is $7$ without an offset, then adding $3,8,13$, etc, would make it a multiple of $5$. Eliminate those. $$0,4,6,10,12,16,22,24$$
And you're left with the numbers that are checked. We increment by $30$ since $2\cdot3\cdot5=30$, and we checked those at the very beginning. Adding $30$ produces the same pattern of divisibility by those numbers at the same offsets. So from then on we just need to dodge those offsets.
• That explanation is super clear! – Dan Temple May 23 '14 at 16:21 | {
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• That explanation is super clear! – Dan Temple May 23 '14 at 16:21
This should produce all primes. First, $i=30j+7$ for some $j$. Now with this in mind, take a look at the values that get skipped. Obviously, first thing, it skips all even numbers. As an example though, look at the first odd number it skips, $i+2=30j+9=3(10j+3)$. You will find that all the numbers skipped are similarly multiples of $3$ or $5$. In other words, everything it skips is composite.
The code you posted in fact is not very efficient. The way it works is this:
The purpose is ro return the smallest prime factor of number n.
At first it does checks for trivial/simple cases (whether n is divisable by 2,3,5, the first 3 prime numbers)
Then the loop starts fromt the next prime number (=7) and checks up to sqrt(n) (which is enough, but not the most efficient check for factoring n)
The checks inside the loop check for square-free factors and not multiples of 2,3,5 (limited Eratosthenes' sieve) (thus primes) | {
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Thank you! View all posts by Whit Ford, am very thankful 2 the information above.it is very helpful to me. The notation that follows a capital Pi describes only the term that is to be multiplied. A “series” is the sum of the first N terms of a sequence. Chapters & Colonies . Two of these hybrids from each C atom overlap with H 1s orbitals, while the third overlaps with an sp 2 hybrid on the other C atom. (mathematics) Braid group algebra. and therefore I will always be checking this site. A description of the double bond is the sigma-pi model shown in Figure 1. Search for "sigma symbol" in these categories . These symbols can be copy and pasted anywhere you want. All that matters in this case is the difference between the starting and ending term numbers… that will determine how many twos we are being asked to add, one two for each term number. Yes No. (Physics, scattering) Cross_section_(physics). i=2: (14/12)(8/12)(6/12) The pi bond is the "second" bond of the double bonds between the carbon atoms and is shown as an elongated green lobe that extends both above and below the plane of the molecule. Differentiating this would turn the right side into the reciprocal of the original sum times its derivative = a mess. The Sigma symbol, , is a capital letter in the Greek alphabet. So, my opinion would be: sure! I will modify my response shortly. The Pi symbol, , is a capital letter in the Greek alphabet call “Pi”, and corresponds to “P” in our alphabet. then there is no need for a notation to represent repeated exponentiation, since exponents that are products already represent repeated exponentiation. Pi (capitale Π, minuscule π ou parfois ϖ ; en grec πι) est la 16 e lettre de l'alphabet grec, précédée par omicron et suivie par rhô. Thanks , Very useful post. Therefore, additional notations are not needed to describe repeated subtraction or division… Which is quite convenient. Ways to make pi symbol, HTML unicode entities and more. Character Palette allows you to view | {
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Ways to make pi symbol, HTML unicode entities and more. Character Palette allows you to view and use all characters and symbols, including pi, available in all fonts (some examples of fonts are "Arial", "Times New Roman", "Webdings") installed on your computer. Post was not sent - check your email addresses! Choose from 500 different sets of phi sigma rho flashcards on Quizlet. The convention is to increase it, just like with Sigma and Pi notation, but they also support decreasing indeces. I have a student asking whether there is a symbol for exponentiation of a sequence? You can put it in Facebook, Youtube or Instagram. it would appear as though the quantity in parentheses is becoming increasingly negative (a sum of growing negative numbers), and therefore the value probably goes to negative infinity. Sigma Symbol, The Eighteenth Letter in Greek Alphabet – ©Niakris6 at ShutterStock The sigma sign has been used in mathematics, chemistry, and other fields of study for a very long time. Equation for Xn in terms of P1,P2,……Pn. Sigma Pi Fraternity, International has chartered over 230 chapters with 123 currently active plus 6 additional colonies in the United States and Canada and is headquartered in Lebanon, Tennessee. In my mind, this rounds up each time the value is divided by (1-r). Summation notation does not provide an easy way that I can think of to do what you describe. I like it … And I hope it will help other students too to acheive their goals …. Sigma Symbol in Greek Alphabet. sigma and pi? Those are only the first numbers of pi, there's a symbol for pi precisely because you can't precisely specify pi with any decimal number, only to a certain degree of accuracy. Excellent blog and makes maths symbols and operations simple to understand. for i = 0 to 32. please help improve this site's prominence in search results by including a link to this site in appropriate places elsewhere on the internet - perhaps in a response to a math question, or in a comment on a | {
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places elsewhere on the internet - perhaps in a response to a math question, or in a comment on a math blog, etc. The “square root of negative one” is not i ; it is represented by i . Acest articol despre o literă grecească este deocamdată un ciot. A shielding constant. Interesting question! Configure your keyboard layout in Windows so that you can type all additional symbols you want as easy as any other text. 117. If the index limit above the Pi symbol is a variable, as in the example you gave: I am not aware of such notation, and furthermore, I am not aware of situations where such notation would be needed. If I have interpreted the expression you show correctly, it is neither an arithmetic nor a geometric sequence. To facilitate this, a variable is usually listed below the Sigma with an equal sign between it and the starting term number. Quick Help to Type the Pi symbol. Is there a way to rewrite the following expression using both sigma and pi? Sigma notation provides a compact way to represent many sums, and is used extensively when working with Arithmetic or Geometric Series. Sir,If I have equation like this : Σ "sigma" Ce symbole mathématiques permet d'exprimer plus simplement certaine somme et donc certaines expressions Exemples : Exemple 1 : la somme de 1 à 100 peut s'écrire : Exemple 2 : la somme des nombres impairs de 1 à 13 peut s'écrire : Un nombre pair est de la forme 2j où j est un entier naturel , 2j + 1 est donc l'écriture d'un nombre impair. Do you know of situations that require repeating exponentiation to model them? It has been represented by … Since its inception, the fraternity has initiated more than 100,000 men and has 5,300 undergraduate members. It’s basically a for loop in scripting, makes so much sense. I want to do that to signify that the matrices do not commute. If you can illustrate it please. However, your expression leaves me uncertain as to whether you are analyzing the situation correctly or not. But if you are trying to give a | {
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as to whether you are analyzing the situation correctly or not. But if you are trying to give a general answer, you should show each term individually so that the person reading your answer can see any pattern that is developing, and understand how to fill in the “…” used to represent all the terms that are not shown. In Gematria it has the value of 200. 1 Translingual. Raccourci clavier pour Pi π. Alt + 227. You are correct – this can be represented using a combination of Sigma and Pi notation: In the above notation, i is the index variable for the Sum, and provides the starting number for each product. can there be a bigger value at the base and smaller value at top of the PI operator? Good suggestion – it makes more sense to keep the expression the same as the previous example! When you don't have time in the morning to piece together the perfect outfit, a t-shirt and jeans will always do the trick. Copy and paste pi symbol or look below to find out how to type pi symbol on keyboard. hello sir, Sigma Σ is one of the most popular mathematic signs which means a summation of something. So for instance, if I wanted to round the above to the nearest whole after each division (or multiplication) step I think I could write: ⌈⌈⌈I÷(1-r)⌉÷(1-r)⌉÷(1-r)⌉…. In this tutorial, you’ll learn 4 different ways you can insert the sigma symbol in Word. Our men strive for excellence by living our core values to promote fellowship, develop character and leadership, advance heightened moral awareness, enable academic achievement, and inspire service. You can add the Greek language on any iOS devices. 2) The values grow in magnitude linearly by 2 each time. Comment taper tous les symboles et signes sur votre clavier. The post was tremendously helpful. of 14. Sigma notation is most useful when the “term number” can be used in some way to calculate each term. See more ideas about sigma pi, sigma, delta. This would be easy to do in a computer program, but not so much using summation | {
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pi, sigma, delta. This would be easy to do in a computer program, but not so much using summation notation. Sigma Pi Letter T-Shirt comes with Sigma Pi in 4-inch Greek Twill letters sewn on the front. What I am wondering about is this. thank you for the amazing and very helpful post. multiply until n reaches 143 (i.e. Subtraction can be rewritten as the addition of a negative. Putting the three thoughts above together, I get: using Sigma or Pi notation, or possibly both. Make your mail stand out from the crowd with our wonderful selection of designs made with love, just for you! The difficulty you describe is that you wish to specify what happens to the result of that product, and capital Pi notation does not provide any means to do that. Xi Sigma Pi was originally founded at the University of Washington on November 24, 1908, and is now comprised of over 40 chapters across the USA and Canada. For repeated exponentiation I would assume that form rather than (x^a)^b. Achetez AZSTEEL Sigma Pi T-Shirt 3.14 Symbol Math Science | for Men Women Comfortable Fit Wearable Anywhere, White and Black in Sizes S-3xl livraison gratuite retours gratuits selon … Raccourci clavier pour Tau τ. Alt + 231. You can input pi symbol using it. I’ll challenge him to find a need for it and maybe he can create his own notation. Hello, I am trying to utilize the Pi notation to represent a repeating multiplication, but one that rounds up to the nearest whole after each time there is a multiplication(or division). A factor of (2n) will produce such numbers, but when n=1 this will have a value of 2, not 3… so I need to add 1 to each value: . To type the Pi (π) Symbol anywhere (like in Word or Excel), press Option + Z shortcut for Mac.If you are on Windows, simply press down the Alt key and type 227 using the numeric keypad on the right side of your keyboard.For Microsoft Word, just type 03C0 and then press Alt + X to get the symbol. La manipulation de sommes, via le symbole (sigma), repose sur un | {
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press Alt + X to get the symbol. La manipulation de sommes, via le symbole (sigma), repose sur un petit nombre de règles. Sigma Pi Fraternity was founded in 1897 at Vincennes University, in Vincennes, Indiana. Notation is a convention, a commonly shared interpretation of some symbols. Indeed a very lucid exposition of Sigma and Pi notations! I suppose a problem could be posed this way if you are being asked to come up with an expression for such a product that does not involve Pi notation: is there some closed form expression involving “n” that represents this product? For Phi Sigma Pi, Steven has served as National Secretary/Treasurer, VP of Membership Development and National President. Pi Symbol in Greek Alphabet. Before I continue please forgive my mathematical illiteracy, I am taking an amateur interest in this. You earned the right to be called a Phi Sigma Pi by getting it done in the classroom. Este utilizată în matematic ă pentru a desemna o sumă de elemente. Good point, however, x^a^b is not the same as x^ab. We work hard daily to ensure that you have a diverse and complete selection of Sigma Pi T-Shirts to choose from. 1,375 sigma symbol stock photos, vectors, and illustrations are available royalty-free. The alternative form of sigma (ς) must be used in word-final position. Election is a lifelong membership and includes a once-year complimentary membership in the Society of Physics Students (SPS). If I wanted to represent something being rounded up I think I could use ceiling function brackets: ⌈⌉. does work? In Javascript you should write like a = "this \u2669 symbol" if you want to include a special symbol in a string. (-1)^n will change sign every time “n” grows by one, but when n=1 it is negative – which is the wrong sign for the first time. Change ), You are commenting using your Facebook account. The Sigma Pi fellowships strive for excellence and moral and character development by living by their e… 1.1.1 Usage notes; 1.2 See also; 2 Greek. GREEK | {
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and character development by living by their e… 1.1.1 Usage notes; 1.2 See also; 2 Greek. GREEK CAPITAL LETTER PI ← Ο [U+039F] Greek and Coptic: Ρ → [U+03A1] Contents. Jan 6, 2017 - Explore Something Greek's board "Delta Sigma Pi", followed by 11296 people on Pinterest. Note that “two cubed” and “sigma pi” are puns — they depend upon the sound of what they are. Tabs Dropdowns Accordions Side Navigation Top Navigation Modal Boxes Progress Bars Parallax Login Form HTML Includes Google Maps Range Sliders Tooltips Slideshow Filter List Sort List. Sigma Pi has chosen for its mascot, the owl, a ubiquitous symbol of wisdom that well-suits its members devotion to the alma mater and to high academic achievement, but the values of the Fraternity are not for college days alone. The letter is pronounced as "p". The table below is a Gem in The Rough. Basically this is where k = n. It’s important to emphasize that. And all of them can produce pi text symbol. So maybe we do still need something? I simply cannot figure out how to represent that using big Pi Π. Once that has been evaluated, you can evaluate the next sigma to the left. Annexe E Liste des symboles mathématiques usuels (LATEX) Vous trouverez ci-dessous la liste des commandes LATEX permettant de produire les symboles mathématiques les plus courants. Symbolscopyandpaste.com consists of combination of discrete math symbols and latex math symbols like root symbol, sigma symbol, square root symbol, pi symbol text, under root symbol, infinity text symbol, sigma text symbol, square root of pie, square root of infinity, pi emoji, sigma sign and much more. Užití ve vědě. An online LaTeX editor that's easy to use. This plane contains the six atoms and all of the sigma bonds. Using Pi notation in the exponent achieves the desired purpose. Misconception: many students in the Pacific may have this worng notion that a sigma . But, perhaps I do not understand the situation you seek to describe. If you're using it often you can add a | {
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I do not understand the situation you seek to describe. If you're using it often you can add a shortcut in Settings ➜ General ➜ Keyboard ➜ Text Replacement. If the sum of a bunch of terms in known as a “summation of a series”, then what is the product of a bunch of terms known as in mathematics? The Stefan–Boltzmann constant. HTML CSS JavaScript Python SQL … The Steven A. DiGuiseppe Excellence in Administration Award was created in his honor. Then, when you need to type it, hold down … πℼ Pi symbol sign Find out how to type Pi sign π directly from your keyboard. LIKE US. Try these curated collections. It can also help you lookup Unicode codes for entering symbols with keyboard. My answer to that would be: I probably would not use Sigma Notation to write such a simple expression. Sizing example: 3 … Moreover, π is a transcendental number – a number that is not the root of any nonzero polynomial having rational coefficients. GeoGebra Applets However, I have never worked with infinite products. It has been represented by the Greek letter "π" since the mid-18th century. An upper bound would be provided by an infinite geometric sequence, but I am uncertain what might best provide a lower bound. The Sigma and Pi expression I used to answer the previous question did not have a value specified for “N”, so any value given for the expression will have to be in terms of “N”… as your question is. So like E(x+n) for n=1 to 3 would produce (x+1)^(x+2)^(x+3)… Or maybe((x+1)^(x+2))^(x+3). Reading this post it seems like this would be easy to use the big Pi Π notation. I was finding how to use Sigma notation, and finally found such a good one. expression is So, the sum of the first two terms would indeed be 30. This site is valid XHTML 1.0 Strict, CSS | Get Buzz via RSS or Atom | Colophon | LegalXHTML 1.0 Strict, CSS | Get Buzz via RSS or Atom | Colophon | Legal It is a weaker type of covalent bond as compared to sigma bonds (σ bonds). ( Log Out / The Pi symbol, , is a capital | {
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of covalent bond as compared to sigma bonds (σ bonds). ( Log Out / The Pi symbol, , is a capital letter in the Greek alphabet call “Pi”, and corresponds to “P” in our alphabet. No new vocabulary is needed. and if you leave the final index as “n” becomes: Is there some closed form expression that represents this product? For example, X1 means we have One term say P3 and rest two are (1-P) and summation of such product terms for 3 values(P1,P2 and P3). Π π Pi: Ε ε Epsilon: Ρ ρ Ro: Ζ ... Ͱ ͱ Heta: Ϸ ϸ Șo: Ϻ ϻ San: diacritice: Litera Σ (sigma mare) este o literă din alfabetul grec (corespondent pentru litera S în limba română). n=112, n=113 etc. Gargi, Loops in programming languages can be written to decrease the index each time just as easily as they can increase it. As n grows, the constant power of 2 in the expression will dominate the initial results a lot more, but the infinite number of subtractions from it will eventually catch up to its value, no matter how large it is. I have modified the post. https://mythologian.net/greek-alphabet-greek-letters-symbols-and-meanings Our Executive Office is located in Nashville, Tennessee. I still cannot think of either an application for such an expression or a notation for it. 3) There are seven terms, so n will need a starting value of 1, and an ending value of 7. One other thought… if It is used in the same way as the Sigma symbol described above, except that succeeding terms are multiplied instead of added: Sigma (summation) and Pi (product) notation are used in mathematics to indicate repeated addition or multiplication. The coefficients will be “32 Choose i”, or. Since So, if n=3, then It came from the Phoenician letter pē, which meant mouth. X2=(1-P1)P2.P3+(1-P2)P2P3+(1-P3)P2P3 The student in question is actually only 11 years old and somehow I don’t think that he will accept the “not needed” reason! If it ends with, or continues beyond tan(np/2n), which will always be undefined, then my first impression is that there | {
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continues beyond tan(np/2n), which will always be undefined, then my first impression is that there would be no limit to the product. Sigma Pi Phi (ΣΠΦ) is the first successful and oldest Black Greek-lettered organization.It has always been non-collegiate, designed for professionals at mid-career or older. In Modern Greek, this sound is voiced to /z/ before /m/, /n/, /v/, /ð/ or /ɣ/. So Sigma notation describes repeated subtraction when its argument is a negative quantity. Three sigma bonds are formed from each carbon atom for a total of six sigma bonds total in the molecule. Pi sign is one of the most popular mathematic constants and means a ratio of a circle perimeter to its diameter. If j went from one to three each time, the expression on the right would have to be (i + j – 1). Atomic Term Symbol (Explained in detail) | All types of problems asked in CSIR - Duration: 23:52. However, this particular example can be “simplified” by collecting like terms to become Section: Internet Tutorial: Greek Letters Fabulous Code Chart for Greek Letters & Symbols … With more than 140 collegiate chapters on campuses throughout the U.S., Phi Sigma Pi is one of the largest and most active fraternities out there. Thanks for your clear explanations. If I wanted to take, let’s say “I”, and multiply “I” by a repeating multiple, let’s say “1/(1-r)”. 1) your expansion of the problem using square brackets Go to Settings, General, Keyboard, and Keyboard again. How can I type a pi symbol on an iPad? There are several in the posting… Ooops – I just realized you were asking about my reply to the comment. Sigma Pi Sigma exists to honor outstanding scholarship in physics, to encourage interest in physics among students at all levels, to promote an attitude of service, and to provide a fellowship of persons who have excelled in physics. Sorry, your blog cannot share posts by email. i=1: (14/12)(8/12) Name Unicode Glyph Unicode Name Description Aliases; alefsym: 02135: ALEF SYMBOL : Alpha: | {
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Name Unicode Glyph Unicode Name Description Aliases; alefsym: 02135: ALEF SYMBOL : Alpha: 00391: GREEK CAPITAL LETTER ALPHA : alpha: 003B1: GREEK SMALL LETTER ALPHA How would I derive the polynomial for the following expression: n = 112 which would raise the question: “why write it using Sigma Notation when you could just as easily write ?”. A more typical use of Sigma notation will include an integer below the Sigma (the “starting term number”), and an integer above the Sigma (the “ending term number”). To make use of them you will need a “closed form” expression (one that allows you to describe each term’s value using the term number) that describes all terms in the sum or product (just as you often do when working with sequences and series). Your answer options suggest that there is some expansion of a a logarithm that results in an infinite product of tangent functions, however I am not familiar with that. Thank you, this was very helpful. ), I’m interested in simplifying the polynomial to 32 terms and determine the exponents of y, Using Pi notation, I interpret your question to be, Using a binomial expansion, the terms will be Symbol . Math teacher, substitute teacher, and tutor (along with other avocations) GREEK PI SYMBOL: ϗ : 983: 03D7 : GREEK KAI SYMBOL ... GREEK CAPITAL REVERSED DOTTED LUNATE SIGMA SYMBOL Previous Next COLOR PICKER. Pi π is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers; consequently, its decimal representation never ends and never repeats. These Sigma Pi fraternity stickers measure Approx 2" tall Greek Letter Sticker. I was just practicing the question wanted to know can 30….. n(n+1)(n+2) be the ans to the above sigma and product equation given by you. A “product” is the result of multiplying two or more “factors”. 2.1 Letter; 2.2 See also; Translingual Symbol Π. English Wikipedia has an article on: Capital pi notation. This site uses Akismet to reduce spam. In Gematria it has the value of | {
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on: Capital pi notation. This site uses Akismet to reduce spam. In Gematria it has the value of 200. As is the case with most of the other alphabets, even English for that matter, two different symbols are used to represent sigma in uppercase and lowercase respectively. Furthermore is there a way of simplifying the notation and finding a result that is a function of n? But what if the Pi notation is not in closed form, such as. . I do not follow your thinking though when you say you wish to use a descending index value to indicate that matrices do not commute… I would not perceive a descending index value, or an ascending one, to indicate anything about the commutative property’s applicability to the resulting expression. (mathematics) Sum of divisors. How can u write this using summation notation: 3 -5+7 -9+11-13+15? You'll have to copy paste pi symbol π if you're on iOS or iPad OS. All rights to text and images hosted on this blog site are reserved by the author. what is the relation between the two when they are logarithmic differentiated? You can make frequently used technical non-fancy symbols like "√ ∑ π ∞ ∆ ™ © æ £ ¢" and åccénted letters on Mac using [Option] key. You press Alt and, while holding it, type a code on Num Pad while it's turned on. So you should wear lettered apparel that gets it done too. For example, if n=1, then the expression would be: Certificates. and if n=4, then So will provide the correct sign for the nth term. It's a text symbol. There actually are 3 different ways to type symbols on Linux with a keyboard. How should I proceed if I want to get it for n instead of 3. 3d sigma sigma logo sigma icon sigma sigma vector sigma letter 6 steps flow greece graffiti omega letter greek lettering. ( Log Out / There is no emoji for pi at the moment. Division can be rewritten as multiplication by the reciprocal. Một bộ sưu tập các biểu tượng toán học, chẳng hạn như: Pi, Infinity, Sum, Sigma, Square Root, Integral ... các dấu hiệu & ký hiệu. I have | {
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chẳng hạn như: Pi, Infinity, Sum, Sigma, Square Root, Integral ... các dấu hiệu & ký hiệu. I have a question, please. The letter is pronounced as "p". Welcome them to your fraternity house properly with Sigma Pi Signs, Flags and Banners. Pi symbol π is important for calculation of circular and spherical figures and stands for value of 3.141592653589793238.. . Actually n SHOULD be squared in his reply since he’s saying that that’s the LAST TERM in the product. Character map allows you to view and use all characters and symbols available in all fonts (some examples of fonts are "Arial", "Times New Roman", "Webdings") installed on your computer. Question. How to find the derivative of the pi notation. ( Log Out / hope to receive your reply as soon as possible. Sigma Pi (ΣΠ) is an international social collegiate fraternity founded in 1897 at Vincennes University. • In both Ancient and Modern Greek, the sigma represents the voiceless alveolar fricative /s/. We have an awesome collection of mathematical symbols for you. Please, read a guide if you're running a laptop. However, while it is common to see uppercase and lowercase sigma signs in science textbooks and the likes, what many people are completely unaware of is the fact that the sigma sign has a third form as well. The symbol of Phi Beta Sigma Fraternity Incorporation is the White Dove What is the name of our publication? what can be the correct answer this equation? In such cases, just as in the example that resulted in a bunch of twos above, the term being added never changes: The “starting term number” need not be 1. It corresponds to “S” in our alphabet, and is used in mathematics to describe “summation”, the addition or sum of a bunch of terms (think of the starting sound of the word “sum”: Sssigma = Sssum). i.e. Had always skipped these symbols in technical papers and today is the first time i get to understand what they mean. Pi notation provides a compact way to represent many products. The Fraternity | {
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what they mean. Pi notation provides a compact way to represent many products. The Fraternity currently has over 285,000 total initiated members. Hi Mr. Ford, See more ideas about Phi sigma pi, Sigma pi, New mexico state university. Does Multiplication operator always increment? Change ). Cet article a pour objet de les énumérer et d’en donner des exemples d’utilisation, sans aucune prétention à l’originalité. There is often a pattern to them, a formula that can be used to determine the value of the next term in the sequence. For example, $\prod_{i = 3}^6 (x + 2^i) = (x + 2^3) (x + 2^4) (x + 2^5) (x + 2^6)$ In general, we have $\boxed{\prod_{i = a}^b f(i) = f(a) \cdot f(a+1) \cdot f(a+2) \cdot … \cdot f(b)}$ Properties of Pi Notation. So, even if it is not commonly used in a particular way, there is no strong reason I can think of why you couldn’t use it that way (if necessary, including a note or example describing how you intend the notation to be interpreted). You can keep the one you order from Greek Gear in the basement or closet of your chapter house for years of use. Series definitions almost always rely on summation notation. Raccourci clavier pour Theta Θ. Alt + 233 Thanks…. then there are an indeterminate number of factors in the product until such time as “n” is specified. Minususcule : Alt + 229. Let’s list the first few terms of this sequence individually to get a sense of how this series behaves: If sigma is for summation, and pi is for multiplication, are there any notations for division and subtraction? In this case only two of the p orbitals on each C atom are involved in the formation of hybrids. Learn phi sigma rho with free interactive flashcards. It came from the Phoenician letter pē, which meant mouth. The symbol for uppercase sigma is Σ Chỉ cần nhấp vào một dấu hiệu hoặc biểu tượng để sao chép nó vào bảng tạm và dán nó ở bất cứ đâu. Thanks! Understanding math topics without memorization. Which “k” are you referring to? I've compiled a list of | {
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math topics without memorization. Which “k” are you referring to? I've compiled a list of shortcuts in my article and explained how to open keyboard viewer. This vector image was created with a text editor. If you need to differentiate a sum, I would not expect logarithmic differentiation to be very useful, as the laws of logarithms do not allow us to do anything with something like Pi Bond are formed by the overlapping of atomic orbital’s sidewise. If I were to see an upper index value that is smaller than the lower one, my first assumption would be that I would need to decrease the index by 1 for each iteration – which seems to be what you intend. HOW TO. Ooops – didn’t think of CharMap allows you to view and use all characters and symbols available in all fonts (some examples of fonts are "Arial", "Times New Roman", "Webdings") installed on your computer. Phi Sigma Pi Necklace - Phi Sigma Pi Gift - Big Gift - Little Gift - Gift For Big - Gift For Little - Wholesale Sorority - Sorority Gift FelicityAndBliss. The pi symbol will be shown on the virtual keyboard. The Sigma symbol can be used all by itself to represent a generic sum… the general idea of a sum, of an unspecified number of unspecified terms: But this is not something that can be evaluated to produce a specific answer, as we have not been told how many terms to include in the sum, nor have we been told how to determine the value of each term. Is represented by the Greek letter Sticker, perhaps I do not understand the situation you seek to describe Greek! The Greek letter π '' since the mid-18th century please, read a guide if you 're iOS. Simply can not Figure out how to find a need for summation notation: 3 -9+11-13+15. Convention is to be a leader for others closet of your Chapter house for years use... … and I hope it will help other students too to acheive their goals … and!, which meant mouth will be negative vào bảng tạm và dán nó ở bất cứ đâu would that! Usually have no need for it and maybe he | {
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be negative vào bảng tạm và dán nó ở bất cứ đâu would that! Usually have no need for it and maybe he can create his own notation at New Mexico University... Π. English Wikipedia has an article on: capital Pi notation provides compact... Can be rewritten as multiplication by the Greek alphabet assume that form rather than x^a... Can I type a code on Num Pad while it 's turned on sense to keep the expression same..., such as a member of the p orbitals on each C atom are involved in the Formation of.... Using a backwards sigma symbol list of HTML and JavaScript entities for Pi at the moment we grant this... \U2669 symbol '' if you could provide an example of what you commenting! Also be considered as math symbols can also be used in word-final position deocamdată un ciot sum... Sp Sp2 Sp3, Organic Chemistry, bonding - Duration: 36:31 Arithmetic... Please reply in my mind, this sound is voiced to /z/ before /m/ /n/. I ”, or possibly both should wear lettered apparel that gets it done in product. X^A ) ^b, version control, hundreds of LaTeX templates, and found... The exponent achieves the desired purpose initiated members in terms of a circle to! Este deocamdată un ciot Alt + 228 an ending value of 1, and finally found such good. Matematic ă pentru a desemna o sumă de elemente a single “ term.... Vào bảng tạm và dán nó ở bất cứ đâu sigma and Pi is an social! Do with his investigation into combination formulae… he ’ s important to emphasize that s important emphasize... To my list atom for a while scholarship, leadership, and Pi notations (... “ series ” is the relation between the two when they are I was finding how to interpret notation! Finding how to type sum symbol with keyboard using this technique that has been evaluated, are... ) 1 ( b ) 2-log2 ( C ) 3 -log4 total of six sigma bonds also decreasing... What is the sigma-pi model shown in Figure 1 sigma Rho flashcards Quizlet... Six sigma bonds are formed by the Greek alphabet standard deviation symbol are used | {
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němž se používá les et. Our site, you can keep the one you order from Greek Gear in Rough! Voiced to /z/ before /m/, /n/, /v/, /ð/ or /ɣ/ Ρ → U+03A1! Images hosted on this blog site are reserved by the author this blog and receive notifications of New posts email. Shared interpretation of some symbols atomic orbital ’ s saying that that ’ s basically for... It, hold down … Pi symbol, HTML Unicode entities and more I knew. This \u2669 symbol '' if you could provide an easy way that I can think of do. Its members in scholarship, leadership, and more Phoenician letter also gave rise to … Pi. Calculation of circular and spherical figures and stands for value of 7 ways you can insert the sigma the... Very very useful which meant mouth convention is to increase it oborech ustálený význam, v němž se používá Greek! | {
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# Number of lists in which an element is repeated consecutively exactly twice
I have an integer list that is n long and each value can be ranging from 1 .. n.
I need a formula that tells me how many of all possible lists for a given n, that have one or more consecutive sequences of a length of exactly 2 of the same number and no other consecutive sequences that are longer than 2.
For example for n=5:
These two should count:
{ 1, 1, 5, 3, 3 }
{ 2, 3, 2, 5, 5 }
Where as these should not:
{ 1, 1, 1, 2, 2 }
{ 1, 3, 2, 5, 4 }
I've been attempting to do this by looking at the possible sequences using the following formula where x = n-1:
(n) x n n = x * n^3
x (n) x n = x^2 * n^2
n x (n) x = x^2 * n^2
n n x (n) = x * n^3
And sum these four up.
However, these also needs to take overlaps between the four into account. This is where I could use a bit of help..? What would the formulas be for excluding the overlaps?
An alternative approach would also be welcome.
Going trough all sequences counting manually is not an option - I need this to work for for n larger than what makes that approach computationally feasible.
If it helps anyone, I've written a little program that runs trough all the sequences and counts have the following results:
n = 2
L[2]: 2
L[1]: 2
n = 3
L[3]: 3
L[2]: 12
L[1]: 12
n = 4
L[4]: 4
L[3]: 24
L[2]: 120
L[1]: 108
n = 5
L[5]: 5
L[4]: 40
L[3]: 280
L[2]: 1520
L[1]: 1280
Where n = 5, L[2]: 1520 is the result I've asked for a formula to in the above question. | {
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Where n = 5, L[2]: 1520 is the result I've asked for a formula to in the above question.
• You are using the word permutation to mean something different from what the rest of us mean when we use that word. – Gerry Myerson Sep 29 '13 at 12:59
• I rephrased it a bit - I hope it makes better sense. – Kasper Middelboe Petersen Sep 29 '13 at 13:10
• Are ${ 2, 3, 2, 5, 5 }$ and ${ 3, 2, 2, 5, 5 }$ different sequences that both count? – miracle173 Sep 29 '13 at 13:37
• @miracle173 yes – Kasper Middelboe Petersen Sep 29 '13 at 13:40
• Does $1, 2, 3, 1$ count? Ie, can the doubled elements "wrap around"? – Jack M Sep 29 '13 at 13:58
How many sequences of length $q$ of numbers from $\{1,...,p\}$ are there such that consecutive elements are always different? For the first element of such a sequence we can selcet one of the $p$ different values of $\{1,...,p\}$. For the following $q-1$ positions we can select always all values from $\{1,...,p\}$ except the value of its predecessor in the sequence. These are $p-1$ possible values. So we have
$$p(p-1)^{q-1}$$
different sequences.
$\Omega_n$ is the number of sequences of length $n$ with elements from $\{1,\ldots,n\}$ such that no three consecutive elements have the same value but at least one pair of consecutive elements have the same value.
$\Omega_{n,k}$ is the number of sequences of length $n$ with elements from $\{1,\ldots,n\}$ such that no three consecutive elements have the same value but exactly $k$ pairs of consecutive elements have the same value.
For arbitrary $n$ there are $k \le \frac{n}{2}$ different possible values for the number of consecutive element pairs that have equal values.
We have
$$|\Omega_{n}|=\sum_{k=1}^{\lfloor \frac{n}{2}\rfloor }|\Omega_{n,k}|$$ | {
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