# Necklace combinations with three group of beads

I have a hard question about a way how many different necklaces can be made.

Suppose that we have the following restrictions:

1. We have 3 groups of beads:

All the beads in one group are completely identical. This means that if you put two triangle beads next to each other and then switch their positions this counts as one necklace because the beads are identical

1. Necklaces are identical if they are identical under symmetric operations just as rotate them (𝑟) or turning them around (𝑠).

So if we have a necklace ordered in one way and we rotate it 180 deg or just flip a side this is count as one necklace.

1. We need to use all the 18 beads in each and every new necklace. We can not create a necklace from 17, 16 or less than 18 beads.

I read all the topics here but could not find a question about a group of identical beads. I also read Burnside lemma and Pólya_enumeration_theorem and Necklace_(combinatorics) in wikipedia, but could not find a way how to solve this and what is the correct answer.

From Burnside lemma, I found that the answer should be 57, but is this correct?

I used directly the formula from Burnside lemma, but it does not seem quite right for me, because I do not take into account that the three groups are with different numbers of beads.

$$\frac{1}{24} * (n^6 + 3 * n^4 + 12 * n^3 + 8 * n^2)$$

where n is 3 from three groups.

$$\frac{1}{24} * (3^6 + 3 * 3^4 + 12 * 3^3 + 8 * 3^2) = 57$$

However, as I said earlier despide the fact that the result looks some kind realisting I am not sure that this is the right answer, because I do not use in the formula that we have 4 triangle, 6 square and 8 circle beads.

It looks like Pólya enumeration theorem weighted version is the thing that I need. However, I am not sure how to get to the right answer

• Welcome to MSE! Pls show your work for arriving at $57$ so someone can check its correctness. You can do that by editing your question and e.g. putting your work at the end. May 6 '20 at 20:53
• Thanks @antkam, for the answer. I modified the questions and added the formula that I used and a short description why I think that despide the fact that the result looks like a good number I think that this is not quite right. May 7 '20 at 5:54
• May 7 '20 at 16:19
• Does this answer your question? Circular permutations with indistinguishable objects May 7 '20 at 16:19
• I think you should be using the word "bracelet" and not "necklace". The answer in the link provided by @Vepir gives an answer to your problem for necklaces, which means objects with a cyclic symmetry group, when you want a dihedral symmetry group. This being said you should be able to follow the same line of argument to find the formula you need. May 11 '20 at 20:04

I manage to answer the question and this is the process that I followed:

I consider the 18-bead necklace in the first part of the problem. Here are the eighteen rotations expressed in cycle form where we assume that the slots are numbered from 1 to 18 in clockwise order. The first is the identity (e: no rotation) and the second is the generator g—a rotation by a single position which, when repeated, generates all the elements of the group:

$$e = g^0 \text{ = (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18)}$$

$$g^1 \text{ = (1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18) }$$

$$g^2 \text{= (1 3 5 7 9 11 13 15 17) (2 4 6 8 10 12 14 16 18)}$$

$$g^3 \text{= (1 4 7 10 13 16) (2 5 8 11 14 17) (3 6 9 12 15 18)}$$

$$g^4 \text{= (1 5 9 13 17 3 7 11 15) (2 6 10 14 18 4 8 12 16)}$$

$$g^5 \text{= (1 6 11 16 3 8 13 18 5 10 15 2 7 12 17 4 9 14)}$$

$$g^6 \text{= (1 7 13) (2 8 14) (3 9 15) (4 10 16) (5 11 17) (6 12 18)}$$

$$g^7 \text{= (1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)}$$

$$g^8 \text{= (1 9 17 7 15 5 13 3 11) (2 10 18 8 16 6 14 4 12)}$$

$$g^9 \text{= (1 10) (2 11) (3 12) (4 13) (5 14) (6 15) (7 16) (8 17) (9 18)}$$

$$g^{10} \text{= (1 11 3 13 5 15 7 17 9) (2 12 4 14 6 16 8 18 10)}$$

$$g^{11} \text{= (1 12 5 16 9 2 13 6 17 10 3 14 7 18 11 4 15 8)}$$

$$g^{12} \text{= (1 13 7) (2 14 8) (3 15 9) (4 16 10) (5 17 11) (6 18 12)}$$

$$g^{13} \text{= (1 14 9 4 17 12 7 2 15 10 5 18 13 8 3 16 11 6)}$$

$$g^{14} \text{= (1 15 11 7 3 17 13 9 5) (2 16 12 8 4 18 14 1 6)}$$

$$g^{15} \text{= (1 16 13 10 7 4) (2 17 14 11 8 5) (3 18 15 12 9 6)}$$

$$g^{16} \text{= (1 17 15 13 11 9 7 5 3) (2 18 16 14 12 10 8 6 4)}$$

$$g^{17} \text{= (1 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2)}$$

After that I found the GCD for all cycle form's with and group them in a table:

| Cycle length | Permutations | GCD with 18 |

| 1 | $$g^0$$ | GCD(0, 18)=18 |

| 2 | $$g^9$$ | GCD(9, 18)=9 |

| 3 | $$g^6$$, $$g^{12}$$ | GCD(6, 18)=GCD(12, 18)=6 |

| 6 | $$g^3$$, $$g^{15}$$ | GCD(3, 18)=GCD(15, 18)=3 |

| 9 | $$g^2$$, $$g^4$$, $$g^8$$, $$g^{10}$$, $$g^{14}$$, $$g^{16}$$ | GCD(2, 18)=GCD(4, 18)=GCD(8, 18)=GCD(10, 18)=GCD(14, 18)=GCD(16, 18)=2 |

| 18 | $$g^1$$, $$g^5$$, $$g^7$$, $$g^{11}$$, $$g^{13}$$, $$g^{17}$$ | GCD(1, 18)=GCD(5, 18)=GCD(7, 18)=GCD(11, 18)=GCD(13, 18)=GCD(17, 18)=1 |

We have 18 permutations for rotation and lets name cycle 1 with $$f_1$$, cycle 2 with $$f_2$$ .. cycle n with $$f_n$$

Than the formula for Cycling index is:

$$\frac{f_1^{18} + f_2^9 + 2f_3^6 + 2f_6^3 + 6f_9^2 + 6f_{18}^1}{18}$$

If we solve all the possible necklaces with three colors the result should be (for the moment we do not solve for the three colors with 4, 6 and 8 beads in respective groups):

$$\frac{3^{18} + 3^9 + 2*3^6 + 2*3^3 + 6*3^2 + 6*3^1}{18} = \text{21 524 542}$$

From here because turn around is allowed we need to add and the necklace(bracelet if we follow the right terms) is with even beads we should add the symethric turn arounds.

$$\frac{f_1^{18} + f_2^9 + 2f_3^6 + 2f_6^3 + 6f_9^2 + 6f_{18}^1 + 9f_1^2f_2^8 + 9f_2^9}{2 * 18}$$

and again for three colors without including the different weight:

$$\frac{3^{18} + 3^9 + 2*3^6 + 2*3^3 + 6*3^2 + 6*3^1 + 9 * 3^8 + 9 * 3^9}{2 * 18} = \text{10 781 954}$$

For the moment we have all the possible necklaces and bracelets with three colors. However, in order to find the necklaces and bracelets with three colors (4 reds, 6 greens and 8 blues) we need to replace:

$$f_1 = (x + y + z)$$

$$f_2 = (x^2 + y^2 + z^2)$$

$$f_3 = (x^3 + y^3 + z^3)$$

$$f_6 = (x^6 + y^6 + z^6)$$

$$f_9 = (x^9 + y^9 + z^9)$$

$$f_{18} = (x^{18} + y^{18} + z^{18})$$

and if we replace in the formula it becomes:

$$\frac{(x + y + z)^{18} + (x^2 + y^2 + z^2)^9 + 2(x^3 + y^3 + z^3)^6 + 2(x^6 + y^6 + z^6)^3 + 6(x^9 + y^9 + z^9)^2 + 6(x^{18} + y^{18} + z^{18}) + 9(x + y + z)^2(x^2 + y^2 + z^2)^8 + 9(x^2 + y^2 + z^2)^9}{36}$$

Then we need to find which expressions can expand to $$x^4y^6z^8$$.

After then by using multinominal coeficient I managed to calculate the following results

9 189 180

1260

11 340

11 340

Then I sum all of them and divide them to 36. This gives me the answer of 255 920 which is the answer of the question. We can create 255 920 bracelets with 4 red 6 green and 8 blue beads.


# Is $V$ isomorphic to direct sum of subspace $U$ and $V/U$?

Given a vector space $$V$$ and a subspace $$U$$ of $$V$$. $$V \cong U \oplus(V/U)$$ Does the above equation always hold? Where $$\oplus$$ is external direct sum. For finite dimensional vector space $$V$$, here is my attemp of prove:

Let dimension of $$U$$ be m, dimension of $$V$$ be $$n$$. Find a basis of $$U$$ : $$\{ \mathbf{ u_1, u_2, \cdots ,u_m}\}$$ and extend it to a basis for $$V$$ : $$\{ \mathbf{ u_1, u_2, \cdots ,u_m, v_1, \cdots,v_{n-m} } \}$$.

For every vector $$\mathbf{x} \in V$$, we can write $$\mathbf{x}= c_1 \mathbf{u_1}+ \cdots+ c_m \mathbf{u_m} + d_1 \mathbf{v_1}+ \cdots d_{n-m} \mathbf{v_{n-m}}$$ uniquely. Define a linear map $$T$$ as $$T(\mathbf{x})=(c_1 \mathbf{u_1}+ \cdots+ c_m \mathbf{u_m}, [d_1 \mathbf{v_1}+ \cdots d_{n-m} \mathbf{v_{n-m}}])$$ ,where $$[]$$ is used to express the equivalent class. We claim $$T$$ is an isomorphism.

Surjectivity is obvious. As for injectivity,
if $$T(\mathbf{x})=(\mathbf{0},[\mathbf{0}])$$, then $$c_1 \mathbf{u_1}+ \cdots+ c_m\mathbf{u_m}= \mathbf{0}$$
$$\Rightarrow c_1=0, c_2=0, \cdots,c_m=0$$
$$\Rightarrow x=d_1 \mathbf{v_1}+ \cdots d_{n-m} \mathbf{v_{n-m}}$$
Since $$[d_1 \mathbf{v_1}+ \cdots d_{n-m} \mathbf{v_{n-m}}] = [\mathbf{0}]$$, we have $$(d_1 \mathbf{v_1}+ \cdots d_{n-m} \mathbf{v_{n-m}}-\mathbf{0})\in U$$, which means $$d_1=0, d_2=0, \cdots,d_{n-m}=0$$
$$\Rightarrow \mathbf{x}=\mathbf{0}$$, so $$T$$ is injective.

Is the above proof correct? Does this mean $$V \cong \ker F \oplus (V/ \ker F) \cong \ker F \oplus\mathrm{im}F$$ for any linear map $$F$$, because $$\ker F$$ is a subspace of $$V$$ ?

The final question is about how should I prove it when the dimension of $$V$$ is infinite?

It is true that for every finite dimensional vector space $$V$$ with $$U$$ a vector subspace that $$V \cong U \oplus (V / U)$$ I think your proof is essentially correct. And yes it is true that for $$T: V \rightarrow W$$ any linear that we have $$V \cong \operatorname{ker}(T) \oplus \operatorname{Im}(T)$$ The rank-nullity theorem is a direct consequence of this.

In more technical language, we say that every "short exact sequence" of finite dimensional vector spaces over a field $$k$$ $$\textit{splits}$$. What this means is that if $$T : U \rightarrow V$$, $$S : V \rightarrow W$$ are linear maps such that $$T$$ is injective, $$\operatorname{ker}(S) = \operatorname{Im}(T)$$, and $$S$$ is surjective, then $$V \cong U \oplus W$$.

Note then that this directly gives us our result since if $$U$$ is a subspace of $$V$$, then the inclusion map $$\iota : U \rightarrow V$$ and projection map $$\pi : V \rightarrow (V / U)$$ set up exactly a short exact sequence.

In terms of whether or not this extends to infinite dimensional vector spaces, the result does hold again (assuming the axiom of choice), and the proof is essentially the same. All your proof relies on is the ability to extend a basis of a subspace to a basis of your entire space. We can do this with the axiom of choice.

• I am wondering if the following statement is false for infinite dimensional spaces. "If $V$ is a vector space and $U$ is a subspace of $V$, then there exists another subspace of $V$ called $U^\perp$ such that every element of $V$ can be uniquely expressed as the sum of an element from $U$ with an element from $U^\perp$." (I am thinking a counterexample would be if $U$ was the set of real number sequences with finite support (i.e. eventually zero) and $V$ is the set of all real number sequences.) – irchans Jan 3 at 19:03
• @irchans If we take the axiom of choice, then every subspace of a vector space has a direct sum complement (what you call the perpendicular space, but this language is typically reserved for a space equipped with some bi-linear form). The proof is pretty simple. Let $V$ be a $k$-vector space, with $U$ a vector subspace. Let $\mathcal{B}_{U}$ be a basis for $U$ and extend it to a basis $\mathcal{B}_{V}$ (using the axiom of choice) for $V$. Then let $W = \operatorname{Span}_{k}\left( \mathcal{B}_{V} \backslash \mathcal{B}_{U} \right)$. Then $V = U \oplus W$. – Adam Higgins Jan 3 at 19:13
• @irchans Perhaps the reason you think that your example is a counter example is because of the $\textit{weirdness}$ of bases of infinite dimensional vector spaces. Notice that a subset $S$ of a vector space $V$ is said to be a basis if and only if every element $v \in V$ can be written as a $\textbf{finite}$ linear combination of the elements of $S$, and that there is no finite non-trivial linear relation amongst the elements of $S$. – Adam Higgins Jan 3 at 19:19
• Thank you very much ! – irchans Jan 3 at 19:27
• This set of notes seems relevant math.lsa.umich.edu/~kesmith/infinite.pdf – irchans Jan 3 at 19:38

Your proof works. The answer to your second question is yes, that is true. For an infinite dimensional vector space, take any linear map $$F: V -> W$$. Then $$U = \ker F$$ is a subspace of $$V$$. Note that we have a short exact sequence (if you don't know what that means, don't worry, the explanation is coming) $$0\to U\to V\to V/U\to 0$$

(That is, there's an injective map $$U \to V$$ (inclusion, I'll call it $$i$$) and a surjective map $$V \to V/U$$ (the quotient map, I'll call it $$q$$) such that the image of the injection is the kernel of the surjection).

But there's also a surjective map $$V \to U$$ (projection onto $$U$$, I'll call it $$p$$), and note that for any $$u \in U$$, $$pi(u) = u$$ (since $$p$$ fixes $$u$$).

Now, we're going to show that $$V$$ is the (internal) direct sum of the kernel of $$p$$ and the image of $$i$$. First, note that it's the sum of the two: for any $$v \in V$$, $$v = (v - ip(v)) + ip(v)$$, $$ip(v)$$ is obviously in the image of $$i$$, and $$p(v - ip(v)) = p(v) - pip(v) = 0$$ (with the last equality being due to our note about $$p$$, since $$p(v)\in U$$.

And further, the intersection is trivial: if $$v \in \ker(p)\cap\mathrm{im}(i)$$ then there is some $$u\in U$$ such that $$i(u) = v$$, and $$pi(u) = p(v) = 0$$, but $$pi(u) = u$$, so $$u = 0$$, hence $$v = 0$$. Thus, $$V = \ker(p)\oplus \mathrm{im}(i)$$.

Now, it's clear that $$\mathrm{im}(i)\cong U$$ since it's the image of $$U$$ under an injective map, so we need only show that $$\ker(p)\cong V/U$$.

For that purpose, since $$q$$ is surjective, for any $$w \in V/U$$, there is some $$v \in V$$ such that $$w = q(b)$$. But since $$V = \ker(p)\oplus\mathrm{im}(i)$$, there are unique $$u \in U$$, $$x \in \ker(p)$$ such that $$v = i(u) + x$$, so $$w = q(b) = q(i(u)+x) = qi(u) + q(x)=q(x)$$ (since the image of $$i$$ is the kernel of $$q$$, in particular $$qi = 0$$). Thus, $$q|_{\ker(p)}: \ker(p)\to V/U$$ is surjective. But also, if $$q(v) = 0$$, and $$v \in \ker(p)$$. But $$\ker(q) = \mathrm{im}(i)$$, and $$p$$ fixes $$\mathrm{im}(i)$$, so the only element that it sends to $$0$$ is $$0$$ itself, so we must have $$v = 0$$, hence $$q|_{\ker(p)}$$ is also injective, so is an isomorphism.

Thus, we have $$V \cong \mathrm{im}(i)\oplus\ker(p) \cong U\oplus (V/U)$$, as required.

This is precisely a special case of the Splitting Lemma, and my proof is essentially just one part of the proof of that, translated: there are easier ways to prove it, but I thought that it would be useful to see it in a more broadly applicable form.


# Query on a Solution to the Problem: $\gcd(5a+2,7a+3)=1$ for all integer $a$.

I wish to show that the numbers $$5a+2$$ and $$7a+3$$ are relatively prime for all positive integer $$a$$.

Here are my solutions.

Solution 1. I proceed with Euclidean Algorithm. Note that, for all $$a$$, $$|5a+2|<|7a+3|$$. By Euclidean Algorithm, we can divide $$7a+3$$ by $$5a+2$$. To have

$$7a+3=(5a+2)(1)+(2a+1)$$ continuing we have,

$$5a+2=(2a+1)(2)+a$$

$$2a+1=(2)(a)+1$$

$$2=(1)(2)+0$$

Since the last nonzero remainder in the Euclidean Algorithm for $$7a+3$$ and $$5a+2$$ is 1, we conclude that they are relatively prime.

Solution 2. Suppose that $$d=\gcd(5a+2,7a+3)$$. Since $$d=\gcd(5a+2,7a+3)$$ then the following divisibility conditions follow:

(1) $$d\mid (5a+2)$$

(2) $$d\mid (35a+14)$$

(3) $$d\mid (7a+3)$$

(4) $$d\mid (35a+15)$$.

Now, (2) and (4) implies that $$d$$ divides consecutive integers. The only (positive) integer that posses this property is $$1$$. Thus, $$d=1$$ and that $$7a+3$$ and $$5a+2$$ are relatively prime.

Here are my questions:

1. Is the first proof correct or needs to be more specific? For instance cases for $$a$$ must be considered.

2. Which proof is better than the other?

Thank you so much for your help.

• I think both approaches are good. Nor are they that different really...in both cases you are trying to find smaller and smaller multiples of $d$.
– lulu
Sep 24 '18 at 15:23
• Thank you very much for the kind comment @lulu. Got a follow up question. By "you are trying to find smaller and smaller multiples of $d$, you mean the process of continously dividing the divisor to remainder so that $r$ decreases? Sep 24 '18 at 15:27
• Is there a typo in (4), i.e. shouldn't 35 be multiplied by $a$?
– user431008
Sep 24 '18 at 15:28
• I meant something less precise than that. Euclid provides a somewhat systematic way to find new numbers that $d$ divides...the second method is less systematic, but faster as you seek out convenient expressions. Working by hand, I prefer the second method...were I trying to automate the process, I'd prefer the systematic approach.
– lulu
Sep 24 '18 at 15:28
• I agree @marmot. Thank you for pointing it out. Sep 24 '18 at 15:29

In the first solution you're not using, strictly speaking, the Euclidean algorithm, but a looser version thereof:

Let $$a$$, $$b$$, $$x$$ and $$y$$ be integers; if $$a=bx+y$$, then $$\gcd(a,b)=\gcd(b,y)$$.

The proof consists in showing that the common divisors of $$a$$ and $$b$$ are the same as the common divisors of $$b$$ and $$y$$.

There is no requirement that $$a\ge b$$ or that $$y$$ is the remainder of the division. Indeed your argument actually has a weakness, because $$7a+3\ge 5a+2$$ only if $$2a\ge-1$$, so it doesn't hold for $$a\le-2$$. But $$7a+3\ge 5a+2$$ is not really needed for the argument.

Since successive application of the statement above show that $$\gcd(2a+1,2)=1$$ and the gcd has never changed in the various steps, you can indeed conclude that $$\gcd(5a+2,7a+3)=1$$.

The second solution is OK as well.

You can simplify it by noting that if $$d$$ is a common divisor of $$5a+2$$ and $$7a+3$$, then it divides also $$5(7a+3)-7(5a+2)=1$$

• Thanks for the clarification in solution 1 and by giving a simplified version for solution 2. It is very clear to me now. @egreg Sep 24 '18 at 15:35

Your first proof is correct. I would perhaps complete it, writing that\begin{align}1&=(2a+1)-2a\\&=2a+1-2\bigl(5a+2-2(2a+1)\bigr)\\&=5(2a+1)-2(5a+2)\\&=5\bigl(7a+3-(5a+2)\bigr)-2(5a+2)\\&=5(7a+3)-7(5a+2).\end{align}

Your second proof also works, but it doesn't generalize easily to other situations.

• Thank you very much for completing the proof of solution 1 Prof. I am interested in a case that solution 2 wont work. Thank you Prof. Sep 24 '18 at 15:41
• @JrAntalan Concerning the second proof, I only meant that you have to think about it in a case-by-case basis. Sep 24 '18 at 15:43
• Noted Prof. and Thank you again. Sep 24 '18 at 15:44

Here is a different rendering of the same arguments.

Let $$u=5a+2$$, $$v=7a+3$$. Then $$\pmatrix{ u \\ v} = \pmatrix{ 5 & 2 \\ 7 & 3} \pmatrix{ a \\ 1}$$ and so $$\pmatrix{ a \\ 1} = \pmatrix{ 5 & 2 \\ 7 & 3}^{-1} \pmatrix{ u \\ v} = \pmatrix{ \hphantom- 3 & -2 \\ -7 & \hphantom-5} \pmatrix{ u \\ v}$$ This gives $$1 = -7u+5v = -7(5a+2)+5(7a+3)$$ The key point here is that the matrix has determinant $$1$$ and so its inverse has integer entries.


# For any arrangment of numbers 1 to 10 in a circle, there will always exist a pair of 3 adjacent numbers in the circle that sum up to 17 or more

I set out to solve the following question using the pigeonhole principle

Regardless of how one arranges numbers $1$ to $10$ in a circle, there will always exist a pair of three adjacent numbers in the circle that sum up to $17$ or more.

My outline

[1] There are $10$ triplets consisting of adjacent numbers in the circle, and since each number appears thrice, the total sum of these adjacent triplets for all permutations of the number in the circle, is $3\cdot 55=165$.

[2] If we consider that all the adjacent triplets sum to 16 , and since there are $10$ such triplets, the sum accordingly would be $160$, but we just said the invariant sum is $165$ hence there would have to be a triplet with sum of $17$ or more.

My query

Could someone polish this into a mathematical proof and also clarify if I did make use of the pigeonhole principle.

• Yes, you used pigeonhole. Should say if all sums are $\le 16$, then the sum is $\le 160$. – André Nicolas May 25 '13 at 4:42
• @AndréNicolas noted and aha , the syntax part is exactly why I asked the question on here. – metric-space May 25 '13 at 4:44
• @nerorevenge Note: You can actually show that the 3 adjacent numbers sum to 18 or more. – Calvin Lin May 25 '13 at 8:37

Yes, you used the Pigeonhole Principle. As a very mild correction, you should say that of all sums of three consecutives are $\le 16$, then the sum is $\le 160$.

The proof (with the small correction) is already fully mathematical. Conceivably you might want to explain the $55$ further. It is clear to you and to most users of this site why $55$, but imagine the reader to be easily puzzled.

The part about "all the permutations" is vague, and technically incorrect. You are finding the sum of a consecutive triple, and summing all these sums. Permutations have nothing to do with it, we are talking about a particular fixed arrangement of the numbers.

Remark: We could use a lot of symbols. Starting at a particular place, and going counterclockwise, let our numbers be $a_1,a_2,\dots,a_{10}$. For any $i$, where $1\le i\le 10$, let $S_i=a_i+a_{i+1}+a_{i+2}$, where we use the convention that $a_{10+1}=a_1$, $a_{10+2}=a_2$, and $a_{9+2}=a_1$.

Then $S_1+S_1+\cdots+S_{10}=165$, since each of $1$ to $10$ appears in $3$ of the $S_i$, and $1+2+\cdots+10=55$.

But if all the $S_i$ are $\le 16$, then $\sum_{i=1}^{10}S_i\le 160$. This contradicts the fact that $\sum_{i=1}^{10}S_i=165$.

I prefer your proof, mildly modified.

• eh, is that all? the reason I'm asking is that, inb the past my so called 'proofs' have been criticised for being too informal – metric-space May 25 '13 at 4:45
• @nerorevenge: I am an advocate of the informal but clear. I hope that someone more formal-minded will add an answer. The only issue is that you perhaps left an easily filled gap or two. – André Nicolas May 25 '13 at 4:53
• I have added another criticism. You should really fix the wording there. – André Nicolas May 25 '13 at 4:58
• I agree.That definitely is vaguely worded. – metric-space May 25 '13 at 4:59
• Presumably the "pair" wording came with the problem. But it looks funny to call a trio a pair. – André Nicolas May 25 '13 at 5:23

We will show something stronger, namely that there exists 3 adjacent numbers that sum to 18 or more.

Let the integers be $\{a_i\}_{i=1}^{10}$. WLOG, $a_1 = 1$. Consider

$$a_2 + a_3 + a_4, a_5 + a_6 + a_7, a_8 + a_9 + a_{10}$$

The sum of these 3 numbers is $2+3 +\ldots + 10 = 54$. Hence, by the pigeonhole principle, there exists one number which is at least $\lfloor \frac{54}{3} \rfloor = 18$.

I leave it to you to show that there is a construction where no 3 adjacent numbers sum to 19 or more, which shows that 18 is the best that we can do.

• definitely a stronger proof ,and I have admit it sure is nice, I wonder why cut-the-knot.org didn't ask people to solve for what you proved. – metric-space May 25 '13 at 16:45
• @nerorevenge I used this as a problem on Brilliant.org, and most could only get to 17, and didn't understand why 18 must be true. The 18 case arises quite easily by ignoring the small values which would otherwise reduce our sum. – Calvin Lin May 25 '13 at 23:37


# To find number of real roots

Consider the equation $$x^5-5x=c$$ where c is a real number.

Determine all c such that this equation has exactly 3 real roots.

I know that between consecutive real roots of $$f$$ there is a real root of $$f'$$. Now $$f'$$ in this case is $$5x^4-5$$ which always has two real roots. So the claim should be true for all c.

But I KNOW IT IS NOT TRUE. Where am I messing up?

• Note that while it is necessary for $f'$ to have two real roots (in order for $f$ to have exactly three real roots), it is not sufficient. Of course an odd degree polynomial will always have one real root. – hardmath Dec 12 '19 at 16:12
• ok so how do I find all such c? – Angry_Math_Person Dec 12 '19 at 16:14
• Consider the graph of $p(x) = x^5 - 5x$. Changing the constant $c$ in your equation amounts to moving a horizontal line up or down across this graph. – hardmath Dec 12 '19 at 16:15

Yes, between any two roots of $$f$$, there is a root of $$f'$$. However, just because $$f'$$ has a root, that doesn't mean that $$f$$ has a root on either side. Consider $$f(x)=x^2+1$$.

As for solving this problem, the derivative has only two roots, so we can at most have three roots. For some values of $$c$$ we have three roots, for some values of $$c$$ we have a single root, and for exactly two values of $$c$$, there are two roots. The three-root region is exactly the interval between the two two-root values of $$c$$.

And finding the values of $$c$$ that gives two roots is easier than one might think. They happen exactly when one root of $$f$$ coincides with a root of $$f'$$. So find the roots of $$f'$$, and find the values of $$c$$ that make each of them a root of $$f$$, and you have found the interval of $$c$$-values that gives three roots.

As you know, the given equation has extrema at $$x=\pm1$$. These correspond to values of the polynomial

$$1-5-c$$ and $$1+5-c$$ (the RHS was moved to the left).

Hence the polynomial will grow from $$-\infty$$, reach the maximum, then the minimum and continue growing to $$\infty$$. There are three roots when $$0$$ is in the range $$(-4-c,6-c)$$.

Where am I messing up? Just look at a graph where it fails.

$$x^5-5 x -5$$

I know that between consecutive real roots of f there is a real root of f′. Now f′ in this case is 5x4−5 which always has two real roots. So the claim should be true for all c.

$$A \implies B$$ does not mean $$B \implies A$$.

The two real roots of $$5x^4 - 5$$ are the two roots at $$x = \pm 1$$.

If $$x^5 - 5x=c$$ has three roots then they will be at $$x < -1; -1 < x < 1;$$ and at $$x > 1$$ by your condition.

But there won't be three real roots if there is no root for any $$x< -1$$, or no root between $$-1$$ and $$1$$, or no root for any $$x < -1$$.

$$x=\pm 1$$ are extreme points and if one, the max, is $$>0$$ and the other $$<0$$ then there will be three real roots. But if both are "on the same side of $$0$$" there is no root between them and no root to "the other side".

$$x^5-5x -c|_{-1} = 4-c$$ and $$x^5 - 5x -c|1 = -4-c$$ so $$x =-1$$ is a max and $$x = 1$$ is a min.

If $$f(-1) = 4-c \le 0$$ is a max there will be no root for $$x < -1$$ or for $$-1 < x \le 1$$. If $$f(1) = -4-c\ge 0$$ is a min there will be no root for $$x > 1$$ or for $$-1 \le x < 1$$. So if either $$c \ge 4$$ or if $$c\le 4$$ then there are fewer than three real roots. But if $$-4 < c < 4$$ then there will be three.

Alternatively: we know what the shape of an odd polynomial $$x^5 -5x$$ looks like. It's that polynomial curve with a twisty bit in the middle, goes off to infinity as $$x \to \infty$$, goes to negative infinity as $$x \to -\infty$$. It has roots were the x-axis crosses it (or where it crosses the x-axis-- everything is relative). If we shift it up or shift it down by $$c$$ we can force the x-axis to avoid the twisty bits in the middle and have it have only one root. Or we can deliberately shift it so that the x-axis goes smack through the twisty bits and we have a maximum number of roots. So if $$c$$ is between the max and mins we maximize the number of roots and the x-axis goes through the twisty bits. If $$c$$ is beyond the max an mins we've shoved the twisty bits below or above the x-axis and there is only one root.


# Difference between continuity and uniform continuity

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1. May 7, 2015

### Yunjia

I noticed that uniform continuity is defined regardless of the choice of the value of independent variable, reflecting a function's property on an interval. However, if on a continuous interval, the function is continuous on every point. It seems that the function on that interval must be uniformly continuous. Is this correct? Is there a counterexample for the statement?

2. May 7, 2015

### jbunniii

What you said is almost true. If, on a closed, bounded interval, a function is continuous at every point, then the function is uniformly continuous on that interval.

Counterexample on a non-closed interval: $f(x) = 1/x$ on the interval $(0,1)$.

Counterexample on a closed but unbounded interval: $f(x) = x^2$ on the interval $[0,\infty)$.

3. May 8, 2015

### Svein

Even more general: If a function is continuous at every point in a compact set, it is uniformly continuous on that set.
The proof is simple: Let the compact set be K and take an ε>0. Since the function is continuous at every point x in the set, there is a δx for every x∈K such that |f(w)-f(x)|<ε for every w in <x-δx, x+δx>. Let Ox=<x-δx, x+δx>, then K is contained in $\bigcup_{x\in K}O_{x}$. Since K is compact, it is contained in the union of a finite number of the Ox, say $\bigcup_{n=1}^{N}O_{n}$. Take δ to be the minimum of the δn and |f(w)-f(x)|<ε whenever |w-x|<δ.

4. May 8, 2015

### jbunniii

And here's the proof that a closed, bounded interval is compact, so you can apply Svein's proof.

Let $[a,b]$ be a closed, bounded interval, and let $\mathcal{U}$ be any collection of open sets which covers $[a,b]$. Let $S$ be the set of all $x \in [a,b]$ such that $[a,x]$ can be covered by finitely many of the sets in $\mathcal{U}$. Clearly $a \in S$. This means that $S$ is nonempty and is bounded above (by $b$), so it has a supremum, call it $c$. Since $c \in [a,b]$, it is contained in some $U_c \in \mathcal{U}$, hence there is some interval $(c - \epsilon, c + \epsilon)$ contained in $U_c$. Since only finitely many sets from $\mathcal{U}$ are needed to cover $[a,c - \epsilon/2]$, those sets along with $U_c$ form a finite cover of $[a,c+\epsilon/2]$. This shows that $c \in S$ and moreover, that $c$ cannot be less than $b$. Therefore $c=b$, so all of $[a,b]$ can be covered by finitely many sets in $\mathcal{U}$.

Last edited: May 8, 2015
5. May 8, 2015

### Svein

Excellent! And, of course, since one closed and bounded interval is compact, the union of a finite number of closed and bounded intervals is again compact.

6. May 8, 2015

### jbunniii

I don't think this will work if $w$ and $x$ are not contained in the same $O_n$. I think you need to take $O_x = (x - \delta_x/2, x + \delta_x/2)$ and $\delta$ to be $\min\{\delta_n/2\}$ in order to ensure that $|w-x| < \delta$ implies $|f(w) - f(x)| < \epsilon$.

7. May 8, 2015

### Svein

Possibly. I need to think about that. The crux of the matter is that there is a finite number of intervals that cover K, which means that the minimum of the (finite number of) δ's exist and is greater than 0.

8. May 8, 2015

### HallsofIvy

Here is the fundamental difference between "continuous" and "uniformly continuous":

A function is said to be continuous at a point, x= a, if and only if, given any $\epsilon> 0$ there exist $\delta> 0$ such that if $|x- a|< \delta$ then $|f(x)- f(a)|<\epsilon$.

A function is said to be continuous on a set, A, if and only if, given any $\epsilon> 0$ there exist $\delta> 0$ such that if $|x- a|< \delta$ then $|f(x)- f(a)|<\epsilon$ for all a in set A.

That is, a function is uniformly continuous on a set A if and only if it is continuous at every point in A and, given $\epsilon> 0$ the same $\delta$ can be used for ever point in A.

9. May 8, 2015

### Svein

Agree. I was sloppy and overlooked the simple fact that that the length of the interval Ox is 2⋅δx.

10. May 13, 2015

### Yunjia

The defition of uniformly continuous is perplexing for me. Is it possible for you to illustrate it with a proof about a function which is not uniformly continuous but continuous on a set so that I can compare the two? Thank you.

11. May 14, 2015

### Svein

OK. Take the function $f(x)=\frac{1}{x}$ on the open interval <0, 1>. It is continuous (and even differentiable) on that interval, but not uniformly continuous.

12. May 17, 2015

### Svein

After using pencil an paper for a bit, I came to the conclusion that I should have used ε/2 and δ/2. But - I recall a proof in "Complex analysis in several variables" that ended up in "... less than 10000ε, which is small when ε is small".

13. May 17, 2015

### HallsofIvy

First note an important point about "uniform continuity" and "continuity" you may have overlooked: we define "continuous" at a single point and then say that a function is "continuous on set A" if and only if it is continuous at every point on A. We define "uniformly continuous" only on a set, not at a single points of a set.

A function is uniformly continuous on any closed set on which it is continuous and so on any set contained in a closed set on which it is continuous.
To give an example of a function that is continuous but not uniformly continuous, we need to look at f(x)= 1/x on the set (0, 1).

To show that it is continuous on (0, 1), let a be a point in (0, 1) and look at $|f(x)- f(a)|= |1/x- 1/a|= |a/ax- x/ax|= |(a- x)/ax|= |x- a|/ax< \epsilon$.
We need to find a number, $\delta> 0$ such that if $|x- a|< \delta$, then $|f(x)- f(a)|> \epsilon$. We already have $|x- a| ax\epsilon$ so we need an upper bound on ax. If we start by requiring that $\delta< a/2$ then $|x- a|< \delta< a/2$ so that $-a/2< x- a< a/2[itex] or [itex]a/2< x< 3a/2$ so an upper bound on ax is $3a^2/2$. If $|x- a|< a/2$ and $|x- a|< 3a/2$ then $|f(x)- f(a)|< |x- a|/ax< |x- a|/(3a^2/2)= 2|x- a|/3a^2$ which will be less than $\epsilon$ as long as $|x- a|< 3a^2\epsilon/2$

So we can take $\delta$ to be the smaller of $a/2$ and $3a^2/\epsilon$. Therefore, 1/x is continuous at any point a in (0, 1) so continuous on (0, 1).

Now the point is that this $\delta$ depends on a. It is a decreasing function of a and will go to 0 as a goes to 0. That's fine for "continuity" but for uniform continuity we must be able to use the same $\delta> 0$ for a given $\epsilon$ no matter what the "a" is and we cannot do that. If the problem were to prove uniform continuity on the set [p, 1), which is contained in the closed set [p, 1], we could use, for any a in that set, the $\delta$ that we get for a= p, the smallest x value in the set. But with (0, 1), the smallest $\delta$ is 0 which we cannot use since we must have $\delta> 0$.


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Currently he is on leave from UT Austin and heads the Amazon Research Lab in Berkeley, California, where he is developing and deploying state-of-the-art machine learning methods for Amazon Search. for automated market making. Grading: 3 homeworks 60%, 2 quizzes 20%, 1 project 20%. edu ABSTRACT Accelerating machine learning (ML) over relational data is a. , Bengio, Y. This repository contains all the quizzes/assignments for the specialization "Mathematics for Machine learning" by Imperial College of London on Coursera. Some recent tutorials by Christos and Co. Lawrence [email protected] Welcome to Data analysis with Python - 2020¶. The aim of these notebooks is to help beginners/advanced beginners to grasp linear algebra concepts underlying deep learning and machine learning. A vector in Rn will be denoted as: ~x. Enabling and Optimizing Non-linear Feature Interactions in Factorized Linear Algebra Side Li University of California, San Diego [email protected] HarvardX Biomedical Data Science Open Online Training In 2014 we received funding from the NIH BD2K initiative to develop MOOCs for biomedical data science. Now I’m checking back in with 9 weeks under my belt. Introduction to Machine Learning for Data Science, Udemy. Learn linear algebra. The first part covers basics and preliminaries. Incorporating machine learning capabilities into software or apps is quickly becoming a necessity. Some recent tutorials by Christos and Co. pyplot as plt # pyplot from collections import defaultdict , Counter from functools import partial , reduce. pyplot as plt # pyplot from collections import defaultdict , Counter from functools import partial , reduce. Introduction The aim of this set of lectures is to review some central linear algebra algorithms that we. In case of deep learning algorithms, linear algebra is the driving force. We study fast algorithms for linear algebraic problems that are ubiquitous in data analysis and machine learning. After understanding the nature of both the problem and the field we are dealing with, and before learning how to prepare a data for your machine learning and do the cleaning and preparation for a selected problem. If you're a data scientist studying linear algebra, chances are you are interested in understanding how machine learning algorithms work. Linear Algebra and Learning from Data by Gilbert Strang; twitter github. 06 Linear Algebra - The video lectures are on web. I will list some resources for learning linear algebra. Andrzej Cichocki, Anh-Huy Phan, Qibin Zhao, Namgil Lee, Ivan Oseledets, Masashi Sugiyama, and Danilo Mandic. Linear Algebra for Machine Learning Book. You will also learn how you should use linear algebra in your Python code. Probability and Statistics: Learn Probability and Statistics Through Interactive Visualizations: Seeing Theory was created by Daniel Kunin while an undergraduate at Brown University. As a core programmer, I love taking challenges and love being part of the solution. Linear regression is one of the most popular machine learning algorithms. , and Courville, A. The elements of statistical learning: data mining, inference, and prediction, Springer, 2009 Linear Algebra and Probability Review (part 1 Linear Algebra, part 2 Probability) Assignment 1: Apr 10. Those equations may or may not have a solution. There is no doubt that linear algebra is important in machine learning. View on GitHub mlcourse. Run in Google Colab View source on GitHub Download notebook In this post, we will explore the ways of doing linear algebra only using tensorflow. Read more Tagged as : R linear algebra classification linear discriminant analysis. My work includes researching, developing and implementing novel computational and machine learning algorithms and applications for big data integration and data mining. You can use it as a main text, as a supplement, or for independent study. I’m interested in applying non-standard tools form abstract algebra and topology to the study of neural networks. About data set: Square feet is the Area of house. uk November 1, 2018 Abstract Development systems for deep learning (DL), such as Theano, Torch, TensorFlow, or MXNet, are. Linear algebra has had a marked impact on the field of statistics. In some cases, functions are provided for concepts available elsewhere in R, but where the function call or name is not obvious. GF2] = One Zero Zero Zero Zero Zero One Zero Zero Zero Zero Zero One Zero Zero Zero Zero Zero One Zero Zero Zero Zero Zero One scala> a + a res0: breeze. Siefken, J. APPLICATION: This can be pretty much applied in a generic way to all programs. In general, statistical problems have to do with the estimation of some characteristic derived from data - this can be a point estimate, an interval, or an entire function. Linear Algebra: Foundations to frontiers – edx. Linear Algebra for Machine Learning Discover the Mathematical Language of Data in Python. data in homework problems. Machine learning. Anyone can view the notebooks online by clicking on the links in the readme Table of Contents. In this day, we are going to make the dirty work. In this course on Linear Algebra we look at what linear algebra is and how it relates to vectors and matrices. In other cases, functions are provided to show or. Linear algebra (Systems of linear equations, least-square) [Matrix cookbook] 4: 9/16/2019: Linear Algebra (Vector space, linear independence) 9/18/2019: Linear algebra (Eigendecomposition and matrix factorization) Homework 1 (extended) 5: 9/23/2019: Linear algebra (Eigendecomposition and matrix factorization) 9/25/2019: Linear algebra (Best fit. Acquiring these skills can boost your ability to understand and apply various data science algorithms. The basic mathematics prerequisites for understanding Machine Learning are Calculus-I,II,III, Linear Algebra, and, Probability and Statistics. Making statements based on opinion; back them up with references or personal experience. Randomized numerical linear algebra. It lacks the ability of distributed linear algebra computation in its local interactive shell. You can use it as a main text, as a supplement, or for independent study. NVIDIA CUDA-X GPU-Accelerated Libraries NVIDIA® CUDA-X, built on top of NVIDIA CUDA®, is a collection of libraries, tools, and technologies that deliver dramatically higher performance—compared to CPU-only alternatives— across multiple application domains, from artificial intelligence (AI) to high performance computing (HPC). Lek-Heng Lim. taco is versatile. However, I think that the chapter on linear algebra from the book is a bit tough for beginners. A few weeks ago, I wrote about how and why I was learning Machine Learning, mainly through Andrew Ng’s Coursera course. scala> val a = DenseMatrix. This code has been posted to GitHub under a MIT license, so feel free to modify and deal with code without any restrictions or limitations (no guarantees of any kind. world Overview of scikit-learn Python and Excel Scaling, Centering, Noise with kNN, Linear Regression, Logit Sentiment Analysis with Twitter Time Series Analysis Vectors and Arrays (Linear Algebra) Vectors and Arrays (Linear Algebra) Table of contents. Lesson 1 (April 7): Machine learning pipeline and course overview: video; slides. Also,it would be of much help if they have big set of problems and examples. Machine Learning course (Andrew Ng) is a basic machine learning course. It begins with linear algebra—matrix factorizations A= QR. therefore precede our in tro duction to deep learning with a fo cused presen tation of. Linear algebra cheat sheet for deep learning – Towards Data Science – Medium. They give you better intuition for how algorithms really work under the hood, which enables you to make better decisions. It’s all vectors and matrices of numbers. Today, I will be sharing with you my C# implementation of basic linear algebra concepts. In an image classification problem, we often use neural networks. Matrices and Linear Algebra The Wolfram Language automatically handles both numeric and symbolic matrices, seamlessly switching among large numbers of highly optimized algorithms. R is a widely-used statistical programming language in the data science community. Artificial Neural Networks. Grading (tentative) Quizzes 20%; Course project. Together with your editor or Jupyter notebook these packages allow you to rapidly develop scalable, high-performance analytics and visualizations using succinct, type-safe, production-ready code. eye[GF2](5) a: breeze. Windows-64 (64-bit linear algebra for large data) Unless your computer has more than ~32GB of memory and you need to solve linear algebra problems with arrays containing more than ~2 billion elements, this version will offer no advantage over the recommended Windows-64 version above. For beginning practitioners (i. View on GitHub mlcourse. Tibshirani, J. Open Library is an initiative of the Internet Archive, a 501(c)(3) non-profit, building a digital library of Internet sites and other cultural artifacts in digital form. Another resource is the book with the funny title “No Bullshit Guide to Linear Algebra” by Ivan Savov. It introduces some common tools in machine learning to resolve real applications (e. It calls them tensors. Linear Algebra Examines basic properties of systems of linear equations, vector spaces, inner products, linear independence, dimension, linear transformations, matrices, determinants, eigenvalues, eigenvectors and diagonalization. You will also learn how you should use linear algebra in your Python code. mathematics-for-machine-learning-cousera. We start with representing a fully connected layer as a form of matrix multiplication: - In this example, the weight matrix has a size of $4 \times 3$, the input vector has a size of $3 \times 1$ and the output vector has a of size $4 \times 1$. The code (from. Understanding API Security (Free chapters from a Manning. It turns out, however, that all of those operations can be written in terms of big matrix-matrix or matrix-vector multiplications. 1x Scalable Machine Learning. AMD adopted BLIS as its new BLAS library. After reading this quickstart, you can go to other wiki pages, especially Linear Algebra Cheat-Sheet and Data Structures. I'm studying towards a PhD degree at the University of Chicago, in the department of Statistics. SciPy is built to work with NumPy arrays and provides many. They are full of explanations, code samples, pictures, interesting links, and exercises for you to try. linear; algebra; matrix;. import re , math , random # regexes, math functions, random numbers import matplotlib. By Hadrien Jean, Machine Learning Scientist. Compressed Linear Algebra for Declarative Large-Scale Machine Learning Ahmed Elgohary2, Matthias Boehm1, Peter J. What we did here by attaching the variable mlr to the MyLinearRegression class is to create an instance, a specific object called mlr, which will have its own data and "functions". In the second part, we discuss how deep learning differs from classical machine learning and explain why it is effective in dealing with complex problems such as image and natural language processing. Plotting is based on OpenGL and supports both 2D and 3D plots. Introduction to linear algebra (Fourth Edition). DenseMatrix[X. This content is part of a series following the chapter 2 on linear algebra from the Deep Learning Book by Goodfellow, I. therefore precede our in tro duction to deep learning with a fo cused presen tation of. You (the student) should have taken a mathematical course on linear algebra that covers vector spaces as well as a numerical analysis course that covers computer implementations of numerical algorithms. Fitting Lines to Data. net/?Q73_jQ 2020-01-24T14:53:56+01:00 2020-01-24T15:29:05+01:00. Linear Algebra for Data Science using Python Play all 13:42 Math For Data Science | Practical reasons to learn math for Machine/Deep Learning - Duration: 13 minutes, 42 seconds. The mentors for this project are: @dpshelio @mbobra @drsophiemurray @samaloney. Randy Lao's site for free Machine Learning and Data Science resources and materials. How to Learn Advanced Mathematics Without Heading to University - Part 1 subjects to learn for a prospective quant or data scientist. The Cuckoo linear algebra implementation is based on libcuckoo library1. to map the pixel values of an image to the confidence score of each class. This library holds the principal work done as part of the OpenAstonomy Google Summer of Code 2020 project, Solar Weather Forecasting using Linear Algebra. library with basic linear algebra routines, and the SciPy library adorns NumPy arrays with many important primitives, from numerical optimizers and signal processing to statistics and sparse linear algebra. Download R for Windows 5. Linear Algebra Preliminaries¶ Since I have documented the Linear Algebra Preliminaries in my Prelim Exam note for Numerical Analysis, the interested reader is referred to for more details (Figure. Intro to Data Science / UW Videos. Now we extend linear algebra to convolutions, by using the example of audio data analysis. The Linear Algebra Chapter in Goodfellow et al is a nice and concise introduction, but it may require some previous exposure to linear algebra concepts. AMD adopted BLIS as its new BLAS library. Each point correspondence generates one constraint on F. I had to formulate an algorithm to convert an image of some resolution - say L x M and crop / re-size the image into a new resolution say P x R such that I cover the maximum amount of points/pixels from the original image. Again, the class MyLinearRegression provides instructions on how to build a linear regression model. Linear Algebra: Video: Professor Gilbert Strang's Video Lectures on linear algebra. Above, I created 4 matrices. In addition to this, you’ll be able to perform operations such as addition, subtraction and dot product. It introduces some common tools in machine learning to resolve real applications (e. Conncect between Geometry and Algebra. As of October 2019, I am a senior algorithms scientist at PathAI, where I work on computational pathology. California Housing Price Prediction. Linear Optimization in Python 6. The aim of these notebooks is to help beginners/advanced beginners to grasp linear algebra concepts underlying deep learning and machine learning. 6 (376 ratings) Course Ratings are calculated from individual students’ ratings and a variety of other signals, like age of rating and reliability, to ensure that they reflect course quality fairly and accurately. View on GitHub. They give you better intuition for how algorithms really work under the hood, which enables you to make better decisions. Calculate The Trace Of A. pdf CarletonU- hamidreza sadreazami- radar based fall detection with supervised learning. However, it is currently not easy to implement many basic machine learning primitives in. " Our homework assignments will use NumPy arrays extensively. We start with representing a fully connected layer as a form of matrix multiplication: - In this example, the weight matrix has a size of $4 \times 3$, the input vector has a size of $3 \times 1$ and the output vector has a of size $4 \times 1$. In forecasting, Yuyang has worked on all aspects ranging from practical applications to theoretical foundations. I'm a Data Science practitioner and computer programmer with an avid interest in Exploratory Data Analysis, Statistics, & Machine Learning. edu Arun Kumar University of California, San Diego [email protected] DenseMatrix[X. SciPy is open-source software for mathematics, science, and engineering which includes modules for statistics, optimisation, integration, linear algebra, Fourier transforms, signal and image processing, ODE solvers, and more. Now I’m checking back in with 9 weeks under my belt. Mining the social web: Data mining Facebook, Twitter, LinkedIn, Google+, GitHub, and more (2nd edition) by Matthew A. The Deep Learning Book - Goodfellow, I. Most importantly, the online version of the book is completely free. Based on the second linear algebra course taught by Professor Strang, whose lectures on the training data are widely known, it starts from scratch (the four fundamental subspaces) and is fully accessible without the first text. Siefken, J. then this is the book for you. The aim of these notebooks is to help beginners/advanced beginners to grasp linear algebra concepts underlying deep learning and machine learning. How to Learn Advanced Mathematics Without Heading to University - Part 1 subjects to learn for a prospective quant or data scientist. Rich Ott leads you through two days of intensive learning that include a review of linear algebra essential to machine learning, an introduction to TensorFlow, and a dive into neural networks. All publications, sorted by year. This book provides the conceptual understanding of the essential linear algebra of vectors and matrices for modern engineering and science. n Gilbert Strang. As a motivating example, let us consider image classification. The collection of all linear combinations is called a linear subspace of $\RR^n$, denoted by We will say that the $\bb{v}_i$’s span the linear subspace $\mathcal{L}$. If you have more time to dedicate to your projects and you're also passionate about math, consider to contribute to the library!. Roadmap to begin with Machine Learning: The place to start is to learn (and/or) revise linear algebra. Find the smallest value of k such that the rank-k approximation of the matrix uses the same or more amount of data as the original picture. Bourbaki resulted from similar currents of thought that produced fascism and totalitarian communism: moral panics leading to revolutions, and ultimately “final solutions”, all terrible and evil in. Videos and textbooks with relevant details on linear algebra and singular value decomposition (SVD) can be found by searching Alfredo’s Twitter, for example type linear algebra (from:alfcnz) in the search box. Machine Learning/Data Science. , less sensitive to noisy data) than the Euclidean norm The following result is fundamental in linear algebra: Theorem. This is the world beyond R and Python! Breeze is a library for numerical processing, like probability and statistic functions, optimization, linear algebra, etc. Mathematics for machine learning - I totally recommend this book! If you want to learn the bits and pieces of how linear algebra and calculus is used to develop algorithms like principal component analysis, backpropagation etc. a person's height and you switch from meter to centimeter. Linear Algebra and Learning from Data by Gilbert Strang; twitter github. You NEED Linear Algebra for Machine Learning Whether you want to learn Machine Learning for your work or research or you want to become a master, so the others pay you to do it, you need to know how it works. Linear Algebra: Video: Professor Gilbert Strang's Video Lectures on linear algebra. You will also learn how you should use linear algebra in your Python code. There is no doubt that linear algebra is important in machine learning. Tensor networks for dimensionality reduction and large-scale optimization: part 2 applications and future perspectives. Understanding of calculus, linear algebra, and programming is essential. to map the pixel values of an image to the confidence score of each class. Start here. PBblas - Parallel Block Linear Algebra Subsystem HPCC Systems ML_Core repository on GitHub Installation: ecl bundle install https:. We emphasize that this document is not a. References for "practical" machine learning: Python for data analysis by Wes McKinney. The Vector class imitates the m x 1 vector from linear algebra and contains many useful functions for dealing and interacting with Vectors. The free video lectures of this course are made available as part of Harvard Extension School's Opening Learning Initiative. After reading this quickstart, you can go to other wiki pages, especially Linear Algebra Cheat-Sheet and Data Structures. You can think of an r t i m e s c r times c r t i m e s c matrix as a set of r r r row vectors, each having c c c elements; or you can think of it as a set of c c c column vectors, each having r r r elements. " Our homework assignments will use NumPy arrays extensively. This is a straightforward course to learn Linear Algebra Fundamentals for Data Science in Python. linear; algebra; matrix;. 1-11, [Online-Edition. Row Reduction We row reduce a matrix by performing row operations, in order to find a simpler but equivalent system for which the solution set is easily read off. Support Stability of Maximizing Measures for Shifts of Finite Type Journal of Ergodic Theory and Dynamical Systems (accepted) Calkins, S. If you want to fit a model of higher degree, you can construct polynomial features out of the linear feature data and fit to the model too. Supratim Haldar Lead Data Scientist at Head Digital Works Pvt. Popular Courses Popular with our users in the last month Introduction to Linear Algebra. Made for sharing. Linear Algebra The Rank of a Matrix. For simple linear regression, one can choose degree 1. Linear algebra provides a way of compactly representing and operating on sets of linear equations. This repository contains the learning material for the Nuclear TALENT course Learning from Data: Bayesian Methods and Machine Learning, in York, UK, June 10-28, 2019. Omoju: What I do at GitHub is I build data models, often deep learning models on GitHub data to help GitHub probably build things like a recommendation engine so we can recommend repositories to people. The first part covers basics and preliminaries. , hackers, coders, software engineers, and people working as data scientists in business and industry) you don't need to know that much calculus, linear algebra, or other college-level math to get things done. Grannan, S. This is the site for any aspiring data scientists that want to learn in a quick way. Decision theory, game theory, operational research, … (source: lecture video from The Machine Learning Summer School by Zoubin Ghahramani, Univ. Machine Learning is built on prerequisites, so much so that learning by first principles seems overwhelming. A computer science student that is interested in Machine Learning would be well advised to get a minor in Mathematics (or just get a degree in Mathematics instead!). 1 Positive Semide nite (PSD) and Positive De nite (PD) matrices. 08 Apr 2016 » Naive Bayes Classifiers in Rust Adding NB Classifiers to rusty-machine. References for "practical" machine learning: Python for data analysis by Wes McKinney. provide a summary of the mathematical background needed for an introductory class in machine learning, which at UC Berkeley is known as CS 189/289A. Machine Learning is not just writing Python or R. I still plan to go through it but mildly disappointed. 8 Web Framework. I'm a Data Science practitioner and computer programmer with an avid interest in Exploratory Data Analysis, Statistics, & Machine Learning. This will allow us to introduce some central programming features of high-level languages like Python and compiled languages like C++ and/or Fortran. In my opinion, it is one of the bedrock of machine learning, deep learning and data science. Many universities use the textbook Introduction to Linear Algebra. Implementation [ edit ] Scikit-learn is largely written in Python, and uses numpy extensively for high-performance linear algebra and array operations. This content is part of a series following the chapter 2 on linear algebra from the Deep Learning Book by Goodfellow, I. Probability and Statistics: Learn Probability and Statistics Through Interactive Visualizations: Seeing Theory was created by Daniel Kunin while an undergraduate at Brown University. In addition to this, you'll be able to perform operations such as addition, subtraction and dot product. As a core programmer, I love taking challenges and love being part of the solution. Students will learn and practice fundamental ideas of linear algebra and simultaneously be exposed to and work with real-world applications of these ideas. We will assume mathematical maturity and comfort with algorithms, probability, and linear algebra. Neural networks rely on it heavily, but so do linear regression, factor analysis, and lots of other methods. This Word Mover’s Distance (WMD) can be seen as a special case of Earth Mover’s Distance (EMD), or Wasserstein distance, the one people talked about in Wasserstein GAN. These subjects include matrix algebra, vector spaces, eigenvalues and eigenvectors, symmetric matrices, linear transformations, and more. In this course, you’ll learn how to work with vectors and matrices, solve matrix-vector equations, perform eigenvalue/eigenvector analyses and use principal component analysis to do dimension reduction on real-world datasets. In the field of data science, however, being familiar with linear algebra and statistics is very important to statistical analysis and prediction. Lawrence [email protected] Then last year I learned how he morphed his delightful mathematics book into a brand new title (2019) designed for data scientists - "Linear Algebra and Learning from Data. Choi et al. Now that you understand the key ideas behind linear regression, we can begin to work through a hands-on implementation in code. Linear Algebra The Rank of a Matrix. Linear Algebra. Gradient Descent with Linear Regression - GitHub Pages. The training data is used to find the optimal model but the model should ultimately work for the test data! Conclusion. Mathematics for machine learning - I totally recommend this book! If you want to learn the bits and pieces of how linear algebra and calculus is used to develop algorithms like principal component analysis, backpropagation etc. https://shaarli. Linear Algebra for Data Science using Python Play all 13:42 Math For Data Science | Practical reasons to learn math for Machine/Deep Learning - Duration: 13 minutes, 42 seconds. Twitter: @mpd37, @AnalogAldo, @ChengSoonOng. com) Bangalore, India * Working as a Lead Data Scientist at Head Digital Works Pvt. Using least-squares linear approximation techniques to find the best linear fit to a set of data points results in the equation of a line which minimizes the sum of the squares of the vertical distances from the given points to the line: Note that, unless the line is horizontal, the vertical distance will be slightly larger than the actual distance, which is measured in the direction. I recently released an efficient linear algebra library for Javascript. LINEAR ALGEBRA. Learn how to solve challenging machine learning problems with TensorFlow, Google’s revolutionary new software library for deep learning. import re , math , random # regexes, math functions, random numbers import matplotlib. Statistical Machine Learning (S2 2017) Deck 6 This lecture • Notes on linear algebra ∗Vectors and dot products ∗Hyperplanes and vector normals • Perceptron ∗Introduction to Artificial Neural Networks ∗The perceptron model ∗Stochastic gradient descent 2. A few weeks ago, I wrote about how and why I was learning Machine Learning, mainly through Andrew Ng’s Coursera course. Deep Learning Book Series · 2. Overview - Khan Academy Vectors and Spaces; Matrix Transformations; Python. This course develops the mathematical basis needed to deeply understand how problems of classification and estimation work. pdf; Basic Set Notation & Terminology. Machine learning yearning. hdf5 is a file. If you have more time to dedicate to your projects and you’re also passionate about math, consider to contribute to the library!. Module 0: Introduction & Outline About What is Data Science? Module 1: Required Background Math: Stats, Calculus, Linear Algebra Programming: Basics, Data Structures, Algorithms Databases: Relational Algebra, SQL Important Concepts: Regular expressions, Information Entropy, Distance measurements, OLAP, ETL, BI VS BA and CAP. NOTE: please check for the course practicalities, e. implemented vector and matrix classes with reST-formatted docstrings in Python 3+ General Layout. Linear Algebra Can Help You Choose Your Stock Portfolio Correlation is a very fundamental and viseral way of understanding how the stock market works and how strategies perform. Some recent tutorials by Christos and Co. Python is one of the most commonly used programming languages by data scientists and machine learning engineers. for automated market making. Udacity is the world’s fastest, most efficient way to master the skills tech companies want. I'm interested in applying non-standard tools form abstract algebra and topology to the study of neural networks. We will learn scientific computing basics, topics in numerical linear algebra, mathematical probability (probability spaces, expectation, conditioning, common distributions, law of large numbers and the central limit theorem), statistics (point estimation, confidence intervals, hypothesis testing, maximum likelihood estimation, density. This book provides the conceptual understanding of the essential linear algebra of vectors and matrices for modern engineering and science. Introduction Theunprecedentedadvanceindigitaltechnologyduringthesecondhalfofthe20thcenturyhas producedameasurementrevolutionthatistransformingscience. Now we are ready to see how matrix algebra can be useful when analyzing data. The linux command line: A complete introduction. As we will see, we can do all the common linear algebra operations without using any other library. The elements of statistical learning: data mining, inference, and prediction, Springer, 2009 Linear Algebra and Probability Review (part 1 Linear Algebra, part 2 Probability) Assignment 1: Mar 17. Many popular machine learning methods, including XGBOOST, use matrices to store inputs and process data. In this day, we are going to make the dirty work. Linear Algebra (Michael Damron and Tasho Kaletha) Introduction to Linear Algebra (Strang) Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra (Matousek) Linear Algebra Done Right (Axler) Advanced Linear Algebra (Roman). TS CH9 Hypothesis Testing. Research labs and companies have data to analyze and understand, and this deep learning approach has become widespread. You can find all the notebooks on Github. He was a research fellow with Michael Jordan and Peter Bartlett, University of California at Berkeley, from 2003, and with Bernhard Schoelkopf, Max Planck Institute for Intelligent Systems, Tuebingen, Germany, from 2005. Gilbert Strang is a Professor of Mathematics at Massachusetts Institute of Technology and an Honorary Fellow at Balliol College in Oxford. pdf; TS CH1 Exploratory Data Analysis. The final exam is 9am on Friday May 15 in JNSN-Ice Rink. Just want some books to go deeper than a introductory course. This course develops the mathematical basis needed to deeply understand how problems of classification and estimation work. Learn how to solve challenging machine learning problems with TensorFlow, Google's revolutionary new software library for deep learning. In the second part, we discuss how deep learning differs from classical machine learning and explain why it is effective in dealing with complex problems such as image and natural language processing. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. This is a straightforward course to learn Linear Algebra Fundamentals for Data Science in Python. i recently bought Gilbert Strang's linear algebra book. then this is the book for you. Also, this OpenGL tutorial has useful explanations. A good video series on the topic that allows you to visualize many concepts is Essence of linear algebra. Welcome to the 18. Our goal is to give the beginning student, with little or no prior exposure to linear algebra, a good ground-ing in the basic ideas, as well as an appreciation for how they are used in many applications, including data tting, machine learning and arti cial intelligence, to-. Before that, I was a software engineer at Google where I worked 80% time with the Hotels team on data analytics and 20% time with the operations research team on linear program solvers. Thank you for your interest in Linear Algebra and Learning from Data. In this course, you’ll learn how to work with vectors and matrices, solve matrix-vector equations, perform eigenvalue/eigenvector analyses and use principal component analysis to do dimension reduction on real-world datasets. Learn linear algebra. Then, in Section 2, we quickly bring you up to speed on the prerequisites required for hands-on deep learning, such as how to store and manipulate data, and how to apply various numerical operations based on basic concepts from linear algebra, calculus, and probability. Communication The vast majority of questions about homework, the lectures, or the course should be asked on our Piazza forum, as others will benefit from the responses. I still plan to go through it but mildly disappointed. You can think of an r t i m e s c r times c r t i m e s c matrix as a set of r r r row vectors, each having c c c elements; or you can think of it as a set of c c c column vectors, each having r r r elements. Building on centuries of statistics and mathematics, Data Science uses computational techniques to help the most innovative companies in the world scale. Open Library is an initiative of the Internet Archive, a 501(c)(3) non-profit, building a digital library of Internet sites and other cultural artifacts in digital form. Math is a crucial skill for people who are interested in Data Science and Machine Learning. Data Science and Linear Algebra Fundamentals with Python, SciPy, & NumPy Math is relevant to software engineering but it is often overshadowed by all of the exciting tools and technologies. [Online book] n Andrew Ng. The elements of statistical learning: data mining, inference, and prediction, Springer, 2009 Linear Algebra and Probability Review (part 1 Linear Algebra, part 2 Probability) Assignment 1: Apr 10. Linear Algebra: Video: Professor Gilbert Strang's Video Lectures on linear algebra.