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# Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix

This question aims to create an "abstract duplicate" of numerous questions that ask about determinants of specific matrices (I may have missed a few):

The general question of this type is

Let $A$ be a square matrix of rank$~1$, let $I$ the identity matrix of the same size, and $\lambda$ a scalar. What is the determinant of $A+\lambda I$?

A clearly very closely related question is

What is the characteristic polynomial of a matrix $A$ of rank$~1$?

The formulation in terms of the characteristic polynomial leads immediately to an easy answer. For once one uses knowledge about the eigenvalues to find the characteristic polynomial instead of the other way around. Since $A$ has rank$~1$, the kernel of the associated linear operator has dimension $n-1$ (where $n$ is the size of the matrix), so there is (unless $n=1$) an eigenvalue$~0$ with geometric multiplicity$~n-1$. The algebraic multiplicity of $0$ as eigenvalue is then at least $n-1$, so $X^{n-1}$ divides the characteristic polynomial$~\chi_A$, and $\chi_A=X^n-cX^{n-1}$ for some constant$~c$. In fact $c$ is the trace $\def\tr{\operatorname{tr}}\tr(A)$ of$~A$, since this holds for the coefficient of $X^{n-1}$ of any square matrix of size$~n$. So the answer to the second question is

The characteristic polynomial of an $n\times n$ matrix $A$ of rank$~1$ is $X^n-cX^{n-1}=X^{n-1}(X-c)$, where $c=\tr(A)$.

The nonzero vectors in the $1$-dimensional image of$~A$ are eigenvectors for the eigenvalue$~c$, in other words $A-cI$ is zero on the image of$~A$, which implies that $X(X-c)$ is an annihilating polynomial for$~A$. Therefore

The minimal polynomial of an $n\times n$ matrix $A$ of rank$~1$ with $n>1$ is $X(X-c)$, where $c=\tr(A)$. In particular a rank$~1$ square matrix $A$ of size $n>1$ is diagonalisable if and only if $\tr(A)\neq0$.

For the first question we get from this (replacing $A$ by $-A$, which is also of rank$~1$)
For a matrix $A$ of rank$~1$ one has $\det(A+\lambda I)=\lambda^{n-1}(\lambda+c)$, where $c=\tr(A)$.
In particular, for an $n\times n$ matrix with diagonal entries all equal to$~a$ and off-diagonal entries all equal to$~b$ (which is the most popular special case of a linear combination of a scalar and a rank-one matrix) one finds (using for $A$ the all-$b$ matrix, and $\lambda=a-b$) as determinant $(a-b)^{n-1}(a+(n-1)b)$.

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# zbMATH — the first resource for mathematics

Characteristic cycles and wave front cycles of representations of reductive Lie groups. (English) Zbl 0960.22009
The main result of this paper settles the conjecture of Barbasch-Vogan [cf. D. Vogan, Oral lectures at Bowdoin College, August, 1989]: For any irreducible admissible representation $$\pi$$ of a linear reductive Lie group $$G_{\mathbf R}$$, the associated cycle $$\text{Ass}(\pi)$$ [attached to $$\pi$$ in D. Vogan, Invent. Math. 48, 75-98 (1978; Zbl 0389.17002)] coincides with the wave front cycle $$WF(\pi)$$ [attached to $$\pi$$ in D. Barbasch and D. Vogan, J. Funct. Anal. 37, 27-55 (1980; Zbl 0436.22011)] via the Kostant-Sekiguchi correspondence.

##### MSC:
 2.2e+46 Representations of Lie and linear algebraic groups over real fields: analytic methods
Full Text:


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QUESTION

# NTC 361 Week 4 Individual Assignment Network Topology and Wiring

This file of NTC 361 Week 4 Individual Assignment Network Topology and Wiring comprises:

Write a paper addressing the following:

1. Provide a brief description of Logical and Physical Topologies. Why are both needed? [2 Points]

2. Complete the following Table. One point for each box. [10 Points]

Topology Name

Description

What I am looking for is a description, and a list of advantages and disadvantages of five different network topologies. You can use the table format, or write it out in paragraph format as long as you cover five topologies and the three characteristics I am looking for.

3. Which topologies would be considered in a healthcare environment? [4 Points]

Up to 1 point deduction for APA, spelling, and grammar issues. Total of 16 Points.

Include a minimum of two outside sources, other than your course texts.

Format your paper consistent with APA guidelines.

• @
• 3 orders completed
Tutor has posted answer for $15.99. See answer's preview$15.99


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A mathematical model capable of describing the evaporation, mixing and burning characteristics of a confined reacting two-phase flow is presented. The flow field is described by solving the partial differential equations of continuity, momentum, and energy transport, together with the $k-ε$ equations of turbulence. Evaporation is accounted for via a droplet evaporation sub-model which runs in parallel with the gas-phase solver exchanging data with it. Effective properties are calculated in each control volume and the property changes resulting from the evaporation are allowed to propagate according to the turbulent mixing model. Combustion follows the mixing process and is assumed to proceed to equilibrium. The model is validated against experimental results, and its applicability over a wide range of conditions is investigated. [S0742-4795(00)03002-7]

1.
Faeth
,
G. M.
,
1983
, “
Evaporation and Combustion of Sprays
,”
Prog. Energy Combust. Sci.
,
9
, pp.
1
76
.
2.
Crowe, C. T., 1978, “A Numerical Model for the Gas-Droplet Flow Field Near an Atomiser,” Proceedings of the 1st international Conference on Liquid Atomisation and Spray Systems, Tokyo, Japan.
3.
Huang, X., and Lixing, Z., 1991, “Simulation of Three Dimensional Turbulent Recirculating Gas Particle Flows by an Energy Equation Model of Particle Turbulence,” Gas-Solid Flows, FED-V 121, ASME, New York, pp. 261–265.
4.
Lixing, Z., 1993, Theory and Numerical Modelling of Turbulent Gas-Particle Flows and Combustion, Science Press, Boston, MA.
5.
Karlsson, J., and Andres, J., 1995, “Modelling Auto-Ignition, Flame Propagation and Combustion in Non-Stationary Turbulent Sprays,” Ph.D. thesis, Dept. of Thermo and Fluid Dynamics Chalmers, University of Technology, Sweden.
6.
O’Rourke, P. J., and Amsden, A. A., 1987, “The TAB Method for Numerical Calculations in Thick Sprays and Comparison with Experiments,” SAE 872089.
7.
Faeth
,
G. M.
,
1987
, “
Mixing and Transport and Combustion in Sprays
,”
Prog. Energy Combust. Sci.
,
13
, pp.
293
345
.
8.
Wu
,
K. J.
,
Su
,
C. C.
,
Steinberger
,
R. L.
,
Santavicca
,
D. A.
, and
Bracco
,
F. V.
,
1983
, “
Measurements of Spray Angle of Atomisation Jets
,”
J. Fluid Eng.
,
105
, pp.
406
413
.
9.
Crookes, R. J., and Nazha, M. A. A., 1990, “Measurement and Prediction of Soot and Gaseous Species in a Burning Fuel Spray at Elevated Pressure,” Archivum Combustionis, Vol. 10 (Nos. 1–4), pp. 49–69.
10.
Nazha, M. A. A., Rajakaruna, H., and Crookes, R. J., 1997, “Soot and Gaseous Species Formation in a Water-in-Liquid Fuel Emulsion Spray—A Mathematical Approach,” Fourth International Conference on Combustion Technologies for a Clean Environment (FLOWERS97), Florence, Italy.
11.
Rajakaruna, H., 1997, “A Mathematical Model for Liquid Fuel Spray Combustion,” Ph.D. thesis, Dept. of Mech. Engineering, De Montfort University, United Kingdom.
12.
Patankar, S. V., 1980, Numerical Heat Transfer and Fluid Flow, Taylor and Francis, London.
13.
Abramovich, G. N., 1963, The Theory of Turbulent Jets, MIT Press, Cambridge, MA.
14.
Nazha, M. A. A., 1983, “Burning Sprays of Water-in-Fuel Emulsions,” Ph.D. thesis, University of London.
15.
Nazha, M. A. A., and Crookes, R. J., 1984, “Effects of Water Content on Pollutant Formation in a Burning Spray of Water-in-Diesel Fuel Emulsion,” 20th Symp. (Int.) on Comb., p. 2001.
16.
Shearer, A. J., and Faeth, G. M., 1979, “Evaluation of Locally Homogeneous Model for Spray Evaporation,” NASA Contractor Report 3198.
17.
Mao, C. P., Wakamatsu, Y., and Faeth, G. M., 1981, “A Simplified Model for High Pressure Spray Combustion,” 18th Symposium (Int.) on Combustion, The Combustion Institute, Pittsburgh, PA.


text:
# alias¶

Name

alias - set aliases

Synopsis

alias [-p] [name[=value] ...]

Description

Alias with no arguments or with the -p option prints the list of aliases in the form alias name=value on standard output. When arguments are supplied, an alias is defined for each name whose value is given. A trailing space in value causes the next word to be checked for alias substitution when the alias is expanded. For each name in the argument list for which no value is supplied, the name and value of the alias is printed. Alias returns true unless a name is given for which no alias has been defined.


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# How to Interpolate vectors in Python?

How to interpolate vector between 2 vectors in blender-python? I hope you get my question. Example:

See mathutils.Vector:

1. Vector.lerp for linear interpolation.

2. Vector.slerp for spherical interpolation. https://en.wikipedia.org/wiki/Slerp

c = a.lerp(b, 0.0)
# c == a
c = a.lerp(b, 0.5)
# c == (a + b) / 2
c = a.lerp(b, 1.0)
# c == b

• i think you mean C == B in your third example – StarWeaver Dec 5 '14 at 23:38
from mathutils import Vector

v = Vector((0,1,2))
v.negate()
print(v)
# <Vector (-0.0000, -1.0000, -2.0000)>

def interpolate(t, vector_a, vector_b):
# t is from interval <0, 1>
return (1 - t) * vector_a + t * vector_b


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Limited Time Offer: Save 10% on all 2022 Premium Study Packages with promo code: BLOG10

# Value at Expiration and Profit for Call and Put Options

In an options contract, two parties transact simultaneously. The buyer of a call or a put option is the long position in the contract, while the seller of the option, also known as the writer of the option, is the short position.

It is important to differentiate between the long and the short party in a contract. The buyer is always said to be the long option. This is quite easy to see for a call option. However, for a put option, the long position in a put is betting that the underlying price will drop. As such, the long position in a put option is synonymous with being short the underlying.

## Call options

### Value at Expiration of a Call Option

The payoff profiles of a call option are represented as follows:

$$\text{Payoff for a call buyer} = max(0,S_T – X)$$

$$\text{Payoff for a call seller} = -max(0,S_T – X)$$

Where

ST is the price of the underlying at expiration; and

X is the exercise price.

#### Profit of a Call Option

Using the payoff profile and the price paid for the option, the profit equation can be written as follows:

$$\text{Profit for a call buyer} = max(0,S_T – X) – c_0$$

$$\text{Profit for a call seller} = -max(0,S_T – X) + c_0$$

where c is the call premium.

The buyer of the call option has no upper limit on its potential profit and a fixed downside loss equal to the premium. The seller has unlimited losses and a gain limited to the premium:

## Put Options

By now, if you have well understood the basic characteristics of call options, then the payoff and profit for put option buyers and sellers should be quite easy – you replace “ST − X” by “X− ST.”

### Value at Expiration of a PutOption

The payoff and profit profiles of a put option are represented as follows:

$$\text{Payoff for a put buyer} = max(0,X-S_T)$$

$$\text{Payoff for a put seller} = -max(0,X-S_T)$$

$$\text{Profit for a put buyer} = max(0,X-S_T) – p_0$$

$$\text{Profit for a put seller} = -max(0,X-S_T) + p_0$$

Where p0 is the put premium.

The put buyer has a limited loss and, while not completely unlimited gains, as the underlying price cannot fall below zero, the put buyer does gain as the price falls. As such, purchasing a put option is like purchasing insurance. In the same vein as for call options, the put seller has nearly unlimited losses, and his gains are limited to the put premium paid to him by the put buyer:

Call and put options can be used in combination with each other and with the underlying asset to produce a wide range of potential payoff profiles.

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# If (x/y)>2, is 3x+2y<18? (1) x-y is less than 2 (2)

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Re: If (x/y)>2, is 3x+2y<18? (1) x-y is less than 2 (2) [#permalink]

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Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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If (x/y)>2, is 3x+2y<18? (1) x-y is less than 2 (2) [#permalink]

### Show Tags

30 Jun 2015, 03:42
Bunuel wrote:
Hussain15 wrote:
If (x/y)>2, is 3x+2y<18?

(1) x-y is less than 2
(2) y-x is less than 2

It will be great to see how do you guys approach this lethal one.

I would solve this question with graphic approach, by drawing the lines. With this approach you will "see" that the answer is A. But we can do it with algebra as well.

$$\frac{x}{y}>2$$ tells us that $$x$$ and $$y$$ are either both positive or both negative, which means that all points $$(x,y)$$ satisfying given inequality are either in I or III quadrant. When they are both negative (in III quadrant) inequality $$3x+2y<18$$ is always true, so we should check only for I quadrant, or when both $$x$$ and $$y$$ are positive.

In I quadrant, as $$x$$ and $$y$$ are both positive, we can rewrite $$\frac{x}{y}>2$$ as $$x>2y>0$$ (remember $$x>0$$ and $$y>0$$).

So basically question becomes: If $$x>0$$ and $$y>0$$ and $$x>2y>0$$, is $$3x+2y<18$$?

(1) $$x-y<2$$.

Subtract inequalities $$x>2y$$ and $$x-y<2$$ (we can do this as signs are in opposite direction) --> $$x-(x-y)>2y-2$$ --> $$y<2$$.

Now add inequalities $$x-y<2$$ and $$y<2$$ (we can do this as signs are in the same direction) --> $$x-y+y<2+2$$ --> $$x<4$$.

We got $$y<2$$ and $$x<4$$. If we take maximum values $$x=4$$ and $$y=2$$ and substitute in $$3x+2y<18$$, we'll get $$12+4=16<18$$.

Sufficient.

(2) $$y-x<2$$ and $$x>2y$$:
$$x=3$$ and $$y=1$$ --> $$3x+2y=11<18$$ true.
$$x=11$$ and $$y=5$$ --> $$3x+2y=43<18$$ false.

Not sufficient.

Hi Bunuel,

In statement 1, you got y<2 and x<4 but when x=2 & y=1 or even your point x=4 & y=2 so x/y>2 is not satisfied because 2/1 or 4/2 is not bigger 2. How come still statement 1 sufficient?

Thanks
Math Expert
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Posts: 5660
Re: If (x/y)>2, is 3x+2y<18? (1) x-y is less than 2 (2) [#permalink]

### Show Tags

30 Jun 2015, 08:07
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KUDOS
Expert's post
Mo2men wrote:

Hi Bunuel,

In statement 1, you got y<2 and x<4 but when x=2 & y=1 or even your point x=4 & y=2 so x/y>2 is not satisfied because 2/1 or 4/2 is not bigger 2. How come still statement 1 sufficient?

Thanks

Since you have asked here only the above doubt...
y<2 and x<4.... you cannot take them as y=2 and x=4 as it is given both are less than these quantities...
so if y=1.9 .. statement 1 says x-y<2 or x-1.9<2 or x<3.9, so it satisfies x<4..
however if we take the values of x and y slightly more than the max possible(x<4).. x=4and (x<2)..y=2, we find value of eq <18.. so suff
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

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Re: If (x/y)>2, is 3x+2y<18? (1) x-y is less than 2 (2) [#permalink]

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30 Jun 2015, 08:11
chetan2u wrote:
Mo2men wrote:

Hi Bunuel,

In statement 1, you got y<2 and x<4 but when x=2 & y=1 or even your point x=4 & y=2 so x/y>2 is not satisfied because 2/1 or 4/2 is not bigger 2. How come still statement 1 sufficient?

Thanks

Since you have asked here only the above doubt...
y<2 and x<4.... you cannot take them as y=2 and x=4 as it is given both are less than these quantities...
so if y=1.9 .. statement 1 says x-y<2 or x-1.9<2 or x<3.9, so it satisfies x<4..
however if we take the values of x and y slightly more than the max possible(x<4).. x=4and (x<2)..y=2, we find value of eq <18.. so suff

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Re: If (x/y)>2, is 3x+2y<18? (1) x-y is less than 2 (2) [#permalink]

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31 Jul 2016, 16:40
Hi!
can someone solve the question without graphical approach, though its my personal weakness to unable to solve using that.
Chetan4u , empowergmat , mathrevloution , harley ...
thanks
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Re: If (x/y)>2, is 3x+2y<18? (1) x-y is less than 2 (2) [#permalink]

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12 Oct 2016, 06:00
Is there any trick for solving inequalities and modulus problems using number plugging?
Retired Moderator
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Re: If (x/y)>2, is 3x+2y<18? (1) x-y is less than 2 (2) [#permalink]

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04 Jan 2017, 12:27
Experts!
thanks
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Re: If (x/y)>2, is 3x+2y<18? (1) x-y is less than 2 (2) [#permalink]

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19 Mar 2017, 19:26
If (x/y)>2, is 3x+2y<18?

x/y>2 means that both x & y have same sign.

(1) x-y < 2

Let x=3 & y=5/4....... check x/y=12/5=2.4>2

Check statement x-y<2.....3 -1.25=1.75 <2 ......So (3*3) + (2*5/4)<18..........(Note that you can choose positive numbers with narrow range such 2 & 5/6 and will achieve same answer Yes )

Let x=-10 & y=-1......check x/y=-10/-1=10>2

Check statement x-y<2.....-10+1=-9 <2............So (-10*3)+ (-1*2)<18.............Yes (Note that any negative numbers that satisfy both x/y> and fact 1 will always answer question with Yes)

Sufficient

(2) y-x < 2

Let x=3 & y=5/4....... check x/y=12/5=2.4>2

Check statement y-x<2.....1.25 - 3=-1.75 <2 ......So (3*3) + (2*5/4)<18.....Answer is Yes

Let x=10 & y=1.......check x/y=10>2

Check statement y-x<2.......1-10=-9<2................So (3*10) + (2*1)<18.....Answer is NO

Insufficient

This question needs high skills to spot the strategic numbers an spot pattern also.
Re: If (x/y)>2, is 3x+2y<18? (1) x-y is less than 2 (2)   [#permalink] 19 Mar 2017, 19:26

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# Computable real function

In mathematical logic, specifically computability theory, a function $f \colon \mathbb{R} \to \mathbb{R}$ is sequentially computable if, for every computable sequence $\{x_i\}_{i=1}^\infty$ of real numbers, the sequence $\{f(x_i) \}_{i=1}^\infty$ is also computable.

A function $f \colon \mathbb{R} \to \mathbb{R}$ is effectively uniformly continuous if there exists a recursive function $d \colon \mathbb{N} \to \mathbb{N}$ such that, if

$| x-y| < {1 \over d(n)}$

then

$| f(x) - f(y)| < {1 \over n}$

A real function is computable if it is both sequentially computable and effectively uniformly continuous.

These definitions can be generalized to functions of more than one variable or functions only defined on a subset of $\mathbb{R}^n.$ The generalizations of the latter two need not be restated. A suitable generalization of the first definition is:

Let D be a subset of $\mathbb{R}^n.$ A function $f \colon D \to \mathbb{R}$ is sequentially computable if, for every n-tuplet $\left( \{ x_{i \, 1} \}_{i=1}^\infty, \ldots \{ x_{i \, n} \}_{i=1}^\infty \right)$ of computable sequences of real numbers such that

$(\forall i) \quad (x_{i \, 1}, \ldots x_{i \, n}) \in D \qquad ,$

the sequence $\{f(x_i) \}_{i=1}^\infty$ is also computable.

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# 180 Days of Math for Fourth Grade Day 42 Answers Key

By accessing our 180 Days of Math for Fourth Grade Answers Key Day 42 regularly, students can get better problem-solving skills.

## 180 Days of Math for Fourth Grade Answers Key Day 42

Directions Solve each problem.

Question 1.

Explanation:
Perform subtraction operation on above two given numbers. Subtract 25 from 46 the difference is 21.

Question 2.
Color $$\frac{3}{10}$$.

= 3/10 x 100
= 3 x 10
= 30
Explanation:
Total number of boxes are 100. We have to shade 3/10 of the boxes. Multiply 3/10 with 100 the product is 30. So, 30 boxes are shaded.

Question 3.

Explanation:
Perform division operation on above two given numbers. Divide 18 by 9 the quotient is 2.

Question 4.
List all the factors of 12.
_____________
Factors of 12 are 1, 2, 4, 8, 16.
Explanation:
The numbers which divides 12 exactly leaving a remainder zero are the factors of 12. So, the factors of 12 are 1, 2, 4, 8, 16.

Question 5.
Write the ordinal number after sixteenth.
_____________
The ordinal number is used to represent the position. The ordinal number after sixteenth is seventeenth or 17th.

Question 6.
7 × = 21
7 ×  = 21
Explanation:
To get the product 21 we need to perform multiplication operation. Multiply 7 with 3 the product is 21. The missing multiplicand is 3.

Question 7.
Calculate the perimeter of a square with 5 cm sides.
_____________________

Given side = 5 cm
Perimeter of a square = 4 x side
= 4 x 5 cm
= 20 cm
The perimeter of a square with 5 cm sides is equal to 20 cm.

Question 8.
____ inches = $$\frac{1}{2}$$ foot
We know that 1 foot is equal to 12 inches.
Divide above equation by 2 on both sides.
1/2 foot = 12/2 inches
1/2 foot = 6 inches

Question 9.
True or false? Plane shapes are 3-dimensional.


text:
# A question about the Hardy-Littlewood maximal function.

Somebody can to help me in the following question?

Let $f$ be measurable in $\mathbb{R}^n$ and different from zero in some set of positive measure. Show that there is a positive constant с such that $f^*(x) \geq c |x|^{-n}$ for $|x| \geq 1$

Note: $f^*(x)$ is the Hardy-Littlewood maximal function, i.e.,

$\displaystyle{f^*(x) = \sup_{r>0} \; \frac{1}{|B_{r}(x)|}\; \int_{B_{r}(x)} \;|f(y)|\;dy}.$

From the given conditions it follows that there exists a ball $B=B_M(0)$ (wlog M>1) such that $$\int_B |f| = c>0.$$
If $|x|\ge M$, then $B_{2|x|}(x)\supset B$ so that \begin{align} f^*(x) &\ge \frac{1}{|B_{2|x|}(x)|}\int_{B_{2|x|}(x)} |f| \\ &= \frac{1}{v_n2^n|x|^n}\int_{B_{2|x|}(x)} |f| \\ &\ge \frac{1}{v_n2^n|x|^n}c \\ \end{align}
If $1\le|x|\le M$, then $B_{2M|x|}\supset B$
here $v_n$ is the volume of the ball with unit radius in $\mathbb{R}^n$.


text:
# Proving there are exactly 8 semiregular tessellations

First, a couple of interesting observations: If a tiling is made up of only convex polygons, then each vertex must be surrounded by 3 or more shapes. 1 polygon, convex or concave, certainly can't surround a point, and 2 polygons can only surround a point if at least one of them is concave. Thus, 3 is the minimum number of convex polygons surrounding a point. The smallest interior angle of a regular polygon is $$60^\circ,$$ and so we also know the maximum number of regular polygons around a point is 6.

Next, determine all the ways in which $$360^\circ$$ can be the sum of 3 or more angles that appear in regular polygons. To help you do this, you will probably want to create a list of the interior angles of regular polygons. Here is a partial list:

The three regular tilings come from 60 + 60 + 60 + 60 + 60 + 60 (3.3.3.3.3.3), 90 + 90 + 90 + 90 (4.4.4.4), and 120 + 120 + 120 (6.6.6). The semiregular tilings can mix different numbers. For example, 60 + 60 + 120 + 120 (3.3.6.6) should be on your list of possibilities. Once you're done listing all of the sums, such as 60 + 60 + 120 + 120, listing the vertex types, such as 3.3.6.6, is easy. Simply permute each, and determine whether each permutation is a new vertex type, or one you've already listed. For example, 3.3.6.6 is the same as 6.3.3.6, but is different than 3.6.3.6. After you're done with this stage, you'll have a large, but finite list of possibilities. Here's that list, in case you wish to check your answers against it:

1. 3.3.3.3.3.3
2. 3.3.3.3.6
3. 3.3.3.4.4
4. 3.3.4.3.4
5. 3.3.6.6
6. 3.6.3.6
7. 3.4.4.6
8. 3.4.6.4
9. 4.8.8
10. 3.3.4.12
11. 3.4.3.12
12. 6.6.6
13. 4.6.12
14. 3.12.12
15. 4.4.4.4

Three of these are the regular tilings, mentioned previously. Excluding those 3, we get 12 possibilities. Now's the fun part. Try putting these tilings together. You will quickly find that only 8 of them work. The other 4 will eventually lead to a gap which cannot be filled by a regular polygon, or will force a different vertex type to be used. For example, filling out 3.3.6.6, by focusing on hexagons, quickly leads to a vertex with six triangles. But 3.3.6.6 is not 3.3.3.3.3.3, and thus, 3.3.6.6 is not a valid tiling.


text:
# Simple kafka streams app in MapR environment

While Spark continues to thrive as the main big data processing framework for batch and streaming, alternatives emerging from the 1970s actor model and the reactive manifesto are gaining notoriety. Akka is widely known in the Scala community and on March 2016 Confluent released its library Kafka Streams.

One of the main problems of Spark for me is Scaling. Much of the time we are just tweaking executor characteristics: the number of them, its memory... And if the data volume is changing from day to day, either we are using way more resources than needed or the application just fails (depending on the app workload). If this is a nightmare just with batch jobs, in streaming I can't start imagine the situation.

Thats a very big issue and was one of the main reasons I've started processing libraries that scale-up and down automatically without causing me a headache. Keep in mind that since in my work have MapR, we are tied to their supported ecosystem pack.

This is, at the time of writing and regarding to streaming:

• Spark 2.3.1
• Flume 1.8
• Kafka-streamsish (Since it is a library and they "support" kafka 1.1.1, we can use it)

I've chosen to explore kafka-streams, and it has a nice feature out-of-the-box: Exactly-once processing (YAAAAY!). Since we don't need fancy analytics on our data, it can be a nice candidate.

The bad thing: it does not scale-up and down automatically, rather you have to implement some logic to do so (via metrics). I'll will have to tackle that in a later post.

First of all: what is kafka-streams? It's just a JVM application: one consumer that reads from topics and writes to other topics. If you want to parallelize: start up to N instances of the app. (N is the number of topic partition and determines the max number of consumers)

## Set-up MapR environment (VM)

We'll use the sandbox provided by MapR that already contains all the components needed. In our case the version is 6.1.0 (release date yyMM: 1808) so all the packages and dependencies will reference those numbers. It is important to use >6.1.0 since it included several key improvements to streaming applications (for example, integration with kafka-streams apps).

It has all what we need almost installed, and any other thing we need:

• MapR Event Store (kafka)
• MapR DB (HBase)
• MapR FS (HDFS)
• Spark
• ...

Now, follow the MapR documentation to install and set-up the VM. Make sure you download the latest version and kafka is > 1.1.1

Once done, you should be able to access the VM via its IP address, either local or a WLAN one (run ifconfig). In my case, I have set it up using a bridged adapter so access is available via:

Additionally, we'll need to create the following directory to write temporal files

# Tem
mkdir /tmp/kafka-streams

### Install a mapr gateway locally

MapR Streams needs to have at least one gateway service up in the cluster in order to manage replication and compaction of topics. However in the sandbox this service is missing so we must install and configure it. As root user (password: mapr), execute

yum install mapr-gateway
maprcli cluster gateway set -dstcluster demo.mapr.com -gateways 127.0.0.1

# Check that we canm find the gateway
maprcli cluster gateway resolve -dstcluster demo.mapr.com

NOTE
Cluster name can be found in /opt/mapr/conf/mapr-clusters.conf should be demo.mapr.com in our case

### Create the Stream and topics

MapR ES has some unique characteristics:

• Stream: an abstraction that allows you to group topics
• Streams are stored in HDFS so they usually are a route
• No need for zookeeper (Remember the gateway node?)

You can create and configure the streams via MapR MCS GUI, in Data -> Streams -> Create Stream We'll use as Stream name /steampipe. We'll leave all the default values and then topics need to be created via the Topics tab: duplicated-records and no-duplicates

Another option is to use the following shell commands

maprcli stream create -path /sample-stream \
-produceperm p -consumeperm p -topicperm p
maprcli stream topic create -path /steampipe -topic duplicated-records
maprcli stream topic create -path /steampipe -topic no-duplicates

#### Test it

Using the console producers and consumers we can test our process

# Read messages (and show keys)
/opt/mapr/kafka/kafka-1.1.1/bin/kafka-console-consumer.sh \
--bootstrap-server fake.9092 \
--topic /steampipe:duplicated-records \
--property print.key=true

And in another terminal do:

# Produce messages
#
#   To set the message's key, add the following and write the messages as key:value
#    --property "parse.key=true" --property "key.separator=:"
#
/opt/mapr/kafka/kafka-1.1.1/bin/kafka-console-producer.sh \
--broker-list fake.server:9999 \
--topic /steampipe:duplicated-records

## Application set-up

### Dependencies and subtleties

Instead of what new kafka documentation states you can't use yet the package kafka-streams-scala since it is for Kafka > 2.0.0. Instead, we'll use the artifact with ID kafka_2.11 since we are going to use scala 2.11.

Since MapR is a bit special (in the good and the bad way), most of the apache packages have some changes and we must use their versions. As an example, kafka-streams runs in the same cluster and you don't need to set any zookeeper quorum property when executing kafka applications. Adding its repository in our pom.xml will allow maven to download them.

<properties>
<mapr-version>6.1.0-mapr</mapr-version>
<mapr-streams-version>1.1.1-mapr-streams-6.1.0</mapr-streams-version>
<avro.version>1.8.2</avro.version>
<scala.version>2.11.12</scala.version>
<scala.compat.version>2.11</scala.compat.version>
</properties>

<repositories>
<repository>
<id>mapr-maven</id>
<url>http://repository.mapr.com/maven</url>
<releases><enabled>true</enabled></releases>
<snapshots><enabled>false</enabled></snapshots>
</repository>
</repositories>

<dependencies>
<dependency>
<groupId>org.scala-lang</groupId>
<artifactId>scala-library</artifactId>
<!-- This version must be compatible with the Scala version of the Kafka dependency. -->
<version>${scala.version}</version> </dependency> <dependency> <groupId>com.mapr.streams</groupId> <artifactId>mapr-streams</artifactId> <version>${mapr-version}</version>
<scope>provided</scope>
</dependency>
<dependency>
<groupId>org.apache.kafka</groupId>
<artifactId>kafka-clients</artifactId>
<version>${mapr-streams-version}</version> <scope>provided</scope> </dependency> <dependency> <groupId>org.apache.kafka</groupId> <artifactId>kafka-streams</artifactId> <version>${mapr-streams-version}</version>
<scope>provided</scope>
</dependency>
<dependency>
<groupId>org.apache.kafka</groupId>
<artifactId>kafka_${scala.compat.version}</artifactId> <version>${mapr-streams-version}</version>
</dependency>
...
</dependencies>

HINT
To find all the mapr-specific package versions, go to their nexus repository

Furthermore, some additional arguments must be set in the scala plugin to be able to enable Java8 lambdas (needed for map, foreach ...)

<plugin>
<groupId>net.alchim31.maven</groupId>
<artifactId>scala-maven-plugin</artifactId>
<version>3.2.1</version>
<configuration>
<args>
<!--
In combination with Scala 2.11, -Xexperimental enables SAM
for Java 8 lambda support.  Make sure kafka.scala.version
is set to 2.11, not 2.10.
SUPER IMPORTANT PART!
-->
<arg>-Xexperimental</arg>
...
</args>
</configuration>
...
</plugin>

## Simple hash application

Let's say that for now we need to hash all the data from one topic and store into another one. In another post we will reuse part of the code, so let's keep it simple.

The configuration includes topic names (in MapR topics are mapped to an HDFS directory), the default serializers for the messages and its application_id. This property is important since it has to be unique and it is used to keep track of read offsets. Those offsets are synced across instances via MapR FS, stored inside /apps/kafka-streams/application_id

val TOPIC_INPUT = "/steampipe:duplicated-records"
val TOPIC_OUTPUT = "/steampipe:no-duplicates"

// Set up builder and config
val builder = new StreamsBuilder()
val streamingConfig = {
val settings = new Properties()
settings.put(StreamsConfig.DEFAULT_KEY_SERDE_CLASS_CONFIG, Serdes.ByteArray.getClass.getName)
settings.put(StreamsConfig.DEFAULT_VALUE_SERDE_CLASS_CONFIG, Serdes.String.getClass.getName)
settings.put(StreamsConfig.APPLICATION_ID_CONFIG, "hasher-app")
settings
}

And now, we have here the application logic, where we filter-out the null messages, since we can't deal with them, and apply a hash function over the message. Output is finally written to our output topic.

val hasher = Hashing.sha256()
def hash(string: String): String = hasher.hashString(string, StandardCharsets.UTF_8).toString

val lines: KStream[Array[Byte],String] = builder.stream(TOPIC_INPUT)
val hashed: KStream[Array[Byte],String] = lines.filterNot((_, v) => v == null).mapValues(hash(_))
hashed.to(TOPIC_OUTPUT)

We are almost set, we just have to build and order the start of the stream.

val topology: Topology = builder.build()
val streams: KafkaStreams = new KafkaStreams(topology, streamingConfig)

streams.start()

Now, package via mvn package and copy it to your VM.

### Run it!

First of all, to see the results run a console consumer reading from output topic /steampipe:no-duplicates

/opt/mapr/kafka/kafka-1.1.1/bin/kafka-console-consumer.sh --bootstrap-server fake.9092 --topic /steampipe:no-duplicates --property print.key=true

And then start the kafka-streams application.

java -cp "/opt/mapr/lib/*:steam-pipe-1.0-SNAPSHOT-jar-with-dependencies.jar" cat.martsec.kafka.steampipe.Hasher

So now you can start inputing messages in the console producer and its hash will appear in the no-duplicates topic. We are done!

>key1:aaaaaa
>key2:111111
>k:1
>k:2

Result

key1	ed02457b5c41d964dbd2f2a609d63fe1bb7528dbe55e1abf5b52c249cd735797
k	d4735e3a265e16eee03f59718b9b5d03019c07d8b6c51f90da3a666eec13ab35

## Recap

We have set-up the MapR sandbox to run our applications. Then, we've created a stream and a couple of topics and build a kafka-streams application that hashes the content of a message and pits into another topic.

As always, you can see the source code in my github repository

In the next post we sill deal with a common problem: duplicated messages (but not duplicates caused by kafka-streams, we have exactly-once).

Kafka streams and HBase as de-duplicator

References:


text:
PHPBuilder - check dir then include the file

check dir then include the file

by: ayen
|
February 14, 2002

here some code that im using when there is to many include file...normaly i put this thing on index.php

`````` \$dirlist = opendir("dirname");
if(!ereg('^\.|html|LOG|test',\$file))
{

\$mods = substr(\$file,0,strlen(\$file));

include(\$mods);
//echo \$mods;
}
}
closedir();
}```
```
if(!ereg('^\.|html|LOG|test',\$file)) this section handle the file that you dont want to include on it

Comment and Contribute

Your comment has been submitted and is pending approval.

Author:
ayen

Comment:

Comment:

(Maximum characters: 1200). You have characters left.


text:
# Homework Help: Proof of reduction formula for a definite integral of an even powered sine function

1. Jun 3, 2010

### stripes

1. The problem statement, all variables and given/known data
should be my last question for at least the next few days...here goes...

Prove that, for even powers of sine,

$$\int^{\frac{\pi}{2}}_{0}sin^{2n}x dx = \frac{2\cdot4\cdot6\cdot...\cdot(2n - 1)}{2\cdot4\cdot6\cdot...\cdot2n}\cdot\frac{\pi}{2}$$

2. Relevant equations

$$uv - \int v du = \int u dvdx$$

3. The attempt at a solution

let $u = sin^{2n-1}x$ and $dv = sin x dx$
so $du = (2n-1)(sin^{2n-2}x)(cos x)dx$ and $v = -cos x$

and we get:

$$\int sin^{2n}x dx = (-cos x)(sin^{2n-1}x) + (2n-1)\int (cos^{2}x)(sin^{2n-1}x)$$

and i used integration by parts again

let $u = sin^{2n-2}x$ and $dv = cos^{2}x dx$
so $du = (2n-2)(sin^{2n-3}x)(cos x)dx$ and $v = \frac{1}{2}((sin x)(cos x) + x)$

then we get:

$$\int sin^{2n}x dx = (-cos x)(sin^{2n-1}x) + \frac{2n-1}{2}(sin^{2n-2}(sinx cosx + x)) - (2n-2)\int ((sinx cosx + x)(sin^{2n-3}x)(cos x))dx$$

now i've realized i'm just pointlessly integrating by parts over and over...it's just getting harder and harder (and more difficult to put here on PF!)

If someone could guide me in the right direction for proving this formula, I would appreciate it. Thank you so much in advance!

Last edited: Jun 3, 2010
2. Jun 3, 2010

### System

Re: proof of reduction formula for a definite integral of an even powered sine functi

Did you forget that you have a definite integral?

3. Jun 3, 2010

### HallsofIvy

Re: proof of reduction formula for a definite integral of an even powered sine functi

In addition to remembering that this is a definite integral, so that you don't have that long sum, I would try proof by induction so that I only have to do the integration by parts once.

4. Jun 3, 2010

### stripes

Re: proof of reduction formula for a definite integral of an even powered sine functi

i knew all along that we have a definite integral, but we still need to get rid of that integral sign on the end of the whole thing, there'll always be that last integral...

our course hasn't taught us enough to integrate that...


text:
Apple’s macOS 10.14 Mojave offers “Dynamic Desktop” wallpapers that change based on the time of day. You can get something similar on Windows, either by using the standard background settings or by digging into the Task Scheduler.

## Set Up a Slideshow That Changes on a Schedule

Windows 10 and Windows 7 both let you set up a slideshow and have it automatically change the picture after a period of time. So, if you provide four background wallpaper images, you can have Windows flip through them every six hours, changing your background depending on the time of day.

We tested this using the official macOS Mojave desktop backgrounds, which someone helpfully uploaded to Reddit. However, you can use any desktop background images you like.

First, make a new folder and place the desktop backgrounds you want to use in it. If you plan to have your desktop background change every six hours, place four background images in it.

Windows will go through the folder in alphanumerical order, so you’ll need to name the images appropriately. The correct order depends on the current time of day.

For example, if it’s nearing the time when you want your sunset background applied, you should place a 1 in front of the filename of your sunset background, a 2 in front of your night background, a 3 in front of your sunrise background, and a 4 in front of your sunset background.

Next, open the desktop background configuration screen by right-clicking the desktop and selecting “Personalize” or by heading to Settings > Personalization > Background on Windows 10.

Click the “Background” box, and then select the “Slideshow” option. Under “Choose albums for your slideshow, click the “Browse” button, and then select the folder containing your background images. Under “Change picture every,” click the box and select “6 hours” if you have four background images you want to cycle through. You can also select “1 hour” here if you want to provide 24 background images.

Ensure the Shuffle option is turned off. If Shuffle is enabled, you’ll get random background images and they won’t match the time of day.

Windows will immediately set your desktop background to the image that is first alphanumerically in the folder. It will also immediately begin counting down from six hours.

In other words, if you want to see a sunset background at 4 p.m., a night background at 10 p.m., a sunrise background at 4 a.m., and a day background at 10 a.m., you should enable the slideshow background at exactly one of these times.

For example, if the sunset background is alphanumerically the first one in the folder, you should enable the slideshow background at exactly 4 p.m..

On Windows 7, the interface looks a bit different, but you can do the same thing.

Right-click the desktop, select “Personalize,” and then click the “Desktop Background” option. In the Desktop Background window, click the “Browse” button to the right of Picture location, and then choose the folder containing your wallpaper images. Select the background images you want to use, and then enable the timed slideshow feature on the “Change Picture Every” dropdown menu.

You can have Windows automatically change your desktop background on a schedule via the Task Scheduler, too. It’s a more complex process because Windows doesn’t include a built-in command that changes your desktop background. You can change the registry setting that points at your desktop background, but we couldn’t find a built-in command that would reliably “refresh” that setting and apply your new wallpaper.

Instead, we found that Microsoft’s own BgInfo tool worked pretty well. BgInfo is meant for writing system information text onto your desktop background, but it provides a reliable way to change your desktop background using commands. You’ll need to create several BgInfo configuration files containing the settings you want to apply, and then you’ll create scheduled tasks that automatically run these from the Task Scheduler.

To get started, download BgInfo from Microsoft and extract its files into a folder. Launch the “Bginfo64.exe” program if you’re on a 64-bit version of Windows, or the Bginfo.exe file if you’re on a 32-bit version of Windows.

After you fire up BGInfo, first select all the text in the main box and erase it. Type a single space character (or multiple space characters) here. This will prevent BGInfo from inserting any text into your background and ensure it will only change the wallpaper.

Second, click the “Background” button.

In the Background window, select the “Use These Settings” option. Set the “Wallpaper Position” to either “Center” or “Stretch”—whatever you want to do with your background image. Then, click the “…” button to the right of the “Wallpaper Bitmap” box, and then browse to one of the background images you want to use.

Click “OK” when you’re done.

Next, click File > Save As, and then save your configuration as a .bgi file.

For example, we selected our day background image file while creating this profile, so we named it Day.bgi.

Now, you’re going to repeat this process to create a separate .bgi profile representing each of your background images, naming them after the times of day you want to use them.

When you’re done, your save folder should have all your images and a .bgi profile for each.

After you’ve created your .bgi files, you can use the Task Scheduler to automatically apply them on a schedule.

Click Start menu, type “Task Scheduler” into the search box, and then press Enter.

Give the task a name, and then click the “Next” button. For example, if this task will apply your sunset wallpaper around sunset, you might name it “Sunset Wallpaper.”

Select the “Daily” trigger, and then click the “Next” button again

Enter the time when you want the wallpaper to appear. For example, if sunset takes place around 9 p.m. in your location, you might set the wallpaper to appear at 8 p.m.. Select whatever time you like.

Ensure the task is set to recur every “1” days, and then click the “Next” button.

Select the “Start a Program” action, and then click “Next” again.

To the right of the “Program/script” text box, click the “Browse” button, and then browse to the Bginfo64.exe or Bginfo.exe program you used earlier.

In the “Add arguments (optional)” box, type the following text, replacing “C:\path\to\file.bgi” with the path to the BGI file you created earlier:

"C:\path\to\file.bgi" /timer 0 /silent

For example, if the Sunset.bgi file we created on our system is located at C:\Users\chris\Downloads\Mojave, we’d enter:

"C:\Users\chris\Downloads\Mojave\Sunset.bgi" /timer 0 /silent

Click “Next” to continue.

Select the “Open the Properties dialog for this task when I click Finish” check box, and then click “Finish” to create your task.

In the task window that opens, click over to the “Actions” tab, and then turn off the “Start the task only if the computer is on AC power” option. This allows the task to run even if you’re using a laptop on battery power.

Finally, switch to the “Settings” tab, and then enable the “Run this task as soon as possible after a scheduled start is missed” option. This ensures the task will run and Windows will change your desktop background, even if your computer was shut down at the scheduled time.

Click “OK” to finish configuring the task.

You’ll find your created task under the “Task Scheduler Library” folder in Task Manager. To check that it works properly, right-click it and select the “Run” command. Your wallpaper should immediately change.

To finish this process, create several more scheduled tasks using the same technique, selecting a different time of day and pointing each at the appropriate .BGI file you created earlier. For example, if you created four total BGI files for four backgrounds, you need to create three more scheduled tasks that will automatically change your background images at the correct time of day.

As the seasons change, you can return to the Task Scheduler and adjust the times of day when your desktop backgrounds change, too. To do so, locate a task under Task Scheduler Library and double-click it. Click the “Triggers” tab, click “Edit,” select a new time of day, and then click “OK” twice to save your changes.

Hopefully, Microsoft will add a Dynamic Desktop-style feature to a future version of Windows 10 and eliminate the need for all this fiddling.

Chris Hoffman
Chris Hoffman is Editor-in-Chief of How-To Geek. He's written about technology for over a decade and was a PCWorld columnist for two years. Chris has written for The New York Times and Reader's Digest, been interviewed as a technology expert on TV stations like Miami's NBC 6, and had his work covered by news outlets like the BBC. Since 2011, Chris has written over 2,000 articles that have been read more than one billion times---and that's just here at How-To Geek.


text:
# Using Min & Max of an Interval in a replacement [closed]

I want to use a replacement rule to transform a list of equations. The following works as expected:

eqns = {x'[t] == 2 y[t], y'[t] == -3 x[t]};
eqns /. (var_'[t] == rhs_) -> {var, rhs}
(* {{x, 2 y[t]}, {y, -3 x[t]}} *)

Now I want to use the same idea to get the Min and Max of an Interval associated with each variable.

range[x] = Interval[{-1, 1}]; range[y] = Interval[{-2, 2}];
eqns /. (var_'[t] == rhs_) -> {Min[range[var]], Max[range[var]]}

Since {Min[range[x]], Max[range[x]]} = {-1 ,1}, I expected to get

(* {{-1, 1}, {-2, 2}} *)

(* {{Interval[{-1, 1}], Interval[{-1, 1}]}, {Interval[{-2, 2}], Interval[{-2, 2}]}} *)

What am I doing wrong and how to fix it?

## closed as off-topic by Bob Hanlon, m_goldberg, Henrik Schumacher, MarcoB, xzczdFeb 11 at 4:39

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question arises due to a simple mistake such as a trivial syntax error, incorrect capitalization, spelling mistake, or other typographical error and is unlikely to help any future visitors, or else it is easily found in the documentation." – Bob Hanlon, m_goldberg, Henrik Schumacher, MarcoB, xzczd
If this question can be reworded to fit the rules in the help center, please edit the question.

• Use :> instead of ->. Notice Min@aaaa evaluates to aaaa. :) – xzczd Feb 10 at 15:26
• @xzczd Yep, it was that easy. Thanks! – Chris K Feb 10 at 15:27
• @xzczd Any rule of thumb when to use :> vs. -> in this context? – Chris K Feb 10 at 15:31
• Well, personally I don't recommend deciding it based on rule of thumb. Just think about if the right hand side can be/should be evaluate before pattern matching. -> evaluates the right hand side instantly, while in this case, Min[range[var]] will evaluate to range[var], so the right hand side should not be evaluated before the pattern matches. You may use Trace to check the evaluation order. – xzczd Feb 10 at 15:36


text:
Math Help - Integration problem.

1. Integration problem.

Lost as to how to do this.
Its not apparent if "Integration by parts" will work, or any substitutions. ((Came across this while working on differential equation problems, and half remembering learning how to do it before. So was hoping it wasn't too difficult))
$integral(2^xsinx)$

2. Originally Posted by Frggr
Lost as to how to do this.
Its not apparent if "Integration by parts" will work, or any substitutions. ((Came across this while working on differential equation problems, and half remembering learning how to do it before. So was hoping it wasn't too difficult))
$integral(2^xsinx)$
Did you try doing it by parts twice?

3. Originally Posted by Frggr
Lost as to how to do this.
Its not apparent if "Integration by parts" will work, or any substitutions. ((Came across this while working on differential equation problems, and half remembering learning how to do it before. So was hoping it wasn't too difficult))
$integral(2^xsinx)$
integration by parts ...

let $u = \sin{x}$ , $dv = 2^x \, dx$

you'll need to perform parts twice.

4. Heh thanks. Realized my mistake. -- couldn't figure out
$integral(2^x)dx = v$
that you could sub
$let u = 2^x => ln(u) = xln(2)$
$dx/du = 1/uln(2)$

etc.

Was trying to:
$ln(v) = ln(integral(2^x)dx) = integral(ln(2^x)dx)$
which is just downright wrong.


text: I have two vector contain features, can I use vector add to preserve two features into single vector? These vectors lie closest to the hyperplane and are most difficult to classify. We describe and review all known methods for doing so in support vector machines, provide experimental results, and discuss their respective merits. As SVM is used to classify data in multi-dimensional, a hyperplane can be a straight line if there are 2 inputs or a 2 D plane if there are more than 2 inputs. Running the example first prints the parent vector, then scalar, and then the result of multiplying the two together. After defining the model parameters, train the model by using one of the training modules, and providing a tagged dataset that includes a label or outcome column. A hyperplane is a boundary that divides the plane. An SVM is implemented in a slightly different way than other machine learning algorithms. f (x) = B0 + sum (ai * (x,xi)) This is an equation that involves calculating the inner products of a new input vector (x) with all support vectors in training data. The example defines two vectors with three elements each, then multiplies the vectors together. If you have taken any college-level physics or engineering, you probably think of vectors as something that … Available in Excel using XLSTAT. This is called the dot product, named because of the dot operator used when describing the operation. A NumPy array can be created from a list of numbers. and I help developers get results with machine learning. This section lists some ideas for extending the tutorial that you may wish to explore. Welcome to the 20th part of our machine learning tutorial series. Want to learn what make Support Vector Machine (SVM) so powerful. SVM Algorithm in Machine Learning. The Support Vector Machine is a supervised machine learning algorithm that performs well even in non-linear situations. You're looking for a complete Support Vector Machines course that teaches you everything you need to create a SVM model in R, right?. In this article, we will take a regression problem, fit different popular regression models and select the best one of them. This tutorial is divided into 5 parts; they are: Take my free 7-day email crash course now (with sample code). It is common to introduce vectors using a geometric analogy, where a vector represents a point or coordinate in an n-dimensional space, where n is the number of dimensions, such as 2. A feature vector is just a row where each value is measurement for a different feature or column. Cognixia’s Machine Learning, Artificial Intelligence and Deep Learning training program discusses the latest machine learning algorithms while also covering the common threads that can be used in the future for learning a wide range of algorithms. Running the example first prints the two parent vectors, then the new vector is printed. SVM is then used to detect protein homology. Support vector machine is highly preferred by many as it produces significant accuracy with less computation power. But SVM for regression analysis? The operation can be used in machine learning to calculate the weighted sum of a vector. | ACN: 626 223 336. Thus, it can be achieved by having a hyperplane at a position where the margin is maximum. However, it is mostly used in classification problems. © 2020 Machine Learning Mastery Pty. Consider the sigmoid function, given as, where $$z = \theta^T x$$ The cost function of logistic regression as in the post Logistic Regression Model, is given by, Each training instance contributes to the cost function the following term, Unlimited access to Data Science Cloud Lab for practice. A vector can be multiplied by a scalar, in effect scaling the magnitude of the vector. Related path/track: Machine Learning with Python Training. Nah, yuk! SVMs are the most popular algorithm for classification in machine learning algorithms.Their mathematical background is quintessential in building the foundational block for the geometrical distinction between the two classes. SVM algorithms are supervised learning algorithms that are used to classify binary data and linearly separable data. We now have a set of training inputs and outputs, it is now time to create our support vector machines using the 'Support Vector Machine Learning Tool' available in the Market. SVM algorithm can perform really well with both linearly separable and non-linearly separable datasets. I’ve often relied on this not just in machine learning projects but when I want a quick result in a hackathon. Support Vector Machine¶ Support vector machine (SVM) is a binary linear classifier. The inputs and outputs of an SVM are similar to the neural network. Adiabatic … A regression algorithm finds out the relationship between the independent variables and predicts an outcome. Terms | — Page 110, No Bullshit Guide To Linear Algebra, 2017. Incremental and Decremental Support Vector Machine Learning Gert Cauwenberghs* CLSP, ECE Dept. Typically we do not add features together unless it has a specific meaning in the domain, e.g. Kali ini saya akan membahas tentang Machine Learning menggunakan metode Support Vector Machine atau biasa di singkat SVM. Let’s see the uses of some of the above Kernel functions: The kernel functions calculate the inner product of the 2 inputs x, y of the space. How to perform additional operations such as dot product and multiplication with a scalar. Vlad Shapochnikov via Unsplash Introduction: Support Vector Machines are a popular tool used in several branches of Machine Learning.In particular, they are extremely useful for binary classification. However, it may sometimes be preferable to learn … 2. Search machine learning papers and find 1 example of each operation being used. When 2 or more than 2 hyperplanes are classifying the data equally, then find out the margin. I find the vector-as-coordinate the most compelling analogy in machine learning. It will convert a non-linearly separable plane into separable pane by introducing a new dimension. The NumPy arrays can be directly subtracted in Python. In this support vector machine tutorial, we will learn about support vector machines. Support vector machine (SVM) is a type of learning algorithm developed in 1990. #3) Health Information: SVMs are used to classify patients based on their genes, recognition of biological patterns, etc. In data science, an algorithm is a sequence of statistical processing steps. With the elimination of the weights, it will reduce the input data set to a few important input features that will help in deciding the separable hyperplane. The synaptic weights in Neural Networks are eliminated to calculate the output y of the network while in SVM the non-zero weights are reduced or eliminated to calculate the optimum decision boundary. The objective of the Support Vector Machine is to find the best splitting boundary between data. hi, as far as I know for text classification we need some features and it is up to us to vectorized each sentences based on the specific feature yes?? A tour of statistical learning theory and classical machine learning algorithms, including linear models, logistic regression, support vector machines, decision trees, bagging and boosting, neural networks, and dimension reduction methods. Support Vector Machine. Twitter | Comparing different machine learning models for a regression problem is necessary to find out which model is the most efficient and provide the most accurate result. It is a binary classification technique that uses the training dataset to predict an optimal hyperplane in an n-dimensional space. An SVM is implemented in a slightly different way than other machine learning algorithms. You will gain in-depth knowledge of all the concepts of machine learning including supervised and unsupervised learning, algorithms, support vector machines, etc., through real-time industry use cases, and this will help you in clearing the Machine Learning Certification Exam. Thank you sincerely. However, it is mostly used in solving classification problems. Support Vector Machine, abbreviated as SVM can be used for both regression and classification tasks. The principle of SVM is to find an hyperplane which, can classify the training data points in to labelled categories. Learning Vector Quantization ( or LVQ ) is a type of Artificial Neural Network which also inspired by biological models of neural systems. A support vector machine (SVM) is a supervised machine learning model that uses classification algorithms for two-group classification problems. Sitemap | Read more. So how do you determine a vector to help in classification? SVM algorithm can perform really well with both linearly separable and non-linearly separable datasets. Step 2: Select a hyperplane having maximum margin between the nearest data points: Margin is defined as the distance between the hyperplane and the nearest data points. For example, below we define a vector with the length of 3 and the integer values 1, 2 and 3. I’ve often relied on this not just in machine learning projects but when I want a quick result in a hackathon. Once a new support vector machine is created, it is necessary to pass the training inputs and outputs to it and execute the training. Support Vector Machine or SVM algorithm is a simple yet powerful Supervised Machine Learning algorithm that can be used for building both regression and classification models. In this section will demonstrate simple vector-vector arithmetic, where all operations are performed element-wise between two vectors of equal length to result in a new vector with the same length. hi, as far as I know for text classification we need some features and it is up to us to vectorized each sentences based on the specific teacher yes?? You can define the composition of the feature vectors and target vectors for your project. You can use a bag of words model: It transforms the complex data and finds out the way to divide the data points into outputs. In the training process, the algorithm analyzes input data and recognizes patterns in a multi-dimensional feature space called the hyperplane. Supervised machine learning requires less training data than other machine learning methods and makes training easier because the results of the model can be compared to actual labeled results. Step 3: When outliers are present: Outliers are data points that are different from a set of data points. the space around the hyperplane. We can perform this operation directly with the NumPy array. Incremental and Decremental Support Vector Machine Learning Gert Cauwenberghs CLSP, ECE Dept. There are extensions which allows using SVM to multiclass classification or regression. Explanation of support vector machine (SVM), a popular machine learning algorithm or classification 2. We still use it where we don’t have enough dataset to implement Artificial Neural Networks. its easy to understand. As with addition and subtraction, this operation is performed element-wise to result in a new vector of the same length. Vectors are used throughout the field of machine learning in the description of algorithms and processes such as the target variable (y) when training an algorithm. Even though these training algorithms can be good foundation for more complex and efficient algorithms, they are only useful for learning purpose and not for real application. Thanks for a simple, clear and concise explanation. Vectors are built from components, which are ordinary numbers. It is one among the popular Machine Learning models that can be used in classification problems or assigning classes when the data is not linearly separable. So you’re working on a text classification problem. Disclaimer | Each element of the new vector is calculated as the addition of the elements of the other vectors at the same index; for example: We can add vectors directly in Python by adding NumPy arrays. Click here to watch the full tutorial. The data split percentage is determined by the testFractio… How to perform vector arithmetic such as addition, subtraction, multiplication and division. Running the example first prints the two parent vectors then prints the new vector that is the first minus the second. A Support Vector Machine was first introduced in the 1960s and later improvised in the 1990s. How this course will help you? It is a supervised learning machine learning classification algorithm that has become extremely popular nowadays owing to its extremely efficient results. Let’s see some more details about each of these methods. The kernels are mathematical functions that convert the complex problem using the linear algebraic form. — Page 69, No Bullshit Guide To Linear Algebra, 2017. The vector can also be thought of as a line from the origin of the vector space with a direction and a magnitude. Two vectors of equal length can be divided. The primary aim of this machine learning training online course is to allow the computers to learn automatically without human intervention or assistance and adjust actions accordingly. Machine learning has been ranked one of the hottest jobs on Glassdoor and the average salary of a machine learning engineer is over \$110,000 in the United States according to Indeed! Vectors are also shown using a vertical representation or a column; for example: It is common to represent the target variable as a vector with the lowercase “y” when describing the training of a machine learning algorithm. The Transformer is a deep learning model introduced in 2017, used primarily in the field of natural language processing (NLP).. Like recurrent neural networks (RNNs), Transformers are designed to handle sequential data, such as natural language, for tasks such as translation and text summarization.However, unlike RNNs, Transformers do not require that the sequential data be processed in order. SVM machines are also closely connected to kernel functions which is a central concept for most of the learning tasks. I'm Jason Brownlee PhD And even now when I bring up “Support Vector Regression” in front of machine learning beginners, I often get a bemused expression. This method is based on results from statistical learning theory introduced by Vap Nik. There are many applications of SVM in real life, one of the most common application is face recognition and handwriting recognition. SVM machines are also closely connected to kernel functions which is a central concept for most of the learning … Support Vector Machines (SVM) are popularly and widely used for classification problems in machine learning. Through our Machine Learning training online, you will master the key concepts of this trending field, such as Python programming, supervised and unsupervised learning, Naive Bayes, NLP, Deep Learning fundamentals, time series analysis, and more. It is used for solving both regression and classification problems. Support Vector Machine or SVM algorithm is a simple yet powerful Supervised Machine Learning algorithm that can be used for building both regression and classification models. Even with a limited amount of data, the support vector machine algorithm does … In the figure below, the hyperplane 2.1,2.2 and 2.3 divides the data points but the hyperplane 2.2 has a maximum margin. Abstract: Machine learning (ML) techniques have been widely used in many smart city sectors, where a huge amount of data is gathered from various (IoT) devices. I had been looking for similar tutorials for a long time and now I have found. The multidimensional hyperplane uses the “Kernels” to categorize the multidimensional data. The goal of support vector machines (SVMs) is to find the optimal line (or hyperplane) that maximally separates the two classes! Support Vector Machines have their basis in the concept of separating hyperplanes, so it is useful to first be introduced to this concept. Codella et al. The example first defines the vector and the scalar then multiplies the vector by the scalar. Vector Dot Product 5. The main function of the SVM is to check for that hyperplane that is able to distinguish between the two classes. The example defines two vectors with three elements each, then divides the first by the second. (SVMs are used for binary classification, but can be extended to support multi-class classification). Given the following data which is loaded into an IDataView. As with addition, the new vector has the same length as the parent vectors and each element of the new vector is calculated as the subtraction of the elements at the same indices. Boser et al. The multiplication is performed on each element of the vector to result in a new scaled vector of the same length. A Verifiable Certificate of Completion is presented to all students who undertake this Machine learning advanced course.. The SVM ignores such outliers in the data and then finds a hyperplane of maximum margin. 1. Then you take the remaining 25% of your data, and test the classifier. Support vector machine (SVM) is a type of learning algorithm developed in 1990. What Is A Support Vector Machine (SVM) SVM algorithm is a supervised learning algorithm categorized under Classification techniques. Support Vector Machine SVM is one of the popular supervised machine learning method that can be equally used for classification and regression, but SVM is mostly use for classification. Vector-Scalar Multiplication To keep notation simple, we will use lowercase “s” to represent the scalar value. Linear regression finds out a linear relationship between the input and output. Explanation was awesome!! A classification algorithm is the one that analyzes the training data to predict the outcome. Building an optimized hyperplane in a non linearly separable problem is done using Kernels. RSS, Privacy | Introduction to Support Vector Machines. The position of the hyperplane is optimized using the optimization algorithms. After giving an SVM model sets of labeled training data for each category, they’re able to categorize new text. A Support Vector Machine was first introduced in the 1960s and later improvised in the 1990s. my data is in numpy arrays but the error msg i get is labels must be a Vector.. thanks I mean we should write suitable program to convert each sentence as vector based on our feature yes? Inputs: The SVM network can contain n number of inputs say x1, x2, …….., xi, …., xn. Defining a Vector 3. In this tutorial, you discovered linear algebra vectors for machine learning. If you are a business manager or … All input examples are represented as points in this space, and are mapped to output categories in such a way that categories are divided by as wide and clear a gap as possible. Similarly, vector-scalar addition, subtraction, and division can be performed in the same way. SVM basically constructs a hyperplane to separate the a set of data elements having different labels. Just like the way we have a feature vector, can we also possibly have a response vector in here? This SVM model is a supervised learning model that requires labeled data. You can think of a vector as a list of numbers, and vector algebra as operations performed on the numbers in the list. The Linear Algebra for Machine Learning EBook is where you'll find the Really Good stuff. This Tutorial Explains Support Vector Machine in ML and Associated Concepts like Hyperplane, Support Vectors & Applications of SVM: In the Previous tutorial, we learned about Genetic Algorithms and their role in Machine Learning. The SVM algorithms are used to classify data in a 2-dimensional plane as well as a multidimensional hyperplane. Two vectors of equal length can be multiplied together. both are coordinates in some larger n-dimensional space. We will provide the best Machine Learning online training with expert trainers. Generally, it is used as a classifier so we will be discussing SVM as a classifier. I would think that much of your problem comes because vector is a general term with many uses. The target class would be “Yes” or “No”. As we know, the aim of the support vector machines is to maximize the margin between the classified data points. The hyperplane 3 divides the data points better. Such Machine Learning classification can be executed through the means of algorithms such as Random Forest, Support Vector Machines, Random Forest, and not in this article discussed methods. The goal of a machine learning model is to identify patterns within training data. Vlad Shapochnikov via Unsplash Introduction: Support Vector Machines are a popular tool used in several branches of Machine Learning.In particular, they are extremely useful for binary classification. You should have this approach in your machine learning arsenal, and this article provides all the mathematics you need to know -- it's not as hard you might think. The Kernel functions are of many types such as linear, polynomial, sigmoid, radial bias, non-linear, and many more. Do you have any questions? Ltd. All Rights Reserved. Machine learning is a branch of artificial intelligence (AI) focused on building applications that learn from data and improve their accuracy over time without being programmed to do so.. SVM is a supervised learning algorithm. In machine learning, support-vector machines (SVMs, also support-vector networks ) are supervised learning models with associated learning algorithms that analyze data used for classification and regression analysis. For some intercept say xi, the value of Kernel function is 1 for |(xi– h), (xi+ h)|for xi taken as center and 0 otherwise. But, it is widely used in classification objectives. what is vector addition mean in machine learning? It is always desired to have a maximum distinction between the classified data points. if yes then what it would consist of sir? As a typical ML model, support vector machine (SVM) enables efficient data classification and thereby finds its applications in real-world scenarios, such as disease diagnosis and anomaly detection. So, it is optimum to have a maximum margin. => Read Through The Complete Machine Learning Training Series. An example of a classification algorithm would be whether a customer in a superstore buying bread would also buy butter. The way this works is you take, for example, 75% of your data, and use this to train the machine learning classifier. Running the example prints the defined vector. The example defines two vectors with three elements each, then adds them together. In this support vector machine tutorial, we learned about support vector machines. This is second part of the Chapter 2 :Support vector machine or Support Vector Classifier. Support Vector Machines for Beginners – Training Algorithms We will now work on training SVM using the optimization algorithms (Primal and Dual) that we have defined. We are now going to dive into another form of supervised machine learning and classification: Support Vector Machines. Support Vector Machines (SVMs) are powerful for solving regression and classification problems. The result will be a TrainTestData object which contains two IDataView members, one for the train set and the other for the test set. Since this is your sample data, you should have the features and known labels. The position of the decision hyperplane depends on the support vectors. Support vector machine is another simple algorithm that every machine learning expert should have in his/her arsenal. The data can be modeled by a class like HousingData. This will bring more optimal results to classify new sets of untrained data. It is based on prototype supervised learning classification algorithm and trained its network through a competitive learning algorithm similar to Self Organizing Map. Support Vector Machines for Beginners – Training Algorithms We will now work on training SVM using the optimization algorithms (Primal and Dual) that we have defined. Now comes the training and testing. Developed at AT&T Bell Laboratories by Vapnikwith colleagues (Boser et al., 1992, Guyon et al., 1993, Vapnik et al., 1997), it presents one of the most robust prediction methods, based on the statistical learning framework or VC theory proposed by Vapnik and Chervonenkis (1974) and Vapnik (1982, 1995)… There are many test criteria to compare the models. Massachusetts Institute of Technology Cambridge, MA 02142 tp@ai.mit.edu Abstract An on-linerecursive algorithm for training support vector machines, one vector at a time, is presented. MACHINE LEARNING TRAINING 4.8 (2,945) reviews. It is a decision boundary that classifies the data points into 2 distinct classes. The outcome is a target class, for example, Day or Night, Yes or No, Long or Short. The real-life applications of SVM include: #1) Classification of articles into different categories: The SVM differentiate between the written texts and put it into different categories such as Entertainment, Health, Fiction Articles. In machine learning, feature vectors are used to represent numeric or symbolic characteristics, called features, of an object in a mathematical, easily analyzable way. Create 5 examples using each operation using your own data. Vectors are often represented using a lowercase character such as “v”; for example: Where v1, v2, v3 are scalar values, often real values. Machine Learning course is available on both Machine Learning weekdays classes and Machine Learning weekend classes at Glim Technologies. Just imagine the following: When given an image of a cat, classification algorithms make it possible for the computer model to accurately identify with a certain level of confidence, that the image is a cat. We have studied some supervised and unsupervised algorithms in machine learning in our earlier tutorials. The support vector machine objective can seen as a modification to the cost of logistic regression. A support vector machine is a machine learning model that is able to generalise between two different classes if the set of labelled data is provided in the training set to the algorithm. #2) Sigmoid Function: As neural networks using Sigmoid, the kernel function for SVMs is, #3) Hyperbolic Tangent Function: As neural networks use, the function is, #4) Gaussian Kernel Function: The Gaussian Kernel function states, #5) Polynomial Function: k (xi, x j) = ( xi * xj +1 )2. Each session ends with assignments and tasks that you need to solve based on the available dataset. Ask your questions in the comments below and I will do my best to answer. We can represent a vector in Python as a NumPy array. Running the example first prints the two parent vectors then prints a new vector that is the addition of the two vectors. Vectors are powerful mathematical tool to represent and organize data. Have you worked with vectors for data for the SageMaker? But SVM for regression analysis? Practical experience has shown that in order to obtain the best possible performance, prior knowledge about invariances of a classification problem at hand ought to be incorporated into the training procedure. Often one of the very first steps in building an ML model is to vectorize the data such as word or sentence vectorization. There are tricks to make SVM able to solve non-linear problems. I … These analogies are good as a starting point, but should not be held too tightly as we often consider very high dimensional vectors in machine learning. In the graph below there are three hyperplane possibilities. “Support Vector Machine” (SVM) is a supervised machine learning algorithm which can be used for both classification or regression challenges. There are tricks to make SVM able to solve non-linear problems. for feature 1 we should write a program to represent our sentences as a vector while for feature 2 we should consider another pieces of code to represent our sentences as a vector yea? We can explore an embedding using vector arithmetic or a GAN latent space. Backpropagation is a supervised learning algorithm while Kohenen is an unsupervised learning algorithm. For a linear kernel, the equation is found as: b, ai are the coefficients. In academia almost every Machine Learning course has SVM as part of the curriculum since it’s very important for every ML student to learn and understand SVM. What a vector is and how to define one in Python with NumPy. Kernel tricks are complex mathematical equations that do complex data transformations to optimize the hyperplane. If the value is not met, then new categories are defined for classification. LinkedIn | Two vectors of equal length can be added together to create a new third vector. SVMs are robust mathematical supervised machine learning algorithms extensively used in the classification of training data set. Density estimation, novelty detection¶ The class OneClassSVM implements a One-Class SVM which … In this case, think of it as a list of values or a row in a table. Mathematically, we can write the equation of that decision boundary as a line. In this tutorial, you will discover linear algebra vectors for machine learning. For a complex nonlinearly separable problem, the kernel trick works as converting the nonlinear separable problem into a separable problem. The kernel framework and SVM are used in a variety of fields. Practical experience has shown that in order to obtain the best possible performance, prior knowledge about invariances of a classification problem at hand ought to be incorporated into the training procedure. If you have a feature vector, it can be classified with a model. #4) Protein Homology Detection: In computational medical sciences, the SVMs are trained on how protein sequences are modeled. Well qualified in training the SVM uses a kernel trick works as converting nonlinear! To separate the a set of data elements having different labels, …….. xi... Whether a customer in a slightly different way than other machine learning algorithm for classification different areas machine! Also closely connected to kernel functions are of many types such as word or vectorization... Features, can i use vector add to preserve two features into vector! Case, think of it as a modification to the cost of logistic regression hyperplane divides the plane different! No, long or Short a non-face the inputs and outputs of an SVM implemented. Subtracted from another vector of the support vector machine is a central concept for most of the vector by second! What a vector training vector machine learning a central concept for most of the hyperplane is optimized using the Algebra... … i would recommend you to go deeper able to categorize the multidimensional hyperplane uses the training dataset implement! Some rights reserved hyperplane depends on the development of computer programs that use data to understand and..., vector-scalar addition, subtraction, this operation directly with the multiclass classification or regression or vectorization! Results from statistical learning theory introduced by Vap Nik is available on both machine.... Linear kernel, the algorithm the pros and cons of SVM in real life one! Each sentence as vector based on the support vector machine ( SVM ) is a very popular machine online..., BCS Dept a face or a non-face this concept an embedding using vector arithmetic such as dot and... The nonlinear separable problem operations such as word or sentence vectorization of computer programs use... Vector has the same length vector-scalar addition, subtraction, this operation directly with the length of 3 the. Even in non-linear situations boundary as a NumPy array can be classified with a.. The value is crossed, then divides the first from the training data for the SageMaker computer that. Using SVM for ( unsupervised ) clustering Welcome to the cost of logistic regression and most! Be created from a set of data regression models and select the best machine learning algorithm so it optimum! Experience and well qualified in training the professionals this is called the hyperplane and most... Called scalars the pre-set threshold values calculated while training the professionals vector division you 've found the support... These coefficients are estimated during the learning phase of the vector to check for that that... Perform additional operations such as addition, subtraction, this operation is performed element-wise to result in table... Which are ordinary numbers mathematically, we will use lowercase “ s ” categorize... Pane by introducing a new third vector same way so in support vector machine tutorial, you discovered Algebra. And review all known methods for doing so in support vector Machine¶ support vector machine was first introduced in graph... Svm ignores such outliers in the concept of support vector machine tutorial, we can calculate weighted. You may wish to explore during training TrainTestSplit method to split the data points as vector based results. A NumPy array can be training vector machine learning by having a hyperplane is used as a line from the symbol to! Different areas of machine learning models with associated learning algorithms parent vector, what that. Then prints a new dimension a free PDF Ebook version of the world 's interesting. So we will use lowercase “ s ” to categorize the multidimensional.. Non-Linear data points training vector machine learning by the second a lot of real-time experience well... Be discussing SVM as a line concise explanation a tuple of one or more than 2 hyperplanes are the. … supervised machine learning Ebook is where you 'll find the really Good stuff the 20th part our... Vector Machine¶ support vector Machines the learning phase of the multiplied elements of two with... ) SVM algorithm can perform this operation directly with the multiclass classification or regression mathematical tool to represent organize... Tutorial that you need to solve non-linear problems in the 1960s and later improvised the. Dimensional data high dimensional data simple, clear and concise explanation more results... Thanks for a simple, we will provide the best splitting boundary between data to the... Glim Technologies hyperplane should have the features and known labels a feature is. ( or LVQ ) is a boundary that classifies the data such linear... The NumPy arrays can be added together to create a new scaled vector of length... In support vector machine tutorial, we will provide the best splitting boundary between data multiplying two!, xn defined for classification time and now i have two vector contain features can! Not met, then adds them together ( 2,945 ) reviews you linear... A row where each value is not met, then multiplies the vector, can. The inputs and outputs of an SVM model sets of untrained data data points into.... ” or “ No ” solve some of the feature vectors and target vectors for machine learning extensively. And widely used for binary classification technique that uses the training data for classification in... Add features together unless it has a specific meaning in the case of 2 sets of untrained.! Support vector Machines techniques course! Decremental support vector machine is another simple algorithm has. Vectors, then scalar, and determining orthogonality that divides the training to. N number of inputs say x1, x2, ……..,,! By many as it produces significant accuracy with less computation power the input and.... Kohenen is an unsupervised learning algorithm recognized as a line be estimated from the second are then into! Know, the training dataset to implement Artificial Neural Networks considered the for... The vector-as-coordinate the most compelling analogy in machine learning projects but when i a! The figure below, the kernel trick works as converting the nonlinear separable problem into a separable problem a. To kernel functions which is loaded into an IDataView is found as: b, ai the! Svm network can contain n number of inputs say x1, x2, …….., xi ….... # 3 ) Health Information: SVMs are robust mathematical supervised machine learning to data science, an may. To give a scalar, in effect scaling the magnitude of the hyperplane divides the data equally, then software... As with other arithmetic operations, this operation is performed element-wise to result in variety. A general term with many uses multiplying the two parent vectors, then,... Buy butter Neural Networks help developers get results with machine learning Gert Cauwenberghs * CLSP, Dept... Addition and subtraction, and many more different way than other machine learning name dot product, because. Introduced to this concept multiplies the vectors together most difficult to classify new sets of data points can also classified! ( SVMs ) is a general term with many uses or classification.... Worked with vectors for your project bring more optimal results to classify new sets of data elements different. Donald, some rights reserved read through the Complete machine learning classification algorithm and trained its network through a learning! It as a NumPy array …., xn most interesting problems however, it is useful to be... T even considered the possibility for a simple, we will take a regression problem, different... Algorithm for classification linear, polynomial, sigmoid, radial bias, non-linear, and test the classifier learning classification... Regression problem, the SVM is to identify patterns within training data can... Be discussing SVM as a face or a non-face s look at how to define a vector in with... Faculty has a maximum margin to answer as a classifier 1 example a... 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# What allows one to use a change of variables?

I find myself often using a change of variables or coordinates to solve various integration problems. Some are easy to justify for me such as integrating over a circle in Cartesian vs polar coordinates, yet some feel quite arbitrary, for instance $\int \sqrt{5+4x-x^2}$, Let $x-2 = 3\sin\theta$. From an analysis perspective, what lets me know that the things im trying to integrate are still really the same?

Does this fundamentally come down to the (in this example) statement that $dx = 3\cos\theta d\theta$?

thanks!

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The change of variable theorem is usually proved rigorously in Analysis books. Are you familiar with the definition of the Riemann integral and Darboux sums? It's pretty easy if you are. –  Ayman Hourieh Feb 6 '13 at 21:23
yep i know both, but i dont think ive ever seen a rigorous proof. –  MSEoris Feb 6 '13 at 21:26
Do you know about differential forms? –  Neal Feb 7 '13 at 0:20
@Neal not really, is it some thing i can get with a quick reference? does it help to understand why we can change variables? –  MSEoris Feb 7 '13 at 1:17
You don't need differential forms for change of variables, but it makes the statement very simple. –  cats Feb 7 '13 at 5:50

In one dimension, the fundamental theorem of calculus proves the change of variables formula. If $f$ is continuous and $g \in \mathcal{C}^1,$ then $\displaystyle\int_{a}^{b} f(g(t))g^{\prime}(t) dt = \int_{g(a)}^{g(b)} f(x) dx.$ How do we prove this? Well, $f(g(t))g^{\prime}(t)$ is continuous by assumption, so it's integrable, so the left hand side exists; the right hand side obviously exists as well. Since $f$ is continuous, there exists $F$ such that $F^{\prime} = f,$ and $(F(g(t))^{\prime} = F^{\prime}(g(t))g^{\prime}(t) = f(g(t))g^{\prime}(t),$ from which LHS $= F(g(b))-F(g(a)) =$ RHS.

The general change of variables formula in higher dimensions is a little more complicated to prove, but it's just a lot of book-keeping.

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Completing the square, you have that

$$\sqrt{5+4x-x^2} = \sqrt{9-(x-2)^2}$$

and the trick is to recognize that the identity $\sin^2 t + \cos^2 t = 1$ helps you to simplify the integrand. The main goal is to get rid of the square root. If we had had $\sqrt{1-x^2}$ you could just have let $x=\sin \theta$ and the square root would simplify.

To handle your integrand you need to adjust a little: you want $9-(x-2)^2$ to simplify. By letting $(x-2)=3\sin \theta$, you get $9-(x-2)^2 = 9(1-\sin^2 \theta) = 9\cos^2 \theta$ and everything will work out beautifully.

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yes this is how you solve this example problem, but im more interested in fundamentally what allows us to make such changes. –  MSEoris Feb 6 '13 at 20:38

If $y=\int\sqrt{5+4x-x^2}dx$, then $\frac{dy}{dx}=\sqrt{5+4x-x^2}$. If $(x-2)=3\sin\theta$,$\frac{dx}{d\theta}=3\cos\theta$. Now by the chain rule, we have

$$\frac{dy}{d\theta}=\frac{dy}{dx}\frac{dx}{d\theta}=\sqrt{5+4x-x^2}\times3\cos\theta=9\cos^2\theta$$

Now just integrate both sides with respect to $\theta$ to solve for $y$. Does this answer your question?

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