Monday, 26 August 2019

Vector Spaces

What are vector spaces?

vector spaces,vector space examples,vector,vector space in hindi,vector space model,vector spaces and subspaces,linear algebra vector spaces,basis of vector space,vector space and subspace,space,vector space linear algebra,vector space (literature subject),vector spaces and subspaces linear algebra,vector space basis,vector space mit,vector spaces examples


Let (F,+,.) be a field. The element of F will be called scalars. Let V be a non-empty set whose elements will be called vector. Then V is a vector space over the field F.
If an internal composition in V called the addition of vectors and denoted by'+'.Also for this composition Vis an abelian group.
An external composition in V over F called scalar multiplication and denoted by multiplicatively so that ax⋿V for all a⋿ F and for all 𝞪⋿V(V is close to scalar multiplication) 

Sunday, 25 August 2019

Rings-Algebra

What are Rings?

abstract algebra,algebra,rings,modern algebra,rings abstract algebra,rings in abstract algebra,modern algebra rings,ring,what are rings in abstract algebra,abstract algebra (field of study),algebraic structure,ring theory algebra,ring theory,abstract algebra rings and fields,ring theory abstract algebra,ring theory in abstract algebra

Let R is non-empty set with two binary operations called addition and multiplication. It is denoted by(+,.),  for all a,b⋿R we have a+b⋿Rand a.b⋿R  then this algebraic structure(R,+,.) is called a ring.

Ring with unity

If ring R exists by1 such that 1.a=a=a.1 all a⋿R, then R is called a ring with the unit element.

Commutative Ring

If a ring R, the multiplication composition is also commutative, if we have a.b=b.a all a,b⋿R, then Ris called commutative ring.

Properties of Rings

If R is a ring for all a,b,c⋿R

a0=0a=0
a(-b)=-(ab)=(-a)b
(-a)(-b)=ab
a(b-c)=ab-ac
(b-c)a=ba-ca

How to prove the intersection of two subrings is a subring?


Let S1 and S2  be two subrings of a ring R. Then S1∩S2 is not empty since at least 0⋿ S1∩S2.
Now in order to prove that S1∩S2 is subring, it is sufficient to prove that
1. a⋿ S1∩S2,b⋿ S1∩S2
a-b⋿ S1∩S2
2. a⋿ S1∩S2,b⋿ S1∩S2
ab⋿ S1∩S2
We have 
a⋿ S1∩S2
a⋿ S1,a⋿ S2
b⋿ S1∩S2
b⋿ S1,b⋿ S2

Now S1 and S2  are both subrings.

such that 
a⋿ S1,b⋿ S1
a-b⋿ S1 and ab⋿ S1
and a⋿ S2,b⋿ S2
ab⋿ S2.
Now a-b ⋿ S1,a-b⋿ S2
a-b⋿ S1∩S2
ab⋿ S1,ab⋿ S2
ab⋿ S1∩S2.
Thus 
a⋿ S1∩S2,
b⋿ S1∩S2
a-b⋿ S1∩S2.and
ab⋿ S1∩S2.
so that
 S1∩S2 is a subring of R.

Thursday, 22 August 2019

Groups

What are the groups?

group,algebra,abstract algebra,modern algebra,group theory,groups,abelian group,algebraic structure,algebra group theory,algebra proof,group thery,group example,group type,group examples,group theory bsc math,group isomorphism,algebra (field of study),symmetric groups,mathematics,semi group,group theory in hindi,abstract algebra (field of study),finite simple groups,examples of algebraic structure,factor group,cyclic group

Groups are the branch of abstract algebra.

How does it help?

It has helped in developing chemistry, physics, and computer science.
we define groups
1.Binary Operation
Let S be a non-empty set.Any function*:SxS→S is called binary operation on S.
Example-A binary operation *  on S and (a,b)⋿S xS,we denote *(a,b) by a*b
Now* binary operation on a set S.We say that
1.* is closed on a subset T of S if a*b⋿ T ∀ a,b ⋿ T
2. * is associative if for all a,b,c ⋿ S,(a*b)*c =a*(b*c)
3.* is commutative if for all a,b ⋿ S, a*b=b*a

What are semi Group?

An algebraic structure(G,*) is called a semigroup if the binary operation* is associative in G,if(a*b)*c=a*(b*c)  all a,b,cɛG.
The set of N of all-natural number is a semigroup with respect to the operation of addition of natural numbers. Addition is an associative operation on N.

The algebraic structures(N,*),(I,+) and(R,+) are also semigroup.

How many generators are of the cyclic group G of order 8?

Let ɑ be a generator of G.Then o(ɑ)=8, so we can write
G={ɑ,ɑ^2,ɑ^3,ɑ^4,ɑ^5,ɑ^6,ɑ^7,ɑ^8}.
7is prime to 8, therefore ɑ^7 is also a generator of G.
5 is prime to 8, therefore ɑ^5 is also a generator of G.
3 is prime to 8, therefore ɑ^3 is also a generator of G.

So there are only four generators of G i.e.,ɑ,ɑ^3,ɑ^5,ɑ.



Monday, 12 August 2019

How to generate the two time series

 How to generate the two time series of length 10000

 #fix the random seed

> set.seed(20140629)

> #define length of simulation

> N <- 10000
> #simulate a normal random walk
> x <- cumsum(rnorm(N))
> #set an initial parameter value
> gamma <- 0.6
> #simulate the cointegrating series
> y <- gamma * x + rnorm(N)
> #plot the two series
> plot(x, type='l')
> lines(y,col="red")
https://www.mathclasstutor.online

# Augmented Dickey-Fuller Test Unit Root Test #

###############################################

Test regression none


Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)

Residuals:
    Min      1Q  Median      3Q     Max
-3.7583 -0.6800 -0.0105  0.6604  3.6463

Coefficients:
             Estimate Std. Error t value
z.lag.1    -0.0008567  0.0004221  -2.030
z.diff.lag  0.0287582  0.0099989   2.876
           Pr(>|t|) 
z.lag.1     0.04241 *
z.diff.lag  0.00403 **
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9956 on 9996 degrees of freedom
Multiple R-squared:  0.001212, Adjusted R-squared:  0.001012
F-statistic: 6.067 on 2 and 9996 DF,  p-value: 0.002328


Value of test-statistic is: -2.0297

Critical values for test statistics:
      1pct  5pct 10pct
tau1 -2.58 -1.95 -1.62

Augmented Dickey-Fuller Test Unit Root Test # 

###############################################

Test regression none


Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)

Residuals:
    Min      1Q  Median      3Q     Max
-5.4824 -0.9477 -0.0207  0.9182  5.1281

Coefficients:
             Estimate Std. Error t value
z.lag.1    -0.0032685  0.0009824  -3.327
z.diff.lag -0.4303026  0.0090285 -47.661
           Pr(>|t|) 
z.lag.1    0.000881 ***
z.diff.lag  < 2e-16 ***
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.392 on 9996 degrees of freedom
Multiple R-squared:  0.1875, Adjusted R-squared:  0.1874
F-statistic:  1154 on 2 and 9996 DF,  p-value: < 2.2e-16


Value of test-statistic is: -3.3269

Critical values for test statistics:
      1pct  5pct 10pct
tau1 -2.58 -1.95 -1.62

 #take a linear combination of the series

> z = y - gamma*x
> plot(z,type='l')
https://www.mathclasstutor.online
summary(ur.df(z,type="none"))

###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################

Test regression none


Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)

Residuals:
    Min      1Q  Median      3Q     Max
-3.6372 -0.6856 -0.0036  0.6590  3.5004

Coefficients:
           Estimate Std. Error t value
z.lag.1    -0.99967    0.01423 -70.250
z.diff.lag -0.01232    0.01000  -1.232
           Pr(>|t|) 
z.lag.1      <2e-16 ***
z.diff.lag    0.218 
---
Signif. codes:
  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’
  0.1 ‘ ’ 1

Residual standard error: 0.9983 on 9996 degrees of freedom
Multiple R-squared:  0.5061, Adjusted R-squared:  0.506
F-statistic:  5121 on 2 and 9996 DF,  p-value: < 2.2e-16


Value of test-statistic is: -70.2499

Critical values for test statistics:
      1pct  5pct 10pct
tau1 -2.58 -1.95 -1.62

#Estimate the cointegrating relationship

> #regression without intercept

> coin <- lm(y ~ x -1)

> #obtain the residuals

> coin$resid 

Tuesday, 6 August 2019

How to draw stock charts

How to draw stock charts


getSymbols("BMW.DE", env = bmw_stock, src = "yahoo", from =
+                as.Date("2010-01-01"), to = as.Date("2019-08-05"))
[1] "BMW.DE"
> BMW<-bmw_stock$BMW.DE
> head(BMW)
           BMW.DE.Open BMW.DE.High
2010-01-04      31.820      32.455
2010-01-05      31.960      32.410
2010-01-06      32.450      33.040
2010-01-07      32.650      33.200
2010-01-08      33.335      33.430
2010-01-11      32.995      33.050
           BMW.DE.Low BMW.DE.Close
2010-01-04     31.820       32.050
2010-01-05     31.785       32.310
2010-01-06     32.360       32.810
2010-01-07     32.380       33.100
2010-01-08     32.515       32.655
2010-01-11     32.110       32.170
           BMW.DE.Volume
2010-01-04       1808170
2010-01-05       1564182
2010-01-06       2218604
2010-01-07       2026145
2010-01-08       1925894
2010-01-11       2157825
           BMW.DE.Adjusted
2010-01-04        22.94292
2010-01-05        23.12905
2010-01-06        23.48697
2010-01-07        23.69456
2010-01-08        23.37601
2010-01-11        23.02882
> chartSeries(BMW,multi.col=TRUE,theme="white")
> addMACD()
> addBBands()
www.mathclasstutor.online

> BMW_return <-log(BMW$BMW.DE.Close/BMW$BMW.DE.Open)
> qqnorm(BMW_return, main = "Normal Q-Q Plot of BMW daily log return",
+        xlab = "Theoretical Quantiles",
+        ylab = "Sample Quantiles", plot.it = TRUE, datax = FALSE
+ )
> qqline(BMW_return, col="red")
www.mathclasstutor.online

Latin Cubes

What is Latin Cubes? The practical applications of Latin crops and related designs are factorial experiment.factorial experiments for me...