text:
# Math Help - Area Increasing

1. ## Area Increasing

The length of a rectangle is increasing at a rate of 2 ft. per second, while the width is increasing at a rate of 3 ft. per second. When the length is 4 ft. and the width is 6 ft., how fast, in square feet per second, is the area increasing?

I solved:

A = LW
dA/dt = LdW/dt + WdL/dt
dA/dt = L*1 + W*2
at L = 4, W = 6, dA/dt = 4+6*2 = 16 ft²/sec

...but that's not an answer choice. The answer choices are 144, 24, 5, 26, and 6.

I'd really appreciate some some on this, possibly clear steps on how to solve.

2. ## Re: Area Increasing

Originally Posted by JenniferC
The length of a rectangle is increasing at a rate of 2 ft. per second, while the width is increasing at a rate of 3 ft. per second. When the length is 4 ft. and the width is 6 ft., how fast, in square feet per second, is the area increasing?

I solved:

A = LW
dA/dt = LdW/dt + WdL/dt
dA/dt = L*1 + W*2
at L = 4, W = 6, dA/dt = 4+6*2 = 16 ft²/sec

...but that's not an answer choice. The answer choices are 144, 24, 5, 26, and 6.

I'd really appreciate some some on this, possibly clear steps on how to solve.
I think you meant

$\frac{dA}{dt}=L\cdot 3+W\cdot 2$

at L=4 and W=6

$\frac{dA}{dt}=4\cdot 3+6\cdot 2=12+12=24$


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## G = D7×Q16order 224 = 25·7

### Direct product of D7 and Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D7×Q16
 Chief series C1 — C7 — C14 — C28 — C4×D7 — Q8×D7 — D7×Q16
 Lower central C7 — C14 — C28 — D7×Q16
 Upper central C1 — C2 — C4 — Q16

Generators and relations for D7×Q16
G = < a,b,c,d | a7=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 238 in 60 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, Q8, Q8, D7, C14, C2×C8, Q16, Q16, C2×Q8, Dic7, Dic7, C28, C28, D14, C2×Q16, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, C7×Q8, C8×D7, Dic28, C7⋊Q16, C7×Q16, Q8×D7, D7×Q16
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, D14, C2×Q16, C22×D7, D4×D7, D7×Q16

Smallest permutation representation of D7×Q16
On 112 points
Generators in S112
(1 102 21 12 86 40 50)(2 103 22 13 87 33 51)(3 104 23 14 88 34 52)(4 97 24 15 81 35 53)(5 98 17 16 82 36 54)(6 99 18 9 83 37 55)(7 100 19 10 84 38 56)(8 101 20 11 85 39 49)(25 74 43 59 70 91 109)(26 75 44 60 71 92 110)(27 76 45 61 72 93 111)(28 77 46 62 65 94 112)(29 78 47 63 66 95 105)(30 79 48 64 67 96 106)(31 80 41 57 68 89 107)(32 73 42 58 69 90 108)
(1 54)(2 55)(3 56)(4 49)(5 50)(6 51)(7 52)(8 53)(9 13)(10 14)(11 15)(12 16)(17 86)(18 87)(19 88)(20 81)(21 82)(22 83)(23 84)(24 85)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 97)(40 98)(41 72)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(57 61)(58 62)(59 63)(60 64)(73 94)(74 95)(75 96)(76 89)(77 90)(78 91)(79 92)(80 93)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)
(1 25 5 29)(2 32 6 28)(3 31 7 27)(4 30 8 26)(9 62 13 58)(10 61 14 57)(11 60 15 64)(12 59 16 63)(17 47 21 43)(18 46 22 42)(19 45 23 41)(20 44 24 48)(33 90 37 94)(34 89 38 93)(35 96 39 92)(36 95 40 91)(49 110 53 106)(50 109 54 105)(51 108 55 112)(52 107 56 111)(65 87 69 83)(66 86 70 82)(67 85 71 81)(68 84 72 88)(73 99 77 103)(74 98 78 102)(75 97 79 101)(76 104 80 100)

G:=sub<Sym(112)| (1,102,21,12,86,40,50)(2,103,22,13,87,33,51)(3,104,23,14,88,34,52)(4,97,24,15,81,35,53)(5,98,17,16,82,36,54)(6,99,18,9,83,37,55)(7,100,19,10,84,38,56)(8,101,20,11,85,39,49)(25,74,43,59,70,91,109)(26,75,44,60,71,92,110)(27,76,45,61,72,93,111)(28,77,46,62,65,94,112)(29,78,47,63,66,95,105)(30,79,48,64,67,96,106)(31,80,41,57,68,89,107)(32,73,42,58,69,90,108), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,13)(10,14)(11,15)(12,16)(17,86)(18,87)(19,88)(20,81)(21,82)(22,83)(23,84)(24,85)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,97)(40,98)(41,72)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(57,61)(58,62)(59,63)(60,64)(73,94)(74,95)(75,96)(76,89)(77,90)(78,91)(79,92)(80,93), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,62,13,58)(10,61,14,57)(11,60,15,64)(12,59,16,63)(17,47,21,43)(18,46,22,42)(19,45,23,41)(20,44,24,48)(33,90,37,94)(34,89,38,93)(35,96,39,92)(36,95,40,91)(49,110,53,106)(50,109,54,105)(51,108,55,112)(52,107,56,111)(65,87,69,83)(66,86,70,82)(67,85,71,81)(68,84,72,88)(73,99,77,103)(74,98,78,102)(75,97,79,101)(76,104,80,100)>;

G:=Group( (1,102,21,12,86,40,50)(2,103,22,13,87,33,51)(3,104,23,14,88,34,52)(4,97,24,15,81,35,53)(5,98,17,16,82,36,54)(6,99,18,9,83,37,55)(7,100,19,10,84,38,56)(8,101,20,11,85,39,49)(25,74,43,59,70,91,109)(26,75,44,60,71,92,110)(27,76,45,61,72,93,111)(28,77,46,62,65,94,112)(29,78,47,63,66,95,105)(30,79,48,64,67,96,106)(31,80,41,57,68,89,107)(32,73,42,58,69,90,108), (1,54)(2,55)(3,56)(4,49)(5,50)(6,51)(7,52)(8,53)(9,13)(10,14)(11,15)(12,16)(17,86)(18,87)(19,88)(20,81)(21,82)(22,83)(23,84)(24,85)(25,105)(26,106)(27,107)(28,108)(29,109)(30,110)(31,111)(32,112)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,97)(40,98)(41,72)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(57,61)(58,62)(59,63)(60,64)(73,94)(74,95)(75,96)(76,89)(77,90)(78,91)(79,92)(80,93), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112), (1,25,5,29)(2,32,6,28)(3,31,7,27)(4,30,8,26)(9,62,13,58)(10,61,14,57)(11,60,15,64)(12,59,16,63)(17,47,21,43)(18,46,22,42)(19,45,23,41)(20,44,24,48)(33,90,37,94)(34,89,38,93)(35,96,39,92)(36,95,40,91)(49,110,53,106)(50,109,54,105)(51,108,55,112)(52,107,56,111)(65,87,69,83)(66,86,70,82)(67,85,71,81)(68,84,72,88)(73,99,77,103)(74,98,78,102)(75,97,79,101)(76,104,80,100) );

G=PermutationGroup([[(1,102,21,12,86,40,50),(2,103,22,13,87,33,51),(3,104,23,14,88,34,52),(4,97,24,15,81,35,53),(5,98,17,16,82,36,54),(6,99,18,9,83,37,55),(7,100,19,10,84,38,56),(8,101,20,11,85,39,49),(25,74,43,59,70,91,109),(26,75,44,60,71,92,110),(27,76,45,61,72,93,111),(28,77,46,62,65,94,112),(29,78,47,63,66,95,105),(30,79,48,64,67,96,106),(31,80,41,57,68,89,107),(32,73,42,58,69,90,108)], [(1,54),(2,55),(3,56),(4,49),(5,50),(6,51),(7,52),(8,53),(9,13),(10,14),(11,15),(12,16),(17,86),(18,87),(19,88),(20,81),(21,82),(22,83),(23,84),(24,85),(25,105),(26,106),(27,107),(28,108),(29,109),(30,110),(31,111),(32,112),(33,99),(34,100),(35,101),(36,102),(37,103),(38,104),(39,97),(40,98),(41,72),(42,65),(43,66),(44,67),(45,68),(46,69),(47,70),(48,71),(57,61),(58,62),(59,63),(60,64),(73,94),(74,95),(75,96),(76,89),(77,90),(78,91),(79,92),(80,93)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112)], [(1,25,5,29),(2,32,6,28),(3,31,7,27),(4,30,8,26),(9,62,13,58),(10,61,14,57),(11,60,15,64),(12,59,16,63),(17,47,21,43),(18,46,22,42),(19,45,23,41),(20,44,24,48),(33,90,37,94),(34,89,38,93),(35,96,39,92),(36,95,40,91),(49,110,53,106),(50,109,54,105),(51,108,55,112),(52,107,56,111),(65,87,69,83),(66,86,70,82),(67,85,71,81),(68,84,72,88),(73,99,77,103),(74,98,78,102),(75,97,79,101),(76,104,80,100)]])

D7×Q16 is a maximal subgroup of
SD32⋊D7  Q32⋊D7  D28.30D4  D8.10D14  D28.44D4
D7×Q16 is a maximal quotient of
Dic74Q16  Dic7.1Q16  Dic7⋊Q16  Dic7.Q16  D144Q16  D14.Q16  D14⋊Q16  Dic286C4  C562Q8  Dic142Q8  D14.2Q16  D142Q16  C56.26D4  Dic73Q16  D145Q16  D143Q16

35 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A 8B 8C 8D 14A 14B 14C 28A 28B 28C 28D ··· 28I 56A ··· 56F order 1 2 2 2 4 4 4 4 4 4 7 7 7 8 8 8 8 14 14 14 28 28 28 28 ··· 28 56 ··· 56 size 1 1 7 7 2 4 4 14 28 28 2 2 2 2 2 14 14 2 2 2 4 4 4 8 ··· 8 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 D4 D4 D7 Q16 D14 D14 D4×D7 D7×Q16 kernel D7×Q16 C8×D7 Dic28 C7⋊Q16 C7×Q16 Q8×D7 Dic7 D14 Q16 D7 C8 Q8 C2 C1 # reps 1 1 1 2 1 2 1 1 3 4 3 6 3 6

Matrix representation of D7×Q16 in GL4(𝔽113) generated by

 0 1 0 0 112 79 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 112 0 0 0 0 112
,
 1 0 0 0 0 1 0 0 0 0 0 14 0 0 8 62
,
 112 0 0 0 0 112 0 0 0 0 49 65 0 0 83 64
G:=sub<GL(4,GF(113))| [0,112,0,0,1,79,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,0,8,0,0,14,62],[112,0,0,0,0,112,0,0,0,0,49,83,0,0,65,64] >;

D7×Q16 in GAP, Magma, Sage, TeX

D_7\times Q_{16}
% in TeX

G:=Group("D7xQ16");
// GroupNames label

G:=SmallGroup(224,112);
// by ID

G=gap.SmallGroup(224,112);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,116,86,297,159,69,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^2=c^8=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽

format: plain


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# RSA solving for $p$ and $q$ knowing $\phi(pq)$

I am trying to find primes $p$ and $q$ in the RSA algorithm given $n = pq$ and the value of $\phi(n)$. I know the following:

1. $\phi(n) = (p-1)(q-1) = pq - p - q + 1$
2. Solving for $p+q = pq - \phi(n) + 1$
3. Take $(p-q)^2 = p^2 - 2pq + q^2$
4. Solve $(p-q) = \sqrt{p^2 -2pq + q^2}$
5. Simplify to $(p-q) = \sqrt{p^2 + 2pq -4pq + q^2} = \sqrt{(p+q)^2 -4pq}$

Now we can solve for $p$ and $q$:

$p = \dfrac{(p+q) +(p-q)}{2}$ and $q = \dfrac{(p+q)-(p-q)}{2}$

Is there a better or faster way to solve for $p$ and $q$ given $pq$ and $\phi(pq)$

-
Did you see this posting? math.stackexchange.com/questions/12328/… –  Amzoti Sep 24 '12 at 2:29

There is, based on the simple fact that $$(X-p)(X-q)=X^2-2\frac{p+q}{2}X+pq.$$
All you need to do is compute $\displaystyle m=\frac{p+q}{2}=\frac{n-\phi(n)+1}{2}$, then $$p,q=m\pm\sqrt{m^2-n}.$$


text:
Code:
for(k=0;k<10;k++)
{
c+=shift;
dedispersion(c, displacement, fbk, result);
Here, if I use "c=shift", the program runs well but the c will not change.
If I keep the code like the above, then it can be compiled and linked but it will be closed when running. I am not able to change the parameter "c" in a loop.
The whole code is followed:
Code:
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#define PI 3.14159265358979323846264338327950288419716939937510
#define TWO_PI (6.2831853071795864769252867665590057683943L)
#define NF 33554432  //total number of data.
#define NT 512   //number of channels.
#define NTAO 32    // NTAO is the number that we use to creat filter bank.
#define FS 14000100

/* function prototypes */
void checkN (int N);
void dedispersion(double DM, int *displacement, double (*fbk)[NF/(NT*NTAO)], double (*result)[NF/(NT*NTAO)]);

int
main()
{
int i, j, k, *displacement;
double DMbase, shift, (*fbk)[NF/(NT*NTAO)], (*result)[NF/(NT*NTAO)];
double c=0.0;
FILE *fp;
char file[FILENAME_MAX];  // name of data file

DMbase=50000000000000.0;
shift=DMbase*0.1;

/*initiation */
checkN(NF);          // Check that NF = 2^n for some integer n >= 1.
checkN(NT);
if(!((fbk = malloc( NF/NTAO * sizeof(double)))&&
(result = malloc( NF/NTAO * sizeof(double)))&&
(displacement = malloc( NT * sizeof(int)))))
printf("memory error \n");

if(!(fp = fopen("fbkwithdis.txt", "r")))
{
printf(" fbkwithdis.txt could not be opened!");
exit(EXIT_FAILURE);
}
for(i=0;i<NT;i++)
{
for(j=0;j<NF/(NTAO*NT);j++)
fscanf(fp, "%lf ", &fbk[i][j]);
fprintf(fp, "\n");
}
fclose(fp);

/*de-dispersion*/
for(k=0;k<10;k++)
{
c+=shift;
dedispersion(c, displacement, fbk, result);

/*output data to txt file*/

sprintf(file, "%d.txt", k);

if(!(fp = fopen(file, "w")))
{
printf("   File \'%s\' could not be opened!", file);
exit(EXIT_FAILURE);
}
for(i=0;i<NT;i++)
{
for(j=0;j<NF/(NTAO*NT);j++)
fprintf(fp, "%e ", result[i][j]);
fprintf(fp, "\n");
}
fclose(fp);

}

free(fbk);
free(result);
free(displacement);
}

/* Check that N = 2^n for some integer n >= 1. */
void
checkN (int N)
{
int i;

if(N >= 2)
{
i = N;
while(i==2*(i/2))
i = i/2;  /* While i is even, factor out a 2. */
}  /* For N >=2, we now have N = 2^n iff i = 1. */
if(N < 2 || i != 1)
{
printf("NF, which does not equal 2^n for an integer n >= 1.");
exit(EXIT_FAILURE);
}
}

void dedispersion(double DM, int *displacement, double (*fbk)[NF/(NT*NTAO)], double (*result)[NF/(NT*NTAO)])
{
int i, j;
int m=0, differ=0, k=0;
double time, frequency;

/*find the displacement for each channel*/

for(i=0;i<NT;i++)
{

if(i<NT/2)
frequency=(i*1.0)*FS/(NT-1);
else
frequency=(i*1.0)*FS/(NT-1)-FS;

if(frequency-0.0<0.5&&frequency-0.0>-0.5)
frequency=0.5;

time=DM/((frequency*frequency)*TWO_PI);

displacement[i]=(time/(1.0/FS))/(NT*NTAO);

}

for(i=0;i<NT;i++)
if(displacement[i]>(NF/(NTAO*NT)))
displacement[i]=NF/(NTAO*NT);

/*	for(i=0;i<NT;i++)
printf("%d, ", displacement[i]);
printf("\n");
*/

/*de-dispersion*/
for(i=0;i<NT;i++)
{
for(j=0;j<NF/(NTAO*NT);j++)
{
result[i][j]=0.0;
if((i!=0)&&(i<NT/2))
differ=displacement[i-1]-displacement[i];
else if((i!=NT-1)&&(i>NT/2-1))
differ=displacement[i+1]-displacement[i];
else
differ=0;

if(differ<0)
printf("error! differ is less than zero!\n");

for(m=0;m<differ+1;m++)
{
k=j+m+displacement[i];
if(k>NF/(NTAO*NT)-1)
break;

result[i][j]+=fbk[i][k];

}

}

}

}

 PhysOrg.com science news on PhysOrg.com >> Front-row seats to climate change>> Attacking MRSA with metals from antibacterial clays>> New formula invented for microscope viewing, substitutes for federally controlled drug
 for update: I found that if I set a small value for NF, NT, then it works perfectly. for example, NF=8192 NT=32
 Recognitions: Homework Help Do you have access to a source level debugger that would let you step through the code and see how the variables are changing? Note that all of your defines are integers. Any math you do with those values will be done using integer math, unless you specifically cast them to (double) in mathematical expressions. I don't know if this is an issue for your program, as I didn't study it in detail.

I don't think its the value of c not changing that is the problem

Example with a simplified version of your code:
Code:
#include <stdio.h>
#include <stdlib.h>
#include <math.h>

int
main()
{
int k;
double DMbase, shift;
double c=0.0;

DMbase=50000000000000.0;
shift=DMbase*0.1;

for(k=0;k<10;k++) {
c+=shift;
printf("Loop iter: %d Value of c: %0.2f\n", k, c);
}

return 0;
}
The output I get is:
Loop iter: 0 Value of c: 5000000000000.00
Loop iter: 1 Value of c: 10000000000000.00
Loop iter: 2 Value of c: 15000000000000.00
Loop iter: 3 Value of c: 20000000000000.00
Loop iter: 4 Value of c: 25000000000000.00
Loop iter: 5 Value of c: 30000000000000.00
Loop iter: 6 Value of c: 35000000000000.00
Loop iter: 7 Value of c: 40000000000000.00
Loop iter: 8 Value of c: 45000000000000.00
Loop iter: 9 Value of c: 50000000000000.00

When you say closed when running what do you mean by this? Is it crashing in some form? A segmentation fault?

yes,
A segmentation fault.
The c is not the problem.
I try some constant instead of c, the segmentation fault appears also.

 Quote by gbeagle I don't think its the value of c not changing that is the problem Example with a simplified version of your code: Code: #include #include #include int main() { int k; double DMbase, shift; double c=0.0; DMbase=50000000000000.0; shift=DMbase*0.1; for(k=0;k<10;k++) { c+=shift; printf("Loop iter: %d Value of c: %0.2f\n", k, c); } return 0; } The output I get is: Loop iter: 0 Value of c: 5000000000000.00 Loop iter: 1 Value of c: 10000000000000.00 Loop iter: 2 Value of c: 15000000000000.00 Loop iter: 3 Value of c: 20000000000000.00 Loop iter: 4 Value of c: 25000000000000.00 Loop iter: 5 Value of c: 30000000000000.00 Loop iter: 6 Value of c: 35000000000000.00 Loop iter: 7 Value of c: 40000000000000.00 Loop iter: 8 Value of c: 45000000000000.00 Loop iter: 9 Value of c: 50000000000000.00 When you say closed when running what do you mean by this? Is it crashing in some form? A segmentation fault?

 Mentor You're not giving us enough information to help you with your problem. Your code is complex enough that you should be using a debugger to find where and why you're getting a segmentation fault. I suspect that the call to dedispersion is causing problems, for the reason that rcgldr gave in his earlier post.
 update: In linux system, the code works well. I can get the correct result. In windows system, it does not work sometimes. I can not find the reason. Now, I just use it because my work environment is linux.


text:
# Let w i 2 y i then the w i and x i are iid and

• Notes
• 465

This preview shows page 140 - 143 out of 465 pages.

(Let W i = 2 Y i . Then the W i and X i are iid and Corollary 4.6 applies.)
CHAPTER 4. SUFFICIENT STATISTICS 130 Rule of thumb 4.1: A k –parameter minimal sufficient statistic for a d –dimensional parameter where d<k will not be complete. In the following example d = 1 < 2 = k. (A rule of thumb is something that is frequently true but can not be used to rigorously prove something. Hence this rule of thumb can not be used to prove that the minimal sufficient statistic is not complete.) Warning: Showing that a minimal sufficient statistic is not complete is of little applied interest since complete sufficient statistics are rarely used in applications; nevertheless, many qualifying exams in statistical inference contain such a problem. Example 4.14, Cox and Hinckley (1974, p. 31). Let X 1 ,...,X n be iid N ( μ,γ 2 o μ 2 ) random variables where γ 2 o > 0 is known and μ> 0 . Then this family has a one dimensional parameter μ , but f ( x | μ ) = 1 radicalbig 2 πγ 2 o μ 2 exp parenleftbigg - 1 2 γ 2 o parenrightbigg exp parenleftbigg - 1 2 γ 2 o μ 2 x 2 + 1 γ 2 o μ x parenrightbigg is a two parameter exponential family with Θ = (0 , ) (which contains a one dimensional rectangle), and ( n i =1 X i , n i =1 X 2 i ) is a minimal sufficient statistic. (Theorem 4.5 applies since the functions 1 and 1 2 do not satisfy a linearity constraint.) However, since E μ ( X 2 ) = γ 2 o μ 2 + μ 2 and n i =1 X i N ( nμ,nγ 2 o μ 2 ) implies that E μ [( n summationdisplay i =1 X i ) 2 ] = 2 o μ 2 + n 2 μ 2 , we find that E μ [ n + γ 2 o 1 + γ 2 o n summationdisplay i =1 X 2 i - ( n summationdisplay i =1 X i ) 2 ] = n + γ 2 o 1 + γ 2 o 2 (1 + γ 2 o ) - ( 2 o μ 2 + n 2 μ 2 ) = 0 for all μ so the minimal sufficient statistic is not complete. Notice that Ω = { ( η 1 2 ) : η 1 = - 1 2 γ 2 o η 2 2 } and a plot of η 1 versus η 2 is a quadratic function which can not contain a 2–dimensional rectangle. Notice that ( η 1 2 ) is a one to one function of μ ,
CHAPTER 4. SUFFICIENT STATISTICS 131 and thus this example illustrates that the rectangle needs to be contained in Ω rather than Θ . Example 4.15. The theory does not say that any sufficient statistic from a REF is complete. Let Y be a random variable from a normal N (0 2 ) dis- tribution with σ 2 > 0 . This family is a REF with complete minimal sufficient statistic Y 2 . The data Y is also a sufficient statistic, but Y is not a function of Y 2 . Hence Y is not minimal sufficient and (by Bahadur’s theorem) not complete. Alternatively E σ 2 ( Y ) = 0 but P σ 2 ( Y = 0) = 0 < 1 , so Y is not complete. Theorem 4.9. Let Y 1 ,...,Y n be iid. a) If Y i U ( θ 1 2 ) , then ( Y (1) ,Y ( n ) ) is a complete sufficient statistic for ( θ 1 2 ) . See David (1981, p. 123.) b) If Y i U ( θ 1 2 ) with θ 1 known, then Y ( n ) is a complete sufficient statistic for θ 2 . c) If Y i U ( θ 1 2 ) with θ 2 known, then Y (1) is a complete sufficient statistic for θ 1 .


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# Percolation

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## How to define "space"

The concept of "space" is a rather flexible one in mathematics. We call many things spaces: vector spaces, Euclidian spaces, polish spaces, Banach spaces, Hilbert spaces, ... all of which share some aspect of our intuition about what "space" "is". Colloquially and intuitively, the most prominant aspect of space is our ability to move in it. Space is effectively one-dimensional for a a car in a tunnel -- it can only move forward or backward. We live most of our lives walking in two dimensions, being able to move forward and backward and also being able to turn left and right. Nature itself is concretely 3 dimensional. Many animals like fish and birds exploit this and we ourselves have invented devices from ramps and stools to roller coasters and rockets that allow us to enjoy and exploit this third dimension.

When we build a model, we have to decide if we want to include space. In our pharmacokinetics models, when we used different compartments to represent different organs like the stomach and liver, we were actually making a spatial model of the body. In that case, the use of "space" was implicit -- we didn't use spatial coordinates, and some compartments like blood have a very diffuse spatial position. In our classical mechanics models of ballistic motion and the pendulum position and velocity where dependent variables of time, so space base again part of the state of the systems we were studying, but again in a rather implicit manner.

Explicit spatial models, or simply "spatial models" for short, are models where spatial location is an independent variable, and system state is a function of spatial location, either discretely or continuously, as well as time. A good example of how we do this is found in cellular automata models. A cellular automata model treats space discretely like a tiled floor. Suppose you stand on the floor and occupy one tile, or "cell". From your cell, you can take a step into each of the neighboring cells, and by repeatedly stepping from one cell to the other, you can move long distances.

But exactly how you move depends on the layout of tiles on the floor. You will have different numbers of neighbors to choose from, depending on the layout. We do not worry about this in real life -- the idea of moving from tile-to-tile is only a rough approximation of real movement. But when building a cellular automata model, these details are crucial.

In theory, any graph of edges and nodes can be used as a discrete explicit representation of space, with each node representing a location, and each edge representing the possibilitie of movement between the nodes it connects. But in practice, cellular automata models almost always represent locations as elements of a rectangular array that can be accessed using Cartesian coordinates. Once we have locations, we next need a rule that defines each locations "neighbors", for movement. There are 4 conventional ways to define neighborhoods in a rectangular array. Each of these can be described with a "stencil" connecting each location node to each of its "neighboring" nodes.

• A von Neumann neighborhood contains 4 nodes, creating a lattice of squares.
• A hexagonal neighborhood contians 6 nodes creates a triangular lattice.
• A Moore neighborhood contains 8 nodes and creates an overlapping square lattice.
• A triangular neighborhood contains 3 nodes and creates a hexagonal lattice.

Some of the mathematical properties of these lattices will be explored in the Exercises below.

## Percolation

During summers as a little kid, my friends and I liked to build sand castles at the beach. We would sometimes decorate the castles by putting little dribble towers on the top. We did this using a plastic funnel that would drip out sand and water from the bottom. However, getting the funnel to work right was a little tricky. If the sand I used was too fine and closely packed, the water would not drain down from the top through the funnel and no sand or water would come out the bottom.

The funnel of sand and water is one small example of a very general question -- how does liquid flow through a porous medium? Solutions of this question can be useful for fields from the design of sponges and insulation to well water, and oil and gas extraction. Solutions can even help us understand the spread of malware over the internet and the Ebola virus in Africa. But in order to make sense of flows through porous media, we have to develop new kinds of models.

Percolation is the process by which a fluid or similar substance moves through a porous medium. Hot water, for instance, is percolated through ground beans to make coffee. We call a medium "permeable" if the fluid can move through it, and "impermeable" if it can not. The permeability of the medium depends on the properties of the medium. If the medium is too dense and tightly packed, the fluid cannot get through, but if the medium is loose and lite, then the fluid can pass through. When talking about rubber foams, this condition distinguishes closed-cell and open-cell foams respectively. What we'd like to understand is how the small-scale structure of the medium -- the number, size, and arrangement of its gaps and fills -- relates to the permeability we observe at large scales. In one case, you get a sponge. In the other, you get a life preserver. This may not seem so interesting at first, but the math turns out to be surprising and instructive, and will also give us a chance to exercise our concepts of space described above.

## Site percolation

In any real setting, permeability depend on the chemical and physical composition of the medium, the viscosity of the fluid, temperature, pressure, and even erosion and contamination. But the core concepts of percolation can be qualitatively modeled in very simple and illuminating discrete spatial model. Imagine we want to make a simple model of a porous sponge. Instead of dealing with all three dimensions, suppose we take a planar cross-section of the sponge and look at the connectivity of that. Start with a rectangular array of cells. Each cell corresponds to a node connected to other cells according to some neighborhood. Each cell is colored black or white randomly. Let $$p$$ be the probability that a given site is a void in the media (marked white) while $$1-p$$ is the probability that a given site is filled and marked black. The variable $$p$$ is called the "void fraction", and $$1-p$$ the "fill fraction". When the state of each site is independent of each other sites, such an array can be generated with just a few lines of python code.

[Show code]
p = 0.9
n = 16
A = ceil(rand(n,n) - p)
spy(A)

We can use this code to create a menagerie of random media by tuning the void fraction $$p$$ and the size $$n$$. Let's start by looking at some examples of random arrays. Starting at the top, can we move through the white void-space and reach the bottom?

[Show code]
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31  #!/usr/bin/env python2 from scipy import * from matplotlib.pyplot import * for n in [16,32]: figure(1) clf() i = 1 for p in [0.1, 0.3, 0.4, 0.5, 0.6, 0.9]: A = floor(rand(n,n)+p) subplot(2,3, i) i += 1 spy(A) xlabel('p=%.2f'%(1-p), fontsize=18) gca().get_xaxis().set_ticks([]) gca().get_yaxis().set_ticks([]) savefig('siteperc_n=%02d.png'%n, bbox_inches = 'tight', transparent=True) n=64 i=1 for p in [0.3, 0.4, 0.5]: A = floor(rand(n,n)+p) subplot(1,3, i) i += 1 spy(A) xlabel('p=%.2f'%(1-p), fontsize=18) gca().get_xaxis().set_ticks([]) gca().get_yaxis().set_ticks([]) savefig('siteperc_n=%02d.png'%n, bbox_inches = 'tight', transparent=True)

Well, if $$p$$ is close to 1, it is easy -- there is lots of open white space to move through. And for $$p$$ close to zero, the answer is clearly no. But for values of $$p$$ in the middle, things are a little trickier. what directions can we move in? Can we go diagonal? At this point, we have to make a modelling assumption about movement and which voids are neighbors. Let us propose we can only travel left, right, up, or down, in a von Neumann neighborhood around each site. Then if $$p=0.5$$, we can make it from top to bottom, but if $$p=0.4$$, we can not.

We can compute how far down percolation goes -- the "depth" -- by hand, but that can get hard when $$n$$ gets large. It is better to compute the depth algorithmically.

[Show code]
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59  #!/usr/bin/env python from scipy import * def printbinarymatrix(A): m,n = A.shape newline='\n' f = lambda i : (" " if i < 0.5 else "#") s = newline.join([ "|%s|"%"".join([f(i) for i in row]) for row in A]) s = newline.join(["+%s+"%("-"*n),s,"+%s+"%("-"*n)]) print(s) return def calc_percolation_depth(A, i0=None): """ Given an array A of {0,1}'s, find the farthest across the matrix we can walk on 0-entries, when using a von Neumann neighborhood, if we start at the left side in row i0. If i0 == None, return the maximum over all initial rows. """ upper_bound = A.shape[1] if i0 == None: return max([calc_percolation_depth(A,i) for i in range(upper_bound)]) neighbors = [(1,0), (-1,0), (0,1), (0,-1)] # to track which sites we have visited ... visited = 0*A def next(i,j): if i < 0 or i >= A.shape[0] or j < 0: return 0 if j >= upper_bound: return upper_bound if visited[i,j] == 1: return 0 visited[i,j] = 1 if A[i,j] > 0.5: return j return max([next(i+dx,j+dy) for (dx,dy) in neighbors]) return next(i0,0) def test(): p = 0.45 N = 12 A = floor(rand(N,N) + p) assert set([0,1]) == set(map(int,A.flatten())) print("Here we have a %dx%d matrix, with fill fraction %f"%(N,N,p)) printbinarymatrix(A.T) print("If we start on the top, how far toward the bottom can we move on blanks?") for i in range(N): d = calc_percolation_depth(A,i) print("From column %2d, we go %2d steps down"%(i,d)) print("Max depth: ", calc_percolation_depth(A)) print("Now, double-check with your own eyes. Are all of these correct?") test()

As an example application of this algorithm, above we have a 12x12 matrix, with fill fraction 0.450000 If we start on the top, how far toward the bottom can we move on blanks?

From column  0, we go  2 steps down
From column  1, we go  2 steps down
From column  2, we go  2 steps down
From column  3, we go  0 steps down
From column  4, we go  0 steps down
From column  5, we go  0 steps down
From column  6, we go 12 steps down
From column  7, we go 12 steps down
From column  8, we go  0 steps down
From column  9, we go  3 steps down
From column 10, we go  3 steps down
From column 11, we go  3 steps down
Max depth:  12

Now, double-check with your own eyes. Are all of these correct?

The examples given so far are particular realizations. There are $$16 \times 16$$ array with void fractions of $$p=0.5$$ where the array does not percolation from top to bottom, and there are $$16 \times 16$$ arrays with $$p=0.4$$ where the arrays do percolate from top to bottom. But in practice, the arrays are much larger also. Let's consider the same question for some progressively larger arrays.

Note above that the fluid will not get through for small void fraction $$p$$, and that it will for large void fractions. And interestingly, it seems that as the we look at larger matrices, the chance of getting through becomes less-random and more predictable based on $$p$$. Very important question: For very large systems, is there a specific value $$p=p _ c$$ when the water stop flowing?

To explore this question, we can perform some simulations. Suppose for a given value of the void fraction $$p$$, we draw a thousand $$40 \times 40$$ sample arrays, and use the search algorithm above to calculate the percolation depth of each. Then, we plot what fraction of those arrays percolate to each possible depth as an estimate of the probability that another randomly drawn array will percolate to that depth. The result we see is given below. When the void fraction is small, only a small fraction of the sampled arrays percolate deeper than 10 steps. But when the void fraction is larger than $$0.6$$, the majority of sampled arrays percolate from the top all the way to the bottom of their array. There's a change in the character of the curves someplace between $$p = 0.5$$ and $$p = 0.7$$.

To better see what is happening with percolation process for void fractions between $$0.5$$ and $$0.7$$, we plot the fraction of arrays that percolate the full array height $$n$$. As the array size increases, we see that the curves of fraction percolating verse void fraction become less line-like and more s-shaped. As we approach the limit of infinitely large arrays, these curves converge to a Heaviside function with a jump around $$p = 0.59$$. For very large arrays, very few samples will percolate for void fractions smaller than $$0.59$$, but almost all arrays we sample will percolate for void fractions greater than $$0.59$$.

This steep change in the depth characteristic of these random arrays is called a "phase transition". Phase transitions are discontinuous-changes in the properties of very large systems. In this case, the phase transition is between impermeability and permeability. The most familiar examples may be the boiling of water into steam, or the freezing of water into ice. Steam, water, and ice are three different phases of a system with a very a large number of water molecules. The changes in the properties of the system are abrupt, and occur at special "critical" temperatures; $$0^{\circ}$$ Celcius for freezing and $$100^{\circ}$$ Celcius for boiling. (these temperatures depend on the atmospheric pressure).

The permeability phase transition occurs when the void fraction is near certain percolation threshold $$p _ c$$. Percolation threshold for the model we've described here of site-percolation with von-Neumann neighborhoods is $$p _ c \approx 0.5927$$. Even though the von Neumann neighborhood is one of the simplest spatial models, no closed-form formula is known for its percolation threshold -- it must be estimated using Monte Carlo approaches related to those we've used above. The challenge to find an efficient formula for this percolation threshold remains greatly vexing for statistical mechanics.

But we should also emphasize that the value of the threshold would change if we changed the neighborhood of allowed movement. If we could move in diagonal directions as well, like a Moore neighborhood, then we wouldn't need as much empty space to move through the medium, and the percolation threshold would be smaller.

## Bond percolation with von Neumann neighborhoods

Bond percolation is a different geometric approach to the same questions, and a great illustration of the merits of alternative models of a phenomena. In bond percolation, the void fraction $$p$$ controls the probability that two adjacent nodes are connected along an edge. So, it's not the cells themselves that are empty or filled, but rather the edges connecting the cells.

Instead of each cell being empty or filled, in bond percolation, each edge is present with some probability $$p$$ or absent with probability $$1-p$$. Here is an example of an $$80 \times 80$$ matrix with $$p=0.45$$.

Can you travel from top to bottom along the bonds?

Despite the similarities to site percolation, the difference in geometry has a huge impact on our ability to answer this question. For von Neumann bond percolation, we can compute $$p_c$$ exactly as the lattice becomes infinitely large. The threshold below which you can't travel from top to bottom $$p_c=1/2$$.

To explain why $$p_c$$ must be $$1/2$$, we need to introduce the concept of our bond lattice's dual lattice. Suppose we create a random lattice with von Neuman neighborhoods and void fraction $$p$$. We'll call this our principle lattice. Now, create a new set of nodes, shifted a half-step in both directions for the nodes of the principle lattice. We will call this set of new nodes the "dual nodes". The neighbors of the dual nodes are also given by the von Neumann neighborhood. Now, for any two neighboring dual nodes, connect the nodes with an edge if and only if the new edge will not cross any of the edges in our principle lattice. The lattice made up of the dual nodes and their edges is called the dual lattice. Because of the conditions for edge existence, the dual lattice will have edges present with probability $$1-p$$. Here is an example.

Now, let's consider percolation on our lattice from top-to-bottom, simultaneously with percolation of the dual lattice from left-to-right. If our principle lattice is permeable, then there will be atleast one unbroken path of bonds from top to bottom. This implies that the dual lattice can not have an unbroken path of bonds from left to right, and hence, the dual lattice will be impermeable. On the other hand, if the dual lattice is permeable from left to right, then our principle lattice will be impermeable from top to bottom.

Imagine that $$p _ c < 1/2$$ for bond percolation with von Neumann neighborhoods. Pick the void fraction $$p$$ so $$p _ c< p < 1/2$$. Then both the principle and the dual lattices must be permeable. But this is a contradiction to our observation that only one lattice can be permeable at a time.

On the other hand, if $$p _ c > 1/2$$, then if we pick $$p$$ so that $$1/2 < p < p _ c$$, the lattice will be below the percolation threshold, and thus will not contain a connected path from top to bottom. This implies there is a connected path in the dual lattice separating top from bottom. Thus, the dual lattice is permeable. But since $$1/2 < p < p _ c$$, $$1-p < 1/2 < p _ c$$, so the dual lattice is below percolation threshold and should not be permeable either. Thus, a contradiction. It follows then that the only possibility is $$p _ c = 1/2$$.

## Discussion

• Asymptotic approximations can have jumps

• The distinction between permeable and impermeable is not one of degrees but one of kind -- there is a sharp cut-off below which media are impermeable.

• The solvability of a model is much more fragile than it's general properties. Both site and bond percolation are reasonable idealize models of percolation phenomena. But bond-percolation with von Neumann neighborhoods is easy to solve, while site-percolation with von Neumann neighborhoods remains unsolved. When trying to solve a modelling problem, be open to alternative perspectives.

## Asside: space and the Jordan curve theorem

Our proof that the critical void fraction for bond percolation on a von Neuman lattice is a half relies on an important relationship that the boundary of any simply connected cluster of nodes in our principle lattice is a closed loop of nodes in our dual lattice.

• in site percolation, boundary is not necessary a connected path, so both the principle and dual lattices might both not percolate.

• boundary has a different character than the connected components in discrete spatial models. For example, for site percolation with von Neuman neighborhoods, the boundary of a cluster is NOT a path in von Neuman neighborhoods.

• The Jordan curve theorem is a result in topology of the plane that says any continuous bounded closed curve (like a circle) splits the plane into two disconnected components. The Jordan curve theorem is notorious, as being something intuitively obvious to the lay observer, while being a great headache to prove formally. These percolation examples give some intuition of why it might be hard -- boundaries are NOT always the same as clusters, and so, they might NOT divide the space in the way you expect.

• Moore neighborhoods, being non-planar, allow weird things to happen. For example, a simply connected set can have a boundary set with many holes in it, rather than just 1 hole.

# Exercises

1. The "neighborhood" in a cellular automata model defines the geometry of the space. In class, we discussed 4 different kinds of neighborhoods for 2-dimensional cellular automata on a Cartesian grid -- 8-neighbor Moore, the 6-neighbor, the 4-neighbor von Neumann, and the 3-neighbor.

1. Let $$N_k(x,y)$$ be the set of neighbors of site $$(x,y)$$ where $$x$$ and $$y$$ are integers. If $$k = 3$$, then $N_3(x,y) = \begin{cases} \{ (x+1,y), (x,y+1), (x,y-1) \} & \text{ if x+y is even}, \\ \{ (x-1,y), (x,y+1), (x,y-1) \} & \text{ if x+y is odd}. \end{cases}$ Find similar formulas for $$N_4(x,y)$$, $$N_6(x,y)$$, and $$N_8(x,y)$$.

2. In a regular lattice, the atomic loop length is the smallest number of neighboring edges in loop from $$(0,0)$$ back to itself where the same edge between two neighbors is never traversed more than once. Each such loop with this minimal loop length is called an "atomic loop". For each $$k \in \{ 3,4,6,8 \}$$, find the atomic loop length and the number of atomic loops containing $$(0,0)$$ for lattices with neighborhoods $$N_k(x,y)$$.

3. Given a neighborhood $$N_k(x,y)$$, we can define a metric $$d_k( (x,y), (u,v))$$ to measure the distance between points $$(x,y)$$ and $$(u,v)$$ recursively as follows: $d_k( (x,y), (u,v) ) = \begin{cases} 0 & \text{if (x,y) = (u,v)}, \\ 1 + \min \{ d_k( (w,z), (u,v) ) : (w,z) \in N_k( (x,y) ) \} & \text{otherwise}. \end{cases}$ For each $$k \in \{ 3,4,6,8 \}$$, draw the set of points $$\{ (u,v) : d_k( (0,0), (u,v) ) \leq 2\}$$ on the appropriate lattice.

4. For each $$k \in \{ 3,4,6,8 \}$$, find $$d_k( ( 0, 0), (3,3) )$$.

5. Our choice of neighborhood also has some unexpected implications for the topology of space. Suppose we define a set of nodes as simply connected if and only if between any two nodes in the set, there is a path of neighboring nodes also entirely in the set, and the interior of every closed loop in the set is also in the set. The outer boundary of a simply connected set is the set of nodes that are neighbors of nodes in the given set, but are themselves not in the set. For which neighborhoods, if any, is the boundary of a simply connected set a closed loop?

2. In class, we were able to visually calculate percolation depth because of our awesome pattern analysis wet-ware. Calculating percolation depth algorithmically is a little more complicated, but can be done recursively. See here

1. Let $$A(p,N)$$ be a random $$0/1$$ matrix with shape $$N \times N$$ where sites are filled ($$1$$) with probability $$1-p$$, and void ($$0)$$ with probability $$p$$. Recall that in class, we generated example matrices like this with the python code $A = \text{floor(rand(N,N) + p)}$ For each value of $$p \in \{ 0.7, 0.6, 0.55, 0.5, 0.4 \}$$, simulate $$1000$$ $$40 \times 40$$ matrices. Let $$d(x,p)$$ be the fraction of these matrices with percolation depth greater than $$x$$. Assuming a hexagonal neighborhood, plot $$d(x,p)$$ for each value of $$p$$, all in one plot. Remember to label your plot axes and include a legend.

2. What particular feature of your plot changes qualitatively as $$p$$ changes. What does this change mean?

3. Suppose we define $$h(p,N)$$ as the fraction of $$N\times N$$ $$A(p,N)$$ matrices with hexagonal neighborhoods which percolate all the way through. Plot $$h(p,N)$$ as a function of $$p \in [0.3,0.6]$$ (this is an INTERVAL -- make sure to include values between the extremes.) for $$N \in \{ 5, 10, 20, 50, 100 \}$$. Use atleast 1000 matrices for each. (Warning: This calculation may take you a long time.)

4. Extrapolating from your plot, what do you think will happen to $$\lim_{N\rightarrow \infty} h(p,N)$$?

5. Reality, of course, is often three-dimensional, rather than two-dimensional. Describe how you think the percolation threshold will change when we switch from two dimensions to three dimensions and why it will change.

3. Estimate the percolation threshold of the 3-dimensional equivalent of the von Neumann neighborhood.

4. What is the dual lattice of bond percolation on a hexagonal neighborhood? What does this imply about the percolation threshold of the hexagonal lattice?

5. Show that our bond percolation proof can not be generalized to spaces using Moore neighborhoods by constructing paths from top-to-bottom and left-to-right that do not intersect each other.

6. The concept of percolation can be generalized to continuous space. Suppose we drop $$n$$ circles of radius $$r$$ randomly into a unit square (the locations of their centers is drawn from a two-dimensional Poisson distribution), and that water can not flow through these circles.

1. Find a function $$f(n,r)$$ such that if $$f(n,r) < 0$$, water can flow from top to bottom. (Consider why one choice of $$f$$ might be better than another, and choose the best form of $$f$$ you can find.)
2. Explain why there is no function $$g(n,r)$$ such that if $$g(n,r) < 0$$, water can not flow from top to bottom.
7. To complete: Random graphs are an extreme case commonly referred to as the mean-field limit where space exists but is unstructured. Study the percolation threshold in random graphs. (this is erdo's result, but it is rather un-interesting from a spatial perspective)

( previous, home, next )


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# Information to Energy equations?

Are there are any known formulas or equations that can calculate information from energy or energy from information?

One of them, I believe, is the Bekenstein bound which is the maximum information that can be in a given space before becoming a black hole.

https://en.wikipedia.org/wiki/Bekenstein_bound

The equations involved can be converted for both mass and energy.

So I am wondering if there are other equations involving this kind of conversion from energy to information.

• This is a bit like asking for an equation to calculate an objects speed given its mass. The concepts are not totally unrelated, but there is a lot more going on and so it not possible to reduce it to a single formula. The relationship is very context dependent (and in many contexts essentially non-existent) and is too broad a subject to cover in a single answer – By Symmetry Sep 23 '18 at 8:47
• I am aware that such conversions are not simple, I mean look at the Bekenstein Limit, I am simply asking, just like the Bekenstein Limit, is there any other equations similar to that – C. Jordan Sep 23 '18 at 20:09

The key bound is the Landauer bound: a process that erases 1 bit of information has to spend $$k_B T \ln(2)$$ J of energy as waste heat to carry away the entropy.
Beside that link and the Bekenstein bound, the other main links between energy and information are the various bounds on how fast quantum states can change, such as the Margolus-Levitin bound (an "operation" must take at least time $$\pi\hbar/2E$$ where $$E$$ is the system energy), the Mandelstam-Tamm bound (the time is at least $$\pi \hbar/2\sqrt{\sigma^2}$$ where $$\sigma^2$$ is the variance of system energy) and their many relatives.
• @C.Jordan - You can always use mass-energy equivalence, $E=mc^2$. – Anders Sandberg Sep 25 '18 at 19:54


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# An introduction to SALT

The speech engine that I was talking about in my last article about speech synthesis is an add-in for Internet Explorer. Follow these instructions to install it yourself (you can get Microsoft Speech Application Software Development Kit (SASDK) Version 1.1 now though). To actually create speech enabled web pages, you need to use SALT.Speech Application Language Tags is now fairly standard and being supported by Microsoft means it is almost guaranteed to surive in some form. SALT is an XML markup so you would generally embed it straight into a HTML (or more usually an XHTML) page. The first requirement is to add the SALT namespace to your XHTML document: <html xmlns:salt="http://www.saltforum.org/2002/SALT"> This probably isn’t a requirement but you should do it anyway. The next thing you need to do is to load the add-in and tell it what to handle. This is definitely vendor specific and only applies to the add-in for IE: <?import namespace="salt" implementation="#saltobject" /> All the code does is create an object and then tell it what namespace to look for SALT tags in (in this case the “salt” namespace). There is one potential sticking point. The standalone IE add-in is not the same as the one you get if you install the whole SDK so for debugging purposes the classid will be different. After that it’s just a matter of adding the SALT tags for handling the speech. In this article I’ll just deal with the simplest one, prompts. A prompt is just a piece of speech. Just write your speech inside a <salt:prompt> tag: <salt:prompt id="InstructionsPrompt"> Hello. Thank you for using this salt demo. </salt:prompt> The id can be anything you want as it is only used to uniquely identify the prompt. We now have to get around the semantic seperation of form and function: this simply defines a prompt and doesn’t actually do anything with it. To actually make it do something we have to use JavaScript. Prompt objects all have a Start() function accessible from JavaScript so to make it play, just call InstructionsPrompt.Start(). Although it’s not an ideal solution for testing purposes just attach it to the onload event of the body tag. You can see (and hear) the whole SALT demo. A final note about voices. Windows XP comes with a good voice called Microsoft Mike, but this may not be the default. Some of the others sound really bad. To set Mike as the default: Start -> Control Panel -> Speech -> Text-to-speech Windows Vista will come with new voices (the ones made from sampled sounds I talked about before), one of which is called Anna. SALT, SSML, text-to-speech


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# MSCS Seminar Calendar

Tuesday January 22, 2019
Quantum Topology / Hopf Algebra Seminar
Smooth Generic Position
Jonathan Schneider (UIC)
3:00 PM in 612 SEO
We generalize classical knot diagrams and moves to higher dimensions.
Smooth maps are generic when they cannot be fundamentally altered by small perturbations.
Codimension-1 generic maps may be decomposed into simple structures of finitely many types.
We catalog these structures for dimensions 0 through 3, and also catalog their moves.

Louise Hay Logic Seminar
IP, SOP, and instability (1)
Paolo Degiorgi
3:30 PM in 427 SEO
Wednesday January 23, 2019
Mathematics Education Colloquium
Student perceptions of MyMathLab in Developmental Courses
Aida Alibek (UIC)
10:45 AM in 612 SEO
We will discuss a paper by D.Holt, W.Holt & R.Lumadue "At Cross-Purposes with a Developmental Mathematics Course: Perceptions of Students on the Use of MyMathLab".
This is a discussion seminar, so although prior reading is recommended it is not required. Feel free to come and join the conversation! Everyone is welcome!

Special Colloquium
Too much data!
John Stufken (Arizona State University)
3:00 PM in 636 SEO
The enormous amounts of data that are collected in applications in a wide variety of fields create challenges and opportunities for statisticians. One of the challenges is that traditional statistical methods for data of smaller size may no longer be applicable in the new “big data” environment, for computational reasons or otherwise. The corresponding opportunity lies in the need to develop methods that are applicable for big data. The simplest such methods, and often the most elegant ones, are based on innovations that allow familiar techniques to be applied in this new environment in a computationally feasible way. Adapting existing methods for this new environment can typically not be accomplished by putting “old wine in new bottles”, but requires clever innovations. Traditionally, it goes against a statistician’s core principles to “discard” some of the data. Yet, some data sets are so large that exploration and analysis must proceed by using only some of the data. This leads to the idea of selecting subdata from big data and drawing conclusions from an analysis of the subdata. While this idea brings traditional statistical analysis methods potentially back into the picture, there are the immediate questions of how to select the subdata and, if needed, how to adjust analysis methods. Innovations to accomplish this are the focus of this presentation. We discuss subdata selection methods, with special emphasis on information-based subdata selection, as well as challenges and shortcomings associated with these methods.

Artin-Mumford’s “Some elementary examples of unirational varieties which are not rational” explanation
Fumiaki Suzuki (UIC)
3:00 PM in 712 SEO
We review Artin-Mumford’s construction of a 3-fold which is unirational and non-rational. We start from the motivation, give a construction, and explain it in terms of Severi-Brauer schemes

Graduate Geometry, Topology and Dynamics Seminar
Spring Organizational Meeting
Keaton Quinn (UIC)
3:00 PM in 612 SEO

Algebraic Geometry Seminar
The stable cohomology of moduli spaces of sheaves on surfaces
Izzet Coskun (UIC)
4:00 PM in 427 SEO
Moduli spaces of Gieseker semistable sheaves on surfaces play a central role in mathematics and have many applications to cycles and linear systems on surfaces, Donaldson's 4-manifold invariants and mathematical physics. In this talk, I will describe a conjecture with Matthew Woolf on the cohomology of these moduli spaces. We conjecture that the Betti numbers of these moduli spaces stabilize as the discriminant tends to infinity and that the stable numbers are independent of the rank and the first Chern class. In particular, calculations of Gottsche determine the stable numbers. I will give some evidence for the conjecture. This is joint work with Matthew Woolf.

Introduction to the work of "7 samurais"
Alex Furman (UIC)
4:00 PM in 612 SEO
This semester we plan to discuss the recent work of Abert-Bergeron-Biringer-Gelander-Nikolov-Raimbault-Samet) and to (hereafter 7s) that, among other things, discusses the asymptotic growth of the Betti numbers of compact locally symmetric manifolds $b_i(M)/vol(M)$ as $vol(M)\to \infty$, where $M=\Gamma\backslash X$ are quotients of a fixed (higher rank) irreducible symmetric space.
To understand these results (and to put them in perspective) we need to discuss a variety of important and cool math topics: - L^2-Betti numbers - Luck's Approximation Theorem - Benjaminy-Schramm convergence - Invariant Random Subgroups - Stuck-Zimmer theorem - and more... There is some topology, geometry, dynamics, and number theory in this all.
In this first talk we will give a general overview, and discuss soem organizational topics.
Thursday January 24, 2019
Logic Seminar
TBA
Carlos Arreche (University of Texas)
3:30 PM in 427 SEO
Monday January 28, 2019
Geometry, Topology and Dynamics Seminar
TBA
Kevin Schreve (University of Chicago)
3:00 PM in 636 SEO

Analysis and Applied Mathematics Seminar
Interaction of modulated water waves of finite depth
Yannis Giannoulis (University of Ioannina, Greece)
4:00 PM in 636 SEO
In this talk we consider the water wave problem of finite depth as a nonlinear dispersive system. Motivated by this feature, we are interested in the macroscopic dynamics of the envelopes of small, macroscopically amplitude-modulated fixed carrier waves, the latter being plane wave solutions of the linearized problem. More specifically, we want to know whether some sort of interaction of different (modulated) carrier waves can be observed macroscopically. For pure gravity waves such a macroscopic interaction can be observed only for the next-to-leading order corrections of the macroscopic amplitudes and a relevant system of modulation equations is derived. This system is then justified by employing the stability of the original water wave problem, as established by Lannes in his 2013 book. Time permitting, we discuss also the completely different situation in the case of capillary-gravity water waves, where for resonant carrier waves macroscopic interactions can be observed for the leading order amplitudes and where a stability result for the original water wave problem has to take into account the second-order differential operator of the surface tension.
Tuesday January 29, 2019
Special Colloquium
Individualized Multi-Directional Variable Selection
Dr. Annie Qu (UIUC)
3:00 PM in 636 SEO
In this paper we propose a heterogeneous modeling framework which achieves the individual-wise feature selection and the covariate-wise subgrouping simultaneously. In contrast to conventional model selection approaches,the key component of the new approach is to construct a separation penalty with multi-directional shrinkages, which facilitates individualized modeling to distinguish strong signals from noisy ones and selects different relevant variables for different individuals. Meanwhile, the proposed model identifies subgroups among which individuals share similar covariates’ effects, and thus improves individualized estimation efficiency and feature selection accuracy. Moreover, the proposed model also incorporates within individual correlation for longitudinal data. We provide a general theoretical foundation under a double-divergence modeling framework where the number of individuals and the number of individual-wise measurements can both diverge, which enables the inference on both an individual level and a population level. In particular, we establish the population-wise oracle property for the individualized estimator to ensure its optimal large sample property under various conditions. Simulation studies and an application to HIV longitudinal data are illustrated to compare the new approach to existing variable selection methods.
Tea at 4:00-4:30 PM at SEO 300.
Wednesday January 30, 2019
Ein-Lazarsfeld's Theorem on Singularities of Theta Divisors
Gregory Taylor (UIC)
3:00 PM in 712 SEO

Introduction to the water wave problem of finite depth and modulated waves
Yannis Giannoulis
4:00 PM in 1227 SEO
We will discuss the derivation of the water wave equations for finite depth, their different features depending on the relation between their characteristic lengths, and the notion of modulated waves, and aim to give an outline of the proof of Lannes’s well-posedness result.

Statistics Seminar
Sufficient dimension folding via distance covariance
Wenhui Sheng (Marquette University)
4:00 PM in 636 SEO
We propose a new sufficient dimension folding method using distance covariance for regression in which the predictors are matrix- or array-valued. The method works efficiently without strict assumptions on the predictor. It is model-free and neither smoothing techniques or selection of tuning parameters is needed. Moreover, it works for both univariate and multivariate response cases. We use two approaches to estimate the structural dimensions: bootstrap method and a new method of local search. Simulations and real data analysis support the efficiency and effectiveness of the method.

TBD
Sam Dodds (UIC)
5:00 PM in 636 SEO
Thursday January 31, 2019
Louise Hay Logic Seminar
IP, SOP, and instability (2)
Paolo Degiorgi
4:00 PM in 427 SEO
Monday February 4, 2019
Analysis and Applied Mathematics Seminar
TBA
Helena J. Nussenzveig Lopes (IM-UFRJ)
4:00 PM in 636 SEO
TBA
Tuesday February 5, 2019
Logic Seminar
TBA
Paul Larson (Miami University)
3:30 PM in 427 SEO
Wednesday February 6, 2019
Mathematics Education Colloquium
Something is rotten in the state of Calc 1
Scott Baldridge (Louisiana State University)
10:45 AM in 612 SEO
TBA

TBD
Jay Kopper (UIC)
3:00 PM in 712 SEO

TBD
Joel Stapleton (UIC)
5:00 PM in 636 SEO
Monday February 11, 2019
Analysis and Applied Mathematics Seminar
TBA
Shawn Walker (Louisiana State University)
4:00 PM in 636 SEO
Wednesday February 13, 2019
TBD
See-Hak Seong (UIC)
3:00 PM in 712 SEO

Statistics Seminar
TBA
Chuanshu Ji (University of North Carolina at Chapel Hill)
4:00 PM in 636 SEO
Monday February 18, 2019
Analysis and Applied Mathematics Seminar
Simulating Multilayer Plasmonic Devices with Domain Decomposition Methods: High-Order Perturbation of Surfaces Implementations
David Nicholls (University of Illinois at Chicago)
4:00 PM in 636 SEO
The faithful modeling of the propagation of linear waves in a layered, periodic structure is of paramount importance in many branches of the applied sciences, in particular, in the simulation and design of multilayer plasmonic devices. In this talk we present a novel numerical algorithm for the simulation of such problems which is free of the artificial singularities present in related approaches. We advocate for a non-overlapping domain decomposition method (DDM) phrased in terms of Impedance-Impedance Operators that are immune to the Dirichlet eigenvalues which plague the Dirichlet-Neumann Operators that appear in classical formulations. We demonstrate a High-Order Spectral algorithm to simulate these operators based upon a High-Order Perturbation of Surfaces methodology which is rapid, robust, and highly accurate. We demonstrate the validity and utility of our approach with a sequence of numerical simulations.
Wednesday February 20, 2019
TBD
Ben Gould (UIC)
3:00 PM in 712 SEO

Statistics Seminar
TBA
Subhashis Ghoshal (North Carolina State University)
4:00 PM in 636 SEO
Monday February 25, 2019
Geometry, Topology and Dynamics Seminar
The Extended Haagerup fusion categories
Emily Peters (Loyola University)
3:00 PM in 636 SEO
TBA

Analysis and Applied Mathematics Seminar
TBA
Michael Greenblatt (University of Illinois at Chicago)
4:00 PM in 636 SEO
Tuesday February 26, 2019
Logic Seminar
TBA
Iian Smythe (Rutgers University)
3:30 PM in 427 SEO
Wednesday February 27, 2019
TBD
Shravan Patankar (UIC)
3:00 PM in 712 SEO

Statistics Seminar
TBA
William Li (Shanghai Advanced Institute of Finance)
4:00 PM in 636 SEO
Friday March 1, 2019
Departmental Colloquium
Part of UIC Distinguished Lecture Series
Peter Sarnak (Princeton)
3:00 PM in 636 SEO
Monday March 4, 2019
Analysis and Applied Mathematics Seminar
TBA
Gideon Simpson (Drexel University)
4:00 PM in 636 SEO
Wednesday March 6, 2019
Statistics Seminar
TBA
Yue Niu (University of Arizona)
4:00 PM in 636 SEO
Friday March 8, 2019
Departmental Colloquium
TBA
Bhargav Narayanan (Rutgers)
3:00 PM in 636 SEO
Monday March 11, 2019
Geometry, Topology and Dynamics Seminar
TBA
Mikolaj Fraczyk (Renyi Institute)
3:00 PM in 636 SEO

Analysis and Applied Mathematics Seminar
TBA
Paula Vasquez (University of South Carolina)
4:00 PM in 636 SEO
Wednesday March 13, 2019
Statistics Seminar
TBA
Dr. Xiao Zhang (Michigan Technological University)
4:00 PM in 636 SEO
TBA

TBD
Tom Dean (UIC)
5:00 PM in 636 SEO
Friday March 15, 2019
Departmental Colloquium
TBA
Kathryn Mann (Brown University)
3:00 PM in 636 SEO
Monday March 18, 2019
Geometry, Topology and Dynamics Seminar
TBA
3:00 PM in 636 SEO
Tuesday March 19, 2019
Logic Seminar
TBA
Joshua Wiscons (Sacramento State )
3:30 PM in 427 SEO
Wednesday March 20, 2019
Statistics Seminar
TBA
Mengjia Yu (UIUC)
4:00 PM in 636 SEO

TBD
Fumiaki Suzuki (UIC)
5:00 PM in 636 SEO
Friday March 22, 2019
Departmental Colloquium
TBA
Hans Mueller (UC Davis)
3:00 PM in 636 SEO
Wednesday March 27, 2019
Statistics Seminar
TBA
Xianyang Zhang (Texas A&M University)
4:00 PM in 636 SEO
Monday April 1, 2019
Analysis and Applied Mathematics Seminar
On a dissipative Gross-Pitaevskii-type model for exciton-polariton condensates
Ryan Obermeyer (University of Illinois at Chicago)
4:00 PM in 636 SEO
We study a generalized dissipative Gross-Pitaevskii-type model arising in the description of exciton-polariton condensates. We derive rigorous existence and uniqueness results for this model posed on the one dimensional torus and derive various a-priori bounds on its solution. Then, we analyze in detail the long time behavior of spatially homogenous solutions and their respective steady states. In addition, we will present numerical simulations in the case of more general initial data. We also study the corresponding adiabatic regime which results in a single damped-driven Gross-Pitaveskii equation and compare its dynamics to the one of the full coupled system.
Joint work with C. Sparber, P. Antonelli, P. Markowich, and J. Sierra
Wednesday April 3, 2019
TBD
3:00 PM in 712 SEO

Statistics Seminar
TBA
Marco Ferreira (Virginia Tech)
4:00 PM in 636 SEO

TBD
Aida Alibek (UIC)
5:00 PM in 636 SEO
Friday April 5, 2019
Departmental Colloquium
TBA
Phillip Griffiths (Institute for Advanced Study)
3:00 PM in 636 SEO
Monday April 8, 2019
Mathematical Computer Science Seminar
TBA
Andrew Suk (UCSD)
3:00 PM in 427 SEO
Wednesday April 10, 2019
TBD
Tian Wang (UIC)
3:00 PM in 712 SEO

Algebraic Geometry Seminar
TBA
Alex Perry (Columbia University)
4:00 PM in 427 SEO

Statistics Seminar
TBA
Stacey Tannenbaum (Astellas)
4:00 PM in 636 SEO
Friday April 12, 2019
Departmental Colloquium
TBD
Nikhil Srivastava (UC Berkeley)
3:00 PM in 636 SEO
Monday April 15, 2019
Analysis and Applied Mathematics Seminar
TBA
Ying Wang (University of Oklahoma)
4:00 PM in 636 SEO
Wednesday April 17, 2019
TBD
Jacob Mayle (UIC)
3:00 PM in 712 SEO

Statistics Seminar
TBA
Yehua Li (University of California at Riverside)
4:00 PM in 636 SEO

TBD
Nick Reynolds (UIC)
5:00 PM in 636 SEO
Friday April 19, 2019
Algebraic Geometry Seminar
Double ramification cycles for target varieties
Rahul Pandharipande (ETH)
2:00 PM in 427 SEO
A basic question in the theory of algebraic curves is whether a divisor represents the zeros and poles of a rational function. An explicit solution in terms of periods was given by the work of Abel and Jacobi in the 19th century. In the past few years, a different approach to the question has been pursued: what is the class in the moduli of pointed curves of the locus of such divisors? The answer in Gromov-Witten theory is given by Pixton's formula for the double ramification cycle. I will discuss recent work with F. Janda, A. Pixton, and D. Zvonkine which considers double ramification cycles for target varieties X (where Pixton's original question is viewed as the X=point case). I will also discuss the associated relations studied by Y. Bae.

Departmental Colloquium
TBA
Martina Bode and Jenny Ross (UIC)
3:00 PM in 636 SEO
Monday April 22, 2019
Analysis and Applied Mathematics Seminar
TBA
Tristan Buckmaster (Princeton University)
4:00 PM in 636 SEO
TBA
Wednesday April 24, 2019
Statistics Seminar
TBA
Leonard Stefanski (North Carolina State University)
4:00 PM in 636 SEO
Friday April 26, 2019
Departmental Colloquium
TBA
Richard Samworth (University of Cambridge)
3:00 PM in 636 SEO
Monday April 29, 2019
Geometry, Topology and Dynamics Seminar
TBA
Yulan Qing (University of Toronto)
3:00 PM in 636 SEO

Analysis and Applied Mathematics Seminar
TBA
Colette Guillope (Universite de Paris Est)
4:00 PM in 636 SEO
Wednesday May 1, 2019
Statistics Seminar
TBA
Solomon Harrar (University of Kentucky )
4:00 PM in 636 SEO

Algebraic Geometry Seminar
TBA
Antoni Rangachev (University of Chicago)
4:00 PM in 427 SEO
Monday May 6, 2019
Analysis and Applied Mathematics Seminar
TBA
Theodor Drivas (Princeton University)
4:00 PM in 636 SEO
TBA
Monday September 9, 2019
Geometry, Topology and Dynamics Seminar
TBA
Abdellah Lahdili (Montreal)
3:00 PM in 636 SEO
Friday October 11, 2019
Departmental Colloquium
TBA
Moon Duchin (Tufts)
3:00 PM in 636 SEO
Friday November 1, 2019
Departmental Colloquium
Midwest Dynamical Systems Conference
TBA (TBA)
3:00 PM in 636 SEO
Monday November 11, 2019
Geometry, Topology and Dynamics Seminar
TBA
Xiaowei Wang (Rutgers)
3:00 PM in 636 SEO
UIC LAS MSCS > seminars > seminar calendar


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• ### High temperature antiferromagnetism proximate to a Kondo insulator in $\alpha$-YbAl$_{1-x}$Mn$_{x}$B$_{4}$(1803.09366)

March 25, 2018 cond-mat.str-el
Conventionally, the parallelism between the physical properties of Ce and Yb based magnets and heavy fermions has been considered to arise as a consequence of the electron-hole symmetry known for the band theory in condensed matter physics. Namely, the trivalent state of Ce has $4f^{1}$ configuration, while $4f^{13}$ configuration for Yb$^{3+}$ can be viewed as a state carrying one 4$f$ hole. However, in the light of this parallelism, the magnetism of Yb based intermetallic compounds has been a mystery over decades; the transition temperature of the Yb based compounds is normally very small, as low as $\sim$ 1 K or even lower, while Ce counterparts may often have the transition temperature well exceeding 10 K. Here, we report our experimental discovery of the transition temperature reaching 20 K for the first time in an Yb based compound. Our single crystal study using transport, thermomagnetic and photoemission spectroscopy measurements reveals that the Mn substitution at the Al site in an intermediate valence state of $\alpha$-YbAlB$_{4}$ not only induces antiferromagnetic transition at a record high temperature of 20 K but also transform the heavy fermion liquid state in $\alpha$-YbAlB$_{4}$ into a highly resistive metallic state proximate to a Kondo insulator. The high temperature antiferromagnetism in a system close to a Kondo insulator is unusual and suggest a novel type of itinerant magnetism in the Yb based heavy fermion systems.
• ### Orbital order and fluctuations in the two-leg ladder materials BaFe$_2X_3$ ($X$ = S and Se) and CsFe$_2$Se$_3$(1704.04864)

The electronic structure of BaFe$_2X_3$ ($X$ = S and Se) and CsFe$_2$Se$_3$ in which two-leg ladders are formed by the Fe sites are studied by means of x-ray absorption and resonant inelastic x-ray scattering spectroscopy. The x-ray absorption spectra at the Fe L edges for BaFe$_2X_3$ exhibit two components, indicating that itinerant and localized Fe 3$d$ sites coexist. Substantial x-ray linear dichroism (XLD) is observed in polarization dependent spectra, indicating the existence of orbital order or fluctuation in the Fe-ladder even above the N\'{e}el temperature $T_N$. Direct exchange interaction along the legs of the Fe-ladder stabilizes the orbital and antiferromagnetic orders in BaFe$_2$S$_3$, while the ferromagnetic molecular orbitals are realized between the rungs in CsFe$_2$Se$_3$.
• ### Capturing ultrafast magnetic dynamics by time-resolved soft x-ray magnetic circular dichroism(1701.03156)

Experiments of time-resolved x-ray magnetic circular dichroism (Tr-XMCD) and resonant x-ray scattering at a beamline BL07LSU in SPring-8 with a time-resolution of under 50 ps are presented. A micro-channel plate is utilized for the Tr-XMCD measurements at nearly normal incidence both in the partial electron and total fluorescence yield (PEY and TFY) modes at the L2,3 absorption edges of the 3d transition-metals in the soft x-ray region. The ultrafast photo-induced demagnetization within 50 ps is observed on the dynamics of a magnetic material of FePt thin film, having a distinct threshold of the photon density. The spectrum in the PEY mode is less-distorted both at the L2,3 edges compared with that in the TFY mode and has the potential to apply the sum rule analysis for XMCD spectra in pump-probed experiments.
• ### Thickness dependence and dimensionality effects of charge and magnetic orderings in La1/3Sr2/3FeO3 thin films(1703.09995)

March 29, 2017 cond-mat.str-el
We investigate the thickness effects on charge and magnetic orderings in Fe perovskite oxide La1/3Sr2/3FeO3/SrTiO3 thin films by hard x-ray and resonant soft x-ray scattering (RSXS) with changing thin film thickness systematically. We found that the correlation lengths of the magnetic ordering along the in-plane and out-of-plane directions are comparable and proportional to the thickness, and shows stronger thickness dependence than those of charge orderg. %the thickness dependence of correlation length of charge ordering is smaller than that of magnetic orderings. The magnetic ordered states disappear when the correlation length of magnetic ordering decreases to that of charge ordering through the intrinsic thickness effects. Surface sensitive grazing-incident RSXS revealed that the orderings exist even in the surface region, which indicates that the observed orderings is not affected by surface effect like oxygen vacancies. Critical thickness is in 5-15 nm, which corresponds to 4-11 antiferromagnetic ordering period. This critical value seems to be common to other ferromagnetic oxide thin films.


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# Math Help - trigonometric equations

1. ## trigonometric equations

So I need help with two different types of equations the first is:

Sec^2(x)-2tan^2(x)=2/3

And I need to find all solutions in the interval [0,2pi)
If you can show me how to start solving this equation would be awesome

The second equation is:

Tan(x)=-srt3/3

I understand cos and sin but not really tan I need to find all solutions and write the equation as
X=?+pik, where k is any integer

Consider this $\displaystyle \sec ^2x = 1+ \tan ^2x$


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# Understanding interest rates

When a central bank 'pays' interest rate X / year, what does it mean exactly? Is that pure money printing? Where does that money (interest rate) come from? does it increase the debt of the country?

A popular way to fight high inflation is to increase interest rate, but doesn't that mean printing more money -> more inflation?

Thank you!

• "A popular way to fight high inflation is to increase interest rate" Can you back this up with some sources please? Oct 14 '18 at 8:20
• Oct 14 '18 at 8:26

The usual modern practice is for central banks to set a policy rate, which is where it wants short-term (often overnight) interbank lending rates to be. This then becomes the reference rate for other interest rates. An interest rate is just the cost of borrowing (expressed as a percentage per year); it is itself not money.

The rate of interest is set to achieve policy objectives, such as controlling inflation. (I will not attempt to answer the part of the question whether raising the rate of interest itself causes inflation, as that us extremely controversial.)

There is no need for the central bank to actually pay that rate of interest on anything, as I will discuss later.

It is possible for private banks to borrow at that interest rate from the central bank. In the United States, that would be discount window borrowing. Banks would do this if they need funding and cannot get it elsewhere (it is normally frowned upon). The discount rate is another policy rate that is set as a spread to the main policy rate (the fed funds rate). However, since the private bank is borrowing from the central bank, money is flowing from the private sector back to the central bank, reducing the money supply.

Pre-2008, the Federal Reserve did not pay interest on balances held at it by banks (“reserves”). That is, it could keep interest rates at a target level without paying interest to do so.

However, it now pays interest on reserves. It needs to do so now in order to keep inter-bank rates at its target rate as a result of the decision to create large amounts of excess reserves (“quantitative easing”). The payment of interest by the central bank creates “money,” since balances at the central bank are part of the money supply. (This might be termed “money printing,” but that phrase is somewhat vaguely defined.) The increased balances are created by the central bank adjusting upwards private bank balances (in the same way private banks do to their customers), so the money “appears out of nowhere.”

Does this payment increase the debt of the country? Not by standard definitions, since they usually equate governmental debt to the amount of bonds/bills issued by the fiscal authority (the Treasury in the United States). However, it increases the money issued by the central bank, and money is a liability of the central bank. Since the central government owns the central bank, this means that governmental liabilities are higher.

When a central bank 'pays' interest rate X / year, what does it mean exactly?

It means that by lending to the central bank \$100/(1+X) at the beginning of the year, the central bank will pay \$100 at the end of that year.

A popular way to fight high inflation is to increase interest rate, but doesn't that mean printing more money -> more inflation?

Not necessarily. Increasing the interest rate may attract cash flows from foreign lenders/investors. The effects of it on the exchange rate (appreciation of the local currency), currency reserves, and/or trade balance tend to alleviate the need to merely print more money in the local economy.

You need a clear understanding of interest rate.

Interest rate is nothing but the opportunity cost of holding money. It is the price you pay for the cash in your wallet. It is the price of present consumption.

Now think a level deeper. You have some cash in your wallet or in your house. What were the other opportunities of using that cash? You would have invested that cash and let's say you would have done a small business and earned a profit.

You plan to keep it in the bank and authorise the bank to use that cash for a stipulated time. Now your bank should compensate you with a part of productivity that your cash would generate. Your cash is actually a capital that earns a return. So rate of interest is nothing but rate of return on your investable cash. If there is no investment there would not be any rate of interest because further value could not be generated without investment.

Now inflation could be controlled by increasing the rate of interest in a closed economy. Higher rate of interest would discourage investor to take loans and invest. Higher rate of interest would also discourage households to hold liquid cash and decrease spending thereby decrease their demand which would exert a downward pressure on prices.

The higher interest rate never means printing more money instead it means printing less money.

See, the interest rate is nothing but the price of money which is determined from the intersection of money demand and money supply curves. When you increase the money supply that is you print more money you tend to decrease the price of money that is the interest rate and vice verca.

Hope this explanation brings in conceptual clarity.

• There is some confusion here. Yes, higher rates will exert the deflationary effect you describe, which is in line with other effects I outlined in my answer. But these effects aside, if lender paid \$95 to the central bank for a zero-coupon bond with face value of \$100, the central bank will eventually need to print \$5 in order to honor both its debt (\$95) and its promise (\$5) at the end of year. Printing of money need not happen at the same time the rates are raised, but consumer's decision not to spend \$95 on consumption today does not void the bank's duty to pay the extra \$5 later on. Oct 14 '18 at 21:38 • You're right:$95 in year 2018 is worth $100 in year 2019. When bank sells bonds in an open market it is contractionary monetary policy ie it decreases purchasing power and income and when it repurchases the bonds it is expansionary fiscal policy. The effect you describe is first monetary expansion then subsequent contraction. You are confusing between nominal money and real money. When you are printing money, the output also increases on the supply side with the productivity you generate. Hence$ 5 may be printed, but it is backed by real output which your $95 generated. Oct 15 '18 at 1:25 • Whether the \$95 lent to the central bank generated any productivity or real output (with which to back the extra \$5) is debatable. First, because those \$95 are neither spent on consumption nor invested, but removed from the economy, thereby rendering the \$95 unproductive. Second, bc the purpose of a central bank is to procure and maintain currency stability rather than to stimulate the economy. It is true that the U.S. Federal Reserve also pursues (or pursued at least in the Great Recession) full employment, but that hybrid or dual mission departs from the strict purpose of a central bank. Oct 15 '18 at 21:13 • When you think in an macroeconomic framework, the central bank does not inject all the money it has in the economy all at once. Some people would have purchased bond worth$95 for 1 year, some for 2 years and some for even 10 years. When central bank is repaying you your money, it is parallelly issuing more bonds to suck more money. So inflationary impact depends on the relative magnitudes of money injected or leaked and also on how supply side is reacting. Inflation is multidimensional. You cannot distil only one factor, and evaluate its inflationary impact. Oct 16 '18 at 4:55

So, a central bank sets the interest rate of the instrument of money that it deals in. Example, the Reserve Bank of India (RBI) sets the repurchase rate (repo rate) at which it will buy or sell money against treasury bonds as collateral. So, if the repo rate is 5%, it means that Commercial Banks (bank) can borrow funds from the RBI (by giving treasury bonds as collateral) at 5%.

This is separate from the Open Market Operations (OMO) that a Central Bank performs to control money supply (liquidity) in an economy. So, a Central Bank can announce bond sales or redemptions. If there is a bond sale, the Central Bank will sell the treasury bonds that it holds as assets and accept dollars from the purchasers (typically Banks). In this case the money supply in the economy goes down. The opposite effect occurs when bonds are redeemed/purchased back. The bonds are purchased by the Central Bank and it pays money to the bondholders releasing money into the economy.

In the second case of OMO where the Central Bank pays money to the bondholders, it actually prints money. Similarly, if the Commercial Banks borrow from the Central Bank using the repo mechanism, the latter will print money to lend at that rate.

In reality, Commercial Banks opt for Central Bank borrowing as a last option and opt to borrow money from other Banks where collateral free loans are available (albeit at a higher rate).

Further Reading: There is a reverse repo rate at which the central bank can borrow money from the bank, say at 4%. So, effectively this sets a floor interest rate of 4% for the bank since it will at least get that much and a ceiling rate of 5% at which bank can borrow from the Central bank. The difference 1% is Central Bank's profit.