text:
# Integration of $|f(x)|$

I would like to learn how to use the sign function $\operatorname{sgn}(x)$ to integrate the absolute value of an arbitrary function. I know that integrating piecewise should give me the same result, but sometimes trigonometric functions can become cumbersome. I have tried searching on the internet, but I can't seem to find anything. Perhaps the theory isn't complete, and the methods for integration aren't known for the absolute value of all functions, but I would appreciate if someone could give me one or two places where I can learn a general approach. What I would like to learn is integration for the absolute value of at least a second degree function, and the "basic" trigonometric functions (sin, cos, tan, csc, sec, cot). Links or references to information on the internet will be a lot more helpful to me than references to books. Thank you.

-
The most general advice for integrating $\int|f(x)|\mathrm dx$ is to find where $f(x)$ crosses the horizontal axis, and split your integral accordingly... at least, I know that computing environments with the ability to do this also use this approach. –  Ｊ. Ｍ. Aug 25 '11 at 3:19
The sign function approach not only gives the same result as integrating piecewise, there's also no deep difference between the two approaches, since the sign function, like the absolute value, is defined piecewise using a case distinction; it's basically just $\text{sgn}(x)=|x|/x$ (glossing over the details at $0$). You can express it algebraically e. g. as $\text{sgn}(x)=\sqrt{x^2}/x$ , but you can do the same thing with $|x|=\sqrt{x^2}$ directly. –  joriki Aug 25 '11 at 4:48
@ J.M., joriki: I understand what you mean, but it's just that integration of certain trigonometric functions can become very, very cumbersome. See, if the integral of a trig function has multiple roots, it can be easy to integrate by finding the area under of it it's "humps" and then multiplying that by the number of "humps". However, some trig functions have the property that their "humps" aren't uniform. Then it becomes very time consuming (unless there's an integration method I'm missing). What I mean by "hump" is the area under the curve of a sine function from two successive roots. –  Hautdesert Aug 27 '11 at 3:11
Sure. Sometimes you just have to bite the bullet and do the tear-inducing algebra... –  Ｊ. Ｍ. Aug 28 '11 at 17:52

I looked through some of my stuff last night and found the following references. However, all except one are not "links or references to information on the internet" (unless you have access to a university library), but perhaps by googling some of the titles and names you can come up with some useful items on the internet. I've listed them in the order (highest to lowest) that I'd guess as to how useful they'd be to you (with the exception of the last one, for which I've only seen the Nature summary).

Kenneth O. May, "The calculus of absolute values", American Mathematical Monthly 62 #9 (November 1955), 651-653. (here)

Margaret R. Wiscamb, "A formula for the derivative of the absolute value of a polynomial", Pi Mu Epsilon Journal 4 #9 (Fall 1968), 367-369.

C. Walmsley, "On the area of a square", Mathematical Gazette 38 #325 (September 1954), 210-211. (here)

John L. Spence, "Step function notation", School Science and Mathematics 60 #3 (March 1960), 179-180. (here)

Daryl E. Doan and Joshua Grusd, "Derivative and indefinite integral", Reader Reflections column, Mathematics Teacher 98 #5 (Dec./Jan. 2004/2005), 292.

"prove d|x|/dx = x/|x| without piecewise", MathKB discussion groups, Mathematics - General Topics, 27 January 2006. (here)

Nature (journal) 93 #2315 (12 March 1914), p. 39: Midway down the left column is a brief description of a Russian paper by I. N. Kouchnereff and D. Riabouchinsky that: "...treats this quantity [absolute value operation] as a function of the variable $x$, and shows how this method leads to interesting formulae involving the solution of equations, differentiation, and integration. A number of elegant geometrical applications are also given, such as equations of broken lines [polygonal curves?], and equations of limited portions [bounded?] of planes, such as a square area."

-
These references were helpful. Thank you. –  Hautdesert Aug 27 '11 at 3:13


text:
# Positive Integers n Such That σ(φ(n)) = σ(n)

14

Loading....

Loading....

Loading....

Loading....

Loading....

## Full text

(1)

23 11

Article 08.1.5

Journal of Integer Sequences, Vol. 11 (2008), 2

3 6 1 47

## (n)

### fluca@matmor.unam.mx

Abstract

In this paper, we investigate those positive integersnfor which the equalityσ(φ(n)) = σ(n) holds, whereσ is the sum of the divisors function andφ is the Euler function.

### Introduction

For a positive integer n we writeσ(n) and φ(n) for the sum of divisors function and for the Euler function of n, respectively. In this note, we study those positive integers n such that

σ(φ(n)) = σ(n)

holds. This is sequence A033631 in Sloane’s Online Encylopedia of Integer Sequences. Let A be the set of all such positive integers n and for a positive real number x we put A(x) = A ∩[1, x]. Our result is the following.

(2)

Theorem 1. The estimate

#A(x) =O

x (logx)2

holds for all real numbers x >1.

The above upper bound might actually be the correct order of magnitude of #A(x). Indeed, note that if m is such that

σ(2φ(m)) = 2σ(m) (1)

and if q > m is a Sophie Germain prime, that is a prime number q such that p= 2q+ 1 is also prime, then n =mp ∈ A. The numbers m = 2318, 2806, 5734, 5937, 7198, 8097, . . . all satisfy relation (1), and form Sloane’s sequence A137733. More generally, ifk and m are positive integers such that

σ(2kφ(m)) = 2kσ(m) (2)

and q1 < . . . < qk are primes with pi = 2qi + 1 also primes for i = 1, . . . , k and q1 > m,

then n = p1. . . pkm ∈ A. Now recall that the Prime K-tuplets Conjecture of Dickson (see, for instance, [2, 4, 8]) asserts that, except in cases ruled out by obvious congruence conditions,K linear formsain+bi,i= 1, . . . , K, take prime values simultaneously for about cx/(logx)K integers n x, where c is a positive constant which depends only on the given linear forms. Under this conjecture (applied withK = 2 and the linear formsn and 2n+ 1), we obtain that there should be x/(logx)2 Sophie Germain primes q x, which suggests

that #A(x)x/(logx)2. We will come back to the Sophie Germain primes later.

Throughout, we use the Vinogradov symbols and and the Landau symbols O and o with their regular meanings. We use log for the natural logarithm and p, q and r with or without subscripts for prime numbers.

### Preliminary Results

In this section, we point out a subset B(x) of all the positive integers n x of cardinality O(x/(logx)2). For the proof of Theorem 1 we will work only with the positive integers

n ∈ A(x)\B(x). Further, x0 is a sufficiently large positive real number, where the meaning

of sufficiently large may change from a line to the next. We put

y= exp

logx log logx

.

For a positive integer n we write P(n) for the largest prime factor of n. It is well known that

Ψ(x, y) = #{nx | P(n)y}=xexp((1 +o(1))ulogu) (u→ ∞), (3) where u= logx/logy, provided that u y1/2 (see [1], Corollary 1.3 of [6], or Chapter III.5

of [9]). In our case, u = log logx, so, in particular, the condition u y1/2 is satisfied for

x > x0. We deduce that

(3)

Thus, if we set B1(x) = {n≤x | P(n)≤y},then

#B1(x) = Ψ(x, y) =xexp (−(1 +o(1))(log logx)(log log logx))

< x

(logx)10 forx > x0. (4)

We now let

z = (logx)26

and for a positive integer n we write ρ(n) for its largest squarefull divisor. Recall that a positive integer m is squarefull ifp2 |m whenever p is a prime factor ofm. It is well known

that if we write S(t) ={m t | m is squarefull}, then #S(t) = ζ(3/2)

ζ(3) t

1/2+O(t1/3),

where ζ is the Riemann Zeta-Function (see, for example, Theorem 14.4 in [7]). By partial summation, we easily get that

X

m≥t m squarefull

1 m ≪

1

t1/2. (5)

We now let B2(x) be the set of positive integers n ≤ x such that one of the following

conditions holds: (i) ρ(n)z,

(ii) p|n for some primep such that ρ(p±1)z,

(iii) there exist primes r and p such that p|n,p≡ ±1 (mod r) and ρ(r±1)z.

We will find an upper bound for #B2(x). Let B2,1(x) be the set of thosen ∈ B2(x) for which

(i) holds. We note that for every n∈ B2,1(x) there exists a squarefull positive integer d≥z

such thatd |n. For a fixed d, the number of such nx does not exceed x/d. Hence, #B2,1(x)≤

X

d≥z d squarefull

x d ≪

x

(logx)13, (6)

where we have used estimate (5) with t =z. Now let B2,2(x) be the set of those n ∈ B2(x)

for which (ii) holds. We note that each n∈ B2,2(x) has a prime divisor p such that p≡ ±1

(mod d), where d is as above. Given d and p, the number of such n x does not exceed x/p. Summing up over all choices of pand d we get that

#B2,2(x) ≤

X

d≥z dsquarefull

X

p≡±1 (modd)

p≤x x p ≪x

X

d≥z dsquarefull

log logx φ(d)

≪ x(log logx)2 X d≥z dsquarefull

1 d ≪

x(log logx)2

(4)

where in the above estimates we used aside from estimate (5), the fact that the estimate

X

p≡a (modb)

p≤x 1 p ≤

1 p1(a, b)

+O

log logx φ(b)

(8)

holds uniformly in a, b and x when b x, where a and b are coprime and p1(a, b) is the

smallest prime numberpa (mod b) (note thatp1(1, b)≥b+1 andp1(−1, b) = p1(b−1, b)≥

b1φ(b) for allb 2), together with the well known minimal orderφ(n)/n1/log logx, valid for n in the interval [1, x].

Let B2,3(x) be the set of those n ∈ B2(x) for which (iii) holds. Then there existsr such

that r ≡ ±1 (mod d) for some d as above, as well as p | n such that r | p1 or r | p+ 1. Givend, r andp, the number of suchn xdoes not exceedx/p, and now summing up over all choices ofd, r and p, we get that

#B2,3(x) ≤

X

d≥z dsquarefull

X

r≡±1 (modd)

r≤x

X

p≡±1 (modr)

p≤x x p

≪ x X

d≥z dsquarefull

X

r≡±1 (modd)

r≤x

log logx φ(r) ≪ x(log logx)2 X

d≥z dsquarefull

X

r≡±1 (modd)

r≤x 1 r ≪ x(log logx)3 X

d≥z dsquarefull

1 φ(d) ≪ x(log logx)4 X

d≥z dsquarefull

1 d ≪

x(log logx)4

(logx)13 , (9)

where in the above estimates we used again estimate (5), estimate (8) twice as well as the minimal order of the Euler function on the interval [1, x].

Hence, using estimates (6)–(9), we get

#B2(x)≤#B2,1(x) + #B2,2(x) + #B2,3(x)<

x

(logx)10 for x > x0. (10)

We now put

w= 10 log logx and set

S(w, x) = X ω(m)≥w

m≤x 1 m,

(5)

whereω(m) denotes the number of distinct prime factors of the positive integerm. Note that, using the fact thatX

p≤t 1

p = log logt+O(1) and Stirling’s formulak! = (1 +o(1))k

ke−k√2πk, we have

S(w, x) = X k≥w

X

ω(m)=k m≤x 1 m = X k≥w 1 k! X p≤x X α≥1 1 pα !k = X k≥w 1 k! X p≤x 1 p +O

X p≥2 1 p2 !!k ≪X k≥w 1

k!(log logx+O(1))) k

≤ X

k≥w

elog logx+O(1) k

k

≤X k≥w

elog logx+O(1) w

k

elog logx+O(1) w

w

≪ 1

(logx)10 log(10/e) <

1

(logx)11 (11)

for x > x0 because 10 log(10/e)>11.

We now let B3(x) be the set of positive integers n ≤ x such that one of the following

conditions holds: (i) ω(n)w,

(ii) p|n for some primep for which ω(p±1)w,

(iii) there exist primes r and p such that p|n,p≡ ±1 (mod r) and ω(r±1)w.

Let B3,1(x), B3,2(x) and B3,3(x) be the sets of n ∈ B3(x) for which (i), (ii) and (iii) hold,

respectively.

To bound the cardinality of B3,1(x), note that, using (11), we have

#B3,1(x) =

X

ω(n)≥w n≤x

1 X ω(n)≥w

n≤x x

n =xS(w, x)< x

(logx)11 (12)

for x > x0. To bound the cardinality of B3,2(x), note that each n ∈ B3,2(x) admits a prime

divisorpsuch thatω(p±1)w. Fixing such a p, the number of suchnxdoes not exceed x/p. Summing up over all such p we have, again in light of (11),

#B3,2(x) ≤

X

ω(p±1)≥w p≤x

x p ≤x

   

X

ω(p+1)≥w p+1≤x+1

2 p+ 1 +

X

ω(p−1)≥w p−1≤x

1 p1

   

≤ x(2S(w, x+ 1) +S(w, x))<3xS(w, x) + 2

(6)

for x > x0. To bound the cardinality of B3,3(x), note that for each n ∈ B3,3(x) there exist

some prime r with ω(r±1) w and some prime p | n such that p ≡ ±1 (mod r). Given such r and p, the number of such n x does not exceed x/p. Summing up over all choices of r and p given above we get, again using (11),

#B3,3(x) ≤

X

ω(r±1)≥w r≤x

X

p≡±1 (modr)

p≤x x

p =x(log logx)

X

ω(r±1)≥w r≤x

1 φ(r) = x(log logx) X

ω(r±1)≥w r≤x

1 r1

≤ x(log logx)

   

X

ω(r−1)≥w r−1≤x

1 r1 +

X

ω(r+1)≥w r+1≤x

3 r+ 1

   

≤ x(log logx)(S(w, x) + 3S(w, x+ 1))

≤ 4x(log logx)S(w, x) +O(log logx) x(log logx)

(logx)11 (14)

for x > x0.

Hence, using estimates (12) to (14), we get

#B3(x)≤#B3,1(x) + #B3,2(x) + #B3,3(x)<

x

(logx)10 for x > x0. (15)

We now let

B4(x) = {n≤x | n6∈(B1(x)∪ B2(x)) and p2 |φ(n) for some p > z}.

Letn ∈ B4(x) and letp2 |φ(n) for some primep. Then it is not possible thatp2 |n (because

n 6∈ B2(x)), nor is it possible that p2 | q−1 for some prime factor q of n (again because

n 6∈ B2(x)). Thus, there must exist distinct primes q and r dividing n such that q ≡ 1

(mod p) and r 1 (mod p). Fixing such p, q and r, the number of acceptable values of such n x does not exceed x/(qr). Summing up over all the possible values of p, q and r we arrive at

#B4(x) ≤

X

z≤p≤x

X

q≡1 (modp)

r≡1 (modp)

q<r, qr≤x x qr ≤x

X z≤p≤x 1 2     X

q≡1 (modp)

q≤x 1 q     2

≪ x X

z≤p≤x

(log logx)2

(p1)2 ≪x(log logx)

2 X

z≤p≤x 1 p2

≪ x(log logx)

2

(7)

where we used estimates (8) and (5). Hence, #B4(x)<

x

(logx)10 for all x > x0. (16)

We now let

τ = exp (log logx)2

and let B5(x) stand for the set of n ≤ x which are multiples of a prime p for which either

p1 orp+ 1 has a divisord > τ with P(d)< z. Fix such a pair dand p. Then the number of nx divisible by p is at most x/p. This shows that

#B5(x)≤

X

P(d)<z τ <d≤x

X

p≡±1 (modd)

p≤x x

p ≪xlog logx

X

P(d)<z τ <d≤x

1

d. (17)

It follows easily by partial summation from the estimates (3) for Ψ(x, v), that if we write v = logτ /logz, then

S= X P(d)<z τ <d≤x

1 d ≤

logx

exp((1 +o(1))vlogv). Since v = (log logx)/26, we get that

vlogv = (1/26 +o(1))(log logx)(log log logx) and therefore that

S logx

exp((1 +o(1))vlogv) < 1

(logx)11 (18)

for all x > x0, which together with estimate (17) gives

#B5(x)<

x

(logx)10 for x > x0. (19)

Thus, setting

B(x) =

5

[

i=1

Bi(x), (20)

we get, from estimates (4), (10), (15), (16) and (19) that #B(x)

5

X

i=1

#Bi(x)≪ x

(logx)10 for all x > x0. (21)

### 1

(8)

Theorem 2. Let a and b be any fixed positive integers. Setting

Aa,b={n | σ(aφ(n)) =bσ(n)},

then the estimate

#Aa,b(x)≪a,b x (logx)2 holds for all x3.

Proof. Letxbe large and let B(x) be as in (20). We assume thatnx is a positive integer not inB(x). We letA1(x) be the set ofn∈ Aa,b(x)\B(x) for which (P(n)−1)/2 is not prime but such that there exists a prime number r > z and another prime number q | P(n)1 for which r | gcd(P(n) + 1, q+ 1). To count the number of such positive integers n x, let r and q be fixed primes such that r | q+ 1, and let P be a prime such that q | P 1 and r | P + 1. The number of positive integers n x such that P(n) =P does not exceed x/P. Note that the congruences P ≡ −1 (mod r) and P 1 (mod q) are equivalent to P aq,r (mod qr), where aq,r is the smallest positive integerm satisfying m≡ −1 (mod r) and m1 (mod q). We distinguish two instances:

Case 1: qr < P. Let A

1(x) be the set of such integers n∈ A1(x). Then

#A1(x) X z<r≤x

X

q≡−1 (modr)

q≤x

X

P≡aq,r (modqr)

qr<P≤x x P ≪ xlog logx X

z<r≤x

X

q≡−1 (modr)

q≤x

1 φ(qr) ≪ xlog logx X

z<r≤x 1 r

X

q≡−1 (modr)

q≤x 1 q ≪ x(log logx)2 X

z<r≤x 1 rφ(r) ≪ x(log logx)2X

z<r 1 r2 ≪

x(log logx)2

(logx)11 , (22)

where in the above inequalities we used estimates (8) and (5).

Case 2: qrP. Let A′′

1(x) be the set of such integers n ∈ A1(x). Here we write n = P m. Note that

P > P(m) because y > z for large xand n6∈ B1(x)∪ B2(x). Furthermore, sincer |q+ 1, we

may writeq =rℓ1. Sinceq |P1, we may writeP =sq+ 1 =s(rℓ1) + 1 =srℓ+ 1s. Since r|P + 1, we get that 1s≡ −1 (mod r), and therefore that s 2 (mod r). Hence, there exists a nonnegative integer λ such that s = λr+ 2. If λ = 0, then s = 2 leading to

(9)

P = 2q+ 1, which is impossible. Thus, λ >0. Let us fix the value of λ as well as that of r. Then

rℓ1 = q and (λr2+ 2r)ℓ(λr+ 1) =P (23) are two linear forms in the variableℓ which are simultaneously primes. Note that P x/m and since P ℓr(λr+ 2), we get that ℓ x/(mr(λr+ 2)). In particular, mr(λr+ 2) x. Recall that a typical consequence of Brun’s sieve (see for example Theorem 2.3 in [3]), is that if

L1(m) =Am+B and L2(m) =Cm+D

are linear forms inm with integer coefficients such thatADBC 6= 0 and if we writeE for the product of all primespdividingABCD(ADBC), then the number of positive integers my such that L1(m) and L2(m) are simultaneously primes is

≤ Ky (logy)2

E φ(E)

2

for some absolute constant K. Applying this result for our linear forms in ℓ shown at (23) for which A = r, B = 1, C = λr2+ 2r and D = (λr+ 1), we get that the number of

acceptable values for ℓ does not exceed

K x

mr(λr+ 2) (log(x/(mr(λr+ 2)))2

(λr+ 2)(λr+ 1)r φ((λr+ 2)(λr+ 1)r)

2

≪ x(log logx)

2

mr(λr+ 2),

where for the rightmost inequality we used again the minimal order of the Euler function on the interval [1, x]. Here, K is some absolute constant. Summing up over all possible values of λ, r and m, we get

#A′′1(x) X z<r≤x

X

1≤λ≤x

X

1≤m≤x

x(log logx)2

mr(λr+ 2) < x(log logx)2 X

z<r≤x 1 r2

! X

1≤λ≤x 1 λ

! X

1≤m≤x 1 m

!

≪ x(log logx)

2(logx)2

(logx)13 =

x(log logx)2

(logx)11 , (24)

where we used again estimate (5).

From estimates (22) and (24), we get #A1(x)≤#A′1(x) +A′′1(x)<

x

(logx)10 for all x > x0. (25)

Now letA2(x) be the set of those n∈ Aa,b(x)\(A1(x)∪ B(x)) and such that (P(n)−1)/2

(10)

y > z for large x and n 6∈ B1(x)∪ B2(x). Then bσ(n) = bσ(m)(P + 1). Let d1 be the

largest divisor of P + 1 such that P(d1) ≤ z. Then P + 1 = d1ℓ1, and bσ(n) = bσ(m)d1ℓ1.

Furthermore, φ(n) = φ(m)(P 1), so that aφ(n) = aφ(m)(P 1). Let d be the largest divisor of P 1 which is z-smooth; that is, with P(d) z. Then P 1 = dℓ. Note that ℓ is squarefree (because n6∈ B2(x)) and ℓ and φ(m) are coprime (for if not, there would exist

a prime r > z such that r | ℓ and r | φ(m), so that r2 | φ(n), which is impossible because

n 6∈ B4(x)). Since z > a for x > x0, we get that aφ(m)d and ℓ are coprime, and therefore

that σ(φ(n)) =σ(aφ(m)d)σ(ℓ). We thus get the equation σ(aφ(m)d)σ(ℓ) =bσ(m)d1ℓ1.

Now note that ℓ1 and σ(ℓ) are coprime. Indeed, if not, since ℓ is squarefree, there would

exist a prime factor r of ℓ1 (necessarily exceeding z) dividing q+ 1 for some prime factor q

of ℓ, that is of P 1. But this is impossible because n6∈ A1(x).

Thus, ℓ1 |σ(aφ(m)d). Note now that P m ≤ x, so that m ≤ x/P ≤ x/y. Furthermore,

max{d, d1} ≤ τ because n 6∈ B5(x). Let us now fix m, d and d1. Then ℓ1 | σ(aφ(m)d),

and therefore the number of choices forℓ1 does not exceedτ(σ(aφ(m)d)), where τ(k) is the

number of divisors of the positive integer k. The above argument shows that if we write M for the set of such acceptable values for m, then

#A2(x)≤

xτ2

y max{τ(σ(aφ(m)d))| m ∈ M, d≤τ, d1 ≤τ}. (26) To get an upper bound on τ(σ(aφ(m)d)), we write aφ(m)d as

aφ(m)d=AB,

where A is squarefull,B is squarefree, and A and B are coprime. Clearly, σ(aφ(m)d) = σ(AB) = σ(A)σ(B),

so that

τ(σ(aφ(m)d))τ(σ(A))τ(σ(B)). Since B is squarefree, it is clear that

σ(B)

Y

q|aφ(m)d

(q+ 1). Furthermore,

ω(aφ(m)d)ω(aφ(n))ω(a) +ω(n) +X p|n

ω(p1)w2+w+O(1), (27)

so that we haveω(aφ(m)d)2w2 ifx > x

0. The above inequalities follow from the fact that

n6∈ B3(x). Also, for each one of the at most 2w2prime factorsq ofaφ(m)d, the numberq+ 1

(11)

n 6∈ B3(x). In conclusion, σ(B) | C, where C is a number with at most 2w3 prime factors

whose squarefull part does not exceed z2w2

. Thus,

τ(σ(B))τ(C)2ω(C)τ(ρ(C)), where ω(C)2w3 and ρ(C)z2w2

. Clearly,

τ(ρ(C))ρ(C)zw2 = exp(w2logz) = exp(O(log logx)3), and since also

2ω(C)22w3 = exp(O(log logx)3), we finally get that

τ(σ(B)) = exp(O(log logx)3). (28) We now deal with σ(A). We first note that P(A) z for x > x0. Indeed, if q > z divides

A, thenq2 |aφ(n). If x > x

0, then z > a, in which case the above divisibility relation forces

q2 | φ(n) which is not possible because n 6∈ B

3(x). By looking at the multiplicities of the

prime factors appearing in A, we easily see that

A

ρ(a)ρ(d) Y

p|m

ρ(p1)

   

Y

q|aφ(m)d q≤z

q

   

ω(aφ(m)d)

.

As we have seen at estimate (27),ω(aφ(m)d)2w2, in which case the above relation shows

that

Aρ(a)zω(aφ(m)d)+ω(aφ(m)d)2 z4w4+2w2 = exp(O(log logx)5). But since σ(A)< A2 and τ(σ(A))σ(A)A2, we get that

τ(σ(A)) = exp(O(log logx)5), (29) which together with estimate (28) gives

τ(σ(aφ(m)d))τ(σ(A))τ(σ(B)) = exp(O(log logx)5). Returning to estimate (26), we get that

#A2(x) ≤

xτ2

y exp(O(log logx)

5) =xexp

log loglogxx +O((log logx)5)

< x

(logx)10 for all x > x0. (30)

Thus, writing C(x) =Aa,b(x)\(A1(x)∪ A2(x)∪ B(x)), we get that

Aa,b(x) = #C(x) +O

x (logx)10

(12)

Moreover, if n ∈ C(x), then n=P m, with P > P(m) for x > x0, and (P −1)/2 is a prime.

Hence, P 1 = 2q, where q is a Sophie Germain prime. Since also P x/m, it follows by Brun’s method that the number of such values for P is

m(logxx/m)2 ≤

x

m(logy)2 =

x(log logx)2

m(logx)2 .

In the above inequalities we used the fact that x/mP y. Summing up over allm x, we get

#C(x) X m≤x

x(log logx)2

m(logx)2 ≪

x(log logx)2

logx . (32)

In particular,

#Aa,b(x)≪

x(log logx)2

logx . (33)

This is weaker than the bound claimed by our Theorem 2. However, it implies, by partial summation, that

X

n∈Aa,b(x)

1

n ≤ 1 +

Z x

2

1

td(#Aa,b(t)) = 1 + #Aa,b(t)

t

t=x t=2 +O

Z x

2−

t(log logt)2

t2logt dt

= O((log logx)3). (34)

To get some improvement, we return toC(x). Letn ∈ C(x) and write it asn =P m, where P > P(m). Write also P = 2q+ 1. Then φ(n) = 2φ(m)q. Moreover, q > (y1)/2 > z for x > x0 so that q does not divide aφ(m) (because q > a and n 6∈ B3(x)). Hence,

σ(aφ(n)) = σ(2aφ(m))(q+ 1). On the other hand,bσ(n) =bσ(m)(P + 1) = 2bσ(m)(q+ 1). Thus, the equation σ(aφ(n)) = σ(bn) forces σ(2aφ(m)) = 2bσ(m), implying that m A2a,2b(x/P) ⊂ A2a,2b(x/y). The argument which leads to estimate (32) now provides the better estimate

#C(x) X m∈A2a,2b(x)

x(log logx)2

m(logx)2 ≪

x(log logx)5

(logx)2 .

In particular, we get

#Aa,b(x)≪

x(log logx)5

(logx)2 . (35)

This is still somewhat weaker than what Theorem 2 claims. However, it implies that the sum of the reciprocals of the numbers in Aa,b is convergent. In fact, by partial summation,

(13)

it follows, as in estimates (34), that

X

n∈Aa,b

n≥y 1 n ≤

Z ∞

y− 1

td(#Aa,b(t)) = #Aa,b(t)

t

t=∞ t=y−+O

Z x y−

t(log logt)5

t2(logt)2 dt

≪ (log logy)

5

(logy)2 +

Z ∞

y− 1 t(logt)3/2dt

(log1y)1/2. (36)

We now take another look at C(x). Let again n ∈ C(x) and write n =P m. Fixing m and using the fact that P x/m is a prime such that (P 1)/2 is also a prime, we get that the number of choices for P is

m(log(x/m))x 2.

Hence,

#C(x) X m∈A2a,2b(x/y)

x

m(log(x/m))2.

We now split the above sum atm =x1/2 and use estimate (36) to get

#C(x) X

m∈A2a,2b(x1/2)

x

m(log(x/m))2 +

X

m∈A2a,2b

x1/2

≤m≤x/y

x m(log(x/m))2

(logxx1/2)2

X

m∈A2a,2b

1 m +

x (logy)2

X

m∈A2a,2b

m≥x1/2

1 m

(logxx)2 +

x (logy)5/2 =

x (logx)2 +

x(log logx)5/2

(logx)5/2

(logxx)2, (37)

which together with estimate (31) leads to the desired conclusion of Theorem 2.

### Acknowledgments

This work was done in April of 2006, while the second author was in residence at the Centre de recherches math´ematiques of the Universit´e de Montr´eal for the thematic year Analysis

and Number Theory. This author thanks the organizers for the opportunity of participating

in this program. He was also supported in part by grants SEP-CONACyT 46755, PAPIIT IN104505 and a Guggenheim Fellowship. The first author was supported in part by a grant from NSERC.

(14)

### References

[1] E. R. Canfield, P. Erd˝os and C. Pomerance, On a problem of Oppenheim concerning ”Factorisatio Numerorum”,J. Number Theory 17 (1983), 1–28.

[2] L. E. Dickson, A new extension of Dirichlet’s theorem on prime numbers,Messenger of

Math. 33 (1904), 155–161.

[3] H. Halberstam and H.-E. Richert,Sieve Methods, Academic Press, London, 1974. [4] G. H. Hardy and J. E. Littlewood, Some problems on partitio numerorum III. On the

expression of a number as a sum of primes, Acta Math. 44 (1923), 1–70.

[5] A. Hildebrand, On the number of positive integers x and free of prime factors > y,

J. Number Theory 22 (1986), 289–307.

[6] A. Hildebrand and G. Tenenbaum, Integers without large prime factors, J. de Th´eorie

des Nombres de Bordeaux 5(1983), 411–484.

[7] A. Ivi´c, The Riemann Zeta-Function, Theory and Applications, Dover Publications, 2003.

[8] A. Schinzel and W. Sierpi´nski, Sur certaines hypoth`eses concernant les nombres pre-miers, Acta Arith.4 (1958), 185–208. Erratum, Acta Arith. 5 (1959), 259.

[9] G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge Univ. Press, 1995.

2000 Mathematics Subject Classification: Primary 11A25; Secondary 11N37.

Keywords: Euler function, sum of divisors.

(Concerned with sequences A033631 and A137733.)

Received June 26 2007; revised version received February 9 2008. Published in Journal of

Integer Sequences, February 9 2008.

Updating...

format: plain


text:
# Changeset 4491 for docs/Working/icGrep

Ignore:
Timestamp:
Feb 10, 2015, 6:12:53 PM (4 years ago)
Message:

Minor edits

Location:
docs/Working/icGrep
Files:
2 edited

### Legend:

Unmodified
 r4490 \end{tabular} \end{center} \caption{Matching with Bitwise Data Parallelism}\label{fig:bitwisematch} \caption{Matching {\tt d[a-z]*ed} Using Bitwise Data Parallelism}\label{fig:bitwisematch} \end{figure} In the first step the character class stream {\tt [d]} is matched and the results shifted one position (Advance) to produce marker bitstream $M_1$. Four matches indicated by marker bits are now in play simultaneously. Five matches indicated by marker bits are now in play simultaneously. The next step applies the  MatchStar operation to find all the matches that may then be reached with the Kleene-* repetition is no need to consider these matches one at a time using lazy or greedy matching strategies. Rather, the full marker stream $M_3$ of remaining possibilites after matching {\tt [e]} is easily computed using a shift and bitwise and. The final step produces marker stream $M_4$ indicating that single position computed using bitwise logic and shift. The final step produces marker stream $M_4$ indicating the single position at which the entire regular expression is matched.


text:
How to convert parametric form of tangent line to Cartesian equivalent

I have calculated the tangent line of the following curve $$c(t)=(2cost−cos2t,2sint−sin2t$$) @ $$t=\pi/4$$ to be

$$l(t)=(\sqrt{2}, \sqrt{2}-1)+ (t- \frac{\pi}{4})(2-\sqrt{2}, \sqrt{2})$$

How do I convert this to $$y-y_{0}=f'(x_{0})(x-x_{0})$$?

From $$(\sqrt{2}, \sqrt{2}-1)+ \left(t- \frac{\pi}{4}\right)(2-\sqrt{2}, \sqrt{2})$$ we get $$\begin{cases} x=\left(2-\sqrt{2}\right) \left(t-\frac{\pi }{4}\right)+\sqrt{2}\\ y=\sqrt{2} \left(t-\frac{\pi }{4}\right)+\sqrt{2}-1\\ \end{cases}$$ from the second equation we have

$$t-\frac{\pi}{4}=\frac{y}{\sqrt{2}}+\frac{1}{\sqrt{2}}-1$$

Plug in the first, expand and simplify $$\sqrt{2} x+\left(\sqrt{2}-2\right) y=6-3 \sqrt{2}$$

in another form $$y+3=\left(\sqrt{2}+1\right) x$$ edit

look at the image below. The curve and the tangent at $$t=\pi/4$$

$$...$$

• Thanks for the clarity. Somehow your answer is different to that given in the book. I have checked with wolfram wolframalpha.com/input/… and wolframalpha.com/input/… Jan 31, 2021 at 20:28
• @user9106985 $y-\frac{1}{\sqrt{2}}+1=\frac{x-\sqrt{2}}{\sqrt{2}-1}$ is not tangent to the curve Jan 31, 2021 at 20:46
• Ok... the 2nd link I gave you is the answer I see in the book...so I guess it is a typo in the book. How would you explicitly write your answer in the Cartesian form as in my post? Jan 31, 2021 at 20:50
• And did you reproduce that plot in wolfram? Thanks in advance. Jan 31, 2021 at 20:51
• @user9106985 $$y+3=\left(\sqrt{2}+1\right) x$$ it is even simpler in this form Jan 31, 2021 at 20:58


text:
spots(ChainComplex) -- which spots does the given chain complex has a module.

Synopsis

• Function: spots
• Usage:
s = spots L
• Inputs:
• L, ,
• Outputs:

Description

Returns a list of all the spots where the given chain complex has a module.


text:
WIKISKY.ORG
 Home Getting Started To Survive in the Universe News@Sky Astro Photo The Collection Forum Blog New! FAQ Press Login

# NGC 4298 (Abolesco glacies)

Contents

### Images

DSS Images   Other Images

### Related articles

 A Virgo high-resolution Hα kinematical survey - II. The AtlasA catalogue of ionized gas velocity fields for a sample of 30 spiral andirregular galaxies of the Virgo cluster has been obtained by using 3Doptical data. The aim of this survey is to study the influence ofhigh-density environments on the gaseous kinematics of local clustergalaxies. Observations of the Hα line by means of Fabry-Perotinterferometry have been performed at the Canada-France-HawaiiTelescope, European Southern Observatory 3.6-m telescope, Observatoirede Haute-Provence 1.93-m telescope and Observatoire du montMégantic telescope at angular and spectral samplings from 0.4 to1.6arcsec and 7 to 16kms-1. A recently developed, automaticand adaptive spatial binning technique is used to reach a nearlyconstant signal-to-noise ratio (S/N) over the whole field of view,allowing us to keep a high spatial resolution in high-S/N regions andextend the detection of signal in low-S/N regions. This paper is part ofa series and presents the integrated emission-line and velocity maps ofthe galaxies. Both Hα morphologies and kinematics exhibit signs ofperturbations in the form of, for example, external filaments, inner andnuclear spiral- and ring-like structures, inner kinematical twists,kinematical decoupling of a nuclear spiral, streaming motions alongspiral arms and misalignment between kinematical and photometricorientation axes. A Comparison of Hα and Stellar Scale Lengths in Virgo and Field SpiralsThe scale lengths of the old stars and ionized gas distributions arecompared for similar samples of Virgo Cluster members and field spiralgalaxies via Hα and broad R-band surface photometry. While theR-band and Hα scale lengths are, on average, comparable for thecombined sample, we find significant differences between the field andcluster samples. While the Hα scale lengths of the field galaxiesare a factor of 1.14+/-0.07 longer, on average, than their R-band scalelengths, the Hα scale lengths of Virgo Cluster members are, onaverage, 20% smaller than their R-band scale lengths. Furthermore, inVirgo, the scale length ratios are correlated with the size of thestar-forming disk: galaxies with smaller overall Hα extents alsoshow steeper radial falloff of star formation activity. At the sametime, we find no strong trends in scale length ratio as a function ofother galaxy properties, including galaxy luminosity, inclination,morphological type, central R-band light concentration, or bar type. Ourresults for Hα emission are similar to other results for dustemission, suggesting that Hα and dust have similar distributions.The environmental dependence of the Hα scale length placesadditional constraints on the evolutionary process(es) that cause gasdepletion and a suppression of the star formation rate in clusters ofgalaxies. Ultraluminous X-Ray Sources in Nearby Galaxies from ROSAT High Resolution Imager Observations I. Data AnalysisX-ray observations have revealed in other galaxies a class ofextranuclear X-ray point sources with X-ray luminosities of1039-1041 ergs s-1, exceeding theEddington luminosity for stellar mass X-ray binaries. Theseultraluminous X-ray sources (ULXs) may be powered by intermediate-massblack holes of a few thousand Msolar or stellar mass blackholes with special radiation processes. In this paper, we present asurvey of ULXs in 313 nearby galaxies withD25>1' within 40 Mpc with 467 ROSAT HighResolution Imager (HRI) archival observations. The HRI observations arereduced with uniform procedures, refined by simulations that help definethe point source detection algorithm employed in this survey. A sampleof 562 extragalactic X-ray point sources withLX=1038-1043 ergs s-1 isextracted from 173 survey galaxies, including 106 ULX candidates withinthe D25 isophotes of 63 galaxies and 110 ULX candidatesbetween 1D25 and 2D25 of 64 galaxies, from which aclean sample of 109 ULXs is constructed to minimize the contaminationfrom foreground or background objects. The strong connection betweenULXs and star formation is confirmed based on the striking preference ofULXs to occur in late-type galaxies, especially in star-forming regionssuch as spiral arms. ULXs are variable on timescales over days to yearsand exhibit a variety of long term variability patterns. Theidentifications of ULXs in the clean sample show some ULXs identified assupernovae (remnants), H II regions/nebulae, or young massive stars instar-forming regions, and a few other ULXs identified as old globularclusters. In a subsequent paper, the statistic properties of the surveywill be studied to calculate the occurrence frequencies and luminosityfunctions for ULXs in different types of galaxies to shed light on thenature of these enigmatic sources. The Structural Properties of Isolated Galaxies, Spiral-Spiral Pairs, and Mergers: The Robustness of Galaxy Morphology during Secular EvolutionWe present a structural analysis of nearby galaxies in spiral-spiralpairs in optical BVRI bands and compare them with the structures ofisolated spiral galaxies and galaxies in ongoing mergers. We use thesecomparisons to determine how galaxy structure changes during galaxyinteractions and mergers. We analyze light concentration (C), asymmetry(A), and clumpiness (S) parameters, and use the projections of CASparameter space to compare these samples. We find that the CASparameters of paired galaxies are correlated with the projectedseparations of the pair. For the widest and closest pairs, the CASparameters tend to be similar to those of isolated and ongoing majormergers (e.g., ultraluminous infrared galaxies), respectively. Ourresults imply that galaxy morphology is a robust property that onlychanges significantly during a strong interaction or major merger. Thetypical timescale for this change in our paired sample, based ondynamical friction arguments, is short, τ~0.1-0.5 Gyr. We findaverage enhancement factors for the spiral-pair asymmetries andclumpiness values of ~2.2 and 1.5. The S parameter, which is related tostar formation (SF) activity, has a moderate level of enhancement,suggesting that this activity in modern spirals depends more on internalprocesses than on external conditions. We further test the statisticalcriterion for picking up interacting galaxies in an automated way byusing the A-S projection plane. The diversity of our spiral-pair samplein the CAS space suggests that structural/SF/morphological properties ofinteracting galaxies change abruptly only when the interaction becomesvery strong and the criteria given previously by Conselice for findinggalaxies involved in major mergers are effective. Completing H I observations of galaxies in the Virgo clusterHigh sensitivity (rms noise  0.5 mJy) 21-cm H I line observationswere made of 33 galaxies in the Virgo cluster, using the refurbishedArecibo telescope, which resulted in the detection of 12 objects. Thesedata, combined with the measurements available from the literature,provide the first set of H I data that is complete for all 355 late-type(Sa-Im-BCD) galaxies in the Virgo cluster with mp ≤ 18.0mag. The Virgo cluster H I mass function (HIMF) that was derived forthis optically selected galaxy sample is in agreement with the HIMFderived for the Virgo cluster from the blind HIJASS H I survey and isinconsistent with the Field HIMF. This indicates that both in this richcluster and in the general field, neutral hydrogen is primarilyassociated with late-type galaxies, with marginal contributions fromearly-type galaxies and isolated H I clouds. The inconsistency betweenthe cluster and the field HIMF derives primarily from the difference inthe optical luminosity function of late-type galaxies in the twoenvironments, combined with the HI deficiency that is known to occur ingalaxies in rich clusters.Tables \ref{t1, \ref{sample_dat} and Appendix A are only available inelectronic form at http://www.edpsciences.org On the alignment between binary spiral galaxiesWe show some significance against the null hypothesis of randominteractions of binary spiral galaxies, and in favour of the alternativethat more interactions than expected occur for axes either nearlyparallel (spins being parallel or anti-parallel) or nearly orthogonal.We discuss this in the context of similar prior studies, using adifferent statistical focus in such a way that we are able toincorporate additional data. A multibeam HI survey of the Virgo cluster - two isolated HI clouds?We have carried out a fully sampled large area (4°× 8°)21-cm HI line survey of part of the Virgo cluster using the Jodrell Bankmultibeam instrument. The survey has a sensitivity some three timesbetter than the standard HIJASS (HI Jodrell All Sky Survey) and HIPASS(HI Parkes All Sky Survey) surveys. We detect 31 galaxies, 27 of whichare well-known cluster members. The four new detections have beenconfirmed in the HIPASS data and by follow-up Jodrell Bank pointedobservations. One object lies behind M86, but the other three have noobvious optical counterparts upon inspection of the digital sky surveyfields. These three objects were mapped at Arecibo with a smaller3.6-arcmin half power beam width (HPBW) and a four times bettersensitivity than the Jodrell Bank data, which allow an improveddetermination of the dimensions and location of two of the objects, butsurprisingly failed to detect the third. The two objects are resolved bythe Arecibo beam, giving them a size far larger than any optical imagesin the nearby field. To our mass limit of 5 ×107(Δv/50 km s-1) Msolar andcolumn density limit of 3 × 1018(Δv/50 kms-1) atom cm-2, these new detections representonly about 2 per cent of the cluster atomic hydrogen mass. Ourobservations indicate that the HI mass function of the cluster turnsdown at the low-mass end, making it very different to the field galaxyHI mass function. This is quite different to the Virgo cluster opticalluminosity function, which is much steeper than that in the generalfield. Many of the sample galaxies are relatively gas-poor compared withHI selected samples of field galaxies, confirming the anaemic spirals'view of Virgo cluster late-type galaxies. The velocity distribution ofthe HI detected galaxies is also very different to that of the clusteras a whole. There are relatively more high-velocity galaxies in the HIsample, suggesting that they form part of a currently infallingpopulation. The HI sample with optical identifications has a minimum HIcolumn density cut-off more than an order of magnitude above thatexpected from the sensitivity of the survey. This observed columndensity is above the normally expected level for star formation tooccur. The two detections with no optical counterparts have very muchlower column densities than that of the rest of the sample, below thestar formation threshold. Hα Morphologies and Environmental Effects in Virgo Cluster Spiral GalaxiesWe describe the various Hα morphologies of Virgo Cluster andisolated spiral galaxies and associate the Hα morphologies withthe types of environmental interactions that have altered the clustergalaxies. The spatial distributions of Hα and R-band emission areused to divide the star formation morphologies of the 52 Virgo Clusterspiral galaxies into several categories: normal (37%), anemic (6%),enhanced (6%), and (spatially) truncated (52%). Truncated galaxies arefurther subdivided on the basis of their inner star formation rates intotruncated/normal (37%), truncated/compact (6%), truncated/anemic (8%),and truncated/enhanced (2%). The fraction of anemic galaxies isrelatively small (6%-13%) in both environments, suggesting thatstarvation is not a major factor in the reduced star formation rates ofVirgo spiral galaxies. The majority of Virgo spiral galaxies have theirHα disks truncated (52%), whereas truncated Hα disks arerarer in isolated galaxies (12%). Most of the Hα-truncatedgalaxies have relatively undisturbed stellar disks and normal toslightly enhanced inner disk star formation rates, suggesting thatintracluster medium-interstellar medium (ICM-ISM) stripping is the mainmechanism causing the reduced star formation rates of Virgo spiralgalaxies. Several of the truncated galaxies are peculiar, with enhancedcentral star formation rates, disturbed stellar disks, and barlikedistributions of luminous H II complexes inside the central 1 kpc but nostar formation beyond, suggesting that recent tidal interactions orminor mergers have also influenced their morphology. Two highly inclinedHα-truncated spiral galaxies have numerous extraplanar H IIregions and are likely in an active phase of ICM-ISM stripping. Severalspiral galaxies have one-sided Hα enhancements at the outer edgeof their truncated Hα disks, suggesting modest local enhancementsin their star formation rates due to ICM-ISM interactions. Low-velocitytidal interactions and perhaps outer cluster H I accretion seem to bethe triggers for enhanced global star formation in four Virgo galaxies.These results indicate that most Virgo spiral galaxies experienceICM-ISM stripping, many experience significant tidal effects, and manyexperience both. Radio and Far-Infrared Emission as Tracers of Star Formation and Active Galactic Nuclei in Nearby Cluster GalaxiesWe have studied the radio and far-infrared (FIR) emission from 114galaxies in the seven nearest clusters (<100 Mpc) with prominentX-ray emission to investigate the impact of the cluster environment onthe star formation and active galactic nucleus (AGN) activity in themember galaxies. The X-ray selection criterion is adopted to focus onthe most massive and dynamically relaxed clusters. A large majority ofcluster galaxies show an excess in radio emission over that predictedfrom the radio-FIR correlation, the fraction of sources with radioexcess increases toward cluster cores, and the radial gradient in theFIR/radio flux ratio is a result of radio enhancement. Of theradio-excess sources, 70% are early-type galaxies, and the same fractionhost an AGN. The galaxy density drops by a factor of 10 from thecomposite cluster center out to 1.5 Mpc, yet galaxies show no change inFIR properties over this region and show no indication of masssegregation. We have examined in detail the physical mechanisms thatmight impact the FIR and radio emission of cluster galaxies. Whilecollisional heating of dust may be important for galaxies in clustercenters, it appears to have a negligible effect on the observed FIRemission for our sample galaxies. The correlations between radio and FIRluminosity and radius could be explained by magnetic compression fromthermal intracluster medium pressure. We also find that simple delayedharassment cannot fully account for the observed radio, FIR, and mid-IRproperties of cluster galaxies. Properties of isolated disk galaxiesWe present a new sample of northern isolated galaxies, which are definedby the physical criterion that they were not affected by other galaxiesin their evolution during the last few Gyr. To find them we used thelogarithmic ratio, f, between inner and tidal forces acting upon thecandidate galaxy by a possible perturber. The analysis of thedistribution of the f-values for the galaxies in the Coma cluster leadus to adopt the criterion f ≤ -4.5 for isolated galaxies. Thecandidates were chosen from the CfA catalog of galaxies within thevolume defined by cz ≤5000 km s-1, galactic latitudehigher than 40o and declination ≥-2.5o. Theselection of the sample, based on redshift values (when available),magnitudes and sizes of the candidate galaxies and possible perturberspresent in the same field is discussed. The final list of selectedisolated galaxies includes 203 objects from the initial 1706. The listcontains only truly isolated galaxies in the sense defined, but it is byno means complete, since all the galaxies with possible companions underthe f-criterion but with unknown redshift were discarded. We alsoselected a sample of perturbed galaxies comprised of all the diskgalaxies from the initial list with companions (with known redshift)satisfying f ≥ -2 and \Delta(cz) ≤500 km s-1; a totalof 130 objects. The statistical comparison of both samples showssignificant differences in morphology, sizes, masses, luminosities andcolor indices. Confirming previous results, we found that late spiral,Sc-type galaxies are, in particular, more frequent among isolatedgalaxies, whereas Lenticular galaxies are more abundant among perturbedgalaxies. Isolated systems appear to be smaller, less luminous and bluerthan interacting objects. We also found that bars are twice as frequentamong perturbed galaxies compared to isolated galaxies, in particularfor early Spirals and Lenticulars. The perturbed galaxies have higherLFIR/LB and Mmol/LB ratios,but the atomic gas content is similar for the two samples. The analysisof the luminosity-size and mass-luminosity relations shows similartrends for both families, the main difference being the almost totalabsence of big, bright and massive galaxies among the family of isolatedsystems, together with the almost total absence of small, faint and lowmass galaxies among the perturbed systems. All these aspects indicatethat the evolution induced by interactions with neighbors would proceedfrom late, small, faint and low mass Spirals to earlier, bigger, moreluminous and more massive spiral and lenticular galaxies, producing atthe same time a larger fraction of barred galaxies but preserving thesame relations between global parameters. The properties we found forour sample of isolated galaxies appear similar to those of high redshiftgalaxies, suggesting that the present-day isolated galaxies could bequietly evolved, unused building blocks surviving in low densityenvironments.Tables \ref{t1} and \ref{t2} are only available in electronic form athttp://www.edpsciences.org Radio continuum spectra of galaxies in the Virgo cluster regionNew radio continuum observations of galaxies in the Virgo cluster regionat 4.85, 8.6, and 10.55 GHz are presented. These observations arecombined with existing measurements at 1.4 and 0.325 GHz. The sampleincludes 81 galaxies where spectra with more than two frequencies couldbe derived. Galaxies that show a radio-FIR excess exhibit centralactivity (HII, LINER, AGN). The four Virgo galaxies with the highestabsolute radio excess are found within 2° of the centerof the cluster. Galaxies showing flat radio spectra also host activecenters. There is no clear trend between the spectral index and thegalaxy's distance to the cluster center.Figure 3 is only available in electronic form athttp://www.edpsciences.orgTable 3 is only available in electronic form at the CDS via anonymousftp to cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/418/1 Spectrophotometry of galaxies in the Virgo cluster. II. The dataDrift-scan mode (3600-6800 Å) spectra with 500 Minor-axis velocity gradients in disk galaxiesWe present the ionized-gas kinematics and photometry of a sample of 4spiral galaxies which are characterized by a zero-velocity plateau alongthe major axis and a velocity gradient along the minor axis,respectively. By combining these new kinematical data with thoseavailable in the literature for the ionized-gas component of the S0s andspirals listed in the Revised Shapley-Ames Catalog of Bright Galaxies werealized that about 50% of unbarred galaxies show a remarkable gasvelocity gradient along the optical minor axis. This fraction rises toabout 60% if we include unbarred galaxies with an irregular velocityprofile along the minor axis. This phenomenon is observed all along theHubble sequence of disk galaxies, and it is particularly frequent inearly-type spirals. Since minor-axis velocity gradients are unexpectedif the gas is moving onto circular orbits in a disk coplanar to thestellar one, we conclude that non-circular and off-plane gas motions arenot rare in the inner regions of disk galaxies.Based on observations carried out at the European Southern Observatoryin La Silla (Chile) (ESO 69.B-0706 and 70.B-0338), with the MultipleMirror Telescope which is a joint facility of the SmithsonianInstitution and the University of Arizona, and with the ItalianTelescopio Nazionale Galileo (AOT-5, 3-18) at the Observatorio del Roquede los Muchachos in La Palma (Spain).Table 1 is only available in electronic form athttp://www.edpsciences.org. Table 5 is only available in electronic format the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) orvia http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/416/507 The PDS versus Markarian starburst galaxies: comparing strong and weak IRAS emitter at 12 and 25 μm in the nearby UniverseThe characteristics of the starburst galaxies from the Pico dos Diassurvey (PDS) are compared with those of the nearby ultraviolet (UV)bright Markarian starburst galaxies, having the same limit in redshift(vh < 7500 km s-1) and absolute B magnitude(MB < -18). An important difference is found: theMarkarian galaxies are generally undetected at 12 and 25 μm in IRAS.This is consistent with the UV excess shown by these galaxies andsuggests that the youngest star-forming regions dominating thesegalaxies are relatively free of dust.The far-infrared selection criteria for the PDS are shown to introduce astrong bias towards massive (luminous) and large size late-type spiralgalaxies. This is contrary to the Markarian galaxies, which are found tobe remarkably rich in smaller size early-type galaxies. These resultssuggest that only late-type spirals with a large and massive disc arestrong emitters at 12 and 25 μm in IRAS in the nearby Universe.The Markarian and PDS starburst galaxies are shown to share the sameenvironment. This rules out an explanation of the differences observedin terms of external parameters. These differences may be explained byassuming two different levels of evolution, the Markarian being lessevolved than the PDS galaxies. This interpretation is fully consistentwith the disc formation hypothesis proposed by Coziol et al. to explainthe special properties of the Markarian SBNG. The IRAS Revised Bright Galaxy SampleIRAS flux densities, redshifts, and infrared luminosities are reportedfor all sources identified in the IRAS Revised Bright Galaxy Sample(RBGS), a complete flux-limited survey of all extragalactic objects withtotal 60 μm flux density greater than 5.24 Jy, covering the entiresky surveyed by IRAS at Galactic latitudes |b|>5°. The RBGS includes629 objects, with median and mean sample redshifts of 0.0082 and 0.0126,respectively, and a maximum redshift of 0.0876. The RBGS supersedes theprevious two-part IRAS Bright Galaxy Samples(BGS1+BGS2), which were compiled before the final(Pass 3) calibration of the IRAS Level 1 Archive in 1990 May. The RBGSalso makes use of more accurate and consistent automated methods tomeasure the flux of objects with extended emission. The RBGS contains 39objects that were not present in the BGS1+BGS2,and 28 objects from the BGS1+BGS2 have beendropped from RBGS because their revised 60 μm flux densities are notgreater than 5.24 Jy. Comparison of revised flux measurements forsources in both surveys shows that most flux differences are in therange ~5%-25%, although some faint sources at 12 and 25 μm differ byas much as a factor of 2. Basic properties of the RBGS sources aresummarized, including estimated total infrared luminosities, as well asupdates to cross identifications with sources from optical galaxycatalogs established using the NASA/IPAC Extragalactic Database. Inaddition, an atlas of images from the Digitized Sky Survey with overlaysof the IRAS position uncertainty ellipse and annotated scale bars isprovided for ease in visualizing the optical morphology in context withthe angular and metric size of each object. The revised bolometricinfrared luminosity function, φ(Lir), forinfrared-bright galaxies in the local universe remains best fit by adouble power law, φ(L)~Lα, withα=-0.6(+/-0.1) and α=-2.2(+/-0.1) below and above thecharacteristic'' infrared luminosityL*ir~1010.5Lsolar,respectively. A companion paper provides IRAS High Resolution (HIRES)processing of over 100 RBGS sources where improved spatial resolutionoften provides better IRAS source positions or allows for deconvolutionof close galaxy pairs. A search for Low Surface Brightness galaxies in the near-infrared. I. Selection of the sampleA sample of about 3800 Low Surface Brightness (LSB) galaxies wasselected using the all-sky near-infrared (J, H and Ks-band)2MASS survey. The selected objects have a mean central surfacebrightness within a 5'' radius around their centre fainter than 18 magarcsec-2 in the Ks band, making them the lowestsurface brightness galaxies detected by 2MASS. A description is given ofthe relevant properties of the 2MASS survey and the LSB galaxy selectionprocedure, as well as of basic photometric properties of the selectedobjects. The latter properties are compared to those of other samples ofgalaxies, of both LSBs and classical'' high surface brightness (HSB)objects, which were selected in the optical. The 2MASS LSBs have aBT_c-KT colour which is on average 0.9 mag bluerthan that of HSBs from the NGC. The 2MASS sample does not appear tocontain a significant population of red objects.All tables and Figs. 2a-c are only available in electronic form athttp://www.edpsciences.org A new catalogue of ISM content of normal galaxiesWe have compiled a catalogue of the gas content for a sample of 1916galaxies, considered to be a fair representation of normality''. Thedefinition of a normal'' galaxy adopted in this work implies that wehave purposely excluded from the catalogue galaxies having distortedmorphology (such as interaction bridges, tails or lopsidedness) and/orany signature of peculiar kinematics (such as polar rings,counterrotating disks or other decoupled components). In contrast, wehave included systems hosting active galactic nuclei (AGN) in thecatalogue. This catalogue revises previous compendia on the ISM contentof galaxies published by \citet{bregman} and \citet{casoli}, andcompiles data available in the literature from several small samples ofgalaxies. Masses for warm dust, atomic and molecular gas, as well asX-ray luminosities have been converted to a uniform distance scale takenfrom the Catalogue of Principal Galaxies (PGC). We have used twodifferent normalization factors to explore the variation of the gascontent along the Hubble sequence: the blue luminosity (LB)and the square of linear diameter (D225). Ourcatalogue significantly improves the statistics of previous referencecatalogues and can be used in future studies to define a template ISMcontent for normal'' galaxies along the Hubble sequence. The cataloguecan be accessed on-line and is also available at the Centre desDonnées Stellaires (CDS).The catalogue is available in electronic form athttp://dipastro.pd.astro.it/galletta/ismcat and at the CDS via anonymousftp to\ cdsarc.u-strasbg.fr (130.79.128.5) or via\http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/405/5 12CO(J= 2->1) and CO(J= 3->2) observations of Virgo Cluster spiral galaxies with the KOSMA telescope: Global propertiesWe present 12CO (J= 2->1) and CO (J= 3->2) observationsof quiescent Virgo Cluster spiral galaxies with the KOSMA 3 m submmtelescope. The beam sizes of 80\arcsec at 345 GHz and 120 arcsec at 230GHz are well suited for the investigation of global properties of VirgoCluster galaxies. The observed sample was selected based on previous12CO (J=1->0) detections by Stark et al. (\cite{Stark86}),performed with the AT&T Bell Laboratory 7 m telescope (beam size ~100\arcsec). We were able to detect 18 spiral galaxies in12CO (2->1) and 16 in 12CO (3->2). Beammatched observations of the lowest three 12CO transitionsallow us to compare our results with previous high spatial resolutionstudies of (moderate) starburst galaxies and galactic core regions. Wediscuss the global excitation conditions of the ISM in these quiescentspiral galaxies. The resulting CO (3-2)/(1-0) integrated line ratiosvary over a relatively narrow range of values from 0.35 to 0.14 (on a Kkm s-1-scale) with increasing CO (2-1)/(1-0) ratio (from 0.5to 1.1). The line ratios between the three lowest rotational transitionsof CO cannot be fitted by any radiative transfer model with a singlesource component. A two-component model, assuming a warm, dense nuclearand a cold, less dense disc component allows us to fit the observed lineratios for most of the galaxies individually by selecting suitableparameters. The two-component model, however, fails to explain theobserved correlation of the line ratios. This is due to a variation ofthe relative filling factor of the warm gas alone, assuming a typicalset of parameters for the two components common for all galaxies. A Search for Active Galactic Nuclei in Sc Galaxies with H II SpectraWe have searched for nuclear radio emission from a statisticallycomplete sample of 40 Sc galaxies within 30 Mpc that are opticallyclassified as star-forming objects, in order to determine whether weakactive galactic nuclei might be present. Only three nuclear radiosources were detected, in NGC 864, NGC 4123, and NGC 4535. Thesegalaxies have peak 6 cm radio powers of ~1020 WHz-1 at arcsecond resolution, while upper limits of thenondetected galaxies typically range from 1018.4 to1020 W Hz-1. The three nuclear radio sources areall resolved and appear to have diffuse morphologies, with linear sizesof ~300 pc. This strongly indicates that circumnuclear star formationhas been detected in these three H II galaxies. Comparisons withprevious 20 cm Very Large Array (VLA) results for the detected galaxiesshow that the extended nuclear radio emission has a flat spectrum in twoobjects and is almost certainly generated by thermal emission from gasionized by young stars in the centers of those galaxies. The 6 cm radiopowers are comparable to predictions for thermal emission that are basedon the nuclear Hα luminosities and imply nuclear star formationrates of 0.08-0.8 Msolar yr-1, while thelow-resolution NRAO VLA Sky Survey implies galaxy-wide star formationrates of 0.3-1.0 Msolar yr-1 in stars above 5Msolar. In a few of the undetected galaxies, the upper limitsto the radio power are lower than predicted from the Hαluminosity, possibly because of overresolution of central star-formingregions. Although the presence of active nuclei powered by massive blackholes cannot be definitively ruled out, the present results suggest thatthey are likely to be rare in these late-type galaxies with H IIspectra. The Three-dimensional Structure of the Virgo Cluster Region from Tully-Fisher and H I DataThe distances and H I contents of 161 spiral galaxies in the region ofthe Virgo cluster are used to gain insight into the complicatedstructure of this galaxy system. Special attention has been paid to theinvestigation of the suggestion presented in an earlier work that someperipheral Virgo groups may contain strongly gas-deficient spiralgalaxies. The three-dimensional galaxy distribution has been inferredfrom quality distance estimates obtained by averaging distance modulibased on the Tully-Fisher relationship taken from eight published datasets previously homogenized, resulting in a relation with a dispersionof 0.41 mag. Previous findings that the spiral distribution issubstantially more elongated along the line of sight than in the planeof the sky are confirmed by the current data. In addition, an importanteast-west disparity in this effect has been detected. The overallwidth-to-depth ratio of the Virgo cluster region is about 1:4, with themost distant objects concentrated in the western half. The filamentarystructure of the spiral population and its orientation are alsoreflected by the H I-deficient objects alone. The H I deficiency patternshows a central enhancement extending from ~16 to 22 Mpc inline-of-sight distance; most of this enhancement arises from galaxiesthat belong to the Virgo cluster proper. However, significant gasdeficiencies are also detected outside the main body of the cluster in aprobable group of galaxies at line-of-sight distances ~25-30 Mpc, lyingin the region dominated by the southern edge of the M49 subcluster andclouds W' and W, as well as in various foreground galaxies. In the Virgoregion, the H I content of the galaxies then is not a straightforwardindicator of cluster membership. Bar Galaxies and Their EnvironmentsThe prints of the Palomar Sky Survey, luminosity classifications, andradial velocities were used to assign all northern Shapley-Ames galaxiesto either (1) field, (2) group, or (3) cluster environments. Thisinformation for 930 galaxies shows no evidence for a dependence of barfrequency on galaxy environment. This suggests that the formation of abar in a disk galaxy is mainly determined by the properties of theparent galaxy, rather than by the characteristics of its environment. The UZC-SSRS2 Group CatalogWe apply a friends-of-friends algorithm to the combined Updated ZwickyCatalog and Southern Sky Redshift Survey to construct a catalog of 1168groups of galaxies; 411 of these groups have five or more members withinthe redshift survey. The group catalog covers 4.69 sr, and all groupsexceed the number density contrast threshold, δρ/ρ=80. Wedemonstrate that the groups catalog is homogeneous across the twounderlying redshift surveys; the catalog of groups and their membersthus provides a basis for other statistical studies of the large-scaledistribution of groups and their physical properties. The medianphysical properties of the groups are similar to those for groupsderived from independent surveys, including the ESO Key Programme andthe Las Campanas Redshift Survey. We include tables of groups and theirmembers. Hα surface photometry of galaxies in the Virgo cluster. IV. The current star formation in nearby clusters of galaxiesHα +[NII] imaging observations of 369 late-type (spiral) galaxiesin the Virgo cluster and in the Coma/A1367 supercluster are analyzed,covering 3 rich nearby clusters (A1367, Coma and Virgo) and nearlyisolated galaxies in the Great-Wall. They constitute an opticallyselected sample (mp<16.0) observed with ~ 60 %completeness. These observations provide us with the current(T<107 yrs) star formation properties of galaxies that westudy as a function of the clustercentric projected distances (Theta ).The expected decrease of the star formation rate (SFR), as traced by theHα EW, with decreasing Theta is found only when galaxies brighterthan Mp ~ -19.5 are considered. Fainter objects show no orreverse trends. We also include in our analysis Near Infrared data,providing information on the old (T>109 yrs) stars. Puttogether, the young and the old stellar indicators give the ratio ofcurrently formed stars over the stars formed in the past, orbirthrate'' parameter b. For the considered galaxies we also determinethe global gas content'' combining HI with CO observations. We definethe gas deficiency'' parameter as the logarithmic difference betweenthe gas content of isolated galaxies of a given Hubble type and themeasured gas content. For the isolated objects we find that b decreaseswith increasing NIR luminosity. In other words less massive galaxies arecurrently forming stars at a higher rate than their giant counterpartswhich experienced most of their star formation activity at earliercosmological epochs. The gas-deficient objects, primarily members of theVirgo cluster, have a birthrate significantly lower than the isolatedobjects with normal gas content and of similar NIR luminosity. Thisindicates that the current star formation is regulated by the gaseouscontent of spirals. Whatever mechanism (most plausibly ram-pressurestripping) is responsible for the pattern of gas deficiency observed inspiral galaxies members of rich clusters, it also produces the observedquenching of the current star formation. A significant fraction of gashealthy'' (i.e. with a gas deficiency parameter less than 0.4) andcurrently star forming galaxies is unexpectedly found projected near thecenter of the Virgo cluster. Their average Tully-Fisher distance isfound approximately one magnitude further away (muo = 31.77)than the distance of their gas-deficient counterparts (muo =30.85), suggesting that the gas healthy objects belong to a cloudprojected onto the cluster center, but in fact lying a few Mpc behindVirgo, thus unaffected by the dense IGM of the cluster. Based onobservations taken at the Observatorio Astronómico Nacional(Mexico), the OHP (France), Calar Alto and NOT (Spain) observatories.Table \ref{tab4} is only available in electronic form athttp://www.edpsciences.org Hα surface photometry of galaxies in the Virgo cluster. II. Observations with the OHP and Calar Alto 1.2 m telescopesWe present Hα line imaging observations of 122 galaxies obtainedwith the 1.20 m telescopes of the Observatoire de Haute Provence (OHP)and of Calar Alto. The observed galaxies are mostly Virgo clustermembers (95), along with 10 objects in the Coma/A1367 supercluster, 6 inthe clusters A2197 and A2199, and 11 nearby galaxies taken as fillers.Hα +[NII] fluxes and equivalent widths, as well as images of allthe detected targets, are presented. Based on observations taken at theObservatoire de Haute Provence (OHP) (France), operated by the FrenchCNRS, and Calar Alto Observatory (Spain), operated by theMax-Planck-Institut für Astronomie (Heidelberg) jointly with theSpanish National Commission for Astronomy. Figure 1 is only available inelectronic form at http://www.edpsciences.org Star formation and dust extinction in nearby star-forming and starburst galaxiesWe study the star formation rate and dust extinction properties of asample of nearby star-forming galaxies as derived from Hα and UV (~ 2000 Å) observations and we compare them to those of a sample ofstarburst galaxies. The dust extinction in Hα is estimated fromthe Balmer decrement and the extinction in UV using the FIR to UV fluxratio or the attenuation law for starburst galaxies of Calzetti et al.(\cite{calzetti5}). The Hα and UV emissions are stronglycorrelated with a very low scatter for the star-forming objects and witha much higher scatter for the starburst galaxies. The Hα to UVflux ratio is found to be larger by a factor ~ 2 for the starburstgalaxies. We compare both samples with a purely UV selected sample ofgalaxies and we conclude that the mean Hα and UV properties ofnearby star-forming galaxies are more representative of UV-selectedgalaxies than starburst galaxies. We emphasize that the Hα to UVflux ratio is strongly dependent on the dust extinction: the positivecorrelation found between FHα/FUV andFFIR/FUV vanishes when the Hα and UV fluxare corrected for dust extinction. The Hα to UV flux ratiosconverted into star formation rate and combined with the Balmerdecrement measurements are tentatively used to estimate the dustextinction in UV. An Atlas of Hα and R Images and Radial Profiles of 63 Bright Virgo Cluster Spiral GalaxiesNarrowband Hα and broadband R images and radial profiles arepresented for 63 bright spiral galaxies in the Virgo Cluster. The sampleis complete for Sb-Scd galaxies with B0T<=12and inclination <=75°. Isophotal radii, disk scale lengths,concentration parameters, and integrated fluxes are derived for thesample galaxies. The Frequency of Active and Quiescent Galaxies with Companions: Implications for the Feeding of the NucleusWe analyze the idea that nuclear activity, either active galactic nuclei(AGNs) or star formation, can be triggered by interactions by studyingthe percentage of active, H II, and quiescent galaxies with companions.Our sample was selected from the Palomar survey and avoids selectionbiases faced by previous studies. This sample was split into fivedifferent groups, Seyfert galaxies, LINERs, transition galaxies, H IIgalaxies, and absorption-line galaxies. The comparison between the localgalaxy density distributions of the different groups showed that in mostcases there is no statistically significant difference among galaxies ofdifferent activity types, with the exception that absorption-linegalaxies are seen in higher density environments, since most of them arein the Virgo Cluster. The comparison of the percentage of galaxies withnearby companions showed that there is a higher percentage of LINERs,transition galaxies, and absorption-line galaxies with companions thanSeyfert and H II galaxies. However, we find that when we consider onlygalaxies of similar morphological types (elliptical or spiral), there isno difference in the percentage of galaxies with companions amongdifferent activity types, indicating that the former result was due tothe morphology-density effect. In addition, only small differences arefound when we consider galaxies with similar Hα luminosities. Thecomparison between H II galaxies of different Hα luminositiesshows that there is a significantly higher percentage of galaxies withcompanions among H II galaxies with L(Hα)>1039 ergss-1 than among those with L(Hα)<=1039ergs s-1, indicating that interactions increase the amount ofcircumnuclear star formation, in agreement with previous results. Thefact that we find that galaxies of different activity types have thesame percentage of companions suggests that interactions betweengalaxies is not a necessary condition to trigger the nuclear activity inAGNs. We compare our results with previous ones and discuss theirimplications. BVRI surface photometry of (S+S) binary galaxies I. The dataWe present multicolour broad band (BVRI) photometry for a sample of 33spiral-spiral (S+S) binary galaxies drawn from the KarachentsevCatalogue of Isolated Pairs of Galaxies (KPG). The data is part of ajoint observational programme devoted to systematic photometric study ofone of the most complete and homogeneous pair samples available in theliterature. We present azimuthally averaged colour and surfacebrightness profiles, colour index (B-I) maps, B band and sharp/filteredB band images as well as integrated magnitudes, magnitudes at differentcircular apertures and integrated colours for each pair. Internal andexternal data comparisons show consistency within the estimated errors.Two thirds of the sample have total aperture parameters homogeneouslyderived for the first time. After reevaluating morphology for all thepairs, we find a change in Hubble type for 24 galaxies compared to theoriginal POSS classifications. More than half of our pairs showmorphological concordance which could explain, in part, the strongcorrelation in the (B-V) colour indices (Holmberg Effect) between paircomponents. We find a tendency for barred galaxies to show grand designmorphologies and flat colour profiles. The measurements will be used ina series of forthcoming papers where we try to identify and isolate themain structural and photometric properties of disk galaxies at differentstages of interaction. Table A.1 is only available in electronic form atthe CDS via anonymous ftp cdsarc.u-strasbg.fr (130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/379/54 and Tables 2-4 arealso available in electronic form at the CDS. Full Fig. 5 is onlyavailable in electronic form at the http://www.edpsciences.org Based ondata obtained at the 2.1 m telescope of the Observatorio Guillermo Haroat Cananea, Sonora, México, operated by the Instituto Nacional deAstrofísica, Optica y Electrónica. Nearby Optical Galaxies: Selection of the Sample and Identification of GroupsIn this paper we describe the Nearby Optical Galaxy (NOG) sample, whichis a complete, distance-limited (cz<=6000 km s-1) andmagnitude-limited (B<=14) sample of ~7000 optical galaxies. Thesample covers 2/3 (8.27 sr) of the sky (|b|>20deg) andappears to have a good completeness in redshift (97%). We select thesample on the basis of homogenized corrected total blue magnitudes inorder to minimize systematic effects in galaxy sampling. We identify thegroups in this sample by means of both the hierarchical and thepercolation `friends-of-friends'' methods. The resulting catalogs ofloose groups appear to be similar and are among the largest catalogs ofgroups currently available. Most of the NOG galaxies (~60%) are found tobe members of galaxy pairs (~580 pairs for a total of ~15% of objects)or groups with at least three members (~500 groups for a total of ~45%of objects). About 40% of galaxies are left ungrouped (field galaxies).We illustrate the main features of the NOG galaxy distribution. Comparedto previous optical and IRAS galaxy samples, the NOG provides a densersampling of the galaxy distribution in the nearby universe. Given itslarge sky coverage, the identification of groups, and its high-densitysampling, the NOG is suited to the analysis of the galaxy density fieldof the nearby universe, especially on small scales. Luminosity versus Phase-Space-Density Relation of Galaxies RevisitedWe reexamined the correlation between the BTmagnitude and the phase-space-density parameterw=(D225vc)-1 of galaxies forthe Virgo, the Coma, the Fornax, and the Perseus clusters in an effortto better understand the physical underpinning of the fundamental plane.A tight correlation (BT=alog w+b) common to differentmorphological types of galaxies (E, S0, S) was found for the Virgo andthe Coma clusters, with a=1.87+/-0.10 and 1.33+/-0.11, respectively. Aninvestigation using only E galaxies was made for the four clusters. Theresults indicated that the empirical linear relation might be commonamong the Coma, the Fornax, and the Perseus clusters, with the VirgoCluster showing deviation. This relation, which is another way toproject the fundamental plane, has an expression insensitive to themorphology and may be suitable for treating galaxies of differentmorphological types collectively.
Submit a new article

• - No Links Found -


text:
# What is the efficiency of the new Crown Sterling semiprime factoring method?

In their press release a company called Crown Sterling describes they are working a paper that

includes four different geometric and arithmetic methods for public key (semiprime) factorization and one of the methods titled, “The Reciprocal Factoring Method” includes an analysis of reciprocal values of public keys and their embedded private keys (prime factors) found within their period decimal extensions.

According to the CEO's Instagram, the factoring method is quite simple: search for the primes in the decimal expansion of the reciprocal of the semi prime. It doesn't hold true for all primes, and I get that its not even remotely efficient enough, but while the controversy is ongoing, it would be helpful to know just how inefficient their method is.

• Without actually knowing the full details of the method it's really hard to predict a scaling / efficiency behavior. – SEJPM Sep 26 '19 at 18:03
• However, the press release cites the paper “Accurate and Infinite Prime Number Prediction from Novel Quasi-Prime Analytical Methodology.” also written by the coauthors; that paper is extremely rudimentary (apparently, the observation that primes > 3 are either 1 or 5 mod 6 is novel); that wouldn't give me a great deal of hope for any further paper from the authors... – poncho Sep 26 '19 at 19:19
• The arXiv draft also bears no resemblance to the Instagram bloviation by my reading. – Squeamish Ossifrage Sep 26 '19 at 19:31
• @Ella Rose: I don't agree that the question is primarily opinion-based (even if currently the only answer is). Crown Sterling has released falsifiable material, including this and articles. If we have a snake-oil cryptography tag, Crown Sterling's assertions are a match for it. – fgrieu Sep 27 '19 at 14:13
• Re-opened due to fgrieu's comment. Please ensure that answers stick to answering the question "What is the efficiency of the proposed factoring algorithm". – Ella Rose Sep 27 '19 at 14:37

The bozos at Clown Sterling have adapted the advanced technology of bogosort to factoring: randomly blow a candidate solution out your nose and check whether it works. For a semiprime $$n$$ chosen by modern RSA key generation methods, there are approximately $$\sqrt n$$ candidate solutions to check, which is also the expected cost of this method. For instance, with RSA-2048, it will cost approximately $$2^{1024}$$ trial boogers.

The technical term for everything about these clowns is ‘uncastrated male bovine stool’.* There is no more merit to the technical claims of Crown Sterling than there is to Treadwell Stanton DuPont. The company's method is quite simple: the CEO Robert Grant is a scam artist who presumably uses technobabble like this to defraud unwary investors, and who sank some of the money into buying the prestige of a BlackHat talk slot (which is something that BlackHat really shouldn't offer) in order to drum up more hype around their scam.

And then they act the victim in a bogus narrative of persecution by the dominant establishment when subjected to universal derision by any competent cryptographers who bother to look at the nonsense.

So, what did these scam artists claim?

• In their arXiv draft (which they proudly claim Cornell University published even though anyone can upload a draft to the arXiv with no peer review), Robert Grant and Talal Ghannam:

• (§II) made the remarkable observation that every prime above 3 is congruent to $$\pm1$$ modulo 6 (proof: $$2 \mid 6k \pm 2$$ and $$3 \mid 6k \pm 3$$; this is the newest result they recycled, several centuries old—first published no later than 1509 and first proven no later than 1603, according to Dickson pp. 7–9, but quite possibly known earlier because it's so obvious);
• (§III) reinvented the sieve of Eratosthenes several thousand years late;
• (§IV) discovered that $$x \mapsto x \bmod 9$$ is a ring homomorphism (a technique known since antiquity and more commonly called ‘casting out nines’), and figured out that if $$x \equiv 3 \pmod 9$$, $$x \equiv 6 \pmod 9$$, or $$x \equiv 9 \pmod 9$$, then $$x$$ can't be prime (proof: duh);
• (§VI) deluded themselves into thinking if $$pq \equiv 1 \pmod{10}$$ then either $$\{p \bmod 10, q \bmod 10\} = \{3,7\}$$ or $$p \equiv q \equiv 1 \pmod{10}$$ (counterexample: $$19\cdot 29 = 551$$; perhaps they used ‘proof by trying numbers until I got bored’); and
• (Appendix) proved that $$p \equiv \pm 1 \pmod 6$$ implies $$p^2 \equiv 1 \pmod{24}$$, a trivial consequence of the Chinese remainder theorem first stated no later than nearly two millennia ago (proof: $$p \equiv \pm1 \pmod 2$$ so $$p \bmod 8 \in \{1,3,5,7\}$$ so $$p^2 \equiv 1 \pmod 8$$, and $$p \equiv \pm1 \pmod 3$$ so $$p^2 \equiv 1 \pmod 3$$; hence by CRT, $$p^2 \equiv 1 \pmod{3\cdot 8}$$).

Earthshattering stuff, I'm sure you'll agree.

• In March, they published a press release as Strathspey Crown LLC announcing their findings with no additional technical claims.

• In August, at BlackHat, they spent 115 000 USD for an unreviewed talk slot to debut Time AI, which

uses five dimensions of encryption, leveraging biometrics, music’s infinite variability and dynamic change to secure your data

and is sometimes quantum and which was predictably laughed off the stage at considerable cost to BlackHat's reputation as a purveyor of fine corporate stoogery.

(Crown Sterling have since sued BlackHat because people were mean to them and BlackHat did nothing to stop people from pointing out their idiocy when they presented it at the BlackHat podium they bought.)

• In September, they held a live stunt where they alleged to factor a 256-bit semiprime on a laptop on stage.

For perspective, this record was surpassed no less than a quarter century ago by Dixon & Lenstra, who reported solving a 110-digit/364-bit factoring challenge. The 129-digit/426-bit factoring challenge announced in Scientific American by Martin Gardner in 1977 with the advent of RSA was solved in 1994 by Atkins, Graff, Lenstra, & Leyland revealing the magic words Squeamish Ossifrage. Today you can factor a 512-bit integer for 100 USD using CADO-NFS and msieve on Amazon EC2.

Well, actually, Grant walked back the claim that it was a laptop (apologies for Twitter link to screenshot of Instagram; I don't deal in Instagram so I can't link to the page directly). But randos on Twitter showed how you, too, can break this record in the privacy of your own living room on your own laptop in comparable time.

The fraudsters gave no details about the alleged novel factoring algorithm actually used in the demonstration. A poor soul—pour one out for him!—who actually bothered to watch the video (which I'm not linking because I don't want to give these fraudsters any clicks) noted that some of the output of the alleged program appears to be output from CADO-NFS.

• What about the Instagram post? (Always a highly reputable venue for mathematical and cryptologic publication!) The claim seems to be that you can find the factors of $$pq$$ in the decimal expansion of $$1/pq$$.

First, it doesn't always work. For example, $$1/(11\cdot13) = 0.\overline{006993}$$ in which neither factor 11 nor 13 appears. But one might wonder whether it probably happens.

The period of the decimal expansion of $$1/pq$$ for prime $$p$$ and $$q$$ is $$\operatorname{lcm}(\operatorname{ord}_p 10, \operatorname{ord}_q 10)$$, which is usually not much smaller than $$pq$$. If we model the expansion as a uniform random digit string $$pq$$ (not $$\log_{10} pq$$, but $$pq$$) digits long, what's the probability of finding particular a $$\log_b p$$-digit string, namely $$p$$, in it? The expected number of times that any particular $$m$$-character needle appears as a substring of a uniform random $$h$$-character haystack in a $$b$$-symbol alphabet is $$(h - m + 1) b^{-m}$$ (proof: exercise for reader). Here $$m \approx \log_b p$$ and $$h \approx pq$$, so this is about $$(pq - \log_b p + 1) b^{-\log_b p} \approx q.$$ Cursory random experiments suggest that this is a reasonable model for the substring search, and I know of no theory that would suggest otherwise; it is up to the fraudsters to put up or shut up if they can prove anything better than this.

In other words, in this model, there's a reasonable probability that the the method works if you're willing to search through about (say) $$2^{1024}$$ possible candidate factors of the 2048-bit semiprime in question. Which, as it happens, is essentially random trial division. The fraudsters make it look easy by showing a screenshot with the answer already highlighted in a selectively cropped fragment that skipped over thousands of digits—about as many digits as the value of the numbers we're searching for.

In brief, these clowns—who are trumpeting thousand-year-old mathematical results on a preprint host open to the world as publications by Cornell University—appear to have at best caught up to the state of the art in factoring records from a quarter century ago by copying an alleged algorithm into software that looks just like off-the-shelf factoring software…or they're just using the off-the-shelf factoring software to wow an audience of chumps.

* Stack Exchange may have a profanity filter that has the unfortunate effect of interfering with using certain precise technical terminology.

• Excellent answer, but I do have to remark that "male cow" is technically an oxymoron, since (outside of colloquial speech) cows are by definition female. I imagine the scientifically more accurate term "male cattle" would also suffice to make your (indeed very valid) point? – yyyyyyy Sep 26 '19 at 22:30

TLDR: the efficiency of what they demo is below state of the art a decade ago. What they describe would be plain abyssal.

In the event described in the press release, Crown Sterling made a public demonstration, which can be watched there. The meat of it is a demonstration of the factorization of a 77-digit (256-bit) public modulus of an RSA key, 83473593554391843334619428139045661537976651941410655062632649869770538548577

Numbers of that size were already factorizable in the late 1980s, see Status Report on Factoring (At the Sandia National Laboratories) in proceedings of Eurocrypt 1984. Their factorization lasts 52s with a reported 32 cores, which would have been nice 15 years ago. Googling their number brings well-substantiated assertions that the program used in the demo has output reminiscent of CADO-NFS, and comparable runtime.

If you have a modern browser and one core of your CPU to spare for few minutes, then you can cut-and-paste that number (or any of that size) on this website. A fraction of your own computer will factor the number with ECM, at respectable speed thanks to WebAssembly.

This shows that the technology demonstrated is currently not a threat to modern-size RSA keys, and fuels the suspicions in an other answer.

I could not locate the paper "Novel Geometric Methods for Semiprime Factorization" announced in the press release, and thus can only comment two others sources:

A) An earlier paper of the same authors, Robert E. Grant and Talal Ghannam: Accurate and Infinite Prime Prediction from Novel Quasi-Prime Analytical Methodology, posted to arXiv on 2019-03-20. Only section VI discus prime factorization, on about half a page. It does not precisely state an algorithm, nor quantitatively characterize efficiency.

B) The method illustrated by the Instagram post linked in the question. It is based on the premise that (the decimal expression of) one of the factor of a semi-prime to be factored will likely appear in (the decimal expression of) its inverse. The factorization technique seems to be: compute that inverse over a sufficient length, try substrings of the appropriate size (perhaps restricting to those not obviously composite), and check that factor by trial division.

That works. But under the default assumption that a factor has no tendency to appear in the inverse earlier than a random digit sequence of similar size, in order to find a $$b$$-bit factor of a $$2b$$-bit semiprime, we are expected to test more than $$2^{b-4}$$ integers. That ends up with a method comparable to trial division by random numbers, grossly inefficient and not applicable to RSA moduli larger than 100-bit using the hardare and time in the demo (and certainly not for the 256-bit number in the demo).

Note: in the example picture on Instagram, the first factor found (5987) would be about the 1220's one tested, consistent with the above estimation. The picture is cropped, with the effect of showing considerably less digits.

• Time elapsed: 0d 0h 4m 3.2s – kelalaka Sep 27 '19 at 17:59
• I think the "Crown Sterling semiprime factoring method" would take a bit longer than 4 minutes. – gnasher729 Sep 28 '19 at 23:21


text:
Anindya wrote:

>for all \$$(x, y)\$$ and \$$(a, b)\$$. Unwrapping the left and right hand sides, this is equivalent to

>$\mathcal{X}(a, x) \otimes \mathcal{Y}(a, b) \leq \Phi(x, y) \multimap \Phi(a, b)$

>for all \$$a, x, y, b\$$. But the adjunction definition of \$$\multimap\$$ tells us that this is equivalent to

>$\Phi(x, y) \otimes \mathcal{X}(a, x) \otimes \mathcal{Y}(a, b) \leq \Phi(a, b)$

I think there is a typo. \$$\mathcal{Y}(a, b)\$$ should be \$$\mathcal{Y}(y, b)\$$?


text:
Question

Assertion :Electromagnetic waves exert radiation pressure. Reason: Electromagnetic waves carry energy.

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C
Assertion is correct but Reason is incorrect
D
Both Assertion and Reason are incorrect

Solution

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for AssertionAssertion:The electromagnetic wave has some energy and this energy of the waves can do work. This work done by a wave can be a result of radiation pressure. Thus, the Assertion is correct.Reason:The electromagnetic waves are the result of the magnetic and the electric field present in the medium and therefore, they also have the energy associated with the wave.In this case, this pressure exerted by the wave is due to the energy possessed by the wave. Therefore, the Assertion and Reason are correct and Reason is also the correct explanation of Assertion.Option $$(A)$$ is correct.Physics

Suggest Corrections

0

Similar questions
View More


text:
### Global bifurcation branches for radially symmetric Schrödinger equations

Tobias Weth

#### Abstract

We prove a new result on bifurcating branches of bound states for the nonlinear radially symmetric Schrödinger equation $$-\Delta u =w(|x|)|u|^{\sigma}u -\lambda^2u \ \ \text{ on {\mathbb{R}}^N.}$$ We show that, under suitable assumptions on $w$ and $\sigma$, there exist infinitely many continua of nontrivial bound states $u_\lambda$ which emanate from the trivial solution branch at $\lambda=0$. These continua reach arbitrarily large values of $\lambda$, and they are distinguished by the number of nodal domains of the corresponding solutions $u_\lambda$.

#### Article information

Source
Adv. Differential Equations, Volume 10, Number 7 (2005), 721-746.

Dates
First available in Project Euclid: 18 December 2012


text:
# Individuality and universality in the growth-division laws of single E. coli cells

Kennard, A. S.; Osella, M.; Javer, A.; Grilli, J.; Nghe, P.; Tans, S. J.; Cicuta, P.; Lagomarsino, M. Cosentino
Abstract:
The mean size of exponentially dividing Escherichia coli cells in different nutrient conditions is known to depend on the mean growth rate only. However, the joint fluctuations relating cell size, doubling time, and individual growth rate are only starting to be characterized. Recent studies in bacteria reported a universal trend where the spread in both size and doubling times is a linear function of the population means of these variables. Here we combine experiments and theory and use scaling concepts to elucidate the constraints posed by the second observation on the division control mechanism and on the joint fluctuations of sizes and doubling times. We found that scaling relations based on the means collapse both size and doubling-time distributions across different conditions and explain how the shape of their joint fluctuations deviates from the means. Our data on these joint fluctuations highlight the importance of cell individuality: Single cells do not follow the dependence observed for the means between size and either growth rate or inverse doubling time. Our calculations show that these results emerge from a broad class of division control mechanisms requiring a certain scaling form of the "division hazard rate function," which defines the probability rate of dividing as a function of measurable parameters. This "model free" approach gives a rationale for the universal body-size distributions observed in microbial ecosystems across many microbial species, presumably dividing with multiple mechanisms. Additionally, our experiments show a crossover between fast and slow growth in the relation between individual-cell growth rate and division time, which can be understood in terms of different regimes of genome replication control.
Year:
2016
Type of Publication:
Article
Journal:
Phys Rev E
Volume:
93
Number:
1
Pages:
012408
Month:
January
Note:
[DOI:\href{https://dx.doi.org/10.1103/PhysRevE.93.012408}{10.1103/PhysRevE.93.012408}] [PubMed:\href{https://www.ncbi.nlm.nih.gov/pubmed/26871102}{26871102}]
Hits: 117


text:
" /> -->

#### Term 1 Five Mark Model Questions

10th Standard EM

Reg.No. :
•
•
•
•
•
•

Maths

Time : 02:00:00 Hrs
Total Marks : 50
9 x 5 = 45
1. Let A = {3,4,7,8} and B = {1,7,10}. Which of the following sets are relations from A to B?
R1={(3,7), (4,7), (7,10), (8,1)}

2. Let A = {1,2,3,4} and B ={2,5,8,11,14} be two sets. Let f: A ⟶ B be a function given by f(x)=3x−1. Represent this function
(i) by arrow diagram
(ii) in a table form
(iii) as a set of ordered pairs
(iv) in a graphical form

3. The general term of a sequence is defined as
an = {$\begin{matrix} n\left( n+3 \right) ;n\epsilon N\quad is\quad odd \\ { n }^{ 2 }+1;n\epsilon N\quad is\quad even \end{matrix}$
Find the eleventh and eighteenth terms.

4. In an A.P. the sum of first n terms is $\frac { { 5n }^{ 2 } }{ 2 } +\frac { 3n }{ 2 }$. Find the 17th term

5. Find the GCD of 6x3 - 30x2 + 60x - 48 and 3x3 - 12x2 + 21x - 18.

6. Draw the graph of y = 2x2 and hence solve 2x2 - x - 6 = 0

7. Determine the nature of roots for the following quadratic equations
9x2 - 24x + 16 = 0

8. In $\triangle$ADC=, if DE||BC, AD=x, DB=x−2, and EC=x−1 then find the lengths of the sides AB and AC.

9. The line r passes through the points (–2, 2) and (5, 8) and the line θ passes through the points (–8, 7) and (–2, 0). Is the line r perpendicular to θ ?

10. 1 x 8 = 8
11. Find the equation of a line through the given pair of points
(2, 3) and (-7, -1)


text:
Tag Info

2

From what I can tell this is an initial value ODE problem, so you should be able to use solve_ivp. This handles vector equations just fine. I'm not sure exactly how stiff your problem is, so using the defaults for most of the settings should be fine, and you can adjust these if you find the numerical solution isn't satisfactory. There is one slightly ...

2

Pandas does not provide a three-dimensional data structure at this time, at some time I do enjoy this structure, but the option (as of today) is obsolete and has been removed. However, it is possible to express this type of data using the long format (also known as EAV) with three key columns (or index levels). Bearing this in mind; Jan Šimbera from the ...

2

Curve fitting can be very sensitive to your initial guess for each parameter. Because you don't specify a guess in your code, all of these parameters start with a value of 1. Comparing with the converged results for the t fitting, while t is actually pretty close to 1, the other parameters are much further away. Its mostly just luck that the t value didn't ...

1

You are only moving in one direction for each axis. coin_x = random.randint(1,2) # Move on x axis if coin_x == HEAD: b[step,0]+=1 elif coin_x == TAIL: b[step,0]-=1 # Move on y axis coin_y = random.randint(1,2) if coin_y == HEAD: b[step,1]+=1 elif coin_y == TAIL: b[step,1]-=1 `

1

MPI is designed so that the same executable is started on a number of machines, none of which may be the one on which the mpirun program is called. Furthermore, MPI jobs are typically put into a queue, and the actual job may run hours or days later. As a consequence, MPI jobs are not meant to be used in an interactive mode, but all input should be provided ...

1

Input from the console may work on process zero, if that runs locally, but will certainly not work on other ranks. However, I wouldn't even do interactive input on process zero. #!/bin/bash if [ $PMI_RANK -eq 0 ] ; then ( echo 5 ; echo 1 ; echo foo ) |$* else exec $* fi Put that in a script and start that with mpirun: mpirun myscript.sh myprogram ... 1 Ok, so first, you have a problem: when$s=0$both$x(0) = 0$and$\sin\Phi(0) = \sin(0) = 0$. To figure out how to deal with this, try to extend the function $$\frac{\sin\Phi(s)}{x(s)}$$ continuously for$s = 0\$. That means calculate the limit $$L = \lim_{s \to 0} \, \frac{\sin\Phi(s)}{x(s)}$$ The limit exists, because roughly speaking, the numerator is ...

Only top voted, non community-wiki answers of a minimum length are eligible