bluuebunny/crossref_metadata_2025_split huggingface dataset:
PSYCHOSOCIAL CORRELATES OF BODY MASS INDEX IN FOUR GROUPS OF QUEBEC ADULTS
Authors: PLOURDE HUGUES , NOLIN BERTRAND , RECEVEUR OLIVIER , LEDOUX MARIELLE 
Published: June 2010

SummaryThe objective of the present analysis was to study the associations between body weight psychosocial correlates and body mass index (BMI) among four groups of adults in the Quebec population. Data were taken from the Social Lifestyles and Health 1998 Survey performed by the Institut de la Statistique du Québec (ISQ). The suggested guidelines of the ISQ were used to estimate the population's proportions and for statistical analysis. The groups studied were 25- to 44- and the 45- to 64-year-old men and women. In all groups, currently trying to lose weight increased the odds of reporting an excess weight. Better perceived eating habits was associated with lower BMI in most groups except in the 25- to 44-year-old women, where the trend was not significant. Higher number of physical activities related to transport and cigarette smoking were associated with lower BMI in both men groups. In both women groups, more frequent consumption of alcoholic beverages decreased significantly the odds of reporting excess body weight. A university degree was associated with a lower BMI only in the 25- to 44-year-old men. Regular practise of leisure time physical activity was associated with a lower BMI only in 45- to 64-year-old women. Opposite associations were observed between perceived health and BMI. In the 45- to 64-year-old men, better perceived health increased the odds of reporting an excess weight. Conversely, the odds of reporting excess weight decreased with better health in 25- to 44-year-old women. Many correlates differ between age group and sex. The identification of these factors illustrates the need to adapt obesity-related programmes toward specific sub-groups within the general population.
Hellenized Romans and Barbarized Greeks. Reading the End of Dionysius of Halicarnassus,Antiquitates Romanae
Authors: Peirano Irene 
Published: June 2010

ABSTRACTIn theAntiquitates Romanae, Dionysius of Halicarnassus presents the Romans as a nation of Greeks. Throughout his narrative, Dionysius shows how the Romans have surpassed other Greek nations in the quintessentially Greek areas of morality and conduct. However, this assessment of Rome's cultural and ethnic identity proves to be much more nuanced when read side by side with the narrative of the concluding books of theAntiquitates Romanae, a largely unexplored section of the work dealing with the war between the Romans and Pyrrhus. The culturally-based definition of Roman ‘Greekness’ is accompanied, particularly in the last portion of the narrative, by a troubling awareness of its inherent instability as the Romans increasingly display the tendencies which eventually caused the ‘barbarization’ of Greece.
Rapid and continued population decline in the Spoon-billed Sandpiper Eurynorhynchus pygmeus indicates imminent extinction unless conservation action is taken
Authors: ZÖCKLER CHRISTOPH , SYROECHKOVSKIY EVGENY E. , ATKINSON PHILIP W. 
Published: June 2010

SummaryThe Spoon-billed Sandpiper Eurynorhynchus pygmeus (IUCN Category: Critically Endangered) is in rapid decline. Data from across the entire breeding range (Chukotka and Koryakya in the Russian far north-east) and especially from the well-studied southern core breeding area at Meinypilgyno, confirm the continuing strong decline. At four breeding sites, where more than two counts were available for analysis, the decline was estimated at 26% per annum between 2002 and 2009, or an 88% decline over this period. Allowing for unsurveyed areas, this equates to a decline from a total population of approximately 1,000 breeding pairs in 2000 to 120–220 in 2009. Breeding studies at Meinypilgyno in 2003–2007 (not 2006) showed that the proportion of nests hatching at least one chick was 0.65 and once chicks left the nest, the mean brood size of chicks up to one week old was 1.99. Where it was possible to follow broods, 0.61 chicks fledged per nesting attempt. Survival and recruitment analysis of birds ringed at Meinypilgyno indicated that annual adult survival did not significantly differ over the 2003–2009 study but that recruitment in to the adult breeding population was effectively zero in all but one year of the study (2005). Resighting data for the last two years of the study were sparse due to very low numbers of marked adults being recorded and survival rates over the last 2–3 years of the study must therefore be treated with caution. The analysis therefore indicated that after fledging, survival during immaturity must be very low, leading to a low (or no) recruitment into an ageing population. Recent observations collated from the non-breeding areas confirm the declining trend observed in the breeding areas and imply that the main threats to the population lie along the migration route or in the wintering areas. These are poorly known although hunting in the wintering areas has been identified as a major mortality factor. Other threats include major loss of their intertidal habitats, and collection of birds on the breeding areas by specimen collectors. Improved monitoring in both the breeding and non-breeding areas as well as research on juvenile survival is recommended. Concerted international conservation action is essential if this species is to avoid extinction. This requires (i) improved understanding of the main wintering and staging areas and associated threats; (ii) addressing those threats that can be tackled with immediate effect, such as hunting; (iii) continued long-term monitoring on the breeding areas; (iv) an exploration of other potential breeding areas; (v) conservation action at all important stop-over and wintering sites along the entire flyway and (vi) consideration of a captive-breeding programme to ensure the survival of this species.
Multiplicative functions on arithmetic progressions
Authors: Hildebrand Adolf 
Published: June 2010

Let

f
f

be a multiplicative arithmetic function satisfying

|
f
|

≤
1

\left | f \right | \leq 1

, let

x
≥
10

x \geq 10

and

2
≤
Q
≤

x

1

/

3

2 \leq Q \leq {x^{1/3}}

. It Is shown that, with suitable integers

q
1

≥
2

{q_1} \geq 2

and

q
2

≥
2

{q_2} \geq 2

, the estimate
\[

∑

n
≤
x

n
≡
a
mod
q

f
(
n
)
=

1

φ
(
q
)

∑

n
≤
x

(
n
,
q
)
=
1

f
(
n
)
+
O

(

x
q

(

log
⁡

log
⁡
x

log
⁡
Q

)

−
1

/

2

)

\sum \limits _{\begin {array}{*{20}{c}} {n \leq x} \\ {n \equiv a\bmod q} \\ \end {array} } {f(n) = \frac {1}{{\varphi (q)}}} \sum \limits _{\begin {array}{*{20}{c}} {n \leq x} \\ {(n,q) = 1} \\ \end {array} } {f(n) + O\left ( {\frac {x}{q}{{\left ( {\log \frac {{\log x}}{{\log Q}}} \right )}^{ - 1/2}}} \right )}

\]
holds uniformly for

(

a
,
q

)

=
1

\left ( {a,q} \right ) = 1

and all moduli

q
≤
Q

q \leq Q

that are not multiples of

q
1

{q_1}

or

q
2

{q_2}

.
A function space triple of a compact polyhedron into an open set in Euclidean space
Authors: Sakai Katsuro 
Published: June 2010

Let

X
X

be a non-zero dimensional compact Euclidean polyhedron and

Y
Y

an open set in Euclidean space

R

r

(

r
>
0

)

{{\mathbf {R}}^r}\left ( {r > 0} \right )

. The spaces of (continuous) maps, Lipschitz maps and PL maps from

X
X

to

Y
Y

are denoted by

C

(

X
,
Y

)

C\left ( {X,Y} \right )

,

LIP

(

X
,
Y

)

{\text {LIP}}\left ( {X,Y} \right )

and

PL

(

X
,
Y

)

{\text {PL}}\left ( {X,Y} \right )

, respectively. We prove that the triple
\[

(

C

(

X
,
Y

)

,

LIP

(

X
,
Y

)

PL

(

X
,
Y

)

)

\left ( {C\left ( {X,Y} \right ),{\text {LIP}}\left ( {X,Y} \right ){\text {PL}}\left ( {X,Y} \right )} \right )

\]
is an

(

s
,
Σ
,
σ

)

\left ( {s,\Sigma ,\sigma } \right )

-manifold triple, where

s
=

(

−
1
,
1

)

ω

s = {\left ( { - 1,1} \right )^\omega }

,
\[

Σ
=

{

x
∈
s

|

sup

|

x

(
i
)

|

>
1

}

and

σ
=

{

x
∈
s

|

x

(
i
)

=
0

except

for

finitely

many

i

}

.

\Sigma = \left \{ {x \in s|\sup \left | {x\left ( i \right )} \right | > 1} \right \}{\text {and}}\,\sigma = \left \{ {x \in s|x\left ( i \right ) = 0\,{\text {except}}\,{\text {for}}\,{\text {finitely}}\,{\text {many}}\,i} \right \}.

\]
.
Complete monotonicity of modified Bessel functions
Authors: Ismail Mourad E. H. 
Published: June 2010

We prove that if

ν
>
1

/

2

\nu > 1/2

, then

2

ν
−
1

Γ
(
ν
)

/

[

x

ν

/

2

e

x

K
ν

(

x

)
]

{2^{\nu - 1}}\Gamma (\nu )/[{x^{\nu /2}}{e^{\sqrt x }}{K_\nu }(\sqrt x )]

is the Laplace transform of a selfdecomposable probability distribution while

2
ν

Γ

(

ν
+
1

)

x

−
ν

/

2

e

−

x

I
ν

(

x

)

{2^\nu }\Gamma \left ( {\nu + 1} \right ){x^{ - \nu /2}}{e^{ - \sqrt x }}{I_\nu }\left ( {\sqrt x } \right )

is the Laplace transform of an infinitely divisible distribution. The former result is used to show that an estimate of

M

{\text {M}}

. Wong [13] is sharp. We also prove that the roots of the equations
\[

b
3

l

ν
−
1

(

a

z

)

/

I
ν

(

a

z

)

=

a
3

I

ν
−
1

(

b

z

)

/

I
ν

(

b

z

)

,

{b^3}{l_{\nu - 1}}\left ( {a\sqrt z } \right )/{I_\nu }\left ( {a\sqrt z } \right ) = {a^3}{I_{\nu - 1}}\left ( {b\sqrt z } \right )/{I_\nu }\left ( {b\sqrt z } \right ),

\]
and
\[

b
3

K

ν
+
1

(

a

z

)

/

K
ν

(

a

z

)

=

a
3

K

ν
+
1

(

b

z

)

/

K
ν

(

b

z

)

,
ν
>
0
,
z
≠
0
,

{b^3}{K_{\nu + 1}}\left ( {a\sqrt z } \right )/{K_\nu }\left ( {a\sqrt z } \right ) = {a^3}{K_{\nu + 1}}\left ( {b\sqrt z } \right )/{K_\nu }\left ( {b\sqrt z } \right ),\nu > 0,z \ne 0,

\]
lie in a certain sector contained in the open left half plane. This proves and extends a conjecture of

H

{\text {H}}

. Hattori arising from his work in partial differential equations.
On the structure of semiderivations in prime rings
Authors: Chuang Chen-Lian 
Published: June 2010

Let

R
R

be a prime ring. By a semiderivation associated with a function

g
:
R
→
R

g:R \to R

, we mean an additive mapping

f
:
R
→
R

f:R \to R

such that, for all

x
,
y
∈
R
,
f
(
x
y
)
=
f
(
x
)
g
(
y
)
+
x
f
(
y
)
=
f
(
x
)
y
+
g
(
x
)
f
(
y
)

x,y \in R,f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y)

and

f
(
g
(
x
)
)
=
g
(
f
(
x
)
)

f(g(x)) = g(f(x))

. It is known that

g
g

must necessarily be a ring endomorphism. Here it is shown that

f
f

must be an ordinary derivation or of the form

f
(
x
)
=
λ
(
x
−
g
(
x
)
)

f(x) = \lambda (x - g(x))

for all

x
∈
R

x \in R

, where

λ
\lambda

is an element of the extended centroid of

R
R

.
On the theorem of Frullani
Authors: Arias-de-Reyna Juan 
Published: June 2010

We prove that, for every function

f
:

R

+

→

C

f:{{\mathbf {R}}^ + } \to {\text {C}}

such that

(
f
(
a
x
)
−
f
(
b
x
)
)

/

x

(f(ax) - f(bx))/x

is Denjoy-Perron integrable on

[
0
,
+
∞
)

[0, + \infty )

for every pair of positive real numbers

a
,
b

a,b

, there exists a constant

A
A

(depending only on the values of

f
(
t
)

f(t)

in the neighborhood of 0 and

+
∞

+ \infty

) such that
\[

∫
0
∞

f
(
a
x
)
−
f
(
b
x
)

x

d
x
=
A
l
o
g

a
b

.

\int _0^\infty {\frac {{f(ax) - f(bx)}}{x}} dx = Alog \frac {a}{b}.

\]
To prove this assertion, we identify a Denjoy-Perron integrable function

f
:

R

→

C

f:{\mathbf {R}} \to {\text {C}}

with a distribution. In this way, we obtain the main result of this paper: The value at 0 (in Lojasiewicz sense) of the Fourier transform of the distribution

f
f

is the Denjoy-Perron integral of

f
f

. Assuming the Continuum Hypothesis, we construct an example of a non-Lebesgue measurable function that satisfies the hypotheses of the first theorem.
Some rigidity phenomena for Einstein metrics
Authors: Shen Zhong Min 
Published: June 2010

In this note we study the following problem: When must a complete Einstein metric

g
g

on an

n
n

-manifold with

Ric =

(
n
−
1
)
λ
g

{\text {Ric = }}(n - 1)\lambda g

be a constant curvature metric of sectional curvature

λ
\lambda

?
The converse of the Minkowski’s inequality theorem and its generalization
Authors: Matkowski Janusz 
Published: June 2010

Let (

Ω
\Omega

,

Σ
\Sigma

,

μ
\mu

) be a measure space with two sets

A
,
B
∈
Σ

A,B \in \Sigma

such that

0
>
μ
(
A
)
>
1
>
μ
(
B
)
>
∞

0 > \mu (A) > 1 > \mu (B) > \infty

, and let

φ
:

R

+

→

R

+

\varphi :{{\mathbf {R}}_ + } \to {{\mathbf {R}}_ + }

be bijective and

φ

−
1

{\varphi ^{ - 1}}

continuous at 0. We prove that if for all

μ
\mu

-integrable step functions

x
,
y
:
Ω
→

R

x,y:\Omega \to {\mathbf {R}}

,
\[

φ

−
1

(

∫
Ω

φ
∘

|

x
+
y

|

d
μ

)

≤

φ

−
1

(

∫
Ω

φ
∘

|

x

|

d
μ

)

+

φ

−
1

(

∫
Ω

φ
∘

|

y

|

d
μ

)

{\varphi ^{ - 1}}\left ( {\int _\Omega {\varphi \circ |x + y|d\mu } } \right ) \leq {\varphi ^{ - 1}}\left ( {\int _\Omega {\varphi \circ |x|d\mu } } \right ) + {\varphi ^{ - 1}}\left ( {\int _\Omega {\varphi \circ |y|d\mu } } \right )

\]
then

φ
(
t
)
=
φ
(
1
)

t
p

\varphi (t) = \varphi (1){t^p}

for some

p
≥
1

p \geq 1

. In the case of normalized measure we prove a generalization of Minkowski’s inequality theorem. The suitable results for the reversed inequality are also presented.
Evaluation of the infinite series ∑^{∞}_{𝑛=1,(\frac𝑛𝑝)=1}(\frac𝑛𝑝)₄𝑛⁻¹
Authors: Hardy Kenneth , Williams Kenneth S. 
Published: June 2010

Let

p
p

be a prime

≡
1
(
mod
4
)

\equiv 1(\bmod 4)

. The value of the infinite series

∑

(

n
p

)
=
1

n
=
1

∞

(

n
p

)

4

n

−
1

\sum \nolimits _{_{(\tfrac {n}{p}) = 1}^{n = 1}}^\infty {{{(\tfrac {n}{p})}_4}{n^{ - 1}}}

is given in finite terms.
Subordination and 𝐻^{𝑝} functions
Authors: Younis Rahman 
Published: June 2010

Let

ϕ
\phi

and

W
W

be inner functions with

ϕ
(
0
)
=
W
(
0
)
=
0

\phi (0) = W(0) = 0

. It is shown that if

F
F

is an exposed point of the unit ball of

H
1

{H^1}

and
\[

F
(
W
(

e

i
t

)
)

/

F
(
ϕ
(

e

i
t

)
)
>
0

F(W({e^{it}}))/F(\phi ({e^{it}})) > 0

\]
almost everywhere, then

F
∘
W
=
F
∘
ϕ

F \circ W = F \circ \phi

. If

f
=
z
F

f = zF

such that

F
F

and

1

/

F

1/F

are in

H
r

{H^r}

and

H
s

{H^s}

, respectively, where

1

/

r
+
1

/

s
≤
2

1/r + 1/s \leq 2

and

ϕ
\phi

is a finite Blaschke product, then a necessary and sufficient condition is provided in order for

f
(
W
(

e

i
t

)
)

/

f
(
ϕ
(

e

i
t

)
)

f(W({e^{it}}))/f(\phi ({e^{it}}))

to be positive almost everywhere.
Some questions of Erdős and Graham on numbers of the form ∑𝑔_{𝑛}/2^{𝑔_{𝑛}}
Authors: Borwein Peter , Loring Terry A. 
Published: June 2010

Erdös in 1975 and Erdös and Graham in 1980 raised several questions concerning representing numbers as series of the form

Σ

n
=
1

∞

g
n

/

2

g
n

\Sigma _{n = 1}^\infty \;{g_n}/{2^{{g_n}}}

. For example, does the equation
\[

n

2
n

=

∑

k
=
1

T

g
k

2

g
k

,

T
>
1

,

\frac {n}{{{2^n}}} = \sum \limits _{k = 1}^T {\frac {{{g_k}}}{{{2^{{g_k}}}}},\quad T > 1} ,

\]
have a solution for infinitely many n ? The answer to this question is affirmative; in fact, we conjecture that the above equation is solvable for every n. This conjecture is based on a more general conjecture, namely that the algorithm
\[

a

n
+
1

=
2
(

a
n

mod
n
)

{a_{n + 1}} = 2({a_n}\bmod n)

\]
with initial condition

a
m

∈

Z

{a_m} \in {\mathbf {Z}}

always eventually terminates at zero. This, in turn, is based on an examination of how the "greedy algorithm" can be used to represent numbers in the form

∑

g
n

/

2

g
n

\sum {{g_n}/{2^{{g_n}}}}

. The analysis of this, reformulated as a "base change" algorithm, proves surprising. Some numbers have a unique representation, as above, others have uncountably many. Also, from this analysis we observe that

∑

g
n

/

2

g
n

\sum {{g_n}/{2^{{g_n}}}}

is irrational if

lim

sup
n

(
(

g

n
+
1

−

g
n

)

/

log
⁡
(

g

n
+
1

)
)
=
∞

\lim {\sup _n}(({g_{n + 1}} - {g_n})/\log ({g_{n + 1}})) = \infty

and conjecture that this is best possible.
Lower bounds for the discrepancy of inversive congruential pseudorandom numbers
Authors: Niederreiter Harald 
Published: June 2010

The inversive congruential method is a uniform pseudorandom number generator which was introduced recently. For a prime modulus p the discrepancy

D
p

(
k
)

D_p^{(k)}

of k-tuples of successive pseudorandom numbers generated by this method determines the statistical independence properties of these pseudorandom numbers. It was shown earlier by the author that
\[

D
p

(
k
)

=
O
(

p

−
1

/

2

(
log
⁡
p

)
k

)

for

2
≤
k
>
p
.

D_p^{(k)} = O({p^{ - 1/2}}{(\log p)^k})\quad {\text {for}}\;2 \leq k > p.

\]
Here it is proved that this bound is essentially best possible. In fact, for a positive proportion of the admissible parameters in the inversive congruential method the discrepancy

D
p

(
k
)

D_p^{(k)}

is at least of the order of magnitude

p

−
1

/

2

{p^{ - 1/2}}

for all

k
≥
2

k \geq 2

.
On the numerical condition of Bernstein-Bézier subdivision processes
Authors: Farouki R. T. , Neff C. A. 
Published: June 2010

The linear map M that takes the Bernstein coefficients of a polynomial

P
(
t
)

P(t)

on a given interval [a, b] into those on any subinterval

[

a
¯

,

b
¯

]

[\bar a,\bar b]

is specified by a stochastic matrix which depends only on the degree n of

P
(
t
)

P(t)

and the size and location of

[

a
¯

,

b
¯

]

[\bar a,\bar b]

relative to [a, b]. We show that in the

‖
∙
‖

∞

{\left \| \bullet \right \|_\infty }

-norm, the condition number of M has the simple form

κ
∞

(

M

)
=

[
2
f
max
(

u

m
¯

,

v

m
¯

)

]
n

{\kappa _\infty }({\mathbf {M}}) = {[2f\max ({u_{\bar m}},{v_{\bar m}})]^n}

, where

u

m
¯

=
(

m
¯

−
a
)

/

(
b
−
a
)

{u_{\bar m}} = (\bar m - a)/(b - a)

and

v

m
¯

=
(
b
−

m
¯

)

/

(
b
−
a
)

{v_{\bar m}} = (b - \bar m)/(b - a)

are the barycentric coordinates of the subinterval midpoint

m
¯

=

1
2

(

a
¯

+

b
¯

)

\bar m = \frac {1}{2}( {\bar a + \bar b} )

, and f denotes the "zoom" factor

(
b
−
a
)

/

(

b
¯

−

a
¯

)

(b - a)/(\bar b - \bar a)

of the subdivision map. This suggests a practical rule-of-thumb in assessing how far Bézier curves and surfaces may be subdivided without exceeding prescribed (worst-case) bounds on the typical errors in their control points. The exponential growth of

κ
∞

(

M

)

{\kappa _\infty }({\mathbf {M}})

with n also argues forcefully against the use of high-degree forms in computer-aided geometric design applications.
Complete solutions to families of quartic Thue equations
Authors: Pethő Attila 
Published: June 2010

Using a method due to E. Thomas, we prove that if|a|>9.9⋅1027|a| > 9.9 \cdot {10^{27}}then the Diophantine equations\[x4−ax3y−x2y2+axy3+y4=1{x^4} - a{x^3}y - {x^2}{y^2} + ax{y^3} + {y^4} = 1\]and\[x4−ax3y−3x2y2+axy3+y4=±1{x^4} - a{x^3}y - 3{x^2}{y^2} + ax{y^3} + {y^4} = \pm 1\]have exactly twelve solutions, namely(x,y)=(0,±1),(±1,0),(±1,±1),(∓1,±1),(±a,±1),(±1,∓a)(x,y) = (0, \pm 1), ( \pm 1,0), ( \pm 1, \pm 1), ( \mp 1, \pm 1), ( \pm a, \pm 1), ( \pm 1, \mp a)and eight solutions,(x,y)=(0,±1),(±1,0),(±1,±1),(±1,∓1)(x,y) = (0, \pm 1), ( \pm 1,0), ( \pm 1, \pm 1), ( \pm 1, \mp 1), respectively.
On the numerical evaluation of Legendre’s chi-function
Authors: Boersma J. , Dempsey J. P. 
Published: June 2010

Legendre’s chi-function,

χ
n

(
z
)
=

Σ

k
=
0

∞

z

2
k
+
1

/

(
2
k
+
1

)
n

{\chi _n}(z) = \Sigma _{k = 0}^\infty {z^{2k + 1}}/{(2k + 1)^n}

, is reexpanded in a power series in powers of

log
⁡
z

\log z

. The expansion obtained is well suited for the computation of

χ
n

(
z
)

{\chi _n}(z)

in the two cases of real z close to 1, and

z
=

e

i
α

,
α
∈

R

z = {e^{i\alpha }},\alpha \in \mathbb {R}

. For

n
=
2

n = 2

and

n
=
3

n = 3

, the present computational procedure is shown to be superior to the procedure recently proposed by Dempsey, Liu, and Dempsey, which uses Plana’s summation formula.
A method of fractional steps for scalar conservation laws without the CFL condition
Authors: Holden Helge , Risebro Nils Henrik 
Published: June 2010

We present a numerical method for the n-dimensional initial value problem for the scalar conservation law

u

(

x
1

,
…
,

x
n

,
t

)
t

+

∑

i
=
1

n

f
i

(
u

)

x
1

=
0
,
u
(

x
1

,
…
,

x
n

,
0
)
=

u
0

(

x
1

,
…
,

x
n

)

u{({x_1}, \ldots ,{x_n},t)_t} + \sum _{i = 1}^n{f_i}{(u)_{{x_1}}} = 0, u({x_1}, \ldots ,{x_n},0) = {u_0}({x_1}, \ldots ,{x_n})

. Our method is based on the use of dimensional splitting and Dafermos’s method to solve the one-dimensional equations. This method is unconditionally stable in the sense that the time step is not limited by the space discretization. Furthermore, we show that this method produces a subsequence which converges to the weak entropy solution as both the time and space discretization go to zero. Finally, two numerical examples are discussed.
On the coefficients of the minimal polynomials of Gaussian periods
Authors: Gupta S. , Zagier D. 
Published: June 2010

Let l be a prime number and m a divisor of

l
−
1

l - 1

. Then the Gauss period

ω
=
ζ
+

ζ
λ

+

ζ

λ
2

+
⋯
+

ζ

λ

m
−
1

\omega = \zeta + {\zeta ^\lambda } + {\zeta ^{{\lambda ^2}}} + \cdots + {\zeta ^{{\lambda ^{m - 1}}}}

where

ζ
=

e

2
π
i

/

l

\zeta = {e^{2\pi i/l}}

and

λ
\lambda

is a primitive mth root of unity modulo l, generates a subfield K of

Q

(
ζ
)

\mathbb {Q}(\zeta )

of degree

(
l
−
1
)

/

m

(l - 1)/m

. In this paper we study the reciprocal minimal polynomial

F

l
,
m

(
X
)
=

N

K

/

Q

(
1
−
ω
X
)

{F_{l,m}}(X) = {N_{K/\mathbb {Q}}}(1 - \omega X)

of

ω
\omega

over

Q

\mathbb {Q}

. It will be shown that for fixed m and every N we have

F

l
,
m

(
X
)
≡

(

B
m

(
X

)
l

/

(
1
=
m
X
)

)

1

/

m

(
mod

X
N

)

{F_{l,m}}(X) \equiv {({B_m}{(X)^l}/(1 = mX))^{1/m}}\;\pmod {{X^N}}

for all but finitely many "exceptional primes" l (depending on m and N), where

B
m

(
X
)
∈

Z

[
[
X
]
]

{B_m}(X) \in \mathbb {Z}[[X]]

is a power series depending only on m . A method of computation of this set of exceptional primes is presented. The generalization of the results to the case of composite l is also discussed.
Iterated absolute values of differences of consecutive primes
Authors: Odlyzko Andrew M. 
Published: June 2010

Let

d
0

(
n
)
=

p
n

{d_0}(n) = {p_n}

, the nth prime, for

n
≥
1

n \geq 1

, and let

d

k
+
1

(
n
)
=

|

d
k

(
n
)
−

d
k

(
n
+
1
)

|

{d_{k + 1}}(n) = |{d_k}(n) - {d_k}(n + 1)|

for

k
≥
0
,
n
≥
1

k \geq 0,n \geq 1

. A well-known conjecture, usually ascribed to Gilbreath but actually due to Proth in the 19th century, says that

d
k

(
1
)
=
1

{d_k}(1) = 1

for all

k
≥
1

k \geq 1

. This paper reports on a computation that verified this conjecture for

k
≤
π
(

10

13

)
≈
3
×

10

11

k \leq \pi ({10^{13}}) \approx 3 \times {10^{11}}

. It also discusses the evidence and the heuristics about this conjecture. It is very likely that similar conjectures are also valid for many other integer sequences.
Average case error estimates for the strong probable prime test
Authors: Damgård Ivan , Landrock Peter , Pomerance Carl 
Published: June 2010

Consider a procedure that chooses k-bit odd numbers independently and from the uniform distribution, subjects each number to t independent iterations of the strong probable prime test (Miller-Rabin test) with randomly chosen bases, and outputs the first number found that passes all t tests. Let

p

k
,
t

{p_{k,t}}

denote the probability that this procedure returns a composite number. We obtain numerical upper bounds for

p

k
,
t

{p_{k,t}}

for various choices of k, t and obtain clean explicit functions that bound

p

k
,
t

{p_{k,t}}

for certain infinite classes of k, t. For example, we show

p

100
,
10

≤

2

−
44

,

p

300
,
5

≤

2

−
60

,

p

600
,
1

≤

2

−
75

{p_{100,10}} \leq {2^{ - 44}},{p_{300,5}} \leq {2^{ - 60}},{p_{600,1}} \leq {2^{ - 75}}

, and

p

k
,
1

≤

k
2

4

2
−

k

{p_{k,1}} \leq {k^2}{4^{2 - \sqrt k }}

for all

k
≥
2

k \geq 2

. In addition, we characterize the worst-case numbers with unusually many "false witnesses" and give an upper bound on their distribution that is probably close to best possible.