We study a simple model of similarity-based global cumulative imitation in symmetric games with large and ordered strategy sets and a salient winning player. We show that the learning model explains ...behavior well in both field and laboratory data from one such “winner-takes-all” game: the lowest unique positive integer game in which the player that chose the lowest number not chosen by anyone else wins a fixed prize. We corroborate this finding in three other winner-takes-all games and discuss under what conditions the model may be applicable beyond this class of games. Theoretically, we show that global cumulative imitation without similarity weighting results in a version of the replicator dynamic in winner-takes-all games.
Given
α
,
β
,
γ
∈
0
,
1
with
α
≤
β
, we prove that there exists a subset of
N
such that its lower and upper exponential densities and its lower and upper limit ratios are equal to
α
,
β
,
γ
and 1, ...respectively. This result provides an affirmative answer to an open problem posed by Grekos et al. (Unif Distrib Theory 6:117–130, 2011).
Hecke studies the distribution of fractional parts of quadratic irrationals with Fourier expansion of Dirichlet series. This method is generalized by Behnke and Ash–Friedberg, to study the ...distribution of the number of totally positive integers of given trace in a general totally real number field of any degree. When the field is cubic, we show that the asymptotic behavior of a weighted Diophantine sum is related to the structure of the unit group. The main term can be expressed in terms of Grössencharacter L-functions.
Let
p
be an odd prime. By using a lower bound for linear forms in logarithms of two algebraic numbers, we prove that if
p
>
10
24
, 2 is a primitive root module
p
and the least solution
(
u
1
,
v
1
)
...of Pell’s equation
u
2
-
2
(
p
-
1
)
(
p
-
2
)
v
2
=
1
satisfies
log
u
1
+
v
1
2
(
p
-
1
)
(
p
-
2
)
<
p
2
3
, then the equation
x
m
-
1
x
-
1
=
y
n
-
1
y
-
1
has no positive integer solutions
(
x
,
y
,
m
,
n
)
with
x
=
2
,
y
=
p
and
m
>
n
>
2
.
We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system
{
n
=
a
1
+
a
2
+
…
+
a
s
−
1
,
a
1
a
2
…
a
s
−
1
(
a
1
+
a
2
+
…
+
a
s
−
1
)
=
b
s
has positive ...integer or rational solutions
n, b, a
j
,
i
= 1, 2, …,
s
− 1,
s
⩾ 3. Using the theory of elliptic curves, we prove that it has no positive integer solution for
s
= 3, but there are infinitely many positive integers
n
such that it has a positive integer solution for
s
⩾ 4. As a corollary, for
s
⩾ 4 and any positive integer
n
, the above Diophantine system has a positive rational solution. Meanwhile, we give conditions such that it has infinitely many positive rational solutions for
s
⩾ 4 and a fixed positive integer
n
.
This paper shows one of the tales presented to the contest about tales with mathematical
content organized by the Innovation Educative Group “Mathematical Thinking” in 2015.
The tale introduces the ...two digits numbers in a funny way.
En este artículo se muestra uno de los cuentos presentados al concurso de relatos con
contenido matemático organizado por el GIE Pensamiento matemático en 2015 para
alumnos de la ESO, Bachillerato y universitarios. En él se introducen los números de dos
cifras de una forma divertida.
We prove that there is a sequence of trigonometric polynomials with positive integer coefficients, which converges to zero almost everywhere. We also prove that there is a function
f
: ℝ → ℝ such ...that the sums of its shifts are dense in all real spaces
L
p
(ℝ) for 2 ≤
p
< ∞ and also in the real space
C
0
(R).
An (n = 2k + s,k) Suh–Ramchandran code is shown, where s > 0 is a positive integer, is a minimum storage regeneration code, and can also achieve the optimal repair bandwidth for multiple systematic ...nodes failure repair.
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