Let
q
be a prime power,
F
q
be the finite field of order
q
and
F
q
(
x
)
be the field of rational functions over
F
q
. In this paper we classify and count all rational functions
φ
∈
F
q
(
x
)
of ...degree 3 that induce a permutation of
P
1
(
F
q
)
. As a consequence of our classification, we can show that there is no complete permutation rational function of degree 3 unless
3
∣
q
and
φ
is a polynomial.
An
(
r
,
ℓ
)
-good polynomial is a polynomial of degree
r
+
1
that is constant on
ℓ
subsets of
F
q
, each of size
r
+
1
. For any positive integer
r
≤
4
we provide an
(
r
,
ℓ
)
-good polynomial such ...that
ℓ
=
C
r
q
+
O
(
q
)
, with
C
r
maximal. This directly provides an explicit estimate (up to an error term of
O
(
q
)
, with explict constant) for the maximal length and dimension of a Tamo–Barg LRC. Moreover, we explain how to construct good polynomials achieving these bounds. Finally, we provide computational examples to show how close our estimates are to the actual values of
ℓ
, and we explain how to obtain the best possible good polynomials in degree 5. Our results complete the study by Chen et al. (Des Codes Cryptogr 89(7):1639–1660, 2021), providing
(
r
,
ℓ
)
-good polynomials of degree up to 5, with
ℓ
maximal (up to an error term of
q
), and our methods are independent.
Let
m
be a positive integer,
q
be a prime power, and PG(2,
q
) be the projective plane over the finite field
F
q
. Finding complete
m
-arcs in PG(2,
q
) of size less than
q
is a classical problem in ...finite geometry. In this paper we give a complete answer to this problem when
q
is relatively large compared with
m
, explicitly constructing the smallest
m
-arcs in the literature so far for any
m
≥ 8. For any fixed
m
, our arcs
A
q
,
m
satisfy
|
A
q
,
m
|
−
q
→
−
∞
as
q
grows. To produce such
m
-arcs, we develop a Galois theoretical machinery that allows the transfer of geometric information of points external to the arc, to arithmetic one, which in turn allows to prove the
m
-completeness of the arc.
Let <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula> be a prime power and <inline-formula> <tex-math notation="LaTeX">\mathbb {F}_{q} </tex-math></inline-formula> be the ...finite field of size <inline-formula> <tex-math notation="LaTeX">q </tex-math></inline-formula>. In this paper we provide a Galois theoretical framework that allows to produce good polynomials for the Tamo and Barg construction of optimal locally recoverable codes (LRC). Using our approach we construct new good polynomials and therefore optimal LRCs with new parameters. The existing theory of good polynomials fits entirely in our new framework. The key advantage of our method is that we do not need to rely on arithmetic properties of the pair <inline-formula> <tex-math notation="LaTeX">(q,r) </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula> is the locality of the code.
In this paper we provide a complete answer to a question by Heyman and Shparlinski concerning the natural density of polynomials which are irreducible by Eisenstein's criterion after applying some ...shift. The main tool we use is a local to global principle for density computations over a free \mathbb{Z}-module of finite rank.
The geometric sieve for densities is a very convenient tool proposed by Poonen and Stoll (and independently by Ekedahl) to compute the density of a given subset of the integers. In this paper we ...provide an effective criterion to find all higher moments of the density (e.g. the mean, the variance) of a subset of a finite dimensional free module over the ring of algebraic integers of a number field. More precisely, we provide a geometric sieve that allows the computation of all higher moments corresponding to the density, over a general number field
K
. This work advances the understanding of geometric sieve for density computations in two ways: on one hand, it extends a result of Bright, Browning and Loughran, where they provide the geometric sieve for densities over number fields; on the other hand, it extends the recent result on a geometric sieve for expected values over the integers to both the ring of algebraic integers and to moments higher than the expected value. To show how effective and applicable our method is, we compute the density, mean and variance of Eisenstein polynomials and shifted Eisenstein polynomials over number fields. This extends (and fully covers) results in the literature that were obtained with ad-hoc methods.
In this paper we show that the
Z
/
p
m
Z
-module structure of the ring
E
p
(
m
)
is isomorphic to a
Z
/
p
m
Z
-submodule of the matrix ring over
Z
/
p
m
Z
. Using this intrinsic structure of
E
p
(
m
...)
, solving a linear system over
E
p
(
m
)
becomes computationally equivalent to solving a linear system over
Z
/
p
m
Z
. As an application we break the protocol based on the Diffie–Hellman decomposition problem and ElGamal decomposition problem over
E
p
(
m
)
. Our algorithm terminates in a provable running time of
O
(
m
6
)
Z
/
p
m
Z
-operations.
Objective:
The study aimed to explore the psychological symptoms and the readiness to fight the pandemic of the new generation of healthcare professionals: medical and other healthcare degree ...students.
Methods:
We enrolled 509 medical and healthcare-related degree students during the second outbreak of COVID-19 in Italy. We have examined their psychological symptoms using the 12-item General Health Questionnaire (GHQ-12) and their readiness to fight the pandemic together with their academic career status, their relationship with the university, and their emotional reactions to the pandemic with Visual Analog Scales.
Results:
We retrieved a GHQ mean of 21.65 (SD = 40.63) and readiness to fight the pandemic mean of 53.58 (SD = 31.49). Perceived control affects variables: a negative effect on psychological symptoms and a positive effect on the willingness to fight the pandemic. The other variables with an impact were stress, loneliness, and anger that had a significant and positive impact on psychological symptoms. Age and concern for patients had a significant positive impact on readiness to fight for the pandemic, while years of attendance had a significant but negative impact.
Conclusion:
Universities and Institutions should consider the impact of the pandemic on students, in particular, for its effect on their mental health.
PDF
