The number of real eigenvectors of a real polynomialMaccioni, Mauro
2017 Bollettino dell'Unione Matematica Italiana
doi: 10.1007/s405740160112y
I investigate on the number t of real eigenvectors of a real symmetric tensor. In particular, given a homogeneous polynomial f of degree d in 3 variables, I prove that t is greater or equal than
$$2c+1$$
2
c
+
1
, if d is odd, and t is greater or equal than
$$\max (3,2c+1)$$
max
(
3
,
2
c
+
1
)
, if d is even, where c is the number of ovals in the zero locus of f. About binary forms, I prove that t is greater or equal than the number of real roots of f. Moreover, the above inequalities are sharp for binary forms of any degree and for cubic and quartic ternary forms.
On qBessel Fourier analysis method for classical moment problemDhaouadi, Lazhar
2017 Bollettino dell'Unione Matematica Italiana
doi: 10.1007/s4057401601158
In the first part of this paper, we give a sufficient condition for a particular case of the symmetric moment problem to be determinate using standards methods of qBessel Fourier analysis. This condition it cannot be deduced from any other classical criterion of determinacy. In the second part, we study the qStrum–Liouville equation in the nonreal case and we elaborate an analogue of the well known theorem due to Hermann Weyl concerning the Strum–Liouville equation. This emphasizes the connection between the moment problem associated to a particular class of orthonormal polynomials
$$(P_n)$$
(
P
n
)
and the uniqueness of solution which belong to the
$$L^2$$
L
2
space. The third part is devoted to the study of the qStrum–Liouville equation in the real case and the behavior of solutions at infinity, which give more information about this type of orthonormal polynomials.
A nonlinear elliptic boundary value problem relevant in general relativity and in the theory of electrical heating of conductorsCimatti, Giovanni
2017 Bollettino dell'Unione Matematica Italiana
doi: 10.1007/s4057401701215
The elliptic boundary value problem governing the steady electrical heating of a conductor of heat and electricity, the socalled thermistor problem,
$$\begin{aligned}&{\nabla }\cdot ({\sigma }(u){\nabla }\phi )=0\ {\quad \hbox {in}\ {\Omega }}\quad \phi =\phi _b\ {\quad \hbox {on}\ {\Gamma }}\\&{\nabla }\cdot ({\kappa }(u){\nabla }u)={\sigma }(u){\nabla }\phi ^2\ {\quad \hbox {in}\ {\Omega }}\quad u=0\ {\quad \hbox {on}\ {\Gamma }}, \end{aligned}$$
∇
·
(
σ
(
u
)
∇
ϕ
)
=
0
in
Ω
ϕ
=
ϕ
b
on
Γ
∇
·
(
κ
(
u
)
∇
u
)
=

σ
(
u
)

∇
ϕ

2
in
Ω
u
=
0
on
Γ
,
where
$${\sigma }(u)$$
σ
(
u
)
is the temperature dependent electric conductivity and
$${\kappa }(u)$$
κ
(
u
)
the thermal conductivity, admits a reinterpretation in the framework of general relativity if we choose
$${\sigma }(u)=e^u$$
σ
(
u
)
=
e
u
,
$${\kappa }(u)=1$$
κ
(
u
)
=
1
and, in addition,
$${\Omega }$$
Ω
is a domain of
$${\mathbf{R}^3}$$
R
3
axially symmetric whereas the function
$$\phi _b$$
ϕ
b
, in a cylindrical coordinate system
$${\rho },z,{\varphi }$$
ρ
,
z
,
φ
, is independent of
$${\varphi }$$
φ
. The same analytical methods relevant in the thermistor problem can be used in this new context.
Tangents to Chow groups: on a question of Green–GriffithsDribus, Benjamin; Hoffman, J.; Yang, Sen
2017 Bollettino dell'Unione Matematica Italiana
doi: 10.1007/s4057401701233
We examine the tangent groups at the identity, and more generally the formal completions at the identity, of the Chow groups of algebraic cycles on a nonsingular quasiprojective algebraic variety over a field of characteristic zero. We settle a question recently raised by Mark Green and Phillip Griffiths concerning the existence of Bloch–Gersten–Quillentype resolutions of algebraic Ktheory sheaves on infinitesimal thickenings of nonsingular varieties, and the relationships between these sequences and their “tangent sequences,” expressed in terms of absolute Kähler differentials. More generally, we place Green and Griffiths’ concrete geometric approach to the infinitesimal theory of Chow groups in a natural and formally rigorous structural context, expressed in terms of nonconnective Ktheory, negative cyclic homology, and the relative algebraic Chern character.
Uniqueness of solutions to some quasilinear elliptic equations whose Hamiltonian has natural growth in the gradientArtola, Michel
2017 Bollettino dell'Unione Matematica Italiana
doi: 10.1007/s4057401701304
The paper discusses uniqueness of solutions to stationary elliptic problems of the type
$$\begin{aligned} A(u)+H(u)=f\in {\mathcal {D}}'(\Omega ), \end{aligned}$$
A
(
u
)
+
H
(
u
)
=
f
∈
D
′
(
Ω
)
,
where
$$\Omega \ \in R^{N},\ $$
Ω
∈
R
N
,
$$u\in W^{1,p}(\Omega )\ (1\le p\le +\infty ),\ A(u)\ $$
u
∈
W
1
,
p
(
Ω
)
(
1
≤
p
≤
+
∞
)
,
A
(
u
)
is an elliptic operator,
$$H(u)\ $$
H
(
u
)
is an Hamiltonian that grows with
$$\left {\nabla u}\right ^{p}$$
∇
u
p
and f is given. Methods introduced in Artola (Boll UMI 6(5B):51–71, 1986), (Proceedings of the International Conference on Generalized Functions, (ICGF 2000). Cambridge Scientific Publishers, Cambridge, 51–92, 2004), (Ricerche di Matematica XLIV, fasc. 2:400–420, 1995) for quasilinear parabolic or elliptic equations, together with properties for some continuity moduli, are used to improve some results from Barles and Murat (Arch Ration Mech Anal 133(1):77–101, 1995) for bounded solutions and from Barles and Porretta (Ann Scuola Norm Sup Pisa Cl Sci 5(1):107–136, 2006), Lions (J Anal Math 45: 234–254, 1985) for unbounded solutions, when 1
$$\le p\le 2.$$
≤
p
≤
2
.
Unilateral problems are considered and the case where f depends on the solution u is also discussed.