Ricerche di Matematica最新文献

Neural integral equations are deep learning models based on the theory of integral equations, where the model consists of an integral operator and the corresponding equation (of the second kind) which is learned through an optimization procedure. This approach allows to leverage the nonlocal properties of integral operators in machine learning, but it is computationally expensive. In this article, we introduce a framework for neural integral equations based on spectral methods that allows us to learn an operator in the spectral domain, resulting in a cheaper computational cost, as well as in high interpolation accuracy. We study the properties of our methods and show various theoretical guarantees regarding the approximation capabilities of the model, and convergence to solutions of the numerical methods. We provide numerical experiments to demonstrate the practical effectiveness of the resulting model.

We establish the unique existence of a strong solution to a Dirichlet problem involving a second-order elliptic operator in Hardy spaces. The strong solution also enjoys a regularity estimate in Hardy quasi-norms up to the second-order derivatives.

We examine the following weighted degenerate elliptic equation involving the Grushin operator:

where (x'=(x_{1},...,x_{m})in mathbb {R}^m,) (1le mle N,) (vartheta _{s} in C(mathbb {R}^m, mathbb {R})) is a continuous positive function satisfying

and (Delta _s) is an operator of the form

Under some general hypotheses of the functions (s_i,;i=1,dots , k,) we establish some new Liouville type theorems for stable solutions of this equation for a large classe of weights. Our results recover and considerably improve the previous works (Mtiri in Acta Appl Math 174:7, 2021; Farina and Hasegawa in Proc Royal Soc Edinburgh 150:1567, 2020).

Let R be any non-commutative prime ring of char ((R)ne 2), L a non-central Lie ideal of R and F, G be b-generalized skew derivations of R. Suppose that

for all (uin L) and for some fixed integer (nge 1), then one of the following assertions holds:

1. (1)

   there exist (a'',b''in Q_r) such that (F(x)=xa''), (G(x)=b''x) for all (xin R) with (a''-b''in C);

2. (2)

   (Rsubseteq M_2(K),) the algebra of (2times 2) matrices over a field K and

   • either K is a finite field;

   • or there exists (lambda in C) such that ((F+G)(x)=lambda x) for all (xin R);

   • or there exists (lambda in C) and (hin Q_{r}) such that ((F+G)(x)=hx+xh+lambda x) for all (xin R).

The above result, naturally improves the recent result obtained by Carini et al. in [4].

We provide symmetrization results as mass concentration comparisons for solutions to singular parabolic equations in the cylinder (Omega times (0,T)), (T>0). Here, (Omega subset {mathbb {R}}^N) ((N ge 2)) is a bounded open set, featuring a lower order term that is singular in the solution variable s.

The aim of this paper is to defined the quotient gamma nearness rings and to examine its properties. We generalize an important theorem for quotient gamma nearness rings. More clearly, we prove the following theorem: Let (Mne left{ 0_{M}right} ) be a commutative (Gamma )-nearness ring such that (N_{r}(B)^{*}(N_{r}(B)^{*}M)=N_{r}(B)^{*}M), P be a ( Gamma )-nearness ideal of M such that (N_{r}(B)^{*}(N_{r}(B)^{*}P)=N_{r}(B)^{*}P), and (sim _{B_{r}}) be a congruence indiscernibility relation on M. Then, P is a prime (Gamma )-nearness ideal if and only if M/P is a (Gamma )-nearness integral domain.

This paper investigates the impact of spatial factors on epidemic dynamics using a two-stage contagion model with two possible outcomes, whose probability of occurrence depends of a parameter (q in [0,1]). The model considers direct contagion ((q=1)), where contact with an ill individual causes a susceptible individual to become infected, and indirect contagion ((q=0)), where a susceptible individual, in contact with an infected individual, does not become ill but, enters an exposed intermediate stage, weakening their resistance to the disease. To incorporate spatial effects, we first define a rigorous integro-differential model expressed in terms of the densities of each class into which the population can be divided. The time evolution of these variables is determined by integrals that account for the range of influence of the contagion over the susceptible classes. Under certain conditions, this integro-differential system can be reduced to a system of ordinary differential equations for the average population in each class. Nonetheless, even for low-dimensional cases, the solution to this extended system appears to evade analytical methods. Alternatively, we represent the population using a cellular automata model on a two-dimensional square grid. We demonstrate that the outcome is influenced by the neighborhood size r; in the limit of large r, the model converges to the mean field approximation, where interactions follow a law of mass action. We specifically explore the impact of the initial spatial configuration on the population’s asymptotic behavior, highlighting its significance in cases of local bistability. Additionally, we establish that oscillatory spatial behavior can emerge when considering a recovered or death class. These findings lay the groundwork for future studies on multi-stage contagion in complex networks.

Let H be a normal subgroup of a group G. The normal subgroup based power graph (Gamma _H(G)) of G is the simple undirected graph with vertex set (V(Gamma _H(G))= (Gsetminus H)cup {e}) and two distinct vertices a and b are adjacent if either (aH = b^m H) or (bH=a^nH) for some (m,n in mathbb {N}). In this paper, we continue the study of normal subgroup based power graph and characterize all the pairs (G, H), where H is a non-trivial normal subgroup of G, such that the genus of (Gamma _H(G)) is at most 2. Moreover, we determine all the subgroups H and the quotient groups (frac{G}{H}) such that the cross-cap of (Gamma _H(G)) is at most three.

Consider (xin {mathbb {N}}setminus {0}). A QMED(x)-semigroup is a numerical semigroup S such that (S{setminus }{0}={a+kx mid ain {text {msg}}(S) text{ and } kin {mathbb {N}}}) where ({text {msg}}(S)) denotes the minimal system of generators of S. Note that if x is the multiplicity of S then S is a maximal embedding dimension numerical semigroup. In this work, we show that the set of all QMED(x)-semigroups is a Frobenius pseudo-variety giving the associated tree. Furthermore, we give formulas to obtain the Frobenius number, type, and genus of this class of semigroups.

In the present letter, we deal with the nonlocal operators which are the generators of the tempered stable processes. It is shown that the semigroups generated by the nonlocal operators are analytic in (L^p(mathbb {R}^d)) for any (p in [1,infty )). In particular, the result holds for not only exponentially tempered stable processes but also algebraically tempered ones.