In recent years, several efforts have focused on identifying new families of scattered polynomials. Currently, only three families in $$\mathbb {F}_{q^n}[X]$$ are known to exist for infinitely many values of n and q : (i) pseudoregulus-type monomials, (ii) Lunardon–Polverino-type binomials, and (iii) a family of quadrinomials studied in a series of papers. In this work, we provide sufficient conditions under which these quadrinomials, denoted by $$\psi _{m,h,s}$$ , are scattered. Our results both include and generalize those obtained in previous studies. We also investigate the equivalences between the previously known families of scattered polynomials and those in this new class.
In this paper, we study the reproducing Bergman kernel on smooth bounded domains in the eight-dimensional non-associative octonionic algebra. We establish necessary and sufficient conditions on associator terms involving an intrinsic octonionic weight factor that ensure the existence of a Bergman kernel for square-integrable solutions to the octonionic Cauchy-Riemann equation. The kernel is constructed explicitly through a correspondence with the Green’s function of the domain, yielding closed-form representation formulas for concentric annuli, where the well-known formula for the solid ball is obtained as a limit case. In these cases, the kernel is Hermitian and induces the Bergman projection from the L_2 space of monogenic functions onto the Bergman space. In contrast, for half-balls, orthogonal sectors, and annular sectors, the associator condition fails, and no Hermitian Bergman kernel exists when using the standard weight factor x/|x| for the spherical setting. In such cases, we construct a unique non-Hermitian generalized kernel that reproduces all octonionic monogenic functions, though lacking Hermitian symmetry and therefore not serving as a complete L_2 -projection.
We study the microlocal regularity of the analytic/Gevrey vectors for the following class of second order partial differential equations P(x,D) = ∑ _ℓ ,j=1^n a_ℓ ,j(x) D_ℓ D_j + ∑ _ℓ =1^n i b_ℓ(x) D_ℓ +c(x), where a_ℓ ,j(x) = a_j,ℓ(x) , b_ℓ(x) , ℓ ,j ∈{ 1,… , n} , are real valued real Gevrey functions of order s and c(x) is a Gevrey function of order s, s ≥ 1 , on Ω open neighborhood of the origin in ℝ^n . Thus providing a microlocal version of a result due to Derridj in (Complex Anal. Synerg. 6:10, 2020). The class of operators considered in this work generalizes the classes studied by Braun Rodrigues-Chinni-Cordaro-Jahnke (Proc. Am. Math. Soc. 144:5159–5170, 2016) and Chinni-Derridj (Math. Z. 302:1983–2003, 2022).
Let q ≥ 2 be an integer, ζ (s) be the Riemann zeta function, and put T_q (s):= (s+1)(1-s)^-1 (q^s+2-1) ζ (s+2) - 4 π ^2 s^-1 (1-s)^-1(q^3-s-1) ζ (s-2) . In the present paper, we show that the function T_q (s) has Riemann’s functional equation and its zeros only at the negative even integers and satisfies the Lindelöf and Riemann hypotheses. In addition, we give functions satisfy Riemann’s functional equation and an analogue of the Lindelöf hypothesis but do not fulfill an analogue of the Riemann hypothesis.
We prove sharp L^p - L^q -estimates for the Restriction-Extension operator acting on block-radial functions with the aid of new oscillatory integral estimates and interpolation results in mixed Lorentz spaces. We apply this to the Limiting Absorption Principle for elliptic (pseudo-)differential operators with constant coefficients. In this way we obtain a richer existence theory for Helmholtz-type problems on ℝ^d with block-radial right hand sides.
Abstract We study the $$L^2$$ -gradient flows, $$\partial _t u-\operatorname {div}(\textrm{D}f(x,\mathbb {A}u))=0$$ , of functionals of the type $$\int _{\Omega }f(x,\mathbb {A}u)\,\textrm{d}x$$ , where f is a convex function of linear growth and $$\mathbb {A}$$ is some first-order linear constant-coefficient differential operator. To this end, we identify the relaxation of the functional to the space $$BV^\mathbb {A}\cap L^2$$ , identify its subdifferential, and show pointwise representation formulas for the relaxation and the subdifferential, both with and without Dirichlet boundary conditions. The existence and uniqueness then follow from abstract semigroup theory. We further show that our solutions can be obtained as limits of the corresponding flows with p -growth as $$p\searrow 1$$ .
Let X be a real algebraic set. In the literature, a (real-valued) function φ on X is called regulous if its restriction to each algebraic subset of X is a continuous rational function. A function f on X is called quasi-regulous if it is continuous and f^2=φ ^2 for some regulous function φ on X. Assuming X is nonsingular, we prove that a function f on X is quasi-regulous if and only if its restriction to each algebraic curve in X is quasi-regulous. We also prove three other variants of this result.
We investigate the geometry of the bitension field of Lagrangian surfaces in complex space forms. We first give a classification of Hamiltonian stationary biminimal Lagrangian surfaces in complex space forms, which allows us to show that Hamiltonian stationary biharmonic Lagrangian surfaces in the complex plane are minimal. Then we investigate the geometry of biminimal or biconservative Lagrangian surfaces in complex space forms. Some classifications of Lagrangian surfaces with constant mean curvature or parallel normalized mean curvature field are obtained.
In this paper, we establish a comparison theorem between positive solutions of the weighted Poisson equation with nonhomogeneous Neumann boundary conditions and suitable positive solutions of its Schwarz symmetrized problem satisfying appropriate matching boundary conditions. We also prove comparison results for the weighted Poisson equation with homogeneous Neumann boundary conditions. Our main tools include weighted isoperimetric inequalities and symmetrization techniques.
In this article, we study the break-down of smooth and continuous solutions to isentropic Euler system in multi dimension. Sideris (Comm Math Phys 101(4):475-478, 1985) proved the blow up of smooth solutions when initial data satisfies an 'integral condition'. We show that a C1 solution of isentropic Euler equation breaks down if (i) gradient of initial velocity has a negative real eigenvalue at some point x0 is an element of Rd and (ii) Hessian of initial density satisfies a smallness condition in Sobolev space. Our proof also works for the data which fails to satisfy the above-mentioned 'integral condition'. Furthermore, we prove the global existence of smooth solution when (i) eigenvalues of gradient of initial velocity have non-negative real-part and (ii) initial density satisfies a smallness condition. This extends the global existence result of Grassin (Indiana Univ Math J 47(4):1397-1432, 1998). Another goal of this article is to study the breakdown of continuous weak solutions of isentropic Euler equations. We are able to show that the 'integral condition' of Sideris can cause the breakdown of continuous solutions in finite time. This improves the blow up result of Sideris from C1 to continuous space.
We correct an error in a formula in [1] by adding a supplementary natural geometric hypotesis on the considered surfaces.
A new family of stochastic pre-orders is introduced. We focus on the cases in which two variables are not comparable in the usual stochastic order and study how close they are to be comparable in such order. In particular, we require that the difference between the cumulative distribution functions of the random variables does not change sign up to a certain quantile. Then, the higher the order of the quantile, the closer is the relation to the standard usual stochastic order. An analogous definition can be given in terms of upper quantiles as well, and the one associated to the quantile of higher order has to be preferred. We study some properties of the proposed pre-orders and some connections with other stochastic orders. Then, these relations are applied in the context of reliability theory based on residual and past lifetimes. Finally, they are employed in the study of lifetimes of coherent systems under different assumptions on the components’ lifetimes.
We consider planar Hamiltonian systems with Hamiltonian function of the type H(x,y)=F(y)+G(x) having the origin as a global center, and generalize to this setting some known results on the period map. We are thus led to a characterization of isochronism for scalar second order equations involving the p-Laplacian operator.
We investigate a class of fourth-order elliptic problems involving exponential-type nonlinearities and spatial weights of Hénon type. Motivated by the symmetry-breaking phenomena observed in semilinear second-order problems—such as those governed by the Hénon equation—we consider weighted functionals of the form F_m(u) = ∫ _B |x|^α( e^σ |u|^2 - ∑ _k=0^m σ ^k/k! |u|^2k) dx, defined on the unit ball B ⊂ℝ^4 , where m∈ℕ_0 α > 0 , σ >0 are suitable parameters. We first establish an Adams-type inequality with weight, characterizing the sharp threshold for the boundedness of F on the unit sphere of the biharmonic Sobolev space. Then, we prove that for large values of the weight exponent α , radial symmetry of maximizers is broken. These results extend classical findings in the second-order setting (e.g., Trudinger–Moser-type functionals and the weighted Hénon equation) to the biharmonic context and offer new insights into the interplay between weights, nonlinearity, and symmetry in higher-order PDEs.
We work with Besov spaces with Lorentz smoothness B^s _q L_p,r (ℝ^n ) . Here -∞
In this article, we show the existence and uniqueness of a family of complete elliptic Weingarten surfaces of minimal type, which are invariant by a uniparametric group of horizontal translation. Moreover, we give a complete description of the generating curve of each of these surfaces when the elliptic function satisfies f(x)>0 for x 0 .
