Let α_i, α_j, α_k be three distinct algebraic conjugates of an algebraic number α of degree d ≤ 8 over ℚ. In this paper by exploiting the properties of transitive permutation groups we determine all possible linear relations of the form aα_i + bα_j + cα_k = 0 with non-zero rationals a, b, c. We also provide the complete list of transitive groups which can occur as Galois groups for the minimal polynomial of α. For example, if d = 8, then we find that such relation is possible only if a² + b² = c² and the minimal polynomial of α over ℚ has Galois group isomorphic to D₄, the dihedral group of order 8. These findings extend the research of Dubickas and Jankauskas who investigated the relation aα_i + bα_j + cα_k = 0 for d ≤ 8 when a = b = c or a = b = -c.

We consider maps defined on the interior of a normal, closed cone in a real Banach space that are nonexpansive with respect to Thompson's metric. With mild compactness assumptions, we prove that the Krasnoselskii iterates of such maps converge to a fixed point when one exists. For maps that are also order-preserving, we give simple necessary and sufficient conditions in terms of upper and lower Collatz-Wielandt numbers for the fixed point set to be nonempty and bounded in Thompson's metric. When the map is also real analytic, these conditions are both necessary and sufficient for the map to have a unique fixed point and for all iterates of the map to converge to the fixed point. We demonstrate how these results apply to certain nonlinear matrix equations on the cone of positive definite Hermitian matrices.

In this article, we consider the boundedness for a class of parameterized Littlewood-Paley integrals and their commutators. More precisely, let Ω ∈ L² (S^n-1) be a homogeneous function of degree zero. We prove that parameterized Littlewood-Paley area integral µ^ρ_Ω,S, g^∗_λ function µ^∗,ρ_Ω,λ are bounded on two weighted Herz spaces with variable exponents. It is worth noting that above operators in two weighted Herz spaces with variable exponents are more complex than operators themselves. Moreover, let b be a BMO function, the boundedness of commutators generated by b and parameterized Littlewood-Paley operators will also be showed.

Various methods exist to address the challenges posed by moving domain problems, including the transform method, penalty method, and push-forward and pull-back method. These methods play a crucial role in transforming non-cylindrical domain problems into cylindrical domain problems. In [15], the existence of at least one weak solution for a more general extensible beam was established. However, this paper does not specifically address the uniqueness of solutions and explore their relationship within any finite-dimensional space. Our primary objective is to formulate approximate solutions in a finite-dimensional space using the penalty method, with a focus on achieving uniqueness. Furthermore, we present an illustrative example to enhance the understanding of the concepts discussed.

In this paper we will be concerned with the problem ${\Delta}\,(a{\mid}{\Delta}u{\mid}^p){\mid}{\Delta}u{\mid}^{p-2}{\Delta}u)\;=\,f(u)\;in\;{\Omega},\;u\;=\;{\frac{{\partial}u}{{\partial}n}}\;=\;0\;on\;{\partial}{\Omega},$ where Ω ⊂ ℝ^N is a bounded smooth domain with ${1\,<\,p\,<\,{\frac{N}{2}}}$, f : ℝ → ℝ is a superlinear continuous function with exponential subcritical or exponential critical growth and the function a is C¹. We use as our main tool the Nehari manifold method, supplemented by the application of the quantitative deformation lemma and degree theory results. Our results encompass a broad class of problems.

We investigate the existence, multiplicity and nonexistence of positive solutions to n × n cycled systems of nonlinear differential equations with degenerated p-Laplacians and nonlinearities satisfying combined p-linear condition at 0 and combined p-sublinear, p-superlinear or p-linear condition at ∞. In particular, the existence of at least three positive solutions is discussed. This is even new for n × n cycled systems of nonlinear differential equations with degenerated Laplacians. We carry on only one fixed point theorem for proofs of all cases.

In this paper, we investigate a coordinate-free study of the first approximation Matsumoto metric in a more general manner. Namely, for a Finsler metric (M, L) and a one form B, we study some geometric objects associated with the Matsumoto metric ${\tilde{L}}(x,\,y)\;=\;L(x,\,y)\,+\,{\mathfrak{B}}(x,\,y)\,+\,{\frac{{\mathfrak{B}}^2(x,y)}{L(x,y)}}$ in terms of the objects of L. Here we consider L is Finslerian and so we call ${\tilde{L}}$ the generalized Matsumoto metric. We find the metric and Cartan tensors and other geometric objects associated with ${\tilde{L}}$. We characterize the non-degeneracy of the metric tensor of ${\tilde{L}}$. We find the geodesic spray, Barthel connection and Berwald connection of ${\tilde{L}}$(x, y) when the one form 𝕭 is associated to a concurrent π-vector field. Then, we calculate the curvature of the Barthel connection of ${\tilde{L}}$. To illustrate our primary results, one example is given.

Let P₈(x)=3x²-2x. A polynomial of the form a₁P₈(x₁)+a₂P₈(x₂)+· · ·+a_kP₈(x_k) (a_i ∈ ℕ) is called a k-ary octagonal form. An octagonal form is called regular if it represents every positive integer which is locally represented. In this paper, we prove that there are at most 15 regular ternary octagonal forms and establish the regularity of 12 forms.

In this paper, the first thing we prove is that a weakly Einstein cosymplectic 3-manifold is flat and Einstein. Next, we prove that a strictly almost cosymplectic 3-manifold M is weakly Einstein if and only if M has the Ricci tensor of rank one. In particular, if M is strictly H-almost cosymplectic 3-manifolds, then it is locally isomorphic to the Minkowski motion group E₁,₁ equipped with a left invariant almost cosymplectic structure with a² = b². Moreover, we find that there does not exist a weakly Einstein strictly almost cosymplectic 3-manifold with ∇ξh = -2αhφ, for any non-zero constant α.

On foliations, there are two kinds of harmonic maps, that is, transversally harmonic map and (𝓕, 𝓕')-harmonic map which are equivalent when the foliation is minimal. In this paper, we study the properties of transversally f-harmonic and (𝓕, 𝓕')_f-harmonic maps on weighted Riemannian foliations. Here, f is a basic function. And we investigate the relations between transversal stress-energy tensors and transversally f-harmonic maps.

In this paper, we aim to study the relations between dynamical properties of a homeomorphism (or diffeomorphism) f on a Riemannian manifold (M, g) and its induced lifting map ${\widetilde{f}}$ on the universal covering (${\widetilde{M}}$, ${\widetilde{g}}$) of (M, g). We prove that the L-bounded (asymptotic) average shadowing properties for a homeomorphism f on (M, g) and the lifting ${\widetilde{f}}$ of f on (${\widetilde{M}}$, ${\widetilde{g}}$) are equivalent but the two-sided limit shadowing properties with a gap of each space are not equivalent. We also prove that with a gap, the notion of the unique two-sided limit shadowing property is completely obtained by the notion of product Anosov diffeomorphism. It is a generalized version of [8, Theorem A]. Moreover, for the case that M admits a universal covering ${\widetilde{M}}$ with non-positive sectional curvature, we get the set of all points that two-sided limit shadow with a gap as finding fixed points of a map defined on the space of sequences in tangent spaces. Finally we find the sufficient conditions to get points which two-sided limit shadow with a gap.