In this paper, we provide a new invariant for partial differential equations (PDEs) under contact transformations by using nilpotent graded Lie algebras. By virtue of this invariant, various geometric behavior of PDEs can be understood. As a typical class, we clarify geometric behavior of second-order PDEs in terms of our invariant.

Let $P\geq 3$ be an integer. Let $(V_{n})$ denote generalized Lucas sequence defined by $V_{0}=2$, $V_{1}=P$, and $V_{n+1}=PV_{n}-V_{n-1}$ for $n\geq 1$. In this study, when $P$ is odd, we solve the equation $V_{n}=wx^{2}$ for some values of $w$. Moreover, when $P$ is odd, we solve the equation $V_{n}=wkx^{2}$ with $k \mid P$ and $k \gt 1$ for $w=3,11,13$. Lastly, we solve the equation $V_{n}=wV_{m}x^{2}$ for $w=7,11,13$.

We obtain a complete classification of finite groups in which all noncyclic proper subgroups are nonnormal, and we apply this classification to investigate some structures of finite groups.

This paper deals with the global existence of weak solutions to the initial value problem for the Navier-Stokes equations in $\mathbb{R}^{n}$ ($n \in \mathbb{Z}$, $n\geq 2$). Concerning initial data of the form $Ax+v(0)$, where $A \in M_{n}(\mathbb{R})$ and $v(0) \in L^{2}_{\sigma}(\mathbb{R}^{n})$, the weak solutions are properly-defined with the aid of the alternativity of the trilinear from $(Ax\cdot\nabla)v\cdot\varphi$. Furthermore, we construct the Leray-Hopf weak solution which satisfies not only the Navier-Stokes equations but also the energy inequality via the Galerkin approximation. From the viewpoint of quadratic forms, the Gronwall-Bellman inequality admits the uniform boundedness of the approximate solution.

The nonstationary Navier-Stokes system for a viscous, incompressible fluid influenced by a Coriolis force in the whole space ${\mathbb R}^3$ is considered at large distances. The solvability of the corresponding integral equations of these equations in weighted $L^\infty$-spaces is established. Furthermore, the leading terms of the asymptotic profile of the solution at fixed time $t \gt 0$ for $|x| \gt t$ and far from the axis of rotation are investigated.

A tropical polynomial is called $R$-polynomial if it can be realized as the minimum finishing time of a project network. $R$-polynomials satisfy the term extendability condition, and correspond to simple graphs. We give a characterization of $R$-polynomials in terms of simple graphs.

Let $P^{k}(n,p \times q)$ be the set of all pairs of real polynomial map germs $(f, g) : (\mathbb{R}^{n},0) \rightarrow (\mathbb{R}^{p} \times \mathbb{R}^{q} ,0)$ with degree of $ f_1 , \dots, f_p ,$ $g_1 ,\dots, g_q$ less than or equal to $k \in \N$. The main result of this paper shows that the set of equivalence classes of $P^{k}(n,p \times q)$, with respect to bi-$C^{0}$-$\mathcal{K}$-equivalence, is finite.

Noether's First Theorem guarantees conservation laws provided that the Lagrangian is invariant under a Lie group action. In this paper, via the concept of Killing vector fields and the Minkowski metric, we first construct an important Lie group, known as Hyperbolic Rotation-Translation group. Then, according to Gonçalves and Mansfield's method, we obtain the invariantized Euler-Lagrange equations and the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame, for Lagrangians, which are invariant under Hyperbolic Rotation-Translation (or HRT) group action, in the case where the independent variables are not invariant.

In this paper, we prove some rigidity theorems for compact Bach-flat $n$-manifold with the positive constant scalar curvature. In particular, our conditions in Theorem 2 have the additional properties of being sharp.

For a hyperbolic infinite network, it is well-known that Green potentials with finite energy are Dirichlet potentials. Conversely, if a Dirichlet potential has non-positive Laplacian, then it is a Green potential with finite energy. In this paper, we study whether a Dirichlet potential can be expressed as a difference of two Green potentials with finite energy. Comparisons of the Dirichlet sum of a function and that of its Laplacian play important roles in our study. As a by-product, we obtain a Riesz decomposition of a function whose Laplacian is a Dirichlet function.

For an $(n-1)$-dimensional compact orientable smooth metric measure space $\big(M,g,e^{-f}dv_{g}\big)$ embedded in an $n$-dimensional compact orientable Riemannian manifold $N$, we successfully give a lower bound for the first nonzero eigenvalue of the drifting Laplacian on $M$, provided the Ricci curvature of $N$ is bounded from below by a positive constant and the weighted function $f$ on $M$ satisfies two constraints.

On a smooth foliated map from a complete, possibly non-compact, foliated Riemannian manifold into another foliated Riemannian manifold of which transversal sectional curvature is non-positive, we will show that, if it is transversally biharmonic and has the finite energy and finite bienergy, then it is transversally harmonic.

We investigate deformations of functions on affine space, deformations in which the changes specialize to a distinguished point in the zero-locus of the original function. Such deformations – deformations with isolated polar activity – enable us to obtain nice results on the cohomology of the Milnor fiber of the original function.