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Abstract In this paper, we provide an alternative proof for the classical Sz. Nagy inequality in one dimension by a variational method and generalize it to higher dimensions J(h):=(∫Rd|h| dx)a611∫Rd|63h|2 dx(∫Rd|h|m+1 dx)a+1m+1≥β0, where m > 0 for d = 1, 2, for , and . The Euler–Lagrange equation for critical points of in the non-negative radial decreasing function space is given by a free boundary problem for a generalized Lane–Emden equation, which has a unique solution (denoted by h c ) and the solution determines the best constant for the above generalized Sz. Nagy inequality. The connection between the critical mass for the thin-film equation and the best constant of the Sz. Nagy inequality in one dimension was first noted by Witelski et al (2004 Eur. J. Appl. Math. 15 223–56). For the following critical thin film equation in multi-dimension ht+6366(h 63Δh)+6366(h 63hm)=0,x∈Rd, where m = 1 + 2/d, the critical mass is also given by . A finite time blow-up occurs for solutions with the initial mass larger than M c . On the other hand, if the initial mass is less than M c and a global non-negative entropy weak solution exists, then the second moment goes to infinity as or in for some subsequence . This shows that a part of the mass spreads to infinity.

The finite subgroups of $GL_4(\bm{Z})$ are classified up to conjugation in \cite{BBNWZ}; in particular, there exist $710$ non-conjugate finite groups in $GL_4(\bm{Z})$. Each finite group $G$ of $GL_4(\bm{Z})$ acts naturally on $\bm{Z}^{\oplus 4}$; thus we get a faithful $G$-lattice $M$ with ${\rm rank}_\bm{Z} M=4$. In this way, there are exactly $710$ such lattices. Given a $G$-lattice $M$ with ${\rm rank}_\bm{Z} M=4$, the group $G$ acts on the rational function field $\bm{C}(M):=\bm{C}(x_1,x_2,x_3,x_4)$ by multiplicative actions, i.e. purely monomial automorphisms over $\bm{C}$. We are concerned with the rationality problem of the fixed field $\bm{C}(M)^G$. A tool of our investigation is the unramified Brauer group of the field $\bm{C}(M)^G$ over $\bm{C}$. It is known that, if the unramified Brauer group, denoted by ${\rm Br}_u(\bm{C}(M)^G)$, is non-trivial, then the fixed field $\bm{C}(M)^G$ is not rational (= purely transcendental) over $\bm{C}$. A formula of the unramified Brauer group ${\rm Br}_u(\bm{C}(M)^G)$ for the multiplicative invariant field was found by Saltman in 1990. However, to calculate ${\rm Br}_u(\bm{C}(M)^G)$ for a specific multiplicatively invariant field requires additional efforts, even when the lattice $M$ is of rank equal to $4$. There is a direct decomposition ${\rm Br}_u(\bm{C}(M)^G)= B_0(G) \oplus H^2_u(G,M)$ where $H^2_u(G,M)$ is some subgroup of $H^2(G,M)$. The first summand $B_0(G)$, which is related to the faithful linear representations of $G$, has been investigated by many authors. But the second summand $H^2_u(G,M)$ doesn't receive much attention except when the rank is $\le 3$. Theorem 1. Among the $710$ finite groups $G$, let $M$ be the associated faithful $G$-lattice with ${\rm rank}_\bm{Z} M=4$, there exist precisely $5$ lattices $M$ with ${\rm Br}_u(\bm{C}(M)^G)\neq 0$. In these situations, $B_0(G)=0$ and thus ${\rm Br}_u(\bm{C}(M)^G)\subset H^2(G,M)$. The {\rm GAP IDs} of the five groups $G$ are {\rm (4,12,4,12), (4,32,1,2), (4,32,3,2), (4,33,3,1), (4,33,6,1)} in {\rm \cite{BBNWZ}} and in {\rm \cite{GAP}}. Theorem 2. There exist $6079$ finite subgroups $G$ in $GL_5(\bm{Z})$. Let $M$ be the lattice with rank $5$ associated to each group $G$. Among these lattices precisely $46$ of them satisfy the condition ${\rm Br}_u(\bm{C}(M)^G)\neq 0$. The {\rm GAP IDs} (actually the {\rm CARAT IDs}) of the corresponding groups $G$ may be determined explicitly. A similar result for lattices of rank $6$ is found also. Motivated by these results, we construct $G$-lattices $M$ of rank $2n+2, 4n, p(p-1)$ ($n$ is any positive integer and $p$ is any odd prime number) satisfying that $B_0(G)=0$ and $H^2_u(G,M)\neq 0$; and therefore $\bm{C}(M)^G$ are not rational over $\bm{C}$.

in this paper, we construct two new primitive Evans triangles by using the Pell equation x^2-Dy^2=1, and give the two forms of the Evans triangle and the corresponding Evans ratio, we also get a new conclusion that the ratio is an integer from a few different forms of the bottom and high.

In this paper, we compute the stationary states of the multicomponent phase-field crystal model by formulating it as a block constrained minimization problem. The original infinite-dimensional non-convex minimization problem is approximated by a finite-dimensional constrained non-convex minimization problem after an appropriate spatial discretization. To efficiently solve the above optimization problem, we propose a so-called adaptive block Bregman proximal gradient (AB-BPG) algorithm that fully exploits the problem's block structure. The proposed method updates each order parameter alternatively, and the update order of blocks can be chosen in a deterministic or random manner. Besides, we choose the step size by developing a practical linear search approach such that the generated sequence either keeps energy dissipation or has a controllable subsequence with energy dissipation. The convergence property of the proposed method is established without the requirement of global Lipschitz continuity of the derivative of the bulk energy part by using the Bregman divergence. The numerical results on computing stationary ordered structures in binary, ternary, and quinary component coupled-mode Swift-Hohenberg models have shown a significant acceleration over many existing methods.