Abstract Cohomological equation is of special interest because it concerns the study of time change for flows, topological stability and topological conjugacy in dynamical systems, which is also an auxiliary equation to study the problem of linearization. In this paper, we consider a general form of cohomological equation for planar contractions. By using the ideas of invariant manifold and estimations in [W. Zhang and W. Zhang, dollar sign upper C 1 dollar sign $C^1$ C 1 linearization for planar contractions, J. Funct. Anal. 260 (2011), 2043–2063.], we present new criteria on eigenvalues of the linear parts for the existence of dollar sign upper C 1 dollar sign $C^1$ C 1 solutions in the Poincaré domain. Our results are a generalization of dollar sign upper C 1 dollar sign $C^1$ C 1 linearization for contractions.
This paper addresses three key objectives. First, weighted Sobolev trace embeddings are studied for a class of function spaces defined in the upper half-space. Second, Liouville-type theorems are proved, providing essential insights into the nonexistence of solutions for a class of quasilinear elliptic problems with indefinite boundary conditions under specific constraints. Third, by combining the derived embedding results with the fibering method, the existence of solutions for such problems is established. These findings contribute to a deeper understanding of the analytical and variational properties of these elliptic equations.
Two-sort species yield differential equations for functional digraphs of Cayley permutations. From these, we obtain an explicit formula for fixed-point-free Cayley permutations and prove that their proportion tends to 1 slash e $1/e$1/e, as for permutations and endofunctions. Our approach also yields counting formulas when the functional digraph is a tree, forest, or connected.
Let Omega be a lattice in C with invariants g(2), g(3) and let & wp;(z), zeta(z) be the associated Weierstrass elliptic and zeta functions, respectively. In this paper, we prove that if omega is any non-zero period of & wp;(z) and u(1), u(2) complex numbers such that u(1), u(2), omega are Q-linearly independent with (Zu(1) +Zu(2)) boolean AND Omega = (0), then at least two of the numbers g2, g3, omega, eta, u1, u2, & wp;(u(i)), zeta(u(i)) (1 <= i <= 2) are algebraically independent over Q, where eta is the quasi-period of zeta(z) associated with omega.
Cohomological equation is of special interest because it concerns the study of time change for flows, topological stability and topological conjugacy in dynamical systems, which is also an auxiliary equation to study the problem of linearization. In this paper, we consider a general form of cohomological equation for planar contractions. By using the ideas of invariant manifold and estimations in [W. Zhang and W. Zhang, dollar sign upper C 1 dollar sign $C<^>1$ C 1 linearization for planar contractions, J. Funct. Anal. 260 (2011), 2043-2063.], we present new criteria on eigenvalues of the linear parts for the existence of dollar sign upper C 1 dollar sign $C<^>1$ C 1 solutions in the Poincar & eacute; domain. Our results are a generalization of dollar sign upper C 1 dollar sign $C<^>1$ C 1 linearization for contractions.
Let dollar sign rs left parenthesis n right parenthesis dollar sign $r_s(n)$ r s ( n ) denote the number of representations of dollar sign n dollar sign $n$ n as a sum of dollar sign s dollar sign $s$ s squares. Hurwitz established eleven identities expressing the generating function of dollar sign r 3 left parenthesis an plus b right parenthesis dollar sign $r_3(an+b)$ r 3 ( a n + b ) as a simple infinite product. Cooper and Hirschhorn (Discrete Math 274 (1-3):9-24, 2004) proved that for any dollar sign k greater than or equals 0 dollar sign $k\geq0$ k >= 0 , the generating functions dollar sign sigma summation n equals 0 infinity r 3 left parenthesis 32 kn right parenthesis qn dollar sign $\sum_{n=0}<^>\infty r_3\big(3<^>{2k}n\big)q<^>n$ & sum; n = 0 infinity r 3 ( 3 2 k n ) q n and dollar sign sigma summation n equals 0 infinity r 3 left parenthesis 32 k plus 1 n right parenthesis qn dollar sign $\sum_{n=0}<^>\infty r_3\big(3<^>{2k+1}n\big)q<^>n$ & sum; n = 0 infinity r 3 ( 3 2 k + 1 n ) q n can be written as linear combinations of two specified generalized eta-quotients. In this paper, we substantially extend these results to high dimensions. Specifically, we prove that for any dollar sign k greater than or equals 0 dollar sign $k\geq0$ k >= 0 and dollar sign 3 less than or equals s less than or equals 100 dollar sign $3\leq s\leq 100$ 3 <= s <= 100 , the generating functions dollar sign sigma summation n equals 0 infinity rs left parenthesis 32 k plus 1 n right parenthesis qn dollar sign $\sum_{n=0}<^>\infty r_s\big(3<^>{2k+1}n\big)q<^>n$ & sum; n = 0 infinity r s ( 3 2 k + 1 n ) q n and dollar sign sigma summation n equals 0 infinity rs left parenthesis 32 k plus 2 n right parenthesis qn dollar sign $\sum_{n=0}<^>\infty r_s\big(3<^>{2k+2}n\big)q<^>n$ & sum; n = 0 infinity r s ( 3 2 k + 2 n ) q n can also be expressed as linear combinations of certain generalized eta-quotients. Motivated by these results, we conjecture that this phenomenon holds for dollar sign rs left parenthesis n right parenthesis dollar sign $r_s(n)$ r s ( n ) for all dollar sign s greater than or equals 3 dollar sign $s\geq3$ s >= 3 , and further that dollar sign rs left parenthesis n right parenthesis dollar sign $r_s(n)$ r s ( n ) satisfies an infinite family of internal congruences modulo high powers of dollar sign 3 dollar sign $3$ 3 .
Let M be a compact three-dimensional Riemannian manifold with non-negative Ricci curvature and a non-empty boundary partial derivative M. Fraser and Li [2] established a compactness theorem for the space of compact, properly embedded minimal surfaces of fixed topological type in M with a free boundary on partial derivative M, assuming that partial derivative M is strictly convex with respect to the inward unit normal. In this paper, we show that the strict convexity condition on partial derivative M cannot be relaxed.
We study complete noncompact spacelike mean curvature flow solitons (SMCFS) in a standard static spacetime obeying a suitable constraint on the sectional curvature. In this context, we prove a version of the Omori-Yau generalized maximum principle and apply it to deduce that such an SMCFS must be maximal in the sense that its mean curvature vanishes identically. Next, we use other maximum principles which deal with the notions of convergence to zero at infinity and polynomial volume growth to prove rigidity results for SMCFS. Furthermore, we apply our previous results to establish nonexistence results concerning entire Killing graphs constructed over the Riemannian base of a standard static spacetime. Finally, we also exhibit an example showing the relevance of key hypotheses in our results.
In this paper, we are mainly devoted to the limiting behaviour of smooth inertial manifolds for a class of random retarded differential equations with a singular parameter $\delta$ and their Galerkin approximations, which have not been considered before. Under appropriate conditions, we show not only that the inertial manifolds for this class of random retarded equations converge pointwise to those of the corresponding stochastic equations driven by white noise as $\delta\rightarrow 0$, but also that the inertial manifolds of their Galerkin approximations converge pointwise to those of stochastic equations driven by white noise described above under the simultaneous limits $\delta\rightarrow 0$ and $M\rightarrow +\infty$, where $M$ denotes the dimension index of the orthogonal projection operator $P_M$ in the Galerkin scheme.
We provide four equivalent combinatorial conditions for a simple assembly graph (rigid vertex graph where all vertices are of degree 1 or 4) to have the largest number of Hamiltonian sets of polygonal paths relative to its size. These conditions serve to prove the conjecture that such a maximum, which is equal to $F_{2n+1}-1$, where $F_k$ denotes the $k$th Fibonacci number, is achieved only for special assembly graphs, called tangled cords.
A set $S \subseteq \mathbb{R}$ is almost Erd & odblac;s if, for every $\varepsilon \gt 0$, there exists a set $E \subseteq \mathbb{R}$ of positive Lebesgue measure such that $\{x \in S : ax+b \notin E\}$ is nonempty for all $|a| \gt \varepsilon$ and $b \in \mathbb{R}$. In this note, we show that any decreasing null sequence $(x_n)$ with decay rate greater than $1/2$ is an almost Erd & odblac;s set.
In his seminal work on partitions and divisor functions, MacMahon introduced the following two q-series: \begin{align*} A_k(q):= \sum_{0 \lt s_1 \lt s_2 \lt \cdots \lt s_k} \frac{q<^>{s_1+s_2+\cdots +s_k}}{(1-q<^>{s_1})<^>2(1-q<^>{s_2})<^>2\cdots (1-q<^>{s_k})<^>2},\qquad\quad\quad\quad\\[6pt] C_k(q):= \sum_{0 \lt s_1 \lt s_2 \lt \cdots \lt s_k} \frac{q<^>{2(s_1+s_2+\cdots +s_k)-k}}{(1-q<^>{2s_1-1})<^>2(1-q<^>{2s_2-1})<^>2\cdots (1-q<^>{2s_k-1})<^>2}. \end{align*}In 2013, Andrews and Rose proved that $A_k(q)$ and $C_k(q)$ are quasimodular forms of weight $\leq 2k$. Recently, Ono and Singh proved two interesting identities involving $A_k(q)$ and $C_k(q)$ and showed that the generating functions for the three-coloured partition function $p_3(n)$ and the overpartition function $\overline{p}(n)$ have infinitely many closed formulas in terms of MacMahon's quasimodular forms $A_k(q)$ and $C_k(q)$. In this paper, we introduce the finite forms $A_{k,n}(q)$ and $C_{k,n}(q)$ of MacMahon's q-series $A_k(q)$ and $C_k(q)$ and prove two identities which generalize Ono-Singh's identities. We also prove some new identities involving $A_{k,n}(q)$, $C_{k,n}(q)$ and certain infinite products based on two Bailey pairs. Those identities are analogous to Ono-Singh's identities.
In this paper, we study the existence of $k$ - $11$ -representations of graphs. Inspired by work on permutation patterns, these representations are ways of representing graphs by words where adjacencies between vertices are captured by patterns in the corresponding letters. Our main result is that all graphs are $1$ - $11$ -representable, answering a question originally raised by Cheon et al. in 2018 and repeated in several follow-up papers – including a very recent paper, where it was shown that all graphs on at most $8$ vertices are $1$ - $11$ -representable. Moreover, we prove that all graphs are permutationally $1$ - $11$ -representable – that is representable as the concatenation of permutations of the vertices – answering the existence question in extremely strong fashion. Our construction leads to nearly optimal bounds on the length of the words, as well. It can, moreover, be adapted to represent all acyclic orientations of graphs; this generalizes the fact that word-representations capture semi-transitive orientations of graphs. Our construction also adapts easily to other $k \geq 2$ as well, giving representations using a linear number of permutations when the best known previous bounds used a quadratic number. Finally, we also consider the (non-)existence of ‘even–odd’-representations of graphs. This answers a question raised by Wanless after a conference talk in 2018.
In this paper, we are concerned with the following Dirichlet problems for nonlinear equations involving the fractional $p$-Laplacian: \begin{equation*}\begin{cases} (-\Delta)_p<^>\alpha u=f(x,u,\nabla u),\ \ u \gt 0,\ \ \text{in}\ \ E,\\ \ \ \ \ \ \ u\equiv0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{in}\ \ \mathbb{R}<^>{n}\setminus E, \end{cases}\end{equation*}where $E \subseteq \mathbb{R}<^>{n}$ is a coercive epigraph, i.e., there exists a continuous function $\phi: \, \mathbb{R}<^>{n-1} \rightarrow \mathbb{R}$ satisfying \begin{equation*} \lim_{|x'|\rightarrow+\infty}\phi(x')=+\infty, \end{equation*}such that $E:=\{x=(x',x_{n}) \in \mathbb{R}<^>{n}|\,x_{n} \gt \phi(x')\}$, where $x':= (x_{1},...,x_{n-1}) \in \mathbb{R}<^>{n-1}$. Under some mild assumptions on the nonlinearity $f(x,u,\nabla u)$, we prove strict monotonicity of positive solutions to the above Dirichlet problems involving fractional $p$-Laplacian in coercive epigraph $E$.
In this paper, we study the validity of the sub-supersolution method for the equation \begin{equation*} \begin{cases} -\mbox{div}(K(x)\nabla u)=K(x)|x|<^>{\alpha-2}f(x,u) \,\mbox{in } {\mathbb{R}}<^>{N},\\ u \gt 0 \,\mbox{in } {\mathbb{R}}<^>{N}, \end{cases} \end{equation*} where $N \geq 3$, $K(x)=exp(|x|<^>{\alpha}/4)$, $\alpha\geq 2$ and $f$ is a continuous function, with hypotheses that will be given later. We apply the method to cases where $f$ is singular, where $f$ behaves like a logistic function, showing in both cases the existence and uniqueness of a positive solution.
In this paper, we will investigate the following inequality \begin{equation*} (a_{n+1}a_{n+2} - a_{n}a_{n+3})<^>2 -(a_{n+1}<^>2 -a_{n}a_{n+2})(a_{n+2}<^>2 - a_{n}a_{n+4}) \gt 0, \end{equation*}which arises from the iterated Laguerre operator on functions. We will prove the sequence $\{a_n\}$ of a unified form given by Griffin, Ono, Rolen and Zagier asymptotically satisfies this inequality while the Maclaurin coefficients of the functions in Laguerre-P & oacute;lya class have not to possess this inequality. We also prove the companion version of this inequality. As a consequence, we show the Maclaurin coefficients of the Riemann Xi-function asymptotically satisfy this property. Moreover, we make this approach effective and give the exact thresholds for the positivity of this inequalityfor the partition function, the overpartition function and the smallest part function.
Let $\mathcal{G}$ be the class of all connected simple graphs. The Hoffman program of graphs with respect to a spectral invariant $\lambda(G)$ consists of determining all the limit points of the set $\{\lambda(G)\,\vert\, G\in\mathcal{G}\}$ and characterising all $G$'s such that $\lambda(G)$ does not exceed a fixed limit point. In this paper, we study the Hoffman program for Laplacian matching polynomials of graphs in regard to their largest Laplacian matching roots. Precisely, we determine all the limit points of the largest Laplacian matching roots of graphs less than $\tau = 2+\omega<^>{\frac{1}{2}}+\omega<^>{-\frac{1}{2}}(=4.38+)$, and then characterise the connected graphs with the largest Laplacian matching roots less than $2+\sqrt{5}$, where $\omega=\frac{1}{3}(\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}+1)$.
Let f(z) be the normalized primitive holomorphic Hecke eigenforms of even integral weight k for the full modular group $SL(2,\mathbb{Z})$ and denote $L(s,\mathrm{sym}<^>{2}f)$ be the symmetric square L-function attached to f(z). Suppose that $\lambda_{\mathrm{sym}<^>{2}f}(n)$ be the $\mathrm{Fourier}$ coefficient of $L(s,\mathrm{sym}<^>{2}f)$. In this paper, we investigate the sum $\sum\limits_{n\leqslant x} \lambda<^>{j}_{\mathrm{sym}<^>{2}f }(n) $ for $j\geqslant 3$ and obtain some new results which improve on previous error estimates. We also consider the sum $\sum\limits_{n\leqslant x}\lambda<^>{j}_{f }(n<^>{2})$ and get some similar results.
In this paper, the upper bounds of non-real eigenvalues of indefinite Sturm-Liouville (S-L) problems with boundary conditions depend on the eigenparameter are studied. The upper bounds of real parts, imaginary parts and absolute values of non-real eigenvalues are given under the condition that the coefficients are integrable.
