We prove that there exists no nontrivial L-p (p>1) harmonic 1-form on a complete noncompact minimal hypersurface Sigma(n) in a product manifold S-n(root 2(n-1))& times;R (n >= 3) if the square of L-n-norm of the second fundamental form is less than 4n alpha(2)/C(0)p(2)(n-1)(4 )or the square of the length of the second fundamental form is less than n alpha(2)/2(n-1) , where alpha is an angle function and C0 is the Sobolev constant depending only on n.
The homogenization of trajectory attractors for 3D incompressible Navier–Stokes equations with random perforated domain is derived. By means of the homogenized approach, the trajectory attractors for 3D incompressible Navier–Stokes equations with random perforated domain converge almost surely to the trajectory attractors for 3D incompressible Navier–Stokes equations without obstacles. This paper doesn’t assume that the 3D incompressible Navier–Stokes equations is uniquely solvable.
While considerable work has been devoted to q-supercongruences modulo cyclotomic polynomials Phi n(q) , results concerning higher powers of the q-integer [n] remain scarce. To address this gap, we present a proof of the Guo-Schlosser conjecture modulo [n]q24 Phi n(q2) in this paper. Our proof employs Watson's transformation and Bailey's transformation to introduce auxiliary parameters into the congruence, followed by an analysis involving specific parameter specializations at every primitive m-th root of unity, where m divides n. Additionally, we establish a new q-supercongruence modulo [n]3 that extends the one formulated by Wei.
We model and analyze pursuit-evasion differential games governed by a countably infinite system of two-coupled first-order ordinary differential equations (ODEs) in the ℓ ^2(ℕ; ℝ^2) space. Both players’ control functions are subject to geometric constraints, with the pursuer holding the larger control bound. The pursuer seeks to drive the state trajectory of the system from the initial state z^0 to the zero state within finite time, while the evader seeks to avoid this indefinitely. The system is inherently unstable because the parameters α _j are positive and bounded, causing the uncontrolled dynamics to drift away from the origin. Through mathematical analysis, we derive a closed-form formula for a guaranteed capture time and construct a strategy that achieves capture by that time. A key finding is that for the constructed pursuit strategy, capture is guaranteed in finite time if and only if the pursuer’s resource advantage satisfies a stronger condition ρ >σ +α‖ z^0‖ than the classical requirement ρ >σ ; the classical condition remains necessary but no longer sufficient. Dually, we establish an explicit expression for a guaranteed escape time and exhibit an evasion control that keeps the state away from the origin up to that time. When certain coordinates satisfy a specific inequality, the evader can escape indefinitely even though ρ >σ . We also establish the natural consistency that the guaranteed capture time is never less than the guaranteed escape time. Several illustrative examples demonstrate the sharpness of the obtained conditions, including scenarios where the two guaranteed times coincide and where a finite escape time coexists with the nonexistence of a finite capture time arising from the formulated pursuit strategy. The results reveal how system instability can be exploited by the evader, effectively supplementing its control resources and enabling evasion under conditions less restrictive than those required in stable systems.
The enhanced power graph of a group G , denoted by P_e(G) , is the graph whose vertex set is G , in which two distinct elements are adjacent if and only if they generate a cyclic subgroup. We say a group G is determined by the enhanced power graph if for any group H , the isomorphism P_e(H) ≅ P_e(G) implies that H ≅ G . In this paper, we investigate the structure of the enhanced power graph of a finite nilpotent group in terms of the enhanced power graphs of its Sylow subgroups. We prove that two finite nilpotent groups have isomorphic enhanced power graphs if and only if their corresponding Sylow subgroups have isomorphic enhanced power graphs. Furthermore, we identify specific nilpotent groups that are determined by their enhanced power graphs. Finally, we extend these results to the contexts of power graphs and cyclic graphs.
