Narrow Results

We study derivatives on an interval of length ℓ (or the associated circle of the same length), and certain pseudo-differential operators that arise as their fractional powers. We compare different translations across the interval (around the circle) that are characterized by a twisting angle. These results have application in

In this work, multilayer graphene was chosen and modeled in ANSYS software. In the software, it was subjected to bending forces through various bend angles. In order to test for bending, the model is subjected to rigid support on the base side. Then a force is applied on the side on which the angle is provided to simulate Bending. Stress–strain plots were analyzed for various bending angles. The graphene sheet was also subjected to twist forces within the software. In order to test for torsion, the model is subjected to rigid support on both sides. Then a couple is applied on the opposite sides to simulate torsion. Torsional stress and strain measurements were analyzed. The trends were studied and correlations with the material properties of graphene were established. The quest is to identify a magic angle for bend and twist.

The ℂP²-genus of a knot K is the minimal genus over all isotopy classes of smooth, compact, connected and oriented surfaces properly embedded in ℂP² - B⁴ with boundary K. We compute the ℂP²-genus and realizable degrees of (-2,q)-torus knots for 3 ≤ q ≤ 11 and (2,q)-torus knots for 3 ≤ q ≤ 17. The proofs use gauge theory and twisting operations on knots.

Some composite knots are known to be trivialized by twisting. However, the bridge index of the prime factors in known examples is two. In this note, for any integer n ≥ 1 we will construct composite knots which can be trivialized by twisting and which consist of two prime factors with bridge index greater than n.

In this note, we extend the idea of G-Frobenius algebras (G-FAs) for G a finite group to the case where G is replaced by a finite groupoid. These new structures, which we call groupoid Frobenius algebras, have twists that are entirely analogous to the universal G-FA twists by Z²(G, k^×). The usefulness of these new structures comes from recognizing that there is a large collection of G-FAs that can be regarded as non-trivial groupoid Frobenius algebras. As a consequence of this, the twists associated to groupoid Frobenius algebras can be brought to bear on the problem of twisting G-FAs. In particular, we show that for any integer n ≥ 2, there exists a class of G-FAs with twists by Zⁿ(G, k^×).

Let ${𝒜}$ be a $G{L}_d(ℝ)$-valued cocycle over a subshift of finite type. Under a certain twisting assumption, we prove that ${𝒜}$ has a uniform Lyapunov exponent if and only if the largest Lyapunov exponent of ${𝒜}$ at all periodic points equals. Under the typicality assumption, we give two checkable criteria for deciding whether ${𝒜}$ has uniform singular value exponents.

Given a (quasi-)twilled pre-Lie algebra, we first construct a differential graded Lie algebra ($L_{\infty }$-algebra). Then we study the twisting theory of (quasi-)twilled pre-Lie algebras and show that the result of the twisting by a linear map on a (quasi-)twilled pre-Lie algebra is also a (quasi-)twilled pre-Lie algebra if and only if the linear map is a solution of the Maurer–Cartan equation of the associated differential graded Lie algebra ($L_{\infty }$-algebra). In particular, the relative Rota–Baxter operators (twisted relative Rota–Baxter operators) on pre-Lie algebras are solutions of the Maurer–Cartan equation of the differential graded Lie algebra ($L_{\infty }$-algebra) associated to the certain quasi-twilled pre-Lie algebra. Finally, we use the twisting theory of (quasi-)twilled pre-Lie algebras to study quasi-pre-Lie bialgebras. Moreover, we give a construction of quasi-pre-Lie bialgebras through symplectic Lie algebras, which is parallel to that a Cartan $3$-form on a semi-simple Lie algebra gives a quasi-Lie bialgebra.

Creation of novel $\pi$-conjugated molecules is an important research topic. I describe in this account an approach to this aim that is based on the use of the distorted conformation of porphyrins. Planarization of distorted molecules enables the synthesis of heteroatom-containing porphyrin derivatives. Furthermore, dearomatization reaction proves effective to construct distorted conformations from planar $\pi$-conjugated molecules under mild reaction conditions. According to this protocol, we have succeeded in the synthesis of heteroatom-containing curved-$\pi$ conjugated molecules that had never been achieved by conventional protocols. In particular, a nitrogen-embedded buckybowl is the first example of a buckybowl having a heteroatom in its central position, which exhibits unique properties due to the incorporation of the heteroatom in its curved $\pi$-surface.

Let F be a non-Archimedean local field of characteristic zero, let (π, V) be an irreducible, admissible representation of GSp(4, F) with trivial central character, and let χ be a quadratic character of F^× with conductor c(χ) > 1. We define a twisting operator T_χ from paramodular vectors for π of level n to paramodular vectors for χ ⊗ π of level max(n + 2c(χ), 4c(χ)), and prove that this operator has properties analogous to the well-known GL(2) twisting operator.

Let S_k(Γ^para(N)) be the space of Siegel paramodular forms of level N and weight k. Fix an odd prime p ∤ N and let χ be a nontrivial quadratic Dirichlet character mod p. Based on [Twisting of paramodular vectors, Int. J. Number Theory10 (2014) 1043–1065], we define a linear twisting map 𝒯_χ : S_k(Γ^para(N)) → S_k(Γ^para(Np⁴)). We calculate an explicit expression for this twist, give the commutation relations of this map with the Hecke operators and Atkin–Lehner involution for primes ℓ ≠p, and prove that the L-function of the twist has the expected form.