We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph which is reducible by some finite sequence of these moves, to a graph with no edges is called a knot graph. We show that the class of knot graphs strictly contains the set of delta-wye graphs. We prove that the dimension of the intersection of the cycle and cocycl...

We prove the orthogonality (in the sense of Birkhoff) of the range and the kernel of an important class of elementary operators with respect to the Schatten p-class. 1. Introduction. Let H be a separable infinite-dimensional complex Hilbert space and let B(H) denote the algebra of all bounded operators on H into itself. Given A, B ∈ B(H), we define the generalized derivation δ A,B : B(H) B(H) b...

The fixed point x of (1) is ”stable” provided that given any ball B(x, ε) = {x ∈ Ω/‖x− x‖ < ε}, there is a ball B(x, δ) = {x ∈ Ω/‖x− x‖ < δ} such that if x ∈ B(x, δ) then g(x) ∈ B(x, ε), for k = 0, 1, 2, ... [4]. If in addition there is a ball B(x, r) such that g(x) → x as k → ∞ for all x ∈ B(x, r) then the fixed point x is ”asymptotically stable”.[4]. The domain of attraction DA(x) of the asym...

In this Letter, we report our recent experimental results on the energy gap of the ν=1 quantum Hall state (Δ(ν=1)) in a quantum antidot array sample, where the effective disorder potential can be tuned continuously. Δ(ν=1) is nearly constant at small effective disorders, and collapses at a critical disorder. Moreover, in the weak disorder regime, Δ(ν=1) shows a B(total)(1/2) dependence in tilte...

We consider a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = R+, with drift r0 ∈ R and Hurst parameterH ∈ ( 1 2 , 1). Under a natural stability condition on the drift vector r0 and reflection directions, we establish a return time result for theRFBMprocessZ; that is, for some δ, κ > 0, supx∈B Ex [τB(δ)] < ∞, where B = {x ∈ S : |x| ≤ κ} and τB(δ) = ...

Proposition 1.3. Let X be a subset of R, let x0 be a limit point of X, let f : X → R be a function, and let L be a real number. Then the following two statements are equivalent. • f is differentiable at x0 on X with derivative L. • For every ε > 0, there exists a δ = δ(ε) > 0 such that, if x ∈ X satisfies |x− x0| < δ, then |f(x)− [f(x0) + L(x− x0)]| ≤ ε |x− x0| . Corollary 1.4 (Mean Value Theor...

Very recently, a new concept called multiplicative differential and the corresponding c-differential uniformity were introduced by Ellingsen et al. (IEEE Trans. Inform. Theory 66(9), 5781–5789 2020). A function F(x) over finite field GF(pn) to itself is said have δ, or equivalent, differentially (c,δ)-uniform, when maximum number of solutions x ∈GF(pn) F(x + a) − cF(x) = b, a,b,c ∈GF(pn), c≠ 1 ...

Let f be a function on a bounded domain Ω ⊆ R and δ be a positive function on Ω such that B(x, δ(x)) ⊆ Ω. Let σ(f)(x) be the average of f over the ball B(x, δ(x)). The restricted mean-value theorems discuss the conditions on f, δ, and Ω under which σ(f) = f implies that f is harmonic. In this paper, we study the stability of harmonic functions with respect to the map σ. One expects that, in gen...

and Applied Analysis 3 Example 1.4. Let X, ‖ · ‖ be a normed space, a ∗ b ab, and a b min{a b, 1} for all a, b ∈ 0, 1 . For all x ∈ X and every t > 0, consider μ x, t ⎧ ⎪⎨ ⎪⎩ t t ‖x‖ if t > 0, 0 if t ≤ 0; ν x, t ⎧ ⎪⎨ ⎪⎩ ‖x‖ t |x| if t > 0, 1 if t ≤ 0. 1.1 Then X, μ, ν, ∗, is an IFNS. Remark 1.5 see 3 . Let X, μ, ν, ∗, be an IFNS with the condition μ x, t > 0, ν x, t < 1 implies x 0 ∀t ∈ . 1.2 L...