The generalization of p-frame in Banach spaces is considered in this paper. The concepts of an $tilde{X}$-frame and a system conjugate to $tilde{X}$-frame were introduced. Analogues of the results on the existence of conjugate system were obtained. The stability of $tilde{X}$-frame having a conjugate system is studied.

Two algorithms for constructing a class of compactly supported complex tight wavelet frames with conjugate symmetry are provided. Firstly, based on a given complex refinable function φ, an explicit formula for constructing complex tight wavelet frames is presented. If the given complex refinable function φ is compactly supported conjugate symmetric, then we prove that there exists a compactly s...

1. Throughout P = C ∪ {∞} denotes the Riemann sphere, H denotes the upper half plane, C∗ denotes the multiplicative group of complex numbers, and P = (C \ {0})/C∗ denotes n dimensional complex projective space. For w ∈ C \{0} let [w] := wC∗ denote the corresponding point of P. For A ∈ GLn+1(C) let MA denote the corresponding automorphism of projective space so that MA([w]) = [Aw]. Identify P an...

We introduce a new technique to study pattern avoidance in dynamical systems, namely the use of a commuter function between non-conjugate dynamical systems. We investigate the properties of such a commuter function, specifically h : [0, 1]→ [0, 1] satisfying T1 ◦ h = h ◦ Tμ, where Tμ denotes a symmetric tent map of height μ. We make use of this commuter function to prove strict inclusion of the...

We recall from Chapter 1 that the computation of the equilibrium state of a clamped membrane amounts to the solution of the convex minimization problem (7.1) J(u) = inf v∈H 1 0 (Ω) J(v) , 1 0 (Ω) → lR stands for the convex functional (7.2) J(v) := 1 2 Ω |a grad v| 2 dx − Ω f v dx , v ∈ H 1 0 (Ω). As we observed in Chapter 1, (7.1) is a particular example of a more general convex optimization pr...

Stability of the ;-; stacking interactions in the substituted-coronene||cyclooctatetraene complexes wasstudied using the computational quantum chemistry methods (where || denotes ;–; stackinginteraction, and substituted-coronene is coronene which substituted with four similar X groups; X =OH, SH, H, F, CN, and NO). There are meaningful correlations between changes of geometricalparameters and t...