Proof of (2.17) From (2.10) we have with high probability for large n and uniformly in x and y √ n(F n (x, y) − ˆ F X (x) ˆ G(y)) ≤ α n x, y + log 2 n n − G(y)α n (x, ∞) − ˆ F X (x) α n ∞, y − log 2 n n + 2C log 2 n √ n , √ n(F n (x, y) − ˆ F X (x) ˆ G(y)) ≥ α n x, y − log 2 n n − G(y)α n (x, ∞) − ˆ F X (x) α n ∞, y + log 2 n n − 2C log 2 n √ n. Set V n,0 = √ n(F n − ˆ F X ˆ G). From (2.12) and...

This paper studies the behavior of sequences in the maximal ideal space of the algebra of bounded analytic functions on an arbitrary domain. The main result states that for any such sequence, either the sequence has an interpolating subsequence or infinitely many elements of the sequence lie in the same Gleason part. Introduction Fix a positive integer N and fix a nonempty open subset Q. of C ....

where Λ is the physical scale of the theory related to the running coupling αs = αs(Q ) at the scale Q. Since Λ has no expansion in αs, perturbative contributions are present only in W0, which from power counting is quartically divergent in the UV region. The term with the power Λ/Q is absent since there are no gauge invariant operators of dimension two. In the lattice theory all frequencies ar...

Low-temperature series have been derived for the q-state Potts model on the Kagomé lattice (q = 3 and 4) and for the 3-state honeycomb lattice. The series are derived by the finitelattice method, with many additional terms obtained by noting the structure of the correction terms. Accurate estimates of the critical points are found, which confirm exact, but not rigorous results in the case of th...

Keywords: Sturm–Liouville Spectral parameter Taylor monomials Time scale a b s t r a c t We will derive formulas for finding two linearly independent solutions of the Sturm–Liouville dynamic equation. We will give several examples. In particular, the q-difference equation which has important applications in quantum theory will be presented. In this paper we obtain a spectral parameter power ser...

We give Littlewood–Paley type characterizations for Besov–Triebel–Lizorkin–type spaces $$\mathscr {B}_{pq}^{s\tau },\mathscr {F}_{pq}^{s\tau }$$ and Besov-Morrey $$\mathcal N_{uqp}^s$$ on a special Lipschitz domain $$\Omega \subset \mathbb R^n$$ : suitable sequence of Schwartz functions $$(\phi _j)_{j=0}^\infty $$ , $$\begin{aligned} \Vert f\Vert _{\mathscr }(\Omega )}&\textstyle \approx \sup _...

Let α,β ,x,y be positive integers. Fix a line L : Y = αX + β , and a lattice point Q(x,y) on L. It is well known that the number of lattice paths from the origin to Q which touches L only at Q is given by β x + y ( x + y x )

The Yang-Lee zeros of the Q-state Potts model on recursive lattices are studied for noninteger values of Q. Considering one-dimensional (1D) lattice as a Bethe lattice with coordination number equal to 2, the location of Yang-Lee zeros of 1D ferromagnetic and antiferromagnetic Potts models is completely analyzed in terms of neutral periodical points. Three different regimes for Yang-Lee zeros a...