Possible probability and irreducibility of balanced non-transitive dice

We construct irreducible balanced non-transitive sets of $n$-sided dice for any positive integer $n$, which was raised in \cite[Question 5.2]{SS17}. One main tool of the construction is to study so-called fair sets of dice. Furthermore, we also study the distribution of the probabilities of balanced non-transitive sets of dice. For a lower bound, we show that the probability could be arbitrarily close to $\frac{1}{2}$ and for a upper bound, we construct a balanced non-transitive set of dice whose probability is $\frac{1}{2} + \frac{13-\sqrt{153}}{24} \approx \frac{1}{2} + \frac{1}{9.12}.

Some Classes of Jacobi Matrices and Schrödinger Operators

This dissertation addresses two classes of Jacobi matrices and Schrödinger operators. First, we consider Jacobi matrices and Schrödinger operators that are reflectionless on an interval. We give a systematic development of a certain parametrization of this class, in terms of suitable spectral data, that is due to Marchenko. Then some applications of these ideas are discussed. In the second half, we study structural properties of the Lyapunov exponent $\gamma$ and the density of states $k$ for ergodic (or invariant) Jacobi matrices in a general framework. In this analysis, a central role is played by the function $w=-\gamma+i\pi k$ as a conformal map between certain domains. This idea goes back to Marchenko and Ostrovskii, who used this device in their analysis of the periodic problem

Remarks on conjugation and antilinear operators and their numerical range (Research on structure of operators by order and related topics)

In this paper, we investigate the numerical ranges of conjugations and antilinear operators on a Hilbert space, which will be shown to be annuli in general. This result proves that Toeplitz-Hausdorff Theorem, which says the convexity on the numerical ranges of linear operators, does not hold for the ones of antilinear operators. Moreover, we extend these results to a Banach space

Complex symmetric operators and isotropic vectors in Banach spaces via linear functionals (Research on structure of operators by order and related topics)

We generalize the concept of complex symmetric operators to Banach spaces via their dual spaces. With this extension we show the existence of isotropic vectors on Banach spaces whose dimension is at least two and the relation between the simplicity of an eigenvalue and the non-existence of its isotropic eigenvectors. All this work is based on [M. Cho, I. Hur and J.E. Lee, Complex symmetric operators and isotropic vectors on Banach spaces, J. Math. Anal. Appl. 479 (2019), no. 1, 752-764.]

Ergodic Jacobi matrices and conformal maps

We study structural properties of the Lyapunov exponent $\gamma$ and the density of states $k$ for ergodic (or just invariant) Jacobi matrices in a general framework. In this analysis, a central role is played by the function $w=-\gamma+i\pi k$ as a conformal map between certain domains. This idea goes back to Marchenko and Ostrovskii, who used this device in their analysis of the periodic problem

Schrödinger operators and canonical systems via spectral theory (Research on structure of operators using operator means and related topics)

In this survey article we explore Schrödinger operators and canonical systems via (inverse) spectral theory. After reviewing some basic materials, we summarize several well-known results on spectra of Schrödinger operators. Then (inverse) spectral theory for Schrödinger operators and canonical systems will be investigated

Possible Probability and Irreducibility of Balanced Nontransitive Dice

We construct irreducible balanced nontransitive sets of n-sided dice for any positive integer n. One main tool of the construction is to study so-called fair sets of dice. Furthermore, we also study the distribution of the probabilities of balanced nontransitive sets of dice. For a lower bound, we show that the winning probability can be arbitrarily close to 1/2. We hypothesize that the winning probability cannot be more than 1/2+1/9, and we construct a balanced nontransitive set of dice whose probability is 1/2+13−153/24≈1/2+1/9.12