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We consider the convergence rate of the alternating projection method for the nontransversal intersection of a semialgebraic set and a linear subspace. For such an intersection, the convergence rate is known as sublinear in the worst case. We study the exact convergence rate for a given semialgebraic set and an initial point, and investigate when the convergence rate is linear or sublinear. As a consequence, we show that the exact rates are expressed by multiplicities of the defining polynomials of the semialgebraic set, or related power series in the case that the linear subspace is a line, and we also decide the convergence rate for given data by using elimination theory. Our methods are also applied to give upper bounds for the case that the linear subspace has the dimension more than one. The upper bounds are shown to be tight by obtaining exact convergence rates for a specific semialgebraic set, which depend on the initial points., 22 pages, 1 figure

Let $ G $ be a connected, simply connected semisimple algebraic group over the complex number field, and let $ K $ be the fixed point subgroup of an involutive automorphism of $ G $ so that $ (G, K) $ is a symmetric pair. We take parabolic subgroups $ P $ of $ G $ and $ Q $ of $ K $ respectively and consider the product of partial flag varieties $ G/P $ and $ K/Q $ with diagonal $ K $-action, which we call a \emph{double flag variety for symmetric pair}. It is said to be \emph{of finite type} if there are only finitely many $ K $-orbits on it. In this paper, we give a parametrization of $ K $-orbits on $ G/P \times K/Q $ in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of $ P \subset G $ or $ Q \subset K $ is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of $ K $-spherical flag varieties $ G/P $ and $ G $-spherical homogeneous spaces $ G/Q $., 47 pages, 3 tables; add all the details of the classification

The non-commutative harmonic oscillator is a 2×2-system of harmonic oscillators with a non-trivial correlation. We write down explicitly the special value at s=2 of the spectral zeta function of the non-commutative harmonic oscillator in terms of the complete elliptic integral of the first kind, which is a special case of a hypergeometric function.

We consider the connection problem for the Heun differential equation, which is a Fuchsian differential equation that has four regular singular points. We consider the case in which the parameters in this equation satisfy a certain set of conditions coming from the eigenvalue problem of the non-commutative harmonic oscillators. As an application, we describe eigenvalues with multiplicities greater than 1 and the corresponding odd eigenfunctions of the non-commutative harmonic oscillators. The existence of a rational or a certain algebraic solution of the Heun equation implies that the corresponding eigenvalues has multiplicities greater than 1.

For a prime p and for the number a(n) of solutions of xp = 1 in the symmetric group on n letters, ordp\( (a(n)) \geq [n/p] - [n/p^2] \), and especially, ordp\( (a(n)) = [n/p] - [n/p^2] \) provided \( n \equiv 0 \) mod p2. Let r be an integer with \( 1 \leq r \leq p^2 - 1 \). If ordp\( (a(r)) \leq [r/p] + 1 \), then, for each positive integer m, ordp\( (a(mp^2 + r)) = m(p-1)+ ord_p (a(r)) \). Assume that ordp\( (a(r)) = [r/p] + 2 \). If \( a(p^2 + r) \equiv -p^{p-1} a(r) mod \quad p^{p+[r/p]+2} \), then ordp\( (a(mp^2 + r)) = m(p-1) + [r/p] + 2 \); otherwise, there exists a p-adic integer b such tha ordp\( (a(mp^2 + r)) = m(p-1) + [r/p] + 2 + ord_p(m-b) \).

The spectral problem for non-commutative harmonic oscillators is shown to be equivalent to solve Fuchsian ordinary differential equations with four regular singular points in a complex domain.