The bounds of the solutions to generalized modular equations

In this paper the authors study a monotonicity of several functions involving ma(r) and μa(r). By using these results, the authors obtain some new bounds of the solutions of Ramanujan's generalized modular equations.     

On modular solutions of certain meromorphic modular differential equations

Kaneko and Koike gave the “extremal” quasimodular forms of depth 1 for PSL₂(ℤ) and modular differential equations they satisfy. In this paper, we study modular solutions of their modular differential equations.     

Geometric properties of solutions of Hamilton-Jacobi equations

The method of moving parallel planes, previously used for elliptic and parabolic PDE, is adapted to study solutions of the Cauchy problem for Hamilton-Jacobi equations. This is possible in the framework of the theory of viscosity solutions, using the comparison theorem for such solutions as a kind of maximum principle. One of the main results states that if the initial data are nonnegative and compact supported, the Hamiltonian radial and the level sets expanding, then the level sets become asymptotically spherical as t → ∞, the convergence taking place in the Lipschitz norm.     

Geometric theory of singularities of solutions of nonlinear differential equations

The article is devoted to singularities of integral manifolds which realize solutions of nonlinear partial differential equations and to the algebraic geometric and topological questions related to them.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 20, pp. 207–247, 1988.     

Common solutions to the Lyapunov equations

Let A be an n×n matrix with complex entries. A necessary and sufficient condition is established for the existence of a Hermitian solution H to the equations

$\text{AH+HA}{}^1\text{=HA+A}{}^1\text{H=I}$

.     

Defining equations of modular curves

We obtain defining equations of modular curves X₀(N), X₁(N), and X(N) by explicitly constructing modular functions using generalized Dedekind eta functions. As applications, we describe a method of obtaining a basis for the space of cusp forms of weight 2 on a congruence subgroup. We also use our model of X₀(37) to find explicit modular parameterization of rational elliptic curves of conductor 37.     

Birkhoff-recurrent solutions to algebraic equations

The paper considers algebraic equations whose coefficients are continuous functions given on a connected topological group. It is proven that if the vector function made up of the coefficients is recurrent in the sense of Birkhoff and if the discriminant of the equation never vanishes, then each continuous solution is recurrent. The proof is based on the theory of extensions of dynamic systems.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 617–623, April, 1973.     

Multiple solutions to Hessian equations

This paper concerns with the multiple solutions of Hessian equations ?? _k (??(D ² u))?=?f (x, u) in a (k ? 1)-convex domain ${\Omega\subset \mathbb{R}^{n}}$ . Using the methods of degree theory and a priori estimates we prove the existence of two or more solutions to the Hessian equations.     

Multi-valued solutions to Hessian equations

In this paper, we use the Perron method to prove the existence of bounded multi-valued viscosity solutions to Hessian equations and interior Lipschitz continuity of the multi-valued solutions.     

Positive diagonal solutions to the Lyapunov equations

We study various stability type conditions on a matrix A related to the consistency of the Lyapunov equation AD+DA^t positive definite, where D is a positive diagonal matrix. Such problems arise in mathematical economics, in the study of time-invariant continuous-time systems and in the study of predator-prey systems. Using a theorem of the alternative, a characterization is given for all A satisfying the above equation. In addition, some necessary conditions for consistency and some related ideas are discussed. Finally, a method for constructing a solution D to the equation is given for matrices A satisfying certain conditions.