Issue 4, Volume 45, December 2020

2020

1. Congruences for Partition Quadruples with $t$-Cores M. S. Mahadeva Naika , S. Shivaprasada Nayaka PDF

   ▸ Abstract

   Let $C_t(n)$ denote the number of partition quadruples of $n$ with $t$-cores for $t = 3,5,7,25$. We establish some Ramanujan type congruences modulo $5, 7, 8$ for $C_t(n)$. For example, $n\ge  0$, we have $$\begin{array}{@{}rcl@{}} C_{5}(5n+4)&\equiv& 0\pmod{5},\\ C_{7}(7n+6)&\equiv& 0\pmod{7},\\ C_{3}(16n+14)&\equiv& 0\pmod{8}. \end{array}$$

   795-806
2. The Inverse Scattering Problem for the Perturbed String Equation and Its Application to Integration of the Two-dimensional Generalization from Korteweg-de Vries Equation Pham Loi Vu PDF

   ▸ Abstract

   The inverse scattering problem for the perturbed string equation in characteristic variables on the whole axis is studied. Using the generalized Lax equation generated by the perturbed string equation, we derive the time-evolution of the scattering operator and the two-dimensional generalization from the one-dimensional Korteweg-de Vries equation. This enables us to solve the system of time-dependent fundamental equations in the inverse problem. Then, the solution of the two-dimensional generalization from the Korteweg-de Vries equation is found that is expressed through the found solution of this system.

   807-831
3. Analytic Extensions of Functions of Cayley-Dickson Variables Sergey V. Ludkowski PDF

   ▸ Abstract

   Analytic extensions and approximations of functions of Cayley-Dickson variables are investigated. The case of functions of complexified Cayley-Dickson variables is also encompassed. In particular, functions of octonion variables are studied.

   833-847
4. Calderón’s Problem for Some Classes of Conductivities in Circularly Symmetric Domains Mai Thi Kim Dung , Dang Anh Tuan PDF

   ▸ Abstract

   In this note, we study Calderón’s problem for certain classes of conductivities in domains with circular symmetry in two and three dimensions. Explicit formulas are obtained for the reconstruction of the conductivity from the Dirichlet-to-Neumann map. As a consequence, we show that the reconstruction is Lipschitz stable.

   849-863
5. Conjunction Probability of Smooth Centered Gaussian Processes Viet-Hung Pham PDF

   ▸ Abstract

   In this paper we provide an upper bound for the conjunction probability of independent Gaussian smooth processes, and then, we prove that this bound is a good approximation with exponentially smaller error. Our result confirms the heuristic approximation by Euler characteristic method of Worsley and Friston and also implies the exact value of generalized Pickands constant in a special case. Some results for conjunction probability of correlated processes are also discussed.

   865-874
6. Convergence in Monge-Wasserstein Distance of Mean Field Systems with Locally Lipschitz Coefficients Dung Tien Nguyen , Son Luu Nguyen , Nguyen Huu Du PDF

   ▸ Abstract

   This paper focuses on stochastic systems of weakly interacting particles whose dynamics depend on the empirical measures of the whole populations. The drift and diffusion coefficients of the dynamical systems are assumed to be locally Lipschitz continuous and satisfy global linear growth condition. The limits of such systems as the number of particles tends to infinity are studied, and the rate of convergence of the sequences of empirical measures to their limits in terms of $p^{\rm{th}}$ Monge-Wasserstein distance is established. We also investigate the existence, uniqueness, and boundedness, and continuity of solutions of the limiting McKean-Vlasov equations associated to the systems.

   875-896
7. The Isomorphism Problem for a Family of One-Relator Groups Vu The Khoi PDF

   ▸ Abstract

   In this paper, we use the Alexander ideals of groups to solve the isomorphism problem for a family of groups $G(m, n, p)$ which coincide with the Baumslag-Solitar groups when $p = 1$.

   897-902
8. Long-Time Behavior of a Stochastic SIQR Model with Markov Switching Nguyen Huu Du , Nguyen Thanh Dieu , Tran Quan Ky , Vu Hai Sam PDF

   ▸ Abstract

   In this paper, we consider a stochastic SIQR model with a perturbed disease transmission coefficient. We determine the threshold $\lambda$ that is used to classify the extinction and permanence of the model. Concretely, $\lambda < 0$ implies that the disease-free equilibrium $K(0, 0, 0)$ is globally asymptotically stable; that is, the disease will disappear and the entire population will become susceptible individuals. If $\lambda > 0$, the model is strongly stochastically permanent. Our findings are considered significant improvements over the results in Liu et al. (Appl. Math. Comput. 316, 310–325 2018).

   903-915
9. Feedback Control of Navier-Stokes-Voigt Equations by Finite Determining Parameters Nguyen Thi Ngan , Vu Manh Toi PDF

   ▸ Abstract

   We study the stabilization of stationary solutions to Navier-Stokes-Voigt equations by finite-dimensional feedback control scheme introduced by Azouani and Titi (Evol. Equ. Control Theory 3, 579–594 2014). The designed feedback control scheme is based on the finite number of determining parameters (degrees of freedom), namely, finite number of determining Fourier modes, determining nodes, and volume elements.

   917-930
10. Pseudo Wavelet Frames T. C. Easwaran Nambudiri , K. Parthasarathy PDF

    ▸ Abstract

    We seek to alleviate the unpleasantness of the fact that the class of wavelet frames is not closed under canonical duals by introducing and studying a new class of frames that we call pseudo wavelet frames, which includes all wavelet frames and is ‘self-dual’ in terms of canonical duals. The class also has the ‘extension property’ and is invariant under the Fourier transform.

    931-941
11. On K-Polystability for Log Del Pezzo Pairs of Maeda Type Kento Fujita PDF

    ▸ Abstract

    We give an algebraic proof for which log del Pezzo pairs of Maeda type are K-polystable or not. If the base field is the complex number field, then the result is already known by Li and Sun.

    943-965
12. On the Number of Determining Volume Elements for 3D Navier-Stokes-Voigt Equations Nguyen Duc Huy , Nguyen Thi Ngan , Vu Manh Toi PDF

    ▸ Abstract

    In this paper, we give an upper bound on the number of determining volume elements for the 3D Navier-Stokes-Voigt equations with periodic boundary conditions. Here the bound is estimated explicitly in terms of flow parameters, such as viscosity, smoothing length, forcing, and domain size.

    967-980