We consider a 1-dimensional reaction diffusion equation with the following boundary conditions arising in a theory of the thermal explosion {-u"(t) = λf(u(t)), t ∈ (0, l), -u'(0) + C(0)u(0) = 0, u'(l) + C(l)u(l) = 0, where C : [0, ∞) → (0, ∞) is a continuous and nondecreasing function, λ > 0 is a parameter and f : [0, ∞) → (0, ∞) is a continuous function. We establish the extension of Quadrature method introduced in [8]. Using this extension, we provide numerical results for models with a typical function of f and C in a thermal explosion theory, which verify the existence, uniqueness and multiplicity results proved in [6].

We study divisibilities of elements in the q-commuting matrix C^(q). We first make a coefficient matrix Ĉ of C^(q) which is independent of q, study divisibilities over Ĉ and then retrieve our findings to C^(q). Finally we partition the C^(q) into 2 × 2 block matrices.

Recently, a new type of encryption called Homomorphic Authenticated Encryption (HAE) has been proposed. This combines the functionality of homomorphic encryption with authentication. Several concrete HAE schemes have been developed and security results for homomorphic authenticated encryption, designed by combining a homomorphic message authentication scheme with a homomorphic secret-key encryption, have been partially reported. In this paper, we analyze the security of a design method that combines homomorphic message authentication and homomorphic encryption, with a focus on the encryption after authentication (EAA) type. The results of our analysis show that while non-forgeability and indistinguishability are maintained, strong non-forgeability is not.

The aim of this note is to provide three derivations of the well-known Gregory-Leibniz series for π via a hypergeometric series approach.

We establish the existence and uniqueness of Green functions in Lipschitz domains for stationary Stokes systems with Neumann boundary conditions. For the uniqueness, we impose a different normalization condition from that in Choi et al. (J. Math. Fluid Mech., 20(4):1745-1769, 2018).

In this paper, we consider a stochastic higher-order Kirchhoff-type equation with nonlinear damping and source terms. We prove the blow-up of solution for a stochastic higher-order Kirchhoff-type equation with positive probability or explosive in energy sense.

We present an identity on moments of a compound random variable by using an auxiliary counting random variable. Based on this identity, we develop a new recurrence formula for obtaining the raw and central moments of any order for a given compound random variable.

The motivation in this paper comes from the recent results about Bell inequalities and topological insulators from group theory. Symmetries which are interested in group theory could be mainly used to find material structures. In this point of views, we study group extending by adding one relator which is easily called an equation. So a relative group extension by a adding relator is aspherical if the natural injection is one-to-one and the group ring has no zero divisor. One of concepts of asphericity means that a new group by a adding relator is well extended. Also, we consider that several equations and relative presentations over torsion-free groups are related to zero divisors.

Extending [14], we establish the existence of multiple positive solutions for a Schrödinger-type singular elliptic equation: $$\{{-{\Delta}u+V(x)u={\lambda}{\frac{f(u)}{u^{\beta}}},\;x{\in}{\Omega}, \atop u=0,\;x{\in}{\partial}{\Omega},$$ where 0 ∈ Ω is a bounded domain in ℝ^N, N ≥ 1, with a smooth boundary ∂Ω, β ∈ [0, 1), f ∈ C[0, ∞), V : Ω → ℝ is a bounded function and λ is a positive parameter. In particular, when f(s) > 0 on [0, σ) and f(s) < 0 for s > σ, we establish the existence of at least three positive solutions for a certain range of λ by using the method of sub and supersolutions.

In this paper, we provide Sawyer type conditions on a pair of weights so that the dyadic paraproduct π_b is bounded from L²(u) into L²(v). The conditions can be obtained by checking the boundedness of the dyadic paraproduct on a collection of test functions.