Neutral stochastic differential equations with constant delay (NSDDEs) arise from describing many natural and social phenomena, such as those from medicine, physics, chemistry, epidemiology, neural science and economy. These equations are non-solvable in most cases. In this paper that class of equations is analyzed under the polynomial conditions on their coefficients. Approximate solution to initial NSDDE is obtained as the solution of the equation acquired by the Taylor expansion with respect to the delayed argument of the coefficients of that equation. The existence and uniqueness of newly formed approximate equation is shown and the closeness between solutions of the approximate and initial equations is estimated in the L^p and almost sure sense. The rate of convergence increases as the orders of the Taylor expansions of the coefficients increase. Theoretical results are derived from various elementary, integral, probability and martingale-related inequalities. Examples are provided in order to support theoretical results.

Let E be an elliptic curve defined over ℚ, and let p be an odd prime at which E has good ordinary reduction. Consider the p-primary Selmer group Sel_p^∞(ℚ_∞, E) over the cyclotomic ℤ_p-extension of ℚ, and denote its (algebraic) 𝜇-invariant by 𝜇_p(E). Let ${\bar{\rho}}_{E,p}\;:\;Gal({\bar{\mathbb{Q}}}/{\mathbb{Q}}){\rightarrow}GL_2({\mathbb{Z}}/p{\mathbb{Z}})$ be the Galois representation arising from the action of the absolute Galois group on the p-torsion of E. Greenberg conjectured that if ${\bar{\rho}}_{E,p}$ is reducible, then there exists a rational isogeny E → E' of p-power degree such that 𝜇_p(E') = 0. In this article, we investigate this conjecture by establishing sufficient Galois-theoretic criteria for its validity, formulated in terms of the representation ${\bar{\rho}}_{E,p}$. Our approach relies on a fundamental result of Coates and Sujatha concerning the fine Selmer group. Furthermore, in the case where ${\bar{\rho}}_{E,p}$ is irreducible, we show that our conditions imply 𝜇_p(E) = 0 under the additional assumption that the classical Iwasawa 𝜇-invariant vanishes for the splitting field ${\mathbb{Q}}(E[p])={\bar{\mathbb{Q}}}^{ker\,{\bar{{\rho}}}_{E,p}}$.

In this note, we organize minimal annuli in a slab based on the winding number of the circles that foliate them and study the area of minimal annuli with given winding number. Specifically, we deduce some results regarding the convexity of the length function of corresponding level curves, and apply them to estimate the area of the annuli by comparing to the area of waist of covers of catenoids.

In this paper we study rectifying submanifolds of a Riemannian manifold endowed with an anti-torqued vector field. For this, we first determine a necessary and sufficient condition for the ambient space to admit such a vector field. Then we characterize submanifolds for which an anti-torqued vector field is always assumed to be tangent or normal. A similar characterization is also done in the case of the torqued vector fields. Finally, we obtain that the rectifying submanifolds with antitorqued axis are the warped products whose warping function is a first integration of the conformal scalar of the axis.

Let N be a positive integer and 𝚪 be a subgroup of SL₂(ℤ) containing 𝚪₁(N). Let K be an imaginary quadratic field and 𝒪 be an order of discriminant D_𝒪 in K. Under some assumptions, we show that 𝚪 induces a form class group of discriminant D_𝒪 (or of order 𝒪) and level N if and only if there is a certain canonical model of the modular curve for 𝚪 defined over a suitably small number field. In this way we can find an interesting link between two different subjects.

We present a method of control system design for n bodies in the real line ℝ¹ and on the unit circle S¹, to be collision-free and controllable. The problem reduces to designing a driftless control-affine system in ℝⁿ and in n-torus Tⁿ, respectively, that avoids certain obstacles. We prove the controllability of the system by showing that the vector fields that define the control-affine system, together with their brackets, span the whole tangent space of the state space, and then by applying the Rashevsky-Chow theorem.

We propose an Euler alignment model for target tracking and study its asymptotic behaviors. The proposed model is compressible Euler-type equations with local and nonlocal forcing terms and it can be derived from the kinetic model for target tracking in [3] using velocity moment system and closure condition a.k.a. mono-kinetic ansatz. We also present a suitable sufficient framework leading to a global solvability and exponential tracking toward a prescribed continuum target. For the exponential target tracking, we employ a generalized energy method incorporated together with hypo-coercivity estimate.

Employing the refined moving plane method within the rigorous framework of elliptic geometry, we derive some new monotonicity properties for the solutions of elliptic systems, thereby extending the scope of existing results confined to single-scale equations in the literature.

Homomorphic encryption (HE) is in the spotlight as a solution for privacy-related issues in various real-world scenarios. However, the limited types of operations supported by each HE scheme have been a major drawback in applications. While HE schemes based on learning-with-error (LWE) problem provide efficient lookup table (LUT) evaluation in terms of latency, they have downsides in arithmetic operations and low throughput compared to HE schemes based on ring LWE (RLWE) problem. The use of HE on circuits containing LUT has been partly limited if they contain arithmetic operations or their computational width is large. In this paper, we propose homomorphic algorithms for batched queries on LUTs by using RLWE-based HE schemes. To look up encrypted LUTs of size n on encrypted queries, our algorithms use O(log n) homomorphic comparisons and O(n) multiplications. For unencrypted LUTs, our algorithms use O(log n) comparisons, $O({\sqrt{n}})$ ciphertext multiplications, and O(n) scalar multiplications. We provide a proof-of-concept implementation based on CKKS scheme (Asiacrypt 2017). The amortized running time for an encrypted (resp., unencrypted) LUT of size 512 is 0.041 (resp., 0.025) seconds. Our implementation reported roughly 2.4-6.0x higher throughput than the current implementation of LWE-based schemes, with more flexibility on the structure of the LUTs.

We establish bounds on expected values of various geometric quantities that describe the size of the convex hull spanned by a path of the standard planar Brownian motion. Expected values of the perimeter and the area of the Brownian convex hull are known explicitly, and satisfactory bounds on the expected value of its diameter can be found in the literature as well. In this work we investigate circumradius and inradius of the Brownian convex hull and obtain lower and upper bounds on their expected values. Our other goal is to find bounds on the related inverse processes (that correspond to the perimeter, area, diameter, circumradius and inradius of the convex hull) which provide us with some information on the speed of growth of the size of the Brownian convex hull.

The Fourier coefficients c₁(n) of the elliptic modular j-function are always even for n ≢ 7 (mod 8). In contrast, for n ≡ 7 (mod 8), it is conjectured that "half " of the coefficients take odd values. In this article, we first observe in detail when c₁(8n - 1) is odd and show that the coefficients share the same parity as the coefficients c_𝜇2(n) of the 2nd order mock theta function 𝜇₂(q). Furthermore, we prove that this phenomenon also holds among several Hauptmoduln and between Hauptmoduln and even-order mock theta functions.