Let G = (V, E) be a connected finite graph. We study the Bogomol'nyi equation $${\Delta}u=e^u-4{\pi}{\sum_{s=1}^k}n_s{\delta}_{z_s}\;on\;G,$$, where z₁, z₂, . . . , z_k are arbitrarily chosen distinct vertices on the graph, n_j is a positive integer, j = 1, 2, . . . , k and 𝛿_zs is the Dirac mass at z_s. We obtain a necessary and sufficient condition for the existence and uniqueness of solutions to the Bogomol'nyi equation.

We investigate Krull dimensions of semirings and semifields dealt with in tropical geometry. For a congruence C on a tropical Laurent polynomial semiring 𝕋[X^±₁, . . . , X^±_n], a finite subset T of C is called a finite congruence tropical basis of C if the congruence variety V (T) associated with T coincides with V (C). For C proper, we prove that the Krull dimension of the quotient semiring 𝕋[X^±₁, . . . , X^±_n]/C coincides with the maximum of the dimension of V (C) as a polyhedral complex plus one and that of V (C_𝔹) when both C and C_𝔹 have finite congruence tropical bases, respectively. Here C_𝔹 is the congruence on 𝕋[X^±₁, . . . , X^±_n] generated by {(f_𝔹, g_𝔹) | (f, g) ∈ C} and f_𝔹 is defined as the tropical Laurent polynomial obtained from f by replacing the coefficients of all non-∞ terms of f with the real number zero. With this fact, we also show that rational function semifields of tropical curves that do not consist of only one point have Krull dimension two.

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In this work we investigate semi-Riemannian submanifolds with vanishing extrinsic curvature. Equivalently, we can say that such submanifolds inherit sectional curvature from their ambient space. We prove that a ruled surface in a three-dimensional manifold inherits curvature if and only if its rulings are lines of curvature. Among our principal results, we prove that in the case that a non-degenerate hypersurface with diagonalizable shape operator inherits curvature then at most one of its principal curvatures does not vanish. Moreover, if a hypersurface is the graph of a function then we explicitly calculate the curvature it inherits. If a surface is the graph of a function then it inherits curvature if and only if the determinant of its Hessian vanishes. It is known that the graph of an eikonal function on a surface inherits curvature. We give some characterizations of this case.

We find explicit equations of a new pair of fake projective planes, labeled by (C18, p = 3, {2I}) in the Cartwright-Steger classification. Our method involves starting with known equations of a commensurable fake projective plane (C18, p = 3, ∅, d₃D₃) and working through a chain of cyclic covers and quotients to get to the new one.

We consider compact convex hypersurfaces contracting by the curvature flow with speed equal to the mean curvature minus its reciprocal in Euclidean space. Given a smooth compact strictly convex initial hypersurface, we obtain a pinching estimate for the curvature of the evolving hypersurfaces. We also show that the rescaled hypersurfaces converge smoothly to a round sphere.

Let R(q) denote the Rogers-Ramanujan continued fraction for |q| < 1. Using expressions involving the theta function f(-q) := (q; q)∞ given in modular relations due to Yi, we introduce evaluations for $R(e^{-{\pi}{\sqrt{r}}})$, for r ∈ ℚ_>0. We prove our closed forms using the icosahedral equation for R together with closed forms for and modular relations associated with Ramanujan's G- and g-functions.

Let (𝓐, 𝓑, 𝓒) be a recollement of extriangulated categories. We show that left Frobenius pairs in 𝓐 and 𝓒 induce a left Frobenius pair in 𝓑, and establish a partial converse under mild conditions. Furthermore, by leveraging recollements of extriangulated categories and left Frobenius pairs, we construct a recollement of right triangulated categories.

Following Spivey's seminal recurrence relation for Bell numbers, numerous generalizations have been developed for various families of Bell-type numbers and polynomials. In particular, Kim and Kim introduced degenerate Bell polynomials and established several Spivey-type recurrence relations using analytic and probabilistic techniques. In this work, we introduce a new class of polynomials, called the probabilistic degenerate poly-Bell polynomials, which naturally arise from a combination of degenerate exponential structures and probabilistic generating mechanisms associated with a random variable whose moment generating function exists in a neighborhood of the origin. By combining generating function methods with an operator-based approach, we derive new Spivey-type recurrence relations that extend classical identities in a unified framework. Several special cases are discussed, including reductions to probabilistic degenerate Bell polynomials and classical poly-Bell polynomials. These reductions not only confirm the consistency of the proposed framework but also provide a clearer interpretation of how the new polynomials connect analytic and probabilistic structures. Our results extend and unify earlier works on degenerate and probabilistic Bell-type polynomials.