A series of recent articles has shown that there exist only three monogenic cyclic quartic trinomials in $$\mathbb {Z}[x]$$ , and they are all of the form $$x^4+bx^2+d$$ . In this article, we conduct an analogous investigation for cubic trinomials in $$\mathbb {Z}[x]$$ . Two irreducible cyclic cubic trinomials are said to be equivalent if their splitting fields are equal. We show that there exist two infinite families of non-equivalent monogenic cyclic cubic trinomials of the form $$x^3+Ax+B$$ . We also show that there exist exactly four monogenic cyclic cubic trinomials of the form $$x^3+Ax^2+B$$ , all of which are equivalent to $$x^3-3x+1$$ .

We demonstrate a new method to prove the exact boundary controllability for the wave equation on an interval. The method uses a dynamical argument to demonstrate the shape and the velocity controllability, thereby solving their associated moment problems. This permits solving the moment problem related to exact controllability. The method is illustrated by numerical examples.

We use thin position of Heegaard splittings to give a new proof of Haken’s Lemma that a Heegaard surface of a reducible manifold is reducible and of Scharlemann’s “Strong Haken Theorem”: a Heegaard surface for a 3-manifold may be isotoped to intersect a given collection of essential spheres and discs in a single loop each. We also give a reformulation of Casson and Gordon’s theorem on weakly reducible Heegaard splittings, showing that they exhibit additional structure with respect to certain incompressible surfaces. This article could also serve as an introduction to the theory of generalized Heegaard surfaces and it includes a careful study of their behavior under amalgamation.

AbstractIn this paper, we consider the Diophantine equation $$\lambda _1U_{n_1}+\cdots +\lambda _kU_{n_k}=wp_1^{z_1} \ldots p_s^{z_s},$$ λ 1 U n 1 + ⋯ + λ k U n k = w p 1 z 1 … p s z s , where $$\{U_n\}_{n\ge 0}$$ { U n } n ≥ 0 is a fixed non-degenerate linear recurrence sequence of order greater than or equal to 2; w is a fixed non-zero integer; $$p_1,\dots ,p_s$$ p 1 , ⋯ , p s are fixed, distinct prime numbers; $$\lambda _1,\dots ,\lambda _k$$ λ 1 , ⋯ , λ k are strictly positive integers; and $$n_1,\dots ,n_k,z_1,\dots ,z_s$$ n 1 , ⋯ , n k , z 1 , ⋯ , z s are non-negative integer unknowns. We prove the existence of an effectively computable upper-bound on the solutions $$(n_1,\dots ,n_k,z_1,\dots ,z_s)$$ ( n 1 , ⋯ , n k , z 1 , ⋯ , z s ) . In our proof, we use lower bounds for linear forms in logarithms, extending the work of Pink and Ziegler (Monatshefte Math 185(1):103–131, 2018), Mazumdar and Rout (Monatshefte Math 189(4):695–714, 2019), Meher and Rout (Lith Math J 57(4):506–520, 2017), and Ziegler (Acta Arith 190:139–169, 2019).

Many mathematical statements have the following form. If something is true for all finite subsets of an infinite set I, then it is true for all of I. This paper describes some old and new results on infinite sets of linear and polynomial equations with the property that solutions for all finite subsets of the set of equations imply the existence of a solution for the infinite set of equations.

Let $$(P_n)_{n\ge 0}$$ and $$(R_n )_{n\ge 0}$$ be the Padovan and Perrin sequences, respectively. Let $$b\ge 2$$ be an integer. In this paper, we study the Diophantine equations $$P_{n}=b^{d}R_{m}+R_{k}$$ and $$R_{n}=b^{d}P_{m}+P_{k}$$ in non-negative integers (n, m, k),  where d denotes the number of digits of $$R_k$$ and $$P_k$$ in base b,  respectively. Furthermore, we will see that in the range $$2\le b\le 100$$ the number 170,625 is the largest Padovan number which can be represented as a concatenation of two Perrin numbers, on the other hand the number 101,639 is the highest Perrin number which can be a concatenation of two Padovan numbers.

The topological (resp. geodesic) complexity of a topological (resp. metric) space is roughly the smallest number of continuous rules required to choose paths (resp. shortest paths) between any points of the space. We prove that the geodesic complexity of a 3-dimensional cube exceeds its topological complexity by exactly 2. The proof involves a careful analysis of cut loci of the cube.

AbstractWe enumerate the 188 3-orbit skeletal polyhedra in $${\mathbb {E}}^3$$ E 3 with irreducible symmetry group. The analysis is carried out by determining the polyhedra having each irreducible finite group of isometries as their symmetry group. Relevant information of every polyhedron is also organized in tables.

Trivalent 2-stratifolds X with finite homology groups can be encoded as bicolored trivalent graphs $$\Gamma $$ . We show that the homeomorphism problem for such stratifolds can be solved with complexity O(||V||), where V is the set of vertices of $$\Gamma $$ .

The starting point for this work is the family of functions $$\overline{p}_{-t}(n)$$ which counts the number of t-colored overpartitions of n. In recent years, several infinite families of congruences satisfied by $$\overline{p}_{-t}(n)$$ for specific values of $$t\ge 1$$ have been proven. In particular, in his 2023 work, Saikia proved a number of congruence properties modulo powers of 2 for $$\overline{p}_{-t}(n)$$ for $$t=5,7,11,13$$ . He also included the following conjecture in that paper: Conjecture: For all $$n\ge 0$$ and primes t, we have $$\begin{aligned} \overline{p}_{-t}(8n+1)\equiv & {} 0 \pmod {2}, \\ \overline{p}_{-t}(8n+2)\equiv & {} 0 \pmod {4}, \\ \overline{p}_{-t}(8n+3)\equiv & {} 0 \pmod {8}, \\ \overline{p}_{-t}(8n+4)\equiv & {} 0 \pmod {2}, \\ \overline{p}_{-t}(8n+5)\equiv & {} 0 \pmod {8}, \\ \overline{p}_{-t}(8n+6)\equiv & {} 0 \pmod {8}, \\ \overline{p}_{-t}(8n+7)\equiv & {} 0 \pmod {32}. \end{aligned}$$ Using a truly elementary approach, relying on classical generating function manipulations and dissections, as well as proof by induction, we show that Saikia’s conjecture holds for all odd integers t (not necessarily prime).

We begin by recalling Enrique Ramirez’s days at the University of Göttingen in the late 1960s, and the roots of my connection to him and his work. Starting with the classical Cauchy Integral Formula, we then review some of the key results in the development of integral representations in multidimensional complex analysis up to the early 1960s. Next, we discuss the key steps in the construction of the Henkin–Ramirez kernel and explain the impact that it and its subsequent variations and generalizations had on the field in the 1970s and 1980s. We conclude by briefly discussing some unsolved problems that relate to this work, including some suggestions to encourage the next generation to attempt to make progress on them.

This paper describes an algorithm for obtaining subelliptic estimates on pseudoconvex complex manifolds and CR manifolds. It includes a brief account of the microlocal analysis that is involved. The technique is illustrated on three-dimensional CR manifolds defined by triangular systems constructed to test the effectiveness of previous algorithms. The method introduced here is a “deconstruction” of the earlier technique; the estimates for forms are obtained by piecing together estimates for microlocalized components of the forms.

AbstractA method for successive synthesis of a Weyl matrix (or Dirichlet-to-Neumann map) of an arbitrary quantum tree is proposed. It allows one, starting from one boundary edge, to compute the Weyl matrix of a whole quantum graph by adding on new edges and solving elementary systems of linear algebraic equations in each step.

There are several characterizations of rings over which the modules admit certain covers (like injective, absolutely pure) or envelopes (like flat, torsion-free), not in the usual relation with cotorsion pairs. In this note we discuss commutative rings whose modules have divisible, h-divisible, or weak-injective covers, resp. commutative rings with weak-dimension 1 or projective dimension 1 preenvelopes. Subperfect rings and tight systems that are needed in the discussion are also dealt with in details.

We prove that graphs following the model of Barabasi–Albert tree with n vertices are hypoenergetic in the large n limit.

Bornology is a study of bounded sets, the spaces of such objects, and the naturally associated maps between the spaces of such objects. Previously, the setting for such a study has most often been in the presence of another type of structure such as metrics, topologies, uniformities, and/or, some sort of an algebraic structure with bounded sets being considered only as these were useful in the study of the main objects of concern. Here we carry out a study of bornology with its objects and maps and with these as the focus of the study. We describe the nature of the objects and maps of such things as subspaces, products, quotients, sums, etc., of the resulting category of those spaces. Mathematicians working on research projects centered around various kinds of topology related objects and maps will no doubt find the contents of the present paper very useful and efficient in that these mathematicians will not need to define boundedness related concepts or prove theorems about bounded sets since these items are within this present paper.

In this paper, we find all the Padovan and Perrin numbers which are product of two repdigits.

Let $$k\ge 2$$ be an integer. A generalization of the well-known Fibonacci sequence is the k-Fibonacci sequence. For this sequence, the first k terms are $$0,\ldots ,0,1$$ and each term afterwards is the sum of the preceding k terms. The goal of this paper is to investigate the Fermat and Mersenne numbers having representation as product of two k-Fibonacci numbers.

González-Acuña showed that Artin presentations characterize closed, orientable 3-manifold groups. Winkelnkemper later discovered that each Artin presentation determines a smooth, compact, simply connected 4-manifold. We utilize triangle groups to find all Artin presentations on two generators that present the trivial group. We then determine all smooth, closed, simply connected 4-manifolds with second betti number at most two that appear in Artin presentation theory.

The authors consider the time delay systems both with and without a perturbation term $$\begin{aligned} \dot{x}(t)=-Dx(t)+ C\int _{t-h}^{t}x(s)\mathrm{d}s + P(t,x(t)) \end{aligned}$$ and $$\begin{aligned} \dot{x}(t)= Dx(t) + C\int _{t-h}^{t}x(s)\mathrm{d}s, \end{aligned}$$ where $$x(t)\in {\mathbb {R}}^n$$ is the state vector, D and $$C\in {\mathbb {R}}^{n\times n}$$ are constant matrices, $$P\in C({\mathbb {R}}^{+}\times {\mathbb {R}}^{n},{\mathbb {R}}^{n})$$ , and $$h>0$$ is a constant time delay. They use the Razumikhin method to obtain some new conditions for the uniform asymptotic stability, instability, and exponential stability of the zero solution, the square integrability of the norms of all solutions of the unperturbed equation, and the boundedness of solutions of the perturbed equation. In the process, they are able to give a much simpler version of a recent result by Tian et al. (Appl Math Lett 101:106058, 2020).

We study the differential equation that corresponds to the one-dimensional frictionless unsteady flow model of a cylindrical draining tank. We survey previous results, solve the equation applying new changes of variables and procedures, and present new exact elementary solutions. The problem provides an excellent example of application that is accessible to undergraduate students after a first course on differential equations.

We define the notion of near-geodesic between points of a metric space when no geodesic exists, and use this to extend Recio-Mitter’s notion of geodesic complexity to non-geodesic spaces. This has potential application to topological robotics. We determine explicit near-geodesics and geodesic complexity in a variety of cases.

Universality, a desirable feature in any system. For decades, elusive measurements of three-phase flows have yielded countless permeability models that describe them. However, the equations governing the solution of water and gas co-injection has a robust structure. This universal structure stands for Riemann problems in green oil reservoirs. In the past we established a large class of three phase flow models including convex Corey permeability, Stone I and Brooks–Corey models. These models share the property that characteristic speeds become equal at a state somewhere in the interior of the saturation triangle. Here we construct a three-phase flow model with unequal characteristic speeds in the interior of the saturation triangle, equality occurring only at a point of the boundary of the saturation triangle. Yet the solution for this model still displays the same universal structure, which favors the two possible embedded two-phase flows of water-oil or gas-oil. We focus on showing this structure under the minimum conditions that a permeability model must meet. This finding is a guide to seeking a purely three-phase flow solution maximizing oil recovery.

A theorem of Rihane, Hernane and Togbé on a parametric family of simultaneous Pell equations is proved using classical results on quartic Diophantine equations due to Cohn and to Ljunggren.