We show that the homology torsion growth of a free-by-cyclic group with polynomially growing monodromy vanishes in every dimension independently of the choice of Farber chain. It follows that the integral torsion ${\rho }^{\mathbb{Z}}$ equals the ${\ell }^2$-torsion ${\rho }^{(2)}$ verifying a conjecture of Lück for these groups.

We establish two supercongruences for ${\text{bi}}^s$nomial coefficients. We give a supercongruence similar to Jacobsthal’s binomial congruence, and as a consequence, we confirm the following conjecture for trinomial coefficients:

$${\left(\genfrac{}{}{0.0pt}{}{n{p}^r}{k{p}^r}\right)}_2\equiv {\left(\genfrac{}{}{0.0pt}{}{n{p}^{r-1}}{k{p}^{r-1}}\right)}_2(\mathrm{mod}{p}^{2r}),$$

where $n,k$ be nonnegative integers and $r\ge 1$ is an integer with $p>3$ is an odd prime number, which were posed by G.-S. Mao [On some congruences involving trinomial coefficients. Rocky Mountain J. Math. 50(5) (2020), 1759–1771]. We also generalize the Ljunggren congruence for binomial coefficients to ${\text{bi}}^s$nomial coefficients.

We prove an existence theorem for the solution of a nonlinear integral equation of Volterra type in Hölder space using the technique of relative compactness combined with fixed point theorem. Our main tool is the Schauder fixed point theorem. We provide a suitable example to show the efficiency of our main result.

It is a well-known fact that every path-connected Hausdorff space is arcwise connected. Typically, this result is viewed as a consequence of a sequence of fairly technical results from continuum theory. In this expository note, we present a direct proof of this statement, which constructs an arc from a path by “deleting” a maximal collection of subloops. By breaking the proof into multiple steps, we examine generalizations of this result and the limits of our line of argument. In particular, we identify a modest extension to a class of spaces relevant to algebraic topology.

A ZPUI domain $D$ is an integral domain with property $(\mathrm{\#})$: every nonzero proper ideal $I$ of $D$ can be written as $I=J{P}_1\cdots P_n$, where $J$ is an invertible ideal of $D$ and $\{P_1,\dots ,P_n\}$ is a nonempty collection of pairwise comaximal prime ideals of $D$. Among other things, we study two types of natural generalizations of ZPUI domains: (i) the $J$ in the property $(\mathrm{\#})$ is principal and (ii) the property $(\mathrm{\#})$ holds for all nonzero principal ideals of $D$. For example, we show that (1) $D$ satisfies (i) if and only if $D$ is a ZPUI domain whose invertible ideals are principal and (2) $D$ satisfies (ii) if and only if $D$ is an $h$-local domain in which each maximal ideal is invertible. We also study the $w$-operation analogs of these two properties.

We derive a useful result about the zeros of the $k$-polar polynomials on the unit circle; in particular, we obtain a circular region containing all the zeros of these polynomials. Some examples are presented.

In a different way than in the literature, we define the concept of a quasicoincident using the bipolar fuzzy soft points we previously proposed (2021) and investigate its basic properties. We introduce the notion of a bipolar fuzzy soft net (for short BFS-net) and give convergence of the BFS-nets in a bipolar fuzzy soft topological space with useful results. We show how a BFS-net is derived from a BFS-filter and obtain a characterization about bipolar fuzzy soft Hausdorff spaces. Based on the idea of quasicoincident, we give a new kind of bipolar fuzzy soft continuity and analyze its relationship with the BFS-nets. We put forward the idea of compactness in the setting of bipolar fuzzy soft sets and characterize it through the contribution of the BFS-subnets. Finally, we present some examples to illustrate the defined concepts.

Kirigami is the art of cutting paper to make it articulated and deployable, allowing for it to be shaped into complex two and three-dimensional geometries. The mechanical response of a kirigami sheet when it is pulled at its ends is enabled and limited by the presence of cuts that serve to guide the possible non-planar deformations. Inspired by the geometry of this art form, we ask two questions: (i) What is the shortest path between points at which forces are applied? (ii) What is the nature of the ultimate shape of the sheet when it is strongly stretched? Mathematically, these questions are related to the nature and form of geodesics in the Euclidean plane with linear obstructions (cuts), and the nature and form of isometric immersions of the sheet with cuts when it can be folded on itself, and are related to a class of questions associated with isometric immersions in differential geometry and geodesics in discrete and computational geometry. We provide a constructive proof that the geodesic connecting any two points in the plane is piecewise polygonal. We then prove that the family of polygonal geodesics can be simultaneously rectified into a straight line by flat-folding the sheet so that its configuration is a (non-unique) piecewise affine planar isometric immersion.

Given ${\mathrm{C}}^{\ast }$-algebras $A$ and $B$ and a ${}^{\ast }$-homomorphism $\varphi :A\to B$, we adopt the portrait of the relative $K$-theory $K_{\ast }(\varphi )$ due to Karoubi using Banach categories and Banach functors. We show that the elements of the relative groups may be represented in a simple form. We prove the existence of two six-term exact sequences, and we use these sequences to deduce the fact that the relative theory is isomorphic, in a natural way, to the $K$-theory of the mapping cone.

We provide a rigorous mathematical study for global asymptotic stability (GAS) of a recognized fractional-order hepatitis B epidemic model, which was proposed in a recent work. We use a simple approach to establish the GAS of the fractional-order hepatitis B model. This approach is based on extensions of the Lyapunov stability theory in combination with some nonstandard techniques for fractional dynamical systems. As an important consequence, the GAS of disease free and disease endemic equilibrium points is determined fully. The obtained results not only improve but also generalize some existing works. A set of numerical experiments is performed to support and illustrate the constructed theoretical results.

The module cancellation problem asks whether, given modules $X$, $X^{\prime }$ and $Y$ over a ring $\mathrm{\Lambda }$, the existence of an isomorphism $X\oplus Y\cong X^{\prime }\oplus Y$ implies that $X\cong X^{\prime }$. When $q$ is prime we prove a strong cancellation property for certain modules over $\mathbb{Z}[C_{\infty }\times C_q]$, generalizing, in part, the strong cancellation property for modules over $\mathbb{Z}[C_q]$ established by R. Wiegand (1984).

The minimal excludant of an integer partition, first studied prominently by Andrews and Newman from a combinatorial viewpoint, is the smallest positive integer missing from a partition. Several generalizations of this concept are being explored by mathematicians nowadays. We analogously consider the second minimal excludant of a partition and analyze its relationship with the minimal excludant. This leads us to the notion of a mex sequence and we derive two neat identities involving the number of partitions whose mex sequence has length at least $r$.

A real, rational function $r(t)$ determines an argument function $𝜗(t)$, whose increment $\mathrm{\Delta }𝜗(t)$ relates the numbers of distinct, real zeros and poles of $r(t)$. The present expository note derives such a result, and explains its relevance to the index for a certain class of planar singularities.

We prove some results of restricted sums of four squares using arithmetic of quaternions in the ring of Lipschitz integers. In particular, we show that every nonnegative integer $n$ can be written as $x^2+y^2+z^2+t^2$, where $x$, $y$, $z$, $t$ are integers and $x+3y+3z$ is a power of 2.

Keimel and Lawson (2009) proposed a set of conditions for proving the reflectivity of a category of topological spaces in the category of all $T_0$ spaces. Recently, these conditions were used to prove the reflectivity of the category of all well-filtered spaces. We prove that, in certain sense, these conditions are not only sufficient but also necessary for a category of $T_0$ spaces to be reflective. By applying this general result, we can easily deduce that several categories proposed in domain theory are not reflective, thereby answering a few open problems.

Let $\mathcal{\mathscr{H}}$ be a complex Hilbert space with $\text{dim }\mathcal{\mathscr{H}}\ge 2$ and $\mathcal{ℬ}{(\mathcal{\mathscr{H}})}^{+}$ the positive cone of the algebra of all bounded linear operators on $\mathcal{\mathscr{H}}$. For $A,B\in \mathcal{ℬ}{(\mathcal{\mathscr{H}})}^{+}$, $A$ is called a positive truncation of $B$ if $A=P_{A}B{P}_A$, where $P_A$ denotes the orthogonal projection onto the closure of $R(A)$. We determine structures of all bijections preserving the positive truncations of operators in both directions on the positive cone $\mathcal{ℬ}{(\mathcal{\mathscr{H}})}^{+}$.

The purpose of this manuscript is to consider the controllability and observability of quaternion-valued impulsive differential equations (QIDEs). First, sufficient and necessary criteria for the controllability of linear QIDEs are investigated. Then, by constructing suitable control functions and employing fixed-point theorems, the controllability of nonlinear QIDEs is acquired. Next, sufficient and necessary conditions for the observability of linear QIDEs are derived. Most important of all, theoretical results obtained in the sense of complex-valued and quaternion-valued are equivalent to each other due to the adjoint matrix of quaternion matrix and the isomorphism between quaternion vector space and complex variables space. Finally, two examples are provided to indicate the feasibility of our results achieved.

Let ${\psi }_{q,n}={(-1)}^{n-1}{\psi }_q^{(n)}$ for $n\in \mathbb{N}$, where ${\psi }_q^{(n)}$ are the $q$-polygamma functions. In this paper, by the monotonicity rules for the ratio of two power series, it is proved that, for $q\in (0,1)$ and $n\in \mathbb{N}$, the function

$$x\mapsto F_{q,n}(x;\alpha )=\frac{q^{x+\alpha }-1}{\text{ln }q}\frac{{\psi }_{q,n+1}(x)}{{\psi }_{q,n}(x)},$$

is decreasing (increasing) on $(0,\infty )$ if and only if $\alpha \le {\text{log }}_q(2^n∕(q+1))$ ($\alpha \ge 0$). The conditions for which several relevant functions are monotonic or completely monotonic on $(0,\infty )$ are obtained. Moreover, several relations involving the $q$-polygamma functions are established.