Based on the stubbed coupled line for matching a load, we design an impedance transformer matched the load of ${{\text{Z}}_{\text{L}}} = 100{\Omega }$, working at four frequencies in this paper. This proposed transformer comprises two sections of coupled line constructed from parallel-coupled transmission lines. Two different types of the proposed transformer are given and analyzed, and we derive carefully about the open-circuit form equations for impedance transformer. We fabricate an impedance transformer to certify the validity of design equations theoretically, and the frequency is measured at 0.7/2.4/3.6/5.3 GHz. The measured results are in good agreement with the simulated results at each frequency.

In the context of dependent type theory, we show that coinductive predicates have an equivalent topological counterpart in terms of coinductively generated positivity relations, introduced by G. Sambin to represent closed subsets in point-free topology. Our work is complementary to a previous one with M. E. Maietti, where we showed that, in dependent type theory, the well-known concept of wellfounded trees has a topological counterpart in terms of proof-relevant inductively generated formal covers used to provide a predicative and constructive representation of complete suplattices. The proofs performed within Martin–Löf’s type theory and the Minimalist Foundation have been checked in the Agda proof assistant.

The burgeoning critical scholarship on space in International Relations (IR) overwhelmingly recognises space as a socially produced set of performances, practices, and discourses, converging into meaningful organisations of located experience. Drawing on the writings of Deleuze and Guattari on the related concepts of nomadism and the war machine, I argue that this productive emphasis betrays a continued statist methodology that proceeds by binding, or partitioning, space into finished outcomes. I present a conceptual challenge to the normative emphasis on socially produced space by following nomadism, the immanent tendency to variation in the process of spatial becoming. Working with nomadic potentials brings to the fore smooth space, which includes the continuous possibilities and intensities existing unencumbered beneath concretised productions of organised space. I follow the spatial movement of violence in Punjab during the Indian Partition of 1947 as the emergence of a war machine which deployed the nomadism of smooth space to decompose and upend striations. My objectives are first, to argue for spatial possibilities beyond the normative positivity of produced space, and secondly, to register the fundamental methodological and analytical shifts that these possibilities demand. These shifts can in turn deepen ongoing disciplinary inquiries into indeterminacy.

Let $f $ be a normalized Hecke eigenform of even weight $k \geq 2$ for $SL_2(\mathbb {Z})$. In this article, we establish an asymptotic formula for the shifted convolution sum of a general divisor function, where the sum involves the Fourier coefficients of a multi-folded L-function weighted with a kernel function.

For any integers x and y, let $(x, y)$ and $[x, y]$ stand for the greatest common divisor and the least common multiple of x and y, respectively. Let $a,b$ and n be positive integers, and let $S=\{x_1, \ldots , x_n\}$ be a set of n distinct positive integers. We denote by $(S^a)$ and $[S^a]$ the $n\times n$ matrices having the ath power of $(x_i,x_j)$ and $[x_i,x_j]$, respectively, as the $(i,j)$-entry. Bourque and Ligh [‘On GCD and LCM matrices’, Linear Algebra Appl. 174 (1992), 65–74] showed that if S is factor closed (that is, S contains all positive divisors of any element of S), then the GCD matrix $(S)$ divides the LCM matrix $[S]$ (written as $(S)\mid [S]$) in the ring $M_n({\mathbb Z})$ of $n\times n$ matrices over the integers. Hong [‘Divisibility properties of power GCD matrices and power LCM matrices’, Linear Algebra Appl. 428 (2008), 1001–1008] proved that $(S^a)\mid (S^b)$, $(S^a)\mid [S^b]$ and $[S^a]\mid [S^b]$ in the ring $M_{n}({\mathbb Z})$ when $a\mid b$ and S is a divisor chain (namely, there is a permutation $\sigma $ of order n such that $x_{\sigma (1)}\mid \cdots \mid x_{\sigma (n)}$). In this paper, we show that if $a\mid b$ and S is factor closed, then $(S^a)\mid (S^b)$, $(S^a)\mid [S^b]$ and $[S^a]\mid [S^b]$ in the ring $M_{n}({\mathbb Z})$. The proof is algebraic and p-adic. Our result extends the Bourque–Ligh theorem. Finally, several interesting conjectures are proposed.