We investigate van Rijsbergen’s -measure, break-even point, and 11-point averaged precision, all of which can be translated into 1-dimensional scalar quantity from the precision and the recall. These investigations can be done by comparing to tetrachoric (four-fold) correlation coefficient and phi (four-fold point) coefficient, which are often used as the index of statistical association in a 2...

Let G be a locally compact abelian group with Haar measure m $m_G$ and Φ 1 , 2 $\Phi _1,\,\Phi _2$ Young functions. A bounded measurable function on is called Fourier ( ) $(\Phi _2)$ -multiplier if T f γ = ∫ x ̂ d $$\begin{equation*}\hskip7pc T_m (f)(\gamma )= \int _{G} m(x) \hat{f}(x) \gamma (x) dm_G(x),\hskip-7pc \end{equation*}$$ defined for functions in ∈ L $f\in L^1(\hat{G})$ such that $\ha...

In this paper, we study a class of dispersive wave equations on the Heisenberg group $$H^n$$ . Based Fourier transform , properties Laguerre functions and stationary phase lemma, establish decay estimates for semigroup given by $$e^{\textrm{it}\phi ({\mathscr {L}})}$$ where $$\phi : {\mathbb {R}}^+ \rightarrow {R}}$$ is smooth, $${\mathscr {L}}$$ sub-Laplacian Finally, using duality arguments, ...

We consider pairs of automorphisms $(\phi,\sigma)$ acting on fields Laurent or Puiseux series: shift operators $(\phi\colon x\mapsto x+h\_1, \sigma\colon x+h\_2)$, $q$-difference q\_1x$, $\sigma\colon q\_2x)$, and Mahler x^{p\_1},\ x^{p\_2})$. Given a solution $f$ to linear $\phi$-equation $g$ an algebraic $\sigma$-equation, both transcendental, we show that are algebraically independent over t...

We study the Bernoulli property for a class of partially hyperbolic systems arising from skew products. More precisely, we consider map $(T,M,\mu )$, where $\mu$ is Gibbs measure, an aperiodic Hölder continuous cocycle $\phi :M\to \mathbb {R}$ with zero mean and zero-entropy flow $(K_t,N,\nu )$. then product \begin{equation*} T_\phi (x,y)=(Tx,K_{\phi (x)}y), \end{equation*} acting on $(M\times...

In the present paper, we investigate differential geometric properties soliton surface $M$ associated with Betchov-Da Rios equation. Then, give derivative formulas of Frenet frame unit speed curve $\Phi=\Phi(s,t)$ for all $t$. Also, discuss linear map Weingarten type in tangent space that generates two invariants: $k$ and $h$. Moreover, obtain necessary sufficient conditions equation to be a mi...

A non-minimal coupling $\eta$ has been attracting growing interest particularly in the context of inflation models, though its quantum nature is not clear yet. We study renormalization a scalar electrodynamics (QED). find ${\it no}$ inhomogeneous term group equation (RGE) at two-loop level. This similar to other theories, where an RGE appears only higher-loop order: e.g., four-loop order $\lamb...