Friedmann cosmology with decaying vacuum density in Brans–Dicke theory

In this paper, we study Friedmann cosmology with time-varying vacuum energy density in the context of Brans–Dicke theory. We consider an isotropic and homogeneous flat space, filled with a matter-dominated perfect fluid and a dynamical cosmological term $$\varLambda (t) $$ Λ ( t ) , obeying the equation of state of the vacuum. As the exact nature of a possible time-varying vacuum is yet to be found, we explore $$\varLambda (t)$$ Λ ( t ) given by the phenomenological law $$\varLambda (t)=\lambda +\sigma H$$ Λ ( t ) = λ + σ H , where $$\lambda $$ λ and $$\sigma $$ σ are positive constants. We solve the model and then focus on two different cases $$\varLambda _{H1}$$ Λ H 1 and $$\varLambda _{H2}$$ Λ H 2 by assuming $$\varLambda =\lambda $$ Λ = λ and $$\varLambda =\sigma H$$ Λ = σ H , respectively. Notice that $$\varLambda _{H1}$$ Λ H 1 is the analog of the standard $$\varLambda $$ Λ CDM, but within the Brans–Dicke cosmology. We find the analytical solution of the main cosmological functions such as the Hubble parameter, the scale factor, deceleration and equation of state parameters for these models. In order to test the viability of the cosmological scenarios, we perform two sets of joint observational analyses of the recent Type Ia supernova data (Pantheon), observational measurements of Hubble parameter data, Baryon acoustic oscillation/Cosmic microwave background data and Local Hubble constant for each model. For the sake of comparison, the same data analysis is performed for the $$\varLambda $$ Λ CDM model. Each model shows a transition from decelerated phase to accelerated phase and can be viewed as an effective quintessence behavior. Using the model selection criteria AIC and BIC to distinguish from existing dark energy models, we find that the Brans–Dicke analog of the $$\varLambda $$ Λ -cosmology (i.e. our model $$\varLambda _{H1}$$ Λ H 1 ) performs at a level comparable to the standard $$\varLambda $$ Λ CDM, whereas $$\varLambda _{H2}$$ Λ H 2 is less favoured.

Global existence and convergence of a flow to Kazdan–Warner equation with non-negative prescribed function

We consider an evolution problem associated to the Kazdan–Warner equation on a closed Riemann surface $$(\Sigma ,g)$$ ( Σ , g ) $$\begin{aligned} -\Delta _{g}u=8\pi \left( \dfrac{he^{u}}{\int _{\Sigma }he^{u}\mathop {}\mathrm {d}\mu _{g}}-\dfrac{1}{\int _{\Sigma }\mathop {}\mathrm {d}\mu _{g}}\right) \end{aligned}$$ - Δ g u = 8 π h e u ∫ Σ h e u d μ g - 1 ∫ Σ d μ g where the prescribed function $$h\ge 0$$ h ≥ 0 and $$\max _{\Sigma }h>0$$ max Σ h > 0 . We prove the global existence and convergence under additional assumptions such as $$\begin{aligned} \Delta _{g}\ln h(p_0)+8\pi -2K(p_0)>0 \end{aligned}$$ Δ g ln h ( p 0 ) + 8 π - 2 K ( p 0 ) > 0 for any maximum point $$p_0$$ p 0 of the sum of $$2\ln h$$ 2 ln h and the regular part of the Green function, where K is the Gaussian curvature of $$\Sigma $$ Σ . In particular, this gives a new proof of the existence result by Yang and Zhu (Pro Am Math Soc 145:3953–3959, 2017) which generalizes existence result of Ding et al. (Asian J Math 1:230–248, 1997) to the non-negative prescribed function case.

Limits on the charged Higgs parameters in the two Higgs doublet model using CMS $$\sqrt{s}=13$$ TeV results

The latest CMS results on the upper limits on $$\sigma _{H^\pm }$$σH±BR($$H^\pm \rightarrow \tau ^\pm \nu )$$H±→τ±ν) and $$\sigma _{H^\pm }$$σH±BR($$H^+ \rightarrow t{\bar{b}}$$H+→tb¯) for $$\sqrt{s}=13$$s=13 TeV at an integrated luminosity of 35.9 $$\hbox {fb}^{-1}$$fb-1 are used to impose constraints on the charged Higgs $$H^\pm $$H± parameters within the Two Higgs Doublet Model (2HDM). The 2HDM is the simplest extension of the Standard Model (SM) under the same gauge symmetry to contain charged Higgs and is relatively little constrained compared to the Minimal Supersymmetric Standard Model (MSSM). The latest results lead to much more stringent constraints on the charged Higgs parameter space than for the earlier 8 TeV results. The CMS collaboration also studied the exotic bosonic decays $$H^\pm \rightarrow W^\pm A$$H±→W±A and $$A \rightarrow \mu ^+ \mu ^-$$A→μ+μ- for the first time and put upper limits on the BR($$t\rightarrow H^+ b$$t→H+b) for the light charged Higgs boson. These constraints lead to the exclusion of parameter space which is not excluded by the $$\tau \nu $$τν channel. For comparison the exclusion regions from flavor physics constraints are also discussed.

Transitivity of The delta^n-Relation in Hypergroups

The $\delta^n$-relation was introduced by Leoreanu-Fotea et. al.\cite{130}. In this article, we introduce the concept of$\delta^{n}$-heart of a hypergroup and we determine necessary andsufficient conditions for the relation $\delta^{n}$ to betransitive. Moreover, we determine a family $P_{\sigma}(H)$ ofsubsets of a hypergroup $H$ and we give sufficient conditionssuch that the geometric space $(H, P_{\sigma}(H))$ is stronglytransitive and the relation $\delta^n$ is transitive.

VISIBLE ACTIONS ON FLAG VARIETIES OF TYPE B AND A GENERALISATION OF THE CARTAN DECOMPOSITION

We give a generalisation of the Cartan decomposition for connected compact Lie groups of type B motivated by the work on visible actions of Kobayashi [‘A generalized Cartan decomposition for the double coset space $(U(n_{1})\times U(n_{2})\times U(n_{3})) \backslash U(n)/ (U(p)\times U(q))$’, J. Math. Soc. Japan59 (2007), 669–691] for type A groups. Suppose that $G$ is a connected compact Lie group of type B, $\sigma $ is a Chevalley–Weyl involution and $L$, $H$ are Levi subgroups. First, we prove that $G=LG^{\sigma }H$ holds if and only if either (I) both $H$ and $L$ are maximal and of type A, or (II) $(G,H)$ is symmetric and $L$ is the Levi subgroup of an arbitrary maximal parabolic subgroup up to switching $H$ and $L$. This classification gives a visible action of $L$ on the generalised flag variety $G/H$, as well as that of the $H$-action on $G/L$ and of the $G$-action on $(G\times G)/(L\times H)$. Second, we find an explicit ‘slice’ $B$ with $\dim B=\mathrm {rank}\, G$ in case I, and $\dim B=2$ or $3$ in case II, such that a generalised Cartan decomposition $G=LBH$holds. An application to multiplicity-free theorems of representations is also discussed.