pp 1-18
On Counting Twists of a Character Appearing in its Associated Weil Representation
Consider an irreducible, admissible representation 𝜋 of $GL(2,F)$ whose restriction to $GL(2,F)^+$ breaks up as a sum of two irreducible representations $𝜋_+ +𝜋_-$. If $𝜋=r_θ$, the Weil representation of $GL(2,F)$ attached to a character θ of $K^∗$ does not factor through the norm map from 𝐾 to 𝐹, then $𝜒\in \widehat{K^∗}$ with $(𝜒\cdot p^{θ^{-1}})|F^∗=𝜔 K/F$ occurs in $r_{θ+}$ if and only if $\in(θ𝜒^{-1},\psi_0)=\in(\overline{θ}𝜒^{-1},\psi_0)=1$ and in $r_{θ−}$ if and only if both the epsilon factors are $-1$. But given a conductor 𝑛, can we say precisely how many such 𝜒 will appear in 𝜋? We calculate the number of such characters at each given conductor 𝑛 in this work.
pp 19-26
Projective Normality of Weyl Group Quotients
In this note, we prove that for the standard representation 𝑉 of the Weyl group 𝑊 of a semi-simple algebraic group of type $A_n,B_n,C_n,D_n,F_4$ and $G_2$ over $\mathbb{C}$, the projective variety $\mathbb{P}(V^m)/W$ is projectively normal with respect to the descent of $\mathcal{O}(1)^{\otimes|W|}$, where $V^m$ denote the direct sum of 𝑚 copies of 𝑉.
pp 27-35
Quillen Bundle and Geometric Prequantization of Non-Abelian Vortices on a Riemann Surface
In this paper we prequantize the moduli space of non-abelian vortices. We explicitly calculate the symplectic form arising from 𝐿^{2} metric and we construct a prequantum line bundle whose curvature is proportional to this symplectic form. The prequantum line bundle turns out to be Quillen’s determinant line bundle with a modified Quillen metric. Next, as in the case of abelian vortices, we construct line bundles over the moduli space whose curvatures form a family of symplectic forms which are parametrized by $\Psi_0$, a section of a certain bundle. The equivalence of these prequantum bundles are discussed.
pp 37-44
Nonsmoothable Involutions on Spin 4-Manifolds
Let 𝑋 be a closed, simply-connected, smooth, spin 4-manifold whose intersection form is isomorphic to $n(-E_8)\oplus mH$, where 𝐻 is the hyperbolic form. In this paper, we prove that for 𝑛 such that $n≡ 2\mathrm{mod} 4$, there exists a locally linear pseudofree $\mathbb{Z}_2$-action on 𝑋 which is nonsmoothable with respect to any possible smooth structure on 𝑋.
pp 45-75
On 𝑔-Functions for Laguerre Function Expansions of Hermite Type
We examine weighted $L^p$ boundedness of 𝑔-functions based on semi-groups related to multi-dimensional Laguerre function expansions of Hermite type. A technique of vector-valued Calderón–Zygmund operators is used.
pp 77-81
Lacunary Fourier Series and a Qualitative Uncertainty Principle for Compact Lie Groups
We define lacunary Fourier series on a compact connected semisimple Lie group 𝐺. If $f\in L^1(G)$ has lacunary Fourier series and 𝑓 vanishes on a non empty open subset of 𝐺, then we prove that 𝑓 vanishes identically. This result can be viewed as a qualitative uncertainty principle.
pp 83-91
This paper deals with the existence and stability properties of positive weak solutions to classes of nonlinear systems involving the $(p,q)$-Laplacian of the form
\begin{equation*}\begin{cases}-𝛥_p u=𝜆 a(x)v^𝛼-c, < x\in𝛺,\\ -𝛥_qv=𝜆 b(x)u^𝛽-c, < x\in𝛺,\\ u=0=v, < x\in𝜕𝛺,\end{cases}\end{equation*}
where $𝛥_p$ denotes the 𝑝-Laplacian operator defined by $𝛥_pz=\mathrm{div}(|\nabla z|^{p-2}\nabla z),p>1,𝜆$ and 𝑐 are positive parameters, 𝛺 is a bounded domain in $R^N(N≥ 1)$ with smooth boundary, 𝛼, 𝛽 > 0 and the weights $a(x),b(x)$ satisfying $a(x)\in C(𝛺),b(x)\in C(𝛺)$ and $a(x)>a_0>0,b(x)>b_0>0$, for $x\in𝛺$. We first study the existence of positive weak solution by using the method of sub-super solution and then we study the stability properties of positive weak solution.
pp 93-109
Analysis of a Malaria Model with Mosquito-Dependent Transmission Coefficient for Humans
G C Hazarika Anuradha Bhattacharjee
In this paper, we discuss an ordinary differential equation mathematical model for the spread of malaria in human and mosquito population. We suppose the human population to act as a reservoir. Both the species follow a logistic population model. The transmission coefficient or the interaction coefficient of humans is considered to be dependent on the mosquito population. It is seen that as the factors governing the transmission coefficient of humans increase, so does the number of infected humans. Further, it is observed that as the immigration constant increases, it leads to a rise in infected humans, giving an endemic shape to the disease.
Current Issue
Volume 127 | Issue 5
November 2017
© 2017 Indian Academy of Sciences, Bengaluru.