• Volume 112, Issue 3

      August 2002,   pages  367-475

    • The Determinant Bundle on the Moduli Space of Stable Triples over a Curve

      Indranil Biswas N Raghavendra

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      We construct a holomorphic Hermitian line bundle over the moduli space of stable triples of the form (𝐸1, 𝐸2, 𝜙), where 𝐸1 and 𝐸2 are holomorphic vector bundles over a fixed compact Riemann surface 𝑋, and 𝜙 : 𝐸2 → 𝐸1 is a holomorphic vector bundle homomorphism. The curvature of the Chern connection of this holomorphic Hermitian line bundle is computed. The curvature is shown to coincide with a constant scalar multiple of the natural Kähler form on the moduli space. The construction is based on a result of Quillen on the determinant line bundle over the space of Dolbeault operators on a fixed 𝐶 Hermitian vector bundle over a compact Riemann surface.

    • A Geometric Characterization of Arithmetic Varieties

      Kapil Hari Paranjape

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      A result of Belyi can be stated as follows. Every curve defined over a number field can be expressed as a cover of the projective line with branch locus contained in a rigid divisor. We define the notion of geometrically rigid divisors in surfaces and then show that every surface defined over a number field can be expressed as a cover of the projective plane with branch locus contained in a geometrically rigid divisor in the plane. The main result is the characterization of arithmetically defined divisors in the plane as geometrically rigid divisors in the plane.

    • Principal Bundles on the Projective Line

      V B Mehta S Subramanian

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      We classify principal 𝐺-bundles on the projective line over an arbitrary field 𝑘 of characteristic ≠ 2 or 3, where 𝐺 is a reductive group. If such a bundle is trivial at a 𝑘-rational point, then the structure group can be reduced to a maximal torus.

    • Gao's Conjecture on Zero-Sum Sequences

      B Sury R Thangadurai

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      In this paper, we shall address three closely-related conjectures due to van Emde Boas, W D Gao and Kemnitz on zero-sum problems on $\mathbf{Z}_p \oplus \mathbf{Z}_p$. We prove a number of results including a proof of the conjecture of Gao for the prime 𝑝 = 7 (Theorem 3.1). The conjecture of Kemnitz is also proved (Propositions 4.6, 4.9, 4.10) for many classes of sequences.

    • A Basic Inequality for Submanifolds in Locally Conformal almost Cosymplectic Manifolds

      Mukut Mani Tripathi Jeong-Sik Kim Seon-Bu Kim

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      For submanifolds tangent to the structure vector field in locally conformal almost cosymplectic manifolds of pointwise constant 𝜑-sectional curvature, we establish a basic inequality between the main intrinsic invariants of the submanifold on one side, namely its sectional curvature and its scalar curvature; and its main extrinsic invariant on the other side, namely its squared mean curvature. Some applications including inequalities between the intrinsic invariant 𝛿𝑀 and the squared mean curvature are given. The equality cases are also discussed.

    • Homogenization of a Parabolic Equation in Perforated Domain with Dirichlet Boundary Condition

      A K Nandakumaran M Rajesh

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      In this article, we study the homogenization of the family of parabolic equations over periodically perforated domains

      \begin{align*}𝜕_t b\left(\frac{x}{d_{𝜀}},u_{𝜀}\right)-\mathrm{div} a(u_{𝜀},\nabla u_{𝜀}) & = f(x,t)\quad\text{in}\quad𝛺_{𝜀}×(0,T),\\ u_{𝜀} & = 0\quad\text{on}\quad𝜕𝛺_{𝜀}×(0,T),\\ u_{𝜀}(x,0) & = u_0(x)\quad\text{in}\quad𝛺_{𝜀}.\end{align*}

      Here, $𝛺_{𝜀} = 𝛺 \backslash S_{𝜀}$ is a periodically perforated domain and $d_{𝜀}$ is a sequence of positive numbers which goes to zero. We obtain the homogenized equation. The homogenization of the equations on a fixed domain and also the case of perforated domain with Neumann boundary condition was studied by the authors. The homogenization for a fixed domain and $b(\frac{x}{d_{𝜀}}, u_{𝜀}) ≡ b(u_{𝜀})$ has been done by Jian. We also obtain certain corrector results to improve the weak convergence.

    • Explosive Solutions of Elliptic Equations with Absorption and Non-Linear Gradient Term

      Marius Ghergu Constantin Niculescu Vicenţiu Rădulescu

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      Let 𝑓 be a non-decreasing $C^1$-function such that $f > 0$ on $(0, ∞), f(0) = 0, \int_1^∞ 1/\sqrt{F(t)}dt < ∞$ and $F(t)/f^{2/a}(t)→ 0$ as $t →∞$, where $F(t) = \int_0^t f(s)ds$ and $a \in (0,2]$. We prove the existence of positive large solutions to the equation $𝛥 u + q(x)|\nabla u|^a = p(x) f(u)$ in a smooth bounded domain $𝛺\subset R^N$, provided that 𝑝, 𝑞 are non-negative continuous functions so that any zero of 𝑝 is surrounded by a surface strictly included in 𝛺 on which 𝑝 is positive. Under additional hypotheses on 𝑝 we deduce the existence of solutions if 𝛺 is unbounded.

    • A Result Concerning the Stability of some Difference Equations and its Applications

      Hark-Mahn Kim

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      In this paper, we investigate the Hyers–Ulam stability problem for the difference equation $f(x + p, y + q) - \varphi(x, y)f(x, y)-\psi(x, y) = 0$.

    • Stokes Flow with Slip and Kuwabara Boundary Conditions

      Sunil Datta Satya Deo

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      The forces experienced by randomly and homogeneously distributed parallel circular cylinder or spheres in uniform viscous flow are investigated with slip boundary condition under Stokes approximation using particle-in-cell model technique and the result compared with the no-slip case. The corresponding problem of streaming flow past spheroidal particles departing but little in shape from a sphere is also investigated. The explicit expression for the stream function is obtained to the first order in the small parameter characterizing the deformation. As a particular case of this we considered an oblate spheroid and evaluate the drag on it.

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