pp 139-147
In this article, we prove that infinite number of integers satsify Erdős–Woods conjecture. Moreover, it follows that the number of natural numbers ≤ 𝑥 satisfies Erdős–Woods conjecture with 𝑘 = 2 is at least 𝑐𝑥/(log 𝑥) for some positive constant 𝑐 > 2.
pp 149-160
Subintegrality, invertible modules and Laurent polynomial extensions
Let 𝐴 ⊆ 𝐵 be a commutative ring extension. Let $\mathcal{I}$(𝐴, 𝐵) be the multiplicative group of invertible 𝐴-submodules of 𝐵. In this article, we extend a result of Sadhu and Singh by finding a necessary and sufficient condition on an integral birational extension 𝐴 ⊆ 𝐵 of integral domains with dim 𝐴 ≤ 1, so that the natural map $\mathcal{I}$(𝐴, 𝐵) $→$ $\mathcal{I}(A[X,X^{-1}],B[X,X^{-1}])$ is an isomorphism. In the same situation, we show that if dim 𝐴 ≥ 2, then the condition is necessary but not sufficient. We also discuss some properties of the cokernel of the natural map $\mathcal{I}(A,B)→ \mathcal{I}[X,X^{-1}], B[X,X^{-1}])$ in the general case.
pp 161-165
Let 𝑆 be a scheme. Assume that we are given an action of the one dimensional split torus $\mathbb{G}_{m,S}$ on a smooth affine 𝑆-scheme $\mathfrak{X}$. We consider the limit (also called attractor) subfunctor $\mathfrak{X}_{𝜆}$ consisting of points whose orbit under the given action `admits a limit at 0’. We show that $\mathfrak{X}_{𝜆}$ is representable by a smooth closed subscheme of $\mathfrak{X}$. This result generalizes a theorem of Conrad et al. (Pseudo-reductive groups (2010) Cambridge Univ. Press) where the case when $\mathfrak{X}$ is an affine smooth group and $\mathbb{G}_{m,S}$ acts as a group automorphisms of $\mathfrak{X}$ is considered. It also occurs as a special case of a recent result by Drinfeld on the action of $\mathbb{G}_{m,S}$ on algebraic spaces (Proposition 1.4.20 of Drinfeld V, On algebraic spaces with an action of $\mathfrak{G}_{m}$, preprint 2013) in case 𝑆 is of finite type over a field.
pp 167-172
Revisiting the Zassenhaus conjecture on torsion units for the integral group rings of small groups
In recent years several new restrictions on integral partial augmentations for torsion units of $\mathbb{Z}G$ have been introduced, which have improved the effectiveness of the Luthar–Passi method for checking the Zassenhaus conjecture for specific groups 𝐺. In this note, we report that the Luthar–Passi method with the new restrictions are sufficient to verify the Zassenhaus conjecture with a computer for all groups of order less than 96, except for one group of order 48 – the non-split covering group of 𝑆_{4}, and one of order 72 of isomorphism type (𝐶 × 𝐶) × 𝐷_{8}. To verify the Zassenhaus conjecture for this group we give a new construction of normalized torsion units of $\mathbb{Q}G$ that are not conjugate to elements of $\mathbb{Z}G$.
pp 173-179
On 𝑝-supersolvability of finite groups
A number of authors have studied the structure of a group 𝐺 under the assumption that some subgroups of 𝐺 are well located in 𝐺. We will obtain some new criteria of 𝑝-supersolvability and 𝑝-nilpotency of groups.
pp 181-186
Class-preserving automorphisms of some finite 𝑝-groups
Let 𝐺 be a finite 𝑝-group of order 𝑝^{5}, where 𝑝 is a prime. We give necessary and sufficient conditions on 𝐺 such that 𝐺 has a non-inner class-preserving automorphism. As a consequence, we give short and alternate proofs of results of Yadav (§5 of Proc. Indian Acad. Sci. (Math. Sci.) 118 (2008) 1–11) and Kalra and Gumber (Theorem 4.2 of Indian J. Pure Appl. Math. 44 (2013) 711–725).
pp 187-198
Sesquilinear uniform vector integral
Ion Chiţescu Loredana Ioana Radu Miculescu Lucian Niţă
We introduce and study an integral of Hilbert valued functions with respect to Hilbert valued measures. The integral is sesquilinear (bilinear in the real case) and takes scalar values. Basic properties of this integral are studied and some examples are introduced.
pp 199-207
(2𝑛-1)-Ideal amenability of triangular banach algebras
Let $\mathcal{A}$ and $\mathcal{B}$ be two unital Banach algebras and let $\mathcal{M}$ be an unital Banach $\mathcal{A}$, $\mathcal{B}$-module. Also, let $\mathcal{T}=\left[\begin{smallmatrix} \mathcal{A} & \mathcal{M}\\ & \mathcal{B}\end{smallmatrix}\right]$ be the corresponding triangular Banach algebra. Forrest and Marcoux (Trans. Amer. Math. Soc. 354 (2002) 1435–1452) have studied the 𝑛-weak amenability of triangular Banach algebras. In this paper, we investigate (2𝑛-1)-ideal amenability of $\mathcal{T}$ for all 𝑛 ≥ 1. We introduce the structure of ideals of these Banach algebras and then, we show that (2𝑛-1)-ideal amenability of $\mathcal{T}$ depends on (2𝑛-1)-ideal amenability of Banach algebras $\mathcal{A}$ and $\mathcal{B}$.
pp 209-219
Volume sums of polar Blaschke–Minkowski homomorphisms
In this article, we establish Minkowski and Aleksandrov–Fenchel type inequalities for the volume sum of polars of Blaschke–Minkowski homomorphisms.
pp 221-225
Limit law of the iterated logarithm for 𝐵-valued trimmed sums
Ke-Ang Fu Yuyang Qiu Yeling Tong
Given a sequence of i.i.d. random variables $\{X,X_{n};n≥ 1\}$ taking values in a separable Banach space $(B,\|\cdot \|)$ with topological dual 𝐵^{*}, let $X^{(r)}_{n}=X_{m}$ if $\| X_{m}\|$ is the 𝑟-th maximum of $\{\| X_{k}\|; 1≤ k≤ n\}$ and $^{(r)}S_{n}=S_{n}-(X^{(1)}_{n}+\cdots+X^{(r)}_{n})$ be the trimmed sums when extreme terms are excluded, where $S_{n}=\sum^{n}_{k=1}X_{k}$. In this paper, it is stated that under some suitable conditions,
$$ \lim\limits_{n→ ∞}\frac{1}{\sqrt{2\log \log n}}\max\limits_{1≤ k≤ n}\frac{\| {}^{(r)}S_{k}\|}{\sqrt{k}}=𝜎(X)\quad\text{a.s.,} $$
where $𝜎^{2}(X)=\sup_{f\in B^{*}_{1}}\text{\sf E}f^{2}(X)$ and $B^{*}_{1}$ is the unit ball of 𝐵^{*}.
pp 227-238
Morozov-type discrepancy principle for nonlinear ill-posed problems under 𝜂-condition
For proving the existence of a regularization parameter under a Morozov-type discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems, it is required to impose additional nonlinearity assumptions on the forward operator. Lipschitz continuity of the Freéchet derivative and requirement of the Lipschitz constant to depend on a source condition is one such restriction (Ramlau P, Numer. Funct. Anal. Optim. 23(1&22) (2003) 147–172). Another nonlinearity condition considered by Scherzer (Computing, 51 (1993) 45–60) was by requiring the forward operator to be close to a linear operator in a restricted sense. A seemingly natural nonlinear assumption which appears in many applications which attracted attention in various contexts of the study of nonlinear problems is the so-called 𝜂-condition. However, a Morozov-type discrepancy principle together with 𝜂-condition does not seem to have been studied, except in a recent paper by the author (Bull. Aust. Math. Soc. 79 (2009) 337–342), where error estimates under a general source condition is derived, by assuming the existence of the parameter. In this paper, the existence of the parameter satisfying a Morozov-type discrepancy principle is proved under the 𝜂-condition on the forward operator, by assuming the source condition as in the papers of Scherzer (Computing, 51 (1993) 45–60) and Ramlau (Numer. Funct. Anal. Optim. 23(1&22) (2003) 147–172). This source condition is, in fact, a special case of the source condition in the author’s paper (Bull. Aust. Math. Soc. 79 (2009) 337–342).
pp 239-270
P Dutt Akhlaq Husain A S Vasudeva Murthy C S Upadhyay
This is the first of a series of papers devoted to the study of ℎ- 𝑝 spectral element methods for solving three dimensional elliptic boundary value problems on non-smooth domains using parallel computers. In three dimensions there are three different types of singularities namely; the vertex, the edge and the vertex-edge singularities. In addition, the solution is anisotropic in the neighbourhoods of the edges and vertex-edges. To overcome the singularities which arise in the neighbourhoods of vertices, vertex-edges and edges, we use local systems of coordinates. These local coordinates are modified versions of spherical and cylindrical coordinate systems in their respective neighbourhoods. Away from these neighbourhoods standard Cartesian coordinates are used. In each of these neighbourhoods we use a geometrical mesh which becomes finer near the corners and edges. The geometrical mesh becomes a quasi-uniform mesh in the new system of coordinates. We then derive differentiability estimates in these new set of variables and state our main stability estimate theorem using a non-conforming ℎ- 𝑝 spectral element method whose proof is given in a separate paper.
Current Issue
Volume 127 | Issue 4
September 2017
© 2017 Indian Academy of Sciences, Bengaluru.