arXiv subjects
Publications and source records attributed to C. P. Anil Kumar.
In this article, we prove a representation theorem that any generic line arrangement in the plane over an ordered field which has global cyclicity can be represented isomorphically by a line arrangement with a given set of distinct slopes of the same cardinality.
For any positive integer $k>1$, we classify the antipodal point arrangements on the sphere $S^k$ up to an isomorphism, by associating a finite complete set of cycle invariants.
In this article we prove in main Theorem A that any infinity type real hyperplane arrangement $\mathcal{H}_n^m$ (Definition 2.11) with the associated normal system $\mathcal{N}$ (Definitions [2.2,2.4] can be represented isomorphically (Definition 2.6) by another infinity type hyperplane arrangement $\tilde{\mathcal{H}}_n^m$ with a given associated normal system $\tilde{\mathcal{N}}$ if and only if the normal systems $\mathcal{N}$ and $\tilde{\mathcal{N}}$ are isomorphic, that is, there is a convex positive bijection (Definition 2.5) between a pair of associated sets of normal antipodal pairs of vectors of $\mathcal{N}$ and $\tilde{\mathcal{N}}$.
In this article we introduce generalized projective spaces (Definitions $[2.1, 2.5]$) and prove three main theorems in two different contexts. In the first context we prove, in main Theorem $A$, the surjectivity of the Chinese remainder reduction map associated to the generalized projective space of an ideal with a given factorization into mutually co-maximal ideals each of which is contained in finitely many maximal ideals, using the key concept of choice multiplier hypothesis (Definition $4.11$) which is satisfied. In the second context of surjectivity of the map from $k$-dimensional special linear group to the product of generalized projective spaces of $k$-mutually co-maximal ideals associating the $k$-rows or $k$-columns, we prove remaining two main Theorems $[Ω,Σ]$ under certain conditions either on the ring or on the generalized projective spaces. Finally in the last section we pose open Questions $[9.1, 9.2]$ whose answers in a greater generality is not known.
In this article we consider an open conjecture about coherently labelling a polyhedron in three dimensions. We exhibit all the forty eight possible coherent labellings of a tetrahedron. We also exhibit that some simplicial polyhedra like bipyramids, Kleetopes, gyroelongated bipyramids are coherently labellable. Also we prove that pyramids over $n$-gons for $n\geq 4$, which are not simplicial polyhedra, are coherently labellable. We prove that among platonic solids, the cube and the dodecahedron are not coherently labellable, even though, the tetrahedron, the octahedron and the icosahedron are coherently labellable. Unlike the case of a tetrahedron, in general for a polyhedron, we show that a coherent labelling need not induce a coherent labelling at a vertex. We prove the main conjecture in the affirmative for a certain class of polyhedra which are constructible from tetrahedra through certain types of edge and face vanishing tetrahedron attachments. As a consequence we conclude that a cube cannot be obtained from only these type of tetrahedron attachments. We also give an obstruction criterion for a polyhedron to be not coherently labellable and consequentially show that any polyhedron obtained from a pyramid with its apex chopped off is not coherently labellable. Finally with the suggestion of the affirmative results we prove the main theorem that any simplicial polyhedron is coherently labellable.
For two natural numbers $1<p_1<p_2$, with $α= \frac{\log(p_1)}{\log(p_2)}$ irrational, we describe, in main Theorem $Ω$ and in Note $1.5$, the factorization of two adjacent numbers in the multiplicatively closed subset $S = \{p_1^ip_2^j\mid i,j\in \mathbb{N}\cup\{0\}\}$ using primary and secondary convergents of $α$. This suggests general Question $1.2$ for more than two generators which is still open.
We prove in this article the surjectivity of three maps. We prove in Theorem $1.6$ the surjectivity of the Chinese remainder reduction map associated to the projective space of an ideal with a given factorization into ideals whose radicals are pairwise distinct maximal ideals. In Theorem $1.7$ we prove the surjectivity of the reduction map of the strong approximation type for a ring quotiented by an ideal which satisfies unital set condition. In Theorem $1.8$ we prove for Dedekind type domains which include Dedekind domains, for $k\geq 2$, the map from $k$-dimensional special linear group to the product of projective spaces of $k$ mutually co-maximal ideals associating the $k$-rows or $k$-columns is surjective. Finally this article leads to three interesting questions $[1.9, 1.10, 1.11]$ mentioned in the introduction section.
We prove in Theorem $2.2$ that the multiplicatively closed subset generated by at most two elements in the set of natural numbers $\mathbb{N}$ has arbitrarily large gaps by explicitly constructing large integer intervals with known prime factorization for the end points, which do not contain any element from the multiplicatively closed set apart from the end points, which belong to the multiplicatively closed set. An Example $4.6$ is also illustrated. We also give a criterion in Theorems $7.8,7.12$ by using a geometric correspondence between maximal singly generated multiplicatively closed sets and points of the space $\mathbb{PF}^{\infty}_{\mathbb{Q}\geq 0}$ (refer to Theorem $7.5$) as to when a finitely generated multiplicatively closed set gives rise to a doubly multiplicatively closed line (refer to Definition $7.4$). We answer a similar Question $5.1$ partially about gaps in a multiply-generated multiplicative closed set, when it is contained in a doubly multiplicative closed set using Theorem $7.8$ and Theorem $7.17$. In the appendix Section $8$ we discuss another constructive proof (refer to Theorem $8.6$) for arbitrarily large gap intervals, where the prime factorization is not known for the right end-point unlike the constructive proof of the main result of the article in the case of multiplicatively closed set $\{p_1^ip_2^j\mid i,j\in \mathbb{N}\cup\{0\}\}$ with $\{p_1<p_2,Log_{p_1}(p_2)\}$ irrational for which the prime factorization is known for both the end-points of the gap interval via the stabilization sequence of the irrational $\frac {1}{Log_{p_1}(p_2)}$.
We find the exact size of a maximal non-commuting set in unipotent uppertriangular linear group $UU_4(\mathbb{F}_q)$ in terms of a non-commuting geometric structure (Refer Definition [10]), where $\mathbb{F}_q$ is the finite field with $q$ elements. Then we get bounds on the size of such a set by explicitly finding certain non-commuting sets in the non-commuting structure.
In this paper we prove that the weighted linear combination of products of the k-subsets of an n-set of positive real numbers with weight being the harmonic mean of their reciprocal sets is less than or equal to uniformly weighted sum of products of the k-subsets with weight being the harmonic mean of the whole reciprocal set.
Consider a discrete valuation ring $R$ whose residue field is finite of cardinality at least $3$. For a finite torsion module, we consider transitive subsets $O$ under the action of the automorphism group of the module. We prove that the associated permutation representation on the complex vector space $C[O]$ is multiplicity free. This is achieved by obtaining a complete description of the transitive subsets of $O\times O$ under the diagonal action of the automorphism group.
Many questions about triangles and quadrilaterals with rational sides, diagonals and areas can be reduced to solving certain Diophantine equations. We look at a number of such questions including the question of approximating arbitrary triangles and quadrilaterals by those with rational sides,diagonals and areas. We transform these problems into questions on the existence of infinitely many rational solutions on a two parameter family of quartic curves. This is further transformed to a two parameter family of elliptic curves to deduce our main result concerning density of points on a line which are at a rational distance from three collinear points (Theorem 4). We deduce from this a new proof of density of rational quadrilaterals in the space of all quadrilaterals (Theorem 39). The other main result (Theorem 3) of this article is on the density of rational triangles which is related to analyzing rational points on the unit circle. Interestingly, this enables us to deduce that parallelograms with rational sides and area are dense in the class of all parallelograms. We also give a criterion for density of certain sets in topological spaces using local product structure and prove the density Theorem 6 in the appendix section. An application of this proves the density of rational points as stated in Theorem 31.
In this paper we prove the "Tiling implies Spectral" part of Fuglede's paper for the case of three intervals. Then we prove the "Spectral implies Tiling" part of the conjecture for the case of three equal intervals as also when the intervals have lengths 1/2, 1/4, 1/4. For the general case we change our approach to get information on the structure of the spectrum for the n-interval case. Finally, we use symbolic computations on Mathematica, and prove this part of the conjecture with an additional assumption on the spectrum.