For a positive integer n≥ 3 , the sides and diagonals of a convex n- gon divide the interior of the convex n- gon into finitely (polynomial in n) many regions bounded by them. In this article, we associate to every region a unique n- cycle in the symmetric group S_n of a certain type (defined as 2-standard consecutive cycle) by studying point arrangements in the plane. Then we find that there are more (exponential in n) number of such cycles leading to the conclusion that not every region labelled by a cycle appears in every convex n- gon. In fact most of them do not occur in any given single convex n- gon. Later in the main theorem of this article we characterize combinatorially those cycles (defined as definite cycles) whose corresponding regions occur in every convex n- gon and those cycles (defined as indefinite cycles) whose corresponding regions do not occur in every convex n- gon. As a consequence we characterize those one point extensions of a uniform rank 3 convex oriented matroid for which the one point extension is reducible by the, one point, when it lies inside the convex hull.

Investigations on air pollutants and air quality were undertaken during the COVID-19 pandemic and pre-pandemic period in Alandur (13.03° N, 80.21° E), Tamil Nadu, India. This comparative study of atmospheric pollutants focuses on the variation between the COVID-19 pandemic period (lockdowns imposed to control the spread of novel coronavirus infection) and the pre-pandemic period with ground-based atmospheric pollution monitoring instruments. The observations were obtained from the Centre for Pollution Control Board (CPCB). The observations from 2018 to 2021 period are used for this investigation. First, the normalizing of the time series revealed the general trends of the atmospheric pollutants present in the specified timeline. Later, spectral estimations were done to unveil the hidden peaks and periodicities during the pre-pandemic period and the COVID-19 period. The power spectral analysis is carried out to comprehend the dispersion and variability of atmospheric pollutants by examining and comparing the power spectrum of a specific station using Blackman–Tukey window analysis, aiming to elucidate the impact of the COVID-19 pandemic on atmospheric pollution dynamics. The results indicate a particular trend at peak business hours (ISTs), with notable periodicities evident throughout each timeline. A significant decrease ( 42.5

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 $\tilde{\mathcal{N}}$.

IFor a positive rational $l$, we define the concept of an $l$-elliptic and an $l$-hyperbolic rational set in a metric space. In this article we examine the existence of (i) dense and (ii) infinite $l$-hyperbolic and $l$-ellitpic rationals subsets of the real line and unit circle. For the case of a circle, we prove that the existence of such sets depends on the positivity of ranks of certain associated elliptic curves. We also determine the closures of such sets which are maximal in case they are not dense. In higher dimensions, we show the existence of $l$-ellitpic and $l$-hyperbolic rational infinite sets in unit spheres and Euclidean spaces for certain values of $l$ which satisfy a weaker condition regarding the existence of elements of order more than two, than the positivity of the ranks of the same associated elliptic curves. We also determine their closures. A subset $T$ of the $k$-dimensional unit sphere $S^k$ has an antipodal pair if both $x,-x\in T$ for some $x\in S^k$. In this article, we prove that there does not exist a dense rational set $T\subset S^2$ which has an antipodal pair by assuming Bombieri-Lang Conjecture for surfaces of general type. We actually show that the existence of such a dense rational set in $S^k$ is equivalent to the existence of a dense $2$-hyperbolic rational set in $S^k$ which is further equivalent to the existence of a dense 1-elliptic rational set in the Euclidean space $\mathbb{R}^k$.

We investigated the external magnetic potential due to solar forcing, with nine years of data during 2001-2009, covering the deep solar minimum (2006-2009), at two stations: one is in the polar cap-Vostok (78 degrees 27'S, 106 degrees 52'E; mag. lat 83 degrees S) and another is in the subauroral region -Maitri (70 degrees 45'S, 11 degrees 43'E: mag. lat 67 degrees S) in Antarctica. The significance of the work is associated with space weather prediction and its impact on planet Earth. We used Advance Composition Explorer (ACE) satellite data for the aforesaid period for a thorough understanding of influences due to solar wind origin and to compare the parameter observed in these regions. We used the spherical cap harmonic analysis (SCHA) function as a tool. The inference indicates that at Vostok the magnitude is enhanced throughout and depicts a broad ambient external magnetic potential. It seems to be essentially the intensification of the region 1 currents whereas at Maitri intense electric fields are produced during geomagnetic perturbations which drive a system of disturbed time Region 2 currents over the quiet time currents. During this scenario in Maitri there are noticeable peaks or enhancements in the magnetic potential that can be observed mainly during geomagnetic disturbances. Hence the regression relation developed for external magnetic potential calculation, in terms of solar wind parameters agrees well with polar cap region and the area is relatively less explored earlier, the present investigation can be expected to add knowledge about that regime.

For an odd prime $p$ and a positive integer $n$, it is well known that there are two types of extra-special $p$-groups of order $p^{2n+1}$, first one is the Heisenberg group which has exponent $p$ and the second one is of exponent $p^2$. In this article, a new way of representing the extra-special $p$-group of exponent $p^2$ is given. These representations facilitate an explicit way of finding formulae for any endomorphism and any automorphism of an extra-special $p$-group $G$ for both the types. Based on these formulae, the endomorphism semigroup $End(G)$ and the automorphism group $Aut(G)$ are described. The endomorphism semigroup image of any element in $G$ is found and the orbits under the action of the automorphism group $Aut(G)$ are determined. As a consequence it is deduced that, under the notion of degeneration of elements in $G$, the endomorphism semigroup $End(G)$ induces a partial order on the automorphism orbits when $G$ is the Heisenberg group and does not induce when $G$ is the extra-special $p$-group of exponent $p^2$. Finally we prove that the cardinality of isotropic subspaces of any fixed dimension in a non-degenerate symplectic space is a polynomial in $p$ with non-negative integer coefficients. Using this fact we compute the cardinality of $End(G)$.

We got unique opportunity to carry out the globally improved air pollution-based atmospheric electricity experiments from 2020 to 2021. This period composes of lockdown phases of the Covid-19 pandemic. Tirunelveli (8.7 degrees N,77.8 degrees E) in Tamil Nadu, is one of the southern Indian peninsular stations for the observations of atmospheric electric parameters (AEP). A comparative study of AEP has been made during the pre-pandemic period and the lockdown period imposed to control the spread of Novel Corona virus infection with the help of ground-based indigenously developed atmospheric electric instruments. The analysis of both 2020 and 2021 data showed a marked difference in electric field and air-Earth (AE) current density. The difference in the AEP pattern is attributed to some extent to the decrease in aerosol loading caused by minimum human activities, drastically reduced emissions from industry, building construction, quarrying and mining, cement plants, vehicular emission, and stoppage of particulate matters, etc, which seem to have reduced the resistivity load in a global electric circuit (GEC).

We give an exact criterion of a conjecture of L. M. Kelly to hold true which is stated as follows. If there is a finite family $$\Sigma $$ of mutually skew lines in $$\mathbb {R}^d,d\ge 4$$ such that the 3-flat spanned by every two lines in $$\Sigma $$ , contains at least one more line of $$\Sigma $$ , then we have that all of the lines of $$ \Sigma $$ are contained in a single 3-flat if and only if the arrangement of 3-flats is central. Finally, this article leads to an analogous question for higher dimensional skew affine spaces, where we prove that, for (2, 5)-representations of Sylvester–Gallai designs in $$\mathbb {R}^6$$ , the analogous statement does not hold.

In this article, we prove in the main theorem that, there is a bijection between the isomorphism classes of a certain type of real hyperplane arrangements on the one hand, and the antipodal pairs of convex cones of an associated discriminantal arrangement on the other hand. The type of hyperplane arrangements considered and the isomorphism classes have been defined precisely. As a consequence, we enumerate such isomorphism classes by computing the characteristic polynomial of the discriminantal arrangement. With a certain restriction, the enumerated value is shown to be independent of the discriminantal arrangement. Later we observe that the restriction we impose on the type of hyperplane arrangements is a mild one and that this conditional restriction is quite generic. Moreover the restriction is defined in terms of a normal system being concurrency free which is a generic condition. We also discuss two examples of normal systems which are not concurrency free in the last section and enumerate the number of isomorphism classes.

For non-negative integers k≤ n, we prove a combinatorial identity for the p-binomial coefficient nk_p based on abelian p-groups. A purely combinatorial proof of this identity is not known. While proving this identity, for r∈ℕ∪{0},s∈ℕ and p a prime, we present a purely combinatorial formula for the number of subgroups of ℤ^s of finite index p^r with quotient isomorphic to the finite abelian p-group of type λ, which is a partition of r into at most s parts. This purely combinatorial formula is similar to that for the enumeration of subgroups of a certain type in a finite abelian p-group obtained by Lynne Marie Butler. As consequences, this combinatorial formula gives rise to many enumeration formulae that involve polynomials in p with non-negative integer coefficients.

In this paper, we combinatorially describe the triangles that are present in two types of line arrangements, those which have global cyclicity and those which are infinity type line arrangements. A combinatorial nomenclature has been described for both the types and some properties of the nomenclature have been proved. Later, using the nomenclature, we describe the triangles present in both types of line arrangements in main Theorems A and B. We also prove that the set of triangles uniquely determines, in a certain precise sense, the line arrangements with global cyclicity and not the infinity type line arrangements, where counter examples have been provided. In Theorem 9.1, given a nomenclature, we characterize when a particular line symbol in the nomenclature is a line at infinity for the arrangement determined by the nomenclature.

In this article, for positive integers n≥ m≥ 1, the parameter spaces for the isomorphism classes of the generic point arrangements of cardinality n, and the antipodal point arrangements of cardinality 2n in the Eulidean space ℝ^m are described using the space of totally nonzero Grassmannian Gr^tnz_mn(ℝ). A stratification 𝒮^tnz_mn(ℝ) of the totally nonzero Grassmannian Gr^tnz_mn(ℝ) is mentioned and the parameter spaces are respectively expressed as quotients of the space 𝒮^tnz_mn(ℝ) of strata under suitable actions of the symmetric group S_n and the semidirect product group (ℝ^*)^n⋊ S_n. The cardinalities of the space 𝒮^tnz_mn(ℝ) of strata and of the parameter spaces S_n\𝒮^tnz_mn(ℝ), ((ℝ^*)^n⋊ S_n)\𝒮^tnz_mn(ℝ) are enumerated in dimension m=2. Interestingly enough, the enumerated value of the isomorphism classes of the generic point arrangements in the Euclidean plane is expressed in terms of the number theoretic Euler-totient function. The analogous enumeration questions are still open in higher dimensions for m≥ 3.

For a partition λ = (λ_1^ρ_1>λ_2^ρ_2>λ_3^ρ_3>…>λ_k^ρ_k) and its associated finite abelian p-group 𝒜_λ=i=1k⊕ (ℤ/p^λ_iℤ)^ρ_i, where p is a prime, we consider two actions of its automorphism group 𝒢_λ on 𝒜_λ. The first action is the natural action g∙ a= ^ga for all g∈𝒢_λ and a∈𝒜_λ where the action map is denoted by Λ_1=Id_𝒢_λ:𝒢_λ⟶𝒢_λ and the second action is the trivial action g∙ a=a for all g∈𝒢_λ and a∈𝒜_λ where the action map is denoted by Λ_2:𝒢_λ⟶{e}⊂𝒢_λ the trivial map. For the natural action Λ_1, we show that the first and second cohomology groups H_Λ_1^i(𝒢_λ,𝒜_λ),i=1,2 vanish for any partition λ for an odd prime p. For the trivial action Λ_2 we show that, for an odd prime p, the first cohomology group H_Λ_2^1(𝒢_λ,𝒜_λ) and for an odd prime p≠ 3, the second cohomology group H_Λ_2^2(𝒢_λ,𝒜_λ) vanish if and only if the difference between two successive parts of the partition λ is at most one. This is done by using the modp cohomologies H^i(𝒢_λ,ℤ/pℤ),i=1,2.

The interaction of high velocity plasma with Earth’s magnetic field is fundamental and offer many questions on high latitude electrodynamics. The problems associated with influence of electric field and Field Aligned Current (FAC) generation is investigated with the aid of spherical cap harmonic analysis at 830 Mag. Lat. in southern hemispheres. The investigation is done on the cases with different Interplanetary Magnetic Field (IMF) conditions after the earth directed solar events. The helio-plasma parameters viz., density, velocity, energy, electron temperature are also noted during the field aligned current studies. It seems that, due to external magnetic field influence polarization of plasma electric field take place (reorientation of the convective cells). It happens with different orientation as per the magnitude and direction of By and Bz component and the horizontal currents. It is noted that the FAC value also depends on kinetic energy of the plasma streams and conductivity of external loading. As the plasma decelerates by force Jsw X Esw, the resultant current may extend along the field lines. Increases in the FAC density are seemed to be proportional to the transmission function.

A study of the global electric circuit can help us to understand the electrical environment of the Earth's atmosphere. This approach provides a good frame work for exploring interconnections and coupling of various regions of the Earth's upper atmosphere. With the aim of understanding the behaviour of air-Earth current system during severe meteorological disturbances, we carried out observations of Maxwell's current (air-earth current) for a short period during lightning hours and fair weather days of 2019 at an equatorial station Tirunelveli (8.7 degrees N, 77.8 degrees E), located in southern part of India. Unusual lighting activity was noted during the peak summer of 2019 over this location and corresponding electric variability in air-earth current amplitude and phase charge in electric field were measured. During fair weather days, the current density is only a few pico amps; however, a tenfold increase in the current density was noted during disturbed weather conditions. Our analysis indicates that a rise in temperature led to enhanced convection during mid-day hours, which in turn, contributed to source activity. Possibly, this is the first report that brings out that the rise in temperature is covariant to source activity. We found the wind flow to be moderately southwesterly during the period of observation.

For two natural numbers $$1<p_1<p_2$$, with $$\alpha =\frac{\log (p_1)}{\log (p_2)}$$ irrational, we describe in the Main Theorem $${{\Omega }}$$ 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 $$\alpha $$. This suggests the general Question 1.2 for more than two generators which is still open.

In this article we prove two main results. Firstly, we show that any six-line arrangement, consisting of three pairs of mutually perpendicular lines, does not give rise to a "very generic or sufficiently general" discriminantal arrangement in the sense of C. A. Athanasiadis . We give two proofs of the first result. The second result is as follows. The codimension-one boundary faces of (a region) a convex cone of a very generic discriminantal arrangement has not been characterized and is not known even though the intersection lattice of a very generic discriminantal arrangement is known. So secondly, we show that the number of simplex cells of the very generic hyperplane arrangement ℋ^m_n={H_i:j=1m∑a_ijx_j=c_i,1≤ i≤ n} may not be not precisely equal to the number of codimension-one boundary hyperplanes of ℝ^n of the convex cone C containing (c_1,c_2,…,c_n) in the associated very generic discriminantal arrangement. That is, for 1≤ i_1

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 only a 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 [Omega, Sigma] 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 are not known.