Odd Square Research Articles

On the largest prime factor of non-zero Fourier coefficients of Hecke eigenforms

Abstract Let τ denote the Ramanujan tau function. One is interested in possible prime values of τ function. Since τ is multiplicative and τ ⁢ ( n ) {\tau(n)} is odd if and only if n is an odd square, we only need to consider τ ⁢ ( p 2 ⁢ n ) {\tau(p^{2n})} for primes p and natural numbers n ≥ 1 {n\geq 1} . This is a rather delicate question. In this direction, we show that for any ϵ > 0 {\epsilon>0} and integer n ≥ 1 {n\geq 1} , the largest prime factor of τ ⁢ ( p 2 ⁢ n ) {\tau(p^{2n})} , denoted by P ⁢ ( τ ⁢ ( p 2 ⁢ n ) ) {P(\tau(p^{2n}))} , satisfies P ⁢ ( τ ⁢ ( p 2 ⁢ n ) ) > ( log ⁡ p ) 1 8 ⁢ ( log ⁡ log ⁡ p ) 3 8 - ϵ P(\tau(p^{2n}))>(\log p)^{\frac{1}{8}}(\log\log p)^{\frac{3}{8}-\epsilon} for almost all primes p. This improves a recent work of Bennett, Gherga, Patel and Siksek. Our results are also valid for any non-CM normalized Hecke eigenforms with integer Fourier coefficients.

On large partial ovoids of symplectic and Hermitian polar spaces

AbstractIn this paper we provide constructive lower bounds on the sizes of the largest partial ovoids of the symplectic polar spaces , odd square, , and of the Hermitian polar spaces , even or odd square, , , .

Sums of triangular numbers and sums of squares

For non-negative integers a,b, and n, let N(a,b;n) be the number of representations of n as a sum of squares with coefficients 1 or 3 (a of ones and b of threes). Let N⁎(a,b;n) be the number of representations of n as a sum of odd squares with coefficients 1 or 3 (a of ones and b of threes). We have that N⁎(a,b;8n+a+3b) is the number of representations of n as a sum of triangular numbers with coefficients 1 or 3 (a of ones and b of threes). It is known that for a and b satisfying 1≤a+3b≤7, we haveN⁎(a,b;8n+a+3b)=22+(a4)+abN(a,b;8n+a+3b) and for a and b satisfying a+3b=8, we haveN⁎(a,b;8n+a+3b)=22+(a4)+ab(N(a,b;8n+a+3b)−N(a,b;(8n+a+3b)/4)). Such identities are not known for a+3b>8. In this paper, for general a and b with a+b even, we prove asymptotic equivalence of formulas similar to the above, as n→∞. One of our main results extends a theorem of Bateman, Datskovsky, and Knopp where the case b=0 and general a was considered. Our approach is different from Bateman-Datskovsky-Knopp's proof where the circle method and singular series were used. We achieve our results by explicitly computing the Eisenstein components of the generating functions of N⁎(a,b;8n+a+3b) and N(a,b;8n+a+3b). The method we use is robust and can be adapted in studying the asymptotics of other representation numbers with general coefficients.

Congruence properties of indices of triangular numbers multiple of other triangular numbers

For any non-square integer multiplier \(k\), there is an infinity of triangular numbers multiple of other triangular numbers. We analyze the congruence properties of indices \(\xi\) of triangular numbers multiple of triangular numbers. Remainders in congruence relations \(\xi\) modulo \(k\) come always in pairs whose sum always equal \((k-1)\), always include 0 and \((k-1)\), and only 0 and \((k-1)\) if \(k\) is prime, or an odd power of a prime, or an even square plus one or an odd square minus one or minus two. If the multiplier \(k\) is twice the triangular number of \(n\), the set of remainders includes also \(n\) and \((n^{2}-1)\) and if \(k\) has integer factors, the set of remainders include multiples of a factor following certain rules. Algebraic expressions are found for remainders in function of \(k\) and its factors, with several exceptions. This approach eliminates those \(\xi\) values not providing solutions.

104.25 Proof without Words: Every central octagonal number is an odd square

104.25 Proof without Words: Every central octagonal number is an odd square

Groundwater Depletion and Water Crisis in Delhi

Groundwater governance entails synergetic acts of political, legal social, economic and administrative systems, which equitably and efficiently distribute and manage the resource. It involves the formulation, establishment and implementation of water legislation and policies and creation of institutional framework for water administration. It emphasises the need for clarification of the roles and responsibilities of the government, civil society and private sector. Delhi has complex governance structure due to the simultaneous presence of the Union as well as state government agencies. The complexity of the situation is further compounded with the expansion of the Delhi as National Capital Region and the 1,400 odd square kilometres of the National Capital Territory (NCT) of Delhi. This article deals with the later. It is an attempt to understand and analyse the evolving legal regime on groundwater, the policy framework and institutional structure, as well as the role of courts to manage and regulate the groundwater situation in the city of Delhi. The article has highlighted that the rapid depletion of groundwater in Delhi is the fundamental reason for water scarcity in the city despite efforts of multiple agencies, existence of model bills, acts and regulations for the management of groundwater. For a highly urbanised city like Delhi, we need to think about incentives to discourage people from abstracting groundwater. There are no easy answers to these questions any more than the efforts to find solutions and effectively implement them to overcome the water crisis. This article is a modest attempt to find reasons for water crisis in the NCT of Delhi.

One-Factorisations of Complete Graphs Constructed in Desarguesian Planes of Certain Odd Square Orders

A geometric construction of one-factorisations of complete graphs $K_{q(q-1)}$ is provided for the case when either $q=2^d +1$ is a Fermat prime, or $q=9$. This construction uses the affine group $\mathrm{AGL}(1,q)$, points and ovals in the Desarguesian plane $\mathrm{PG}(2,q^{2})$ to produce one-factorisations of the complete graph $K_{q(q-1)}$.

On a quadratic Waring's problem with congruence conditions

For each positive integer $n$, let $g_\Delta(n)$ be the smallest positive integer $g$ such that every complete quadratic polynomial in $n$ variables which can be represented by a sum of odd squares is represented by a sum of at most $g$ odd squares. In this paper, we analyze $g_\Delta(n)$ by studying representations of integral quadratic forms by sums of squares with certain congruence condition. We prove that the growth of $g_\Delta(n)$ is at most an exponential of $\sqrt{n}$, which is the same as the best known upper bound on the $g$-invariants of the original quadratic Waring's problem. We also determine the exact value of $g_\Delta(n)$ for each positive integer less than or equal to $4$.

A New Modified Approach to Improve PSNR, NC and MSE via Invisible Embedding of Binary Information using HAAR Transformation

A new modified approach of digital watermarking using HAAR wavelet function is presented in this paper. An invisible watermark or message is embedded into original data for protection. At the user end image segmentation approach is used to separate the object of interest from message and cover. The message is inserted in watermark on choosing the odd square of LH and even square of HL band and proposed embedding rule set. A set of images, two of family members and 4 already existing are as used as cover image to improve PSNR, NC and MSE.

Packing of odd squares revisited

It is known that $$\sum \nolimits _{i =1}^\infty {1/ (2i+1)^2}={\pi ^2/8}-1$$ . We can ask what is the smallest $$\epsilon \ge 0$$ such that all squares of sides 1 / 3, 1 / 5, 1 / 7,...can be packed into a rectangle of area $${\pi ^2/8}-1+\epsilon $$ . We show that the proof of Paulhus $$'$$ key Lemma for the best known result is false and we give new upper estimate $$\epsilon <4.43\times 10^{-10}$$ .

二项式系数级数连带奇数倒数平方和

二项式系数级数连带奇数倒数平方和

Planar arcs

Let p denote the characteristic of Fq, the finite field with q elements. We prove that if q is odd then an arc of size q+2−t in the projective plane over Fq, which is not contained in a conic, is contained in the intersection of two curves, which do not share a common component, and have degree at most t+p⌊logp⁡t⌋, provided a certain technical condition on t is satisfied.This implies that if q is odd then an arc of size at least q−q+q/p+3 is contained in a conic if q is square and an arc of size at least q−q+72 is contained in a conic if q is prime. This is of particular interest in the case that q is an odd square, since then there are examples of arcs, not contained in a conic, of size q−q+1, and it has long been conjectured that if q≠9 is an odd square then any larger arc is contained in a conic.These bounds improve on previously known bounds when q is an odd square and for primes less than 1783. The previously known bounds, obtained by Segre [26], Hirschfeld and Korchmáros [17][18], and Voloch [32][33], rely on results on the number of points on algebraic curves over finite fields, in particular the Hasse–Weil theorem and the Stöhr–Voloch theorem, and are based on Segre's idea to associate an algebraic curve in the dual plane containing the tangents to an arc. In this paper we do not rely on such theorems, but use a new approach starting from a scaled coordinate-free version of Segre's lemma of tangents.Arcs in the projective plane over Fq of size q and q+1, q odd, were classified by Segre [25] in 1955. In this article, we complete the classification of arcs of size q−1 and q−2.The main theorem also verifies the MDS conjecture for a wider range of dimensions in the case that q is an odd square. The MDS conjecture states that if 4⩽k⩽q−2, a k-dimensional linear MDS code has length at most q+1. Here, we verify the conjecture for k⩽q−q/p+2, in the case that q is an odd square.

Proof Without Words: Products of Odd Squares and Triangular Numbers

Proof Without Words: Products of Odd Squares and Triangular Numbers

Proof Without Words: On Sums of Squares and Triangles

Remarks. If the diagram above is extended to contain the odd squares too, then it becomes clear that ∑nk = 1k2 = ∑k = 1nTk + ∑n − 1k = 1Tk, which can also be derived from the results given in eithe...

Tight sets in finite classical polar spaces

Abstract We show that every i-tight set in the Hermitian variety H(2r + 1, q) is a union of pairwise disjoint (2r + 1)-dimensional Baer subgeometries PG ( 2 r + 1 , q ) $\text{PG}(2r+1,\,\sqrt{q})$ and generators of H(2r + 1, q), if q ≥ 81 is an odd square and i &lt; (q 2/3 − 1)/2. We also show that an i-tight set in the symplectic polar space W(2r + 1, q) is a union of pairwise disjoint generators of W(2r + 1, q), pairs of disjoint r-spaces {Δ, Δ ⊥}, and (2r + 1)-dimensional Baer subgeometries. For W(2r + 1, q) with r even, pairs of disjoint r-spaces {Δ, Δ ⊥} cannot occur. The (2r + 1)-dimensional Baer subgeometries in the i-tight set of W(2r + 1, q) are invariant under the symplectic polarity ⊥ of W(2r + 1, q) or they arise in pairs of disjoint Baer subgeometries corresponding to each other under ⊥. This improves previous results where i &lt; q 5 / 8 / 2 + 1 $i \lt q^{5/8} / \sqrt{2} +1$ was assumed. Generalizing known techniques and using recent results on blocking sets and minihypers, we present an alternative proof of this result and consequently improve the upper bound on i to (q 2/3 − 1)/2. We also apply our results on tight sets to improve a known result on maximal partial spreads in W(2r + 1, q).

Unitals in shift planes of odd order

Abstract A finite shift plane can be equivalently defined via abelian relative difference sets as well as planar functions. In this paper, we present a generic way to construct unitals in finite shift planes of odd square order. We investigate various geometric and combinatorial properties of these planes, such as the self-duality, the existence of O’Nan configurations, Wilbrink’s conditions, the designs formed by circles and so on. We also show that our unitals are inequivalent to the unitals derived from unitary polarities in the same shift planes. As designs, our unitals are also not isomorphic to the classical unitals (the Hermitian curves).

Square-root cancellation for the signs of Latin squares

Let L(n) be the number of Latin squares of order n, and let Leven(n) and Lodd(n) be the number of even and odd such squares, so that L(n)=Leven(n)+Lodd(n). The Alon-Tarsi conjecture states that Leven(n) ź Lodd(n) when n is even (when n is odd the two are equal for very simple reasons). In this short note we prove that $$\left| {{L^{even}}\left( n \right) - {L^{odd}}\left( n \right)} \right| \leqslant L{\left( n \right)^{\frac{1}{2} + o\left( 1 \right)}}$$|Leven(n)źLodd(n)|źL(n)12+o(1), thus establishing the conjecture that the number of even and odd Latin squares, while conjecturally not equal in even dimensions, are equal to leading order asymptotically. Two proofs are given: both proceed by applying a differential operator to an exponential integral over SU(n). The method is inspired by a recent result of Kumar-Landsberg.

On the classification of weakly integral modular categories

We classify all modular categories of dimension 4m, where m is an odd square-free integer, and all ranks 6 and 7 weakly integral modular categories. This completes the classification of weakly integral modular categories through rank 7. Our results imply that all integral modular categories of rank at most 7 are pointed (that is, every simple object has dimension 1). All strictly weakly integral (weakly integral but non-integral) modular categories of ranks 6 and 7 have dimension 4m, with m an odd square free integer, so their classification is an application of our main result. The classification of rank 7 integral modular categories is facilitated by an analysis of actions on modular categories by two groups: the Galois group of the field generated by the entries of the S-matrix and the group of isomorphism classes of invertible simple objects. The interplay of these two actions is of independent interest, and we derive some valuable arithmetic consequences from their actions.

English

A &#119901;. &#119902; graph G = &#119881;, &#119864; is said to be a square graceful graph ifthere exists an injective function f: V &#119866; → 0,1,2,3, … , &#119902;2 such that the induced mapping &#119891;&#119901; : E &#119866; → 1,4,9, … , &#119902;2 defined by &#119891;&#119901; &#119906;&#119907; = &#119891; &#119906; − &#119891; &#119907; is an injection. The function f is called a square graceful labeling of G. In this paper the square graceful labeling of the caterpillar S &#119883;1, &#119883;2, … , &#119883;&#119899; , the graphs &#119875;&#119899; −1 1,2, … &#119899; ,m&#119870;1,&#119899; ∪ &#119904;&#119870;1,&#119905; , &#119870;1,&#119894; , &#119899;&#119894; =1 &#119875;&#119899; ⨀&#119870;1 − &#119890;,H graph and some other graphsare studied. A new parameter called star square graceful deficiency number of a graph is defined and the star square graceful deficiency number of the cycle &#119862;3 is determined. Two new definitions namely, odd square graceful labeling and even square graceful labeling of a graph are defined with example.

Proof of the List Edge Coloring Conjecture for Complete Graphs of Prime Degree

We prove that the list-chromatic index and paintability index of $K_{p+1}$ is $p$, for all odd primes $p$. This implies that the List Edge Coloring Conjecture holds for complete graphs with less then 10 vertices. It also shows that there are arbitrarily big complete graphs for which the conjecture holds, even among the complete graphs of class 1. Our proof combines the Quantitative Combinatorial Nullstellensatz with the Paintability Nullstellensatz and a group action on symmetric Latin squares. It displays various ways of using different Nullstellensätze. We also obtain a partial proof of a version of Alon and Tarsi's Conjecture about even and odd Latin squares.