Odd Sum Research Articles

Odd and Co – Odd Sum Degree Domination in Graphs with Algorithm

Odd and Co – Odd Sum Degree Domination in Graphs with Algorithm

Application of Partitions of Odd Numbers and their Odd Sums to Prove the Nonexistence of Odd Perfect Numbers

Perfect numbers, which are integers equal to the sum of their proper divisors, excluding themselves, have intrigued mathematicians for centuries. While it is established that even perfect numbers can be expressed as 2p-1(2p-1), where p and 2p-1 are prime numbers (Mersenne primes), the existence of odd perfect numbers remains an unsolved problem. This study aims to prove the nonexistence of odd perfect numbers by utilizing an algorithm which demonstrates that a positive even integer can be partitioned into all pairs of odd numbers. Using this approach, it is proven that any positive odd number 2n+1 can be partitioned into all pairs of both odd and even numbers and from the set of these partitions, we show that there exist a proper subset containing all proper divisors of 2n+1. Using these results, and the facts that there exist infinitely many odd numbers and the odd sums of odd numbers is always odd, we prove the nonexistence of odd perfect numbers contributing to the conjecture that they do not exist.

A Study on Co – odd (even) Sum Degree Edge Domination Number in Graphs

An edge dominating set of a graph is said to be an odd (even) sum degree edge dominating set (osded (esded) - set) of G if the sum of the degree of all edges in X is an odd (even) number. The odd (even) sum degree edge domination number is the minimum cardinality taken over all odd (even) sum degree edge dominating sets of G and is defined as zero if no such odd (even) sum degree edge dominating set exists in G. In this paper, the odd (even) sum degree domination concept is extended on the co-dominating set E-T of a graph G, where T is an edge dominating set of G. The corresponding parameters co-odd (even) sum degree edge dominating set, co-odd (even) sum degree edge domination number and co-odd (even) sum degree edge domination value is defined. Further, the exact values of the above said parameters are found for some standard classes of graphs. The bounds of the co-odd (even) sum degree edge domination number are obtained in terms of basic graph terminologies. The co-odd (even) sum degree edge dominating sets are characterized. The relationships with other edge domination parameters are also studied.

Direct proof through an equilateral hyperbola to find the infinite Pythagorean triples and a direct proof to find the sum of all odd numbers through a telescopic series

In this article a prove new properties like a connection with Pythagorean triples and a hiperbolic equation. I prove a direct proof about a serie, a telescopic series about the even sum and odd sum. I had an article that use this properties but this results is mine. I discovered this theorem and prove in this article.

The center of the twisted Heisenberg category, factorial Schur Q-functions, and transition functions on the Schur graph

We establish an isomorphism between the center of the twisted Heisenberg category and the subalgebra \(\Gamma \) of the symmetric functions generated by odd power sums. We give a graphical description of the factorial Schur Q-functions and inhomogeneous power sums as closed diagrams in the twisted Heisenberg category and show that the bubble generators of the center correspond to two sets of generators of \(\Gamma \) which encode data related to up/down transition functions on the Schur graph. Finally, we describe an action of the trace of the twisted Heisenberg category, the W-algebra \(W^-\subset W_{1+\infty }\), on \(\Gamma \).

Melham's Conjecture on Odd Power Sums of Fibonacci Numbers

Ozeki and Prodinger showed that the odd power sum of the first several consecutive Fibonacci numbers of even order is equal to a polynomial evaluated at a certain Fibonacci number of odd order. We prove that this polynomial and its derivative both vanish at 1, and will be an integer polynomial after multiplying it by a product of the first consecutive Lucas numbers of odd order. This presents an affirmative answer to a conjecture of Melham.

Excitation of low-frequency residual currents at combination frequencies of an ionising two-colour laser pulse

We have studied the processes of excitation of low-frequency residual currents in a plasma produced through ionisation of gases by two-colour laser pulses in laser-plasma schemes for THz generation. We have developed an analytical approach that allows one to find residual currents in the case when one of the components of a two-colour pulse is weak enough. The derived analytical expressions show that the effective generation of the residual current (and hence the effective THz generation) is possible if the ratio of the frequencies in the two-colour laser pulse is close to a rational fraction with a not very big odd sum of the numerator and denominator. The results of numerical calculations (including those based on the solution of the three-dimensional time-dependent Schrödinger equation) agree well with the analytical results.

On Sums of Odd and Even Terms of the Fibonacci Sequence

On Sums of Odd and Even Terms of the Fibonacci Sequence

Proof Without Words: Sums of Odd and Even Squares

(2015). Proof Without Words: Sums of Odd and Even Squares. Mathematics Magazine: Vol. 88, No. 4, pp. 278-279.

Odd Sum Labeling of Graphs Obtained by Duplicating Any Edge of Some Graphs

An injective function $f:V(G)\rightarrow \{0,1,2,\dots,q\}$ is an odd sum labeling if the induced edge labeling $f^*$ defined by $f^*(uv)=f(u)+f(v),$ for all $uv\in E(G),$ is bijective and $f^*(E(G))=\{1,3,5,\dots,2q-1\}.$ A graph is said to be an odd sum graph if it admits an odd sum labeling. In this paper we study the odd sum property of graphs obtained by duplicating any edge of some graphs.

Group flow, complex flow, unit vector flow, and the [formula omitted]-flow conjecture

If F is a (possibly infinite) subset of an abelian group Γ, then we define f(F,Γ) as the smallest natural number such that every f(F,Γ)-edge-connected (finite) graph G has a flow where all flow values are elements in F. We prove that f(F,Γ) exists if and only if some odd sum of elements in F equals some even sum. We discuss various instances of this problem. We prove that every 6-edge-connected graph has a flow whose flow values are the three roots of unity in the complex plane. If the edge-connectivity 6 can be reduced, then it can be reduced to 4, and the 3-flow conjecture follows. We prove that every 14-edge-connected graph has a flow whose flow values are the five roots of unity in the complex plane. Any such flow is balanced modulo 5. So, if the edge-connectivity 14 can be reduced to 9, then the 5-flow conjecture follows, as observed by F. Jaeger. We use vector flow to prove that, for each odd natural number k⩾3, every (3k−1)-edge-connected graph has a collection of k even subgraphs such that every edge is in precisely k−1 of them. Finally, the flow result gives a considerable freedom to prescribe the flow values in the (2+ϵ)-flow conjecture by L. Goddyn and P. Seymour. For example, if k is a natural number and G is a 6k-edge-connected graph, then G has a flow with flow values 1, 1+1/k. It also has, for any irrational number ϵ, a flow with flow values 1, 1+ϵ, 1+ϵ+1/k.

Odd sum labeling of some subdivision graphs

An injective function ƒ : V(G) →{0,1, 2,..., q} is an odd sum labeling if the induced edge labeling ƒ* defined by ƒ*(uv) = ƒ (u) + ƒ(v), for all uvє E(G), is bijective and ƒ*(E(G)) = {1, 3, 5,..., 2q - 1}. A graph is said to be an odd sum graph if it admits an odd sum labeling. In this paper, we have studied the odd sum property of the subdivision of the triangular snake, quadrilateral snake, slanting ladder, Cp ○ K1, H ○ K1, Cm@Cn, the grid graph Pm x Pn, duplication of a vertex of a path and duplication of a vertex of a cycle.

Derivation Power Sums of Even Integer Number Formula

This paper included derivative method for the even r power sums of even integer numbers formula to approach high even (r+2) power sums of even integer numbers formula so on we can approach from derivative odd r power sums of even integer numbers formula to high odd (r+2) power sums of even integer numbers formula this derivative excellence have ability to used by computer programming language or any application like Microsoft Office Excel. Also this research discovered the relationship between r power sums of even integer numbers formula and both formulas for same power sums of odd integer numbers formula and for r power sums of all integer numbers formula in another way.

Derivation Power Sums of Even Integer Number Formula

This paper included derivative method for the even r power sums of even integer numbers formula to approach high even (r+2) power sums of even integer numbers formula so on we can approach from derivative odd r power sums of even integer numbers formula to high odd (r+2) power sums of even integer numbers formula this derivative excellence have ability to used by computer programming language or any application like Microsoft Office Excel. Also this research discovered the relationship between r power sums of even integer numbers formula and both formulas for same power sums of odd integer numbers formula and for r power sums of all integer numbers formula in another way.

Derivative-Based Midpoint Quadrature Rule

A new family of numerical integration formula is presented, which uses the function evaluation at the midpoint of the interval and odd derivatives at the endpoints. Because the weights for the odd derivatives sum to zero, the derivative calculations cancel out for the interior points in the composite form, so that these derivatives must only be calculated at the endpoints of the overall interval of integration. When using N subintervals, the basic rule which uses the midpoint function evaluation and the first derivative at the endpoints achieves fourth order accuracy for the cost of N/2 function evaluations and 2 derivative evaluations, whereas the three point open Newton-Cotes method uses 3N/4 function evaluations to achieve the same order of accuracy. These derivative-based midpoint quadrature methods are shown to be more computationally efficient than both the open and closed Newton-Cotes quadrature rules of the same order. This family of derivative-based midpoint quadrature rules are derived using the concept of precision, along with the error term. A theorem concerning the order of accuracy of quadrature rule using the concept of precision is provided to justify its use to determine the leading order error term.

Power sums of Fibonacci and Lucas numbers

Polynomial representation formulae for power sums of the extended Fibonacci-Lucas numbers are established, which include, as special cases, four formulae for odd power sums of Melham type on Fibonacci and Lucas numbers, obtained recently by Ozeki and Prodinger (2009).

Faulhaber’s theorem on power sums

We observe that the classical Faulhaber’s theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression a + b , a + 2 b , … , a + n b is a polynomial in n a + n ( n + 1 ) b / 2 . While this assertion can be deduced from the original Fauhalber’s theorem, we give an alternative formula in terms of the Bernoulli polynomials. Moreover, by utilizing the central factorial numbers as in the approach of Knuth, we derive formulas for r -fold sums of powers without resorting to the notion of r -reflective functions. We also provide formulas for the r -fold alternating sums of powers in terms of Euler polynomials.

Odd Ramanujan Sums of Complex Roots of Unity

A special class of odd-symmetric length-4N periodic signals is studied, and it is shown how the odd Ramanujan sums are used as weighting coefficients to compute their pure imaginary discrete Fourier transform (DFT) integer-valued coefficients. The odd Ramanujan sum, being the sums of complex roots of unity, can be calculated either using closed-form formulas or computed recursively through the impulse response of a derived infinite impulse response (IIR) filter. This special class of odd-symmetric signals and odd Ramanujan sums can be combined together with the previous even-symmetric special class signals and the well-known even Ramanujan sums as a useful tool for signal processing

Three Topics in Additive Prime Number Theory

We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yildirim on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: work of Tao and the author on linear equations in primes.

Exotic Hadron in Pole-Dominated QCD Sum Rules

We study pentaquark Θ(I = 0, J = 1/2) in the QCD sum rules emphasizing that we can not extract any properties of the pentaquark outside of the Borel window. To find the appropriate Borel window, we prepare a favorable set up of the correlators and carry out the operator product expansion up to dimension 15 within factorization hypothesis. Our procedures reduce the unwanted high energy contaminations and enhance the low energy correlation. In the Borel window, independent analyses for the chiral-even and odd sum rules give the consistent values of the Θ mass, 1.68±0.22 GeV, and the residue. The parity is found to be positive.