Kumaran1987

My Omnidimensional AP Research: From a Simple Number Pattern to a General Power-Sum Formula By Kumaran Kanniappan My OMNIBAL research explores a question: If we know exactly how a sequence is generated, can we calculate its powered total without visiting every term? For arithmetic progressions, we can express that calculation through a midpoint, symmetry, and reusable moment formulas. In this article, I explain the approach from a simple example to multidimensional arrangements and higher powers, then describe how I would evaluate its practical usefulness. 1. BEGIN WITH A FAMILIAR PATTERN Consider: 3, 5, 7, 9, 11, 13, 15, 17 Each number increases by two. This is an arithmetic progression, or AP. Write it as: a_i = F + h*i, for i = 0,...,X-1 The parameters are: - F: first value. - h: common difference. - X: number of values. For this sequence: F = 3, h = 2, X = 8 Its last value is: L = F + h*(X-1) = 17 Its midpoint is: M = (F+L)/2 = 10 2. THE MIDPOINT REVEALS THE STRUCTURE Pair the valu...

From Raw Data to Formula-Compressed Computing: OMNIBAL, OmniFit, HORN and the Search for O(1) Algorithms Modern computing usually treats data as individual elements. If a system contains one million values, we commonly store one million values. If we need their total, we may scan them, preprocess them, index them, or maintain an additional data structure. But what happens when those million values are not truly independent? What if they follow a mathematical structure? This question leads to an interesting direction in OMNIBAL and Omnidimensional Formula research: Instead of storing every element, can we store the mathematical rule that generates the elements? And if we can identify that rule, can some operations that traditionally depend on N become independent of N? That is the idea behind combining: OmniFit + OmniSegment + OMNIBAL + HORN into a formula-oriented data-structures-and-algorithms architecture. 1. The Four Omnidimensional Formula Families The mathematical foundation begin...

OMNIBAL — Omnidimensional Sequence Lab OMNIBAL Omnidimensional engine Machine-verified calculator AP · GP · HP / Interactive research lab Shape a sequence. See every dimension. Enter a starting number, a spacing or ratio, a power, and any dimensional arrangement. OMNIBAL generates the run, maps every cell, and returns the final computation. AP · Add GP · Multiply HP · Reciprocal 01 / Define the run Arithmetic progression Add equally spaced values after raising each one to the chosen power. Starting number Spacing difference (Δ) Dimensional arrangement 4 cells ...

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