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| viXra.org > Combinatorics and Graph Theory > 2601 |
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[3] viXra:2601.0113 [pdf] replaced on 2026-02-01 00:12:41
Authors: Bassam Abdul-Baki
Comments: 4 Pages.
In this paper, we improve the lower-bounds for optimal Golomb rulers.
Category: Combinatorics and Graph Theory
[2] viXra:2601.0079 [pdf] submitted on 2026-01-20 20:04:22
Authors: Andrzej Gecow, Laszlo Barna Iantovics
Comments: 17 Pages.
Half-chaos was detected using simulations of complex, autonomous, fixed-size Kauffman networks, but it also occurs in open, growing networks. Mathematicians dealing with deterministic chaos expect its description in mathematical terms. This article (read at least to ch.3.5) can significantly help to create such a description. Kauffman, studying the statistical properties of a set of completely random systems, identifies two states of the system: either it is ordered, where a small disturbance practically always dies out, or it is chaotic, where a small disturbance practically always causes significant damage (a change in functioning). In between, there is a narrow (in the system parameters) phase transition. The assumption of a small attractor causes the system to cease to be fully random, and a third state is revealed - half-chaotic, in which small and large damage have a similar share, despite the system parameters are chaotic. If this system were fully random, it would almost always give strong chaos. A mechanism for increasing system stability following a permanent disturbance has been identified. It involves limiting secondary initiations by shortening the attractor. Leaving in a half-chaotic system only changes that give small damage do not lead out of half-chaos. In the distribution of the damage size, there is a large gap between small and large damages, which naturally defines ‘small changes’. Leaving one change which causes large damage leads to a practically irreversible transition to ordinary chaos. Unique stability of half-chaos extends the range of allowed parameters for models of human-created and living systems, previously limited to the edge of chaos by the famous Kauffman hypothesis. Half-chaos explains the essence of the life process.
Category: Combinatorics and Graph Theory
[1] viXra:2601.0056 [pdf] replaced on 2026-04-04 00:24:39
Authors: Christopher C. Mbakwe
Comments: 32 Pages.
This paper presents a novel proof of the non-existence of odd perfect numbers using the framework of algebraic circuit theory and spectral graph theory. We construct a specialized resistive network, Γm(n), where the topology is uniquely determined by the divisor structure of an integer n. By embedding the arithmetic properties of the sum-of-divisors function σ(n) into the Kirchhoff Laplacian L(Γ),we demonstrate that the potential distribution of the network satisfies a discrete harmonic extension if and only if n satisfies specific divisor identities. We then generalize the result to odd k perfect numbers for k > 1.
Category: Combinatorics and Graph Theory
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