P versus NP

Vega, Frank

doi:10.20944/preprints201908.0037.v2

Cited by 3 publications

(3 citation statements)

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“…In this way, we obtain that L B / ∈ REG, since it is not possible that L B ∈ NSPACE(o(log log n)) under the result of L V B / ∈ NSPACE(S(n)) for all S(n) = o(log n) as a consequence of Theorem 15 when L = N L. Consequently, we obtain a contradiction just assuming that the Beal's conjecture is false and L B ∈ REG when L = N L. In contraposition, we have the Beal's conjecture is true or this has an infinite number of counterexamples both for a specific choice of exponents (x, y, z), since L B uses a specific choice of exponents (x, y, z) when L = N L. The Darmon-Granville theorem uses Faltings's theorem to show that for every specific choice of exponents (x, y, z), there are at most finitely many co-prime solutions for (A, B, C) [9], [11]. In conclusion, we obtain that necessarily the Beal's conjecture is true for this specific choice of exponents (x, y, z) as the remaining only option when L = N L. Since we took arbitrarily the exponents (x, y, z), then the Beal's conjecture will be true for every specific choice of exponents (x, y, z) when L = N L. On November 2019, Frank Vega proves that L = N P which also implies that L = N L [28], [29]. In this way, the Beal's conjecture is true.…”

Section: Theorem 16 the Beal's Conjecture Is Truementioning

confidence: 96%

The Complexity of Number Theory

Vega¹

2020

Preprint

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The Goldbach's conjecture has been described as the most difficult problem in the history of Mathematics. This conjecture states that every even integer greater than 2 can be written as the sum of two primes. This is known as the strong Goldbach's conjecture. The conjecture that all odd numbers greater than 7 are the sum of three odd primes is known today as the weak Goldbach conjecture. A major complexity classes are L, NL and NSPACE(S(n)) for some S(n). Whether L = NL is a fundamental question that it is as important as it is unresolved. We show if the weak Goldbach's conjecture is true, then the problem PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n). This proof is based on the assumption that if some language belongs to NSPACE(S(n)), then the unary version of that language belongs to NSPACE(S(log n)) and vice versa. However, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples. Since Harald Helfgott proved that the weak Goldbach's conjecture is true, then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples, where the case of infinite number of counterexamples statistically seems to be unlikely. In addition, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the Beal's conjecture is true when L = NL. On November 2019, Frank Vega proves that L = NP which also implies that L = NL. In this way, the Beal's conjecture is true and since the Beal's conjecture is a generalization of Fermat's Last Theorem, then this is also a simple and short proof for that Theorem.

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Section: Theorem 16 the Beal's Conjecture Is Truementioning

confidence: 96%

The Complexity of Number Theory

Vega¹

2020

Preprint

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“…¿Cuándo es oportuna? Pedro de Vega sostuvo con brillantez que en «toda situación límite no cabe otro dilema que el de falsear la Constitución o reformarla», lo que equivale a indicar que «la necesidad jurídica…determina su conveniencia política» 69 . Pero me temo que no sea así muchas veces y no estoy seguro de que podamos seguir contemplando esta cuestión desde la razón teórica y el idealismo antes que desde la razón práctica.…”

Section: Una Teoría De Una Constitución Realizadaunclassified

De la revisión de las Constituciones: Constituciones nuevas y viejas

Roca¹

2018

TYRC

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El liberalismo fue demasiado prudente al regular los procedimientos. El constitucionalismo democrático debe seguir requisitos más accesibles y desdramatizar la reforma como una herramienta ordinaria. Pero la cuestión no depende de la rigidez sino de unas condiciones materiales que hagan posible la reforma. Rígida no es irreformable. La desaparición de la resiliencia constitucional marca el momento de la reforma. Sin embargo, este problema central no es jurídico: la capacidad de una mayoría de encontrar aliados y erigir un pacto constitucional. La reforma es una decisión política. La tramitación demanda mayorías cualificadas, deliberación y participación, pero el poder de reforma es constituido y el referendo no resulta indefectible, si la cuestión no afecta a decisiones políticas básicas. Las constituciones escritas son hoy rígidas: la verdadera distinción surge entre constituciones nuevas y viejas. Las viejas o estáticas son simbólicas y dejan su especificación a mutaciones judiciales, interpretaciones manipuladoras o a vías de hecho. La reforma recuerda al pueblo su soberanía y revitaliza la democracia representativa; mas donde no existe una cultura de la negociación tampoco puede haberla de la reforma. El escenario de pluralismo de constituciones está transformando nuestras constituciones nacionales y debería tener consecuencias en correlativas reformas constitucionales al ritmo en que avanza la integración supranacional.Liberal constitutionalism has been too cautious in regulation of constitutional reform proceedings. Democratic constitutionalism should follow easier requirements and simple procedures. It is necessary to normalise constitutional reform as an ordinary tool. However, constitutional review does not depend on legal rigidity but on certain material conditions that make reforms possible. When constitutional resilience disappears the moment of the reform seems to have arrived. Yet this is not a legal issue, but depends on the capacity of a majority to gain allies in order to reach a constitutional arrangement. Reform is always a political decision. However, legal proceedings require qualified majorities, democratic deliberation and political engagement, but being the reform a pouvoir constitué a referendum is not always necessary, if essential political decisions are not involved. Written constitutions are nowadays rigid, the true difference comes between new and old constitutions. Old or static constitutions are symbolic and leave their development either to judicial review, or even to some forced interpretations by public powers or the impact of simple facts in realities. Constitutional reform reminds citizens their popular sovereignty and strengthens representative democracy. Nevertheless, in those countries were a culture of negotiation does not exist, a culture of constitutional reform cannot exist too. The present-day scenario of multilevel constitutional pluralism is transforming domestic constitutions, and this phenomenon must have consequences in the promotion of regular internal constitutional reforms at the same rhythm as the advance of supranational integration.

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“…On the other hand, seasonal variations of d( 18 O) and 17 O excess (D( 17 O) ¼ d( 17 O) À 0.52 Â d( 18 O) described by Barkan and Luz (2007)) depend on the nitric acid (HNO 3 ) production pathway from atmospheric NO x , with higher d( 18 (Hastings et al, 2004;Morin et al, 2008;Frey et al, 2009;Morin et al, 2012). Using bi-weekly measurements of d( 15 N) and D( 17 O), Morin et al (2008Morin et al ( , 2012 found a connection between low d( 15 N) and high D( 17 O) values as a consequence of NO x produced by local photochemical emissions from the snowpack that were later oxidized to NO 3 À by reactive halogens (e.g., BrO…”

Section: Introductionmentioning

confidence: 99%

Composition, isotopic fingerprint and source attribution of nitrate deposition from rain and fog at a Sub-Arctic Mountain site in Central Sweden (Mt Åreskutan)

Vega¹,

Mårtensson²,

Wideqvist³

et al. 2019

Tellus B: Chemical and Physical Meteorology

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While dry and rain deposition of nitrate (NO 3 À ) and ammonium (NH 4 þ ) are regularly assessed, fog deposition is often overlooked. This work assesses summer fog events contribution to nitrogen deposition and availability for forest ecosystems. Rain and fog samples were collected at Mt Åreskutan, Sweden, during CAEsAR (Cloud and Aerosol Characterization Experiment), in 2014. NH 4 þ þ NO 3 À represent (31 ± 25) % of total rain ion amount, and (31 ± 42) % in fog. Based on ion concentrations and the nitrate stable isotope signatures d( 15 N) and d( 18 O), it was possible to detect the plume generated by the V€ astmanland forest fire; NO x emissions from oil rigs and Kola Peninsula; and the plume of Bardarbunga volcano, Iceland. Scavenging of ions by fog was more efficient than by rain. Rain NH 4 þ and NO 3 À deposition was (26 ± 36)lmol m À2 d À1 and (23 ± 27) lmol m À2 d À1 , respectively. Fog NH 4 þ and NO 3 À contributed (77 ± 80) % to total wet deposition of these species. Upscaling rain deposition fluxes to 1 year gave an inorganic nitrogen deposition of (18 ± 16) mmol m À2 a À1 ((252 ± 224) mg m À2 a À1 N equivalents), whereas fog deposition was estimated as (59 ± 47) mmol m À2 a À1 ((826 ± 658) mg m À2 a À1 N equivalents). Annual fog deposition was four times higher than previously reported for the area which only considered rain deposition. However, great uncertainty on the calculation of fog deposition need to be bear in mind. These findings suggest that fog should be considered in deposition estimates of inorganic nitrogen and major ions. If fog deposition is not accounted for, ion wet deposition may be greatly underestimated. Further sampling of wet and dry deposition is important for understanding the influence of nitrogen deposition on forest and vegetation development, as well as soil major ion loads.

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P versus NP

Cited by 3 publications

References 9 publications

The Complexity of Number Theory

The Complexity of Number Theory

De la revisión de las Constituciones: Constituciones nuevas y viejas

Composition, isotopic fingerprint and source attribution of nitrate deposition from rain and fog at a Sub-Arctic Mountain site in Central Sweden (Mt Åreskutan)

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