Recent studies by mathematicians and physicists have identified a close association between the distribution of prime numbers and quantum mechanical laws governing the subatomic dynamics of quantum systems such as the electron or the photon. It is now recognised that Cantorian fractal space-time fluctuations characterise dynamical systems of all space-time scales ranging from the microscopic su...

Hu, Kriz and May recently reexamined ideas implicit in Priddy's elegant homotopy theoretic construction of the Brown-Peterson spectrum at a prime p. They discussed May's notions of nuclear complexes and of cores of spaces, spectra, and commutative S-algebras. Their most striking conclusions, due to Hu and Kriz, were negative: cores are not unique up to equivalence, and BP is not a core of M U c...

Previous work constructed a generalized truncated Brown-Peterson spectrum of chromatic height 2 at the prime 2 as an E∞-ring spectrum, based on the study of elliptic curves with level-3 structure. We show that the natural map forgetting this level structure induces an E∞-ring map from the spectrum of topological modular forms to this truncated Brown-Peterson spectrum, and that this orientation ...

6 Introduction to Affine Schemes 2 6.1 Rings and Modules of Fractions . . . . . . . . . . . . . . . . . 2 6.2 The Spectrum of a Unital Commutative Ring . . . . . . . . . 5 6.3 The Spectrum of a Quotient Ring . . . . . . . . . . . . . . . . 7 6.4 The Spectrum of a Ring of Fractions . . . . . . . . . . . . . . 9 6.5 Intersections of Prime Ideals . . . . . . . . . . . . . . . . . . . 11 6.6 Topolo...

We study the “q-commutative” power series ring R := kq[[x1, . . . , xn]], defined by the relations xixj = qijxjxi, for mulitiplicatively antisymmetric scalars qij in a field k. Our results provide a detailed account of prime ideal structure for a class of noncommutative, complete, local, noetherian domains having arbitrarily high (but finite) Krull, global, and classical Krull dimension. In par...

Given a finite set of eigenvalues regular Sturm-Liouville problem for the equation -y{\prime}{\prime}+q(x)y={\lambda}y, potential q(x) which is unknown. We show possibility to compute more without any additional information on q(x). Moreover, considering with boundary conditions y{\prime}(0)-hy(0)=0 and y{\prime}({\pi})+Hy({\pi})=0, where h, H are some constants, we complete its spectrum neithe...

in this work, by employing the krasnosel'skii fixed point theorem, we study the existence of positive solutions of a three-point boundary value problem for the following fourth-order differential equation begin{eqnarray*} left { begin{array}{ll} u^{(4)}(t) -f(t,u(t),u^{prime prime }(t))=0 hspace{1cm} 0 leq t leq 1, & u(0) = u(1)=0, hspace{1cm} alpha u^{prime prime }(0) - beta u^{prime prime...

In this paper, a combination of algebraic and topological methods is applied to obtain new structural results on Gelfand residuated lattices. It demonstrated that Gelfand’s lattices strongly tied up with the hull–kernel topology. Particularly, it shown lattice if only its prime spectrum, equipped topology, normal. The class soft introduced, semisimple. are characterized using pure part filters....

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if R P is an artinian valuation ring for each maximal prime ideal P. Commutative rings for which each indecomposable module has a local endomorphism ring are studied...

We prove that for the iterated Poisson polynomial rings known as nilpotent algebras (or Poisson-CGL extensions), prime spectrum is catenary, i.e., all saturated chains of inclusions ideals between any two given have same length.