Artin–Schelter regular algebras of dimension five with three generators

2021 ◽  
pp. 2250235
Author(s):  
Jun Li
Keyword(s):  
Dimension Formula ◽  
Regular Algebras ◽  
Generating Relations ◽  
Degree Type

In this paper, we investigate Artin–Schelter regular algebras of dimension [Formula: see text] with three generators in degree [Formula: see text] under the hypothesis that [Formula: see text], in which the degree types of the relations for the number of the generating relations less than five can be determined. We prove that the only possible degree type of three generating relations is [Formula: see text] and the only possible degree type of four generating relations is [Formula: see text].

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

scholarly journals Dynamic Łukasiewicz logic and its application to immune system

2021 ◽  
Author(s):  
Antonio Di Nola ◽  
Revaz Grigolia ◽  
Nunu Mitskevich ◽  
Gaetano Vitale
Keyword(s):  
Immune System ◽  
Scalar Multiplication ◽  
Kripke Semantics ◽  
Łukasiewicz Logic ◽  
Lukasiewicz Logic ◽  
Regular Algebras

AbstractIt is introduced an immune dynamic n-valued Łukasiewicz logic $$ID{\L }_n$$ I D Ł n on the base of n-valued Łukasiewicz logic $${\L }_n$$ Ł n and corresponding to it immune dynamic $$MV_n$$ M V n -algebra ($$IDL_n$$ I D L n -algebra), $$1< n < \omega $$ 1 < n < ω , which are algebraic counterparts of the logic, that in turn represent two-sorted algebras $$(\mathcal {M}, \mathcal {R}, \Diamond )$$ ( M , R , ◊ ) that combine the varieties of $$MV_n$$ M V n -algebras $$\mathcal {M} = (M, \oplus , \odot , \sim , 0,1)$$ M = ( M , ⊕ , ⊙ , ∼ , 0 , 1 ) and regular algebras $$\mathcal {R} = (R,\cup , ;, ^*)$$ R = ( R , ∪ , ; , ∗ ) into a single finitely axiomatized variety resembling R-module with “scalar” multiplication $$\Diamond $$ ◊ . Kripke semantics is developed for immune dynamic Łukasiewicz logic $$ID{\L }_n$$ I D Ł n with application in immune system.

LATTICE DYNAMICS OF THE BINARY APERIODIC CHAINS OF ATOMS I: FRACTAL DIMENSION OF PHONON SPECTRA

1995 ◽  
Vol 09 (12) ◽  
pp. 1429-1451 ◽  
Author(s):  
WŁODZIMIERZ SALEJDA
Keyword(s):  
Fractal Dimension ◽  
Lattice Dynamics ◽  
Nearest Neighbor ◽  
Average Distance ◽  
Local Maximum ◽  
Fractal Dimensions ◽  
Dimension Formula ◽  
Phonon Spectra ◽  
Wide Range ◽  
Silver Mean

The microscopic harmonic model of lattice dynamics of the binary chains of atoms is formulated and studied numerically. The dependence of spring constants of the nearest-neighbor (NN) interactions on the average distance between atoms are taken into account. The covering fractal dimensions [Formula: see text] of the Cantor-set-like phonon spec-tra (PS) of generalized Fibonacci and non-Fibonaccian aperiodic chains containing of 16384≤N≤33461 atoms are determined numerically. The dependence of [Formula: see text] on the strength Q of NN interactions and on R=mH/mL, where mH and mL denotes the mass of heavy and light atoms, respectively, are calculated for a wide range of Q and R. In particular we found: (1) The fractal dimension [Formula: see text] of the PS for the so-called goldenmean, silver-mean, bronze-mean, dodecagonal and Severin chain shows a local maximum at increasing magnitude of Q and R>1; (2) At sufficiently large Q we observe power-like diminishing of [Formula: see text] i.e. [Formula: see text], where α=−0.14±0.02 and α=−0.10±0.02 for the above specified chains and so-called octagonal, copper-mean, nickel-mean, Thue-Morse, Rudin-Shapiro chain, respectively.

KINETIC RESPONSE OF SURFACES DEFINED BY FINITE FRACTALS

Fractals ◽  
2010 ◽  
Vol 18 (04) ◽  
pp. 461-476 ◽  
Author(s):  
PRADEEP R. NAIR ◽  
MUHAMMAD A. ALAM
Keyword(s):  
Scaling Laws ◽  
Finite Size ◽  
Critical Dimension ◽  
The Novel ◽  
Scale Invariant ◽  
Fractal Surfaces ◽  
Dimension Formula ◽  
Diffusion Limited ◽  
Engineered Systems ◽  
Size Limits

Historically, fractal analysis has been remarkably successful in describing wide ranging kinetic processes on (idealized) scale invariant objects in terms of elegantly simple universal scaling laws. However, as nanostructured materials find increasing applications in energy storage, energy conversion, healthcare, etc., one must reexamine the premise of traditional fractal scaling laws as it only applies to physically unrealistic infinite systems, while all natural/engineered systems are necessarily finite. In this article, we address the consequences of the 'finite-size' problem in the context of time dependent diffusion towards fractal surfaces via the novel technique of Cantor-transforms to (i) illustrate how finiteness modifies its classical scaling exponents; (ii) establish that for finite systems, the diffusion-limited reaction is decelerated below a critical dimension [Formula: see text] and accelerated above it; and (iii) to identify the crossover size-limits beyond which a finite system can be considered (practically) infinite and redefine the very notion of 'finiteness' of fractals in terms of its kinetic response. Our results have broad implications regarding dynamics of systems defined by the same fractal dimension, but differentiated by degree of scaling iteration or morphogenesis, e.g. variation in lung capacity between a child and adult.

scholarly journals The dimension formula of the space of cusp forms of weight one for Γ0(p)

1988 ◽  
Vol 111 ◽  
pp. 115-129 ◽  
Author(s):  
Yoshio Tanigawa ◽  
Hirofumi Ishikawa
Keyword(s):  
Cusp Forms ◽  
Dimension Formula ◽  
Weight One ◽  
Space Of Cusp Forms

The purpose of this paper is to study the dimension formula for cusp forms of weight one, following the series of Hiramatsu [2] and Hiramatsu-Akiyama [3]. We define as usual the subgroup Γ0(N) of SL2(Z) by.

A Proposed Experiment Using a Sphere with an Eccentric Cavity for Hypothetical Long-Range Interactions

1997 ◽  
Vol 12 (08) ◽  
pp. 1465-1482 ◽  
Author(s):  
G. L. Klimchitskaya ◽  
V. M. Mostepanenko ◽  
C. Romero ◽  
Ye. P. Krivtsov ◽  
A. Ye. Sinelnikov
Keyword(s):  
Long Range ◽  
High Sensitivity ◽  
Experimental Setup ◽  
Interaction Range ◽  
Steel Sphere ◽  
Long Range Interactions ◽  
Torsional Pendulum ◽  
Additional Interaction ◽  
Homogeneous Gravitational Field ◽  
Degree Type

The constraints are examined which may be obtained for the parameters of long-range hypothetical interactions by the use of the precise experimental setup created originally for the calibration of accelerometers. This setup includes the large rotating steel sphere with a nonconcentric spherical cavity in which the strictly homogeneous gravitational field arises. The field of additional interaction produced by the atoms of the sphere, however, is not homogeneous. The essential features required of the detector of additional interaction which is the torsional pendulum of high sensitivity are discussed. Both the cases of the Yukawa-type and degree-type hypothetical interactions are investigated. It is shown that the known-to-date constraints for Yukawa-type interactions may be strengthened by a factor of 400 in the appropriate interaction range. For the degree-type hypothetical forces decreasing with distance as r-3, r-4 and r-5 correspondingly the known constraints may be strengthened by the factors of 90, 35 and 20. The conclusion is made that with the use of the specially created related setup it will be conceivable to strengthen the constraints for Yukawa-type interactions up to 4500 times over a wide interaction range.

scholarly journals A note on a certain class of functions related to Hurwitz zeta function and Lambert transform

2000 ◽  
Vol 31 (1) ◽  
pp. 49-56
Author(s):  
R. K. Raina ◽  
T. S. Nahar
Keyword(s):  
Zeta Function ◽  
Hurwitz Zeta Function ◽  
Multiple Series ◽  
Generating Relations ◽  
Class Of Functions

In this paper we obtain multiple-series generating relations involving a class of function $ \theta_{(p_n)}^{(\mu_n)}(s,a;x_1,\ldots,x_n)$ which are connected to the Hurwitz zeta function. Also, a new generalization of Lambert transform is introduced, and its relationship with the above class of functions further depicted.

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