Cyclic Extension Research Articles

Genus theory and ε-conjectures on p-class groups

We suspect that the “genus part” of the class number of a number field K may be an obstruction for an “easy and unconditional proof” of the classical p-rank ε-conjecture for p-class groups and, a fortiori, for a proof of the “strong ε-conjecture”: #(CℓK⊗Zp)≪d,p,ε(|DK|)ε for all K of degree d. We analyze the weight of genus theory in this inequality by means of an infinite family of degree p cyclic fields with many ramified primes, then we prove the p-rank ε-conjecture: #(CℓK⊗Fp)≪d,p,ε(|DK|)ε, for d=p and the family of degree p cyclic extensions (Theorem 2.5) then sketch the case of arbitrary base fields. The possible obstruction for the strong form, in the degree p cyclic case, is the order of magnitude of the set of “exceptional” p-classes given by a well-known non-predictible algorithm, but largely controlled thanks to recent density results due to Koymans–Pagano. Then we compare the ε-conjectures with some p-adic conjectures, of Brauer–Siegel type, about the torsion group TK of the Galois group of the maximal abelian p-ramified pro-p-extension of totally real number fields K. We give numerical computations with the corresponding PARI/GP programs.

The Biomechanical Basis of the Claw Finger Deformity: A Computational Simulation Study

Claw finger deformity occurs during attempted finger extension in patients whose intrinsic finger muscles are weakened or paralyzed by neural impairments. The deformity is generally not acutely present after intrinsic muscle palsy. The delayed onset, with severity progressing over time, suggests soft tissue changes that affect the passive biomechanics of the hand exacerbate and advance the deformity. Clinical interventions may be more effective if such secondary biomechanical changes are effectively addressed. Using a computational model, we simulated these altered soft tissue biomechanical properties to quantify their effects on coordinated finger extension. To evaluate the effects of maladaptive changes in soft tissue biomechanical properties on the development and progression of the claw finger deformity after intrinsic muscle palsy, we completed 45 biomechanical simulations of cyclic index finger flexion and extension, varying the muscle excitation level, clinically relevant biomechanical factors, and wrist position. We evaluated to what extent (1) increased joint laxity, (2) decreased mechanical advantage of the extensors about the proximal interphalangeal joint, and (3) shortening of the flexor muscles contributed to the development of claw finger deformity in an intrinsic-minus hand model. Of the mechanisms studied, shortening (or contracture) of the extrinsic finger flexors was the factor most associated with the development of claw finger deformity in simulation. These simulations suggest that adaptive shortening of the extrinsic finger flexors is required for the development of claw finger deformity. Increased joint laxity and decreased extensor mechanical advantage only contributed to the severity of the deformity in simulations when shortening of the flexor muscles was present. In both the acute and chronic stages of intrinsic finger paralysis, maintaining extrinsic finger flexor length should be an area of focus in rehabilitation to prevent formation of the claw finger deformity and achieve optimal outcomes after surgical interventions.

Stably rational surfaces over a quasi-finite field

Let be a field and a smooth, projective, stably -rational surface. If is split by a cyclic extension (for example, if the field is finite or, more generally, quasi-finite), then the surface is -rational.

ON THE DIMENSION OF PERMUTATION VECTOR SPACES

Let $K$ be a field that admits a cyclic Galois extension of degree $n\geq 2$. The symmetric group $S_{n}$ acts on $K^{n}$ by permutation of coordinates. Given a subgroup $G$ of $S_{n}$ and $u\in K^{n}$, let $V_{G}(u)$ be the $K$-vector space spanned by the orbit of $u$ under the action of $G$. In this paper we show that, for a special family of groups $G$ of affine type, the dimension of $V_{G}(u)$ can be computed via the greatest common divisor of certain polynomials in $K[x]$. We present some applications of our results to the cases $K=\mathbb{Q}$ and $K$ finite.

On cyclic essential extensions of simple modules over differential operator rings

In this paper, we discuss under which conditions cyclic essential extensions of simple modules over a differential operator ring R[θ;δ] are Artinian. In particular, we study the case when R is either δ-simple or δ-primitive. Furthermore, we obtain important results when R is an affine algebra of Kull dimension 2. As an application, we characterize the differential operator rings for which cyclic essential extensions of simple modules are Artinian.

Diophantine definability of nonnorms of cyclic extensions of global fields

We show that, for any square-free natural number n n and any global field K K with ( char ( K ) , n ) = 1 (\textrm {char}(K), n)=1 containing a primitive n n th root of unity, the pairs ( x , y ) ∈ K × × K × (x,y)\in K^{\times }\times K^{\times } such that x x is not a relative norm of K ( y n ) / K K(\sqrt [n]{y})/K form a diophantine set over K K . We use the Hasse norm theorem, Kummer theory, and class field theory to prove this result. We also prove that, for any n ∈ N n\in {\mathbb {N}} and any global field K K with char ( K ) ≠ n \textrm {char}(K)\neq n , K × ∖ K × n K^{\times }\setminus K^{\times n} is diophantine over K K . For a number field K K , this is a result of Colliot-Thélène and Van Geel, proved using results on the Brauer–Manin obstruction. Additionally, we prove a variation of our main theorem for global fields K K without the n n th roots of unity, where we parametrize varieties arising from norm forms of cyclic extensions of K K without any rational points by a diophantine set.

Nanopillared Surfaces Disrupt Pseudomonas aeruginosa Mechanoresponsive Upstream Motility.

Pseudomonas aeruginosa is an opportunistic, multidrug-resistant, human pathogen that forms biofilms in environments with fluid flow, such as the lungs of cystic fibrosis patients, industrial pipelines, and medical devices. P. aeruginosa twitches upstream on surfaces by the cyclic extension and retraction of its mechanoresponsive type IV pili motility appendages. The prevention of upstream motility, host invasion, and infectious biofilm formation in fluid flow systems remains an unmet challenge. Here, we describe the design and application of scalable nanopillared surface structures fabricated using nanoimprint lithography that reduce upstream motility and colonization by P. aeruginosa. We used flow channels to induce shear stress typically found in catheter tubes and microscopy analysis to investigate the impact of nanopillared surfaces with different packing fractions on upstream motility trajectory, displacement, velocity, and surface attachment. We found that densely packed, subcellular nanopillared surfaces, with pillar periodicities ranging from 200 to 600 nm and widths ranging from 70 to 215 nm, inhibit the mechanoresponsive upstream motility and surface attachment. This bacteria-nanostructured surface interface effect allows us to tailor surfaces with specific nanopillared geometries for disrupting cell motility and attachment in fluid flow systems.

Cyclic extensions of prime degree and their p-adic regulators

Cyclic extensions of prime degree and their p-adic regulators

Annihilators of the Ideal Class Group of a Cyclic Extension of an Imaginary Quadratic Field

AbstractThe aim of this paper is to study the group of elliptic units of a cyclic extension $L$ of an imaginary quadratic field $K$ such that the degree $[L:K]$ is a power of an odd prime $p$. We construct an explicit root of the usual top generator of this group, and we use it to obtain an annihilation result of the $p$-Sylow subgroup of the ideal class group of $L$.

Stably rational surfaces over a quasi-finite field

Let $k$ be a field and $X$ a smooth, projective, stably $k$-rational surface. If $X$ is split by a cyclic extension, for instance if the field $k$ is finite or more generally quasi-finite, then the surface $X$ is $k$-rational. Bibliography: 22 titles.

Composantes isotypiques de pro-$p$-extensions de corps de nombres et $p$-rationalite

Soit p un nombre premier et soit K/k une extension galoisienne finie de corps de nombres de groupe de Galois ∆ d’ordre etranger a p. Soit S un ensemble fini de places finies de k contenant l’ensemble S p des places p-adiques et soit KS la pro-p-extension maximale de K non-ramifiee en dehors de S. Soi tG:“GS{Hun quotient de GS:“Gal KS/K) sur lequel ∆ agit trivialement. Posons X:“H{rH, Hs. Dans ce travail,nous etudions la φ-composante X φ de X pour les caracteres Qp-irreductibles φ de ∆ et nous montrons entre autres, sous la conjecture de Leopoldt et pour tout caractere non trivial φ, que X φ est ZpvGw-libre si et seulement si la φ-composante de la Zp-torsion de Gs /{Gs, Gs est triviale. Nous faisons egalement une etude numerique sur la liberte de Xφ dans des extensions cycliques K{Qde degre 3 et de degre 4 (`a partir de familles de polynomes donnees par Balady, Lecacheux et plus recemment par Balady et Washington),et ainsi que dans des extensions diedrales de degre 6 de Q. Les resultats de nos calculs renforcent une recente conjecture de Gras

On Cyclic Descents for Tableaux

AbstractThe notion of descent set, for permutations as well as for standard Young tableaux (SYT), is classical. Cellini introduced a natural notion of cyclic descent set for permutations, and Rhoades introduced such a notion for SYT—but only for rectangular shapes. In this work we define cyclic extensions of descent sets in a general context and prove existence and essential uniqueness for SYT of almost all shapes. The proof applies nonnegativity properties of Postnikov’s toric Schur polynomials, providing a new interpretation of certain Gromov–Witten invariants.

Cyclic behavior of unit bucket for tripod foundation system supporting offshore wind turbine via model tests

AbstractOffshore wind energy has been growing up as a promising renewable energy source. Recently, tripod suction bucket foundation is rapidly expanding as a foundation system supporting the offshore wind turbine. In offshore environment, wind turbine foundation structures should be designed considering cyclic loading which can lead to permanent deformation of structure, tilting problem, and overall degradation of soil stiffness. However, it is technically difficult to predict the cyclic behavior of the tripod accurately because the cyclic behaviors of the tripod bucket can be inferred from vertical pullout and compression behaviors of each single bucket elements. In this paper, a series of model tests was performed by applying cyclic vertical compression and extension loadings to a single bucket element that is one element of the tripod foundation. Loading directions, level, and rate were controlled for investigating of cyclic behavior of tripod foundation. On the basis of testing results, the permanent deformation and cyclic stiffness response of tripod suction caisson were discussed. Based on the test results, it was confirmed that the cyclic behavior of the single bucket is affected by the load level and rate. In addition, the behavior showed quite different trends with the loading directions: compression, pullout, and two‐way.

Signal Modulation and Processing in Nonlinear Fibre Channels by Employing the Riemann–Hilbert Problem

Most of the nonlinear Fourier transform (NFT) based optical communication systems studied so far deal with the burst mode operation that substantially reduce achievable spectral efficiency. The burst mode requirement emerges due to the very nature of the commonly used version of the NFT processing method: it can process only rapidly decaying signals, requires zero-padding guard intervals for processing of dispersion-induced channel memory, and does not allow one to control the time-domain occupation well. Some of the limitations and drawbacks imposed by this approach can be rectified by the recently introduced more mathematically demanding periodic NFT processing tools. However, the studies incorporating the signals with cyclic prefix extension into the NFT transmission framework have so far lacked the efficient digital signal processing (DSP) method of synthesizing an optical signal, the shortcoming that diminishes the approach flexibility. In this paper, we introduce the Riemann–Hilbert problem (RHP) based DSP method as a flexible and expandable tool that would allow one to utilize the periodic NFT spectrum for transmission purposes without former restrictions. First, we outline the theoretical framework and clarify the implementation underlying the proposed new DSP method. Then we present the results of numerical modelling quantifying the performance of long-haul RHP-based transmission with the account of optical noise, demonstrating the good performance quality and potential of RHP-based optical communication systems.

Nonassociative cyclic extensions of fields and central simple algebras

We define nonassociative cyclic extensions of degree m of both fields and central simple algebras over fields. If a suitable field contains a primitive mth (resp., qth) root of unity, we show that suitable nonassociative generalized cyclic division algebras yield nonassociative cyclic extensions of degree m (resp., qs). Some of Amitsur's classical results on non-commutative associative cyclic extensions of both fields and central simple algebras are obtained as special cases.

On the Reduced Group of Principal Units in Cyclic Extensions of Local Fields

The reduced group of principal units of cyclic extensions of local fields is considered as a Galois module. This module is studied by examining the Jordan canonical form of the generating automorphism of the Galois group.

Construction of a Cyclic Extension of Degree p2 for a Complete Field

The aim of the paper is to construct an embedding of a given cyclic extension of degree p of a complete discrete valuation field of characteristic 0 with an arbitrary residue field of characteristic p > 0 into a cyclic extension of degree p2. The result extends the construction obtained by S. V. Vostokov and I. B. Zhukov in terms of Witt vectors, to a wider interval of values for the ramification jump of the original field extension.

Average bounds for the $$\ell $$ ℓ -torsion in class groups of cyclic extensions

For all positive integers $$\ell $$ l , we prove non-trivial bounds for the $$\ell $$ l -torsion in the class group of K, which hold for almost all number fields K in certain families of cyclic extensions of arbitrarily large degree. In particular, such bounds hold for almost all cyclic degree-p-extensions of F, where F is an arbitrary number field and p is any prime for which F and the pth cyclotomic field are linearly disjoint. Along the way, we prove precise asymptotic counting results for the fields of bounded discriminant in our families with prescribed splitting behavior at finitely many primes.

Electrochemical Detection of SNP in Human Mitochondrial DNA Using Cyclic Primer Extension with Biotinylated Nucletides and Enzymatic Labeling at Disposable Pencil Graphite Electrodes

AbstractA novel method of SNP typing in human mitochondrial DNA utilizing enzymatic labeling and electrochemical detection at disposable pencil graphite electrodes is described. The procedure is based on amplification of DNA stretches by cyclic primer extension (PEx) of SNP‐specific diagnostic primers in a mixture of biotinylated and natural nucleotides. The diagnostic primers are designed to recognize, by its 3'‐terminal nucleotide, the SNP‐site in target template. Under optimized conditions of the PEx reaction, efficient polymerase synthesis of biotin‐labeled strands takes place only in the case of full complementarity between the diagnostic primer and the target SNP site. There is also benefit from introducing many biotin molecules per extended DNA strand, resulting in another level of signal amplification. After adsorption of biotinylated PEx products at the electrode surface, streptavidin‐alkaline phosphatase conjugate was bound to the biotin tags, 1‐naphthol was enzymatically produced and electrochemically detected. Several critical steps and parameters of the assay, including termination of 3’‐OH ends of residual amplification primers, temperature for annealing of diagnostic primers, relative amount of biotinylated deoxynucleoside triphosphate in the PEx mixture and number of PEx cycles were optimized in this study to attain best SNP resolution, and reduction of time needed for the analysis.

Isomorphism problems for Hopf–Galois structures on separable field extensions

Let L/K be a finite separable extension of fields whose Galois closure E/K has Galois group G. Greither and Pareigis use Galois descent to show that a Hopf algebra giving a Hopf–Galois structure on L/K has the form E[N]G for some group N of order [L:K]. We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as K-algebras. By applying our results, we complete a detailed analysis of the distinct Hopf algebras and K-algebras that appear in the classification of Hopf–Galois structures on a cyclic extension of degree pn, for p an odd prime number.