Nottingham Group Research Articles

Automata and finite order elements in the Nottingham group

The Nottingham group at 2 is the group of (formal) power series t+a2t2+a3t3+⋯ in the variable t with coefficients ai from the field with two elements, where the group operation is given by composition of power series. The depth of such a series is the largest d⩾1 for which a2=…=ad=0.Only a handful of power series of finite order (forcedly a power of 2) are explicitly known through a formula for their coefficients. We argue in this paper that it is advantageous to describe such series in closed computational form through automata, based on effective versions of proofs of Christol's theorem identifying algebraic and automatic series.Up to conjugation, there are only finitely many series σ of order 2n with fixed break sequence (i.e. the sequence of depths of σ∘2i). Starting from Witt vector or Carlitz module constructions, we give an explicit automaton-theoretic description of: (a) representatives up to conjugation for all series of order 4 with break sequence (1,m) for m<10; (b) representatives up to conjugation for all series of order 8 with minimal break sequence (1,3,11); and (c) an embedding of the Klein four-group into the Nottingham group at 2.We study the complexity of the new examples from the algebro-geometric properties of the equations they satisfy. For this, we generalise the theory of sparseness of power series to a four-step hierarchy of complexity, for which we give both Galois-theoretic and combinatorial descriptions. We identify where our different series fit into this hierarchy. We construct sparse representatives for the conjugacy class of elements of order two and depth 2μ±1(μ⩾1). Series with small state complexity can end up high in the hierarchy. This is true, for example, for a new automaton we found, representing a series of order 4 with 5 states (the minimal possible number for such a series).

Prediction of Prognostic Factors and Genotypes in Patients With Breast Cancer Using Multiple Mathematical Models of MR Diffusion Imaging.

PurposeTo explore the clinical value of apparent diffusion coefficient (ADC), intravoxel incoherent motion (IVIM), and diffusion kurtosis imaging (DKI) based on diffusion-weighted MRI (DW-MRI) for predicting genotypes and prognostic factors of breast cancer.Materials and MethodsA total of 227 patients with breast cancer confirmed by pathology were reviewed retrospectively. Diffusion-weighted imaging (DWI), IVIM, and DKI were performed in all patients. The corresponding ADC, true diffusion coefficient (D), perfusion-related diffusion coefficient (D*), perfusion fraction (f), mean diffusion rate (MD), and mean kurtosis value (MK) were measured. Multivariate logistic regression analysis and receiver operating characteristic (ROC) curve were used to analyze the diagnostic efficacy in predicting the Nottingham prognostic index (NPI), the expression of antigen Ki-67, and the molecular subtypes of breast cancer. The nomogram of the combined genotype-prediction model was established based on the multivariate logistic regression model results.ResultsD* and MK values were significantly higher in the high-grade Nottingham group (NPI ≥ 3.4) than the low-grade Nottingham group (NPI < 3.4) (p < 0.01). When D* ≥ 30.95 × 10−3 mm2/s and MK ≥ 0.69, the NPI tended to be high grade (with areas under the curve (AUCs) of 0.712 and 0.647, respectively). The combination of D* and MK demonstrated the highest AUC of 0.734 in grading NPI with sensitivity and accuracy of 71.7% and 77.1%, respectively. Additionally, higher D*, f, and MK and lower ADC and D values were observed in the high Ki-67 than low Ki-67 expression groups (p < 0.05). The AUC of the combined model (D + D* + f + MK) was 0.755, being significantly higher than that of single parameters (Z = 2.770~3.244, p = 0.001~0.006) in distinguishing high from low Ki-67 expression. D* and f values in the Luminal A subtype were significantly lower than in other subtypes (p < 0.05). Luminal B showed decreased D value compared with other subtypes (p < 0.05). The HER-2-positive subtype demonstrated increased ADC values compared with the Luminal B subtype (p < 0.05). Luminal A/B showed significantly lower D, D*, MD, and MK than the non-Luminal subtypes (p < 0.05). The combined model (D + D* + MD + MK) showed an AUC of 0.830 in diagnosing the Luminal and non-Luminal subtypes, which is significantly higher than that of a single parameter (Z = 3.273~4.440, p < 0.01). f ≥ 54.30% [odds ratio (OR) = 1.038, p < 0.001] and MK ≥ 0.68 (OR = 24.745, p = 0.012) were found to be significant predictors of triple-negative subtypes. The combination of f and MK values demonstrated superior diagnostic performance with AUC, sensitivity, specificity, and accuracy of 0.756, 67.5%, 77.5%, and 82.4%, respectively. Moreover, as shown in the calibration curve, strong agreements were observed between nomogram prediction probability and actual findings in the prediction of genotypes (p = 0.22, 0.74).ConclusionDWI, IVIM, and DKI, as MR diffusion imaging techniques with different mathematical models showed potential to identify the prognosis and genotype of breast cancer. In addition, the combination of these three models can improve the diagnostic efficiency and thus may contribute to opting for an appropriate therapeutic approach in clinic treatment.

Diamond distances in Nottingham algebras

Nottingham algebras are a class of just-infinite-dimensional, modular, [Formula: see text]-graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. Homogeneous components of a Nottingham algebra have dimension one or two, and in the latter case they are called diamonds. The first diamond occurs in degree [Formula: see text], and the second occurs in degree [Formula: see text], a power of the characteristic. Many examples of Nottingham algebras are known, in which each diamond past the first can be assigned a type, either belonging to the underlying field or equal to [Formula: see text]. A prospective classification of Nottingham algebras requires describing all possible diamond patterns. In this paper, we establish some crucial contributions towards that goal. One is showing that all diamonds, past the first, of an arbitrary Nottingham algebra [Formula: see text] can be assigned a type, in such a way that the degrees and types of the diamonds completely describe [Formula: see text]. At the same time we prove that the difference in degrees of any two consecutive diamonds in any Nottingham algebra equals [Formula: see text]. As a side-product of our investigation, we classify the Nottingham algebras where all diamonds have type [Formula: see text].

Translation in Ukrainian and adaptation of the English questionnaires: scales of meaningful auditory integration (MAIS) and scales of meaningful language use (MUSS) scales for assessing language development in children with deafness after cochlear implantation

Introduction: The world community is concerned with the assessment of the speech and hearing development of children with sensorineural deafness after cochlear implantation, since the most important result of rehabilitation is the success of social integration of such children. In most countries of the world, the Meaningful Auditory Integration Scale (MAIS) and the Meaningful Use of Language Scale (MUSS) are used to assess the auditory and language integration of children with sensorineural hearing loss after cochlear implantation (CI). Unfortunately, in Ukraine there are still no Ukrainian questionnaires for assessing the auditory development of children with sensorineural hearing loss after cochlear implantation. The aim of our study was to translate MAIS and MUSS manuals into Ukrainian, to adapt and validate them culturally. Materials and methods: Materials for translation and cultural adaptation were English-language versions of the Meaningful Auditory Integration Scale (MAIS) and Meaningful Use of Language Scale (MUSS), developed by the Nottingham Group of Cochlear Implantation, which are recognized and recommended for use by all cochlear implant manufacturers. Results obtained: The evaluation consisted of the following stages: preparatory stage – collection and analysis of literature; translation of original texts from English into Ukrainian; reversed translation by an independent translator of Ukrainian text into English; collation of the translation and comparison with the original questionnaire by an independent linguist; Cognitive analysis; field-testing of textbooks translated into Ukrainian with a small group of children after the CI; corrections, final revision of translations to identify and correct any grammatical, orthographic and other errors. The Ukrainian version of the questionnaire demonstrated a high level of internal compliance, testability and validity. Conclusions: Given the considerable relevance and demand ability of the materials, the tests can be recognized as valid and recommended for use in cochlear implant rehabilitation process.

The impact of tumour biology on the management of primary breast cancer in older women—based on a research programme in Nottingham

: The incidence of breast cancer increases with age. Average life expectancies are increasing; the older population is expanding globally. This presents a huge challenge on an international scale in the coming years as more older people are living with breast cancer. Despite this, most research in this field remains focused in younger patients. In this article, we outline the current issues facing understanding of the biology of primary breast cancer in older women with regards to treatment decision making. The main treatment dilemmas concern (I) primary treatment [surgery versus non-operative therapies in estrogen receptor (ER) positive and negative tumours] and (II) adjuvant treatment (such as endocrine therapy or chemotherapy). We then discuss work in this field from the Nottingham Breast Cancer Research Centre, which includes biological assessment of a large (N=1,758) cohort of older (aged ≥70 years) women with primary breast cancer with long-term follow-up data. At a biological level, we understand breast cancer belongs to four main subtypes [luminal A, luminal B, human epidermal growth factor receptor 2 (HER2) over-expression, or triple negative breast cancer (TNBC)], with treatment plans based upon these. The Nottingham group have found a biological cluster unique to older women with primary breast cancer (low ER luminal type), which is not seen in their younger (<70 years) counterparts, as well as differences between age and clinical outcome in patients with ER-positive, HER2-positive and TNBC. This adds further evidence that the biology of breast cancer in older women is different to that of younger women and therefore, should be treated as such.

Verbal width in the Nottingham group and related Lie algebras

In [1], B. Klopsch proved that the Nottingham group over a finite field is verbally elliptic. We prove a similar result for fields of zero characteristic. We also prove that the Virasoro Lie algebra and some its subalgebras are polynomially elliptic.

Estimate of the superradiance spectrum in dispersive media.

In 2016, the Nottingham group detected the rotational superradiance effect. While this experiment demonstrated the robustness of the superradiance process, it still lacks a complete theoretical description due to the many effects at stage in the experiment. In this paper, we shine new light on this experiment by deriving an estimate of the reflection coefficient in the dispersive regime by means of a Wentzel-Kramers-Brillouin analysis. This estimate is used to evaluate the reflection coefficient spectrum of counter-rotating modes in the Nottingham experiment. Our finding suggests that the vortex flow in the superradiance experiment was not purely absorbing, contrary to the event horizon of a rotating black hole. While this result increases the gap between this experimental vortex flow and a rotating black hole, it is argued that it is in fact this gap that is the source of novel ideas. This article is part of a discussion meeting issue 'The next generation of analogue gravity experiments'.

A cohomological treatise of HKG-covers with applications to the Nottingham group

We characterize Harbater-Katz-Gabber curves in terms of a family of cohomology classes satisfying a compatibility condition. Our construction is applied to the description of finite subgroups of the Nottingham Group.

Torsion elements of order 𝑝2 in the Nottingham group

Abstract Let κ be a characteristic p finite field of q elements and 𝔑 κ {\mathfrak{N}_{\kappa}} the Nottingham group over κ. Lubin associated to every conjugacy class of torsion element of 𝔑 κ {\mathfrak{N}_{\kappa}} a type. We establish an upper bound B ⁢ ( q ; l , m ) {B(q;l,m)} on the number of conjugacy classes of order p 2 {p^{2}} torsion elements u of 𝔑 κ {\mathfrak{N}_{\kappa}} of type 〈 l , m 〉 {\langle l,m\rangle} . In the case where l &lt; p {l&lt;p} , the bound B ⁢ ( q ; l , m ) {B(q;l,m)} is the exact number of conjugacy classes. Moreover, we give a criterion on when u and u n {u^{n}} are conjugate.

Maximal deviation of large powers in the Nottingham group

Abstract The group of formal power series under substitution over 𝔽 p {\mathbb{F}_{p}} , the so-called Nottingham group, is a pro-p group so it is a metric space. The intertwining of these two structures is an important object of study. This paper is concerned with how the power map influences the distance of elements. Suppose that d ⁢ ( f , g ) = d 1 {d(f,\,g)=d_{1}} while d ⁢ ( f , 1 ) = d 2 ≥ d 1 {d(f,1)=d_{2}\geq d_{1}} . In this paper we provide a sharp bound for d ⁢ ( f j , g j ) {d(f^{j},\,g^{j})} in terms of d 1 , d 2 {d_{1},\,d_{2}} and the exponent j. This bound confirms a conjecture by K. Keating.

New uniform diameter bounds in pro-$p$ groups

We give newupper bounds for the diameters of finite groups which do not depend on a choice of generating set. Our method exploits the commutator structure of certain profinite groups, in a fashion analogous to the Solovay–Kitaev procedure from quantum computation. We obtain polylogarithmic upper bounds for the diameters of finite quotients of groups with an analytic structure over a pro-$p$ domain (with exponent depending on the dimension); Chevalley groups over a pro-$p$ domain (with exponent independent of the dimension) and the Nottingham group of a finite field. We also discuss some consequences of our results for random walks on groups.

On the Heights of elements in the Nottingham Group

ABSTRACTIn this note, we define the height of an element in the Nottingham group and use heights to determine which abelian subgroups of the Nottingham group correspond to -extensions of characteristic 0 local fields by the field of norms functor. As an application, we use the result to estimate the size of the normalizer of these abelian subgroups.

The pro-p group of upper unitriangular matrices

We study the pro-p group G whose finite quotients give the prototypical Sylow p-subgroup of the general linear groups over a finite field of prime characteristic p. In this article, we extend the known results on the subgroup structure of G. In particular, we give an explicit embedding of the Nottingham group as a subgroup and show that it is selfnormalising. Holubowski [13–15] studies a free product Cp⁎Cp as a (discrete) subgroup of G and we prove that its closure is selfnormalising of infinite index in the subgroup of 2-periodic elements of G. We also discuss change of rings: field extensions and a variant for the p-adic integers, this latter linking G with some well known p-adic analytic groups. Finally, we calculate the Hausdorff dimensions of some closed subgroups of G and show that the Hausdorff spectrum of G is the whole interval [0,1] which is obtained by considering partition subgroups only.

Beauville structures in finite p-groups

We study the existence of (unmixed) Beauville structures in finite p-groups, where p is a prime. First of all, we extend Catanese's characterisation of abelian Beauville groups to finite p-groups satisfying certain conditions which are much weaker than commutativity. This result applies to all known families of p-groups with a good behaviour with respect to powers: regular p-groups, powerful p-groups and more generally potent p-groups, and (generalised) p-central p-groups. In particular, our characterisation holds for all p-groups of order at most pp, which allows us to determine the exact number of Beauville groups of order p5, for p≥5, and of order p6, for p≥7. On the other hand, we determine which quotients of the Nottingham group over Fp are Beauville groups, for an odd prime p. As a consequence, we give the first explicit infinite family of Beauville 3-groups, and we show that there are Beauville 3-groups of order 3n for every n≥5.

Powers of elements of the series substitution group [formula omitted]

For the Jennings group J(k) of substitutions of formal power series with coefficients in a field k of positive characteristic (the Nottingham group), the depth of the powers of its elements is studied. In particular, it is shown that the case of a field with characteristic 2 is completely different from the case of a field with odd prime characteristic. It is also shown that the case of the field k=Z2 differs from the case of other fields with characteristic 2. Explicit embeddings of the groups Zpm and Zp⊕Zp in J(Zp) are constructed.

Artin-Hasse exponential mapping, algebraic groups in positive characteristic, and the Nottingham group

Connected commutative subgroups of the prounipotent radical of the group of formal power series in one variable over an algebraically closed field of positive characteristic with respect to the operation of substitution are studied.

The elephant in the room

Models for out-of-hours coverage of intensive care units (ICUs) post-cardiac surgery vary widely across Europe and the world. In the UK and in the major units in the USA, where there are high volume units, there tend to be a high number of trainees or middle grades who are able to provide 24-h resident coverage to the ICU and the wards. Classically in the UK, 9–10 trainees would be required to provide such cover and also attend theatres and outpatients. The model is very different in smaller units. As there are not enough trainee surgeons to provide 24-h coverage, an alternative method must be used to cover the patients on the night of their surgery or at weekends. This model may use general intensivists, placing the patients on general ICUs after their surgery, or they may indeed have non-resident senior surgeons providing the support to non-doctors in the unit. The Nottingham group have led the way in the UK in demonstrating how you can safely set up your service without the requirement of a large number of resident surgeons. This paper is an outstanding example of how careful implementation of this novel system has demonstrated perfect patient safety, and also how it has shown additional benefits that may not have been predicted, including cost reductions and improved surgical training. So, how does this affect you in your unit? First, if you are in a large unit with multiple trainees providing 24-h cover, consider this. If you provided the exemplary training to a group of nurse practitioners as was done here and placed them on call instead releasing your trainees from night time commitment, then if you reduced middle grade surgeon numbers you would save a significant amount of money ( just under £200 000 in this study). But the remaining trainees (which would presumably be the highest quality ones) would then have a much greater exposure to your theatres and thus much better training. This would include more theatre time but also being able to select the most appropriate theatres for their level of ability. This ability to provide a better level of training to fewer trainees would further attract better trainees to your unit, indeed this would be training Nirvana, with the ability to choose your theatre, with minimal competition from other trainees! The result for your unit is financial savings, and better quality trainees for no risk in the provision of care to your patients. So, what about smaller units? Here is the Elephant in the room in cardiac surgery. I must declare an interest in this subject in that I provide training to all levels of clinicians in protocols to manage the patient who arrest after cardiac surgery, including emergency resternotomy (www.csu-als.com). We have provided training to many senior nurses from small units, including many from small units in the USA, where having your cardiac surgery in a small local unit is very popular. I have frequently talked to nurses on these units that are covered by non-resident surgeons about their experiences of cardiac arrests. Many recount massaging a patient with tamponade for 30–40 min, while the surgeon comes in from home, and there is a universally fatal outcome from this. More alarmingly, I talked to three nurses who have been talked through emergency resternotomy on the phone as the surgeon drives in, again universally with a fatal outcome. Recurrent figures from units across the world put the incidence of arrest after cardiac surgery at around 1%, so it is not an infrequent event in terms of the annual experience of a unit. However, I am still surprised that lawyers have not yet picked up on the fact that a delayed resternotomy in a patient who arrests after cardiac surgery might be an avoidable contributing cause of their client’s mortality. Thus for smaller units without resident surgical cover, the Nottingham group has demonstrated the perfect model to ensure complete patient safety including from the not infrequent emergencies that we experience in cardiac surgery. There are other models of outstanding good practice in the provision of non-medical cover in cardiac surgery across the world, including at centres in the UK and in the USA, such as the nurse practitioner programme at Duke University Hospital where excellent training and regular moulage practice for emergencies are a cornerstone of their practice. Thus, I encourage all large units to consider following the Nottingham model in order to attain cost savings and to improve your resident’s training, and all small units to adopt their model in order to improve your patient’s safety and to keep the lawyers away from your door!

Normal Zeta Functions of Some pro-p Groups

We give an explicit formula for the number of normal subgroups of index p n of the pro-p group , and for the normal zeta function associated with this group. Let 𝒬1(s, r) be the subgroups of the Nottingham group discovered by Ershov. We observe that these groups are isospectral with . This provides us with an infinite family of noncommensurable normally isospectral groups.

Torsion in the Nottingham group

From local class-field theory and higher ramification theory one gets a classification up to conjugacy of the torsion elements of arbitrary order in the Nottingham group over a finite field, in terms of continuous characters on the multiplicative group of principal units in the ring of formal power series over the field. An essential part of this classification is to partition the torsion elements according to the depths of their successive p-power iterates. The classification described here is good enough to permit an independent proof of Klopsch's theorem on torsion elements of order p, but not good enough to give a full description of the finite set of classes of torsion elements with a given depth-sequence. The final section exhibits an efficient method of calculating the first few hundred terms of a series of order pn, limited only by the capabilities of the computation package used, but gives no idea of any formula for describing the coefficients.

On the commensurator of the Nottingham group

Let p > 5 be a prime number. We prove that the abstract commensurator of the Nottingham group N(F p ) coincides with its automorphism group, which is known to be a finite extension of N(F p ). As a corollary we deduce that the Nottingham group cannot be embedded as an open subgroup of a topologically simple group.