Inertia Groups Research Articles

A characterization of ramification groups via jet algebras

AbstractWe present a functorial method to define ramification groups, identifying them as inertia groups of an induced action on composite jet algebras. This framework lays the foundation for defining higher ramification groups for actions involving group schemes. To achieve this, we introduce Taylor maps within the category of commutative unitary rings at prime ideals of an $$R$$ R -algebra and compute their kernels for algebras of finite type over a field with separably generated residue fields.

On the hereditarity and maximality of weak crossed product orders over discrete valuation rings

Consider a weak crossed product order [Formula: see text] in the crossed product algebra [Formula: see text], where [Formula: see text] is the integral closure of a discrete valuation ring [Formula: see text] in a tamely ramified Galois extension [Formula: see text] of the field of fractions of [Formula: see text]. Assume that [Formula: see text] is local. In this paper, we develop a method that can help to determine whether [Formula: see text] is a maximal order by looking for certain ideals in [Formula: see text] where [Formula: see text] is the inertia group. We demonstrate this method by classifying specific examples of hereditary weak crossed product orders as being either maximal or nonmaximal among the [Formula: see text]-orders of [Formula: see text] and we give a criterion for determining maximality of [Formula: see text] when [Formula: see text] is cyclic.

Skeletal filtrations of the fundamental group of a non-archimedean curve

In this paper we study skeleta of residually tame coverings of a marked curve over a non-archimedean field. We first prove a simultaneous semistable reduction theorem for residually tame coverings, which we then use to construct a tropicalization functor from the category of residually tame coverings of a marked curve (X,D) to the category of tame coverings of a metrized complex Σ associated to (X,D). We enhance the latter category by adding a set of gluing data to every covering and we show that this yields an equivalence of categories. We use this skeletal interpretation to define the absolute decomposition and inertia group of a curve, which can be seen as the first subgroups in a ramification filtration of the fundamental group of the curve. We prove that the cyclic coverings that arise from the corresponding decomposition and inertia quotients coincide with the coverings that arise from the toric and connected parts of the analytic Jacobian of the curve.

The Relationship between Ferritin Levels and Uterine Inertia in Labor Women

Objective: To determine the relationship between ferritin levels and the incidence of uterine inertia in pregnant women. Method: This study uses a prospective cohort design. The study was conducted on all pregnant women aged 20-40 years who had antenatal care and be in labor at the Teaching Hospital in the Department of Obstetrics and Gynecology, Faculty of Medicine, Hasanuddin University. Data were analyzed with the chi-square test and the Mann-Whitney test. Results: Obtained 76 pregnant women where 44.7% of them had uterine inertia. The average ferritin level in the uterine inertia group was significantly lower (3.80 ± 2.84 ng/mL) than without the uterine inertia group (12.99 ± 11.40 ng/mL) with a value p<0.001. The average ferritin level in the mild anemia group was significantly higher (9.83 ± 10.35 ng/mL) than in the moderate anemia group (3.82 ± 2.51 ng/mL) with a p-value <0.05. Pregnant women with moderate anemia were found to be 4.68 more susceptible uterine inertia than pregnant women with mild anemia. Conclusion: There was a relationship between ferritin levels in pregnant women and the incidence of uterine inertia where low ferritin levels was risk factor of uterine inertia.

On the existence of special nonlinear invariants for round functions of XSL-ciphers

Abstract Nonlinear invariants of round transformations in XSL-schemes are studied. The emphasis is on invariants which may be found by means of the approach suggested at the conference ASIACRYPT 2016. Some known results on the inertia groups of decomposable functions are used to describe conditions on S-boxes and matrices of XSL-schemes which are necessary for the existence of such invariants. It is shown that for a number of schemes these conditions are not satisfied.

Secondary prevention and follow-up of patients with ACS and not-at-target LDL: An Italian real-world retro-prospective analysis by the INERTIA group

BackgroundIn patients with recent ACS, the latest ESC/EAS guidelines for management of dyslipidaemia recommend intensification of LDL-C-lowering therapy. ObjectiveReport a real-world picture of lipid-lowering therapy prescribed and cholesterol targets achieved in post-ACS patients before and after a specific educational program. MethodsRetrospective data collection prior to the educational course and prospective data collection after the course of consecutive very high-risk patients with ACS admitted in 2020 in 13 Italian cardiology departments, and with a non-target LDL-C level at discharge. ResultsData from 336 patients were included, 229 in the retrospective phase and 107 in the post-course prospective phase. At discharge, statins were prescribed in 98.1% of patients, alone in 62.3% of patients (65% of which at high doses) and in combination with ezetimibe in 35.8% of cases (52% at high doses). A significant reduction was obtained in total and LDL cholesterol (LDL-C) from discharge to the first control visit. Thirty-five percent of patients achieved a target LDL-C <55 mg/dL according to ESC 2019 guidelines. Fifty percent of patients achieved the <55 mg/dL target for LDL-C after a mean of 120 days from the ACS event. ConclusionsOur analysis, though numerically and methodologically limited, suggests that management of cholesterolaemia and achievement of LDL-C targets are largely suboptimal and need significant improvement to comply with the lipid-lowering guidelines for very high CV risk patients. Earlier high intensity statin combination therapy should be encouraged in patients with high residual risk.

Towards the generalized purely wild inertia conjecture for product of alternating and symmetric groups

We obtain new evidence for the Purely Wild Inertia Conjecture posed by Abhyankar and for its generalization. We show that this generalized conjecture is true for any product of simple Alternating groups in odd characteristics, and for any product of certain Symmetric or Alternating groups in characteristic two. We also obtain important results towards the realization of the inertia groups which can be applied to more general set up. We further show that the Purely Wild Inertia Conjecture is true for any product of perfect quasi p-groups (groups generated by their Sylow p-subgroups) if the conjecture is established for individual groups.

On ternary Diophantine equations of signature over number fields

AbstractIn this paper, we prove results about solutions of the Diophantine equation $x^p+y^p=z^3$ over various number fields using the modular method. First, by assuming some standard modularity conjecture, we prove an asymptotic result for general number fields of narrow class number one satisfying some technical conditions. Second, we show that there is an explicit bound such that the equation $x^p+y^p=z^3$ does not have a particular type of solution over $K=\mathbb {Q}(\sqrt {-d})$ , where $d=1,7,19,43,67$ whenever p is bigger than this bound. During the course of the proof, we prove various results about the irreducibility of Galois representations, image of inertia groups, and Bianchi newforms.

Anchoring and Sleep Inertia.

Many occupational settings require individuals to make important decisions immediately after awakening. Although a plethora of psychological research has separately examined both sleep and anchoring effects on decision-making, little is known about their interaction. In the present study, we seek to shed light on the link between sleep inertia, the performance impairment immediately after awakening, and individuals' susceptibility to the anchoring bias. We proposed that sleep inertia would moderate participants' adjustment from anchors because sleep inertia leads to less cognitive effort invested, resulting in a stronger anchoring effect. One hundred four subjects were randomly assigned to an experimental group that answered anchoring tasks immediately after being awakened at nighttime or a control group that answered anchoring tasks at daytime. Our findings replicated the well-established anchoring effect in that higher anchors led participants to higher estimates than lower anchors. We did not find significant effects of sleep inertia. While the sleep inertia group reported greater sleepiness and having invested less cognitive effort compared to the control group, no systematic anchoring differences emerged, and cognitive effort did not qualify as a mediator of the anchoring effect. Bayesian analyses provide empirical evidence for these null findings. Implications for the anchoring literature and future research are discussed.

Trace and Künneth formulas for singularity categories and applications

We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base monoidal differential graded (dg)-category. Moreover, we prove Künneth formulas for dg-category of singularities and for inertia-invariant vanishing cycles. As an application, we prove a categorical version of Bloch's conductor conjecture (originally stated by Spencer Bloch in 1985), under the additional hypothesis that the monodromy action of the inertia group is unipotent.

Inertia groups and smooth structures on quaternionic projective spaces

Abstract This paper deals with certain results on the number of smooth structures on quaternionic projective spaces, obtained through the computation of inertia groups and their analogues, which in turn are computed using techniques from stable homotopy theory. We show that the concordance inertia group is trivial in dimension 20, but there are many examples in high dimensions where the concordance inertia group is non-trivial. We extend these to computations of concordance classes of smooth structures. These have applications to 3-sphere actions on homotopy spheres and tangential homotopy structures.

Technology for predicting abnormal labor

BACKGROUND: Abnormal labor is the most common complication of labor. It occurs in 10-15% cases of labor and is an indication for operative delivery in a third of all cases. Until now, there are no effective ways to predict abnormal labor. Meanwhile, the use of high-tech forecasting methods is not available for a wide range of obstetric institutions in the Russian Federation.&#x0D; AIM: The aim of this study was to create a technology for predicting abnormal labor, based on generally available methods of laboratory and instrumental research.&#x0D; MATERIALS AND METHODS: Based on the data collected in the Regional Clinical Hospital Perinatal Center, Chita, Russia in 2018-2021, the retrospective analysis of 200 cases of labor was carried out. The total sample was divided into four study groups: 100 women with normal labor activity (group 1), 30 women with uterine inertia (group 2), 30 women with incoordinate uterine activity (group 3), and 50 women with excessive uterine activity (group 4). The groups were comparable in terms of age, anthropometric parameters and extragenital pathology. All women on the eve of labor (1-2 days) underwent general clinical and ultrasound examination. Statistical processing of the results was carried out using the IBM SPSS Statistics version 25.0 software.&#x0D; RESULTS: The technology for predicting abnormal labor is implemented based on a multilayer perceptron, with the percentage of incorrect predictions being 21.3%. The structure of the trained neural network included nine input neurons: labor parity, gestational age, leukocyte count, erythrocyte sedimentation rate, total protein concentration, amniotic fluid index, biparietal size, as well as fetal head and abdomen circumference.&#x0D; CONCLUSIONS: An integrated approach based on generally available laboratory and instrumental research methods, such as complete blood count and biochemical blood test, as well as ultrasound examination, on the eve of labor allows for predicting the abnormal labor development with an accuracy of up to 70%. The use of this technology in clinical practice will help, in the future, not only to prevent abnormal labor, but also to reduce the incidence of adverse obstetric and perinatal outcomes.

Aerobic Exercise-Assisted Cardiac Regeneration by Inhibiting Tryptase Release in Mast Cells after Myocardial Infarction.

Background Cardiovascular disease (CVD) contributes critically to the mortality, morbidity, and economic problem of illness globally. Exercise is a share of everyone's life. Some evidence-based studies have frequently shown a progressive correlation between physical activity and good health. Objective The effects of daily exercise on cardiomyocyte size, collagen content (fibrosis), and releasing mast cells (MCsʼ) tryptase of the model of myocardial infarction (MI) were assessed. Methods 40 rats were coincidentally spread into sham+inertia (control), sham+exercise, infarction+inertia, and infarction+exercise groups. An experimental model of acute MI was induced in infarction groups. One week after surgery, exercising groups were allowed to an aerobic exercise program for six weeks. At the endpoint of the study, all examinations were performed. Results We found lesser fibrosis in sham+exercise and infarction+exercise groups compared to sham+inertia and infarction+inertia groups, respectively (p = 0.023, p = 0.001). Also, infarction groups were significantly lower than sham groups (p < 0.05) and the infarction+exercise group was significantly lower than the infarction+inertia group (p < 0.05). The effect of exercise on MCs while increased MC density and degranulation occur at the site of fibrosis, we demonstrated that exercise decreases both total MC density and degranulation in both sham and infarction groups (p < 0.05). Immunohistochemistry examinations were significantly higher expression of MCsʼ tryptase in infarction groups than sham groups (p < 0.05, p < 0.0001). Conclusion Exercise improves fibrosis and cardiac function in both healthy and MI rats by inhibiting released MCsʼ tryptase.

Inter-universal Teichmüller Theory I: Construction of Hodge Theaters

The present paper is the first in a series of four papers, the goal of which is to establish an arithmetic version of Teichmüller theory for number fields equipped with an elliptic curve – which we refer to as "inter-universal Teichmüller theory" – by applying the theory of semi-graphs of anabelioids , Frobenioids , the étale theta function , and log-shells developed in earlier papers by the author. We begin by fixing what we call "initial \Theta -data" , which consists of an elliptic curve E_F over a number field F , and a prime number l\ge 5 , as well as some other technical data satisfying certain technical properties. This data determines various hyperbolic orbicurves that are related via finite étale coverings to the once-punctured elliptic curve X_F determined by E_F . These finite étale coverings admit various symmetry properties arising from the additive and multiplicative structures on the ring \mathbb{F}_l=\mathbb{Z}/l\mathbb{Z} acting on the l -torsion points of the elliptic curve. We then construct " \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theaters" associated to the given \Theta -data. These \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theaters may be thought of as miniature models of conventional scheme theory in which the two underlying combinatorial dimensions of a number field – which may be thought of as corresponding to the additive and multiplicative structures of a ring or, alternatively, to the group of units and value group of a local field associated to the number field – are, in some sense, "dismantled" or "disentangled" from one another. All \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theaters are isomorphic to one another, but may also be related to one another by means of a " \Theta -link" , which relates certain Frobenioid-theoretic portions of one \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theater to another in a fashion that is not compatible with the respective conventional ring/scheme theory structures . In particular, it is a highly nontrivial problem to relate the ring structures on either side of the \Theta -link to one another. This will be achieved, up to certain "relatively mild indeterminacies" , in future papers in the series by applying the absolute anabelian geometry developed in earlier papers by the author. The resulting description of an "alien ring structure" [associated, say, to the domain of the \Theta -link] in terms of a given ring structure [associated, say, to the codomain of the \Theta -link] will be applied in the final paper of the series to obtain results in diophantine geometry . Finally, we discuss certain technical results concerning profinite conjugates of decomposition and inertia groups in the tempered fundamental group of a p -adic hyperbolic curve that will be of use in the development of the theory of the present series of papers, but are also of independent interest.

Inter-universal Teichmüller Theory II: Hodge–Arakelov-Theoretic Evaluation

In the present paper, which is the second in a series of four papers, we study the Kummer theory surrounding the Hodge–Arakelov-theoretic evaluation – i.e., evaluation in the style of the scheme-theoretic Hodge–Arakelov theory established by the author in previous papers – of the [reciprocal of the l -th root of the] theta function at l -torsion points [strictly speaking, shifted by a suitable 2-torsion point], for l\ge 5 a prime number. In the first paper of the series, we studied "miniature models of conventional scheme theory" , which we referred to as \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theaters , that were associated to certain data, called initial \Theta -data , that includes an elliptic curve E_F over a number field F , together with a prime number l\ge 5 . The underlying \Theta -Hodge theaters of these \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theaters were glued to one another by means of " \Theta -links" , that identify the [reciprocal of the l -th root of the] theta function at primes of bad reduction of E_F in one \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theater with [ 2l -th roots of] the q -parameter at primes of bad reduction of E_F in another \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theater. The theory developed in the present paper allows one to construct certain new versions of this " \Theta -link". One such new version is the \Theta^{\times\mu}_{\mathrm{gau}} -link , which is similar to the \Theta -link, but involves the theta values at l -torsion points , rather than the theta function itself. One important aspect of the constructions that underlie the \Theta^{\times\mu}_{\mathrm{gau}} -link is the study of multiradiality properties, i.e., properties of the "arithmetic holomorphic structure" – or, more concretely, the ring/scheme structure – arising from one \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theater that may be formulated in such a way as to make sense from the point of view of the arithmetic holomorphic structure of another \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theater which is related to the original \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theater by means of the [ non-scheme-theoretic !] \Theta^{\times\mu}_{\mathrm{gau}} -link. For instance, certain of the various rigidity properties of the étale theta function studied in an earlier paper by the author may be interpreted as multiradiality properties in the context of the theory of the present series of papers. Another important aspect of the constructions that underlie the \Theta^{\times\mu}_{\mathrm{gau}} -link is the study of "conjugate synchronization" via the \mathbb{F}^{\rtimes\pm}_l -symmetry of a \Theta^{\pm\mathrm{ell}}\mathrm{NF} -Hodge theater. Conjugate synchronization refers to a certain system of isomorphisms – which are free of any conjugacy indeterminacies ! – between copies of local absolute Galois groups at the various l -torsion points at which the theta function is evaluated. Conjugate synchronization plays an important role in the Kummer theory surrounding the evaluation of the theta function at l -torsion points and is applied in the study of coricity properties of [i.e., the study of objects left invariant by] the \Theta^{\times\mu}_{\mathrm{gau}} -link. Global aspects of conjugate synchronization require the resolution, via results obtained in the first paper of the series, of certain technicalities involving profinite conjugates of tempered cuspidal inertia groups.

Implementing an Action Over Inertia Group Program in Community Residential Rehabilitation Services: Group Participant and Facilitator Perspectives.

Introduction: A time-use focused intervention, Action Over Inertia (AOI) designed to address restricted activity patterns and support recovery, was adapted for use in Australian community residential mental health services.Method: Qualitative case study research explored the use of AOI groups across three Community Care Units from the perspectives of group participants with enduring mental illness and group facilitators. Fifteen interviews were conducted: five group participants were interviewed twice 4 weeks apart, and five group facilitators on completion of the group intervention. Interview data were analyzed thematically using constant comparative methods.Findings: Two overarching themes, “Making Change” and “Facilitating Change” were identified. Efforts to make change in their lives were supported by participants recognising the value of personally meaningful activities for well-being and of activity experiences that fostered hope and recovery, whereas a sense of “stuckness,” time for activities and life events could disrupt “getting me going.” For the facilitators, facilitating change involved recognizing inertia as a challenge; getting people going; and looking at how AOI intervention works to impact inertia.Conclusion: AOI in a group format supports participants to identify barriers to more active living; to appreciate how time-use and well-being interrelate; and to reframe and take steps to overcome inertia. Further research should evaluate AOI groups as a means of providing individualized support for activity re-engagement as part of recovery oriented mental health rehabilitation.

On the Inertia Conjecture for Alternating group covers

Let G1 and G2 be perfect groups such that there exist connected G1-Galois and G2-Galois étale covers of the affine line over an algebraically closed field of characteristic p>0 with the cyclic p-groups P1 and P2 as the inertia groups above ∞, respectively. Then we show that there is a connected G1×G2-Galois étale cover of the affine line with an inertia group I above ∞ where I is a cyclic subgroup of P1×P2 of index p. As a consequence, it is shown that the wild part of the Inertia Conjecture is true for any product of Alternating groups, each of degree p or coprime to p. For d a multiple of p, a new étale Ad-cover of the affine line is obtained using an explicit equation, and it is shown that this cover has the minimal possible upper jump.

Explicit resolution of weak wild quotient singularities on arithmetic surfaces

A weak wild arithmetic quotient singularity arises from the quotient of a smooth arithmetic surface by a finite group action, where the inertia group of a point on a closed characteristicppfiber is app-group acting with smallest possible ramification jump. In this paper, we give complete explicit resolutions of these singularities using deformation theory and valuation theory, taking a more local perspective than previous work has taken. Our descriptions answer several questions of Lorenzini. Along the way, we give a valuation-theoretic criterion for a normal snc-model ofP1\mathbb {P}^1over a discretely valued field to be regular.

Unramified extensions over low degree number fields

For various nonsolvable groups G, we prove the existence of extensions of the rationals Q with Galois group G and inertia groups of order dividing ge(G), where ge(G) is the smallest exponent of a generating set for G. For these groups G, this gives the existence of number fields of degree ge(G) with an unramified G-extension. The existence of such extensions over Q for all finite groups would imply that, for every finite group G, there exists a quadratic number field admitting an unramified G-extension, as was recently conjectured. We also provide further evidence for the existence of such extensions for all finite groups, by proving their existence when Q is replaced with a function field k(t) where k is an ample field.

Constructing hyperelliptic curves with surjective Galois representations

In this paper we show how to explicitly write down equations of hyperelliptic curves over Q \mathbb {Q} such that for all odd primes ℓ \ell the image of the mod ℓ \ell Galois representation is the general symplectic group. The proof relies on understanding the action of inertia groups on the ℓ \ell -torsion of the Jacobian, including at primes where the Jacobian has non-semistable reduction. We also give a framework for systematically dealing with primitivity of symplectic mod ℓ \ell Galois representations. The main result of the paper is the following. Suppose n = 2 g + 2 n=2g+2 is an even integer that can be written as a sum of two primes in two different ways, with none of the primes being the largest primes less than n n (this hypothesis is expected to hold for all g ≠ 0 , 1 , 2 , 3 , 4 , 5 , 7 , g\neq 0,1,2,3,4,5,7, and 13 13 ). Then there is an explicit N ∈ Z N\in \mathbb {Z} and an explicit monic polynomial f 0 ( x ) ∈ Z [ x ] f_0(x)\in \mathbb {Z}[x] of degree n n , such that the Jacobian J J of every curve of the form y 2 = f ( x ) y^2=f(x) has Gal ⁡ ( Q ( J [ ℓ ] ) / Q ) ≅ GSp 2 g ⁡ ( F ℓ ) \operatorname {Gal}(\mathbb {Q}(J[\ell ])/\mathbb {Q})\cong \operatorname {GSp}_{2g}(\mathbb {F}_\ell ) for all odd primes ℓ \ell and Gal ⁡ ( Q ( J [ 2 ] ) / Q ) ≅ S 2 g + 2 \operatorname {Gal}(\mathbb {Q}(J[2])/\mathbb {Q})\cong S_{2g+2} , whenever f ( x ) ∈ Z [ x ] f(x)\in \mathbb {Z}[x] is monic with f ( x ) ≡ f 0 ( x ) mod N f(x)\equiv f_0(x) \bmod {N} and with no roots of multiplicity greater than 2 2 in F ¯ p \overline {\mathbb {F}}_p for any p ∤ N p\nmid N .