Pure Extensions Research Articles

On the ideal discriminant of some relative pure extensions

Let L = K(α) be an extension of a number field K where α satisfies the monic irreducible polynomial P(X) = Xp −a ∈ R[X] of prime degree p and such that a is pth power free in R := OK (the ring of integers of K). The purpose of this paper is to give an explicit formula for the ideal discriminant DL/K of L over K involving only the prime ideals dividing the principal ideals aR and pR. As an illustration, we compute the discriminant DL/K of a family of septic and quintic pure fields over quadratic fields. Hence a slightly simpler computation of discriminant DL/K is obtained.

Hybrid finite-amplitude periodic modes for two uniformly magnetized spheres.

We analyze a system of two uniformly magnetized spheres, one fixed and the other free to slide in frictionless contact with the surface of the first. The centers of the two magnets, and their magnetic moments, are restricted to a plane. We search for sets of initial conditions that yield finite-amplitude oscillatory periodic solutions. We extend two small-amplitude base modes, one with orbital and spin motions that are in phase and the other out of phase, to finite amplitudes and show that the motion for arbitrary oscillatory solutions can be considered to be a nonlinear superposition of these base modes. Some solutions are pure periodic finite-amplitude extensions of one base mode, while others are hybrid finite-amplitude superpositions of the two modes. Hybrid modes with rational frequency ratios are periodic and come in families defined by their frequency ratios. We further characterize hybrid periodic modes by identifying two symmetry classes that describe their relative phases. We see continuous transitions between one finite-amplitude base mode and the other, with one mode gradually transforming into the other. We also calculate frequency spectra of nonperiodic modes, show that the two base modes have well-defined frequencies even for nonperiodic states, and show that periodic solutions can give clues about the behavior of nearby nonperiodic solutions. In the limit of small amplitudes, we confirm that the computed frequencies of these modes agree with small-amplitude analytical results. We also generate a Lyapunov exponent heatmap that reflects periodic and nonperiodic regions of state space.

The 2-fold pure extensions need not split

In this paper, we give an example of locally compact abelian groups $A$ and $C$ such that ${\rm Pext}^{2}(C,A)\neq 0$.

On the monogenity of pure quartic relative extensions of {{mathbb {Q}}}(i)

We consider pure quartic relative extensions of the number field {{mathbb {Q}}}(i) of type K={{mathbb {Q}}}(root 4 of {a+bi}), where a,bin {{mathbb {Z}}} and bne 0, such that a+biin {{mathbb {Z}}}[i] is square-free. We describe integral bases of these fields. The index form equation is reduced to a relative cubic Thue equation over {{mathbb {Q}}}(i) and some corresponding quadratic form equations. We consider monogenity of K and relative monogenity of K over {{mathbb {Q}}}(i). We shall show how our former method based on the factors of the index form can be used in the relative case to exclude relative monogenity in some cases.

BIG COHEN–MACAULAY TEST IDEALS IN EQUAL CHARACTERISTIC ZERO VIA ULTRAPRODUCTS

AbstractUtilizing ultraproducts, Schoutens constructed a big Cohen–Macaulay (BCM) algebra $\mathcal {B}(R)$ over a local domain R essentially of finite type over $\mathbb {C}$ . We show that if R is normal and $\Delta $ is an effective $\mathbb {Q}$ -Weil divisor on $\operatorname {Spec} R$ such that $K_R+\Delta $ is $\mathbb {Q}$ -Cartier, then the BCM test ideal $\tau _{\widehat {\mathcal {B}(R)}}(\widehat {R},\widehat {\Delta })$ of $(\widehat {R},\widehat {\Delta })$ with respect to $\widehat {\mathcal {B}(R)}$ coincides with the multiplier ideal $\mathcal {J}(\widehat {R},\widehat {\Delta })$ of $(\widehat {R},\widehat {\Delta })$ , where $\widehat {R}$ and $\widehat {\mathcal {B}(R)}$ are the $\mathfrak {m}$ -adic completions of R and $\mathcal {B}(R)$ , respectively, and $\widehat {\Delta }$ is the flat pullback of $\Delta $ by the canonical morphism $\operatorname {Spec} \widehat {R}\to \operatorname {Spec} R$ . As an application, we obtain a result on the behavior of multiplier ideals under pure ring extensions.

A characterization of multiplier ideals via ultraproducts

In this paper, using ultra-Frobenii, we introduce a variant of Schoutens’ non-standard tight closure (Schoutens in Manuscr Math 111:379–412, 2003), ultra-tight closure, on ideals of a local domain R essentially of finite type over $$\mathbb {C}$$ . We prove that the ultra-test ideal $$\tau _{\textrm{u}}(R,\mathfrak {a}^t)$$ , the annihilator ideal of all ultra-tight closure relations of R, coincides with the multiplier ideal $$\mathcal {J}({\text {Spec}}R,\mathfrak {a}^t)$$ if R is normal $$\mathbb {Q}$$ -Gorenstein. As an application, we study a behavior of multiplier ideals under pure ring extensions.

Optimisation of satellite geometries in Very Low Earth Orbits for drag minimisation and lifetime extension

The utilisation of the Very Low Earth Orbit (VLEO) region offers significant application specific, technological, operational, and cost benefits. However, attaining sustained and economically viable VLEO flight is challenging, primarily due to the significant, barely predictable and dynamically changing drag caused by the residual atmosphere, which leads to a rapid deterioration of any spacecraft’s orbit unless mitigated by a combination of active and passive techniques. This article addresses one passive method by optimising satellite shapes in order to achieve a minimisation of the atmospheric drag force and thus extension of operational lifetime. Contrary to previous investigations in the field, a constant internal volume is maintained to account for the placement of satellite instruments and payload inside the structure. Moreover, the satellite geometry is not varied heuristically but optimised, via a numerical 2D profile optimisation specifically developed for this purpose. From the resulting optimal satellite profiles, 3D satellite bodies are derived, which are then verified via the Direct Simulation Monte Carlo method within the open-source particle code PICLas. In addition, rather unconventional designs, i.e. ring geometries, which are based on the assumption of fully specular particle reflections, are proposed and assessed. The optimised satellite geometries offer pure passive lifetime extensions of up to 46% compared to a slender reference body, while the above-mentioned ring geometries achieve passive lifetime extensions of more than 3000%. Finally, the article presents design recommendations for VLEO satellites in dependence of different surface properties.

Electroweak phase transition triggered by fermion sector

To realize first-order electroweak phase transition, it is necessary to generate a barrier in the thermal Higgs potential, which is usually triggered by scalar degree of freedom. We instead investigate phase transition patterns in pure fermion extensions of the standard model, and find that additional fermions with mass hierarchy and mixing could develop such a barrier and realize a strongly first-order phase transition in such models. In the Higgs potential with polynomial parametrization, the barrier can be generated in the following two patterns by fermionic reduction effects: (I) positive quadratic term, negative cubic term and positive quartic term or (II) positive quadratic term, negative quartic term and positive higher dimensional term, such as dimensional 6 operator.

Differential characterization of quasi-concave functions without twice differentiability

This paper presents a necessary and sufficient condition for a real-valued function defined on an open and convex subset of a Banach space to be quasi-concave, and a sufficient condition for such a function to be strictly quasi-concave. These conditions are applicable to continuously differentiable functions that satisfy a mild additional assumption, and do not require the functions to be twice differentiable. Because this additional assumption is trivially satisfied for twice continuously differentiable functions, our results are pure extensions to classical results.

On left ideal essential extensions of rings

The main goal of this paper is to extend Flanigan’s theorem in [4] concerning ideal essential extensions of rings to left ideal essential extensions. Moreover, we give new proofs of two Flanigan’s theorems and answer a question raised by M. Petrich in [5] which is related to pure extensions of rings.

Integrability in Finite Terms and Actions of Lie Groups

According to Liouville's Theorem, an indefinite integral of an elementary function is usually not an elementary function. In this notes, we discuss that statement and a proof of this result. The differential Galois group of the extension obtained by adjoining an integral does not determine whether the integral is an elementary function or not. Nevertheless, Liouville's Theorem can be proved using differential Galois groups. The first step towards such a proof was suggested by Abel. This step is related to algebraic extensions and their finite Galois groups. A significant part of this notes is dedicated to a second step, which deals with pure transcendent extensions and their Galois groups which are connected Lie groups. The idea of the proof goes back to J.Liouville and J.Ritt.

Partial actions and an embedding theorem for inverse semigroups

We give a simple construction involving partial actions which permits us to obtain an easy proof of a weakened version of L. O’Carroll’s theorem on idempotent pure extensions of inverse semigroups.

Abelian C-minimal valued groups

We classify abelian C-minimal valued groups up to pure and elementary extensions and small modifications. We define first the notion of almost regularity for abelian valued groups. Then we introduce on the chain of valuations the structure which enables us to characterize C-minimal abelian valued groups as the almost regular ones with an o-minimal chain.

Weak positivity for Hodge modules

We prove the weak positivity of the kernels of Kodaira-Spencertype maps for pure Hodge module extensions of generically defined variations of Hodge structure.

Calculating Power Integral bases by Solving Relative Thue Equations

Abstract In our recent paper I. Gaál: Calculating “small” solutions of relative Thue equations, J. Experiment. Math. (to appear) we gave an efficient algorithm to calculate “small” solutions of relative Thue equations (where “small” means an upper bound of type 10500 for the sizes of solutions). Here we apply this algorithm to calculating power integral bases in sextic fields with an imaginary quadratic subfield and to calculating relative power integral bases in pure quartic extensions of imaginary quadratic fields. In both cases the crucial point of the calculation is the resolution of a relative Thue equation. We produce numerical data that were not known before.

Rational solutions of certain Diophantine equations involving norms

In this note we present some results concerning the unirationality of the algebraic variety $\cal{S}_{f}$ given by the equation \begin{equation*} N_{K/k}(X_{1}+\alpha X_{2}+\alpha^2 X_{3})=f(t), \end{equation*} where $k$ is a number field, $K=k(\alpha)$, $\alpha$ is a root of an irreducible polynomial $h(x)=x^3+ax+b\in k[x]$ and $f\in k[t]$. We are mainly interested in the case of pure cubic extensions, i.e. $a=0$ and $b\in k\setminus k^{3}$. We prove that if $\op{deg}f=4$ and the variety $\cal{S}_{f}$ contains a $k$-rational point $(x_{0},y_{0},z_{0},t_{0})$ with $f(t_{0})\neq 0$, then $\cal{S}_{f}$ is $k$-unirational. A similar result is proved for a broad family of quintic polynomials $f$ satisfying some mild conditions (for example this family contains all irreducible polynomials). Moreover, the unirationality of $\cal{S}_{f}$ (with non-trivial $k$-rational point) is proved for any polynomial $f$ of degree 6 with $f$ not equivalent to the polynomial $h$ satisfying the condition $h(t)\neq h(\zeta_{3}t)$, where $\zeta_{3}$ is the primitive third root of unity. We are able to prove the same result for an extension of degree 3 generated by the root of polynomial $h(x)=x^3+ax+b\in k[x]$, provided that $f(t)=t^6+a_{4}t^4+a_{1}t+a_{0}\in k[t]$ with $a_{1}a_{4}\neq 0$.

Radical extensions for the Carlitz module

Let L/K be a finite extension of congruence function fields. We say that L/K is a radical extension if L is generated by roots of polynomials CM(u)−α∈K[u], where CM(u) is the action of Carlitz. We study a special class of these extensions, the pure coradical extensions. We prove that any pure coradical extension has order a power of the characteristic of K. We also give bounds for the Carlitz torsion of these extensions.

State-dependent representation independence

Mitchell's notion of representation independence is a particularly useful application of Reynolds' relational parametricity -- two different implementations of an abstract data type can be shown contextually equivalent so long as there exists a relation between their type representations that is preserved by their operations. There have been a number of methods proposed for proving representation independence in various pure extensions of System F (where data abstraction is achieved through existential typing), as well as in Algol- or Java-like languages (where data abstraction is achieved through the use of local mutable state). However, none of these approaches addresses the interaction of existential type abstraction and local state. In particular, none allows one to prove representation independence results for generative ADTs -- i.e. ADTs that both maintain some local state and define abstract types whose internal representations are dependent on that local state. In this paper, we present a syntactic, logical-relations-based method for proving representation independence of generative ADTs in a language supporting polymorphic types, existential types, general recursive types, and unrestricted ML-style mutable references. We demonstrate the effectiveness of our method by using it to prove several interesting contextual equivalences that involve a close interaction between existential typing and local state, as well as some well-known equivalences from the literature (such as Pitts and Stark's "awkward" example) that have caused trouble for previous logical-relations-based methods. The success of our method relies on two key technical innovations. First, in order to handle generative ADTs, we develop a possible-worlds model in which relational interpretations of types are allowed to grow over time in a manner that is tightly coupled with changes to some local state. Second, we employ a step-indexed stratification of possible worlds, which facilitates a simplified account of mutable references of higher type.

Pure Extensions, Proof Rules, and Hybrid Axiomatics

In this paper we argue that hybrid logic is the deductive setting most natural for Kripke semantics. We do so by investigating hybrid axiomatics for a variety of systems, ranging from the basic hybrid language (a decidable system with the same complexity as orthodox propositional modal logic) to the strong Priorean language (which offers full first-order expressivity). We show that hybrid logic offers a genuinely first-order perspective on Kripke semantics: it is possible to define base logics which extend automatically to a wide variety of frame classes and to prove completeness using the Henkin method. In the weaker languages, this requires the use of non-orthodox rules. We discuss these rules in detail and prove non-eliminability and eliminability results. We also show how another type of rule, which reflects the structure of the strong Priorean language, can be employed to give an even wider coverage of frame classes. We show that this deductive apparatus gets progressively simpler as we work our way up the expressivity hierarchy, and conclude the paper by showing that the approach transfers to first-order hybrid logic.

Complex CKM from spontaneous CP violation without flavor changing neutral current

We analyse the general constraints on unified gauge models with spontaneous CP breaking that satisfy the conditions that (i) CP violation in the quark sector is described by a realistic complex CKM matrix, and (ii) there is no significant flavor changing neutral current effects in the quark sector. We show that the crucial requirement in order to conform to the above conditions is that spontaneous CP breaking occurs at a very high scale by complex vevs of standard model singlet Higgs fields. Two classes of models are found, one consisting of pure Higgs extensions and the other one involving fermionic extensions of the standard model. We give examples of each class and discuss their possible embeddings into higher unified theories. One of the models has the interesting property that spontaneous CP violation is triggered by spontaneous P violation, thereby linking the scale of CP violation to the seesaw scale for neutrino masses.