Ideal Class Research Articles

Specialization of integral closure of ideals by general elements

In this paper, we prove a result similar to results of Itoh [J. Algebra 150 (1992), pp. 101–117] and Hong-Ulrich [J. Lond. Math. Soc. (2) 90 (2014), pp. 861–878], proving that integral closure of an ideal is compatible with specialization by a general element of that ideal for ideals of height at least two in a large class of rings. Moreover, we show integral closure of sufficiently large powers of the ideal is compatible with specialization by a general element of the original ideal. In a polynomial ring over an infinite field, we give a class of squarefree monomial ideals for which the integral closure is compatible with specialization by a general linear form.

The Ptolemy-Euler Theorem via Reflection

Summary Although there are several generalizations of the Ptolemy-Euler theorem in plane geometry, its higher dimensional cases are still interesting enough to deserve attention. In this note, we give a new proof of the Ptolemy-Euler theorem in R n , via a modified matrix version (reflection transformation) of a classical algebraic identity.

Indian Classical Dancers in the Diaspora: How Free Improvisation and Contact Improvisation Programs Help Enhance Dance Proficiency, Improvisation Competence, and Forming Multigroup Ethnic Identities

ABSTRACT This article reports a quasi-experiment on the impact of contact improvisation (CI) sessions and free improvisation (FI) interfaces on dance proficiency, improvisation competence/output quality, and multigroup ethnic identity formation of Indian classical dancers in the United Kingdom, United States, Singapore, and Australia. FI participants reported higher scores on dance proficiency, improvisation competence, and multigroup ethnic identity formation compared to those who participated in the CI sessions. FI was most effective for diaspora female Indian classical dancers, Bharatanatyam dancers, dancers with 9–12 years of dance education, who had an advanced/higher diploma formal education in dance, who attended 21–40 (>50%) FI interfaces, and completed 21–40 (>50%) FI homework sessions. FI is a performance technique foregrounding spontaneity and a modality to augment the repertoire of Indian classical dances to speak to an international audience by adapting western aesthetics. It is a way to bring contemporariness into Indian classical dances and shape postcolonial dancer identities.

Flexible Linker-Based Triazine-Functionalized 2D Covalent Organic Frameworks for Supercapacitor and Gas Sorption Applications.

Covalent organic frameworks (COFs) having a large surface area, porosity, and substantial amounts of heteroatom content are recognized as the ideal class of materials for energy storage and gas sorption applications. In this work, we have synthesized four different porous COF materials by the polycondensation of a heteroatom-rich flexible triazine-based trialdehyde linker, namely 2,4,6-tris(4-formylphenoxy)-1,3,5-triazine (TPT-CHO), with four different triamine linkers. Triamine linkers were chosen based on differences in size, symmetry, planarity, and heteroatom content, leading to the synthesis of four different COF materials named IITR-COF-1, IITR-COF-2, IITR-COF-3, and IITR-COF-4. IITR-COF-1, synthesized within 24 h from the most planar and largest amine monomer, exhibited the largest Brunauer-Emmett-Teller (BET) surface area of 2830 m2 g-1, superior crystallinity, and remarkable reproducibility compared to the other COFs. All of the synthesized COFs were explored for energy and gas storage applications. It is shown that the surface area and redox-active triazene rings in the materials have a profound effect on energy and gas storage enhancement. In a three-electrode setup, IITR-COF-1 achieved an electrochemical stability potential window (ESPW) of 2.0 V, demonstrating a high specific capacitance of 182.6 F g-1 with energy and power densities of 101.5 Wh kg-1 and 298.3 W kg-1, respectively, at a current density of 0.3 A g-1 in 0.5 M K2SO4 (aq) with long-term durability. The symmetric supercapacitor of IITR-COF-1//IITR-COF-1 exhibited a notable specific capacitance of 30.5 F g-1 and an energy density of 17.0 Wh kg-1 at a current density of 0.12 A g-1. At the same time, it demonstrated 111.3% retention of its initial specific capacitance after 10k charge-discharge cycles. Moreover, it exhibited exceptional CO2 capture capacity of 25.90 and 10.10 wt % at 273 and 298 K, respectively, with 2.1 wt % of H2 storage capacity at 77 K and 1 bar.

Ideal classes of orders in quaternion algebras

We provide an algorithm that, given any order O in a quaternion algebra over a global field, computes representatives of all right equivalence classes of right O-ideals, including the non-invertible ones. The theory is developed for a more general kind of algebras.

Some Identities Involving $q$-Stirling Numbers of the Second Kind in Type B

The recent interest in type B $q$-Stirling numbers of the second kind prompted us to give a type B analogue of a classical identity connecting the $q$-Stirling numbers of the second kind and Carlitz's major $q$-Eulerian numbers, which turns out to be a $q$-analogue of an identity due to Bagno, Biagioli and Garber. We provide a combinatorial proof of this identity and an algebraic proof of a more general identity for colored permutations. In addition, we prove some $q$-identities about the $q$-Stirling numbers of the second kind in types A, B and D.

Strong Dimensional and Structural Dependencies of Hot Carrier Effects in InGaAs Nanowires: Implications for Photovoltaic Solar Cells.

III-V nanowire structures are among the promising material systems with applications in hot carrier solar cells. These nanostructures can meet the requirements for such photovoltaic devices, i.e., the suppression of thermalization loss, an efficient hot carrier transport, and enhanced photoabsorption thanks to their unique one-dimensional (1D) geometry and density-of-states. Here, we investigate the effects of spatial confinement of photogenerated hot carriers in InGaAs-InAlAs core-shell nanowires, which presents an ideal class of hot carrier solar cell materials due to its suitable electronic properties. Using steady-state photoluminescence spectroscopy, our study reveals that by increasing the degree of spatial confinement and Auger recombination, the effects of hot carriers increase, which is in good agreement with theoretical modeling. However, for thin nanowires, the temperature of hot carriers decreases as the effects of crystal disorder increase. This observation is confirmed by probing the extent of the disorder-induced Urbach tail and linked to the presence of a higher density of stacking defects in the limit of thin nanowires. These findings expand our knowledge of hot carrier thermalization in nanowires, which can be applied for designing efficient 1D hot carrier absorbers for advanced-concept photovoltaic solar cells.

A New Notion of Fuzzy Function Ideal Convergence

P.M. Pu and Y.M. Liu extended Moore-Smith’s convergence of nets to fuzzy topology and Y.M. Liu provided analogous results to J. Kelley’s classical characterization theorem of net convergence by introducing the notion of fuzzy convergence classes. In a previous paper, the authors of this study provided modified versions of this characterization by using an alternative notion of convergence of fuzzy nets, introduced by B.M.U. Afsan, named fuzzy net ideal convergence. Our main scope here is to generalize and simplify the preceding results. Specifically, we insert the concept of a fuzzy function ideal convergence class, L, on a non-empty set, X, consisting of triads (f,e,I), where f is a function from a non-empty set, D, to the set FP(X) of fuzzy points in X, which we call fuzzy function, e∈FP(X), and I is a proper ideal on D, and we provide necessary and sufficient conditions to establish the existence of a unique fuzzy topology, δ, on X, such that (f,e,I)∈L iff fI-converges to e, relative to the fuzzy topology δ.

On the capitulation problem of some pure metacyclic fields of degree 20 II

Let $n$ be a $5^{th}$ power-free natural number and $k_0\,=\,\mathbb{Q}(\zeta_5)$ be the cyclotomic field generated by a primitive $5^{th}$ root of unity $\zeta_5$. Then $k\,=\,\mathbb{Q}(\sqrt[5]{n},\zeta_5)$ is a pure metacyclic field of absolute degree $20$. In the case that $k$ possesses a $5$-class group $C_{k,5}$ of type $(5,5)$ and all the classes are ambiguous under the action of $Gal(k/k_0)$, the capitulation of $5$-ideal classes of $k$ in its unramified cyclic quintic extensions is determined.

Quasi-inner automorphisms of Drinfeld modular groups

Let A be the set of elements in an algebraic function field K over \mathbb{F}_{q} which are integral outside a fixed place \infty . Let G=\mathrm{GL}_{2}(A) be a Drinfeld modular group . The normalizer of G in \mathrm{GL}_{2}(K) , where K is the quotient field of A , gives rise to automorphisms of G , which we refer to as quasi-inner . Modulo the inner automorphisms of G , they form a group \mathrm{Quinn}(G) which is isomorphic to \mathrm{Cl}(A)_{2} , the 2 -torsion in the ideal class group \mathrm{Cl}(A) . The group \mathrm{Quinn}(G) acts on all kinds of objects associated with G . For example, it acts freely on the cusps and elliptic points of G . If \mathcal{T} is the associated Bruhat–Tits tree, the elements of \mathrm{Quinn}(G) induce non-trivial automorphisms of the quotient graph G \backslash \mathcal{T} , generalizing an earlier result of Serre. It is known that the ends of G \backslash \mathcal{T} are in one-to-one correspondence with the cusps of G . Consequently, \mathrm{Quinn}(G) acts freely on the ends. In addition, \mathrm{Quinn}(G) acts transitively on those ends which are in one-to-one correspondence with the vertices of G \backslash \mathcal{T} whose stabilizers are isomorphic to \mathrm{GL}_{2}(\mathbb{F}_{q}) .

On Ф-s-1-absorbing δ-primary ideals of commutative rings

Let R be a commutative ring with unity (1 ? 0) and let J(R) be the set of all ideals of R. Let ? : J(R) ? J(R) ? {?} be a reduction function of ideals of R and let ? : J(R) ? J(R) be an expansion function of ideals of R. We recall that a proper ideal I of R is called a ? -1-absorbing ?-primary ideal of R, if whenever abc ? I ? ? (I) for some nonunit elements a, b, c ? R, then ab ? I or c ? ?(I). In this paper, we introduce a new class of ideals that is a generalization to the class of ?-1-absorbing ?-primary ideals. Let S be a multiplicative subset of R such that 1 ? S and let I be a proper ideal of R with S ? I = ?, then I is called a ?-S-1-absorbing ?-primary ideal of R associated to s ? S, if whenever abc ? I? ?(I) for some nonunit elements a, b, c ? R, then sab ? I or sc ? ?(I). In this paper, we have presented a range of different examples, properties, characterizations of this new class of ideals.

A Systematic Literature Review: Training Model, Learning Concept, and Ideal Class Approach for Generation Z Based on Addie Model

This research is a literature study with a descriptive qualitative type of research with a literature review that attempts to describe the training model, learning concept, and ideal class form for Generation Z in Indonesia using the ADDIE Model approach. In this research, the author used various written sources such as articles, journals, and documents relevant to the study. This research shows that Generation Z are familiar with technology, like modern learning models with discussions in the form of dialogue, direct practice, and using digital access to deepen the material. This requires educators including trainers to adapt to create training with an environment suitable for the acceptance of Generation Z. The next research that is interesting to develop is the effectiveness of implementing the ADDIE Model in various training for Generation Z.

Moments of ideal class counting functions

We consider the counting function of ideals in a given ideal class of a number field of degree d. This describes, at least conjecturally, the Fourier coefficients of an automorphic form on GL(d), typically not a Hecke eigenform and not cuspidal. We compute its moments, and also investigate the moments of the corresponding cuspidal projection.

A Novel Ferroelectric Rashba Semiconductor.

Fast, reversible, and low-power manipulation of the spin texture is crucial for next generation spintronic devices like non-volatile bipolar memories, switchable spin current injectors or spin field effect transistors. Ferroelectric Rashba semiconductors (FERSC) are the ideal class of materials for the realization of such devices. Their ferroelectric character enables an electronic control of the Rashba-type spin texture by means of the reversible and switchable polarization. Yet, only very few materials are established to belong to this class of multifunctional materials. Here, Pb1- xGexTe is unraveled as a novel FERSC system down to nanoscale. The ferroelectric phase transition and concomitant lattice distortion are demonstrated by temperature dependent X-ray diffraction, and their effect on electronic properties are measured by angle-resolved photoemission spectroscopy. In few nanometer-thick epitaxial heterostructures, a large Rashba spin-splitting is exhibiting a wide tuning range as a function of temperature and Ge content. This work defines Pb1- xGexTe as a high-potential FERSC system for spintronic applications.

On Quasi Maximal Ideals of Commutative Rings

Let $$R$$ be a commutative ring with $$1\ne 0$$. A proper ideal $$I$$ of $$R$$ is said to be a quasi maximal ideal if for every $$a\in R-I$$, either $$I+Ra=R$$ or $$I+Ra$$ is a maximal ideal of $$R$$. This class of ideals lies between 2-absorbing ideals and maximal ideals which is different from prime ideals. In addition to give fundamental properties of quasi maximal ideals, we characterize principal ideal UN-rings with $$\sqrt{0}^2=(0)$$, direct product of two fields, and Noetherian zero dimensional modules in terms of quasi maximal ideals.

Three-wave and four-wave interactions in the 4d Einstein Gauss-Bonnet (EGB) and Lovelock theories

We derive the conformal constraints satisfied by classical vertices of a (Einstein) Gauss-Bonnet theory around flat space, in general dimensions and at d=4 (4d EGB). In 4d EGB they are obtained by a singular limit of the integral of the Euler-Poincarè density. Our analysis exploits the relation between this theory and the conformal anomaly action, which allows to uncover some interesting features of the GB vertex at cubic and quartic level. If we introduce a conformal decomposition of the metric, the resulting theory can be formulated in two different versions, which are regularization dependent, a local one which is quartic in the dilaton field, and a nonlocal one, with a quadratic dilaton. The nonlocal version is derived by a finite redefinition of the GB density with the inclusion of a (d−4)R2 correction, before performing the singular d→4 limit. In the local version of the theory, we show how the independent dynamics of the metric and of the dilaton are interwinded by a classical trace identity. Three-gravitational wave interactions can be organised in a nontrivial way by using directly the nonlocal 4d EGB version of the theory. This is possible thanks to the consistency of such formulation - only up to 3-point functions - directly inherited from the conformal anomaly (Riegert) action. The constraints satisfied by the vertices are classical, hierarchical Ward identities. At quartic level, similar relations are derived, borrowing from the analysis of the counterterms of the 4T correlators of the conformal anomaly action, as defined by a perturbative expansion. For d≠4 these constraints hold also for Lovelock actions. They can be extended to higher order topological invariants in such class of theories.

Recent Progress of Remediating Heavy Metal Contaminated Soil Using Layered Double Hydroxides as Super-Stable Mineralizer.

Heavy metal contamination in soil, which is harmful to both ecosystem and mankind, has attracted worldwide attention from the academic and industrial communities. However, the most-widely used remediation technologies such as electrochemistry, elution, and phytoremediation. suffer from either secondary pollution, long cycle time or high cost. In contrast, in situ mineralization technology shows great potential due to its universality, durability and economical efficiency. As such, the development of mineralizers with both high efficiency and low-cost is the core of in situmineralization. In 2021, the concept of 'Super-Stable Mineralization' was proposed for the first time by Kong et al.[1] The layered double hydroxides (denoted as LDHs), with the unique host-guest intercalated structure and multiple interactions between the host laminate and the guest anions, are considered as an ideal class of materials for super-stable mineralization. In this review, we systematically summarize the application of LDHs in the treatment of heavy metal contaminated soil from the view of: 1) the structure-activity relationship of LDHs in in situ mineralization, 2) the advantages of LDHs in mineralizing heavy metals, 3) the scale-up preparation of LDHs-based mineralizers and 4) the practical application of LDHs in treating contaminated soil. At last, we highlight the challenges and opportunities for the rational design of LDH-based mineralizer in the future.

MacMahon's partition analysis XIV: Partitions with n copies of n

We apply the methods of partition analysis to partitions with n copies of n. This allows us to obtain multivariable generating functions related to classical Rogers-Ramanujan type identities. Also, partitions with n copies of n are extended to partition diamonds yielding numerous new results including a natural connection to overpartitions and a variety of partition congruences.

The Ostrowski quotient of an elliptic curve

For [Formula: see text] a finite Galois extension of number fields, the relative Pólya group [Formula: see text] is the subgroup of the ideal class group of [Formula: see text] generated by all the strongly ambiguous ideal classes in [Formula: see text]. The notion of Ostrowski quotient [Formula: see text], as the cokernel of the capitulation map into [Formula: see text], has been recently introduced in [E. Shahoseini, A. Rajaei and A. Maarefparvar, Ostrowski quotients for finite extensions of number fields, Pacific J. Math. 321 (2022) 415–429, doi: 10.2140/pjm.2022.321.415]. In this paper, using some results of [C. D. González-Avilés, Capitulation, ambiguous classes and the cohomology of the units, J. Reine Angew. Math. 613 (2007) 75–97], we find a new approach to define [Formula: see text] and [Formula: see text] which is the main motivation for us to investigate analogous notions in the elliptic curve setting. For [Formula: see text] an elliptic curve defined over [Formula: see text], we define the Ostrowski quotient [Formula: see text] and the coarse Ostrowski quotient [Formula: see text] of [Formula: see text] relative to [Formula: see text], for which in the latter group we do not take into account primes of bad reduction. Our main result is a nontrivial structure theorem for the group [Formula: see text] and we analyze this theorem, in some details, for the class of curves [Formula: see text] over quadratic extensions [Formula: see text].

Apéry Sets and the Ideal Class Monoid of a Numerical Semigroup

The aim of this article is to study the ideal class monoid {mathscr {C}}ell (S) of a numerical semigroup S introduced by V. Barucci and F. Khouja. We prove new bounds on the cardinality of {mathscr {C}}ell (S). We observe that {mathscr {C}}ell (S) is isomorphic to the monoid of ideals of S whose smallest element is 0, which helps to relate {mathscr {C}}ell (S) to the Apéry sets and the Kunz coordinates of S. We study some combinatorial and algebraic properties of {mathscr {C}}ell (S), including the reduction number of ideals, and the Hasse diagrams of {mathscr {C}}ell (S) with respect to inclusion and addition. From these diagrams, we can recover some notable invariants of the semigroup. Finally, we prove some results about irreducible elements, atoms, quarks, and primes of ({mathscr {C}}ell (S),+). Idempotent ideals coincide with over-semigroups and idempotent quarks correspond to unitary extensions of the semigroup. We show that a numerical semigroup is irreducible if and only if {mathscr {C}}ell (S) has at most two quarks.