In this paper, we show that the torsion-free rank of [Formula: see text] has finite upper bound for [Formula: see text], where [Formula: see text] runs through the pro-[Formula: see text] subgroups of finite index in a pro-[Formula: see text] group [Formula: see text] that is (nilpotent of class [Formula: see text])-by-abelian such that [Formula: see text] is of type [Formula: see text].
The paper is devoted to model-theoretic properties of Kac–Moody groups with the focus on elementary equivalence of Kac–Moody groups. We show that elementary equivalence of (untwisted) affine Kac–Moody groups implies coincidence of their generalized Cartan matrices and the elementary equivalence of their ground fields. We study also the Diophantine problem in affine Kac–Moody groups. We show that for the loop group the Diophantine problem is polynomial time equivalent (more precisely, Karp equivalent) to the problem in the ground ring. Finally, we show that in affine Kac–Moody groups over finite fields the Diophantine problem is undecidable.
Primitive positive constructions between relational structures are motivated by computational complexity of constraint satisfaction problems. They induce a partial order on equivalence classes of finite relational structures. We show that this order has an infinite third layer, given by the equivalence class of an oriented graph and of infinitely many permutation groups that are abstractly isomorphic to all the finite simple groups in a one-to-one correspondence. More concretely, the two topmost layers of the primitive positive constructability poset are known to consist of a single element each. We consider [Formula: see text], a representative of the element in the second layer, and its polymorphism clone [Formula: see text]. We show that any clone over a finite domain that admits a quasi-Maltsev operation and fully symmetric operations of all arities admits an incoming minion homomorphism from [Formula: see text]. We use this result to show that in the primitive positive contructability poset, the lower covers of [Formula: see text] are represented by the transitive tournament on three vertices and for each finite simple group by the disjoint union of all its primitive group actions.
In this paper, we show that, for a Lie-solvable Novikov algebra, the Baer radical is precisely the set of all right nilpotent elements, whereas the Andrunakievich radical is the largest left quasiregular ideal. We investigate the extent to which certain properties of associative and commutative algebras equipped with a derivation are preserved under the Gel’fand–Dorfman construction.
Let [Formula: see text] be a free partially commutative monoid with involution and [Formula: see text] its quotient group (for example, a right-angled Artin or Coxeter group). We show that for any system of word equations over [Formula: see text] with recognizable constraints, the solution set — in [Formula: see text] or in [Formula: see text] — is an EDT0L language. It is given by an NFA [Formula: see text] recognizing endomorphisms over some extended monoid. Furthermore, if the input size is [Formula: see text], then [Formula: see text] can be constructed effectively by an [Formula: see text]-transducer. As a consequence, both Satisfiability (whether the system admits a solution) and Finiteness (whether the solution set is infinite) are decidable in [Formula: see text]. For a natural subclass of constraints, we conjecture that these problems are [Formula: see text]-complete.
In this paper, we show that the group of all tame automorphisms of the free metabelian Lie algebra [Formula: see text] of rank [Formula: see text] over a field is an amalgamated free product of its subgroups. Moreover, we prove that the group of all almost tame automorphisms — that is, the subgroup generated by all Chein automorphisms (or one row transformations) of [Formula: see text] — also admits an amalgamated free product structure. By using these results, we provide simpler proofs of the wildness of certain automorphisms.
This study employed flame atomic absorption spectrophotometry to determine the levels of selected heavy metals in Swiss chard. Samples were collected from three vicinities of Addis Ababa: Akaki, Sebeta, and Kotebe. A 0.5-g dried and powdered sample was analyzed using the wet digestion method with 69%-72% HNO3 and 70% HClO4, with optimized digestion. The calibration curves and coefficient (r) value were between 0.996 and 0.999, showing very good linearity. The accuracy of the optimized procedure was tested using samples that had a known amount of the substance added. The recovery percentages ranged from 95.89% to 100%, which is a good range. The mean concentrations (mg/kg) of nickel (0.017) and zinc (0.088) in the Swiss chard were determined . The mean concentrations of metals in Swiss chard from the three areas indicated a higher concentration of zinc in Kotebe compared to Akaki and Sebeta. A higher concentration of nickel (Ni) was found in Akaki's Swiss chard compared to Sebeta and Kotebe. The Pearson correlation coefficients of metals from the Swiss chard between nickel and zinc showed a very strong correlation. The best approach combines immediate risk reduction such as cleaner irrigation and consumer warnings, with long-term remediation such as soil treatment, pollution control, and policy enforcement to protect both farmers and consumers.
We present a general description of the notion of a retract of an algebra with a set of binary operations and describe an algorithm for finding the retract. We apply the result to fill a gap and give a correct definition of multipermutation degenerate set-theoretical solutions of the Yang-Baxter equation which has not been established so far.
Let d >= 2 be an integer, K0 a perfect field such that char(K-0)d, n > d an integer prime to d, f(x) is an element of K-0[x] a degree n monic polynomial without repeated roots, and & Cscr;(f,d )a smooth projective model of the affine curve y(d) = f(x). Let J(& Cscr;(f,d)) be the Jacobian of the K-0-curve & Cscr;(f,d). We identify & Cscr;(f,d) with its canonical image in J(& Cscr;(f,d)) (such that the infinite point of & Cscr;(f,d) goes to the zero of the group law on J(& Cscr;(f,d))). We say that an integer m > 1 is (n,d)-reachable over K-0 if there exists a polynomial f(x) as above such that & Cscr;(f,d)(K-0) contains a torsion point of order m.Earlier we proved that if m is (n,d)-reachable, then either m = d or m >= n (in addition, both d and n are (n,d)-reachable).In this paper, we prove the following: center dot If n < m < 2n and if m is (n, d)-reachable over K-0, then either d m or m-n mod d. center dot If either char(K-0) = 0, or K0 is infinite and char(K-0) > n, then d & centerdot; [(n + d)/d] is (n, d)-reachable if and only if n-(d-1) & centerdot; [(n + d)/d] > 0. center dot If char(K-0) = 0, then n + d is (n, d)-reachable if and only if d2-2d < n. center dot If d = 2 (the hyperelliptic case) and char(K-0) = 0, then m is (n, d)-reachable if n + 1 < m < 2n + 1 and m not equal 2n. (The case when n < m < 3(n-1)/2 was done earlier by E. V. Flynn.)
In this paper, we investigate the structure of Malcev algebras satisfying the J(xy,z,u) = 0. These algebras correspond precisely to the tangent algebras of left automorphic analytic Moufang loops. They can be characterized as the variety J(4) in which every Jacobian of length four vanishes identically. We also prove that the larger variety J(5), defined by the vanishing of all Jacobians of length five, consists of special algebras.
Let R be a Noetherian UFD containing Q and B = R[X,Y,Z]. We show that, under certain conditions, the kernel of the following type of R-derivations D of B is generated by at most three elements over R: (a) D is an extension of an R-elementary derivation D(0 )of R[X, Y ] such that D(Z) E ker(D-0) or D(Z) is a homogeneous polynomial in X and Y with respect to the standard grading. (b) D is a nice R-derivation satisfying D(R[X, Y ]) C R[X, Y ] and D(X), D(Y) form a regular sequence or are comaximal in R[X, Y ]. (c) D is triangular monomial. (d) D is an extension of an irreducible triangular R-derivation D(0)of R[X, Y ] such that D(Z) E ker(D-0).
Growth functions of modules encode important algebraic properties. Conversely, we address the inverse problem of realizing functions that satisfy necessary conditions as growth functions of modules with specific algebraic properties, exploring concavity, polynomial growth, regular elements and polynomial identities.
Piecewise testability of a regular language can be decided by checking whether its minimal deterministic automaton contains no nontrivial cycles and whether for every subset of the input alphabet, all computations on words over this subalphabet are confluent. In this paper, it is proved that if such an automaton contains no simple path of length greater than k, then the accepted language is k-piecewise testable. Furthermore, it is proved that the problem of deciding k-piecewise testability of regular languages given by deterministic finite automata is coNP-complete for every k >= 4, while this problem is known to be solvable in polynomial time for k <= 3.
Let G be a group, and let F-n = < x(1), ... , x(n)> be the free group of rank n. For every word w is an element of F-n, the word map is defined as (w) over tilde : G(n) -> G, where (w) over tilde (g(1), ... , g(n)) = w(g(1), ... , g(n)). The set W-w := (w) over tilde (-1)(e), where e is the identity of G, is the variety of representations of the finitely generated group Gamma(w) = with one relation w. In this paper we consder some properties of such varieties for the case when G is a simple algebraic group, in particular, G = SL2(C). Also, we consider here the question of the surjectivity of a word map (w) over tilde : SL2(C)(2) -> SL(C)) for w is an element of F-2.
This paper presents a method combining a thermal desorption (TD) unit with gas chromatography-combustion-isotope ratio mass spectrometry (GC-C-IRMS) to determine compound specific δ13C values of seven VOCs (benzene, toluene, ethylbenzene, m/p-xylene, o-Xylene, styrene, and cumene). The main innovation lies in the systematic validation of TD-GC-C-IRMS for atmospheric VOC analysis, which enables reliable δ13C determination with minimal isotopic bias. The optimal TD conditions were 300°C and 5 min. Advantages include high precision (SD ≤ 0.4‰), good reproducibility, long-term sample stability (up to 50 days), and suitability for on-site air sampling. Limitations are the coverage of only seven compounds and a wider range of VOC species and more complex matrices should be addressed in future work. The method was successfully applied to ambient air samples, and isotopic signatures indicated that VOCs were mainly derived from traffic-related emissions. This approach provides a robust and practical tool for source identification of atmospheric VOCs and supports source apportionment studies in environmental research.
The integration of artificial intelligence (AI) plays a crucial role in modern analytical chemistry, offering solutions to long-standing challenges. Conventional techniques, such as spectrophotometric analysis and chromatography, often face issues like spectral overlap, matrix interference, and extensive experimental optimization. AI and machine learning (ML) approaches address these limitations by enabling spectral deconvolution, pattern recognition, prediction of retention factors, and automated optimization of separation conditions. Beyond enhancing traditional methods, AI supports the development of innovative analytical platforms. Modern analytical chemistry increasingly relies on smartphone- and paper-based sensors for on-site detection of biomarkers and pollutants. These portable, low-cost systems generate complex datasets requiring advanced computational tools, where AI can improve reliability and sensitivity when validated. AI also plays a vital role in synthesizing and optimizing nanomaterials such as carbon quantum dots (CQDs), accelerating experimental fine-tuning through predictive modeling and optimization algorithms. Moreover, AI facilitates the interpretation of large-scale data, providing deeper insights while reducing human error and analysis time. Despite these advancements, challenges remain regarding model interpretability and the integration of heterogeneous datasets. Addressing these requires explainable ML methods that bridge computational predictions with chemical reasoning. This review highlights current AI applications in chromatographic analysis, drug stability studies, and modern analytical chemistry, discusses implementation challenges, and explores future trends shaping the next generation of intelligent analytical systems.
Metformin, a widely prescribed antihyperglycemic agent, plays a crucial role in the management of Type 2 diabetes by reducing hepatic glucose production and increasing insulin sensitivity. Its effectiveness in glucose regulation has made it a cornerstone in diabetes care, necessitating precise monitoring of drug levels to optimize therapeutic outcomes and minimize potential adverse effects. This review explores contemporary biosensor technologies designed for the sensitive detection of metformin, emphasizing their significance in clinical and pharmaceutical settings. We analyze various biosensor platforms, including electrochemical, optical, and piezoelectric systems, highlighting their principles, advantages, and challenges. Additionally, we discuss the integration of nanomaterials to enhance detection sensitivity and specificity. Given the rising prevalence of diabetes globally, the development of innovative biosensing strategies for metformin detection is paramount in ensuring effective patient management and improving treatment adherence. The insights gained from this review aim to propel further research and development in this vital area of biomedical engineering.
Volatile organic compounds (VOCs) emitted in human matrices have gained attention for their potential in noninvasive disease detection, utilizing canine olfaction and chemical analysis. However, the stability of these VOCs under various storage conditions remains poorly understood, presenting a challenge to accurate diagnostics. This study investigates the effects of storage temperature and time on VOC conservation using three types of sorbents, Sorbstars, Twisters and Getxent, without the aim of directly comparing sorbent performance. Initially, synthetic sweat-like mixtures were analyzed on Sorbstars and Getxent, revealing unexpected increases in signal intensity after 1 month and 2 months, which led to a shift toward human sweat samples for a more realistic assessment. Long-term storage of human sweat samples collected on Sorbstars resulted in a marked decrease in total VOC signal intensity, with losses exceeding 80% after 18 months. Short-term studies (2-3 months, Sorbstar and Twisters) revealed temperature-dependent changes in VOC signal intensity that varied by VOC and sorbent. Overall, these results highlight the importance of storage temperature and duration in VOC conservation. As a preliminary study, these findings emphasize that VOC stability should be characterized for the specific VOCs of interest and that further investigations are required to establish robust and compound-specific conservation protocols across different storage conditions.
A fast, simple, and precise stability-indicating high-performance liquid chromatographic (HPLC) method was developed for analysis of triclabendazole (TBZ) and levamisole (LEV) in oral suspension. Using a mobile phase composed of acetonitrile, methanol, and water (50:40:10, v/v/v), with the pH adjusted to 4.6 with 0.1 M phosphoric acid. The mobile phase was filtered, degassed, and pumped at a flow rate of 1.0 mL/min. Detection was performed at 245 nm. The proposed method was validated with respect to specificity, linearity, limit of detection (LOD) and limit of quantification (LOQ), interday and intraday precision, robustness, and accuracy. The retention times for TBZ and LEV were found to be about 3.116 and 1.385 min, respectively. Calibration plots were linear with correlation coefficient of 0.9999 in the concentration range of 200-800 and 150-600 μg/mL for TBZ and LEV, respectively. The method was verified to be stability indicating by separating the active ingredients from their stressed testing degradation products. The procedure proved acceptable robustness to variations in flow rate, wavelength, and column temperature. The method was conveniently used for the analysis of TBZ and LEV in oral suspension.
A Pisot numeration system U = (u(n))(n >= 0) for N is one where the sequence of positive integers (u(n))(n >= 0), with un = 1, is generated by a recurrence whose polynomial is the minimal polynomial of a Pisot number. The Zeckendorf numeration Z is the simplest such example. We define generalized equations of Z-Mahler type, and we show that if a sequence over a commutative ring is Z-regular, then it is the sequence of coefficients of a series which is a solution of a Z-Mahler equation. Conversely, if the Z-Mahler equation is isolating, then its solutions define Zregular sequences. This is a generalization of results of Becker and Dumas. We provide an example to show that there exist non-isolating Z-Mahler equations whose solutions do not define Zregular sequences. Our proof yields a new construction of weighted automata that generate classical q-regular sequences. Our results can be generalized to numeration systems generated by recurrences whose characteristic polynomial is the minimal polynomial of a Pisot number.
