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The purpose of this short note is to present a correction of the proof of the main result given in the paper “Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings,” Demonstr. Math. 53 (2020), 38–43.

ISO 15189 requires laboratories to estimate the uncertainty of their quantitative measurements and to maintain them within relevant performance specifications. Furthermore, it refers to ISO TS 20914 for instructions on how to estimate the uncertainty and what to take into consideration when communicating uncertainty of measurement with requesting clinicians. These instructions include the responsibility of laboratories to verify that bias is not larger than medically significant. If estimated to be larger than acceptable, such bias first needs to be eliminated or (temporarily) corrected for. In the latter case, the uncertainty of such correction becomes part of the estimation of the total measurement uncertainty. If small enough to be acceptable, bias becomes part of the long term within laboratory random variation. Sources of possible bias are (not limited to) changes in reagent or calibrator lot variation or calibration itself. In this paper we clarify how the rationale and mathematics from an EFLM WG ISO/A position paper on allowable between reagent lot variation can be applied to calculate whether bias can be accepted to become part of long-term imprecision. The central point of this rationale is to prevent the risk that requesting clinicians confuse changes in bias with changes in the steady state of their patients. [ABSTRACT FROM AUTHOR]

We deduce new properties of the orbicyclic function of several variables investigated in a recent paper by V. A. Liskovets. We point out that the function and its connection to the number of solutions of certain linear congruences occur in the literature in a slightly different form. We investigate another similar function considered by Deitmar, Koyama and Kurokawa by studying analytic properties of some zeta functions of Igusa type. Simple number-theoretic proofs for some known properties are also given. [ABSTRACT FROM AUTHOR]

The article informs about case study of a serious and multiple ethical misconduct after an author submitted a paper to five journals for publication and committed some form of plagiarism. Topics include mathematics and in sciences in general, multiple submissions of the same results or parallel papers are prohibited as serious ethical misconduct; and mathematics simultaneous or concurrent submission of a manuscript describing the same research publication in the electronic online systems.

In this paper, we employ the techniques in [C. Cavaterra, S. Dipierro, Z. Gao and E. Valdinoci, Global gradient estimates for a general type of nonlinear parabolic equations, J. Geom. Anal. 32 2022, 2, Paper No. 65] and the approach in [H. T. Dung and N. T. Dung, Sharp gradient estimates for a heat equation in Riemannian manifolds, Proc. Amer. Math. Soc. 147 2019, 12, 5329–5338] to derive sharp gradient estimates for a positive solution to the heat equation u t = Δ ⁢ u + a ⁢ u ⁢ log ⁡ u in a complete noncompact Riemannian manifold (where a is a real constant). This is an extension of the gradient estimates of Dung and Dung. [ABSTRACT FROM AUTHOR]

This paper studies the inhomogeneous defocusing coupled Schrödinger system i ⁢ u ˙ j + Δ ⁢ u j = | x | - ρ ⁢ (∑ 1 ≤ k ≤ m a j ⁢ k ⁢ | u k | p ) ⁢ | u j | p - 2 ⁢ u j , ρ > 0 , j ∈ [ 1 , m ] . The goal of this work is to prove the scattering of energy global solutions in the conformal space made up of f ∈ H 1 ⁢ (ℝ N) such that x ⁢ f ∈ L 2 ⁢ (ℝ N) . The present paper is a complement of the previous work by the first author and Ghanmi [T. Saanouni and R. Ghanmi, Inhomogeneous coupled non-linear Schrödinger systems, J. Math. Phys. 62 2021, 10, Paper No. 101508]. Indeed, the supplementary assumption x ⁢ u 0 ∈ L 2 enables us to get the scattering in the mass-sub-critical regime p 0 < p ≤ 2 - ρ N + 1 , where p 0 is the Strauss exponent. The proof is based on the decay of global solutions coupled with some non-linear estimates of the source term in Strichartz norms and some standard conformal transformations. Precisely, one gets | t | α ⁢ ∥ u ⁢ (t) ∥ L r ⁢ (ℝ N) ≲ 1 for some α > 0 and a range of Lebesgue norms. The decay rate in the mass super-critical regime is the same one as of e i ⋅ Δ ⁢ u 0 . This rate is different in the mass sub-critical regime, which requires some extra assumptions. The novelty here is the scattering of global solutions in the weighted conformal space for the class of source terms p 0 < p < 2 - ρ N - 2 + 1 . This helps to better understand the asymptotic behavior of the energy solutions. Indeed, the source term has a negligible effect for large time and the above non-linear Schrödinger problem behaves like the associated linear one. In order to avoid a singular source term, one assumes that p ≥ 2 , which restricts the space dimensions to N ≤ 3 . In a paper in progress, the authors treat the same problem in the complementary case ρ < 0 . [ABSTRACT FROM AUTHOR]

Let 퐺 be a finite reductive group such that the derived subgroup of the underlying algebraic group is a product of quasi-simple groups of type 햠. In this paper, we give an explicit description of the action of automorphisms of 퐺 on the set of its irreducible complex characters. This generalizes a recent result of M. Cabanes and B. Späth [Equivariant character correspondences and inductive McKay condition for type 햠, J. Reine Angew. Math.728 (2017), 153–194] and provides a useful tool for investigating the local sides of the local-global conjectures as one usually needs to deal with Levi subgroups. As an application we obtain a generalization of the stabilizer condition in the so-called inductive McKay condition [B. Späth, Inductive McKay condition in defining characteristic, Bull. Lond. Math. Soc.44 (2012), 3, 426–438; Theorem 2.12] for irreducible characters of 퐺. Moreover, a criterion is given to explicitly determine whether an irreducible character is a constituent of a given generalized Gelfand–Graev character of 퐺. [ABSTRACT FROM AUTHOR]

The paper is concerned with Lane–Emden and Brezis–Nirenberg problems involving the affine p-Laplace nonlocal operator Δ p 풜 , which has been introduced in [J. Haddad, C. H. Jiménez and M. Montenegro, From affine Poincaré inequalities to affine spectral inequalities, Adv. Math. 386 2021, Article ID 107808] driven by the affine L p energy ℰ p , Ω from convex geometry due to [E. Lutwak, D. Yang and G. Zhang, Sharp affine L p Sobolev inequalities, J. Differential Geom. 62 2002, 1, 17–38]. We are particularly interested in the existence and nonexistence of positive C 1 solutions of least energy type. Part of the main difficulties are caused by the absence of convexity of ℰ p , Ω and by the comparison ℰ p , Ω ⁢ (u) ≤ ∥ u ∥ W 0 1 , p ⁢ (Ω) generally strict. [ABSTRACT FROM AUTHOR]

In [S. Yang and Y. Lin, Multilinear strongly singular integral operators with generalized kernels and applications, AIMS Math. 6 2021, 12, 13533–13551], the authors of the present paper further weaken the smoothness condition of kernel functions with multilinear strongly singular Calderón–Zygmund operators of [Y. Lin, Multilinear theory of strongly singular Calderón–Zygmund operators and applications, Nonlinear Anal. 192 2020, Article ID 111699]. They defined a new class of multilinear strongly singular integral operators, and studied its weighted L p boundedness, variable exponent L p ⁢ (⋅) boundedness and endpoint estimates. In this paper, we naturally consider the boundedness of multilinear commutators and multilinear iterated commutators, which are generated by multilinear strongly singular integral operators with generalized kernels and Lipschitz functions. Our results include the corresponding results of multilinear strongly singular Calderón–Zygmund operators and classical multilinear Calderón–Zygmund operators, respectively. [ABSTRACT FROM AUTHOR]

no abstract available [ABSTRACT FROM AUTHOR]

This paper clarifies and discusses Imre Lakatos' claim that mathematics is quasi-empirical in one of his less-discussed papers A Renaissance of Empiricism in the Recent Philosophy of Mathematics. I argue that (1) Lakatos' motivation for classifying mathematics as a quasi-empirical theory is epistemological; (2) what can be called the quasi-empirical epistemology of mathematics is not correct; (3) analysing where the quasi-empirical epistemology of mathematics goes wrong will bring to light reasons to endorse a pluralist view of mathematics. [ABSTRACT FROM AUTHOR]

Let k ≥ 2. A generalization of the well-known Pell sequence is the k-Pell sequence whose first k terms are 0,..., 0, 1 and each term afterwards is given by the linear recurrence p n (k) = 2 P n − 1 (k) + P n − 2 (k) + ⋯ + P n − k (k) . The goal of this paper is to show that 11, 33, 55, 88 and 99 are all repdigits expressible as sum or difference of two k-Pell. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a modified version of Baker-Davenport reduction method (due to Dujella and Pethő). This extends a result of Bravo and Herrera [Repdigits in generalized Pell sequences, Arch. Math. (Brno) 56(4) (2020), 249–262]. [ABSTRACT FROM AUTHOR]

In this paper, we focused on a one-dimensional elastic system with a double porosity structure and frictional damping acting on both porous equations. We introduce two stability numbers χ 0 {\chi_{0}} and χ 1 {\chi_{1}} and prove that the solution of the system decays exponentially provided that χ 0 = 0 {\chi_{0}=0} and χ 1 ≠ 0 {\chi_{1}\neq 0} . Otherwise, we prove the absence of exponential decay. Our results improve the results of [N. Bazarra, J. R. Fernández, M. C. Leseduarte, A. Magaña and R. Quintanilla, On the thermoelasticity with two porosities: Asymptotic behaviour, Math. Mech. Solids 24 2019, 9, 2713–2725] and [A. Nemsi and A. Fareh, Exponential decay of the solution of a double porous elastic system, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 83 2021, 1, 41–50]. [ABSTRACT FROM AUTHOR]

Neyman’s 1923 paper introduced the potential outcomes framework and the foundations of randomization-based inference. We discuss the influence of Neyman’s paper on four introductory to intermediate-level textbooks by Berkeley faculty members (Scheffé; Hodges and Lehmann; Freedman, Pisani, and Purves; and Dunning). These examples illustrate that Neyman’s key insights can be explained in intuitive and interesting ways to audiences at all levels, including undergraduates in introductory statistics courses. We have found Freedman, Pisani, and Purves’s box-of-tickets model to be a valuable expository tool, and we also find their intuitive explanation of Neyman’s variance result helpful: It is a “minor miracle” that in randomized experiments, the two-sample zz-test is conservative because of “two mistakes that cancel.” All four books take a more positive view of Neyman’s results than Neyman himself did. We encourage educators and researchers to explore ways to communicate Neyman’s ideas that are helpful for their own audiences.

In this paper, we study the higher-order Schwarzian derivative Sn(f) proposed by H. Tamanoi [Higher Schwarzian operators and combinatorics of the Schwarzian derivative, Math. Ann. 305 (1996), 127–151]. For the strongly starlike functions of order α and strongly convex functions of order α, the sharp bound of |S3(f)(0)| is obtained. When n ∈ [2, 7], we prove that the higher Bers maps induced by Sn(f) on Weil-Petersson Teichmüller space and BMO-Teichmüller space are holomorphic. [ABSTRACT FROM AUTHOR]

We continue the study of the Chow ring of the moduli stack 픐 g , n of prestable curves begun in [Y. Bae and J. Schmitt, Chow rings of stacks of prestable curves I, Forum Math. Sigma 10 2022, Paper No. e28]. In genus 0, we show that the Chow ring of 픐 0 , n coincides with the tautological ring and give a complete description in terms of (additive) generators and relations. This generalizes earlier results by Keel and by Kontsevich and Manin for the spaces of stable curves. Our argument uses the boundary stratification of the moduli stack together with the study of the first higher Chow groups of the strata, in particular providing a new proof of the results of Kontsevich and Manin. [ABSTRACT FROM AUTHOR]

In the mid-first century BC Geminus of Rhodes, a scientist and philosopher close to Posidonius, composed a comprehensive Theory of Mathematical Sciences, in the surviving fragments of which the numerous characters are referred to plainly by name, with some of them being namesakes of other, more well-known mathematicians and philosophers. This paper tries to set apart the namesakes of Geminus, of which there are four in his fragments: Theodorus, Hippias, Oenopides, and Menaechmus. [ABSTRACT FROM AUTHOR]

Let F q be a finite field of order 푞. Iosevich and Rudnev [Erdős distance problem in vector spaces over finite fields, Trans. Amer. Math. Soc.359 (2007), 12, 6127–6142] proved that, for any set A ⊂ F q d , if | A | ≫ q d + 1 2 , then the distance set Δ ⁢ (A) contains a positive proportion of all distances. Although this result is sharp in odd dimensions, it is conjectured that the right exponent should be d 2 in even dimensions. During the last 15 years, only some improvements have been made in two dimensions, and the conjecture is still wide open in higher dimensions. To fill the gap, we need to understand more about the structures of the distance sets; the main purpose of this paper is to provide some structural theorems on the distribution of square and non-square distances. [ABSTRACT FROM AUTHOR]

In this paper, we establish the sharp Calderón–Vaillancourt theorem on L p spaces for bi-parameter and bilinear pseudo-differential operators with symbols of critical order by deriving a sufficient and necessary condition on its symbol. This sharpens the result of [G. Lu and L. Zhang, Bi-parameter and bilinear Calderón–Vaillancourt theorem with subcritical order, Forum Math. 28 2016, 6, 1087–1094] which was only proved for symbols of subcritical order. [ABSTRACT FROM AUTHOR]

The article "How many points does your star have?" by Max Hoffmann describes a guide for folding and cutting a star from a square sheet of paper. By folding and cutting, a star with a specific number of points is created. The author suggests using colored paper and using the star as a decoration for the room. The question is also raised whether it is possible to create stars with more or fewer points. The author, Dr. Max Hoffmann, works at the Institute for Mathematics at the University of Paderborn. [Extracted from the article]

In this paper, we prove Strichartz estimates for many body Schrödinger equations in the periodic setting, specifically on tori 핋 d {\mathbb{T}^{d}} , where d ≥ 3 {d\geq 3} . The results hold for both rational and irrational tori, and for small interacting potentials in a certain sense. Our work is based on the standard Strichartz estimate for Schrödinger operators on periodic domains, as developed in [J. Bourgain and C. Demeter, The proof of the l 2 l^{2} decoupling conjecture, Ann. of Math. (2) 182 2015, 1, 351–389]. As a comparison, this result can be regarded as a periodic analogue of [Y. Hong, Strichartz estimates for N-body Schrödinger operators with small potential interactions, Discrete Contin. Dyn. Syst. 37 2017, 10, 5355–5365] though we do not use the same perturbation method. We also note that the perturbation method fails due to the derivative loss property of the periodic Strichartz estimate. [ABSTRACT FROM AUTHOR]

At Aristotle, Metaphysics E.1, 1026a14, Schwegler's conjectural emendation of the manuscript reading ἀχώριστα to χωριστά has been widely adopted. The objects of physical science are therefore here 'separate', or 'independently existent'. By contrast, the manuscripts make them 'not separate', construed by earlier commentators as dependent on matter. In this paper, I offer a new defense of the manuscript reading. I review past defenses based on the internal consistency of the chapter, explore where they have left supporters of the emendation unpersuaded, and attempt to strengthen their appeal. I challenge Schwegler's central case, developed by Ross and others, that the construction μὲν ἀλλ' οὐκ demands an implausible 'logical antithesis' between inseparability and mobility. This is arguably the fundamental obstacle to the manuscript reading, and counterexamples have not to date convinced emenders. I offer a new, systematic review of Aristotle's use of the phrase, including relevant cases from the Rhetoric, to show how its usual meaning in Aristotle supports the transmitted text; I also reply to possible objections. Finally, I explore the implications of this defense for the classification of the sciences in E.1. [ABSTRACT FROM AUTHOR]

In a recent paper [JUKIĆ, D.: A necessary and sufficient criterion for the existence of the global minima of a continuous lower bounded function on a noncompact set, J. Comput. Appl. Math. 375 (2020)], a new existence level was introduced and then was used to obtain a necessary and sufficient criterion for the existence of the global minima of a continuous lower bounded function on a noncompact set. In this paper, we determined that existence level for the residual sum of squares of the power-law regression with an unknown location parameter, and so we obtained a necessary and sufficient condition which guarantee the existence of the least squares estimate. [ABSTRACT FROM AUTHOR]

In a recent paper, Sokół et al. [Applications of k-Fibonacci numbers for the starlike analytic functions, Hacet. J. Math. Stat. 44(1) (2015), 121{127] obtained an upper bound for the Fekete-Szegö functional ϕλ when λ 2 R of functions belong to the class SLk connected with k-Fibonacci numbers. The main purpose of this paper is to obtain sharp bounds for ϕλ both λ 2 R and λ 2 C. [ABSTRACT FROM AUTHOR]

In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation (−Δ)su+μu=(Iα*F(u))F′(u) inRN, ${\left(-{\Delta}\right)}^{s}u+\mu u=\left({I}_{\alpha }{\ast}F\left(u\right)\right){F}^{\prime }\left(u\right)\quad \text{in} {\mathbb{R}}^{N},$ (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), Iα∼1|x|N−α ${I}_{\alpha }\sim \frac{1}{\vert x{\vert }^{N-\alpha }}$ is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u∈Hs(RN) $u\in {H}^{s}\left({\mathbb{R}}^{N}\right)$ , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫RNu2=m>0 ${\int }_{{\mathbb{R}}^{N}}{u}^{2}=m{ >}0$ prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, “Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities,” Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, “Nonlinear scalar field equations II: existence of infinitely many solutions,” Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C 1-regularity.

In this paper, we consider the general dual fractional parabolic problem ∂tαu(x,t)+Lu(x,t)=f(t,u(x,t))inRn×R. ${\partial }_{t}^{\alpha }u\left(x,t\right)+\mathcal{L}u\left(x,t\right)=f\left(t,u\left(x,t\right)\right) \text{in} {\mathbb{R}}^{n}{\times}\mathbb{R}.$ We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f. Our result here actually solves a well-known problem known as Gibbons’ conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator L $\mathcal{L}$ and the Marchaud time derivative ∂tα ${\partial }_{t}^{\alpha }$ , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator L $\mathcal{L}$ , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons’ conjecture. It is worth noting that our results are novel even for a special case of L $\mathcal{L}$ , the fractional Laplacian (−Δ)s, and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators.

In this paper, we consider a class of functionals subject to a duality restriction. The functional is of the form J(Ω,Ω*)=∫Ωf+∫Ω*g $\mathcal{J}\left({\Omega},{{\Omega}}^{{\ast}}\right)={\int }_{{\Omega}}f+{\int }_{{{\Omega}}^{{\ast}}}g$ , where f, g are given nonnegative functions in a manifold. The duality is a relation α(x, y) ≤ 0 ∀ x ∈ Ω, y ∈ Ω*, for a suitable function α. This model covers several geometric and physical applications. In this paper we review two topological methods introduced in the study of the functional, and discuss possible extensions of the methods to related problems.

Š. Schwarz in his paper [SCHWARZ, Š.: Zur Theorie der Halbgruppen, Sborník prác Prírodovedeckej fakulty Slovenskej univerzity v Bratislave, Vol. VI, Bratislava, 1943, 64 pp.] proved the existence of maximal subgroups in periodic semigroups and a decade later he brought into play the maximal subsemigroups and thus he embodied the idempotents in the structural description of semigroups [SCHWARZ, Š.: Contribution to the theory of torsion semigroups, Czechoslovak Math. J. 3(1) (1953), 7–21]. Later in his papers he showed that a proper description of these structural elements can be used to (re)prove many useful and important results in algebra and number theory. The present paper gives a survey of selected results scattered throughout the literature where an semigroup approach based on tools like idempotent, maximal subgroup or maximal subsemigroup either led to a new insight into the substance of the known results or helped to discover new approach to solve problems. Special attention will be given to some disregarded historical connections between semigroup and ring theory. [ABSTRACT FROM AUTHOR]

In this paper we establish general weighted Hardy identities for several subelliptic settings including Hardy identities on the Heisenberg group, Carnot groups with respect to a homogeneous gauge and Carnot–Carathéodory metric, general nilpotent groups, and certain families of Hörmander vector fields. We also introduce new weighted uncertainty principles in these settings. This is done by continuing the program initiated by [N. Lam, G. Lu and L. Zhang, Factorizations and Hardy's-type identities and inequalities on upper half spaces, Calc. Var. Partial Differential Equations 58 2019, 6, Paper No. 183; N. Lam, G. Lu and L. Zhang, Geometric Hardy's inequalities with general distance functions, J. Funct. Anal. 279 2020, 8, Article ID 108673] of using the Bessel pairs introduced by [N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Math. Surveys Monogr. 187, American Mathematical Society, Providence, 2013] to obtain Hardy identities. Using these identities, we are able to improve significantly existing Hardy inequalities in the literature in the aforementioned subelliptic settings. In particular, we establish the Hardy identities and inequalities in the spirit of [H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 1997, 443–469] and [H. Brezis and M. Marcus, Hardy's inequalities revisited. Dedicated to Ennio De Giorgi, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 1–2, 217–237] in these settings. [ABSTRACT FROM AUTHOR]

In this paper, we will establish the existence and the Hölder regularity of the local time of the Brownian bridge. Our results are obtained by using a result on Malliavin calculus in [K. Es-Sebaiy, D. Nualart, Y. Ouknine and C. A. Tudor, Occupation densities for certain processes related to fractional Brownian motion, Stochastics 82 2010, 1–3, 133–147] for a Gaussian process with an absolutely continuous random drift, jointly with the classical approach based on the concept of local nondeterminism for Gaussian processes introduced by Berman [S. M. Berman, Local nondeterminism and local times of Gaussian processes, Indiana Univ. Math. J. 23 1973/74, 69–94]. [ABSTRACT FROM AUTHOR]

In this paper, we study the angle estimate of distance functions from minimal hypersurfaces in manifolds of Ricci curvature bounded from below using Colding's method in [T. H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 1997, 3, 477–501]. With Cheeger–Colding theory, we obtain the Laplacian comparison for limits of distance functions from minimal hypersurfaces in the version of Ricci limit space. As an application, if a sequence of minimal hypersurfaces converges to a metric cone C ⁢ Y × ℝ n - k ( 2 ≤ k ≤ n ) in a non-collapsing metric cone C ⁢ X × ℝ n - k obtained from ambient manifolds of almost nonnegative Ricci curvature, then we can prove a Frankel property for the cross section Y of CY. Namely, Y has only one connected component in X. [ABSTRACT FROM AUTHOR]

The paper has a form of a survey on basics of logical geometry and consists of three parts. It is focused on the relationship between many-sorted theory, which leads to logical geometry and one-sorted theory, which is based on important model-theoretic concepts. Our aim is to show that both approaches go in parallel and there are bridges which allow to transfer results, notions and problems back and forth. Thus, an additional freedom in choosing an approach appears. A list of problems which naturally arise in this field is another objective of the paper. [ABSTRACT FROM AUTHOR]

The main result of this paper is a formula, in terms of characters, for the number of elements of a normal subgroup H of a finite group G which are not commutators in G. [ABSTRACT FROM AUTHOR]

Being informed that one of our articles is cited in the paper mentioned in the title, we downloaded it, and we were surprised to see that, practically, all the results from our paper were reproduced in Section 3 of Meghea and Stamin’s article. Having in view the title of the article, one is tempted to think that the remarks mentioned in the paper are original and there are examples given as to where and how (at least) some of the reviewed results are effectively applied. Unfortunately, a closer look shows that most of those remarks in Section 3 are, in fact, extracted from our article, and it is not shown how a specific result is used in a certain application. So, our aim in the present note is to discuss the content of Section 3 of Meghea and Stamin’s paper, emphasizing their Remark 8, in which it is asserted that the proof of Lemma 7 in our article is “full of errors.”

Plato used mathematics extensively in his account of the cosmos in the Timaeus, but as he did not use equations, but did use geometry, harmony and according to some, numerology, it has not been clear how or to what effect he used mathematics. This paper argues that the relationship between mathematics and cosmology is not atemporally evident and that Plato's use of mathematics was an open and rational possibility in his context, though that sort of use of mathematics has subsequently been superseded as science has progressed. I argue that there is a philosophically and historically meaningful space between 'primitive' or unreflective uses of mathematics and the modern conception of how mathematics relates to cosmology. Plato's use of mathematics in the Timaeus enabled the cosmos to be as good as it could be, allowed the demiurge a rational choice (of which planetary orbits and which atomic shapes to instantiate) and allowed Timaeus to give an account of the cosmos (where if the demiurge did not have such a rational choice he would not have been able to do so). I also argue that within this space it is both meaningful and important to differentiate between Pythagorean and Platonic uses of number and that we need to reject the idea of 'Pythagorean/Platonic number mysticism'. Plato's use of number in the Timaeus was not mystical even though it does not match modern usage. [ABSTRACT FROM AUTHOR]

This paper classifies the Dirac series for complex E 7 . As applications, we verify a few conjectures raised in 2011, 2019 and 2020 for this exceptional Lie group. In particular, according to Conjecture 1.1 of Barbasch and Pandžić [Dirac cohomology and unipotent representations of complex groups, Noncommutative Geometry and Global Analysis, Contemp. Math. 546, American Mathematical Society, Providence (2011), 1–22], our classification should be helpful for understanding the unitary dual of complex E 7 . [ABSTRACT FROM AUTHOR]

The central idea of Lakatos' quasi-empiricism view of the philosophy of mathematics is that truth values are transmitted bottom-up, but only falsity can be transmitted from basic statements. As it is falsity but not truth that flows bottomup, Lakatos emphasizes that observation and induction play no role in both conjecturing and proving phases in mathematics. In this paper, I argue that Lakatos' view that one cannot obtain primitive conjectures by induction contradicts the history of mathematics, and therefore undermines his quasi-empiricism theory. I argue that his misconception of induction causes this view of Lakatos. Finally, I propose that Wittgenstein's view that "mathematics does have a grammatical nature, but it is also rooted in empirical regularities" suggests the possibility to improve Lakatos' view by maintaining his quasi-empiricism while accepting the role induction plays in the conjecturing phase. [ABSTRACT FROM AUTHOR]

The same Teacher 2 was involved and additionally Teacher 3, a 20-year-old Multilingual German who is born and raised in Germany with German and Lebanese Arabic (as Lebanon is part of Levantine, her dialect is very similar to the students' Syrian dialect, cf. [9] 1980). This is where the following transcript starts. source: corpus MuM-Multi II © Redder Teacher 2 admits that the students' solution is also correct (score area 245 f.) and then wants to make sure the students solve the next task accordingly. These two aspects of multilingual profiles will not be object of our analysis in this paper, but still have to be mentioned as teachers have to be aware that possibly traumatized students as well as students that never attended school could participate in their classrooms. For this purpose, [13] et al. (2017) have developed the "Bilingual Student Identification and Profile" ([13] et al. 2017: 31) to support teachers to "systematically collect information about who their students are, the languages they speak, their cultural practices, their experiences, and the world that they know" ([13] et al. 2017: 31). Hence, it is important to create possibilities for the students in which they are allowed to use any language they know without limitations, to enable the students to access their full mental capacities and conceptual knowledge, whilst maintaining a classroom discourse that allows the teachers to assess the students' knowledge and foster them individually. [Extracted from the article]

In this paper, a common fixed point theorem is demonstrated for a sequence of multivalued mappings which satisfy certain requirements in complete metric spaces. The results proved here will generalize and extend the results due to Ćirić [1]. Suitable examples are given at the end to support the results proved herein. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sharp damping estimates for a class of one-dimensional oscillatory integral operators with real-analytic phases. By establishing endpoint estimates for suitably damped oscillatory integral operators, we are able to give a new proof of the sharp L p {L^{p}} estimates, which have been proved by Xiao in [Endpoint estimates for one-dimensional oscillatory integral operators, Adv. Math. 316 2017, 255–291]. The damping estimates obtained in this paper are of independent interest. [ABSTRACT FROM AUTHOR]