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This study aims at understanding the role of the tools chosen throughout the processes of solving a non-routine mathematical problem and communicating its solution. In assuming that problem-solving is a synchronous activity of mathematization and expression of mathematical thinking we take our proposed Mathematical Problem Solving with Technology (MPST) model to analyze the processes of solving-and-expressing-problems. Resorting to qualitative methods for data collection and analysis, we report on the case of an 8th grader working on a covariation problem to examine the role that paper-and-pencil and digital tools play in the development of a conceptual model of the situation. We found that the resources used throughout the solving-and-expressing activity influenced the depth of the conceptual model developed, within a process of progressive mathematization. Whereas paper-and-pencil led to the emergence of a conceptual model based on exploring particular cases, the digital transformation of the solution was triggered by the process of communicating its mathematical justification and expanded the previous model. Moreover, the complexity of this activity is evidenced by its multiple sequences of processes. Finally, the integration process seems crucial as the concomitant use of technological and mathematical resources precedes major advancements in the expansion of the conceptual model.

We investigated the influence of mental imagery expertise in 15 pen and paper role-players as an expert group compared to the gender-matched control group of computer role-players in the difficult Vandenberg and Kuse mental rotation task. In this task, the participants have to decide which two of four rotated figures match the target figure. The dependent measures were performance speed and accuracy. In our exploratory investigation, we further examined midline frontal theta band activation, parietal alpha band activation, and parietal alpha band asymmetry in EEG as indicator for the chosen rotation strategy. Additionally, we explored the gender influence on performance and EEG activation, although a very small female sample section was given. The expected gender difference concerning performance accuracy was negated by expertise in pen and paper role-playing women, while the gender-specific difference in performance speed was preserved. Moreover, gender differences concerning electro-cortical measures revealed differences in rotation strategy, with women using top-down strategies compared to men, who were using top-down strategies and active inhibition of associative cortical areas. These strategy uses were further moderated by expertise, with higher expertise leading to more pronounced activation patters, especially during successful performance. However, due to the very limited sample size, the findings of this explorative study have to be interpreted cautiously.

Investigating the research trends within a scientific domain by analyzing semantic information extracted from scientific journals has been a topic of interest in the natural language processing (NLP) field. A research trend evaluation is generally based on the time evolution of the term occurrence or the term topic, but it neglects an important aspect—research publication latency. The average time lag between the research and its publication may vary from one month to more than one year, and it is a characteristic that may have significant impact when assessing research trends, mainly for rapidly evolving scientific areas. To cope with this problem, the present paper is the first work that explicitly considers research publication latency as a parameter in the trend evaluation process. Consequently, we provide a new trend detection methodology that mixes auto-ARIMA prediction with Mann–Kendall trend evaluations. The experimental results in an electronic design automation case study prove the viability of our approach.

Paper waste in the mockups design with regular, irregular, and amorphous patterns is a critical problem in digital printing presses. Paper waste reduction directly impacts production costs, generating business and environmental benefits. This problem can be mapped to the two-dimensional irregular bin-packing problem. In this paper, an iterated local search algorithm using a novel neighborhood structure to detect overlaps between amorphous shapes is introduced. This algorithm is used to solve the paper waste problem, modeled as one 2D irregular bin-packing problem. The experimental results show that this approach works efficiently and effectively to detect and correct the overlaps between regular, irregular, and amorphous figures.

Erratum to the paper [Konovalov, Alexander; Smoktunowicz, Agata; Vendramin, Leandro. On skew braces and their ideals. Exp. Math. 30 (2021), no. 1, 95–104. DOI: 10.1080/10586458.2018.1492476.].

We introduce our efforts within the project ‘The science of deep specification’ to work out the key formal underpinnings of industrial-scale formal specifications of software and hardware components, anticipating a world where large verified systems are routinely built out of smaller verified components that are also used by many other projects. We identify an important class of specification that has already been used in a few experiments that connect strong component-correctness theorems across the work of different teams. To help popularize the unique advantages of that style, we dub itdeep specification, and we say that it encompasses specifications that arerich,two-sided,formalandlive(terms that we define in the article). Our core team is developing a proof-of-concept system (based on the Coq proof assistant) whose specification and verification work is divided across largely decoupled subteams at our four institutions, encompassing hardware microarchitecture, compilers, operating systems and applications, along with cross-cutting principles and tools for effective specification. We also aim to catalyse interest in the approach, not just by basic researchers but also by users in industry.This article is part of the themed issue ‘Verified trustworthy software systems’.

This paper is a corrigendum to the article ``On the ideal theorem for number fields''. The main result of this paper proves to be untrue and is replaced by an estimate of a weighted sum with an improved error term.

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.

This comment is devoted to the paper “Lie Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation” (Symmery, 2020, vol.12, 170), in which several results are either incorrect, or incomplete, or misleading.

We correct a mistake in a lemma in the paper cited in the title and show that it did not affect any of the other results of the paper. To this end, we prove results on linearly disjoint field extensions that do not seem to be commonly known. We give an example to show that a separability assumption in one of these results cannot be dropped (doing so had led to the mistake). Further, we discuss recent generalizations of the original classification of defect extensions.

We give common generalizations of the Menon-type identities by Sivaramakrishnan (1969) and Li, Kim, Qiao (2019). Our general identities involve arithmetic functions of several variables, and also contain, as special cases, identities for gcd-sum type functions. We point out a new Menon-type identity concerning the lcm function. We present a simple character free approach for the proof., Comment: 14 pages

Inspired by the article “Anomalous relaxation model based on the fractional derivative with a Prabhakar-like kernel” (Z. Angew. Math. Phys. (2019) 70:42) whose authors Zhao and Sun studied the integro-differential equation with the kernel given by the Prabhakar function $$e^{-\gamma }_{\alpha , \beta }(t, \lambda )$$ , we provide the solution to this equation which is complementary to that obtained up to now. Our solution is valid for effective relaxation times whose admissible range extends the limits given in Zhao and Sun (Z Angew Math Phys 70:42, 2019, Theorem 3.1) to all positive values. For special choices of parameters entering the equation itself and/or characterizing the kernel, the solution comprises to known phenomenological relaxation patterns, e.g., to the Cole–Cole model (if $$\gamma = 1, \beta =1-\alpha $$ ) or to the standard Debye relaxation.

Presented in this paper are the findings of the panel entitled Outlets for undergraduate research as delivered at the Trends in Undergraduate Research in Mathematical Sciences (TURMS) in Chicago on October 27, 2012. We specifically discuss venues and best practices for student papers, posters, and presentations.

The aim of this paper is to study a new equivalent form of the weak maximum principle for a large class of differential operators on Riemannian manifolds. This new form has been inspired by the work of Berestycki, Hamel and Rossi for trace operators, and allows us to shed new light on it and to introduce a new sufficient bounded Khas’minskii type condition for its validity. We show its effectiveness by applying it to obtain some uniqueness results in a geometric setting.

In his paper "On a construction of semigroups", M. Kolibiar gives a construction for a semigroup $T$ (beginning from a semigroup $S$) which is said to be derived from the semigroup $S$ by a $\theta$-construction. He asserted that every semigroup $T$ can be derived from the factor semigroup $T/\theta (T)$ by a $\theta$-construction, where $\theta (T)$ is the congruence on $T$ defined by: $(a, b)\in \theta (T)$ if and only if $xa=xb$ for all $x\in T$. Unfortunately, the paper contains some incorrect part. In our present paper we give a revision of the paper., Comment: Version v5 differs from the published and the earlier versions in the following: some part of the proof of Theorem 2 are clarified, and some typing errors are corrected

A record mentioned in the paper by Moree (Abh Math Sem Univ Hamburg 77:109–127, 2007) was recently improved on by Akbary and Scholten. However, the record mentioned was not the then record. The then record, due to Gallot (2004), actually slightly improves on that obtained recently by Akbary and Scholten.

We correct the proof of Theorem 3.8 in [GP2].

This note concerns the paper "On the controllability of a coupled system of two Korteweg–de Vries equations" by Micu et al. [2]. They study a nonlinear coupled system of two Korteweg–de Vries equations and prove that the system is controllable by using four boundary controls. Here, we prove that in some cases it is possible to get the controllablity of the system by using only two controls. This can be done depending on both the spatial domain and the control time.

International audience; In a previous paper, we have described the construction of an automaton from a rational expression which has the property that the automaton built from an expression which is itself computed from a co-deterministic automaton by the state elimination method is co-deterministic. It turned out that the definition on which the construction is based was inappropriate, and thus the proof of the property was flawed. We give here the correct definition of the broken derived terms of an expression which allow to define the automaton and the detailed full proof of the property.

The spectral gap conjecture of M. van den Berg [2, formula (65)] asserts that λ2 − λ1 ≥ 3π for all convex euclidean domains of diameter 1, where λ1 and λ2 denote the first two eigenvalues of the Dirichlet Laplacian. Notice that equality holds for the 1-dimensional unit interval, which can be regarded also as a degenerate n-dimensional rectangular box. The gap estimate is conjectured to hold more generally for Schrodinger operators with convex potentials, under Dirichlet boundary conditions; see the work of S.-T. Yau and collaborators [9, 11]. This Schrodinger gap conjecture was proved some time ago in 1 dimension by R. Lavine [8], and more recently in all dimensions by B. Andrews and J. Clutterbuck [1]. The proof in this journal by H. Donnelly [3] of the original gap conjecture in 2 dimensions (for the Dirichlet Laplacian with zero potential) is not correct. The Editors of Mathematische Zeitschrift have asked us to describe the flaws in the proof, in order to clarify the state of the literature. Donnelly’s approach to the problem is a natural one: first perform a shape optimization to rule out a non-degenerate minimizing domain, and then analyze the spectral gap for a sequence of domains degenerating to an interval, with the help of results by D. Jerison [5]. (For some history on this approach, and on the gap conjecture more generally, see the report on the AIM meeting “Low Eigenvalues of Laplace and Schrodinger Operators” [10], especially page 12 of the open problems list.) The error lies in the proof of the shape optimization step, as we now explain. Donnelly wishes to prove that no minimizing domain can exist for

This note is related to a nice short paper of M. Ochiai. We prove in a very fast way that the two-parameter family of Heegaard diagrams, constructed by Ochiai, encodes the genuine three-sphere. The result is obtained, up to isotopy, by using a sequence of only three moves in this order: a Whitehead–Zieschang reduction, a band sum and a cancellation of a handle.

We generalize and give a simple proof of a Theorem of Blasco. For a topological space X , C(X) denotes the algebra of real valued continuous functions on X under the operation (αf + gh)(x) := αf(x) + h(x)g(x) for each f, g, h ∈ C(X) and α ∈ R. An algebra A on X means a subalgebra of C(X) which contains the constant functions. By a ∗-algebra A on X we mean an algebra on X and all bounded functions in A separate points from the closed sets. Let A be an algebra on X and π : A −→ R be a (algebra) homomorphism and α be a cardinal number. π is said to be α-evaluating if for each subset B of A with cardinality at most α there exits xB ∈ X such that π(f) = f(xB) for each f ∈ B. If α = |N|(=the cardinal number of the set of the natural numbers N), the α-evaluating homomorphism is called countable evaluating. Let α be a cardinal number. Recall that a topological space X is called αLindelof if for each open cover U has a subcover V such that the cardinality of V is at most α. A lindelof space means a |N|-Lindelof space. The following theorem is given in [1] as one of the main results. Theorem 1. Let X be a completely regular Hausdorff space. Then X is Lindelof if and only if each countable evaluating homomorphism π on a ∗-algebra A on X is point evaluating. Received June 9, 2004; accepted June 30, 2004. Communicated by Bor-Luh Lin. 2000 Mathematics Subject Classification: Primary 54C35, 46E25.

Most of the known continued fraction expansions of special functions are limit periodic. This means that the classical approximants S n (0) are normally not the best ones to use for approximations. In this paper we suggest a number of approximants S n (w n ) which converge faster. The estimation of the improvement and bounds for the error |f - Sn(wn)| (which we still call the truncation error) are mainly obtained by means of Thron's parabola sequence theorem and the oval sequence theorem.

Let H(Dn) be the space of holomorphic functions on the unit polydisk Dn, and let , where p, q> 0, α = (α1,…,αn) with αj > -1, j =1,..., n, be the class of all measurable functions f defined on Dn such thatwhere Mp(f,r) denote the p-integral means of the function f. Denote the weighted Bergman space on . We provide a characterization for a function f being in . Using the characterization we prove the following result: Let p> 1, then the Cesàro operator is bounded on the space .

Three papers from the early history of efficient parametric estimation are reprinted with commentary: (1) Fisher (1912), "On an absolute criterion for fitting frequency curves"; (2) Engledow and Yule (1914), "The determination of the best value of the coupling-ratio from a given set of data"; and (3) Fisher (1922), "The systematic location of genes by means of crossover observations."

We consider the problem of optimal partisan gerrymandering: a legislator in charge of redrawing the boundaries of equal-sized congressional districts wants to ensure the best electoral outcome for his own party. The so-called gerrymanderer faces two issues: the number of districts is finite and there is uncertainty at the level of each district. Solutions to this problem consists in cracking favorable voters in as many districts as possible to get tight majorities, and in packing unfavorable voters in the remaining districts. The optimal payoff of the gerrymanderer tends to increase as the uncertainty decreases and the number of districts is large. With an infinite number of districts, this problem boils down to concavifying a function, similarly to the optimal Bayesian persuasion problem. We introduce a measure of fairness and show that optimal gerrymandering is accordingly closer to uniform districting (full cracking), which is most unfair, than to community districting (full packing), which is very fair.

In this paper, we analyze the complexity of an eco-epidemiological model for phytoplankton–zooplankton system in presence of toxicity and time delay. Holling type II function response is incorporated to address the predation rate as well as toxic substance distribution in zooplankton. It is also presumed that infected phytoplankton does recover from the viral infection. In the absence of time delay, stability and Hopf-bifurcation conditions are investigated to explore the system dynamics around all the possible equilibrium points. Further, in the presence of time delay, conditions for local stability are derived around the interior equilibria and the properties of the periodic solution are obtained by applying normal form theory and central manifold arguments. Computational simulation is performed to illustrate our theoretical findings. It is explored that system dynamics is very sensitive corresponding to carrying capacity and toxin liberation rate and able to generate chaos. Further, it is observed that time delay in the viral infection process can destabilize the phytoplankton density whereas zooplankton density remains in its old state. Incorporation of time delay also gives the scenario of double Hopf-bifurcation. Some control parameters are discussed to stabilize system dynamics. The effect of time delay on (i) growth rate of susceptible phytoplankton shows the extinction and double Hopf-bifurcation in the zooplankton population, (ii) a sufficiently large value of carrying capacity stabilizes the chaotic dynamics or makes the whole system chaotic with further increment.

We formulate a simple susceptible‐infectious‐recovery (SIR) model to describe the spread of the coronavirus under strict social restrictions. The transmission rate in this model is exponentially decreasing with time. We find a formula for basic reproduction function and estimate the maximum number of daily infected individuals. We fit the model to induced death data in Italy, United States, Germany, France, India, Spain, and China over the period from the first reported death to August 7, 2020. We notice that the model has excellent fit to the disease death data in these countries. We estimate the model's parameters in each of these countries with 95% confidence intervals. We order the strength of social restrictions in these countries using the exponential rate. We estimate the time needed to reduce the basic reproduction function to one unit and use it to order the quality of social restrictions in these countries. The social restriction in China was the strictest and the most effective and in India was the weakest and the least effective. Policy‐makers may apply the Chinese successful social restriction experiment and avoid the Indian unsuccessful one. [ FROM AUTHOR] Copyright of Mathematical Methods in the Applied Sciences is the property of John Wiley & Sons, Inc. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full . (Copyright applies to all s.)

This short paper concerns the analysis of the M/M/k queueing system with customer abandonment. In this system, service managers provide a finite buffer space, which is a waiting area that prevents customers from abandoning the system. Abandonment of the system can occur from reneging (exiting from the queue while waiting), and/or balking (leaving the system without waiting). We derive an analytical expression to represent the impact of the buffer space capacity on the delay probability and the abandonment probability for a system with deferred abandonment. The result indicates the provision of the buffer space in a large system could only increase the delay probability while the abandonment probability remains unchanged. Despite the benevolent intentions of service managers, providing a buffer space may exacerbate the performance of larger systems.

In the current article, we aim to study in detail a novel coronavirus (2019-nCoV or COVID-19) mathematical model for different aspects under Caputo fractional derivative. First, from analysis point of view, existence is necessary to be investigated for any applied problem. Therefore, we used fixed point theorem's due to Banach's and Schaefer's to establish some sufficient results regarding existence and uniqueness of the solution to the proposed model. On the other hand, stability is important in respect of approximate solution, so we have developed condition sufficient for the stability of Ulam-Hyers and their different types for the considered system. In addition, the model has also been considered for semianalytical solution via Laplace Adomian decomposition method (LADM). On Matlab, by taking some real data about Pakistan, we graph the obtained results. In the last of the manuscript, a detail discussion and brief conclusion are provided.

In this paper we develop new tensor methods for unconstrained convex optimization, which solve at each iteration an auxiliary problem of minimizing convex multivariate polynomial. We analyze the simplest scheme, based on minimization of a regularized local model of the objective function, and its accelerated version obtained in the framework of estimating sequences. Their rates of convergence are compared with the worst-case lower complexity bounds for corresponding problem classes. Finally, for the third-order methods, we suggest an efficient technique for solving the auxiliary problem, which is based on the recently developed relative smoothness condition (Bauschke et al. in Math Oper Res 42:330–348, 2017; Lu et al. in SIOPT 28(1):333–354, 2018). With this elaboration, the third-order methods become implementable and very fast. The rate of convergence in terms of the function value for the accelerated third-order scheme reaches the level $$O\left( {1 \over k^4}\right) $$ O 1 k 4 , where k is the number of iterations. This is very close to the lower bound of the order $$O\left( {1 \over k^5}\right) $$ O 1 k 5 , which is also justified in this paper. At the same time, in many important cases the computational cost of one iteration of this method remains on the level typical for the second-order methods.

This paper introduced the use of two new adsorbents, Akrosorb soda-lime and Bentonite clay, for refinery flare flue gas capture and storage. This study also developed a novel pilot plant model with 409.7149 kg/h capacity refinery flare emission capture with a novel adsorption column configuration using Akrosorb soda-lime and Bentonite clay adsorbents. The flare flue gas adsorption unit was designed, fabricated, test run, and commissioned. The adsorption column temperature is 28 ± 10 °C and has a pressure of 131.7 kPa. The novel plant RSM optimization result shows that 93.24% of CO2 and 62.18% of CO were absorbed, while 86.14% of NOx and 55.87% of HC were absorbed. The established optimum conditions of CO2, NOx, HC, and CO removal efficiency are 22 °C, 2 atm, and 60 min. The variation in flare gas emission could impact the removal efficiency of the plant. The results show the maximum adsorption ability or capacity of 314.30 mg/g, and 68.90 mg/g was reached at 60 min for Akrosorb soda-lime and molded Bentonite adsorbents. Therefore, the developed novel technology for CO2 and other GHG capture is technically feasible and friendly. The combined usage of both adsorbents will enhance the capture of GHG at a low cost compared to using Akrosorb alone as an adsorbent.

COVID-19 pandemic has affected all aspects of people's lives and disrupted the economy. Forecasting the number of cases infected with this virus can help authorities make accurate decisions on the interventions that must be implemented to control the pandemic. Investigation of the studies on COVID-19 forecasting indicates that various techniques such as statistical, mathematical, and machine and deep learning have been utilized. Although deep learning models have shown promising results in this context, their performance can be improved using auxiliary features. Therefore, in this study, we propose two hybrid deep learning methods that utilize the statistical features as auxiliary inputs and associate them with their main input. Specifically, we design a hybrid method of the multihead attention mechanism and the statistical features (ATT_FE) and a combined method of convolutional neural network and the statistical features (CNN_FE) and apply them to COVID-19 data of 10 countries with the highest number of confirmed cases. The results of experiments indicate that the hybrid models outperform their conventional counterparts in terms of performance measures. The experiments also demonstrate the superiority of the hybrid ATT_FE method over the long short-term memory model.

The class of log-elliptical distributions is well used and studied in risk measurement and actuarial science. The reason is that risks are often skewed and positive when they describe pure risks, i.e., risks in which there is no possibility of profit. In practice, risk managers confront a system of mutually dependent risks, not only one risk. Thus, it is important to measure risks while capturing their dependence structure. In this short paper, we compute the multivariate risk measures, multivariate tail conditional expectation, and multivariate tail covariance measure for the family of log-elliptical distributions, which captures the dependence structure of the risks while focusing on the tail of their distributions, i.e., on extreme loss events. We then study our result and examine special cases, as well as the optimal portfolio selection using such measures. Finally, we show how the given multivariate tail moments can also be computed for log-skew elliptical models based on similar approaches given for the log-elliptical case.

For modified Szasz operators, S. M. Mazhar and V. Totik gave a direct approximation theorem for continuous functions. In this paper we extend this direct result to combinations of these operators. An inverse theorem to this direct estimate is given. An equivalent relation between the derivatives of these operators and smoothness of functions is also presented.

The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples, the first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data.

The article deals with the analysis of the fractional COVID‐19 epidemic model (FCEM) with a convex incidence rate. Keeping in view the fading memory and crossover behavior found in many biological phenomena, we study the coronavirus disease by using the noninteger Caputo derivative (CD). Under the Caputo operator (CO), existence and uniqueness for the solutions of the FCEM have been analyzed using fixed point theorems. We study all the basic properties and results including local and global stability. We show the global stability of disease‐free equilibrium using the method of Castillo‐Chavez, while for disease endemic, we use the method of geometrical approach. Sensitivity analysis is carried out to highlight the most sensitive parameters corresponding to basic reproduction number. Simulations are performed via first‐order convergent numerical technique to determine how changes in parameters affect the dynamical behavior of the system.

Let G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice $$\mathcal {L}(H)$$ L ( H ) embeds in $$\mathcal {L}(G)$$ L ( G ) . It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.