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In 2019, Bagul et al. posed two open problems dealing with inequalities involving trigonometric and hyperbolic functions and an adjustable parameter. This article is an attempt to solve these open problems. The results are supported by three-dimensional graphics, taking into account the variation of the parameter involved.

The present article is an essay about mathematical intuition and Artificial intelligence (A.I.), followed by a guided excursion to Robbins' Problem of Optimal Stopping. The latter is still open. The first part of the essay (Sect. 1 to Sect. 4), is written in a narrative style with well-known examples and an easily accessible terminology. It is easy to read. Its goal is to motivate mathematicians to think about certain aspects of A.I., and to prepare people in A.I. for the special features of Robbins' problem. The major objective is presented in the second part (Sect. 5 to Sect. 13). It is to guide readers through the probabilistic intuition behind Robbins' problem and to show why A.I., and in particular Deep Learning, may contribute an essential part in its solution. This article is an essay, because it contains no new mathematical results, and no implementation of deep learning either. However it presents a clear mission statement which seems important to us. We substantiate it by proposing two concrete challenges to view and solve the main part of the problem. [ABSTRACT FROM AUTHOR]

In this paper, based on the experimental parameters of polar permafrost formation, the mathematical module, solid mechanics module, and Darcy's law module of COMSOL Multiphysics software are used to realize the coupled solution of PDE of temperature field, moisture field, and stress field considering the ice-water phase transition and moisture convection, and the relationship between the permafrost moisture convection and temperature change is analyzed by using the heat-flowsolid-moisture mutual coupling, which reveals that the law of action of low temperature intrusion into rock formation by polar drilling. [ABSTRACT FROM AUTHOR]

The advent of the sixth generation of mobile communications (6G) ushers in an era of heightened demand for advanced network intelligence to tackle the challenges of an expanding network landscape and increasing service demands. Deep Learning (DL), as a crucial technique for instilling intelligence into 6G, has demonstrated powerful and promising development. This paper provides a comprehensive overview of the pivotal role of DL in 6G, exploring the myriad opportunities and challenges that arise. Firstly, we present a detailed vision for DL in 6G, emphasizing areas such as adaptive resource allocation, intelligent network management, robust signal processing, ubiquitous edge intelligence, and endogenous security. Secondly, this paper reviews how DL models leverage their unique learning capabilities to solve complex service demands in 6G. The models discussed include Convolutional Neural Networks (CNN), Generative Adversarial Networks (GAN), Graph Neural Networks (GNN), Deep Reinforcement Learning (DRL), Transformer, Federated Learning (FL), and Meta Learning. Additionally, we examine the specific challenges each DL model faces within the 6G context. Moreover, we delve into the rapidly evolving field of Artificial Intelligence Generated Content (AIGC), examining its development and impact within the 6G framework. Finally, this paper culminates in a detailed discussion of ten critical open problems in integrating DL with 6G, setting the stage for future research and development in this field.

Problems constitute the starting point of all scientific research. The essay reflects on the different kinds of problems that scientists address in their research and discusses a list of 10 problems for the field of computational historical linguistics, that was proposed throughout 2019 in a series of blog posts (see http://phylonetworks.blogspot.com/). In contrast to problems identified in different contexts, these problems were considered to be solvable, but no solution could be proposed back then. By discussing the problems in the light of developments that have been made in the field during the past five years, a modified list is proposed that takes new insights into account but also finds that the majority of the problems has not yet been solved.

Open mathematical modelling problems that can be solved with multiple methods and have multiple possible results are an important part of school curricula in mathematics and science. Solving open modelling problems in school should prepare students to apply their mathematical knowledge in their current and future lives. One characteristic of these problems is that information that is essential for solving the problems is missing. In the present study, we aimed to analyze students' cognitive barriers while they solved open modelling problems, and we evaluated the effects of instructional prompts on their success in solving such problems. A quantitative experimental study (N = 263) and a qualitative study (N = 4) with secondary school students indicated that identifying unknown quantities and making numerical assumptions about these quantities are important cognitive barriers to solving open modelling problems. Task-specific instructional prompts helped students overcome these barriers and improved their solution rates. Students who were given instructional prompts included numerical assumptions in their solutions more often than students who were not given such prompts. These findings contribute to theories about solving open modelling problems by uncovering cognitive barriers and describing students' cognitive processes as they solve these problems. In addition, the findings contribute to improving teaching practice by indicating the potential and limitations of task-specific instructional prompts that can be used to support students' solution processes in the classroom. [ABSTRACT FROM AUTHOR]

Problems constitute the starting point of all scientific research. The essay reflects on the different kinds of problems that scientists address in their research and discusses a list of 10 problems for the field of computational historical linguistics, that was proposed throughout 2019 in a series of blog posts. In contrast to problems identified in different contexts, these problems were considered to be solvable, but no solution could be proposed back then. By discussing the problems in the light of developments that have been made in the field during the past five years, a modified list is proposed that takes new insights into account but also finds that the majority of the problems has not yet been solved.

This paper addresses the entire functions via theory of the tempered xi function. The zeros of the entire functions by the Fourier sine and cosine integrals are studied in detail. The method for the heat-diffusion equation is proposed to structure the connection between Fourier analysis and heat-diffusion equation show the open problems in analytic number theory. [ABSTRACT FROM AUTHOR]

After the launch of STEREO twin spacecraft, and most recently of Solar Orbiter and Parker Solar Probe spacecraft, the next mission that will explore Sun-Earth interactions and how the Sun modulates the Heliosphere will be the “Lagrange” mission, which will consist of two satellites placed in orbit around L1 and L5 Sun-Earth Lagrangian points. Despite the significant novelties that will be provided by such a double vantage point, there will be also missing information, that are briefly discussed here. For future heliospheric missions, an alternative advantageous approach that has not been considered so far would be to place two twin spacecraft not in L1 and L5, but in L4 and L5 Lagrangian points. If these two spacecraft will be equipped with in situ instruments, and also remote sensing instruments measuring not only photospheric but also coronal magnetic fields, significant advancing will be possible. In particular, data provided by such a twin mission will allow to follow the evolution of magnetic fields from inside the Sun (with stereoscopic helioseismology), to its surface (with classical photospheric magnetometers), and its atmosphere (with spectro-polarimeters); this will provide a tremendous improvement in our physical understanding of solar activity. Moreover, the L4-L5 twin satellites will take different interesting configurations, such as relative quadrature, and quasi-quadrature with the Earth, providing a baseline for monitoring the Sun-to-Earth propagation of solar disturbances.

In this paper, the authors prove some inequalities and completely monotonic properties of polygamma functions. As an application, we give lower bound for the zeta function on natural numbers. Partially, we answer the fifth and sixth open problems listed in [F. Qi and R. P. Agarwal, On complete monotonicity for several classes of functions related to ratios of gamma functions, J. Inequalities Appl.2019(36) (2019) 42]. We propose two open problems on completely monotonic functions related to polygamma functions. [ABSTRACT FROM AUTHOR]

Problems constitute the starting point of all scientific research. The essay reflects on the different kinds of problems that scientists address in their research and discusses a list of 10 problems for the field of computational historical linguistics, that was proposed throughout 2019 in a series of blog posts (see http://phylonetworks.blogspot.com/). In contrast to problems identified in different contexts, these problems were considered to be solvable, but no solution could be proposed back then. By discussing the problems in the light of developments that have been made in the field during the past five years, a modified list is proposed that takes new insights into account but also finds that the majority of the problems has not yet been solved., Competing Interests: No competing interests were disclosed., (Copyright: © 2024 List JM.)

Previous research demonstrated that some children inappropriately solve multiplicative missing-value word problems additively, while others inappropriately solve additive missing-value word problems multiplicatively. Besides lacking skills, children's preference for additive or multiplicative relations has been shown to contribute to those errors. The present research, using a mixed-method approach, investigated the nature of upper primary school children's relational preference by empirically examining characteristics of intuitions that had been postulated previously. After administering a pre-test, selected children who preferred additive or multiplicative relations further participated in one of two studies using open problems for which both types of relations were appropriate: either a reaction time study (n = 110) in which children's acceptance behavior and reaction times were measured or a semi-structured individual interview study (n = 18) in which their answers, verbalizations, and conviction scores were collected. Results of both studies revealed that relational preference was perseverant and exerted a coercive effect on children's reasoning: Children mostly considered only the preferential type of relation as an appropriate answer in open problems and rejected alternative answers. Furthermore, relational preference appeared as immediate, self-evident, and certain: Children rejected the non-preferential answer more quickly than an irrelevant distractor of comparable size, experienced difficulties in justifying why they gave their preferential answer, and were very convinced of this preferential answer. While this characterization held for both relational preferences, it was especially prominent for the multiplicative one. These results not only have implications for research on and educational practice in multiplicative and additive reasoning but also for the measurement of relational preference. [ABSTRACT FROM AUTHOR]

In this paper, we propose the answer to an open problem recently posed. In addition, we derive some inequalities by introducing certain parameters λ, p and q. [ABSTRACT FROM AUTHOR]

Some integral inequalities related to the open problem of F. Qi are recently obtained. In the present work we general- ize these inequalities. First, is given the answer to an open problem. Secondly, these inequalities are generalized for many functions and different parameters. In addition, we consider two other theorems by introducing a certain parameter λ. [ABSTRACT FROM AUTHOR]

Predictive modeling of the evolutionary dynamics of cancer is a challenging issue in computational cancer biology. In this paper, we propose a general mathematical model framework for the evolutionary dynamics of cancer, including plasticity and heterogeneity in cancer cells. Cancer is a group of diseases involving abnormal cell growth, during which abnormal regulation of stem cell regeneration is essential for the dynamics of cancer development. In general, the dynamics of stem cell regeneration can be simplified as a G0 phase cell cycle model, which leads to a delay differentiation equation. When cell heterogeneity and plasticity are considered, we establish a differential-integral equation based on the random transition of epigenetic states of stem cells during cell division. The proposed model highlights cell heterogeneity and plasticity; connects the heterogeneity with cell-to-cell variance in cellular behaviors (for example, proliferation, apoptosis, and differentiation/senescence); and can be extended to include gene mutation-induced tumor development. Hybrid computational models are developed based on the mathematical model framework and are applied to the processes of inflammation-induced tumorigenesis and tumor relapse after chimeric antigen receptor (CAR)-T cell therapy. Finally, we propose several mathematical problems related to the proposed differential-integral equation. Solutions to these problems are crucial for understanding the evolutionary dynamics of cancer. [ABSTRACT FROM AUTHOR]

Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry and partly from p-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some of the motivating problems and some of the recent progress in the field of arithmetic dynamics. [ABSTRACT FROM AUTHOR]

Abstract Subdivision has been an active research topic for about 40 years, and it has seen commercial exploitation in animated Computer Graphics. The question has to be asked: 'Is there anything left to do research-wise?' This note contains the authors' responses. Yes, much has been learnt and does not need to be relearnt, but there are still results that we cannot prove, properties of specific schemes that we cannot compute exactly, and there are still ways in which we need to improve the technology to make it useful in the more demanding contexts of CAD, CAM, and CAE. We have approached the subject by looking first at curves, then at surfaces, and finally at the trivariate structures which are required for CAE. For each dimension we identify first what is already established, and then proceed to suggest directions for further research. Three different labels indicate, for each useful piece of work, the relative difficulty and hence the prospects of success: Of course, these ratings are just our personal estimates. We will only know for sure how hard the questions were when the work has been done. Further, we do not claim completeness of the list of topics discussed here. Also the list of references selects only a few samples from the vast literature. [ABSTRACT FROM AUTHOR]

An effect algebra (EA) is an algebraic structure useful in quantum logic. A sequential effect algebra (SEA) is an EA with a sequential product additionally defined. SEAs are useful in the study of quantum measurement theory and by extension quantum computation. A paper published by Gudder in 2005 posited multiple open problems about sequential effect algebras. Among them was the following question: if a and b are elements of a sequential effect algebra, and a is orthogonal to b , is it the case that the sequential product of a and b is orthogonal to itself? In this paper, we prove his assertion. [ABSTRACT FROM AUTHOR]

Dynamical systems appear in many models in all sciences and in technology. They can be either discrete or continuous, finite or infinite dimensional, deterministic or with random terms. Many theoretical results, the related algorithms and implementations for careful simulations and a wide range of applications have been obtained up to now. But still many key questions remain open. They are mainly related either to global aspects of the dynamics or to the lack of a sufficiently good agreement between qualitative and quantitative results. In these notes a sample of questions, for which the author is not aware of the existence of a good solution, are presented. Of course, it is easy to largely extend the list. [ABSTRACT FROM AUTHOR]

Life in today’s world is characterized by complexity and rapid change. Twenty-first century skills and especially mathematical understanding are supposed to crucially contribute to meeting the demands of our world since mathematics offers strategies to structure or simplify complex problems. An open question is which teaching practices are appropriate to provide all students with such skills and to broaden the participation of underprivileged students. The present article explores these aspects by focusing on complex tasks, a practice that can be considered highly accepted in the context of mathematics education all over the world. We will concentrate on the perspective of the German mathematics education community as the foundation of our considerations. Based on an analytical investigation of mathematical literacy and twenty-first century skills (such as creativity, critical thinking, or problem-solving), we will address central ideas and characteristics of complex mathematical tasks. To complement the analytical approach, we will illustrate their characteristics as well as possible intersections with twenty-first century skills by presenting an elementary school teaching experiment. Finally, we will critically discuss the potentials and pitfalls of complex mathematical tasks from an abstract perspective and conclude by debating practical consequences for organizing mathematical learning-teaching-processes.

We have selected problems that may not yet be well known, but have the potential to push the research in interesting directions. In particular, we state problems that do not require specific knowledge outside the standard circle of ideas in discrete geometry. Despite the relatively simple statements, these problems are related to current research and their solutions are likely to require new ideas and approaches. We have chosen problems from different fields to make this short paper attractive to a wide range of specialists.The article is published in the author’s wording.

While previous studies mainly focused on children’s additive and multiplicative reasoning abilities, we studied third to sixth graders’ preference for additive or multiplicative relations. This was investigated by means of schematic problems that were open to both types of relations, namely arrow schemes containing three given numbers and a fourth missing one. In study 1, children had to fill out the missing number, while in study 2, children had to indicate all possibly correct answers among a set of given alternatives. Both studies explicitly showed the existence of a preference for additive relations in some children, while others preferred multiplicative relations. Mainly younger children preferred additive relations, whereas mainly children in upper primary education preferred multiplicative relations. Number ratios also impacted children’s preference, especially in fifth grade. Moreover, the results of study 2 provided evidence for the strength of children’s preference and showed that calculation skills do not coincide with preference, and hence, that preference and calculation skills are two distinct child characteristics. The results of both studies using these open problems resembled previous research results using classical multiplicative or additive word problems. This supports the hypothesis that children’s preferred type of relations may be at play in solving classical word problems as well—besides their abilities—and may hence be an additional factor explaining the mistakes that children make in those word problems. This research line thus seems promising for further research as well as educational practice. [ABSTRACT FROM AUTHOR]

In this paper, the task “Number Triangle” stands for problem fields. The problem field in question is planned by the author to be used in the Finnish comprehensive school. With its aid, our purpose is to show how open teaching acts in practice. The most important in these problem tasks is the way they are presented in teaching: A problem field should be offered to pupils little by little. And the continuation in the problem field depends always on pupils' answers. The answers of problems are not given here, since they are not as important as pupils' independent solving of problems. How far the teacher proceeds with the problem field in question depends on answers given by pupils. [ABSTRACT FROM AUTHOR]

Humans are able to rapidly and accuratelyrecognize familiar faces and identities under widelyvarying and difficult viewing conditions, such asilluminations changing, occlusion, scaling or rotation.Motivated by its importance in human-to-humancommunication and leading to various applications, frombiometrics to human – computer interaction, the facerecognition task is a major issue in the Computer Visionfield and more. This paper presents a brief survey onstate-of-the art methods used to cope with thischallenging task.

The present didactical propose is referred about biological transformation of biomass that include CTS perspective and the “school science” whose obetives are related to the diary life in the context of usefuls and believed theoretical models that are going to evolute and the methodology used is referred to solving open problems that makes more susceptibility and dialog in the classroom.