Theorem Research Articles

Information entropy-assisted intuitionistic fuzzy rough feature subset selection

Fuzzy-rough set (FRS) has been effectively implemented as a powerful pre-processing tool to cope with the issues such as vagueness, imprecision, noise and uncertainty in high-dimensional data analysis. However, FRS fails to handle the uncertainty due to identification, which is very much common in heterogeneous data. Moreover, it is always challenging to handle different issues such as vagueness, uncertainty, and noise during elimination of irrelevant and redundant features. This paper investigates an intuitionistic fuzzy rough set (IFRS)-aided feature selection method by using information entropy notion to tackle these issues simultaneously. We establish an intuitionistic fuzzy (IF) similarity relation. Then IFRS model is discussed based on this relation. Next, we present a novel IF granular structure. Based on the granular structure, we present lambda-conditional entropy for IF framework. Further, relevant mathematical theorems are proved to justify the theoretical aspect of the models. Moreover, a positive region preserved attribute selection method is proposed. A comprehensive experimental study is discussed to demonstrate the effectiveness and practical validation of the proposed method. Finally, a methodology is developed based on proposed models, which increases accuracy for the minority RNA silencing suppressors against majority class non-suppressors as evident from better sensitivity and G-means metrics.

Games in Mathematics Education: Trends and Methodological Approaches in the Context of the Congress of European Research in Mathematics Education (CERME) Review

This study aimed to examine the studies on digital games presented in The Congress of European Research in Mathematics Education (CERME) using a systematic analysis approach in terms of research method, data collection tool, and how they define digital games. Sixteen studies related to games were selected from the proceedings. The findings of the research revealed that the number of studies on games has increased in recent years, and the interest of the mathematics education community in this subject has gradually increased. It was seen that qualitative research approaches were used in most of these studies. As a data collection tool, a semi-structured or clinical interview form was mainly used. To reach more comprehensive results, it is necessary to give importance to quantitative research approaches and even to prefer the use of both approaches together. It was indicated that the tendency towards mathematical reasoning and puzzle games has continued predominantly in recent years. Although it was seen that the number of studies on digital games has increased in the last five years, only six studies in total were presented at the CERME. Moreover, three digital games developed by using the tools in the context of dynamic geometry environments were found. Another significant finding was that most of these studies did not define games. Only a few studies that defined the game considered the game as a tool used to acquire a mathematical concept or knowledge. Based on the findings, it can be argued that digital games are not among the interests of the mathematics education community. Finally, it was observed that mathematical theorems and approaches were central to the studies reviewed. Games were generally defined as tools within these studies. Therefore, future research should particularly consider fundamental game features such as game mechanics to create more integrated research environments with mathematics education.

When will Two Agents Agree on a Quantum Measurement Outcome? Intersubjective Agreement in QBism

In the QBist approach to quantum mechanics, a measurement is an action an agent takes on the world external to herself. A measurement device is an extension of the agent and both measurement outcomes and their probabilities are personal to the agent. According to QBism, nothing in the quantum formalism implies that the quantum state assignments of two agents or their respective measurement outcomes need to be mutually consistent. Recently, Khrennikov has claimed that QBism’s personalist theory of quantum measurement is invalidated by Ozawa’s so-called intersubjectivity theorem. Here, following Stacey, we refute Khrennikov’s claim by showing that it is not Ozawa’s mathematical theorem but an additional assumption made by Khrennikov that QBism is incompatible with. We then address the question of intersubjective agreement in QBism more generally. Even though there is never a necessity for two agents to agree on their respective measurement outcomes, a QBist agent can strive to create conditions under which she would expect another agent’s reported measurement outcome to agree with hers. It turns out that the assumptions of Ozawa’s theorem provide an example for just such a condition.

Perception, Evolution, and the Explanatory Scope of Scientific Theories

According to the interface theory of perception, our perceptual systems have evolved to provide a species-specific interface to guide adaptive behaviour, and not to provide veridical representations of an observer-independent world. Results of simulations of evolutionary resource games, genetic algorithms, and multiple mathematical theorems have supported and fleshed out this claim in various ways. They indicate that the probability is zero that any perceptual system has been shaped by natural selection to represent the true structure of an observer-independent world. Bagwell (2023) thinks that this argument is self-defeating because it implies that organisms, resources, genes, etc. that are essential constituents of Darwin's theory do not provide an insight into the nature of observerindependent reality. Bagwell's argument relies on a false premise, however — namely, that the mathematical formulation of a scientific theory cannot dethrone its fundamental concepts. More broadly, his argument involves a fundamental misunderstanding of the nature of scientific theories — of their explanatory scope and their necessary limits.

The gaussian sequence 3th order in finite groups

Recent research examines Gaussian numbers in third-order recurrent sequences in connection with finite groups in the field of Abstract Algebra. Thus, some definitions were introduced aiming to transform infinite sequences into finite ones. In this regard, the Fourier transform was used as a visualization technique, explored in Google Colab. The mathematical theorems presented were established with the purpose of examining the patterns of these sequences and their corresponding cycles. As a future perspective, it is intended to investigate other mathematical theorems to generalize the sequences into finite groups.

Self-correlated spatial random variables: From an auto- to a sui- model respecification

This paper marks the 50-year publication anniversary of Besag's seminal spatial auto- models paper. His classic article synthesizes generic autoregressive specifications (i.e., a response variable appears on both sides of a regression equation and/or probability function equal sign) for the following six popular random variables: normal, logistic (i.e., Bernoulli), binomial, Poisson, exponential, and gamma. Besag dismisses these last two while recognizing failures of both as well as the more scientifically critical counts-oriented auto-Poisson. His initially unsuccessful subsequent work first attempted to repair them (e.g., pseudo-likelihood estimation), and then successfully revise them within the context of mixed models, formulating a spatially structured random effects term that effectively and efficiently absorbs and accounts for spatial autocorrelation in geospatial data. One remaining weakness of all but the auto-normal is a need to resort to Markov chain Monte Carlo (MCMC) techniques for legitimate estimation purposes. Recently, Griffith succeeded in devising an innovative uniform distribution genre—sui-uniform random variables—that accommodates spatial autocorrelation, too. Its most appealing feature is that, by applying two powerful mathematical statistical theorems (i.e., the probability integral transform, and the quantile function), it redeems Besag's auto- model failures. This paper details conversion of Besag's initial six modified variates, exemplifying them with both simulation experiments and publicly accessible real-world georeferenced data. The principal outcome is valuable spatial statistical advancements, with special reference to Moran eigenvector spatial filtering.

Finiteness theorems and counting conjectures for the flux landscape

In this paper, we explore the string theory landscape obtained from type IIB and F-theory flux compactifications. We first give a comprehensive introduction to a number of mathematical finiteness theorems, indicate how they have been obtained, and clarify their implications for the structure of the locus of flux vacua. Subsequently, in order to address finer details of the locus of flux vacua, we propose three mathematically precise conjectures on the expected number of connected components, geometric complexity, and dimensionality of the vacuum locus. With the recent breakthroughs on the tameness of Hodge theory, we believe that they are attainable to rigorous mathematical tools and can be successfully addressed in the near future. The remainder of the paper is concerned with more technical aspects of the finiteness theorems. In particular, we investigate their local implications and explain how infinite tails of disconnected vacua approaching the boundaries of the moduli space are forbidden. To make this precise, we present new results on asymptotic expansions of Hodge inner products near arbitrary boundaries of the complex structure moduli space.

Dressing the cusp: how paraxial sharp-edge diffraction theory solves a basic issue in catastrophe optics

The description of light diffraction using catastrophe optics is one of the most intriguing theoretical inventions in the field of classical optics of the last four decades. Its practical implementation has faced some resistance over the years, mainly due to the difficulty of decorating the different (topologically speaking) types of optical singularities (caustics) that concur to build the skeleton on which diffraction patterns stem. Such a fundamental dressing problem has been solved in the past only for the so-called fold, which lies at the bottom of the hierarchy of structurally stable caustics. Climbing this hierarchy implies considerably more challenging mathematical problems to be solved. An ancient mathematical theorem is employed here to find the complete solution of the dressing problem for the cusp, which is placed, in the stable caustic hierarchy, immediately after the fold. The other ingredient used for achieving such an important theoretical result is the paraxial version of the boundary diffraction wave theory, whose tight connection with catastrophe optics has recently been emphasized [Opt. Lett. 41, 3114 (2016)OPLEDP0146-959210.1364/OL.41.003114]. A significant example of the developed algorithm, aimed at demonstrating its effectiveness and ease of implementation, is also presented.

Analytical Solution to the Steady Navier-Stokes Equation for a Supersonic Cone in the Area of Boundary Layer Behind the Shock Wave by Means Tunnel Mathematics

The means of tunnel mathematics (the theory of functions of a spatial complex variable) allow for an analytical solution to the problem of supersonic flow around a cone in the area of the boundary layer and beyond. The peculiar feature of the Navier-Stokes equations is that they allow for the determination of analytical velocity fields of fluids only for a small number of simple problems. Of course, the problem of the supersonic motion of fluid around a cone is not included in this number. Tunnel mathematics is a method for finding analytical vector velocity fields for steady flows of fluids with axial symmetry. The Navier-Stokes equations are then used to determine pressure and temperature distributions. The main theorem of tunnel mathematics allows for the determination of these distributions for planes z = const (similar to constructing slices of a brain in an MRI procedure). By collecting these “slices,” we can obtain full space distributions of pressure and temperature around a supersonic cone. At this stage of investigation, the conclusions obtained through tunnel mathematics make it possible to qualitatively assess the thickness of the boundary layer on the cone's surface, the shape of the shock wave, and whether the shock wave intersects the boundary layer. First of all, we focused on ensuring that the resulting solutions corresponded to the physical pattern of phenomena. No doubt, solutions obtained through tunnel mathematics must be confirmed experimentally.

Self-supervised tomographic image noise suppression via residual image prior network

Computed tomography (CT) denoising is a challenging task in medical imaging that has garnered considerable attention. Supervised networks require a lot of noisy-clean image pairs, which are always unavailable in clinical settings. Existing self-supervised algorithms for suppressing noise with paired noisy images have limitations, such as ignoring the residual between similar image pairs during training and insufficiently learning the spectrum information of images. In this study, we propose a Residual Image Prior Network (RIP-Net) to sufficiently model the residual between the paired similar noisy images. Our approach offers new insights into the field by addressing the limitations of existing methods. We first establish a mathematical theorem clarifying the non-equivalence between similar-image-based self-supervised learning and supervised learning. It helps us better understand the strengths and limitations of self-supervised learning. Secondly, we introduce a novel regularization term to model a low-frequency residual image prior. This can improve the accuracy and robustness of our model. Finally, we design a well-structured denoising network capable of exploring spectrum information while simultaneously sensing context messages. The network has dual paths for modeling high and low-frequency compositions in the raw noisy image. Additionally, context perception modules capture local and global interactions to produce high-quality images. The comprehensive experiments on preclinical photon-counting CT, clinical brain CT, and low-dose CT datasets, demonstrate that our RIP-Net is superior to other unsupervised denoising methods.

The Banach Fixed Point Theorem: selected topics from its hundred-year history

On June 24, 1920 Stefan Banach presented his doctoral dissertation titled O operacjach na zbiorach abstrakcyjnych i ich zastosowaniach do równañ całkowych (On operations on abstract sets and their applications to integral equations) to the Philosophy Faculty of Jan Kazimierz University in Lvov. He passed his PhD examinations in mathematics, physics and philosophy, and in January 1921 he became a doctor. A year later, he published the results of his doctorate in Fundamenta Mathematicae. Among them there was the theorem known today as the Banach Fixed Point Theorem or the Banach Contraction Principle. It is one of the most famous theorems in mathematics, one of many under the name of Banach. It concerns certain mappings (called contractions) of a complete metric space into itself and it gives the conditions sufficient for the existence and uniqueness of a fixed point of such mapping. In 2022 we had a centenary of publishing this theorem. In the paper, we want to present its most important modifications and generalizations, several contractive conditions, the converse theorems and some applications. It is not possible to provide complete information about what has been written during the last hundred years about the Banach Fixed Point Theorem and we are just trying to touch on some breakthrough moments in the development of the metric fixed point theory. The main purpose of this article is to organize the knowledge on this subject and to elaborate a broad bibliography which all interested persons can refer to.

Computing Edge Irregularity Strength of Star and Banana Trees Using Algorithmic Approach

After the Chartrand definition of graph labeling, since 1988 many graph families have been labeled through mathematical techniques. A basic approach in those labelings was to find a pattern among the labels and then prove them using sequences and series formulae. In 2018, Asim applied computer-based algorithms to overcome this limitation and label such families where mathematical solutions were either not available or the solution was not optimum. Asim et al. in 2018 introduced the algorithmic solution in the area of edge irregular labeling for computing a better upper-bound of the complete graph \(es(K_n)\) and a tight upper-bound for the complete \(m\)-ary tree \({es(T}_{m,h})\) using computer-based experiments. Later on, more problems like complete bipartite and circulant graphs were solved using the same technique. Algorithmic solutions opened a new horizon for researchers to customize these algorithms for other types of labeling and for more complex graphs. In this article, to compute edge irregular \(k\)-labeling of star \(S_{m,n}\) and banana tree \({BT}_{m,n}\), new algorithms are designed, and results are obtained by executing them on computers. To validate the results of computer-based experiments with mathematical theorems, inductive reasoning is adopted. Tabulated results are analyzed using the law of double inequality and it is concluded that both families of trees observe the property of edge irregularity strength and are tight for \(\left\lceil \frac{|V|}{2} \right\rceil\)-labeling.

Aksiologi Sains pada Matematika

This research investigates the relationship between the axiology of science and mathematics, focusing on values, ethics, and norms in the context of scientific research. The study underscores the importance of integrating the axiology of science into mathematics, highlighting that values and ethics should be integral considerations in the development of mathematical knowledge. The research methodology employed is a comprehensive literature review, exploring the contributions of ethical values to the development of mathematics and the ethical implications in mathematical practice. The results of the study reveal that mathematics is not a separate entity from values and ethics; rather, it is closely intertwined with them. The research details how personal values, scientific values, and social values influence the mathematical research process and its application in various scientific disciplines. Ethical implications in the development of mathematical theory are also a significant focus, considering how ethics affect axiomatic considerations, definitions, and theorems in mathematics. This research suggests that integrating the axiology of science into mathematics is highly relevant for the advancement of mathematical knowledge. It helps mitigate biases, enhances understanding of virtues, raises ethical awareness, and improves the abilities of mathematicians to adhere to values, ethics, and norms in their research. The research findings hold significance in the contexts of mathematical education and practice and in promoting ethical considerations in the use of mathematics in science and society.

Covering assisted intuitionistic fuzzy bi-selection technique for data reduction and its applications

The dimension and size of data is growing rapidly with the extensive applications of computer science and lab based engineering in daily life. Due to availability of vagueness, later uncertainty, redundancy, irrelevancy, and noise, which imposes concerns in building effective learning models. Fuzzy rough set and its extensions have been applied to deal with these issues by various data reduction approaches. However, construction of a model that can cope with all these issues simultaneously is always a challenging task. None of the studies till date has addressed all these issues simultaneously. This paper investigates a method based on the notions of intuitionistic fuzzy (IF) and rough sets to avoid these obstacles simultaneously by putting forward an interesting data reduction technique. To accomplish this task, firstly, a novel IF similarity relation is addressed. Secondly, we establish an IF rough set model on the basis of this similarity relation. Thirdly, an IF granular structure is presented by using the established similarity relation and the lower approximation. Next, the mathematical theorems are used to validate the proposed notions. Then, the importance-degree of the IF granules is employed for redundant size elimination. Further, significance-degree-preserved dimensionality reduction is discussed. Hence, simultaneous instance and feature selection for large volume of high-dimensional datasets can be performed to eliminate redundancy and irrelevancy in both dimension and size, where vagueness and later uncertainty are handled with rough and IF sets respectively, whilst noise is tackled with IF granular structure. Thereafter, a comprehensive experiment is carried out over the benchmark datasets to demonstrate the effectiveness of simultaneous feature and data point selection methods. Finally, our proposed methodology aided framework is discussed to enhance the regression performance for IC50 of Antiviral Peptides.

A novel intuitionistic fuzzy rough instance selection and attribute reduction with kernelized intuitionistic fuzzy C-means clustering to handle imbalanced datasets

Due to advancement of internet and lab based technologies, large volume of high dimensional data are generated every day. These data usually consisted of several issues such as class imbalance, noise, later uncertainty, irrelevant and/or redundant features, and redundancy in size. These issues degrade the overall performance measures of the various machine learning algorithms. An efficient method to cope with such issues for a large sized datasets is to apply efficient data reduction techniques. In the recent years, numerous data reduction techniques have been presented based on fuzzy rough set theory to tackle these obstacles. However, building such a method that can handle all the above mentioned issues simultaneously is still a challenging task. In this study, we handle all these obstacles simultaneously by introducing a novel approach to obtain the reduced dataset by combining kernelized intuitionistic fuzzy C-means with intuitionistic fuzzy rough set model. Intuitionistic fuzzy rough set handles the vagueness and uncertainty in a better way than fuzzy set aided models as it takes use of membership, non-membership as well as hesitancy to capture the uncertainty of real-valued datasets. To generate the membership and non-membership grades, kernelized intuitionistic fuzzy C- means based notion is introduced. Further, an intuitionistic fuzzy rough set model is established by addressing lower and upper approximations based on a novel similarity relation. Moreover, all the necessary conditions are justified with the relevant mathematical theorems. Next, this model is employed for dimensionality reduction based on the concept of discernibility matrix by using the idea of different classes’ ratios to avoid the noise. Thereafter, positive region is defined with the help of lower approximation. The positive region information is applied to tackle problematic instances available in both minority and majority class for the imbalanced dataset after generation of artificial samples by synthetic minority oversampling technique (SMOTE). A comprehensive experimental study is added to show the effectiveness of the proposed technique. Finally, a framework is established to improve the prediction of animal toxin peptides.

Slow dissipation and spreading in disordered classical systems: A direct comparison between numerics and mathematical bounds.

We study the breakdown of Anderson localization in the one-dimensional nonlinear Klein-Gordon chain, a prototypical example of a disordered classical many-body system. A series of numerical works indicate that an initially localized wave packet spreads polynomially in time, while analytical studies rather suggest a much slower spreading. Here, we focus on the decorrelation time in equilibrium. On the one hand, we provide a mathematical theorem establishing that this time is larger than any inverse power law in the effective anharmonicity parameter λ, and on the other hand our numerics show that it follows a power law for a broad range of values of λ. This numerical behavior is fully consistent with the power law observed numerically in spreading experiments, and we conclude that the state-of-the-art numerics may well be unable to capture the long-time behavior of such classical disordered systems.

Gauging non-invertible symmetries: topological interfaces and generalized orbifold groupoid in 2d QFT

Gauging is a powerful operation on symmetries in quantum field theory (QFT), as it connects distinct theories and also reveals hidden structures in a given theory. We initiate a systematic investigation of gauging discrete generalized symmetries in two-dimensional QFT. Such symmetries are described by topological defect lines (TDLs) which obey fusion rules that are non-invertible in general. Despite this seemingly exotic feature, all well-known properties in gauging invertible symmetries carry over to this general setting, which greatly enhances both the scope and the power of gauging. This is established by formulating generalized gauging in terms of topological interfaces between QFTs, which explains the physical picture for the mathematical concept of algebra objects and associated module categories over fusion categories that encapsulate the algebraic properties of generalized symmetries and their gaugings. This perspective also provides simple physical derivations of well-known mathematical theorems in category theory from basic axiomatic properties of QFT in the presence of such interfaces. We discuss a bootstrap-type analysis to classify such topological interfaces and thus the possible generalized gaugings and demonstrate the procedure in concrete examples of fusion categories. Moreover we present a number of examples to illustrate generalized gauging and its properties in concrete conformal field theories (CFTs). In particular, we identify the generalized orbifold groupoid that captures the structure of fusion between topological interfaces (equivalently sequential gaugings) as well as a plethora of new self-dualities in CFTs under generalized gaugings.

Doumou Douala Therapeutic Theorems for Modulo 3 Congruence Memory

Three mathematical theorems invented by Doumou in the medical field, aimed at curing and relieving people. Its proposition can be demonstrated by logical reasoning based on established facts or justifiable hypotheses, which is of crucial importance in the treatment of illnesses, aiming to cure them. This theorem enables the individual to find a perfect fit, a harmony between his thoughts, feelings, words and actions.

Hybrid similarity relation based mutual information for feature selection in intuitionistic fuzzy rough framework and its applications

Fuzzy rough entropy established in the notion of fuzzy rough set theory, which has been effectively and efficiently applied for feature selection to handle the uncertainty in real-valued datasets. Further, Fuzzy rough mutual information has been presented by integrating information entropy with fuzzy rough set to measure the importance of features. However, none of the methods till date can handle noise, uncertainty and vagueness simultaneously due to both judgement and identification, which lead to degrade the overall performances of the learning algorithms with the increment in the number of mixed valued conditional features. In the current study, these issues are tackled by presenting a novel intuitionistic fuzzy (IF) assisted mutual information concept along with IF granular structure. Initially, a hybrid IF similarity relation is introduced. Based on this relation, an IF granular structure is introduced. Then, IF rough conditional and joint entropies are established. Further, mutual information based on these concepts are discussed. Next, mathematical theorems are proved to demonstrate the validity of the given notions. Thereafter, significance of the features subset is computed by using this mutual information, and corresponding feature selection is suggested to delete the irrelevant and redundant features. The current approach effectively handles noise and subsequent uncertainty in both nominal and mixed data (including both nominal and category variables). Moreover, comprehensive experimental performances are evaluated on real-valued benchmark datasets to demonstrate the practical validation and effectiveness of the addressed technique. Finally, an application of the proposed method is exhibited to improve the prediction of phospholipidosis positive molecules. RF(h2o) produces the most effective results till date based on our proposed methodology with sensitivity, accuracy, specificity, MCC, and AUC of 86.7%, 90.1%, 93.0% , 0.808, and 0.922 respectively.

Couplings and Matchings: Combinatorial notes on Strassen’s theorem

Some mathematical theorems represent ideas that are discovered again and again in different forms. One such theorem is Hall’s marriage theorem. This theorem is equivalent to several other theorems in combinatorics and optimization theory, in the sense that these results can easily be derived from each other. Remarkably, this equivalence extends to Strassen’s theorem, a celebrated result on couplings of probability measures. In this paper the equivalence between Strassen’s theorem and Hall’s theorem is investigated from a combinatorial perspective. A novel combinatorial lemma will be introduced that can be used to deduce both Hall’s theorem and a finite version of Strassen’s theorem, providing a simple proof of their equivalence.