The study shows that the volumes and surfaces of $n$-balls, $n$-simplices, and $n$-orthoplices are holomorphic functions of $n$, which makes those objects omnidimensional, that is well defined in any complex dimension. Applications of these formulas to the omnidimensional polytopes inscribed in and circumscribed about $n$-balls reveal previously unknown properties of these geometric objects. In particular, for $0 < n < 1$, the volumes of the omnidimensional polytopes are larger than those of circumscribing $n$-balls, and both their volumes and surfaces are smaller than those of inscribed $n$-balls. The surface of an $n$-simplex circumscribing unit diameter $n$-ball is spirally convergent to zero with real $n$ approaching negative infinity but first has a local maximum at $n=-3.5$. The surface of an $n$-orthoplex circumscribing unit diameter $n$-ball is spirally divergent with real $n$ approaching negative infinity but first has a local minimum at $n=-1.5$, where its real and imaginary parts are equal to each other; similarly, as its volume, where the similar local minimum occurs at $n=-3.5$. Reflection functions for volumes and surfaces of these polytopes inscribed in and circumscribed about $n$-balls are proposed. symmetries of products and quotients of volumes in complex dimensions $n$ and $-n$ and of surfaces in complex dimensions $n$ and $2-n$ are shown to be independent of the metric factor and the gamma function. Specific symmetries also hold between volumes and surfaces in dimensions $n = -1/2$ and $n = 1/2$.

In this paper we provide some important and significant observations on invo-regular rings appeared in Ann. Univ. Mariae Curie-Sklodowska Sect. A Mathematica (2018). In addition we exhibit that strongly invo-regular rings, invo-regular rings and quasi invo-regular rings all coincide with the well known and well characterized tripotent rings and all these rings further coincide with the well known unit regular rings.

Constructing new graph from the graph's parameters and related notions in the way that, the study on the new graph and old graph in their parameters could be facilitated. As graph, new graph has some characteristics and results which are related to the structure of this graph. For this purpose, regular graph is considered so the internal relation and external relation on this new graph are studied. The kind of having same number of edges when this number is originated by common number of graphs like maximum degree, minimum degree, domination number, coloring number and clique number, is founded in the word of having regular graph

The study examines n-balls, regular n-simplices, and n-orthoplices in real dimensions using novel recurrence relations that removed indefiniteness present in known formulas. They show that in the negative, integer dimensions volumes of n-balls are zero if n is even, positive if n = -4k - 1, and negative if n = -4k - 3, for natural k. Volumes and surfaces of n-cubes inscribed in n-balls in negative dimensions are complex, wherein for negative, integer dimensions they are associated with integral powers of the imaginary unit. The relations are continuous for n Î ℝ and show that the constant of π is absent for 0 ≤ n < 2. For n < -1 self-dual n-simplices are undefined in the negative, integer dimensions, and their volumes and surfaces are imaginary in the negative, fractional ones, and divergent with decreasing n. In negative, integer dimensions n-orthoplices reduce to the empty set, and their real volumes and imaginary surfaces are divergent in negative, fractional ones with decreasing n. Out of three regular, convex polytopes present in all natural dimensions, only n-orthoplices, n-cubes (and n-balls) are defined in the negative, integer dimensions.

A closed chain of oscillators can be considered as a model of ring-shaped ecosystems, such as atolls or coastal zones of inland reservoirs. As an oscillator model, we use the logistic map that often referred to as an archetypical example of how complex dynamics can arise from very simple nonlinear equations. We investigate the influence of the model parameters both on the nature of oscillations in the oscillator ring and on the spatial structures that arise in this case. Namely, we demonstrate a variety of emerging spatial structures depending on the initial conditions.

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. We prove the Riemann hypothesis using the Complexity Theory. Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. The Goldbach's conjecture is one of the most important and unsolved problems in number theory. Nowadays, it is one of the open problems of Hilbert and Landau. We show the Goldbach's conjecture is true using the Complexity Theory as well. An important complexity class is 1NSPACE(S(n)) for some S(n). These mathematical proofs are based on if some unary language belongs to 1NSPACE(S(log n)), then the binary version of that language belongs to 1NSPACE(S(n)) and vice versa.

A fast algorithm is established to transform points of the unit sphere into fundamental region symmetrically. With the resulting algorithm, a flexible form of invariant mappings is achieved to generate aesthetic patterns with symmetries of the regular polyhedra. This method avoids the order restriction of symmetry groups, which can be similarly extended to treat regular polytopes in n-dimensional space for n>=4.

In this article some important observations have been reported on recent works related to weakly tripotent rings and locally invo-regular rings. Our findings give additional results as well as correct some recent results on weakly tripotent rings and locally invo-regular rings appeared in Rendiconti Sem. Mat. Univ. Pol. Torino (2021) and Azerbaijan Journal of Mathematics (2021) respectively. In addition we exhibit that if the Jacobson radical J(A) of a ring is strongly involution t-clean then the characteristic of J(A) need not be four. This finding improves an important result appeared in Eur. J. Pure Appl. Math (2022).

In this paper, we give the Galois adjunction form of the regular relation, and by using residuated mapping, we find the connection between it and order-generating set. Furthermore, we obtain the equivalent conclusion of "regular". In addition, by using the specialization order of topological space and the lower approximate operator in the rough set, we find a continuous homeomorphic mapping from a regular space to a Alexandrov topological space

New setting is introduced to study the global powerful alliance. Global powerful alliance is about a set of vertices which are applied into the setting of neutrosophic graphs. Neighborhood has the key role to define this notion. Also, neighborhood is defined based on strong edges. Strong edge gets a framework as neighborhood and after that, too close vertices have key role to define global powerful alliance based on strong edges. The structure of set is studied and general results are obtained. Also, some classes of neutrosophic graphs excluding empty, path, star, and wheel and containing complete, cycle and r-regular-strong are investigated in the terms of set, minimal set, number, and neutrosophic number. Neutrosophic number is used in this way. It’s applied to use the type of neutrosophic number in the way that, three values of a vertex are used and they’ve same share to construct this number. It’s called “modified neutrosophic number”. Summation of three values of vertex makes one number and applying it to a set makes neutrosophic number of set. This approach facilitates identifying minimal set and optimal set which forms minimal-global-powerful-alliance number and minimal-global-powerful-alliance-neutrosophic number. Two different types of sets namely global-powerful alliance and minimal-global-powerful alliance are defined. Global-powerful alliance identifies the sets in general vision but minimal-global-powerful alliance takes focus on the sets which deleting a vertex is impossible. Minimal-global-powerful-alliance number is about minimum cardinality amid the cardinalities of all minimal-global-powerful alliances in a given neutrosophic graph. New notions are applied in the settings both individual and family. Family of neutrosophic graphs has an open avenue, in the way that, the family only contains same classes of neutrosophic graphs. The results are about minimal-global-powerful alliance, minimal-global-powerful-alliance number and its corresponded sets, minimal-global-powerful-alliance-neutrosophic number and its corresponded sets, and characterizing all minimal-global-powerful alliances, minimal-t-powerful alliance, minimal-t-powerful-alliance number and its corresponded sets, minimal-t-powerful-alliance-neutrosophic number and its corresponded sets, and characterizing all minimal-t-powerful alliances. The connections amid t-powerful-alliances are obtained. The number of connected components has some relations with this new concept and it gets some results. Some classes of neutrosophic graphs behave differently when the parity of vertices are different and in this case, cycle, and complete illustrate these behaviors. Two applications concerning complete model as individual and family, under the titles of time table and scheduling conclude the results and they give more clarifications and closing remarks. In this study, there’s an open way to extend these results into the family of these classes of neutrosophic graphs. The family of neutrosophic graphs aren’t study deeply and with more results but it seems that analogous results are determined. Slight progress is obtained in the family of these models but there are open avenues to study family of other models as same models and different models. There’s a question. How can be related to each other, two sets partitioning the vertex set of a graph? The ideas of neighborhood and neighbors based on strong edges illustrate open way to get results. A set is global powerful alliance when two sets partitioning vertex set have uniform structure. All members of set have more amount of neighbors in the set than out of set and reversely for non-members of set with less members in the way that the set is simultaneously t-offensive and(t-2)-defensive. A set is global if t=0. It leads us to the notion of global powerful alliance. Different edges make different neighborhoods but it’s used one style edge titled strong edge. These notions are applied into neutrosophic graphs as individuals and family of them. Independent set as an alliance is a special set which has no neighbor inside and it implies some drawbacks for these notions. Finding special sets which are well-known, is an open way to purse this study. Special set which its members have only one neighbor inside, characterize the connected components where the cardinality of its complement is the number of connected components. Some problems are proposed to pursue this study. Basic familiarities with graph theory and neutrosophic graph theory are proposed for this article.

In this note we generalize the regularity concept for semigroups in two ways simultaneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions.

In this lecture notes, we will introduce Asymptotics, then we will give a short glimpse on Perturbation theory (regular versus singular), which plays a crucial role especially in theoretical physics. Our goal is to find Asymptotic series that approximates the values of integrals depending on some parameter or the solutions of differential equations. These methods that lead to obtain more effective algorithms of numerical evaluation are called: Asymptotic Methods.

The three-dimensional morphology of frictional sub-surfaces holds significant importance for studying tribological issues. However, the uniformity of horizontal datum in 3D scanning is limited for curved surfaces, resulting in the inability to obtain accurate contour characterization parameters from the scanning results. This study aims to address this issue by constructing regular surface equations and normalizing the 3D profiler scanning results. By fitting the data, a "plane" surface representative of the surface features is obtained, and the paper demonstrates this approach on the surface morphology of different worn parts in the frictional area of spherical bearings located in the specific environment of the Three Gorges gate. The results indicate that the obtained "plane" effectively reconstructs the three-dimensional morphology map of the regular surface. Moreover, this reconstructed plane not only clearly illustrates the surface characteristics but also provides the foundation for analyzing the wear mechanism.

Let N = N₀+ N₁+ N₂ be a Z₃-graded Novikov algebra. The main goal of the paper is to prove that over a field of characteristic not equal to 3 the algebra N is solvable if N₀ is solvable. We also show that a $Z_2$-graded Novikov algebra N=N₀+ N₂ over a field of characteristic not equal to 2 is solvable if N₀ is solvable. This implies that for every n of the form n=2^k3^l, any Z_n-graded Novikov algebra N over a field of characteristic not equal to 2,3 is solvable if N₀ is solvable.

A lot of research studies have been surveyed the completed genomes of prokaryotic and eukaryotic and focused on the correlation between the percentage of microsatellite sequences in completed genomes and the whole size of the organism genomes. There are fewer studies made in repetitive sequences otherwise simple sequence repeats or long tandem repeats of virus genomes. simple sequence repeats (SSRs) are the most important regions for recombination and moving repeats blocks from site to another site in the genomes. A tool was programmed and designed by visual basic 6.0 to find the long tandem repeats in DNA sequences of the small genomes. The tool named “Repeater Finder Regular Expression”, (RFRE) Version 1.0, 2016. The tool was utilized to discover different pattern of long tandem repeats (LTR) motifs on the completed genomes of human corona virus strains by using a joined regular expression language. In this study, a twenty-nine accession numbers of human coronavirus completed genomes, (hku1) strains were retrieved from the Genbank. The researcher can write a different regular expression patterns and joined regular expression patterns through the designed tool to search and find a specific motif of nucleotide sequences inside the complete genomes. The RFRE tool searched and found three different total lengths of a perfect long tandem repeats (240bp, 300bp and 480bp). A Dot plot gave a picture view for the long tandem repeat sequences in the completed genome sequence (KF430201.1) of human coronavirus. The genomic dot plot tool YASS was used as a genomic similarity searching tool to check for the uninterrupted repeats and confirm the sensitivity of the (RFRE) tool. To identify the recombination site in the genomes of human coronavirus the RAT tool was applied to find the recombination sites between the completed genomes of human corona viruses .The RAT tool recognized the recombination site in the nucleotide position (3012) and at the same time this recombination site position (3012) was also recognized as a beginning position of a long tandem repeat. A precise motif was predicted from the translated repeats of Human Corona Virus which found by PRATT tool. There was a relationship between the total length of long tandem repeats and genome size of Human Corona Virus and the correlation value R² was equal to (0,451). In conclusion, this study presented the importance of finding the long tandem repeats of human coronavirus and gives a relationship between the completed genome size of human coronavirus types and long tandem repeats. The nucleotide position (3012) was a hot spot site for a recombination among the complete genomes of human coronavirus and also identified as a repetitive site in the genomes of human corona virus (hku1). The repeats in human coronavirus (hku1) were predicated to be a main major role of virus evolution.

We consider Pisot family substitution tilings in $\R^d$ whose dynamical spectrum is pure point. There are two cut-and-project schemes(CPS) which arise naturally: one from the Pisot family property and the other from the pure point spectrum respectively. The first CPS has an internal space $\R^m$ for some integer $m \in \N$ defined from the Pisot family property, and the second CPS has an internal space $H$ which is an abstract space defined from the property of the pure point spectrum. However it is not known how these two CPS's are related. Here we provide a sufficient condition to make a connection between the two CPS's. In the case of Pisot unimodular substitution tiling in $\R$, the two CPS's turn out to be same due to [5, Remark 18.5].

Acute intense exercise causes significant oxidative stress and consequently an increase in total antioxidant capacity; however, the mechanisms and combined effects of intense exercise and smoking on oxidative stress among active and non-active smokers are not clear. The aim of this study was to investigate the effect of acute intense exercise on antioxidant enzyme activity responses in active and non-active individuals exposed to cigarette smoke. The study included 40 subjects who were equally classified as: smokers that did exercise (SE), smokers that did not do exercise (SnE), non-smokers that did exercise (NSE), and non-smokers that did not do exercise (NSnE). The adjusted Astrand test was used to exhaust the subjects. Salivary enzymes of peroxidase (POX), catalase (CAT), and superoxide dismutase (SOD) were measured, by spectrophotometry methods, at 3 different time points: pre-test (TP1), post-test (TP2), and one hour after finishing the test (TP3). Significant (p<0.05) group x time interactions were found for the three enzymes. Salivary POX, CAT and SOD increased in all groups from TP1 to TP2 and decreased from TP2 to TP3. Only the NSE showed a significant difference between TP1 to TP3 in POX and SOD by +0.011 ± 0.007 and +0.075 ± 0.02 (U/ml), respectively. The NSE showed significantly higher levels of POX, CAT and SOD in TP2 compared to the other groups. Furthermore, NSE and NSnE had higher levels of POX, CAT and SOD in TP1 and TP3 (p<0.05) compared with SE and SnE. Only in the NSnE, were no differences observed in CAT compared with SE and SnE in TP3. These results showed that the antioxidant level at rest and in the recovery time after the acute intense exercise was lower in SE and SnE compared with NSE and NSnE, suggesting that smoking habit may reduce the ameliorating effect of regular physical activity on acute exercise-induced oxidative stress.

General characteristics of K changes together with their fine structure associated with ground flashes in Sri Lanka in the tropics are presented. It is found that on average there are about 2 K changes associated with each return stroke. Analysis of the fine structure of the K changes shows that the K change is a chaotic pulse burst. Some of these chaotic pulse bursts start and the others end as a regular pulse bursts. Sometimes the chaotic part occurs in between two regular pulse bursts. This is in agreement with the recent published results that claim that chaotic pulse bursts are a random superposition of regular pulse bursts. The results show that the small step fields identified in the literature as K changes are the static fields associated with these pulse bursts.

In this note we consolidate and give a brief description of several recently published results in ring theory having disproof. These results have been published during 2016 to 2021 in the so called non-predatory reputed mathematical journals indexed in the well known database like Scopus. We have considered results on rings in which each element is a sum of two idempotents appeared in Canad. Math. Bull. (2016), weakly tripotent rings appeared in Bull. Korean Math. Soc. (2018) and Rendiconti Sem. Mat. Univ. Pol. Torino (2021), involution t-clean rings appeared in Eur. J. Pure Appl. Math (2022) and strongly 2-nil clean rings with units of order two appeared in Eur. J. Pure Appl. Math (2023).

. Background: regular exercise elicits adaptive changes from several organs and physiological processes, including erythrocyte properties. Methods: in a group of 79 subjects (62 men and 17 women; mean age 31.37 +/- 10.19 years) who trained several times a week as they practiced amateur sports, we evaluated the elongation index, marker of erythrocyte deformability, the red blood cell distribution width (RDW), indicator of erythrocyte anisocytosis, hematocrit, hemoglobin and the main erythrocyte indices (MCV, MCH, MCHC) in basal conditions. Results: in comparison with a group of healthy not trained volunteers, the values of the elongation index, and not the RDW, are increased and this datum is accompanied by an increase of MCV and MCHC, likely related to an increased presence of circulating young erythrocytes in trained subjects. We also divided the same group according to the median of the VO2max, observing that the subgroup above the median shows both an increase in the elongation index values and a decrease in MCH and MCHC. Conclusions: in trained subjects there is no correlation between the values of elongation index and the RDW, while of particular interest and of a certain complexity seem the interrelations among elongation index, RDW and the main erythrocyte indices.

Background: Sickle cell disease (SCD) is a public health problem in the Democratic Republic of Congo. While reference sickle cell centers have been implemented in capital cities of African countries and have proven to be beneficial for SCD patients, they have never been set up in rural areas for families with very low sources of income. Method: A cohort of 143 children with SCD aged 10 years old (IQR (interquartile range): 6–15 years) (sex ratio male/female = 1.3) were clinically followed for 12 months without any specific intervention aside from the management of acute events, and then for 12 months with a monthly medical visit, biological follow-up, and regular prophylaxis. Results: The median age of patients at the diagnosis of SCD was 2 years (IQR: 1–5). The implementation of standardized and regular follow-ups in a new sickle cell referral center in a remote city showed an increase in the annual mean hemoglobin level from 50 to 70 g/L (p = 0.001), and a decrease in the lymphocyte count and spleen size (p < 0.001). A significant decrease (p < 0.001) in the average annual number of hospitalizations and episodes of vaso-occlusive crises, blood transfusions, infections, and acute chest syndromes were also observed. Conclusions: Creation of a sickle cell referral center and the regular follow-up of children with SCD are possible and applicable in the context of a remote city of an African country and represent simple and accessible measures that can reduce the morbimortality of children with sickle cell disease.

In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty principle. They both express Fourier dualities within the space of tempered distributions and these dualities are furthermore the inverses of one another. While Poisson’s summation formula expresses a duality between discretization and periodization, Heisenberg’s uncertainty principle expresses a duality between regularization and localization. We define regularization and localization on generalized functions and show that the Fourier transform of regular functions are local functions and, vice versa, the Fourier transform of local functions are regular functions.

The paper discusses the classification of fuzzy metrics based on their continuity conditions, dividing them into Erceg, Deng, Yang-Shi, and Chen metrics. It explores the relationships between these types of fuzzy metrics, concluding that a Deng metric in [0,1]-topology must also be Erceg, Chen, and Yang-Shi metrics. The paper also proves that the product of countably many Deng pseudo-metric spaces remains a Deng pseudo-metric space, and demonstrates some σ-locally finite properties of Deng metric space. Additionally, the paper constructs two interrelated mappings based on normal space and concludes that if a [0,1]-topological space is T1 and regular, and its topology has a σ-locally finite base, then it is Deng metrizable, and thus Erceg, Yang-Shi, and Chen metrizable as well.

Binary mixtures of surfactants build a binary mixed micelle in which the ratio of surfactants usually differs from the initial ratio of surfactants in their binary mixture. The thermodynamic stabilization of the binary mixed micellar pseudophase about the hypothetical ideal state (intermolecular interactions between the different particles and the conformational states of the particles are identical to those of monocomponent states) is described by the molar excess Gibbs free energy (gE). The dependence of gE on the molar fraction of surfactant i (xi) from the binary mixed micelle can be described by a symmetric function (symmetry is described to the line parallel to the y-axis and passes through xi = 0.5) or by an asymmetric function. Theoretical analysis (canonical partition function, conformational analysis) examines how the presence of different polar functional groups, some of which are sterically shielded from the steroid skeleton of bile salt (surfactant), affect the symmetry of the function gE of the binary mixed micelle of the cholic acid anion (bile salt) and classical cationic surfactant (hydrophobic tail and polar head). Suppose the steroid skeleton of the bile salt contains non-sterically shielded polar groups (or the temperature is relatively high). In that case, gE is a symmetric function. At the same time, if the steroid skeleton also contains sterically shielded polar groups, then the gE function is asymmetric.

Age-related Macular Degeneration (AMD) is the major cause of elders’ vision loss, early screening and treatment are the most efficient way to reduce the rate of blindness. AI-based methods based on ophthalmic images play great potential for AMD diagnosis. However, low levels of accuracy, robustness, and explainability are challenges for AI approaches applied in clinics. Thus, this study proposed a multi-type of data source fusion method and a multi-model fusion approach for AMD detection. Typical unsupervised (Hierarchical Clustering and K-Means), typical supervised (SVM, VGG-16, and ResNet) methods, and proposed methods (multi-source data fusion-based method and multi-model fusion-based approach) are compared based on Optical Coherence Tomography (OCT), Fundus Autofluorescence (FAF), regular color fundus photography (CFP) and Ultra-Wide field Fundus (UWF) images. Data preprocessing and enhancements of each type of data are discussed. A feature extraction based on unsupervised ML models, feature combination and normalization, and multi-layer perception (MLP) algorithm are involved in the proposed multi-source data fusion-based method. Supervised ML and DL models and a voting mechanism are involved in the multi-model fusion-based approach. Findings show that the proposed methods present a high performance of accuracy and robustness. A real-world UWF database is involved from Shenzhen Aier Hospital. Practical and theoretical contributions are delivered. A reference value for medical diagnosis based on multiple digital images is contributed.

Most of the research in data mining concentrated on finding the positive associations that exist in frequent patterns. Frequent patterns have no bearing on the time duration and, therefore, lose interestingness. The Regularity of an itemset is related to the time at which the transaction occurred and the time distance between the transactions. Frequent Regular item sets can be generated considering the Frequency and the Regularity of the occurrence of the Item Sets. The frequent and regular Item sets will be so huge that they cannot be handled in real time. The count of Frequent and regular item sets gets reduced drastically when the properties “closed” and “Maximality” are applied. Finding negative associations among such patterns are important in the field of medicine. Many drugs administered to cure some diseases have counter reactions leading to many other diseases. Negative associations among the mined patterns are important as they reveal significant contradictions frequently among medical drugs administered by medical practitioners to cure some diseases. This paper proposes a method that mines medical databases to find regular, frequent patterns applied with closed and maximal properties and find negative associations among those patterns. The proposed algorithm is accurate to 98% and performs 9% more efficiently than the other algorithms presented in the literature.