University of IsfahanMathematics and Society2345-649310420260220Bayesian Neural Networks; why and how?Bayesian Neural Networks; why and how?1242881410.22108/msci.2024.141722.1668FAMoeinMonemiSchool of Engineering Science, College of Engineering, University of Tehran, Tehran, IranS. MahmoudTaheriSchool of Engineering Science, College of Engineering, University of Tehran Tehran, IranMortezaAminiDepartment of Algorithms and Computation, School of Engineering Sciences, University of TehranJournal Article20240605One of the main challenges in utilizing neural networks is the problem of overfitting. This occurs when a neural network model fits the training data too precisely, but fails to generalize to data outside the training set. This lack of generalization is often observed when the number of training samples is smaller than the number of features being analyzed, and the complexity of the model — that is, the number of weights and biases in the neural network — is high. In such situations, ensemble learning, and more specifically bagging methods, are commonly employed. These methods use resampling techniques to incorporate uncertainty into the model, thereby improving the model’s ability to generalize. However, when the training sample size is extremely limited, resampling becomes less effective, and the uncertainty introduced in the model is very limited. Bayesian neural networks address this by quantifying parameter uncertainty and considering parameter states that may not have been observed in the existing data. This leads to a significant improvement in model generalization. In addition to mitigating overfitting, this approach also provides access to the posterior predictive distribution, allowing for the calculation of prediction intervals. In this article, we briefly review Bayesian neural networks, explain how they are trained, and then analyze data and compare these models with standard neural networks.One of the main challenges in utilizing neural networks is the problem of overfitting. This occurs when a neural network model fits the training data too precisely, but fails to generalize to data outside the training set. This lack of generalization is often observed when the number of training samples is smaller than the number of features being analyzed, and the complexity of the model — that is, the number of weights and biases in the neural network — is high. In such situations, ensemble learning, and more specifically bagging methods, are commonly employed. These methods use resampling techniques to incorporate uncertainty into the model, thereby improving the model’s ability to generalize. However, when the training sample size is extremely limited, resampling becomes less effective, and the uncertainty introduced in the model is very limited. Bayesian neural networks address this by quantifying parameter uncertainty and considering parameter states that may not have been observed in the existing data. This leads to a significant improvement in model generalization. In addition to mitigating overfitting, this approach also provides access to the posterior predictive distribution, allowing for the calculation of prediction intervals. In this article, we briefly review Bayesian neural networks, explain how they are trained, and then analyze data and compare these models with standard neural networks.https://math-sci.ui.ac.ir/article_28814_b5f97f869fe61bad8ebf7de954a9e543.pdfUniversity of IsfahanMathematics and Society2345-649310420250701The scientific collaboration of researchers in the field of entrepreneurship management using vertex coloring and co-authored polygraphsThe scientific collaboration of researchers in the field of entrepreneurship management using vertex coloring and co-authored polygraphs25522896410.22108/msci.2024.141644.1666FARezaRostameiBusiness Administration, Payam Noor University, P. O. Box 19395-4697, Tehran, Iran.FarzadShaveisiDepartment of Mathematics, Faculty of Science, Razi University, P. O. Box 67144-14971, Kermanshah, Iran.MostafaAminiDepartment of Mathematics, Faculty of Sciences, Payame Noor University, P. O. Box 19395-4697, Tehran, Iran.Journal Article20240528The purpose of the current research is to study and compare graphs co-authored by Iranian researchers in Persian scientific research journals in the field of entrepreneurship management using vertex and edge coloring of graphs. In this research, the data related to the scientific-research documents of 308 Iranian researchers in 5 Persian scientific research journals in the field of entrepreneurship management from 2019 to 2023 by using the issues printed in these journals and the graphs related to the authors of these journals were extracted using they are drawn from mathematical software and then compared and analyzed using algebraic parameters such as the average degree of vertices, independence, chromatic and matching number. In the main results, it was checked that the average grade of Smart Business Management Studies Journal is 4.3 and higher than other journals. Also, the largest research group (clique number) is 6 people and the most research variety (independence number) is 25 and it is related to entrepreneurship development magazines. Among the discussed journals in the field of entrepreneurship management, according to the average grades, the amount of research cooperation of the Journal of Smart Business Management Studies is at a higher level than the other four journals. Also, according to the independence number, the variety of research fields in the journal of entrepreneurship development is more than other journals discussed in this field in the country.The purpose of the current research is to study and compare graphs co-authored by Iranian researchers in Persian scientific research journals in the field of entrepreneurship management using vertex and edge coloring of graphs. In this research, the data related to the scientific-research documents of 308 Iranian researchers in 5 Persian scientific research journals in the field of entrepreneurship management from 2019 to 2023 by using the issues printed in these journals and the graphs related to the authors of these journals were extracted using they are drawn from mathematical software and then compared and analyzed using algebraic parameters such as the average degree of vertices, independence, chromatic and matching number. In the main results, it was checked that the average grade of Smart Business Management Studies Journal is 4.3 and higher than other journals. Also, the largest research group (clique number) is 6 people and the most research variety (independence number) is 25 and it is related to entrepreneurship development magazines. Among the discussed journals in the field of entrepreneurship management, according to the average grades, the amount of research cooperation of the Journal of Smart Business Management Studies is at a higher level than the other four journals. Also, according to the independence number, the variety of research fields in the journal of entrepreneurship development is more than other journals discussed in this field in the country.https://math-sci.ui.ac.ir/article_28964_4f60661267dd1e9d908add25c0946a0f.pdfUniversity of IsfahanMathematics and Society2345-649310420250629Calabi flow on Riemann surfacesCalabi flow on Riemann surfaces53602896310.22108/msci.2024.142651.1687FARezaSeyyedaliSchool of Mathematics, Institute for research in fundamental sciences, P.O. Box 19395-5746 Tehran, IranJournal Article20240831An important theorem in complex analysis is the uniformization theorem. As a result of uniformizationn theorem, any compact Riemann surface admits a metric of constant Gaussian curvature. A modern way to prove the uniformization theorem is to use geometric flows. Since seminal work of Perleman, many authors prove it using curvature flows. In an important work, Tian used Ricci flow to prove the uniformization theorem. Later Chen used the Calabi flow to give another proof of the Theorem. In contrast to Ricci flow, Calabi flow is a fourth order parabolic PDE. It is quite difficult to deal with fourth order PDEs partially due to lack of any maximum principle. Even proving the long time existence of Calabi flow is hard and it is still open in complex dimension $n \geq 2.$ In a breakthrough, Chen-Cheng provide a strong tool in order to deal with certain fourth order nonlinear PDEs. In this article, appealing to the results of Chen-Cheng, we give a different proof for the long time existence of the Calabi flow on compact Riemann surfaces of positive genus.An important theorem in complex analysis is the uniformization theorem. As a result of uniformizationn theorem, any compact Riemann surface admits a metric of constant Gaussian curvature. A modern way to prove the uniformization theorem is to use geometric flows. Since seminal work of Perleman, many authors prove it using curvature flows. In an important work, Tian used Ricci flow to prove the uniformization theorem. Later Chen used the Calabi flow to give another proof of the Theorem. In contrast to Ricci flow, Calabi flow is a fourth order parabolic PDE. It is quite difficult to deal with fourth order PDEs partially due to lack of any maximum principle. Even proving the long time existence of Calabi flow is hard and it is still open in complex dimension $n \geq 2.$ In a breakthrough, Chen-Cheng provide a strong tool in order to deal with certain fourth order nonlinear PDEs. In this article, appealing to the results of Chen-Cheng, we give a different proof for the long time existence of the Calabi flow on compact Riemann surfaces of positive genus.https://math-sci.ui.ac.ir/article_28963_63a155f4dee7af7b5a005d0fff110412.pdfUniversity of IsfahanMathematics and Society2345-649310420260220Of Cheese and Crust: A Proof of the Pizza Conjecture and Other Tasty ResultsOf Cheese and Crust: A Proof of the Pizza Conjecture and Other Tasty Results61842941810.22108/msci.2025.143345.1722FAHassanHaghighiFaculty of Mathematics, K. N. Toosi University of Technology Tehran, IranJournal Article20250116<strong>Translator's abstract: </strong>For a given natural number $N$, a circular pizza is divided into $2N$ equiangular slices by means of $N$ straight, concurrent cuts at an arbitrary point $P$ of the interior of the pizza. Then, each slice of the pizza is shared by two individuals (``Gray" and ``White"), who alternate slices. A natural question that may arise is that does the area of the gray slices exceed that of white slices? When $N$ is even, an answer to this question has already been given. In this paper, whenever $N$ is an odd number, a complete answer to this problem is given. Moreover, the method of proof are generalized to three dimensional pizzas, so called ``calzones", that are some type of pizzas which are obtained by filling the space above the circular pizzas with cheese surrounded by means of upper surfaces such as paraboloid, semi-ellipsoid or cone, and both volumes and surface areas of the Gray and White slices are computed.<strong>Translator's abstract: </strong>For a given natural number $N$, a circular pizza is divided into $2N$ equiangular slices by means of $N$ straight, concurrent cuts at an arbitrary point $P$ of the interior of the pizza. Then, each slice of the pizza is shared by two individuals (``Gray" and ``White"), who alternate slices. A natural question that may arise is that does the area of the gray slices exceed that of white slices? When $N$ is even, an answer to this question has already been given. In this paper, whenever $N$ is an odd number, a complete answer to this problem is given. Moreover, the method of proof are generalized to three dimensional pizzas, so called ``calzones", that are some type of pizzas which are obtained by filling the space above the circular pizzas with cheese surrounded by means of upper surfaces such as paraboloid, semi-ellipsoid or cone, and both volumes and surface areas of the Gray and White slices are computed.https://math-sci.ui.ac.ir/article_29418_9b2f9581da53cc51ae52bd11bcb32def.pdfUniversity of IsfahanMathematics and Society2345-649310420250903Some asymptotically Euclidean tangent bundles and their ADM massesSome asymptotically Euclidean tangent bundles and their ADM masses851042917110.22108/msci.2025.143588.1713FASajjadLakzianDepartment of Mathematical Sciences, Isfahan University of Technology, 8415683111, Isfahan, IranJournal Article20241203Geometric relativity is mostly referred to the study of spatial manifolds in space-time manifolds as main objects in general relativity. Our goal in this paper, is to study spatial models of the universe that are themselves tangent bundles. In Arnowitt, Deser and Misner's formulation of general relativity as a Hamiltonian system, the main quantities that are studied are total energy and mass of a system and in this paper, we will look at the ADM mass for some asymptotically Euclidean tangent bundles. First, we will completely characterize asymptotically Euclidean interpolation metrics on tangent bundles of simple manifolds. We will then consider a family of interpolation metrics that we will call admissible metrics. We will define the notion of lower ADM mass (that is weaker than ADM mass) and estimate the lower ADM mass of admissible metrics. From the said estimate, we show that a stronger version of Schoen and Yau's positive mass and rigidity of positive mass holds for admissible metrics.Geometric relativity is mostly referred to the study of spatial manifolds in space-time manifolds as main objects in general relativity. Our goal in this paper, is to study spatial models of the universe that are themselves tangent bundles. In Arnowitt, Deser and Misner's formulation of general relativity as a Hamiltonian system, the main quantities that are studied are total energy and mass of a system and in this paper, we will look at the ADM mass for some asymptotically Euclidean tangent bundles. First, we will completely characterize asymptotically Euclidean interpolation metrics on tangent bundles of simple manifolds. We will then consider a family of interpolation metrics that we will call admissible metrics. We will define the notion of lower ADM mass (that is weaker than ADM mass) and estimate the lower ADM mass of admissible metrics. From the said estimate, we show that a stronger version of Schoen and Yau's positive mass and rigidity of positive mass holds for admissible metrics.https://math-sci.ui.ac.ir/article_29171_92e16fa1cde35aecbac23da82f2c91c8.pdfUniversity of IsfahanMathematics and Society2345-649310420260220New results on Wolstenholmes theorem and its applicationsNew results on Wolstenholmes theorem and its applications1051312921810.22108/msci.2025.141789.1669FADanielYaqubiDepartment of Computer Engineering, University of Torbat e Jam, , IranMadjidMirzavaziriDepartment of Mathematics, Ferdowsi University of Mashhad, Iran.Journal Article20240610In 1862, Wolstenholme proved that for primes $p\geq 5$, $\binom{2p-1}{p-1}\equiv 1\pmod{p^3}$. This theorem is equivalent to the divisibility of the coefficients of the harmonic series \[1+\frac{1}{2}+\cdots+\frac{1}{p-1}\] by $p^2$. The far-reaching implications of Wolstenholme's theorem ignited the curiosity of mathematicians in the 19th century, leading to a surge of investigations into the divisibility properties of coefficients in rational fractions and binomial coefficients involving powers of primes. The emergence of Bernoulli numbers in this theorem and their intricate connection to binomial coefficients firmly established the roots of this mathematical domain in analytic number theory. In this paper, we embark on a journey to explore Wolstenholme's theorem for higher powers of primes and delve into the intricacies of these diverse proofs. The converse of Wolstenholme's theorem, first proposed by Jones, asserts that a natural number n satisfying the congruence $\binom{2n-1}{n-1}\equiv 1\pmod{p^3}$ must be prime. We conclude our exploration by examining the conditions of the converse of Wolstenholme's theorem and presenting several intriguing problems that beckon further investigation.In 1862, Wolstenholme proved that for primes $p\geq 5$, $\binom{2p-1}{p-1}\equiv 1\pmod{p^3}$. This theorem is equivalent to the divisibility of the coefficients of the harmonic series \[1+\frac{1}{2}+\cdots+\frac{1}{p-1}\] by $p^2$. The far-reaching implications of Wolstenholme's theorem ignited the curiosity of mathematicians in the 19th century, leading to a surge of investigations into the divisibility properties of coefficients in rational fractions and binomial coefficients involving powers of primes. The emergence of Bernoulli numbers in this theorem and their intricate connection to binomial coefficients firmly established the roots of this mathematical domain in analytic number theory. In this paper, we embark on a journey to explore Wolstenholme's theorem for higher powers of primes and delve into the intricacies of these diverse proofs. The converse of Wolstenholme's theorem, first proposed by Jones, asserts that a natural number n satisfying the congruence $\binom{2n-1}{n-1}\equiv 1\pmod{p^3}$ must be prime. We conclude our exploration by examining the conditions of the converse of Wolstenholme's theorem and presenting several intriguing problems that beckon further investigation.https://math-sci.ui.ac.ir/article_29218_9f068ee0c77ec2cb1b14e6c2295e2058.pdf