Classical Results Research Articles

Absolute concentration robustness and multistationarity in reaction networks: Conditions for coexistence

Abstract Many reaction networks arising in applications are multistationary, that is, they have the capacity for more than one steady state, while some networks exhibit absolute concentration robustness (ACR), which means that some species concentration is the same at all steady states. Both multistationarity and ACR are significant in biological settings, but only recently has attention focused on the possibility for these properties to coexist. Our main result states that such coexistence in at-most-bimolecular networks (which encompass most networks arising in biology) requires at least three species, five complexes and three reactions. We prove additional bounds on the number of reactions for general networks based on the number of linear conservation laws. Finally, we prove that, outside of a few exceptional cases, ACR is equivalent to non-multistationarity for bimolecular networks that are small (more precisely, one-dimensional or up to two species). Our proofs involve analyses of systems of sparse polynomials, and we also use classical results from chemical reaction network theory.

Generalized orthogonal measures on the space of unital completely positive maps

Abstract A classical result by Effros connects the barycentric decomposition of a state on a C*-algebra to the disintegration theory of the GNS representation of the state with respect to an orthogonal measure on the state space of the C*-algebra. In this note, we take this approach to the space of unital completely positive maps on a C*-algebra with values in B ⁢ ( H ) {B(H)} , connecting the barycentric decomposition of the unital completely positive map and the disintegration theory of the minimal Stinespring dilation of the same. This generalizes Effros’ work in the non-commutative setting. We do this by introducing a special class of barycentric measures which we call generalized orthogonal measures. We end this note by mentioning some examples of generalized orthogonal measures.

NEW REGION FOR ZEROS OF A POLAR DERIVATIVE OF POLYNOMIAL

Let P(z) := Pn j=0 aj z j be a polynomial of degree n. Then by Cauchy’s Classical result, all zeros of P(z) lie in |z| ≤ 1 + max 0≤j≤n−1 | aj an | . In this paper, we shall extend such results to polar derivative of an algebraic polynomial.

АСИМПТОТИЧНІ ВЛАСТИВОСТІ ВИПАДКОВИХ МАТРИЦЬ

The work explores the asymptotic properties of large-dimensional stochastic matrices N under the condition of independence of matrix elements or rows (columns). An analysis of the main properties of eigenvalues of stochastic matrices is conducted. The work is dedicated to investigating the asymptotic characteristics of random matrices under the absence of the second moment and also considers the presence of "heavy tails" in the corresponding transitions in the adjacency matrices of the respective graph. The main result of the work is formulated in terms of the transition matrix of a discrete Markov chain and its eigenvalues. In proving the theorem, a non-degenerate Markov chain is considered, describing a mathematical model of random processes that do not change over time with known transition probabilities between states and a corresponding stochastic matrix P, one of whose eigenvalues is unity, and all eigenvalues of the stochastic matrix do not exceed it in absolute value. The proof of this fact follows from the Perron–Frobenius theorem, which concerns the properties of positive matrices and their eigenvalues. Thus, the theorem considered manages to expand the class of random matrices A for which convergence of eigenvalues of the matrix can be applied under the conditions imposed on the elements of the adjacency matrix. Moreover, the imposed conditions are relaxed compared to classical results, where the existence of a finite second moment for the elements of the adjacency matrix is required. This result generalizes both classical results for the normal distribution and similar results of other authors. The obtained result can be used in graph clustering problems to choose the optimal number of clusters, namely, they can be applied to determine the optimal number of clusters in a Grid system, complex networks, in investigations of the structure of the web space, etc.

Life Cycle Assessment (LCA) analysis on the dyeing process to increase sustainable production in batik SMEs

Satrio Utomo SME is a batik small and medium enterprise located in Kedunggudel, Sukoharjo, which was established in 2010. This business specializes in producing two types of batik, written batik and stamped batik. Both production processes utilize Remazol as the raw material for synthetic dyes. Notably, while written batik exclusively employs 100% Remazol dye, stamped batik incorporates recycled Remazol dye. This study aims to calculate, analyze, and offer recommendations for enhancing eco-efficiency in the utilization of Remazole dye by employing the Life Cycle Assessment method with SimaPro 9.3 software to evaluate the environmental impact of both batik production methods. The results indicate that the eco- efficiency index for written batik stands at 1.03, whereas stamped batik boasts an index of 1.07, signifying superior eco-efficiency. This underscores the greater environmental impact of written batik, while stamped batik demonstrates a more favorable balance between economic and ecological considerations. The study proposes an enhancement strategy: increasing the production of stamped batik employing recycled Remazol dye due to its higher net value. For written batik, the recommendation involves transitioning from synthetic dyes to natural dyes, known for their environmental friendliness and ability to yield classic batik results.

Development and Innovation of Chinese Classical Dance Teaching Mode in the Background of Internet

Abstract Since the development of Chinese classical dance, its dance technology and dance skills have been maturing for a long time, but the innovation and development of classical dance teaching mode needs to be improved. In this paper, OptiTrack is used to capture the dance movement data of classical dancers, transform the data into three-dimensional coordinates, and extract the features of classical dance data through direction normalization. The Hidden Markov Model obtains the time series information in the features and constructs the correlation model. Using the skeletal model data to get the center of mass of the human body, the convolution-gated recurrent unit network structure is used to improve the accuracy of the classical dance score generation results. Combining the above methods, a new teaching model framework for classical dance is proposed from three aspects and empirically analyzed. The empirical results show that the mean value of each dimension of classical dance teachers’ teaching ability ranges from 3.4562-4.3621, which is an excellent overall performance. In the comparison between students’ classical dance movements and standard movements, there existed higher scores than traditional movements in the two time periods within 20 minutes, which were 30 and 5 points higher than the conventional scores, respectively. The rest of the time was slightly lower than the regular movements, which shows that the innovation effect of classical dance teaching is better, and optimizes students’ classical dance movements to a certain extent.

On dominated multivalued operators involving nonlinear contractions and applications

<abstract> <p>The objective of this research is to establish new results for set-valued dominated mappings that meet the criteria of advanced locally contractions in a complete extended <italic>b</italic>-metric space. Additionally, we intend to establish new fixed point outcomes for a couple of dominated multi-functions on a closed ball that satisfy generalized local contractions. In this study, we present novel findings for dominated maps in an ordered complete extended <italic>b</italic>-metric space. Additionally, we introduce a new concept of multi-graph dominated mappings on a closed ball within these spaces and demonstrate some original results for graphic contractions equipped with a graphic structure. To demonstrate the uniqueness of our new discoveries, we verify their applicability in obtaining a joint solution of integral and functional equations. Our findings have also led to modifications of numerous classical and contemporary results in existing research literature.</p> </abstract>

Radial solutions and a local bifurcation result for a singular elliptic problem with Neumann condition

We study the problem − Δ u = λ u − u − 1 with a Neumann boundary condition; the peculiarity being the presence of the singular term − u − 1 . We point out that the minus sign in front of the negative power of u is particularly challenging, since no convexity argument can be invoked. Using bifurcation techniques we are able to prove the existence of solution ( u λ , λ ) with u λ approaching the trivial constant solution u = λ − 1 / 2 and λ close to an eigenvalue of a suitable linearized problem. To achieve this we also need to prove a generalization of a classical two-branch bifurcation result for potential operators. Next we study the radial case and show that in this case one of the bifurcation branches is global and we find the asymptotical behavior of such a branch. This results allows to derive the existence of multiple solutions u with λ fixed.

Voronoi's formulae and the Gauss problem

We present classical and new results on the remainder term in the circle problem. The proofs are based on the application of various versions of Voronoi's formula. Bibliography: 54 titles.

On the Cardinality and Power Set of Fuzzy Sets and Multiple Sets

A multiple set is a generalized version of a fuzzy set that deals with the uncertainty of an object along with its multiplicity. This work is a comparative analysis of the fundamental conclusions based on the cardinality of the classical set and fuzzy set. Several classical results based on fuzzy cardinality are then generalized to multiple set theory. An introduction to the concept of the power set of a multiple set is made, along with some basic results that support it. Furthermore, an extensive analysis of the power set properties of the fuzzy set is also carried out.

Analytical Insights into the Modified Fractional Bell Polynomial with Mittag-Leffler Parameter

In this paper, we will define the Modified Fractional Bell Polynomial by incorporating the Mittag Leffler function of one parameter. The Existence and convergence of the Modified Fractional Bell Polynomial will be established by extending the classical results of Bell Polynomial and Mittag Leffler function of one parameter in the fractional calculus. Additionally, we explore the inverse of the Modified Fractional Bell Polynomial, providing a step-by-step proof of its existence. This result enhances the applicability of the polynomial by allowing a unique mapping from each output to a set of input values. The introduction of the Modified Fractional Bell Polynomial, with its well-established properties, opens avenues for further research and applications in diverse mathematical contexts. The generality of the polynomial makes it a powerful tool for modeling complex phenomena.

On Schur-type theorem for Leibniz 3-algebras

One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group G/ζ(G) of a group G is finite, then its derived subgroup [G,G] is also finite. This theorem was proved by B. Neumann in 1951. This result has numerous generalizations and modifications in group theory. At the same time, similar investigations were conducted in other algebraic structures, namely in modules, linear groups, topological groups, n-groups, associative algebras, Lie algebras, Lie n-algebras. In 2016, L.A. Kurdachenko, J. Otal and O.O. Pypka proved an analogue of Schur theorem for Leibniz algebras: if central factor-algebra L/ζ(L) of Leibniz algebra L has finite dimension, then its derived ideal [L,L] is also finite-dimensional. Moreover, they also proved a slightly modified analogue of Schur theorem: if the codimensions of the left ζ^l (L) and right ζ^r (L) centers of Leibniz algebra L are finite, then its derived ideal [L,L] is also finite-dimensional. One of the generalizations of Leibniz algebras is the so-called Leibniz n-algebras. It is worth noting that Leibniz n-algebra theory is currently much less developed than Leibniz algebra theory. One of the directions of development of the general theory of Leibniz n-algebras is the search for analogies with other types of algebras. Therefore, the question of proving analogs of the above results for this type of algebras naturally arises. In this article, we prove the analogues of the two mentioned theorems for Leibniz n-algebras for the case n=3. The obtained results indicate the prospects of further research in this direction.

The Cheeger problem in abstract measure spaces

AbstractWe consider nonnegative ‐finite measure spaces coupled with a proper functional that plays the role of a perimeter. We introduce the Cheeger problem in this framework and extend many classical results on the Cheeger constant and on Cheeger sets to this setting, requiring minimal assumptions on the pair measure space perimeter. Throughout the paper, the measure space will never be asked to be metric, at most topological, and this requires the introduction of a suitable notion of Sobolev spaces, induced by the coarea formula with the given perimeter.

On the Newton-Broyden method for solving systems of nonlinear equations

The Newton-Broyden method for solving systems of nonlinear equations with operator decomposition is studied in this paper. A local convergence analysis is provided under classical Lipschitz conditions and results of numerical experiments are presented. The effectiveness of this method in solving various types of nonlinear systems is demonstrated.

Symplectic rational blow-ups on rational 4-manifolds

We prove that if a symplectic 4-manifold X X becomes a rational 4-manifold after applying rational blow-down surgery, then the symplectic 4-manifold X X is originally rational. That is, a symplectic rational blow-up of a rational symplectic 4 4 -manifold is again rational. As an application we show that a degeneration of a family of smooth rational complex surfaces is a rational surface if the degeneration has at most quotient surface singularities, which generalizes slightly a classical result of Bădescu [J. Reine Angew. Math. 367 (1986), pp. 76–89] in algebraic geometry under a mild additional condition.

On the Filippov-Ważewski relaxation theorem for a certain class of fractional differential inclusions

The purpose of this text is to propose an attempt of an extension of the Filippov-Ważewski Relaxation Theorem for a certain class of fractional differential inclusions. The classical result devoted to ordinary differential inclusions is a part of the qualitative theory: a description of the relationship between the solutions to the differential inclusion and the convexified differential inclusion was given. There exist several generalizations of that result and in this note a method is proposed so that the range of some parameters is extended. An example of a possible application may arise in control theory and the question is whether it is possible to have the same reachable set economizing the set of controls.

An Improved Fractional Moment Maximum Entropy Method With Polynomial Fitting

Abstract The moment method is commonly used in reliability analysis, in which the maximum entropy method (MEM) and polynomial fitting (PF) have been widely used due to their advantages in accuracy and efficiency, respectively. In this paper, we propose a novel reliability analysis method by combining MEM and PF. The probability density function is preliminarily estimated using the fractional moment maximum entropy method (FM-MEM), based on which PF is then used to further improve the accuracy. The proposed method can avoid the phenomenon of the negative probability density and function oscillations in PF effectively. Moreover, the order of the exponential polynomial in the FM-MEM is adaptively selected in the preliminary solution calculation process. An iterative process for the number of exponential polynomial terms is also proposed, using the integral of the moment error function and the integrals of the local and global negative probability density as the convergence criteria. Four numerical examples and one engineering example are tested, and the results are compared with those of the Monte Carlo simulation and the classical FM-MEM results, respectively, demonstrating the good performance of the proposed method.

Global stability of Oldroyd-B fluids in plane Couette flow

We prove conditions for global nonlinear stability of Oldroyd-B viscoelastic fluid flows in the Couette shear flow geometry. Global stability is inferred by analysing a new functional, called a perturbation entropy, to quantify the magnitude of the polymer perturbations from their steady-state values. The conditions for global stability extend, in a physically natural manner, classical results on global stability of Newtonian Couette flow.

Correspondence of Donaldson-Thomas and Gopakumar-Vafa invariants on local Calabi-Yau 4-folds over V5 and V22

We compute Gromov-Witten (GW) and Donaldson-Thomas (DT) invariants (and also descendant invariants) for local CY 4-folds over Fano 3-folds, V5 and V22 up to degree 3. We use torus localization for GW invariants computation, and use classical results for Hilbert schemes on V5 and V22 for DT invariants computation. From these computations, one can check correspondence between DT and Gopakumar-Vafa (GV) invariants conjectured by Cao-Maulik-Toda in genus 0. Also we can compute genus 1 GV invariants via the conjecture of Cao-Toda, which turned out to be 0. These fit into the fact that there are no smooth elliptic curves in V5 and V22 up to degree 3.

Machine learning-assisted high precision predictive modelling of convective heat transfer in fluid channels fabricated by laser powder bed fusion

Laser powder bed fusion (LPBF) has proven to be an effective tool in fabricating heat transfer devices with improved efficiency. However, the accurate prediction of convective heat transfer in LPBF-fabricated fluid channels remains a challenge. The classical Gnielinski model is regarded as the most accurate correlation for predicting forced convective heat transfer in traditional pipes. However, whether it is applicable to LPBF-fabricated pipes is yet to be determined. To address this challenge, in this study, pipe samples with diameters of 3 mm, 4 mm, and 5 mm were designed and fabricated using LPBF along the building angles of 0°, 45°, and 90°. The pressure loss and heat transfer characteristics of these samples were experimentally measured. Results showed that there was a maximum prediction error of 72.1 % between the classical Gnielinski model and experimental results. To improve the prediction accuracy, a corrected Colebrook model with improved prediction accuracy of the friction factor was developed, introduced into the classical Gnielinski model, and reduced the maximum prediction error down to 36.8 %. To further improve the prediction accuracy, a gradient descent-based machine learning method was adopted to reconstruct the Gnielinski model, which achieved a high prediction accuracy with prediction errors of below 10 %. With the assistance of machine learning, a high precision predictive model of convective heat transfer in LPBF-fabricated fluid channels was established.