Classical Results Research Articles

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.

Enhancing functional efficiency in information-extreme machine learning with logistic regression ensembles

The subject matter of the article is the application of supervised machine learning for the task of object class recognition. The goal is enhancing functional efficiency in information-extreme technology (IET) for object class recognition. The tasks to be solved are: to analyse possible ways of increasing the functional efficiency IET approach; implement an ensemble of models that include logistic regression for prioritizing recognition features and an IEI learning algorithm; compare the functional efficiency of the resulting ensemble of models on well-known dataset with classic approach and results of other researchers. The methods: The method is developed within the framework of the functional approach to modelling natural intelligence applied to the problem of object classification. The following results were obtained: The study tries to augment existing IET to support feature prioritization as part of the object class recognition algorithm. The classical information-extreme algorithm treats all input features equivalently important in forming the decisive rule. As a result, the object features with strong correlation are not prioritized by the algorithm's decisive mechanism – resulting in decreasing functional efficiency in exam mode. The proposed approach is solving this problem by applying a two-stage approach. In the first stage the multiclass logistic regression applied to the input training features vectors of objects to be classified – formed the normalized training matrix. To prevent overfitting of the logistic regression, a model the L2(ridge) regularization method was used. On the second stage, the information-extreme method as input takes the result of the first stage. The geometrical parameters of class containers and the control tolerances on the recognition features were considered as the optimization parameters. Conclusions. The proposed approach increases MNIST (Modified National Institute of Standards and Technology) dataset classification accuracy compared with the classic information-extreme method by 26,44%. The proposed approach has a 3.77% lower accuracy compared to neural-like approaches but uses fewer resources in the training phase and allows retraining the model, as well as expanding the dictionary of recognition classes without model retraining.

Stability and moduli space of generalized Ricci solitons

The generalized Einstein–Hilbert action is an extension of the classic scalar curvature energy and Perelman’s F-functional which incorporates a closed three-form. The critical points are known as generalized Ricci solitons, which arise naturally in mathematical physics, complex geometry, and generalized geometry. Through a delicate analysis of the group of generalized gauge transformations, and implementing a novel connection, we give a simple formula for the second variation of this energy which generalizes the Lichnerowicz operator in the Einstein case. As an application we show that all Bismut-flat manifolds are linearly stable critical points, and admit nontrivial deformations arising from Lie theory. Furthermore, this leads to extensions of classic results of Koiso [1–4] and Podesta, Spiro, Kröncke [5–8] to the moduli space of generalized Ricci solitons. To finish we classify deformations of the Bismut-flat structure on S3 and show that some are integrable while others are not.

Towards Homological Methods in Graphic Statics

Recent developments in applied algebraic topology can simplify and extend results in graphic statics – the analysis of equilibrium forces, dual diagrams, and more. The techniques introduced here are inspired by recent developments in cellular cosheaves and their homology. While the general theory has a few technical prerequisites (including homology and exact sequences), an elementary introduction based on little more than linear algebra is possible. A few classical results, such as Maxwell's Rule and 2D graphic statics duality, are quickly derived from core ideas in algebraic topology. Contributions include: (1) a reformulation of statics and planar graphic statics in terms of cosheaves and their homology; (2) a new proof of Maxwell's Rule in arbitrary dimensions using Euler characteristic; and (3) derivation of a novel relationship between mechanisms of the form diagram and obstructions to the generation of force diagrams. This last contribution presages deeper results beyond planar graphic statics.

Information Bottleneck and Aggregated Learning.

We consider the problem of learning a neural network classifier. Under the information bottleneck (IB) principle, we associate with this classification problem a representation learning problem, which we call "IB learning". We show that IB learning is, in fact, equivalent to a special class of the quantization problem. The classical results in rate-distortion theory then suggest that IB learning can benefit from a "vector quantization" approach, namely, simultaneously learning the representations of multiple input objects. Such an approach assisted with some variational techniques, result in a novel learning framework, "Aggregated Learning", for classification with neural network models. In this framework, several objects are jointly classified by a single neural network. The effectiveness of this framework is verified through extensive experiments on standard image recognition and text classification tasks.

Note on Some New (n, m)-Type Bounds for Graph Energy

The classical results by McClelland (1971) and Koolen & Moluton (2001) provide upper bounds for graph energy in terms of number of vertices (n) and number of edges (m). Recently, in MATCH Commun. Math. Comput. Chem. 79 (2018) and 91 (2024), new such (n, m)-type bounds were communicated. In this paper, we analyze these bounds and find that one is identical to the Koolen– Moulton bound, whereas the other is inferior to it.

Clinical Manifestations.

The California Cognitive Assessment Battery (CCAB; ccabstudy.com) is a remotely monitored, at-home, computerized behavioral test and questionnaire suite. Here we evaluate the interpretation of 25 tests within the US English version of the CCAB using both an exploratory factor analysis (EFA) of 155 healthy adults and a confirmatory factor analysis (CFA) of 312 healthy older adults participating in a longitudinal study. We also estimated cross-sectional age gradients for best fitting latent factors [1]. First, one set of "classic" latent factors (from 1 factor to 8 factors) was extracted from the literature [2][3][4] for assignment of CCAB tests into traditional categories. Second, 155 adult participants (age 22-86 y.o., mn 54, 57% male, education 8-20 yrs., md 14) participated in a prefinal version of the CCAB. EFA was performed on their primary CCAB results using the 'psych' (v. 1.9.12.31-1) toolbox in R (v. 3.6.3) with 11 different rotations to extract a second set of CFA latent factors. Third, above two sets of latent factors were used in the 'lavaan' CFA toolbox (v. 0.6.5-1) in R and applied to the first timepoint (enrollment and session 1) of CCAB longitudinal data from an older adult group (age 56-89 y.o., mn 71, 58% male, education 0-20 yrs., mn 14). The EFA and CFA model fit characteristics (Fig 1) show moderately good fits for the EFA and best fitting CFA models. The EFA produced reasonable models from 2 factors to 6 factors. The best fitting "classic" CFA along with its latent factor age regressions is shown in Figure 2. The best fitting EFA-derived CFA along with its latent factor age regressions is shown in Figure 3. The best-fit classic CFA results showed moderately good fit, and visual tests of both speed and memory showed steeper cross-sectional age gradients than did respective verbal response tests. The EFA-derived latent factors were better fitting than were the classic factors as they were derived from the CCAB itself. In both cases, however, CCAB test speed measures showed steeper cross-sectional age gradients than did memory measures. Ongoing work will characterize longitudinal latent factor age gradients.

Nowhere coexpanding functions.

We define a family of C1 functions, which we call "nowhere coexpanding functions," that is closed under composition and includes all C3 functions with non-positive Schwarzian derivatives. We establish results on the number and nature of the fixed points of these functions, including a generalization of a classic result of Singer.

Supermodular and directionally convex comparison results for general factor models

This paper provides comparison results for general factor models with respect to the supermodular and directionally convex order. These results extend and strengthen previous ordering results from the literature concerning certain classes of mixture models as mixtures of multivariate normals, multivariate elliptic and exchangeable models to general factor models. For the main results, we first strengthen some known orthant ordering results for the multivariate ▪ -product of the specifications, which represents the copula of the factor model, to the stronger notion of the supermodular ordering. The stronger comparison results are based on classical rearrangement results and in particular are rendered possible by some involved constructions of transfers as arising from mass transfer theory. The ordering results for ▪ -products are then extended to factor models with general conditional dependencies. As a consequence of the ordering results, we derive worst case scenarios in relevant classes of factor models allowing, in particular, interesting applications to deriving sharp bounds in financial and insurance risk models. The results and methods of this paper are a further indication of the particular effectiveness of Sklar‘s copula notion.

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

On minimum spanning trees for random Euclidean bipartite graphs

AbstractWe consider the minimum spanning tree problem on a weighted complete bipartite graph $K_{n_R, n_B}$ whose $n=n_R+n_B$ vertices are random, i.i.d. uniformly distributed points in the unit cube in $d$ dimensions and edge weights are the $p$ -th power of their Euclidean distance, with $p\gt 0$ . In the large $n$ limit with $n_R/n \to \alpha _R$ and $0\lt \alpha _R\lt 1$ , we show that the maximum vertex degree of the tree grows logarithmically, in contrast with the classical, non-bipartite, case, where a uniform bound holds depending on $d$ only. Despite this difference, for $p\lt d$ , we are able to prove that the total edge costs normalized by the rate $n^{1-p/d}$ converge to a limiting constant that can be represented as a series of integrals, thus extending a classical result of Avram and Bertsimas to the bipartite case and confirming a conjecture of Riva, Caracciolo and Malatesta.

A Memoir on Pseudo-Variational Techniques for Parabolic PDE’s Incorporating Boundary Value Constraints

To enrich existing literature and extend the reach of relevant apt theoretical techniques, an alternative for proxy pseudo-variational analysis of a fundamental class of parabolic Partial Differential Equations (PDE’s) is brought to the fore in this paper. Classical results in Sobolev Space theory are extrapolated with the aid of more current, standard tools from developments in fractional calculus. The high potency of inference from data obtained via the trace boundary operator is proffered, with the simple heat equation in one spatial dimension as a case study.

Graphical methods and rings of invariants on the symmetric algebra

Abstract Let G be a complex classical group, and let V be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of G-invariant polynomial functions on the space $\mathcal P^m(V)$ of degree-m homogeneous polynomial functions on V. In this paper, we replace $\mathcal P^m(V)$ with the full polynomial algebra $\mathcal P(V)$ . As a result, the invariant ring is no longer finitely generated. Hence, instead of seeking generators, we aim to write down linear bases for bigraded components. Indeed, when G is of sufficiently high rank, we realize these bases as sets of graphs with prescribed number of vertices and edges. When the rank of G is small, there arise complicated linear dependencies among the graphs, but we remedy this setback via representation theory: in particular, we determine the dimension of an arbitrary component in terms of branching multiplicities from the general linear group to the symmetric group. We thereby obtain an expression for the bigraded Hilbert series of the ring of invariants on $\mathcal P(V)$ . We conclude with examples using our graphical notation, several of which recover classical results.

Remarks on the inverse Galois problem over function fields

In this paper, we prove new instances of the inverse Galois problem over global function fields for finite groups of Lie type. This is done by constructing compatible systems of ℓ-adic Galois representations valued in a semisimple group G using Galois theoretic and automorphic methods, and then proving that the Galois images are maximal for a set of primes of positive density using a classical result of Larsen on Galois images for compatible systems.

Noetherian spectrum condition and the ring 𝒜[X

Let [Formula: see text] be an ascending chain of commutative rings with identity and let [Formula: see text] be the ring of polynomials with coefficient of degree [Formula: see text] in [Formula: see text] for each [Formula: see text] In the first part of this paper, we give a necessary and sufficient conditions for the ring [Formula: see text] to satisfy Noetherian spectrum (a ring [Formula: see text] is said to have Noetherian spectrum if [Formula: see text] satisfies the Ascending Chain Condition (ACC) on radical ideals). In particular, we extend classical results proved by Ohm and Pendleton in [J. Ohm and R. Pendleton, Rings with Noetherian spectrum, Duke Math. J. 35 (1968) 631–639] and by Hizem in [S. Hizem, Chain conditions in rings of the form A + XB[X] and A + XI[X], in Commutative Algebra and its Applications (de Gruyter, 2009), pp. 259–274]. The second part of this paper is motivated by the interesting results proved on uniformly [Formula: see text]-Noetherian rings by Kim et al. in [W. Qi, H. Kim, F. Wang, M. Chen and W. Zhao, Uniformly S-Noetherian rings (submitted)]. We introduce the concept of uniformly [Formula: see text]-Noetherian spectrum and study its properties. Let [Formula: see text] be a commutative ring and [Formula: see text] a multiplicative subset of [Formula: see text] We say that [Formula: see text] has uniformly[Formula: see text]-Noetherian spectrum if that there exists an [Formula: see text] such that for any ideal [Formula: see text] of [Formula: see text], [Formula: see text] for some finitely generated subideal [Formula: see text] of [Formula: see text] We give the Eakin–Nagata–Formanek theorem for the uniformly [Formula: see text]-Noetherian spectrum condition. We also study the uniformly [Formula: see text]-Noetherian uniformly properties on several ring constructions (Nagata’s idealization and amalgamated algebras).