Extremal Problems Research Articles

A new approach for solving the multidimensional control optimization problems with first‐order partial differential equations constraints

AbstractThe purpose of this paper is to provide the linearization technique to solve the multidimensional control optimization problem (MCOP) involving first‐order partial differential equation (PDEs) constraints. Firstly, we use the modified objective function approach for simplifying the aforesaid extremum problem (MCOP) and show that the solution sets of the original control optimization problem and its modified control optimization problem (MCOP) are equivalent under convexity assumptions. Further, we use the absolute value exact penalty function method to transform (MCOP) into a penalized control problem (MCOP) . Then, we establish the equivalence between a minimizer of the modified penalized optimization problem (MCOP) and a saddle point of the Lagrangian defined for the modified optimization problem (MCOP) under appropriate convexity hypotheses. Moreover, the results established in the paper are illustrated by some examples of MCOPs involving first‐order PDEs constraints.

Exact Results for Generalized Extremal Problems Forbidding an Even Cycle

Exact Results for Generalized Extremal Problems Forbidding an Even Cycle

Extremal problems in BMO and VMO involving the Garsia norm

Extremal problems in BMO and VMO involving the Garsia norm

Positivity-preserving pseudo arc-length method for solving extreme explosive shock wave flow field problems in hyperbolic conservation law equations

Positivity-preserving pseudo arc-length method for solving extreme explosive shock wave flow field problems in hyperbolic conservation law equations

The Theoretical Gap in the Study of New Quality Productive Forces and the Economic Analytical Perspective of “Heterogeneity”

Abstract Economic studies on the new quality productive forces NQPF are mushrooming, however, on the whole, the majority of existing studies primarily focus on revealing and summarizing the economic laws guiding the development of NQPF based on the“general”characteristics of individual organizations at the microlevel and institutions at the macro level. This paper argues that the background of “getting great” and the competition scenario between countries determine that “heterogeneity” is the most relevant and theoretically challenging issue in the NQPF, and also the most likely to yield significant academic outcomes. In view of the fact that the existing problem setting, conceptual framework, and analytical tools of existing eccomics fail to effectively address the unique prescriptive phenomena of NQPF, including direct competition between countries, power countries, and heterogeneity, future academic research on new quality productive forces should go beyond the current “general” issues orientation, directly address the extreme phenomena and problems in the “genesis” stage in the latest round of scientific and technological revolution, and strive to push forward the frontiers of theoretical research.

Skin disease classification using two path deep transfer learning models

Skin diseases are among the most common diseases that affect millions of lives per year, yet diagnosing these has several complexities even for trained dermatologists due to overlapping symptoms and features in several diseases. A myriad of deep learning models have been proposed as a solution for diagnosing but a clinically useful model with high accuracy multi disease classification and lower computational complexity is still unavailable. This study focuses on comparing different image pre-processing techniques, transfer learning models and ensemble learning techniques to build a computationally cheap model for 8-class identification of skin diseases. A two path model with EfficientNet and MobileNetV2 transfer learning models as base feature extractors and a final model that stacks the two model results and classifies the images into one of the eight classes is used. The model is trained and tested on ISIC-2019 dataset for 8 class image classification that involves the three types of skin cancers as well. The dataset has an extreme class imbalance problem which leads to favored prediction of the classes with more image, for this first we run simple image augmentation. Secondly, two distinctly processed images are created from each initial image. The two path model takes the two images, gives each to a base model and combines the two outputs, enabling the classifier to consider different features that become prominent due to dissimilar preprocessing techniques. The model is tested with new images on multiple standard metrics to get a final overview of its performance, it gives a diagnosing accuracy of 70% which is close to some state of the art models that consume higher computational power. The classification results imply that by further improving the data gathering and preprocessing techniques along with exploring other base transfer learning models the results of the final model can be reliable while maintaining a low computational requirement, making the diagnoses accessible. This also highlights that such two path algorithms that employ simpler models could be useful for multi class classification tasks where differently processed images might be required to extract features of distinct diseases.

CLASSES OF CLOSE-TO-STARLIKE FUNCTIONS BASED ON THE REFERENCE FUNCTION OF A GENERAL FORM

The purpose of the article is to introduce and explore a wide class of doubly close-to-starlike functions, while demonstrating a unified approach to solving a certain range of extreme problems. The article defines a reference function of a general form – a starlike function, on the basis of which classes of close-to-starlike and doubly close-to-starlike functions can be constructed. On the basis of a general support function containing three parameters and new estimates of analytical functions, a generalization of various classes of close-to-starlike and doubly close-to-starlike functions is introduced, considered in a number of articles published in recent years, including the introduced class contains a generalized class of typically real functions. The properties of the introduced class of functions are studied, for example, the growth theorem, estimates of the modulus of the logarithmic derivative of the function and the radius of starlikeness are obtained, in particular cases leading to previously known results and representing new results. All the results of the article are accurate.

Extremal Problems for a Matching and Any Other Graph

Extremal Problems for a Matching and Any Other Graph

A Two-Parameter Ridge Estimator for Handling Extreme Multicollinearity Problems in Logistic Regression

This paper introduces a robust two-parameter ridge estimator that is customized for logistic regression models, which tend to be sensitive to extreme multicollinearity problems. Inflated standard errors and unreliability in the results stem from the problem of multicollinearity characterized by high correlations among predictor variables in logistic regression model. Traditional approaches like the Maximum Likelihood Estimator (MLE) and one-parameter ridge-type estimators often perform poorly under these settings, thus calling for the development of more robust approaches. This new proposal is called New Biased Two Parameter (NBTP), which extends the ridge regression framework by introducing additional biasing parameters customized for an extreme multicollinearity problem. The paper combines a theoretical analysis with extensive Monte Carlo simulations and real application to Pena data. It demonstrates that the new estimator, New Biased Two Parameter (NBTP), provides much more stable and accurate parameter estimates than previous methods. The results underscore the importance of using robust estimation methods within logistic regression, especially when multicollinearity may be widespread in fields such as medical research, finance, and the social sciences.

ҚАТТЫ ЦЕНТРІ БАР ДӨҢГЕЛЕК ОСЬТІК СИММЕТРИЯЛЫҚ МЕМБРАНАНЫҢ ФИЗИКАЛЫҚ ЖӘНЕ ГЕОМЕТРИЯЛЫҚ ПАРАМЕТРЛЕРІН ОҢТАЙЛАНДЫРУ

The mathematical model of an isotropic circular plate-membrane with a non-deformable fixed central disk loaded by a uniformly distributed static load is considered in a linear formulation. On this basis, the extreme problem of determining the rational physical and geometrical characteristics of an elastic element from the condition of the maximum of the target function of the thrust (permutation) force with two coupling equations in the form of the classical Huber-Genke-Mises strength hypothesis and a geometrical relation limiting the membrane thickness in accordance with the known fundamental Kirchhoff assumptions of the technical theory of small plate bending is solved. To illustrate the proposed algorithm and calculation procedure, a numerical example of the selection of optimum parameters of a manganese steel diaphragm with given dimensions of thickness and outer radius is presented. The results of the work can be used in the process of designing high-precision membrane-type pressure gauges, widely used in mechanical engineering, aviation, instrumentation, and construction in the design of pressure tanks with controlled excess pressure of gas or liquid.

A subgradient projection method for quasiconvex multiobjetive optimization

We discuss a subgradient projection method for dealing with the nonconvex nonsmooth multiobjective optimization problem when every component of the vector-valued function is strongly quasiconvex in the sense of Polyak [Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Soviet Math. 1966;7:72–75]. Under mild assumptions, we ensure that the generated sequence converges to an efficient solution point of the multiobjective optimization problem and we provide a kind of linear convergence rate. The algorithm is based on a specific generalized subdifferential that was recently introduced for strongly quasiconvex functions, this approach provides new and valuable information on the generated sequence and its convergent point. Finally, we present numerical experiments for classes of quadratic fractional programming problems.

Best approximations for the weighted combination of the Cauchy-Szegö kernel and its derivative in the mean

In this paper, we study an extremal problem involving best approximation in the Hardy space $H^1$ on the unit disk $\mathbb D$. Specifically, we consider weighted combinations of the Cauchy-Szegö kernel and its derivative, parameterized by an inner funtion $\varphi$ and a complex number $\lambda$, and provide explicit formulas for the best approximation $e_{\varphi,z}(\lambda)$ by the subspace $H^1_0$. We also describe the extremal functions associated with this approximation. Our main result gives the form of $e_{\varphi,z}(\lambda)$ as a function of $\lambda$ and shows that, for a sufficiently large module of $\lambda$, the extremal function is linear in $\lambda$ and unique. We apply this result to establish a sharp inequality for holomorphic functions in the unit disk, leading to a new version of the Schwarz-Pick inequality.

Some stability results for spectral extremal problems of graphs with bounded matching number

Some stability results for spectral extremal problems of graphs with bounded matching number

BEST APPROXIMATION OF FUNCTIONS IN THE HARDY SPACE

We solve extremal problems of best polynomial approximation of functions which lie the Hardy space

Assessing Quality of Life Following Radical Cystectomy and Ileal Conduit for Bladder Cancer: A Study Utilizing EuroQol 5-Dimensional 3-Level

Abstract Background: Urinary bladder cancer continues to be a significant health challenge and often requires radical cystectomy with ileal conduit (RCIC) as a primary therapeutic intervention in Stage II/III disease. The impact of this surgery on patients’ quality of life (QoL) is a critical aspect of evaluating the overall success of bladder cancer management. Therefore, we conducted a study to assess patients’ QoL following RCIC in bladder cancer patients using EuroQol 5-dimensional 3-level (EQ-5D-3L) instrument. Materials and Methods: In this cross-sectional study, patients of urinary bladder cancers from tertiary care centers, who underwent RCIC as a part of their curative treatment, were evaluated 1 month following surgery for QoL using the EQ-5D-3L instrument, in which patients were asked to report problems in 5 dimensions of this questionnaire. Results: We found that RCIC has a substantial immediate negative impact on QoL, primarily due to postoperative discomfort, anxiety/depression, and other urinary diversion-related issues. Nearly 36% of patients had some or extreme problems in 3 or more dimensions out of 5 in EQ-5D-3L. Conclusion: This study highlights the significance of EQ-5D-3L as a tool for evaluating QoL in patients of bladder cancer following RCIC, providing a useful framework for comprehending the multidimensional nature of their experiences. The findings emphasize that to maximize QoL, bladder cancer treatment requires individualized interventions, psychosocial support, and collaborative decision-making in bladder cancer management.

Pure-positivity-preserving methods with an optimal sufficient CFL number for fifth-order MR-WENO schemes on structured meshes

In this paper, one-dimensional and two-dimensional pure-positivity-preserving (PPP) methods are proposed for fifth-order finite volume multi-resolution WENO (MR-WENO) schemes to solve some extreme problems on structured meshes. The MR-WENO spatial reconstruction procedures only require one five-cell, one three-cell, and one one-cell stencils for achieving uniform fifth-order accuracy in smooth regions and keeping essentially non-oscillatory property in non-smooth regions in one dimension. One redefines five new cell averages vectors after performing such spatial reconstructions and design one quartic polynomials vector and three quadratic polynomials vectors based on them. After that, a new detective process is used to examine the positivity of density and pressure of three quadratic polynomials vectors inside the whole target cell. If the negativity happens, a new compression limiter is carried out to enable the positivity of density and pressure of three quadratic polynomials vectors over the whole target cell and the positivity of density and pressure of one quartic polynomials vector at the midpoint of the target cell. It is a new way to design the positivity-preserving methods to keep fifth-order accuracy and the positivity over the target cell instead of only at some discrete Gauss-Lobatto quadrature points, since the precise minimum values of the density and pressure are now available. Then a theoretically proof is given to increase the optimal sufficient CFL number from 1/12 to 1/6 for the fifth-order WENO schemes. This methodology can be expanded to multi-dimensions easily. Unlike some classical positivity-preserving methods, the PPP methods could apply a special four-point Gauss-Lobatto quadrature formula or any other quadrature formulas on condition that their numerical precision is no smaller than four. Since the optimal CFL number of 1/6 is a sufficient but not necessary condition, the novelty PPP methods for fifth-order finite volume MR-WENO schemes with a larger practical CFL number of 0.6 are also available and robust enough when simulating some extreme problems without timely halving its value.

Effect of ginkgo diterpene lactone meglumine on the quality of life in patients with acute ischemic stroke

ObjectiveGinkgo diterpene lactone meglumine (GDLM) could improve the functional outcome after acute ischemic stroke (AIS). This study aimed to investigate the efficacy of GDLM on the quality of life in patients with AIS in China.MethodsThis is a post hoc analysis of Efficacy and Safety of Ginkgo Diterpene Lactone Meglumine in Acute Ischemic Stroke trial. The quality of life was measured using the EuroQoL questionnaire, including EQ-5D and EQ visual analogue scale (EQ-VAS). The primary outcomes were changes in EQ-5D and EQ-VAS from baseline to day 14 and day 90 after randomization.ResultsA total of 3219 patients with completed data on outcomes were enrolled, with median age of 63 years (interquartile range, 55–70) and 2,067 (64.2%) men. GDLM was associated with a significant decrease in scores of ED-5Q components (from 0 [no problem] to 3[extreme problem]), the mean difference between GDLM and placebo group was -0.14 for mobility, -0.11 for usual activities and self-care, -0.09 for pain/discomfort, and -0.34 for anxiety/depression on day 14, respectively. Similar results were observed on day 90. Additionally, there was statistically significant difference of changes in EQ-VAS between the GDLM group and the placebo group from baseline to day 14 (mean difference, 1.70; 95% confidence interval [CI], 0.78–2.62; P = 0.0003) and to day 90 after randomization (mean difference, 3.29; 95% CI, 2.37–4.22; P < 0.001).ConclusionsIn this analysis of Chinese patients with AIS, GDLM could improve the 14-day and 90-day quality of life compared with the placebo.Trial registrationURL: https://www.clinicaltrials.gov. Unique identifier: NCT02526225. Registration Date: 2016–02-01.

Extremal interpolation of convex scattered data in ℝ3 by smooth edge convex minimum L∞‐norm networks: Characterization and solution

We consider the extremal problem of interpolation of convex scattered data in by smooth edge convex curve networks with minimal ‐norm of the second derivative for . The problem for was set and solved by Andersson et al. (1995). Vlachkova (2019) extended the results of Andersson et al. (1995) and solved the problem for . The minimum edge convex ‐norm network for is obtained from the solution to a system of nonlinear equations with coefficients determined by the data. The solution in the case is unique for strictly convex data. The corresponding extremal problem for remained open. The case is of particular interest in the context of applications since it has a solution which is a smooth curve network consisting of quadratic splines, that is, a smooth curve network of the lowest possible computational complexity. Here, we show that the extremal interpolation problem for always has a solution. We give a characterization of this solution. We show that a solution to the problem for can be found by solving a system of nonlinear equations in the case where it exists.

A Subclass of Bi-univalent Functions Defined by aSymmetric q-Derivative Operator and Gegenbauer Polynomials

This paper introduces a novel subclass of bi-univalent analytic functions by utilizing a symmetric q-derivative operator in conjunction with Gegenbauer polynomials. Within this newly defined subclass, we derive bounds for the first two Maclaurin coefficients and address the Fekete-Szeg o problem. By varying the parameters in our results between 0 and 1, we obtain a range of new insights and rediscover some previously established results. This approach not only broadens the scope of bi-univalent function theory but also deepens the understanding of coefficient bounds and extremal problems within this context.

Health-related quality of life and nutrient intake of the elderly with type 2 diabetes according to comorbidity burden: a cross-sectional study

Objectives: This study aimed to explore the cross-sectional association between health-related quality of life (HRQoL) according to the number of comorbidities in older adults with type 2 diabetes mellitus (T2DM) using the Euro Quality of Life-5 Dimensions (EQ-5D) index. Methods: This study included 3,553 participants aged ≥ 65 years from the 2008–2020 Korea National Health and Nutrition Examination Survey. Dietary data were collected through 24-hour recall interviews by trained researchers, and demographic and lifestyle information via self-administered questionnaires. HRQoL was measured using a modified EQ-5D scale. Multivariable linear regression analyzed the associations between EQ-5D scores, nutrients and comorbidity, controlling for sociodemographic and health variables. Results: Most participants reported ‘no problems’ in the EQ-5D scores, although approximately 17% to 47% of participants reported ‘some problems’ or ‘extreme problems,’ depending on the dimension. As comorbidities increased, significant declines were observed across all dimensions, particularly in mobility, usual activities, pain/discomfort, and anxiety/ depression. Nutrient intake analysis revealed that participants with three or more comorbidities consumed less carbohydrates, but more fat. Conclusion: Our findings demonstrate that among older adults with T2DM, a higher number of comorbidities is associated with decreased HRQoL. Additionally, there are differences in nutrient intake patterns among those with more comorbidities, specifically decreased carbohydrate intake and increased fat intake. These results emphasize the need for comprehensive and tailored management strategies that consider both diabetes and the co-occurring health conditions. By addressing the complex healthcare needs of individuals with multiple comorbidities, it is possible to enhance their HRQoL and overall well-being.