Extremal Problems Research Articles

On a sharp inequality of Adimurthi–Druet type and extremal functions

Our main purpose is to establish the existence and nonexistence of extremal functions for sharp inequality of Adimurthi–Druet type for fractional dimensions on the entire space. Precisely, we extend the sharp Trudinger–Moser type inequality in de Oliveira and do Ó (Calc Var Part Differ Equ 52:125–163, 2015) for the entire space. In addition, we perform the two-step strategy due to L. Carleson and S-Y.A. Chang together with blow up analysis method to ensure the existence of maximizers for the associated extremal problems for both subcritical and critical regimes. We also present a nonexistence result under a subcritical regime for some special cases.

Coloring directed hypergraphs

Inspired by earlier results about proper and polychromatic coloring of hypergraphs, we investigate such colorings of directed hypergraphs, that is, hypergraphs in which the vertex set of each hyperedge is partitioned into two parts, a tail and a head. We present a conjecture of D. Pálvölgyi and the author, which states that directed hypergraphs with a certain restriction on their pairwise intersections can be properly colored with two colors. Besides other contributions, our main result is a proof of this conjecture for 3-uniform directed hypergraphs. This result can be phrased equivalently such that if a 3-uniform directed hypergraph avoids a certain directed hypergraph with two hyperedges, then it admits a proper 2-coloring. Previously, only extremal problems regarding the maximum number of edges of directed hypergraphs that avoid a certain hyperedge were studied.

A robust fifth order finite difference Hermite WENO scheme for compressible Euler equations

In this paper, we develop a robust fifth order finite difference Hermite weighted essentially non-oscillatory (HWENO) scheme for compressible Euler equations following the HWENO with limiter (HWENO-L) scheme (Zhang and Zhao, 2023). The HWENO-L scheme reduced storage and increased efficiency by using restricted derivatives only for time discretizations. However, it cannot control spurious oscillations well when facing strong shocks since the derivatives are directly used in spatial discretizations without any restrictions. To address such an issue, our proposed HWENO scheme performs flux reconstructions in the finite difference framework without using the derivative value of a target cell, which can result in a simpler and more robust scheme. The resulting scheme is simpler while achieving fifth order accuracy, making it more efficient. Besides, numerically we find it is very robust for some extreme problems even without positivity-preserving limiters. The proposed scheme also inherits advantages of previous HWENO schemes, including arbitrary positive linear weights in flux reconstructions, compact reconstructed stencils, and high resolution. Extensive numerical tests are performed to demonstrate the fifth order accuracy, efficiency, robustness, and high resolution of the proposed HWENO scheme.

Large Independent Sets from Local Considerations

The following natural problem was raised independently by Erdős–Hajnal and Linial–Rabinovich in the early ’90 s. How large must the independence number alpha (G) of a graph G be whose every m vertices contain an independent set of size r? In this paper, we discuss new methods to attack this problem. The first new approach, based on bounding Ramsey numbers of certain graphs, allows us to improve the previously best lower bounds due to Linial–Rabinovich, Erdős–Hajnal and Alon–Sudakov. As an example, we prove that any n-vertex graph G having an independent set of size 3 among every 7 vertices has alpha (G) ge Omega (n^{5/12}). This confirms a conjecture of Erdős and Hajnal that alpha (G) should be at least n^{1/3+varepsilon } and brings the exponent halfway to the best possible value of 1/2. Our second approach deals with upper bounds. It relies on a reduction of the original question to the following natural extremal problem. What is the minimum possible value of the 2-density (The 2-density of a graph H is defined as m_2(H):=max _{H' subseteq H, |H'| ge 3} frac{e(H')-1}{|H'|-2}) of a graph on m vertices having no independent set of size r? This allows us to improve previous upper bounds due to Linial–Rabinovich, Krivelevich and Kostochka-Jancey. As part of our arguments, we link the problem of Erdős–Hajnal and Linial–Rabinovich and our new extremal 2-density problem to a number of other well-studied questions. This leads to many interesting directions for future research.

The Problem of Complex Heat Transfer with Cauchy-Type Conditions on a Part of the Boundary

The paper considers a boundary value problem for stationary equations of complex heat transfer with an undetermined boundary condition for the radiation intensity on a part of the boundary and an overdetermined condition on another part of the boundary. An optimization method for solving this problem is proposed, and an analysis of the corresponding problem of boundary optimal control is presented. It is shown that the sequence of solutions of extremum problems converges to the solution of a problem with Cauchy-type conditions. The efficiency of the algorithm is illustrated by numerical examples.

A Review on Prediction and Analysis of Forest Fires Using AI and ML Algorithms

Abstract: Forest fires or wildfires are major catastrophes that occur in forests, grasslands, or prairies (Grassland areas). Wildfires mostly occur either due to natural factors or human activities such as smoking cigarettes, campfires, or arson, etc. Forest fire has become one of the most drastic problems that cause damage to several forests around the globe. To prevent forest fires, analysis, and predictions should be made on land that is affected by forest fires based on temperature, wind, humidity, etc. Depending upon the above factors, analysis, and prediction are done, which region has a high possibility of wildfires’ dangerous effects. Fire detection will help in finding and controlling an extreme problem in forests, this will help in reducing the forest fire in the future. Many Algorithms are available for the detection of the fire, Based on the use of Algorithms analysis and prediction of the forest fires are made.

The impact of Philosophy and Critical Thinking as ‘Intellectual Immunity’ in Deterring Violence Thought

This research proposes philosophy as an intellectual immunity and a way to open up to others in order to prevent violence discussed. Therefore, the research is a serious philosophical attempt to discuss one of the dangerous issues that contemporary man suffers from, which is terrorism and violence extremism. The philosophical approach to this issue is analysis and criticism, and then the alternative available to it. Secondly, the research deals with the interpretation of the role of philosophy in how to deal with crises by presenting keyconcepts and ideas stemming from the great philosophers. The new interpretation that the researcher offers to philosophy is that philosophy is an intellectual and preventive deterrent to extremism and terrorism. In other words; Philosophical thinking is tantamount to digging up the wrong intellectual and logical foundations on which extremism and terrorism are based. By presenting the alternative, and the effort is not focused only on denouncing or rhetorically denouncing the issue. The research is an answer to the questions that are often asked: What is the role of philosophy in dealing with the crises that contemporary man is going through? Can critical philosophical thought be an intellectual and mental skill to reduce the effects of extremist thought or weaken the tendency toward extremism? Can the extremist person be intellectually deformed and mentally ill, or are there other things behind him? Through this question, the researcher goes deep into Greek philosophy, especially Stoicism, and defines philosophy through the role or function it plays in dealing with the problem of terrorism and extremism.

Extremal problems of double stars

In a generalized Tur\'an problem, two graphs $H$ and $F$ are given and the question is the maximum number of copies of $H$ in an $F$-free graph of order $n$. In this paper, we study the number of double stars $S_{k,l}$ in triangle-free graphs. We also study an opposite version of this question: what is the maximum number edges/triangles in graphs with double star type restrictions, which leads us to study two questions related to the extremal number of triangles or edges in graphs with degree-sum constraints over adjacent or non-adjacent vertices.

Extremal problems of Turán-type for a univariate complex coefficient polynomial

Extremal problems of Turán-type for a univariate complex coefficient polynomial

Health-related quality of life and medication use among individuals with Angelman syndrome.

The primary goal of this analysis is to describe the health-related quality of life (HRQoL), medical history, and medication use among adolescents and adults individuals with Angelman syndrome (AS). The analysis uses baseline data collected during the STARS study, a double-blind placebo controlled trial of gaboxadol (OV101) in adolescents and adults with AS. The HRQoL was estimated using EuroQoL 5-Dimension 5-Level (EQ-5D) health questionnaire proxy 1 version, which was completed by the caregivers. EQ-5D consists of two parts, a 5-dimension descriptive and a visual analogue scale (VAS) component. The utility score derived from EQ-5D ranges from 0 to 1 (perfect health) and VAS ranges from 0 to 100 (perfect health). 87 individuals with AS were included in the present analysis. The mean utility score was 0.44 ± 0.20 and VAS score was 84 ± 1.5. The EQ-5D data indicated that the self-care, mobility and daily activities were most impacted. All adolescents (100%) and most adults (93%) had at least moderate problems with self-care activities, such as washing or dressing themselves. More than half (55%) of the adolescents and adults had at least moderate issues with mobility and usual activities. Approximately, 30% of adolescents and adults had moderate to extreme problems with anxiety/depression. High baseline concomitant use of medications was observed across both age groups with an average of 5 medications being used per person. This study highlights the impact of AS on HRQoL and medication utilization among adolescents and adults individuals with AS.

The maximum spectral radius of {C3,C5}-free graphs of given size

In this paper we consider the extremal problem on adjacency spectral radius of {C3,C5}-free graphs. Assume that G is a graph with m edges having no isolated vertices, and let λ be the spectral radius of its adjacency matrix. Firstly, by using the method of characterizing {C3,C5}-free non-bipartite graphs whose second largest eigenvalue is less than 54, we show that, if G is a {C3,C5}-free non-bipartite graph of size m, thenλ⩽∑u∈VGd(u)2−m+4f−54. Equality holds if and only if G≅C7, where d(u) is the degree of vertex u and f denotes the number of 4-cycles in G. Secondly, we show that, if G is a {C3,C5}-free non-bipartite graph of odd size m, then λ⩽θ(m) with equality if and only if G≅RK2,m−32, where θ(m) is the largest root ofx4−x3−(m−3)x2+(m−4)x+m−5=0 and RK2,m−32 is obtained by replacing an edge of the complete bipartite graph K2,m−32 with P5.

The Impact of External Dynamics of the Pamir Crisis in 2018-2022

Despite the fact that the Pamir region has been experiencing turbulence for decades, experts agree that lately the central government's course towards Pamir has intensified to an unprecedented degree. The author's hypothesis is that one of the main reasons for this trend is a change in the configuration of external actors' interests. In this regard, the research question is: “What is the impact of key external actors on the dynamics of the Pamir crisis of 2018-2022?” The methodological basis of the study includes a critical analysis of primary and secondary sources, case studies and comparative analysis. The scientific novelty of the study lies in the fact that this topic remains marginal both in Russia and abroad, and in Tajikistan it is often taboo. Another factor preventing this topic from leaving the periphery of scientific knowledge is the inaccessibility of reliable information sources. Based on the results of the study, the author comes to the conclusion that the Pamir crisis is an extremely multifactorial conflict that is becoming increasingly internationalized. Pamir is an extremely resource-rich and promising region, which borders on Afghanistan, ruled by a terrorist regime, and on the Chinese XUAR, where the problems of terrorism, separatism and extremism remain unresolved. All this makes the Pamirs a point of interest intertwining of various external players in economics, security and the military-industrial complex and makes the dynamics of the internal crisis in Pamir dependent on the configuration of these interests.

Willingness to Pay for Medicines used in the Management of Hypertension and Diabetes Mellitus in Plateau State, Nigeria.

Background: Due to poverty, in low- and middle-income countries the increase in cost leading to unwillingness to pay for life long treatment of most chronic diseases like hypertension and diabetes in Nigeria is always on the increase resulting to increase in morbidity and mortality. To fifind out the solution, the willingness to pay for essential medicines by hypertensive and diabetic patients in Jos and environs in 2019 was used as a tool. Method: A cross sectional descriptive study using the health valuation method with the 5 dimension 3 level version of European Quality of Life 5 Dimensions 3 Level Version EQ-5D (EQ-5D-3L) is a generic tool for Patient Reported Outcomes (PRO) measurement that can assess patients' quality of life, irrespective of the disease introduced in 1990 by the European quality of life (EuroQol) group questionnaire was adopted the fifive dimensions namely: Mobility, Self-care, Usual activities, Pain / discomfort, and Anxiety / depression while the 3 Levels include: No problems/ pain/ anxiety, Some problems/ pain/ anxiety, and Extreme problems/ pain/anxiety. Results: Out of the 360 respondents of which majority were females 192 (53.3%), respondents aged 55 years were in majority 303 (84.2%). Those with hypertension were higher 257 (71.40%) than those with diabetes and those with both diseases. Majority of the respondents earn between NGN61,000 to NGN140,000 23 (64.5%) while only 5 (1.4%) of them earn NGN181,000 and above. Conclusion: Most of the respondents (75.4%) where willing to pay between NGN3,000.00 and less for their treatment while only a few (1.2%) where willing to pay NGN15,000.00 and above. Over 70% of the respondents indicated no problems with mobility, self-care, carrying out usual activities, and anxiety/depression while 98.2% had no problems of taking care of themselves. There was signifificant relationship between gender and anxiety/depression (P-0.001). Married respondents reported problems with mobility more than others (P=0.0001). Those who earn between NGN61,000 and NGN100,000 signifificantly did not have any problems with mobility (P=0.0001) and self-care (P=0.031) respectively compared to other income categories.&nbsp

The Practice of Sufism And Religious Moderation In The Kauman Pesantren Communities, Central Java, Indonesia

This article deals with efforts to find solutions to the problems of extremism and religious radicalism. This was done by looking at the tolerant, peaceful, and harmonious practice of living together in the pesantren of Kauman and ethnic Chinese community Islamic boarding schools in Central Java, Indonesia. Data were obtained through interviews, observation, and documentation with qualitative methods. The founder of the pesantren instilled Sufistic values in the students. Sufistic values that are instilled include; mahabbah (love), tawazun (balance), tasamuh (tolerance), musawah (equality). The tasawuf doctrine applied by the pesantren of Kauman is very prospective and becomes a spiritual force in the present and the future. By implementing the Sufistic doctrines and religious moderation values, the threat of this disintegration and problems of extremism and religious radicalism that have been attempted by various groups can be totally removed, and hopefully, people would live in peace and harmony with plural, diverse and heterogenous ethnic.

Some Upper Bounds of Maximum <I>E</I>-Eigenvalues of Uniform Hypergraphs

A hypergraphs, as a generalization of a general graph, is often used as an effective tool to describe complex structures in discrete mathematics, computer science and other fields. Hypergraph theory and related parameters of hypergraph are important research topics in hypergraph theory. In particular, the problem of spectral extremum of graphs has been widely concerned. This problem originates from the problem proposed by Brualdi and Solheid in 1986. That is to find the upper and lower bounds of spectral radius of a given graph class and characterize the polar graph that reaches the upper and lower bounds. Let <I>H </I>be a uniform hypergraph. Let <I>A</I>(<I>H</I>) be the adjacency tensor of <I>H</I>. In this work, by using Perron-Frobenius theorem, Hölder’s inequality and inequality of arithmetic and geometric means, we establish some upper bounds for the maximum <I>E</I>-eigenvalue of a uniform hypergraph instead of the degrees of vertices and edge number of hypergraph H. In addition, we characterize the extremal hypergraphs that reach the upper bounds.

Investigation of load-bearing capacity of rigid-plastic reinforced ellipsoidal shells of rotation

For ellipsoidal fiber-reinforced shells of rotation, the upper (kinematic) limit of load-bearing capacity is determined. The structures are clamped along the lower support contour and have rigid polar plate. The material of the components is rigid-plastic with asymmetry in tension and compression. The matrix is orthotropic. The principal axes of matrix anisotropy coincide with meridional, circumferential and transverse directions. Two-layer shell model and refined relations of structural model with two-dimensional stress state in components are used. Extreme problem is formulated and numerically solved for spherical and two different ellipsoidal shells. Influence of reinforcement parameters on limit load is investigated.

Localized versions of extremal problems

We generalize several classical theorems in extremal combinatorics by replacing a global constraint with an inequality which holds for all objects in a given class. In particular we obtain generalizations of Turán’s theorem, the Erdős-Gallai theorem, the LYM-inequality, the Erdős-Ko-Rado theorem and the Erdős-Szekeres theorem on sequences.

On a Conjecture of Cai–Zhang–Shen for Figurate Primes

A conjecture of Cai–Zhang–Shen for figurate primes says that every integer k&gt;1 is the sum of two figurate primes. In this paper, we give an equivalent proposition to the conjecture. By considering extreme value problems with constraints about the conjecture in the cases of odd and even integers and using the method of Lagrange multipliers, the Cardano formula for cubic equations, and the contradiction, we prove the conjecture.

Association of physical activity, smoking, and socioeconomic factors on health checkup participation in community-dwelling stroke survivors aged 50 years or older

BackgroundWe investigated the sociodemographic and health-related factors associated with health checkup participation in community-dwelling stroke survivors.MethodsAmong participants of the Korea National Health and Nutrition Examination Survey, 642 stroke survivors were included. We investigated the sociodemographic, medical, and health-related quality of life factors—evaluated by the EuroQol 5-Dimension Questionnaire (EQ-5D)—associated with participation in any type of health checkup. To explore the associations between multiple variables and health checkup participation, a multivariable complex-sample logistic regression model was used.ResultsOne-third of the community-dwelling stroke survivors did not receive a health checkup in the past two years. Insufficient physical activity (OR: 0.5, 95% CI: 0.3–0.9), current smoking (OR: 0.4, 95% CI: 0.2–0.8), low education level (OR: 0.5, 95% CI: 0.3–0.9), living alone (OR: 0.5, 95% CI: 0.3–0.998), and no occupation (OR: 0.5, 95% CI: 0.3–0.9) showed independent negative associations with health checkup participation. Among the five EQ-5D dimensions, mobility, self-care, usual activities, and pain/discomfort dimensions were associated with health checkup participation rate.ConclusionPolicies and further research are needed to promote health checkups for stroke survivors who are physically inactive, currently smoking, living alone, unemployed, less educated, or having extreme problems in their daily lives.

On the Stability of Solutions to Control Problems for a Nonlinear Reaction–Diffusion–Convection Model

The problems of multiplicative control for the reaction–diffusion–convection model with coefficients that are nonlinearly dependent on the solution and also dependent on spatial variables are studied. In the case of a power-law dependence of the model coefficients on the solution, optimality systems are derived for extremum problems. With their help, estimates of the local stability of solutions of specific control problems with respect to small perturbations of both the cost functionals and one of the given functions of the boundary value problem are obtained.