Open Questions Research Articles

Numerical simulation of an effective transform mechanism with convergence analysis of the fractional diffusion-wave equations

In the current study, we solve two very important mathematical models, such as the time fractional-order space-fractional telegraph and diffusion-wave equations using a reliable technique called the Adomian decomposition natural method (ADNM), which combines Adomian decomposition and natural transform. The diffusion wave equation describes the flood wave propagation, which is used in solving overland and open channel flow problems. For this reason, it is critical to fully understand and effectively solve the diffusion wave equations. Because telegraph equations are crucial for modeling and developing voltage or frequency transmission, they are widely used in physics and engineering. Furthermore, the designing process is greatly impacted by the uncertainty in the system parameters. For nonlinear ordinary differential equations based on the theorem of Banach fixed point, we provide existence and uniqueness theorem proofs. The present approach has been successfully used to explore exact solutions for time fractional-order and space fractional-order applications. The results show how effective and valuable the ADNM. This paper presents a methodology that will be used in future work to address similar nonlinear problems related to fractional calculus.

On the closure of the Airault–Mckean–Moser locus for elliptic KdV potentials via Darboux transformations

We study the general elliptic KdV potentials, which can be expressed (up to adding a constant) as q_{\mathbf{p}}(z)\coloneq\sum_{j=1}^{n}m_{j}(m_{j}+1)\wp(z-p_{j}),\quad m_{j}\in\mathbb{N}. We give an elementary proof of the theorem that the singularity \mathbf{p}=( \overset{m_{1}( m_{1}+1) /2}{\overbrace{p_{1},\dots ,p_{1}}},\dots,\overset{m_{n}(m_{n}+1) /2}{\overbrace{p_{n},\dots,p_{n}}% }) is contained in the closure of the elliptic Airault–Mckean–Moser locus, which was proved previously by Treibich and Verdier in the late 1980s using purely algebro-geometric methods. Our proof is based on Darboux transformations and does not use algebraic geometry. This solves an open problem posed by Gesztesy, Unterkofler, and Weikard [Trans. Amer. Math. Soc. 358 (2006), 603–656]. Some applications are also given.

Problems on [formula omitted]

This is an update on, and expansion of, our paper Open problems on βω in the book Open Problems in Topology.

On the hyperspaces of meager and regular continua

Given a metric continuum X, we consider the collection of all regular subcontinua of X and the collection of all meager subcontinua of X, these hyperspaces are denoted by D(X) and M(X), respectively. It is known that D(X) is compact if and only if D(X) is finite.In this way, we find some conditions related about the cardinality of D(X) and we reduce the fact to count the elements of D(X) to a Graph Theory problem, as an application of this, we prove in particular that | D ( X ) | ∉ { 2,3,4,5,8 , 9 } h t ) for any continuum X. Also, we prove that D(X) is never homeomorphic to ℕ . On the other hand, given a point p ∈ X , we consider the meager composant and the filament composant of p in X, denoted by MXp and FcsX(p), respectively, and we study some relations between MXp and FcsX(p) such as the equality of them as a subset of X. Also, we construct examples showing that the collection Fcs ( X ) = { FcsX ( p ) : p ∈ X } can be homeomorphic to: any finite discrete space, the harmonic sequence, the closure of the harmonic sequence and the Cantor set. Finally, we study the contractibility of M(X); we prove the arc of pseudo-arcs, which is a no contractible continuum, satisfies that its hyperspace of meager subcontinua is contractible, given a solution to an open problem. Also, we rise open problems.

Rational noncrossing Coxeter–Catalan combinatorics

AbstractWe solve two open problems in Coxeter–Catalan combinatorics. First, we introduce a family of rational noncrossing objects for any finite Coxeter group, using the combinatorics of distinguished subwords. Second, we give a type‐uniform proof that these noncrossing Catalan objects are counted by the rational Coxeter–Catalan number, using the character theory of the associated Hecke algebra and the properties of Lusztig's exotic Fourier transform. We solve the same problems for rational noncrossing parking objects.

Closing the Gap Between Theory and Practice During Alternating Optimization for GANs.

Synthesizing high-quality and diverse samples is the main goal of generative models. Despite recent great progress in generative adversarial networks (GANs), mode collapse is still an open problem, and mitigating it will benefit the generator to better capture the target data distribution. This article rethinks alternating optimization in GANs, which is a classic approach to training GANs in practice. We find that the theory presented in the original GANs does not accommodate this practical solution. Under the alternating optimization manner, the vanilla loss function provides an inappropriate objective for the generator. This objective forces the generator to produce the output with the highest discriminative probability of the discriminator, which leads to mode collapse in GANs. To address this problem, we introduce a novel loss function for the generator to adapt to the alternating optimization nature. When updating the generator by the proposed loss function, the reverse Kullback-Leibler divergence between the model distribution and the target distribution is theoretically optimized, which encourages the model to learn the target distribution. The results of extensive experiments demonstrate that our approach can consistently boost model performance on various datasets and network structures.

Geodesic connectivity and rooftop envelopes in the Cegrell classes

AbstractThis study examines geodesics and plurisubharmonic envelopes within the Cegrell classes on bounded hyperconvex domains in $$\mathbb {C}^n$$ C n . We establish that solutions possessing comparable singularities to the complex Monge–Ampère equation are identical, affirmatively addressing a longstanding open question raised by Cegrell. This achievement furnishes the most general form of the Bedford–Taylor comparison principle within the Cegrell classes. Building on this foundational result, we explore plurisubharmonic geodesics, broadening the criteria for geodesic connectivity among plurisubharmonic functions with connectable boundary values. Our investigation also delves into the notion of rooftop envelopes, revealing that the rooftop equality condition and the idempotency conjecture are valid under substantially weaker conditions than previously established, a finding made possible by our proven uniqueness result. The paper concludes by discussing the core open problems within the Cegrell classes related to the complex Monge–Ampère equation.

Antipsychotic Drugs: A Concise Review of History, Classification, Indications, Mechanism, Efficacy, Side Effects, Dosing, and Clinical Application.

The introduction of the first antipsychotic drug, chlorpromazine, was a milestone for psychiatry. The authors review the history, classification, indications, mechanism, efficacy, side effects, dosing, drug initiation, switching, and other practical issues and questions related to antipsychotics. Classifications such as first-generation/typical versus second-generation/atypical antipsychotics are neither valid nor useful; these agents should be described according to the Neuroscience-based Nomenclature (NbN). Antipsychotic drugs are not specific for treating schizophrenia. They reduce psychosis regardless of the underlying diagnosis, and they go beyond nonspecific sedation. All currently available antipsychotic drugs are dopamine blockers or dopamine partial agonists. In schizophrenia, effect sizes for relapse prevention are larger than for acute treatment. A major unresolved problem is the implausible increase in placebo response in antipsychotic drug trials over the decades. Differences in side effects, which can be objectively measured, such as weight gain, are less equivocal than differences in rating-scale-measured (subjective) efficacy. The criteria for choosing among antipsychotics are mainly pragmatic and include factors such as available formulations, metabolism, half-life, efficacy, and side effects in previous illness episodes. Plasma levels help to detect nonadherence, and once-daily dosing at night (which is possible with many antipsychotics) and long-acting injectable formulations are useful when adherence is a problem. Dose-response curves for both acute treatment and relapse prevention follow a hyperbolic pattern, with maximally efficacious average dosages for schizophrenia of around 5 mg/day risperidone equivalents. Computer apps facilitating the choice between drugs are available. Future drug development should include pharmacogenetics and focus on drugs for specific aspects of psychosis.

Non-Invasive Assessment of Coronary Microvascular Dysfunction Using Vascular Deformation-Based Flow Estimation.

Non-invasive computation of the index of microcirculatory resistance from coronary computed tomography angiography (CTA), referred to as IMR[Formula: see text], is a promising approach for quantitative assessment of coronary microvascular dysfunction (CMD). However, the computation of IMR[Formula: see text] remains an important unresolved problem due to its high requirement for the accuracy of coronary blood flow. Existing CTA-based methods for estimating coronary blood flow rely on physiological assumption models to indirectly identify, which leads to inadequate personalization of total and vessel-specific flow. To overcome this challenge, we propose a vascular deformation-based flow estimation (VDFE) model to directly estimate coronary blood flow for reliable IMR[Formula: see text] computation. Specifically, we extract the vascular deformation of each vascular segment from multi-phase CTA. The concept of inverse problem solving is applied to implicitly derive coronary blood flow based on the physical constraint relationship between blood flow and vascular deformation. The vascular deformation constraints imposed on each segment within the vascular structure ensure sufficient individualization of coronary blood flow. Experimental studies on 106 vessels collected from 89 subjects demonstrate the validity of our VDFE, achieving an IMR[Formula: see text] accuracy of 82.08 %. The coronary blood flow estimated by VDFE has better reliability than the other four existing methods. Our proposed VDFE is an effective approach to non-invasively compute IMR[Formula: see text] with excellent diagnostic performance. The VDFE has the potential to serve as a safe, effective, and cost-effective clinical tool for guiding CMD clinical treatment and assessing prognosis.

Equivariant insight into hyperspaces of convex bodies

Let cc(Rn) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space Rn endowed with the Hausdorff metric, and let cc(Bn)={A∈cc(Rn)|A⊂Bn}, where Bn is the closed unit ball of Rn. Let cb(Rn) be the subset of cc(Rn) consisting of all convex bodies, i.e., of sets A∈cc(Rn) with non-empty interior. In this survey article we reveal fundamental properties of the natural action Aff(n)↷cb(Rn) of the affine group that leads to structure results for cb(Rn) and for several important subsets of it. Orbit spaces of cb(Rn) and its subsets by some appropriate subgroups G<Aff(n) are homeomorphic to such important objects like the Hilbert cube or the Banach-Mazur compacta. Among other hyperspaces studied here are cw(Rn) - the hyperspace of convex bodies of constant width, Bp - the hyperspace of convex bodies with smooth boundary of positive Gaussian curvature, L(n) - the hyperspace of convex bodies for which the Euclidean unit ball is the Löwner ellipsoid, cˇ(n) - the hyperspace of convex sets for which the Euclidean unit ball is the Chebyshev ball, etc. The paper contains also several new results concerning equivariant extension properties of the hyperspaces in question, as well as the homeomorphism type of the hyperspace cb(Bn)=cb(Rn)∩cc(Bn) and its orbit spaces. Equivariant extension properties of these hyperspaces are applied to provide a very short proof of the classical Grünbaum Conjecture from Convex Geometry. Along the paper seventeen open problems are raised.

Topologies related to (I)-envelopes

We investigate the question whether the (I)-envelope of any subset of a dual to a Banach space X may be described as the closed convex hull in a suitable topology. If X contains no copy of ℓ1 then the weak topology generated by functionals of the first Baire class in the weak⁎ topology works. On the other hand, if X contains a complemented copy of ℓ1 or X=C([0,1]) no locally convex topology works. If we do not require the topology to be locally convex, the problem is still open. We further introduce and compare several natural intermediate closure operators on a dual Banach space. Finally, we collect several intriguing open problems related to (I)-envelopes.

On invariant means and pre-syndetic subgroups

Beyond the locally compact case, equivalent notions of amenability diverge, and some properties no longer hold, for instance amenability is not inherited by topological subgroups. This investigation is guided by some amenability-type properties of groups of paths and loops. It is shown that a version of amenability called skew-amenability is inherited by pre-syndetic subgroups in the sense of Basso and Zucker (in particular, by co-compact subgroups). It follows that co-compact subgroups of amenable topological groups whose left and right uniformities coincide are amenable. We discuss a version of amenability belonging to P. Malliavin and M.-P. Malliavin: the existence of a mean on bounded Borel functions that is invariant under the left action of a dense subgroup. We observe that this property is in general strictly stronger than amenability, and establish for it Reiter- and Følner-type criteria. Finally, there is a review of open problems.

Rigorously proven chaos in chemical kinetics.

This study addresses a longstanding question regarding the mathematical proof of chaotic behavior in kinetic differential equations. Following the numerous numerical and experimental results in the past 50 years, we introduce two formal chemical reactions that rigorously demonstrate this behavior. Our approach involves transforming chaotic equations into kinetic differential equations and then realizing them through formal chemical reactions. The findings present a novel perspective on chaotic dynamics within chemical kinetics, thereby resolving a longstanding open problem.

Distinct polytropic behavior of plasma during ICME-HSS interaction

Interplanetary Coronal Mass Ejections (ICMEs) and High-Speed Streams (HSSs) are significant drivers of space disturbance in interplanetary space. The interaction between these structures can lead to various phenomena, including the generation of waves, enhanced geo-effectiveness, particle acceleration, and more. However, understanding the plasma thermodynamic properties during an ICME-HSS interaction remains an open problem. In this study, we utilize observations from STEREO and WIND to investigate the thermodynamic behavior of plasma within an ICME and a HSS. Here, we used the thermal pressure (Pth) and proton number density (Np) data from spacecraft. By fitting the linear model to the scattered data of lnPth vs lnNp we estimated the polytropic index (α). Our analysis reveals that the ICME observed before its interaction with the HSS by STEREO-A exhibits a polytropic index α∼1.0, indicating an isothermal process. Conversely, WIND observations of the same ICME before its interaction with the HSS show a nearly isothermal behavior, with α=0.7. The HSS observed by WIND demonstrates α=1.8 before its interaction with the ICME. Further analysis of WIND observations within the ICME-HSS interaction region determines a polytropic index of α=2.5. These findings suggest that the HSS region displays a nearly adiabatic behavior, the ICME region manifests closely isothermal characteristics, and the ICME-HSS interaction region exhibits super-adiabatic behavior. We propose that the expansion of ICME increases due to its propagation in the vicinity of HSS, potentially leading to additional cooling of the observed ICME magnetic cloud, which differs from the ambient solar wind.

Dinamika Harga Beras di Kota Kendari: Analisis Fluktuasi 2015-2023

Food price stability is one of the essential issues in the direction of government policy. Price uncertainty at the farm level and the guarantee of food price stability at the consumer level are unresolved problems. This study aims to analyze rice price fluctuations in Kendari City from 2015 to 2024. The research was conducted in Kendari City in March - May 2024. The research used secondary data, namely time series data on rice prices in Kendari City from 2015 to 2023. The research data collection technique is a literature study, namely collecting time series data on rice prices in Kendari City sourced from the Central Bureau of Statistics and related agencies. The research variable is the price of rice in Kendari City from 2015 to 2023. Data analysis used to analyze rice price fluctuations in Kendari City is the Coefficient of Variation (CV). The results showed that rice price fluctuations in Kendari City from 2015 to 2023 were low, with a coefficient of variation (CV) of 3%. The highest rice price condition in Kendari City occurred in 2023, which amounted to IDR 13,321 per kilogram. At the same time, the lowest rice price condition occurred in 2015, which amounted to IDR 9,738 per kilogram. Therefore, government policy intervention is needed to stabilize food prices, especially rice in Kendari City.

Modern features and problems of forming balanced economic growth and development of the Republic of Tatarstan

This paper examines the theoretical foundations and provisions for the formation of balanced economic growth and development of the region considering its territorial, socio-economic, resource, and structural features and using the Republic of Tatarstan as an example. The paper analyses the theoretical and methodological apparatus proposed by scientists and specialists in ensuring the economic growth and development of the region. The purpose is to identify the main risks, factors and problems in the economic sectors of the Republic of Tatarstan, which to one degree or another are capable of influencing the processes of forming the balanced economic growth and development of the region. Based on the analysis, the paper identifies common problems for the regions and formulates the main directions for improving the economic situation in the region, and also submits for consideration the authors’ model of interaction between enterprises, state and municipal entities in order to reduce the negative impact of previously identified threats and problems, which considers new global challenges, transformations and restructures in economic processes associated with the introduction of restrictive measures, political isolation, various economic sanctions, and with the unresolved internal problems in the field of the economic growth and development of the Republic of Tatarstan.

A comprehensive review of combating EDoS attacks in cloud services with deep learning and advanced network security technologies including DDoS protection and intrusion prevention systems

This comprehensive review examines the current state of research and practice in combating Economic Denial of Sustainability (EDoS) attacks in cloud services, with a focus on deep learning approaches and advanced network security technologies. The paper provides an in-depth analysis of EDoS attack characteristics, their impact on cloud economics, and the challenges faced in mitigation efforts. It explores the application of deep learning techniques in EDoS detection and prevention, highlighting recent advancements in neural network architectures and feature extraction methods. The review also covers the integration of advanced network security technologies, including next-generation firewalls, software-defined networking, and cloud-native security solutions, in the context of EDoS protection. Furthermore, it discusses the adaptation of Distributed Denial of Service (DDoS) mitigation strategies for EDoS attacks, emphasizing traffic analysis and anomaly detection techniques. The role of Intrusion Prevention Systems (IPS) in EDoS mitigation is examined, comparing signature-based and behavior-based approaches and exploring their integration with other security components. The paper concludes by identifying emerging threats, regulatory considerations, and open research problems in EDoS protection, providing valuable insights for researchers and practitioners in the field of cloud security. This review aims to serve as a comprehensive resource for understanding the current landscape of EDoS attacks and defense mechanisms, while also highlighting future directions for research and development in this critical area of cloud computing security.

Column generation approach for 1.5-dimensional cutting stock problem with technical constraints

In this study, the 1.5-dimensional cutting stock problem with technical constraints is considered. In the literature, this problem is also defined as a strip packing or open dimension problem. When given a strip of infinite length and bounded width, the problem is to define a packing of rectangular objects into a strip that minimizes its final length. Technical constraints, such as the order type and the number of strips, are indispensable in real life; however, they are often neglected in the literature because they make the problem difficult to solve. Only one study was reached in the literature that took into account technical constraints, but in that mentioned study, only a mathematical model was proposed for the problem. In this context, our aim is to solve the problem with a more effective approach. The research question in this study is the usability of the column generation technique to solve the 1.5-dimensional cutting stock problem. In this study, the column generation approach was proposed for the first time for the considered problem. To demonstrate the performance of the proposed solution method, randomly generated test problems were solved with GAMS/Cplex. As we report the results, proposed column generation approach (CG) reaches very close (such as %1 and %2 error) solutions to integrated mathematical model (IM) for small sized problems in a second. On the other hand, while CG solved all the problems in a reasonable time, IM could not produce a feasible solution to some problems. Numerical experiments showed that the column generation algorithm outperforms the integrated mathematical model for the problem.

Discrete Geometry

A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as convex geometry, combinatorics, or topology. Two open problem sessions highlighted the abundance of open questions and many of the results presented were obtained by young researchers, confirming the vitality of the field.

Mini-Workshop: Bridging Number Theory and Nichols Algebras via Deformations

Nichols algebras are graded Hopf algebra objects in braided tensor categories. They appeared first in a paper by Nichols in 1978 in the search for new examples of Hopf algebras. Rediscovered later several times, they also provide a conceptual explanation of the construction of quantum groups. The aim of the workshop is to review recent developments in the field, initiate collaborations, and discuss new approaches to open problems.