Uniqueness Results Research Articles

New Results for Existence, Uniqueness, and Ulam Stable Theorem to Caputo–Fabrizio Fractional Differential Equations with Periodic Boundary Conditions

New Results for Existence, Uniqueness, and Ulam Stable Theorem to Caputo–Fabrizio Fractional Differential Equations with Periodic Boundary Conditions

Broken Ray Transform for Twisted Geodesics on Surfaces with a Reflecting Obstacle

We prove a uniqueness result for the broken ray transform acting on the sums of functions and 1-forms on surfaces in the presence of an external force and a reflecting obstacle. We assume that the considered twisted geodesic flows have nonpositive curvature. The broken rays are generated from the twisted geodesic flows by the law of reflection on the boundary of a suitably convex obstacle. Our work generalizes recent results for the broken geodesic ray transform on surfaces to more general families of curves including the magnetic flows and Gaussian thermostats.

Theory of majorana zero modes in unconventional superconductors

Abstract Majorana fermions are spin-1/2 neutral particles that are their own antiparticles and were initially predicted by Ettore Majorana in particle physics but their observation still remains elusive. The concept of Majorana fermions has been borrowed into condensed matter physics, where, unlike particle physics, Majorana fermions emerge as zero-energy quasiparticles that can be engineered by combining electrons and holes and have therefore been coined Majorana zero modes. In this review, we provide a pedagogical explanation of the basic properties of Majorana zero modes in unconventional superconductors and their consequences in experimental observables, putting a special emphasis on the initial theoretical discoveries. In particular, we first show that Majorana zero modes are self-conjugated and emerge as a special type of zero energy surface Andreev bound states at the boundary of unconventional superconductors. We then explore Majorana zero modes in one-dimensional spin-polarized p-wave superconductors, where we address the formation of topological superconductivity and the physical realization in superconductor-semiconductor hybrids. In this part we highlight that Majorana quasiparticles appear as zero-energy edge states, exhibiting charge neutrality, spin-polarized, and spatial nonlocality as unique properties that can be already seen from their energies and wavefunctions. Next, we discuss analytically obtained Green’s functions of p-wave superconductors and demonstrate that the emergence of Majorana zero modes is always accompanied by the formation of odd-frequency spin-triplet pairing as a unique result of the self-conjugate nature of Majorana zero modes. We finally address the signatures of Majorana zero modes in tunneling spectroscopy, including the anomalous proximity effect, and the phase-biased Josephson effect.

On the dynamic Coulomb's frictional contact problem for thermo‐piezoelectric materials

AbstractIn this work, we study, from a variational point of view, a dynamic contact problem between a thermo‐piezoelectric body and a thermally conductive foundation. The normal compliance contact condition and Coulomb's friction law are employed to model the contact. We provide existence and uniqueness results of a weak solution to the model using adequate auxiliary problems, an abstract result on nonlinear first‐order evolution inequalities and Banach fixed point argument. Finally, the continuous dependence of the solution on the surface traction force and the surface electrical charge is studied.

Reflected BSDEs driven by G-Brownian motion with time-varying Lipschitz coefficients

In this paper, we consider the reflected backward stochastic differential equations driven by [Formula: see text]-Brownian motion (reflected [Formula: see text]-BSDEs) with time-varying Lipschitz coefficients. We obtain the uniqueness result by establishing a priori estimates. For the existence, the solution can be approximated by a family of reflected [Formula: see text]-BSDEs with Lipschitz conditions and by penalized [Formula: see text]-BSDEs with time-varying coefficients. The latter approximation is useful to get the comparison theorem. Finally, we study the reflected [Formula: see text]-BSDEs with infinite time horizon.

Flow of Bingham fluid through a three dimensional thin layer

The paper is devoted to the study of asymptotic behavior of the solution of three dimensional steady flow of Bingham fluid through a three dimensional thin layer with Dirichlet boundary conditions. We are interested in the asymptotic behavior, to this aim we prove some convergence results concerning the velocity and pressure when the thickness tends to zero. The limit problem obtained after transforming the original problem into one posed over a fixed reference domain and the parameter representing the thickness of the layer tend to zero is studied. The lower-dimensional constitutive law and the differential equation satisfied by the limit variables in the non rigid zone are obtained. In addition, the uniqueness of limit solution has been also established. Existence and uniqueness results and a lower-dimensional constitutive law are obtained. An identical study of a two-dimensional problem yields a one dimensional model prevalent in engineering literature.

Partial data inverse problems for magnetic Schrödinger operators on conformally transversally anisotropic manifolds

We study inverse boundary problems for the magnetic Schrödinger operator with Hölder continuous magnetic potentials and continuous electric potentials on a conformally transversally anisotropic Riemannian manifold of dimension n ⩾ 3 with connected boundary. A global uniqueness result is established for magnetic fields and electric potentials from the partial Cauchy data on the boundary of the manifold provided that the geodesic X-ray transform on the transversal manifold is injective.

Uniqueness of phase retrieval with short-time linear canonical transform

In this paper, we study the problem of phase retrieval with short-time linear canonical transform (STLCT). The relation between signal, window and their STLCT is provided through Fourier transform. Based on this theorem, a uniqueness result is established for all square integrable functions. For nonseparable real continuous signal, we prove the uniqueness theorems under some weaker conditions. In complex bandlimited and cardinal [Formula: see text]-spline spaces, uniqueness results are provided with magnitude-only STLCT.

Local near-field scattering data enables unique reconstruction of rough electric potentials

The focus of this paper is the study of the inverse point-source scattering problem, specifically in relation to a certain class of electric potentials. Our research provides a novel uniqueness result for the inverse problem with local data, obtained from the near field pattern. Our work improves the work of Caro and Garcia, who investigated both the direct problem and the inverse problem with global near field data for critically singular and δ -shell potentials. The primary contribution of our research is the introduction of a Runge approximation result for the near field data on the scattering problem which, in combination with an interior regularity argument, enables us to establish a uniqueness result for the inverse problem with local data. Additionaly, we manage to consider a slightly wider class of potentials.

Metal-organic framework as nanocarriers for agricultural applications: a review

Agrochemicals are substances used to promote plant growth and eradicate pests; and are essential to meeting the world food demand both qualitatively and quantitatively. Unfortunately, their extensive usage has resulted in certain significant problems, i.e., soil contamination, bioaccumulation, and the development of pest resistance. Consequently, there is a need for a reduced and more regulated usage of agrochemicals. The development of sustainable nanomaterials is encouraged in agricultural nanotechnology to minimize the usage of conventional fertilizers, herbicides, and pesticides. Metal-organic Framework (MOF)s are porous materials made of functional organic ligands and metal nodes or clusters connected by covalent bonds. Researchers are becoming increasingly interested in using MOF in a variety of sectors. The features of engineered MOFs have the potential to be an optimal strategy for producing impressively unique results in agricultural research. They also have advantages in terms of biocompatibility and biodegradability, which can be leveraged to boost the efficiency of traditional agrochemicals administration. Controlled release of agrochemicals offers an innovative solution to decrease the nutrient uptake rates and contamination of the environment. The application of MOF nanoparticles as nanocarriers to modify the loading and release properties of agrochemicals is particularly highlighted in this review.

Existence and Uniqueness Result for Fuzzy Fractional Order Goursat Partial Differential Equations

In this manuscript, we discuss fractional fuzzy Goursat problems with Caputo’s gH-differentiability. The second-order mixed derivative term in Goursat problems and two types of Caputo’s gH-differentiability pose challenges to dealing with Goursat problems. Therefore, in this study, we convert Goursat problems to equivalent systems fuzzy integral equations to deal properly with the mixed derivative term and two types of Caputo’s gH-differentiability. In this study, we utilize the concept of metric fixed point theory to discuss the existence of a unique solution of fractional fuzzy Goursat problems. For the useability of established theoretical work, we provide some numerical problems. We also discuss the solutions to numerical problems by conformable double Laplace transform. To show the validity of the solutions we provide 3D plots. We discuss, as an application, why fractional partial fuzzy differential equations are the generalization of usual partial fuzzy differential equations by providing a suitable reason. Moreover, we show the advantages of the proposed fractional transform over the usual Laplace transform.

Scalar BSDEs of iterated-logarithmically sub-linear generators with integrable terminal values

We establish a general existence and uniqueness result of integrable adapted solutions to scalar backward stochastic differential equations with integrable parameters, where the generator g has an iterated-logarithmically uniform continuity in the second unknown variable z. The result improves our previous one in Fan et al. (2023), and can be applied to the (conditional) g-expectation and the star-shaped risk measure.

Anisotropic equation based on fractional diffusion tensor for image noise removal

In this paper, we propose an anisotropic diffusion equation based on fractional‐order tensor diffusion applied on noisy images. We start by investigating the existence and uniqueness results of the proposed model. We also provide a discretization scheme based on the finite difference method. In order to prove the efficacy of our proposed approach, we deliver some simulations, which show robustness and remarkable results against noise while preserving edges and corners, compared to some well‐known models in literature. For a fair comparison, we manage to obtain the optimal parameters of every model thanks to PSNR and SSIM. We demonstrate the robustness efficacy of our model against high level of noise.

Theoretical and mathematical codynamics of nonlinear tuberculosis and COVID-19 model pertaining to fractional calculus and probabilistic approach

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is a novel virus known as coronavirus 2 (SARS-CoV-2) that affects the pulmonary structure and results in the coronavirus illness 2019 (COVID-19). Tuberculosis (TB) and COVID-19 codynamics have been documented in numerous nations. Understanding the complexities of codynamics is now critically necessary as a consequence. The aim of this research is to construct a co-infection model of TB and COVID-19 in the context of fractional calculus operators, white noise and probability density functions, employing a rigorous biological investigation. By exhibiting that the system possesses non-negative and bounded global outcomes, it is shown that the approach is both mathematically and biologically practicable. The required conditions are derived, guaranteeing the eradication of the infection. Sensitivity analysis and bifurcation of the submodel are also investigated with system parameters. Furthermore, existence and uniqueness results are established, and the configuration is tested for the existence of an ergodic stationary distribution. For discovering the system’s long-term behavior, a deterministic-probabilistic technique for modeling is designed and operated in MATLAB. By employing an extensive review, we hope that the previously mentioned approach improves and leads to mitigating the two diseases and their co-infections by examining a variety of behavioral trends, such as transitions to unpredictable procedures. In addition, the piecewise differential strategies are being outlined as having promising potential for scholars in a range of contexts because they empower them to include particular characteristics across multiple time frame phases. Such formulas can be strengthened via classical technique, power-law, exponential decay, generalized Mittag–Leffler kernels, probability density functions and random procedures. Furthermore, we get an accurate description of the probability density function encircling a quasi-equilibrium point if the effect of TB and COVID-19 minimizes the propagation of the codynamics. Consequently, scholars can obtain better outcomes when analyzing facts using random perturbations by implementing these strategies for challenging issues. Random perturbations in TB and COVID-19 co-infection are crucial in controlling the spread of an epidemic whenever the suggested circulation is steady and the amount of infection eliminated is closely correlated with the random perturbation level.

Correlates of teachers’ classroom management self-efficacy: A systematic review and meta-analysis

This meta-analysis examined literature from the last two decades to identify factors that correlate with teachers’ classroom management self-efficacy (CMSE) and to estimate the effect size of these relationships. Online and reference list searches from international and Chinese databases yielded 1085 unique results. However, with a focus on empirical research the final sample consisted of 87 studies and 22 correlates. The findings cluster the correlates of CMSE into three categories: teacher-level factors (working experience, constructivist beliefs, teacher stress, job satisfaction, teacher commitment, teacher personality, and teacher burnout), classroom-level factors (classroom climate, classroom management, students’ misbehaviour, students’ achievement, classroom interaction, and student-teacher relationship), and school-level factors (principal leadership and school culture). The results of this meta-analysis show small to large correlations between these 15 factors with CMSE. How these factors are associated with teachers’ CMSE and recommendations for future CMSE research are discussed.

Interventions for behavioral health comorbidities in the hospice setting: a scoping review.

Behavioral health (BH) comorbidities in hospice patients are widespread and impact important outcomes, including symptom burden, quality of life, and caregiver wellbeing. However, evidencebased BH interventions tailored for the hospice setting remain understudied. We conducted a scoping review with the objective of mapping studies of interventions for BH comorbidities in the hospice setting. We included empirical studies among hospice patients of interventions with BH outcomes. We abstracted data on study design, intervention type, and patient characteristics. Our search generated 7,672 unique results, of which 37 were ultimately included in our analysis. Studies represented 16 regions, with the United Kingdom (n=13) most represented. The most frequent intervention type was complementary and alternative interventions (n=13), followed by psychotherapeutic interventions (n=12). Most of the studies were either pilot or feasibility investigations. Fifteen studies employed a randomized controlled trial design. The most frequently utilized measurement tools for BH outcomes included the Hospital Anxiety and Depression Scale and the Edmonton Symptom Assessment Scale. Seventeen studies demonstrated statistically significant results in a BH outcome measure. BH conditions prevalent among hospice patients that were the focus of intervention efforts included depression symptoms, anxiety symptoms, and general psychological distress. No study focused on trauma-related disorders or substance use disorders. This scoping review reveals a concerning gap in research regarding evidence-based BH interventions in hospice settings, especially in the U.S. Despite extensive utilization of hospice care services and the high prevalence of BH conditions among hospice patients, randomized controlled trials focused on improving BH outcomes remain scant. The current BH practices, like the widespread use of benzodiazepines and antipsychotics, may not be rooted in robust evidence, underscoring an urgent need for investment in hospice research infrastructure and tailored clinical trials to test behavioral approaches to mitigate mental health outcomes at the end of life.

Analysis of coupled system of q$$ q $$‐fractional Langevin differential equations with q$$ q $$‐fractional integral conditions

In this dissertation, we study the coupled system of ‐fractional Langevin differential equations involving ‐Caputo derivative having ‐fractional integral conditions. With the help of some adequate conditions, we investigate the uniqueness and existence of mild solution of the aforementioned system. We also analyze various kinds of Ulam's stability. Banach fixed point theorem and Leray–Schauder of cone type are used to illustrate the existence and uniqueness results. We also used non‐linear functional analysis methods to explore variety of stability types. An example is provided to clearly demonstrate our theoretical outcomes.

Policy Formulation for an Optimal Level of Savings in a Dynamic Setting

Objective: Previous literature on optimal savings relies on specific utility and production technology functional forms which might not be able to produce robust results as different utility / production functions may lead to dramatically different or opposing results. This paper derives an optimal savings policy based on parameters (such as slopes of demand and supply curves) which can be empirically estimated and provides a unique and robust result irrespective of shape and form of individual utilities / production functions. Furthermore, existing literature does not consider welfare loss when savings market is adjusting to final equilibrium after a shock while deriving an optimal savings policy. In addition to that, number of savers (all public and private saving entities, including households, firms, etc.) and saving rate are vital parameters for savings in an economy, and while deriving an optimal savings policy, it is necessary to take into consideration these parameters to ensure that quantum of savings due to interest rate movement gets adjusted in target time duration, without which there may be additional efficiency loss than that envisioned while deriving an optimal savings policy for an economy. Methods: This research project designs a dynamical model for savings market and extends that to a three-dimensional savings system in an economy by taking into consideration number of savers, saving rate, and interest rate; and based on that derives an optimal comprehensive savings policy while accounting for efficiency losses when savings market, saving rate, and number of savers are adjusting to final equilibrium, in addition to the welfare loss on account of equilibrium shift. Results: Without consideration of welfare loss / gain while savings market is adjusting after implementation of a savings policy, welfare picture remains incomplete, and the optimal savings policy based on partial welfare cannot be considered as optimal in true sense. Traditionally, welfare of only producer and consumer is taken into consideration without accounting for welfare of production factors. An expression of efficiency loss / gain as a result of savings policy based on welfare including those of production factors has been presented and optimal savings policies have been derived by minimizing efficiency losses and presented as a final result in the form of mathematical expressions. This paper demonstrates that both supply and demand shocks operate through a common channel, i.e., inventory of funds in savings market as both kinds of shock affect inventory of funds and hence can be categorized just as an inventory shock. Conclusion: For optimal welfare gains, practitioners / policy makers must estimate theoretically derived optimal savings policies based on the dynamic model developed in this paper and presented in the form of mathematical expressions from real world relevant data for implementation.

Large-N Principal Chiral Model in Arbitrary External Fields.

We report the explicit solution for the vacuum state of the two-dimensional SU(N) principal chiral model at large N for an arbitrary set of chemical potentials and any interaction strength, a unique result of such kind for an asymptotically free quantum field theory. The solution matches one-loop perturbative calculation at weak coupling, and in the opposite strong-coupling regime exhibits an emergent spacial dimension from the continuum limit of the SU(N) Dynkin diagram.

Uniqueness of solutions to some classes of anisotropic and isotropic curvature problems

In this paper, we apply various methods to establish the uniqueness of solutions to some classes of anisotropic and isotropic curvature problems. Firstly, by employing integral formulas derived by S. S. Chern [18], we obtain the uniqueness of smooth admissible solutions to a class of Orlicz-(Christoffel)-Minkowski problems. Secondly, inspired by Simon's uniqueness result [62], we then prove that the only smooth strictly convex solution to the following isotropic curvature problem(0.1)(Pk(W)Pl(W))1k−l=ψ(u,r)onSn must be an origin-centred sphere, where W=(∇2u+ug0), ∂1ψ≥0,∂2ψ≥0 and at least one of these inequalities is strict. As an application, we establish the uniqueness of solutions to the isotropic Gaussian-Minkowski problem. Finally, we derive the uniqueness result for the following isotropic Lp dual Minkowski problem(0.2)u1−prq−n−1det⁡(W)=1onSn, where −n−1<p≤−1 and n+1≤q≤n+12+14−(1+p)(n+1+p)n(n+2). This result utilizes the method developed by Ivaki and Milman [39] and generalizes a result due to Brendle, Choi and Daskalopoulos [10].