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We study an inverse boundary value problem for a polyharmonic operator in two dimensions. We show that the Cauchy data uniquely determine all the anisotropic perturbations of orders at most m - 1 {m-1} and several perturbations of orders m to 2 ⁢ m - 2 {2m-2} under some restriction. The uniqueness proof relies on the ∂ ¯ {\bar{\partial}} -techniques and the method of stationary phase. [ABSTRACT FROM AUTHOR]

Let S 1 , S 2 , S 3 {S_{1},S_{2},S_{3}} be oriented non-flat minimal surfaces immersed in ℝ n + 1 {{\mathbb{R}}^{n+1}} ( n ≥ 2 ) {(n\geq 2)} with the Gauss maps G 1 , G 2 , G 3 {G_{1},G_{2},G_{3}} into ℙ n ⁢ ( ℂ ) {{\mathbb{P}}^{n}({\mathbb{C}})} , respectively. Assume that there are conformal diffeomorphisms Φ 2 , Φ 3 {\Phi_{2},\Phi_{3}} of S 1 {S_{1}} onto S 2 , S 3 {S_{2},S_{3}} respectively and let f 1 = G 1 {f^{1}=G_{1}} , f 2 = G 2 ∘ Φ 2 {f^{2}=G_{2}\circ\Phi_{2}} , f 3 = G 3 ∘ Φ 3 {f^{3}=G_{3}\circ\Phi_{3}} . In this paper, we will show that f 1 , f 2 , f 3 {f^{1},f^{2},f^{3}} are algebraic dependence, i.e., f 1 ∧ f 2 ∧ f 3 ≡ 0 {f^{1}\wedge f^{2}\wedge f^{3}\equiv 0} , if they have the same inverse images for a few hyperplanes of ℙ n ⁢ ( ℂ ) {{\mathbb{P}}^{n}({\mathbb{C}})} in general position with some certain conditions. [ABSTRACT FROM AUTHOR]

In addition to considering the influence of a species on other species (centrality), the identification of keystone species should also consider whether it has a unique network position (uniqueness). Clarifying the correlation between species centrality and uniqueness, or whether central species exhibit positional redundancy in food webs, can enhance the understanding of mechanisms stabilizing food webs. However, the findings from analyses utilizing different measures of uniqueness are contentious. In this study, we constructed the food web of the eutrophic Zhangze Lake (northern China) using stable isotopes to estimate the contribution of food sources to consumers. We then calculated trophic field overlap index to assess the uniqueness of species using this quantified food web information. Additionally, we analyzed how the weak interaction and the threshold value under which interactions are considered as weak influence the relationship between species uniqueness and centrality. The findings revealed that, in contrast to the traditional trophic field overlap index, the improved index identified the basal species with the strongest trophic interactions with other species and the filter‐feeding fishes with the fewest interactors as the most unique species, while the basal species were considered the least unique under the traditional index. The ranking results of the traditional index and the centrality showed a significant negative correlation, whereas the improved index displayed a positive correlation, which was caused by the comprehensive consideration of threshold and weak interaction. In addition, based on the data analysis, the suggestion was given for the selection of indices used to identify unique species, and whether central species have positional redundancy was explored. The methodology used in study provides a practical solution for the identification of the most unique species, points out and solves the problems faced in the uniqueness analysis, and advances the understanding of the mechanisms underlying food web robustness. [ABSTRACT FROM AUTHOR]

While the closure of a language family ℒ under certain language operations is the least family of languages which contains all members of ℒ and is closed under all of the operations, a kernel of ℒ is a maximal family of languages which is a sub-family of ℒ and is closed under all of the operations. Here we investigate properties of kernels of general language families and operations defined thereon as well as kernels of (deterministic) (linear) context-free languages with a focus on Boolean operations. While the closures of language families are unique, this uniqueness is not obvious for kernels. We consider properties of language families and of operations that yield unique and non-unique, i.e. a set, of kernels. For the latter case, the question whether the union of all kernels coincides with the language family, or whether there are languages that do not belong to any kernel is addressed. Additionally, languages that are mandatory for each (Boolean) kernel and languages that are optional for (Boolean) kernels are studied. That is, we consider the intersection of all Boolean kernels as well as their union. The expressive capacities of these families are addressed leading to a hierarchical structure. Further closure properties are considered. Furthermore, we study descriptional complexity aspects of these families, where languages are represented by context-free grammars with proofs attached. It turns out that the size trade-offs between all families in question and deterministic context-free languages are non-recursive. That is, one can choose an arbitrarily large recursive function f, but the gain in economy of description eventually exceeds f when changing from the latter system to the former. [ABSTRACT FROM AUTHOR]

Social recommendation (SR) based on Graph Neural Networks (GNNs) presents a promising avenue to significantly improve user experience by leveraging historical behavior and social data, which benefits from capturing user preferences through higher-order relationships. Although two socially connected users will prefer certain specific items, their preferences in other items are likely to be inconsistent. We argue that current GNNs-based social recommendation methods only focus on the commonalities of user preferences, but ignore the uniqueness. In addition, GNNs also suffers from the data sparsity problem commonly observed in recommender system. To address these limitations, we propose the Integrating Commonalities and Uniqueness of users and items method, namely SocialCU, which combines GNNs and contrastive learning to gain commonalities and uniqueness for SR. To be specific, we firstly model the original data as the user-item interaction graph and user-user social graph and use GNNs to obtain the commonalities of nodes (users or items). Then, we design the adaptive data augmentation to build dual contrastive learning to refine the uniqueness of nodes and mitigate data sparsity by extracting supervised signals. We have conducted extensive experiments on three real-world datasets to demonstrate the performance advantages of SocialCU over current state-of-the-art recommendation methods and the rationality of the model design. [ABSTRACT FROM AUTHOR]

The article investigates a Dirichlet-type problem with conjugation conditions for a degenerate equation of high even order with variable coefficients, including the Riemann–Liouville fractional derivative, in a rectangular region consisting of two subdomains ( y > 0 and y < 0 ), in each of which the equation has various kind.The solution is constructed as a series in terms of eigenfunctions of the one-dimensional problem, the existence of eigenfunctions is proved by the method of the theory of integral equations with symmetric kernels. The theorem of expansion in terms of the system of obtained eigenfunctions is proved. Sufficent conditions are found for boundary functions under which the solution in the form of a series converges uniformly. When justifying the convergence of a series, the problem of "small denominators" arises.This problem has been successfully solved in the article. The uniqueness of the solution is proved by the spectral method. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate limit cycles of the generalized Rayleigh–Liénard oscillator x ˙ = φ (y) , y ˙ = - g (x) - f (x , y) y . A criterion on the uniqueness and stability of limit cycles is given. We apply the criterion to two generalized Rayleigh–Liénard systems and get the uniqueness of limit cycles. Moreover, the stability and locations of limit cycles are obtained if they exist. [ABSTRACT FROM AUTHOR]

In this work, we deal with a more general form of fractional Langevin equation. The equation’s nonlinearity term f is relevant to fractional integral and fractional derivative. By using the fixed point theorems, we study the existence and uniqueness of solutions of initial value problem for the nonlinear fractional Langevin equation and obtain some new results. Further, by using the technique of nonlinear functional analysis, we study the stability of Ulam-Hyers, Ulam-Hyers-Rassias and semi-Ulam-Hyers-Rassias for the initial value problem of nonlinear Langevin equation. Finally, some examples are given to show the effectiveness of theoretical results. [ABSTRACT FROM AUTHOR]

Existence of solutions to a Φ -Laplacian singular system is obtained via shifting method and variational methods. A priori estimates are furnished through De Giorgi’s technique, Talenti’s rearrangement argument, and exploiting the weak Harnack inequality, while decay of solutions is obtained via comparison with radial solutions to auxiliary problems. Finally, uniqueness is investigated, and a Díaz-Saá type result is provided. [ABSTRACT FROM AUTHOR]

This is a sequel work on the existence of the solution to the free boundary problem on injection of fluid from a slot into a uniform stream with two free boundaries by Stojanovic (IMA J Appl Math 41:237–253, 1988). However, the uniqueness of the solution to the two-phase fluids problem with two free boundaries remains unresolved. In this paper, we will establish the asymptotic behavior of the flow in the upstream and prove the uniqueness of the solution to this problem. [ABSTRACT FROM AUTHOR]

By constructing various Pohozaev identities, we study the uniqueness of minimizers for L2$$ {L}&#x0005E;2 $$‐subcritical inhomogeneous variational problems with spatially decaying nonlinear terms, which contains x=0$$ x&#x0003D;0 $$ as a singular point. [ABSTRACT FROM AUTHOR]

We study the global bifurcation diagram of the positive solutions to the problem { Δ u + λ f (u) = 0 in B , u = 0 on ∂ B , where B is the unit ball in R N with N ≥ 3. Under general supercritical growth conditions on f (u) , we show that an unbounded bifurcation curve has no turning point, which indicates the existence of the singular extremal solution. In particular, our theory can be applied to the super-exponential cases of f (u) , and we exhibit that a bifurcation curve for Δ u + λ f (u) = 0 has the same qualitative property as a classical Gel'fand problem Δ u + λ e u = 0 for N ≥ 3 except N = 10. Main technical tools are intrinsic transformations for semilinear elliptic equations and ODE techniques. [ABSTRACT FROM AUTHOR]

Each year, millions of people die from the airborne infectious illness tuberculosis (TB). Several drug-susceptible (DS) and drug-resistant (DR) forms of the causative agent, Mycobacterium tuberculosis (MTB), are currently common in the majority of affluent and developing nations, particularly in Bangladesh, and completely drug-resistant strains are beginning to arise. The main purpose of this research is to develop and examine a non-integer-order mathematical model for the dynamics of tuberculosis transmission using the fractal fractional operator. By demonstrating characteristics such as the boundedness of solutions, positivity, and reliance of the solution on the original data, the biological well-posedness of the mathematical model formulation was investigated for TB cases from 2002 to 2017 in KPK Pakistan. Ulam-Hyres stability is also used to assess both local and global aspects of TB bacterial infection. Sensitivity analysis of the TB model with therapy was also examined. The advanced numerical technique is used to find the solution of the fractional-order system to check the impact of fractional parameters. Simulation highlights that all classes have converging qualities and retain established positions with time, which shows the actual behavior of bacterial infection with TB. [ABSTRACT FROM AUTHOR]

This paper deals with some existence and uniqueness results for a class of problems systems for nonlinear k$$ k $$‐generalized ψ$$ \psi $$‐Hilfer fractional differential equations with periodic conditions. The arguments are based on Mawhin's coincidence degree theory. Furthermore, an illustration is presented to demonstrate the plausibility of our results. [ABSTRACT FROM AUTHOR]

In this paper, we present a uniqueness theorem obtained by using direct calculation. This theorem is applicable to stability problems of functional equations whose solutions are monomial or generalized polynomial mappings of degree n. The advantage of this uniqueness theorem is that it simplifies the proof by eliminating the need to repeatedly and cumbersomely prove uniqueness in stability studies. [ABSTRACT FROM AUTHOR]

This work studies an inhomogeneous generalized Hartree equation with inverse square potential. The purpose is to prove the existence and strong instability of inter-critical standing waves. This means that there are infinitely many data near to the ground state, such that the associated solution blows-up in finite time. The proof combines a variational analysis with the standard variance identity. The challenge is to deal with three difficulties: the singular potential | x | − 2 , an inhomogeneous term | x | − λ , and a non-local source term. [ABSTRACT FROM AUTHOR]

We consider a nonlinear coefficient inverse problem of reconstructing the density and the memory matrix of a viscoelastic medium by probing the medium with a family of wave fields excited by moment tensor point sources. A spatially non-overdetermined formulation is investigated, in which the manifolds of point sources and detectors do not coincide and have a total dimension equal to three. The requirements for these manifolds are established to ensure the unique solvability of the studied inverse problem. The results are achieved by reducing the problem to a chain of connected systems of linear integral equations of the M. M. Lavrentiev type. [ABSTRACT FROM AUTHOR]

This paper is concerned with discontinuous inverse problem generated by complex-valued weight Sturm–Liouville differential operator with λ-dependent boundary conditions. We establish some properties of spectral characteristic and prove that the potential on the whole interval can be uniquely determined by the Weyl-type function or two spectra. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of time-periodic population model with dispersal. It is well known that the existence of the periodic traveling fronts has been established. However, the uniqueness and stability of such fronts remain unsolved. In this paper, we first prove the uniqueness of non-critical periodic traveling fronts. Then, we show that all non-critical periodic traveling fronts are exponentially asymptotically stable. [ABSTRACT FROM AUTHOR]

In the paper, we study the uniqueness of meromorphic function with few poles sharing two pairs of small functions IM with its th derivatives and obtain two results which improve as well as generalize the recent result due to Huang et al. [1] in a large extend. [ABSTRACT FROM AUTHOR]

This work studies a coupled non-linear Schrödinger system with a singular source term. First, we investigate the question of the local existence of solutions. Second, one proves the existence of global solutions which scatter in some Sobolev spaces. Finally, one establishes the existence of non-global solutions. The main difficulty here is to overcome the regularity problem in the non-linearity. Indeed, because of the singularity of the source term, the classical contraction method in the energy space fails in such a regime. So, this paper is to fill such a gap in the literature. The argument follows ideas in T. Cazenave and I. Naumkin (Comm. Contemp. Math. , 19 (2017), 1650038). This consists to remark that the singularity problem is only near the origin. So, one needs to impose that the solution stays away from zero. This is not trivial, since there is no maximum principle for the Schrödinger equation. The existence of global solutions which scatter follows with the pseudo-conformal transformation via the existence of local solutions. Finally, the existence of non-global solutions follows with the classical variance method. [ABSTRACT FROM AUTHOR]

This article is concerned with the problem of source identification for a distributed-order time-fractional diffusion equation (DTFDE). The uniqueness, ill-posedness and conditional stability estimate for the inverse source problem are demonstrated. Our main objective is to reconstruct the stable source term utilizing an iterative generalized quasi-reversibility method(IGQRM). In theory, an a priori and an a posteriori regularization parameter selection strategies are proposed to obtain the convergence estimates between the regularized solution and the exact solution. In numerical experiment, some numerical examples are presented to describe the stability and validity of our proposed regularization method. [ABSTRACT FROM AUTHOR]

The Hybrid NAR-RBFs Networks for COVID-19 fractional order model is examined in this scientific study. Hybrid NAR-RBFs Networks for COVID-19, that is more infectious which is appearing in numerous areas as people strive to stop the COVID-19 pandemic. It is crucial to figure out how to create strategies that would stop the spread of COVID-19 with a different age groups. We used the epidemic scenario in the Hybrid NAR-RBFs Networks as a case study in order to replicate the propagation of the modified COVID-19. In this research work, existence and stability are verified for COVID-19 as well as proved unique solutions by applying some results of fixed point theory. The developed approach to investigate the impact of Hybrid NAR-RBFs Networks due to COVID-19 at different age groups is relatively advanced. Also obtain solutions for a proposed model by utilizing Atanga Toufik technique and fractal fractional which are the advanced techniques for such type of infectious problems for continuous monitoring of spread of COVID-19 in different age groups. Comparisons has been made to check the efficiency of techniques as well as for finding the reliable solutions to understand the dynamical behavior of Hybrid NAR-RBFs Networks for non-linear COVID-19. Finally, the parameters are evaluated to see the impact of illness and present numerical simulations using Matlab to see actual behavior of this infectious disease for Hybrid NAR-RBFs Networks of COVID-19 for different age groups. [ABSTRACT FROM AUTHOR]

In this paper, we consider the 2D hyperbolic Prandtl equations on the half plane. Firstly, we obtain the global existence of solutions by using the classical energy methods in an analytic framework. Then, we prove the uniqueness of solutions. Besides, we also obtain a time exponential decay in analytic regularity norm of the solutions for any time t≥0$$ t\ge 0 $$. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the existence and uniqueness of solutions for a sixth‐order differential equation involving nonlocal and integral boundary conditions. Firstly, we obtain the properties of the relevant Green's functions. The existence result of at least one nontrivial solution is obtained by applying the Krasnoselskii–Zabreiko fixed point theorem. Moreover, we also establish the existence of unique solution to the considered problem via Hölder and Minkowski inequalities and Rus's theorem. Finally, two numerical examples are included to show the applicability of our main results. [ABSTRACT FROM AUTHOR]

In this paper, the existence and the uniqueness of an evolutionary anisotropic $ p_i(x) $ p i (x) -Laplacian equation with a damping term are studied. If the damping term is with a subcritical index, by the Di Giorgi iteration technique, the $ L^{\infty } $ L ∞ -estimate of the weak solutions can be obtained. The existence of weak solution is proved by the renormalized solution method, and how the anisotropic characteristic of the considered equation affect the $ L^{\infty } $ L ∞ -estimate of the weak solutions is revealed. The uniqueness is true strongly depending on subcritical index of the damping term, and this result goes beyond previous efforts in the literature (Bertsch M, Dal Passo R, Ughi M: Discontinuous viscosity solutions of a degenerate parabolic equation. Trans Amer Math Soc. 1990;320:779–798; Li Z, Yan B, Gao W. Existence of solutions to a parabolic $ p(x)- $ p (x) − Laplace equation with convection term via $ L^{\infty }- $ L ∞ − Estimates. Electron J Differ Equ. 2015;46:1–21; Zhang Q, Shi P. Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms. Nonlinear Anal. 2010;72:2744–2752; Zhou W, Cai S. The continuity of the viscosity of the Cauchy problem of a degenerate parabolic equation not in divergence form. J Jilin University (Natural Sci.). 2004;42:341–345), etc. [ABSTRACT FROM AUTHOR]

Existence, non-existence, and uniqueness of mass-conserving weak solutions to the continuous collision-induced nonlinear fragmentation equations are established for the collision kernels Φ satisfying Φ (x , y) = x λ 1 y λ 2 + y λ 1 x λ 2 , (x , y) ∈ (0 , ∞) 2 , with λ 1 ≤ λ 2 ≤ 1 , and non-integrable fragment daughter distributions. In particular, global existence of mass-conserving weak solutions is shown when 1 ≤ λ : = λ 1 + λ 2 ≤ 2 with λ 1 ≥ k 0 , the parameter k 0 ∈ (0 , 1) being related to the non-integrability of the fragment daughter distribution. The existence of at least one mass-conserving weak solution is also demonstrated when 2 k 0 ≤ λ < 1 with λ 1 ≥ k 0 but its maximal existence time is shown to be finite. Uniqueness is also established in both cases. The last result deals with the non-existence of mass-conserving weak solutions, even on a small time interval, for power law fragment daughter distribution when λ 1 < k 0. It is worth mentioning that the previous literature on the nonlinear fragmentation equation does not treat non-integrable fragment daughter distribution functions. [ABSTRACT FROM AUTHOR]

The original KWC system is widely used in materials science. It was proposed in [R. Kobayashi, J. A. Warren, and W. C. Carter, Phys. D, 140 (2000), pp. 141--150] and is based on the phase field model of planar grain boundary motion. This model suffers from two key challenges. First, it is difficult to establish its relation to physics, in particular a variational model. Second, it lacks uniqueness. The former has been recently studied within the realm of BV theory. The latter only holds under various simplifications. This article introduces a pseudo-parabolic version of the KWC system. A direct relationship with variational model (as gradient flow) and uniqueness are established without making any unrealistic simplifications. Namely, this is the first KWC system which is both physically and mathematically valid. The proposed model overcomes the well-known open issues. [ABSTRACT FROM AUTHOR]

We introduce a class of fully nonlinear mean field games posed in [0,T] × Rd. We justify that they are related to controlled local or nonlocal diffusions, and more generally in our setting, to a new control interpretation involving time change rates of stochastic (L\'evy) processes. The main results are the existence and uniqueness of solutions under general assumptions. These results are applied to nondegenerate equations--including both local second-order and nonlocal with fractional Laplacians. Uniqueness holds under the monotonicity of couplings and convexity of the Hamiltonian, but neither monotonicity nor convexity need to be strict. We consider a rich class of nonlocal operators and processes and develop tools to work in the whole space without explicit moment assumptions. [ABSTRACT FROM AUTHOR]

We explore the existence, uniqueness, and multiplicity of positive solutions to a system of Hadamard fractional differential equations that contain fractional integral terms. Defined on a finite interval, this system is subject to general coupled nonlocal boundary conditions encompassing Riemann–Stieltjes integrals and Hadamard fractional derivatives. To establish the main results, we employ several fixed-point theorems, namely the Banach contraction mapping principle, the Schauder fixed-point theorem, the Leggett–Williams fixed-point theorem, and the Guo–Krasnosel'skii fixed-point theorem. [ABSTRACT FROM AUTHOR]

Under consideration is a nonlinear transport equation containing two nonlinearities and a coefficient of a lower-order nonlinear term depending on two or three space variables. We study the direct problem with the data on a part of the lateral surface of a cylindrical domain, explicitly construct a solution, and prove the uniqueness of the solution. Also, we state the problem of recovering the coefficient on some information about a solution to the direct problem and demonstrate that the inverse problem reduces to an X-ray tomography problem. This opens a way to its efficient numerical solution. [ABSTRACT FROM AUTHOR]

In this work we study extremals on Lie groups G endowed with a left invariant polyhedral Finsler structure. We use the Pontryagin’s Maximal Principle (PMP) to find curves on the cotangent bundle of the group, such that its projections on G are extremals. Let g and g ∗ be the Lie algebra of G and its dual space respectively. We represent this problem as a control system a ′ (t) = - ad ∗ (u (t)) (a (t)) of Euler-Arnold type equation, where u (t) is a measurable control in the unit sphere of g and a (t) is an absolutely continuous curve in g ∗ . A solution (u (t) , a (t)) of this control system is a Pontryagin extremal and a (t) is its vertical part. In this work we show that for a fixed vertical part of the Pontryagin extremal a (t) , the uniqueness of u (t) such that (u (t) , a (t)) is a Pontryagin extremal can be studied through an asymptotic curvature of a (t) . [ABSTRACT FROM AUTHOR]

As the tourism industry navigates post-pandemic recovery, understanding customer perceptions of uniqueness in tourism experiences is critical. It enables businesses to tailor offerings that stand out and attract travelers seeking novel and distinct experiences after a period of limited travel opportunities. The current study bridges this knowledge gap by employing the Bidirectional Encoder Representations from Transformers (BERT) model to analyze online reviews of Airbnb Experiences. BERT, a tool for contextual and sentiment analysis, aids us in identifying and categorizing experiences that contribute to value creation for tourists. We propose four dimensions of uniqueness, grounded in the service-dominant (S-D) logic framework. Our research enriches academic discourse surrounding the role of uniqueness in value creation and provides organizations with strategic insights into enhancing the distinctiveness of their offerings. [ABSTRACT FROM AUTHOR]

In this paper, we study the uniqueness problems of fn(z)L(g) and gn(z)L(f) when they share a non-zero small function α(z) with finite weights, where L(h) represents any one of h(k)(z), h(z + c), h(z + c) - h(z) and h(k)(z + c), k ≥ 1 and c is a non-zero constant. Here f(z) and g(z) are transcendental meromorphic (or entire) functions and α(z) is a small function with respect to both f(z) and g(z). Our results improve and supplement the recent results due to Gao and Liu [Bull. Korean Math. Soc. 59 (2022), 155-166]. [ABSTRACT FROM AUTHOR]

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This paper deals with the existence and uniqueness results for a class of non-coercive Dirichlet elliptic problems whose model example is - div (1 (m - u) β (1 + | u |) q | ∇ u | p - 2 ∇ u + c (x) | u | p - 2 u sin (u - m)) + g (u) = f in Ω , u = 0 on ∂ Ω , where Ω is a bounded open subset of R N (N ≥ 2) , 1 < p < N , m > 0 , 0 < β < 1 , q > 0 , |c| belongs to L N p - 1 (Ω) and g is a continuous function in R which satisfies the sign condition and the data f belongs to L 1 (Ω) . [ABSTRACT FROM AUTHOR]

This paper examines the effects of handmade (vs. machine-made) gifts on social relationships. Across three studies, we find that handmade gifts promote social relationships. This effect can be explained by the perceived uniqueness of such gifts. Furthermore, these effects are moderated by social class (upper vs. lower). Specifically, membership in the upper class enhances the recipient's perceived uniqueness of handmade gifts, which in turn enhances the recipient's evaluation of social relationships. However, for members of the lower class, their perception of the uniqueness of handmade gifts becomes weaker, to the detriment of social relationships. These novel findings have both theoretical and practical significance for establishing harmonious interpersonal relationships and the consumption of handmade gifts. [ABSTRACT FROM AUTHOR]

This work studies a coupled non-linear Schrödinger system with a singular source term. First, we investigate the question of the local existence of solutions. Second, one proves the existence of global solutions which scatter in some Sobolev spaces. Finally, one establishes the existence of non-global solutions. The main difficulty here is to overcome the regularity problem in the non-linearity. Indeed, because of the singularity of the source term, the classical contraction method in the energy space fails in such a regime. So, this paper is to fill such a gap in the literature. The argument follows ideas in T. Cazenave and I. Naumkin (Comm. Contemp. Math., 19 (2017), 1650038). This consists to remark that the singularity problem is only near the origin. So, one needs to impose that the solution stays away from zero. This is not trivial, since there is no maximum principle for the Schrödinger equation. The existence of global solutions which scatter follows with the pseudo-conformal transformation via the existence of local solutions. Finally, the existence of non-global solutions follows with the classical variance method.

Introduction. The emerging digital society demonstrates the problem of justifying the value, worth and price of digital products and services, which requires consideration of the specificity of value in this society. Theoretical analysis. Following the financial and art market, the boundary between “value” and “worth” is blurred in the digital society. Value, including the value of an individual, becomes variable and is conditioned by the presence of unique attributes and unique experiences that obtain market value. At the same time, under standard digital protocols and technologies, the true uniqueness of an individual experience can only be achieved through the creation of customized digital ecosystems in the form of robotic swarms that support human life activities. A set of digital devices that have sufficient resources to continuously self-renew and form a closed environment around the individual that provides a unique experience can be called a chrysalic swarm. The chrysalic swarm creates an opportunity for the individual to escape from the logic of mutual competition dictated by society. Conclusion. Studying the value problematization in the digital society of swarm intelligence opens a theoretical perspective on the problem of the crisis of social elites in the context of the fulfillment of this scenario of society development.

Abstract The Hybrid NAR-RBFs Networks for COVID-19 fractional order model is examined in this scientific study. Hybrid NAR-RBFs Networks for COVID-19, that is more infectious which is appearing in numerous areas as people strive to stop the COVID-19 pandemic. It is crucial to figure out how to create strategies that would stop the spread of COVID-19 with a different age groups. We used the epidemic scenario in the Hybrid NAR-RBFs Networks as a case study in order to replicate the propagation of the modified COVID-19. In this research work, existence and stability are verified for COVID-19 as well as proved unique solutions by applying some results of fixed point theory. The developed approach to investigate the impact of Hybrid NAR-RBFs Networks due to COVID-19 at different age groups is relatively advanced. Also obtain solutions for a proposed model by utilizing Atanga Toufik technique and fractal fractional which are the advanced techniques for such type of infectious problems for continuous monitoring of spread of COVID-19 in different age groups. Comparisons has been made to check the efficiency of techniques as well as for finding the reliable solutions to understand the dynamical behavior of Hybrid NAR-RBFs Networks for non-linear COVID-19. Finally, the parameters are evaluated to see the impact of illness and present numerical simulations using Matlab to see actual behavior of this infectious disease for Hybrid NAR-RBFs Networks of COVID-19 for different age groups.

Customers frequently gravitate toward unique products, and firms increasingly utilize mass customization strategies allowing customers to self-customize products according to their unique preferences. While existing research shows that customers are willing to pay extra for this uniqueness, the present investigation points to a potential cost of self-customization that has been largely overlooked thus far. Specifically, the authors argue that what creates value for the individual consumer-designer (i.e., the original customer of the self-customized product) might conversely be detrimental to potential customers on the secondhand market, particularly in the context of aesthetic (vs. functional) customization. Results of three distinct data sets (including an analysis of more than 500,000 preowned car sales listings) support this uniqueness-hurts-resale hypothesis and provide a series of more nuanced findings. Consistent with the theorizing and empirical studies, three follow-up experiments show that although consumer-designers' valuations are positively affected by uniqueness, uniqueness indeed negatively affects secondhand-market customers' willingness to pay. This is because the more unique a given configuration is to a given consumer-designer, the lower the likelihood that said design will meet secondhand-market customers' taste preferences. The findings point to a tension between maximizing utility at first purchase and minimizing the related cost of aesthetic customization at resale. [ABSTRACT FROM AUTHOR]

We prove the uniqueness of positive solutions to the BVP$ \begin{align} \left\{ \begin{array}{c} -\Delta u = \lambda f(u)\ \ \ \text{in }\Omega , \\ \frac{\partial u}{\partial n}+bu = 0\ \ \ \text{on }\partial \Omega , \end{array} \right. \end{align} $when the parameter $ \lambda $ is large independent of $ b\in \mathbb{(} 0, \infty) $. Here, $ \Omega $ is a bounded domain in $ \mathbb{R}^{n} $ with smooth boundary $ \partial \Omega , \ f:[0, \infty)\rightarrow [ 0, \infty) $ is continuous, sublinear at $ \infty $, and satisfies a concavity-like condition for $ u $ large. [ABSTRACT FROM AUTHOR]

We show that the Cauchy problem for the Camassa–Holm equation has a unique, global, weak, and dissipative solution for any initial data u 0 ∈ H 1 (R) , such that u 0 , x is bounded from above almost everywhere. In particular, we establish a one-to-one correspondence between the properties specific to the dissipative solutions and a solution operator associating to each initial data exactly one solution. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the following Schrödinger–Kirchhoff‐type problem with fourth‐order dispersion: γΔ2u−a+b∫ℝN|∇u|2dxΔu+u=|u|2σu,x∈ℝN,u∈H2(ℝN),$$ \left\{\begin{array}{c}\gamma {\Delta}&amp;#x0005E;2u-\left(a&amp;#x0002B;b{\int}_{{\mathrm{\mathbb{R}}}&amp;#x0005E;N}{\left&amp;#x0007C;\nabla u\right&amp;#x0007C;}&amp;#x0005E;2\mathrm{d}x\right)\Delta u&amp;#x0002B;u&amp;#x0003D;{\left&amp;#x0007C;u\right&amp;#x0007C;}&amp;#x0005E;{2\sigma }u,x\in {\mathrm{\mathbb{R}}}&amp;#x0005E;N,\\ {}u\in {H}&amp;#x0005E;2\left({\mathrm{\mathbb{R}}}&amp;#x0005E;N\right),\end{array}\right. $$where γ,a,b>0$$ \gamma, a,b&gt;0 $$ are constants and σ≥1$$ \sigma \ge 1 $$. We make use of Nehari manifold technique together with concentration‐compactness principle to prove that the above equation has at least a ground state solution for 1≤σ<4N−4$$ 1\le \sigma &lt;\frac{4}{N-4} $$ if N=5$$ N&amp;#x0003D;5 $$, 6, and 7, and for 1≤σ<+∞$$ 1\le \sigma &lt;&amp;#x0002B;\infty $$ if N≤4$$ N\le 4 $$. Moreover, we also investigate the asymptotic behaviors of ground state solutions when some coefficients tend to zero. Among them, a uniqueness result about ground state solutions is obtained by implicit function theorem, and a blow‐up result is established by Pohozaev identity if dimension N=3$$ N&amp;#x0003D;3 $$. [ABSTRACT FROM AUTHOR]

In this paper, we study a generalized Biot–Savart law for suitable velocity that possibly diverges at infinity, and then show its application to the 2D general incompressible inviscid fluids. We first prove the generalized Biot–Savart law for the active scalar equations in a discrete rotational symmetry framework, which allows the velocity grow almost linearly at infinity. Based on this, we further obtain a unique global symmetric solution to Euler equation under the Yudovich type regularity. Additionally, we investigate the local well-posedness for the Boussnesq equation, SQG equation, and especially for the IPM equation which enjoys particular symmetric property in our setting. The proof mainly relies on the Fourier model analysis and some refined estimates to the singular integral operator. [ABSTRACT FROM AUTHOR]

Danaher and Nyholm (Philosophy & Technology 37:106, 2024) explore whether avatar technology makes humans less valuable by making them less scarce. They identify two forms of human scarcity, intrinstic scarcity and instrumental scarcity, and explore how each is impacted by avatar representation. Here I argue that avatars cannot make humans less scarce but that, nevertheless, the use of avatar technology can undermine the value of human uniqueness. [ABSTRACT FROM AUTHOR]

In this work, we consider a more general version of the nonlocal thermistor problem, which describes the temperature diffusion produced when an electric current passes through a material. We investigate the doubly nonlinear problem where the nonlocal term is present on the right-hand side of the equation that describes the temperature evolution. Specifically, we employ topological degree theory to establish the existence of a solution to the considered problem. Furthermore, we separately address the uniqueness of the obtained solution. Additionally, we establish a priori estimates to demonstrate the convergence of a developed finite volume scheme used for the discretization of the continuous parabolic problem. Finally, to numerically simulate the proposed finite volume scheme, we use the Picard-type iteration process for the fully implicit scheme and approximate the nonlocal term represented by the integral with Simpson's rule to validate the efficiency and robustness of the proposed scheme. [ABSTRACT FROM AUTHOR]

We consider the non-local equations with non-negative L 1 -data in the fractional Musielak–Sobolev spaces. Utilizing approximation and energy methods, we establish the existence and uniqueness of non-negative renormalized solutions for such problems. The operators discussed in this work include the fractional Orlicz operators with variable exponents, the fractional double-phase operators with variable exponents, and the anisotropic fractional p-Laplacian operators, among others. [ABSTRACT FROM AUTHOR]