Critical Point Research Articles

Federated Gaussian Process: Convergence, Automatic Personalization and Multi-fidelity Modeling.

In this paper, we propose FGPR: a Federated Gaussian process ( GP) regression framework that uses an averaging strategy for model aggregation and stochastic gradient descent for local computations. Notably, the resulting global model excels in personalization as FGPR jointly learns a shared prior across all devices. The predictive posterior is then obtained by exploiting this shared prior and conditioning on local data, which encodes personalized features from a specific dataset. Theoretically, we show that FGPR converges to a critical point of the full log-marginal likelihood function, subject to statistical errors. This result offers standalone value as it brings federated learning theoretical results to correlated paradigms. Through extensive case studies on several regression tasks, we show that FGPR excels in a wide range of applications and is a promising approach for privacy-preserving multi-fidelity data modeling.

Structural Analysis and Optimization of Washing Machine Spider, Drum and Shaft

Abstract: This project's thesis centers on identifying diverse loading scenarios and leveraging their outcomes to optimize two components of a washing machine. The initial phase of this report involves generating a numerical model of a washing machine. This encompasses assembling it in a mechanical software program (ANSYS) and defining specific boundary conditions and material properties for the problem. Decisions on simulation parameters, such as mesh size, types of elements, and the number of iterations in the calculations, will also be made. An examination of various loading scenarios is conducted to gain an overall understanding of the problem and select four specific cases for application in reconstructing the target parts (Spider and front cover). This examination involves calculating the worst relative angle between the Spider and the resultant force for two opposing loads inside the drum. The resulting angle is found to be 35º between the arm and the direction of the resultant force. By applying these diverse loading conditions to the model, the mechanical behavior of the parts will be determined, and this information will be utilized in the reconstruction. Mechanical behavior, in this context, refers to understanding the maximum levels of stress (tension, compression, and shear) and deformation (displacement and strain). Once the critical points of stress in the Spider and front cover are identified, various new shapes are created. The reconstruction parameters considered include volume reduction, stress level reduction, and deformation. Two of these shapes are selected and tested in the numerical model as a validation process.

Isotope effects in supercooled H2O and D2O and a corresponding-states-like rescaling of the temperature and pressure.

Water shows anomalous properties that are enhanced upon supercooling. The unusual behavior is observed in both H2O and D2O, however, with different temperature dependences for the two isotopes. It is often noted that comparing the properties of the isotopes at two different temperatures (i.e., a temperature shift) approximately accounts for many of the observations-with a temperature shift of 7.2K in the temperature of maximum density being the most well-known example. However, the physical justification for such a shift is unclear. Motivated by recent work demonstrating a "corresponding-states-like" rescaling for water properties in three classical water models that all exhibit a liquid-liquid transition and critical point [Uralcan et al., J. Chem. Phys. 150, 064503 (2019)], the applicability of this approach for reconciling the differences in the temperature- and pressure-dependent thermodynamic properties of H2O and D2O is investigated here. Utilizing previously published data and equations-of-state for H2O and D2O, we show that the available data and models for these isotopes are consistent with such a low temperature correspondence. These observations provide support for the hypothesis that a liquid-liquid critical point, which is predicted to occur at low temperatures and high pressures, is the origin of many of water's anomalies.

Isotope effect on the anomalies of water: A corresponding states analysis.

Light and heavy water show similar anomalies in thermodynamic and dynamic properties, with a consistent trend of anomalies occurring at higher temperatures in heavy water. Viscosity also increases faster upon cooling in heavy water, causing a giant isotope effect, with a viscosity ratio near 2.4 at 244K. While a simple temperature shift apparently helps in collapsing experimental data for both isotopes, it lacks a clear justification, changes value with the property considered, and requires additional ad hoc scaling factors. Here, we use a corresponding states analysis based on the possible existence of a liquid-liquid critical point in supercooled water. This provides a coherent framework that leads to the collapse of thermodynamic data. The ratio between the dynamic properties of the isotopes is strongly reduced. In particular, the decoupling between viscosity η and self-diffusion D, measured as a function of temperature T by the Stokes-Einstein ratio Dη/T, is found to collapse after applying the corresponding states analysis. Our results are consistent with simulations and suggest that the various isotope effects mirror the one on the liquid-liquid transition.

Scaling and Universality at Ramped Quench Dynamical Quantum Phase Transitions.

The nonequilibrium dynamics of a periodically driven extended XY model, in the presence of
linear time dependent magnetic filed, is investigated using the notion of dynamical quantum phase
transitions (DQPTs). Along the similar lines to the equilibrium phase transition, the main purpose
of this work is to search the fundamental concepts such as scaling and universality at the ramped
quench DQPTs. We have shown that the critical points of the model, where the gap closing occurs,
can be moved by tuning the driven frequency and consequently the presence/absence of DQPTs
can be flexibly controlled by adjusting the driven frequency. We have uncovered that, for a ramp
across the single quantum critical point, the critical mode at which DQPTs occur is classified into
three regions: the Kibble-Zurek (KZ) region, where the critical mode scales linearly with the square
root of the sweep velocity, pre-saturated (PS) region, and the saturated (S) region where the critical
mode makes a plateau versus the sweep velocity. While for a ramp that crosses two critical points,
the critical modes disclose just KZ and PS regions. On the basis of numerical simulations, we find
that the dynamical free energy scales linerly with time, as approaches to DQPT time, with the
exponent ν = 1 ± 0.01 for all sweep velocities and driven frequencies.

Spontaneous breaking of the SO(2N) symmetry in the Gross-Neveu model

The canonical Gross-Neveu model for N two-component Dirac fermions in 2+1 dimensions suffers a continuous phase transition at a critical interaction gc1∼1/N at large N, at which its continuous symmetry SO(2N) is preserved and a discrete (Ising) symmetry becomes spontaneously broken. A recent mean-field calculation, however, points to an additional transition at a different critical gc2∼−Ngc1, at which SO(2N)→SO(N)×SO(N). To study the latter phase transition we rewrite the Gross-Neveu interaction g(ψ¯ψ)2 in terms of three different quartic terms for the single (L=1) 4N-component real (Majorana) fermion, and then extend the theory to L>1. This allows us to track the evolution of the fixed points of the renormalization group transformation starting from L≫1, where one can discern three distinct critical points which correspond to continuous phase transitions into (1) SO(2N)-singlet mass-order-parameter, (2) SO(2N)-symmetric-tensor mass-order-parameters, and (3) SO(2N)-adjoint nematic-order-parameters, down to L=1 value that is relevant to the standard Gross-Neveu model. Below the critical value of Lc(N)≈0.35N for N≫1 only the Gross-Neveu critical point (1) still implies a diverging susceptibility for its corresponding (SO(2N)-singlet) order parameter, whereas the two new critical points that existed at large L ultimately become equivalent to the Gaussian fixed point at L=1. We interpret this metamorphosis of the SO(2N)-symmetric-tensor fixed point from critical to spurious as an indication that the transition at gc2 in the original Gross-Neveu model is turned first-order by fluctuations. Published by the American Physical Society 2024

Magneto-induced topological phase transition in inverted InAs/GaSb bilayers

We report a magneto-induced topological phase transition in inverted InAs/GaSb bilayers from a quantum spin Hall insulator to a normal insulator. We utilize a dual-gated Corbino device in which the degree of band inversion, or equivalently the electron and hole densities, can be continuously tuned. We observe a topological phase transition around the magnetic field where a band crossing occurs, accompanied by a bulk-gap closure characterized by a bulk conductance peak (BCP). In another set of experiments, we study the transition under a tilted magnetic field (tilt angle ). We observe the characteristic magnetoconductance around BCP as a function of , which dramatically depends on the density of the bilayers. In a relatively deep inversion (hence a higher density) regime, where the electron-hole hybridization dominates the excitonic interaction, the BCP grows with . On the contrary, in a shallowly inverted (a lower density) regime, where the excitonic interaction dominates the hybridization, the BCP is suppressed indicating a smooth crossover without a gap closure. This suggests the existence of a low-density, correlated insulator with spontaneous symmetry breaking near the critical point. Our highly controllable electron-hole system offers an ideal platform to study interacting topological states as proposed by recent theories. Published by the American Physical Society 2024

“We can do this”: Physical educators’ role breadth self-efficacy to be involved with CSPAPs from an occupational socialization perspective

In the United States, the Comprehensive School Physical Activity Program (CSPAP) is a whole-of-school framework to ensure youth meet physical activity guidelines. Physical education teachers (PETs) are poised to be CSPAP leaders, but implementation is low. PETs’ involvement with CSPAPs may be better understood through the lens of their role breadth self-efficacy, which captures how professionals feel about their ability to undertake expanded workplace roles. Furthermore, investigating PETs’ CSPAP-related role breadth self-efficacy from the perspective of occupational socialization theory could help to identify critical intervention points in PETs’ career development to support CSPAP implementation. The purpose of this study, therefore, was to explore PETs’ perceptions of how the phases of occupational socialization theory influence their role breadth self-efficacy to be involved with CSPAPs. Stratified random sampling and convenience sampling were used to recruit 25 PETs ( n = 16 female) in the United States. Data were collected using individual, semi-structured interviews and analyzed using constant comparison. The findings are represented in three overarching themes, each accompanied by corresponding subthemes: (a) modeling and encouragement from socializing agents, (b) learn by doing: mastery experiences, and (c) agent of change: “I believe in this.” The PETs’ childhood physical activity experiences, physical activity leadership development within teacher education, mentorship from experienced educators who exemplify effective strategies, and the capacity to foster social capital within the workplace are key factors impacting PETs’ CSPAP-related role breadth self-efficacy. Further investigation into mentorship programs for novice PETs to lead CSPAPs and best practices for CSPAP training in teacher education is recommended.

Linear Convergence of the Derivative-Free Proximal Bundle Method on Convex Nonsmooth Functions, with Application to the Derivative-Free VU\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{VU}$\\end{document}-Algorithm

Proximal bundle methods are a class of optimisation algorithms that leverage the proximal operator to address nonsmoothness in the objective function efficiently. This study focuses on a derivative-free (DFO) proximal bundle method and one of its applications called the DFO VU\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{VU}$\\end{document}-algorithm. These algorithms incorporate approximate proximal points as subprocedures in order to optimise convex nonsmooth functions based on approximated subdifferential information. Interestingly, the classical VU\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{VU}$\\end{document}-algorithm, which operates on true subgradient values, achieves superlinear convergence. At each iteration, the algorithm divides the whole space into two: the smooth U\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{U}$\\end{document}-space and the nonsmooth V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{V}$\\end{document}-space. It takes a Newton-like step on the U\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{U}$\\end{document}-space and a proximal-point step on the V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{V}$\\end{document}-space, enabling it to handle both smooth and nonsmooth parts effectively and converge faster. In this work, we reveal the worst possible convergence rate for the DFO VU\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{VU}$\\end{document}-method by showing the linear convergence of the DFO proximal bundle method. This will be done by presenting a suitable framework and using the subdifferential-based error bound on the distance to critical points.

Describing the Phase Equilibrium Line of Perfluorooctane, Based on the Clapeyron–Clausius Equation in the Range of Temperatures from the Triple Point to the Critical Point

Describing the Phase Equilibrium Line of Perfluorooctane, Based on the Clapeyron–Clausius Equation in the Range of Temperatures from the Triple Point to the Critical Point

Was postdevelopment too much? Autonomous struggle, academic coloniality & the radical roots of the pluriverse

ABSTRACT Where did postdevelopment thought go? Was its anti-development message too much for academia? While acknowledging some overlap between postdevelopment and mainstream academic decolonial thought, we argue that postdevelopment, and its conceptualization of the pluriverse directly challenges extractivism, statism, and capitalism or, in a word, development. After discussing aspects of mainstream decolonial thought, seven main points of postdevelopment criticism are reviewed and debunked. We demonstrate that resistance and ‘attack’ are enduring feature of postdevelopment praxis from the Zapatistas to the countless other (socio)ecological struggles across the world. Responding to critique, this article presents three postdevelopment practices: the Organización Popular Francisco Villa de Izquierda Independiente (OPFVII) in the Acapatzingo community, Mexico City; the Zone-to-Defend (Zone à Défendre, ZAD) concept formalized in France; and the Global Tapestry of Alternatives (GTA) initiative. The conclusion stresses the importance of postdevelopment and a pluriverse working towards anti-capitalism/statism/extractivism/patriarchal world to avoid (neo)colonial recuperations of anti-colonial/statist struggle.

On the Identity and Nomenclature of some Indian Plants

The paper deals with the nomenclature, synonymy and identity of some Indian species of. Randia Linn. (sensu lato), Butea Koen. ex Roxb., and Gymnosporia Bth. & Hk. f. Recent work has shown that nomenclatural changes are required in these genera and a few critical points have to be elucidated and corrected. These are discussed in the following pages. Two new combinations are also proposed.

Comparative Study of the Stilbenes and Other Phenolic Compounds in Cabernet Sauvignon Wines Obtained from Two Different Vinifications: Traditional and Co-Inoculation

From grape cultivation to ripening and harvest timing to processing, each step of the winemaking process can be a critical point when it comes to wine quality and phenolic composition. In this study, the influence of winemaking technology on resveratrol and quercetin content, as well as other polyphenolic compounds, was investigated. Resveratrol is a non-flavonoid polyphenolic stilbene synthesized by grape skin when damaged by infectious diseases or ionizing radiation. Quercetin is a phenol found in grape skins and stems and is produced to protect grapes from UV light damage. Trans-resveratrol and quercetin are known to act as antioxidants, reduce the risk of atherosclerosis and type 2 diabetes, inhibit the growth of cancer cells, and prevent the release of allergic and inflammatory molecules. However, the question was whether red wine could be enriched with these phenols using a co-inoculation winemaking technology. The main new idea was to completely replace the cold maceration process with maceration with the addition of wild yeast (Torulaspora delbrueckii, Td). Maceration with the addition of wild yeast (Td) offers the following advantages over traditional cold maceration: (1) higher concentrations of trans-resveratrol (>35–40%) and quercetin (>35–40%) in the final wine, (2) the new wine has a higher potential for human health, (3) the wine has better aroma and stability due to the higher mannoprotein content, and (4) better energy efficiency in the production process. The study of stability during storage and aging also included derivatives of benzoic acid and hydroxycinnamic acid, piceid, catechin, naringenin, rutin, kaempherol, hesperetin, and anthocyanins. This study found that younger wines had higher phenolic content, while storage of the wine resulted in a decrease in total phenolic content, especially monomeric stilbenes and quercetin. This study represents a small part of the investigation of the influence of non-Saccharomyces yeasts on the phenolic profile of wine, which still requires extensive research with practical application. In addition, non-Saccharomyces yeasts such as Kluyveromyces thermotolerans, Candida stellata, and Metschnikowia pulcherrima could also be used in future studies.

Temperate phage-antibiotic synergy across antibiotic classes reveals new mechanism for preventing lysogeny.

A recent demonstration of synergy between a temperate phage and the antibiotic ciprofloxacin suggested a scalable approach to exploiting temperate phages in therapy, termed temperate phage-antibiotic synergy, which specifically interacted with the lysis-lysogeny decision. To determine whether this would hold true across antibiotics, we challenged Escherichia coli with the phage HK97 and a set of 13 antibiotics spanning seven classes. As expected, given the conserved induction pathway, we observed synergy with classes of drugs known to induce an SOS response: a sulfa drug, other quinolones, and mitomycin C. While some β-lactams exhibited synergy, this appeared to be traditional phage-antibiotic synergy, with no effect on the lysis-lysogeny decision. Curiously, we observed a potent synergy with antibiotics not known to induce the SOS response: protein synthesis inhibitors gentamicin, kanamycin, tetracycline, and azithromycin. The synergy results in an eightfold reduction in the effective minimum inhibitory concentration of gentamicin, complete eradication of the bacteria, and, when administered at sub-optimal doses, drastically decreases the frequency of lysogens emerging from the combined challenge. However, lysogens exhibit no increased sensitivity to the antibiotic; synergy was maintained in the absence of RecA; and the antibiotic reduced the initial frequency of lysogeny rather than selecting against formed lysogens. Our results confirm that SOS-inducing antibiotics broadly result in temperate-phage-specific synergy, but that other antibiotics can interact with temperate phages specifically and result in synergy. This is the first report of a means of chemically blocking entry into lysogeny, providing a new means for manipulating the key lysis-lysogeny decision.IMPORTANCEThe lysis-lysogeny decision is made by most bacterial viruses (bacteriophages, phages), determining whether to kill their host or go dormant within it. With over half of the bacteria containing phages waiting to wake, this is one of the most important behaviors in all of biology. These phages are also considered unusable for therapy because of this behavior. In this paper, we show that many antibiotics bias this behavior to "wake" the dormant phages, forcing them to kill their host, but some also prevent dormancy in the first place. These will be important tools to study this critical decision point and may enable the therapeutic use of these phages.

Topological interpretation of extremal and Davies-type phase transitions of black holes

Topological arguments are currently being used as a novel scheme to discern the properties of black holes while ignoring their detailed structure and specific field equations. Among various avenues of black hole physics, where this novel approach is being utilized, the phase transition in black hole thermodynamics lies at the forefront. There are several types of phase transition in black holes; such as the van der Waals type phase transition, Davies-type phase transition, extremal phase transition, and Hawking-Page (HP) transition. So far, the topological interpretation, where the critical point has been identified with the non-zero topological charge, has been obtained only for the van der Waals type phase transition and HP transition in different spacetimes. To complete the picture, here we provide the same interpretation for two other phase transitions: Davies-type phase transition and extremal phase transition. The entire analysis is general and is valid for any spacetime where these types of phase transitions are observed. More importantly, our analysis suggests that amid the apparent differences in these phase transitions, they share the same topological characteristics, i.e. non-zero topological charge arising from different thermodynamic potentials in different types of phase transition.

Crossover from gas-like to liquid-like molecular diffusion in a simple supercritical fluid.

According to textbooks, no physical observable can be discerned allowing to distinguish a liquid from a gas beyond the critical point. Yet, several proposals have been put forward challenging this view and various transition boundaries between a gas-like and a liquid-like behaviour, including the so-called Widom and Frenkel lines, and percolation line, have been suggested to delineate the supercritical state space. Here we report observation of a crossover from gas-like (Gaussian) to liquid-like (Lorentzian) self-dynamic structure factor by incoherent quasi-elastic neutron scattering measurements on supercritical fluid methane as a function of pressure, along the 200 K isotherm. The molecular self-diffusion coefficient was derived from the best Gaussian (at low pressures) or Lorentzian (at high pressures) fits to the neutron spectra. The Gaussian-to-Lorentzian crossover is progressive and takes place at about the Widom line intercept (59 bar). At considerably higher pressures, a liquid-like jump diffusion mechanism properly describes the supercritical fluid on both sides of the Frenkel line. The present observation of a gas-like to liquid-like crossover in the self dynamics of a simple supercritical fluid confirms emerging views on the unexpectedly complex physics of the supercritical state, and could have planet-wide implications and possible industrial applications in green chemistry.

Critical Exponents and Universality for Fractal Time Processes above the Upper Critical Dimensionality

We study the critical behaviors of systems undergoing fractal time processes above the upper critical dimension. We derive a set of novel critical exponents, irrespective of the order of the fractional time derivative or the particular form of interaction in the Hamiltonian. For fractal time processes, we not only discover new universality classes with a dimensional constant but also decompose the dangerous irrelevant variables to obtain corrections for critical dynamic behavior and static critical properties. This contrasts with the traditional theory of critical phenomena, which posits that static critical exponents are unrelated to the dynamical processes. Simulations of the Landau–Ginzburg model for fractal time processes and the Ising model with temporal long-range interactions both show good agreement with our set of critical exponents, verifying its universality. The discovery of this new universality class provides a method for examining whether a system is undergoing a fractal time process near the critical point.

“Foul Abolition Calumny”: Reframing Racial Violence in William Wells Brown’s Clotel & the Antebellum Press

Abstract Focusing on Brown’s Clotel (1853), this essay reconstructs the surprising print history of an infamous antebellum lynching that became a national sensation twice over: first, in 1842 when the Mississippi Free Trader published a graphic report legitimating white mob violence that went viral in the Northern press and, later, when Brown reframed that report in his 1853 novel. Brown’s remix was subsequently reported in the New-York Dialy Tribune, the nation’s leading daily, which attributed the report solely to the Mississippi paper. The Free Trader in turn lambasted the Tribune for its “foul abolition calumny,” not realizing that this sly critique of slavery and lynching originated with an African American author. Besides bringing to light a new source and context—an important discovery demonstrating that Clotel had a much greater presence in the antebellum US than previously assumed—I situate Brown within an earlier gallows tradition and a later practice of antilynching activism (à la Ida B. Wells) to argue that Clotel constitutes a critical inflection point: an early African American intervention in public discourses about criminalized Black people and white mob violence that played out before the Civil War—the standard starting point for scholarship on anti-Black lynching.

Quantum scaling of the spin lattice relaxation rate in the checkerboard J-Q model.

Motivated by recent progress on the experimental realization of proximate deconfined quantum critical point in a frustrated quantum magnet, we study the low-energy spin dynamics of a related checkerboard J-Q model by using quantum Monte Carlo simulations. The ground state of this model undergoes a weakly first-order quantum phase transition with an emergent O(4) symmetry between an antiferromagnetic state and a plaquette valence bond solid. The calculated spin lattice relaxation rate of nuclear magnetic resonance, 1/T1, exhibits distinct low-temperature behaviors depending on the ground states. With decreasing the temperature, 1/T1 rises up on the antiferromagnetic side, characterizing a crossover to the renormalized classical regime, whereas 1/T1 drops exponentially on the side of valence bond solid, reflecting the gap opening in the plaquette ordered phase. The extracted spin gap scales with the distance to the transition point as a power-law with an exponent φ ≈ 0.3, consistent with the scaling ansatz φ = νz with ν ≈ 0.3 and z = 1. Near the quantum phase transition, the temperature dependent 1/T1 shows a broad crossover regime where a universal scaling 1/T1 ∼ T η with η ≈ 0.6 is found. Our results suggest a quantum scaling regime associated with the emergent enhanced symmetry near this first-order quantum phase transition.&#xD.

Existence Results Related to a Singular Fractional Double-Phase Problem in the Whole Space

In this paper, we will study a singular problem involving the fractional (q1(x,.)-q2(x,.))-Laplacian operator in the whole space RN,(N≥2). More precisely, we combine the variational method with monotonicity arguments to prove that the associated functional energy admits a critical point, which is a weak solution for such a problem.