Regular Theory Research Articles

Parallel (competitive) reactions of micelle formation and cyclodextrin–surfactant inclusion complex formation in aqueous solution: A consequence of the Gibbs–Duhem equation—invariance of the mass action law and the possibility of determining the equilibrium constant of the inclusion complex

The Gibbs–Duhem equation results in the independence between the standard Gibbs free energy of micelle formation and the standard Gibbs free energy of monocomponent surfactant’s and cyclodextrin’s (CD’s) inclusion complex formation in aqueous solution. The experimental data show that the surfactant’s critical micellar concentration increases in the presence of CD. At a constant temperature, the standard Gibbs free energy of micelle formation remains constant, irrespective of the presence of parallel competitive processes such as inclusion complex formation. This indicates that the critical micellar concentration remains unchanged at the corresponding concentration scale. As a result, the equilibrium constant for surfactant–CD inclusion complex formation can be determined by changing the critical micellar concentration in the presence and absence of CD. A graphical method for determining the equilibrium constant of the inclusion complex is also presented. An analytical approach for analyzing the influence of the formation of the inclusion complex with CD on the formation of the binary mixed micellar pseudophase is presented. Suppose that different values exist for the equilibrium constants of the formation of the inclusion complex for two surfactants from their binary mixture. In this case, there is a change in the initial mole fraction from the initial mixture of surfactants, which should be considered when applying the regular solution theory protocol when determining the parameters of a binary mixed micelle in the presence of CD.

Interfaces, Free Boundaries and Geometric Partial Differential Equations

Partial differential equations arising in the context of interfaces and free boundaries encompass a flourishing area of research. The workshop focused on new developments and emerging new themes. At the same time also new interesting results on more traditional areas like, e.g. regularity theory and classical numerical approaches have been addressed. By convening experts from various disciplines related to modeling, analysis, and numerical methods concerning interfaces and free boundaries, the workshop facilitated progress on longstanding open questions and paved the way for novel research directions.

Constraint Maps with Free Boundaries: the Obstacle Case

This paper revives a four-decade-old problem concerning regularity theory for (continuous) constraint maps with free boundaries. Dividing the map into two parts, the distance part and the projected image to the constraint, one can prove various properties for each component. As has already been pointed out in the literature, the distance part falls under the classical obstacle problem, which is well-studied by classical methods. A perplexing issue, untouched in the literature, concerns the properties of the projected image and its higher regularity, which we show to be at most of class C2,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{2,1}$$\\end{document}. In arbitrary dimensions, we prove that the image map is globally of class W3,BMO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{3,BMO}$$\\end{document}, and locally of class C2,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{2,1}$$\\end{document} around the regular part of the free boundary. The issue becomes more delicate around singular points, and we resolve it in two dimensions. In the appendix, we extend some of our results to what we call leaky maps.

Lipschitz regularity of sub-elliptic harmonic maps into CAT(0) space

Abstract We prove the local Lipschitz continuity of sub-elliptic harmonic maps between certain singular spaces, more specifically from the 𝑛-dimensional Heisenberg group into CAT ⁡ ( 0 ) \operatorname{CAT}(0) spaces. Our main theorem establishes that these maps have the desired Lipschitz regularity, extending the Hölder regularity in this setting proven in [Y. Gui, J. Jost and X. Li-Jost, Subelliptic harmonic maps with values in metric spaces of nonpositive curvature, Commun. Math. Res. 38 (2022), 4, 516–534] and obtaining same regularity as in [H.-C. Zhang and X.-P. Zhu, Lipschitz continuity of harmonic maps between Alexandrov spaces, Invent. Math. 211 (2018), 3, 863–934] for certain sub-Riemannian geometries; see also [N. Gigli, On the regularity of harmonic maps from RCD ⁢ ( K , N ) \mathrm{RCD}(K,N) to CAT ⁢ ( 0 ) \mathrm{CAT}(0) spaces and related results, preprint (2022), https://arxiv.org/abs/2204.04317; and A. Mondino and D. Semola, Lipschitz continuity and Bochner–Eells–Sampson inequality for harmonic maps from RCD ⁡ ( k , n ) \operatorname{RCD}(k,n) spaces to CAT ⁡ ( 0 ) \operatorname{CAT}(0) spaces, preprint (2022), https://arxiv.org/abs/2202.01590] for the generalisation to RCD spaces. The present result paves the way for a general regularity theory of sub-elliptic harmonic maps, providing a versatile approach applicable beyond the Heisenberg group.

Deriving measurement collapse using zeta function regularisation and speculative measurement theory

This paper shows how an application of zeta function regularisation to a physical model of quantum measurement yields a solution to the problem of wavefunction collapse. Realistic measurement dynamics based on a particle becoming non-isolated are introduced and, based on this, an outcome function is derived using the method of maximum entropy. It is shown how regularisation of an information theoretic quantity related to this outcome function leads to apparent collapse of the wavefunction. The physical principles and key assumptions that underlie this theory are discussed. Some possible experimental approaches are described.

Optimal CHSH values for regular polygon theories in generalized probabilistic theories

Optimal CHSH values for regular polygon theories in generalized probabilistic theories

Damage identification method based on interval regularization theory

In the field of damage identification, traditional regularization methods neglect the impact of uncertainty factors on the selection of regularization parameters, leading to a decrease in the accuracy of damage identification. Therefore, this study proposes a damage identification based on interval truncated singular value decomposition (DI-ITSVD) method that considers the uncertainty in the selection of regularization parameter. This method treats model errors and measurement noise as interval uncertainties, and introduces the quantified uncertainties into the damage identification solutions through uncertainty propagation methods to determine the interval boundary. Uncertainty regularization parameters are selected to balance residuals and solutions using interval and generalized cross-validation methods. The key aspect of the proposed method in this paper is the integration of interval uncertainty propagation with the truncated singular value decomposition method to ensure the accuracy and stability of the damage identification equation solution. A numerical example of a 29-bar planar truss has been performed to test the effectiveness of the proposed method. The superiority of this method is verified by comparing the identification results with other improved truncated singular value decomposition methods. Finally, the practical application effect of the proposed method was also verified through an experimental work.

Morrey regularity theory of Rivière’s equation

This note is devoted to developing Morrey regularity theory for the following system of Rivière − Δ u = Ω ⋅ ∇ u + f in B 2 , \begin{equation*} -\Delta u=\Omega \cdot \nabla u+f \qquad \text {in }B^{2}, \end{equation*} under the assumption that f f belongs to some Morrey space. Our results extend the L p L^p regularity theory of Sharp and Topping [Trans. Amer. Math. Soc. 365 (2013), pp. 2317–2339], and also generalize a Hölder continuity result of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24] on harmonic mappings. Potential applications of our results are also possible in second order conformally invariant geometrical problems as that of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24].

Existence of weak solutions to a Cahn–Hilliard–Biot system

We prove existence of weak solutions to a diffuse interface model describing the flow of a fluid through a deformable porous medium consisting of two phases. The system non-linearly couples Biot’s equations for poroelasticity, including phase-field dependent material properties, with the Cahn–Hilliard equation to model the evolution of the solid, and is further augmented by a visco-elastic regularization of Kelvin–Voigt type. To obtain this result, we approximate the problem in two steps, where first a semi-Galerkin ansatz is employed to show existence of weak solutions to regularized systems, for which later on compactness arguments allow limit passage. Notably, we also establish a maximal regularity theory for linear visco-elastic problems.

A Lewisian regularity theory

In this paper, we develop a non-reductive variant of the regularity theory of causation proposed in Andreas and Günther (Pacific Philosophical Quarterly 105: 2–32, 2024). The variant is motivated as a refinement of Lewis’s (Journal of Philosophy 70:556–567, 1973) regularity theory. We do not pursue a reductive theory here because we found a challenge for Baumgartner's (Erkenntnis 78:85–109, 2013) regularity theory which applies to our previous theory as well. The challenge is sidestepped by a framework of law-like propositions resembling structural equations. We furthermore improve the deviancy condition of our previous theory. Finally, we show that the present theory can compete with the most advanced regularity and counterfactual accounts.

Self-assembly of mixed surfactants sodium dodecylsulfate and polyethylene glycol dodecyl ether in aqueous solutions

The thermodynamic behavior of mixed systems containing the anionic surfactant sodium dodecyl sulfate (SDS) and the nonionic surfactant polyethylene glycol dodecyl ether (Brij L4) in aqueous solutions was investigated. Electrical conductivity and interfacial tension measurements were employed to investigate the concentration-dependent properties of these surfactant mixtures. Two main experiments were conducted: i) constant ratio experiment: the overall surfactant concentration was varied while maintaining a fixed ratio between SDS and Brij L4. It was shown that the critical micelle concentration (CMC) determined from electrical conductivity measurements does not indicate the formation of the first mixed micelles as observed using tensiometry, but the point from which the adsorption of counterions by the existing micelles became important. By using the Regular Solution Theory (RST), it was found that the interaction between SDS and Brij L4 is synergistic, driven by dipole-ion interactions of the hydrophilic regions of the two surfactants. ii) Fixed SDS concentration while increase the Brij L4 concentration: the resulting electrical conductivity exhibited non-trivial behavior which complexity arose from simultaneous phenomena of incorporation of free SDS molecules into the micelles as Brij L4 concentration increased, leading to a decrease in electrical conductivity; liberation of some adsorbed counterions from the mixed micelles to the solution, which increases the conductivity and increase of viscosity due to Brij L4 further addition that leads to a sequential decrease in electrical conductivity.

Canonical lifts in multisymplectic De Donder–Weyl Hamiltonian field theories

We define canonical lifts of vector fields to the multisymplectic multimomentum bundles of De Donder–Weyl Hamiltonian (first-order) field theories and to the appropriate premultisymplectic embedded constraint submanifolds on which singular field theories are studied. These new canonical lifts are used to study the so-called natural Noether symmetries present in both regular and singular Hamiltonian field theories along with their associated conserved quantities obtained from Noether’s theorem. The Klein–Gordon field, the Polyakov bosonic string, and Einstein–Cartan gravity in 3+1 dimensions are analyzed in depth as applications of these concepts; as a peripheral result obtained in the analysis of the bosonic string, we provide a new geometrical interpretation of the well-known Virasoro constraint.

Regularity theory for parabolic operators in the half-space with boundary degeneracy

Regularity theory for parabolic operators in the half-space with boundary degeneracy

Fixed-time control and estimation of discontinuous fuzzy leakage-delayed networks: Indefinite and economical conditions of Filippov systems

This paper considers the fixed-time stability of Filippov systems and discontinuous networks described by the Filippov systems. Firstly, the more generalized and economical inequality condition on Lyapunov function is proposed, which can not only improve the previous negative definite conditions, but also the indefinite conditions. Based on the definition of the Lyapunov stability and finite-time convergence, new fixed-time stability lemmas are proposed and the settling-time is also estimated. In order to apply the new fixed-time stability lemmas to the neural networks, fixed-time stabilization of a class of fuzzy leakage-delayed neural networks with discontinuous activations is considered which is rarely studied since the delay is leakage and activations are discontinuous and they are not easy to deal with simultaneously. By means of the Filippov regularization and differential inclusions theory, new criteria on the stabilization of the considered networks are derived via the new modified non-chattering controller, which can improve the previous ones with parameter and symbolic function. Finally, example and numerical simulations further examine the correctness of the main results.

An accelerated inexact Newton regularization scheme with a learned feature-selection rule for non-linear inverse problems

With computational inverse problems, it is desirable to develop an efficient inversion algorithm to find a solution from measurement data through a mathematical model connecting the unknown solution and measurable quantity based on the first principles. However, most of mathematical models represent only a few aspects of the physical quantity of interest, and some of them are even incomplete in the sense that one measurement corresponds to many solutions satisfying the forward model. In this paper, in light of the recently developed iNETT method in (2023 Inverse Problems 39 055002), we propose a novel iterative regularization method for efficiently solving non-linear ill-posed inverse problems with potentially non-injective forward mappings and (locally) non-stable inversion mappings. Our approach integrates the inexact Newton iteration, the non-stationary iterated Tikhonov regularization, the two-point gradient acceleration method, and the structure-free feature-selection rule. The main difficulty in the regularization technique is how to design an appropriate regularization penalty, capturing the key feature of the unknown solution. To overcome this difficulty, we replace the traditional regularization penalty with a deep neural network, which is structure-free and can identify the correct solution in a huge null space. A comprehensive convergence analysis of the proposed algorithm is performed under standard assumptions of regularization theory. Numerical experiments with comparisons with other state-of-the-art methods for two model problems are presented to show the efficiency of the proposed approach.

Synergistic effects of an amphiphilic drug(propranolol hydrochloride) with cationic surfactants in an aqueous medium: A physicochemical study

The interactions of amphiphilic drug(propranolol hydrochloride, PPL) with cationic surfactants, such as dodecyltrimethylammoniumbromide(DTAB), tetradecyltrimethylammonium bromide (TTAB), trimethyltetradecylammonium chloride(TTAC) and tetyltrimethylammonium bromide(CTAB), in an aqueous medium were evaluated. The surface tension measurement was used to examine the mixed micellization behavior and surface properties of drug-surfactant mixtures. The nonideal or synergistic behaviors of the mixed systems were explored by using several physical models, such as Clint, Rubingh, Rosen, and Motomura models, based on regular solution theory and pseudophase approximation. In addition, the synergism between PPL and cationic surfactants was also confirmed using ultraviolet–visible spectroscopy. The stoichiometric ratios, binding constants, and free energy changes of the drug–surfactant mixtures were computed using Benesi–Hildebrand (B–H) theory.

Can a dialetheist stay regular?

For a dialetheist, it is rational to believe that true contradictions exist. However, we argue that a dialetheist faces a complex dilemma given some bridge principles for rational beliefs that connect possibility and probability, including the so-called ‘Regularity Principle’. Either her belief is not doxastically possible even for her, or she must assign positive credence to the proposition that dialetheia exists. The former makes her belief prima facie self-defeating. The latter seems to compel her to choose between several more fine-grained options, each of which seems undesirable for a regular and rational probability theory. It is unclear how a dialetheist would stay rationally regular, so we pose this challenge, seeking a response from a dialetheist.

The forward self-similar solution of fractional incompressible Navier-Stokes system: The critical case

In this paper, we study the regularity and pointwise estimates of forward self-similar solutions of fractional Navier-Stokes system under the critical case. By employing a Caffarelli,Kohn and Nirenberg-type iteration, L∞ estimates of the self-similar solution's profile are established, which is a key ingredient to ensure that the global weighted energy estimate procedure used in [28] is performed under the critical case. As a product, its natural pointwise bounds are recovered. Moreover, to obtain the optimal spatial decay estimate of self-similar solution's profile, a new technique is required due to lack of the related regularity theory.

Convergent Regularization in Inverse Problems and Linear Plug-and-Play Denoisers

AbstractRegularization is necessary when solving inverse problems to ensure the well-posedness of the solution map. Additionally, it is desired that the chosen regularization strategy is convergent in the sense that the solution map converges to a solution of the noise-free operator equation. This provides an important guarantee that stable solutions can be computed for all noise levels and that solutions satisfy the operator equation in the limit of vanishing noise. In recent years, reconstructions in inverse problems are increasingly approached from a data-driven perspective. Despite empirical success, the majority of data-driven approaches do not provide a convergent regularization strategy. One such popular example is given by iterative plug-and-play (PnP) denoising using off-the-shelf image denoisers. These usually provide only convergence of the PnP iterates to a fixed point, under suitable regularity assumptions on the denoiser, rather than convergence of the method as a regularization technique, thatis under vanishing noise and regularization strength. This paper serves two purposes: first, we provide an overview of the classical regularization theory in inverse problems and survey a few notable recent data-driven methods that are provably convergent regularization schemes. We then continue to discuss PnP algorithms and their established convergence guarantees. Subsequently, we consider PnP algorithms with learned linear denoisers and propose a novel spectral filtering technique of the denoiser to control the strength of regularization. Further, by relating the implicit regularization of the denoiser to an explicit regularization functional, we are the first to rigorously show that PnP with a learned linear denoiser leads to a convergent regularization scheme. The theoretical analysis is corroborated by numerical experiments for the classical inverse problem of tomographic image reconstruction.

Singular bifurcations and regularization theory.

Nonlinear sciences are present today in almost all disciplines, ranging from physics to social sciences. A major task in nonlinear science is the classification of different types of bifurcations (e.g., pitchfork and saddle-node) from a given state to another. Bifurcation analysis is traditionally based on the assumption of a regular perturbative expansion, close to the bifurcation point, in terms of a variable describing the passage of a system from one state to another. However, it is shown that a regular expansion is not the rule due to the existence of hidden singularities in many models, paving the way to a new paradigm in nonlinear science, that of singular bifurcations. The theory is first illustrated on an example borrowed from the field of active matter (phoretic microswimers), showing a singular bifurcation. We then present a universal theory on how to handle and regularize these bifurcations, bringing to light a novel facet of nonlinear sciences that has long been overlooked.