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When using quantum chemical methods to study electronically excited states of open-shell molecules, it is often beneficial to start with wave functions that are spin eigenfunctions. For excited states of molecules containing heavy elements, spin–orbit coupling (SOC) is important and needs to be included as well. An efficient approach is to include SOC perturbatively on top of a restricted open-shell Kohn–Sham (ROKS) time-dependent density functional theory, which can be combined with the Tamm–Dancoff approximation (TDA) to suppress numerical instabilities. We implemented and assessed the potential of such a ROKS-TDA-SOC method, also featuring the possibility of calculating transition dipole moments between states to allow for full spectrum simulation. Our study shows that the ROKS-TDA-SOC formalism yields a clear and easy-to-use method to obtain electronically excited states of open-shell molecules that are of moderate size and contain heavy elements. [ABSTRACT FROM AUTHOR]

The theory of polymer dynamics describes the intermediate scattering function for a polymer molecule in terms of relaxation modes defined by normal coordinates for the corresponding coarse-grained model. However, due to the difficulty of defining the normal coordinates for arbitrary polymer molecules, it is generally challenging to express the intermediate scattering function for a polymer molecule in terms of relaxation modes. To overcome this challenge, we propose a general method to calculate the intermediate scattering function for a polymer molecule on the basis of a relaxation mode analysis approach [Takano and Miyashita, J. Phys. Soc. Jpn. 64, 3688 (1995)]. In the proposed method, relaxation modes defined by eigenfunctions in a Markov process are evaluated on the basis of the simulation results for a polymer molecule and used to calculate the intermediate scattering function for that molecule. To demonstrate the effectiveness of the present method, we simulate the dynamics of a linear polymer molecule in a dilute solution and apply it to the calculation of the intermediate scattering function for the polymer molecule. The evaluation results regarding the relaxation modes reasonably describe the intermediate scattering function on the length scale of the radius of gyration of the polymer molecule. Accordingly, we examine the contributions of the pure relaxation and oscillatory relaxation processes to the entire intermediate scattering function. [ABSTRACT FROM AUTHOR]

We study global solutions to the classical one-phase free boundary problem that have finite Morse index relative to the Alt-Caffarelli functional. We show that such solutions are stable outside a compact set and characterize the index as the maximal number of linearly independent L 2 integrable eigenfunctions of the corresponding Robin eigenvalue problem, associated to negative eigenvalues. As an application, we obtain a complete classification of global solutions of finite Morse index in the plane. Our results are counterparts to the minimal surface theorems of Fischer-Colbrie and Gulliver. [ABSTRACT FROM AUTHOR]

We consider the Paneitz type equation Δ 2 u − α Δ u + β (u − u q) = 0 on a closed Riemannian manifold (M n , g) of dimension n ≥ 3. We reduce the equation to a fourth order ordinary differential equation assuming that (M , g) admits a proper isoparametric function. Assuming that q > 1 , α and β are positive and α 2 > 4 β , we prove that the global nonconstant solutions of this ordinary differential equation only have nondegenerate critical points. Applying global bifurcation theory we then prove multiplicity results for positive solutions of the equation when q < p ⁎ , where p ⁎ = n + 4 n − 4 if n > 4 and p ⁎ = ∞ if n = 3 , 4. As an application and motivation we prove multiplicity results for conformal constant Q -curvature metrics. For example, consider closed positive Einstein manifolds (M n , g) and (X m , h) of dimensions n , m ≥ 3. Assuming that M admits a proper isoparametric function (with a symmetry condition) we prove that as δ > 0 gets close to 0, the number of constant Q -curvature metrics conformal to g δ = g + δ h goes to infinity. [ABSTRACT FROM AUTHOR]

We numerically investigate the generalized Steklov problem for the modified Helmholtz equation and focus on the relation between its spectrum and the geometric structure of the domain. We address three distinct aspects: (i) the asymptotic behavior of eigenvalues for polygonal domains; (ii) the dependence of the integrals of eigenfunctions on domain symmetries; and (iii) the localization and exponential decay of Steklov eigenfunctions away from the boundary for smooth shapes and in the presence of corners. For this purpose, we implemented two complementary numerical methods to compute the eigenvalues and eigenfunctions of the associated Dirichlet-to-Neumann operator for planar bounded domains. We also discuss applications of the obtained results in the theory of diffusion-controlled reactions and formulate conjectures with relevance in spectral geometry. [ABSTRACT FROM AUTHOR]

In this work, the solution is justified using the method of separation of variables for initial-boundary value problems of the heat conduction equation with piecewise constant thermal conductivity coefficient under Sturm-type boundary conditions (separated boundary conditions). Various cases are considered. A system of eigenfunctions of the corresponding spectral problem is constructed and it is proved that this system of eigenfunctions forms a Riesz basis. The theorem of the existence of a unique classical solution to the problem is proved. [ABSTRACT FROM AUTHOR]

We study the eigenvalue problem for a self-adjoint 1D Dirac operator. It is known that, near an energy level where the square of the potential makes a simple well, the eigenvalues are approximated by a Bohr–Sommerfeld type quantization rule. A remarkable difference from the Schrödinger case appears in the Maslov correction term. This fact was recently found by Hirota [Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov–Shabat operator, J. Math. Phys. 58 (2017) 102–108] under the analyticity condition of the potential using a complex WKB method. In this paper, we approach this problem with a microlocal technique focusing on the asymptotic behavior of the eigenfunction along the characteristic set to generalize the result to C ∞ potentials. [ABSTRACT FROM AUTHOR]

We study the cubic Szegö equation which is an integrable nonlinear nondispersive and nonlocal evolution equation. In particular, we present a direct approach for obtaining the multiphase and multisoliton solutions as well as a special class of periodic solutions. Our method is substantially different from the existing one which relies mainly on the spectral analysis of the Hankel operator. We show that the cubic Szegö equation can be bilinearized through appropriate dependent variable transformations and then the solutions satisfy a set of bilinear equations. The proof is carried out within the framework of an elementary theory of determinants. Furthermore, we demonstrate that the eigenfunctions associated with the multiphase solutions satisfy the Lax pair for the cubic Szegö equation, providing an alternative proof of the solutions. Last, the eigenvalue problem for a periodic solution is solved exactly to obtain the analytical expressions of the eigenvalues. [ABSTRACT FROM AUTHOR]

Cathepsins, a family of lysosomal peptidases, play a crucial role in maintaining cellular homeostasis by regulating protein turnover and degradation as well as many specific regulatory actions that are important for proper cell function and human health. Alterations in the activity and expression of cathepsins have been observed in many diseases such as cancer, inflammation, neurodegenerative disorders, bone remodelling-related conditions and others. These changes are not exclusively harmful, but rather appear to be a compensatory response on the lack of one cathepsin in order to maintain tissue integrity. The upregulation of specific cathepsins in response to the inhibition or dysfunction of other cathepsins suggests a fine-tuned system of proteolytic balance and understanding the compensatory role of cathepsins may improve therapeutic potential of cathepsin's inhibitors. Selectively targeting one cathepsin or modulating their activity could offer new treatment strategies for a number of diseases. This review emphasises the need for comprehensive research into cathepsin biology in the context of disease. The identification of the specific cathepsins involved in compensatory responses, the elucidation of the underlying molecular mechanisms and the development of targeted interventions could lead to innovative therapeutic approaches. [Display omitted] [ABSTRACT FROM AUTHOR]

An algorithm is given to express a proper rational function (i.e., the degree of the denominator exceeds that of the numerator) over any field as an alternating sum of reciprocals of polynomials of increasing degree. This is analogous to existing algorithms for representing proper rational numbers. [ABSTRACT FROM AUTHOR]

We study the effects of perturbing the boundary of a rectangle on the nodal sets of eigenfunctions of the Laplacian. Namely, for a rectangle of a given aspect ratio N, we identify the first Dirichlet mode to feature a crossing in its nodal set and perturb one of the sides of the rectangle by a close to flat, smooth curve. Such perturbations will often "open" the crossing in the nodal set, splitting it into two curves, and we study the separation between these curves and their regularity. The main technique used is an approximate separation of variables that allows us to restrict study to the first two Fourier modes in an eigenfunction expansion. We show how the nature of the boundary perturbation provides conditions on the orientation of the opening and estimates on its size. In particular, several features of the perturbed nodal set are asymptotically independent of the aspect ratio, which contrasts with prior works. Numerical results supporting our findings are also presented. [ABSTRACT FROM AUTHOR]

Anyons with a statistical phase parameter α ∈ (0 , 2) are a kind of quasi-particles that, for topological reasons, only exist in a 1D or 2D world. We consider the dimensional reduction for a 2D system of anyons in a tight wave guide. More specifically, we study the 2D magnetic gauge picture model with an imposed anisotropic harmonic potential that traps particles much stronger in the y-direction than in the x-direction. We prove that both the eigenenergies and the eigenfunctions are asymptotically decoupled into the loose confining direction and the tight confining direction during this reduction. The limit 1D system for the x-direction is given by the impenetrable Tonks–Girardeau Bose gas, which has no dependency on α , and no trace left of the long-range interactions of the 2D model. [ABSTRACT FROM AUTHOR]

The quality of deep soil mixing (DSM) columns in fine-grained soils remains challenging and needs to be addressed unequivocally to solve the existing practical problems. Breakthroughs can be achieved through field investigations; hence in the present study, the results of a field study on 16 DSM elements with various mix designs in cohesive clayey soils are presented. The rig data, continuous full-depth coring, and unconfined compressive strength (UCS) tests were used to evaluate the strength and uniformity of DSM columns. The results shed light on the fact that the occurrence of entrained mixing phenomena, an issue in which cohesive soils stick the blades, could be estimated using rig data, which might be solved by implementing a water predrilling phase. Furthermore, by following the fluctuations in the pull-down, rotary, and hoist pressures of the drilling rig data, the uniformity of DSM elements could be assessed indirectly. Based on the finding, it is noticeable that a conspicuous discrepancy has been observed between the Federal Highway Administration (FHWA) equation for predicting the UCS of DSM columns and the captured results. Furthermore, the correlations between elastic modulus and UCS of DSM elements were about the half of values by FHWA. Practical Applications: The implementation of offshore mega projects, such as refinery sites, can pose significant challenges, particularly when dealing with cohesive clayey soils with inadequate strength. In such circumstances, conducting deep soil mixing as a soil improvement method can emerge as a viable solution. The present study illustrates the detailed process of selecting the best mix design using the results of a comprehensive field validation program in Hormozgan. Drilling in stiff clayey soils is highly prone to sticking soils to the drilling auger, which hinders the proper functions of auger blades. The problem can be remarkably controlled by adding a predrilling phase to the mix design. Moreover, continuous monitoring of the critical drilling machine parameters, such as induced pressures, can shed light on the quality of deep soil mixing columns, without conducting extra field tests that play a key role in the project costs. The findings of the study can serve as a benchmark for engineers, contractors, and clients who are involved in soil improvement projects that utilize the deep soil mixing method. [ABSTRACT FROM AUTHOR]

It is well-known that(see e.g. Proposition 1.3.17 in [10]) the normalized principal eigenfunction $ \phi_1 $ of single elliptic eigenvalue problem$ -d\Delta \phi -c(x)\phi = \lambda\phi $with Robin boundary condition converges to $ 0 $ as $ d\rightarrow 0^+ $ locally uniformly in $ \{x\in\bar\Omega|\; c(x)<\max_{\bar{\Omega}} c(x)\} $. The method used in [10] is designed for single equation and seems difficult to be applied to the elliptic systems directly. In this paper, we extend the conclusion above to elliptic system by introducing a different approach. [ABSTRACT FROM AUTHOR]

The article discusses the use of MRI scans from the ADNI dataset to develop biomarkers for early detection of Alzheimer's disease. The study focuses on data pre-processing, data augmentation, and the experimental setup of convolutional neural networks. Results show the effectiveness of specific augmentation strategies and the importance of model depth in achieving optimal performance. The study also evaluates the impact of dropout and generalization across image resolution, providing insights for future research in this area. [Extracted from the article]

The extended state pertains to the dispersion of the system's eigenfunctions across the whole lattice. Recent studies have shown that the non-Hermitian skin effect (NHSE) can reshape the wavefunction of topological modes. The localized states of topological modes within the bandgap gradually delocalized into extended states through the manipulation of NHSE. Here, we clarify the NHSE direction using the Bloch spectral winding numbers and reestablish the bulk-boundary correspondence through the non-Bloch winding numbers in the generalized Brillouin zone. We elucidate the formation of extended state by employing the localized decay length. Then, we have designed non-Hermitian topological circuits for experimental verification based on the voltage follower. The corner states, edge states, and extended states in 1D, 2D, and 3D circuits were observed through the measurement of node voltage. Our work can achieve the sustainable extended mode and provides significant cases for the analysis of topolectrical circuits. [ABSTRACT FROM AUTHOR]

The properties of activation functions profoundly affect the approximation performance of neural networks. It is always one of the most important tasks to find proper activation functions in approximation by neural networks for some specific purpose. In this paper, we introduce a new activation function which is generated by the Joukowski transformation, and construct some new neural network operators activated by the new activation function. We show that the new neural network operators can be used to approximate the continuous functions on some compact sets. In fact, we give the approximation rate by using the modulus of continuity of the objective function (see Theorem 1). Finally, we provide some specific numerical examples to verify the fitting effects of the neural network operators with different parameters for continuous functions. [ABSTRACT FROM AUTHOR]

The morphological characteristics of a product serve as essential carriers for conveying design intentions. These characteristics directly affect users' comprehension of the product's functions and proper usage, which are critical to the safety of product utilization and the overall comfort of the user experience. Incorporating prior experience to predict users' cognitive intentions regarding product form characteristics can provide valuable evaluation and decision-making references for design. This approach effectively reduces product development risks and contributes to enhancing user acceptance and experience. The study established intention discrimination indicators for form characteristics, covering six dimensions: functional orientation, behavioral intention, recognizability, cognitive load, attention distribution, and experiential feeling. Combining multidimensional scaling (MDS) and systematic clustering, samples were screened, and the morphological decomposition method was used to categorize and extract form characteristic categories and feature factors. The entropy weight method was applied to assign weights to the feature categories, and a feedforward neural network (FNN) was employed to construct a prediction model for cognitive intentions regarding product form characteristics. The model was tested using leave-one-out cross-validation, yielding a mean squared error (MSE) of 0.0089 and an R correlation coefficient of 0.9998, indicating high reliability. Finally, the feasibility and effectiveness of this method were validated through a case study on earthquake science experience facilities. [ABSTRACT FROM AUTHOR]

We address the linear eigenvalue problem associated with reaction-diffusion models for patchy linear domains with any finite number of unique patches. Interface conditions between adjacent patches may be chosen independently to accurately reflect organism behavior. We show that there exists a real principal eigenvalue with a corresponding non-negative eigenfunction that is non-zero interior to the domain. We demonstrate that our model is a generalization of several existing two-patch models. We consider simple three- and four-patch landscapes representing fragmented reserves to illustrate how our model can be used to analyze population persistence, spatial distribution, and migration dynamics. Fragmented landscapes with increased heterogeneity capture migration dynamics not found in landscapes with only two patch types, such as favorable habitat patches that experience net immigration. Conditions for the existence and non-existence of zero-flux conditions interior to three-patch systems are shown, which provide insight to the location of sources and direction(s) of migration. • Principal eigenvalues are studied for patchy domain problems, and corresponding eigenfunctions are positive on the interior. • An implicit equation is used to obtain numerical solutions to eigenvalue problems on previously intractable patchy domains. • Conditions are derived for the existence of extrema for principal eigenfunctions. • Results are used to simulate different landscapes with fragmented reserves. • Predictions are made for migration across landscapes, including when favorable patches might receive net immigration. [ABSTRACT FROM AUTHOR]

The singularities of the coupled fields near the apex of a piezoelectric/piezomagnetic composite wedge under antiplane deformation were studied. With the complex variable function theory and the eigenfunction expansion method, the explicit expressions of the eigenfunction equations for singularity orders were derived for 16 combinations of mechanically free or clamped, electrically closed or open and magnetically closed or open ones. Based on the obtained eigenequations, some numerical results were given to show the influences of the wedge angles, boundary conditions and material combination types on the singular behaviors. The results show that, the singularities of the coupled fields in a piezoelectric/piezomagnetic composite wedge are more complicated than those in a piezoelectric counterpart. [ABSTRACT FROM AUTHOR]

The primary acoustic signal of the voice is generated by the complex oscillation of the vocal folds (VFs), whereby physicians can barely examine the medial VF surface due to its anatomical inaccessibility. In this study, we investigated possibilities to infer medial surface dynamics by analyzing correlations in the oscillatory behavior of the superior and medial VF surfaces of four human hemilarynges, each in 24 different combinations of flow rate, VF adduction, and elongation. The two surfaces were recorded synchronously during sustained phonation using two high-speed camera setups and were subsequently 3D-reconstructed. The 3D surface parameters of mean and maximum velocities and displacements and general phonation parameters were calculated. The VF oscillations were also analyzed using empirical eigenfunctions (EEFs) and mucosal wave propagation, calculated from medial surface trajectories. Strong linear correlations were found between the 3D parameters of the superior and medial VF surfaces, ranging from 0.8 to 0.95. The linear regressions showed similar values for the maximum velocities at all hemilarynges (0.69–0.9), indicating the most promising parameter for predicting the medial surface. Since excessive VF velocities are suspected to cause phono-trauma and VF polyps, this parameter could provide added value to laryngeal diagnostics in the future. [ABSTRACT FROM AUTHOR]

In this review, we explore the intricate relationship between glucose metabolism and mechanotransduction pathways, with a specific focus on the role of the Hippo signaling pathway in chondrocyte pathophysiology. Glucose metabolism is a vital element in maintaining proper chondrocyte function, but it has also been implicated in the pathogenesis of osteoarthritis (OA) via the induction of pro-inflammatory signaling pathways and the establishment of an intracellular environment conducive to OA. Alternatively, mechanotransduction pathways such as the Hippo pathway possess the capacity to respond to mechanical stimuli and have an integral role in maintaining chondrocyte homeostasis. However, these mechanotransduction pathways can be dysregulated and potentially contribute to the progression of OA. We discussed how alterations in glucose levels may modulate the Hippo pathway components via a variety of mechanisms. Characterizing the interaction between glucose metabolism and the Hippo pathway highlights the necessity of balancing both metabolic and mechanical signaling to maintain chondrocyte health and optimal functionality. Furthermore, this review demonstrates the scarcity of the literature on the relationship between glucose metabolism and mechanotransduction and provides a summary of current research dedicated to this specific area of study. Ultimately, increased research into this topic may elucidate novel mechanisms and relationships integrating mechanotransduction and glucose metabolism. Through this review we hope to inspire future research into this topic to develop innovative treatments for addressing the clinical challenges of OA. [ABSTRACT FROM AUTHOR]

A bicluster consists of a subset of rows and columns of a given matrix, whose intersection defines the region (bicluster) of values of precisely defined condition. Through the decades, a variety of biclustering techniques have been successfully developed. Recently, it was proved that many possible patterns defined in two-dimensional data could be found with the application of Boolean reasoning. The provided theorems showed that any existing pattern in the data could be unequivocally encoded as an implicant of a proper Boolean function. Moreover, a prime implicant of that function encoded the inclusion-maximal (non-extendable) pattern. On the other hand, the definition of some two-dimensional patterns may be easily extended to three-dimensional patterns (triclusters) as well as to any number of dimensions (n-clusters). This paper presents a new approach for searching for three- and higher-dimensional simple patterns in continuous data with Boolean reasoning. Providing the definition of the Boolean function for this tasks, it is shown that the similar correspondence—implicants encode patterns, and prime implicants encode inclusion-maximal patterns—has a strong mathematical background: the proofs of appropriate theorems are also presented in this paper. [ABSTRACT FROM AUTHOR]

Koopman operator theory offers a basis for the systematic transformation and linearization of complex dynamical systems. We propose a method to approximate eigenfunctions of the Koopman operator for sufficiently smooth, deterministic and autonomous dynamical systems with hyperbolic fixed points in an equation‐based context. Approximations of the eigenfunctions are obtained in form of a rational ansatz whose coefficients are determined by minimizing a residual through a bi‐quadratic optimization problem. In addition, we consider an extension of the Hartman‐Grobman theorem, which was first proposed by Lan and Mezić in 2013, as a linear constraint. The implementation for a damped pendulum shows that the approach works in general, however, the optimization problem is non‐convex and thus sensible w. r. t. initial conditions, and increases proportional to the number of ansatz functions to the power of four. [ABSTRACT FROM AUTHOR]

The information contained in this article is suitable for clinicians practicing in the United States desiring a general overview of the assessment and management of spinal cord injury (SCI), focusing on initial care, assessment, acute management, complications, prognostication, and future research directions. SCI presents significant challenges, affecting patients physically, emotionally, and financially, with variable recovery outcomes ranging from full functionality to lifelong dependence on caregivers. Initial care aims to minimize secondary injury through thorough neurological evaluations and imaging studies to assess the severity of the injury. Acute management prioritizes stabilizing respiratory and cardiovascular functions and maintaining proper spinal cord perfusion. Patients with unstable or progressive neurological decline benefit from timely surgical intervention to optimize neurological recovery. Subacute management focuses on addressing common complications affecting the respiratory, gastrointestinal, and genitourinary systems, emphasizing a holistic, multidisciplinary approach. Prognostication is currently based on neurological assessments and imaging findings, but emerging biomarkers offer the potential to refine outcome predictions further. Additionally, novel therapeutic interventions, such as hypothermia therapy and neuroprotective medications are being explored to mitigate secondary damage and enhance recovery. This paper serves as a high-yield refresher for clinicians for the assessment and management of acute spinal cord injury during index admission. [ABSTRACT FROM AUTHOR]

In a recent study, the eigenvalue numbers of Case eigenfunctions were investigated for the situation of anisotropic scattering. The results of this study is applied to the criticality problem. The investigation is performed with two different methods: the singular value decomposition (SVD) method and the H N method. The former was recently improved using the SVD to solve the matrix of the problem, and the latter is based on the usage of the Case method for proper boundary conditions. We have two different eigenvalues for the given number of secondary neutrons and the scattering parameter. Either these eigenvalues are both complex, or one is complex and one is real. The investigation is to show the effect of the second eigenvalues over the critical thickness values. [ABSTRACT FROM AUTHOR]

On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space D 0 1 , p into L q and the summability properties of the distance function. We prove that, in the superconformal case (i.e. when 푝 is larger than the dimension), these two facts are equivalent, while in the subconformal and conformal cases (i.e. when 푝 is less than or equal to the dimension), we construct counterexamples to this equivalence. In turn, our analysis permits to study the asymptotic behavior of the positive solution of the Lane–Emden equation for the 푝-Laplacian with sub-homogeneous right-hand side, as the exponent 푝 diverges to ∞. The case of first eigenfunctions of the 푝-Laplacian is included, as well. As particular cases of our analysis, we retrieve some well-known convergence results, under optimal assumptions on the open sets. We also give some new geometric estimates for generalized principal frequencies. [ABSTRACT FROM AUTHOR]

The aim of this note is to study the problem $ -{\rm div}(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u=0 $ − div (| ∇ u | p − 2 ∇u) + V | u | p − 2 u = 0 in Ω, where Ω is a bounded domain in $ \mathds {R}^n $ R n and the potential V is assumed to be not equivalent to zero and lies in $ L_p(\Omega) $ L p (Ω). Also, we establish the strong unique continuation property of the eigenfunctions for the p-Laplacian operator in the case where $ V\in L_p(\Omega) $ V ∈ L p (Ω). [ABSTRACT FROM AUTHOR]

The article examines spectral problems in a domain $ \Omega _{\varepsilon } $ Ω ϵ , which is the union of a domain $ \Omega _{0} $ Ω 0 and a lot of thin trees that are ε-periodically situated along a manifold on the boundary of $ \Omega _{0} $ Ω 0 . The trees possess a finite number of branching levels. At the boundaries of branches from each branching level, the perturbed Steklov spectral condition $ \partial _{\boldsymbol {\nu }}u^{\varepsilon }=\varepsilon ^{\alpha } \lambda ^\varepsilon \varrho _{i,m}u^{\varepsilon } $ ∂ ν u ϵ = ϵ α λ ϵ ϱ i , m u ϵ is given, where $ \alpha \in \mathbb {R} $ α ∈ R . For a fixed ε, the problem possesses a discrete spectrum, and we analyze the asymptotic behavior of the eigenvalues and eigenfunctions as $ \varepsilon \to 0 $ ϵ → 0 , i.e. when the number of thin trees infinitely increases and their thickness vanishes. Three qualitatively distinct cases of the asymptotic behavior of the spectrum are identified depending on the parameter α. For each case, the study provides proof of the Hausdorff convergence of the spectrum, construction of leading asymptotic terms, and justification of the asymptotic estimates for the eigenvalues and eigenfunctions. For a point in the essential spectrum of the homogenized problem, we discover finite-energy eigenoscillations localized in one or some of the thin trees, and prove the asymptotic estimates. Cases, where the Steklov condition contains α depending on the branching level, are also discussed. [ABSTRACT FROM AUTHOR]

SUMMARY: The accessory proteins Hyponastic‐like 1 (HYL1) and Serrated (SE) enhance the precise and efficient processing of miRNAs by Dicer‐like 1 (DCL1), which is important for proper miRNA function. However, other factors determining the precision and efficiency of miRNA biogenesis are not well‐known. Here, we found that an asymmetric bulge (AB) at the 3′ end of miR‐5p (produced from the 5′ arm of the pre‐miRNA) reduced the precision of the second cleavage, whereas an AB at other sites of miR‐5p mainly affected the accumulation level of miR‐5p in transient expression in Nicotiana benthamiana. In contrast, many ABs in miR‐3p (produced from the 3′ arm of the pre‐miRNA) impose strong negative impact on the processing precision and the accumulation level of miR‐5p in N. benthamiana. Arabidopsis DCL1/SE/HYL1 complex‐mediated miRNA processing was reconstituted in Saccharomyces cerevisiae to further investigate AB‐mediated interference with DCL1 processing. With this system, the positional effect of AB on miRNA processing was tested. The results showed that ABs on the middle of miR‐5p have less of an impact on DCL1 cleavage efficiency and precision, whereas those on miR‐3p or near the ends of miR‐5p strongly reduce DCL1 cleavage activity, precision or both. Studies using the yeast miRNA processing system and transgenic Arabidopsis also revealed the importance of the interaction between the 2‐nt 3′ overhang of pre‐miRNA and the 3′ overhang binding pocket (3′BP) on the precision of the second cleavage reaction for many endogenous miRNAs. These findings provide new insights into the mechanism of miRNA biogenesis. Significance Statement: Factors determining the precision and efficiency of miRNA biogenesis are not well‐known. This study found that an asymmetric bulge at different positions of miR‐5p and miR‐3p had different impacts on miRNA levels and processing precision and revealed the importance of the interaction between the 2‐nt 3′ overhang of pre‐miRNA and the 3′ overhang binding pocket (3'BP) on the precision of the second cleavage reaction for many endogenous miRNAs. [ABSTRACT FROM AUTHOR]

In this paper, we examine the problems of existence and computation of proper and stable stabilising compensators in the feedback path of a special class of linear time-invariant multivariable systems characterised by a strictly proper transfer function matrix. For the class of systems that are 'square', i.e. have the same number of inputs and outputs and have no zeros in the closed right half complex plane and the class of 'non-square' systems that include a 'square' subsystem that has no zeros in the closed right half complex plane we present a sufficient condition for the existence and computation of proper stabilising (pole placing) compensators which in certain cases that depend on the transfer function matrix of the system can be chosen to be stable. [ABSTRACT FROM AUTHOR]

In response to the challenges posed by complex boundary conditions and singularities in molecular systems and quantum chemistry, accurately determining energy levels (eigenvalues) and corresponding wavefunctions (eigenfunctions) is crucial for understanding molecular behavior and interactions. Mathematically, eigenvalues and normalized eigenfunctions play crucial role in proving the existence and uniqueness of solutions for nonlinear boundary value problems (BVPs). In this paper, we present an iterative procedure for computing the eigenvalues (μ ) and normalized eigenfunctions of novel fractional singular eigenvalue problems, D 2 α y (t) + k t α D α y (t) + μ y (t) = 0 , 0 < t < 1 , 0 < α ≤ 1 , with boundary condition, y ′ (0) = 0 , y (1) = 0 , where D α , D 2 α represents the Caputo fractional derivative, k ≥ 1 . We propose a novel method for computing Lagrange multipliers, which enhances the variational iteration method to yield convergent solutions. Numerical findings suggest that this strategy is simple yet powerful and effective. [ABSTRACT FROM AUTHOR]

We prove that the real parts of equivariant (but non-invariant) eigenfunctions of generic bundle metrics on non-trivial principal S1 bundles over manifolds of any dimension have connected nodal sets and exactly 2 nodal domains. This generalizes earlier results of the authors in the 3-dimensional case. The failure of the results on for non-free S¹ actions is illustrated on even-dimensional spheres by one-parameter subgroups of rotations whose fixed point set consists of two antipodal points. [ABSTRACT FROM AUTHOR]

We consider an analytic family of Riemannian metrics on a compact smooth manifoldM. We assume the Dirichlet boundary condition for the η-Laplacian and obtain Hadamardtype variation formulas for analytic curves of eigenfunctions and eigenvalues. As an application, we show that for a subset of all Cr Riemannian metrics Mr on M, all eigenvalues of the η-Laplacian are generically simple, for 2 ≤ r - ∞. This implies the existence of a residual set of metrics in Mr that makes the spectrum of the η-Laplacian simple. Likewise, we show that there exists a residual set of drifting functions η in the space Fr of all Cr functions on M, that again makes the spectrum of the η-Laplacian simple, for 2 ≤ r < ∞. Besides, we provide a precise information about the complement of these residual sets as well as about the structure of the set of deformations of a Riemannian metric (respectively, of the set of deformations of a drifting function) which preserves double eigenvalues. Moreover, we consider a family of perturbations of a domain in a Riemannian manifold and obtain Hadamard-type formulas for the eigenvalues of the η-Laplacian in this case. We also establish generic properties of eigenvalues in this context. [ABSTRACT FROM AUTHOR]

Medically relevant biodegradable materials assist in the recovery of damaged tissue in the bones. Oral, cardiovascular, and orthopedic implants frequently use metallic implants. The incorporation of biodegradable metals (BMs) in medical applications is growing as a result of their suitability for use as temporary implants and their capacity to degrade once they accomplished their intended use. BMs have strong biocompatibility, which is the capacity of the substance to function with the proper host interaction in a specific situation. The implant will disintegrate during the healing process, preventing long‐term negative effects and the requirement for surgical removal. Because magnesium (Mg) is the body's fourth‐most plentiful element, its broken‐down by‐products can either be stored in new bone or calluses, or they can be released into circulation and eliminated through the urine. Mg is a biodegradable substance, although it has far lower corrosion resistance than other orthopedic implants. Mg‐based biomaterials are the most thoroughly researched and often used BMs due to their elastic modulus matching that of human bones, as well as their biosafety, biodegradability, and radiographic visibility. Thus, under this review, BM alloys, specifically Mg‐based metal alloys, and their properties, advantages, and challenges of Mg alloy were reviewed. [ABSTRACT FROM AUTHOR]

Introduction: Among hyperthermophilic organisms, in vivo protein localization is challenging due to the high growth temperatures that can disrupt proper folding and function of mostly mesophilic-derived fluorescent proteins. While protein localization in the thermophilic model archaeon S. acidocaldarius has been achieved using antibodies with fluorescent probes in fixed cells, the use of thermostable fluorescent proteins for live imaging in thermophilic archaea has so far been unsuccessful. Given the significance of live protein localization in the field of archaeal cell biology, we aimed to identify fluorescent proteins for use in S. acidocaldarius. Methods: We expressed various previously published and optimized thermostable fluorescent proteins along with fusion proteins of interest and analyzed the cells using flow cytometry and (thermo-) fluorescent microscopy. Results: Of the tested proteins, thermal green protein (TGP) exhibited the brightest fluorescence when expressed in Sulfolobus cells. By optimizing the linker between TGP and a protein of interest, we could additionally successfully fuse proteins with minimal loss of fluorescence. TGP-CdvB and TGP-PCNA1 fusions displayed localization patterns consistent with previous immunolocalization experiments. Discussion: These initial results in live protein localization in S. acidocaldarius at high temperatures, combined with recent advancements in thermomicroscopy, open new avenues in the field of archaeal cell biology. This progress finally enables localization experiments in thermophilic archaea, which have so far been limited to mesophilic organisms. [ABSTRACT FROM AUTHOR]

In this paper we ascertain the singular point-wise behavior of the positive solutions of a semilinear elliptic boundary value problem (1) at the critical value of the parameter, λ , where it begins its metasolution regime. As the weight function m (x) changes sign in Ω, our result is a substantial extension of a previous, very recent, result of Li et al. [8] , where it was imposed the (very strong) condition that m ≥ 0 on a neighborhood of b − 1 ({ 0 }). In this paper, we are simply assuming that m (x 0) > 0 for some x 0 ∈ b − 1 ({ 0 }). • Theorem 1.1 proves that the behavior of the solutions proved by Li et al. [8] also occurs with much weaker hypotheses. • Theorem 3.1 is a substantial extension of Theorem 2.1 of López-Gómez and Sabina de Lis [12]. • Lemma 2.1 provides a useful estimate of eigenfunctions associated to an eigenvalue problem with sign changing weight. [ABSTRACT FROM AUTHOR]

Soils contain large amounts of carbon stored as organic carbon and carbonates. These carbon pools can contribute to climate regulation and are of primary importance in ensuring proper soil functioning. However, their accurate quantification remains a complex task. Rock-Eval® thermal analysis has emerged as an alternative to classic dry combustion and wet methods due to its ability to simultaneously provide organic and inorganic carbon measurements on the same subsample. However, it has been observed that Rock-Eval® systematically underestimates the soil organic carbon (SOC) while overestimating the soil inorganic carbon (SIC). In this technical note, we propose a validated correction of both SOC and SIC based on a machine-learning model and using a diverse dataset of 240 soil samples. We show that the proposed correction significantly increases the accuracy of the Rock-Eval® method when compared to reference SOC and SIC values and applied to the dataset used for training and testing and that it can be successfully applied to data originating from different Rock-Eval® machines without changing the routine analytical protocol. The transferability of the model allows for its future implementation in the Geoworks software so that Rock-Eval® machines can routinely provide accurate SIC and SOC measurements. [ABSTRACT FROM AUTHOR]

Collagen posttranslational processing is crucial for its proper assembly and function. Disruption of collagen processing leads to tissue development and structure disorders like osteogenesis imperfecta (OI). OI-related collagen processing machinery includes prolyl 3-hydroxylase 1 (P3H1), peptidyl-prolyl cis-trans isomerase B (PPIB), and cartilage-associated protein (CRTAP), with their structural organization and mechanism unclear. We determine cryo-EM structures of the P3H1/CRTAP/PPIB complex. The active sites of P3H1 and PPIB form a face-to-face bifunctional reaction center, indicating a coupled modification mechanism. The structure of the P3H1/CRTAP/PPIB/collagen peptide complex reveals multiple binding sites, suggesting a substrate interacting zone. Unexpectedly, a dual-ternary complex is observed, and the balance between ternary and dual-ternary states can be altered by mutations in the P3H1/PPIB active site and the addition of PPIB inhibitors. These findings provide insights into the structural basis of collagen processing by P3H1/CRTAP/PPIB and the molecular pathology of collagen-related disorders. Collagen requires complicated modifications for proper assembly. Here, the authors show the structural basis of human collagen processing by the P3H1/CRTAP/PPIB complex, revealing a 'face-to-face' catalytic site configuration, collagen binding sites, and transition between trimer and hexamer states. [ABSTRACT FROM AUTHOR]

The Koopman operator framework holds promise for spectral analysis of nonlinear dynamical systems based on linear operators. Eigenvalues and eigenfunctions of the Koopman operator, the so-called Koopman eigenvalues and Koopman eigenfunctions, respectively, mirror global properties of the system's flow. In this paper, we perform the Koopman analysis of the singularly perturbed van der Pol system. First, we show the spectral signature depending on singular perturbation: how two Koopman principal eigenvalues are ordered and what distinct shapes emerge in their associated Koopman eigenfunctions. Second, we discuss the singular limit of the Koopman operator, which is derived through the concatenation of Koopman operators for the fast and slow subsystems. From the spectral properties of the Koopman operator for the singularly perturbed system and the singular limit, we suggest that the Koopman eigenfunctions inherit geometric properties of the singularly perturbed system. These results are applicable to general planar singularly perturbed systems with stable limit cycles. [ABSTRACT FROM AUTHOR]

In the field of nonlinear mathematical physics, Ablowitz et al.'s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs are all constant-coefficient models. In this study, we establish a fractional-order KdV (fKdV)-type equation with variable coefficients and show that the IST is capable of solving the variable-coefficient fKdV (vcfKdV)-type equation. Firstly, according to Ablowitz et al.'s fractional-order algorithm and the anomalous dispersion relation, we derive the vcfKdV-type equation contained in a new class of integrable fNLEEs, which can be used to describe the dispersion transport in fractal media. Secondly, we reconstruct the potential function based on the time-dependent scattering data, and rewrite the explicit form of the vcfKdV-type equation using the completeness of eigenfunctions. Thirdly, under the assumption of reflectionless potential, we obtain an explicit expression for the fractional n-soliton solution of the vcfKdV-type equation. Finally, as specific examples, we study the spatial structures of the obtained fractional one- and two-soliton solutions. We find that the fractional soliton solutions and their linear, X-shaped, parabolic, sine/cosine, and semi-sine/semi-cosine trajectories formed on the coordinate plane have power–law dependence on discrete spectral parameters and are also affected by variable coefficients, which may have research value for the related hyperdispersion transport in fractional-order nonlinear media. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with global existence of large solutions for a dissipative model arising from electro-hydrodynamics, which is the nonlinear nonlocal system coupled by the Poisson–Nernst–Planck equations and the incompressible Navier–Stokes equations through charge transport and external forcing terms. By introducing some proper weighted functions and fully using the algebraic structure of the system, we prove that, under some conditions imposed on the indices p, p1, q, r, α, there exist two positive constants c0, C0 such that if the initial data u 0 = ( u 0 h , u 0 3 ) and (v0, w0) satisfy ‖ u 0 h ‖ B ̇ p 1 , ∞ − 1 + 3 p 1 + ‖ u 0 h ‖ B ̇ p 1 , ∞ − 1 + 3 p 1 α ‖ u 0 3 ‖ B ̇ p 1 , ∞ − 1 + 3 p 1 1 − α + K 0 ≤ c 0 with K 0 ≔ ‖ v 0 ‖ B ̇ q , 1 − 2 + 3 q ⁡ exp C 0 ‖ u 0 ‖ B ̇ p , 1 − 1 + 3 p + C 0 ‖ w 0 ‖ B ̇ r , 1 − 2 + 3 r + 1 exp C 0 ‖ u 0 ‖ B ̇ p , 1 − 1 + 3 p , then the system admits a unique global solution. Moreover, the global existence of large solution was also established in two dimensional case. [ABSTRACT FROM AUTHOR]

Body temperature is primarily regulated by the hypothalamus, ensuring proper metabolic function. Envenomation by Phoneutria nigriventer can cause symptoms such as hypothermia, hyperthermia, sweating, and shivering, all related to thermoregulation. This study aims to analyze and identify components of the venom that affect thermoregulation and to evaluate possible mechanisms. Rats were used for thermoregulation analysis, venom fractionation by gel filtration and reverse-phase chromatography (C18), and sequencing by Edman degradation. The venom exhibited hypothermic effects in rats, while its fractions demonstrated both hypothermic (pool II) and hyperthermic (pool III) effects. Further separations of the pools with C18 identified specific peaks responsible for these effects. However, as the peaks were further purified, their effects became less significant. Tests on U87 human glioblastoma cells showed no toxicity. Sequencing of the most active peaks revealed masses similar to those of the Tachykinin and Ctenotoxin families, both known to act on the nervous system. The study concludes that molecules derived from venom can act synergistically or antagonistically. Additionally, toxins that affect thermoregulation are poorly studied and require further characterization. These toxins could potentially serve as sources for the development of new thermoregulatory drugs. [ABSTRACT FROM AUTHOR]

The skin barrier is essential for maintaining the body's internal homeostasis, protecting against harmful external substances, and regulating water and electrolyte balance. Traditional Chinese Medicine (TCM) offers notable advantages in restoring skin barrier function due to its diverse components, targets, and pathways. Recent studies have demonstrated that active ingredients in TCM can safely and effectively repair damaged skin barriers, reinstating their proper functions. This review article provides a comprehensive overview of the mechanisms underlying skin barrier damage and explores how the bioactive constituents of TCM contribute to skin barrier repair, thereby offering a theoretical framework to inform clinical practices. [ABSTRACT FROM AUTHOR]

We examine the gauge transformations between the three-component Kadomtsev–Petviashvili (KP) hierarchy and the three-component modified Kadomtsev–Petviashvili (mKP) hierarchy. We introduce the auto-Bäcklund transformation of the three-component KP hierarchy and describe the Miura transformation of the three-component KP and mKP hierarchies, that is, . Additionally, the auto-Bäcklund transformation of the three-component mKP hierarchy is also given. This provides more powerful evidence that the gauge transformations generate Miura and auto-Bäcklund transformation on the eigenfunctions of the three-component KP and mKP hierarchies. [ABSTRACT FROM AUTHOR]

For a -component Camassa–Holm equation, as well as a -component generalization of the modified Camassa–Holm equation, nonlocal infinitesimal symmetries quadratically dependent on eigenfunctions of linear spectral problems are constructed from functional gradients of spectral parameters. With appropriate pseudopotentials, these nonlocal infinitesimal symmetries are prolonged to enlarged systems, and then explicitly integrated to generate symmetry transformations in finite form for the enlarged systems. As implementations of these finite symmetry transformations, some kinds of nontrivial solutions and Bäcklund transformations are derived for both equations. [ABSTRACT FROM AUTHOR]

The aim of this work is to characterize the asymptotic behaviour of the first eigenfunction of the generalised p-Laplace operator with mixed (Dirichlet and Neumann) boundary conditions in cylindrical domains when the length of the cylindrical domains tends to infinity. This generalises an earlier work of Chipot et al. (Asymptot. Anal.85(3–4) (2013) 199–227) where the linear case p = 2 is studied. Asymptotic behavior of all the higher eigenvalues of the linear case and the second eigenvalues of general case (using topological degree) for such problems is also studied. [ABSTRACT FROM AUTHOR]

The Pauli–Schrodinger equation for a non-relativistic position dependent mass with spin 1/2 and magnetic momentum μ0 in a generalized Morse potential and permanent magnetic field is solved using the Nikiforov–Uvarov method. The energy eigenvalues and the corresponding eigenfunctions are obtained analytically. It is also shown that the results established in a previous work appear to be a special case. [ABSTRACT FROM AUTHOR]

This paper studies the convergence of the proximal point method for quasiconvex functions in finite dimensional complete Riemannian manifolds. We prove initially that, in the general case, when the objective function is proper and lower semicontinuous, each accumulation point of the sequence generated by the method, if it exists, is a limiting critical point of the function. Then, under the assumptions that the sectional curvature of the manifold is bounded above by some non negative constant and the objective function is quasiconvex we analyze two cases. When the constant is zero, the global convergence of the algorithm to a limiting critical point is assured and if it is positive, we prove the local convergence for a class of quasiconvex functions, which includes Lipschitz functions. [ABSTRACT FROM AUTHOR]

Bivariate extreme value distributions can be used to model dependence of observations from random variables in extreme levels. There is no finite dimensional parametric family for these distributions, but they can be characterized by a certain one-dimensional function which is known as Pickands dependence function. In many applications the general approach is to estimate the dependence function with a non-parametric method and then conduct further analysis based on the estimate. Although this approach is flexible in the sense that it does not impose any special structure on the dependence function, its main drawback is that the estimate is not available in a closed form. This paper provides some theoretical results which can be used to find a closed form approximation for an exact or an estimate of a twice differentiable dependence function and its derivatives. We demonstrate the methodology by calculating approximations for symmetric and asymmetric logistic dependence functions and their second derivatives. We show that the theory can be even applied to approximating a non-parametric estimate of dependence function using a convex optimization algorithm. Other discussed applications include a procedure for testing whether an estimate of dependence function can be assumed to be symmetric and estimation of the concordance measures of a bivariate extreme value distribution. Finally, an Australian annual maximum temperature dataset is used to illustrate how the theory can be used to build semi-infinite and compact predictions regions. [ABSTRACT FROM AUTHOR]