Homogeneous Functions Research Articles

Asymptotic expansions for harmonic functions at conical boundary points

We prove three theorems about the asymptotic behavior of solutions u to the homogeneous Dirichlet problem for the Laplace equation at boundary points with tangent cones. First, under very mild hypotheses, we show that the doubling index of u either has a unique finite limit, or goes to infinity; in other words, there is a well-defined order of vanishing. Second, under more quantitative hypotheses, we prove that if the order of vanishing of u is finite at a boundary point 0 , then locally u(x) = |x|^{m} \psi(x/|x|) + o(|x|^{m}) , where |x|^{m} \psi(x/|x|) is a homogeneous harmonic function on the tangent cone. Finally, we construct a convex domain in three dimensions where such an expansion fails at a boundary point, showing that some quantitative hypotheses are necessary in general. The assumptions in all of the results only involve regularity at a single point, and in particular are much weaker than what is necessary for unique continuation, monotonicity of Almgren’s frequency, Carleman estimates, or other related techniques.

Assessing regional homogeneity and cognitive function alterations in pediatric brain tumor patients: a resting-state functional magnetic resonance imaging study.

Cognitive impairment mechanisms in children with preoperative brain tumors are not well understood. This study aimed to determine the correlation between the changes of resting-state functional magnetic resonance imaging (rs-fMRI) and the fourth edition of the Wechsler Intelligence Scale for Children (WISC-IV) in patients with brain tumors before surgery and in healthy controls (HCs). rs-fMRI data were acquired using 3-T magnetic resonance imaging (MRI) scanners for 21 patients with pediatric brain tumor and 19 age- and gender-matched HCs. The data of WISC-IV were collected by psychiatrists. We used chi-square tests and two-sample t-tests to identify clinical features with significant associations before surgery. A two-sample t-tests was used to identify brain regions with significant changes in regional homogeneity (ReHo) before surgery in patients. Pearson correlation coefficients were used to assess the relationship between changes in ReHo and the five measures in the WISC-IV. The ReHo values were significantly decreased in the left anterior cingulate (T=-4.391) and right middle frontal gyrus (MFG) (T=-5.130) in patients compared to controls. Notably, ReHo values in the right MFG showed a positive correlation with the Perceptual Reasoning Index (R=0.471; P=0.031) and Working Memory Index (R=0.531; P=0.013) of the WISC-IV. The study identified significant ReHo alterations in patients with pediatric brain tumor, primarily in brain regions associated with cognitive processing, and revealed a positive correlation between these alterations and specific cognitive functions. These findings contribute to understanding cognitive impairments in this patient group and suggest potential areas for targeted intervention.

Differences of regional homogeneity and cognitive function between psychotic depression and drug-naïve schizophrenia

BackgroundPsychotic depression (PD) and schizophrenia (SCZ) share overlapping symptoms yet differ in etiology, progression, and treatment approaches. Differentiating these disorders through symptom-based diagnosis is challenging, emphasizing the need for a clearer understanding of their distinct cognitive and neural mechanisms.AimThis study aims to compare cognitive impairments and brain functional activities in PD and SCZ to pinpoint distinguishing characteristics of each disorder.MethodsWe evaluated cognitive function in 42 PD and 30 SCZ patients using the Repeatable Battery for the Assessment of Neuropsychological Status (RBANS) and resting-state functional magnetic resonance imaging (rs-fMRI). Regional homogeneity (ReHo) values were derived from rs-fMRI data, and group differences in RBANS scores were analyzed. Additionally, Pearson correlation analysis was performed to assess the relationship between cognitive domains and brain functional metrics.Results(1) The SCZ group showed significantly lower RBANS scores than the PD group across all cognitive domains, particularly in visuospatial/constructional ability and delayed memory (p < 0.05); (2) The SCZ group exhibited a significantly higher ReHo value in the left precuneus compared to the PD group (p < 0.05); (3) A negative correlation was observed between visuospatial construction, delayed memory scores, and the ReHo value of the left precuneus.ConclusionCognitive impairment is more pronounced in SCZ than in PD, with marked deficits in visuospatial and memory domains. Enhanced left precuneus activity further differentiates SCZ from PD and correlates with cognitive impairments in both disorders, providing neuroimaging-based evidence to aid differential diagnosis and insights into cognitive dysfunction mechanisms, while also paving a clearer path for psychiatric research.

Output-feedback control design for Takagi-Sugeno fuzzy systems through Lyapunov functions depending polynomially on the states

This paper proposes a new strategy based on linear matrix inequalities (LMIs) for the synthesis of state- and output-feedback controllers through parallel distributed compensation for continuous-time Takagi-Sugeno fuzzy systems. The main novelty of the proposed design technique is the use of homogeneous polynomial Lyapunov functions of degree larger than two on the states, generalizing the results based on quadratic functions. Parametrized in terms of a constant matrix, those homogeneous polynomial Lyapunov functions provide less conservative results in terms of stability and decay rate of trajectories when dealing with membership functions whose time-derivatives have unknown bounds. The synthesis procedure is formulated in terms of a locally convergent iterative algorithm, where a set of LMIs defined in an augmented parameter space is solved at each iteration. Numerical examples are used to compare the proposed method with quadratic stabilizability techniques available in the literature, illustrating the advantages of higher degree Lyapunov functions.

Necessary and Sufficient Conditions for the Boundedness of Multiple Integral Operators with Super-Homogeneous Kernels in Weighted Lebesgue Space

Super-homogeneous functions including homogeneous functions, quasi-homogeneous functions, and several non-homogeneous functions are considered. Using the weight function method, the construction conditions of Hilbert-type multiple integral inequalities with super-homogeneous kernels are first discussed. Then, using the obtained results, the construction problem of bounded multiple integral operators with super-homogeneous kernels in weighted Lebesgue space is discussed, and the necessary and sufficient conditions for operator boundedness and the operator norm formula are obtained.

A remark on the obstacle problem with lower regularity

We construct a monotone quantity for the classical obstacle problem with non-smooth obstacle and show that the blow-ups are homogeneous functions of degree α < 2 .

Higher spins and Finsler geometry

Finsler geometry is a natural generalization of (pseudo-)Riemannian geometry, where the line element is not the square root of a quadratic form but a more general homogeneous function. Parameterizing this in terms of symmetric tensors suggests a possible interpretation in terms of higher-spin fields. We will see here that, at linear level in these fields, the Finsler version of the Ricci tensor leads to the curved-space Fronsdal equation for all spins, plus a Stueckelberg-like coupling. Nonlinear terms can also be systematically analyzed, suggesting a possible interacting structure. No particular choice of spacetime dimension is needed. The Stueckelberg mechanism breaks gauge transformations to a redundancy that does not change the geometry. This creates a serious issue: non-transverse modes are not eliminated, at least for the versions of Finsler dynamics examined in this paper.

A nonlinear spectral core–periphery detection method for multiplex networks

Core–periphery detection aims to separate the nodes of a complex network into two subsets: a core that is densely connected to the entire network and a periphery that is densely connected to the core but sparsely connected internally. The definition of core–periphery structure in multiplex networks that record different types of interactions between the same set of nodes on different layers is non-trivial since a node may belong to the core in some layers and to the periphery in others. We propose a nonlinear spectral method for multiplex networks that simultaneously optimizes a node and a layer coreness vector by maximizing a suitable non-convex homogeneous objective function by a provably convergent alternating fixed-point iteration. We derive a quantitative measure for the quality of a given multiplex core–periphery structure that allows the determination of the optimal core size. Numerical experiments on synthetic and real-world networks illustrate that our approach is robust against noisy layers and significantly outperforms baseline methods while improving the latter with our novel optimized layer coreness weights. As the runtime of our method depends linearly on the number of edges of the network, it is scalable to large-scale multiplex networks.

Integrating the Relationship Between Exponents of the Utility Function and the Proportion of the Substitution and Income Effects of the Slutsky Equation into the Microeconomics Literature

The primary purpose of this paper is to explore the relationship between exponents of a linearly homogeneous (Cobb-Douglass type ) utility function and magnitude of the substitution effect (SE) and income effect (IE) in both the Slutsky and Hicks approaches. It is explored that when the price of a good with a larger exponent in the utility function increases the income effect is greater than the substitution effect, and the converse holds when the price of a good with a relatively smaller exponent increase. When exponents of both goods in the utility function are of the same magnitude, the income and substitution effects are equal irrespective of whether the price of the good or good changes. This concept has not been explored before and thus, it should be included in Microeconomics literature. It has also been shown that for a given utility function the ratios of the utility elasticities of the goods equal the ratio of the income and substitution effects. Also, the Marshallian demand curve is more price elastic than the Slutsky demand curve which is more elastic than the Hicks demand curve.

Classification of Solutions to the Anisotropic N-Liouville Equation in ℝN

Abstract Given $N\geq 2$, we completely classify solutions to the anisotropic $N$-Liouville equation $$ \begin{align*} &amp;-\Delta_N^H\,u=e^u \quad\textrm{in}\ \mathbb{R}^N,\end{align*} $$ under the finite mass condition $\int _{\mathbb{R}^{N}} e^{u}\,dx&amp;lt;+\infty $. Here $\Delta _{N}^{H}$ is the so-called Finsler $N$-Laplacian induced by a positively homogeneous function $H$. As a consequence in the planar case $N=2$, we give an affirmative answer to a conjecture made in [ 53].

Adaptive least-squares methods for convection-dominated diffusion-reaction problems

This paper studies adaptive least-squares finite element methods for convection-dominated diffusion-reaction problems. The least-squares methods are based on the first-order system of the primal and dual variables with various ways of imposing outflow boundary conditions. The coercivity of the homogeneous least-squares functionals are established, and the a priori error estimates of the least-squares methods are obtained in a norm that incorporates the streamline derivative. All methods have the same convergence rate provided that meshes in the layer regions are fine enough. To increase computational accuracy and reduce computational cost, adaptive least-squares methods are implemented and numerical results are presented for some test problems.

Minimally fine exhausters

In this article, we study upper exhausters of positively homogenous functions defined on locally convex space. We prove the existence of minimally fine exhauster finer than the given one. Such exhauster has to be unique for two-element exhauster. We show that a positively homogenous function is upper semicontinuous if and only if it has an upper exhauster. We prove that Clarke subdifferential of a positively homogenous function is the closed convex hull of the union of its minimally fine exhauster. We also present a number of appropriate examples.

An Electrolyte Engineered Homonuclear Copper Complex as Homogeneous Catalyst for Lithium-Sulfur Batteries.

Lithium-sulfur (Li-S) batteries suffer from severe polysulfide shuttle, retarded sulfur conversion kinetics and notorious lithium dendrites, which has curtailed the discharge capacity, cycling lifespan and safety. Engineered catalysts act as a feasible strategy to synchronously manipulate the evolution behaviors of sulfur and lithium species. Herein, a chlorine bridge-enabled binuclear copper complex (Cu-2-T) is in situ synthesized in electrolyte as homogeneous catalyst for rationalizing the Li-S redox reactions. The well-designed Cu-2-T provides completely active sites and sufficient contact for homogeneously guiding the Li2S nucleation/decomposition reactions, and stabilizing the lithium working interface according to the synchrotron radiation X-ray 3D nano-computed tomography, small angle neutron scattering and COMSOL results. Moreover, Cu-2-T with the content of 0.25wt% approaching saturated concentration in electrolyte further boosts the homogeneous optimization function in really operated Li-S batteries. Accordingly, the capacity retention of the Li-S battery is elevated from 51.4% to 86.3% at 0.2 C, and reaches 77.0% at 1.0 C over 400 cycles. Furthermore, the sulfur cathode with the assistance of Cu-2-T realizes the stable cycling under the practical scenarios of soft-packaged pouch cell and high sulfur loading (6.5mg cm-2 with the electrolyte usage of 4.5µL mgS -1).

Some characterizations on isoquants of homogeneous and quasi-sum production functions in microeconomics

Production function is an important concept in neoclassical economics and of great significance in economics and management. In the theory of geometric variations, the minimality of surfaces describes the optimal solutions of energy equations. Due to the nature of minimum energy, beautiful shape and stable structure, the theory of minimal surfaces is widely utilized in the fields of engineering, architecture, physics and molecular chemistry. In this article, we investigate homogeneous production functions and quasi-sum production functions with minimal isoquants, which solves two problems proposed by Neacşu et al. After meticulous calculations, the complete classifications of homogeneous production functions and quasi-sum production functions are reached. The research results interestingly indicate that homogeneous production functions with minimal isoquants are closely related to harmonic functions, which is a classical concept in mathematics and have widely applications.

Classification of anisotropic local Hardy spaces and inhomogeneous Triebel–Lizorkin spaces

This paper provides a characterization of when two expansive matrices yield the same anisotropic local Hardy and inhomogeneous Triebel–Lizorkin spaces. The characterization is in terms of the coarse equivalence of certain quasi-norms associated to the matrices. For nondiagonal matrices, these conditions are strictly weaker than those classifying the coincidence of the corresponding homogeneous function spaces. The obtained results complete the classification of anisotropic Besov and Triebel–Lizorkin spaces associated to general expansive matrices.

Bent functions satisfying the dual bent condition and permutations with the (Am) property

The concatenation of four Boolean bent functions f=f1||f2||f3||f4 is bent if and only if the dual bent condition f1∗+f2∗+f3∗+f4∗=1 is satisfied. However, to specify four bent functions satisfying this duality condition is in general quite a difficult task. Commonly, to simplify this problem, certain relations between fi are assumed, as well as functions fi of a special shape are considered, e.g., fi(x,y)=x·πi(y)+hi(y) are Maiorana-McFarland bent functions. In the case when permutations πi of F2m have the (Am) property and Maiorana-McFarland bent functions fi satisfy the additional condition f1+f2+f3+f4=0, the dual bent condition is known to have a relatively simple shape allowing to specify the functions fi explicitly. In this paper, we generalize this result for the case when Maiorana-McFarland bent functions fi satisfy the condition f1(x,y)+f2(x,y)+f3(x,y)+f4(x,y)=s(y) and provide a construction of new permutations with the (Am) property from the old ones. Combining these two results, we obtain a recursive construction method of bent functions satisfying the dual bent condition. Moreover, we provide a generic condition on the Maiorana-McFarland bent functions f1,f2,f3,f4 stemming from the permutations of F2m with the (Am) property, such that the concatenation f=f1||f2||f3||f4 does not belong, up to equivalence, to the Maiorana-McFarland class. Using monomial permutations πi of F2m with the (Am) property and monomial functions hi on F2m, we provide explicit constructions of such bent functions; a particular case of our result shows how one can construct bent functions from APN permutations, when m is odd. Finally, with our construction method, we explain how one can construct homogeneous cubic bent functions, noticing that only very few design methods of these objects are known.

Evaluation and Analysis of Cement Raw Meal Homogenization Characteristics Based on Simulated Equipment Models.

In recent years, the variability in the composition of cement raw materials has increasingly impacted the quality of cement products. However, there has been relatively little research on the homogenization effects of equipment in the cement production process. Existing studies mainly focus on the primary functions of equipment, such as the grinding efficiency of ball mills, the thermal decomposition in cyclone preheaters, and the thermal decomposition in rotary kilns. This study selected four typical pieces of equipment with significant homogenization functions for an in-depth investigation: ball mills, pneumatic homogenizing silos, cyclone preheaters, and rotary kilns. To assess the homogenization efficacy of each apparatus, scaled-down models of these devices were constructed and subjected to simulated experiments. To improve experimental efficiency and realistically simulate actual production conditions in a laboratory setting, this study used the uniformity of the electrical capacitance of mixed powders instead of compositional uniformity to analyze homogenization effects. The test material in the experiment consisted of a mixture of raw meal from a cement factory with a high dielectric constant and Fe3O4 powder. The parallel plate capacitance method was employed to ascertain the capacitance value of the mixed powder prior to and subsequent to treatment by each equipment model. The fluctuation of the input and output curves was analyzed, and the standard deviation (S), coefficient of variation (R), and homogenization multiplier (H) were calculated in order to evaluate the homogenization effect of each equipment model on the raw meal. The findings of the study indicated that the pneumatic homogenizer exhibited an exemplary homogenization effect, followed by the ball mill. For the ball mill, a higher proportion of small balls in the gradation can significantly enhance the homogenization effect without considering the grinding efficiency. The five-stage cyclone preheater also has a better homogenization effect, while the rotary kiln has a less significant homogenization effect on raw meal. Finally, the raw meal processed by each equipment model was used for clinker calcination and the preparation of cement mortar samples. After curing for three days, the compressive and flexural strengths of the samples were tested, thereby indirectly verifying the homogenization effect of each equipment model on the raw meal. This study helps to understand the homogenization process of raw materials by equipment in cement production and provides certain reference and data support for equipment selection, operation optimization, and quality control in the cement production process.

Stability analysis of a multilayer diffusion-reaction heat transfer problem with a very large number of layers

The stability of thermal conduction problems is of much practical interest in the context of Li-ion batteries, power electronic devices, chemical reactors and other engineering systems. While past work in predicting the thermal stability of a multilayer diffusion-reaction problem has been carried out using pole analysis or by determining imaginary eigenvalues of the problem, such techniques are impractical when the number of layers in the problem is very large. This work presents an exact analysis of the stability of a multilayer diffusion-reaction problem, in which the number of layers may be very large. Using homogenization and Heaviside functions based representation of thermophysical properties and parameters, it is shown that the stability of such a system may be easily predicted by computing the determinant of a matrix containing contributions from each layer/interface. Results from the present work are shown to correctly reduce to results from past work for pure-diffusion and single-layer special cases, as well as agree well with numerical simulations. Threshold for thermal stability of a representative Li-ion battery pack containing 1000 Li-ion cells is determined. It is shown that stability of the system is governed by a balance between temperature-dependent heat generation, diffusion and heat removal from the boundary. A key result of practical interest is that even small improvement in thermal diffusivity of Li-ion cells may greatly help reduce the risk of thermal instability. Additionally, practical guidelines for the maximum permissible operating time of a battery pack to avoid thermal runaway are discussed. This work advances the theoretical understanding of multilayer diffusion-reaction problems, and helps understand the thermal stability of Li-ion battery packs containing a very large number of cells, which is very difficult to analyze using traditional techniques.

Buckling mode constraints for topology optimization using eigenvector aggregates

Buckling-constrained structural design problems have conventionally prioritized optimizing the buckling load factor with less consideration given to the buckling mode shape. In this work, mode shape constraints are imposed within a topology optimization problem using an eigenvector aggregate constraint that is a weighted sum of homogeneous quadratic functions of the linearized buckling eigenvectors. A generalized formulation of the eigenvector aggregate is introduced, extending previous work. A new adjoint-based derivative evaluation technique is derived that is valid even in the presence of repeated eigenvalues. Numerical examples, including a clamped beam, a compressed column, and a square plate, demonstrate the effectiveness of the proposed approach. The results show the ability of the eigenvector aggregate to handle repeated eigenvalues, enable design space exploration, and capture mode shape switching.

Further results on positively homogeneous subadditive functions by using Csiszár f-divergence

Further results on positively homogeneous subadditive functions by using Csiszár f-divergence