Search Results (164)

Existing studies on transverse phonon-polaritons in one-dimensional piezoelectric superlattices, based on the assumption of infinite lateral dimensions (perpendicular to the periodic direction of ferroelectric domains), have shown that only transverse superlattice vibrations with a strain component along the periodic direction can couple with electromagnetic waves to generate transverse phonon-polaritons. Real samples, however, inevitably have finite lateral dimensions, indicating that the infinite-lateral-size model requires modification. In this study, we find that in laterally finite systems, pure transverse superlattice vibrations (those without any strain component along the periodic direction) can also couple with electromagnetic waves, giving rise to a new class of pure transverse phonon-polaritons. Theoretical analysis reveals that the energy of this mode is primarily confined to the crystal surface and propagates as surface waves. Experimental verification confirms the existence of this polariton, and this result provides a new degree of freedom for the design of microwave devices based on piezoelectric superlattices. Full article

In this paper, we study matrix transformations on spaces of generalized almost convergent double sequences with powers. Extending classical results of Lorentz, Maddox, and Nanda, we characterize several classes of infinite matrices that map between Maddox’s double sequence spaces and spaces of almost convergent (to zero) double sequences with powers. Our results generalize earlier characterizations for single sequence spaces obtained by the authors in previous work, providing a structured framework for studying summability and convergence in higher dimensions. Full article

Quantiles are among the most fundamental constructs in probability theory and statistics, intrinsically linked to order structures, stochastic dominance, and the principles of robust statistical inference. Although the univariate theory of quantiles is by now classical and well developed, their generalization to multivariate settings remains mathematically subtle and methodologically demanding. In particular, extending the notion of “location within a distribution” beyond one dimension raises delicate questions of geometry, ordering, and equivariance. Within this landscape, the spatial—or geometric—formulation of multivariate quantiles has emerged as a rigorous and conceptually unifying framework capable of reconciling these issues. In this work we advance this paradigm by introducing a kernel-based estimation procedure for nonparametric conditional geometric quantiles of a multivariate response $Y\in {\mathbb{R}}^q$ ($q\ge 2$) given a functional covariate X that takes values in an infinite-dimensional space. The data are assumed to form a strictly stationary and ergodic process, while the responses may be subject to a missing-at-random mechanism, a feature of substantial practical relevance. Our analysis establishes strong consistency of the proposed estimator, characterizes its optimal convergence rate, and derives its asymptotic distribution. These limit theorems, in turn, provide the theoretical foundation for constructing asymptotically valid confidence regions and for performing inference in multivariate quantile regression with functional covariates. The theoretical developments rest on natural complexity conditions for the involved functional classes together with mild smoothness and regularity assumptions. This balance between generality and mathematical precision ensures that the resulting methodology is not only robust in a rigorous probabilistic sense but also widely applicable to contemporary problems in high-dimensional and functional data analysis. The proposed methodology is numerically investigated through simulations and is implemented in a real data application. Full article

In the design of cellular structures using unit cell-based modeling, idealized structures with infinite dimensions and negligible boundary conditions are often assumed in order to simplify the analysis. However, such treatments also result in significant errors for the performance predictions of actual cellular components with finite dimensions. In this study, the pattern size effects resulting from finite-sized cellular designs were investigated systematically for various cellular designs. Two types of size effects, namely, lateral and along-stress size effects, were defined and investigated using simulation-based studies. It was found that different cellular designs exhibit significantly different size effects, which are also dependent on factors including Poisson’s ratio, structural symmetry, and the unit cell dimensional aspect ratio. The coupling effect between the two size effects was also discussed. This study provides a more systematic understanding of the size effects of cellular structures that can be used to guide future designs. Full article

Daoist female alchemy (nüdan 女丹) texts articulate a bodily paradigm in which humans and nature mutually enfold one another, and in which yin and yang interact in harmonious complementarity. Through an analysis of three key dimensions, the yin-yang cosmology embedded in these texts, the ways menstruation, desire, and the female breasts are reconceived in the course of cultivation, and the ideal of gestating an a priori (xiantian 先天) embryo, this article argues that nüdan writings present a gender-symmetrical, pre-sexual symbolic culture. This culture both acknowledges gender difference and ultimately transcends it, seeking a return to the undifferentiated, yin-yang combined condition of primordial Dao. These texts reveal that women and men possess complementary yin and yang attributes that must be reintegrated in order to return to the a priori state and attain infinite vitality. They likewise suggest that both women and men harbor active, originary desire, and that only through equivalent processes of bodily transformation, reverting the sexualized, adult bodies into the unsexualized bodies of the girl and boy, can practitioners acquire the power to gestate the inner elixir, symbolizing inexhaustible vitality. In this sense, nüdan writings develop a pre-sexual narrative centered on vitality, offering a resonant response to concerns within postmodern feminism regarding how to dismantle centralized, phallogocentric narratives while enriching non-gender-centralized symbolic cultures. They thus provide a special path to reconsider gender not by advancing forward, but by stepping back into a more primordial, integrated ideal. Full article

Most primitive religions originated from the devout worship of celestial deities, earthly spirits, and ghosts. In oracle bone inscriptions, rituals related to praying for rain, temple worship, river deity worship, and the worship of great deities were referred to as “fang” 方 or “yi fang” 以方. The Supreme God was the paramount deity of the Yin Shang Dynasty people; by the early Zhou Dynasty, the Supreme God and ancestral spirits began to merge. The hexagram of Contemplation 觀卦 (guan gua) establishes instruction through the concept of “contemplation” fully presenting the entire process of shamans, sorcerers, or ritual hosts participating in temple sacrifices, and completing the hand-washing ritual 盥 (guan) and the sacrifice-offering ritual 薦 (jian). It emphasizes the sincere communication between humans and Heaven. When a monarch performs the guan ritual, he embodies inner “sincerity and clarity” 誠明 (chengming); in response, the celestial deities will “show trust” 有孚 (youfu). Thus, it can be verified that deities exist in Heaven, and an interactive, responsive relationship is formed between Heaven and humans. The nine in the fifth place (the dominant line) possesses great inspiring power. The two fundamental dimensions for interpreting the hexagram structure are “the great view is above” 大觀在上 (da guan zai shang) and “[t]hose below look toward him and are transformed” 下觀而化 (xia guan er hua). These dimensions not only highlight the infinite transcendence, charisma, and appeal of the worshipped deities but also underscore humans’ profound reverence and faith in deities and the absolute existence. Sages 聖人 (sheng ren), as intermediaries between humans and deities, established religion for the sake of human life but did not regard themselves as religious leaders. However, from the Shang and Zhou dynasties to the Spring and Autumn period, a transition occurred in the spiritual life of the Chinese people: from shamanism to ritual propriety 禮 (li), and from theistic culture to humanistic culture. This transition laid the fundamental direction for the development and evolution of Chinese culture over the following 2500 years. Confucius attempted to replace or eliminate the shamanistic elements in early Confucians with personalized moral experience and ethical consciousness. Full article

The electronic band structure of two-dimensional lithium is calculated using the Dirac equation. Lithium is modeled as a two-dimensional square lattice in which the two strongly bound inner electrons and the fixed nucleus are treated as a positively charged ion (+e), while the outer electron is assumed to be uniformly distributed within the cell. The electronic potential is obtained by considering Coulomb-type interactions between the charges inside the unit cell and those in the surrounding cells. A numerical method that divides the unit cell into small pieces is employed to calculate the potential and then the Fourier coefficients are obtained. The Bloch method is used to determine the energy bands, leading to an eigenvalue matrix equation (in momentum space) of infinite dimension, which is truncated and solved using standard matrix diagonalization techniques. Convergence is analyzed with respect to the key parameters influencing the calculation: the lattice period, the dimension of the eigenvalue matrix, the unit-cell partition used to compute the potential’s Fourier coefficients, and the number of neighboring cells that contribute to the electronic interaction. Full article

We develop a topological-combinatorial framework applying classical Ramsey theory to systems of arcs connecting points on Jordan curves and their higher-dimensional analogues. A Jordan curve Λ partitions the plane into interior and exterior regions, enabling a canonical two-coloring of every arc connecting points on Λ according to whether its interior lies in Int(Λ) or Ext(Λ). Using this intrinsic coloring, we prove that any configuration of six points on Λ necessarily contains a monochromatic triangle, and that this property is invariant under all homeomorphisms of the plane. Extending the construction by including arcs lying on Λ itself yields a natural three-coloring, from which the classical value $R\left(3,3.3\right)=17$ guarantees the appearance of monochromatic triangles for sufficiently large point sets. For infinite point sets on Λ, the infinite Ramsey theorem ensures the existence of infinite monochromatic cliques, which we likewise show to be preserved under arbitrary topological deformations. The framework extends to Jordan surfaces and Jordan–Brouwer hypersurfaces in higher dimensions, where interior, exterior, and boundary regions again generate canonical colorings and Ramsey-type constraints. These results reveal a general principle: the separation properties of codimension-one topological boundaries induce universal combinatorial structures—such as monochromatic triangles and infinite monochromatic subsets—that are stable under continuous deformations. The approach offers new links between geometric topology, extremal combinatorics, and the analysis of constrained networks and interfaces. Full article

In this paper, we introduce a generalized framework for conditional probability in stochastic processes taking values in infinite-dimensional normed spaces. Classical definitions, based on measurability with respect to a conditioning $\sigma$-algebra, become inadequate when the available information is restricted to a $\sigma$-algebra generated by a finite or countable family of random variables. In such settings, many events of interest are not measurable with respect to the conditioning $\sigma$-field, preventing the standard definition of conditional probability. To overcome this limitation, we propose an extension of the coherent conditioning model through the use of Hausdorff measures. The key idea is to exploit the non-equivalence of norms in infinite-dimensional spaces, which gives rise to distinct metric structures and corresponding Hausdorff dimensions for the same events. Conditional probabilities are then defined relative to families of Hausdorff outer measures parameterized by their dimensional exponents. This geometric reformulation allows the notion of conditionality to depend explicitly on the underlying metric and topological properties of the space. The resulting model provides a flexible and coherent framework for analyzing conditioning in infinite-dimensional stochastic systems, with potential implications for Bayesian inference in functional spaces. Full article

We present an investigation into the band structure of acoustic waves in surface phononic crystals (SPnC), which comprise a square lattice of shallow cylinders on a mechanically isotropic semi-infinite substrate, utilizing the finite element method (FEM). The introduction of crystal periodicity to the surface modifies Rayleigh modes from non-dispersive to dispersive, thereby enabling the transformation of these modes into radiative or leaky forms. This spatial dispersion may facilitate the emergence of bound states in the continuum (BIC) by providing conditions appropriate for closing the radiative channels. A symmetry-protected BIC appears at the $\Gamma$ point only when the periodicity of the crystal extends in the two dimensions of the surface plane. The decoupling from the radiative channels is due to symmetry incompatibility. An accidental BIC emerges in both one- and two-dimensional SPnCs at finite wave vectors. The partial-wave model applied to the empty lattice approximation shows that the underlying mechanism giving rise to the emergence of the accidental BIC is related to the simultaneous fulfillment of the nullification condition of the transverse radiative channel amplitude and the dispersion equation. Furthermore, the presence of the accidental BIC is not compromised by structural alterations that preserve the crystal symmetry, with only its frequency being influenced. Full article

Drug delivery from cylindrical tablets of arbitrary dimensions is discussed here, using the analytical solution of diffusion equation. Utilizing dimensionless quantities, we show that the release profiles are determined by a unique parameter, represented by the aspect ratio of the cylindrical formulation. Fractional release curves are presented for different values of the aspect ratio, covering a range of many orders of magnitude. The corresponding release profiles lie in between the two opposite limits of release from thin slabs and two-dimensional radial release, pertinent to the cases of thin and long cylinders, respectively. In a quest for a part of the delivery process closer to a zero-order release, the release rate is calculated, which is found to exhibit the typical behavior of purely diffusional release systems. Two simple fitting formulae, containing two parameters each, are considered to approximate the infinite series of the exact solution: The stretched exponential (Weibull) function and a recently suggested expression interpolating between the correct time dependencies at the initial and final stages of the process. The latter provides a better fitting in all cases. The variation of the fitting parameters with the aspect ratio of the device is presented for both fitting functions. We also calculate the characteristic release time, which is found to correspond to an amount of fractional release between 64% and around 68% depending on the cylindrical aspect ratio. We discuss how the last quantities can be used to estimate the drug diffusion coefficient from experimental release profiles and apply these ideas to published drug delivery data. Full article

Let $(X_i,d_i)$ be a compact metric space with metric $d_i, i=1,2\dots ,k$, and G be a discrete infinitely countable amenable group. This paper is based on continuous actions $G\curvearrowright X_i$ on compact metric spaces $(X_i,d_i)$. Firstly, we introduce the concept of weighted Bowen balls, and then use the concept of weighted Bowen balls to introduce the corresponding lower (upper) weighted local entropy, as well as propose the concept of weighted Bowen topological entropy defined in terms of Hausdorff dimension by weighted Bowen balls, and prove Billingsley-type theorem between these two types of entropies by using the equivalent definition of weighted Bowen topological entropy. Full article

The fact that the Ising model in higher dimensions than 1D features a phase transition at the critical temperature $T_c$ despite its apparent simplicity is one of the main reasons why it has lost none of its fascination and remains a central benchmark in modeling physical systems. Building on our previous work, where an approximative analytic free-energy expression for finite 2D-Ising lattices was introduced, we investigate different extrapolation strategies for estimating $T_c$ of the infinite system from exactly solvable small lattices. Finite square lattices of linear dimension N with free and periodic boundary conditions were analyzed, exploiting their exactly accessible density of states to compute the heat capacity profiles $C\left(T\right)$. Different approaches were compared, including scaling models for the peak temperature $T_{\max }\left(N\right)$ and an envelope construction across the set of $C\left(T\right)$-profiles. We find that both approaches converge to the same asymptotic value and compare favorably to the established Binder cumulant method. Remarkably, a model for $T_{\max }$ with a single model parameter following an $N/(N+1)$-law provides robust convergence, with a physical analogy motivating this proportionality. Our findings highlight that surprisingly few, but highly accurate, finite-size results are sufficient to obtain a precise extrapolation. Full article

We propose a Ramsey approach to the dimensional analysis of physical systems, which complements the seminal Buckingham theorem. Dimensionless constants describing a given physical system are represented as vertices of a graph, referred to as a dimensions graph. Two vertices are connected by an aqua-colored edge if they share at least one common dimensional physical quantity and by a brown edge if they do not. In this way, a bi-colored complete Ramsey graph is obtained. The relations introduced between the vertices of the dimensions graph are non-transitive. According to the Ramsey theorem, a monochromatic triangle must necessarily appear in a dimensions graph constructed from six vertices, regardless of the order of the vertices. Mantel–Turán analysis is applied to study these graphs. The proposed Ramsey approach is extended to graphs constructed from fundamental physical constants. A physical interpretation of the Ramsey analysis of dimensions graphs is suggested. A generalization of the proposed Ramsey scheme to multi-colored Ramsey graphs is also discussed, along with an extension to infinite sets of dimensionless constants. The introduced dimensions graphs are invariant under rotations of reference frames, but they are sensitive to Galilean and Lorentz transformations. The coloring of the dimensions graph is independent of the chosen system of units. The number of vertices in a dimensions graph is relativistically invariant and independent of the system of units. Full article

Various authors have suggested that Kaluza–Klein variants of traversable wormholes might to some extent ameliorate the defocussing properties (the curvature condition violations, and implied energy condition violations) inherent in positing the existence of a traversable wormhole throat. Unfortunately such a hope is ill-founded. We shall show that in a traditional Kaluza–Klein context the price paid for completely eliminating the defocussing properties of the wormhole throat is extremely high—to completely eliminate curvature condition violations the 5th dimension has to become truly enormous (formally infinite) in the vicinity of the wormhole throat, in a manner that is fundamentally incompatible with the traditional Kaluza–Klein ansatz. At best, the extra dimensions allow one to move the curvature condition violations around, they cannot be eliminated except at prohibitive cost. While traversable Kaluza–Klein wormholes might be interesting for other reasons, it must be emphasized that adding a 5th dimension is not particularly useful in terms of ameliorating violations of the curvature conditions. Full article