Search Results (161)

Traditional frameworks of innovation diffusion, such as epidemic and Lotka–Volterra–Gause models, treat technological substitution primarily as a population-driven or social communication process, frequently overlooking the critical constraints imposed by external factor scarcities. To address this fundamental economic gap, this study performs a qualitative analysis of exploitative competition between two distinct technologies sharing two complementary resources, modeled via a non-linear system of chemostat-type consumer–resource ordinary differential equations. Technologies are represented as homogeneous populations of elemental firms, where individual output is governed by a ratio-dependent, fixed-proportions Leontief production function integrated with a hyperbolic clearing response. Operating within an open industrial system, the model accounts for resource supply rates and firm exit dynamics. We analytically derive the coordinates of both boundary and interior fixed points within the non-negative orthant of the phase space. By investigating the eigenvalues of the associated Jacobian matrix, we establish necessary and sufficient conditions for local asymptotic stability, competitive exclusion, and technological coexistence, demonstrating that efficiency is determined by a break-even resource availability threshold. Our results reveal that structural reconfigurations of the industry supply plane trigger bifurcations between local dominance and multistability. The latter manifests as a path-dependent, Quastlerian selection of initial conditions rather than inherent technological superiority. Finally, we establish the geometric boundaries of the stable assemblage niche, proving that technological diversity is regulated by resource supply rates. By explicitly incorporating resource scarcity into a dynamical predator–prey framework, the proposed model offers a more robust economic and mathematical foundation for innovation diffusion, providing policymakers with structural insights into the resource allocation strategy and the long-term management of industrial diversity. Full article

We study the unsteady mechanical response of an incompressible viscoelastic polymeric fluid in a flat channel, governed by the Vinogradov–Pokrovskii rheological model. The motion arises from an electrohydrodynamic reduction of Poiseuille type, after which the mechanical subsystem decouples from the electric field; the velocity then depends on time and on the transverse coordinate only. Treating the rheological parameter as small, we reduce the governing system in the leading-order approximation to a non-autonomous second-order evolution equation whose stiffness coefficient relaxes exponentially in time, so that the nonstationarity is driven by the internal relaxation of the normal stress rather than by an external force. For spatially homogeneous initial normal stress, we diagonalize the Galerkin system in the sine basis and obtain an explicit modal representation in which each mode satisfies a Bessel equation whose order depends on the mode number. This yields a critical index that splits the modes into three regimes—real order, zero order, and purely imaginary order—a structure absent from the classical UCM and Oldroyd-B solutions. Using the explicit representation, we prove convergence of the modal series and show that the solution decays in the long-time limit, so that the rest state is asymptotically stable in the natural energy phase space. The analytical solution is confirmed numerically. Full article

Neural metric learning is often assessed by retrieval accuracy, but a learned dissimilarity can rank examples well while failing to have norm-like algebraic structure. This paper studies a precise finite question: within a Euclidean-anchored residual family of neural dissimilarities, can one reduce sampled defects of homogeneity, subadditivity, and dyadic reconstruction on latent differences without destroying retrieval performance? The construction is inspired by asymptotically metrically normable (AMN) vector spaces, but its claims are finite, sampled, and latent: it does not prove global AMN rigidity or certify a metric on the input space. The framework is motivated by the observation that many learned similarities have the form $K=\exp (-E/\tau )$ and therefore encode an unbounded distance-like quantity or squared distance-like quantity behind a bounded affinity. The AMN-relevant object is this cost, not the bounded kernel value. We formalize bounded-perturbation stability of the large-scale specific energy $E(nv,0)/n$, the conversion of subadditivity into multiplicative affinity consistency, and the quotient interpretation in which directions of zero large-scale cost are collapsed. The mathematical development then introduces finite dyadic diagnostics, learned-gauge and convex-unit-ball interpretations, finite norm-envelope witnesses, dyadic stability bounds, and refinement towers of witness norms. The empirical part reports full official Fashion-MNIST experiments with supervised-contrastive and proxy-anchor-style Euclidean baselines, post hoc audits for shrinkage, residual flexibility, off-training scales, and latent extrapolation, and a ten-seed full-query/full-gallery UCI Human Activity Recognition benchmark. The results show that Euclidean objectives can be stronger for Recall@1, whereas AMN-inspired residual regularization substantially reduces finite norm-like defects inside the residual family. The contribution is therefore a finite diagnostic and regularization framework for learned latent dissimilarities, not a state-of-the-art retrieval objective. Full article

This study extends the Absement Method to leaky-aquifer pumping-test analysis by time integrating the Hantush–Jacob governing equation and deriving four complementary operators. Time integrating the Hantush–Jacob equation yields S·s = T∇²A − C·A, with storativity S, drawdown s, transmissivity T, the time (t)-integrated drawdown A(t) (absement), and leakance C. The four operators, A(t), time-averaged A(t)/t, windowed ΔA/Δt, and the normalized absement derivative (NAD), are applied jointly across all available observation wells. In a homogeneous aquifer, the fitted operators and NAD diagnostic provide mutually consistent parameter and flow-regime signatures. In a heterogeneous aquifer, systematic differences between operators become part of the interpretation: T-related variation appears as changes in the ΔA/Δt sliding profile across wells, whereas the leakage factor B = √(T/C)-related variation is identified by divergent A(t)/t asymptotes and NAD type-curve crossing. Monte Carlo assessment under composite noise (N = 50) confirms near-zero parameter bias, with T and S standard deviations approximately 3–4 times smaller for A(t)/t and ΔA/Δt than for A(t). The three field cases are identified: a 14% outward T decline with spatially uniform B (sandstone aquifer); approximately homogeneous T with outward-declining B flagged by NAD type-curve crossing before fitting (sandy aquifer); and T–B coupling resolution through the windowed ΔA/Δt profile (medium-grained sandstone aquifer). The outputs supported sustainable-yield assessment directly from routine pumping-test records. Full article

In the present work, we study a class of nonhomogeneous linear systems with fractional derivatives in Caputo’s sense of incommensurate order and distributed delays, satisfying the Perron condition. More precisely we study the impact of the Perron condition on the boundedness of the fundamental and extended fundamental matrices of the corresponding homogeneous system. We prove that if the nonhomogeneous system satisfies the Perron condition, then the fundamental matrix $C(t,s)$ and the extended fundamental matrix $Q(t,s)$ of the homogeneous system are uniformly bounded. As a consequence we obtain that the boundedness of the matrix $Q(t,s)$ is a necessary and sufficient condition for the uniform stability of the zero solution of the homogeneous system. Furthermore, we also prove that the extended fundamental matrix $Q(t,s)$ tends to zero when $t\to \infty$, which is a necessary and sufficient condition for the uniform asymptotic stability of the zero solution of the homogeneous system under study. Full article

This article introduces and develops a comprehensive theory of the Mixed Degenerate Gould–Hopper–Appell Type Polynomials $MDGHA$-$TPs$, constructed by embedding an Appell factor into the framework of degenerate Gould–Hopper generating functions. Beginning with the generating function formulation, we derive explicit series representations, monomial-type operational identities, recurrence relations, and a determinantal form that encodes the algebraic structure of the family. Summation identities expressed via Stirling numbers of the first kind and addition-type formulas are established. A detailed numerical investigation of the zero distributions of these polynomials is then carried out, with graphical illustrations revealing symmetry patterns and geometric arrangements in the complex plane. Connections with classical sequences of Appell, Hermite, and Gould–Hopper are explored throughout. The article concludes with remarks on open problems including the orthogonality of the $MDGHA$-$TPs$ with respect to suitable weight functions, the asymptotic behaviour of their zeros as the degree tends to infinity, and potential applications to boundary-value problems in heat diffusion, perturbation expansions in quantum mechanics, and signal processing in non-homogeneous media. Full article

We applied an algebraic approach, developed within the framework of the theory of a commutative monoid of self-dual symmetric polynomials, to the problem of effective isotropic conductivity ${\sigma }_e({\sigma }_1,\dots ,{\sigma }_n)$ in two-dimensional n-phase symmetric composites with partial isotropic conductivities ${\sigma }_j$. The upper $\mathsf{\Omega }({\sigma }_1,\dots ,{\sigma }_n)$ and lower $\omega ({\sigma }_1,\dots ,{\sigma }_n)$ bounds for ${\sigma }_e({\sigma }_1,\dots ,{\sigma }_n)$, found by the algebraic approach for $n=3,4$, are universal (independent of the composite microstructure) and possess all algebraic properties of ${\sigma }_e({\sigma }_1,\dots ,{\sigma }_n)$ that follow from physics: first-order homogeneity, full permutation invariance, Keller’s self-duality, positivity, and monotony. The bounds are compatible with the trivial solution ${\sigma }_e(\sigma ,\dots ,\sigma )=\sigma$ and satisfy Dykhne’s ansatz. Their comparison with previously known numerical calculations, asymptotic analysis, and exact results for the effective isotropic conductivity ${\sigma }_e({\sigma }_1,\dots ,{\sigma }_n)$ of two-dimensional three- and four-phase composites showed complete agreement. The bounds $\mathsf{\Omega }({\sigma }_1,\dots ,{\sigma }_n)$ and $\omega ({\sigma }_1,\dots ,{\sigma }_n)$ in both cases $n=3,4$ are stronger than the currently known variational bounds. Full article

In the ElastoHydrodynamic Lubrication (EHL) meshing contact model for rough interfaces with convex–concave textured micro-asperities, the geometric morphology of the meshing interface exhibits pronounced multiscale characteristics: the macroscale manifests as the correlation between Interface-Enriched Lubrication (IEL) performance and meshing Anti-Scuffing Load-Bearing Capacity (ASLBC), while the microscale corresponds to the textured morphology of rough interfaces. In numerical simulations of EHL meshing contact, such cross-scale disparities necessitate solving large-scale systems of analytical solution equations. Assuming periodicity or quasi-periodicity at the microscale, various established methods enable decoupling the macroscopic and microscopic scales, such formalized approaches constitute homogenization theory. However, classical asymptotic assumptions may introduce considerable approximation errors. This study proposes a micro-texture-informed homogenized contact model based on multiscale characterization that incorporates the coupled effects of gear interface meshing forces and thermo-elastic deformations, effectively extending the applicability of classical asymptotic homogenization methods. Full article

An analytical model of frictional heat transfer during the uniform sliding of two layers is proposed. One layer is composed of a functionally graded material (FGM) with a thermal conductivity coefficient that varies exponentially across its thickness, while the second layer is homogeneous, with constant thermophysical properties. The thermal problem of friction is formulated as an initial boundary value problem of heat conduction, accounting for the thermal contact conductance and convective heat exchange with the environment. An exact solution for constant friction power was obtained using the Laplace integral transform, supplemented by an asymptotic form for the initial stage of heating. Based on these analytical solutions, a comprehensive study was carried out for a frictional system comprising a ceramic–metal FGM composite in contact with a homogeneous friction material. A dimensional analysis allowed for both a qualitative and quantitative investigation into the influence of contact conductance, convective heat exchange, layer thickness and the FGM gradient parameter on the temperature evolution and distribution, as well as the time to reach the steady state. It was demonstrated that the implementation of an appropriately graded material can substantially improve thermal operating conditions by enhancing heat dissipation into the material bulk and intensifying convective cooling. Full article

The Fourier transform is continuous in the weak sense of tempered distribution; this ensures the weak stability of Fourier pairs. This article investigates a stronger form of stability of the pair of homogeneous profiles $\left(\right|x{|}^{-\alpha },c_d\left|\xi {|}^{d-\alpha }\right)$ on ${\mathbb{R}}^d$ that encompasses the case where the homogeneous profiles exist only on a large but finite range. In this case, largely overlooked in the literature, we provide precise error estimates in terms of the size of the tails outside the homogeneous range. We also prove a series of refined properties of the Fourier transform on related questions including criteria that ensure an approximate homogeneous behavior asymptotically near the origin or at infinity. The sharpness of our results is checked with numerical simulations. We also investigate briefly how these results consolidate the mathematical foundations of turbulence theory. Full article

The density power divergence (DPD) is a well-studied member of the Bregman divergence family and forms the basis of widely used minimum divergence estimators that balance efficiency and robustness. In this paper, we introduce and study a new sub-class of Bregman divergences, termed the exponentially weighted divergence (EWD), designed to generate competitive and practically interpretable inference procedures. The EWD is constructed so that its associated weight function remains bounded within the interval $[0, 1]$, which facilitates a transparent interpretation of robustness through controlled downweighting of low-density observations and avoids excessive influence from high-density points. We develop minimum EWD estimators (MEWDEs) within a general framework accommodating independent but non-homogeneous data, thereby extending classical minimum divergence theory beyond the i.i.d. setting. Under standard regularity conditions, we establish Fisher consistency and asymptotic normality, and we analyze robustness properties through influence function calculations. The EWD framework is further extended to parametric hypothesis testing, for which we derive the asymptotic null distribution of a Bregman divergence-based test statistic. Extensive simulation studies and real-data applications demonstrate that the proposed estimators perform comparably to, and often more robustly than, existing DPD-based procedures, particularly under moderate to heavy contamination, while retaining high efficiency under clean data. Overall, the EWD provides a tractable and interpretable alternative within the Bregman divergence class for robust parametric estimation and testing. Full article

If our observable Universe is only a tiny region of a vastly larger and conformally older spacetime, then the usual formulations of the classical flatness and horizon problems of the Hot Big Bang can be reinterpreted as artifacts manifesting an observational selection effect; we occupy a small causal domain of a much larger causally-connected and possibly non-flat spacetime. A sufficiently large positive cosmological constant, $\Lambda$, sets the future asymptotic horizon scale of the observable Universe, ∼${\Lambda }^{-1/2}$, thereby implying that the observable Universe may simply be a minute patch of a far larger pre-existing one, hereafter a Small Patch Hypothesis. Importantly, this observational bound is purely geometric; regardless of when the Universe is observed, the maximum accessible scale is finite and fixed by $\Lambda$, independent of inflationary dynamics, anthropic arguments, or assumptions about the global hosting spacetime. The externally possibly frozen past-eternal state implied by a pre-existing, causally connected spacetime motivates, but does not strictly require, viewing the perturbation field as being in (or arbitrarily close to) a coarse-grained maximum-entropy—equilibrium—configuration. Conditionalizing only on fixed mean and variance, a Gaussian distribution uniquely emerges, while the absence of entropy gradients corresponds to adiabaticity. In this work these features are therefore treated as plausible maximum-ignorance priors for super-horizon perturbations, rather than as rigorously derived consequences of a fully developed microscopic notion of gravitational entropy. In this sense, inflation becomes one viable realization of the proposed Small Patch Hypothesis. Here, one particular non-inflationary alternative is considered for illustrative purposes in which a primordial spectrum $P_{\zeta }\left(k\right)$ of the gauge-invariant perturbation $\zeta$ that pre-dates the Big Bang grows logarithmically toward large scales, $k\to 0$, and in fact diverges at some finite $k_c$. If $k_c\ll {\Lambda }^{-1/2}$, then our local cosmic patch probes only the regime where $\zeta \ll 1$ and appears exceptionally smooth. Over the comparatively narrow observable window, this $P_{\zeta }\left(k\right)$ mimics a slightly red-tilted, inflation-like spectrum. Rather than introducing high-energy new fields, this perspective frames large-scale homogeneity, isotropy, Gaussianity, adiabaticity, and the observed thermodynamic Arrow of Time as possible consequences of restricted observational access to a much larger Universe in equilibrium, rather than signatures of a unique early-Universe mechanism. Current observations cannot distinguish this logarithmically running spectrum from the standard power-law one, but future probes—for example high-resolution 21-cm measurements of the Dark Ages—may be able to falsify it. Full article

We calculate the mechanical response $r\left(x,t\right)$ of an initially quiescent semi-infinite homogeneous medium to a pulse applied at the origin, and this is achieved within the framework of the Kelvin–Voigt model. Although this problem has been extensively studied in the literature because of its wide range of applications—particularly in seismology—here, we present a solution in a novel integral form. This integral solution avoids the numerical computation of the solution in terms of the inverse Laplace transform; that is, numerical integration in the complex plane. In particular, we derive integral form expressions for both delta-pulse and step-pulse excitations which are simpler and more computationally efficient than those previously reported in the literature. Furthermore, the obtained expressions allow us to obtain simple asymptotic formulas for $r\left(x,t\right)$ as $x,t\to 0,\infty$ for both step- and delta-type pulses. Full article

In this paper, we consider a general two-prey one-predator reaction–diffusion system with prey competition and double prey-taxis. We first present some preliminary results, including global-in-time existence and a priori estimates of classical solutions to this system with ratio-dependent and non-ratio-dependent predator functional responses. Our main concern is the global stability of spatially homogeneous coexistence steady states. For generalized prey-dependent models, we show that the global asymptotic stability of the coexistence steady state relates to two prey-taxis coefficients provided that the predator functional response, the conversion, and diffusion rates are given. Since there is little prospect of establishing a unified Lyapunov functional for the systems with a predator-dependent functional response, we propose an explicit predator-dependent model and construct the corresponding Lyapunov functional. The theoretical results can cover most three-species chemotaxis systems. Full article

In this paper, we investigate the asymptotic behavior of solutions to a diffusive forest kinematic model, which describes the interactions among young trees, old trees, and airborne seeds. Our study focuses on the stability of the positive equilibrium, the occurrence of Hopf bifurcation yielding spatially homogeneous periodic solutions, and the subsequent Turing instability induced by diffusion in these periodic states. The analysis highlights that the juvenile tree mortality rate, represented by a quadratic function of mature tree density, plays a central dynamical role. Specifically, the parameter corresponding to the mature tree density at which juvenile mortality is minimized serves as a key Hopf bifurcation parameter. This indicates that the system’s transition to periodic solutions and later to diffusion-driven pattern formation can be effectively regulated through this parameter. From an ecological perspective, these results suggest that forest management strategies capable of indirectly influencing factors related to this critical parameter could help control the emergence of spatial patterns, such as forest patches. Furthermore, the functional form of the mortality rate offers a useful foundation for future studies examining how different assumptions regarding tree interaction morphology may influence ecosystem patterning. Full article