Polynomial Growth Research Articles

Polynomial log-volume growth and the GK-dimensions of twisted homogeneous coordinate rings

Let f be a zero entropy automorphism of a compact Kähler manifold X . We study the polynomial log-volume growth \mathrm{Plov}(f) of f in light of the dynamical filtrations introduced in our previous work with T.-C. Dinh. We obtain new upper bounds and lower bounds of \mathrm{Plov}(f) . As a corollary, we completely determine \mathrm{Plov}(f) when \dim X = 3 , extending a result of Artin–Van den Bergh for surfaces. When X is projective, \mathrm{Plov}(f) + 1 coincides with the Gelfand–Kirillov dimensions \mathrm{GKdim}(X,f) of the twisted homogeneous coordinate rings associated to (X,f) . Reformulating these results for \mathrm{GKdim}(X,f) , we improve Keeler’s bounds of \mathrm{GKdim}(X,f) and provide effective upper bounds of \mathrm{GKdim}(X,f) which only depend on \dim X .

Providing Insight for Pediatric Ear Surgery: Analysis of Middle Ear Development via HRCT.

Providing insight for pediatric ear surgery via investigations on the development patterns of ossicles, mastoid, and external auditory canal (EAC). This retrospective study analyzed high-resolution computed tomography (HRCT) scans of 191 healthy temporal bones ranging from infants to adults. Subjects were grouped by 1-year intervals for developmental regression models and 3-year intervals for stage comparisons using t-tests or Mann-Whitney U tests. The size of auditory ossicles and tympanic cavity (TC) remained stable during development, while the minimum diameter of the tympanic sinus (TS) entrance was reduced. Regarding mastoid pneumatization, the air cells can be observed at birth, became pronounced at 2 years old, and were fully developed around the age of 5, with subsequent growth primarily involving radial expansion. Furthermore, the EAC demonstrated significant growth with age: the width of EAC increased linearly ( = 0.12x + 4.01, R2 = 0.85), while the length of EAC followed a polynomial growth pattern ( = -0.03x2 + 1.15x + 6.25, R2 = 0.96). Ossicles and TC remain stable during development. Furthermore, mastoid air cells may have developed in the early stages of life, while their diameter increases synchronously with EAC. All in all, ossicular chain reconstruction surgery and endoscopic ear surgery can be performed in babies. NA Laryngoscope, 2024.

Gradual Replacement of Soybean Meal with Brewer’s Yeast in Fingerling Nile Tilapia (Oreochromis niloticus) Diet, Resulting in a Polynomial Growth Pattern, Independent of Whether Reared in a Biofloc or Clear-Water System

A 60-day feeding experiment was conducted to examine whether (i) soybean meal (SBM) protein in the diet of Nile tilapia (Oreochromis niloticus) can be replaced with protein from spent brewer’s yeast (SBY); (ii) co-rearing with biofloc alters fish growth, feed conversion and protein efficiency compared with rearing in clear water; and (iii) accumulated protein quantity and quality in biofloc acts as a possible feed source for the fish in periods of low feed intake. The fish were reared in either a bio-recirculating aquaculture system (Bio-RAS) or a clear-water RAS (Cw-RAS). In Bio-RAS, the mechanical and biological filters used in Cw-RAS were replaced with an open bioreactor that delivered heterotrophic-based biofloc to the rearing tanks and also acted as a sedimentation trap for effluent water before recirculating it back into the rearing unit. The fish were fed four iso-nitrogenous and iso-energetic diets (~28% crude protein, ~19 MJ kg−1 gross energy) in which SBM protein was replaced with increasing levels of SBY, with triplicate tanks per inclusion level. The results revealed that average fish growth was greater in a biofloc environment compared with clear water and also greater at higher inclusion levels of SBY. However, in both rearing environments, fish growth displayed a second-degree polynomial distribution with increasing SBY inclusion level, with a peak between 30% and 60% inclusion. Fish in the biofloc environment showed better feed conversion ratio and protein retention, likely through ingesting both given feed and biofloc. Biofloc contained a significant amount of accumulated protein with a high biological profile, thereby constituting a possible feed reserve for the fish. A conclusion underlined by the apparent improved feed conversion of Bio-RAS reared fish, where that ingestion of biofloc will reduce the need for external feed per unit growth.

Cumulative stressor exposure predicts menstrual cycle affective changes in a transdiagnostic outpatient sample with past-month suicidal ideation.

Affective responses to the menstrual cycle vary widely. Some individuals experience severe symptoms like those with premenstrual dysphoric disorder, while others have minimal changes. The reasons for these differences are unclear, but prior studies suggest stressor exposure may play a role. However, research in at-risk psychiatric samples is lacking. In a large clinical sample, we conducted a prospective study of how lifetime stressors relate to degree of affective change across the cycle. 114 outpatients with past-month suicidal ideation (SI) provided daily ratings (n = 6187) of negative affect and SI across 1-3 menstrual cycles. Participants completed the Stress and Adversity Inventory (STRAIN), which measures different stressor exposures (i.e. interpersonal loss, physical danger) throughout the life course, including before and after menarche. Multilevel polynomial growth models tested the relationship between menstrual cycle time and symptoms, moderated by stressor exposure. Greater lifetime stressor exposure predicted a more pronounced perimenstrual increase in active SI, along with marginally significant similar patterns for negative affect and passive SI. Additionally, pre-menarche stressors significantly increased the cyclicity of active SI compared to post-menarche stressors. Exposure to more interpersonal loss stressors predicted greater perimenstrual symptom change of negative affect, passive SI and active SI. Exploratory item-level analyses showed that lifetime stressors moderated a more severe perimenstrual symptom trajectory for mood swings, anger/irritability, rejection sensitivity, and interpersonal conflict. These findings suggest that greater lifetime stressor exposure may lead to heightened emotional reactivity to ovarian hormone fluctuations, elevating the risk of psychopathology.

EXISTENCE OF THE GLOBAL ATTRACTOR FOR A HYPERBOLIC PHASE FIELD SYSTEM OF CAGINALP TYPE WITH RELAXATION, GOVERNED BY A POLYNOMIAL GROWTH POTENTIAL OF DEGREE

Phase field systems have attracted the attention of many researchers in physics and mathematics for several years. They have a wide range of applications, including materials science, thermal welding, crystal formation, and others. In such phenomena, perturbations can occur. The industrial world seeks to understand the behavior of the perturbed system while seeking better control of the perturbation. Our goal in this paper is to study the asymptotic behavior of the solution, proving the existence of the global attractor for perturbed phase field systems with a regular potential and polynomial growth typically of degree .

On almost polynomial growth of proper central polynomials

To any associative algebra A A is associated a numerical sequence c n δ ( A ) c_n^{\delta }(A) , n ≥ 1 n\ge 1 , called the sequence of proper central codimensions of A A . It gives information on the growth of the proper central polynomials of the algebra. If A A is a PI-algebra over a field of characteristic zero it has been recently shown that such a sequence either grows exponentially or is polynomially bounded. Here we classify, up to PI-equivalence, the algebras A A for which the sequence c n δ ( A ) c_n^{\delta }(A) , n ≥ 1 n\ge 1 , has almost polynomial growth. Then we face a similar problem in the setting of group-graded algebras and we obtain a classification also in this case when the corresponding sequence of proper central codimensions has almost polynomial growth.

Adiabatic gauge potential and integrability breaking with free fermions

We revisit the problem of integrability breaking in free fermionic quantum spin chains. We investigate the so-called adiabatic gauge potential (AGP), which was recently proposed as an accurate probe of quantum chaos. We also study the so-called weak integrability breaking, which occurs if the dynamical effects of the perturbation do not appear at leading order in the perturbing parameter. A recent statement in the literature claimed that integrability breaking should generally lead to an exponential growth of the AGP norm with respect to the volume. However, afterwards it was found that weak integrability breaking is a counter-example, leading to a cross-over between polynomial and exponential growth. Here we show that in free fermionic systems the AGP norm always grows polynomially, if the perturbation is local with respect to the fermions, even if the perturbation strongly breaks integrability. As a by-product of our computations we also find, that in free fermionic spin chains there are operators which weakly break integrability, but which are not associated with known long range deformations.

Central limit theorem for linear eigenvalue statistics of the adjacency matrices of random simplicial complexes

AbstractWe study the adjacency matrix of the Linial–Meshulam complex model, which is a higher‐dimensional generalization of the Erdős–Rényi graph model. Recently, Knowles and Rosenthal proved that the empirical spectral distribution of the adjacency matrix is asymptotically given by Wigner's semicircle law in a diluted regime. In this article, we prove a central limit theorem for the linear eigenvalue statistics for test functions of polynomial growth that is of class on a closed interval. The proof is based on higher‐dimensional combinatorial enumerations and concentration properties of random symmetric matrices. Furthermore, when the test function is a polynomial function, we obtain the explicit formula for the variance of the limiting Gaussian distribution.

Independence of linear spectral statistics and the point process at the edge of Wigner matrices

Let [Formula: see text] be a Wigner matrix of dimension [Formula: see text] with eigenvalues [Formula: see text] and [Formula: see text] be an analytic function on [Formula: see text] with polynomial growth. It is known that Tr[[Formula: see text])]-E[Tr[[Formula: see text])]] converges in distribution to a normal random variable with mean [Formula: see text] and a finite variance depending on [Formula: see text]. On the other hand, it is also known that [Formula: see text] converges in distribution to the GOE Tracy widom law. In this paper we prove that whenever the entries of the Wigner matrix are sub-Gaussian, Tr[[Formula: see text])]-E[Tr[[Formula: see text])]] is asymptotically independent of the point process at the edge of the spectrum. Hence, one gets that [Formula: see text] and Tr[[Formula: see text])]-E[Tr[[Formula: see text])]] are asymptotically independent. The main ingredient of the proof is based on a recent paper by Banerjee [A new combinatorial approach for tracy–widom law of wigner matrices, preprint (2022), arXiv:2201.00300 ]. The result of this paper can be viewed as a first step to find the joint distribution of eigenvalues in the bulk and the edge.

Polynomial growth of Betti sequences over local rings

Polynomial growth of Betti sequences over local rings

Planar minimal surfaces with polynomial growth in the Sp(4,ℝ)-symmetric space

abstract: We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the $\Sp(4,\R)$-symmetric space. We describe a homeomorphism between the "Hitchin component" of wild $\Sp(4,\R)$-Higgs bundles over $\CP^1$ with a single pole at infinity and a component of maximal surfaces with light-like polygonal boundary in $\h^{2,2}$. Moreover, we identify those surfaces with convex embeddings into the Grassmannian of symplectic planes of $\R^4$. We show, in addition, that our planar maximal surfaces are the local limits of equivariant maximal surfaces in $\h^{2,2}$ associated to $\Sp(4,\R)$-Hitchin representations along rays of holomorphic quartic differentials.

Lie semisimple algebras of derivations and varieties of PI-algebras with almost polynomial growth

We consider associative algebras with an action by derivations by some finite dimensional and semisimple Lie algebra. We prove that if a differential variety has almost polynomial growth, then it is generated by one of the algebras U T 2 ( W λ ) UT_2(W_\lambda ) or E n d ( W μ ) End(W_\mu ) for some integral dominant weight λ , μ \lambda ,\mu with μ ≠ 0 \mu \neq 0 . In the special case L = s l 2 L=\mathfrak {sl}_2 we prove that this is a sufficient condition too.

Schwartz correspondence for real motion groups in low dimensions

For a Gelfand pair (G, K) with G a Lie group of polynomial growth and K a compact subgroup, the Schwartz correspondence states that the spherical transform maps the bi-K-invariant Schwartz space S(K\\G/K)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {S}}}(K\\backslash G/K)$$\\end{document} isomorphically onto the space S(ΣD)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {S}}}(\\Sigma _{{\\mathcal {D}}})$$\\end{document}, where ΣD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma _{{\\mathcal {D}}}$$\\end{document} is an embedded copy of the Gelfand spectrum in Rℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {R}}}^\\ell $$\\end{document}, canonically associated to a generating system D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {D}}}$$\\end{document} of G-invariant differential operators on G/K, and S(ΣD)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {S}}}(\\Sigma _{{\\mathcal {D}}})$$\\end{document} consists of restrictions to ΣD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma _{{\\mathcal {D}}}$$\\end{document} of Schwartz functions on Rℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {R}}}^\\ell $$\\end{document}. Schwartz correspondence is known to hold for a large variety of Gelfand pairs of polynomial growth. In this paper we prove that it holds for the strong Gelfand pair (Mn,SOn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(M_n,SO_n)$$\\end{document} with n=3,4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=3,4$$\\end{document}. The rather trivial case n=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=2$$\\end{document} is included in previous work by the same authors.

Controller design and energy analysis for stochastic differential systems with one-sided polynomial growth under finite-time attraction constraint

This article presents a novel controller design approach for stochastic nonlinear systems exhibiting one-sided polynomial growth. The objective is to achieve finite-time attractiveness and estimate control energy consumption. By constructing an appropriate controller and employing inequality techniques, sufficient conditions for achieving finite-time attractiveness in stochastic nonlinear systems are established. Moreover, the article introduces the concept of expected energy consumption for stochastic systems and provides upper bounds on control time and energy consumption. A comprehensive trade-off analysis between control time and energy consumption is conducted to identify the optimal control intensity. The proposed approach is validated through numerical examples, demonstrating its effectiveness in practice.

Strict monotonicity for first passage percolation on graphs of polynomial growth and quasi-trees

Strict monotonicity for first passage percolation on graphs of polynomial growth and quasi-trees

Martingale solutions to stochastic nonlocal Cahn–Hilliard- Navier–Stokes systems with singular potentials driven by multiplicative noise of jump type

Abstract In the diffuse interface theory, the motion of two incompressible viscous fluids and the evolution of their interface are described by the well-known model H. The model consists of the Navier–Stokes equations, nonlinearly coupled with a convective Cahn–Hilliard type equation. Here we consider a stochastic version of the model driven by a noise of Lévy type, and where the standard Cahn–Hilliard equation is replaced by its nonlocal version with a singular (e.g., logarithmic) potential. The case of smooth potentials with arbitrary polynomial growth has been already analyzed in [G. Deugoué, A. Ndongmo Ngana and T. Tachim Medjo, Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type, Phys. D 398 2019, 23–68]. Taking advantage of this previous result, we investigate this more challenging and physically relevant case. Global existence of a martingale solution is proved with no-slip and no-flux boundary conditions in both 2D and 3D bounded domains. In the two-dimensional case, we prove the uniqueness of weak solutions when the viscosity is constant.

Asymptotic behavior of homological invariants of localizations of modules

Let R be a commutative noetherian ring, I an ideal of R, and M a finitely generated R-module. We consider the asymptotic injective dimensions, projective dimensions, Bass numbers, and Betti numbers of localizations of M/InM at prime ideals of R and prove that these invariants are stable or have polynomial growth for large integers n that do not depend on the prime ideals.

Ground state solutions for magnetic Schrödinger equations with polynomial growth

Abstract In this article, we investigate the following nonlinear magnetic Schrödinger equations: ( − i ∇ + A ( x ) ) 2 u + V ( x ) u = f 1 ( x , ∣ v ∣ 2 ) v , ( − i ∇ + A ( x ) ) 2 v + V ( x ) v = f 2 ( x , ∣ u ∣ 2 ) u , \left\{\begin{array}{l}{\left(-i\nabla +A\left(x))}^{2}u+V\left(x)u={f}_{1}\left(x,{| v| }^{2})v,\\ {\left(-i\nabla +A\left(x))}^{2}v+V\left(x)v={f}_{2}\left(x,{| u| }^{2})u,\end{array}\right. where V V is the electric potential and A A is the magnetic potential. Assuming that the nonlinear function f i ( i = 1 , 2 ) {f}_{i}\left(i=1,2) satisfies three types of polynomial growth assumptions: super-quadratic, asymptotically quadratic, and local super-quadratic at ∣ x ∣ → ∞ | x| \to \infty , we prove the existence of the Nehari-Pankov type ground state solutions using critical point theory together with the non-Nehari manifold method. The resulting problem engages two major difficulties: the first one is that the associated functional is strongly indefinite, and the second lies in verifying the link geometry and showing the boundedness of Cerami sequences. Our results extend and complement the present ones in the literature.

Optimal decay rates for nonlinear Moore–Gibson–Thompson equation with memory and Neumann boundary

We consider the initial value problem of Moore–Gibson–Thompson equation with memory effect, being subject to the Neumann boundary condition. Here, the nonlinearity f is conservative and satisfies some polynomial growth conditions. Under the condition that g ( t ) decays exponentially, we obtain the uniform polynomial decaying rate of energy and the solution. Moreover, we show this decay rate is optimal in the sense that there exists a class of solutions decaying exactly at this rate.

Practical stabilization of multi-links highly nonlinear Takagi-Sugeno fuzzy complex networks with Lévy noise based on aperiodically intermittent discrete-time observation control

This paper investigates practical stabilization of multi-links highly nonlinear Takagi-Sugeno fuzzy complex networks with Lévy noise (MHFNLs) via aperiodically intermittent discrete-time observation control (AIDOC). It is worth pointing out that AIDOC is considered into MHFNLs for the first time. Due to the difficulty of satisfying linear conditions in real life, polynomial growth conditions are used instead of linear growth conditions. Meanwhile, a global Lyapunov function is constructed and sufficient conditions which can remain MHFNLs practically stable are gained. Finally, an application and its numerical simulations are presented to verify the effectiveness and availability of the theoretical results.