Abstract The Buchsbaum–Eisenbud–Horrocks conjecture has attracted the attention of many researchers working in Commutative Algebra and Algebraic Geometry in the last fifty years. Quite recently, a variant of this conjecture has been formulated by Lima-Pereira, Nuño-Ballesteros, Orefice-Okamoto, and Tomazella in their study of complete intersection singularities. The purpose of this paper is, on the one hand, to exhibit some cases where this conjecture holds. On the other hand, we show that the conjecture fails in general by exhibiting some counterexamples. Finally, we formulate a conjecture that can be regarded as a local variant of Buchsbaum–Eisenbud–Horrocks’ one.
We show that two simple, separable, nuclear, and ${\mathcal{Z}_0}$ -stable $\mathrm{C}^{*}$ -algebras are isomorphic if they are trace-preservingly homotopy equivalent. This result does not assume the Universal Coefficient Theorem and can be viewed as a tracial stably projectionless analogue of the homotopy rigidity theorem for Kirchberg algebras.
We study pathwise well-posedness of the stochastic heat equation (SHE) with a multiplicative noise on the circle. By combining the convolution Young and rough integration theory, introduced by Gubinelli and Tindel (2010), with the random tensor estimate approach to pathwise well-posedness of stochastic dispersive PDEs with multiplicative noises, introduced by Chapouto and the second and third authors (2026), we establish pathwise well-posedness of SHE in both the Young and rough cases, improving the results in Gubinelli and Tindel (2010). In particular, in the rough case (= the white-in-time case), our result covers the case of almost space-time white noise, thus establishing an optimal result within the framework of one-parameter rough paths.
We prove a quantitative isoperimetric inequality for nearly spherical domains in the Bergman ball in double struck upper C 2. We prove a Fuglede-type theorem for such sets. This result is a counterpart of a similar result obtained for the hyperbolic unit ball, and it provides the first result on the isoperimetric phenomenon in the Bergman ball.
This paper addresses the simultaneous homogenization and dimension reduction of thin composite plates reinforced with rigid substructures, within the framework of non-linear elasticity. The reference (undeformed) configuration consists of a periodically perforated elastic matrix, where the holes are occupied by rigid inclusions. We first establish a decomposition of the deformation for such composite structures. Under the assumption that the thickness parameter delta $\delta$ delta is asymptotically smaller than the periodicity epsilon $\varepsilon$ epsilon , we derive a reduced asymptotic model as both parameters tend to zero. Using rescaled unfolding operators, we characterize the limiting behaviour of the Green-St. Venant strain tensor. Through upper Gamma $\Gamma$ Gamma -convergence, we obtain the homogenized limit energy and establish the existence of a minimizer. The resulting limit model is of constrained K & aacute;rm & aacute;n type, where the limiting displacement satisfies a first-order infinitesimal isometry constraint.
Given a positive, non-increasing sequence a $a$ a with finite sum equal to 1 $1$ 1 , we consider the family of all closed subsets of left bracket 0 comma 1 right bracket $[0,1]$ [ 0 , 1 ] whose complementary open intervals have lengths given by a rearrangement of the sequence a $a$ a . We study the full range of possible theta $ heta$ theta -intermediate dimensions of these sets and, under suitable assumptions on the sequence, we show that this range forms a closed interval, whose endpoints we compute explicitly. This paper fills a gap in the literature concerning the dimensional properties of complementary sets.
The present paper is concerned with the optimal weight of vibrating string equations with the first two eigenvalues $\lambda_1$ and $\lambda_2$ being given. Applying the method of critical equations in $L<^>p[0,1]$ for $p \gt 1$ and the inverse spectral theory of Sturm-Liouville problems with measure coefficients, we find that the optimal weight can be uniquely determined if and only if $\lambda_2 \ge 2\lambda_1$ provided that the weight is non-negative and symmetrical. As an application, we provide an estimation of the extremum for partial trace of the first two eigenvalues on a sphere in $L<^>1[0,1]$.
In this paper, we establish general results for the asymptotic behaviour of solutions of dynamical systems in Banach spaces. We show that if the initial datum possesses a certain decay, then the corresponding solution emanating from the Cauchy problem studied inherits the same behaviour at any further time for which it exists. Our results are applied to a wide class of linear and non-linear models. In particular, we use our main results to show persistence properties for classical linear and non-linear ordinary differential equations (ODEs), the Benjamin-Bona-Mahony (BBM) equation, and the generalized Boussinesq equation.
In this paper, we use a degenerate reaction-diffusion system with free boundaries to model the spatial spread of rabies among foxes, whose population is divided into three sub-populations: susceptible ( $S$), infected ( $E$), and rabid ( $I$). Based on established biological observations, susceptible and infected foxes are assumed to be territorial (random diffusion is ignored for $E$ and $S$ in the model), whereas rabid foxes disperse randomly (random diffusion is assumed for $I$), causing the spread of the disease. While $S$ evolves over the entire real line $\mathbb{R}$, $E$ and $I$ are found only in the infected region represented by an interval $[g(t), h(t)]$, which expands with moving fronts $x = g(t)$ and $x = h(t)$ as time $t$ increases. We show that this system admits a unique global solution and then analyse its dynamics and establish a spreading-vanishing dichotomy in certain natural parameter regimes. Moreover, we supply some simple sufficient conditions for the vanishing and spreading of the rabies, respectively. For example, we show that if the corresponding ODE system has basic reproduction number $\mathcal{R}_0 \gt 1$, then a spreading-vanishing dichotomy holds, and the outcome depends on the initial size of the infected region, while if a certain quantity $\mathcal R_0<^>* \in (\mathcal R_0,\infty)$ is no bigger than 1, then the rabies will always vanish. The degenerate nature of the model, combined with the evolving infected region, causes considerable difficulties in the mathematical treatment, both in proving the well-posedness and in understanding the long-time dynamics. This paper appears to be the first to treat a free boundary model where one reaction-diffusion equation is coupled with two ordinary differential equations.
Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\Gamma=SL(2,\mathbb{Z})$. Denote by $\lambda_{ ext{sym}<^>{2}f}(n)$ the $n$th normalized coefficient of the Dirichlet series expansion of the symmetric square $L$-function $L(s, ext{sym}<^>{2}f)$. In this paper, we are interested in the shifted convolution sum \begin{equation*} \sum_{h\leq H}\sum_{N \lt n\leq 2N}\lambda_{ extrm{sym}<^>{2}f}(n)\lambda_{ extrm{sym}<^>{2}f}(n+h). \end{equation*}We establish a non-trivial bound for $H\gg N<^>{1/4}$.
For every dollar sign 0 less than alpha less than or equals infinity dollar sign $0 \lt \alpha\le\infty$ 0 < α ≤ ∞ $ we construct a continuous pure mixing map (topologically mixing, but not exact) on the Gehman dendrite with topological entropy dollar sign alpha dollar sign $\alpha$ $ α $ . It has been previously shown by Špitalský that there are exact maps on the Gehman dendrite with arbitrarily low positive topological entropy. Together, these results show that the entropy of maps on the Gehman dendrite does not exhibit the paradoxical behaviour reported for graph maps, where the infimum of the topological entropy of exact maps is strictly smaller than the infimum of the entropy of pure mixing maps. The latter result, stated in terms of popular notions of chaos, says that for maps on graphs, lower entropy implies stronger Devaney chaos. The conclusion of this paper says that lower entropy does not force stronger chaos for maps of the Gehman dendrite.
We derive Vorono & iuml; summation formulas for the Liouville function $\lambda(n)$, the M & ouml;bius function $\mu(n)$, and for $d<^>{2}(n)$, where $d(n)$ is the divisor function. The formula for $\lambda(n)$ requires explicit evaluation of certain infinite series for which the use of the Vinogradov-Korobov zero-free region of the Riemann zeta function is indispensable. Several results of independent interest are obtained as special cases of these formulas. For example, a special case of the one for $\mu(n)$ is a famous result of Ramanujan, Hardy, and Littlewood. Cohen-type and Ramanujan-Guinand-type identities are established for $\lambda(n)$ and $\sigma_a(n)\sigma_b(n)$, where $\sigma_s(n)$ is the generalized divisor function. As expected, infinite series over the non-trivial zeros of $\zeta(s)$ now form an essential part of all of these formulas. A series involving $\sigma_a(n)\sigma_b(n)$ and the product of modified Bessel functions occurring in one of our identities has appeared in a recent work of Dorigoni and Treilis in string theory. Lastly, we obtain results on oscillations of Riesz sums associated to $\lambda(n), \mu(n)$ and of the error term of Riesz sum of $d<^>2(n)$ under the assumption of the Riemann Hypothesis, simplicity of the zeros of $\zeta(s)$, the Linear Independence conjecture, and a weaker form of the Gonek-Hejhal conjecture.
This paper is concerned with the existence of normalized solutions for the following class of Hamiltonian elliptic systems: \begin{align*} \left\{\begin{array}{ll} - \Delta u = \lambda u + |v|{q-2}v \quad \text{in } \mathbb{R}{N}, \\ - \Delta v = \lambda v + |u|{p-2}u \quad \text{in } \mathbb{R}{N}, \\ \displaystyle\int_{\mathbb{R}N}(|u|2 + |v|2) = m, \end{array}\right. \end{align*}where $m \gt 0$ and $2 \lt p,q \lt 2{*}=2N/(N-2)$. We prove that a normalized solution exists for different ranges of $p,q$. A typical feature of this class of problems is that the associated energy functionals are strongly indefinite; that is, the domain has a saddle-point geometry in which both positive and negative subspaces of the quadratic form are infinite-dimensional. Another difficulty is the lack of the compact embedding $H{1}(\mathbb{R}N) \hookrightarrow L{2}(\mathbb{R}N)$, which persists also if we restrict ourselves to a radial setting. Our main result is novel for this class of systems.
We study a size-structured tree growth model from [4-6], described by the nonlinear renewal equation $ \phi(t) = \mathfrak{F} \phi_t, \ \phi_t \in L<^>1_\rho(\mathbb{R}_{-}), $ with reproduction, death, and growth rates $\beta$, $\mu$, and $g$. We prove that, under mild conditions on these rates, the equation generates a semiflow in $L<^>1_\rho(\mathbb{R}_{-})$ that is permanent and possesses a compact global attractor $\mathcal{A}$. If $\beta$ is monotone, $\mathcal{A}$ reduces to a single asymptotically stable equilibrium attracting all compact sets with positive initial data. Adapting an approach from [21], originally developed for simpler renewal equations, we investigate stability and persistence in this more complex setting via the one-dimensional recurrence $ b_{n+1} = \mathfrak{F} b_n, $ thereby complementing the functional-analytic framework of [13].
This note corrects an inaccuracy in the paper “Blow-up for the wave equation with nonlinear source and boundary damping terms” by the authors, appeared in Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), no. 4, 759–778.
This study is concerned with nonnegative solutions of the no-flux initial-boundary value problem for the doubly degenerate nutrient taxis system with a logistic source, as given by ( u(t) = del & centerdot; (uv del u) - del & centerdot; (u(2)v del v) + u - u(2) , vt = triangle v - uv, in a smoothly bounded planar domain ohm with (u, v)|t=0 = (u(0), v(0)). It is shown that despite substantially weakened diffusion, the stabilizing effects of logistic-type cell kinetics may overbalance any heterogeneity-supporting tendency of cross-diffusion: Namely, it is seen that for all suitably smooth initial data satisfying v(0) > 0 in ohm(-) and integral(ohm) ln u(0) > -infinity, a corresponding continuous weak solution, known to exist globally according to previous literature, has the property that as t -> infinity, u(-, t) -> 1 in L-1(ohm) and v(-, t) -> 0 in L-infinity(ohm).
Recent research has demonstrated the importance of spatial diffusion and environmental heterogeneity in influencing the transmission dynamics of infectious diseases. At the same time, human mobility patterns have been shown to exhibit scale-free, nonlocal dynamics characterized by an anomalous L & eacute;vy process diffusion, which is mathematically represented by nonlocal equations involving fractional Laplacian operators. To investigate the effects of environmental heterogeneity and long-range geographical disease transmission, we propose a time-periodic susceptible-infectious-susceptible (SIS) epidemic model that incorporates anomalous diffusion and spatial heterogeneity. The key issues of this paper include the existence and stability of both disease-free and endemic periodic equilibria, as well as the impact of diffusion rates and fractional powers on the spatial distribution of these periodic states. Our analytical findings indicate that spatio-temporal heterogeneity promotes disease persistence and that the fractional power can modulate the transmission threshold.
We provide a simplified approach to the the stable Hopf invariant. We provide short elementary proofs of the Cartan Formula, the Composition Formula, and the Transfer formula. In addition, when π is a discrete group, we show how to extend these results to the stable category of π-spaces. We also consider the extent to which the stable Hopf invariant is unique.
In this article, we investigate the higher topological complexity of oriented Seifert fibred manifolds that are Eilenberg-MacLane spaces $K(G,1)$ with infinite fundamental group $G$. We first refine the cohomological lower bounds for higher topological complexity by introducing the notion of higher topological complexity weights. As an application, we show that the $r<^>{\text{th}}$ topological complexity of these manifolds lies in $\{3r-1, 3r, 3r+1\}$, and characterize large families where the value is $3r$ or $3r+1$. Additionally, we establish a sufficient condition for higher topological complexity to be exactly $3r$ when the base surface is orientable and aspherical. Finally, we show that the higher topological complexity of the wedge of finitely many closed, orientable, aspherical $3$-manifolds is exactly $3r+1$.
