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We consider periodic homogenization with localized defects of boundary value problems for semilinear ODE systems of the type $$ \Big((A(x/\varepsilon)+B(x/\varepsilon))u'(x)+c(x,u(x))\Big)'= d(x,u(x)) \mbox{ for } x \in (0,1),\; u(0)=u(1)=0. $$ For small $\varepsilon>0$ we show existence of weak solutions $u=u_\varepsilon$ as well as their local uniqueness for $\|u-u_0\|_\infty \approx 0$, where $u=u_0$ is a given solution to the homogenized problem $$ \Big(A_0u'+c(x,u(x))\Big)'= d(x,u(x)) \mbox{ for } x \in (0,1),\; u(0)=u(1)=0,\; A_0:=\left(\int_0^1A(y)^{-1}dy\right)^{-1} $$ such that the linearized problem $$ \Big(A_0u'+\partial_uc(x,u_0(x))u(x)\Big)'= \partial_ud(x,u_0(x))u(x) \mbox{ for } x \in (0,1),\; u(0)=u(1)=0 $$ does not have weak solutions $u\not=0$. Further, we prove that $\|u_\varepsilon-u_0\|_\infty\to 0$ and, if $c(\cdot,u)\in W^{1,\infty}((0,1);\mathbb{R}^n)$, that $\|u_\varepsilon-u_0\|_\infty=O(\varepsilon)$ for $\varepsilon \to 0$. We assume that $A \in L^\infty(\mathbb{R};\mathbb{M}_n)$ is 1-periodic, $B \in L^\infty(\mathbb{R};\mathbb{M}_n)\cap L^1(\mathbb{R};\mathbb{M}_n)$, $A(y)$ and $A(y)+B(y)$ are positive definite uniformly with respect to $y$, $c(x,\cdot),d(x,\cdot)\in C^1(\mathbb{R}^n;\mathbb{R}^n)$ and $c(\cdot,u),d(\cdot,u) \in L^\infty((0,1);\mathbb{R}^n)$. The main tool of the proofs is an abstract result of implicit function theorem type which has been tailored for applications to nonlinear singular perturbation and homogenization problems., Comment: arXiv admin note: substantial text overlap with arXiv:2309.15611

We show that a locally finite, connected graph has a coarse embedding into a Hilbert space if and only if there exist bond percolations with arbitrarily large marginals and two-point function vanishing at infinity. We further show that the decay of the two-point function is stretched exponential with stretching exponent $\alpha\in[0,1]$ if and only if the $L^1$-compression exponent of the graph is at least $\alpha$, leading to a probabilistic characterization of this exponent. These results are new even in the particular setting of Cayley graphs of finitely generated groups. The proofs build on new probabilistic methods introduced recently by the authors to study group-invariant percolation on Cayley graphs [25,26], which are now extended to the general, non-symmetric situation of graphs to study their coarse embeddability and $L^1$-compression exponents., Comment: Minor improvements in the presentation

Completion problems, of recovering a point from a set of observed coordinates, are abundant in applications to image reconstruction, phylogenetics, and data science. We consider a completion problem coming from algebraic statistics: to describe the completions of a point to a probability distribution lying in a given log-linear model. When there are finitely many completions, we show that these points either have a unique completion or two completions to the log-linear model depending on the set of observed coordinates. We additionally describe the region of points which have a completion to the log-linear model., Comment: This work was conducted as a part of the Algebraic Statistics and Our Changing World Program hosted by the Institute for Mathematical and Statistical Innovation (IMSI). 21 pages, 5 figures

We consider periodic homogenization of boundary value problems for quasilinear second-order ODE systems in divergence form of the type $a(x,x/\varepsilon,u(x),u'(x))'= f(x,x/\varepsilon,u(x),u'(x))$ for $x \in [0,1]$. For small $\varepsilon>0$ we show existence of weak solutions $u=u_\varepsilon$ as well as their local uniqueness for $\|u-u_0\|_\infty \approx 0$, where $u_0$ is a given non-degenerate solution to the homogenized boundary value problem, and we describe the rate of convergence to zero for $\varepsilon \to 0$ of the homogenization error $\|u_\varepsilon-u_0\|_\infty$. In particular, we show that this rate depends on the smoothness of the maps $a(\cdot,y,u,u')$ and $f(\cdot,y,u,u')$. Our assumptions are, roughly speaking, as follows: The maps $a,f:[0,1]\times\mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n$ are continuous, the maps $a(x,y,\cdot,\cdot)$ and $f(x,y,\cdot,\cdot)$ are $C^1$-smooth, the maps $a(x,\cdot,u,u')$ and $f(x,\cdot,u,u')$ are 1-periodic, and the maps $a(x,y,u,\cdot)$ are strongly monotone and Lipschitz continuous uniformly with respect to $x$, $y$ and bounded $u$. No global solution uniqueness is supposed. Because $x$ is one-dimensional, no correctors and no cell problems are needed. But, because the problem is nonlinear, we have to care about commutability of homogenization and linearization. The main tool of the proofs is an abstract result of implicit function theorem type which in the past has been applied to singularly perturbed nonlinear ODEs and elliptic and parabolic PDEs and, hence, which permits a common approach to existence and local uniqueness results for singularly perturbed problems and and for homogenization problems.

We consider periodic homogenization of boundary value problems for second-order semilinear elliptic systems in 2D of the type $$ \partial_{x_i}\left(a_{ij}^{\alpha \beta}(x/\varepsilon)\partial_{x_j}u(x)+b_i^\alpha(x,u(x))\right)=b^\alpha(x,u(x)) \mbox{ for } x \in \Omega. $$ For small $\varepsilon>0$ we prove existence of weak solutions $u=u_\varepsilon$ as well as their local uniqueness for $\|u-u_0\|_\infty \approx 0$, where $u_0$ is a given non-degenerate weak solution to the homogenized boundary value problem, and we estimate the rate of convergence to zero of $\|u_\varepsilon-u_0\|_\infty$ for $\varepsilon \to 0$. Our assumptions are, roughly speaking, as follows: The functions $a_{ij}^{\alpha \beta}$ are bounded, measurable and $\mathbb{Z}^2$-periodic, the functions $b_i^\alpha(\cdot,u)$ and $b^\alpha(\cdot,u)$ are bounded and measurable, the functions $b_i^\alpha(x,\cdot)$ and $b^\alpha(x,\cdot)$ are $C^1$-smooth, and $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^2$. Neither global solution uniqueness is supposed nor growth restrictions of $b_i^\alpha(x,\cdot)$ or $b^\alpha(x,\cdot)$ nor higher regularity of $u_0$, and cross-diffusion is allowed. The main tool of the proofs is an abstract result of implicit function theorem type which in the past has been applied to singularly perturbed nonlinear ODEs and elliptic and parabolic PDEs and, hence, which permits a common approach to existence, local uniqueness and error estimates for singularly perturbed problems and and for homogenization problems.

Let $\mathcal G$ be the Cayley graph of a finitely generated, infinite group $\Gamma$. We show that $\Gamma$ has the Haagerup property if and only if for every $\alpha<1$, there is a $\Gamma$-invariant bond percolation $\mathbb P$ on $\mathcal G$ with $\mathbb E[\mathrm{deg}_{\omega}(g)]>\alpha\mathrm{deg}_{\mathcal G}(g)$ for every vertex $g$ and with the two-point function $\tau(g,h)=\mathbb P\big[g\leftrightarrow h\big]$ vanishing as $d(g,h)\to\infty$. On the other hand, we show that $\Gamma$ has Kazhdan's property (T) if and only if there exists a threshold $\alpha^*<1$ such that for every $\Gamma$-invariant bond percolation $\mathbb P$ on $\mathcal G$, $\mathbb E[\mathrm{deg}_\omega(o)]>\alpha^*\mathrm{deg}(o)$ implies that the two-point function is uniformly bounded away from zero. These results in particular answer questions raised by Lyons (J. Math. Phys. 41. 1099-1126 (2000)) about characterizations of properties of groups beyond amenability through group-invariant percolations. The method of proof is new and is based on a construction of percolations with suitable dependence structures built from invariant point processes on spaces with measured walls. This construction furthermore leads to quantitative bounds on the two-point functions, exhibiting in particular exponential decay of the two-point function in several prominent examples of Haagerup groups, including co-compact Fuchsian groups, co-compact discrete subgroups of $\mathrm{Isom}(\mathbb H^n)$ and lamplighters over free groups. This method also allows us to extend the aforementioned characterization of property (T) to the setting of relative property (T) and provide an application to Bernoulli percolation at the uniqueness threshold., Comment: Introduction and presentation substantially revised

This paper concerns autonomous boundary value problems for 1D semilinear hyperbolic PDEs. For time-periodic classical solutions, which satisfy a certain non-resonance condition, we show the following: If the PDEs are continuous with respect to the space variable $x$ and $C^\infty$-smooth with respect to the unknown function $u$, then the solution is $C^\infty$-smooth with respect to the time variable $t$, and if the PDEs are $C^\infty$-smooth with respect to $x$ and $u$, then the solution is $C^\infty$-smooth with respect to $t$ and $x$. The same is true for appropriate weak solutions. Moreover, we show examples of time-periodic functions, which do not satisfy the non-resonance condition, such that they are weak, but not classical solutions, and such that they are classical solutions, but not $C^\infty$-smooth, neither with respect to $t$ nor with respect to $x$, even if the PDEs are $C^\infty$-smooth with respect to $x$ and $u$. For the proofs we use Fredholm solvability properties of linear time-periodic hyperbolic PDEs and a result of E. N. Dancer about regularity of solutions to abstract equivariant equations., Comment: 22 pages

Let $\Gamma$ be a countable discrete group. Given any sequence $(f_n)_{n\geq 1}$ of $\ell^p$-normalized functions ($p\in [1,2)$), consider the associated positive definite matrix coefficients $\langle f_n, \rho(\cdot) f_n\rangle$ of the right regular representation $\rho$. We construct an orthogonal decomposition of the corresponding {\it Schur multipliers} on the reduced group $C^*$-algebra or the uniform Roe algebra of $\Gamma$. We identify this decomposition explicitly via the limit points of the orbits $(\widetilde f_n)_{n\geq 1}$ in the group-invariant compactification of the quotient space constructed by Varadhan and the first author in [14]. We apply this result and use positive-definiteness to provide two (quite different) characterizations of amenability of $\Gamma$ -- one via a variational approach and the other using group-invariant percolation on Cayley graphs constructed by Benjamini, Lyons, Peres and Schramm [1]. These results underline, from a new point of view to the best of our knowledge, the manner in which Schur multipliers capture geometric properties of the underlying group $\Gamma$., Comment: Improved presentation, to appear in "Journal in Functional Analysis"

This paper concerns the behavior of time-periodic solutions to 1D dissipative autonomous semilinear hyperbolic PDEs under the influence of small time-periodic forcing. We show that the phenomenon of forced frequency locking happens similarly to the analogous phenomena known for ODEs or parabolic PDEs. However, the proofs are essentially more difficult than for ODEs or parabolic PDEs. In particular, non-resonance conditions are needed, which do not have counterparts in the cases of ODEs or parabolic PDEs. We derive a scalar equation which answers the main question of forced frequency locking: Which time shifts of the solution to the unforced equation do survive under which forcing?, Comment: 47 pages

We consider boundary value problems for 1D autonomous damped and delayed semilinear wave equations of the type $$ \partial^2_t u(t,x)- a(x,\lambda)^2\partial_x^2u(t,x)= b(x,\lambda,u(t,x),u(t-\tau,x),\partial_tu(t,x),\partial_xu(t,x)), \; x \in (0,1) $$ with smooth coefficient functions $a$ and $b$ such that $a(x,\lambda)>0$ and $b(x,\lambda,0,0,0,0) = 0$ for all $x$ and $\lambda$. We state conditions ensuring Hopf bifurcation, i.e., existence, local uniqueness (up to time shifts), regularity (with respect to $t$ and $x$) and smooth dependence (on $\tau$ and $\lambda$) of small non-stationary time-periodic solutions, which bifurcate from the stationary solution $u=0$, and we derive a formula which determines the bifurcation direction with respect to the bifurcation parameter $\tau$. To this end, we transform the wave equation into a system of partial integral equations by means of integration along characteristics, and then we apply a Lyapunov-Schmidt procedure and a generalized implicit function theorem to this system. The main technical difficulties, which have to be managed, are typical for hyperbolic PDEs (with or without delay): small divisors and the "loss of derivatives" property. We do not use any properties of the corresponding initial-boundary value problem. In particular, our results are true also for negative delays $\tau$., Comment: 36 pages

The paper concerns boundary value problems for general nonautonomous first order quasilinear hyperbolic systems in a strip. We construct small global classical solutions, assuming that the right hand sides are small. In the case that all data of the quasilinear problem are almost periodic, we prove that the bounded solution is also almost periodic. For the nonhomogeneous version of a linearized problem, we provide stable dissipativity conditions ensuring a unique bounded continuous solution for any smooth right-hand sides. In the autonomous case, this solution is two times continuously differentiable. In the nonautonomous case, the continuous solution is differentiable under additional dissipativity conditions, which are essential. A crucial ingredient of our approach is a perturbation theorem for general linear hyperbolic systems. One of the technical complications we overcome is the "loss of smoothness" property of hyperbolic PDEs., Comment: Final version, to appear in Journal of Evolution Equations

We consider boundary value problems for quasilinear first-order one-dimensional hyperbolic systems in a strip. The boundary conditions are supposed to be of a smoothing type, in the sense that the $L^2$-generalized solutions to the initial-boundary value problems become eventually $C^2$-smooth for any initial $L^2$-data. We investigate small global classical solutions and obtain the existence and uniqueness result under the condition that the evolution family generated by the linearized problem has exponential dichotomy on R. We prove that the dichotomy survives under small perturbations in the leading coefficients of the hyperbolic system. Assuming that the coefficients of the hyperbolic system are almost periodic, we prove that the bounded solution is almost periodic also., Comment: Corrected typos