We study the planar 3-colorablesubgroup $\mathcal{E}$ of Thompson’s group F and its even part ${\mathcal{E}_{\rm EVEN}}$. The latter is obtained by cutting $\mathcal{E}$ with a finite index subgroup of F isomorphic to F, namely the rectangular subgroup $K_{(2,2)}$. We show that the even part ${\mathcal{E}_{\rm EVEN}}$ of the planar 3-colorable subgroup admits a description in terms of stabilisers of suitable subsets of dyadic rationals. As a consequence ${\mathcal{E}_{\rm EVEN}}$ is closed in the sense of Golan and Sapir. We then study three quasi-regular representations associated with ${\mathcal{E}_{\rm EVEN}}$: two are shown to be irreducible and one to be reducible.

We give a simple sufficient condition for Quinn’s ‘bordism-type’ spectra to be weakly equivalent to commutative symmetric ring spectra. We also show that the symmetric signature is (up to weak equivalence) a monoidal transformation between symmetric monoidal functors, which implies that the Sullivan–Ranicki orientation of topological bundles is represented by a ring map between commutative symmetric ring spectra. In the course of proving these statements, we give a new description of symmetric L theory which may be of independent interest.

We study global-in-time dynamics of the stochastic nonlinear beam equations (SNLB) with an additive space-time white noise, posed on the four-dimensional torus. The roughness of the noise leads us to introducing a time-dependent renormalization, after which we show that SNLB is pathwise locally well-posed in all subcritical and most of the critical regimes. For the (renormalized) defocusing cubic SNLB, we establish pathwise global well-posedness below the energy space, by adapting a hybrid argument of Gubinelli-Koch-Oh-Tolomeo (2022) that combines the I-method with a Gronwall-type argument. Lastly, we show almost sure global well-posedness and invariance of the Gibbs measure for the stochastic damped nonlinear beam equations in the defocusing case.

In the present paper, we consider the following fractional double phase problem with nonlocal reaction:

\begin{align*}\qquad \quad \left\{\begin{array}{lll}(- \Delta)_p^s u + (- \Delta)_q ^s u + V(\varepsilon x) (|u|^{p - 2} u + |u|^{q - 2} u) = \left(\frac{1}{|x|^\mu} * F(u)\right) f(u) & \ \text{in}\ \mathbb{R}^N, \\u \in W^{s,p}(\mathbb{R}^N) \cap W^{s,q}(\mathbb{R}^N) ,\ u \gt 0 & \ \text{in}\ \mathbb{R}^N,\end{array}\right\}\end{align*}

where ɛ is apositive parameter, $0 \lt s \lt 1$, $2 \leqslant p \lt q \lt \min\{2p, N / s\}$, $0 \lt \mu \lt sp$, $(- \Delta)_t^s$ $(t \in \left\{p,q\right\})$ is the fractional t-Laplace operator, the reaction term $f : \mathbb{R} \mapsto \mathbb{R}$ is continuous, and the potential $V \in C (\mathbb{R}^N , \mathbb{R})$ satisfying a local condition. Using a variational approach and topological tools (the non-standard C1-Nehari manifold analysis and the abstract category theory), multiplicity of positive solutions and concentration properties for the above problem are established. Our results extend and complement some previous contributions related to double phase variational integrals.

In this paper, we establish a new version of one-dimensional discrete improved Hardy’s inequality with shifts by introducing a shifting discrete Dirichlet’s Laplacian. We prove that the general discrete Hardy’s inequality as well as its variants in some special cases admit improvements. Further, it is proved that two-variable discrete $p$-Hardy inequality can also be improved via improved discrete $p$-Hardy inequality in one dimension. The result is also extended to the multivariable cases.

In this article, we consider some critical Brézis-Nirenberg problems in dimension $N \geq 3$ that do not have a solution. We prove that a supercritical perturbation can lead to the existence of a positive solution. More precisely, we consider the equation:

\begin{equation*}\left\{\begin{array}{rllll}-\Delta u & = & \lambda u^{q-1} + u^{2^*+ r^\alpha -1} & \mbox{in} & B, \\u & \gt & 0 & \mbox{in} & B, \\u&=&0 & \mbox{on} & \partial B,\\\end{array}\right.\end{equation*}

$B \subset \mathbb{R}^N$

$N\geq 3$

$r=\vert x \vert$

$\alpha \in (0,\min\{N/2,N-2\})$

$q\in [2,2^*]$

$r \in (0,1)$

To date, the bestmethodsfor estimating the growth of mean values of arithmetic functions rely on the Voronoï summation formula. By noticing a general pattern in the proof of his summation formula, Voronoï postulated that analogous summation formulas for $\sum a(n)f(n)$ can be obtained with ‘nice’ test functions f(n), provided a(n) is an ‘arithmetic function’. These arithmetic functions a(n) are called so because they are expected to appear as coefficients of some L-functions satisfying certain properties. It has been well-known that the functional equation for a general L-function can be used to derive a Voronoï-type summation identity for that L-function. In this article, we show that such a Voronoï-typesummation identity in fact endows the L-function with some structural properties, yielding in particular the functional equation. We do this by considering Dirichlet series satisfying functional equations involving multiple Gamma factors and show that a given arithmetic function appears as a coefficient of such a Dirichlet series if and only if it satisfies the aforementioned summation formulas.

The aim of this article is to study the asymptotic behaviour of non-autonomous stochastic lattice systems. We first show the existence and uniqueness of a pullback measure attractor. Moreover, when deterministic external forcing terms are periodic in time, we show the pullback measure attractors are periodic. We then study the upper semicontinuity of pullback measure attractors as the noise intensity goes to zero. Pullback asymptotic compact for a family of probability measures with respect to probability distributions of the solutions is demonstrated by using uniform a priori estimates for far-field values of solutions.

In a two-dimensional plane, entire solutions of the Allen–Cahn type equation with a finite Morse index necessarily have finite ends. In the case that the nonlinearity is a sine function, all the finite-end solutions have been classified. However, for the classical Allen–Cahn nonlinearity, the structure of the moduli space of these solutions remains unknown. We construct in this paper new finite-end solutions to the Allen–Cahn equation, which will be called fence of saddle solutions, by gluing saddle solutions together. Our construction can be generalized to the case of gluing multiple four-end solutions, with some of their ends being almost parallel.

Consider a flow in $\mathbb{R}^3$ and let K be the biggest invariant subset of some compact region of interest $N \subseteq \mathbb{R}^3$. The set K is often not computable, but the way the flow crosses the boundary of N can provide indirect information about it. For example, classical tools such as Ważewski’s principle or the Poincaré–Hopf theorem can be used to detect whether K is non-empty or contains rest points, respectively. We present a criterion that can establish whether K has a non-trivial homology by looking at the subset of the boundary of N along which the flow is tangent to N. We prove that the criterion is as sharp as possible with the information it uses as an input. We also show that it is algorithmically checkable.

In this work, we are interested in studying the following class of problems:(𝒫λμ)

\begin{align}\left\{\begin{array}{ll}-\Delta u=f_\lambda(x,u,v)& \text{in}~~\Omega\\-\Delta v=g_\mu(x,u,v) & \text{in}~~\Omega\\0\not\equiv u\geq 0,\,\,0\not\equiv v\geq 0& \text{in}~~\Omega\\u=v=0&\text{on}~~\partial\Omega\end{array}\right.\end{align}

$\mathbb{R}^N$

$t\mapsto f_\lambda(x,t,t)$

$t\mapsto g_\mu(x,t,t)$

$(\mathcal{P}_{\lambda\mu})$

In this paper, we establish a second main theorem for holomorphic maps with finite growth index on complex discs intersecting arbitrary families of hypersurfaces (fixed and moving) in projective varieties, which gives an above bound of the sum of truncated defects. Our result also generalizes and improves many previous second main theorems for holomorphic maps from ${\mathbb{C}}$ intersecting hypersurfaces (moving and fixed) in projective varieties.

We study the real-valued modified KdV equation on the real line and the circle in both the focusing and the defocusing cases. By employing the method of commuting flows introduced by Killip and Vişan (2019), we prove global well-posedness in Hs for $0\leq s \lt \tfrac{1}{2}$. On the line, we show how the arguments in the recent article by Harrop-Griffiths, Killip, and Vişan (2020) may be simplified in the higher regularity regime $s\geq 0$. On the circle, we provide an alternative proof of the sharp global well-posedness in L2 due to Kappeler and Topalov (2005) and also extend this to the large-data focusing case.

Given a non-negative integer n and a ring R with identity, we construct a hereditary abelian model structure on the category of left R-modules where the class of cofibrant objects coincides with $\mathcal{GF}_n(R)$ the class of left R-modules with Gorenstein flat dimension at most n, the class of fibrant objects coincides with $\mathcal{F}_n(R)^\perp$ the right ${\rm Ext}$-orthogonal class of left R-modules with flat dimension at most n, and the class of trivial objects coincides with $\mathcal{PGF}(R)^\perp$ the right ${\rm Ext}$-orthogonal class of PGF left R-modules recently introduced by Šaroch and . The homotopy category of this model structure is triangulated equivalent to the stable category $\underline{\mathcal{GF}(R)\cap\mathcal{C}(R)}$ modulo flat-cotorsion modules and it is compactly generated when R has finite global Gorenstein projective dimension.

The second part of this paper deals with the PGF dimension of modules and rings. Our results suggest that this dimension could serve as an alternative definition of the Gorenstein projective dimension. We show, among other things, that (n-)perfect rings can be characterized in terms of Gorenstein homological dimensions, similar to the classical ones, and the global Gorenstein projective dimension coincides with the global PGF dimension.

We introduce the notion of the equivariant covering type of a space X on which a finite group G acts and study its properties. The equivariant covering type measures the size of G-equivariant good covers of X and is thus an extension of the covering type of a space, introduced by Karoubi and Weibel. We show that the equivariant covering type is a G-homotopy invariant and describe its relation with other G-invariants, like the equivariant LS-category, G-genus, and the multiplicative structures of equivariant cohomology theories. We also compute the G-covering type of regular G-graphs, give estimates for orientation-preserving actions on surfaces and for the projectivizations of complex representations of G and cohomology spheres. As an application, we derive estimates of sizes of minimal G-triangulations for various G-spaces.

Given a connected reductive algebraic group G over an algebraically closed field, we investigate the Picard group of the moduli stack of principal G-bundles over an arbitrary family of smooth curves.