We provide a fairly self-contained account of the localisation and cofinality theorems for the algebraic $\operatorname K$-theory of stable $\infty$-categories. It is based on a general formula for the evaluation of an additive functor on a Verdier quotient closely following work of Waldhausen. We also include a new proof of the additivity theorem of $\operatorname K$-theory, strongly inspired by Ranicki's algebraic Thom construction, a short proof of the universality theorem of Blumberg, Gepner and Tabuada, and a second proof of the cofinality theorem which is based on the universal property of $\operatorname K$-theory.

We work with polynomial three-dimensional rigid differential systems. Using the Lyapunov constants, we obtain lower bounds for the cyclicity of the known rigid centres on their centre manifolds. Moreover, we obtain an example of a quadratic rigid centre from which is possible to bifurcate 13 limit cycles, which is a new lower bound for three-dimensional quadratic systems.

The existence of isometric embedding of $S_q^m$ into $S_p^n$, where $1\leq p\neq q\leq \infty$ and $m,n\geq 2$, has been recently studied in [6]. In this article, we extend the study of isometric embeddability beyond the above-mentioned range of $p$ and $q$. More precisely, we show that there is no isometric embedding of the commutative quasi-Banach space $\ell _q^m(\mathbb {R})$ into $\ell _p^n(\mathbb {R})$, where $(q,p)\in (0,\infty )\times (0,1)$ and $p\neq q$. As non-commutative quasi-Banach spaces, we show that there is no isometric embedding of $S_q^m$ into $S_p^n$, where $(q,p)\in (0,2)\setminus \{1\}\times (0,1)$ $\cup \, \{1\}\times (0,1)\setminus \left \{\!\frac {1}{n}:n\in \mathbb {N}\right \}$ $\cup \, \{\infty \}\times (0,1)\setminus \left \{\!\frac {1}{n}:n\in \mathbb {N}\right \}$ and $p\neq q$. Moreover, in some restrictive cases, we also show that there is no isometric embedding of $S_q^m$ into $S_p^n$, where $(q,p)\in [2, \infty )\times (0,1)$. A new tool in our paper is the non-commutative Clarkson's inequality for Schatten class operators. Other tools involved are the Kato–Rellich theorem and multiple operator integrals in perturbation theory, followed by intricate computations involving power-series analysis.

Let ${\mathcal {D}}$ and $T$ be, respectively, a $C^1$ distribution of $k$-planes and a normal $k$-current on ${\mathbb {R}}^n$. Then ${\mathcal {D}}$ has to be involutive at almost every superdensity point of the tangency set of $T$ with respect to ${\mathcal {D}}$.

We study fluctuations of the error term for the number of integer lattice points lying inside a three-dimensional Cygan–Korányi ball of large radius. We prove that the error term, suitably normalized, has a limiting value distribution which is absolutely continuous, and we provide estimates for the decay rate of the corresponding density on the real line. In addition, we establish the existence of all moments for the normalized error term, and we prove that these are given by the moments of the corresponding density.

This paper mainly concerns the KAM persistence of the mapping $\mathscr {F}:\mathbb {T}^{n}\times E\rightarrow \mathbb {T}^{n}\times \mathbb {R}^{n}$ with intersection property, where $E\subset \mathbb {R}^{n}$ is a connected closed bounded domain with interior points. By assuming that the frequency mapping satisfies certain topological degree condition and weak convexity condition, we prove some Moser-type results about the invariant torus of mapping $\mathscr {F}$ with frequency-preserving under small perturbations. To our knowledge, this is the first approach to Moser's theorem with frequency-preserving. Moreover, given perturbed mappings over $\mathbb {T}^n$, it is shown that such persistence still holds when the frequency mapping and perturbations are only continuous about parameter beyond Lipschitz or even Hölder type. We also touch the parameter without dimension limitation problem under such settings.

This paper proves the energy equality for distributional solutions to fractional Navier-Stokes equations, which gives a new proof and covers the classical result of Galdi [Proc. Amer. Math. Soc. 147 (2019), 785–792].

Let $R$ be a strongly $\mathbb {Z}^2$-graded ring, and let $C$ be a bounded chain complex of finitely generated free $R$-modules. The complex $C$ is $R_{(0,0)}$-finitely dominated, or of type $FP$ over $R_{(0,0)}$, if it is chain homotopy equivalent to a bounded complex of finitely generated projective $R_{(0,0)}$-modules. We show that this happens if and only if $C$ becomes acyclic after taking tensor product with a certain eight rings of formal power series,  the graded analogues of classical Novikov rings. This extends results of Ranicki,  Quinn and the first author on Laurent polynomial rings in one and two indeterminates.

For a finitely dominated Poincaré duality space $M$, we show how the first author's total obstruction $\mu _M$ to the existence of a Poincaré embedding of the diagonal map $M \to M \times M$ in [17] relates to the Reidemeister trace of the identity map of $M$. We then apply this relationship to show that $\mu _M$ vanishes when suitable conditions on the fundamental group of $M$ are satisfied.

We study anti-symmetric solutions about the hyperplane $\{x_n=0\}$ for the following fractional Hardy–Hénon system:

\[ \left\{\begin{array}{@{}ll} (-\Delta)^{s_1}u(x)=|x|^\alpha v^p(x), & x\in\mathbb{R}_+^n, \\ (-\Delta)^{s_2}v(x)=|x|^\beta u^q(x), & x\in\mathbb{R}_+^n, \\ u(x)\geq 0, & v(x)\geq 0,\ x\in\mathbb{R}_+^n, \end{array}\right. \]

$0< s_1,s_2<1$

$n>2\max \{s_1,s_2\}$

$(p,q)$

$\alpha,\beta$

$s_1=s_2$

$(p,q)$

$\alpha, \beta$

We introduce a Morse theory for posets of Bestvina–Brady type combining matchings and height functions. This theory generalizes Forman's discrete Morse theory for regular CW-complexes and extends previous results on Morse theory for $h$-regular posets to all finite posets. We also develop a relative version of Morse theory which allows us to compare the topology of a poset with that of a given subposet.

For linear stochastic differential equations with bounded coefficients, we establish the robustness of nonuniform mean-square exponential dichotomy (NMS-ED) on $[t_{0},\,+\infty )$, $(-\infty,\,t_{0}]$ and the whole ${\Bbb R}$ separately, in the sense that such an NMS-ED persists under a sufficiently small linear perturbation. The result for the nonuniform mean-square exponential contraction is also discussed. Moreover, in the process of proving the existence of NMS-ED, we use the observation that the projections of the ‘exponential growing solutions’ and the ‘exponential decaying solutions’ on $[t_{0},\,+\infty )$, $(-\infty,\,t_{0}]$ and ${\Bbb R}$ are different but related. Thus, the relations of three types of projections on $[t_{0},\,+\infty )$, $(-\infty,\,t_{0}]$ and ${\Bbb R}$ are discussed.

We discuss how countable subadditivity of operators can be derived from subadditivity under mild forms of continuity, and provide examples manifesting such circumstances.

We develop a Thom–Mather theory of frontals analogous to Ishikawa's theory of deformations of Legendrian singularities but at the frontal level, avoiding the use of the contact setting. In particular, we define concepts like frontal stability, versality of frontal unfoldings or frontal codimension. We prove several characterizations of stability, including a frontal Mather–Gaffney criterion, and of versality. We then define the method of reduction with which we show how to construct frontal versal unfoldings of plane curves and show how to construct stable unfoldings of corank 1 frontals with isolated instability which are not necessarily versal. We prove a frontal version of Mond's conjecture in dimension 1. Finally, we classify stable frontal multigerms and give a complete classification of corank 1 stable frontals from $\mathbb {C}^3$ to $\mathbb {C}^4$.

The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schrödinger equation

$$-\Delta u+\lambda u+V(x)u=|u|^{p-2}u,\quad u\in H^1_0(\Omega),\quad\int_\Omega u^2{\rm d}\,x=\rho^2,\quad\lambda\in\mathbb{R},$$

$\Omega =\mathbb {R}^N$

$\mathbb {R}^N\setminus \Omega$

$\rho >0$

$V\ge 0$

$V\equiv 0$

$p\in (2,2+\frac 4 N)$

$\bar u$

$V$

$\|V\|_{L^q}$

$q\ge \frac N2$

$q>1$

$N=2$

$V$

$\mathbb {R}^N\setminus \Omega$

$\bar u$

$V\equiv 0$

$\Omega =\mathbb {R}^N$

The exponential ordering is exploited in the context of nonautonomous delay systems, inducing monotone skew-product semiflows under less restrictive conditions than usual. Some dynamical concepts linked to the order, such as semiequilibria, are considered for the exponential ordering, with implications for the determination of the presence of uniform persistence or the existence of global attractors. Also, some important conclusions on the long-term dynamics and attraction are obtained for monotone and sublinear delay systems for this ordering. The results are then applied to almost periodic Nicholson systems and new conditions are given for the existence of a unique almost periodic positive solution which asymptotically attracts every other positive solution.

This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. The magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology, respectively, for finite metric spaces, first introduced by Leinster and Hepworth–Willerton. Several authors study them restricting to graphs with path metric, and some properties which are similar to the ordinary homology theory have come to light. However, the whole picture of their behaviour is still unrevealed, and it is expected that they catch some geometric properties of graphs. In this article, we show that the girth of graphs partially determines the magnitude homology, that is, the larger girth a graph has, the more homologies near the diagonal part vanish. Furthermore, applying this result to a typical random graph, we investigate how the diagonality of graphs varies statistically as the edge density increases. In particular, we show that there exists a phase transition phenomenon for the diagonality.

This paper is concerned with singular matrix difference equations of mixed order. The existence and uniqueness of initial value problems for these equations are derived, and then the classification of them is obtained with a similar classical Weyl's method by selecting a suitable quasi-difference. An equivalent characterization of this classification is given in terms of the number of linearly independent square summable solutions of the equation. The influence of off-diagonal coefficients on the classification is illustrated by two examples. In particular, two limit point criteria are established in terms of coefficients of the equation.

In this paper, we mainly prove the following conjectures of Z.-W. Sun (J. Number Theory 133 (2013), 2914–2928): let $p>2$ be a prime. If $p=x^2+3y^2$ with $x,y\in \mathbb {Z}$ and $x\equiv 1\ ({\rm {mod}}\ 3)$, then

\[ x\equiv\frac14\sum_{k=0}^{p-1}(3k+4)\frac{f_k}{2^k}\equiv\frac12\sum_{k=0}^{p-1}(3k+2)\frac{f_k}{({-}4)^k}\ ({\rm{mod}}\ p^2), \]

$p\equiv 1\pmod 3$

\[ \sum_{k=0}^{p-1}\frac{f_k}{2^k}\equiv\sum_{k=0}^{p-1}\frac{f_k}{({-}4)^k}\ ({\rm{mod}}\ p^3), \]

$f_n=\sum _{k=0}^n\binom {n}k^3$

$n$