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We establish two nonlinear compactness theorems in Lp(0,T;B) with hypothesis on time translations, which are nonlinear counterparts of two results by Simon (1987) [1]. The first theorem sharpens a result by Maitre (2003) [10] and is important in the study of doubly nonlinear elliptic–parabolic equations. Based on this theorem, we then obtain a time translation counterpart of a result by Dubinskii˘ (1965) [5], which is supposed to be useful in the study of some nonlinear kinetic equations (e.g. the FENE-type bead–spring chains model).

In 1975, P. Erd\"{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq n+36t$$ for $t=1260r+169 \,\ (r\geq 1)$ and $n \geq 540t^{2}+\frac{175811}{2}t+\frac{7989}{2}$. Consequently, $\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + {2 \over 5}},$ which is better than the previous bounds $\sqrt 2$ (see [2]), $\sqrt {2+{2562\over 6911}}$ (see [7]).

Recent work explores the candidate phases of the 4d adjoint quantum chromodynamics (QCD$_4$) with an SU(2) gauge group and two massless adjoint Weyl fermions. Both Cordova-Dumitrescu and Bi-Senthil propose possible low energy 4d topological quantum field theories (TQFTs) to saturate the higher 't Hooft anomalies of adjoint QCD$_4$ under a renormalization-group (RG) flow from high energy. In this work, we generalize the symmetry-extension method [arXiv:1705.06728] to higher symmetries, and formulate higher group cohomology and cobordism theory approach to construct higher-symmetric TQFTs. We prove that the symmetry-extension method saturates certain anomalies, but also prove that neither $A \mathcal{P}_2(B_2)$ nor $\mathcal{P}_2(B_2)$ can be fully trivialized, with the background 1-form field $A$, Pontryagin square $\mathcal{P}_2$ and 2-form field $B_2$. Surprisingly, this indicates an obstruction to constructing a fully 1-form center and 0-form chiral symmetry (full discrete axial symmetry) preserving 4d TQFT with confinement, a no-go scenario via symmetry-extension for specific higher anomalies. We comment on the implications and constraints on deconfined quantum critical points (dQCP), quantum spin liquids (QSL) or quantum fermionic liquids in condensed matter, and ultraviolet-infrared (UV-IR) duality in 3+1 spacetime dimensions.