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In this paper we derive the best constant for the following $L^{\mathrm{\infty <\#comment/>}}$-type Gagliardo-Nirenberg interpolation inequality $\Vert <\#comment/>u{\Vert <\#comment/>}_{L^{\mathrm{\infty <\#comment/>}}}\le <\#comment/>C_{q,\mathrm{\infty <\#comment/>},p}\Vert <\#comment/>u{\Vert <\#comment/>}_{L^{q+1}}^{1-<\#comment/>\mathrm{\theta <\#comment/>}}\Vert <\#comment/>\mathrm{\nabla <\#comment/>}u{\Vert <\#comment/>}_{L^p}^{\mathrm{\theta <\#comment/>}},\mathrm{\theta <\#comment/>}=\frac{pd}{dp+(p-<\#comment/>d)(q+1)},$where parameters $q$and $p$satisfy the conditions $p>d\ge <\#comment/>1$, $q\ge <\#comment/>0$. The best constant $C_{q,\mathrm{\infty <\#comment/>},p}$is given by $C_{q,\mathrm{\infty <\#comment/>},p}={\mathrm{\theta <\#comment/>}}^{-<\#comment/>\frac{\mathrm{\theta <\#comment/>}}{p}}(1-<\#comment/>\mathrm{\theta <\#comment/>}{)}^{\frac{\mathrm{\theta <\#comment/>}}{p}}{M}_c^{-<\#comment/>\frac{\mathrm{\theta <\#comment/>}}{d}},M_c≔{\int <\#comment/>}_{{\mathbb{R}}^d}{u}_{c,\mathrm{\infty <\#comment/>}}^{q+1}dx,$where $u_{c,\mathrm{\infty <\#comment/>}}$is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when $u=A{u}_{c,\mathrm{\infty <\#comment/>}}\left(\mathrm{\lambda <\#comment/>}\right(x-<\#comment/>x_0\left)\right)$for any real numbers $A$, $\mathrm{\lambda <\#comment/>}>0$and $x_0\in <\#comment/>{\mathbb{R}}^d$. In fact, the generalized Lane-Emden equation in ${\mathbb{R}}^d$contains a delta function as a source and it is a Thomas-Fermi type equation. For $q=0$or $d=1$, $u_{c,\mathrm{\infty <\#comment/>}}$have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show that $u_{c,m}\to <\#comment/>u_{c,\mathrm{\infty <\#comment/>}}$and $C_{q,m,p}\to <\#comment/>C_{q,\mathrm{\infty <\#comment/>},p}$as $m\to <\#comment/>+\mathrm{\infty <\#comment/>}$for $d=1$, where $u_{c,m}$and $C_{q,m,p}$are the function achieving equality and the best constant of $L^m$-type Gagliardo-Nirenberg interpolation inequality, respectively. 

Abstract The stone marten (Martes foina) is an important species for cytogenetic studies in the order Carnivora. ZooFISH probes created from its chromosomes provided a strong and clean signal in chromosome painting experiments and were valuable for studying the evolution of carnivoran genome architecture. The research revealed that the stone marten chromosome set is similar to the presumed ancestral karyotype of the Carnivora, which added an additional value for the species. Using linked-read and Hi-C sequencing, we generated a chromosome-length genome assembly of a male stone marten (Gansu province, China) from a primary cell line. The stone marten assembly had a length of 2.42 Gbp, scaffold N50 of 144 Mbp, and a 96.2% BUSCO completeness score. We identified 19 chromosomal scaffolds (2n = 38) and assigned them chromosome ids based on chromosome painting data. Annotation identified 20,087 protein-coding gene models, of which 18,283 were assigned common names. Comparison of the stone marten assembly with the cat, dog, and human genomes revealed several small syntenic blocks absent on the published painting maps. Finally, we assessed the heterozygosity and its distribution over the chromosomes. The detected low heterozygosity level (0.4 hetSNPs/kbp) and the presence of long runs of homozygosity require further research and a new evaluation of the conservation status of the stone marten in China. Combined with available carnivoran genomes in large-scale synteny analysis, the stone marten genome will highlight new features and events in carnivoran evolution, hidden from cytogenetic approaches. 

The ruminants are one of the most successful mammalian lineages, exhibiting morphological and habitat diversity and containing several key livestock species. To better understand their evolution, we generated and analyzed de novo assembled genomes of 44 ruminant species, representing all six Ruminantia families. We used these genomes to create a time-calibrated phylogeny to resolve topological controversies, overcoming the challenges of incomplete lineage sorting. Population dynamic analyses show that population declines commenced between 100,000 and 50,000 years ago, which is concomitant with expansion in human populations. We also reveal genes and regulatory elements that possibly contribute to the evolution of the digestive system, cranial appendages, immune system, metabolism, body size, cursorial locomotion, and dentition of the ruminants.