The founding emperor of the Han envisioned the noble rank (lie hou) as a system rewarding “merit” (gong) that mainly referred to military achievements. However, the criteria for granting the noble rank changed considerably throughout the Han. This is reflected by the various categories of nobles in the Han shu tables: meritorious ministers (gongchen hou), the kings’ sons (wangzi hou), and the imperial affines and favorites (waiqi enze hou), as well as the new category of eunuch nobles (huanzhe hou) in the Eastern Han. This article argues that the Han shu tables should be read as one of the multiple narratives about the noble rank and merit during the Han rather than an objective statistical summary. Whereas the Han shu tables emphasize Gaozu’s original definition of merit, the imperial edicts granting the noble rank kept reinterpreting merit to serve the court’s contemporary needs. Recently excavated Han manuscripts provide a third way of viewing merit based on the length of service.

Let $E/\mathbb {Q}(T)$ be a nonisotrivial elliptic curve of rank r. A theorem due to Silverman [‘Heights and the specialization map for families of abelian varieties’, J. reine angew. Math. 342 (1983), 197–211] implies that the rank $r_t$ of the specialisation $E_t/\mathbb {Q}$ is at least r for all but finitely many $t \in \mathbb {Q}$. Moreover, it is conjectured that $r_t \leq r+2$, except for a set of density $0$. When $E/\mathbb {Q}(T)$ has a torsion point of order $2$, under an assumption on the discriminant of a Weierstrass equation for $E/\mathbb {Q}(T)$, we produce an upper bound for $r_t$ that is valid for infinitely many t. We also present two examples of nonisotrivial elliptic curves $E/\mathbb {Q}(T)$ such that $r_t \leq r+1$ for infinitely many t.

A remarkable difference between the concept of rank for matrices and that for three-way arrays has to do with the occurrence of non-maximal rank. The set of n × n matrices that have a rank less than n has zero volume. Kruskal pointed out that a 2 × 2 × 2 array has rank three or less, and that the subsets of those 2 × 2 × 2 arrays for which the rank is two or three both have positive volume. These subsets can be distinguished by the roots of a certain polynomial. The present paper generalizes Kruskal's results to 2 × n × n arrays. Incidentally, it is shown that two n × n matrices can be diagonalized simultaneously with positive probability.

An assertion that the parameters of a covariance structure are locally identified at a certain point only if the rank of the Jacobian matrix at that point equals the number of parameters, is shown to be false by means of a counterexample.

A weaker generalized inverse (Rao's g-inverse; Graybill's c-inverse) can be used in place of the Moore-Penrose generalized inverse to obtain multiple and canonical correlations from singular covariance matrices.

We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. The result covers both matrices over finite fields with independent non-zero entries and $\{0,1\}$-matrices over the rationals. The sufficient condition is generally necessary as well.

We introduce a family of local ranks $D_Q$ depending on a finite set Q of pairs of the form $(\varphi (x,y),q(y)),$ where $\varphi (x,y)$ is a formula and $q(y)$ is a global type. We prove that in any NSOP$_1$ theory these ranks satisfy some desirable properties; in particular, $D_Q(x=x)<\omega $ for any finite tuple of variables x and any Q, if $q\supseteq p$ is a Kim-forking extension of types, then $D_Q(q)<D_Q(p)$ for some Q, and if $q\supseteq p$ is a Kim-non-forking extension, then $D_Q(q)=D_Q(p)$ for every Q that involves only invariant types whose Morley powers are -stationary. We give natural examples of families of invariant types satisfying this property in some NSOP$_1$ theories.

We also answer a question of Granger about equivalence of dividing and dividing finitely in the theory $T_\infty $ of vector spaces with a generic bilinear form. We conclude that forking equals dividing in $T_\infty $, strengthening an earlier observation that $T_\infty $ satisfies the existence axiom for forking independence.

Finally, we slightly modify our definitions and go beyond NSOP$_1$ to find out that our local ranks are bounded by the well-known ranks: the inp-rank (burden), and hence, in particular, by the dp-rank. Therefore, our local ranks are finite provided that the dp-rank is finite, for example, if T is dp-minimal. Hence, our notion of rank identifies a non-trivial class of theories containing all NSOP$_1$ and NTP$_2$ theories.

Bogomolov and Tschinkel [‘Algebraic varieties over small fields’, Diophantine Geometry, U. Zannier (ed.), CRM Series, 4 (Scuola Normale Superiore di Pisa, Pisa, 2007), 73–91] proved that, given two complex elliptic curves $E_1$ and $E_2$ along with even degree-$2$ maps $\pi _j\colon E_j\to \mathbb {P}^1$ having different branch loci, the intersection of the image of the torsion points of $E_1$ and $E_2$ under their respective $\pi _j$ is finite. They conjectured (also in works with Fu) that the cardinality of this intersection is uniformly bounded independently of the elliptic curves. The recent proof of the uniform Manin–Mumford conjecture implies a full solution of the Bogomolov–Fu–Tschinkel conjecture. In this paper, we prove a generalisation of the Bogomolov–Fu–Tschinkel conjecture whereby, instead of even degree-$2$ maps, one can use any rational functions of bounded degree on the elliptic curves as long as they have different branch loci. Our approach combines Nevanlinna theory with the uniform Manin–Mumford conjecture. With similar techniques, we also prove a result on lower bounds for ranks of elliptic curves over number fields.

When looking at others, primates primarily focus on the face – detecting the face first and looking at it longer than other parts of the body. This is because primate faces, even without expression, convey trait information crucial for navigating social relationships. Recent studies on primates, including humans, have linked facial features, specifically facial width-to-height ratio (fWHR), to rank and Dominance-related personality traits, suggesting these links’ potential role in social decisions. However, studies on the association between dominance and fWHR report contradictory results in humans and variable patterns in nonhuman primates. It is also not clear whether and how nonhuman primates perceive different facial cues to personality traits and whether these may have evolved as social signals. This review summarises the variable facial-personality links, their underlying proximate and evolutionary mechanisms and their perception across primates. We emphasise the importance of employing comparative research, including various primate species and human populations, to disentangle phylogeny from socio-ecological drivers and to understand the selection pressures driving the facial-personality links in humans. Finally, we encourage researchers to move away from single facial measures and towards holistic measures and to complement perception studies using neuroscientific methods.

The use of gamification to motivate engagement has greatly increased the number of ways in which people compete. Many of these competitions allow individuals to see how they rank as a competition progresses. Our work aims to provide a better understanding of how individuals feel about different rank outcomes in competitions. We do this by applying the principles of expected utility theory to elicit utility curves for over 3,000 people across three studies using hypothetical competition scenarios. We find consistent support for the following generalizations: 1) individuals are risk-seeking when in second place, 2) they are risk-averse when in second-to-last place, and 3) the utility decrease going from first to second place is greater than their decrease going from second-to-last to last place. Our results suggest individuals are both last-place averse and first-place seeking, with an even stronger inclination towards the latter.

For a nonconstant elliptic surface over $\mathbb {P}^1$ defined over $\mathbb {Q}$, it is a result of Silverman [‘Heights and the specialization map for families of abelian varieties’, J. reine angew. Math. 342 (1983), 197–211] that the Mordell–Weil rank of the fibres is at least the rank of the group of sections, up to finitely many fibres. If the elliptic surface is nonisotrivial, one expects that this bound is an equality for infinitely many fibres, although no example is known unconditionally. Under the Bunyakovsky conjecture, such an example has been constructed by Neumann [‘Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten. I’, Math. Nachr. 49 (1971), 107–123] and Setzer [‘Elliptic curves of prime conductor’, J. Lond. Math. Soc. (2) 10 (1975), 367–378]. In this note, we show that the Legendre elliptic surface has the desired property, conditional on the existence of infinitely many Mersenne primes.