Expression of a hypomorphic Pomc allele alters leptin dynamics during late pregnancy

Proopiomelanocortin (POMC) neurons in the hypothalamic arcuate nucleus (ARC) are essential for normal energy homeostasis. Maximal ARC Pomc transcription is dependent on neuronal Pomc enhancer 1 (nPE1), located 12 kb upstream from the promoter. Selective deletion of nPE1 in mice decreases ARC Pomc expression by 70%, sufficient to induce mild obesity. Because nPE1 is located exclusively in the genomes of placental mammals, we questioned whether its hypomorphic mutation would also alter placental Pomc expression and the metabolic adaptations associated with pregnancy and lactation. We assessed placental development, pup growth, circulating leptin and expression of Pomc, Agrp and alternatively spliced leptin receptor (LepR) isoforms in the ARC and placenta of Pomc∆1/∆1 and Pomc+/+ dams. Despite indistinguishable body weights, lean mass, food intake, placental histology and Pomc expression and overall pregnancy outcomes between the genotypes, Pomc ∆1/∆1 females had increased pre-pregnancy fat mass that paradoxically decreased to control levels by parturition. However, Pomc∆1/∆1 dams had exaggerated increases in circulating leptin, up to twice of that of the typically elevated levels in Pomc+/+ mice at the end of pregnancy, despite their equivalent fat mass. Pomc∆1/∆1dams also had increased placental expression of soluble leptin receptor (LepRe), although the protein levels of LEPRE in circulation were the same as Pomc+/+ controls. Together, these data suggest that the hypomorphic Pomc∆1/∆1 allele is responsible for the perinatal super hyperleptinemia of Pomc∆1/∆1 dams, possibly due to upregulated leptin secretion from individual adipocytes.

Maximal Arcs, Codes, and New Links Between Projective Planes of Order 16

In this paper we consider binary linear codes spanned by incidence matrices of Steiner 2-designs associated with maximal arcs in projective planes of even order, and their dual codes. Upper and lower bounds on the 2-rank of the incidence matrices are derived. A lower bound on the minimum distance of the dual codes is proved, and it is shown that the bound is achieved if and only if the related maximal arc contains a hyperoval of the plane. The binary linear codes of length 52 spanned by the incidence matrices of 2-$(52,4,1)$ designs associated with previously known and some newly found maximal arcs of degree 4 in projective planes of order 16 are analyzed and classified up to equivalence. The classification shows that some designs associated with maximal arcs in nonisomorphic planes generate equivalent codes. This phenomenon establishes new links between several of the known planes. A conjecture concerning the codes of maximal arcs in $PG(2,2^m)$ is formulated.

Extremal problems for non-periodic splines on real domain and their derivatives

We consider the Bojanov-Naidenov problem over the set $\sigma_{h,r}$ of all non-periodic splines $s$ of order $r$ and minimal defect with knots at the points $kh$, $k \in \mathbb{Z}$. More exactly, for given $n, r \in \mathbb{N}$; $p, A > 0$ and any fixed interval $[a, b] \subset \mathbb{R}$ we solve the following extremal problem $\int\limits_a^b |x(t)|^q dt \rightarrow \sup$, $q \geqslant p$, over the classes $\sigma_{h,r}^p(A) := \bigl\{ s(\cdot + \tau) \colon s \in \sigma_{h,r}, \| s \|_{p, \delta} \leqslant A \| \varphi_{\lambda, r} \|_{p, \delta}, \delta \in (0, h], \tau \in \mathbb{R} \bigr\}$, where $\| x \|_{p, \delta} := \sup \bigl\{ \| x \|_{L_p[a,b]} \colon a, b \in \mathbb{R}, 0 < b - a \leqslant \delta \bigr\}$, and $\varphi_{\lambda, r}$ is $(2\pi / \lambda)$-periodic spline of Euler of order $r$. In particularly, for $k = 1, ..., r - 1$ we solve the extremal problem $\int\limits_a^b |x^{(k)}(t)|^q dt \rightarrow \sup$, $q \geqslant 1$, over the classes $\sigma_{h,r}^p (A)$. Note that the problems (1) and (2) were solved earlier on the classes $\sigma_{h,r}(A, p) := \bigl\{ s(\cdot + \tau) \colon s \in \sigma_{h,r}, L(s)_p \leqslant AL(\varphi_{n,r})_p, \tau \in \mathbb{R} \bigr\}$, where $L(x)_p := \sup \bigl\{ \| x \|_{L_p[a, b]} \colon a, b \in \mathbb{R}, |x(t)| > 0, t \in (a, b) \bigr\}$. We prove that the classes $\sigma_{h,r}^p (A)$ are wider than the classes $\sigma_{h,r}(A,p)$. Similarly we solve the analog of Erdös problem about the characterisation of the spline $s \in \sigma_{h,r}^p(A)$ that has maximal arc length over fixed interval $[a, b] \subset \mathbb{R}$.

On Sets with Few Intersection Numbers in Finite Projective and Affine Spaces

In this paper we study sets $X$ of points of both affine and projective spaces over the Galois field $\mathop{\rm{GF}}(q)$ such that every line of the geometry that is neither contained in $X$ nor disjoint from $X$ meets the set $X$ in a constant number of points and we determine all such sets. This study has its main motivation in connection with a recent study of neighbour transitive codes in Johnson graphs by Liebler and Praeger [Designs, Codes and Crypt., 2014]. We prove that, up to complements, in $\mathop{\rm{PG}}(n,q)$ such a set $X$ is either a subspace or $n=2,q$ is even and $X$ is a maximal arc of degree $m$. In $\mathop{\rm{AG}}(n,q)$ we show that $X$ is either the union of parallel hyperplanes or a cylinder with base a maximal arc of degree $m$ (or the complement of a maximal arc) or a cylinder with base the projection of a quadric. Finally we show that in the affine case there are examples (different from subspaces or their complements) in $\mathop{\rm{AG}}(n,4)$ and in $\mathop{\rm{AG}}(n,16)$ giving new neighbour transitive codes in Johnson graphs.

On analogue of one problem of Erdös for non-periodic splines on the real domain

We solve the analog of some problem of Erdös about the characterization of the non-periodic spline of order r and of minimal defect, with knots at the points $kh$, $k\in \mathbb{Z}$ and fixed uniform norm that has maximal arc lens over any fixed interval.

Large Incidence-free Sets in Geometries

Let $\Pi = (P,L,I)$ denote a rank two geometry. In this paper, we are interested in the largest value of $|X||Y|$ where $X \subset P$ and $Y \subset L$ are sets such that $(X \times Y) \cap I = \emptyset$. Let $\alpha(\Pi)$ denote this value. We concentrate on the case where $P$ is the point set of $\mathsf{PG}(n,q)$ and $L$ is the set of $k$-spaces in $\mathsf{PG}(n,q)$. In the case that $\Pi$ is the projective plane $\mathsf{PG}(2,q)$, where $P$ is the set of points and $L$ is the set of lines of the projective plane, Haemers proved that maximal arcs in projective planes together with the set of lines not intersecting the maximal arc determine $\alpha(\mathsf{PG}(2,q))$ when $q$ is an even power of $2$. Therefore, in those cases,\[ \alpha(\Pi) = q(q - \sqrt{q} + 1)^2.\] We give both a short combinatorial proof and a linear algebraic proof of this result, and consider the analogous problem in generalized polygons. More generally, if $P$ is the point set of $\mathsf{PG}(n,q)$ and $L$ is the set of $k$-spaces in $\mathsf{PG}(n,q)$, where $1 \leq k \leq n - 1$, and $\Pi_q = (P,L,I)$, then we show as $q \rightarrow \infty$ that \[ \frac{1}{4}q^{(k + 2)(n - k)} \lesssim \alpha(\Pi) \lesssim q^{(k + 2)(n - k)}.\] The upper bounds are proved by combinatorial and spectral techniques. This leaves the open question as to the smallest possible value of $\alpha(\Pi)$ for each value of $k$. We prove that if for each $N \in \mathbb N$, $\Pi_N$ is a partial linear space with $N$ points and $N$ lines, then $\alpha(\Pi_N) \gtrsim \frac{1}{e}N^{3/2}$ as $N \rightarrow \infty$.