Ten ice-sheet models are used to study sensitivity of the Greenland and Antarctic ice sheets to prescribed changes of surface mass balance, sub-ice-shelf melting and basal sliding. Results exhibit a large range in projected contributions to sea-level change. In most cases, the ice volume above flotation lost is linearly dependent on the strength of the forcing. Combinations of forcings can be closely approximated by linearly summing the contributions from single forcing experiments, suggesting that nonlinear feedbacks are modest. Our models indicate that Greenland is more sensitive than Antarctica to likely atmospheric changes in temperature and precipitation, while Antarctica is more sensitive to increased ice-shelf basal melting. An experiment approximating the Intergovernmental Panel on Climate Change’s RCP8.5 scenario produces additional first-century contributions to sea level of 22.3 and 8.1 cm from Greenland and Antarctica, respectively, with a range among models of 62 and 14 cm, respectively. By 200 years, projections increase to 53.2 and 26.7 cm, respectively, with ranges of 79 and 43 cm. Linear interpolation of the sensitivity results closely approximates these projections, revealing the relative contributions of the individual forcings on the combined volume change and suggesting that total ice-sheet response to complicated forcings over 200 years can be linearized.

Microindentation is performed on hot isostatic pressed (HIP) Mg-Al (AM40) alloy samples produced by high-pressure die cast (HPDC) process for the purpose of quantifying the mechanical properties of the α-Mg grains. The process of obtaining elastic modulus and hardness from indentation load-depth curves is well established in the literature. A new inverse method is developed to extract plastic properties in this study. The method utilizes empirical yield strength-hardness relationship reported in the literature together with finite element modeling of the individual indentation. Due to the shallow depth of the indentation, indentation size effect (ISE) is taken into account when determining plastic properties. The stress versus strain behavior is determined for a series of indents. The resulting average values and standard deviations are obtained for future use as input distributions for microstructure-based property prediction of AM40.

We investigate the numbers of complex zeros of Littlewood polynomials $p\left( z \right)$ (polynomials with coefficients {−1, 1}) inside or on the unit circle $\left| z \right|\,=\,1$, denoted by $N\left( p \right)$ and $U\left( p \right)$, respectively. Two types of Littlewood polynomials are considered: Littlewood polynomials with one sign change in the sequence of coefficients and Littlewood polynomials with one negative coefficient. We obtain explicit formulas for $N\left( p \right)$, $U\left( p \right)$ for polynomials $p\left( z \right)$ of these types. We show that if $n\,+\,1$ is a prime number, then for each integer $k,\,0\,⩽\,k\,⩽\,n-1$, there exists a Littlewood polynomial $p\left( z \right)$ of degree $n$ with $N\left( p \right)\,=\,k$ and $U\left( p \right)\,=\,0$. Furthermore, we describe some cases where the ratios $N\left( p \right)/n$ and $U\left( p \right)/n$ have limits as $n\,\to \,\infty $ and find the corresponding limit values.

Periodic arrays of low-aspect ratio silicon nanopillars strongly reduce front surface reflection over a broad wavelength range. In this study, we numerically simulate the reflection of light for thick crystalline silicon substrates nanostructured through a combination of silica nanosphere lithography (SNL) and metal-assisted chemical etching (MaCE), producing ordered arrays of nanopillars with hexagonal periodicity. Using statistical methods, we show that the simulated measurements are in good agreement with the spectrophotometry measurements of the fabricated nanopillars.

Let $\text{ }\!\!\lambda\!\!\text{ }\left( n \right)$ denote the Liouville function. Complementary to the prime number theorem, Chowla conjectured that

*

$$\sum\limits_{n\le x}{\lambda \,\left( f\left( n \right) \right)}=o\left( x \right)$$

for any polynomial $f\left( x \right)$ with integer coefficients which is not of form $bg{{\left( x \right)}^{2}}$.

We report on novel liquid crystals with extremely large flexoelectric coefficients in a range of ultra-fast photonic modes, namely 1) the uniform lying helix, that leads to in-plain switching, birefringence phase devices with 100 μs switching times at low fields, i.e.2-5 V/μm, and analogue or grey scale capability, 2) the uniform standing helix, using planar surface alignment and in-plane fields, with sub ms response times and optical contrasts in excess of 5000:1 with a perfect optically isotropic or black “off state”, 3) the wide temperature range blue phase that leads to field controlled reflective color, 4) chiral nematic optical reflectors electric field tunable over a wide wavelength range and 5) high slope efficiency, wide wavelength range tunable narrow linewidth microscopic liquid crystal lasers.

Let ${{A}_{n}}\,=\,\{{{a}_{0}}+{{a}_{1}}z+\,.\,.\,.\,+\,{{a}_{n-1}}{{z}^{n-1}}\,:\,{{a}_{j}}\,\in \,\{0,1\}\}$ , whose elements are called zero-one polynomials and correspond naturally to the ${{2}^{n}}$ subsets of $\left[ n \right]\,:=\,\{0,\,1,\,.\,.\,.\,,\,n-1\}$. We also let ${{A}_{n,m\,}}\,=\,\{\alpha \left( z \right)\,\in \,{{A}_{n}}\,:\,\alpha \left( 1 \right)\,=\,m\}$ , whose elements correspond to the $\left( _{m}^{n} \right)$ subsets of $\left[ n \right]$ of size $m$, and let ${{B}_{n}}\,=\,{{A}_{n+1}}\,\backslash \,{{A}_{n}}$ , whose elements are the zero-one polynomials of degree exactly $n$.

Many researchers have studied norms of polynomials with restricted coefficients. Using $\|\alpha {{\|}_{p}}$ to denote the usual ${{L}_{p}}$ norm of $\alpha$ on the unit circle, one easily sees that $\alpha \left( z \right)\,=\,{{a}_{0}}+{{a}_{1}}z+.\,.\,.+{{a}_{N}}{{z}^{N}}\,\in \,\mathbb{R}\left[ z \right]$ satisfies $\|\alpha \|_{2}^{2}\,=\,{{c}_{0}}$ and $\|\alpha \|_{4}^{4}\,=\,c_{0}^{2}\,+\,2\left( c_{1}^{2}\,+\,.\,.\,.\,+\,c_{N}^{2} \right)$ , where ${{c}_{k}}\,:=\,\sum _{j=0}^{N-k}\,{{a}_{j}}{{a}_{j+k}}\,\text{for}\,\text{0}\le \,\text{k}\,\le N$.

If $\alpha \left( z \right)\,\in \,{{A}_{n,m}}$ , say $\alpha \left( z \right)\,=\,{{z}^{{{\text{ }\!\!\beta\!\!\text{ }}_{1}}}}\,+\,.\,.\,.\,+\,{{z}^{{{\text{ }\!\!\beta\!\!\text{ }}_{m}}}}$ where ${{\beta }_{1}}\,<\,.\,.\,.\,<\,{{\beta }_{m}}$ , then ${{c}_{k}}$ is the number of times $k$ appears as a difference ${{\beta }_{i}}\,-\,{{\beta }_{j}}$ . The condition that $\alpha \,\in \,{{A}_{n,m}}$ satisfies ${{c}_{k}}\,\in \,\{0,1\}$ for $1\,\le \,k\,\le \,n\,-\,1$ is thus equivalent to the condition that $\{{{\beta }_{1}},\,.\,.\,.\,,\,{{\beta }_{m}}\}$ is a Sidon set (meaning all differences of pairs of elements are distinct).

In this paper, we find the average of $\left\| \alpha \right\|_{4}^{4}$ over $\alpha \,\in \,{{A}_{n}}$, $\alpha \,\in \,{{B}_{n}}$ and $\alpha \,\in \,{{A}_{n,m}}$. We further show that our expression for the average of $\left\| \alpha \right\|_{4}^{4}$ over ${{A}_{n,m}}$ yields a new proof of the known result: if $m\,=\,o\left( {{n}^{1/4}} \right)$ and $B\left( n,\,m \right)$ denotes the number of Sidon sets of size $m$ in $\left[ n \right]$, then almost all subsets of $\left[ n \right]$ of size $m$ are Sidon, in the sense that ${{\lim }_{n\to \infty }}\,B\left( n,\,m \right)/\left( _{m}^{n} \right)\,=\,1$.

Advances in the understanding of appetite are leading to a refined concept of disease cachexia and point to novel therapeutic strategies based on the manipulation of appetite. The complex social and psychological short-term influences on appetite obscure the fact that over the longer term appetite is tightly regulated by physiological considerations; the homeostatic control of energy balance. Like obesity, which is now viewed as a disorder of homeostasis, cachexia can be seen as an adaptive response to the disease state that becomes harmful when prolonged. Several lines of evidence implicate a disorder of appetite regulation in the pathogenesis of cachexia. As the only known circulating mediator of increased appetite the peptide hormone ghrelin has attracted attention as a potential therapy. Trials in patients with various chronic illnesses, including cancer and kidney failure, have demonstrated short-term increases in energy intake. Trials in patients with emphysema and heart failure have also shown benefits in clinical outcomes such as lean body mass and exercise capacity, and longer-term trials using oral analogues are being undertaken. As well as improving nutrition, ghrelin has a number of other actions that may be useful, including an anti-inflammatory effect; of interest since many cachexias are associated with inappropriate immune activation. The manipulation of appetite, in particular by ghrelin agonism, is emerging as an exciting potential therapy for disease cachexia. Future research should focus on the ascertainment of clinically-relevant outcomes, and further characterisation of the non-nutritional effects of this pathway.

Let $q$,$m$,$M\,\ge \,2$ be positive integers and ${{r}_{1}},\,{{r}_{2}},...,\,{{r}_{m}}$ be positive rationals and consider the following $M$ multivariate infinite products

$${{F}_{i}}\,=\,\prod\limits_{j=0}^{\infty }{(1\,+\,{{q}^{-(Mj+i)}}\,{{r}_{1}}\,+\,{{q}^{-2(Mj+i)}}\,{{r}_{2}}\,+\,\cdot \cdot \cdot +\,{{q}^{-m(Mj+i)}}\,{{r}_{m}})}$$

for $i\,=\,0,\,1,\,.\,.\,.\,,\,M\,-\,1$. In this article, we study the linear independence of these infinite products. In particular, we obtain a lower bound for the dimension of the vector space $\mathbb{Q}{{F}_{0}}\,+\,\mathbb{Q}{{F}_{1}}\,+\cdot \cdot \cdot +\,\mathbb{Q}{{F}_{M-1}}\,+\,\mathbb{Q}$ over ℚ and show that among these $M$ infinite products, ${{F}_{0}}\,+\,{{F}_{1}},...,\,{{F}_{M-1}}$ , at least $\sim \,M/m\left( m+1 \right)$ of them are irrational for fixed $m$ and $M\,\to \,\infty $.

We present changes in volume, immersion density, and tensile property observed from accelerated aged plutonium alloys. Accelerated alloys (or spiked alloys) are plutonium alloys enriched with approximately 7.5 weight percent of the faster-decaying 238Pu to accelerate the aging process by approximately 17 times the rate of unaged weapons-grade plutonium. After sixty equivalent years of aging on spiked alloys, the dilatometry shows the samples at 35°C have swelled in volume by 0.15 to 0.17 % and now exhibit a near linear volume increase due to helium in-growth. The immersion density of spiked alloys shows decrease in density, similar normalized volumetric changes (expansion) for spiked alloys. Tensile tests show increasing yield and engineering ultimate strength as spiked alloys are aged.

The main result of this paper gives an explicit computation of the $L_4$ norm of any cyclotomic polynomial of the form \[ f(x)=\Phi_{p_1}(\pm x)\Phi_{p_2}(\pm x^{p_1})\cdots \Phi_{p_r}(\pm x^{p_1 p_2\cdots p_{r-1}}). \] Here $\Phi_{p}$ is the $p$th cyclotomic polynomial and the $p_i$ are primes that are not necessarily distinct.

A corollary of this is the following theorem.

THEOREM. If\[ f(x)=\Phi_{p_1}(\e_1 x)\Phi_{p_2}(\e_2 x^{p_1})\cdots \Phi_{p_r}(\e_r x^{p_1 p_2\cdots p_{r-1}}), \]where N = p1p2···prand εj = ±, then \[ \frac{\| f\|_4^4}{N^2} & \ge & \frac{\big\|\Phi_2(-x)\Phi_2(-x^2)\cdots \Phi_2\big({-}x^{2^{r-1}}\big) \big\|_4^4}{4^r}\\& = & \frac{\big(\frac12 +\frac{5}{34}\sqrt{17}\big)(1+\sqrt{17})^r - \big({-}\frac12 +\frac{5}{34}\sqrt{17}\big)(1-\sqrt{17})^r}{4^r}. \] In particular the minimum possible normalized L4 norm of any polynomial of the above form is attained by \[ \Phi_2(-x)\Phi_2(-x^2)\cdots \Phi_2\big({-}x^{2^{r-1}}\big). \]

Let ${{b}_{1}},...,{{b}_{5}}$ be non-zero integers and $n$ any integer. Suppose that ${{b}_{1}}+\cdot \cdot \cdot +{{b}_{5}}\equiv n$ (mod 24) and $\left( {{b}_{i}},{{b}_{j}} \right)=1$ for $1\le i<j\le 5$. In this paper we prove that

1. (i) if all ${{b}_{j}}$ are positive and $n\gg \max {{\left\{ \left| {{b}_{j}} \right| \right\}}^{41+\varepsilon }}$, then the quadratic equation ${{b}_{1}}p_{1}^{2}+\cdot \cdot \cdot +{{b}_{5}}p_{5}^{2}=n$ is soluble in primes ${{p}_{j}}$, and

2. (ii) if ${{b}_{j}}$ are not all of the same sign, then the above quadratic equation has prime solutions satisfying ${{p}_{j}}\ll \sqrt{\left| n \right|}+\max {{\left\{ \left| {{b}_{j}} \right| \right\}}^{20+\varepsilon }}$.

We give explicit formulas for the ${{L}_{4}}$ norm (or equivalently for the merit factors) of various sequences of polynomials related to the polynomials

$$f\left( z \right):=\,\sum\limits_{n=0}^{N-1}{\left( \frac{n}{N} \right){{z}^{n}}.}$$

and

$${{f}_{t}}(z)\,=\,\sum\limits_{n=0}^{N-1}{\left( \frac{n+t}{N} \right){{z}^{n}}.}$$

where $\left( \frac{.}{N} \right)$ is the Jacobi symbol.

Two cases of particular interest are when $N\,=\,pq$ is a product of two primes and $p\,=\,q\,+\,2$ or $p\,=\,q\,+\,4$. This extends work of Høholdt, Jensen and Jensen and of the authors.

This study arises from a number of conjectures of Erdős, Littlewood and others that concern the norms of polynomials with −1, 1 coefficients on the disc. The current best examples are of the above form when $N$ is prime and it is natural to see what happens for composite $N$.