The distribution of divisors of polynomials

Let $F(x)$ be an irreducible polynomial with integer coefficients and degree at least 2. For $x\ge z\ge y\ge 2$, denote by $H_F(x, y, z)$ the number of integers $n\le x$ such that $F(n)$ has at least one divisor $d$ with $y<d\le z$. We determine the order of magnitude of $H_F(x, y, z)$ uniformly for $y+y/\log^C y < z\le y^2$ and $y\le x^{1-\delta}$, showing that the order is the same as the order of $H(x,y,z)$, the number of positive integers $n\le x$ with a divisor in $(y,z]$. Here $C$ is an arbitrarily large constant and $\delta>0$ is arbitrarily small.Comment: v2. minor edits and correction

Sumoylation regulates lamin A function and is lost in lamin A mutants associated with familial cardiomyopathies

Lamin A mutations cause many diseases, including cardiomyopathies and Progeria Syndrome. The covalent attachment of small ubiquitin-like modifier (SUMO) polypeptides regulates the function of many proteins. Until now, no examples of human disease-causing mutations that occur within a sumoylation consensus sequence and alter sumoylation were known. We show that lamin A is sumoylated at lysine 201 and that two lamin A mutants associated with familial dilated cardiomyopathy, E203G and E203K, exhibit decreased sumoylation. E203 occupies the conserved +2 position in the sumoylation consensus ΨKXE. Lamin A mutants E203G, E203K, and K201R all exhibit a similar aberrant subcellular localization and are associated with increased cell death. Fibroblasts from an individual with the E203K lamin A mutation also exhibit decreased lamin A sumoylation and increased cell death. These results suggest that SUMO modification is important for normal lamin A function and implicate an involvement for altered sumoylation in the E203G/E203K lamin A cardiomyopathies

H\u3csub\u3e3\u3c/sub\u3ePO\u3csub\u3e4\u3c/sub\u3e-Imbibed Three-Dimensional Polyacrylamide/Polyacrylamide Hydrogel as a High-Temperature Proton Exchange Membrane with Excellent Acid Retention

We herein report the use of polyacrylamide/polyacrylamide interpenetrating polymer network (PAM/PAM IPN) hydrogel as a matrix to imbibe proton conducting H3PO4, forming a robust proton exchange membrane (PEM) suitable for high-temperature PEM fuel cells by combining excellent acid retention, simple synthesis, and low cost. Its extraordinary ability to absorb large quantity of aqueous solution is fully utilized to achieve high H3PO4 loading, showing a proton conductivity of 0.0833 S cm−1 at 183 °C in dry air. The synthesized membrane also shows excellent acid retention even under mechanical load and high humidity. These profound advantages along with simple and low-cost synthesis promise the new membrane to be a strong candidate as a high-temperature PEM

Large dynamic range SPR measurements in the visible using a ZnSe prism

Large dynamic index measurement range (n = 1 to n = 1.7) using surface plasmon resonance (SPR) shifts is demonstrated with a ZnSe prism at 632.8 nm, limited by the available high index liquid hosts. In contrast to borosilicate based SPR measurements where angular limitations restrict solvent use to water and requires considerable care dealing with Fresnel reflections, the ZnSe approach allows SPR spectroscopies to be applied to a varied range of solvents An uncertainty in angular resolution between 1.5 and 6 deg, depending on the solvent and SPR angle, was estimated. The refractive index change for a given glucose concentration in water was measured to be n = (0.114 to 0.007) per precentage C6H12O6 conc. Given the transmission properties of ZnSe the processes can be readily extended into the mid infrared.Comment: Submitted to Opt Let

Hydrophobic Hydrogel Caged H\u3csub\u3e3\u3c/sub\u3ePO\u3csub\u3e4\u3c/sub\u3e as a New Class of High-Temperature Proton Exchange Membranes with Enhanced Acid Retention

We herein report a new class of high-temperature proton exchange membranes comprised of poly(acrylic acid-graft-hexadecyltrimethylammonium bromide) (PAA-g-CTAB) or poly(acrylic acid)-graft-poly(ethylene glycol) (PAA-g-PEG) hydrophobic hydrogel caged H3PO4. The membranes exhibit reasonable proton conductivity, enhanced H3PO4 retention ability and low solubility in water, making them promising as potential high performance and robust electrolytes for high-temperature proton exchange membrane fuel cells. Although the proton conductivity is still lower than that of H3PO4 doped PBI membranes, the new concept provides a different approach to proton exchange membranes for acid retention

Approaches to improving consistency of interval fuzzy preference relations

This article introduces a consistency index for measuring the consistency level of an interval fuzzy preference relation (IFPR). An approach is then proposed to construct an additive consistent IFPR from a given inconsistent IFPR. By using a weighted averaging method combining the original IFPR and the constructed consistent IFPR, a formula is put forward to repair an inconsistent IFPR to generate an IFPR with acceptable consistency. An iterative algorithm is subsequently developed to rectify an inconsistent IFPR and derive one with acceptable consistency and weak transitivity. The proposed approaches can not only improve consistency of IFPRs but also preserve the initial interval uncertainty information as much as possible. Numerical examples are presented to illustrate how to apply the proposed approaches

U-type bilateral sacral fracture with spino-pelvic dissociation caused by epileptic seizure

Sacral fracture with spino-pelvic dissociation is a rare and unstable injury caused by high-energy trauma, often with serious haemodynamic and neurological implications. Diagnosis is easily delayed or missed as it is often masked by severe associated injuries. Here, we present an unusual case of spino-pelvic dissociation sustained during a seizure episode in a young epileptic patient on long-term anticonvulsant therapy with previous thoracolumbar spinal arthrodesis. This unique case brings to light the need for clinicians to consider sacral fractures in patients presenting with low back pain with no preceding trauma who otherwise may have risk factors for pathological fractures