On the Asymptotic Existence of Hadamard Matrices

It is conjectured that Hadamard matrices exist for all orders $4t$ ($t>0$). However, despite a sustained effort over more than five decades, the strongest overall existence results are asymptotic results of the form: for all odd natural numbers $k$, there is a Hadamard matrix of order $k2^{[a+b\log_2k]}$, where $a$ and $b$ are fixed non-negative constants. To prove the Hadamard Conjecture, it is sufficient to show that we may take $a=2$ and $b=0$. Since Seberry's ground-breaking result, which showed that we may take $a=0$ and $b=2$, there have been several improvements where $b$ has been by stages reduced to 3/8. In this paper, we show that for all $\epsilon>0$, the set of odd numbers $k$ for which there is a Hadamard matrix of order $k2^{2+[\epsilon\log_2k]}$ has positive density in the set of natural numbers. The proof adapts a number-theoretic argument of Erdos and Odlyzko to show that there are enough Paley Hadamard matrices to give the result.Comment: Keywords: Hadamard matrices, Asymptotic existence, Cocyclic Hadamard matrices, Relative difference sets, Riesel numbers, Extended Riemann hypothesis. (Received 2 August 2008, Available online 18 March 2009

Trades in complex Hadamard matrices

A trade in a complex Hadamard matrix is a set of entries which can be changed to obtain a different complex Hadamard matrix. We show that in a real Hadamard matrix of order $n$ all trades contain at least $n$ entries. We call a trade rectangular if it consists of a submatrix that can be multiplied by some scalar $c \neq 1$ to obtain another complex Hadamard matrix. We give a characterisation of rectangular trades in complex Hadamard matrices of order $n$ and show that they all contain at least $n$ entries. We conjecture that all trades in complex Hadamard matrices contain at least $n$ entries.Comment: 9 pages, no figure

Pathogenic Variants in Fucokinase Cause a Congenital Disorder of Glycosylation

FUK encodes fucokinase, the only enzyme capable of converting L-fucose to fucose-1-phosphate, which will ultimately be used for synthesizing GDP-fucose, the donor substrate for all fucosyltransferases. Although it is essential for fucose salvage, this pathway is thought to make only a minor contribution to the total amount of GDP-fucose. A second pathway, the major de novo pathway, involves conversion of GDP-mannose to GDP-fucose. Here we describe two unrelated individuals who have pathogenic variants in FUK and who presented with severe developmental delays, encephalopathy, intractable seizures, and hypotonia. The first individual was compound heterozygous for c.667T>C (p.Ser223Pro) and c.2047C>T (p.Arg683Cys), and the second individual was homozygous for c.2980A>C (p.Lys994Gln). Skin fibroblasts from the first individual confirmed the variants as loss of function and showed significant decreases in total GDP-[3H] fucose and [3H] fucose-1-phosphate. There was also a decrease in the incorporation of [5,6-3H]-fucose into fucosylated glycoproteins. Lys994 has previously been shown to be an important site for ubiquitin conjugation. Here, we show that loss-of-function variants in FUK cause a congenital glycosylation disorder characterized by a defective fucose-salvage pathway

Preassociative aggregation functions

The classical property of associativity is very often considered in aggregation function theory and fuzzy logic. In this paper we provide axiomatizations of various classes of preassociative functions, where preassociativity is a generalization of associativity recently introduced by the authors. These axiomatizations are based on existing characterizations of some noteworthy classes of associative operations, such as the class of Acz\'elian semigroups and the class of t-norms.Comment: arXiv admin note: text overlap with arXiv:1309.730

Exotic complex Hadamard matrices, and their equivalence

In this paper we use a design theoretical approach to construct new, previously unknown complex Hadamard matrices. Our methods generalize and extend the earlier results of de la Harpe--Jones and Munemasa--Watatani and offer a theoretical explanation for the existence of some sporadic examples of complex Hadamard matrices in the existing literature. As it is increasingly difficult to distinguish inequivalent matrices from each other, we propose a new invariant, the fingerprint of complex Hadamard matrices. As a side result, we refute a conjecture of Koukouvinos et al. on (n-8)x(n-8) minors of real Hadamard matrices.Comment: 10 pages. To appear in Cryptography and Communications: Discrete Structures, Boolean Functions and Sequence

TSPO interacts with VDAC1 and triggers a ROS-mediated inhibition of mitochondrial quality control

The 18-kDa TSPO (translocator protein) localizes on the outer mitochondrial membrane (OMM) and participates in cholesterol transport. Here, we report that TSPO inhibits mitochondrial autophagy downstream of the PINK1-PARK2 pathway, preventing essential ubiquitination of proteins. TSPO abolishes mitochondrial relocation of SQSTM1/p62 (sequestosome 1), and consequently that of the autophagic marker LC3 (microtubule-associated protein 1 light chain 3), thus leading to an accumulation of dysfunctional mitochondria, altering the appearance of the network. Independent of cholesterol regulation, the modulation of mitophagy by TSPO is instead dependent on VDAC1 (voltage-dependent anion channel 1), to which TSPO binds, reducing mitochondrial coupling and promoting an overproduction of reactive oxygen species (ROS) that counteracts PARK2-mediated ubiquitination of proteins. These data identify TSPO as a novel element in the regulation of mitochondrial quality control by autophagy, and demonstrate the importance for cell homeostasis of its expression ratio with VDAC1

Cloning and expression of a mammalian peptide chain release factor with sequence similarity to tryptophanyl-tRNA synthetases

The termination of protein synthesis is encoded by in-frame nonsense (stop) codons. Most organisms use three nonsense codons: UGA, UAG, and UAA. In contrast to sense codons, which are decoded by specific tRNAs, nonsense codons are decoded by proteins called release factors (RFs). Here we report the cloning of a mammalian RF cDNA by the use of monoclonal antibodies specific for rabbit RF. Functional studies showed that, when expressed in Escherichia coli, the protein encoded by this cDNA has in vitro biochemical characteristics similar to those of previously characterized mammalian RFs. DNA sequencing of this eukaryotic RF cDNA revealed a remarkable sequence similarity to bacterial and mitochondrial tryptophanyl-tRNA synthetases, with the greatest similarity confined to the synthetase active site, and no obvious similarity to bacterial RFs

All Teleportation and Dense Coding Schemes

We establish a one-to-one correspondence between (1) quantum teleportation schemes, (2) dense coding schemes, (3) orthonormal bases of maximally entangled vectors, (4) orthonormal bases of unitary operators with respect to the Hilbert-Schmidt scalar product, and (5) depolarizing operations, whose Kraus operators can be chosen to be unitary. The teleportation and dense coding schemes are assumed to be ``tight'' in the sense that all Hilbert spaces involved have the same finite dimension d, and the classical channel involved distinguishes d^2 signals. A general construction procedure for orthonormal bases of unitaries, involving Latin Squares and complex Hadamard Matrices is also presented.Comment: 21 pages, LaTe