Involutions and Trivolutions in Algebras Related to Second Duals of Group Algebras

We define a trivolution on a complex algebra $A$ as a non-zero conjugate-linear, anti-homomorphism $\tau$ on $A$, which is a generalized inverse of itself, that is, $\tau^3=\tau$. We give several characterizations of trivolutions and show with examples that they appear naturally on many Banach algebras, particularly those arising from group algebras. We give several results on the existence or non-existence of involutions on the dual of a topologically introverted space. We investigate conditions under which the dual of a topologically introverted space admits trivolutions

A Laboratory Investigation on Thermal Properties of the Opalinus Claystone

Some aspects of the thermal behavior of the Opalinus claystone are investigated through laboratory tests conducted on a new hollow cylinder triaxial apparatus specially designed for studying the thermo-hydro-mechanical behavior of very low permeable materials. Two hollow cylinder samples are first resaturated under isotropic stress state equal to the mean effective in situ one in order to minimize swelling and induced damage during the resaturation phase. Two drained heating-cooling cycles are performed on the first sample of Opalinus claystone. During the first cycle, a thermo-elasto-plastic response similar to that of plastic clays with low overconsolidation ratio is obtained. The thermal hardening of the sample is demonstrated by the quasi-reversible behavior of the sample during the second heating-cooling cycle. An undrained heating test performed on the second sample of Opalinus claystone induces an excess pore pressure in this sample. This induced pore pressure is attributed to the higher thermal expansion coefficient of pore water compared to that of the solid phase. It is shown that the excess pore pressure generated in the sample by undrained heating cannot be modeled by considering the free water thermal expansion coefficient. The thermal expansion coefficient of the Opalinus claystone water is back-analyzed from the experimental results which show a higher value than free wate

Fillet yield, proximate composition and mineral contents in Indian spiny halibut Psettodes erumei caught from the coastal waters of Bushehr (Persian Gulf)

The objective of this study was to assess fillet yield, proximate composition and mineral contents of the Indian spiny halibut (Psettodes erumei) during different seasons. Fish samples (female = 100 and male = 100) were caught from the coastal waters of Bushehr province and body weight and length were taken to predict fillet weight and yield. Large differences in the fillet yield were observed between seasons. The highest fillet yield (49.4%) was obtained in the samples collected in autumn while samples collected in spring had the lowest yield (42.1%). There was a linear relationship between fish length and fillet weight while no significant correlation was found between fillet yield and body measurements (weight and length). The fat content of Indian spiny halibut was < 1.2% throughout the sampling period. Based on the results, fish collected in all seasons except spring for fillets may lead to a higher production with no significant difference between two sexes

On the difficulty of learning chaotic dynamics with RNNs

Recurrent neural networks (RNNs) are wide-spread machine learning tools for modeling sequential and time series data. They are notoriously hard to train because their loss gradients backpropagated in time tend to saturate or diverge during training. This is known as the exploding and vanishing gradient problem. Previous solutions to this issue either built on rather complicated, purpose-engineered architectures with gated memory buffers, or - more recently - imposed constraints that ensure convergence to a fixed point or restrict (the eigenspectrum of) the recurrence matrix. Such constraints, however, convey severe limitations on the expressivity of the RNN. Essential intrinsic dynamics such as multistability or chaos are disabled. This is inherently at disaccord with the chaotic nature of many, if not most, time series encountered in nature and society. It is particularly problematic in scientific applications where one aims to reconstruct the underlying dynamical system. Here we offer a comprehensive theoretical treatment of this problem by relating the loss gradients during RNN training to the Lyapunov spectrum of RNN-generated orbits. We mathematically prove that RNNs producing stable equilibrium or cyclic behavior have bounded gradients, whereas the gradients of RNNs with chaotic dynamics always diverge. Based on these analyses and insights we suggest ways of how to optimize the training process on chaotic data according to the system's Lyapunov spectrum, regardless of the employed RNN architecture

New parton distributions in fixed flavour factorization scheme from recent deep-inelastic-scattering data

We present our QCD analysis of the proton structure function $F_2^p(x,Q^2)$ to determine the parton distributions at the next-to-leading order (NLO). The heavy quark contributions to $F_2^i(x,Q^2)$, with $i$ = $c$, $b$ have been included in the framework of the `fixed flavour number scheme' (FFNS). The results obtained in the FFNS are compared with available results such as the general-mass variable-flavour-number scheme (GM-VFNS) and other prescriptions used in global fits of PDFs. In the present QCD analysis, we use a wide range of the inclusive neutral-current deep-inelastic-scattering (NC DIS) data, including the most recent data for charm $F_2^c$, bottom $F_2^b$, longitudinal $F_L$ structure functions and also the reduced DIS cross sections $\sigma_{r,NC}^\pm$ from HERA experiments. The most recent HERMES data for proton and deuteron structure functions are also added. We take into account ZEUS neutral current $e^ \pm p$ DIS inclusive jet cross section data from HERA together with the recent Tevatron Run-II inclusive jet cross section data from CDF and D{\O}. The impact of these recent DIS data on the PDFs extracted from the global fits are studied. We present two families of PDFs, {\tt KKT12} and {\tt KKT12C}, without and with HERA `combined' data sets on $e^{\pm}p$ DIS. We find these are in good agreement with the available theoretical models.Comment: 23 pages, 26 figures and 4 tables. V3: Only few comments and references added in the replaced version, results unchanged. Code can be found at http://particles.ipm.ir/links/QCD.ht