On the asymptotic input-output weight distributions and thresholds of convolutional and turbo-like encoders

We present a general method for computing the asymptotic input-output weight distribution of convolutional encoders. In some instances, one can derive explicit analytic expressions. In general, though, to determine the growth rate of the input-output weight distribution for a particular normalized input weight /spl kappa/ and output weight /spl omega/, a system of polynomial equations has to be solved. This method is then used to determine the asymptotic weight distribution of various concatenated code ensembles and to derive lower bounds on the thresholds of these ensembles under maximum-likelihood (ML) decoding.     

Reactivity of high-valent iron-oxo species in enzymes and synthetic reagents: a tale of many states

The Account discusses the phenomenon of two-state reactivity (TSR) or multistate reactivity (MSR) in high-valent metal-oxo reagents, projecting its wide-ranging applicability starting from the bare species, through the reagents made by Que, Nam, and collaborators, to the Mn(V)-oxo substituted polyoxometalate, all the way to Compound I species of heme enzymes. The Account shows how the behaviors of all these variegated species derive from a simple set of electronic structure principles. Experimental trends that demonstrate TSR and MSR are discussed. Diagnostic mechanistic probes are proposed for the TSR/MSR scenario, based on kinetic isotope effect, stereochemical studies, and magnetic- and electric-field effects.     

A Tale of Two Mounts: The History of Chemistry at the Hebrew University of Jerusalem

This is the story of the establishment of the Hebrew University (HU), and within it, the sub-story of the first school of chemistry in the Land of Israel, from 1923. It spans more than 100 years, from the publication of the pamphlet, Eine Jüdische Hochschule, in 1902, all the way to 2015. The story starts with a dream, almost in the realm of science fiction, the audacious vision of building a university in a stateless and impoverished region, and all the way to an Institute of Chemistry of HU that has served as the mother of chemistry in the State of Israel.     

Parity-check density versus performance of binary linear block codes over memoryless symmetric channels

We derive lower bounds on the density of parity-check matrices of binary linear codes which are used over memoryless binary-input output-symmetric (MBIOS) channels. The bounds are expressed in terms of the gap between the rate of these codes for which reliable communications is achievable and the channel capacity; they are valid for every sequence of binary linear block codes if there exists a decoding algorithm under which the average bit-error probability vanishes. For every MBIOS channel, we construct a sequence of ensembles of regular low-density parity-check (LDPC) codes, so that an upper bound on the asymptotic density of their parity-check matrices scales similarly to the lower bound. The tightness of the lower bound is demonstrated for the binary erasure channel by analyzing a sequence of ensembles of right-regular LDPC codes which was introduced by Shokrollahi, and which is known to achieve the capacity of this channel. Under iterative message-passing decoding, we show that this sequence of ensembles is asymptotically optimal (in a sense to be defined in this paper), strengthening a result of Shokrollahi. Finally, we derive lower bounds on the bit-error probability and on the gap to capacity for binary linear block codes which are represented by bipartite graphs, and study their performance limitations over MBIOS channels. The latter bounds provide a quantitative measure for the number of cycles of bipartite graphs which represent good error-correction codes.     

Identity SN2 Reactions. The Relationship Between Transition State Geometry,Selectivity and the Mechanism of Solvent Action

Reactivity trends in the identity S_N2 reaction (X⁻ + CH₃X → XCH₃ + X⁻) are discussed using the state correlation diagram model. The model projects that the activation process can be defined as a distortion effort to transfer a single electron from the nucleophile to the leaving group across the alkyl group (CH₃). Three domains of reactivity are discussed (a) the geometry of the transition state (looseness and tightness), (b) the intrinsic selectivity of substrates, and (c) the mechanism of solvent action. A link is drawn between these three domains, and a physical insight is provided.     

Does Hydride Ion Transfer from Silanes to Carbenium Ions Proceed Via a Rate-Determining Formation of a Silicenium Ion or Via a Rate-Determining Electron Transfer? An Ab Initio Quantum Mechanical Study and a Curve-Crossing Analysis

The hydride transfer reactions from simple silanes to carbenium ions are studied by ab initio calculations. The simplest reaction, H₄Si + CH₃⁺ → H₃Si⁺ + CH₄, is also studied with inclusion of the solvent effect (with the SCRF method) in the ab initio scheme. Under all conditions the preferred mechanism is the synchronous hydride transfer (SHT), which is barrierless in the gas phase but possesses small barriers in solution. The mechanistic alternative involving a rate-determining single electron transfer (SET) step followed by H-atom abstraction is found to be of very high energy. Modelling of the primary isotope effect for the SHT process of H₃SiH(D) + CH₃* → H₃Si⁺ + H₃CH(D) shows that the primary isotope effect is small, between ca. 1.1 and 2.7, for the entire relevant range of Si—H(D) distances (1.5–2.3 Å). Furthermore, the pattern of the computed primary isotope effect shows it to be an insensitive probe of the SHT mechanism. The curve-crossing method is used to model the mechanistic dichotomy. It is shown that the reaction profiles for both SHT and SET arise from an avoided crossing between the ground state and a charge transfer state of the R₃SiH//R′₃C⁺ reactant pair. Thus, in the SHT mechanism a single electron switches sites in synchronicity with bond reorganization, while in SET the electron switch precedes the bond coupling. This avoided bond coupling is the foremost disadvantage of the SET mechanism. The common origin of the avoided crossing elucidates the reason why SHT exhibits characteristics of an electron transfer process without actually being a SET process.     

Bounds on the error probability ofml decoding for block and turbo-block codes

The performance of either structured or random turbo-block codes and binary, systematic block codes operating over the additive white Gaussian noise (Awgn) channel, is assessed by upper bounds on the error probalities of maximum likelihood (Ml) decoding. These bounds on the block and bit error probability which depend respectively on the distance spectrum and the input-output weight enumeration function (Iowef) of these codes, are compared, for a variety of cases, to simulated performance of iterative decoding and also to some reported simulated lower bounds on the performance ofMl decoders. The comparisons facilitate to assess the efficiency of iterative decoding (as compared to the optimalMl decoding rule) on one hand and the tightness of the examined upper bounds on the other. We focus here on uniformly interleaved and parallel concatenated turbo-Hamming codes, and to that end theIowefs of Hamming and turbo-Hamming codes are calculated by an efficient algorithm. The usefulness of the bounds is demonstrated for uniformly interleaved turbo-Hamming codes at rates exceeding the cut-off rate, where the results are compared to the simulated performance of iteratively decoded turbo-Hamming codes with structured and statistical interleavers. We consider also the ensemble performance of ‘repeat and accumulate’ (Ka) codes, a family of serially concatenated turbo-block codes, introduced by Divsalar, Jin and McEliece. Although, the outer and inner codes possess a very simple structure: a repetitive and a differential encoder respectively, our upper bounds indicate impressive performance at rates considerably beyond the cut-off rate. This is also evidenced in literature by computer simulations of the performance of iteratively decodedRa codes with a particular structured interleaver.     

On improved bounds on the decoding error probability of block codesover interleaved fading channels, with applications to turbo-like codes

We derive here improved upper bounds on the decoding error probability of block codes which are transmitted over fully interleaved Rician fading channels, coherently detected and maximum-likelihood (ML) decoded. We assume that the fading coefficients during each symbol are statistically independent (due to a perfect channel interleaver), and that perfect estimates of these fading coefficients are provided to the receiver. The improved upper bounds on the block and bit error probabilities are derived for fully interleaved fading channels with various orders of space diversity, and are found by generalizing some previously introduced upper bounds for the binary-input additive white Gaussian nose (AWGN) channel. The advantage of these bounds over the ubiquitous union bound is demonstrated for some ensembles of turbo codes and low-density parity-check (LDPC) codes, and it is especially pronounced in a portion of the rate region exceeding the cutoff rate. Our generalization of the Duman and Salehi bound (Duman and Salehi 1998, Duman 1998) which is based on certain variations of Gallager's (1965) bounding technique, is demonstrated to be the tightest reported upper bound. We therefore apply it to calculate numerically upper bounds on the thresholds of some ensembles of turbo-like codes, referring to the optimal ML decoding. For certain ensembles of uniformly interleaved turbo codes, the upper bounds derived here also indicate good match with computer simulation results of efficient iterative decoding algorithms     

How to conceptualize catalytic cycles? The energetic span model

A computational study of a catalytic cycle generates state energies (the E-representation), whereas experiments lead to rate constants (the k-representation). Based on transition state theory (TST), these are equivalent representations. Nevertheless, until recently, there has been no simple way to calculate the efficiency of a catalytic cycle, that is, its turnover frequency (TOF), from a theoretically obtained energy profile. In this Account, we introduce the energetic span model that enables one to evaluate TOFs in a straightforward manner and in affinity with the Curtin-Hammett principle. As shown herein, the model implies a change in our kinetic concepts. Analogous to Ohm's law, the catalytic chemical current (the TOF) can be defined by a chemical potential (independent of the mechanism) divided by a chemical resistance (dependent on the mechanism and the nature of the catalyst). This formulation is based on Eyring's TST and corresponds to a steady-state regime. In many catalytic cycles, only one transition state and one intermediate determine the TOF. We call them the TOF-determining transition state (TDTS) and the TOF-determining intermediate (TDI). These key states can be located, from among the many states available to a catalytic cycle, by assessing the degree of TOF control (X(TOF)); this last term resembles the structure-reactivity coefficient in classical physical organic chemistry. The TDTS-TDI energy difference and the reaction driving force define the energetic span (δE) of the cycle. Whenever the TDTS appears after the TDI, δE is the energy difference between these two states; when the opposite is true, we must also add the driving force to this difference. Having δE, the TOF is expressed simply in the Arrhenius-Eyring fashion, wherein δE serves as the apparent activation energy of the cycle. An important lesson from this model is that neither one transition state nor one reaction step possess all the kinetic information that determines the efficiency of a catalyst. Additionally, the TDI and TDTS are not necessarily the highest and lowest states, nor do they have to be adjoined as a single step. As such, we can conclude that a change in the conceptualization of catalytic cycles is in order: in catalysis, there are no rate-determining steps, but rather rate-determining states. We also include a study on the effect of reactant and product concentrations. In the energetic span approximation, only the reactants or products that are located between the TDI and TDTS accelerate or inhibit the reaction. In this manner, the energetic span model creates a direct link between experimental quantities and theoretical results. The versatility of the energetic span model is demonstrated with several catalytic cycles of organometallic reactions.     

An Improved Sphere-Packing Bound for Finite-Length Codes Over Symmetric Memoryless Channels

This paper derives an improved sphere-packing (ISP) bound for finite-length error-correcting codes whose transmission takes place over symmetric memoryless channels, and the codes are decoded with an arbitrary list decoder. We first review classical results, i.e., the 1959 sphere-packing (SP59) bound of Shannon for the Gaussian channel, and the 1967 sphere-packing (SP67) bound of Shannon et al. for discrete memoryless channels. An improvement on the SP67 bound, as suggested by Valembois and Fossorier, is also discussed. These concepts are used for the derivation of a new lower bound on the error probability of list decoding (referred to as the ISP bound) which is uniformly tighter than the SP67 bound and its improved version. The ISP bound is applicable to symmetric memoryless channels, and some of its applications are presented. Its tightness under maximum-likelihood (ML) decoding is studied by comparing the ISP bound to previously reported upper and lower bounds on the ML decoding error probability, and also to computer simulations of iteratively decoded turbo-like codes. This paper also presents a technique which performs the entire calculation of the SP59 bound in the logarithmic domain, thus facilitating the exact calculation of this bound for moderate to large block lengths without the need for the asymptotic approximations provided by Shannon.