Entropy of Operators or why Matrix Multiplication is Hard for Depth-Two Circuits

We consider unbounded fanin depth-2 circuits with arbitrary boolean functions as gates. We define the entropy of an operator f:{0,1} ⁿ →{0,1} ^m as the logarithm of the maximum number of vectors distinguishable by at least one special subfunction of f. Our main result is that every depth-2 circuit for f requires at least entropy(f) wires. This is reminiscent of a classical lower bound of Nechiporuk on the formula size, and gives an information-theoretic explanation of why some operators require many wires. We use this to prove a tight estimate Θ(n ³) of the smallest number of wires in any depth-2 circuit computing the product of two n by n matrices over any finite field. Previously known lower bound for this operator was Ω(n ²log n).     

Accumulation and Toxicity of Superparamagnetic Iron Oxide Nanoparticles in Cells and Experimental Animals

The uptake and distribution of negatively charged superparamagnetic iron oxide (Fe₃O₄) nanoparticles (SPIONs) in mouse embryonic fibroblasts NIH3T3, and magnetic resonance imaging (MRI) signal influenced by SPIONs injected into experimental animals, were visualized and investigated. Cellular uptake and distribution of the SPIONs in NIH3T3 after staining with Prussian Blue were investigated by a bright-field microscope equipped with digital color camera. SPIONs were localized in vesicles, mostly placed near the nucleus. Toxicity of SPION nanoparticles tested with cell viability assay (XTT) was estimated. The viability of NIH3T3 cells remains approximately 95% within 3–24 h of incubation, and only a slight decrease of viability was observed after 48 h of incubation. MRI studies on Wistar rats using a clinical 1.5 T MRI scanner were showing that SPIONs give a negative contrast in the MRI. The dynamic MRI measurements of the SPION clearance from the injection site shows that SPIONs slowly disappear from injection sites and only a low concentration of nanoparticles was completely eliminated within three weeks. No functionalized SPIONs accumulate in cells by endocytic mechanism, none accumulate in the nucleus, and none are toxic at a desirable concentration. Therefore, they could be used as a dual imaging agent: as contrast agents for MRI and for traditional optical biopsy by using Prussian Blue staining.     

Estimation of ambient temperature impact on vertical dynamic behaviour of passenger rail vehicle with damaged wheels

Maintenance and repair specialists of rolling stock face the problem related to accurate and reliable assessment of the vibrations of the car body of passenger rail vehicle, caused by wheel damage, and its impact on passengers. Another important aspect is to deter-mine the permissible speed at which the passenger car with wheel damage could move to the nearest station or depot without badly damaging the rails. The purpose of this research is to examine the dynamic processes of the interaction of the passenger car chassis with the railway track using a newly developed mathematical models called the “Passenger Car - Track” system. During the simulation, the dynamic characteristics of the track, bogies and car body in different season temperature were estimated. The validation of theoretical results was obtained based on the field tests of the test train as well as the results of measurements of changes in track stiffness temperatures. Finally, conclusions and suggestions are presented.     

On the minimum number of negations leading to super-polynomial savings

We show that an explicit sequence of monotone functions can be computed by Boolean circuits with polynomial (in n) number of And, Or and Not gates, but every such circuit must use at least logn−O(loglogn) Not gates. This is almost optimal because results of Markov [J. ACM 5 (1958) 331] and Fisher [Lecture Notes in Comput. Sci., Vol. 33, Springer, 1974, p. 71] imply that, with only small increase of the total number of gates, any circuit in n variables can be simulated by a circuit with at most ⌈log(n+1)⌉ Not gates.     

On P versus NP $ \cap $ co-NP for decision trees and read-once branching programs

It is known that if a Boolean function f in n variables has a DNF and a CNF of size then f also has a (deterministic) decision tree of size exp(O(log n log² N)). We show that this simulation cannot be made polynomial: we exhibit explicit Boolean functions f that require deterministic trees of size exp where N is the total number of monomials in minimal DNFs for f and ?f. Moreover, we exhibit new examples of explicit Boolean functions that require deterministic read-once branching programs of exponential size whereas both the functions and their negations have small nondeterministic read-once branching programs. One example results from the Bruen—Blokhuis bound on the size of nontrivial blocking sets in projective planes: it is remarkably simple and combinatorially clear. Other examples have the additional property that f is in AC⁰. Received: June 5 1997.     

Yet harder knapsack problems

Already 30 years ago, Chvátal has shown that some instances of the zero-one knapsack problem cannot be solved in polynomial time using a particular type of branch-and-bound algorithms based on relaxations of linear programs together with some rudimentary cutting-plane arguments as bounding rules. We extend this result by proving an exponential lower bound in a more general class of branch-and-bound and dynamic programming algorithms which are allowed to use memoization and arbitrarily powerful bound rules to detect and remove subproblems leading to no optimal solution.     

Fast High-Voltage Light Triggered Superconducting Opening Switch

The dynamics of light induced switching from the super-conducting or mixed state to the normal state were investigated in thin Y-Ba-Cu-0 films biased by high-current nanosecond pulses. It was demonstrated that the subnanosecond duration voltage transients with amplitudes of up to several hundred volts which are generated by the light triggered super-conducting opening switch are the result of film heating by light pulses and the bias current.     

Some bounds on multiparty communication complexity of pointer jumping

We introduce the model of conservative one-way multiparty complexity and prove lower and upper bounds on the complexity of pointer jumping.? The pointer jumping function takes as its input a directed layered graph with a starting node and k layers of n nodes, and a single edge from each node to one node from the next layer. The output is the node reached by following k edges from the starting node. In a conservative protocol, the ith player can see only the node reached by following the first i–1 edges and the edges on the jth layer for each j > i. This is a restriction of the general model where the ith player sees edges of all layers except for the ith one. In a one-way protocol, each player communicates only once in a prescribed order: first Player 1 writes a message on the blackboard, then Player 2, etc., until the last player gives the answer. The cost is the total number of bits written on the blackboard.?Our main results are the following bounds on k-party conservative one-way communication complexity of pointer jumping with k layers:? (1) A lower bound of order for any .?(2) Matching upper and lower bounds of order for . received March 22, 1996     

On uncertainty versus size in branching programs

We propose an information-theoretic approach to proving lower bounds on the size of branching programs. The argument is based on Kraft type inequalities for the average amount of uncertainty about (or entropy of) a given input during the various stages of computation. The uncertainty is measured by the average depth of so-called ‘splitting trees’ for sets of inputs reaching particular nodes of the program.

We first demonstrate the approach for read-once branching programs. Then, we introduce a strictly larger class of so-called ‘balanced’ branching programs and, using the suggested approach, prove that some explicit Boolean functions cannot be computed by balanced programs of polynomial size. These lower bounds are new since some explicit functions, which are known to be hard for most previously considered restricted classes of branching programs, can be easily computed by balanced branching programs of polynomial size.     

Spectroscopy With Nanostructured Superconducting Single Photon Detectors

Superconducting single-photon detectors (SSPDs) are nanostructured devices made from ultrathin superconducting films. They are typically operated at liquid helium temperature and exhibit high detection efficiency, in combination with very low dark counts, fast response time, and extremely low timing jitter, within a broad wavelength range from ultraviolet to mid-infrared (up to 6 mum). SSPDs are very attractive for applications such as fiber-based telecommunication, where single-photon sensitivity and high photon-counting rates are required. We review the current state-of-the-art in the SSPD research and development, and compare the SSPD performance to the best semiconducting avalanche photodiodes and other superconducting photon detectors. Furthermore, we demonstrate that SSPDs can also be successfully implemented in photon-energy-resolving experiments. Our approach is based on the fact that the size of the hotspot, a nonsuperconducting region generated upon photon absorption, is linearly dependent on the photon energy. We introduce a statistical method, where, by measuring the SSPD system detection efficiency at different bias currents, we are able to resolve the wavelength of the incident photons with a resolution of 50 nm.