On the Complexity of Hardness Amplification

An exact recursive formula is derived to describe the structure of an ideal first-order /spl Sigma/-/spl Delta/ output sequence as a function of its input. Specifically, it is shown that every /spl Sigma/-/spl Delta/ sequence generated by the constant input x/spl isin/[0, 1] can be decomposed into a shorter E-A subsequence whose input x'/spl isin/[0, 1) may be used to recover that of the original /spl Sigma/-/spl Delta/ sequence. This formula is applied to develop an O(N log N) algorithm for decoding an N-length sequence. Without knowledge of the modulator's initial state, it exhibits an average improvement, over all initial states, of 4.2 dB in output signal-to-noise ratio (SNR) compared with a near-optimal linear finite impulse response (FIR) filter. The regularity of the ideal first-order /spl Sigma/-/spl Delta/ structure with constant inputs permits the algorithm to be extended to bandlimited and noise-corrupted data. A simple error correction procedure is demonstrated, and it is shown that the recursive algorithm can outperform FIR filters on sequences of length N<64 having input SNRs as low as 30 dB.

We study the space complexity of refuting unsatisfiable random k-CNFs in the resolution proof system. We prove that for any large enough /spl Delta/, with high probability a random k-CNF over n variables and /spl Delta/n clauses requires resolution clause space of /spl Omega/(n/spl middot//spl Delta//sup -1+/spl epsiv//k-2-/spl epsiv//), for any 0 >/spl radic/n. This bound is nearly tight. Specifically, we show that with high probability, a random 3-CNF with /spl Delta/n clauses requires tree-like refutation size of exp(/spl Omega/(n//spl Delta//sup 1+/spl epsiv//1-/spl epsiv//)), for any 0</spl epsiv/<1/2. Our space lower bound is the consequence of three main contributions. 1. We introduce a 2-player matching game on bipartite graphs G to prove that there are no perfect matchings in G. 2. We reduce lower bounds for the clause space of a formula F in resolution to lower bounds for the complexity of the game played on the bipartite graph G(F) associated with F. 3. We prove that the complexity of the game is large whenever G is an expander graph. Finally, a simple probabilistic analysis shows that for a random formula F, with high probability G(F) is an expander. We also extend our result to the case of G-PHP, a generalization of the pigeonhole principle based on bipartite graphs G. We prove that the clause space for G-PHP can be reduced to the game complexity on G.

Summary form only given. Relativistic plasma waves (RPW) can accelerate electrons with gradients in excess of a few GeV/m. Theoretical studies involving RPW driven by a two-wavelength laser pulse predict that such waves are most efficiently excited when the plasma frequency /spl omega//sub p/ = /spl Delta//spl omega/, where /spl Delta//spl omega/ is the difference between the two laser frequencies. Experimentally, the amplitude of the waves driven near this resonance condition largely follows the predictions. In these experiments, either a laser pulse was used to scatter off the plasma wave at resonant densities or an injected electron beam was used to sample the plasma wave (and to accelerate electrons). In this paper a collinear Thomson scattering technique using an independent probe beam is applied to detect the RPW. In this range, very low scattering efficiencies and small frequency shifts between the probe and the scattered beam make measurements very challenging. The results reveal that, while the amount of scattered light has a peak near resonance, up to 10 times more light is scattered at plasma densities well above resonance.

Summary form only given. The feasibility of cavity-controlled spontaneous emission of a motionless two level atom is analyzed quantum mechanically and nonperturbatively. It is proposed to control the emission by putting the excited atom into the combined field of two lossless modes with their frequencies /spl omega//sub 2,1/ detuned symmetrically about the atomic one /spl omega//sub /spl alpha//, i.e. /spl omega//sub 2,1/=/spl omega//sub /spl alpha///spl plusmn//spl Delta/. The controlling /spl omega//sub 2,1/ modes are isolated from the free radiation field by an one-dimensional ideal cavity (or by two crossed ones) at zero temperature. These modes are taken to be linearly polarized in the same plane. Because the wavelength and cavity dimensions are assumed to be of the same order, the retardation effects are neglected. The spontaneous emission is treated as a radiative decay of the macromolecule-like atom+/spl omega//sub 2,1/ modes-subsystem (the H/sub s/-one) under its interaction with the free H/sub k/-field.

Pin-efficient bussed network families are discussed that can-in one clock tick-simultaneously shift all data in a k-dimensional grid to neighboring processors in any one of the 3/sup k/-1 'compass directions' x/spl I.oarr//spl rarr/x/spl I.oarr/+/spl delta//spl I.oarr/, for every nonzero vector /spl delta//spl I.oarr/ /spl isin/ {-1,0,1}/sup k/. The networks have the advantages of being simple to describe (using a single 5-state automaton), extendible (the k-dimensional network is obtained by extending the busses of the (k-1)-dimensional network), and provably optimal for k/spl les/3. The networks use only [3/2(/spl radic/3)/sup k/] pins per processor, which is within 3/2 of the theoretical minimum number of pins required. The best previously known family uses 2/sup k/ pins. >

We prove that for any decision tree calculating a Boolean function f : {-1,1}/sup n/ /spl rarr/ {-1, 1}, Var[f] /spl les/ /spl Sigma/ /sub i=1/ /sup n/ /spl delta//sup i/Inf/sub i/(f), i = 1 where /spl delta//sup i/ is the probability that the ith input variable is read and Inf/sub i/(f) is the influence of the ith variable on f. The variance, influence and probability are taken with respect to an arbitrary product measure on {-1, 1}/sup n/n. It follows that the minimum depth of a decision tree calculating a given balanced function is at least the reciprocal of the largest influence of any input variable. Likewise, any balanced Boolean function with a decision tree of depth d has a variable with influence at least 1/d. The only previous nontrivial lower bound known was /spl Omega/(d2/sup -d/). Our inequality has many generalizations, allowing us to prove influence lower bounds for randomized decision trees, decision trees on arbitrary product probability spaces, and decision trees with nonBoolean outputs. As an application of our results we give a very easy proof that the randomized query complexity of nontrivial monotone graph properties is at least/spl Omega/(v/sup 4/3//p/sup 1/3/), where v is the number of vertices and p /spl les/ 1/2 is the critical threshold probability. This supersedes the milestone /spl Omega/(v/sup 4/3//p/sup 1/3/) bound of Hajnal (1991) and is sometimes superior to the best known lower bounds of Chakrabarti-Khot (2001) and Friedgut-Kahn-Wigderson (2002).

Naor and Yung (1989) show that a one-bit-compressing universal one-way hash function (UOWHF) can be constructed based on a one-way permutation. This construction can be iterated to build a UOWHF which compresses by /spl epsiv/n bits, at the cost of /spl epsiv/n invocations of the one-way permutation. The show that this construction is not far from optimal, in the following sense, there exists an oracle relative to which there exists a one-way permutation with inversion probability 2/sup -p(n)/ (for any p(n)/spl isin//spl omega/(log n)), but any construction of an /spl epsiv/n-bit-compressing UOWHF. Requires /spl Omega/(/spl radic/n/p(n)) invocations of the one-way permutation, on average. (For example, there exists in this relativized world a one-way permutation with inversion probability n/sup -/spl omega/(1)/, but no UOWHF that involves it fewer than /spl Omega/(/spl radic/n/log n) times.) Thus any proof that a more efficient UOWHF can be derived from a one-way permutation is necessarily non-relativizing; in particular, no provable construction of a more efficient UOWHF can exist based solely on a black box one-way permutation. This result can be viewed as a partial justification for the practice of building efficient UOWHFs from stronger primitives (such as collision intractable hash functions), rather than from weaker primitives such as one-way permutations.

Summary form only given. The single-particle theory of electron multipactor is now well understood. The multipactor parameter space is spanned by the normalized magnetic and electric field parameters. In particular, for given normalized magnetic field /spl Omega/ multipactor can errupt within a certain range of electric field amplitudes /spl epsi//sub 1/</spl epsi/</spl epsi//sub 2/; to each value /spl epsi/ corresponds a different locked-phase /spl phi/. Once a multipactor has been launched, it is very important to know the saturation level; the latter determines the cavity power losses when the multipactor is a parasitic side effect, or the saturated current when a multipactor is deliberately triggered for the production of short period high current pulsed beams. The collective space charge effects must be included to address multipactor saturation. One proposed approach employs a single charge sheet and yields saturation when the secondary emission coefficient /spl delta/ equals one. Setting /spl delta/=1 fixes the impact parameters and therefore the value of the driving electric field /spl epsi/; the external RF field adjusts to that value through the nonlinear effect of the induced multipactor current on the cavity LC circuit. A limitation of the method is that one must start near the appropriate /spl epsi/ since the range of the voltage self-adjustment is usually small. A most general approach introduced here is based on multilayered space charge cloud involving electrons emitted at slightly different times. There is a continuous distribution of impact energies and interaction phases between the leading and the trailing edge of the bunch.

We first experimentally observed the doubly-excited electronic state 2/sup 1//spl Delta//sub g/ of Na/sub 2/ dissociating into Na (3p) + Na (3p) limit. This state has the deepest interaction potential well among the 12 electronic states formed from this first doubly-excited limit. A single line Ar/sup +/ laser (total of 4 lines) was used to excite the Na/sub 2/ molecules from ground state X/sup 1//spl Sigma//sub g//sup +/ to the intermediate B/sup 1//spl Pi//sub u/ state. A solid-state pumped single mode Ti:sapphire laser was scanning from 12100 to 14300 cm/sup -1/ to search oh the optical-optical double resonance (OODR) signals. Using a combination of filters and photomultiplier tube to detect the total fluorescence from the OODR, more than 150 ro-vibrational levels from the 2/sup 1//spl Delta//sub g/ state were observed. Comparing the detected excitation intensity with the Franck-Condon factors and the observed vibrational separations with theoretical calculations, the lowest observed vibrational quantum number were assigned to /spl nu/ = 3. The T/sub e/, /spl omega//sub e/, and B/sub e/ from Dunham expansion fitting are 32540.669 cm/sup -1/, 124.039 cm/sup -1/ and 0.1186 cm-1, respectively.

Suppose two distant observers, Alice and Bob, share some form of entanglement - quantum correlations - in some bipartite pure quantum states. They may apply local operations and classical communication to convert one form of entanglement to another. Since entanglement is regarded as a resource in quantum information processing, it is an important question to ask how much classical communication, which is also a resource, is needed in the inter-conversion process of entanglement. In this paper, we address this important question in the many-copy case. The inter-conversion process of entanglement is usually divided into two types: concentrating the entanglement from many partially entangled states into a smaller number of maximally entangled states (i.e., singlets) and the reverse process of diluting singlets into partially entangled states. It is known that entanglement concentration requires no classical communication, but the best prior art result for diluting to N copies of a partially entangled state requires an amount of communication on the order of /spl radic/N. Our main result is to prove that this prior art result is optimal up to a constant factor; any procedure for approximately creating N partially entangled states from singlets requires /spl Omega/(/spl radic/N) bits of classical communication. Previously not even a constant bound was known for approximate entanglement transformations. We also prove a lower bound on the inefficiency of the process: to dilute singlets to N copies of a partially entangled state, the entropy of entanglement must decrease by /spl Omega/(/spl radic/N). Moreover, we introduce two new tools - /spl delta/-significant subspaces and the standard form protocol reduction in entanglement manipulations. We hope that these two new tools will be useful in other work in quantum information theory.

In this paper a macroscopic phenomenological model for the microwave properties of superconductors is presented. The model is based on the idea that there are two kinds of current carriers, and instead of the first London's equation a new equation is derived. This model can be applied to both low- and high-temperature superconductors. Using this model, an expression for the microwave surface resistance is derived and the surface resistance versus frequency is calculated. The results show that the relation between resistance and frequency is not R/sub 3//spl sim//spl omega//sup 2/ as indicated by both BCS theory and London model, but R/sub s//spl sim//spl omega//sup a/, where a is between 1 and 2 (e.g. a=1.35) for thin film high-T/sub c/ superconductors YBa/sub 2/Cu/sub 3/O/sub 7-/spl delta//. The temperature dependence of R/sub s/ is simulated using the given model. These relations and the values of the surface resistance agree well with experimental results. A residual resistance may be interpreted from this model.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A special TE/sub 011+/spl delta// mode sapphire resonator and an equivalent perfect conductor calibration technique were developed to measure the surface resistance Rs of a single piece of HTS thin film non-destructively at 10 GHz and 77 K. The resonator is formed by a permanent probe and a replaceable test sample. The probe has Q-factor greater than 280000-good sample load repeatability and sensitivity about 20 /spl mu//spl Omega/ for HTS samples under test with dimension equal to or greater than /spl phi/28 mm. The calibration employs a symmetric configuration which presents an equivalent Rs of zero ohm and ensures the accuracy of the measurement. The measured Rs data of YBCO thin films made by laser ablation and magnetic sputtering are less than 250 /spl mu//spl Omega/. >

The temperature stability of the oscillation frequency, (/spl Delta//spl omega/ / /spl omega/)/ /spl Delta/T, of an S-band feedback oscillator is derived in terms of the temperature stability of the stable resonator and that of the circuits external to the resonator. Conditions have been established for the optimum external circuit to achieve temperature stability of the oscillation frequency.

In 1968, Veselago posed that media with simultaneous negative values of the permittivity and permeability would exhibit a negative index of refraction (Soviet Physics Uspekhi, vol. 10, pp. 509-514, 1968.). Due to the left-handed relationship between the electric field, magnetic field, and wave vector in these materials he termed them left-handed media (LHM). These LHM are expected to have many unusual properties such as inverted Snell's law, Doppler shift and Cherenkov radiation. However, until recently, such materials were unavailable and these predictions remained uncorroborated for nearly thirty years. As with any new field, the topic of the LHM suffers from some misconceptions and misunderstandings. One such misconception has to do with the notion of negative group velocity which has been closely linked with these media (R.A. Shelby et al, Appl. Phys. Lett., vol. 78, pp. 489-91, 2001) and in fact has been used as equivalent to the term LHM. In this paper, the existence and meaning of negative group velocity in LHM are discussed. It is shown that similar to normal right-handed media (RHM), LHM can also exhibit a negative group velocity and group delay within the region of anomalous dispersion (/spl delta/n//spl delta//spl omega/<0). On the other hand, in the experiments presented so far, the researchers have only measured positive group velocity and negative phase velocity corresponding to so called backward waves (BWs).

It is shown that the impulsive (generalized) systems can be combined with usual bang-bang disturbance systems to yield mixed disturbance action systems. The state response is governed by nonlinear differential equations with bounded disturbance action /spl omega/ and impulse disturbance action /spl upsi/. The impulse disturbance action has its structure given that is affine /spl upsi/=/spl Sigma//sub i=1//sup k/ c/sub i//spl delta/(t-t/sub i/), |c/sub i/|/spl les/1. In this case the differential equation solution is a discontinuous function. After changing the original task into k+1 auxiliary combined degrading severity tasks, we get the necessary conditions for the most stressful disturbance of mixed action for some stable systems. Examples are given to illustrate the theory.