Function Ofx Research Articles

Iteration, Inequalities, and Differentiability in Analog Computers

Shannon's general purpose analog computer (GPAC) is an elegant model of analog computation in continuous time. In this paper, we consider whether the set G of GPAC-computable functions is closed under iteration, that is, whether for any function f(x)∈G there is a function F(x, t)∈G such that F(x, t)=ft(x) for nonnegative integers t. We show that G is not closed under iteration, but a simple extension of it is. In particular, if we relax the definition of the GPAC slightly to include unique solutions to boundary value problems, or equivalently if we allow functions xkθ(x) that sense inequalities in a differentiable way, the resulting class, which we call G+θk, is closed under iteration. Furthermore, G+θk includes all primitive recursive functions and has the additional closure property that if T(x) is in G+θk, then any function ofx computable by a Turing machine in T(x) time is also.

The 4f-ligand hybridization in theevolution of heavy-fermion behaviour in the seriesCeRu2-xNixSi2

CeRu2Si2 is a heavy-fermion compound andCeNi2Si2 is a mixed-valence compound. We measured theLIII absorption spectra of CeRu2-xNixSi2 (x = 0, 0.2, 0.4, 0.8, 0.9, 1.0, 1.1, 1.4, 1.6, and 2.0) at 16, 50, 100,200, and 300 K. The linear coefficient of the specific heat γ ofCeRu2-xNixSi2 is enhanced rapidly for Ce valence v<3.08. The structure of CeRu2-xNixSi2 is consistentwith the tetragonal space group I4/mmm. There is no criticallattice constant c separating the strongly mixed-valence regimefrom the heavy-fermion regime for CeRu2-xNixSi2. Thevalue of the lattice constant c is almost a linear function ofx, whereas there is a deviation from linearity for the values ofthe lattice constant a when x<0.4. Judging by the LIIIx-ray absorption spectra of CeRu2-xNixSi2 andCeRu2Si2-xGex, the evolution from heavy-fermion tomixed-valence behaviour might be due just to the hybridizationbetween the Ce 4f-electron wave function and the sp wavefunction of Ru.

Characterizations based on conditional expectations

For given real functionsg andh, first we give necessary and sufficient conditions such that there exists a random variableX satisfying thatE(g(X)|X≥y)=h(y)r x (y),∀y ∈ C x , whereC x andT X are the support and the failure rate function ofX, respectively. These extend the results of Ruiz and Navarro (1994) and Ghitany et al. (1995). Next we investigate necessary and sufficient conditions such thath(y)=E(g(X)|X≥y), for a given functionh.

Enhanced critical current density due to flux pinning from latticedefects in pulsed laser ablated Y1-xDyxBa2Cu3O7-δ thin films

The effect of lattice defects at the unit cell level created bythe stress field when two rare earths with different ionic radii are mixedat the rare earth site, on the flux pinning and critical current density inY1-xDyxBa2Cu3O7-δ thin filmsprepared by the pulsed laser deposition technique, was investigated usingSQUID magnetometry at different temperatures for x = 0 to 1 in steps of0.2. From the isothermal magnetic hysteresis recorded up to 5.5 T at 5, 35and 77 K, the critical current density, pinning force density andirreversibility fields are estimated and their variation as a function ofx analysed. An inverse relation between current density andirreversibility field is observed at all the temperatures except for thefilm having a 20% concentration of Dy in Y which also exhibits the highestcritical current density among the films investigated, suggesting this tobe the optimum composition for effective flux pinning from stress fieldinduced lattice defects.

Spectral mimicry: A method of synthesizing matching time series with different Fourier spectra

Given a stationary time seriesX and another stationary time seriesY (with a different power spectral density), we describe an algorithm for constructing a stationary time series Z that contains exactly the same values asX permuted in an order such that the power spectral density ofZ closely resembles that ofY. We call this methodspectral mimicry. We prove (under certain restrictions) that, if the univariate cumulative distribution function (CDF) ofX is identical to the CDF ofY, then the power spectral density ofZ equals the power spectral density ofY. We also show, for a class of examples, that when the CDFs ofX andY differ modestly, the power spectral density ofZ closely approximates the power spectral density ofY. The algorithm, developed to design an experiment in microbial population dynamics, has a variety of other applications.

On Zeta Functions of Arithmetically Defined Graphs

We study the graphX(n) that is defined as the finite part of the quotient Γ(n)\\T, with T the Bruhat–Tits tree over Fq((1/T)) and Γ(n) the principal congruence subgroup of Γ=GL2(Fq[T]) of leveln∈Fq[T]. We give concrete realizations of theL-functions of the finite part of the halfline Γ\\T for finite unitary representations of Γ that factor over Γ(n),nprime. This allows us to give explicit formulae for the zeta function ofX(n) for smalln. As an application, we show that these graphs are very good concentrators. Moreover, we construct a new unbounded family of Ramanujan graphs, considering regularizations ofX(n).

Tests of significance for the average square deviation from target in a finite lot

Consider lots of discrete items 1, 2, …,N with quality characteristicsx 1,x 2, …,x N . Leta be a target value for item quality. Lot quality is identified with the average square deviation\(z = \mathop - \limits_N^1 \mathop \Sigma \limits_{i = 1}^N (x_1 - a)^2 \) from target per item in the lot (lot average square deviation from target). Under economic considerations this is an appropriate lot quality indicator if the loss respectively the profit incurred from an item is a quadratic function ofx i −a. The present paper investigates tests of significance on the lot average square deviationz under the following assumptions: The lot is a subsequence of a process of production, storage, transport; the random quality characteristics of items resulting from this process are i.i.d. with normal distributionN(μ, σ 2); the target valuea coincides with the process meanμ.

Resolutions of generic points lying on a smooth quadric

For a 0-dimensional schemeX on a smooth quadricQ we define a special type of resolution of its ideal sheaf as a locally freeO Q. These resolutions allow to find, for schemes which are generic inQ, the minimal free resolution ofX as a subscheme of ℙ3. For almost all such schemes the graded Betti numbers in ℙ3 depend only on the Hilbert function ofX in ℙ3.

Multi-prover Encoding Schemes and Three-prover Proof Systems

Suppose two provers agree in a polynomialpand want to reveal a single vaiuey=p(x) to a verifier wherexis chosen arbitrarily by the verifier. Whereas honest provers should be able to agree on any polynomialpthe verifier wants to be sure that with any (cheating) pair of provers the valueyhe receives is a polynomial function ofx. We formalize this question and introduce multi-prover (quasi-)encoding schemes to solve it. Using multi-prover quasi-encoding schemes we are able to develop new results about interactive proofs. The best previous result appears in [BGLR] and states the existence of one-round-four-prover interactive proof systems for the languages in NP achieving any constant error probability withO(logn) random bits andpoly(loglogn) answer size. We improve this result in two respects. First we decrease the number of provers to three, and then we decrease the answer-size to a constant. Using unrelated (parallel repetition) techniques the same was independently and simultaneously achieved by [FK] with only two provers. When the error-probability is required to approach zero, our technique is more efficient in the number of random bits and in the answer size. Showing the fast progress in this central topic of theoretical computer science in the short time since these results were achieved Raz's proof of the parallel repetition conjecture [R] lead to further improvements in the parameters of interactive proofs for NP problems.

The empirical process of a short-range dependent stationary sequence under Gaussian subordination

Consider the stationary sequenceX 1=G(Z 1),X 2=G(Z 2),..., whereG(·) is an arbitrary Borel function andZ 1,Z 2,... is a mean-zero stationary Gaussian sequence with covariance functionr(k)=E(Z 1 Z k+1) satisfyingr(0)=1 and ∑ =1 ∞ |r(k)| m < ∞, where, withI{·} denoting the indicator function andF(·) the continuous marginal distribution function of the sequence {X n }, the integerm is the Hermite rank of the family {I{G(·)≦ x} −F(x):x∈R}. LetF n (·) be the empirical distribution function ofX 1,...,X n . We prove that, asn→∞, the empirical processn 1/2{F n (·)-F(·)} converges in distribution to a Gaussian process in the spaceD[−∞,∞].

The empirical process of a short-range dependent stationary sequence under Gaussian subordination

Consider the stationary sequenceX 1=G(Z 1),X 2=G(Z 2),..., whereG(·) is an arbitrary Borel function andZ 1,Z 2,... is a mean-zero stationary Gaussian sequence with covariance functionr(k)=E(Z 1 Z k+1) satisfyingr(0)=1 and ∑ =1 ∞ |r(k)| m < ∞, where, withI{·} denoting the indicator function andF(·) the continuous marginal distribution function of the sequence {X n }, the integerm is the Hermite rank of the family {I{G(·)≦ x} −F(x):x∈R}. LetF n (·) be the empirical distribution function ofX 1,...,X n . We prove that, asn→∞, the empirical processn 1/2{F n (·)-F(·)} converges in distribution to a Gaussian process in the spaceD[−∞,∞].

Inclusive charged particle distributions in deep inelastic scattering events at HERA

A measurement of inclusive charged particle distributions in deep inelasticep scattering forγ*p centre-of-mass energies 75<W<175 GeV and momentum transfer squared 10<Q 2<160 GeV2 from the ZEUS detector at HERA is presented. The differential charged particle rates in theγ*p centre-of-mass system as a function of the scaled longitudinal momentum,x F, and of the transverse momentum,p* t and 〈p* t 2〉, as a function ofx F, W andQ 2 are given. Separate distributions are shown for events with (LRG) and without (NRG) a rapidity gap with respect to the proton direction. The data are compared with results from experiments at lower beam energies, with the naive quark parton model and with parton models including perturbative QCD corrections. The comparison shows the importance of the higher order QCD processes. Significant differences of the inclusive charged particle rates between NRG and LRG events at the sameW are observed. The value of 〈p* t 2〉 for LRG events with a hadronic massM x, which excludes the forward produced baryonic system, is similar to the 〈p* t 2〉 value observed in fixed target experiments atW≈M x.

Two-photon absorption in Hg1–xCdxTe as a function ofx

Two-photon absorption in Hg1–xCdxTe as a function ofx

Production of neutral strange particles in muon-nucleon scattering at 490 GeV

The production ofK0, Λ and\(\bar \Lambda \) particles is studied in the E665 muon-nucleon experiment at Fermilab. The average multiplicities and squared transverse momenta are measured as a function ofx F andW2. Most features of the data can be well described by the Lund model. Within this model, the data on the K0/π± ratios and on the averageK0 multiplicity in the forward region favor a strangeness suppression factors/u in the fragmentation process near 0.20. Clear evidence for QCD effects is seen in the average squared transverse momentum ofK0 and Λ particles.

Determination of the ratior v =d v u v of the valence quark distributions in the proton from neutrino and antineutrino reactions on hydrogen and deuterium

Based on a QCD analysis of the parton momentum distributions in the proton, the ratior v =d v /u v of thed andu valence quark distributions is determined as function ofx in the range 0.01<x<0.7. The analysis uses data from neutrino and antineutrino charged current interactions on hydrogen and deuterium, obtained with BEBC in the (anti)neutrino wideband beam of the CERN SPS. Since v mainly depends on the deuterium/hydrogen ratios of the normalisedx-y-Q2-distributions many systematic effects cancel. It is found thatr v decreases with increasingx, and drops below the naiveSU(6) expectation of 0.5 forx≳0.3. An extrapolation ofr v tox=1 is consistent with the hypothesisr v (1)=0.

Photo- and hadro-production of? (1020),K *(892)0 and $$\overline {K*} (892)^0 $$ mesons in the energy range 65 to 175 GeV

Inclusive production of ϕ,K *0, and\(\overline {K*^0 } \) mesons has been measured in γp, π± p andK ± p collisions at beam energies of 65 GeV<E γ<175 GeV andE π/K =80 and 140 GeV. Cross sections have been determined over the range 0<x F<1.0 and 0< PT<1.8 GeV/c. Emphasis is put on the comparison of cross sections for different projectiles as a function ofx F so as to study the effects of common quarks between the beam particle and the detected ϕ,K *0 or\(\overline {K*^0 } \). The data are compared with a parton fusion model. Many features of the data are well explained. In detail the strange quark appears to carry a large fraction of the kaon momentum and the contribution of the valence quarks from the proton is small.

On the almost sure (a.s.) central limit theorem for random variables with infinite variance

We give criteria for a sequence (X n ) of i.i.d.r.v.'s to satisfy the a.s. central limit theorem, i.e., $$\mathop {\lim }\limits_{N \to \infty } \frac{1}{{\log N}}\sum\limits_{k \leqslant N} {\frac{1}{k}I\left\{ {\frac{{S_k }}{{a_k }} - b_k< x} \right\} = \phi (x)} a.s{\mathbf{ }}for{\mathbf{ }}all{\mathbf{ }}x$$ a.s. for allx whereS n =X 1+X 2+...+X n , (a n ) and (b n ) are numerical sequences andI denotes indicator function. By a recent result of Lacey and Philipp,(7) and Fisher,(6) and its converse obtained here, the a.s. central limit theorem holds with $$a_n = \sqrt n $$ iffEX 1 2 <+∞ which shows that for $$a_n = \sqrt n $$ the a.s. central limit theorem and the corresponding ordinary (weak) central limit theorem are equivalent. Our main results show that for general (a n ) the situation is radically different and paradoxical: the weak central limit theorem implies the a.s. central limit theorem but the converse is false and in fact the validity of the a.s. central limit theorem permits very irregular distributional behavior ofS n /a n −b n . We also show that the validity of the a.s. central limit theorem is closely connected with the limiting behavior of the classical fractionx 2(1−F(x)+F(−x))/∫|t|≤x t 2 dF(t) whereF is the distribution function ofX 1.

Necessary conditions for characterization of laws via mixed sums

SupposeX andY are independent and identically distributed, and independent ofU which satisfies 0≤U≤1. Recent work has centered on finding the lawsL(X) for whichX ℞U(X+Y) where ℞ denotes equality in law. We show that this equation corresponds to a certain projective invariance property under random rotations. Implicitly or explicitly, it has been assumed that the characteristic function ofX has an expansion property near the origin. We show that solutions may be admitted in the absence of this condition when −logU has a lattice law. A continuous version of the basic problem replaces sums with a Levy process. Instead we consider self-similar processes, showing that a solution exists only whenU is constant, and then all processes of a given order are admitted.

Anderson localization and layered superconductor 2H-NbSe2−x S x

The zero-field critical temperature and the coherence length in the layered superconductor 2H-NbSe2−x S x (x = 0–2.0) were investigated. The zero-field critical temperature decreases with increasing the residual resistivity. This result can be explained in terms of the three-dimensional Anderson localization with the mobility edge below the Fermi level. The coherence length as a function ofx can be explained by the theory of the anisotropic three-dimensional dirty superconductor. However it shows anomalous behavior whenx = 0.6. This may be relevant to the change of the crystal structure or the disappearance of the CDW. The effective mass ratio does not depend onx whenx ≤ 0.4.

Formation of oxide phases in the system Fe2O3-Gd2O3 Part II

The formation of oxide phases in the system (1 -x) Fe2O3 +xGd2O3 was investigated for 0 ⩽x ⩽ 1. On the basis of XRD measurements the distribution of oxide phases, α-Fe2O3, Gd3Fe5O12, GdFeO3 and Gd2O3 was determined, as a function ofx. No solid solutions were observed with certainty even at the very ends of the concentration range. This was also confirmed by57Fe Mossbauer spectroscopy. New accurate crystallographic data for Gd3Fe5O12 are given. The formation of oxide phases in the system Fe2O3- Gd2O3 is compared with the data for analogous system Fe2O3-Eu2O3.