Double Exponential Dependence Research Articles

Neighbourhood complexity of graphs of bounded twin-width

We give essentially tight bounds for, ν(d,k), the maximum number of distinct neighbourhoods on a set X of k vertices in a graph with twin-width at most d. Using the celebrated Marcus–Tardos theorem, two independent works (Bonnet et al., 2022; Przybyszewski, 2022) have shown the upper bound ν(d,k)⩽exp(exp(O(d)))k, with a double-exponential dependence in the twin-width. The work of Gajarsky et al. (2022), using the framework of local types, implies the existence of a single-exponential bound (without explicitly stating such a bound). We give such an explicit bound, and prove that it is essentially tight. Indeed, we give a short self-contained proof that for every d and kν(d,k)⩽(d+2)2d+1k=2d+logd+Θ(1)k,and build a bipartite graph implying ν(d,k)⩾2d+logd+Θ(1)k, in the regime when k is large enough compared to d.

Random Knotting in Fractal Ring Polymers.

Many ring polymer systems of physical and biological interest exhibit both pronounced topological effects and nontrivial self-similarity, but the relationship between these two phenomena has not yet been clearly established. Here, we use theory and simulation to formulate such a connection by studying a fundamental topological property—the random knotting probability—for ring polymers with varying fractal dimension, df. Using straightforward scaling arguments, we generalize a classic mathematical result, showing that the probability of a trivial knot decays exponentially with chain size, N, for all fractal dimensions: P0(N) ∝ exp(−N/N0). However, no such simple considerations can account for the dependence of the knotting length, N0, on df, necessitating a more involved analytical calculation. This analysis reveals a complicated double-exponential dependence, which is well supported by numerical data. By contrast, functional forms typical of simple scaling theories fail to adequately describe the observations. These findings are equally valid for two-dimensional ring polymer systems, where “knotting” is defined as the intersection of any two segments.

Graph Square Roots of Small Distance from Degree One Graphs

Given a graph class \({\mathscr{H}}\), the task of the \({\mathscr{H}}\)-Square Root problem is to decide whether an input graph G has a square root H from \({\mathscr{H}}\). We are interested in the parameterized complexity of the problem for classes \({\mathscr{H}}\) that are composed by the graphs at vertex deletion distance at most k from graphs of maximum degree at most one, that is, we are looking for a square root H such that there is a modulator S of size k such that H − S is the disjoint union of isolated vertices and disjoint edges. We show that different variants of the problems with constraints on the number of isolated vertices and edges in H − S are FPT when parameterized by k by demonstrating algorithms with running time \(2^{2^{\mathcal {O}(k)}}\cdot n^{5}\). We further show that the running time of our algorithms is asymptotically optimal and it is unlikely that the double-exponential dependence on k could be avoided. In particular, we prove that the VC-kRoot problem, that asks whether an input graph has a square root with vertex cover of size at most k, cannot be solved in time \(2^{2^{o(k)}}\cdot n^{\mathcal {O}(1)}\) unless the Exponential Time Hypothesis fails. Moreover, we point out that VC-kRoot parameterized by k does not admit a subexponential kernel unless P = NP.

Excretion of 137Сs from silver Prussian carp (Carassius gibelio) at 5 °С water temperature

After the accidents at Chernobyl and Fukushima-1 nuclear power plants, the radioactive contamination of fish can still exceed the permissible levels of radionuclides in food. In order to predict the radioactive contamination of fish, the parameterization of mathematical models is the basis for radiation protection of humans and the environment. Until recently, there were no parameters of radiocaesium metabolism which is the rate of uptake and excretion of radionuclides from fish at a water temperature below 8 10 ° С during winter season. The aim of this work was to determine the rate of 137 Cs excretion (biological half-life) from the silver Prussian carp (Carassius gibelio) at a water temperature of 5±1 ° C under controlled conditions in aquarium. As a result of experimental studies, the values of the 137 Cs biological half-life were obtained from the silver Prussian carp (392±206 days) in the absence of feeding at a water temperature of 5±1 °C. That corresponds to the values previously obtained by us in similar conditions after the radioactive contamination of fish in the environment of the Chernobyl Exclusion Zone 433±162 days. When describing the dynamics of a decrease of the 137 Cs activity concentration in the body of silver Prussian carp with double exponential dependence, the proportion of the fast component of the radionuclide content decrease was 19±3%. In this case, the fast and slow rates of 137 Cs excretion from the body of the fish were equal to 0.3±0.2 days -1 and 0.0004±0.0003 days -1 , respectively. The results obtained in the work show that the level of radioactive contamination of fish will be constant during the winter season. And its seasonal decrease in winter (according to the sawtooth dependence), usually expected out of projected estimates, does not happen. That was also confirmed by the results and experimental data obtained in the environment of the Chernobyl Exclusion Zone.

Magnetic Response of the Stacks of HTS Tapes

We have performed a numerical analysis of the magnetic response of a single piece of a high-temperature superconducting (HTS) tape and the stacks of rectangular pieces of HTS tapes. The number of tapes in the stacks was varied from $N=1$ to 100. We have carried out the simulation in the framework of the critical state model, taking into account the double-exponential dependence of the critical current density on the external applied field that was proposed by the authors earlier. The results of the calculation of the magnetization of rectangular packages of HTS tapes are presented and discussed. We found that the value of the magnetization per one tape is reduced as the number of tapes in the stack is increased. The decrease in the normalized magnetization was shown to be due to mutual screening among tapes in the stack. We have compared the calculated magnetization with the experimental data and found a good agreement between both results. The novel method of calculations of the magnetic properties of HTS tapes can be successfully used for the calculation of the magnetization of HTS tapes and stacks with arbitrary shape.

Low temperature iron gettering by grown-in defects in p-type Czochralski silicon

Low temperature iron gettering in as-grown boron doped Czochralski silicon (Cz-Si) at temperatures between 220 and 500 °C is studied using microwave-photoconductive decay based minority carrier lifetime measurements. Scanning infrared microscopy technique is used to study the defect density/size distribution in the samples before and after anneal. It is found that the decrease of interstitial iron (Fei) concentration shows a double exponential dependence on annealing time at all temperatures. This suggests the existence of two sinks for Fei. Meanwhile, the observed bulk defect densities and sizes in contaminated and as-grown samples are nearly the same, implying that the grown-in defects could be the gettering centers in this process. The results are important for understanding and controlling low temperature Fei gettering during processing of Cz-Si based devices.

Approximating Vector Scheduling: Almost Matching Upper and Lower Bounds.

We consider the Vector Scheduling problem, a natural generalization of the classical makespan minimization problem to multiple resources. Here, we are given n jobs, represented as d-dimensional vectors in [0,1]^d, and m identical machines, and the goal is to assign the jobs to machines such that the maximum load of each machine over all the coordinates is at most 1. For fixed d, the problem admits an approximation scheme, and the best known running time is n^{f(epsilon ,d)} where f(epsilon ,d) = (1/epsilon )^{tilde{O}(d)} (tilde{O} suppresses polylogarithmic terms in d). In particular, the dependence on d is double exponential. In this paper we show that a double exponential dependence on d is necessary, and give an improved algorithm with essentially optimal running time. Specifically, we let exp (x) denote 2^x and show that: (1) For any epsilon <1, there is no (1+epsilon )-approximation with running time exp left( o(lfloor 1/epsilon rfloor ^{d/3})right) unless the Exponential Time Hypothesis fails. (2) No (1+epsilon )-approximation with running time exp left( lfloor 1/epsilon rfloor ^{o(d)}right) exists, unless NP has subexponential time algorithms. (3) Similar lower bounds also hold even if epsilon m extra machines are allowed (i.e. with resource augmentation), for sufficiently small epsilon >0. (4) We complement these lower bounds with a (1+epsilon )-approximation that runs in time exp left( (1/epsilon )^{O(d log log d)}right) + nd. This gives the first efficient approximation scheme (EPTAS) for the problem.

Subtractive method for obtaining the dark current-voltage characteristic and its types for the residual (nongenerating) part of a multi-junction solar cell

The paper is focused on the fundamental loss in the nongenerating (residual) part of multi-junction solar cells. A method for determining the current-voltage characteristic of the residual part of solar cells is suggested and substantiated. The method is a generalization of the technique applicable to single-junction solar photovoltaic converters. The imbalance of photogenerated currents and the luminescence coupling between subcells is taken into account, which makes it possible to apply the suggested method to multi-junction solar cells. The method is applied to single-junction (InGaP, GaAs, Ge) and triple-junction (InGaP/GaAs/Ge) solar cells. Two types of current-voltage characteristics I–V are revealed and empirical laws for these characteristics are found. The first type shows a monotonic superlinearity, J ∝ Vn (n ≈ 1.3–1.4), and is due to spreading resistance. The second type is only observed for triple-junction solar cells and has the form of a double-exponential dependence with a sublinear initial portion, J ∝ [exp(V/E1) − exp(−V/E2)], with E1 ≈ 0.35 V and E2 ≈ 0.15–0.30 V. As a result, it is found that the charge transport in the residual part of multi-junction solar cells is limited not only by the spreading resistance, but also by other factors, e.g., by isotype heterointerfaces.

A faster FPT algorithm for Bipartite Contraction

The Bipartite Contraction problem is to decide, given a graph G and a parameter k, whether we can obtain a bipartite graph from G by at most k edge contractions. The fixed-parameter tractability of the problem was shown by Heggernes et al. [13], with an algorithm whose running time has double-exponential dependence on k. We present a new randomized FPT algorithm for the problem, which is both conceptually simpler and achieves an improved 2O(k2)nm running time, i.e., avoiding the double-exponential dependence on k. The algorithm can be derandomized using standard techniques.

Iron-boron pair dissociation in silicon under strong illumination

The dissociation of iron-boron pairs (FeB) in Czochralski silicon under strong illumination was investigated. It is found that the dissociation process shows a double exponential dependence on time. The first fast process is suggested to be caused by a positive Fe in FeB capturing two electrons and diffusion triggered by the electron-phonon interactions, while the second slow one would involve the capturing of one electron followed by temperature dependent dissociation with an activation energy of (0.21 ± 0.03) eV. The results are important for understanding and controlling the behavior of FeB in concentrator solar cells.

Atomistic simulations of diffusional creep in a nanocrystalline body-centered cubic material

Molecular dynamics (MD) simulations are used to study diffusion-accommodated creep deformation in nanocrystalline molybdenum, a body-centered cubic metal. In our simulations, the microstructures are subjected to constant-stress loading at levels below the dislocation nucleation threshold and at high temperatures (i.e., T > 0.75 T melt), thereby ensuring that the overall deformation is indeed attributable to atomic self-diffusion. The initial microstructures were designed to consist of hexagonally shaped columnar grains bounded by high-energy asymmetric tilt grain boundaries (GBs). Remarkably the creep rates, which exhibit a double-exponential dependence on temperature and a double power-law dependence on grain size, indicate that both GB diffusion in the form of Coble creep and lattice diffusion in the form of Nabarro–Herring creep contribute to the overall deformation. For the first time in an MD simulation, we observe the formation and emission of vacancies from high-angle GBs into the grain interiors, thus enabling bulk diffusion.

Microbial activity in soils frozen to below −39 °C

Recent research on life in extreme environments has shown that some microorganisms metabolize at extremely low temperatures in Arctic and Antarctic ice and permafrost. Here, we present kinetic data on CO 2 and 14CO 2 release from intact and 14C-glucose amended tundra soils (Barrow, Alaska) incubated for up to a year at 0 to −39°C. The rate of CO 2 production declined exponentially with temperature but it remained positive and measurable, e.g. 2–7 ng CO 2–C cm −3 soil d −1, at −39 °C. The variation of CO 2 release rate ( v) was adequately explained by the double exponential dependence on temperature ( T) and unfrozen water content ( W) ( r 2>0.98): v= A exp( λT+ kW) and where A, λ and k are constants. The rate of 14CO 2 release from added glucose declined more steeply with cooling as compared with the release of total CO 2, indicating that (a) there could be some abiotic component in the measured flux of CO 2 or (b) endogenous respiration is more cold-resistant than substrate-induced respiration. The respiration activity was completely eliminated by soil sterilization (1 h, 121 °C), stimulated by the addition of oxidizable substrate (glucose, yeast extract), and reduced by the addition of acetate, which inhibits microbial processes in acidic soils (pH 3–5). The tundra soil from Barrow displayed higher below-zero activity than boreal soils from West Siberia and Sweden. The permafrost soils (20–30 cm) were more active than the samples from seasonally frozen topsoil (0–10 cm, Barrow). Finding measurable respiration to −39 °C is significant for determining, understanding, and predicting current and future CO 2 emission to the atmosphere and for understanding the low temperature limits of microbial activity on the Earth and on other planets.

Time-resolved measurement of dissipation-induced decoherence in a Josephson junction.

We determined the dissipation-induced decoherence time (DIDT) of a superconducting Josephson tunnel junction by time-resolved measurements of its escape dynamics. Double-exponential behavior of the time-dependent escape probability was observed, suggesting the occurrence of a two-level decay-tunneling process in which energy relaxation from the excited to the ground level significantly affects the escape dynamics of the system. The observation of temporal double-exponential dependence enables direct measurements of the DIDT, a property critical to the study of quantum dynamics and the realization of macroscopic quantum coherence and quantum computing. We found that the DIDT was tau(d) > 11 micros at T = 0.55 K, demonstrating good prospects for implementing quantum computing with Josephson devices.

Evaluation of diffusion coefficient and ionic mobility in (NH4)4Fe(CN)6·1.5 H2O

The diffusion coefficient, D , and the ionic mobility, μ, in the protonic conductor ammonium ferrocyanide hydrate have been determined by the isothermal transient ionic current method. D is also determined from the time dependence of the build up of potential across the samples and theretical expressions describing this build up in terms of double exponential dependence on time are obtained. The values obtained are D =3.875×10 −11 m 2 s −1 and μ=1.65×10 −9 m 2 V −1 s −1 .

Trajectory study of the formation and decay of silicon trimer complexes in monomer–dimer collisions

The dynamics of the formation and dissociation of Si3 complexes in atom–dimer collisions have been investigated using quasiclassical trajectories on a potential-energy surface that is a global fit to the available experimental data and ab initio CI and MP4 results. The fitted surface yields excellent results for the equilibrium geometries of both Si2 and Si3. The rms deviation of the energies from the MP4 calculations is 0.117 eV. The Si3 fundamental frequencies agree with the MP4 results to within 37 cm−1 or less. Formation cross sections for Si3 formation are reported as a function of initial relative translational energy. These cross sections are found to exhibit a near double exponential dependence upon relative translational energy. The thermal rate coefficients are on the order of 1015 cm3/mol s and show a negative temperature dependence at temperatures below 200 K. Microcanonical dissociation rate coefficients for Si3 complexes are reported for six different internal Si3 energies. These values follow an RRK energy dependence with s=2.67. The average Si3 lifetimes are found to lie in the range 4.71×10−13−5.02×10−11 s.

Study of the Anomalous Paramagnetic Relaxation Observed at Low Fields in a Kramers Pair Salt

Measurements of the paramagnetic relaxation rate were made for various concentrations of Fe+++ ions in K3Co(CN)6 (0.2–1.0 at.%) at low-field values (60–650 Oe) and low temperatures (1.24 &amp;lt; T &amp;lt; 1.38 °K) using the direct-observation sampled-recovery technique. The results obtained were found to be concentration dependent and to display at the higher concentrations a double exponential dependence on time. This behavior is at variance with the single-ion Kronig–Van Vleck theory that has been shown to apply at higher fields. It is shown that this behavior can be attributed to the existence of numerous alternate relaxation paths that involve exchange coupled clusters of magnetic complexes.