Search results for "Logarithm"
showing 10 items of 182 documents
Electrophoresis of model colloidal spheres in low salt aqueous suspension
2004
We report on comprehensive measurements of the electrophoretic mobility μ of a highly charged spherical colloid in deionized or low salt aqueous suspensions, where fluid and crystalline order develops with increased packing fraction Φ. We propose the existence of a 'universal' shape of the μ(Φ) showing three distinct regimes: a logarithmic increase, a plateau and a logarithmic decrease. The position and the height of the plateau are found to be influenced by the particle surface properties and the electrolyte concentration. In particular, it starts once the counter-ion concentration becomes equal to the concentration of background electrolyte. This coincides only loosely with the range of Φ…
Computation of flow velocity in rough channels
2006
Accurate estimate of flow-velocity profile is of crucial importance both for scientific purposes and for solving numerous engineering problems that include, among others, sediment transport, contaminant transport, flow resistance evaluation. This paper presents a new empirical equation to represent the vertical velocity profile. The proposed equation is essentially a modified form of the well-known logarithm law of the wall and contains three parameters having a clear physical meaning. The applicability of the equation and its accuracy assessment for different hydraulic conditions, including non-uniform conditions, is verified by using experimental data obtained by different sources. The va…
On the statistics of pairs of logarithms of integers
2022
We study the statistics of pairs of logarithms of positive integers at various scalings, either with trivial weights or with weights given by the Euler function, proving the existence of pair correlation functions. We prove that at the linear scaling, which is not the usual scaling by the inverse of the average gap, the pair correlations exhibit a level repulsion similar to radial distribution functions of fluids. We prove total loss of mass phenomena at superlinear scalings, and constant nonzero asymptotic behavior at sublinear scalings. The case of Euler weights has applications to the pair correlation of the lengths of common perpendicular geodesic arcs from the maximal Margulis cusp nei…
Uncommon Suffix Tries
2011
Common assumptions on the source producing the words inserted in a suffix trie with $n$ leaves lead to a $\log n$ height and saturation level. We provide an example of a suffix trie whose height increases faster than a power of $n$ and another one whose saturation level is negligible with respect to $\log n$. Both are built from VLMC (Variable Length Markov Chain) probabilistic sources; they are easily extended to families of sources having the same properties. The first example corresponds to a ''logarithmic infinite comb'' and enjoys a non uniform polynomial mixing. The second one corresponds to a ''factorial infinite comb'' for which mixing is uniform and exponential.
A subquadratic algorithm for minimum palindromic factorization
2014
We give an $\mathcal{O}(n \log n)$-time, $\mathcal{O}(n)$-space algorithm for factoring a string into the minimum number of palindromic substrings. That is, given a string $S [1..n]$, in $\mathcal{O}(n \log n)$ time our algorithm returns the minimum number of palindromes $S_1,\ldots, S_\ell$ such that $S = S_1 \cdots S_\ell$. We also show that the time complexity is $\mathcal{O}(n)$ on average and $\Omega(n\log n)$ in the worst case. The last result is based on a characterization of the palindromic structure of Zimin words.
Computational Limitations of Affine Automata
2019
We present two new results on the computational limitations of affine automata. First, we show that the computation of bounded-error rational-values affine automata is simulated in logarithmic space. Second, we give an impossibility result for algebraic-valued affine automata. As a result, we identify some unary languages (in logarithmic space) that are not recognized by algebraic-valued affine automata with cutpoints.
Finite state verifiers with constant randomness
2014
We give a new characterization of $\mathsf{NL}$ as the class of languages whose members have certificates that can be verified with small error in polynomial time by finite state machines that use a constant number of random bits, as opposed to its conventional description in terms of deterministic logarithmic-space verifiers. It turns out that allowing two-way interaction with the prover does not change the class of verifiable languages, and that no polynomially bounded amount of randomness is useful for constant-memory computers when used as language recognizers, or public-coin verifiers. A corollary of our main result is that the class of outcome problems corresponding to O(log n)-space …
Exponential sums related to Maass forms
2019
We estimate short exponential sums weighted by the Fourier coefficients of a Maass form. This requires working out a certain transformation formula for non-linear exponential sums, which is of independent interest. We also discuss how the results depend on the growth of the Fourier coefficients in question. As a byproduct of these considerations, we can slightly extend the range of validity of a short exponential sum estimate for holomorphic cusp forms. The short estimates allow us to reduce smoothing errors. In particular, we prove an analogue of an approximate functional equation previously proven for holomorphic cusp form coefficients. As an application of these, we remove the logarithm …
Symmetric logarithmic derivative of Fermionic Gaussian states
2018
In this article we derive a closed form expression for the symmetric logarithmic derivative of Fermionic Gaussian states. This provides a direct way of computing the quantum Fisher Information for Fermionic Gaussian states. Applications ranges from quantum Metrology with thermal states and non-equilibrium steady states with Fermionic many-body systems.
One-Loop Two and Three-Point Functions
2015
In this chapter we present a few relevant calculations of one-loop, one and two-point (scalar, vector and tensor) functions. IR and UV divergences are extensively treated. One example of IR-pole cancellation is presented. The two and three-body phase space integrals in D dimensions, needed for the calculation of IR divergent cross sections are also given. Last, the usage of the generic parametrization ( 6.27) for non-integer powers of propagators (which appear when one needs to integrate over the four-momentum, logarithmic functions that depend on the four-momentum) is shown with a simple two-loop example. With the tools given here, the reader should find straightforward to construct any hi…