Hole probability for non-interacting fermions in a d-dimensional trap

The hole probability, i.e., the probability that a region is void of particles, is a benchmark of correlations in many body systems. We compute analytically this probability P (R) for a sphere of radius R in the case of N noninteracting fermions in their ground state in a d-dimensional trapping potential. Using a connection to the Laguerre-Wishart ensembles of random matrices, we show that, for large N and in the bulk of the Fermi gas, P (R) is described by a universal scaling function of kF R, for which we obtain an exact formula (kF being the local Fermi wave-vector). It exhibits a super exponential tail P (R) / e-κd(kF R)d+1 where κdis a universal amplitude, in good agreement with existing numerical simulations. When R is of the order of the radius of the Fermi gas, the hole probability is described by a large deviation form which is not universal and which we compute exactly for the harmonic potential. Similar results also hold in momentum space.

Power partitions and a generalized eta transformation property

In their famous paper on partitions, Hardy and Ramanujan also raised the question of the behaviour of the number $p_s(n)$ of partitions of a positive integer~$n$ into $s$-th powers and gave some preliminary results. We give first an asymptotic formula to all orders, and then an exact formula, describing the behaviour of the corresponding generating function $P_s(q) = \prod_{n=1}^\infty \bigl(1-q^{n^s}\bigr)^{-1}$ near any root of unity, generalizing the modular transformation behaviour of the Dedekind eta-function in the case $s=1$. This is then combined with the Hardy-Ramanujan circle method to give a rather precise formula for $p_s(n)$ of the same general type of the one that they gave for~$s=1$. There are several new features, the most striking being that the contributions coming from various roots of unity behave very erratically rather than decreasing uniformly as in their situation. Thus in their famous calculation of $p(200)$ the contributions from arcs of the circle near roots of unity of order 1, 2, 3, 4 and 5 have 13, 5, 2, 1 and 1 digits, respectively, but in the corresponding calculation for $p_2(100000)$ these contributions have 60, 27, 4, 33, and 16 digits, respectively, of wildly varying sizes

Rozansky–Witten Theory, Localised Then Tilted

The paper has two parts, in the first part, we apply the localisation technique to the Rozansky–Witten theory on compact hyperkähler targets. We do so via first reformulating the theory as some supersymmetric sigma-model. We obtain the exact formula for the partition function with Wilson loops on $$S^1\times \Sigma _g$$ S 1 × Σ g and the lens spaces, the results match with earlier computations using Feynman diagrams on K3. The second part is motivated by a very curious paper (Gukov in J Geom Phys 168, 104311, 2021), where the equivariant index formula for the dimension of the Hilbert space of the Rozansky–Witten theory is interpreted as a kind of Verlinde formula. In this interpretation, the fixed points of the target hyperkähler geometry correspond to certain ‘states’. We extend the formalism of part one to incorporate equivariance on the target geometry. For certain non-compact hyperkähler geometry, we can apply the tilting theory to the derived category of coherent sheaves, whose objects label the Wilson loops, allowing us to pick a basis for the latter. We can then compute the fusion products in this basis and we show that the objects that have diagonal fusion rules are intimately related to the fixed points of the geometry. Using these objects as basis to compute the dimension of the Hilbert space leads back to the Verlinde formula, thus answering the question that motivated the paper.

Integer Programming Formulations For The Frobenius Problem

The Frobenius number of a set of relatively prime positive integers α1,α2,…,αn such that α1< α2< …< αn, is the largest integer that can not be written as a nonnegative integer linear combination of the given set. Finding the Frobenius number is known as the Frobenius problem, which is also named as the coin exchange problem or the postage stamp problem. This problem is closely related with the equality constrained integer knapsack problem. It is known that this problem is NP-hard. Extensive research has been conducted for finding the Frobenius number of a given set of positive integers. An exact formula exists for the case n=2 and various formulas have been derived for all special cases of n = 3. Many algorithms have been proposed for n≥4. As far as we are aware, there does not exist any integer programming approach for this problem which is the main motivation of this paper. We present four integer linear programming formulations about the Frobenius number of a given set of positive integers. Our first formulation is used to check if a given positive integer is the Frobenius number of a given set of positive integers. The second formulation aims at finding the Frobenius number directly. The third formulation involves the residue classes with respect to the least member of the given set of positive integers, where a residue table is computed comprising all values modulo that least member, and the Frobenius number is obtained from there. Based on the same approach underlying the third formulation, we propose our fourth formulation which produces the Frobenius number directly. We demonstrate how to use our formulations with several examples. For illustrative purposes, some computa-tional analysis is also presented.

The spectral line asymmetry of the Doppler effect in relativistic plasmas

{In this work, we present a comparative study between the relativistic and non- relativistic Doppler effects on spectral line profiles in ultra-hot plasmas at the laboratory system. We have established an exact formula of the relativistic Doppler profile in ultra-high-temperature plasma that is not a Gaussian one (unlike the nonrelativistic Doppler profile that is Gaussian). We have also derived a new FWHM (Full Width at Half Maximum) formula of the corresponding profile that is different from the non-relativistic FWHM (sqrtlog(T=M)). We have also shown that, in the relativistic case, Doppler broadening exhibits an asymmetry of spectral line profile (non- gaussian profile). To ensure the validity of our investigation, we have compared our theoretical calculation with the experimental results that shows a good agreement.

Kurtosis of von Neumann entanglement entropy

In this work, we study the statistical behavior of entanglement in quantum bipartite systems under the Hilbert-Schmidt ensemble as assessed by the standard measure - the von Neumann entropy. Expressions of the first three exact cumulants of von Neumann entropy are known in the literature. The main contribution of the present work is the exact formula of the corresponding fourth cumulant that controls the tail behavior of the distribution. As a key ingredient in deriving the result, we make use of what we refer to as unsimplifiable summation bases leading to a complete cancellation. In addition to providing further evidence of the conjectured Gaussian limit of the von Neumann entropy, the obtained formula also provides an improved finite-size approximation to the distribution.

An Analytical Model for Spatially Varying Clear-Sky CO2 Forcing

Clear-sky CO2 forcing is known to vary significantly over the globe, but the state dependence which controls this is not well understood. Here we extend the formalism of Wilson and Gea-Banacloche (2012) to obtain a quantitatively accurate analytical model for spatially-varying instantaneous CO2 forcing, which depends only on surface temperature Ts, stratospheric temperature, and column relative humidity RH. This model shows that CO2 forcing can be considered a swap of surface emission for stratospheric emission, and thus depends primarily on surface-stratosphere temperature contrast. The strong meridional gradient in CO2 forcing is thus largely due to the strong meridional gradient in Ts. In the tropics and mid-latitudes, however, the presence of H2O modulates the forcing by replacing surface emission with RH-dependent atmospheric emission. This substantially reduces the forcing in the tropics, introduces forcing variations due to spatially-varying RH, and sets an upper limit (with respect to Ts variations) on CO2 forcing which is reached in the present-day tropics.In addition, we extend our analytical model to the instantaneous tropopause forcing, and find that this forcing depends on Ts only, with no dependence on stratospheric temperature. We also analyze the ‘τ = 1’ approximation for the emission level, and derive an exact formula for the emission level which yields values closer to τ = 1/2 than to τ = 1.

Fast and exact quantification of motif occurrences in biological sequences

Background Identification of motifs and quantification of their occurrences are important for the study of genetic diseases, gene evolution, transcription sites, and other biological mechanisms. Exact formulae for estimating count distributions of motifs under Markovian assumptions have high computational complexity and are impractical to be used on large motif sets. Approximated formulae, e.g. based on compound Poisson, are faster, but reliable p value calculation remains challenging. Here, we introduce ‘motif_prob’, a fast implementation of an exact formula for motif count distribution through progressive approximation with arbitrary precision. Our implementation speeds up the exact calculation, usually impractical, making it feasible and posit to substitute currently employed heuristics. Results We implement motif_prob in both Perl and C+ + languages, using an efficient error-bound iterative process for the exact formula, providing comparison with state-of-the-art tools (e.g. MoSDi) in terms of precision, run time benchmarks, along with a real-world use case on bacterial motif characterization. Our software is able to process a million of motifs (13–31 bases) over genome lengths of 5 million bases within the minute on a regular laptop, and the run times for both the Perl and C+ + code are several orders of magnitude smaller (50–1000× faster) than MoSDi, even when using their fast compound Poisson approximation (60–120× faster). In the real-world use cases, we first show the consistency of motif_prob with MoSDi, and then how the p-value quantification is crucial for enrichment quantification when bacteria have different GC content, using motifs found in antimicrobial resistance genes. The software and the code sources are available under the MIT license at https://github.com/DataIntellSystLab/motif_prob. Conclusions The motif_prob software is a multi-platform and efficient open source solution for calculating exact frequency distributions of motifs. It can be integrated with motif discovery/characterization tools for quantifying enrichment and deviation from expected frequency ranges with exact p values, without loss in data processing efficiency.

Quantitative traits under selection: derivations of distributions, higher moments, and the effects of recombination

The mathematical theory of quantitative traits is over one hundred years old but it is still a fertile area for research and analysis. However, the effects of selection on a quantitative trait, while well understood for the effects on the mean and variance, have traditionally been difficult to attack from the perspective of analyzing the probability density of the breeding values and deriving higher (third and fourth) moments as well as analyzing the impact of recombination. In this paper, the exact formula for the breeding value distribution after selection is derived and, using new integral tables, the first four moments are given exact expressions for the first time. In addition, the effects of recombination on the full distribution of breeding values are demonstrated. Finally, the changes of GXE covariance in the selected parent population caused by factors similar to the Bulmer Effect are also investigated in detail.