25,032 research outputs found

    Asymptotics for kk-crank of kk-colored partitions

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    In this paper, we obtain asymptotic formulas for kk-crank of kk-colored partitions. Let Mk(a,c;n)M_k(a, c; n) denote the number of kk-colored partitions of nn with a kk-crank congruent to aa mod cc. For the cases k=2,3,4k=2,3,4, Fu and Tang derived several inequality relations for Mk(a,c;n)M_k(a, c; n) using generating functions. We employ the Hardy-Ramanujan Circle Method to extend the results of Fu and Tang. Furthermore, additional inequality relations for Mk(a,c;n)M_k(a, c; n) have been established, such as logarithmic concavity and logarithmic subadditivity.Comment: 40 pages. arXiv admin note: text overlap with arXiv:1311.4344 by other author

    Chiral active fluids: Odd viscosity, active turbulence, and directed flows of hydrodynamic microrotors

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    While the number of publications on rotating active matter has rapidly increased in recent years, studies on purely hydrodynamically interacting rotors on the microscale are still rare, especially from the perspective of particle based hydrodynamic simulations. The work presented here targets to fill this gap. By means of high-performance computer simulations, performed in a highly parallelised fashion on graphics processing units, the dynamics of ensembles of up to 70,000 rotating colloids immersed in an explicit mesoscopic solvent consisting out of up to 30 million fluid particles, are investigated. Some of the results presented in this thesis have been worked out in collaboration with experimentalists, such that the theoretical considerations developed in this thesis are supported by experiments, and vice versa. The studied system, modelled in order to resemble the essential physics of the experimentally realisable system, consists out of rotating magnetic colloidal particles, i.e., (micro-)rotors, rotating in sync to an externally applied magnetic field, where the rotors solely interact via hydrodynamic and steric interactions. Overall, the agreement between simulations and experiments is very good, proving that hydrodynamic interactions play a key role in this and related systems. While already an isolated rotating colloid is driven out of equilibrium, only collections of two or more rotors have experimentally shown to be able to convert the rotational energy input into translational dynamics in an orbital rotating fashion. The rotating colloids inject circular flows into the fluid, such that detailed balance is broken, and it is not a priori known whether equilibrium properties of colloids can be extended to isolated rotating colloids. A joint theoretical and experimental analysis of isolated, pairs, and small groups of hydrodynamically interacting rotors is given in chapter 2. While the translational dynamics of isolated rotors effectively resemble the dynamics of non-rotating colloids, the orbital rotation of pairs of rotors can be described with leading order hydrodynamics and a two-dimensional analogy of Faxén’s law is derived. In chapter 3, a homogeneously distributed ensemble of rotors (bulk) as a realisation of a chiral active fluid is studied and it is explicitly shown computationally and experimentally that it carries odd viscosity. The mutual orbital translation of rotors and an increase of the effective solvent viscosity with rotor density lead to a non-monotonous behaviour of the average translational velocity. Meanwhile, the rotor suspension bears a finite osmotic compressibility resulting from the long-ranged nature of hydrody- namic interactions such that rotational and odd stresses are transmitted through the solvent also at small and intermediate rotor densities. Consequently, density inhomogeneities predicted for chiral active fluids with odd viscosity can be found and allow for an explicit measurement of odd viscosity in simulations and experiments. At intermediate densities, the collective dynamics shows the emergence of multi-scale vortices and chaotic motion which is identified as active turbulence with a self-similar power-law decay in the energy spectrum, showing that the injected energy on the rotor scale is transported to larger scales, similar to the inverse energy cascade of clas- sical two-dimensional turbulence. While either odd viscosity or active turbulence have been reported in chiral active matter previously, the system studied here shows that the emergence of both simultaneously is possible resulting from the osmotic compressibility and hydrodynamic mediation of odd and active stresses. The collective dynamics of colloids rotating out of phase, i.e., where a constant torque instead of a constant angular velocity is applied, is shown to be qualitatively very similar. However, at smaller densities, local density inhomogeneities imply position dependent angular velocities of the rotors resulting from inter-rotor friction. While the friction of a quasi-2D layer of active colloids with the substrate is often not easily modifiable in experiments, the incorporation of substrate friction into the simulation models typically implies a considerable increase in computational effort. In chapter 4, a very efficient way of incorporating the friction with a substrate into a two-dimensional multiparticle collision dynamics solvent is introduced, allowing for an explicit investigation of the influences of substrate on active dynamics. For the rotor fluid, it is explicitly shown that the influence of the substrate friction results in a cutoff of the hydrodynamic interaction length, such that the maximum size of the formed vortices is controlled by the substrate friction, also resulting in a cutoff in the energy spectrum, because energy is taken out of the system at the respective length. These findings are in agreement with the experiments. Since active particles in confinement are known to organise in states of collective dynamics, ensembles of rotationally actuated colloids are studied in circular confinement and in the presence of periodic obstacle lattices in chapters 5 and 6, respectively. The results show that the chaotic active turbulent transport of rotors in suspension can be enhanced and guided resulting from edge flows generated at the boundaries, as has recently been reported for a related chiral active system. The consequent collective rotor dynamics can be regarded as a superposition of active turbulent and imposed flows, leading to on average stationary flows. In contrast to the bulk dynamics, the imposed flows inject additional energy into the system on the long length scales, and the same scaling behaviour of the energy spectrum as in bulk is only obtained if the energy injection scales, due to the mutual generation of rotor translational dynamics throughout the system and the edge flows, are well separated. The combination of edge flow and entropic layering at the boundaries leads to oscillating hydrodynamic stresses and consequently to an oscillating vorticity profile. In the presence of odd viscosity, this consequently leads to non-trivial steady-state density modulations at the boundary, resulting from a balance of osmotic pressure and odd stresses. Relevant for the efficient dispersion and mixing of inert particles on the mesoscale by means of active turbulent mixing powered by rotors, a study of the dynamics of a binary mixture consisting out of rotors and passive particles is presented in chapter 7. Because the rotors are not self-propelled, but the translational dynamics is induced by the surrounding rotors, the passive particles, which do not inject further energy into the system, are transported according to the same mechanism as the rotors. The collective dynamics thus resembles the pure rotor bulk dynamics at the respective density of only rotors. However, since no odd stresses act between the passive particles, only mutual rotor interactions lead to odd stresses leading to the accumulation of rotors in the regions of positive vorticity. This density increase is associated with a pressure increase, which balances the odd stresses acting on the rotors. However, the passive particles are only subject to the accumulation induced pressure increase such that these particles are transported into the areas of low rotor concentration, i.e., the regions of negative vorticity. Under conditions of sustained vortex flow, this results in segregation of both particle types. Since local symmetry breaking can convert injected rotational into translational energy, microswimmers can be constructed out of rotor materials when a suitable breaking of symmetry is kept in the vicinity of a rotor. One hypothetical realisation, i.e., a coupled rotor pair consisting out of two rotors of opposite angular velocity and of fixed distance, termed a birotor, are studied in chapter 8. The birotor pumps the fluid into one direction and consequently translates into the opposite direction, and creates a flow field reminiscent of a source doublet, or sliplet flow field. Fixed in space the birotor might be an interesting realisation of a microfluidic pump. The trans- lational dynamics of a birotor can be mapped onto the active Brownian particle model for single swimmers. However, due to the hydrodynamic interactions among the rotors, the birotor ensemble dynamics do not show the emergence of stable motility induced clustering. The reason for this is the flow created by birotor in small aggregates which effectively pushes further arriving birotors away from small aggregates, which eventually are all dispersed by thermal fluctuations

    Multidimensional adaptive order GP-WENO via kernel-based reconstruction

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    This paper presents a fully multidimensional kernel-based reconstruction scheme for finite volume methods applied to systems of hyperbolic conservation laws, with a particular emphasis on the compressible Euler equations. Non-oscillatory reconstruction is achieved through an adaptive order weighted essentially non-oscillatory (WENO-AO) method cast into a form suited to multidimensional stencils and reconstruction. A kernel-based approach inspired by Gaussian process (GP) modeling is presented here. This approach allows the creation of a scheme of arbitrary order with simply defined multidimensional stencils and substencils. Furthermore, the fully multidimensional nature of the reconstruction allows a more straightforward extension to higher spatial dimensions and removes the need for complicated boundary conditions on intermediate quantities in modified dimension-by-dimension methods. In addition, a new simple-yet-effective set of reconstruction variables is introduced, as well as an easy-to-implement effective limiter for positivity preservation, both of which could be useful in existing schemes with little modification. The proposed scheme is applied to a suite of stringent and informative benchmark problems to demonstrate its efficacy and utility.Comment: Submitted to Journal of Computational Physics April 202

    Infinitely wide limits for deep Stable neural networks: sub-linear, linear and super-linear activation functions

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    There is a growing literature on the study of large-width properties of deep Gaussian neural networks (NNs), i.e. deep NNs with Gaussian-distributed parameters or weights, and Gaussian stochastic processes. Motivated by some empirical and theoretical studies showing the potential of replacing Gaussian distributions with Stable distributions, namely distributions with heavy tails, in this paper we investigate large-width properties of deep Stable NNs, i.e. deep NNs with Stable-distributed parameters. For sub-linear activation functions, a recent work has characterized the infinitely wide limit of a suitable rescaled deep Stable NN in terms of a Stable stochastic process, both under the assumption of a ``joint growth" and under the assumption of a ``sequential growth" of the width over the NN's layers. Here, assuming a ``sequential growth" of the width, we extend such a characterization to a general class of activation functions, which includes sub-linear, asymptotically linear and super-linear functions. As a novelty with respect to previous works, our results rely on the use of a generalized central limit theorem for heavy tails distributions, which allows for an interesting unified treatment of infinitely wide limits for deep Stable NNs. Our study shows that the scaling of Stable NNs and the stability of their infinitely wide limits may depend on the choice of the activation function, bringing out a critical difference with respect to the Gaussian setting.Comment: 20 pages, 2 figure

    Diverging black hole entropy from quantum infrared non-localities

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    Local higher-derivative corrections to the Einstein-Hilbert action yield sub-leading corrections to the Bekenstein-Hawking area law. Here we show that if the quantum effective action comprises a certain class of infrared non-localities, the entropy of large black holes generally diverges to either positive or negative infinity. In such theories, large spherically symmetric black holes would be either highly chaotic or thermodynamically impossible, respectively. In turn, this puts strong constraints on the Laurent expansion of the form factors in the effective action.Comment: 5 pages + appendi

    Universality in prelimiting tail behavior for regular subgraph counts in the Poisson regime

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    Let NN be the number of copies of a small subgraph HH in an Erd\H{o}s-R\'enyi graph GG(n,pn)G \sim \mathcal{G}(n, p_n) where pn0p_n \to 0 is chosen so that EN=c\mathbb{E} N = c, a constant. Results of Bollob\'as show that for regular graphs HH, the count NN weakly converges to a Poisson random variable. For large but finite nn, and for the specific case of the triangle, investigations of the upper tail P(Nkn)\mathbb{P}(N \geq k_n) by Ganguly, Hiesmayr and Nam (2022) revealed that there is a phase transition in the tail behavior and the associated mechanism. Smaller values of knk_n correspond to disjoint occurrences of HH, leading to Poisson tails, with a different behavior emerging when knk_n is large, guided by the appearance of an almost clique. We show that a similar phase transition also occurs when HH is any regular graph, at the point where kn12/qlogkn=lognk_n^{1 -2/q}\log k_n = \log n (qq is the number of vertices in HH). This establishes universality of this transition, previously known only for the case of the triangle.Comment: 23 pages, 3 figure

    3d mirror symmetry of braided tensor categories

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    We study the braided tensor structure of line operators in the topological A and B twists of abelian 3d N=4\mathcal{N}=4 gauge theories, as accessed via boundary vertex operator algebras (VOA's). We focus exclusively on abelian theories. We first find a non-perturbative completion of boundary VOA's in the B twist, which start out as certain affine Lie superalebras; and we construct free-field realizations of both A and B-twist VOA's, finding an interesting interplay with the symmetry fractionalization group of bulk theories. We use the free-field realizations to establish an isomorphism between A and B VOA's related by 3d mirror symmetry. Turning to line operators, we extend previous physical classifications of line operators to include new monodromy defects and bound states. We also outline a mechanism by which continuous global symmetries in a physical theory are promoted to higher symmetries in a topological twist -- in our case, these are infinite one-form symmetries, related to boundary spectral flow, which structure the categories of lines and control abelian gauging. Finally, we establish the existence of braided tensor structure on categories of line operators, viewed as non-semisimple categories of modules for boundary VOA's. In the A twist, we obtain the categories by extending modules of symplectic boson VOA's, corresponding to gauging free hypermultiplets; in the B twist, we instead extend Kazhdan-Lusztig categories for affine Lie superalgebras. We prove braided tensor equivalences among the categories of 3d-mirror theories. All results on VOA's and their module categories are mathematically rigorous; they rely strongly on recently developed techniques to access non-semisimple extensions.Comment: 158 pages, comments welcome

    Optimal distributed control for a viscous non-local tumour growth model

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    In this paper, we address an optimal distributed control problem for a non-local model of phase-field type, describing the evolution of tumour cells in presence of a nutrient. The model couples a non-local and viscous Cahn-Hilliard equation for the phase parameter with a reaction-diffusion equation for the nutrient. The optimal control problem aims at finding a therapy, encoded as a source term in the system, both in the form of radiotherapy and chemotherapy, which could lead to the evolution of the phase variable towards a desired final target. First, we prove strong well-posedness for the system of non-linear partial differential equations. In particular, due to the presence of a viscous regularisation, we can also consider double-well potentials of singular type and cross-diffusion terms related to the effects of chemotaxis. Moreover, the particular structure of the reaction terms allows us to prove new regularity results for this kind of system. Then, turning to the optimal control problem, we prove the existence of an optimal therapy and, by studying Fr\'echet-differentiability properties of the control-to-state operator and the corresponding adjoint system, we obtain the first-order necessary optimality conditions.Comment: 43 page

    Moduli Stabilisation and the Statistics of Low-Energy Physics in the String Landscape

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    In this thesis we present a detailed analysis of the statistical properties of the type IIB flux landscape of string theory. We focus primarily on models constructed via the Large Volume Scenario (LVS) and KKLT and study the distribution of various phenomenologically relevant quantities. First, we compare our considerations with previous results and point out the importance of Kähler moduli stabilisation, which has been neglected in this context so far. We perform different moduli stabilisation procedures and compare the resulting distributions. To this end, we derive the expressions for the gravitino mass, various quantities related to axion physics and other phenomenologically interesting quantities in terms of the fundamental flux dependent quantities gsg_s, W0W_0 and n\mathfrak{n}, the parameter which specifies the nature of the non-perturbative effects. Exploiting our knowledge of the distribution of these fundamental parameters, we can derive a distribution for all the quantities we are interested in. For models that are stabilised via LVS we find a logarithmic distribution, whereas for KKLT and perturbatively stabilised models we find a power-law distribution. We continue by investigating the statistical significance of a newly found class of KKLT vacua and present a search algorithm for such constructions. We conclude by presenting an application of our findings. Given the mild preference for higher scale supersymmetry breaking, we present a model of the early universe, which allows for additional periods of early matter domination and ultimately leads to rather sharp predictions for the dark matter mass in this model. We find the dark matter mass to be in the very heavy range mχ10101011 GeVm_{\chi}\sim 10^{10}-10^{11}\text{ GeV}
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