Hitting spheres on hyperbolic spaces

For a hyperbolic Brownian motion on the Poincar\'e half-plane $\mathbb{H}^2$, starting from a point of hyperbolic coordinates $z=(\eta, \alpha)$ inside a hyperbolic disc $U$ of radius $\bar{\eta}$, we obtain the probability of hitting the boundary $\partial U$ at the point $(\bar \eta,\bar \alpha)$. For $\bar{\eta} \to \infty$ we derive the asymptotic Cauchy hitting distribution on $\partial \mathbb{H}^2$ and for small values of $\eta$ and $\bar \eta$ we obtain the classical Euclidean Poisson kernel. The exit probabilities $\mathbb{P}_z\{T_{\eta_1}<T_{\eta_2}\}$ from a hyperbolic annulus in $\mathbb{H}^2$ of radii $\eta_1$ and $\eta_2$ are derived and the transient behaviour of hyperbolic Brownian motion is considered. Similar probabilities are calculated also for a Brownian motion on the surface of the three dimensional sphere. For the hyperbolic half-space $\mathbb{H}^n$ we obtain the Poisson kernel of a ball in terms of a series involving Gegenbauer polynomials and hypergeometric functions. For small domains in $\mathbb{H}^n$ we obtain the $n$-dimensional Euclidean Poisson kernel. The exit probabilities from an annulus are derived also in the $n$-dimensional case

On the correlation between critical points and critical values for random spherical harmonics

We study the correlation between the total number of critical points of random spherical harmonics and the number of critical points with value in any interval I ⊂ ℝ. We show that the correlation is asymptotically zero, while the partial correlation, after controlling the random L2-norm on the sphere of the eigenfunctions, is asymptotically one. Our findings complement the results obtained by Wigman (2012) and Marinucci and Rossi (2021) on the correlation between nodal and boundary length of random spherical harmonics

Patch-repetition correlation length in glassy systems

We obtain the patch-repetition entropy Sigma within the Random First Order Transition theory (RFOT) and for the square plaquette system, a model related to the dynamical facilitation theory of glassy dynamics. We find that in both cases the entropy of patches of linear size l, Sigma(l), scales as s_c l^d+A l^{d-1} down to length-scales of the order of one, where A is a positive constant, s_c is the configurational entropy density and d the spatial dimension. In consequence, the only meaningful length that can be defined from patch-repetition is the cross-over length xi=A/s_c. We relate xi to the typical length-scales already discussed in the literature and show that it is always of the order of the largest static length. Our results provide new insights, which are particularly relevant for RFOT theory, on the possible real space structure of super-cooled liquids. They suggest that this structure differs from a mosaic of different patches having roughly the same size.Comment: 6 page

Surface tension fluctuations and a new spinodal point in glass-forming liquids

The dramatic slowdown of glass-forming liquids has been variously linked to increasing dynamic and static correlation lengths. Yet, empirical evidence is insufficient to decide among competing theories. The random first order theory (RFOT) links the dynamic slowdown to the growth of amorphous static order, whose range depends on a balance between configurational entropy and surface tension. This last quantity is expected to vanish when the temperature surpasses a spinodal point beyond which there are no metastable states. Here we measure for the first time the surface tension in a model glass-former, and find that it vanishes at the energy separating minima from saddles, demonstrating the existence of a spinodal point for amorphous metastable order. Moreover, the fluctuations of surface tension become smaller for lower temperatures, in quantitative agreement with recent theoretical speculation that spatial correlations in glassy systems relax nonexponentially because of the narrowing of the surface tension distribution.Comment: 6 pages, 5 figure

A phase-separation perspective on dynamic heterogeneities in glass-forming liquids

We study dynamic heterogeneities in a model glass-former whose overlap with a reference configuration is constrained to a fixed value. The system phase-separates into regions of small and large overlap, so that dynamical correlations remain strong even for asymptotic times. We calculate an appropriate thermodynamic potential and find evidence of a Maxwell's construction consistent with a spinodal decomposition of two phases. Our results suggest that dynamic heterogeneities are the expression of an ephemeral phase-separating regime ruled by a finite surface tension

Numerical simulations of liquids with amorphous boundary conditions

It has recently become clear that simulations under amorphpous boundary conditions (ABCs) can provide valuable information on the dynamics and thermodynamics of disordered systems with no obvious ordered parameter. In particular, they allow to detect a correlation length that is not measurable with standard correlation functions. Here we explain what exactly is meant by ABCs, discuss their relation with point-to-set correlations and briefly describe some recent results obtained with this technique.Comment: Presented at STATPHYS 2

Who has the last word? Understanding How to Sample Online Discussions

In online debates individual arguments support or attack each other, leading to some subset of arguments being considered more relevant than others. However, in large discussions readers are often forced to sample a subset of the arguments being put forth. Since such sampling is rarely done in a principled manner, users may not read all the relevant arguments to get a full picture of the debate. This paper is interested in answering the question of how users should sample online conversations to selectively favour the currently justified or accepted positions in the debate. We apply techniques from argumentation theory and complex networks to build a model that predicts the probabilities of the normatively justified arguments given their location in online discussions. Our model shows that the proportion of replies that are supportive, the number of replies that comments receive, and the locations of un-replied comments all determine the probability that a comment is a justified argument. We show that when the degree distribution of the number of replies is homogeneous along the discussion, for acrimonious discussions, the distribution of justified arguments depends on the parity of the graph level. In supportive discussions the probability of having justified comments increases as one moves away from the root. For discussion trees that have a non-homogeneous in-degree distribution, for supportive discussions we observe the same behaviour as before, while for acrimonious discussions we cannot observe the same parity-based distribution. This is verified with data obtained from the online debating platform Kialo. By predicting the locations of the justified arguments in reply trees, we can suggest which arguments readers should sample to grasp the currently accepted opinions in such discussions. Our models have important implications for the design of future online debating platforms

A potential role of il-6/il-6r in the development and management of colon cancer

Colorectal cancer (CRC) is the third most frequent cancer worldwide and the second greatest cause of cancer deaths. About 75% of all CRCs are sporadic cancers and arise following somatic mutations, while about 10% are hereditary cancers caused by germline mutations in specific genes. Several factors, such as growth factors, cytokines, and genetic or epigenetic alterations in specific oncogenes or tumor-suppressor genes, play a role during the adenoma–carcinoma sequence. Recent studies have reported an increase in interleukin-6 (IL-6) and soluble interleukin-6 receptor (sIL-6R) levels in the sera of patients affected by colon cancer that correlate with the tumor size, suggesting a potential role for IL-6 in colon cancer progression. IL-6 is a pleiotropic cytokine showing both pro-and anti-inflammatory roles. Two different types of IL-6 signaling are known. Classic IL-6 signaling involves the binding of IL-6 to its membrane receptor on the surfaces of target cells; alternatively, IL-6 binds to sIL-6R in a process called IL-6 trans-signaling. The activation of IL-6 transsignaling by metalloproteinases has been described during colon cancer progression and metastasis, involving a shift from membrane-bound interleukin-6 receptor (IL-6R) expression on the tumor cell surface toward the release of soluble IL-6R. In this review, we aim to shed light on the role of IL-6 signaling pathway alterations in sporadic colorectal cancer and the development of familial polyposis syndrome. Furthermore, we evaluate the possible roles of IL-6 and IL-6R as biomarkers useful in disease follow-up and as potential targets for therapy, such as monoclonal antibodies against IL-6 or IL-6R, or a food-based approach against IL-6

An experimental and theoretical approach for an estimation of DKth

The existence of a fatigue threshold value may affect the design process when a damage-tolerant design is considered that uses non-destructive techniques for evaluating the shape and dimensions of the defects inside materials. Obviously it should be possible to estimate the stress field surrounding these defects and this is not generally a problem with modern numerical methods.Many factors are involved in determining the growth rate of a fatigue crack. Some of these are highly significant and it is possible to obtain the coefficients of a correlation function. Some others are not well defined and the only effect is to expand the scatter of experimental data.Consider the sigmoidal curve we obtain when plotting the crack growth rate versus the applied DK_I . A very difficult parameter to measure but very useful for fatigue design is the DK_Ith value, because below this value a crack may be forming, hence, here DK_Ith is defined by the transition between a normal (e.g. 10-10 m/cycle) and a very low range of crack growth rate (&lt;10-10 m/cycle).The DgrKIth value is very difficult to obtain by experimental methods because the growth rate is of the order or less than the atomic lattice span (3 × 10-10 m/cycle), but we can correlate the transition value with the cyclic crack tip plastic zone size and other structural parameters of metallic materials.The aim of this work is to offer a contribution about the parameters which influence DK_Ith in stainless steels and welded joints based on the crack tip plastic zone radius

Marvels and Pitfalls of the Langevin Algorithm in Noisy High-Dimensional Inference

Gradient-descent-based algorithms and their stochastic versions have widespread applications in machine learning and statistical inference. In this work, we carry out an analytic study of the performance of the algorithm most commonly considered in physics, the Langevin algorithm, in the context of noisy high-dimensional inference. We employ the Langevin algorithm to sample the posterior probability measure for the spiked mixed matrix-tensor model. The typical behavior of this algorithm is described by a system of integrodifferential equations that we call the Langevin state evolution, whose solution is compared with the one of the state evolution of approximate message passing (AMP). Our results show that, remarkably, the algorithmic threshold of the Langevin algorithm is suboptimal with respect to the one given by AMP. This phenomenon is due to the residual glassiness present in that region of parameters. We also present a simple heuristic expression of the transition line, which appears to be in agreement with the numerical results