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We consider the problem of estimating the roughness of the volatility in a stochastic volatility model that arises as a nonlinear function of fractional Brownian motion with drift. To this end, we introduce a new estimator that measures the so-called roughness exponent of a continuous trajectory, based on discrete observations of its antiderivative. We provide conditions on the underlying trajectory under which our estimator converges in a strictly pathwise sense. Then we verify that these conditions are satisfied by almost every sample path of fractional Brownian motion (with drift). As a consequence, we obtain strong consistency theorems in the context of a large class of rough volatility models. Numerical simulations show that our estimation procedure performs well after passing to a scale-invariant modification of our estimator.

A variant of the standard notion of branching bisimilarity for processes with discrete relative timing is proposed which is coarser than the standard notion. Using a version of ACP (Algebra of Communicating Processes) with abstraction for processes with discrete relative timing, it is shown that the proposed variant allows of both the functional correctness and the performance properties of the PAR (Positive Acknowledgement with Retransmission) protocol to be analyzed. In the version of ACP concerned, the difference between the standard notion of branching bisimilarity and its proposed variant is characterized by a single axiom schema.

We present a rigorous and precise analysis of the maximum degree and the average degree in a dynamic duplication-divergence graph model introduced by Sol\'e, Pastor-Satorras et al. in which the graph grows according to a duplication-divergence mechanism, i.e. by iteratively creating a copy of some node and then randomly alternating the neighborhood of a new node with probability $p$. This model captures the growth of some real-world processes e.g. biological or social networks. In this paper, we prove that for some $0 < p < 1$ the maximum degree and the average degree of a duplication-divergence graph on $t$ vertices are asymptotically concentrated with high probability around $t^p$ and $\max\{t^{2 p - 1}, 1\}$, respectively, i.e. they are within at most a polylogarithmic factor from these values with probability at least $1 - t^{-A}$ for any constant $A > 0$.

This article study the average conditioning for a random underdetermined polynomial system. The expected value of the moments of the condition number are compared to the moments of the condition number of random matrices. An expression for these moments is given by studying the kernel finding problem for random matrices. Furthermore, the second moment of the Frobenius condition number is computed.

Consider minimizing the entropy of a mixture of states by choosing each state subject to constraints. If the spectrum of each state is fixed, we expect that in order to reduce the entropy of the mixture, we should make the states less distinguishable in some sense. Here, we study a class of optimization problems that are inspired by this situation and shed light on the relevant notions of distinguishability. The motivation for our study is the recently introduced spin alignment conjecture. In the original version of the underlying problem, each state in the mixture is constrained to be a freely chosen state on a subset of $n$ qubits tensored with a fixed state $Q$ on each of the qubits in the complement. According to the conjecture, the entropy of the mixture is minimized by choosing the freely chosen state in each term to be a tensor product of projectors onto a fixed maximal eigenvector of $Q$, which maximally "aligns" the terms in the mixture. We generalize this problem in several ways. First, instead of minimizing entropy, we consider maximizing arbitrary unitarily invariant convex functions such as Fan norms and Schatten norms. To formalize and generalize the conjectured required alignment, we define alignment as a preorder on tuples of self-adjoint operators that is induced by majorization. We prove the generalized conjecture for Schatten norms of integer order, for the case where the freely chosen states are constrained to be classical, and for the case where only two states contribute to the mixture and $Q$ is proportional to a projector. The last case fits into a more general situation where we give explicit conditions for maximal alignment. The spin alignment problem has a natural "dual" formulation, versions of which have further generalizations that we introduce.

We study Lindstrom quantifiers that satisfy certain closure properties which are motivated by the study of polymorphisms in the context of constraint satisfaction problems (CSP). When the algebra of polymorphisms of a finite structure B satisfies certain equations, this gives rise to a natural closure condition on the class of structures that map homomorphically to B. The collection of quantifiers that satisfy closure conditions arising from a fixed set of equations are rather more general than those arising as CSP. For any such conditions P, we define a pebble game that delimits the distinguishing power of the infinitary logic with all quantifiers that are P-closed. We use the pebble game to show that the problem of deciding whether a system of linear equations is solvable in Z2 is not expressible in the infinitary logic with all quantifiers closed under a near-unanimity condition.

The solutions of scalar ordinary differential equations become more complex as their coefficients increase in magnitude. As a consequence, when a standard solver is applied to such an equation, its running time grows with the magnitudes of the equation's coefficients. It is well known, however, that scalar ordinary differential equations with slowly-varying coefficients admit slowly-varying phase functions whose cost to represent via standard techniques is largely independent of the magnitude of the equation's coefficients. This observation is the basis of most methods for the asymptotic approximation of the solutions of ordinary differential equations, including the WKB method. Here, we introduce two numerical algorithms for constructing phase functions for scalar ordinary differential equations inspired by the classical Levin method for the calculation of oscillatory integrals. In the case of a large class of scalar ordinary differential equations with slowly-varying coefficients, their running times are independent of the magnitude of the equation's coefficients. The results of extensive numerical experiments demonstrating the properties of our algorithms are presented.

This note presents a refined local approximation for the logarithm of the ratio between the negative multinomial probability mass function and a multivariate normal density, both having the same mean-covariance structure. This approximation, which is derived using Stirling's formula and a meticulous treatment of Taylor expansions, yields an upper bound on the Hellinger distance between the jittered negative multinomial distribution and the corresponding multivariate normal distribution. Upper bounds on the Le Cam distance between negative multinomial and multivariate normal experiments ensue.

Many mechanisms behind the evolution of cooperation, such as reciprocity, indirect reciprocity, and altruistic punishment, require group knowledge of individual actions. But what keeps people cooperating when no one is looking? Conformist norm internalization, the tendency to abide by the behavior of the majority of the group, even when it is individually harmful, could be the answer. In this paper, we analyze a world where (1) there is group selection and punishment by indirect reciprocity but (2) many actions (half) go unobserved, and therefore unpunished. Can norm internalization fill this `observation gap' and lead to high levels of cooperation, even when agents may in principle cooperate only when likely to be caught and punished? Specifically, we seek to understand whether adding norm internalization to the strategy space in a public goods game can lead to higher levels of cooperation when both norm internalization and cooperation start out rare. We found the answer to be positive, but, interestingly, not because norm internalizers end up making up a substantial fraction of the population, nor because they cooperate much more than other agent types. Instead, norm internalizers, by polarizing, catalyzing, and stabilizing cooperation, can increase levels of cooperation of other agent types, while only making up a minority of the population themselves.

Particle methods based on evolving the spatial derivatives of the solution were originally introduced to simulate reaction-diffusion processes, inspired by vortex methods for the Navier--Stokes equations. Such methods, referred to as gradient random walk methods, were extensively studied in the '90s and have several interesting features, such as being grid free, automatically adapting to the solution by concentrating elements where the gradient is large and significantly reducing the variance of the standard random walk approach. In this work, we revive these ideas by showing how to generalize the approach to a larger class of partial differential equations, including hyperbolic systems of conservation laws. To achieve this goal, we first extend the classical Monte Carlo method to relaxation approximation of systems of conservation laws, and subsequently consider a novel particle dynamics based on the spatial derivatives of the solution. The methodology, combined with asymptotic-preserving splitting discretization, yields a way to construct a new class of gradient-based Monte Carlo methods for hyperbolic systems of conservation laws. Several results in one spatial dimension for scalar equations and systems of conservation laws show that the new methods are very promising and yield remarkable improvements compared to standard Monte Carlo approaches, either in terms of variance reduction as well as in describing the shock structure.