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For many decades, advances in static verification have focused on linear integer arithmetic (LIA) programs. Many real-world programs are, however, written with non-linear integer arithmetic (NLA) expressions, such as programs that model physical events, control systems, or nonlinear activation functions in neural networks. While there are some approaches to reasoning about such NLA programs, still many verification tools fall short when trying to analyze them. To expand the scope of existing tools, we introduce a new method of converting programs with NLA expressions into semantically equivalent LIA programs via a technique we call dual rewriting. Dual rewriting discovers a linear replacement for an NLA Boolean expression (e.g. as found in conditional branching), simultaneously exploring both the positive and negative side of the condition, and using a combination of static validation and dynamic generalization of counterexamples. While perhaps surprising at first, this is often possible because the truth value of a Boolean NLA expression can be characterized in terms of a Boolean combination of linearly-described regions/intervals where the expression is true and those where it is false. The upshot is that rewriting NLA expressions to LIA expressions beforehand enables off-the-shelf LIA tools to be applied to the wider class of NLA programs. We built a new tool DrNLA and show it can discover LIA replacements for a variety of NLA programs. We then applied our work to branching-time verification of NLA programs, creating the first set of such benchmarks (92 in total) and showing that DrNLA's rewriting enable tools such as FuncTion and T2 to verify CTL properties of 42 programs that previously could not be verified. We also show a potential use of DrNLA assisting Frama-C in program slicing, and report that execution speed is not impacted much by rewriting.

We present a finite element scheme for fractional diffusion problems with varying diffusivity and fractional order. We consider a symmetric integral form of these nonlocal equations defined on general geometries and in arbitrary bounded domains. A number of challenges are encountered when discretizing these equations. The first comes from the heterogeneous kernel singularity in the fractional integral operator. The second comes from the dense discrete operator with its quadratic growth in memory footprint and arithmetic operations. An additional challenge comes from the need to handle volume conditions-the generalization of classical local boundary conditions to the nonlocal setting. Satisfying these conditions requires that the effect of the whole domain, including both the interior and exterior regions, can be computed on every interior point in the discretization. Performed directly, this would result in quadratic complexity. To address these challenges, we propose a strategy that decomposes the stiffness matrix into three components. The first is a sparse matrix that handles the singular near-field separately and is computed by adapting singular quadrature techniques available for the homogeneous case to the case of spatially variable order. The second component handles the remaining smooth part of the near-field as well as the far field and is approximated by a hierarchical $\mathcal{H}^{2}$ matrix that maintains linear complexity in storage and operations. The third component handles the effect of the global mesh at every node and is written as a weighted mass matrix whose density is computed by a fast-multipole type method. The resulting algorithm has therefore overall linear space and time complexity. Analysis of the consistency of the stiffness matrix is provided and numerical experiments are conducted to illustrate the convergence and performance of the proposed algorithm.

Consider a random sample $(X_{1},\ldots,X_{n})$ from an unknown discrete distribution $P=\sum_{j\geq1}p_{j}\delta_{s_{j}}$ on a countable alphabet $\mathbb{S}$, and let $(Y_{n,j})_{j\geq1}$ be the empirical frequencies of distinct symbols $s_{j}$'s in the sample. We consider the problem of estimating the $r$-order missing mass, which is a discrete functional of $P$ defined as $$\theta_{r}(P;\mathbf{X}_{n})=\sum_{j\geq1}p^{r}_{j}I(Y_{n,j}=0).$$ This is generalization of the missing mass whose estimation is a classical problem in statistics, being the subject of numerous studies both in theory and methods. First, we introduce a nonparametric estimator of $\theta_{r}(P;\mathbf{X}_{n})$ and a corresponding non-asymptotic confidence interval through concentration properties of $\theta_{r}(P;\mathbf{X}_{n})$. Then, we investigate minimax estimation of $\theta_{r}(P;\mathbf{X}_{n})$, which is the main contribution of our work. We show that minimax estimation is not feasible over the class of all discrete distributions on $\mathbb{S}$, and not even for distributions with regularly varying tails, which only guarantee that our estimator is consistent for $\theta_{r}(P;\mathbf{X}_{n})$. This leads to introduce the stronger assumption of second-order regular variation for the tail behaviour of $P$, which is proved to be sufficient for minimax estimation of $\theta_r(P;\mathbf{X}_{n})$, making the proposed estimator an optimal minimax estimator of $\theta_{r}(P;\mathbf{X}_{n})$. Our interest in the $r$-order missing mass arises from forensic statistics, where the estimation of the $2$-order missing mass appears in connection to the estimation of the likelihood ratio $T(P,\mathbf{X}_{n})=\theta_{1}(P;\mathbf{X}_{n})/\theta_{2}(P;\mathbf{X}_{n})$, known as the "fundamental problem of forensic mathematics". We present theoretical guarantees to nonparametric estimation of $T(P,\mathbf{X}_{n})$.

In the classical communication setting multiple senders having access to the same source of information and transmitting it over channel(s) to a receiver in general leads to a decrease in estimation error at the receiver as compared with the single sender case. However, if the objectives of the information providers are different from that of the estimator, this might result in interesting strategic interactions and outcomes. In this work, we consider a hierarchical signaling game between multiple senders (information designers) and a single receiver (decision maker) each having their own, possibly misaligned, objectives. The senders lead the game by committing to individual information disclosure policies simultaneously, within the framework of a non-cooperative Nash game among themselves. This is followed by the receiver's action decision. With Gaussian information structure and quadratic objectives (which depend on underlying state and receiver's action) for all the players, we show that in general the equilibrium is not unique. We hence identify a set of equilibria and further show that linear noiseless policies can achieve a minimal element of this set. Additionally, we show that competition among the senders is beneficial to the receiver, as compared with cooperation among the senders. Further, we extend our analysis to a dynamic signaling game of finite horizon with Markovian information evolution. We show that linear memoryless policies can achieve equilibrium in this dynamic game. We also consider an extension to a game with multiple receivers having coupled objectives. We provide algorithms to compute the equilibrium strategies in all these cases. Finally, via extensive simulations, we analyze the effects of multiple senders in varying degrees of alignment among their objectives.

Existing theories on deep nonparametric regression have shown that when the input data lie on a low-dimensional manifold, deep neural networks can adapt to the intrinsic data structures. In real world applications, such an assumption of data lying exactly on a low dimensional manifold is stringent. This paper introduces a relaxed assumption that the input data are concentrated around a subset of $\mathbb{R}^d$ denoted by $\mathcal{S}$, and the intrinsic dimension of $\mathcal{S}$ can be characterized by a new complexity notation -- effective Minkowski dimension. We prove that, the sample complexity of deep nonparametric regression only depends on the effective Minkowski dimension of $\mathcal{S}$ denoted by $p$. We further illustrate our theoretical findings by considering nonparametric regression with an anisotropic Gaussian random design $N(0,\Sigma)$, where $\Sigma$ is full rank. When the eigenvalues of $\Sigma$ have an exponential or polynomial decay, the effective Minkowski dimension of such an Gaussian random design is $p=\mathcal{O}(\sqrt{\log n})$ or $p=\mathcal{O}(n^\gamma)$, respectively, where $n$ is the sample size and $\gamma\in(0,1)$ is a small constant depending on the polynomial decay rate. Our theory shows that, when the manifold assumption does not hold, deep neural networks can still adapt to the effective Minkowski dimension of the data, and circumvent the curse of the ambient dimensionality for moderate sample sizes.

The stochastic dynamic matching problem has recently drawn attention in the stochastic-modeling community due to its numerous applications, ranging from supply-chain management to kidney exchange programs. In this paper, we consider a matching problem in which items of different classes arrive according to independent Poisson processes. Unmatched items are stored in a queue, and compatibility constraints are described by a simple graph on the classes, so that two items can be matched if their classes are neighbors in the graph. We analyze the efficiency of matching policies, not only in terms of system stability, but also in terms of matching rates between different classes. Our results rely on the observation that, under any stable policy, the matching rates satisfy a conservation equation that equates the arrival and departure rates of each item class. Our main contributions are threefold. We first introduce a mapping between the dimension of the solution set of this conservation equation, the structure of the compatibility graph, and the existence of a stable policy. In particular, this allows us to derive a necessary and sufficient stability condition that is verifiable in polynomial time. Secondly, we describe the convex polytope of non-negative solutions of the conservation equation. When this polytope is reduced to a single point, we give a closed-form expression of the solution; in general, we characterize the vertices of this polytope using again the graph structure. Lastly, we show that greedy policies cannot, in general, achieve every point in the polytope. In contrast, non-greedy policies can reach any point of the interior of this polytope, and we give a condition for these policies to also reach the boundary of the polytope.

In multivariate time series analysis, the coherence measures the linear dependency between two-time series at different frequencies. However, real data applications often exhibit nonlinear dependency in the frequency domain. Conventional coherence analysis fails to capture such dependency. The quantile coherence, on the other hand, characterizes nonlinear dependency by defining the coherence at a set of quantile levels based on trigonometric quantile regression. Although quantile coherence is a more powerful tool, its estimation remains challenging due to the high level of noise. This paper introduces a new estimation technique for quantile coherence. The proposed method is semi-parametric, which uses the parametric form of the spectrum of the vector autoregressive (VAR) model as an approximation to the quantile spectral matrix, along with nonparametric smoothing across quantiles. For each fixed quantile level, we obtain the VAR parameters from the quantile periodograms, then, using the Durbin-Levinson algorithm, we calculate the preliminary estimate of quantile coherence using the VAR parameters. Finally, we smooth the preliminary estimate of quantile coherence across quantiles using a nonparametric smoother. Numerical results show that the proposed estimation method outperforms nonparametric methods. We show that quantile coherence-based bivariate time series clustering has advantages over the ordinary VAR coherence. For applications, the identified clusters of financial stocks by quantile coherence with a market benchmark are shown to have an intriguing and more accurate structure of diversified investment portfolios that may be used by investors to make better decisions.

Empirical studies suggest a deep intertwining between opinion formation and decision-making processes, but these have been treated as separate problems in the study of dynamical models for social networks. In this paper, we bridge the gap in the literature by proposing a novel coevolutionary model, in which each individual selects an action from a binary set and has an opinion on which action they prefer. Actions and opinions coevolve on a two-layer network. For homogeneous parameters, undirected networks, and under reasonable assumptions on the asynchronous updating mechanics, we prove that the coevolutionary dynamics is an ordinal potential game, enabling analysis via potential game theory. Specifically, we establish global convergence to the Nash equilibria of the game, proving that actions converge in a finite number of time steps, while opinions converge asymptotically. Next, we provide sufficient conditions for the existence of, and convergence to, polarized equilibria, whereby the population splits into two communities, each selecting and supporting one of the actions. Finally, we use simulations to examine the social psychological phenomenon of pluralistic ignorance.

Conventional harvesting problems for natural resources often assume physiological homogeneity of the body length/weight among individuals. However, such assumptions generally are not valid in real-world problems, where heterogeneity plays an essential role in the planning of biological resource harvesting. Furthermore, it is difficult to observe heterogeneity directly from the available data. This paper presents a novel optimal control framework for the cost-efficient harvesting of biological resources for application in fisheries management. The heterogeneity is incorporated into the resource dynamics, which is the population dynamics in this case, through a probability density that can be distorted from the reality. Subsequently, the distortion, which is the model uncertainty, is penalized through a divergence, leading to a non-standard dynamic differential game wherein the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation has a unique nonlinear partial differential term. Here, the existence and uniqueness results of the HJBI equation are presented along with an explicit monotone finite difference method. Finally, the proposed optimal control is applied to a harvesting problem with recreationally, economically, and ecologically important fish species using collected field data.

We consider nonlinear delay differential and renewal equations with infinite delay. We extend the work of Gyllenberg et al, Appl. Math. Comput. (2018) by introducing a unifying abstract framework and derive a finite-dimensional approximating system via pseudospectral discretization. For renewal equations, via integration we consider a reformulation in a space of absolutely continuous functions that ensures that point evaluation is well defined. We prove the one-to-one correspondence of equilibria between the original equation and its approximation, and that linearization and discretization commute. Our most important result is the proof of convergence of the characteristic roots of the pseudospectral approximation of the linear(ized) equations, which ensures that the finite-dimensional system correctly reproduces the stability properties of the original linear equation if the dimension of the approximation is large enough. This result is illustrated with several numerical tests, which also demonstrate the effectiveness of the approach for the bifurcation analysis of equilibria of nonlinear equations.