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We consider a geometric programming problem consisting in minimizing a function given by the supremum of finitely many log-Laplace transforms of discrete nonnegative measures on a Euclidean space. Under a coerciveness assumption, we show that a $\varepsilon$-minimizer can be computed in a time that is polynomial in the input size and in $|\log\varepsilon|$. This is obtained by establishing bit-size estimates on approximate minimizers and by applying the ellipsoid method. We also derive polynomial iteration complexity bounds for the interior point method applied to the same class of problems. We deduce that the spectral radius of a partially symmetric, weakly irreducible nonnegative tensor can be approximated within $\varepsilon$ error in poly-time. For strongly irreducible tensors, we also show that the logarithm of the positive eigenvector is poly-time computable. Our results also yield that the the maximum of a nonnegative homogeneous $d$-form in the unit ball with respect to $d$-H\"older norm can be approximated in poly-time. In particular, the spectral radius of uniform weighted hypergraphs and some known upper bounds for the clique number of uniform hypergraphs are poly-time computable.

This paper studies the convergence of a spatial semidiscretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. For non-smooth initial values, the regularity of the mild solution is investigated, and an error estimate is derived with the spatial $ L^2 $-norm. For smooth initial values, two error estimates with the general spatial $ L^q $-norms are established.

In this article, the stabilizer free weak Galerkin (SFWG) finite element method is applied to the Ciarlet-Raviart mixed form of the Biharmonic equation. We utilize the SFWG solutions of the second elliptic problems to define projection operators, build error equations, and further derive the error estimates. Finally, numerical examples support the results reached by the theory.

We propose a new Riemannian gradient descent method for computing spherical area-preserving mappings of topological spheres using a Riemannian retraction-based framework with theoretically guaranteed convergence. The objective function is based on the stretch energy functional, and the minimization is constrained on a power manifold of unit spheres embedded in 3-dimensional Euclidean space. Numerical experiments on several mesh models demonstrate the accuracy and stability of the proposed framework. Comparisons with two existing state-of-the-art methods for computing area-preserving mappings demonstrate that our algorithm is both competitive and more efficient. Finally, we present a concrete application to the problem of landmark-aligned surface registration of two brain models.

Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. With the help of a discrete fractional Gr\"{o}nwall inequality, global almost optimal error estimates in $L^2$- and $H^1$-norms are derived for the problem with initial data $u_0 \in H_0^1(\Omega)\cap H^2(\Omega)$. The novelty of our approach is based on managing the interaction of the L1 approximation of the fractional derivative and the time discrete elliptic operator to derive the optimal estimate in $H^1$-norm directly. Furthermore, a super convergence result is established when the elliptic operator is self-adjoint with time and space varying coefficients, and as a consequence, an $L^\infty$ error estimate is obtained for 2D problems that too with the initial condition is in $ H_0^1(\Omega)\cap H^2(\Omega)$. All results proved in this paper are valid uniformly as $\alpha\longrightarrow 1^{-}$, where $\alpha$ is the order of the Caputo fractional derivative. Numerical experiments are presented to validate our theoretical findings.

Symplectic integrators are widely implemented numerical integrators for Hamiltonian mechanics, which preserve the Hamiltonian structure (symplecticity) of the system. Although the symplectic integrator does not conserve the energy of the system, it is well known that there exists a conserving modified Hamiltonian, called the shadow Hamiltonian. For the Nambu mechanics, which is one of the generalized Hamiltonian mechanics, we can also construct structure-preserving integrators by the same procedure used to construct the symplectic integrators. In the structure-preserving integrator, however, the existence of shadow Hamiltonians is non-trivial. This is because the Nambu mechanics is driven by multiple Hamiltonians and it is non-trivial whether the time evolution by the integrator can be cast into the Nambu mechanical time evolution driven by multiple shadow Hamiltonians. In the present paper we construct structure-preserving integrators for a simple Nambu mechanical system, and derive the shadow Hamiltonians in two ways. This is the first attempt to derive shadow Hamiltonians of structure-preserving integrators for Nambu mechanics. We show that the fundamental identity, which corresponds to the Jacobi identity in Hamiltonian mechanics, plays an important role to calculate the shadow Hamiltonians using the Baker-Campbell-Hausdorff formula. It turns out that the resulting shadow Hamiltonians have indefinite forms depending on how the fundamental identities are used. This is not a technical artifact, because the exact shadow Hamiltonians obtained independently have the same indefiniteness.

The merit factor of a $\{-1, 1\}$ binary sequence measures the collective smallness of its non-trivial aperiodic autocorrelations. Binary sequences with large merit factor are important in digital communications because they allow the efficient separation of signals from noise. It is a longstanding open question whether the maximum merit factor is asymptotically unbounded and, if so, what is its limiting value. Attempts to answer this question over almost sixty years have identified certain classes of binary sequences as particularly important: skew-symmetric sequences, symmetric sequences, and anti-symmetric sequences. Using only elementary methods, we find an exact formula for the mean and variance of the reciprocal merit factor of sequences in each of these classes, and in the class of all binary sequences. This provides a much deeper understanding of the distribution of the merit factor in these four classes than was previously available. A consequence is that, for each of the four classes, the merit factor of a sequence drawn uniformly at random from the class converges in probability to a constant as the sequence length increases.

We establish an invariance principle for polynomial functions of $n$ independent high-dimensional random vectors, and also show that the obtained rates are nearly optimal. Both the dimension of the vectors and the degree of the polynomial are permitted to grow with $n$. Specifically, we obtain a finite sample upper bound for the error of approximation by a polynomial of Gaussians, measured in Kolmogorov distance, and extend it to functions that are approximately polynomial in a mean squared error sense. We give a corresponding lower bound that shows the invariance principle holds up to polynomial degree $o(\log n)$. The proof is constructive and adapts an asymmetrisation argument due to V. V. Senatov. As applications, we obtain a higher-order delta method with possibly non-Gaussian limits, and generalise a number of known results on high-dimensional and infinite-order U-statistics, and on fluctuations of subgraph counts.

We investigate a convective Brinkman--Forchheimer problem coupled with a heat transfer equation. The investigated model considers thermal diffusion and viscosity depending on the temperature. We prove the existence of a solution without restriction on the data and uniqueness when the solution is slightly smoother and the data is suitably restricted. We propose a finite element discretization scheme for the considered model and derive convergence results and a priori error estimates. Finally, we illustrate the theory with numerical examples.

Confidence intervals based on the central limit theorem (CLT) are a cornerstone of classical statistics. Despite being only asymptotically valid, they are ubiquitous because they permit statistical inference under weak assumptions and can often be applied to problems even when nonasymptotic inference is impossible. This paper introduces time-uniform analogues of such asymptotic confidence intervals, adding to the literature on confidence sequences (CS) -- sequences of confidence intervals that are uniformly valid over time -- which provide valid inference at arbitrary stopping times and incur no penalties for "peeking" at the data, unlike classical confidence intervals which require the sample size to be fixed in advance. Existing CSs in the literature are nonasymptotic, enjoying finite-sample guarantees but not the aforementioned broad applicability of asymptotic confidence intervals. This work provides a definition for "asymptotic CSs" and a general recipe for deriving them. Asymptotic CSs forgo nonasymptotic validity for CLT-like versatility and (asymptotic) time-uniform guarantees. While the CLT approximates the distribution of a sample average by that of a Gaussian for a fixed sample size, we use strong invariance principles (stemming from the seminal 1960s work of Strassen) to uniformly approximate the entire sample average process by an implicit Gaussian process. As an illustration, we derive asymptotic CSs for the average treatment effect in observational studies (for which nonasymptotic bounds are essentially impossible to derive even in the fixed-time regime) as well as randomized experiments, enabling causal inference in sequential environments.