numerical analysts

Abstract: We study how boundary geometry affects $L^q$ approximation by shallow ReLU neural networks on convex bodies $K\subset\mathbb R^d$. The target classes are unit balls of weighted variation spaces that can enlarge classical spaces by incorporating domain geometry into constraints on output coefficients. We establish upper and lower bounds for arbitrary convex bodies and determine the optimal polynomial approximation exponents for all planar convex bodies and their products with polytopes, including cubes and circular cylinders. For $C^2$ convex bodies, we determine the optimal exponents in a dimension-dependent range of $q$ extending beyond $2$. When all principal curvatures are positive, the result extends to every finite $q\ge2$. For these classes, the dependence of the optimal exponent on the domain is captured by a single geometric growth exponent associated with small boundary caps. We also identify sharp weight thresholds for retaining the classical approximation exponent and improve previous weights on Euclidean balls while preserving the classical $L^2$ approximation rate. When the target is given as a finite ReLU network, the upper bounds can be realized by a compressed network using only the original neurons, without increasing the weighted coefficient sum. These results show that boundary geometry can change the optimal exponent even when dimension, error norm, and weighting rule are fixed.

Abstract: Euler's constant $γ$, also known as the Euler-Mascheroni constant, remains one of the most elusive constants in mathematics. Despite extensive study, its irrationality is still unproven, and Apéry-like approximations of it are remarkably scarce. We present a novel identity that expresses $γ$ in terms of a Meijer G-function and its first few derivatives. This identity is then generalized and used to systematically generate infinitely many Apéry-like approximations of $γ$ using a Conservative Matrix Field construction based on the Meijer G-function.

Abstract: We study which parameters of deep, unmasked, single-head attention are determined by its input--output function. For known positive nonconstant real-analytic normalizers, the function generically determines the effective scores and combined value map up to the signs induced by even normalizers. This proves the real-analytic case of a conjecture of Henry--Marchetti--Kohn, including softmax. We then classify exceptional fibers under explicit normalizer conditions, identifying when collapse makes later scores unobservable, and establish sharp Taylor orders for local identification. Near simultaneous query/key collapse, we compute the complete native Jacobian decay spectrum on separating finite input banks. For common first nonconstant normalizer degree $k$, layer $i$ has contact order $2k3^{i-1}-1$, with exact multiplicities and kernel dimension. High-precision and automatic differentiation calculations illustrate the resulting loss of numerical sensitivity.

Abstract: We construct neural approximation classes native to the graph spaces H(div) and H(curl), in two and three dimensions and, for H(div), in any dimension. Every realization lies in the space for all parameter values, and with kinked potentials, such as ReLU networks, the admissible jumps appear at finite width. The classes are images of scalar and componentwise networks under fixed operators of the de Rham complex, and do not involve a mesh or finite element emulation. For H(div) in R^n two native classes are given on an equal footing, with a skew-symmetric potential $A$: $\mathrm{Div}\,A+R_nq+\mathbf{h}$, with the divergence $q$ as an explicit unknown, and $\mathrm{Div}\,A+\mathbf{z}$ with an $H^1$ field $\mathbf{z}$; for H(curl) the analogous classes are $\mathrm{grad}\,φ+Sr+\mathbf{h}$ in two dimensions and $\mathrm{grad}\,φ+\mathbf{z}$ in two and three dimensions. In all of them every interface jump of the field is carried by the potential term, $\mathrm{Div}\,A$ or $\mathrm{grad}\,φ$, while the remaining part has no interface jump (it is an $H^1$ field in the regular-decomposition classes); the classes with $\mathbf{z}$ are the componentwise approach enriched by this term. Known or learned interface geometry enters the potential through factors with trainable amplitudes, and the remaining part if the divergence jumps. The classes lead to the FOSLS-deRhaNN method, first-order system least squares with de Rham neural networks, whose loss is the least-squares functional posed in the natural spaces of the weak formulation; for elliptic equations this includes $H^{-1}$ right-hand sides and $H^{1/2}$ Dirichlet data. Elliptic equations with discontinuous coefficients and curl-curl problems are treated as instances, with the functional equivalent to the error; linear transport with discontinuous solutions and conservation laws with shocks use the same flux classes.

Abstract: Equivalent finite parameterizations can represent the same functions and intrinsic norm yet induce different optimization algorithms. We study this effect in a controlled finite Brownian RKHS with nodal, increment, and spectral coordinates. Classical finite-element, RKHS-interpolation, Brownian-covariance, and mixed-boundary DCT identities make the shared hypothesis class, Brownian energy, approximation operator, and coordinate maps explicit. Our main results concern the optimization geometry of this fixed model. With mapped initialization, identical scalar steps, and identical minibatches, nodal and spectral GD/SGD have exactly the same mapped trajectories. Increment GD is an explicit Euler step for the constant Brownian/Sobolev metric, with factor $1/h$. For Brownian-regularized least squares, $κ_2(\mathbf H_{\mathrm{inc}})\le1+A/ρ$, independently of grid resolution $G$ for fixed $A$, $ρ>0$, and the stated normalization. Under the stated standard-Adam convention, the universal orthogonal equivariance group is exactly the signed permutations; the block DCT-VIII transform is not one. Float64 tests over five grids numerically verify the finite identities, mapped one-layer and recursive trajectories, conditioning predictions, and theorem-matched Adam separation. Thus coordinate effects are isolated without changing the represented functions, intrinsic regularizer, or approximation space.