arXiv subjects

Tsogtgerel Gantumur

Publications and source records attributed to Tsogtgerel Gantumur.

15 recordsLinked to original sources

arXiv2026

Projection geometry and relaxed quasi-orthogonality for inf-sup stable Galerkin methods

We give an elementary, coordinate-free proof that uniformly inf-sup stable nested Petrov-Galerkin methods on Hilbert spaces satisfy relaxed general quasi-orthogonality. More precisely, for any window of $N$ consecutive levels, the sum of squared Galerkin increments is bounded by the squared error at the beginning of the window times $N^σ$, where $σ<1$. The exponent $σ$ and the prefactor can be chosen explicitly in terms of the uniform projection bound. The proof uses Hilbert-space projection estimates, dyadic iteration, and duality, without relying on matrix factorizations. As a corollary, we obtain explicit growth bounds for finite block-LU factors. In the symmetric indefinite case, the positive and negative spectral splittings of the Galerkin detail spaces yield a characterization of full quasi-orthogonality in terms of uniform boundedness of the associated finite-window sign operators. We also construct a self-adjoint involution and a nested, uniformly inf-sup stable Galerkin sequence for which full general quasi-orthogonality fails. For every $0<α<1$, the construction yields a hierarchy whose finite-window constants have growth of exact order $N^α$, along with an explicit target whose Galerkin increments are not square summable.

math.NA↗

arXiv2026

An expository review of the Chebyshev-Sylvester method in prime number theory

This paper gives a self-contained expository and computational account of the elementary methods developed by P.L. Chebyshev and J.J. Sylvester to obtain explicit bounds for the distribution of prime numbers. The method replaces the Mobius function in a basic convolution identity by finitely supported arithmetic functions, or schemes, and relates the resulting auxiliary functions to the Chebyshev function psi(x) through the summatory logarithm T(x)=log([x]!). We examine a number of schemes introduced by Chebyshev and Sylvester, with particular emphasis on Sylvester's iterative refinement procedure. For a fixed choice of terms, this procedure leads to an explicit two-dimensional affine recurrence for upper and lower bounds on psi(x), whose fixed point can be analyzed by elementary linear algebra. We then implement the procedure computationally and introduce a simple one-parameter selection rule, the rho-rule, for choosing which terms to retain in the iteration. Numerical exploration of the historical schemes reproduces Sylvester's bounds and, in several cases, yields modest improvements of the resulting constants. The accompanying Python implementation is publicly available.

math.NT↗

arXiv2026

Exact ReLU realization of binary affine refinement iterates via reflection folding and cone switching

We study vector-valued binary affine refinement operators with finitely supported matrix masks and compactly supported continuous piecewise linear input and forcing data. We prove that every finite refinement iterate admits an exact ReLU realization of fixed width and depth linear in the number of iterations. No separation of the forcing profile from the binary cell seams is required. The main mechanism is universal reflection doubling. Pairing each residual profile with its reflection replaces the two binary transition matrices by one fixed block matrix together with a fixed swap involution. The cell-seam identity makes the two branch candidates agree at the tent fold, while their swap-odd component is bounded linearly by the distance to the fold. This permits exact branch selection by a fixed continuous piecewise linear cone switch, without multiplication by a variable selector. The resulting primal recursion requires the residual orbit in reverse order. We obtain exact backward replay from the residual memory controller developed previously for affine refinement, interpreted here through the reflection quotient of circle doubling. The construction propagates the full vectorized profiles rather than decomposing the input and forcing into reference atoms. We also treat stage-dependent forcing from a fixed finite-dimensional family and show that genuine reflection equivariance reduces the doubled cascade to a single parity sector.

math.OC↗

arXiv2026

Exact ReLU realization of affine one-dimensional refinement iterates via residual memory and offset frames

We study vector-valued affine refinement operators of the form [ (Wγ)(t)=\sum_{j\in\mathbb{Z}} A_jγ(Mt-j)+B(t), ] with finitely supported matrix mask and compactly supported continuous piecewise linear input and forcing data. Building on the homogeneous realization theorem for (B\equiv 0), we prove that, for (M\ge 3), every finite affine iterate (W^nγ) admits an exact fixed-width ReLU realization whose depth is (O(n)). The main new ingredient is a residual memory controller. It replaces the noninvertible residual dynamics by an injective skew-product and permits exact backward replay of the residual states required by a Horner-type evaluation of the affine forcing sum. Offset frames align the forcing atoms away from residual seams, allowing complementary loop readouts to recover their values exactly. The remaining branch-selection ambiguity occurs only where the accumulated affine state has already vanished. For (M\ge 3), the result applies to arbitrary compactly supported continuous piecewise linear forcing terms. For (M=2), the same construction applies to ordinary-frame seam-separated forcing. We also prove a stage-dependent extension for forcing terms in a fixed finite-dimensional continuous piecewise linear span and record the resulting linear-depth upgrade for open-curve, finite-state, and Hilbert- and Morton-type recursive constructions.

cs.LG↗

arXiv2026

Neural network realization of binary refinement iterates via a two-chart atlas selector

Refinement operators generate many functions used in wavelet constructions, subdivision schemes, and geometric modeling. Their finite iterates can develop rapidly increasing numbers of linear pieces, making them a natural test case for the expressive power of deep neural networks. Earlier work showed that, for scalar binary refinement with a finitely supported mask, every compactly supported continuous piecewise linear seed has finite refinement iterates that admit exact ReLU realizations of fixed width and depth growing linearly with the number of refinement steps. The present paper gives a new construction of this known theorem. The difficulty is that the refinement cascade is driven by discontinuous binary digit choices, whereas ReLU networks produce continuous piecewise linear maps. We represent the residual dynamics on a polygonal model of the circle and describe each residual position in two overlapping coordinate systems, one ordinary and one shifted by one half. Their discontinuities occur at different points. The network switches between the two descriptions only where both are valid and the corresponding fixed linear cascade updates agree, so the switch is exact and requires no multiplication by a variable selector. The construction also gives exact readout of every continuous piecewise linear circle function satisfying the natural endpoint compatibility condition. Localized seeds are handled by a two-pass network, and translation covariance, finite decomposition, and gluing extend the result to arbitrary compactly supported continuous piecewise linear seeds in a preserved support window.

math.NA↗

arXiv2026

Projectional continuous data assimilation on the torus: Resonant and kernel-free regimes

We study continuous data assimilation for the two-dimensional incompressible Navier-Stokes equations on the periodic torus using a single signed scalar velocity projection in a prescribed spatially varying direction. We prove exponential synchronization in both L2 and H1 through two complementary mechanisms. In the regular resonant regime, nontrivial invisible currents are controlled through a moving-frame expansion, adapted coordinates, and Farhat-Lunasin-Titi logarithmic estimates, provided the resulting geometric shear defect is viscously absorbable. This recovers the periodic one-component mechanism for constant rational directions and applies to genuinely nonconstant projection fields. In the kernel-free regime, qualitative injectivity and compactness yield observability with arbitrarily small viscous leakage, giving synchronization for every sufficiently large gain without an upper gain restriction. For sufficiently regular projection fields, both mechanisms extend to L2-stable Type-I coarse scalar observations satisfying a first-order approximation property, under the usual gain-resolution condition. A common parabolic smoothing argument then upgrades all four L2-synchronization results to H1-synchronization without additional observation hypotheses.

math.AP↗

arXiv2026

Conditional Euclidean-Hamiltonian reductions for sign-problem toy models

Euclidean Monte Carlo methods are effective when the path-integral weight is real and nonnegative, but finite-density fermion systems often produce sign-changing or complex scalar weights after the fermionic sector is traced out. Hamiltonian formulations avoid this complex-weight sampling problem but face rapid Hilbert-space growth. This paper studies a conditional Euclidean-Hamiltonian (CEH) reduction that combines these two descriptions. The calculation is organized around a Monte Carlo-tractable reference problem and a residual active sector. Instead of tracing the active sector into a determinant or scalar weight, CEH keeps it operator-valued and uses the reference calculation to determine projected correlation or transfer matrices. These matrices define a finite effective Hamiltonian, with the remaining finite-density dependence introduced after projection. At finite rank, the result is an effective model whose accuracy must be tested through basis enlargement, metric conditioning, stochastic matrix-element errors, and number-sector diagnostics. The construction is examined in three finite benchmarks. A two-channel oscillator tests conditional basis compression; a positive-measure stochastic calculation tests correlation-matrix harvesting and GEVP extraction, including a comparison with a PDMS-style estimator; and a four-site Hubbard ring combines a finite-budget determinant-sign stress test with finite-density continuation in non-target active spaces. The benchmarks support the proposed trade from scalar sign reweighting to a monitored active-space approximation in structured finite models, but they do not provide an end-to-end stochastic CEH treatment of the Hubbard sign problem or establish favorable scaling. Usefulness requires both low-rank approximability and efficient extraction of the required projected matrix data.

quant-ph↗

arXiv2026

Rotationally invariant dynamical lattice regulators for Euclidean quantum field theories

We introduce a dynamical lattice regulator for Euclidean quantum field theories on a fixed hypercubic graph $Λ\simeq\mathbb{Z}^d$, in which the embedding $x:Λ\to\mathbb{R}^d$ is promoted to a dynamical field and integrated over subject to shape regularity constraints. The total action is local on $Λ$, gauge invariant, and depends on $x$ only through Euclidean invariants built from edge vectors (local metrics, volumes, etc.), hence the partition function is exactly covariant under the global special Euclidean group SE(d) at any lattice spacing. The intended symmetry restoring mechanism is not rigid global zero modes but short-range local twisting of the embedding that mixes local orientations. Our universality discussion is conditioned on a short-range geometry hypothesis (SR): after quotienting the global SE(d) modes, connected correlators of local geometric observables have correlation length O(1) in lattice units. We prove Osterwalder-Schrader reflection positivity for the coupled system with embedding $x$ and generic gauge and matter fields $(U,Φ)$ in finite volume by treating $x$ as an additional multiplet of scalar fields on $Λ$. Assuming (SR), integrating out $x$ at fixed cutoff yields a local Symanzik effective action in which geometry fluctuations generate only SO(d)-invariant irrelevant operators and finite renormalizations. For example, in d=4 we recover the standard one-loop $β$-function in a scalar $ϕ^4$ test theory. Finally, we describe a practical local Monte Carlo update and report d=2 proof-of-concept simulations showing O(1)-scale geometry correlations, a direct SO(2)-connection diagnostic of short-range local twisting, and evidence for reduced axis-vs-diagonal cutoff artefacts relative to a fixed lattice at matched bare parameters.

hep-lat↗

arXiv2026

Exact ReLU realization of tensor-product refinement iterates

We study scalar dyadic refinement operators on R^2 of the form (Vf)(x,y) = sum_{(j,k) in Z^2} c_{j,k} f(2x-j, 2y-k), where only finitely many mask coefficients c_{j,k} are nonzero. Under a fixed support-window hypothesis, we prove that for every compactly supported continuous piecewise linear seed g:R^2->R, the iterates V^n g admit exact ReLU realizations of fixed width and depth O(n). This gives a first genuinely two-dimensional extension of the exact realization theory for refinement cascades. Using the one-dimensional exact loop-controller framework, the proof transports the tensor-product residual dynamics exactly on the product of two polygonal loops and reduces the remaining seam ambiguity to a final readout and selector step. The matrix cascade is then handled by a fixed-depth recursive block, and general compactly supported continuous piecewise linear seeds are reduced to a finite decomposition together with exact clamped gluing on the support window. This identifies the tensor-product dyadic case as a natural first multivariate instance of the loop-controller method for refinement iterates.

math.CA↗

arXiv2026

Exact Loop Controllers for ReLU Realization of Homogeneous Curve Refinements

We study homogeneous refinement operators $(Vγ)(t)=\sum_{j\in\mathbb Z}A_jγ(Mt-j)$, acting on compactly supported continuous piecewise linear curves $γ:\mathbb R\to\mathbb R^p$, where $M\ge2$ and only finitely many matrices $A_j\in\mathbb R^{p\times p}$ are nonzero. We prove that the iterates $V^nγ$ admit exact ReLU realizations of fixed width and depth $O(n)$. The main new ingredient is an exact loop controller for the residual dynamics. Instead of propagating scalar residual surrogates, the construction transports the residual orbit by a forward-exact state on a polygonal loop. Scalar factors and digit selectors are then recovered from this loop state by complementary CPwL readouts. The loop seam is not removed, but its remaining ambiguity is confined to the final readout/selector stage, where it is harmless because the scalar atom is supported away from the seam. This gives a homogeneous $M$-ary vector-valued extension of the scalar binary refinable-function construction with a more geometric controller architecture. We also record crude exponential bounds on the network weights and biases. Affine forcing terms are handled by expanding affine iterates into finite sums of homogeneous iterates, giving exact fixed-width realizations with depth $O(n^2)$, and anchored open curves reduce to compactly supported defects with affine anchor mismatch. We also describe homogeneous polygonal generators, including dragon-type examples and a self-intersecting Hilbert-type prototype in arbitrary dimension. The extended version includes stage-dependent forcing, finite-state stacking reductions, and further geometric constructions such as Koch-, Gosper-, Morton-, and connector-based Hilbert-type variants.

math.CA↗

arXiv2026

Possible Reforms of the Tibetan Lunisolar Calendar

The family of Tibetan lunisolar calendars operates on a shared arithmetic axiom (67 lunar months = 65 solar months) that provides a rigid structure but causes observable seasonal drift. This study deconstructs the calendar through a progressive analytical sequence, first presenting it as an explicit computational procedure, then isolating its structural core of incidence rules and mean-motion models. This separation distinguishes structurally forced features from tradition-dependent ones, allowing inaccuracies to be rigorously decomposed into internal arithmetic drift, sidereal misalignment, and anomaly-phase defects. Crucially, computational analysis also reveals remarkable historical robustness: the discrete arithmetic of traditional day rules renders boundary tie-cases operationally absent, while large internal temporal buffers and the multi-hour inaccuracy of the classical lunar model insulated the calendar against geographic variation. On this basis, the paper develops a stratified reform space rather than a single replacement proposal. The resulting standards range from conservative rational repairs preserving traditional arithmetic to explicit astronomical reconstructions culminating in fully dynamical models of true solar and lunar motion. The guiding question is how far astronomical correction can be carried without discarding the Tibetan calendrical identity embodied in the structural rules for month and day labeling. Finally, calendric reform requires more than new formulas and constants; it demands precise numerical semantics. The proposed standards are thus formulated not merely as abstract models, but as executable, reproducible specifications suitable for implementation, validation, and long-term transmission across computational environments.

physics.soc-ph↗

arXiv2025

A direct proof of the equivalence between Dirichlet's principle and Perron's method

We give a short proof that for a bounded domain $Ω\subset\mathbb{R}^n$ and continuous boundary data $g\in C(\partialΩ)$ admitting a continuous finite-energy extension $ϕ\in H^{1}(Ω)\cap C(\barΩ)$, the minimizer of the Dirichlet energy \[ E(v) = \int_Ω |\nabla v|^{2}\,dx, \qquad v-ϕ\in H^{1}_{0}(Ω), \] coincides with the Perron solution $h_g$ of the Dirichlet problem $Δu = 0$ in $Ω$ with boundary data $g$. The argument stays entirely in $H^{1}(Ω)$ and uses only strong convergence via strict convexity of the Dirichlet energy, Friedrichs' inequality, Weyl's lemma, and Wiener's exhaustion by regular subdomains. No weak convergence, Poisson problems with distributional right hand sides, or general elliptic theory are needed.

math.AP↗

arXiv2016

Convergence rates of adaptive methods, Besov spaces, and multilevel approximation

This paper concerns characterizations of approximation classes associated to adaptive finite element methods with isotropic h-refinements. It is known from the seminal work of Binev, Dahmen, DeVore and Petrushev that such classes are related to Besov spaces. The range of parameters for which the inverse embedding results hold is rather limited, and recently, Gaspoz and Morin have shown, among other things, that this limitation disappears if we replace Besov spaces by suitable approximation spaces associated to finite element approximation from uniformly refined triangulations. We call the latter spaces *multievel approximation spaces*, and argue that these spaces are placed naturally halfway between adaptive approximation classes and Besov spaces, in the sense that it is more natural to relate multilevel approximation spaces with either Besov spaces or adaptive approximation classes, than to go directly from adaptive approximation classes to Besov spaces. In particular, we prove embeddings of multilevel approximation spaces into adaptive approximation classes, complementing the inverse embedding theorems of Gaspoz and Morin. Furthermore, in the present paper, we initiate a theoretical study of adaptive approximation classes that are defined using a modified notion of error, the so-called *total error*, which is the energy error plus an oscillation term. Such approximation classes have recently been shown to arise naturally in the analysis of adaptive algorithms. We first develop a sufficiently general approximation theory framework to handle such modifications, and then apply the abstract theory to second order elliptic problems discretized by Lagrange finite elements, resulting in characterizations of modified approximation classes in terms of memberships of the problem solution and data into certain approximation spaces, which are in turn related to Besov spaces.

math.NA↗

arXiv2014

On the convergence theory of adaptive mixed finite element methods for the Stokes problem

In the first part of this paper, we establish a conditional optimality result for an adaptive mixed finite element method for the stationary Stokes problem discretized by the standard Taylor-Hood elements, under the assumption of the so-called general quasi-orthogonality. Optimality is measured in terms of a modified approximation class defined through the total error, as is customary since the seminal work of Cascon, Kreuzer, Nochetto and Siebert. The second part of the paper is independent of optimality results, and concerns interrelations between the modified approximation classes and the standard approximation classes (the latter defined through the energy error). Building on the tools developed in the papers of Binev, Dahmen, DeVore, and Petrushev, and of Gaspoz and Morin, we prove that the modified approximation class coincides with the standard approximation class, modulo the assumption that the data is regular enough in an appropriate scale of Besov spaces.

math.NA↗

arXiv2012

Adaptive boundary element methods with convergence rates

This paper presents adaptive boundary element methods for positive, negative, as well as zero order operator equations, together with proofs that they converge at certain rates. The convergence rates are quasi-optimal in a certain sense under mild assumptions that are analogous to what is typically assumed in the theory of adaptive finite element methods. In particular, no saturation-type assumption is used. The main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods, and an inverse-type inequality involving boundary integral operators on locally refined finite element spaces.

math.NA↗