Papers

Kanade and Russell conjectured seventeen Rogers–Ramanujan type identities of modulus 12. Eleven of these identities involving triple sums were proved by Bringmann–Jennings-Shaffer–Mahlburg and by Rosengren. Motivated by these works and using similar methods, we settle all the six remaining conjectures including two triple sum identities labeled $I_{5a}$ and $I_{6a}$ originated from Russell's thesis and four quadruple sum identities labeled 7, 7a, 8 and 8a. Our proof of the triple sum identities combines linear recurrences, $q$-difference equations, and $q$-series summation formulas. For the quadruple sum identities, we represent the sums as contour integrals whose integrands are infinite products and evaluate them by residue calculus. The resulting residues reduce to single sum cubic basic hypergeometric series, and we are able to express them as infinite products.

Markov chain central limit theorems (CLTs) and their associated variances are very important for implementing Markov chain Monte Carlo algorithms among other applications. Häggström and Rosenthal (2007) presented various results regarding the equality of different formulae for this variance and also posed seven open problems. We resolve all seven in this paper. Our main result is a general characterization of Markov chain CLTs: for stationary and ergodic Markov chains, we prove that the normalized partial sums satisfy a $\sqrt{n}$-CLT if and only if the Poisson equation admits a finite solution with square-integrable martingale differences whose second moment equals the CLT variance. In particular, this result does not require reversibility of the chain or square integrability of the function. It implies that Roberts' holding-probability condition always precludes a $\sqrt{n}$-CLT, and for reversible chains, whenever a $\sqrt{n}$-CLT holds, the variance limit is finite if the function is square-integrable, otherwise undefined. In addition, we develop a general principle that expresses Fourier coefficients as the autocovariances of an ergodic nonreversible Markov chain.

Let $\Gamma=(V,E)$ be a distance-biregular graph with even diameter $d$ and girth $2d-2$. On the edge-set $E$ we define relations that form an association scheme; this is done by considering the linear span of the corresponding adjacency matrices, and, by means of intersection diagrams, we prove that this is an algebra. Furthermore, we describe all the irreducible representations of this algebra, and, by considering the multiplicities of the 2-dimensional ones, we show that there are no such distance-biregular graphs with $28\le d\le 46$ and $d\ge 52$.

In this paper we investigate in a systematic way the solution structure of nonnegative solutions for coupled nonlinear Schrödinger systems in the all attractive regime. For all frequencies equal case we obtain results on Morse index of synchronized positive vector solutions and nonnegative semi-vector solutions as well as their kernel of linearized systems at these solutions. We establish a variational characterization of synchronized positive vector solutions and as applications we give a new existence result about positive vector solutions for the general systems with arbitrary frequencies. We also examine the synchronization phenomenon of positive vector solutions, and prove that if the interaction matrix has exactly one positive eigenvalue, any positive vector solution is synchronized. Finally we provide examples of domains for which synchronization of positive solutions fails to hold if the interaction matrix has at least two positive eigenvalues. Our results reveal the effect of the spectral information of the interaction matrix of the couplings and geometry of a domain on the solution structure.

For $(Q_t,P_t)$ governed by suitably damped underdamped Langevin dynamics, we quantify the convergence in KL divergence of the positional marginal to a $\sigma$-strongly log-concave target $\pi(dq) = \frac{1}{Z}e^{-V(q)}dq$ as \begin{align*} \operatorname{KL}(\operatorname{Law}(Q_t) \| \pi) \leq e^{-\sqrt{\sigma} t}\operatorname{KL}(\operatorname{Law}(Q_{0}, P_{0})\, \|\, \Pi_0), \end{align*} where $\Pi_0$ denotes an appropriately selected reference measure. When $\pi$ is merely log-concave, the estimate \begin{align*} \operatorname{KL}(\operatorname{Law}(Q_t) \| \pi) \leq \frac{\tau^2}{t^2}\operatorname{KL}(\operatorname{Law}(Q_{\tau}, P_{\tau})\, \|\, \Pi_{\tau}) \end{align*} is derived, where $\Pi_\tau$ denotes another correspondingly chosen reference measure at time $\tau > 0$. Both rates match precisely the canonical rates of the corresponding accelerated gradient flows in $\mathbb{R}^d$. They are achieved by the novel interpretation of the underdamped Langevin dynamics as a combination of strategies in an online sampling game and by estimating the KL divergence using a cost function informed by fictitious competitors.

Changes in the internal energy of a thermodynamic system are fundamentally described by the first law of thermodynamics. However, the first law has a highly compact form that does not explicitly reveal the dynamics of the system or provide a corresponding equation of motion. While equations of motion can be written explicitly for mechanical and electrical systems, an analogous dynamical formulation for thermodynamic systems has, to our knowledge, not been established directly from the first law. In this work, we show that, through dimensional reduction, the first law can be transformed from its compact thermodynamic form into an explicit equation of motion for an open thermodynamic system. Remarkably, the resulting equation has the same mathematical structure as the equations governing a driven damped harmonic oscillator and a driven series RLC circuit. Detailed dimensional analyses of the derived equation and the effective coefficients further confirm the dimensional consistency of the formulation. We further perform a systematic comparison of the mechanical, electrical, and thermodynamic systems and show that, despite their distinct physical origins, they share the same universal dynamical structure. Finally, we construct a Lagrangian formulation of the thermodynamic system and establish its correspondence with the Lagrangian descriptions of the mechanical and electrical systems. These results demonstrate that an explicit equation of motion can be generated directly from the first law of thermodynamics and reveal a common dynamical framework underlying thermodynamic, mechanical, and electrical systems.

This paper develops a convex tube model predictive control formulation for constrained nonlinear systems. We consider dynamics described by a discrete-time state-space model with parametric and additive uncertainty that admits a difference-of-convex decomposition. Convex directional bounds of the nonlinear dynamics are combined with configuration-constrained polytopic tubes, whose predefined combinatorial structure yields an affine parameterization of their vertices. The resulting finite-dimensional sufficient conditions certify robust one-step tube propagation under parametric uncertainty and additive disturbances, while allowing the tube geometry and an associated vertex control law to be optimized jointly in a single convex program. An implicit terminal condition guarantees recursive feasibility and convergence of the predicted tube to a target robust control invariant set. Numerical results illustrate the closed-loop properties and the trade-off between geometric flexibility and computational complexity.

We prove boundary null controllability in any positive time for a finite system of Euler–Bernoulli beams with possibly non-diagonalizable coupling. For $0<\rho<2$, a Fourier–Jordan reduction yields a vector moment problem with polynomial–exponential modes. Under the Fattorini–Hautus condition, global spectral non-collision, and nonvanishing modal transfer factors, a suitably estimated block-biorthogonal family produces an $H^2$ boundary control. Geometric multiplicities of the coupling matrix determine the minimal control dimension, and Jordan-block lengths determine the degrees of the generalized moments. At $\rho=2$, an exact heat-system reduction gives the same result for $A=A^*\geq0$, despite the time-differential linkage of the reduced boundary inputs.

We give Hodge-theoretic characterizations of characteristic-zero varieties of open $F$-nilpotent and open weakly $F$-nilpotent type, extending a result of Srinivas and Takagi beyond isolated singularities. We prove one direction unconditionally and the converse assuming the weak ordinarity conjecture.

Let $G_{d}$ be a $d \times d$ matrix with independent standard complex Gaussian entries, let $\alpha_{\mathbb{C}}(d)$ be the expected average singular value of $G_{d}/\sqrt{d}$, and set $\Delta_d := \alpha_{\mathbb{C}}(d)-\alpha_{\mathbb{C}}(d+1)$. The statistic $\alpha_{\mathbb{C}}(d)$ admits a variational representation as an expected normalized maximum over the unitary group and governs approximation guarantees for the little Grothendieck problem over the unitary group and related unitary registration problems. We obtain a strictly positive lower bound and an upper bound for $\Delta_d$, both valid in every dimension, together with corresponding bounds for $\alpha_{\mathbb{C}}(d)$ around the Marchenko–Pastur limit. These bounds match the sharp leading behavior of complete asymptotic expansions for both quantities, whose coefficients are explicitly computable. The proof combines a three-term recurrence for $Y_d = d^{3/2}\alpha_{\mathbb{C}}(d)$, obtained from its continuous dual Hahn representation, along with singularity analysis of the underlying Laguerre moment generating function.

We study Sobolev and $BV$-extension properties of planar subsets. In particular, we prove that fat Sierpiński carpets that support a weak $(1,1)$-Poincaré inequality are $W^{1,1}$-extension sets, and we provide an explicit linear extension operator for them. We also show that for planar domains satisfying a weak $(1,1)$-Poincaré inequality the $W^{1,1}$-extension property and the $BV$-extension property are equivalent. Finally, as a consequence of the previous result, we show that if a simply-connected planar domain is Ahlfors regular and supports a weak $(1,1)$-Poincaré inequality, then it is a (linear) $W^{1,1}$-extension domain.

We develop a general discrete approximation scheme for time-changed Brownian motions on $\mathbb{R}^d$. Our approximation scheme works for any smooth measure with full quasi-support on $\mathbb{R}^d$ with suitable initial distributions. Under some mild conditions on the smooth measure, the discrete approximation scheme works for every starting point. Our results in particular give a discrete approximation scheme for Liouville Brownian motions.

Greedy algorithms efficiently approximate combinatorial optimization problems, but their guarantees weaken when feasibility couples combinatorial structure with global physical constraints. We study monotone nondecreasing supermodular minimization over the bases of a graphic greedoid under physics-induced constraints. We model physics-informed selection using a look-ahead oracle that identifies candidates extendable to a feasible basis, yielding the Conditioned Sequential Greedy Algorithm. We derive a closed-form approximation bound, which we call the price of feasibility, based on the variability of oracle-restricted candidate sets and a probabilistic correction for unobserved elements. As a case study, we show that FORWARD, an algorithm for multi-source radial network reconfiguration, instantiates this framework. Numerical results demonstrate the tightness of the bound and quantify the feasibility-optimality trade-off.

Topological fluctuations are essential to nonperturbative gauge theories but become increasingly difficult to sample toward the continuum limit, where Markov chains can freeze in sectors of fixed topological charge. We introduce a generative sampler, a mixture of sector-resolved samplers (MSRS) that explicitly resolves these sectors and exploits a key advantage of generative models, the ability to directly evaluate the domain-restricted partition function and thereby determine the relative weights of disconnected sectors. We train a generative model in a reference topological sector combined with a bijective topological shift that deterministically maps its samples to other sectors. We demonstrate the method in two-dimensional compact U (1) lattice gauge theory, where it reproduces the topological-charge distribution and yields an unbiased susceptibility in a regime where hybrid Monte Carlo is frozen and overrelaxation gives inaccurate estimates. Our approach also outperforms existing flow-based samplers by orders of magnitude. These results demonstrate that explicit sector resolution provides a promising route to overcoming topological barriers in lattice gauge theory.

As AI-code assistant tools become widespread, automatic assessment of the correctness of generated code becomes a significant challenge. Code LLMs are prone to hallucinations, which may lead to code that does not solve the required problem, or even to code with severe security vulnerabilities. In this paper, we introduce CodeTD – the first approach to pre-execution assessment of code correctness based on topological data analysis (TDA) of Code LLMs' attention maps. Our method quantifies prompt-generation mismatch using topological patterns of attention maps. We carry out experiments with common benchmarks (HumanEval, MBPP, BigCodeBench, MultiPL-E), 5 programming languages and 10 Code LLMs of size up to 34B parameters. The experimental results show that the proposed method outperforms recent baselines. Moreover, CodeTD is transferable between coding benchmarks.

For $1\le m<n$, let $c_{m,n}$ be the infimum of the scales $r$ at which the Vietoris-Rips simplicial complex of the unit round sphere $\mathbb{S}^m$, equipped with its geodesic metric, admits a continuous odd map from $\mathbb{S}^n$. Previous work established the lower bound $\frac{1}{2}c_{m,n}\le d_{\mathrm{GH}}(\mathbb{S}^m,\mathbb{S}^n)$ through a quantitative Borsuk-Ulam theorem and raised the question of whether it is always sharp. We show that this bound can underestimate the distance by a factor arbitrarily close to two, the largest possible factor. Combining projective-code upper bounds for $c_{m,n}$ with lower bounds for $d_{\mathrm{GH}}$ obtained from obstructions to homomorphisms between Borsuk graphs, we obtain explicit finite-dimensional bounds. Along sequences of pairs $1\le m<n$, we have $c_{m,n}\to\frac{\pi}{2}$ whenever $\log n/m\to0$, while $d_{\mathrm{GH}}(\mathbb{S}^m,\mathbb{S}^n)\to\frac{\pi}{2}$ whenever $m/n\to0$. When both conditions hold (for example when $n=m^2$) the ratio of the distance to $\frac{1}{2}c_{m,n}$ tends to two. For each fixed $m\ge3$, the inequality is strict for all sufficiently large $n$.

We prove the sharp quantitative Faber–Krahn inequality with a computable dimensional constant, resolving a problem posed by Brasco and De Philippis~\cite[Open Problem~2]{brasco2017spectral}. In contrast with the known proof of the sharp estimate, our argument avoids indirect compactness arguments and selection principles. It runs through a new level-set proof of the Saint–Venant inequality, using that the torsional deficit controls both the average isoperimetric deficit and the oscillation of the gradient magnitude along the boundaries of level sets. An effective local strong isoperimetric estimate for confined sets then yields the result.

We study the high-dimensional asymptotics of Harish-Chandra-Itzykson-Zuber (HCIZ) integrals in the extensive-rank regime. The limit of these integrals is governed by a one-dimensional boundary-value hydrodynamical problem originally derived by Matytsin (1994) and rigorously proved by Guionnet and Zeitouni (2002). Despite its wide-ranging applications, explicit solutions to this problem are known only in a few specific cases. In this work, we introduce an efficient numerical scheme based on a particle discretization and prove its convergence to the continuous boundary-value problem for generic boundary densities. We validate our approach against known analytical solutions and apply it to generic densities, uncovering interesting dynamical phenomena. The high-dimensional limit of HCIZ integrals appears in various contexts, from the large deviations of random matrix spectra to the limiting free energy of disordered systems, high-dimensional statistics, and machine learning. As such, our contribution opens the way towards the numerical exploration of a wide range of high-dimensional models that were previously intractable.

In this paper we establish a necessary and sufficient condition for the solvability of the Dirichlet problem in the unit ball $\mathbb{B}$ for the second order system $\partial_{\underline{x}} f\partial_{\underline{x}}=0$, where $\partial_{\underline{x}}$ stands for the Clifford-algebra valued Dirac operator in $\mathbb{R}^m$. We prove that the problem admits a solution whenever the boundary data belong to $C^1(\partial\mathbb{B})$. Conversely, we construct explicit counterexamples showing that solvability may fail if this smoothness assumption on the boundary is not satisfied. Moreover, in contrast with the standard commutative setting, we show how that solutions exhibit a regularity one order less than the boundary data, when the latter belong to $C^{k,\nu}(\partial\mathbb{B})$.

Deciding whether a system of multivariate quadratic equations over $\mathbb F_2$ has a solution is a classical NP-complete problem, and remains so for square systems, with as many equations as variables. The hardness of this problem is one of the cornerstones of nowadays post-quantum cryptography. Let $\MQ_0(n)$ and $\MQ_1(n)$ denote the sets of square quadratic systems in $n$ variables having respectively no solutions and exactly one solution. $\cup_{n\geq 2} \MQ_0(n)$ is a coNP-complete language, while $\cup_{n\geq 2} \MQ_1(n)$ lies in DP. It is known that $\lim_{n\to \infty} |\MQ_1(n)|/|\MQ_0(n)|=1$. Here we prove the explicit finite-$n$ bounds \[ |\MQ_0(n)|<|\MQ_1(n)| \le \left(1+\frac{1}{2^n-1}\right)|\MQ_0(n)|, \] More generally, let $Q_d$ be the space of polynomial functions $(\FF_2)^n\to\mathbb F_2$ of degree at most $d$, and let $\alpha_k$ count square systems in $(Q_d)^n$ having exactly $k$ solutions. Then \[ \alpha_0<\alpha_1 \le \left(1+\frac{1}{2^n-1}\right)\alpha_0\,, \qquad 2\le d\le n \,. \] The proof combines matroid and coding-theoretic methods. We interpret $(\FF_2)^n$ as the ground set of the evaluation matroid of $Q_d$, express $\alpha_0$ and $\alpha_1$ through characteristic polynomials, and use a Whitney-type sign-reversing involution to show that the only terms that can push $\alpha_1-\alpha_0$ below $\alpha_1/2^n$ come from the elements of a matroid port. These are identified with minimal-support words of the Reed–Muller code $\RM(n-d-1,n)=\RM(d,n)^\perp$; the required estimate then follows from the MacWilliams identity, the minimum-distance bound $2^{d+1}$, and the even-weight structure of the code.

Consider an infinite edge-weighted graph satisfying an isoperimetric inequality of the type $\|\partial A\|\geq C|A|^\alpha$ for some $\alpha,C>0$, where $\|\partial A\|$ denotes the weighted size of the edge boundary of $A$. We prove that, for $C$ large enough depending on $\alpha$, if each edge is open independently with probability given by its weight, then any vertex is connected to infinity with positive probability. The result also holds under weaker isoperimetric assumptions and on finite graphs. The proof brings a new perspective on the recent proof of the Benjamini–Schramm conjecture concerning the same problem with homogeneous weights. The crucial novelty in our proof is that, rather than simply counting cutsets, we introduce a new Peierls argument which takes into account internal and external connectivity costs in addition to the cost of the blocking surface. We provide two applications for the above result. First, we show that every non-summable long-range percolation on $\mathbb{Z}^d$, $d\geq 2$, admits a percolating truncation, solving a conjecture of Sidoravicius, Surgailis and Vares and its generalization by Friedli and de Lima. Secondly, we show that there exists a universal constant $C < \infty$ such that $p_{\mathrm{c}} \leq C/\Delta$ for every transitive graph of superlinear growth and vertex degree $\Delta$, thus proving a conjecture of Easo and Hutchcroft.

This paper studies joint submodular welfare maximization and routing over graphs, where agents select items under diminishing returns and transport them through a network with congestion-dependent costs. Although welfare maximization admits matroid-based approximations and routing reduces to shortest paths under modular costs, their coupling creates supermodular interactions that break separability. We show that, for fixed routing, the objective remains submodular in the allocation variable, enabling a principled decomposition. Building on this property, we propose the Welfare-based Hierarchical Routing Algorithm (WHIRL), which alternates between tractable routing and allocation updates. Routing is initialized through its modular counterpart, while supermodular effects are modeled as bounded perturbations. The method has finite convergence guarantees and approximation bounds that depend explicitly on the deviation from modular routing. Numerical results illustrate the impact of routing-induced coupling and show that WHIRL achieves a favorable tradeoff between solution quality and computational cost.

Let $p$ be an odd prime, and let $a_n$ be the reduction modulo $p$ of the $n$th coefficient of the Artin-Hasse exponential. We study the weighted $p$-section convolutions $W_k$ formed from the coefficients $a_{kp}$. We prove the conjecture of Avitabile and Mattarei for $1<k<p$ and extend it to a single global power-series identity determining the entire sequence $(W_k)_{k\geq 0}$. This identity yields an explicit base-$p$ digit formula; in particular, $(W_k)$ is $p$-automatic and admits an explicit finite-state evaluator. Independently, the exact conjugated $p$-section equation gives a strict $p$-adic fixed-point iteration for the logarithmic derivative of the $p$-section. Big-Witt reconstruction then recovers the $p$-section, and hence all coefficients of the Artin-Hasse exponential, to arbitrary prescribed $p$-adic precision by a purely modular procedure.

Fix an integer $K\ge2$, and let $C_K^n =\{(e^{\frac{2\pi ij}{K}})_{j=0}^{K-1}\}^n$ be the product of cyclic groups of order $K$. For a Fourier character $\chi_\alpha$, let $s(\alpha)$ be the number of active coordinates. We give a self-contained proposed proof that the dimension-free Bohnenblust–Hille constants governed by interaction order grow polynomially: if $p_d=2d/(d+1)$ and \[ \gamma_2=\frac12, \qquad \gamma_K=\frac{K\log(K-1)}{4(K-2)}\quad(K\ge3), \] then the $\ell^{p_d}$ norm of Fourier coefficients $\{\hat f(\alpha)\}$ is bounded by $L^\infty$ norm of $f$ multiplied by $C(K) d^{4\gamma_K+5}$. The constant $C(K)$ is actually at most of the order $K^{5/2}$.

Understanding generalization remains a central challenge in machine learning because it requires jointly considering data, architecture, and training dynamics. In this paper, we develop a theoretical framework that characterizes how these factors jointly shape generalization performance throughout training. More precisely, we study a broad class of neural networks trained under the $\ell^2$ loss by gradient descent (GD) with weight decay, and prove the convergence of GD to a neighbourhood of the global minimizers of the empirical loss. By partitioning the space based on the input data, we then decompose the population error into data error, optimization error, and prediction variation error, and bound them separately. In particular, for the prediction variation error, which measures the oscillations of the learned function, we propose (local) approximate homogeneity and derive explicit cellwise and layerwise bounds for its evolution along the training trajectory. These bounds yield two important implications: a necessary condition of improved generalization explains differences in layerwise generalization behavior; a sufficient condition describes delayed generalization and provides a theoretical characterization of grokking.

Field temporal prediction and source identification constitute canonical problems in dynamical systems. Conventional approaches to these problems depend on a thorough understanding of the governing partial differential equations (PDEs). Recently, deep learning, as represented by neural operators, has provided a data-driven paradigm for addressing such tasks. However, most existing global neural operators for PDEs require large training datasets and many learnable parameters, with limited interpretability and generalization. We propose the local gradient neural operator (LGNO) as a lightweight and interpretable alternative for field temporal evolution prediction and source identification in typical mechanical problems. The method builds on priors from nonlinear gradient discretization and uses multilayer perceptron convolutional layers to learn translation-invariant local kernels that resemble discrete stencils. A zero consistent stencil factorization separates coefficient learning from field reconstruction, rendering the learned operators more transparent. For problems with symmetries, network folding shares equivalent components and reduces parameter counts. We evaluate the method on PDE benchmarks covering linear and nonlinear, static and dynamic, and low and high dimensional cases. Results show that LGNO maintains accuracy, parameter efficiency, and rollout stability across these tasks, and further exhibits wide applicability to mechanical problems including diffusion, flow, and quantum phenomena.

We investigate the geometric fluctuations of principal subspaces for high-dimensional covariance matrices through the squared Frobenius $\sin\Theta$ distance between the sample and population eigenspaces associated with the $r_p$ largest eigenvalues. An explicit first-order expansion and a central limit theorem are established for this subspace distance. The theory allows the subspace dimension to diverge subject to $r_p=o(n)$, where $n$ is the sample size. It also permits a diverging spectral norm of the population covariance matrix, population spikes of different orders, and repeated or closely spaced spikes. This sharp characterisation captures features of the subspace estimation error that are not reflected in existing perturbation bounds. As applications, we derive an explicit asymptotic expansion for the expected PCA excess risk and a refined error bound for distributed PCA. In both cases, existing upper bounds can increase with the spiked-block condition number when some leading spikes become stronger, whereas our results show that the corresponding estimation errors need not increase and may instead decrease. Numerical experiments reproduce this contrasting behaviour and demonstrate the finite-sample accuracy of our theoretical findings.

We classify all natural, conformally covariant, symmetric (0,2)-tensors of conformal weight -2 and differential order less than or equal to 4 in dimension 4, built from the metric, the Schouten tensor, covariant derivatives, and the Kulkarni-Nomizu product. We prove that the space of such tensors is exactly 2-dimensional, spanned by the Bach tensor (order 2) and the Eastwood-Singer tensor (order 4). The algebraic core is formally verified using Lean 4 with Mathlib. The 8x11 constraint matrix and divergence-free condition are verified computationally with exact rational arithmetic. Both English and Russian versions provided; Russian version available as ancillary.

For a fixed graph F, the F-degree of a vertex v in a host graph H is the number of subgraphs of H isomorphic to F that contain v, and H is F-irregular if its F-degrees are pairwise distinct. We show that every finite connected graph F on at least three vertices admits a finite connected F-irregular host. For noncomplete F, the proof builds the host from a threshold graph with one deleted edge; when the minimum degree is at least two, a small incidence gadget with distinct weighted column sums separates the remaining exceptional vertices. The construction also yields infinitely many pairwise non-isomorphic finite connected F-irregular hosts for every noncomplete F. The complete-pattern case follows from a theorem of Chartrand, Holbert, Oellermann and Swart. A Lean 4 formalization of Theorem 1.1 is described, taking the published complete-pattern theorem as its sole custom axiom.

Classical approximation methods are usually improved by changing the sampling set, increasing the number of data points, or selecting a different basis. We advocate a complementary viewpoint: keep the sampled values fixed and modify the representation through a suitable mapping. This viewpoint unifies several constructions that were originally introduced for different purposes. Rescaled radial basis function interpolation maps the interpolation operator through a normalization that enforces exact reproduction of constants. Variably scaled kernels map the geometry by lifting the data to a higherdimensional manifold determined by a scale function. Mapped bases and fake nodes map the approximation space without resampling the data, while rational kernel expansions can be interpreted as nonlinear mappings of the trial space. We develop a common notation for these mechanisms, summarize their approximation and stability properties, and explain how discontinuous and rational variants fit the same framework. Reproducible numerical experiments illustrate constant reproduction, error reduction through data-mimicking scale functions, and the suppression of the Runge phenomenon by mapped polynomial bases. The results support a general principle: mappings provide a flexible way to adapt approximation spaces to data geometry and regularity without modifying the original observations