Papers

We consider a random subgraph $G_p(n,r,<s)$ of the complete distance graph $G(n,r,<s)$ whose vertices are the $r$-element subsets of the set $\{1,\dots,n\}$ and whose edges join pairs of subsets that intersect in fewer than $s$ elements; each edge survives independently of the others with probability $p$. The independence number of the graph $G(n,r,<s)$ equals $C_{n-s}^{r-s}$ – this is the classical Erdos-Ko-Rado theorem. We prove that, for $r=r(n)\to\infty$, $s=s(n)\to\infty$, $s=o(r)$, $r^2=o(n)$ and $p\ge 16\,sr^2\ln(n/r)/n$, with probability tending to 1 the independence number of the random graph $G_p(n,r,<s)$ also equals $C_{n-s}^{r-s}$, i.e., the Erdos-Ko-Rado result is stable under random sparsification of the graph. Thereby, in the range of parameters $s\to\infty$, $s=o(r)$, a recent result of Raigorodskii and Karas is strengthened: the lower bound on the probability $p$ that guarantees stability is lowered by a factor of about $r/s$.

We construct an MF C*-algebra A such that the spatial tensor product of A and the hyperfinite II_1 factor R contains a proper isometry. Consequently, stable finiteness of C*-algebras is not stable under tensor product and the hyperfinite II_1 factor is not a quasidiagonal C*-algebra. The C*-algebra A is non-simple and has only non-faithful tracial states. This result was obtained using OpenAI's Chat GPT Pro 6.0.

An intrinsically knotted graph is one for which every spatial embedding contains a nontrivially knotted cycle. Classifying such graphs is a central problem in spatial graph theory. It is known that every intrinsically knotted graph has at least 21 edges, and the case of 21 edges has been completely resolved. For 22 edges, however, the classification remains incomplete. In particular, exactly eight triangle-free examples with a vertex of degree at least 5 are known, leaving only the case in which all vertices have degree 3 or 4. In this paper, we introduce a method for detecting intrinsic knottedness based on induced subgraphs obtained by deleting pairs of vertices. Using this method, we classify all triangle-free intrinsically knotted graphs with 22 edges having eight vertices of degree~4 and four of degree~3. We prove that there are exactly four: Cousins 43, 105, and 109 in the $E_9\!+\!e$ family and the graph $H_{12}\! +\! e$ in the $H_9\!+\!e$ family.

In this paper, we propose weighted Mathai-Haubold entropy along with their residual and past versions and study their properties. We develop aging classes based on the weighted Mathai-Haubold residual and past entropy measures. Also, some inequalities related to the three proposed measures are discussed. Non-parametric estimators of the proposed measures are introduced and their properties are investigated. Performance of this estimator is evaluated by means of bias and mean squared error using Monte-Carlo simulations.

Let $A\in\mathbb{R}^{m\times n}$ have columns of Euclidean norm at most one. We prove that $\operatorname{disc}(A)\le2395\left(1+\log_+\frac n9\right)^{1/4}+2\sqrt2$. Here $\log_+t=\max\{0,\log t\}$. Building on Bansal and Jiang's affine spectral independence framework, we remove the $(\log\log n)^{7/4}$ factor from their bound. The fourth root comes from balancing the logarithmic decrease in the alive dimension against the fourth power of the row thresholds. Historical exponential sums control the covariance budget across size classes with summable thresholds. An exact threshold-sum certificate gives the coefficient $2395$, and rounding at most eight remaining fractional coordinates costs $2\sqrt2$. The finite construction also gives partial colourings from any prescribed starting point and at any prescribed depth, preserving existing signs. We formalize the partial- and full-colouring theorems in Lean, including the finite trajectory, exact threshold sum and final rounding, with Bansal–Jiang Theorem A.4 as the sole external research theorem assumption.

We consider the level sets of the Gaussian free field and the vacant set of random interlacements, both defined on a typical realization of the infinite cluster of supercritical Bernoulli bond percolation on $\mathbb{Z}^d$, $d \geq 3$. We prove that in the entire supercritical regime of Bernoulli bond percolation, both the level sets of the Gaussian free field and the vacant set of random interlacements undergo non-trivial percolation phase transitions at deterministic critical levels. A key aspect of the proof is the development of certain quenched controls over tree embeddings, permitting the application of a static renormalization scheme in the presence of spatial irregularities, which may be of independent interest.

In this paper, we present a family of high-order cut finite element methods based on the discontinuous Galerkin (DG) framework for scalar hyperbolic conservation laws on complex domains. Building on our previous work, we develop a multidimensional formulation that combines macro-element stabilization with flux limiting to obtain a scheme that preserves the maximum principle and remains robust with respect to arbitrary boundary cuts of the background mesh. The physical domain is embedded in a regular background mesh, which may produce arbitrarily small cut cells. To avoid the severe time step restrictions typically associated with such cells, ghost penalty stabilization terms are added on interior facets of macro-elements. The resulting method exhibits stability and accuracy properties similar to those of standard DG methods on fitted meshes. An $L^2$-stability result is derived for the semi-discrete scheme under both periodic and inflow-outflow boundary conditions. To enforce the maximum principle and suppress nonphysical oscillations, we adapt limiter techniques from standard DG methods to the CutFEM setting by defining limiting parameters on macro-elements. In particular, we present a macro-element-based parameterized flux limiter together with adaptations of the Zhang-Shu bound-preserving limiter and the Barth-Jespersen slope limiter. Numerical experiments in two and three spatial dimensions demonstrate optimal convergence orders, preservation of the maximum principle, and accurate shock capturing without spurious oscillations, even for challenging cut configurations involving very small-cut cell intersections.

This paper studies a time-inconsistent mixed regular-singular control problem in continuous time with non-exponential discount. We seek time-consistent equilibria in an intrapersonal game framework and propose a novel definition of regular-singular equilibrium where both regular and singular controls are unified via control laws. We establish a verification theorem to provide a sufficient condition of the equilibrium. Furthermore, we propose a novel concept of mild equilibrium for which perturbations are applied separately to the regular and singular control and give the mild verification theorem. We illustrate the applicability of the theory to the effort and dividend problem and obtain an explicit equilibrium depending on the effort and dividend thresholds under a mixture of exponential discount functions and a pseudo-exponential discount function by solving the extended Hamilton–Jacobi–Bellman (HJB) system. The convexity of the value function is rigorously established, and the equilibrium conditions are verified. In addition, we establish the existence of the effort and dividend thresholds under a mixture of exponential discount functions. Numerical results reveal the impacts of exogenous parameters on the effort and dividend thresholds and the equilibrium value function, along with their economic implications.

We prove that the maximal derived length of a solvable subgroup of the plane Cremona group over $\mathbb C$ is equal to $5$. The main difficulty is the case of subgroups containing loxodromic elements. To treat this case, we prove that the maximal derived length of a finite solvable subgroup is $4$ and establish a centraliser theorem: the centraliser of a finite solvable subgroup of derived length $4$ contains no loxodromic element. The proof of the centraliser theorem relies on Blanc's classification of maximal algebraic subgroups and then uses equivariant Sarkisov theory, orbit estimates, superrigidity, and fixed curves of genus at least two.

We analyze the covariant phase space of Klusoň's canonical Carroll non-BPS D$_p$-brane actions. The Carroll limit is formulated as a contraction of the canonical phase space: the canonical one-form is invariant under the scaling of conjugate pairs, while the leading Hamiltonian constraint differs between the electric-like and magnetic-like sectors. This difference controls the weak closure of the electric-like constraint algebra and the stronger closure of the magnetic-like Hamiltonian brackets. We separate the generic non-BPS sector, which carries $D-1$ local phase-space degrees of freedom, from the tachyon-vacuum sector, which carries $D-2$ only after imposing second-class background conditions. We also identify the rank condition for the generic electric-like sector and its strengthening in the vacuum sector. Finally, we show that the Carroll–Weyl $\chi$ transformation, the matter-sector analogue of the null-string Carroll–Weyl symmetry, preserves the symplectic form and admits a charge, but is obstructed by the constraints and cannot be promoted to a first-class gauge generator except in highly restricted corners, a global vacuum sector or the singular potential $V\propto T^{-1}$ at $p=1$. The obstruction is algebraic in $\chi$, so no restriction of its spacetime profile can remove it. This contrasts with null string theory, where a restricted $\chi$-symmetry can be completed to an additional first-class constraint.

Sparsity is a powerful structural resource in optimization and statistics. We develop frameworks for leveraging sparsity in sampling problems over the Hamming slice $\mathcal{X}_k^d:=\{\mathbf{x}\in\{\pm 1\}^d:|\{i:\mathbf{x}_i=1\}|=k\}$, in high-dimensional regimes where $k\ll d$ (i.e., where $\mathcal{X}_k^d$ is highly magnetized). We use our frameworks to design improved samplers for canonical problems in the study of Ising models and Bayesian sparse linear regression. Our first main result considers the Sherrington–Kirkpatrick (SK) model restricted to fixed-magnetization slices $\mathcal{X}_k^d$. We give a polynomial-time sampler for fixed-magnetization SK models at any inverse temperature $\beta>0$, under arbitrary external fields, provided that $k\le c_\beta d$ for an appropriate constant $c_\beta$. By combining this result with an annealing strategy for estimating normalizing constants, we obtain polynomial-time samplers for the SK model at arbitrarily low temperatures under a sufficiently strong external field of strength $h$. In the large-$\beta$ limit, our framework permits sampling at field strengths within constant factors of the Almeida–Thouless line delineating the replica-symmetric and replica-symmetry-breaking regions ([dAT78]), improving polynomially over the field strength $h(\beta)$ required by the recent work of [BAR26]. Our second main result concerns the measurement complexity of polynomial-time Bayesian sparse linear regression. Recent work by [KSTZ25] shows how to sample from the canonical Gaussian spike-and-slab posterior with expected sparsity $k$, at any signal-to-noise ratio, given $n\gtrsim k^3\log^3 d$ Gaussian measurements. We improve this requirement to $n\gtrsim k^{3/2}\log^2 d+k\log^3 d$, using a common sparsity-aware framework underlying both our results.

In this paper, we prove pathwise uniform convergence at rates up to $1/2$ for tamed exponential Euler schemes for semilinear hyperbolic stochastic evolution equations with superlinearly growing nonlinearities and multiplicative noise. We take the term hyperbolic to mean that the leading operator generates a contractive $C_0$-semigroup but no parabolic smoothing occurs. Under local Lipschitz, polynomial growth, coercivity, and monotonicity conditions on the nonlinearities, we establish pathwise uniform strong error estimates of the form \begin{equation*} \Big(\mathbb{E}\max_{0\le j \le N} \|U(t_j)-U^j\|_X^p\Big)^{1/p} \lesssim \sqrt{k} \end{equation*} on a Hilbert space $X$ for $p\in [2,\infty)$. Here, $U$ is the mild solution and $U^j$ is the tamed exponential Euler approximation at time $t_j=jk$ with step size $k>0$. This extends previous convergence results for non-parabolic SPDEs from globally to locally Lipschitz nonlinearities, allowing both drift and diffusion to grow polynomially. In a stochastic Kato framework, we further establish local and global well-posedness as well as uniform a priori estimates for the mild solution and its approximation. Applications to nonlinear stochastic transport, Airy, wave-type, and dissipatively damped nonlinear Schrödinger equations are included, covering different nonlinearities-stopped and fractionally tamed schemes. For the Klein-Gordon equation with cubic velocity damping, this complements previous results obtained for additive noise.

We establish a one-to-one correspondence between geodesics associated with the one-parameter family of $\alpha$-connections on the Gaussian statistical manifold and a class of continuous stochastic processes characterized by a time-independent noise intensity. We demonstrate that geodesics in expectation parameters naturally classify into three distinct geometric categories, among which the Boundary-connecting class allows us to construct an explicit linear stochastic realization with constant diffusion, representing a generalized bridge process. This result demonstrates how continuous stochastic processes within this Gaussian class can be extended along geometric curves. Under appropriate operational limits, this generalized bridge process reduces to fundamental stochastic dynamics, either Ornstein-Uhlenbeck (OU) relaxation or free Brownian diffusion. Crucially, the physical restoring force governing the resulting OU relaxation directly determines the underlying connection parameter $\alpha$, providing a concrete physical observable to constrain the manifold geometry. Depending on the chosen affine connection representation, this restoring force can be attributed either to scalar curvature or purely to non-metricity, establishing a direct conceptual analogy with the Geometrical Trinity of Gravity. Furthermore, applying this framework to driven stochastic thermodynamics, we show that the work-minimizing optimal protocol in the slow-driving limit coincides precisely with an expectation geodesic of the statistical manifold with non-metricity equipped with $(g, {}^{(1/2)}\Gamma, {}^{(-1/2)}\Gamma)$, highlighting the active physical role of non-metricity in information geometry.

We study the two-generator one-relator groups defined by $\langle x_0,x_1\mid U(x_0,x_1)U(x_1,x_0)^{-1}\rangle$, and determine exactly which of them fail to contain a free subgroup of rank $2$. We show that every such group is a solvable Baumslag–Solitar group $BS(1,m)$, and that every integer value $m$ can occur. This disproves a conjecture predicting that only $m\in \{-1,0,1\}$ can occur.

We study Turán-type extremal problems for cancellative and locally thin uniform hypergraphs. An $r$-uniform hypergraph is $t$-cancellative if $(\cup_{i=1}^t A_i)\cup B\ne (\cup_{i=1}^t A_i)\cup C$ whenever $A_1,\ldots,A_t,B,C$ are distinct edges. Let $C_t(n,r)$ denote the maximum number of edges in such a hypergraph on $n$ vertices. For all fixed integers $t,k\ge2$, we prove that $C_{2(t-1)}(n,tk)=(1+o(1))\frac{\binom{n}{k}}{\binom{tk-1}{k-1}}$ as $n\to\infty$. In the case $t=2$, this shows that Füredi's 2012 upper bound for $C_2(n,2k)$ is asymptotically sharp. The lower bound uses locally sparse induced packings, while the upper bound follows from double counting and a matching argument. More generally, for integers $s\ge t\ge1$, an $r$-uniform hypergraph is locally $(s,t)$-thin if among any $s$ distinct edges, at least $t$ contain a vertex that lies in none of the other $s-1$ edges. This notion includes cancellative hypergraphs as special cases. We establish general upper and lower bounds for the corresponding extremal numbers and determine their polynomial order of growth under suitable divisibility assumptions.

Let $k=\mathbb F_q$. We classify, up to pre- and post-composition by $k$-Möbius transformations, all separable $k$-indecomposable rational functions $f\in k(X)$ of degree greater than one whose Galois closure has genus zero. The classification is valid in arbitrary characteristic and includes exact arithmetic conditions and class counts over the prescribed field. Semilinear Frobenius descent determines the finite-field forms and which geometric decompositions descend to $k$. For every separable $f\in k(X)$ of degree greater than one with Galois closure of genus zero and every $m\geqslant1$, we prove that $f$ permutes $\mathbf P^1(\mathbb F_{q^m})$ if and only if it is exceptional over $\mathbb F_{q^m}$, meaning that it permutes $\mathbf P^1(L)$ for infinitely many finite extensions $L/\mathbb F_{q^m}$. No indecomposability assumption or lower bound on $q$ is needed. The argument combines fixed-point averaging with ramification on the Galois-closure curve. If the full constant field is $\mathbb F_{q^d}$, these properties depend only on $\gcd(m,d)$. For each classified family we determine the permutation extension degrees explicitly and characterize polynomial representatives.

In this paper, we give the dual characterization of conformal tetrahedra in constant curvature spaces, which develops a geometric technique to obtain the global rigidity of (ideal) hyperbolic, Euclidean, spherical sphere packings on a tetrahedron. Moreover, we introduce ideal hyperbolic sphere packings and extend the variational principle to obtain the global rigidity of (ideal) hyperbolic sphere packings on 3-manifolds. We also provide a characterization of conformal $n(\ge3)$-simplices in constant curvature spaces, which is useful for studying higher-dimensional sphere packings.

Can asymmetry between two interacting networks fundamentally change how they synchronize? We identify double explosive transitions in the forward direction, backward direction, or a combination thereof, with single or double hysteresis loops in an adaptive bilayer multiplex network of Kuramoto oscillators with pairwise and three-body interactions and asymmetric phase lags. Using the Ott-Antonsen reduction, we derive a low-dimensional system and perform a stability analysis. The reduced model accurately captures the full microscopic dynamics and enables analytical expressions for the bifurcation points. Systematic mapping across multiple parameter planes reveals eight distinct synchronization regimes. The relative ordering of saddle-node and pitchfork bifurcation points—controlled by phase-lag asymmetry, cross-layer adaptation, and the higher-order interaction strength—creates two distinct coherent branches (weak and strong), giving rise to double explosive transitions. Crucially, phase-lag asymmetry acts as a robust control knob: while symmetric phase lags suppress explosive transitions, layer-specific differences promote multistability and double explosive transitions. The higher-order interaction strength $K_2$ and adaptation strengths $q$, $p$, and $h$ further modulate these transitions in a complex, parameter-dependent manner. Excellent agreement between analytical predictions and numerical simulations confirms the robustness of our reduced description.

Let $(a_n)_{n\geq0}$ be a sequence of complex numbers, and define \[ b_n=\sum_{k=0}^n \binom{n}{k} a_k, \qquad c_n=\sum_{k=0}^n \binom{n}{k}(-1)^{n-k}a_k. \] Let $\beta>1$, put $\alpha=\sqrt{\beta^2-1}$, and suppose that $b_n,c_n=O(\beta^n)$. Mashreghi and Ransford proved that \[ \limsup_{n\to\infty}\frac{|a_n|}{\alpha^n} \leq \kappa \left(\limsup_{n\to\infty}\frac{|b_n|}{\beta^n}\right)^{\!1/2}\! \left(\limsup_{n\to\infty}\frac{|c_n|}{\beta^n}\right)^{\!1/2} \] with a universal constant satisfying $2/\sqrt{3} \leq \kappa\leq2$. We prove that the optimal constant is indeed $\kappa=2/\sqrt{3}$. The proof, which uses exponential generating functions, Phragmén-Lindelöf estimates, and Cauchy's formula, is compared to the classical one of Mashreghi and Ransford.

We give conditions ensuring that an asymptotically rigid mapping class group, specifically a surface Houghton group $\mathcal{H}(S)$, has finite index in the asymptotically conformal modular group $\text{Mod}_0(X)$, where $X$ is a hyperbolic structure on $S$. These include geometric conditions on the pieces of the underlying rigid structure, as well as the existence in $\mathcal{H}(S)$ of an end-periodic homeomorphism which is asymptotically conformal. As a consequence, if $S$ has $n\geq 3$ ends, then $\text{Mod}_0(X)$ has type $F_{n-1}$ but not $FP_n$. We also establish analogous results for $L^p$ modular groups.

Let $P_t$ denote the induced path on $t$ vertices. Let $\omega(G)$ denote the maximum number of vertices in a clique of a graph $G$. Gyárfás (1987) proved that every $P_t$-free graph $G$ satisfies $\chi(G)\le(t-1)^{\omega(G)-1}$, and Gravier, Hoàng, and Maffray (2003) improved this to $\chi(G)\le (t-2)^{\omega(G)-1}$ for $t\ge4$. We lower the base of the exponential by one: for every $t\ge5$, every $P_t$-free graph $G$ satisfies \[ \chi(G)\le 3\,(t-3)^{\omega(G)+4}. \] The proof combines two refinements of the Gyárfás path argument and was developed with the assistance of Claude Fable 5.1 of Anthropic and GPT Pro of OpenAI.

We study the Longest Edge Bisection (LEB) process as a dynamical system on the projective shape space of simplices. A long-standing conjecture going back to Adler and Rivara-Levin and motivated by finite-element mesh refinement, often taken as a standing assumption, is that this procedure is non-degenerate and, in fact, in a certain way periodic. We prove: \begin{itemize} \item There are 3-dimensional simplices such that the longest edge-bisection algorithm degenerates. \item There is an open set of 4-dimensional simplices on which the longest edge-bisection algorithm degenerates. \item If parametrizing the space of $d$-dimensional simplices by independent standard Gaussian vectors, then as $d$ increases, a random simplex degenerates asymptotically almost surely. \end{itemize} This is realized through exhibiting hyperbolic behaviour of the LEB process. We also exhibit elliptic behaviour that is nonperiodic.

We study modified scattering for the two-dimensional defocusing nonlinear Schrödinger equation (NLS) with the gauge-invariant, scattering-critical nonlinearity $|u|u$. We prove that the remodulated interaction representation has a local $L^2$ limit for general data in $\Sigma=H^1\cap\F^{-1}H^1$. For radial data, we prove convergence in $L^p(\R^2)$ for every $2<p<\infty$.

We study graph-based automata: nondeterministic finite automata obtained from edge-colored or oriented graphs by taking every vertex as both initial and accepting, and every edge as a pair of opposite transitions. The language of these automata corresponds to the set of edge-colored or oriented paths mapping to their corresponding graphs. We develop an analogous notion for trees and characterise the languages recognised by these models. For tree languages we prove the existence of a unique size- minimal graph and, more generally, a homomorphism-minimal graph for both word and tree languages using duality methods. In order to further motivate these models, we showcase a few results at the inter- section between graph theory and automata theory: We relate graph-based automata to reversible automata, give a decomposition of graph-based languages into reversible languages, and introduce the remanent language of an undirected graph as the inter- section of all its orientations. This remanent language captures structural information on the graph such as chromatic number.

We consider the heat equation in an interval of the real line, with null initial condition, and Dirichlet boundary controls. We study the cost of reaching an attainable state at some time $T>0$. Precisely, given a state in the interval that we know is reachable by suitable boundary controls, we estimate the size of the smallest controls that reach it, in terms of an appropriate norm of this state. Our estimate is optimal in some sense as the time horizon goes to $0$.

We prove a trajectory large deviation principle, in the low-temperature limit, for the two-dimensional Wick-ordered defocusing cubic nonlinear wave equation, with initial data drawn from the invariant Gibbs measure. We further show that asymptotically vanishing high-frequency perturbations of the Gibbs initial data, with the same static rate function, generate a continuum of distinct mass-shifted trajectory rate functions.

The aim of this paper is to present a topological duality for the algebraic category of Stonean Hilbert algebras and certain morphisms between them, called $\neg$-morphisms. Also, we apply this duality to characterize the Heyting algebra of $\alpha$-ideals of a Stonean Hilbert algebra.

This paper proposes a shared-radius co-prime circular array for high-resolution, continuous 2D Direction-of-Arrival (DoA) estimation in 3D space, jointly estimating azimuth and elevation angles. The proposed architecture consists of two uniform circular sub-arrays with co-prime antenna counts sharing a common radius RR, a design that intrinsically suppresses mutual coupling leakage compared to dense uniform arrays. Unlike existing works that rely on complex phase-mode transformations to map circular structures to virtual linear arrays, we introduce a hybrid continuous-recovery framework operating directly in the physical spatial domain. By integrating a fast, discrete coarse-grid search with a swarm-intelligence continuous refinement stage, the proposed method completely bypasses discrete grid-mismatch limitations and computationally expensive eigenvalue decompositions. A rigorous theoretical analysis using Nivens Theorem establishes the spatial uniqueness of the true source direction, effectively resolving phase ambiguities. Simulation results demonstrate that this hybrid scheme achieves superior resolution and lower Root Mean Square Error (RMSE) at low Signal-to-Noise Ratios (SNR) compared to uniform configurations, while asymptotically converging to the theoretical Cramer-Rao Bound (CRB) at high SNRs.

Let \(p(n)\) denote the ordinary partition function. Motivated by analogous questions concerning Euler's totient function and its complementary counting function, we study the range of the partition-derived sequence \(p(n)-n\). We give combinatorial interpretations of this sequence and investigate both the attained and omitted positive integers. We obtain exact and asymptotic information about the gaps between consecutive attained values and show that the range is remarkably sparse: its counting function has order \((\log x)^2\), and consequently the range has natural density zero. We also extend the discussion to partitions whose Durfee square has side at least a fixed positive integer.

Kernel spectral algorithms are often summarized by convergence rates, which hide how their risk depends jointly on regularization, noise, the population spectrum, target coefficients, and the chosen filter. Gaussian sequence models arise as a simplified but characteristic setting for studying the interplay of these factors, where the kernel spectral algorithm corresponds to a coordinatewise shrinkage estimator. Under mild assumptions, we show that the risk of a kernel spectral estimator is asymptotically equivalent to that of the corresponding Gaussian sequence model estimator with the same filter. The explicit sequence model risk then yields a full characterization of the risk of kernel spectral algorithms in terms of the population spectrum, target coefficients, and filter. We establish this equivalence for a broad class of spectral filters, covering kernel ridge regression, generalized ridge regression, iterated kernel ridge regression, gradient flow, stable gradient descent, smoothed spectral cutoff, spectral clipping, and Pinsker shrinkage. Our risk equivalence not only holds in the classical fixed-dimensional regime but also applies to the high dimensional regime where the input dimension scales with the sample size. As applications, our risk equivalence recovers the minimax upper rates, and establishes the exact Pinsker constant in RKHS regression.