Papers

An algebra with zero is an algebra that contains an element 0, called zero, such that all fundamental operations have the property that the result is zero whenever at least one of the arguments takes the value 0. We introduce the notion of the higher commutator of ideals and Rees congruences for such algebras. It is shown that the commutator of Rees congruences is equal to the Rees congruence of the commutator of the corresponding ideals. We apply this result to provide characterisations of n-supernilpotency and n-nilpotency in algebras with zero, as well as n-solvability in groupoids with zero.

Let $\Palt=(\C,\T)$ be a $k$-palette. For $0\le t\le k-1$, its $t$th coordinate-extension degree is the minimum, over every choice of $t$ coordinates and every assignment of colors to them, of the proportion of assignments to the remaining $k-t$ coordinates that complete the fixed colors to an admissible $k$-tuple. For a $k$-graph $F$, we define $\pi_t^{\ext}(F)$ as the supremum of this degree over all palettes not admitted by $F$. We prove that \[ \pi_t^{\ext}(F)=0 \quad\text{if and only if}\quad F\text{ is }t\text{-layered}. \] We also relate $t$-layeredness to vanishing orders, min-layeredness, max-layeredness, and layeredness. These results recover and extend previous characterizations of Reiher, Rödl, and Schacht and of Lamaison, and answer a question of Lamaison for $3$-graphs. At $t=0$, the parameter $\pi_0^{\ext}(F)$ is the $(k-2)$-uniform Turán density $\pi_{k-2}(F)$. For every $k\ge3$ and $r\ge2$, we construct a finite $k$-graph $F_{k,r}$ with \( \pi_{k-2}(F_{k,r})=2(r-1)/rk^k. \) Thus $2/k^k$ is an accumulation point for single forbidden $k$-graphs. We also show that the least density of a $k$-graph that fails condition $\Sp$ of Lin, Wang and Zhou is $4/(3k^k)$. Finally, for every admissible matching of size $m$, we construct a $k$-graph that satisfies $\Sp$ for every coordinate pair, has no vanishing order, and has density $2^m/k^k$. This disproves a conjecture of Lin, Wang and Zhou for every $k\ge3$.

We establish the local well-posedness of strong solutions for a thermodynamically consistent diffuse interface model describing two-phase flows with surfactants, specifically the so-called Model C introduced by Garcke, Lam, and Stinner. Because of the non-standard, mixed-order strongly coupled structure of Model C, the classical Agmon-Douglis-Nirenberg theory and conventional Lopatinskii-Shapiro boundary conditions are not applicable due to divergent scaling behaviors. To overcome this, we develop a tailored resolvent analysis, adapted to the temporal regularity of the source term, in a suitable Hilbert space, which yields an exact cancellation of the highest-order cross-couplings in the estimates. This crucial decoupling enables the rigorous application of the operator-valued Mikhlin multiplier theorem to obtain resolvent estimates on $\mathbb{R}$. Furthermore, we analyze a structural variant, denoted as Model D, which achieves a crucial structural decoupling by transferring the interfacial surfactant energy entirely from the gradient part to the potential part, and we readily establish local well-posedness for strong solutions.

Quasi-cyclic low-density parity-check (QC-LDPC) codes are widely used in modern communication standards due to their simple structure and efficient encoder and decoder implementations. However, the commutativity of the circulant permutation matrices used in their construction significantly limits their girth and minimum distance, affecting iterative decoding performance. In this paper, we investigate protograph-based codes constructed using permutation polynomials of higher degree, focusing on a closed family of non-commuting binomial permutation polynomials that enables these constructions to exceed both of these girth and minimum distance upper bounds while maintaining a simple representation. Simulation results demonstrate improved decoding performance compared to constructions based on affine permutation polynomials, which include QC-LDPC codes.

S. Lie was the first to try to classify maximal subalgebras of simple vectorial (i.e., consisting of vector fields) Lie algebras with polynomial coefficients. In Appendix it is shown that, as stated, the problem is wild: describing all subalgebras contains the classification of pairs of matrices up to simultaneous conjugation, so there are too many such subalgebras. Several researchers distinguished various classes of maximal subalgebras important in applications. Here, in simple (or in Cartan prolongations close to simple) vectorial Lie superalgebras with polynomial coefficients considered with gradings associated with Weisfeiler filtrations, we describe maximal simple (and close to simple) graded subalgebras — a superization of what Sophus Lie tackled. Together with I. Shchepochkina's description of the maximal subalgebras of matrix Lie superalgebras, see arXiv:hep-th/9702122, this paper solves the super and infinite-dimensional version of Dynkin's problem — the description of the maximal subalgebras of simple finite-dimensional Lie algebras — except for the 3 exceptional ambients considered elsewhere.

For every fixed $r\ge3$ and all sufficiently large $n$, we determine the largest adjacency spectral radius of an $n$-vertex graph with fewer than $q c_r(n)$ copies of $K_{r+1}$, where $1\le q<n/r$. Here $c_r(n)$ is the number of copies created by adding one edge to a largest part of the Turán graph. We also determine all extremal graphs. In most cases the extremal graph is obtained from an almost balanced complete multipartite graph by adding a star in one part. Two small values of $q$ require separate constructions, and an additional transition occurs when $r=3$ and $n\equiv2\pmod3$. Under the non-strict constraint, the unique extremal graph is obtained by adding a $q$-edge star to a largest part of the Turán graph. We determine the difference between the strict and non-strict values and prove that the sharp matching threshold is $\sqrt2(r-1)/r$. The proof treats separately the ranges $q=o(n)$, $q/m\to\gamma\in(0,1)$, and $q/m\to1$.

We study the allocation of a force of a fixed size across independent battlefields against a known distribution of opposing forces. Formulating battlefield outcomes through loss and survival functions allows us to establish allocation results without assuming a specific system of combat equations. For convex enemy loss functions, an optimal allocation exists with at most one partially saturated battlefield. We prove NP-hardness for every prescribed family of strictly convex, strictly increasing loss functions satisfying an endpoint normalization, and give an exact pseudopolynomial algorithm for the special case when the total force size and saturation thresholds are integers. We also derive allocation rules for maximizing successful engagements, maximizing own survivors, and maximizing the ratio of enemy losses to own losses. Applications include a power-law family containing the classical Lanchester, guerrilla, and mixed combat models, as well as models with implicitly defined outcomes.

We study parallel samplers that reveal each coordinate once and draw the coordinates of each batch independently from their exact conditional marginals. Under the same sampling interface and a hard three-round cap, randomization can improve accuracy beyond every deterministic policy that adapts its batches to previously observed values. For an explicit full-support binary family, the optimal deterministic adaptive forward Kullback-Leibler divergence grows linearly with the dimension, while a uniform mixture of four fixed schedules has exponentially small divergence. Only two independent fair bits are needed to select the schedule. We determine the asymptotic additive constant in the deterministic optimum and show that its optimal total variation tends to one quarter. An outward-rounded finite-sum certificate establishes divergence separation at dimension 389. The mechanism combines an irreversible first-batch dependence loss with posterior concentration in the randomized components. A density-coverage lemma describes how their mixture restores the target. A complementary noisy-matching family exhibits a growing round gap for nonadaptive schedules, although uniform random coordinate order still has linear mixture error. These results concern known finite structures and exact conditional marginals.

We prove Mond's conjecture for $\mathcal A$-finite corank-one map germs $(\mathbb C^3,0)\to(\mathbb C^4,0)$: the $\mathcal A_e$-codimension is at most the image Milnor number, with equality for quasihomogeneous germs. For a quasihomogeneous germ we determine the derivations of its image modulo the conductor fields, in terms of the double-point surface, its cross-cap curve and its normalisation. This yields a closed formula, depending only on the weights and degrees, for the Hilbert series of the graded $\mathcal A_e$-normal space; its value at $t=1$ coincides with Ohmoto's formula for the image Milnor number, which comes from Segre–Schwartz–MacPherson Thom polynomials. For arbitrary germs we show that generic homogeneous corank-one germs of coprime degrees are $\mathcal A$-finite in every source dimension, and apply the reduction theorem of Fernández de Bobadilla, Nuño-Ballesteros and Peñafort Sanchis. In every pair of nice dimensions this reduces Mond's conjecture for corank-one germs to an inequality for generic homogeneous germs.

We develop a parity-sensitive admissibility theory for topologically expansive decreasing Lorenz maps. By embedding them into the standard circle map $m_{-2}$, we realize every decreasing-admissible pair. We then study the negative doubling map $T_{-2}$ with a hole $(a,b)$, where $0<a\leq1/2\leq b<1$. Using the alternating lexicographic order, we classify plateaux of the symbolic survivor spaces. Under weak admissibility and strict shifted ordering, the boundary subshifts are realized as kneading spaces of decreasing Lorenz maps. We also establish entropy correspondences across countable itinerary discrepancies in completed survivor systems. For fixed $a$, we prove that $b\mapsto\dim_{\mathrm H}S_{-2}(a,b)$ is a devil's staircase, possibly constant. Under the pullback-null condition~(P), we obtain the corresponding survivor-entropy result for expansive decreasing Lorenz maps. We verify this condition for the piecewise-linear family in Example 5.1.

This paper develops a deep learning framework for solving high-dimensional stationary Fokker-Planck equations by combining Monte Carlo weak supervision, energy-variational physics constraints, and quasi-Newton refinement. Monte Carlo statistics anchor the scale of the stationary density, while a free-energy variational loss enforces equilibrium physics; the formulation also applies to stationary systems with pointwise orthogonal rotational components and, using a family of test functions, to systems with more general non-gradient drift. Numerical results demonstrate accurate and robust density approximations across a range of high-dimensional stationary Fokker-Planck problems, including rough-potential and noisy-data settings.

The famous and actively studied problem of Brown–Erdős–Sós from 1973 asks for $f^{(r)}(n;s,k)$, the maximum number of edges in an $r$-graph with $n$ vertices in which no $s$ vertices span $k$ or more edges. In this paper, we concentrate on the case $r=3$ and $s=k+2$, with $k\ge2$ fixed and $n\to\infty$; then it is easy to show that the extremal function grows quadratically in $n$. Delcourt and Postle proved that the limit $\pi(k):=\lim_{n\to\infty} f^{(3)}(n;k+2,k)/n^2$ exists for every $k$. While Brown, Erdős and Sós observed that $\pi(2)=1/6$ already in the 1970s, the value of $\pi(k)$ for $3\le k\le 7$ was determined only recently (by various subgroups of Glock, Joos, Kim, Kühn, Lichev, Pikhurko, and Sun). Very recently, Chao, Huang and Liu determined $\pi(k)$ for every odd $k$. Independently of the last result, we show that $\pi(9)=1/5$. Also, we prove that $\pi(8)\le {5053}/{26544}$, which is within $0.0029$ of the best known lower bound $\pi(8)\ge 3/16$. The new upper bounds are obtained by expressing some previous arguments as a linear program and then using a computer to generate and solve its instances. Our proof of the lower bound on $\pi(9)$ is based on a finite field construction combined with existing packing results.

Low-rank approximation can require additional matrix–vector products to verify that its error meets a prescribed tolerance. We characterize this certification cost for both relative matrix error and mean-square output error. For a single approximation matrix candidate, we determine the exact dimension-uniform minimax query constant as the allowed failure probability vanishes. Our main result concerns reusing validation responses as the approximation space expands. For a candidate family constructed independently of validation, one batch supports an entire nested path without increasing the query budget with the number of checks. Across \(W\) paths, a concentration bound exploiting shared residual energy yields a \(\sqrt{\log(W+1)}\) dependence. A matching lower bound establishes its optimality for fixed interior error targets and sufficiently small separation gaps. Finally, we compare two uniformly valid certificates on the same dispersed-spectrum family. Optimizing the validation budget within each rule family yields costs of orders \(N^{1/3}\) and \(N^{2/3}\) for validation and construction beyond the true target. Code is available at https://anonymous.4open.science/r/Low-rank-approximation-1275/

We propose and analyse a fully mixed finite element method for a two-way coupled mechanochemical model of calcium signalling in viscoelastic tissue. The mechanical response follows a Kelvin–Voigt law, with the elastic and viscous stresses, velocity, and spin tensor as unknowns, and the symmetry of the total stress imposed weakly. The reaction-diffusion system for calcium and inactivated inositol triphosphate is also formulated in mixed form, introducing the diffusive flux as an additional unknown. The coupling is bidirectional: calcium generates an active stress, while local dilation feeds back into the calcium dynamics. We establish global existence and uniqueness by reducing the coupled system to an integro-differential problem and subsequently recovering the mixed variables. The semidiscrete scheme is formulated for general conforming finite element spaces and shown to be well posed under a single inf-sup condition. An interpolation operator that commutes with the divergence and preserves the weak symmetry pairing yields optimal a priori error estimates. We specialise the analysis to three families of weakly symmetric finite elements, deriving explicit convergence rates and, for one family, a superconvergent local post-processing of the displacement. Numerical tests in two and three dimensions confirm the predicted convergence rates and illustrate the method through viscoelastic benchmarks and mechanically driven calcium wavetrains.

We develop an approach to study K-stability of Fano varieties by estimating the local stability threshold along a reducible zero-dimensional subscheme on a smooth curve. As an application, we we show that all smooth index-one Fano hypersurfaces in quadrics are K-stable, when the dimension is higher than one. To prove it, we apply the Abban-Zhuang method to reduce the stability problem to lower dimensions. Specifically, we construct suitable admissible flags such that the last term of the Abban-Zhuang estimate is a local stability threshold along multiple points on a smooth curve, and give a convex geometric interpretation of the estimate.

We construct Riemannian metrics of strictly positive sectional curvature on at least 508 pairwise nondiffeomorphic exotic fifteen-spheres, without distinguishing orientations. The construction starts with quaternionic line bundles over the product of the quaternionic projective plane and the four-sphere. Surgery on an eleven-sphere fiber produces homotopy fifteen-spheres with conjugation-equivariant attaching maps. A calculation on the surgery filling gives their Eells–Kuiper invariants and distinguishes 1015 nonstandard oriented smooth types. The required representation has infinitesimal action norm at most one, so the examples also satisfy the hypothesis of Galaz-García's bounded-representation transfer theorem. The topological construction and invariant calculation are independent of the curvature input.

We compute the singular cohomology groups of odd biisotropic symplectic Grassmannians $\operatorname{S_2Gr}(k, 2n+1)$ by presenting an explicit dimension-preserving combinatorial bijection between the Bialynicki-Birula cells in $\operatorname{S_2Gr}(k, 2n+1)$ and the Schubert cells in $\operatorname{Gr}(k, 2n+2-k)$.

Existing convergence analyses of Muon either assume exact orthogonalization or analyze classical Newton–Schulz polynomials, and guarantee only stationarity, so it is unresolved what Muon's five tuned Newton–Schulz steps preserve and whether that suffices to reach a prescribed neural-network training loss. We establish a finite-time training guarantee that accounts for both momentum accumulation before orthogonalization and the tuned finite-step update. For full-batch training of a sufficiently wide two-layer ReLU network with fixed random output weights and a positive-definite limiting neural tangent kernel, we prove that Muon reaches any target empirical squared loss $\varepsilon>0$ with high probability over initialization. For every momentum parameter $\mu\in[0,1)$, a target-dependent constant learning rate proportional to $(1-\mu)\sqrt{\varepsilon}$ yields a hitting-time bound of $O((1-\mu)^{-1}\varepsilon^{-1/2})$, with other problem parameters fixed. The sufficient width is independent of both target accuracy and momentum. The analysis shows that the tuned Newton–Schulz map preserves alignment with the momentum buffer while bounding the update's spectral norm. Control of gradient variation near initialization transfers this alignment to the current gradient, ensuring descent until the target is reached without requiring exact orthogonalization. Numerical experiments support these mechanisms at widths below the sufficient theoretical threshold: gradient-update alignment remains above the analytical reference, and all 30 runs across six widths and five student initializations on a fixed teacher-student dataset reach the target loss while maintaining kernel positivity.

We derive the sub-Riemannian geodesics of the affine-additive group using the Pontryagin Maximum Principle (PMP) and compute the associated cut times. We then proceed to construct the Carnot-Carathéodory sphere of the affine-additive group as well as the flask surface; the latter is the analogue of the Pansu sphere (bubble set) in the first Heisenberg group.

Uncertainty quantification (UQ) for random partial differential equations (PDEs) is ubiquitous in computational science and engineering. However, classical spectral solvers for this class of problems face the curse of dimensionality, and existing neural solvers often ignore the stochastic structure that makes moments and calibration tractable. We introduce a stochastic separable physics-informed neural network, dubbed S$^{2}$-PINN, that represents the solution $u(t,\mathbf{x},\mathbf{Z})$ of a random PDE with a learnable Gaussian spatial dictionary, Fourier temporal features, and a generalized polynomial chaos (gPC) stochastic basis, coupled by a low-rank Canonical Polyadic (CP) tensor decomposition core. The method is trained with a hybrid strong-form and gPC-projected residual loss. Our theoretical analysis establishes that the separable class is dense in $L^2$ under mild conditions, and the projected residual corresponds exactly to a stochastic Galerkin constraint. Furthermore, we show that mini-batch projection coefficients are logarithmically dependent on the number of gPC modes, and that the orthogonality penalty controls the conditioning of the learned spatial dictionary. Using four manufactured random PDE benchmarks, we show that S$^{2}$-PINN outperforms nine baselines in terms of mean and variance accuracy, as well as calibration, while using significantly fewer parameters. Further evaluations on non-manufactured Poisson and Darcy problems, a stochastic Navier–Stokes problem, a diffusion scaling study of higher random dimensions, and two stochastic inverse problems reveal the generalization capabilities of the proposed structure. Together, these results support stochastic separability as an effective design principle for physics-informed neural UQ. The code for the experiments can be found in https://github.com/DMax1314/s2pinn

We show that ample stability suffices to apply Teleman quantisation to the endomorphism bundle of the universal representation on a quiver moduli space, and thus in particular to deduce its infinitesimal rigidity. This answers a question left open in our earlier work with Brecan, Franzen, and Reineke on the sheaf cohomology of universal bundles. This sufficiency makes several recent results applicable in the widest possible range, subject only to ample stability, which for coprime dimension vectors guarantees that the Picard rank of the moduli space is maximal.

We construct a finite cyclic $L_3$ presentation (that is, a cyclic $L_\infty$ algebra with brackets vanishing above arity three) of the variational deformation theory defined by the heterotic $G_2$ superpotential near a torsion-free standard embedding, to first order in $\alpha'$. Auxiliary fields for the inverse coframe, spinor derivative, flux and induced tangent connection give a shifted-cotangent Hamiltonian of degree at most four. Eliminating the auxiliary fields recovers the physical functional. On the flat seven-torus, this functional has a nonzero quintic Taylor coefficient in local physical spinor coordinates. For $Y=S^1\times X$, with $X$ a Calabi–Yau threefold, we restrict to the product locus with horizontal gauge and gerbe data. The functional reduces to $ \frac L4\operatorname{Im}\int_X(H+i\,d\omega)\wedge\Omega, $ where $L$ is the circle length and $\Omega=e^{-2\Phi}\Psi$ is the weighted complex volume. The induced $SU(3)$ structure selects a Hermitian tangent Courant subalgebroid. Transverse isotropic lifts in its complexification encode the holomorphic connection and string equations, together with the induced tangent connection, weighted canonical line and gerbe hierarchy. Their Taylor brackets form an $L_3$ algebra whose string part agrees with the known holomorphic heterotic algebra. We then show that this holomorphic theory is obtained by characteristic Hamiltonian reduction of a finite contractible stabilization of the product variational theory, in every cochain degree. While the quotient captures the holomorphic equations, conformal balance, gauge primitivity and leading-flux admissibility are additional conditions on a real representative.

We investigate the weakly nonlinear evolution of phototaxis-induced bioconvection in an absorbing and isotropically scattering algal suspension subjected to oblique and normal collimated irradiation. Two hydrodynamic configurations are considered, a rigid lower boundary with either a stress-free or a rigid upper surface. A multiple-scales expansion is carried to third order, and the Fredholm solvability condition yields a Stuart–Landau equation for the amplitude of the critical mode. Oblique irradiation affects the onset threshold in opposite ways in the two configurations. It progressively lowers the critical Rayleigh number beneath a stress-free surface and progressively raises it beneath a rigid surface. Beneath a stress-free surface the onset remains stationary, but the pitchfork bifurcation changes from supercritical to subcritical and becomes strongly subcritical at large incidence angles, where the preferred pattern wavelength also increases markedly. Beneath a rigid surface the critical mode changes from stationary to oscillatory at moderate obliquity, giving a supercritical Hopf bifurcation, and returns to stationary at large obliquity, where the pitchfork bifurcation becomes subcritical.

We determine which homotopy nine-spheres admit positive sectional curvature: they are precisely the four smooth types with vanishing real Dirac index. The other four admit no metric of nonnegative sectional curvature. In particular, we construct a strictly positively curved metric on the exotic Kervaire nine-sphere, answering the longstanding nonnegative-curvature existence question in this dimension. The proof keeps explicit attaching maps throughout. A quaternionic commutator realizes the Kervaire sphere, a mixed representation realizes the Hopf image of the exotic eight-sphere, and a four-periodic normal-framing formula identifies the remaining index-zero class. Following the compatible-disk method, we give the curvature and boundary estimates for the first two models, including an explicit lower bound for the Kervaire metric in terms of the curvature and first covariant-derivative norms of a fixed connection, with the base scale and fiber normalization fixed. For the third model, we construct a positive metric on a cylinder joining a quotient metric to its pullback by a marked right Hopf twist. A four-form modification controls the connection terms involving both left and right quaternionic multiplication. Two positive disk fillings close the cylinder with the prescribed attaching map. A local convolution lemma supplies the smooth gluing while preserving a strict sectional-curvature lower bound.

We study global bifurcation for Yamabe-type equations on closed Riemannian manifolds with symmetries. For isometric group actions with interval orbit space and a suitable range of exponents, we prove the existence of pairwise disjoint unbounded continua bifurcating from the positive eigenvalues of the Laplace–Beltrami operator that admit invariant eigenfunctions. All nontrivial solutions in these continua change sign. The results cover critical and some supercritical exponents. On spheres and complex and quaternionic projective spaces, we obtain lower bounds on the number of geometrically distinct continua bifurcating from the same eigenvalue.

In this paper, we study a generalised family of fractional BBM equations on the $d$ dimensional torus $\mathbb{T}^d$ for $d\geq 1$. We establish low regularity local and global well-posedness results on the $H^s(\mathbb{T}^d)$ scale that improve upon the works of Bona and Chen (2003) and Kim and Kwak (2026). We also prove endpoint global well-posedness for the cubic regularised Benjamin-Ono equation in $H^{1/2}(\mathbb{T})$. Furthermore, we obtain a well-posedness result for the generalised fractional BBM in the spaces $L^{k+1}(\mathbb{T}^d)$ where $k$ is order of the polynomial nonlinearity. As an application of this result, we show that for a certain range of values and for a suitable randomisation of the initial data, the model is almost surely globally well-posed in $H^s(\mathbb{T}^d)$ for $s\geq 0$. Lastly, we show that the family exhibits a strong form of ill-posedness in $H^s(\mathbb{T}^d)$ for $s<0$ in the form of an infinite loss of regularity at all initial datum.

We prove that all excited states $-\sin(nx)$, $n\geq2$, of the De Gregorio equation on the torus are nonlinearly unstable in the sense of Hadamard. More precisely, perturbations arbitrarily small in $H^2$ leave a fixed $H^1$ neighborhood of the state. For every $n\geq2$, the linearized operator has a real eigenvalue $\lambda_n\in(0.56,0.6)$. To prove this, we transform the eigenvalue problem into a Fuchsian equation on the complex disk and obtain a matching condition for the eigenvalue. This condition is verified rigorously using interval arithmetic. The nonlinear instability follows from a bootstrap argument in the spirit of Guo, Hallstrom, and Spirn.

We present a space-time finite element formulation for the transient analysis of geometrically exact shear-deformable beams. The formulation treats space and time within a unified finite element framework and employs a mixed approach in which translational and angular velocities are introduced as independent fields. The rotation field is parametrized using nodal rotation (pseudo-)vectors and a suitable objective interpolation of the triad field is employed. The resulting formulation supports generally curved beam structures, including geometric kinks and rigid joints. Furthermore, time-dependent material domains can be represented directly through the geometry of the space-time surface, enabling the treatment of non-material boundaries without modifying the underlying beam formulation. Numerical examples demonstrate convergence to solutions obtained with classical time-stepping schemes, objectivity of the formulation, and the influence of the space-time stabilization. Finally, the sliding spaghetti problem illustrates the applicability of the proposed approach to axially moving continua.

Information Processing Capacity (IPC) is a powerful system-agnostic metric for quantifying a dynamical system's computational capability and is widely used in quantum reservoir computing. However, standard IPC suffers from severe computational limitations. Computing IPC involves estimating how well the system can reconstruct, via linear regression, an (in principle) infinite family of orthogonal polynomial functions of delayed inputs. In practice, this evaluation must be made finite, so we need only to consider polynomials up to a chosen maximum degree. Moreover, the computational complexity for computing IPC grows combinatorially with the considered maximum degree and delay. Here, we introduce Kernel-IPC (KIPC), a scalable framework that uses kernel methods to compute the total IPC without explicitly enumerating basis functions. This allows us to account for the full (infinite) basis (without degree truncation). We also provide a null-hypothesis test for statistically comparing the KIPC of two dynamical systems. We validate KIPC on an echo state network and demonstrate its utility for quantum reservoir computing on a disordered spin chain.

The exact limiting distribution of a volatility target index has recently been established under a log-normal assumption on the risky asset. In this paper, we show that the structural limit remains valid under a broad class of stochastic volatility processes. In addition, we show that, when the limit of the joint distribution of the volatility target index and the driving Brownian motions is considered, an extra independent Brownian motion must be involved. Besides the structural limit theorem, we derive a correction formula for the expected quadratic variation of the discrete rebalancing volatility target index, under the assumption that the volatility of the underlying risky asset is a deterministic function of time. As a consequence of the discrete correction formula, it is shown that, as the rebalancing time step $\Delta t\to 0$ and the observation window parameter $\lambda \to 1$ simultaneously along a path, the expected quadratic variation converges to the target variance if and only if $(1-\lambda)^{-1}\Delta t \to 0$ along the same path. Numerical results are provided to support the structural limit, using the local volatility model and the Heston stochastic volatility model as examples, as well as the effectiveness of the discrete correction formula.