Papers

When computing sub/super-level-set persistent homology (PH), the effect of noise may introduce millions of (short-lived) topological generators, presenting an obstacle to both the computation of PH of large 3D images, and any analysis of PH that incorporates the number of generators. As such, it is often necessary to denoise the data before computing its PH. We analyze the PH of synthetic 3D images of porous media in the presence of spatially uncorrelated noise, and perform a comparative analysis of various topological measures (e.g. bottleneck distance, Wasserstein distance, persistence statistics and persistence images) to assess their robustness to both noise and the denoising process (i.e. adding spatially uncorrelated Gaussian noise, and denoising by either a Gaussian convolution or a machine learning approach).

Communication security has traditionally been built upon one of two external resources: shared secret keys or a communication advantage over the eavesdropper. However, many practical wireless scenarios, including infrastructure-less, emergency, and highly dynamic networks, cannot guarantee either resource, motivating the need for a new communication security principle. This paper introduces a new communication security paradigm that exploits temporal dependency as a security resource. Unlike conventional secrecy techniques that prevent an eavesdropper from recovering transmitted bits, the proposed paradigm allows packet decoding but prevents correct interpretation by making original and dummy packets computationally indistinguishable. Rather than protecting individual transmissions, successive transmissions are intentionally coupled so that future communication depends on correctly interpreting previous ones. As one realization, we develop a state-chained random linear network coding (RLNC) framework in which the synchronization state required to interpret each transmission block is embedded in the previous block. Therefore, synchronization failures propagate across future transmissions, resulting in persistent eavesdropper asynchronization. We analytically characterize the probability and persistence of eavesdropper asynchronization, together with the computational complexity of resynchronization, and develop transmission strategies based on transmit power and intentional-interference optimization. Numerical results demonstrate sub-second eavesdropper asynchronization under a worst- case adversarial model with no channel advantage, no secret assumptions, complete protocol knowledge, and an arbitrarily stronger eavesdropper.

This paper proposes a stage independent multiple sampling plan (SIMSP) to improve inspection efficiency by reducing the number of samples required at each sampling stage. Unlike conventional multiple sampling plans (MSP), the proposed SIMSP eliminates the dependence of each sampling stage on the outcome of the preceding inspection. The proposed approach is developed for non-repairable products sold under a pro-rata warranty policy based on the generalized process capability index $C_{py}$. The SIMSP is designed under a Type-II hybrid censoring scheme (Type-II HCS), which provides greater flexibility in controlling test time and failure information in life testing experiments. The asymptotic distribution of the process capability index estimate is used to compute the operating characteristic (OC) function, and the exact Fisher information matrix (FIM) is obtained for further statistical analysis. A constraint optimization problem is formulated to determine the optimal design by minimizing the total cost subject to the manufacturer's and consumer's risk. Numerical investigations are conducted to examine the effects of model parameters, warranty policy characteristics, and cost factors on the optimal solution. The results demonstrate that the proposed approach provides an economically efficient and feasible method for lot acceptance while satisfying the tolerable risks requirements.

In this study, we propose a Physics-Informed Neural Networks (PINNs) framework that incorporates Tikhonov regularization to solve terminal-state tracking optimal control constrained by parabolic partial differential equations (PDEs). This problem is inherently ill-posed, as infinitely many distributed controls may drive the system to the same desired state, so the regularization guides the optimizer toward the minimum-energy control, restoring numerical stability and yielding a smooth, physically meaningful solution. On the theoretical side, we establish a consistency result showing that PINNs minimizers nearly attain the continuous regularized objective under residual and quadrature approximation assumptions, and a novel error estimate that bounds the deviation of the learned control from the minimum-energy solution in terms of the PINNs training residuals and the regularization parameter. Numerical experiments on the linear heat equation and the nonlinear Burgers'equation demonstrate that the regularized PINNs framework accurately achieves the target terminal state while producing controls with significantly lower energy and smoother profiles compared to unregularized baselines.

In this paper, we construct a Lagrange-type basis for the Aomoto space $AO(\mathcal A)$, naturally indexed by the chambers of the hyperplane arrangement $\mathcal A$. The construction relies on a dimension theorem of Orlik and Terao and yields an interpolation formula for elements of $AO(\mathcal A)$. We use this formula to characterize the extremal configurations in the strong polarization inequality as those arising from finite Coxeter reflection systems. We further show that the interpolation formula gives rise to a family of chamber identities, including identities that were central to our earlier proof of the strong polarization problem and the Gaussian product inequality. Finally, we adapt the recent breakthrough of Ouimet and Greaves to prove a generalized Gaussian Product Inequality for completely monotone functions.

For critical Bernoulli bond percolation on $\Z^d$ in the high-dimensional regime, we prove that the rescaled empirical measure of the cluster of the origin converges, in a suitable $\sigma$-finite sense, to the total occupation measure of super-Brownian motion. Combined with a uniform lower mass bound for the critical cluster with respect to the extrinsic (Euclidean) metric, which we also prove and which is of independent interest, the measure convergence further yields the convergence of the rescaled cluster as a compact set, in the Hausdorff metric. As a consequence, we are able to obtain the sharp one-arm asymptotics $r^2\,\bP(0\leftrightarrow \partial B_r)\to \theta_1\in(0,\infty)$, hence refining a result of Kozma and Nachmias.

We give the first example of a non-kinetic smooth homotopy coherent action of order two on a closed simply connected smooth four-manifold. This action is obtained by restricting a nontrivial smooth homotopy coherent action of the discrete circle group on a stabilized $K3$ surface. We also construct relatively non-kinetic smooth homotopy coherent extensions of boundary involutions over compact smooth $4$-manifolds. In addition, we exhibit boundary involutions that admit locally linear topological extensions but no smooth extensions over the same stabilized fillings. Nevertheless, we prove a Wall-type theorem showing that every free involution on a disjoint union of integral homology spheres extends smoothly over any simply connected smooth filling after sufficiently many stabilizations by $S^2\times S^2$.

In Problem 4 of his 1988 paper on differential posets, Stanley asked whether the weighted $k$-multichain series of an arbitrary differential poset is always a rational multiple of the $k$th power of its rank series. We answer this question in the negative. Already for $k=2$, we construct a locally finite $1$-differential poset $P$ for which $M_{P,2}(q)/F_P(q)^2$ is not rational over any field of characteristic zero. The construction uses Wagner's crown reflection. At each sufficiently high rank there are two valid one-rank extensions with the same required Gram matrix and with new rank sizes differing by one. We show that the first quotient coefficient affected by that rank distinguishes the two choices. Choosing successively between them and diagonalizing against the rational power series produces the required poset. The same coefficient argument also yields continuum many distinct quotient series, of which continuum many are nonrational.

Let $X=\mathbb{R}^n$ be equipped with Lebesgue measure $\lambda$, and let $\mathcal{R}$ be the space of normalized nonnegative densities in $L^1(X,\lambda)$. Each $\rho\in\mathcal{R}$ induces the weighted Hilbert space $H_\rho=L^2(X,\rho\,d\lambda)$. Through the alignment isometry $U_\rho[f]_\rho=\sqrt{\rho}f$, the space $H_\rho$ is identified with the closed observable subspace $V_\rho=\{u\in L^2(X,\lambda):u=0\ \lambda\text{-a.e. on }\{\rho=0\}\}$. The companion density-projection completion theorem identifies the metric completion of the aligned object space with $L^2(X,\lambda)\times\mathcal{R}$. We study bounded operator families $A_\rho:H_\rho\to H_\rho$ and characterize all bounded ambient extensions of the aligned operator $A_\rho^\sharp=U_\rho A_\rho U_\rho^{-1}$. Their collection is an affine space modeled on $\mathcal{B}(Z_\rho,L^2(X,\lambda))$, where $Z_\rho=V_\rho^\perp$ is the zero-density defect subspace. The defect-annihilating extension attains the minimum possible operator norm, and we classify self-adjoint, positive, and orthogonal-projection extensions. We prove that the induced map $(u,\rho)\mapsto(\widetilde A_\rho u,\rho)$ is continuous exactly when the ambient family is strongly continuous, and that global Lipschitz continuity forces density independence. We also characterize convergence of support projections, show that $L^1$-convergence alone does not control support-dependent operators, and construct stable multiplication and density-weighted Hilbert–Schmidt families. The latter are $1/2$-Hölder continuous in operator norm with respect to the $L^1$-distance between densities.

Through a "partial strong convexity" lemma, this paper shows how bounds on subproblem objective value suboptimality can be used in inexact augmented Lagrangian methods and ADMM algorithms. The ADMM result uses a small but important refinement on a long-standing criterion for approximately solving subproblems. The results enable two new approaches to computing Lagrangian bounds on the optimal values of stochastic mixed-integer programming problems, with simpler convergence analysis than the prior state of the art. In each case, the subproblems are solved by variants of the classical Frank-Wolfe algorithm. However, as compared to prior methods of the same type, there is much more freedom in the choice of Frank-Wolfe variant.

We develop a colored Markov-modulated Brownian motion model for diffusion-scale systems in which newly created work preempts the active layer, occupies the top of an ordered stack, and must deplete before suspended work resumes. Such preemption arises whenever new work interrupts whatever is currently active, as in last-come, first-served queues and interrupt-driven scheduling, with diffusive dynamics appropriate when the processing rate itself fluctuates randomly. Colors encode stack position and priority, while a background phase, evolving through first-kind transitions that leave the active color unchanged, controls drift and volatility within each color. Second-kind transitions create higher colors at strictly positive random launch heights. A backward recursion summarizes higher-color excursions through return matrices and occupation-density kernels. We obtain stationary product-form representations under hold-and-jump and regulated-base boundary conventions, given explicitly through matrix-analytic formulae, and give matrix-transform formulas for illustrative launch-height families.

Upon fixing the adiabatic index, the spherically symmetric liquid Lane-Emden stars governed by the Euler-Poisson system with a "stiffened gas" equation of state $p=\rho^\gamma-1$ form a one-parameter family, naturally parametrised by the central density $\kappa=\bar\rho_\kappa(0)\in(1,\infty)$. In contrast to the gaseous case, the liquid free boundary breaks the self-similarity of the family and bends the mass-radius curve, creating extrema of the mass when $\gamma<2(d-1)/d$. We formulate a turning point principle in the spirit of Zel'dovich, Wheeler and Thorne, and of its rigorous relativistic counterpart from the works of Hadžić, Lin and Rein: the number of radial growing modes equals the negative Morse index of the linearised operator, is locally constant along the family, and can change only at the turning points of the mass-radius curve, the bending orientation there dictating whether a growing mode is gained or lost. We describe the large central density limit in which the gaseous tail (read off from a planar dynamical system) determines whether the mass-radius curve spirals, and hence whether the number of growing modes remains bounded or tends to infinity. As corollaries we recover and sharpen the known radial (in)stability results for liquid Lane-Emden stars and, in combination with existing non-radial analysis, obtain a turning point criterion for non-radial stability.

We present a short proof of Feige's conjecture: for $n$ independent nonnegative random variables with expectation one, the probability that their sum is less than $n+1$ is at least $\left(\frac{n}{n+1}\right)^n\ge \frac{1}{e}$. The proof was obtained with the assistance of GPT-5.6 Sol and builds on the recent breakthrough of Vlassis and Thomas establishing Gaffke's conjecture on the finite-sample validity of a distribution-free $p$-value. We also discuss the implications of the subsequent work of Ming, Ramdas, Shen, Wang, and Waudby-Smith.

This work uses the generalized source approach to develop a class of well-conditioned integral operators that are free of internal resonance, without the need for combined formulations. The Generalized source integral equations (GSIEs) kernels are obtained by augmenting the conventional electric field integral equation (EFIE) kernel with auxiliary contributions to enhance the rank deficiency of the corresponding moment matrix blocks. This paper presents the first investigation of the spectra of GSIE operators for auxiliary kernels produced by internal scattering convex shields. Using closed-form expressions for concentric circular scatterers and shields, it is shown that, with shield parameters that are suitable for enhanced compressibility, the transverse magnetic (TM)- and transverse electric (TE)-GSIE operators are free of internal proper and quasi-resonances. The auxiliary components are shown to be compact perturbations of their EFIE counterparts. Hence these GSIESs inherit their dense-discretization breakdown, which remains curable via Calderón-type preconditioning. The mechanisms that govern the high-frequency breakdown are shown to be influenced by the auxiliary component, leading, in some cases, to greater resilience. These observations are shown to remain valid for moment matrices and GSIEs designed with non-circular stencil shields. For both GSIE-dual and Yukawa-kernel preconditioners, the formulations exhibit favorable spectral properties while maintaining their compressibility. This makes the formulations attractive for the design of fast iterative solvers. The results on the two-dimensional shield-based operators provide a foundation for extending the approach to three-dimensional problems and to auxiliary kernels with broader geometric applicability.

We study a three-dimensional charged-particle dynamics model in the nonrelativistic momentum formulation and construct a curve-reference exponential integrator (CREI) for this system. Since the zero eigenvalue of linear section is not semisimple, the system after usual change of variables does not meet the spectral condition used in the classical locally linearized extended exponential integrator (LLEEI). A kernel-range decomposition of the dominant magnetic matrix and a further linear transformation reduce the equation to a system with a semisimple zero block and one constant skew-symmetric oscillatory block. The CREI expands the nonlinear term along the exact oscillatory curve rather than at a fixed point. Retaining monomials through degree k defines CREI(k + 1). We prove uniform local and global error bounds and present reproducible numerical experiments for convergence in h and uniformity in {\epsilon}.

In this paper, we present a novel second-order method called Accelerated Inexact Newton Extragradient (AINE) for convex optimization using $\delta$-inexact Hessians. We show that AINE can find an $\epsilon$-solution in the inexact second-order oracle (ISO) complexity of $\mathcal{O}( (\delta/\epsilon)^{1/2} + (L_2/\epsilon)^{2/7} )$ when the Hessian is $L_2$-Lipschitz continuous, and a better complexity of $\mathcal{O}( (\delta/\epsilon)^{1/2} + (L_3/\epsilon)^{1/5} )$ when the third-order derivative is $L_3$-Lipschitz continuous. Notably, each iteration of our method can be conducted in the same running time as matrix multiplication up to logarithmic factors. In addition, we also establish matching oracle complexity lower bounds for both setups, demonstrating the optimality of our methods.

Symmetric non-negative matrix factorization (SymNMF) recovers latent group structure from a dependence matrix, but its dense, quadratic-memory objective has confined prior work to moderate sizes. We present a large-scale GPU study of seven algorithm families (over 30 configurations) on absolute Pearson correlation and tail pairwise dependence matrices from Extreme Value Theory, two proxies for empirical risk-factor estimation on large portfolios. A trace-identity reformulation eliminates all $n \times n$ intermediates, so a single GPU reaches $n \approx 10^5$ and multi-node distribution scales to $n = 10^6$ and beyond. Under a two-phase protocol, eleven methods converge at moderate scale; six remain efficient enough at $n = 10^5$ (five AdaGrad-family plus ADMM), and five AdaGrad-family methods still converge at $n = 10^6$: AdaGrad, RMSprop, and three we introduce (Piecewise AdaGrad, Row-Stochastic SVRG, Block-SVRG AdaptGrow). At $n = 10^6$ the fastest solver tracks the matrix spectrum: Block-SVRG AdaptGrow wins on the flat, ill-conditioned tail-dependence spectrum, where its lower per-iteration cost decides a long factorization, and full-batch AdaGrad wins on the dominant-low-rank correlation spectrum, where the run is short. We also benchmark spherical K-means as a hard-label baseline: cheaper when angular cluster structure is present, yet provably degenerate once the matrix collapses toward a single common factor, where the soft factorization remains necessary.

This article is part of a broader programme investigating which features of finite-dimensional Riemannian geometry persist in infinite dimensions. The Hopf–Rinow theorem fails even for Hilbert manifolds: metric and geodesic completeness need not agree, and neither property guarantees a length-minimizing geodesic between prescribed endpoints. Despite this failure, our first main result shows that the conformal class of every smooth strong Riemannian metric on a smooth separable Hilbert manifold contains a smooth strong representative that is metrically and geodesically complete and such that every two points in the same connected component are joined by a length-minimizing geodesic. By contrast, our second main result establishes a local flexibility phenomenon for metric completeness that is inherently infinite-dimensional. Given any prescribed Hilbert-norm ball, a metrically complete strong metric admits a conformal deformation which is equal to one outside that ball, preserves geodesic completeness, and destroys metric completeness.

Rotary position embeddings (RoPE) modify attention scores through position-dependent rotations, but their effect on normalized token dynamics is not captured by the vanilla spherical self-attention model. We study the continuous-time dynamics obtained when queries and keys are rotated while values remain on the unit sphere. The resulting attention kernel is reversible and admits a sharp uniform softmax floor, yet the natural RoPE interaction energy has derivatives of both signs within one fixed nontrivial system. Every consensus state remains an equilibrium, and its transverse linearization is a reversible Markov operator whose kernel depends on the consensus point through its energy across RoPE planes. On a resonant single-frequency ring we derive an exact Bessel-aliasing spectrum, including non-coprime frequencies and the correct fixed-ring large-$\beta$ asymptotics. Globally, closed hemispheres are invariant, while pairwise non-obtuse configurations and strict open semicircles contract with explicit half-angle and single-point tail bounds. These regional estimates instantiate a kernel-generic positivity principle with the sharp RoPE softmax floor. RoPE also selects an explicit score-flattening twisted branch; the generic resonant family is non-hyperbolic and linearly unstable, whereas an odd antipodal family becomes a hyperbolic saddle after quotienting global rotation. In multiple dimensions, the local consensus gap can depend non-monotonically on the allocation of energy across frequency planes, so no universal ordering by frequency is valid. Independent matrix, finite-difference, and nonlinear-flow computations cross-check the theorem boundaries and the reported constants.

We construct global weak solutions of the inviscid surface quasi-geostrophic equation in $\mathbb R^2$ and in smooth bounded domains, for arbitrary initial data in the critical Lorentz space $L^{4/3,2}$. The solutions conserve the Hamiltonian $\|\Lambda^{-1/2}\theta(t)\|_{L^2}^2$ for all times. The second Lorentz exponent is determined by the sharp boundedness $\Lambda^{-1/2}:L^{4/3,2}\to L^2$; the corresponding estimate fails for $L^{4/3,q}$ when $q>2$. We use an approximation scheme that is tailored for Lorentz spaces. It smooths the advecting velocity and the initial data, and preserves order in distribution functions. Removing the approximation is made possible by uniform bounds for the Lorentz norms of high amplitude cutoffs of the solutions.

We prove a general theorem on the decomposition into K-mixing components for recurrent, piecewise-smooth hyperbolic maps in two dimensions. In contrast with classical results, we do not assume that the map preserves a probability measure. As an application, we show that every recurrent aperiodic Lorentz gas satisfying certain uniformity assumptions is K-mixing. Finally, we discuss some consequences of K-mixing for the decay of correlations relative to zero-mean $L^1$ observables in aperiodic Lorentz gases.

We study the task of estimating a $d-$dimensional quantum state $\rho$ under the constraint that the measurement is $\alpha-$gentle. Such measurements $M$ do not collapse the state; they issue both a random variable $R^M = \omega$ containing statistical information and a post-measurement state $\rho_{M \to \omega}$ such that $\|\rho_{M\to \omega} - \rho\|_{Tr} \leq \alpha$. We describe gentle measurements and their connection to quantum differential privacy. Our results show that the optimal minimax estimation rate in Frobenius norm is of order $d^3/(n \alpha^2)$, instead of $d^2/n$ for general measurements. Moreover, for rank $r$ states with $r\leq d$ we prove that the optimal minimax rate is $rd^2/(n \alpha^2)$, instead of $rd/n$. Very surprisingly, the loss for gentleness $d/\alpha^2$ scales with the ambient dimension of the Hilbert space, rather than the number of parameters $rd$, typically seen in classical differential privacy. We propose optimal gentle measurements and indicate how they can be physically implemented using an ancillary state and a CNOT gate to entangle it with the initial state. We notice that the resulting random variable has a likelihood that satisfies local differential privacy. Lower bounds are proven through a new quantum information-theoretic inequality applied to well chosen families of states in the manifold of (small-rank) quantum states.

Motivated by Dombrowski and Dresden's work in 2025, we find the exact values of the limits \[\lim_{k\to\infty}\int_0^{\rho\pi}|\sin^n(kx)-\sin^nx|dx\quad\text{and}\quad \lim_{k\to\infty}\int_0^{\rho\pi}|\cos^n(kx)-\cos^nx|dx\] for $k,n\in\mathbb{N}$ and $\rho\ge 0$. In addition, we provide several simple recursive formulas which relate these integrals together. The key technique is to locate the zeros of the integrands explicitly, which allows removal of the absolute value and reduces the problem to evaluating limits of telescoping trigonometric sums via asymptotic analysis.

Let $\Omega$ be a bounded Lipschitz domain of $\mathbb R^N$, $N\geq 2$. In this paper, we study the asymptotic behavior of the first Robin eigenvalue of the $p$-Laplace operator as $\beta$ goes to $0$ and as $\beta$ goes to $+\infty$, deriving sharp asymptotic expansions of the eigenvalue; the expansion in the Dirichlet limit $\beta\to+\infty$ is obtained under the additional assumption that $\partial\Omega$ is of class $C^{1,1}$.

For a compact set $S\subset\R^2$, the local Steiner formula of Hug, Last and Weil expresses the parallel volume $V_S(t)$ through the proximal normal bundle and the truncated fiber lengths $\min\{t,\delta_S\}$. We introduce conic reach, a geometric condition with two requirements: $S$ coincides with a cone near each point of a finite, well-separated singular set, and every proximal normal fiber away from those points has length at least $\rho$. In the plane, these requirements force every link to be a finite union of circular arcs whose complementary gaps have width bounded below. Computing the feet-localized tube of each cone and combining it with the local Steiner formula, we show that $V_S$ is a polynomial of degree at most two on $(0,\rho)$, with explicit coefficients; in particular $\polreach(S)\ge\conreach(S)$. A one-dimensional converse shows that a quadratic wall contribution forces a linear cut function. Compact domains with piecewise-$C^2$ boundary, uniformly wedge-like at their reentrant corners, have positive conic reach, and their volume coefficients are given by a Gauss–Bonnet-type formula. For the L-shaped polygon the three invariants separate: $\reach(L)=0$, $\conreach(L)=\tfrac13$, $\polreach(L)=1$. A cuspidal notch, two overlapping discs and a Cantor fan of segments show that tangential contact, curvature at a reentrant corner and degenerating link gaps each destroy polynomiality.

We solve Bellman's lost-in-a-forest problem for the golden gnomon $G$, the isosceles triangle with equal sides $1$ and apex angle $108^\circ$: the shortest curve guaranteed to reach the boundary of $G$ from an unknown starting position and heading is a symmetric seven-piece path of segments, circular shoulders, and tangents, of exactly determined length $C=1.282676025459\ldots$. To our knowledge, this is the first proved exact optimum for an isosceles triangle whose base angle is below $45^\circ$. The curve's parameters come from one isolated quartic root, and $C$ is transcendental. Equivalently, $C^{-1}G$ is the smallest homothetic golden-gnomon cover of all unit arcs. The proof introduces a balanced support calibration: one weighted family of escape inequalities, built on the linear relation among the triangle's three normals, exactly saturated by the candidate, through eighteen exact support windows, and confronting every shorter competitor at once. Aggregation along the normal fan compresses the calibration to a finite zero-sum family of supported vectors; summation by parts then bounds its total by path length whenever the running suffix balance, the ledger, stays in the unit disk. A local two-gap surgery and cyclic bitonicity force a shortest hypothetical counterexample into exactly the temporal order the ledger tolerates. Lean 4 verifies the two finite algebraic certificate families and the reusable discrete ledger identities and bounds.

A simplicial set is $d$-Segal if and only if it is $(d{+}1)$-coskeletal and satisfies the $d$-Segal condition in the two lowest relevant simplicial dimensions.

What are the minimal requirements for a physically reasonable mathematical model for quantum theory or a potential extension? In addressing this question, rather than relying on the generalized probabilistic theories or other usual approaches, we propose a reset based on the following fundamental features: transition probabilities, which are typical of quantum theory, and continuous reversible dynamical processes (e.g., represented by a Lie group), which play the same crucial role in classical as well as in quantum physics. In doing so, we get back to the very elementary transition probability space framework originally introduced by Bogdan Mielnik in 1968 and combine it with the continuous reversibility condition. The primary class of valid spaces arises from the irreducible atomic JBW algebras, which encompass the irreducible atomic von Neumann algebras and the Jordan matrix algebras. The von Neumann algebras represent the model of common quantum theory. Beyond these, there is a non-standard valid space, where the exceptional Lie group $E_6$ acts transitively. While $E_6$ has been proposed as a candidate for internal symmetries in particle physics, we highlight that this model deviates from standard quantum theory in critical ways. Specifically, it fails to guarantee the existence of post-measurement states in all situations. Though the above postulates are powerful, they still allow for some physically meaningless models. A further feature of quantum theory (the correspondence between the pure states and the minimal projections) leads us to an additional requirement that rules out these models. However, a complete classification of the valid models remains one of the open issues, pointed out in the paper, which is intended for the mathematical and physical experts, inspiring them to tackle these problems.

Takenaka-Malmquist (TM) systems, as generalizations of the trigonometric function system, play a decisive role in approximation theory on the open unit disc and the upper-half complex plane. In the present paper with the setting of boundary weighted Hardy spaces we develop what we call kernel-TM systems and related approximation theory aspects. In such framework, besides adaptive Fourier decomposition (AFD), we also establish double-zero AFD (DAFD)as an advanced sparse representation in the generalized Hardy spaces. The paper also includes brief reviews of several commonly studied RKHSs as subclasses of the classical Hardy space with particular attention to admissibility of kernel-TM systems.

We develop a general framework of identification and estimation for automatic debiased machine learning (DML) where the parameter of interest $\theta_0$ is identified by a moment condition involving a nuisance $\gamma_0$ that may be high dimensional. We establish conditions under which the Riesz representer $\alpha_0$, which is at the core of DML, is identified, and show that the identification occurs precisely when $\alpha_0$ uniquely optimizes a quadratic functional. This characterization enables us to develop a general estimation procedure for $\alpha_0$ that allows for generic $\gamma_0$ including those defined by models with endogeneity and encompasses both classical sieves and modern architectures such as deep neural networks. To improve estimation precision and mitigate the curse of dimensionality, we incorporate shape constraints on $\gamma_0$ by embedding them into a possibly nonlinear parameter space. We illustrate our estimation procedure through simulations and empirical applications.