Papers

More than 50 years ago, Golomb and Welch conjectured that no tilings of $\mathbb{Z}^n$ by Lee balls of radius $r$ exist for $n \geq 3$ and $r \geq 2$. This conjecture remains open even under the additional assumption that the tiling is a lattice tiling. In general, there are two known effective approaches to the lattice case which are based on symmetric polynomials and on the arithmetic of group-ring equations, respectively. In this paper, we develop both approaches. First, we determine all coefficients of the Q-polynomials associated with Lee balls; previously, only leading coefficients and certain special cases were known. Second, for arbitrary radii, we give necessary and sufficient conditions for the existence of lattice tilings by Lee balls in terms of group-ring equations, extending earlier results for radii $r=2$ and $3$. We then derive new necessary conditions for such tilings. For every radius $3\le r\le 50$, we prove that the set of dimensions for which no lattice tiling exists has positive density in \(\mathbb{Z}^+\). We also give an algorithm that combines the available necessary conditions and apply it to all pairs $(n,r)$ for which the corresponding Lee ball of radius $r$ and dimension $n$ has size less than $2^{30}$. Among these $1{,}358$ pairs, the combined method excludes $1{,}356$, leaving only $(n,r)=(35,3)$ and $(805,3)$.

Score-based generative models driven by $\alpha$-stable Lévy processes have been proposed as an alternative to Brownian diffusion models, with empirical evidence suggesting improved sampling and better modeling of heavy-tailed data. Yet theoretical guarantees for these models remain limited. We develop a non-asymptotic convergence analysis for scale-preserving $\alpha$-stable generative dynamics, covering both compactly supported and power-law targets. In p-Wasserstein distance, we quantify the effects of early stopping, initialization, score approximation, and time discretization, and show exponential decay of the initialization error. A comparison with Brownian dynamics reveals that global Wasserstein convergence alone does not fundamentally distinguish the two models: both initialization errors decay exponentially with the forward horizon on the common class of heavy-tailed targets. A sharp separation emerges instead in the tails. At any positive generation time, stable dynamics produces power-law tails with index $\alpha$, whereas Brownian dynamics remains light-tailed. For a target with tail index $\nu$, asymptotically correct extreme-quantile scaling is possible precisely when $\alpha$=$\nu$. These results identify tail-index matching as a principled mechanism behind the ability of stable generative models to capture extreme events.

We study finite-time singularity formation for the scalar defocusing nonlinear wave equation $-\partial_t^2u+\Delta u=u^p$. First, for fixed $d\geq4$ and odd $p\geq3$, we establish a conditional blow-up result under the assumption that a smooth self-similar imploding profile exists for the leading-order relativistic Euler system, with associated normalized coefficients satisfying quantitative analytic radius conditions. Second, we verify both the profile and the analytic radius conditions in dimensions four and five for all sufficiently large odd $p$. Using finite speed of propagation, we obtain a dimension-independent threshold $p_0$ such that, for every $d\geq4$ and every odd $p\geq p_0$, finite-time blow-up occurs for some smooth, compactly supported, real-valued initial data. The proof combines a normalized formulation that separates the leading periodic oscillation from the unknown modulation, growth of the spatial analytic radius induced by an outgoing property, and frequency-dependent matrix multipliers that control the cumulative growth caused by the periodic linearized potential.

Let $(W,S)$ be a Coxeter system of finite rank, let $T$ be its set of reflections and let $c \in W$ be a Coxeter element. For $x$ in the interval $[1,c]_T$ of the absolute order, let $P(x)$ be the subgroup generated by the reflections $t\le_T x$. We prove that $x$ is a Coxeter element of $P(x)$ with respect to the canonical Coxeter generators of $P(x)$, that $P(x)$ is generated by the factors of any reduced reflection factorization of $x$, and that the Hurwitz action on the reduced reflection factorizations of $x$ is transitive. The proof reduces the general case to the case of a Coxeter system whose Coxeter matrix has entries in $\{1,2,3,\infty\}$, which was treated by Hubery and Krause.

This paper establishes global well-posedness for the defocusing energy-critical nonlinear Schrödinger equation on the three-dimensional torus with a class of additive Lévy noises. For arbitrary deterministic initial data in the energy space, we construct probabilistically strong solutions that are pathwise unique in the critical solution class. Continuous dependence on initial data, convergence of finite-activity approximations, and a vanishing-noise limit hold on finite time intervals. When the effective noise is nonzero, quantitative bounds for mass exit times imply almost-sure unboundedness of the mass and exclude free scattering. The expected energy then admits a cubic lower bound in time. The analysis couples pathwise critical perturbation theory with energy estimates adapted to Poisson compensation and stopping-time arguments based on the exact mass balance.

This chapter focuses on finite volume methods. The finite volume method is a discretization method that is well suited for the numerical simulation of various types (for instance, elliptic, parabolic, or hyperbolic) of conservation laws; it has been extensively used in several engineering fields, such as fluid mechanics, heat and mass transfer, or petroleum engineering. Some of the important features of the finite volume method are similar to those of the finite element method: it may be used on arbitrary geometries, using structured or unstructured meshes, and it leads to robust schemes. The finite volume method is locally conservative because it is based on a “balance” approach: a local balance is written on each discretization cell that is often called “control volume;” by the divergence formula, an integral formulation of the fluxes over the boundary of the control volume is then obtained. The fluxes on the boundary are discretized with respect to the discrete unknowns.

In this article we prove the existence of continuous and compact embeddings for a class of modular sequence spaces generated by Orlicz functions. The class under consideration contains sequence spaces of Orlicz-Sobolev type, including certain special cases that have played an important role in a number of applications regarding problems arising in non-equilibrium statistical mechanics, as well as Nakano's class of Lebesgue sequence spaces with variable exponents. In a few exceptional cases we provide necessary and sufficient conditions for such embeddings to hold, in many others only sufficient conditions and in that situation we clearly indicate why these are not necessary by means of additional criteria and specific examples. Our results are complementary to those already available in the literature. We illustrate them with several concrete examples.

We study the 3D rotating primitive equations with gravity in a horizontally periodic channel, with viscous horizontal velocity and no density diffusion. For every Coriolis parameter, stably stratified equilibria are asymptotically stable. Small compatible \(H^4\) perturbations yield global classical solutions with uniform \(H^4\) bounds. The oscillatory density and velocity decay like \((1+t)^{-1}\); the density converges at the same rate to a stratified profile, its horizontal mean faster. For small data, the limit profile is the decreasing vertical rearrangement of the initial density, the unique potential-energy minimizer among rearrangements, independent of rotation, viscosity, and initial velocity. Kinetic plus available potential energy is dissipated exactly by viscosity and decays like \((1+t)^{-2}\), and each fluid particle settles where the limit density equals its own. These rates are optimal for every rotation rate. The linearized problem has real, slowly decaying normal modes, viscously overdamped high-vertical-wavenumber internal gravity waves, whose small solutions show the \(H^r\) density rates \((1+t)^{-(1-r/4)}\), \(0\le r<4\), pressure rate \((1+t)^{-1}\), and energy rate \((1+t)^{-2}\) cannot be improved uniformly for small data. Key difficulties are the loss of a horizontal derivative in the vertical velocity, the failure of \(H^1(\mathbb T^2)\subset L^\infty\), and rotation-induced coupling of divergent and vortical motion, controlled at top order by a positive power of the initial low-order energy. If the stratification is unstable somewhere, every horizontal mode is linearly unstable with real growth rates and the equilibrium is nonlinearly unstable; rotation strictly lowers each growth rate, with an explicit Taylor-number bound depending on the horizontal wavelength, but leaves their supremum unchanged.

In this paper, we establish quantitative central limit theorems for the integrated fluctuations of the intensity of a counting process. Building on the work of Coutin and using the Clark-Ocone formula, we first derive a general upper bound on the Wasserstein distance between a Gaussian random variable and the normalized integral of the centered intensity. We then apply this result to subcritical linear Hawkes processes. In this setting, we prove that the integrated fluctuations of the intensity converge in distribution to a centered Gaussian random variable, with a convergence rate governed by the tail behavior of the resolvent associated with the underlying Volterra equation.

Accurate information-theoretic analysis of multiuser wavelength-division multiplexed (WDM) optical links is complicated by Kerr nonlinearity and chromatic dispersion. Existing capacity bounds for first-order regular perturbative (RP) channel models have relied on per-symbol peak power constraints to simplify the underlying extremal analysis. In this paper, we replace these peak constraints with the more practically relevant average input power constraints and study the resulting RP WDM interference channel. Unlike the peak-constrained formulation, an exact finite signal-to-noise ratio (SNR) analysis under average power leads in general to a nonconvex, high-dimensional optimization problem with no closed-form solution. We show that the problem admits a tractable characterization in the high-SNR regime. Using a genie-aided argument, we derive closed-form single-user high-SNR capacity bounds for the RP interference channel and identify constant-amplitude interferer signaling as extremal. Furthermore, under an empirically validated non-negativity condition on the dominant RP kernels and within a truncated first-order RP model, we establish a finite-SNR extremality result, yielding a single-user capacity characterization within the RP model under average power constraints. Building on these single-user corner points, we construct an all-helpsone achievable region via time sharing, which strictly improves upon treating nonlinear interference as noise. Numerical results and split-step Fourier simulations of the nonlinear Schrodinger equation channel confirm that the qualitative conclusions persist over practical launch powers. The results provide new insight into extremal signaling and multiuser trade-offs for nonlinear WDM systems within the RP approximation.

We determine the large-genus order of the Weil-Petersson diameter of $\mathcal{M}_{g,n}$ for each fixed number of punctures: namely, $\mathrm{diam}(\mathcal{M}_{g,n})\asymp\sqrt g\log g$. The main new ingredient is an integrated Ricci estimate for geodesics that minimize in moduli space and have thick endpoints, with no systole bound along their interiors.

Inserting a resolution of the identity between two operators turns a matrix element of their product into a phase-space integral. For unbounded operators the product need not exist on the vector at hand, while the integral may still converge. We follow this gap through the chain $e^{\mu a^2}e^{\lambda a^{\dagger2}}$ of two-photon exponentials, where it can be traced completely. Five regions of parameter space appear: where the Bochner conditions of the strong formulation hold, where the product exists as a vector, where it exists as a sesquilinear form, where the coherent-state integral converges, and where the Fock-space series converges. The first three are strictly nested inside the intersection of the last two, and these two overlap without either containing the other. A single Hermite-type function agrees with the matrix element where it exists and with the integral and the series wherever they converge, so the integral carries the analytic continuation of the matrix element beyond the domain of the product and beyond the convergence of its Fock series. On the real slice of parameter space, an iterated form of the integral also reaches every point where the series converges. The analysis rests on a weak composition lemma whose hypotheses concern only the bra, the ket and the resolving family. For bounded chains the same identity is unconditional, and worked examples of that kind are given in a companion paper.

Let $G$ be a connected graph. The eccentric subgraph number $\epsilon(v)$ of a vertex $v$ of $G$ is defined as $\epsilon(v)=\min\{f_G(v,u): u\in V(G)\}$, where $f_G(v,u)$ is the number of connected subgraphs of $G$ containing both $v$ and $u$. The eccentric subgraph index $ES(G)$ of $G$ is defined as $ES(G)=\displaystyle\sum_{v\in V(G)}\epsilon(v).$ In this paper, we obtain the unicyclic graph that maximizes the eccentric subgraph index among all unicyclic graphs on $n$ vertices with a fixed girth. The graph that maximizes the eccentric subgraph index among all unicyclic graphs on $n$ vertices is also obtained. We then characterize the trees that maximize the eccentric subgraph index over trees on $n$ vertices with fixed diameter. Among all connected graphs on $n$ vertices with a fixed number of pendant vertices, the graph which maximizes the eccentric subgraph index is also characterized.

Subspace-design codes form a central unifying framework in coding theory, connecting probabilistic existence results and explicit algebraically structured constructions. The two core families are folded Reed-Solomon codes and univariate multiplicity codes, both possessing the subspace-design property. These codes reach list-decoding capacity with polynomial-size lists, but their alphabet size grows with code length and hurts practical efficiency. Goyal, Guruswami, and Hsieh achieved a major breakthrough: they constructed the first explicit subspace-design codes over constant-size alphabets via the Alon-Edmonds-Luby expander framework, settling an open problem and showing subspace-design properties can hold without large alphabets. Despite recent progress, constructing subspace-design codes with low-complexity efficient encoding remains a critical open challenge. This work gives an explicit construction of r-dimensional subspace-design codes over a constant alphabet whose size grows exponentially with r. By comparison, Goyal et al. (FOCS 2026) built r-dimensional subspace-design codes with alphabet size doubly exponential in r squared. Our construction is computationally efficient: encoding scales quadratically with block length and linearly with dimension r in field operations. We use modified algebraic geometry codes, replacing rational evaluation points with higher-degree places to obtain a valid additive code. To confirm parameter bounds, we use Moore matrices to analyze determinants with many zero entries. We also generalize univariate multiplicity codes, originally built from Reed-Solomon codes, to algebraic geometry codes, and prove these AG-multiplicity codes achieve optimal parameters.

Let $k_0$ be a field of characteristic $0$, and let $C/k_0$ be a smooth geometrically integral curve of positive genus with $C(k_0)=\varnothing$. We prove that the regular inverse Galois problem is solvable over every field existentially closed in the class of $k_0$-fields avoiding $C$, including the curve-excluding fields of Johnson and the author. To this end, we show that for every finite group $G$, there is a parameter variety over $\mathbb Q$ whose function field admits a regular realization of $G$ without adding points to $C$. Finally, we establish $\mathrm{RIGP}$ over the variety-excluding fields of Szachniewicz and the author, when the excluded variety is a locally closed subvariety of an abelian variety.

Completely positive programs are known to provide an exact representation to several NP-hard optimization problems. In this work, we look at the maximum of random quadratic functions and compute the maximum expected value over ambiguity sets defined with moment and support information. We propose exact completely positive programs and semidefinite programs when the random variables are Bernoulli, nonnegative and unrestricted in support. Our experimental study demonstrates the utility of these formulations.

We study random walks with geometric or log-gamma increments, reweighted by certain exponential functionals. We establish precise asymptotics for all moments of exponential functionals, revealing a phase transition as the drifts of the walks vary. To describe the large time limit of reweighted random walks, we introduce a discrete $T$ transformation on random walk paths and establish Girsanov relations. As a corollary, we show that the stationary measures of geometric last passage percolation and the log-gamma polymer on a strip converge to their half-space counterparts as the strip width tends to infinity. In the log-gamma case, this provides an alternative proof of a recent result in arXiv:2607.17030.

The maximum likelihood degree of a Gaussian graphical model measures the algebraic complexity of its likelihood equations. This paper studies maximum likelihood degrees of Gaussian graphical models through chordal completion. When a graph becomes chordal after one missing edge is added, its maximum likelihood degree is determined by the maximal cliques and clique-tree separators containing that edge. This recovers Uhler's formula for complete bipartite graphs with one part of size two, and extends it to deleted-spine clique books, whose degree is unchanged when a page is thickened to a clique. For several fill edges, the same method gives a product formula when no clique or separator bag contains two fill edges. We also determine four classes of vertices at which clique blow-up preserves the degree. Two follow from known results, and two are established here.

This short article is devoted to the study of the positive solution of Yamabe-type semi-linear equation with prescribed coefficient on general closed manifolds. Our main result is to provide some sufficient conditions for the existence of the positive solutions and give some examples to show that those conditions are sharp. An application to scalar-flat manifold settles a long standing question in prescribing scalar curvature problem in negative form.

We provide a sufficient condition for a group of homeomorphisms of a compact Hausdorff space to be non-invariably generated, generalizing previous results proved using the dynamics of a group's action. We exhibit new applications among Scott-Röver-Nekrashevych groups and among groups of tree automorphism of with (almost) locally prescribed actions.

We prove the fence conjecture of Dumas and Wolf: for a convex projective polygon with at least five vertices, every zero of the Pick differential lies in the interior of the region obtained by cutting off the triangles at consecutive triples of vertices. The proof gives a stronger estimate for arbitrary properly convex domains. In affine-sphere coordinates normalized at a Pick zero, the domain contains the closed disk of radius $2^{-1/4}$ and its closure lies in the open disk of radius $2^{1/4}$, unless it is a rotated square. The proof compares the canonical Monge-Ampère potential of the domain with that of a normalized square. Their third-order contact gives alternating local sign sectors. A planar separation argument and the maximum principle show that the negative sectors belong to four distinct components, each meeting a different side of the square. Rotation gives the inner disk bound, and conormal duality gives the outer bound. Both constants are optimal for the class of properly convex domains normalized at a Pick zero. The argument requires neither boundary smoothness nor any restriction on the multiplicity of the Pick zero.

This work analyzes a $C^1$-conforming virtual element method for the optimal control of generalized Oseen equations, formulated using the stream-function on simply connected polygonal domains. The method is based on a conforming discretization in $H^2$ of arbitrary order $k \ge 2$. The discrete scheme considers an approximation of the state and co-state equations which involves a specifically designed operator for the convective term and a variational discretization of the control variable. We derive quasi-optimal a priori error estimates in the $H^1$ and $H^2$-norms for the stream-function, and in the $L^2$-norm for the control, under standard assumptions on the family of polygonal meshes. The proposed method recovers the fluid velocity, vorticity, and pressure using suitable projections. Finally, we report several numerical experiments to support the theoretical findings using different polygonal meshes including the cavity problem. We highlight that this analysis also covers OCPs governed by Brinkman and Stokes equation.

We prove that a convex body with $C^\infty$ boundary in an odd dimension $n\ge3$ is algebraically integrable if and only if it is an ellipsoid. This answers Arnold's problem for smooth convex bodies. Together with Vassiliev's even-dimensional nonexistence theorem, it gives the classification in every dimension $n\ge2$. More generally, in odd dimensions, a convex body with $C^\infty$ semialgebraic boundary is an ellipsoid whenever its section volume function satisfies a joint algebraic relation.

We study weighted composition–differentiation operators on mixed-norm, derivative Hardy and F(p,q,s) spaces. Explicit dyadic criteria yield boundedness, compactness and two-sided norm and essential-norm estimates, including endpoint and quasi-Banach cases. We also obtain multiplier and approximation results, Schatten criteria on weighted Dirichlet spaces, and exact singular values for monomial weights and linear symbols.

This paper studies infinite-horizon equilibrium dividend distribution for a firm whose surplus follows a diffusion under observable Markov regime switching and is absorbed at bankruptcy. Dividend payments are modeled as singular controls, and non-exponential discounting leads to time inconsistency. We formulate equilibrium control laws and establish a verification theorem for a general finite-state model with surplus-dependent coefficients. For a two-state model with constant positive drifts and volatilities and bi-exponential discounting, we prove the existence of a regime-dependent equilibrium barrier policy for arbitrary positive switching intensities. The policy retains surplus below the prevailing regime's threshold and distributes any excess immediately. Numerical experiments examine how shareholder patience and economic conditions affect dividend decisions. Over the parameter ranges considered, greater discount-rate dispersion at a fixed initial average rate raises both barriers, while switching speed can generate nonmonotone responses and reverse their ordering.

Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$, and let $1\leq k\leq n$. We consider the space of smooth $k$-admissible potentials introduced in the study of the Hessian-cscK equation by Guo–Smith–Tong. The complex $k$-Hessian equation identifies this space, modulo constants, with the space of smooth positive densities of fixed total mass. We compute the differential of the Hessian density map and use it to pull back the Fisher–Rao metric. The resulting Riemannian metric is \[ G^{\HC,k}_{\varphi}(u,v) =\int_X \bigl(\Delta_{\varphi}^{(k)}u\bigr) \bigl(\Delta_{\varphi}^{(k)}v\bigr) \,\omega_{\varphi}^{k}\wedge\omega^{n-k}. \] We then show that the square-root Hessian density map \[ \varphi\longmapsto 2\sqrt{\frac{\omega_{\varphi}^{k}\wedge\omega^{n-k}}{\omega^n}} \] is a global Riemannian isometry onto the positive part of the $L^2(X,\omega^n)$-sphere of radius $2\sqrt{V}$, where $V=\int_X\omega^n$. For $k=n$, this recovers the spherical description of the classical Calabi metric. We also derive the curvature, geodesic and diameter formulas, and prove a quantitative local comparison between adjacent Hessian–Fisher distances.

We study the sharp Fourier extension inequality from $L^2(\mathbb S^{d-1})$ to the radial–angular mixed space $L^q_{\mathrm{rad}}L^2_{\mathrm{ang}}$. Our work is motivated by and builds upon the foundational analysis of Carneiro, Oliveira e Silva and Sousa \cite{COS}, who initiated the sharp mixed-norm spherical restriction problem from the viewpoint of spherical-harmonic decomposition. Their reduction revealed that the extremal problem is governed by a competition among the individual harmonic modes, and their results established degree-zero extremizers in several important ranges of exponents, including even integer exponents and the endpoint case. The present paper addresses this question by establishing a complete modal ordering result. Building on their spherical-harmonic reduction, the sharp-constant problem is reduced to the weighted Bessel moments \[ I_{\nu,k}(q) := \int_0^\infty |J_{\nu+k}(r)|^q r^{2\nu+1-\nu q}\,dr, \qquad \nu:=\frac d2-1. \] For every integer $d\ge2$ and every finite admissible exponent $q>2d/(d-1)$, we prove \[ I_{\nu,k}(q)<I_{\nu,0}(q), \qquad k\ge1, \] and hence constants are the unique nonzero extremizers of the sharp mixed-norm extension inequality. In dimensions $d\ge3$ we prove the stronger strict ordering \[ I_{\nu,0}(q)>I_{\nu,1}(q)>I_{\nu,2}(q)>\cdots. \] We resolve this difficulty by combining exact Hankel asymptotics with a finite remainder comparison. The modal ordering also yields the optimal quadratic global stability inequality. For every finite admissible $q$, \[ C_{d,q}^2\|f\|_2^2 -\|\widehat{f\sigma}\|_{L^q_{\mathrm{rad}}L^2_{\mathrm{ang}}}^2 \ge \Gamma_{d,q}\|f-P_0f\|_2^2, \] with the best possible coefficient. When $d\ge3$, this coefficient is explicitly \[ \Gamma_{d,q} =(2\pi)^d \left[ I_{\nu,0}(q)^{2/q}-I_{\nu,1}(q)^{2/q} \right]. \]

We study two-player zero-sum stochastic games in which a stream of stage payoffs is evaluated by a translation-invariant charge on the set of stages. Every patient valuation, that is, every diffuse charge that obeys the time value of money principle, is translation-invariant, and by Mertens and Neyman each player has, for every $\varepsilon>0$, a strategy that is $\varepsilon$-optimal for all patient valuations simultaneously. We show that this robustness breaks down for translation-invariant charges. In the Big Match with $0$–$1$ payoffs, whose uniform value is $1/2$, every strategy of Player 1 is worthless: for every strategy of Player 1 and every $\delta>0$, Player 2 has a pure reply, and there is a translation-invariant charge under which Player 1's payoff is less than $\delta$. Consequently, the Big Match has no value with respect to all translation-invariant charges. This contrasts with finite Markov decision processes, where every stationary Blackwell optimal strategy is optimal for every translation-invariant charge. We pose questions on robust guarantees in general stochastic games and on optimality with respect to a fixed translation-invariant charge.

Individualized survival predictions are often summarized by scalar risk scores or event probabilities at selected time horizons, although clinically relevant risk may also differ in its temporal organization. We consider finite-dimensional representations of individual hazard trajectories through interpretable hazard functionals, such as peak timing, cumulative hazard over prespecified intervals, and persistence of elevated risk. A central issue is that the richness of such representations is constrained by the hazard geometry supported by the underlying survival model. We formalize this through model-induced tangent spaces and functional ranks, derive an exact decomposition of model-supported variation into directions retained and lost by a chosen representation, and introduce a model-relative notion of local completeness. We further distinguish intrinsic model capacity from the variation actually activated by covariates. These results motivate a structured extension of AFT time scaling through a positive-affine deformation on the log-time scale, yielding a power-warp mechanism with an additional temporal deformation direction. Combined with a log-logistic reference hazard, this gives a parsimonious likelihood-based working specification for which peak timing and peak height provide complete coordinates over the unimodal regime. We also establish plug-in recovery and asymptotic inference for hazard functionals. Simulations and a lung-cancer application illustrate how model-induced restrictions propagate to hazard-functional representations.

The relationship between classical methods for solving inverse problems and neural network methods is examined. An answer is provided to the question of what inverse problems can be solved using neural networks. The answer is that neural network algorithms can solve only conditionally well-posed problems (well-posed according to Tikhonov). The reason is that the design of standard neural network algorithms violates the well-known Bakushinsky veto from the theory of ill-posed problems. It requires that any method theoretically guaranteeing a stable solution to an ill-posed problem must depend explicitly on the error level of data. The article highlights the distinction between the concept of regularization (RA) in the theory of ill-posed problems and regularization as understood in neural network training, where Bakushinsky's veto is violated. Due to this violation, any changes to the neural network architecture, regularization during training, or training dataset augmentation cannot alter the answer specified above. It is theoretically demonstrated that an ill-posed inverse problem can be solved only by using a parametric set of neural networks that explicitly depend on the known error levels (as parameters) of data. Accordingly, as an alternative to standard neural network algorithms, \neural network based regularization algorithms (NNBRAs) are proposed to solve ill-posed inverse problems. One approach to their construction involves using a regularized dataset. Such a dataset can be generated from perturbed inverse-problem data using a specific base RA explicitly depended on the error level of the data. Test solving the simplest finite-dimensional ill-posed inverse problem demonstrate that the accuracy of the NNBRA approximate solutions is comparable to that of the base RA. At the same time, errors of conventional neural network methods may be unacceptable.