Papers

We prove an all genus full descendant open mirror symmetry for footballs. The B-model is given by the Chekhov-Eynard-Orantin topological recursion on the mirror curve.

The affine Conway group is the group of affine isometries of the Leech lattice. This group is isomorphic to the automorphism group of a standard fundamental domain for the action of the Weyl group on the even hyperbolic lattice II_{1, 25} of rank 26. In this paper, we show that the affine Conway group has exactly nine conjugacy classes of involutions. We investigate their properties and show that, among these nine classes, one class can be regarded as an analogue of the class of Enriques involutions of K3 surfaces. Motivated by possible applications to K3 and Enriques surfaces, we investigate in detail the orthogonal groups of the hyperbolic lattices arising as the invariant sublattices of involutions in II_{1, 25}. This computation is carried out using the Borcherds method. Unlike the examples considered previously, the induced chambers possess infinitely many walls.

We study online search for an unknown affine hyperplane in $\mathbb{R}^D$, for arbitrary fixed finite dimension. Building on a companion self-similar cell reduction and support-function formulation, we ask how the mechanism changes as the normal space grows from $\mathbb{S}^0$ to $\mathbb{S}^{D-1}$. In $D=1$, alternation and productivity yield an equal-ripple principle and the exact stationary constant $9$. In $D=2$, the analogous relative equilibrium is a logarithmic spiral whose bottleneck chord imposes tangency and selects the pitch. For exponential orbits $\Gamma(\sigma)=e^{\kappa\sigma}\omega(\sigma)$, we develop log-directional geometry, exponentially discounted memory, gauges, and recursive hyperspherical parametrizations. Without a shape ansatz, the bottleneck admits a certificate supported by at most $D$ historical suppliers, and at globally worst phases the current point lies on the active face. Within regular chambers we derive exact variation, tangency, pitch, age, and, in $D=3$, delay-system identities. Odd-dimensional obstructions, antipodal subclasses, and harmonic towers provide constraints and explicit candidate families but are not claimed globally optimal. Finally, the N-COMP theorem shows that $C_D^*$ is a computable real for every fixed finite $D$ and that algebraic polygonal $\varepsilon$-optimal cells can in principle be synthesized. Numerical screening through $D=10$ is kept separate from the proved results.

Koopman autoencoders (KAEs) seek a higher-dimensional latent representation in which nonlinear dynamics evolve linearly. However, many interesting systems have multiple basins of attraction, and both theoretical and empirical work has shown these multibasin systems cannot generally admit a single finite-dimensional global Koopman embedding under standard assumptions. We posit that encoders with a sparsity-inducing objective encouraging few active latent coefficients will provide latent supports as an inspectable basin-modeling principle for Koopman autoencoders. We use these encoders producing sparse latents in training Sparse Koopman Autoencoders (SKAEs) without basin labels or other regime annotations, and treat the learned latent supports as model-produced regime variables after training. Across a range of procedurally generated multibasin systems and chaotic flows, we show that SKAEs have superior forecasting performance compared to dense-latent KAEs. We also perform a mechanistic study that shows latent supports produced by SKAEs are both essential for the quality of the representation and useful for identifying basins on basin-interior states, whereas dense-latent KAEs collapse to an uninformative single support family. These results identify sparse latents and their corresponding supports as label-free, interpretable regime variables for Koopman learning in nonlinear systems with multiple local dynamical laws.

Conventional orthogonal frequency division multiplexing (OFDM) channel estimation relies on single-tap estimation and time-frequency (TF) interpolation, which becomes unreliable in high-mobility channels because Doppler-induced inter-carrier interference (ICI) invalidates the underlying element-wise TF model. This paper proposes a neural-network-based delay-Doppler (DD)-assisted channel estimation framework for OFDM over doubly selective channels. We first derive an ICI-aware TF domain input-output relation and formulate channel estimation as a DD recovery problem. Unlike conventional sparse recovery approaches, the proposed framework does not require the equivalent DD domain channel vector to be strictly sparse, thereby accommodating the leakage induced by fractional delay and Doppler shifts. Since the data symbols are unknown during channel estimation, the sensing matrix is constructed using only the known pilot symbols. As a result, data-induced interference is not explicitly modeled, leading to a structured mismatch in the pilot observations. To tackle this challenge, the adopted network iteratively exchanges observation- and channel-domain features through the sensing matrix to learn the mapping from these contaminated observations to the equivalent DD domain channel, which is subsequently used to reconstruct the TF-domain channel. Simulation results show that the proposed method achieves lower normalized mean-square error and bit-error rate than conventional OFDM estimators.

DeepHSIC is introduced as a neural receiver for hybrid downlink IM-NOMA transmission. The considered scheme combines power-domain NOMA with a composite OFDM/OFDM-IM waveform, so that user information is mapped jointly onto constellation symbols, subcarrier-index patterns, and different power levels. Although maximum-likelihood detection can achieve strong reliability for this model, its search space grows rapidly with the number of users and subcarriers. Conventional SIC reduces part of this burden, but its sequential cancellation may still accumulate errors and does not fully exploit the structure of IM-NOMA signals. To address this limitation, the proposed detector embeds dedicated deep neural network modules into the receiver and replaces the most computationally demanding SIC operations with learned inference blocks. The receiver is trained for Rayleigh fading channels and uses preprocessed channel-output features to recover user symbols. Simulation results show that DeepHSIC reaches BER performance close to model-based detectors under both perfect and imperfect CSI while requiring substantially lower detection time. These results indicate that learned SIC-style detection is a practical candidate for scalable hybrid downlink IM-NOMA receivers.

Let $S$ be a minimal compact complex surface of class VII with $b_2(S)=3$. We prove that if $S$ carries two distinct singular holomorphic foliations, then $S$ contains a global spherical shell. Moreover, it is an Inoue-Hirzebruch surface. The proof is based on Teleman's theorem on the existence of a cycle of rational curves and results of Dloussky and is inspired by work of Brunella.

Fox $p$-colorings of knots and links admit a linear-algebraic description in terms of braid representations. For each braid $\beta \in B_n$ we associate a representation $M : B_n \to \mathrm{GL}(n,\mathbb{F}_p)$ such that Fox $p$-colorings of the closed braid $\widehat{\beta}$ correspond to fixed points of $M(\beta)$. As a consequence, $$ \mathrm{Col}_p(\widehat{\beta}) = p^{\dim \ker(M(\beta)-I)}, $$ yielding an explicit formula for the number of Fox $p$-colorings of arbitrary knots and links. We apply the method to torus links and spiral links, obtaining explicit descriptions of their Fox 3-coloring spaces.

It was conjectured that no presilting objects exist in the singularity category of a finite-dimensional algebra over a field except the zero object. In this note, we give counterexamples to this conjecture for algebras over fields of positive characteristic.

An ideal $I$ of a polynomial ring $R=k[x_1,\dots,x_n]$ is called weakly Newton-nondegenerate, or weakly NND, if its integral closure $\overline{I}$ is a monomial ideal. We study weak Newton nondegeneracy for the family of quadratic binomial ideals \[ I=(x_1^2+\epsilon_1x_{a_1}x_{b_1},\ \dots,\ x_n^2+\epsilon_nx_{a_n}x_{b_n}),\qquad \epsilon_i\in\{\pm1\},\ a_i\neq b_i, \] over an algebraically closed field. We prove that $I$ is weakly NND if and only if $\overline I=\mathfrak{m}^2$, if and only if the given generators form a regular sequence, and if and only if an explicit combinatorial condition on the pair (support pattern, sign pattern) holds: no nonempty subset $S\subseteq\{1,\dots,n\}$ is simultaneously closed for the support data and sign-trivial for the associated lattice of relations. The last equivalence rests on a solvability criterion for systems of monomial equations over a divisible abelian group, in the spirit of Eisenbud and Sturmfels. As an application, we classify all such ideals for $n=3$.

A shoreline-search path starts at the origin and must meet an unknown affine line, without knowing either its normal or its distance. We first establish a self-similar reduction theorem for homogeneous search problems whose historical information is a record profile updated by pointwise maximum. Two quasi-returns of the normalized state delimit a block that renews the required profile by itself; a short connector closes this block into a cell. Every finite-ratio path can therefore be approximated, with arbitrarily small loss, by repetitions of a single cell at all scales. The main chain is then made effective. A finite coding of the state space computably bounds the scale factor and normalized length of a nearly optimal cell. For planar Shoreline search, the support function of the convex hull gives an exact cell functional. A one-sided polygonalization then reduces the problem to a computable number of vertices, after which quantifier elimination decides whether a polygonal cell exists below a rational threshold. It follows that the optimal deterministic planar Shoreline value $C_2^*$ is a computable real: for every rational $\epsilon>0$, an algorithm terminates with a rational interval of width at most $\epsilon$ containing $C_2^*$. Additional results—sliding memory, Bellman transitions, deadlines, geometric filters, and relative equilibria—are presented separately as a toolbox for certified computation and for the study of spiral rigidity; they are not used in the computability proof.

Strongly bounded objects in a weakly approximable triangulated category are known to relate closely to global dimension and to play a significant role in the uniqueness problem for triangulated enhancements. In this paper, we investigate when the full subcategory of strongly bounded objects admits a bounded $t$-structure. Our main result, under a finiteness condition termed the finite strong finitistic dimension, states that this happens exactly when the subcategory agrees with the full subcategory of bounded objects in the ambient triangulated category. In that case, the bounded $t$-structure is unique up to equivalence; even more, the uniqueness holds unconditionally on the full subcategory of bounded objects.

In this paper, we focus on the asymptotic behavior of random iterated functions systems, in which both the family of transformations and the distribution of selecting them vary at each step. We consider two types of dynamics: the time-inhomogeneous Markov chain arising from forward iterates and the non-Markovian process generated by backward iterates. For both settings, we provide certain criteria for the existence of a probability measure that is geometrically attracting in the bounded Lipschitz distance, independently of the initial distribution. Finally, we illustrate our results through applications to specific models involving affine transformations.

We determine the sharp restricted isometry threshold for recovery at global minima of the rank-restricted matrix LASSO. For target rank $r_{\star}$, if the rank-$k$ RIP constant satisfies $\delta<\delta_{\mathrm{sharp}}(k/r_{\star})$, where $\delta_{\mathrm{sharp}}(t)=t/(4-t)$ for $0<t<4/3$ and $\delta_{\mathrm{sharp}}(t)=\sqrt{(t-1)/t}$ for $t\ge4/3$, then every global minimizer has Frobenius error $\lesssim\sqrt{r_{\star}}\lambda$ for all $\lambda\gtrsim\|\mathcal{A}^{*}(\xi)\|_{\mathrm{op}}$ and at every search rank $r\ge r_{\star}$. The constants depend only on the RIP constant and $t=k/r_{\star}$, and in particular are independent of the search rank. When the rank restriction is inactive, the result specializes to the ordinary convex matrix LASSO. We also obtain the analogous results for sparsity-restricted vector LASSO. Conversely, we show that the threshold $\delta<\delta_{\mathrm{sharp}}(k/r_{\star})$ cannot be improved, due to the existence of counterexamples whose global minimizers fail to recover the ground truth.

This paper proposes gappy probabilistic manifold decomposition (Gappy PMD), a nonlinear method for reconstructing high-dimensional fields from extremely sparse measurements. Gappy PMD reconstructs the field on the nonlinear manifold learned by probabilistic manifold decomposition (PMD). We further propose a differentiable point selection method for reduced-order model (ROM)-based field reconstruction (DPS). Using differentiable meshless interpolation within the ROM-based reconstruction framework, DPS makes the full-field reconstruction error differentiable with respect to the sampling locations and directly optimizes these locations. In addition, a theoretical error analysis for Gappy PMD is also given. It splits the squared reconstruction error into two orthogonal parts: one normal to the reconstruction manifold and the other induced by sparse sampling and observation noise. Under a stability condition on the sampling operator, this error vanishes with the PMD approximation error and the noise. The Gappy PMD is evaluated on three numerical test cases: flow past a cylinder, lid-driven cavity flow, and backward-facing step flow. For the same reduced dimension and sampling points, Gappy PMD attains mean relative $L^2$ errors one to two orders of magnitude below Gappy POD. Optimizing the sampling points with DPS further improves reconstruction accuracy and robustness.

We define the notion of weighted polyhedron which can be thought of as the geometric realization of a weighted simplicial complex introduced by Dawson. Moreover, we will define a weighted version of singular homology theory for a weighted polyhedron and prove that it is isomorphic to the weighted simplicial homology of the weighted polyhedron. This implies that weighted simplicial homology is an invariant under isomorphisms and more generally under certain type of homotopy equivalences of weighted polyhedra. Moreover, we will generalize the cup product and cap product to weighted singular cohomology. In addition, we will interpret some known theories of orbifolds in terms of our weighted singular homology and cohomology.

Let $$ A=\begin{bmatrix} H_1 & E^* \\ E & H_2 \end{bmatrix} \quad\text{and}\quad \widetilde A=\begin{bmatrix} H_1 & 0 \\ 0 & H_2 \end{bmatrix} $$ be partitioned Hermitian matrices, with $\widetilde A$ obtained by dropping the off-diagonal blocks of $A$. Let $\eta$ be the minimum distance between the spectra of $H_1$ and $H_2$; these spectra may interlace. Define, for $\delta,\epsilon\ge0$, $$ \phi(\delta,\epsilon)= \begin{cases} 2\epsilon/(\delta+\sqrt{\delta^2+4\epsilon^2}), &\quad\mbox{if $(\delta,\epsilon)\ne (0,0)$}, 1, &\quad\mbox{if $(\delta,\epsilon) = (0,0)$}, \end{cases} $$ and let $\widetilde E=A-\widetilde A$ and $\epsilon_2=\|E\|_2=\|\widetilde E\|_2$, the matrix spectral norm. Li and Li [Linear Algebra Appl., 395:183–190, 2005] established a sharp spectral-norm bound on the changes in the eigenvalues of $A$: $$ \big\|\mathrm{diag}\big(\pmb{\lambda}(A)-\pmb{\lambda}(\widetilde A)\big)\big\|_2 \le \phi(\eta,\epsilon_2)\,\|E\|_2, $$ where $\pmb{\lambda}(A)$ is the vector whose components are the eigenvalues of $A$ in descending order and similarly for $\pmb{\lambda}(\widetilde A)$. In this paper, this bound is extended to every unitarily invariant norm $\|\cdot\|_\mathrm{ui}$: $$ \big\|\mathrm{diag}\big(\pmb{\lambda}(A)-\pmb{\lambda}(\widetilde A)\big)\big\|_\mathrm{ui} \le \phi(\eta,\epsilon_2)\,\|A-\widetilde A\|_\mathrm{ui}. $$ A $2\times2$ example attains equality for every such norm. An analogous bound for singular values is also established when the off-diagonal blocks of a rectangular matrix are dropped.

We consider the Stokes operator $A$ on smooth exterior domains $\Omega$ of $\mathbb{R}^n$ in homogeneous Sobolev spaces $\widehat H^{\kappa,q}_w(\Omega)$ with radially symmetric Muckenhoupt weights $w\in \mathscr A_q$. A fundamental property is the existence of a bounded $\mathscr H^\infty$-calculus of the Stokes operator on weighted nonhomogeneous and homogeneous $L^q$ Sobolev spaces. This property implies the existence of uniformly bounded purely imaginary powers $A^{it}$, $t\in\mathbb{R}$, and the characterization of domains of fractional powers $A^\theta$ equipped with nonhomogeneous ($\| u\|_{L^q_w} + \|A^\theta u\|_{L^q_w}$) as well as homogeneous norm ($\|A^\theta u\|_{L^q_w}$) as complex interpolation spaces. The final aim is the identification with homogeneous spaces $\widehat{\mathcal D}((-\Delta_{q,w})^\theta) = [L^q_{w},\widehat{\mathcal D}(-\Delta_{q,w})]_\theta$ intersected by a space of solenoidal vector fields. Moreover, we obtain weighted variational inequalities for weak solutions of the Stokes equations, weighted $L^q$-$L^r$ decay estimates of the Stokes semigroup and $L^p$-maximal regularity on $L^q_{\sigma,w}(\Omega)$.

In this paper we consider the Navier-Stokes equations in exterior domains of $\mathbb{R}^n$, $n\geq 3$, with a periodically in time moving boundary $\partial\Omega(t)$ and external force $f(t)$. For this case we prove the existence of a locally unique mild time periodic solution in weighted function spaces with radially symmetric Muckenhoupt weights. The solutions split into a stationary part controlled by potential theoretic estimates and a purely oscillatory part constructed as mild solution via analytic semigroup theory. To deal with perturbation terms of even second order - coming from a coordinate transform and the moving boundary - in weighted, homogeneous Sobolev spaces a maximal $L^1$ type regularity estimate will be used in weighted Lorentz spaces. To control the convective term an $\mathcal H^\infty$-calculus in weighted spaces of the Stokes operator, its $BIP$ property and embedding estimates of fractional powers are exploited, see a recent paper by the authors: The Stokes operator on exterior domains in homogeneous weighted function spaces: From weak theory to $\mathscr H^\infty$-calculus to fractional domains (2025).

Convex-concave saddle-point problems are ubiquitous across diverse domains, including machine learning, image processing, economics, and equilibrium problems. Primal-dual algorithms provide a highly effective and powerful framework for convex-concave saddle-point problems. Convex combination has become a crucial acceleration technique for primal-dual algorithms, and the integration of this technique has recently made these algorithms a highly active research topic. The choice of the convex combination parameter often has a significant impact on both the theoretical analysis and the numerical performance of the corresponding algorithms. However, the requirements on this parameter imposed by theory are sometimes inconsistent with those suggested by numerical experiments. For instance, theoretical analysis often requires the parameter to be small, while numerical experiments tend sometimes to favor larger values. To address this inconsistency and further advance primal-dual algorithms with convex combination, we develop a novel strategy based on two convex combinations, integrate it into a primal-dual framework, and propose a new primal-dual algorithm with linesearch, termed NPDAL-n, for general convex-concave saddle-point problems. The proposed two convex combinations in NPDAL-n ensure that the permissible range of the convex combination parameters is mainly determined by theoretical considerations, with little regard for numerical performance. Through rigorous Lyapunov energy descent analysis, we establish the global convergence and a sublinear ergodic convergence rate of $\mathcal{O}(1/N)$ for NPDAL-n under standard assumptions. When the primal function is strongly convex, we develop an accelerated version of NPDAL-n that achieves an optimal $\mathcal{O}(1/N^2)$ rate.

Piecewise-linear (PWL) optimization problems arise in many mixed-integer programming (MIP) optimization applications, including portfolio optimization, workforce scheduling, and resource allocation. But solving them to global optimality remains computationally expensive because branch-and-bound repeatedly solves LP relaxation subproblems. Existing solvers are largely CPU-centric, leaving the scalability of modern GPUs underutilized. Few prior GPU-accelerated branch-and-bound either targets neural network which is not suitable for general PWL optimization, or accelerates only auxiliary subroutines such as strong branching heuristics within CPU-centric MIP solvers. To bridge this gap, we propose B$^3$-PWL, a GPU-centric batched branch-and-bound framework for piecewise-linear optimization with Special Ordered Set of type 2 (SOS2) constraints. Our method solves batches of LP relaxation subproblems concurrently on the GPU using a first-order primal-dual solver, enabled by a specialized batched block-tiled sparse matrix kernel. To complement bound computation, we further introduce a unified feasibility search module that combines an SOS2 repair primal heuristic with a batched feasibility pump to rapidly obtain feasible incumbents and improve pruning efficiency. On a benchmark of 43 PWL-MIP instances, B$^3$-PWL achieves a 9.25x geometric-mean speedup over NVIDIA cuOpt while reaching high-quality feasible incumbents on every tested instance. On a public valve-point unit-commitment benchmark, it further outperforms NVIDIA cuOpt and the open-source CPU solvers SCIP and HiGHS, demonstrating the potential of first-order LP methods as the central engine of GPU-accelerated branch-and-bound.

We prove the optimal global regularity of admissible solutions to a transformed very fast diffusion equation in the range $-1<p<0$, posed on smooth bounded domains with zero Dirichlet boundary data and initial data comparable to the distance function. More precisely, we establish existence and uniqueness and show that solutions belong to $C^{1,p+1}(\overline\Omega)$ in space for every positive time and are $C^\infty$ in time uniformly up to the boundary. Moreover, all their time derivatives belong to $C^{1,p+1}(\overline\Omega)$, and the exponent $p+1$ is optimal. These regularity estimates further yield fine long-time asymptotics toward the friendly giant solution, including a first-order expansion in the $C^{1,p+1}(\overline{\Omega})$ topology and an improved convergence rate for the relative error in $C^{p+1}(\overline\Omega)$.

We establish Schauder estimates for linearized very fast diffusion equations with Dirichlet boundary conditions in bounded smooth domains. These estimates provide an important ingredient for deriving optimal boundary regularity and long-time dynamics of solutions to very fast diffusion equations.

We develop a few calculus rules to compute or lower bound the facial distance of a polytope. We illustrate our calculus rules on various popular polytopes. In particular, we provide a provide a lower bound on the facial distance of the Birkhoff polytope.

This paper studies the 2D Navier–Stokes equation on a bounded domain with Navier-slip boundary conditions, driven by spatially localized stationary forcing generated by a stochastic heat equation. We prove exponential mixing for the associated Navier–Stokes–heat system, and consequently exponential convergence of the velocity law under stationary forcing. The main difficulty is that the white noise is confined to a subdomain and reaches the velocity indirectly through the heat component. To address the degeneracy, the proof combines Malliavin calculus with PDE control theory. The key ingredient is a stabilization scheme that converts localized Navier–Stokes controls into time-regular controls compatible with the heat dynamics.

Universal recovery – the existence of a single recovery map that corrects an entire family of error channels – is a central feature of quantum error correction (QEC). In exact QEC, linearity guarantees that a code correcting a given error set also corrects every channel whose Kraus operators lie in its linear span, and that a single recovery map suffices for all such channels. Approximate quantum error correction (AQEC), which relaxes perfect recovery to recovery with controlled error, has traditionally lacked this structure. In a recent paper (arXiv:2607.22995), we developed a theory of approximate quantum error correction showing that a restricted form of linearity persists in the approximate setting, yielding uniform AQEC guarantees for the family of channels controlled by a given error set. In this work, we complete the picture by establishing the second half of universal recovery in the approximate setting: a single recovery map can simultaneously correct every channel controlled by a given error set. The error-set theory we proposed quantifies approximate correctability through two parameters: the environment-leakage distance, governing worst-case performance, and the Knill–Laflamme Hellinger distance, governing average-case performance. We show here that both quantities also control universal decoding. We further study the Petz map naturally associated with an error set as an explicit universal recovery, and obtain uniform average- and worst-case guarantees across the entire family of channels.

In this paper, we find the explicit representation of the set-valued metric projection operator from l1 to the unit closed ball in l1. By using this representation, we investigate the properties of the Mordukhovich derivatives of the set-valued metric projection operator, which is applied to calculate its covering constant. As a special case, we consider the 2-d Banach space. We find the explicit solutions of the set-valued metric projection operator from the considered 2-d Banach space to its unit closed ball. By these solutions, we will calculate its Mordukhovich derivatives in details.

A classical kaleidocycle is a closed cyclic chain of congruent, edge-hinged tetrahedra with successive hinge axes at right angles, capable of continuous inside-out motion. Kaleidocycles composed of six isosceles tetrahedra (disphenoids) form a natural generalization in which consecutive hinge axes need not be perpendicular. Their kinematics are governed by the closure equation for threefold-symmetric Bricard 6R linkages. We derive a new compact closure equation directly from threefold symmetry using the composition of rotations, and show its equivalence to the classical formulation. Furthermore, we establish two equivalent necessary and sufficient criteria for a kaleidocycle to admit continuous, unblocked motion in terms of the geometric parameters of the constituent disphenoid. These criteria yield sharp bounds on the admissible geometric parameters of realizable kaleidocycles. Finally, we study assemblies of eight such kaleidocycles joined face-to-face. We identify a configuration exhibiting full octahedral symmetry, which forms a polyhedral structure closely related to the disdyakis dodecahedron.

We introduce the $\mathfrak M$arseille anti-construction $\mathfrak{Ma}$ for varieties of algebras defined by multilinear identities of degree $3$, obtained by changing the sign of the right-nested part of each defining polynomial. This gives, for any variety $\Var$, a new variety $\mathfrak{Ma}$-$\Var$ that generalizes antiassociative algebras and the recently studied anti-Novikov and anti-right-symmetric $\big($pre-Jacobi–Jordan$\big)$ algebras. We describe free objects in several $\mathfrak{Ma}$-varieties. On the operadic level, over a field of characteristic $\mathrm{char}\,\Bbbk\neq 2$, we show $(\mathfrak{Ma}\text{-}\Var)^{(!)}=\mathfrak{Ma}\text{-}(\Var^{(!)})$ and that the Dong property is preserved by $\mathfrak{Ma}.$ Finally, we obtain the necessary and sufficient condition for $\mathfrak{Ma}\text{-}\Var=\Var$. The second part of the paper is dedicated to another type of anti-algebras, called $\mathcal{M}anin$ anti-algebras. We give the description of this type of algebras via calculation of the white Manin product of suitable operads and present various examples of known $\mathcal{M}anin$ anti-algebras. In the end, we proved that the symmetric affinization of an algebra gives a Jacobi-Jordan algebra if and only if the original algebra is a $\mathcal{M}a$-Novikov algebra, that is, an analog of a well-known result about affinization of Novikov algebras given by Balinskii and Novikov.

We introduce a new class of commutative associative algebras whose structural constants are linear operators acting in a linear topological space. An analogue of the theory of monogenic functions is developed for such algebras. It is shown that the components of monogenic functions satisfy second-order operator-differential equations. This approach provides a constructive method for obtaining solutions of various partial differential and integro-differential equations. Several illustrative examples are presented.