Papers

Permutation optimization arises whenever the components of a system are fixed but their ordering affects performance. We introduce PRISM, a predictive protocol for permutation optimization that measures a fitness landscape before selecting a search strategy. PRISM uses inexpensive landscape diagnostics, including one-step move autocorrelation and fitness-distance correlation, to predict useful mutation operators, identify when structured search is likely to outperform random sampling, and detect regimes in which search provides little advantage. Across synthetic permutation landscapes, neural architecture benchmarks, scientific machine learning pipelines, and large-language-model instruction ordering, the protocol makes testable predictions about search behavior before optimization begins. Exhaustive instruction-ordering experiments reveal substantial performance variation induced solely by permutation, while cross-model experiments show that useful ordering structure can transfer across model families and task difficulty. Additional experiments demonstrate that instruction ordering remains consequential after prompt wording is optimized, indicating that content optimization and ordering optimization are complementary. The results position PRISM not as a universally superior optimizer, but as a framework for determining when permutation search is useful, which representation and operator should be used, and when simpler alternatives are preferable.

In this paper we consider (MIMO) finite dimensional linear and time-invariant systems and irrational single-input single-output (SISO) transfer functions. In both cases our aim is to describe as best as possible the range of the input-output map $u(\cdot) \mapsto y(\cdot)$. The theory in finite dimension is quite satisfying but rather scarce, the first aim of this paper is to collect the main results in this direction in a comprehensive way. Our second aim is to improve a recent result on the tracking problem for irrational SISO transfer functions, allowing one to completely describe the outputs of such system for inputs $u \in L^2(0,\infty)$. An application is given for a heat equation with polynomial coefficients.

We study the spatial regularity of regular Lagrangian flows associated with vector fields in the DiPerna–Lions class \(\boldsymbol b\in L^1((0,T);W^{1,1}_{\loc}(\mathbb R^d)), \) under the standard growth and compressibility assumptions. For vector fields in \(L^1_tW^{1,p}_{\loc,x}\), with \(p>1\), the flow \(\boldsymbol X(t,\cdot)\) is known to satisfy a quantitative local Lipschitz estimate, which implies that it is Lipschitz continuous in the Lusin sense. We prove that, at the endpoint \(p=1\), this estimate admits an Osgood-type counterpart. More precisely, we construct an increasing function \(G\), with \(G(0+)=-\infty\), determined by the integrability of \(D\boldsymbol b\), such that \[ G\bigl(|\boldsymbol X(t,x)-\boldsymbol X(t,y)|\bigr) \leq G\bigl(|\boldsymbol X(s,x)-\boldsymbol X(s,y)|\bigr) + \int_s^t \bigl(k(\tau,x)+k(\tau,y)\bigr)\,\dd\tau, \] where \(k\) is locally integrable. As a consequence, the flow \(\boldsymbol X(t,\cdot)\) is uniformly continuous outside a set of arbitrarily small measure, with an explicit modulus of continuity determined by the integrability properties of \(D\boldsymbol b\). The resulting moduli include Hölder and log-Lipschitz regimes, as well as substantially weaker Osgood moduli. Our approach is based on a new family of weighted maximal operators associated with slowly varying functions in the sense of Karamata. We also provide examples showing that the resulting estimates are sharp in several respects and that the classical Lipschitz-type estimate may fail at the endpoint \(p=1\). Finally, we apply the flow estimates to transport equations, obtaining weighted logarithmic Sobolev regularity for transported scalars and corresponding lower bounds on functional and geometric mixing scales in the \(W^{1,1}\) setting.

We study radial nonlinear instability of compactly supported nonrotating equilibria of the three-dimensional Euler–Poisson system with a physical-vacuum boundary. Let $n^u(\mu)$ denote the radial instability index furnished by the turning-point theory of Lin and Zeng. Under general structural assumptions on the pressure law, suppose that $n^u(\mu)>0$ and that the equilibrium is not a mass extremum, $M'(\mu)\neq0$. Conditional on the existence of a sufficiently regular radial solution on the relevant time interval, we establish two nonlinear escape criteria. First, every perturbation with Hamiltonian strictly below that of the equilibrium exits a fixed neighborhood on a logarithmic time scale controlled by the least unstable linear growth rate. For the subclass of data for which the associated Lyapunov functional is initially nonnegative, we also obtain an explicit exponential lower bound in the weighted displacement norm. Second, sufficiently small perturbations whose Riesz projection onto the finite-dimensional unstable subspace is not too small escape on a logarithmic time scale. The second argument uses the exponential trichotomy of the linearized Hamiltonian flow and an invariant-cone estimate. In the polytropic class, the conditional estimates combine with the radial physical-vacuum local theory to yield unconditional nonlinear instability in the corresponding classical-solution topology. The results complement Jang's nonlinear instability theorem for Lane–Emden stars by treating mechanisms that do not require initial alignment with a leading growing eigenmode and by applying, conditionally, to unstable branches for general equations of state.

We prove that, up to row and column permutations and complement operations, the only complement minimally non-totally unimodular matrices are the cycle matrices $C_3$ and $C_5$. This settles a conjecture of Chervet, Grappe, and Vallée. As a consequence, every simplicial cone generated by the rows of a totally equimodular matrix admits a regular unimodular Hilbert triangulation.

Causal models in the same Markov equivalence class are, in the absence of strong assumptions, statistically indistinguishable at any sample size. For modest sample sizes there is also the possibility that several such classes are compatible with the data, pointing to a confidence set of causal models as the appropriate presentation of evidence. Non-identifiability of Gaussian causal models from the same Markov equivalence class corresponds to a plurality of inverse-covariance representations in the unconstrained parametrisation of Cox and Wermuth (1993). This parametrisation, avoiding conic constraints that would otherwise complicate distributional approximations, facilitates construction of, and theoretical analysis for, a confidence set of causal models based on standard likelihood theory. By drawing on a geometric formulation of Evans (2020), we provide insight into which causal models are most likely to be included in the confidence set when the true parameter values of the generating model are at the borderline of detectability, delineating those models whose inclusion probabilities are stable at the nominal level, slowly decaying with sample size, and quickly decaying with sample size. We also study settings in which two or more causal models are in operation, exploring how the mixture weights, and the geometry of the models in the mixture relative to candidate models, interact with the inclusion probabilities. The purpose of the paper is to probe, from the perspective of structural properties of the true causal mechanism, the limits of what is achievable in causal inferential settings.

When a variety $V_\varphi$ equals the image of a polynomial map $\varphi$ whose coordinate functions are combinatorial generating polynomials (i.e.~polynomials enumerating combinatorial objects), the geometry of $V_\varphi$ reflects identities satisfied by the generating polynomials. The resulting interplay between combinatorics and algebraic geometry can be used to answer questions about $V_\varphi$. A recent technique proposes to do so using a partially ordered set (poset) $P_\varphi$ defined via the coefficient vectors of the polynomials defining $\varphi$. This paper characterizes the poset $P_\varphi$ when the generating polynomials defining $\varphi$ enumerate subgraphs of a directed tree known as treks. The characterization is used to compute the linear span of $V_\varphi$, prove it is toric and deduce a basis for its vanishing ideal. It is also shown that this poset of trek polynomials for a directed tree is a so-called $\pi$-system if and only if the tree satisfies a property characterized via Stanley's P-partitions. As an additional consequence, it is shown that the varieties for two distinct directed trees intersect in a strictly lower-dimensional variety. This solves an instance of the structural identifiability problem in the graphical models program from statistics.

Backward differences are a standard digital realization of derivative action in proportional-integral-derivative control. This note proves that, for multivariable state-space plants, this implementation can be unstable no matter how small the sampling period is. The exact lifted model reveals fast eigenvalues created by the stored previous output sample. Away from boundary spectra, stability is equivalent to ideal loop stability plus Schur stability of the product of the output, input, and derivative gain matrices.

We investigate the properties of multivariate polynomials evaluated at points with commuting coordinates over division rings and octonion algebras. Given a division ring $D$, this set of points is denoted by $D_c^n$, and in the special case of $D=\mathbb{H}$, Alon and Paran showed that its points correspond to the maximal left ideals of $\mathbb{H}[x_1,\dots,x_n]$. Here we show that if a polynomial $f$ vanishes at $\vec{a} \in D_c^n$, then any left multiple $gf$ also vanishes at $\vec{a}$. Consequently, over a central division algebra or an octonion algebra, any root of $f$ in $D_c^n$ is also a root of its (reduced) norm. We apply these evaluation properties to discrete algebraic dynamics, proving that if a point in $D_c^n$ is a fixed point of an $n$-tuple $T=(f_1,\dots,f_n)$ of polynomials in $n$ variables, then it is a fixed point of $T^{\circ m}$ for any positive integer $m$.

This paper presents robust numerical formulations for suppressing low-frequency cancellation errors in the magnetic field integral equation (MFIE) for perfect electric conductors and the Müller integral equations for homogeneous and composite dielectric objects. The proposed approach combines quasi-Helmholtz projectors, which separate the Helmholtz components of the surface currents, with a tailored rescaling procedure that preserves the components essential for accurate far-field evaluation from finite-precision cancellation. The resulting formulations retain the conditioning properties of the mixed MFIE and Müller discretizations in the dense-mesh and low-frequency regimes. Numerical experiments involving simply and multiply connected scatterers, composite dielectric structures, multiscale and high-contrast configurations demonstrate that the proposed methods yield accurate far-field predictions across a broad frequency range, from the full-wave regime to extremely low frequencies.

Bulgarian solitaire is a well-studied, no-choice, no-loss, one-player game involving stacks of cards. More formally, it is a self-map on the set of partitions of a fixed integer $n.$ As a finite dynamical system, its long-term behavior is well understood. Every trajectory ends in a cycle. The partitions that are in a cycle are parameterized by binary vectors, and the cycles by binary necklaces. Call a partition separated if distinct part sizes differ by at least two. The vast majority of partitions belonging to a cycle are not separated. Motivated by this fact, we consider a variant where the player has choices, but is restricted to separated partitions and, if unable to make a legal move, may lose. We prove that for $n>73$, there are cycles, and hence winning initial positions. We analyze the game for small values of $n$ and describe computations which, together with our main result, show that there are cycles for $n \in \{2,6,8,11,14,16,18,21\}$ and for $n \ge 23$, but for no other $n.$

In this paper, we consider an optimal stopping problem with infinite horizon, non-negative pay-offs and non-linear evaluations $\rho_{S,\tau}$ indexed by two indices: $S$ and $\tau$, where $S$ is the time of evaluation and $\tau$ is the time when the pay-off is revealed. The agent's stopping strategies are constrained to be in the set of so-called Bermudan stopping times $\Theta$. Under suitable assumptions on the non-linear evaluations $\rho$ and on the pay-off, we show that a dynamic programming principle holds in this framework. We investigate the existence of $\varepsilon$-optimal stopping times, as well as the existence of optimal stopping times. We show that an $\varepsilon$-optimal stopping time exists. We also prove that the first time when the value family hits the pay-off is optimal if and only if it is finite. We also provide Doob's type convergence for non-negative \emph{$(\Theta, \rho)$}-supermartingales in the case where $\rho_{S,\tau}=\rho_S$ depends on the first index only. We provide an example from BSDEs with infinite horizon.

In this paper, we establish several new bounds for the distributional sectional category ($\mathrm{dsecat}$). We first prove Jauhari's conjecture, thereby establishing a cohomological lower bound for $\mathrm{dsecat}$ with arbitrary coefficients. We then obtain an analogous lower bound with rational coefficients by constructing a natural graded $\mathbb{Q}$-linear section of the homomorphism induced in cohomology by the symmetric diagonal. As applications, we provide several computations of the distributional sectional category. We also establish multiplicative product and composition inequalities for $\mathrm{dsecat}$. Finally, we apply these results to distributional homotopic distance. A sequential version of this invariant was recently introduced by Jauhari and Oprea; we give an equivalent formulation in terms of distributed homotopies and derive new fibration and multiplicative triangle inequalities.

We study the global boundedness of weak solutions to a class of nonlinear elliptic equations in bounded Lipschitz domains, subject to conormal boundary conditions. Using techniques from Morrey space theory, Adams–Mazya trace inequalities, and higher integrability properties of the gradient, we derive uniform a priori bounds for the solutions. A crucial step in our analysis is the construction of appropriate measures associated with the solution and the application of the Hartman–Stampacchia lemma, which enables us to control the solution up to the boundary of the domain.

We completely classify the possible complexity classes of non-closed operator ranges on separable Banach spaces over a Polish non-Archimedean non-trivially valued field. These are precisely $\boldsymbol{\Pi }_{1+\lambda +n+2}^{0}$ for a countable ordinal $\lambda $ that is either zero or a limit ordinal, and a finite ordinal $n$. Considering Fréchet spaces produces the additional complexity classes $\boldsymbol{\Pi }_{\lambda }^{0}$ for a countable limit ordinal $\lambda $.

Horikawa surfaces are minimal complex algebraic surfaces of general type with minimal Chern slope, satisfying either $c_2=5c^2_1+36$ if $c_1^2$ is even, or $c_2=5c^2_1+30$ if $c_1^2$ is odd. We prove that very general Horikawa surfaces with $p_g\ge 5$ contain only finitely many rational or elliptic curves. Moreover, we provide an explicit characterization and count of these curves. Our results also apply to very general Horikawa surfaces of the first kind with $p_g\in \{3,4\}.$

As firms increasingly deploy machine learning for strategic decision-making, understanding algorithmic interactions has become central to operations research and economics. This paper studies learning in infinite-horizon, nonzero-sum linear-quadratic stochastic games under a radically uncoupled information structure, where players are either unaware of opponents or strategically oblivious, observing only a common state and their own action history. Under this minimal information, we analyze an asynchronous decentralized learning process in which each player independently runs a single-agent $\epsilon$-greedy iterated least-squares algorithm. We prove that, despite being unable to identify the system parameters, players' learning dynamics converge almost surely to the complete-information Nash equilibrium and characterize the convergence rate. We then apply the framework to a dynamic Cournot competition with sticky prices. Numerical experiments validate the theoretical results and show that learning under limited information reduces firm profits under both low and high price stickiness, while total surplus declines and market concentration increases when price stickiness is high. Publicly revealing aggregate market output substantially accelerates convergence and mitigates these welfare losses.

Kühne established equidistribution for canonical adelic line bundles on semiabelian varieties, a setting which need not lie in the quasi-canonical range of Yuan-Zhang's theorem. We study adelic line bundles on semiabelian compactifications whose toric part is governed by a toric metrized divisor. The main unconditional result is generic equidistribution when the toric metric is monocritical and arithmetically T-effective. The proof isolates the asymptotic estimates in Kühne's argument and reinterprets them as estimates along explicit compression paths. This gives a comparison mechanism between canonical, quasi-canonical, and more general toric metrics. For the Bogomolov application, the quasi-canonical case is handled by a Kühne local-trivialization transport package. After fixing a single theta-factor convention for the local trivializations, the proof checks the Picard-zero theta factors under Kühne's operations and obtains the Bogomolov theorem for the metric class treated in this paper, namely the monocritical and arithmetically T-effective toric metrics, through the quasi-canonical replacement argument.

We settle two conjectures of Gautschi on the degree dependence of the zeros of the Jacobi polynomials $P_n^{(\alpha,\beta)}$, $\alpha,\beta>-1$, and obtain results substantially stronger than those conjectured. The conjectures stem from a line of questions originating in spherical cubature and hyperinterpolation. A Liouville transformation and Sturm comparison yield affine comparison principles with exact thresholds for the pointwise monotonicity of the rescaled potential. We prove that an increasing affine ordering with a degree-independent shift exists if and only if $|\beta|\leq1/2$. For the spectral scale $n+(\alpha+\beta+1)/2$, we determine the exact parameter regions for the two opposite orderings and show that no uniform spectral ordering is possible outside them. We also characterise all equality cases and derive finite-degree bounds in terms of Bessel zeros. The resulting classifications are exact and cannot be enlarged: outside the stated parameter regions the corresponding uniform zero orderings necessarily fail.

We construct real-analytic diffeomorphisms of \(S^3\times S^3\cong \SU(2)\times\SU(2)\) that lift We study real-analytic word maps of \(S^3\times S^3\cong \SU(2)\times\SU(2)\) obtained by evaluating Nielsen automorphisms of the free group \(F_2\). Fried studied these word maps and computed their topological entropy. For each maximal torus \(T\subset\SU(2)\), the product \(T\times T\) is invariant and carries the corresponding toral dynamics. Their union is the commuting locus. We describe its regular part as a twisted conjugacy-sphere bundle over the punctured pillowcase and determine the local topology at the four central singularities. Periodic toral points give periodic conjugacy two-spheres, which obstruct Anosov hyperbolicity on \(S^3\times S^3\). Normalized images of the coordinate three-cycles converge to stable and unstable Ruelle–Sullivan eigencurrents supported on the commuting locus. The construction extends to \(\Sp(1)^n\). The higher-rank commuting locus has dimension \(n+2\), periodic conjugacy two-spheres persist, and the action on \(H_{3k}((S^3)^n)\) is \(\Lambda^k A\). For \(n\geq3\), IA-automorphisms can act nontrivially on the \(\SU(2)\)-character variety although they are trivial on abelianization. The degree-three eigencurrents again have a Ruelle–Sullivan description; higher exterior-power classes lead to a compactness problem for normal currents. We also relate the nonuniqueness of Nielsen lifts to Fried's computation of the mapping class group of \(S^3\times S^3\), where the group \(\Theta_7\) of homotopy seven-spheres appears in the kernel of the action on homology.

For a commutative Bezout ring R, we give a criterion, in terms of colon ideals with principal radical, for the lattice $\mathring{\mathcal{K}}(\mathrm{Spec}(R))$ of compact open subsets of $\mathrm{Spec}(R)$ to be a Heyting algebra. Bezhanishvili and Tressl showed that $\mathring{\mathcal{K}}(\mathrm{Spec}(C(T)))$ is pseudocomplemented whenever T is a basically disconnected compact Hausdorff space, and asked whether $\mathrm{Spec}(C(\beta\mathbb{N}))$ is actually an Esakia space. We show it is not: applying our criterion to $C(\beta\mathbb{N}) \cong \ell^\infty(\mathbb{N},\mathbb{R})$ produces a diagonal counterexample, and the same obstruction rules out $\beta D$ for every infinite discrete D. A grid-existence theorem for $\sigma$-complete Boolean algebras lets us push the construction to every basically disconnected compact Hausdorff space, settling the Bezhanishvili-Tressl question completely: for no infinite compact Hausdorff space T is $\mathrm{Spec}(C(T))$ an Esakia space.

In previous work with Harman and Snowden, we found explicit formulas for units of rank ideals of the matrix monoid algebra $\mathbb{K}\left[\operatorname{Mat}_n\left(\mathbb{F}_q\right)\right]$, where $\mathbb{K}$ is an algebraically closed field with characteristic not dividing $q$. In this work, we use these explicit formulas to establish a regularized Godement–Jacquet functional equation valid for irreducible representations of $\operatorname{GL}_n\left(\mathbb{F}_q\right)$ and of the algebra $\mathbb{K}\left[\operatorname{Mat}_n\left(\mathbb{F}_q\right)\right]$. Using this functional equation we give an explicit expression for primitive central idempotents corresponding to irreducible representations of $\mathbb{K}\left[\operatorname{Mat}_n\left(\mathbb{F}_q\right)\right]$, where the characteristic of $\mathbb{K}$ does not divide $\left|\operatorname{GL}_n\left(\mathbb{F}_q\right)\right|$. Interestingly, the coefficients in this formula are closely related to degenerate non-abelian Gauss sums.

A new recursive implementation of the equivariant moving frame method for infinite-dimensional Lie pseudo-group actions is presented. It allows pseudo-group normalizations to be performed before the prolonged action is computed, thereby avoiding the introduction of unnecessarily complicated intermediate expressions. Several examples are provided to illustrate the algorithm.

We study real multilinear forms with coefficients in $\{-1,1\}$ on finite-dimensional Hilbert spaces. Every trilinear sign form on $\ell_2^r\times\ell_2^n\times\ell_2^n$ has norm at least $\sqrt n$. Writing $K_{r,n}$ for the least norm divided by $\sqrt n$, we prove that $K_{r,n}=1$ exactly when a Hadamard matrix of order $n$ exists and $r\le\rho(n)$, where $\rho$ is the Hurwitz–Radon function. If equality fails, we obtain an explicit gap above $\sqrt n$. We also prove two asymptotic results. If $1\le m_n\le n$ and $\limsup r_n/\log_2 n<2$, there are sign forms on $\ell_2^{r_n}\times\ell_2^{m_n}\times\ell_2^n$ with norm $(1+o(1))\sqrt n$. In the square case, if $r\ge2\lceil\log_2(8n)\rceil$, then $K_{r,n}-1\ge c(1+\log_2 n)^{-4}$. We also prove a fourth-moment estimate in every fixed multilinear order, characterize equality, and give exact and asymptotic constructions.

We prove that every embeddable $3\times3$ row-stochastic matrix is a product of at most seven elementary row-stochastic matrices. Frydman's example shows that this bound is sharp. The proof combines Frydman's six-factor criterion and spiral theorem with a reduction to a three-parameter critical family and an analysis of the membership certificates that persist along it.

We study O'Neil-type inequalities in Lorentz spaces for two extremal geometric configurations of translation-invariant multilinear convolution operators. For the one-parameter model, we complement the existing weak-kernel theory with estimates for kernels in the Lorentz space $L^{r,s}$, obtaining a stronger target Lorentz space when $s$ is finite. For the full-dimensional model, we prove a Lorentz refinement of Oberlin's multilinear Young inequality in the interior of its admissible range. For nonnegative full-dimensional kernels, we extend Hörmander's necessary condition to the multilinear setting in Lorentz spaces and obtain an additional restriction on the secondary indices in the critical case. We also prove the failure of the full-dimensional weak $L^1$ endpoint, use additive combinatorics to construct one-parameter restricted weak-type counterexamples for $p<1$, and obtain complementary positive results for kernels of special structure.

Let $G$ be a simply connected simple complex algebraic group. It is proved by Oya, Qin, and Yakimov that the quantized coordinate ring $\mathcal{O}_q(G)$ is generated by generalized quantum minors, and therefore carries a quantized cluster algebra structure, for all $G$ but type $F_4$. In this article, we settle the $F_4$ case by an argument uniform across $G_2$, $F_4$, and $E_8$. The main idea is to bootstrap the existing proof in type $E_8$, which relies on Lusztig's canonical basis of the quantum adjoint representation, and replace it with the crystal combinatorics of the quasi-minuscule representation. As a consequence, we prove that $\mathcal{O}_q(F_4)$ also has a quantized cluster algebra structure.

Let (M,g) be a compact n-dimensional Riemannian manifold, n>2. We introduce a restricted variational principle for the Einstein-Hilbert functional by requiring the admissible metric variations to satisfy the harmonic gauge condition. We derive the corresponding Euler-Lagrange equation and show that a metric is critical with respect to all volume-preserving harmonic variations if and only if its Einstein tensor differs from a multiple of the metric by an element of the image of the adjoint Bianchi operator. We prove that every harmonic critical metric determines a compact Ricci soliton whose soliton constant is given by the normalized Einstein-Hilbert functional. By Perelman's theorem, every such metric is in fact the metric of a compact gradient Ricci soliton. Conversely, every compact gradient Ricci soliton satisfies the restricted Euler-Lagrange equation. Thus, a compact Riemannian metric is harmonic critical if and only if it is the metric of a compact gradient Ricci soliton. We further show that the gauge one-form differs from the negative differential of a soliton potential by a Killing one-form. In particular, if the Ricci tensor is negative definite, then the gauge one-form vanishes and the metric is Einstein. Moreover, every non-Einstein harmonic critical metric is necessarily shrinking.

We study the determinant of left multiplication in the 32-dimensional Cayley-Dickson algebra $A_5$. For $x \in A_5$, let $L_x$ denote left multiplication and let $N(x)=|x|^2$. Using the fourfold multiplicity of the eigenspaces of $L_{x^*}L_x$ and Newton identities applied to its trace invariants, we construct a homogeneous polynomial $D_{14}$ of degree 14 and prove the factorization $\det L_x=N(x)^2D_{14}(x)^2$. Consequently, a nonzero element $x \in A_5$ is a left zero divisor if and only if $D_{14}(x)=0$. We also prove the sharp bound $0 \le D_{14}(x) \le N(x)^7$, which yields $0 \le \det L_x \le |x|^{32}$, and characterize the equality case in terms of alternative elements. We compare this factorization with the corresponding structure in the sedenions $A_4$. In $A_4$ an additional norm factor occurs, whereas in $A_5$ we prove, after complexification, that $N$ does not divide $D_{14}$. Thus the passage from $A_4$ to $A_5$ exhibits a genuine change in the determinant factorization. The construction gives a basis-independent polynomial description of the determinant-level spectral structure of $A_5$ and an explicit algebraic criterion for its zero divisors.

We investigate the relationship between the tame factorization property, denoted by $\mathfrak{TF}$ and introduced in the companion paper \cite{CDI}, and the DN-$\Omega$ type linear topological invariants of Fréchet spaces. Combining the basic properties of $\mathfrak{TF}$ with known characterizations of tameness and boundedness, we obtain several results identifying the triples of Fréchet spaces that possess $\mathfrak{TF}$. We further exhibit examples showing that tame factorization property is a strictly weaker condition than tameness, indeed, we construct triples possessing $\mathfrak{TF}$ none of whose individual pairs are tame. We then investigate triples consisting of an arbitrary Fréchet space $X$, a nuclear Fréchet space $Y$ satisfying the properties $\underline{DN}$ and $\Omega$, and a power series space of finite type $\Lambda_1(\mathcal{E})$ or infinite type $\Lambda_\infty(\mathcal{E})$. We show that requiring such a triple to possess the tame factorization property $\mathfrak{TF}$ characterizes the corresponding linear topological invariants of $X$; in some cases this holds without any restriction on $Y$, while in others it requires the coincidence of the approximate diametral dimension of $Y$ with that of $\Lambda_1(\mathcal{E})$ or $\Lambda_\infty(\mathcal{E})$.