Papers

In multi-hop device-to-device (D2D) networks, the optimization of network-level metrics is particularly difficult due to the tight coupling between network-layer routing and physical-layer resource allocation. Departing from traditional average-performance metrics, this paper addresses the joint optimization of routing paths, transmission power, and bandwidth allocation. We formulate a generalized cost function to minimize the maximum transmission time (i.e., the bottleneck delay) alongside the total energy consumption. To tackle the resulting highly non-convex formulation, we propose a novel block coordinate descent (BCD) framework. At the network layer, we develop two adaptive routing algorithms: a matrix-free Frank-Wolfe (MF-FW) algorithm for fast execution in dense topologies, and a low-rank primal-dual interior-point method (LR-PDIPM) that bypasses dense matrix inversions via the Sherman-Morrison formula for high-precision solutions. At the physical layer, we design a parallel dual ascent algorithm leveraging a time-domain perspective transformation to solve the resource allocation subproblem to global optimality. The proposed BCD framework is proven to converge to an {\epsilon}-neighborhood of a stationary point. Through comprehensive experiments, the proposed BCD framework establishes its superiority in achieving the optimal delay-energy trade-off. Specifically, the LR-PDIPM variant achieves a maximum 9.14-fold reduction in total energy consumption and up to an order of magnitude improvement in energy efficiency, while maintaining a bounded maximum delay gap (up to 3.78-fold) relative to the best baseline. Meanwhile, the warm-start MF-FW variant identifies near-optimal solutions in mere seconds, serving as a highly practical engineering approach.

Recently, topological MV-algebras have been investigated by several mathematicians. In this paper, we mainly show that for every Hausdorff topological MV-algebra $A$, there exists a natural topological isomorphism $i_A:A\rightarrow A^\bullet$ of $A$ onto a closed subalgebra of the pathwise connected, locally pathwise connected topological MV-algebra $A^\bullet$. Furthermore, we show that there is an extension to a bounded continuous function on the MV-algebra $A^\bullet$ for each continuous real-valued bounded function on a topological MV-algebra $A$. Finally, we prove that if $\varphi:A_1\rightarrow A_2$ is a continuous homomorphism of topological MV-algebras, then $\varphi$ admits a natural extension to a continuous homomorphism $\varphi^\bullet:A_1^\bullet\rightarrow A_2^\bullet$; in addition, if $\varphi$ is open and onto, then so is $\varphi^\bullet$.

In this paper, we revisit the axisymmetric Nirenberg problem for certain prescribed scalar curvature functions whose critical points include the north and south poles and whose order of flatness at both points is $n-2$. It has been known for some time that the solution set is compact when the parameter points $(K_1,K_2,a_1,a_2)\in\mathbb{R}^4$, determined by the first two leading expansions of the prescribed scalar curvature functions at the poles, stay away from a critical three-dimensional hypersurface $\Sigma^0$, and that loss of compactness occurs near $\Sigma^0$. We construct, in dimensions $n \geq 4$, two-bump axisymmetric solutions for parameter points sufficiently close to $\Sigma^0$ from one side, where the relevant side is determined by the next-order flatness terms. This gives a one-sided refinement of the previously known blow-up phenomenon near the critical hypersurface $\Sigma^0$: two-bump blow-up occurs from this side, whereas the solution set remains compact on the opposite side.

We answer two open questions about weakly o-minimal fields posed by Macpherson, Marker, and Steinhorn: whether weakly o-minimal fields have the exchange property and whether they have generic differentiability. We construct an ordered field $(K,+,\cdot,\le)$ and a function $f : K \to K$ such that the expansion $(K,+,\cdot,\le,f)$ has a weakly o-minimal complete theory but $f$ is nowhere differentiable. In an appendix, we prove that algebraic closure has the exchange property in any weakly o-minimal theory of ordered fields.

We propose a lowest-order \(H(\operatorname{div})\)-conforming mixed extended virtual element method for elliptic interface problems on interface-unfitted polygonal meshes. The flux and pressure are approximated by subdomain-wise extended \(H(\operatorname{div})\)-VEM spaces and by piecewise constants, respectively. On cut elements, the computable polynomial projection is defined on the whole background element and then restricted to the two subdomains. Compared with BDM-type polynomial spaces, the mixed VEM space contains a non-polynomial component, which gives rise to additional consistency terms on cut elements. To control these terms, we use an enhanced kernel stabilization on cut elements and an interface normal-flux average in the mixed coupling. A corrected interface-flux penalty and a local divergence ghost penalty are added to obtain cut-position-independent stability without using a volume div-div augmentation. We prove continuity, a discrete inf-sup condition, and an optimal first-order error estimate in a mesh-dependent norm. The constants are independent of the mesh size and of the position of the interface relative to the background mesh, but may depend on the coefficient contrast.

We study the $t$-uniform hypergraphicality problem under a compressed representation of the degree sequence. Instead of listing all vertex degrees explicitly, the input consists of pairs $$ (\delta_1,n_1),\dots,(\delta_k,n_k), $$ meaning that exactly $n_i$ vertices have degree $\delta_i$. Thus the parameter $k$ denotes the number of distinct degrees. Although deciding $t$-hypergraphicality is NP-complete for every fixed $t>2$, we prove that the problem is fixed-parameter tractable parameterized by $(k,t)$. Our result shows that tractability extends substantially beyond previously known bounded-range regimes: even degree sequences with large overall degree spread can be handled efficiently when the number of distinct degrees is bounded. Our approach decomposes hyperedges according to their types with respect to the degree classes, yielding a bounded-dimension spectrum representation. Using balancing hinge-flips, we show that every feasible spectrum can be transformed into a realization of the prescribed degree sequence. This leads to an integer programming feasibility formulation with $$ \binom{t+k-1}{k-1} $$ variables. Applying Lenstra's theorem yields an FPT algorithm running in time $$ f(k,t)\cdot \mathrm{poly}(L), $$ where $L$ denotes the encoding length of the compressed input.

In this article, a rigorous well-posedness result is established for a moist compressible primitive-equation system with phase changes. The model couples the compressible primitive equations to a bulk microphysics model for vapor water, cloud water, and rain water, including evaporation, condensation, auto-conversion, collection, and sedimentation effects. Global strong well-posedness is proved for initial data sufficiently close to constant equilibria.

The main ingredient of this paper is an edge-local variable-degree projection on a triangle that is uniformly stable in both L2 and H1. We use this two-dimensional operator in two tetrahedral constructions. First, on a reference tetrahedron, we build an H1-stable projection from a high order polynomial space onto a variable-degree space whose degrees are prescribed independently on edges, faces, and in the volume. Since the tetrahedral projection is local and trace-compatible, it also gives an h- and p-uniform stable decomposition, in the weighted energy norm, for conforming hp spaces, and hence a uniform additive Schwarz preconditioner for the conforming Laplace operator. Second, on a uniformly regular mapped tetrahedral mesh with elementwise variable polynomial degrees, the same triangular projection gives the finite-layer edge truncation needed in a p-uniform stable DG-to-CG decomposition for the symmetric IPDG norm. The DG-to-CG decomposition, combined with the conforming splitting, gives the IPDG preconditioner. The constants depend only on reference shapes, the local degree-spread bound within each tetrahedron, the neighbor-degree bound across mesh faces, uniform map-regularity, patch cardinalities, and the coefficient path constants; they are independent of h, of the local polynomial degrees, and of the coefficient contrast.

Let $f(n,k)$ be the minimum size of a collection of bicliques such that (i) every edge of the complete graph $K_n$ is covered by at least one and at most $k$ bicliques in the collection, and (ii) for each edge $\{u,v\}$, the number of bicliques in which $u$ appears in the first class and $v$ in the second class differs by at most one from the number of bicliques in which $u$ appears in the second class and $v$ in the first class. For $k=1$, $f(n,k)$ reduces to the biclique partition number of $K_n$, and the Graham-Pollak theorem gives $f(n,1)=n-1$. For $k=2$, $f(n,k)$ is the ordered biclique partition number of $K_n$, for which it is known that $c_1 n^{1/2} \le f(n,2) \le c_2 n^{1/2+o(1)}$ for some positive constants $c_1$ and $c_2$. In this note, we give almost tight bounds for $f(n,k)$ for fixed $k \ge 2$: \[ (1+o(1))c_1(k)\cdot n^{\frac{1}{\lceil k/2\rceil+1}} \le f(n,k) \le (1+o(1))c_2(k)\cdot n^{\frac{1}{\lfloor k/2\rfloor+1}+o(1)}, \] where $c_1(k)$ and $c_2(k)$ are positive constants.

Let $G$ be a graph. The size and the signless Laplacian spectral radius of $G$ are denoted by $e(G)$ and $q(G)$, respectively. A spanning subgraph $F$ of $G$ is called an $H_b$-factor of $G$ if $d_F(v)\in\{1,3,5,\ldots,b-1,b\}$ for every $v\in V(G)$, where $b\geq2$ is an even integer. Lu and Wang obtained a sufficient condition according to the number of odd components in $G-S$ for a connected graph $G$ of even order to have an $H_b$-factor, where $S$ is a subset of $V(G)$ [H. Lu, D. Wang, On Cui-Kano's characterization problem on graph factors, J. Graph Theory 74 (2013) 335–343]. In this paper, motivated by Lu and Wang's above result, we establish a lower bound for the size in an $n$-vertex connected graph $G$ with given minimum degree to guarantee that $G$ has an $H_b$-factor. Further, we show a lower bound for the signless Laplacian spectral radius in an $n$-vertex 2-connected graph $G$ with given minimum degree to ensure that $G$ has an $H_b$-factor.

This manuscript studies the problem of independence testing between two high-dimensional time series without assuming weak stationarity, that is, allowing their autocovariances to vary over time. To this end, we propose a bimodal weighted-average test statistic that removes the bias induced by temporal dependence under the null hypothesis, thereby avoiding the need to whiten the time series prior to hypothesis testing – a procedure that is challenging in high-dimensional and nonstationary settings. To facilitate statistical inference, we develop a dependent wild bootstrap procedure. On the theoretical side, we derive a concentration inequality for quadratic forms of time series data stemming from a class of high-dimensional, nonlinear, and nonstationary processes. This result enables us to derive the asymptotic null distribution of the proposed test statistic and to establish the validity of the bootstrap algorithm. Numerical results show that the proposed test attains desired size and good power performance even when the dimension exceeds the sample size or when the data-generating process exhibits time-varying autocovariances. In contrast, tests based on whitening time series fail to maintain correct size in the presence of unstable autocovariance structures. Since nonstationary autocovariances commonly arise in real-life time series data, our work offers a robust procedure for independence testing.

This paper investigates the nonexistence of solutions to semilinear parabolic and hyperbolic inequalities with positive potentials on metric graphs, including both nonnegative solutions and sign-changing solutions. The Laplacian under consideration is of a nonstandard type, incorporating contributions from both the vertices and edges of the metric graph. We construct a new pseudo-metric and introduce suitable space-time test functions of either coupled or separated type. Under suitable weighted space-time volume growth conditions on the potential, we establish nonexistence results for very weak solutions. More precisely, we show that all such solutions to the inequality must be identically zero.

Within a three-body system comprised of two celestial bodies and a spacecraft, the dynamical environment near a smaller primary is significantly perturbed, motivating a balance between global insight and model fidelity. While averaged dynamics offer an integrable model to classify solution landscapes, they inherently lack the accuracy of the unaveraged dynamics, such as the Hill Restricted Three-Body Problem and Circular Restricted Three-Body Problem. This work establishes a systematic bridge between the averaged and unaveraged regimes by explicitly linking averaged equilibria to symmetric periodic orbits in the unaveraged three-body systems. A unified frequency framework is introduced to characterize the mapping of invariant tori across the dynamical models. Leveraging the parity of the resonance ratio, an initialization scheme is developed to identify admissible apse configurations, enabling the a priori prediction of solution multiplicity and symmetry types. Furthermore, the global evolution of families derived from averaged equilibria is traced via bifurcation and frequency analysis. These findings are synthesized into archetypical bifurcation diagrams, providing a comprehensive atlas of the symmetric periodic orbit web within the HR3BP and CR3BP. The resulting framework not only clarifies the topological origins of complex periodic orbit families but also offers a versatile tool for trajectory design in cislunar and multi-body environments.

Undirected graphical models provide a fundamental framework for representing conditional independence structures among high-dimensional random variables. While undirected graphical model selection has become a central problem in high-dimensional statistics, most existing methods are restricted to parametric settings. In this paper, we develop a nonparametric approach to undirected graphical model selection based on diffusion models. Recent work has shown that diffusion models can adapt to the unknown graph structure of the underlying distribution, yet utilizing these models for explicit graph estimation remains unexplored. To bridge this gap, we introduce a novel diffusion-based method for nonparametric undirected graphical model selection. We establish the model selection consistency of the proposed method and demonstrate its empirical performance through extensive simulations and two real data analyses.

This letter introduces a novel, full-wave, physics-compliant stochastic dyadic Green's function (SDGF) framework for modeling electromagnetic (EM) multiple-input-multiple-output (MIMO) channels under wavenumber uncertainty. Unlike conventional phenomenological fading models, the proposed approach provides what appear to be the simplest exact random field models of electromagnetic line-of-sight (LoS) propagation that are also exact solutions of Maxwell's equations. Hence, we dub them Maxwellian random field theoretic models. These physically consistent stochastic models, including an analytically tractable wavenumber Gaussian model and a more general stochastic plane wave (SPW) model, serve as fundamental baseline models for stochastic LoS channel characterization. By preserving the vectorial structure of Maxwell's equations and the dispersion relation, the framework naturally incorporates both propagating and evanescent modes. Our analysis of ergodic capacity and degrees of freedom (DoF) reveals that the key results of the complex SPW model can be reproduced by the simpler Gaussian model with limited variance. Furthermore, we provide examples using 2D continuous MIMO systems, illustrating how the model's Maxwell-consistent stochasticity explains observed increases in channel capacity and DoF over the deterministic MIMO capacity baseline. These idealized Maxwellian random field theoretic models offer a physically grounded reference point for understanding fundamental limits in stochastic LoS propagation environments.

We plug two gaps in the constructive proof of Theorem 1 (respectively, Theorem 2) in <cite>dsb</cite>, showing that the property of C-connectedness (respectively, O-connectedness) of a subset S of R is equivalent to S containing the interval [a,b] (respectively, (a,b)) whenever a and b are in S and a < b.

Let $L_\mu(G)=A(G)-\mu D(G)$ be the generalized $\mu$-adjacency matrix of a finite graph $G$. Fan, Xing, Zhang, and Wang constructed pairs of non-degree-similar trees for which the Smith normal forms of the matrices $tI-L_\mu(G)$ over $\mathbb{Q}(\mu)[t]$ coincide, and conjectured that their construction remains valid when the attached rooted path is replaced by an arbitrary rooted tree. We prove this conjecture as a consequence of a more general coalescence theorem: if a finite graph $H$ has two vertices $u$ and $v$ that are cospectral for $L_\mu(H)$, then, for every finite rooted graph $R$ with root $r$, the matrices \[ tI-L_\mu(R(r)\odot H(u)) \quad\text{and}\quad tI-L_\mu(R(r)\odot H(v)) \] have the same Smith normal form over $\mathbb{Q}(\mu)[t]$, where $\odot$ denotes coalescence of rooted graphs. The proof uses an orthogonal intertwiner over a real closed extension field.

In this paper, we propose an operator learning method based on the multiscale Fourier neural operator (MscaleFNO) for inverse medium problems of Helmholtz equations. The MscaleFNO provides a neural surrogate model with reduced spectral bias for the Helmholtz equations, mapping highly oscillatory medium profiles to scattered wavefields. A plug-and-play inversion using elucidated diffusion model is introduced to regularize the inverse solver based on least squares of data misfits. Numerical results for partial aperture inversion of oscillatory two-dimensional media demonstrate the advantage and effectiveness of MscaleFNO for accurate reconstruction of highly oscillatory medium properties.

Let $s$ denote West's stack-sorting map. In 2020, Defant characterized and enumerated the set $s^{n-m}(S_n)$ for $n \geq 2m-3$. While $|s^{n-m}(S_n)| = B_m$ when $n \geq 2m-2$, where $B_m$ denotes the $m$th Bell number, there are additional permutations when $n = 2m-3$. In this paper, we explore the more complex $n = 2m-4$ case, with several forms of additional permutations. We characterize $s^{m-4}(S_{2m-4})$ and find that its size is \[B_m + \frac{m^2 + 7m - 28}{2}\] for $m \geq 5$. This answers Defant's question about the $2m-4$ case. Furthermore, we find some differences in the behavior of the $2m-5$ case compared to the $2m-3$ and $2m-4$ cases.

Sustainable resource management requires harvesting strategies that account for environmental variability and ecological uncertainty. This study investigates optimal harvesting of renewable biological resources within a stochastic framework, where population dynamics are influenced by random environmental fluctuations and modeled using stochastic differential equations. Two complementary approaches are employed: the Hamilton-Jacobi-Bellman (HJB) equation and the Feynman-Kac representation. The HJB framework provides a dynamic optimization rule and characterizes the value function through a nonlinear partial differential equation, while the Feynman-Kac approach offers a probabilistic interpretation of expected returns. A comparative analysis demonstrates the theoretical consistency and practical relevance of both methods for designing economically efficient and ecologically sustainable harvesting policies under uncertainty.

It is a long-standing problem to prove that the number of distinct topological types of Calabi-Yau threefolds is finite. A related proposition, Reid's fantasy, conjectures that all Calabi-Yau threefolds are connected in a single moduli space through extremal transitions. Finiteness of topological types has been proven for the class of elliptic and genus one fibered Calabi-Yau threefolds, which recently have been shown to constitute the vast majority of known Calabi-Yau threefolds; the moduli space of elliptic CY3's is connected. In this letter, we demonstrate that all non-fibered Calabi-Yau threefolds in two of the largest known classes (toric hypersurfaces and complete intersections in products of projective spaces) are connected to fibered Calabi-Yau threefolds through a simple class of geometric transitions involving the shrinking of a single divisor from a fibered geometry. This suggests that non-fibered Calabi-Yau threefolds are rare special cases that are reached by simplifying fibered Calabi-Yau threefolds, and points to a natural path towards proving finiteness and Reid's fantasy for Calabi-Yau threefolds.

We consider nonlinear semidefinite programming problems with complementarity constraints (SDCMPCC), a class of highly degenerate problems where classical optimality conditions typically fail. In this context, weak stationarity conditions have been developed to address their degeneracy. While these notions are well understood, their algorithmic implications remain largely unexplored in semidefinite complementarity settings. We introduce a reformulation based on the spectral decomposition of the complementarity structure, which preserves local solutions and enables a tractable analysis. Within this framework, we analyze an augmented Lagrangian method for SDCMPCC and prove that under a suitable extension of Robinson's condition, every accumulation point of the generated sequence is W-stationary, or C-stationary under a stricter condition for solving the subproblems.

We characterize the equality case in the Brunn-Minkowski inequality for the Monge-Ampère eigenvalue of general bounded convex domains. This together with the author's previous work completely resolves a question raised by Salani (A Brunn-Minkowski inequality for the Monge-Ampère eigenvalue. Adv. Math. 194 (2005)).

The $abc$-conjecture (Masser and Oesterle) has remained open for decades. By measuring $abc$-triples using a particular quality metric, the conjecture may be framed as seeking the asymptotic distribution of triples of sufficient quality. We create new classes of quality metrics to develop variants on the $abc$-conjecture, with each metric based upon the doubly geometric mean of the prime factors of triples. We investigate the behavior of the resulting class of quality metrics; by determining families of triples that yield high quality, we establish several asymptotic results that are analogous to the $abc$-conjecture for our metrics. We also develop sharp phase transitions for the behavior of families of such quality metrics within specified parametrizations for smoothness of primes in $abc$-triples, using heuristics from the Szpiro ratio for associated Frey curves. Finally, we implement algorithms to determine triples with high qualities with sub-linear runtime, an asymptotic speedup over naïve approaches. Our analysis offers robust variations of, and connections to, the $abc$-conjecture that offer independent questions of analytical interest.

For a digraph $D$, let $\alpha_2(D)$ be the largest size of a vertex set no two of whose vertices lie in a common directed $2$-cycle. Let $f_2(a)$ be the least integer $K$ such that every $K$-connected digraph $D$ with $\alpha_2(D)\le a$ has a Hamilton cycle. Jackson proved in 1987 that $f_2(a)\le2^a(a+2)!$, whereas a conjecture of Jackson and Ordaz predicts $f_2(a)\le a+1$. We prove the quadratic bound $f_2(a)\le12000a^2$. We also prove that $\kappa(H)\ge6000(\alpha(H)+r)$ guarantees vertex-disjoint paths joining any prescribed $r$ pairs of distinct vertices and covering $V(H)$.

We develop an explicit PBW normal ordering theory for the $26$ double extension regular algebras of type $(14641)$ in the Zhang-Zhang classification. With respect to the order $x_1\prec x_2\prec y_1\prec y_2$, we obtain closed two-letter formulas for the internal relations and recursive coefficient systems for mixed words, products of PBW monomials, powers of normal blocks, and noncommutative multinomial expressions. The internal coefficients are mostly quantum or skew-commutative, while the Jordan families produce Lah-Whitney, hence Stirling-type, triangular arrays. The symbolic reductions are supported by a SageMath implementation included as an ancillary file.

Prime numbers are traditionally studied through numerical, probabilistic, and analytic frameworks. In this work, we introduce the concept of a prime event language, in which arithmetic phenomena are represented as symbolic event sequences and analyzed using tools from information theory and stochastic processes. Using all primes up to $N = 5 \times 10^9$ (234,954,223 primes), we construct event languages based on twin-prime occurrences and record prime-gap events. We investigate their statistical properties through finite-order Markov models, train/test validation, mutual-information analysis, and information-horizon measurements. For the Twin Prime Event Language, first-order Markov modeling reduces test-set cross entropy from 0.325350 bits to 0.319949 bits, corresponding to an information gain of approximately 0.0054 bits. This gain survives out-of-sample validation and therefore reflects genuine statistical structure rather than overfitting. Mutual-information analysis independently confirms the Markov results and shows that measurable dependence is concentrated almost entirely at lag 1. The mutual information decreases from approximately $5.96 \times 10^{-3}$ bits at lag 1 to approximately $5.07 \times 10^{-7}$ bits at lag 2 (approximately 11,700-fold reduction), representing a reduction of more than four orders of magnitude. Beyond lag 2, residual information fluctuates near the statistical noise floor. These results indicate that prime event languages are neither perfectly memoryless nor strongly predictable. Instead, they exhibit weak but reproducible short-range statistical structure characterized by first-order dependence and an effective information horizon of approximately one event.

We introduce the notion of an “initial condition” for a module over a commutative Noetherian local ring, allowing for a recursive construction of its “solution modules”. If the given module has zero-dimensional support, such as the residue field of the local ring and those encountered in residual complexes, we demonstrate that the solution module is an injective hull of the given module. The construction of the solution module for finitely generated given module is explicit and computable, devoid of the need for Zorn's lemma.

In this paper, we mainly focus on pointwise, arbitrarily high-order interior estimates for the mixed local-nonlocal elliptic equation \begin{equation*} (-\Delta)^{s}u(x)-\Delta u(x)=f(x),\quad x\in B_r(0) \end{equation*} with $0<s<1$. The main challenges in this setting are the absence of an explicit Green function and the ineffectiveness of standard bootstrap arguments. Our approach overcomes these difficulties via the Campanato iteration method, which inductively constructs polynomials approximating the solution. Using the fact that all functions are locally $s$-harmonic up to a small error significantly reduces the computational complexity when studying the regularity of the fractional Laplacian acting on these polynomials. Finally, the regularity estimates are obtained from the analysis of a manageable recursive inequality system.

Muon replaces a matrix gradient $G=U\Sigma V^\top$ by its polar factor $UV^\top$. This keeps the singular directions selected by the gradient, but makes the update spectrum flat. We study the optimization bias created by this operation. Under explicit alignment assumptions, we prove that the polar update is the one-step entropy-maximizing choice among bounded updates that use the gradient singular directions and do not adapt to the current weight spectrum. In an underdetermined regression model, we derive exact singular-value dynamics for continuous-time Muon and identify a measurement-dependent condition under which the normalized spectrum moves toward equal nonzero singular values. This geometry also rules out a common low-rank interpretation: at fixed Frobenius norm, Muon's distinguished state has a flat spectrum, whereas nuclear-norm minimization favors spectral concentration. Controlled matrix-sensing experiments separate the effect from simple gradient rescaling, show that norm-matched gradient descent does not reproduce Muon, and recover the predicted flattening trend across broad ablations. In small NanoGPT pretraining, Muon preserves stable rank, has a broad learning-rate plateau, and improves validation loss relative to AdamW; in a matched small-ViT control, the ranking reverses. The resulting picture is regime-dependent: Muon is not universally superior, but its flat-spectrum bias can help when many spectral directions need to remain active.