Papers

In this paper, we develop a general theory of quasi-Whittaker supermodules over Lie superalgebras induced from an arbitrary ideal. We determine the quasi-Whittaker vectors in universal supermodules, establish an irreducibility criterion, and classify several families of irreducible supermodules. The odd part produces a new irreducibility phenomenon absent from the Lie algebra setting. As applications, we determine all irreducible quasi-Whittaker supermodules over the $N=1$ super Schrödinger algebra and the $N=1$ $\frac{3}{2}$-conformal Galilei superalgebra, and over the complete spectrum-generating superalgebra in a special case.

We study the Cauchy problem for a three-dimensional bipolar Euler–Poisson system in which one fluid is pressureless and undamped, while the other is subject to momentum relaxation. For sufficiently small smooth perturbations of a constant equilibrium, we prove the global existence and uniqueness of smooth solutions under an irrotationality assumption on the initial velocity of the pressureless fluid, together with algebraic time-decay estimates. The main difficulty is that the velocity of the pressureless fluid is dissipated only indirectly through the Poisson coupling, and this mechanism degenerates strongly at high frequencies, leading to a regularity-loss structure. We overcome this difficulty by combining refined Green-function estimates, a low–middle–high frequency decomposition, and high-order nonlinear energy estimates adapted to the asymmetric regularity hierarchy. The result establishes a global small-data theory for this asymmetric regime, in which pressure and damping are simultaneously absent from the same fluid.

We investigate the typical dynamical properties of hypercyclic operators in $\mathcal{L}_M(X)$, the set of all bounded linear operators on $X$ whose norms are at most $M$, when $X=\ell_p$, $1< p<\infty$. We show that, with respect to SOT$^*$, a typical operator $T\in \mathcal{L}_M(X)$ is weakly mixing, is weakly disjoint from a given hypercyclic operator $S$, is not topologically ergodic, and satisfies $(T,T^2,\dotsc,T^k)$ is disjoint hypercyclic for any $k\geq 2$. We also show that the similar typical dynamical properties for the concrete family $\mathcal{M}=\{I+B_w\in \mathcal{L}(X)\colon w\in c_0(\mathbb{Z})\}$, endowed with the norm topology, where $B_w$ is a bilateral weighted backward shift.

We consider the problem of computing rational invariants of nilpotent Lie algebras. We compare two methods that are commonly used for this task: the method of integral curves and the Dixmier map. Given a derivation of a rational function field with polynomial coefficients, we formulate a condition under which the kernel can be recovered from a family of rational integral curves, and we show that triangular derivations satisfy this hypothesis. This yields an explicit description of the kernel as a purely transcendental extension and produces algebraically independent generators. We also show that, in the triangular case, the resulting generators agree with those obtained from the Dixmier map via a local slice. A careful analysis of the generating set obtained from this method leads to an algorithm for computing generators of the rational invariant field of a nilpotent Lie algebra. An implementation of the methods is available in the SageMath system.

We develop a framework for offline full-window branch discovery, optionally followed by online continuation with an ensemble Kalman filter (EnKF). Three mechanisms drive the branch search: adjoint path-kernel (APK) differentiation balances kernel differentiation and correction-stabilized path perturbation, shifting the optimization from exploration to exploitation; an optimized Gaussian initial law broadens the search over initial-state basins; and loss-weighted mixing across independent runs recombines successful path components. We may then select an interior state using a local loss and continue online with an EnKF. In 40-dimensional Lorenz-96 experiments, the mean offline path RMSE of APK is 4.3 times smaller than that of population weak-$\mathrm{4D\text{-}Var}_x$. The resulting APK-EnKF method has a mean online RMSE 64 times smaller than that of ordinary EnKF.

Quasiperiodic order is characterized by irrational frequencies whose rational approximability known as irrationality measure defines distinct arithmetic classes. We establish that this arithmetic classification has direct physical consequences for long-wavelength charge fluctuations. In translation-covariant quasiperiodic systems, the infrared scaling of charge fluctuation is governed by the interplay between the irrationality exponent of irrational frequency and the large-harmonic decay of the hull charge profile: the latter determines the available charge weight, while the former controls how efficiently that weight is transferred to the infrared. Consequently, all algebraic irrational frequencies share the same arithmetic scaling, whereas exceptionally well-approximable transcendental frequencies can exhibit strongly enhanced infrared fluctuation scaling. We further prove that occupied states separated from the Fermi level by a gap stable throughout the hull contribute only an analytic infrared background, leaving the nontrivial scaling to near-Fermi states. Our results extend to general translation-covariant multi-frequency quasiperiodic systems.

We prove that finite-volume locally Hermitian symmetric spaces of noncompact type have nonnegative perverse Euler characteristics. To show this, we obtain a nefness result for the logarithmic cotangent bundle of a smooth toroidal compactification. Combining this with a positivity criterion for Euler characteristics of perverse sheaves, we deduce the nonnegativity result. We further prove that the inequality is strict for perverse sheaves with full support. As applications, we get nonnegativity results for perverse Euler characteristics on various moduli spaces.

In this paper, we study the extensions of KLRW algebras to tensor products of Verma module representations of $\mathfrak{sl}_2$. Our motivation is to construct a theory of Khovanov homology for knot complements in $S^3$ (and which also categorifies the Gukov-Manolescu two-variable series for knot complements), which will be done in the second part of this work. We construct the categorification of R-matrices for Verma modules as functors given by derived tensor products with diagrammatic bimodules and explicitly compute their projective resolutions. We also prove these braiding functors induce an action of the braid group on the relevant categories. Then, we describe how to incorporate strands in finite-dimensional representations of $\mathfrak{sl}_2$, thereby establishing functors that serve as the Khovanov homology on a braid complement. In the case of the unknot, this gives knot homologies in $S^1\times D^2$, which we compare to Annular Khovanov Homology through several examples and show they are very closely related, conjecturing they are of the same dimension. We conclude with a proposal for the categorification of the cups and caps of Verma module colored strands, which we build upon in the next paper.

Minimum-distance bipartite matching on general networks has numerous applications various fields. This paper first focuses on deterministic problems and presents an exact edgewise-separable convex-flow reformulation. By introducing a smooth monotone rearrangement approximation of the edge-wise imbalance profiles, the convex-flow reformulation's can be solved efficiently. If we further conduct a first-order resistance-based approximation of the convex program, a one-step Laplacian-based estimator can be analytically derived in closed forms. The paper also studies random problems where supply and demand points are randomly distributed. We show that the expected optimal matching distance scales with the square root of the number of points if the supply/demand point distributions are identical, or linearly otherwise. In the former case, the optimal flow is proven to be centered, symmetric, and sub-Gaussian. In the latter case, the limiting resistance network characterizes how supply-demand imbalance is redistributed and motivates a fast algorithm that approximate the optimal flow based on the limiting resistance. Numerical experiments show that the proposed estimators closely approximate the exact matching cost while substantially reducing computation time. The proven theoretical properties of the random matching solution are numerically verified by large-scale Monte Carlo simulations.

This paper is devoted to the well-posedness of classical Prandtl equations in a finite order Sobolev space. For a initial data with degenerate critical points and general outflow, we obtain the local-in-time existence and uniqueness of the solution to the Prandtl equations in a Sobolev space, by introducing a new iteration scheme and linear cancelation. This result shows that Oleinik's monotonicity condition is not a necessary condition for the Prandtl equations to be well-posed in Sobolev spaces and provides evidence to demonstrate that zero shear stress does not necessarily lead to boundary layer separation in two-dimensional unsteady boundary layers.

Data assimilation (DA) is, traditionally, a cycling process that relies on time-dependent priors to propagate information from past observations to future cycles. Using denoising diffusion generative modeling for DA is challenging because standard approaches use a fixed training data set, which in turn leads to a static prior that ignores information from past observations. Because past observations are ignored, DA systems with static priors lead to larger posterior errors than cycling DA systems. Incorporating time-dependent priors into generative models, however, requires expensive and frequent retraining. Motivated by linear systems theory - where the dependence of a prediction of a Kalman filter on past observations decays exponentially - we condition diffusion models on short windows of past observations. Specifically, we describe training procedures for two frameworks: a diffusion DA system predicting the current state given a set of past observations, and a diffusion “direct observation prediction” (DOP) system, predicting future observations given a set of past observations. Using a canonical linear system, we show that both systems can achieve the minimal posterior error characteristic of a fully-cycled DA/DOP system, without re-training, provided the time windows are long enough. The linear setup ensures analytical tractability, avoids confounding neural network training errors, and confirms that conditioning on windows of past observations is required for efficient and accurate diffusion-based DA or DOP.

In the Navier–Stokes equations, incompressibility allows rewriting the viscous term in various forms leading to distinct numerical properties and flow descriptions. Furthermore, models accounting for non-Newtonian, thermal or turbulent effects often break the constant-viscosity assumption, thereby producing additional consistency terms. In this context, the present work compares the classical symmetric-gradient diffusion term with more recent variable-viscosity generalizations of the Laplacian and rotational forms. We discuss, analyze and test their differences with respect to implementation, efficiency, numerical stability and outflow boundary conditions. With a focus on time-dependent flows, we consider second-order implicit-explicit (IMEX) temporal discretizations aimed at improving efficiency and numerical stability. Through a rigorous stability analysis, we show how selected explicit treatments can bypass algorithmic nonlinearities without inducing CFL conditions. Our numerical results highlight important differences between the three viscous formulations—especially in the presence of outflow boundaries, for which the generalized Laplacian form proves more suitable in diffusion-dominated regimes. %(as widely known for constant viscosity).

A point set is universal for planar graphs on $n$ vertices if every such graph has a straight-line drawing without crossings whose vertices belong to the set. We construct universal point sets of size $n^{1+o(1)}$, improving the previous quadratic upper bound. Our construction uses the reduction of Bannister, Cheng, Devanny, and Eppstein from universal point sets to superpatterns for $213$-avoiding permutations. We represent these permutations by ordered rooted forests and construct a small family of intervals containing every such forest. The result follows from a straightforward bound on the size of the family of intervals. GPT-6 Astra assisted in developing the construction and proof.

We prove that $M_n(\mathbb{C})$ contains no abelian Kadison–Singer algebra for any $n \geq 3$.

We perform a numerical study of uniform meandric systems via their encoding by pairs of simple random walks and their interpretation as a model of critical percolation on random planar maps. We investigate certain natural box-crossing events and provide numerical evidence for a strong negative association between opposite open crossings and formulate conjectures on their asymptotic behavior. Our second contribution is a simple law-preserving dynamics on meandric systems, by local peak/valley flips in the walk encoding. Our simulations suggest that the percolation structure decorrelates much faster than the underlying random planar map.

For infinitely many sizes we exhibit a cycle of binary units, each taking the exclusive-or of itself and its two clockwise neighbours, that is a complex in every state, no proper subsystem being more integrated, and whose integrated information grows as two raised to a power quadratic in the number of units.

We prove Fuglede's conjecture for finite cyclic groups of square-free order. Specifically, if \(N\) is square-free and \(A\subseteq \mathbb Z_N\) is nonempty with \(|A|=m\), then \(A\) is spectral if and only if it tiles \(\mathbb Z_N\); moreover, these conditions hold precisely when \(m\mid N\) and \(A\) is a complete set of representatives modulo \(m\). This gives a complete structural description of spectral and tiling sets in \(\mathbb Z_N\). The key new ingredient is a prime-coordinate elimination theorem: if \(P\nmid nm\) is an odd prime, then every \(m\)-point spectral pair in \(\mathbb Z_n\times\mathbb Z_P\) projects injectively to a spectral pair in \(\mathbb Z_n\). Its proof combines character-matrix orthogonality, cyclotomic coefficient comparison, and a \(P\)-adic divisibility argument for associated Gram matrices. Iterating this reduction yields the result for arbitrary square-free \(N\), with no restriction on the number or relative sizes of its prime factors.

The least interval diameter needed to realize every attainable cardinality of an $h$-fold sumset of a four-element integer set is $\binom{h+2}{2}+(h\bmod2)$ for $2\le h\le10$ and $d_h=(h-3)(h-1)$ for $h\ge11$. We prove the formula uniformly for $h\ge17$ and use complete finite searches, with a relation-rank coverage argument, for the remaining orders. For the lower bound, we show that every integer four-set with $\binom{h+3}{3}-17$ distinct $h$-fold sums has normalized diameter in $\{d_h,d_h+1,d_h+2\}$ when $h\ge9$, and we construct sets attaining $d_h$. For the upper bound, a relation-lattice index argument restricts larger minimum-diameter representatives to two or three active directions. Two general constructions and four special defects then provide short sets realizing every remaining cardinality.

Recent work of the author investigates certain random processes, valued in abelian p-groups, that naturally arise in the study of Haar random matrices over $\mathbb{Z}_p$. It was found, somewhat surprisingly, that these processes are reversible Markov chains. In this short note, we give a simple alternative derivation of this fact. The new derivation also proves that this phenomenon is not specific to $\mathbb{Z}_p$, but generalizes to Haar random matrices over any profinite local ring.

We prove categorical Torelli theorems for ordinary and weighted hypersurfaces using the singularity category of the completed affine cone. For Fano hypersurfaces we use the Kuznetsov component, whereas for hypersurfaces of general type we use the full category of graded matrix factorizations. When the degree is coprime to the sum of the weights, the internal grading shift can be expressed in terms of the Serre functor and the cohomological shift. Its dg orbit is therefore intrinsic and, after taking the perfect hull, agrees with the dg singularity category of the completed cone. The formal singularity is then recovered by the categorical Mather–Yau theorem. For ordinary homogeneous equations, the resulting formal equivalence induces a projective linear equivalence; in the weighted setting the dual torus action determines the grading of the Tjurina algebra and hence the weighted homogeneous equation. The argument gives a uniform proof for Fano hypersurfaces under the stated numerical assumptions and proves new categorical Torelli theorems for hypersurfaces of general type.

We prove that the Parisi measure of the Sherrington-Kirkpatrick model has connected support at every positive temperature and every positive external field. When the model breaks replica symmetry, its support is an interval contained in $(0,1)$, its restriction to the interior has a smooth density that remains strictly positive up to both endpoints, and both endpoints carry atoms. The proof rests on a strict inequality for moments of spatial derivatives of the Parisi PDE, which both excludes gaps in the support and forces positivity of the density. The same estimates recover the zero-field support description and give a lower bound for the mass of its upper atom.

The solution of the Diophantine equation ay^2=b(x^k-c^k) is studied in different cases of k. Methods are proposed for different cases of k, which are convenient for computer algorithm implementation. The original equation is reduced to lower-degree equations, which can improve the efficiency in searching possible integer solutions. A sufficient condition of the unsolvability of ay^2=b(x^2n-c^2n) is presented.

We study prediction of one unmeasured copy of an unknown finite-dimensional pure quantum state after independently measuring the observed copies with the one-copy Haar POVM. For the resulting fixed separable observation, the Bayes predictive state under quantum relative-entropy loss is the full-rank posterior mean. We evaluate it exactly through permanents of minors of the outcome Gram matrix, prove positivity and normalization of the permanent representation, and establish minimaxity among decision rules based on these fixed outcomes. Using Hilbert–Schmidt projection of the posterior mean and an exactly analyzed linear-inversion competitor, we prove a finite-sample posterior-purity bound and an explicit relative-entropy guarantee with sample count $O((d^2/\epsilon)\log(d/\epsilon))$. Using the exact collective benchmarks proved in a companion paper, we also obtain entropy and posterior-purity comparisons. To connect the fine product data to a regular ray-valued experiment without assuming deterministic finite-sample uniqueness, we draw one random orientation for a finite projective mesh, retain it for the entire sample, and apply a selected finite-alphabet MLE in the resulting reference experiment. A nested recovering mesh uniformly recovers the fine Fisher information. Consequently every sufficiently fine fixed mesh gives a regular finite-alphabet model. Its selected MLE has Haar-averaged inverse-Fisher coefficient $\overline a_k$, and refining the mesh after the fixed-mesh large-sample limit makes $\overline a_k$ approach $d-1$. This yields epsilon-optimal leading relative-entropy and posterior-overlap bounds for the fine posterior and the sharper fixed-dimensional asymptotic scale $(d/\epsilon)\log(d/\epsilon)$. The latter is not asserted as a uniform finite-sample guarantee. No deterministic finite-sample MLE uniqueness or triangular mesh schedule is used.

It has been asked whether a reduced non-McCoy ring can satisfy both the annihilator condition and $(A_n)$ for some $n \geq 2$. In this work, we prove that such rings exist exactly when $n=2$, by constructing a countable example and showing that the annihilator condition together with $(A_3)$ forces a ring to be McCoy.

It was conjectured that every integral domain $D$ satisfies $\dim\operatorname{Int}(D) \leq \dim D[X]$. We disprove this conjecture and, more strongly, classify all pairs for which both dimensions are finite. Writing $m=\dim D[X]$ and $n=\dim\operatorname{Int}(D)$, the pairs that occur are precisely $\{(1,1),(2,2)\}\ \cup\ \{(m,n)\in\mathbb Z_{>0}^{2}:m\geq3,\ n\geq m-1\}$.

A Hermitian completely bounded map $\Phi:\mathscr{A}\to B(\mathcal{H})$ need not have a canonical completely positive analogue of an absolute value. We study this problem through the ordered set $\mathsf{E}(\Phi)$ of completely positive symmetric majorants of $\Phi$, which we call Wittstock envelopes. We show that every Wittstock envelope canonically determines a minimal Wittstock envelope below it. However, minimal envelopes need not be unique. We therefore study the stronger question of when a least completely positive majorant exists. We obtain a criterion, in terms of central supports of spectral projections, for the existence of least majorants below a fixed envelope. In finite dimensions, the Choi representation gives a simple characterization: a least completely positive majorant exists if and only if \(\Phi\) or \(-\Phi\) is completely positive. Finally, we use the envelope-relative modulus to define envelope-relative perspectives for Hermitian completely bounded maps. These constructions extend Okayasu's completely positive map perspectives, reduce to the usual perspective in the completely positive case, and remain dependent in general on the chosen Wittstock envelopes.

In this paper, we establish some new localization sets for Pareto Z-eigenvalues of tensor eigenvalue complementarity problems and discuss the relationships among existing ones. As applications, a sufficient condition for the solution existence of the tensor complementarity problem (TCP) and the tensor generalized eigenvalue complementarity problem (TGEiCP) are presented. In addition, this condition is also used to test the vacuum stability of a general scalar potential

Local anticoncentration controls the conversion of additive output probability estimates to relative estimates in Gaussian boson sampling. We prove a center-uniform quadratic small-ball bound for hafnians of complex Gaussian transpose Gram matrices for $2n$ selected modes, with an explicit finite coefficient that is polynomial when the number of Gaussian rows is at least of order $n^2/\log n$. Letting the Gaussian row count tend to infinity yields anticoncentration for the independent complex symmetric Gaussian ensemble. For the transpose Gram hafnian bound, the proof compresses dependent hafnian cofactors by Fourier interpolation and controls their self-coupled variance through a conditional Wishart identity that preserves the transpose Gram coordinate. Independent perturbations satisfy related bounds, and a positive mixture density shows that the local quadratic power is exact. Together with the uniform hiding theorem in the companion paper, the result gives local anticoncentration for finite Haar interferometers.

We compare return probabilities for random walks stopped at an independent geometric time on Lubotzky–Phillips–Sarnak graphs and on the infinite regular tree, with the McKay measure giving the tree value and the Ihara zeta function describing the additional returns caused by cycles. We estimate this correction when the mean walk length is comparable to the length of the shortest cycle, a regime in which walks long enough to traverse a cycle receive non-negligible weight. Unconditionally, we prove an upper bound with a double logarithmic factor and bounded fixed moments over prime levels after multiplication by the number of vertices. Under the Generalized Riemann Hypothesis for quadratic Dirichlet $L$-functions, the correction is of the same order as the reciprocal of the number of vertices.

We study the problem of counting Italian dominating functions (IDFs) of each possible weight on paths, trees, and cycles. An Italian dominating function on a graph $G=(V,E)$ is a function $f\colon V\to\{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ satisfies $\sum_{u\in N(v)}f(u)\ge 2$; the Italian domination polynomial $D_I(G,x)=\sum_k d_I(G,k)\,x^k$ records the number $d_I(G,k)$ of such functions of weight~$k$. This polynomial is the Italian analogue of the well-studied domination and Roman domination polynomials, and computing its coefficients on a fixed graph class is, in essence, a constrained integer composition problem: passing to the deficit variables $y(v)=2-f(v)$, each coefficient $d_I(G,2n-j)$ counts the number of compositions of $j$ into $n$ parts, each at most~$2$, that additionally satisfy a local adjacency capacity constraint. We make this connection explicit and use it to derive closed-form expressions for $d_I(C_n,2n-j)$, $j=0,\ldots,5$, as trinomial coefficients minus explicitly enumerated forbidden configurations. On the algorithmic side, we give dynamic programming algorithms that compute all coefficients of $D_I$ on path graphs ($O(n^2)$ time, $O(n)$ space), on general trees ($O(n^2)$ time, $O(n)$ space, via a post-order merge with a deficit parameter), and on cycle graphs ($O(n^2)$ time, $O(n)$ space, via boundary conditioning), together with formal correctness proofs and complexity analyses. The known values $\gamma_I(P_n)=\lfloor n/2\rfloor+1$ and $\gamma_I(C_n)=\lceil n/2\rceil$ are recalled with references rather than reproved. All numerical results are verified against brute-force enumeration, and complete tables are provided.