Papers

A finite set of unit vectors is $k$-irreducible if every signed sum of between two and $k$ distinct elements has norm greater than one. Let $\mathcal{R}_k$ be the maximal asymptotic rate of such sets, and let $\kappa(\alpha)$ be the maximal asymptotic rate of spherical codes with pairwise inner products at most $\alpha$. For $k \ge 2$ we show: \begin{align} \mathcal{R}_k \le \min_{1 \le r \le \lfloor k/2 \rfloor} \frac{1}{r} \, \kappa\!\left(1 - \frac{1}{2r}\right) \, . \end{align} Combining this with standard sphere packing bounds, for large $k$ we obtain an almost-tight asymptotic comparison with the known lower bounds: \begin{align} \left(\tfrac{1}{2}-o(1)\right) \, \frac{\log_2 k}{k} \le \mathcal{R}_k \le (1 + o(1)) \, \frac{\log_2 k}{k} \, . \end{align}

Let $X$ be an affinoid rigid analytic space over a $p$-adic field, equipped with a suitable étale map to a unit polydisk. We provide a formalism of imperfect relative period rings over $X$, which lie inside the corresponding perfect relative period rings constructed by Kedlaya–Liu. Then we establish a relative version of the Fontaine–Cherbonnier–Colmez equivalence between $p$-adic local systems on $X$ and étale $(\varphi, \Gamma)$-modules over such imperfect relative period rings, generalizing previous works of Andreatta–Brinon and others. Using this equivalence, we construct $p$-adic differential equations attached to de Rham local systems, carrying both geometric and arithmetic differential operators. This generalizes the work of Berger to the relative geometric setting. Along the way, we study a relative Fontaine–Sen theory on the decompletion of $\Gamma$-modules over the relative $\mathbf{B}_{\mathrm{dR}}^+$-period rings.

Digraphs with constant cut balance admit nearly linear directed cut sparsifiers. This condition requires the total arc weights in the two directions of every cut to be within a constant factor of each other. We ask whether this condition also permits nearly linear spectral sparsification with respect to the energy of the nonlinear directed Laplacian. For a weighted digraph $G=(V,E,w)$, let \[ Q_G^+(x)=\sum_{(u,v)\in E}w_{uv}(x_u-x_v)_+^2, \qquad (t)_+:=\max\{t,0\}. \] This energy agrees with the outgoing-cut function on binary vectors. A spectral sparsifier is a nonnegatively reweighted subgraph that preserves $Q_G^+(x)$ within a factor of $1\pm\varepsilon$ simultaneously for all $x\in\mathbb R^V$. We show that cut balance alone does not yield nearly linear spectral sparsifiers: for constant error, the worst-case support size for simple unweighted Eulerian digraphs is $\widetilde\Theta(n^{3/2})$, although Eulerian digraphs are perfectly cut-balanced and admit nearly linear directed cut sparsifiers. In contrast, we prove that every $n$-vertex tournament has a spectral sparsifier with $\widetilde O(n/\varepsilon^3)$ arcs, without any assumption on its cut balance. This includes the transitive tournament, whose cut balance is unbounded. Thus perfect balance does not guarantee nearly linear spectral sparsification, while unbounded imbalance does not preclude it. Finally, we use convex duality to show that preserving $Q_G^+$ also preserves, for every feasible demand vector, the optimum quadratic cost of a nonnegative flow. Hence the guarantee contains information beyond directed cut values.

In this paper, we give a complete description of the Hilbert schemes of rational curves up to degree $6$ on a prime Fano threefold $X$ of degree $22$. One of the key ingredients in the geometry of these Hilbert schemes is the geometry of bisecant conics associated with rational curves on $X$. As applications, we describe additional geometric features of these moduli spaces and derive Donaldson–Thomas (DT) type invariants.

We show that the overlaps of two independent replicas drawn from the Gibbs measure of the Curie–Weiss model with random field (CWRF) exhibit sub-Gaussian tails when the inverse temperature $ \beta < 1$ and the random field is drawn from a Gaussian distribution that makes the expected magnetization of the CWRF equal to its expected overlap in the thermodynamic limit. To show this, we prove finite-moment concentration via a rigorous Laplace approximation for a certain random rate function, and “bootstrap” it to a uniform bound for the moment-generating function of the squared overlap deviations using a sharp analysis of the associated rate function.

We prove a pair of complementary small-data stability theorems describing the global future and past structure of classical solutions to the isentropic, spherically symmetric $3D$ compressible Euler equations. We allow any equation of state with positive sound speed, except that our shock-formation results do not apply to the Chaplygin gas. Our first theorem concerns open sets of smooth Cauchy data that are perturbations of trivial data with vanishing velocity and constant positive density. The perturbed data have finite kinetic energy, and the perturbed density is equal to a positive constant plus a signed “asymptotically flat” tail. Our main theorem yields a complete description of the existence and uniqueness of the maximal globally hyperbolic development (MGHD) of the data. The boundary of the MGHD contains hypersurfaces that extend to spatial infinity on which our solutions develop gradient-singularities. This is the first MGHD existence-uniqueness-stability result for shock-forming solutions for any multi-$D$ quasilinear wave system. While prior works have yielded local portions of “candidate” MGHDs, neither the existence nor the uniqueness of an MGHD can be inferred from local considerations. Our second theorem considers analogous data with the opposite sign of the $1/r$ density tail. We prove global existence throughout spacetime, to both the future and past. The solution's asymptotic behavior towards null infinity is not linear, but rather is distorted by a logarithmic bending of the sound cones away from the flat Minkowski ones, leading to logarithmically enhanced dispersion. This is the first small-data global existence result in the entire spacetime for any $3D$ quasilinear wave system that fails to satisfy the null condition and the weak null condition. The main mechanism of stabilization is a global-in-spacetime rarefaction effect, tied to the $1/r$ tail.

For an ordered graph $H$ and an integer $q\geq2$, let $r_{<}(H;q)$ denote the $q$-color ordered Ramsey number of $H$. Conlon, Fox, Lee and Sudakov asked whether, for every $q\geq3$, there is a constant $c_q$ such that $r_{<}(M;q)\leq n^{c_q\log n}$ for every ordered matching $M$ on $n$ vertices. We answer this negatively in a strong form: for every $q\geq2$, there is $c_q>0$ such that almost every perfect matching $M$ on $[n]$ satisfies $$r_{<}(M;q)>2^{c_q(\log n)^q/(\log\log n)^{q-1}}.$$ This matches the general upper bound up to a factor of $(\log\log n)^{q-1}$ in the exponent. We also give two applications. First, we prove a lower bound conjectured by Fox, He and Wigderson for multicolor Ramsey numbers of acyclic digraphs. Second, we strengthen a result of Axenovich, Rollin and Ueckerdt by giving a superquasipolynomial lower bound on the maximum chromatic number of $M$-free ordered graphs, for almost every perfect matching $M$ on $[n]$.

We introduce function tables, an entrywise representation of the coefficient functions that appear in the worker responses of a secure distributed matrix multiplication (SDMM) scheme. We work under the outer-product partition, with $K$ row blocks, $L$ column blocks, and privacy against any $T$ colluding workers, in the general model of linear encoding and linear decoding. In this representation, privacy is a rank condition on the data and mask coefficients, and decodability is linear independence of the desired entries modulo the nuisance space. Degree tables, cyclic-addition tables, and algebraic-geometry constructions are the special cases obtained by restricting the coefficient functions to a structured family; we impose no such restriction, so our converses bind every linear scheme. For $T=1$, we determine the exact optimum over every finite field $\mathbb{F}_q$: it is $KL+K+L$ when $q\geq3$, and $KL+K+L+1$ over $\mathbb{F}_2$, where the identity $z^2=z$ forces one more worker. For arbitrary $T$, we prove $N\geq KL+K+L$ and $N\geq\max\{K,L\}+T$ with no MDS hypothesis on the masks; the first is stronger than the previously known bound $KL+\max\{K,L\}+2T-1$ whenever $\min\{K,L\}\geq2T$. We then reduce field feasibility exactly to MDS existence: a scheme exists over $\mathbb{F}_q$ if and only if an $[\max\{K,L\}+T,T]$ linear MDS code does, and whenever it does, a Cartesian construction attains $N=(K+T)(L+T)$ over that same field. For $T=2$ this makes $q\geq\max\{K,L\}+1$ necessary and sufficient, and we give a projective-line construction with $N=KL+K+L+2$ whenever $KL+K+L$ divides $q-1$; for $K,L\geq2$ it matches the best known worker count while requiring only an element of order $KL+K+L$.

Solving fixed-point problems by reducing residual bounds is one of the central approaches in fixed-point iteration, and, in particular, the Halpern iteration is known to effectively reduce the residual. Recently, a new type of duality theorem has been discovered in the convex optimization and fixed-point problem literature and establish a certain symmetry in the convergence behavior of algorithms that are dual to each other. However, these duality theorems rely on a Lyapunov-style proof structure and mostly focus on Hilbert spaces. In this work, we first establish an M-duality theorem for Mann iterations with Lipschitz operators in general normed spaces, assuming only mild conditions on the coefficients of the iterations. Moreover, leveraging this M-duality, we derive new algorithms that attain a strictly smaller worst-case residual bound than that of the Halpern iteration, which was previously known as the state-of-the-art method in general normed spaces.

We consider the full two-dimensional water-wave equations in finite depth, with and without surface tension. Water waves have been studied mathematically for more than two centuries. Reflection symmetry remains a common assumption in constructions of periodic steady waves, although asymmetric profiles have been observed in experiments and numerical computations. We construct families of periodic stationary solutions whose free surfaces have no axis of reflection, in both the pure-gravity and capillary–gravity cases. To the best of our knowledge, this provides the first rigorous construction of asymmetric periodic pure-gravity waves for the full two-dimensional water-wave problem in finite depth. The proof combines localized vorticity laws with a Lyapunov–Schmidt reduction. In the pure-gravity case, we introduce a small background shear flow to obtain an invertible boundary operator and couple the interior and free-surface approximations at leading order.

We prove global existence of mild solutions to the spatially inhomogeneous Fermi–Dirac BGK equation for arbitrary-size Pauli-admissible initial data with finite mass and kinetic energy, allowing both vacuum and locally saturated zero-temperature states. The construction relies on a moment-compatible regularization of the local equilibrium that preserves the Pauli bound and uniform moment control while remaining consistent up to the saturation boundary. The solutions satisfy mass, momentum, and energy conservation and, under an additional finite spatial second-moment assumption, an entropy-gap H-theorem on the full Pauli interval. Under uniform moment and entropy bounds, we further prove that, after extraction of a subsequence, quantum mild solutions converge strongly in phase space, uniformly on bounded time intervals, to a mild solution of the classical BGK equation.

We consider a finite two-dimensional pinned harmonic chain with an interaction range that diverges with the system size and with conservative rotational noise. Under a joint scaling in which the interaction range diverges microscopically but vanishes macroscopically, we prove that the expected em- pirical energy measures converge to the unique measure-valued weak solution of the heat equation.

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence. We show that a multiplicative function $f$ satisfying \[ f(F_n+F_m)=f(F_n)+f(F_m)\qquad(n,m\geq 1) \] need not be the identity function, even when $f$ takes positive integer values. This answers negatively a question posed by Spiro in 1992. The smallest example presented here is obtained from $F_{31}=557\cdot 2417$. We prove the required divisibility equivalence, formulate an abstract prime-signature construction, and give a practical criterion producing further examples. We also describe the AI-assisted search that led to the construction and provide a reproducible certificate checker.

This paper introduces rlaopt, a PyTorch-based package for large-scale optimization and scientific computing using randomized numerical linear algebra (RandNLA). Despite substantial progress in RandNLA-based algorithms, few implementations combine GPU acceleration with a simple interface for specifying optimization problems. rlaopt addresses this gap by providing GPU-enabled solvers for positive-definite linear systems and convex empirical risk minimization with constraints and regularizers. These solvers use RandNLA to accelerate conjugate gradient (NystromPCG), operator splitting (NysADMM), and stochastic gradient methods (SAPPHIRE). Moreover, rlaopt includes a modeling language that lets users specify problems using natural mathematical syntax. rlaopt automatically checks compatibility with the selected solver and performs the required problem decomposition. The solvers also support differentiation through their iterations, enabling applications such as hyperparameter tuning. Experiments on ridge regression, bounded multinomial logistic regression, and bounded elastic net identify when randomized preconditioning improves performance and demonstrate substantial speedups from GPU execution. The package is open-source under an Apache license, with source code at https://github.com/udellgroup/rlaopt and version 0.1.0 available on PyPI.

In nonconvex optimization problems arising in geometric machine learning, data augmentation is commonly used to promote invariance by averaging empirical losses over transformations of the data. Computing the fully augmented objective, however, requires access to every element of the transformation group $G$, which may be prohibitively expensive when $G$ is large or accessible only through sampling. We study whether full augmentation can instead be approximated using a small, fixed sample of transformations acquired before optimization and reused thereafter. Under suitable regularity conditions, we show that, with probability at least $1-\delta$, gradient descent (GD) on the resulting sparsely augmented objective returns an $\varepsilon$-stationary point of the fully augmented objective using $\mathcal{O}\bigl((\log |G|+\log(1/\delta))/\varepsilon^2\bigr)$ group-transformation-oracle queries. By comparison, standard group stochastic gradient descent (group-SGD), which samples a fresh transformation at every iteration, uses $\mathcal{O}(1/\varepsilon^4)$ transformation queries. Therefore, gradient descent with fixed sparse augmentation requires fewer transformation queries than both GD applied to the fully augmented objective and group-SGD. Our proof techniques, which may be of independent interest, establish a uniform approximation of the full group-averaged gradient field by a random group average using spectral properties of group-induced operators and tools from representation theory.

The log-correlated Gaussian field (LGF) on $\mathbb{R}^d$ ($d \geq 2$) is a centered Gaussian random tempered distribution, defined modulo additive constants, whose covariance kernel is $\log(1/|x-y|)$. In two dimensions, the LGF coincides with the whole-plane Gaussian Free Field (GFF). Because the field is a distribution, studying its pointwise behavior requires regularization via convolution with a mollifier. A point is called $\alpha$-thick if the mollified field at that point grows like $\alpha \log(1/\epsilon)$ as the mollification scale $\epsilon \to 0$. A natural question is whether the set of $\alpha$-thick points depends on the choice of mollifier. We prove that for any two admissible mollifiers $\rho$ and $\sigma$ satisfying some mild conditions, the thick point sets coincide almost surely. The result holds in all dimensions $d \geq 2$, and in the special case $d = 2$ extends to the zero-boundary GFF on any open domain with harmonically non-trivial boundary via the Markov property.

We prove Morita invariance of the Campbell–Lind–Malkiewich–Ponto–Zakharevich restriction-system trace after passage to perfect modules. It therefore defines an intrinsic integral restriction trace for an exact endofunctor of a small idempotent-complete stable $\infty$-category. On $\pi_0$, the $m$-th ghost is the laced trace of the $m$-fold iterate, compatibly with Frobenius. For lattice-graded algebras, the ghost targets have twisted cocenters in degree zero; over the open parameter torus, this applies to the cyclic bimodule of Dinkins–Karpov–Krylov. For finite monomial endomorphisms of toric varieties, we construct motivic restriction classes whose ghosts are sums over cones fixed by the iterates. In one example, two classes have the same first ghost, while their second ghosts differ after rational Betti realization.

When modelling a system of interest with a delay differential equation, a distributed delay may be the appropriate modelling choice when the delayed response occurs over a significant range of times rather than with a single constant delay. In systems with a minimum physical transit or processing time, however, the delay kernel should respect a fixed and positive minimum delay. A shifted Erlang kernel is a convenient choice, because the shifted linear chain trick allows one to replace the convolution associated with this type of kernel by a finite auxiliary-variable representation with one constant delay. As we demonstrate with a case study of the Ghil-Zaliapin-Thompson (GZT) model of the El Niño Southern Oscillation with seasonal forcing and distributed delayed oceanic feedback, this reduction makes it possible to perform a bifurcation analysis with numerical continuation techniques for any width of the Erlang distribution. Specifically, we present the bifurcation and resonance structure of the distributed-delay GZT model in the plane of forcing strength and delay for different widths of the distribution. To ensure a like-for-like comparison and determine the influence of delay distribution, we present these results in terms of the effective delay and, moreover, rescale the feedback strength to account for its attenuation with increasing width of the distribution.

Let $\X$ be a Hausdorff second-countable smooth orbifold without boundary, possibly noncompact and ineffective. Suppose $\dim\X\leq n$ and $|G_x|\leq B$ for integers $n\geq0$ and $B\geq1$, where $G_x$ is the full stabilizer at $x$. We construct a smooth Hermitian bundle of explicit rank $R(n,B)$ whose fibre at $x$ is a positive multiple of $\C[G_x]$. Adams operations cancel the Bott obstructions on the boundary spheres of a locally finite good triangulation, and the connectivity of Stiefel manifolds gives actual bundles representing the resulting virtual classes. Smoothing preserves all stabilizer representations. Unitary frames give $\X\simeq[M/U(R(n,B))]$ for a smooth manifold $M$ with a proper locally free action; $M$ is compact exactly when $|\X|$ is compact. Thus every compact smooth orbifold is presentable.

Let $D$ be a smooth divisor in a smooth projective complex variety $X$. For connected curves of arithmetic genus one with prescribed signed contact orders, we prove that the refined punctured cycle of Battistella–Nabijou–Ranganathan and the negative-contact cycle of Fan–Wu–You agree after pushforward to the common moduli space of stable maps with divisor evaluations. Thus the pushed-forward refined punctured cycle is the constant coefficient of the pushed-forward root-stack virtual class, normalized by one power of the root order for each negative contact. The key step is a comparison for the universal target. After restricting to finite-type open substacks determined by the fixed pair $(X,D)$ and numerical data $\Gamma$, we prove that the positive BNR space maps finitely and with generic degree one onto Crumplin's main component. Using Crumplin's genus-one component description and degree formulas, we identify this component's fundamental cycle with the constant coefficient of the universal orbifold virtual class under comparison of root orders. Refined zero-section pullback recovers the negative contacts, and compatible virtual pullbacks and root-forgetting pushforwards transfer the resulting identity to $(X,D)$.

We prove uniform-in-time propagation of chaos and last-iterate consistency for a target-adapted Stein variational gradient descent (SVGD) flow on compact connected manifolds. The target has a smooth positive density, and the particles start independently from a fixed smooth nonnegative density ratio. The construction uses the Green–Bessel operator $Q_{r,\pi}=A_\pi^{-1}(\mathrm{Id}+A_\pi)^{-r}$ of the reversible target Langevin generator. Sufficient Bessel smoothing gives a scalar kernel with finite diagonal, and a positive matrix lift realizes its potential force as a Stein velocity. Population and empirical flows then dissipate the same finite target discrepancy. Population entropy and the target spectral gap give decay of this discrepancy; a finite-time particle comparison reaches a time after which common-energy monotonicity controls every later time. The resulting expected uniform discrepancy is $O((\log N)^{-1/2})$, with a corresponding logarithmic $W_1$ bound and consistency along every sequence $t_N\to\infty$. We also prove an exact finite-mode approximation theorem with an explicit spatial-resolution error and a feature-factorized particle implementation. For confining Euclidean targets, we establish static kernel and moment results and give a conditional dynamical extension under explicit population-regularity and transport hypotheses.

Low-rank tensor modeling has become an effective tool for hyperspectral anomaly detection. However, existing methods still suffer from high computational cost and limited flexibility in characterizing spatially structured anomalies. To address these issues, this paper proposes a hyperspectral anomaly detection method based on group sparse low-rank tensor factorization with automatic anomaly grouping (GSAA). Specifically, the low tubal rank background is characterized by imposing group sparsity on tensor factors, which provides an efficient alternative to direct tensor rank regularization. For anomaly modeling, a latent grouping map is introduced to build an automatic anomaly grouping penalty, allowing anomaly groups to be adaptively inferred from the data rather than predefined at the pixel level. To further exploit complementary spectral and spatial information, GSAA is applied in both domains, and the resulting detection maps are fused to form a spectral–spatial version of GSAA, termed GSAA-SS. An efficient linearized alternating direction method of multipliers algorithm with convergence guarantee is developed to solve the resulting model. Experimental results on five real hyperspectral datasets demonstrate that the proposed method achieves superior detection performance and competitive computational efficiency compared with several state-of-the-art methods.

We construct explicit Riemannian metrics with positive biorthogonal curvature on $S^2\times T^2$, answering a question of Bettiol. In particular, positive biorthogonal curvature on a closed four-manifold does not force its fundamental group to be virtually free.

The Dinew–Popovici functional is an energy functional for Hermitian symplectic metrics in a fixed Aeppli cohomology class. Its vanishing characterizes the Kähler metrics in that class, providing a variational approach to Kähler geometry. Dinew and Popovici proved that every critical point is Kähler in complex dimension three. We give a negative answer to Erfan Soheil's question about higher dimensions by constructing non-Kähler critical metrics on products of two Kähler surfaces $(S_1,\eta_1)$ and $(S_2,\eta_2)$. These metrics are critical under every variation on the product 4-fold. We characterize the critical product metrics and prove that non-Kähler critical products exist in the Aeppli class $[\eta_1+\eta_2]_A$ precisely when the canonical bundles of the two surface factors are smoothly trivial. As a by-product, we develop a geometric flow driven by the torsion tensor and converging to the fixed Kähler background when this background metric has nonnegative holomorphic bisectional curvature.

In this paper, we study the formation, precise asymptotics, and structural stability of neckpinch singularities in mean curvature flow. Motivated by the static rigidity of cylindrical self-shrinkers established by Colding, Ilmanen and Minicozzi, we first prove a dynamical rigidity result: mean curvature flow of hypersurfaces that are initially graphically close to a generalized cylinder on a sufficiently large scale, subject to a localized quadratic upward bending, inevitably develop a neckpinch singularity in finite time. We also establish sharp asymptotic expansions for the profile functions of these locally evolving graphs. We show that the rescaled graphical radius converges to a specific polynomial profile with quadratic bending, proving that the resulting singularities are nondegenerate. Finally, we establish an openness theorem showing that nondegenerate neckpinches are structurally stable under C^2 perturbations of the initial data. Combined with recent density theorems for the 3-dimensional mean curvature flow by Szekelyhidi, our results confirm that nondegenerate neckpinches constitute a generic and stable phenomenon in 3-dimensional mean curvature flow.

We construct the Bogomol'nyi equations for Abelian gauge–Higgs vortices in which the Maxwell gauge sector is replaced by Kruglov nonlinear electrodynamics, a power-law family that interpolates between Maxwell theory, Born–Infeld electrodynamics, and exponential electrodynamics, characterized by a dimensionless exponent $\sigma$. Using the stressless method, we derive a pair of first-order equations directly from the vanishing of the spatial stress tensor, without assuming the Higgs potential a priori. For generic $\sigma$, the gauge and Higgs sectors are coupled through an implicit algebraic relation. We therefore introduce a constitutive map $\Phi(Y;\sigma)$ and analyze its monotonicity and range to determine the conditions for a smooth admissible Bogomol'nyi branch. For $\sigma>1/2$, the constitutive map is strictly monotonic and unbounded, whereas for $0<\sigma<1/2$ it possesses a finite maximum; the marginal case $\sigma=1/2$ is bounded. These properties yield explicit bounds on the nonlinear parameter $\beta$ for the latter cases. We further obtain closed-form constitutive relations, BPS potentials, and gauge-field equations for six representative values of $\sigma$, spanning linear, quadratic, and cubic algebraic structures. The corresponding vortex profiles are then computed numerically. The resulting BPS string tension is purely topological, $\mu_{\rm BPS}=2\pi n$, independent of both $\sigma$ and $\beta$.

In this paper, we propose a new one-point iterative method for finding simple roots of nonlinear equations by modifying Halley's method. Its correction function is positive and bounded on $\mathbb{R}$ and matches the local expansion of Halley's correction function near a simple root. The method avoids the singularity and sign reversal of Halley's correction function while retaining at least third-order local convergence. We also establish sufficient conditions for global convergence. Numerical experiments show that the proposed method achieves high convergence success rates and favorable iteration counts over a wide range of initial guesses. Further experiments near the singularity of Halley's correction function demonstrate robust convergence behavior of the proposed method. An application to the van der Waals equation illustrates the effectiveness of the proposed method.

In this paper, we use the global invariant manifold and the reduced singular cohomology groups method, which is different from those in the literature, to study the dynamic bifurcation from infinity of the nonautonomous evolution equation in terms of invariant sets by applying the Conley index theory (due to Rybakowski). We first establish a nonautonomous global invariant manifold for the abstract equation, which allows us to reduce the original system to this finite-dimensional manifold. Then, a homotopy between the reduced equation and a product flow is constructed. Finally, by considering the reduced singular cohomology theory of the Conley index, we establish our main theorems on dynamic bifurcations from infinity for this nonautonomous equation. As an example, a nonautonomous parabolic equation on unbounded domains is considered. Some new detailed results on bifurcations from infinity of the parabolic equation under an appropriate Landesman-Lazer type condition are proved, including the existence of a nonautonomous Morse decomposition, and improving the earlier works in the literature.

We study the spectral sign implication for quantum logic, defined by the nonnegative spectral projection of the operator obtained by subtracting the antecedent projection from the consequent projection. The construction agrees with classical material implication on commuting projections and compares arbitrary pairs of projections through the spectral structure of their difference. It satisfies Hardegree's four minimal implicative conditions, his law of contraposition, and a falsity condition. It differs from the standard polynomial implications in that its value need not belong to the ortholattice generated by its arguments. Within a uniform class of Borel constructions for pairs of projections, the operations satisfying entailment, contraposition, and the falsity condition correspond exactly to measurable choices of spectral branch. In the continuous subclass, these three conditions determine the spectral sign implication uniquely. In finite dimensions, the operation is also the largest among the acceptance projections of optimal projective tests for Helstrom discrimination between subspace states with priors proportional to rank. These results connect quantum implication with the operator theory of two projections and binary quantum discrimination.

We construct a set $U\leq_T\emptyset''$ whose bounded finite-one degree, ordered internally by one-one reducibility, consists exactly of a countable dense linear order without endpoints. More precisely, \[ \left( \{[B]_1:B\equiv_{\mathrm{bfo}}U\},\leq_1 \right) \cong (\mathbb Q,\leq). \] This exact realization contrasts with two previous results. In earlier work, a bounded finite-one degree was constructed that contains a copy of $(\mathbb Q,\leq)$, but the same degree also contains an infinite antichain of one-one degrees and, more generally, embedded copies of all countable partial orders; thus the dense chain does not exhaust the degree. In a different direction, $m$-rigidity yields an almost-sure and comeager obstruction: for a measure-$1$ and comeager class of sets, the corresponding bounded finite-one degree contains an infinite antichain of one-one degrees and hence is not linearly ordered. The present construction shows that, despite this typical negative behaviour, exact dense linear order can occur. In particular, it gives an affirmative answer to Open Question~3 of Richter, Stephan, and Zhang. The proof has two main parts. A block homogenization construction produces a noncylindrical set $U$, a weak dyadic tower $(U_e)_{e\in\mathbb N}$, computable reservoirs of both colours, and a base absorption property. An abstract absorption-to-exhaustivity theorem then shows that every member of the bounded finite-one degree of $U$ is one-one equivalent to some finite autojoin $mU_e$, thereby yielding the exhaustive classification above.