Papers

We study the temporal error growth of the Strang splitting method for the periodic cubic nonlinear Schrödinger (NLS) equation. The leading error is governed by a forced linearized equation, whose growth depends sharply on dimension and the sign of the nonlinearity. In the 1D defocusing case, we prove a uniform quadratic upper bound $C(1+T^2)\tau^2$ using the global Birkhoff transformation and the resulting degenerate structure of the linearized flow. In the higher-dimensional defocusing case, the exponential growth is constructed using arbitrarily small unstable standing waves. Moreover, to prove the higher-dimensional defocusing upper bound, we use the periodic Strichartz estimates of Killip and Vişan to show that $\int_0^T\|u(t)\|_{L^\infty}^2\,dt\le C(u_0)(1+T)$. This yields an exponential rate independent of the final time, despite possible growth of higher Sobolev norms. The exponential lower bound in the focusing case is obtained from the unstable linearized dynamics around a plane wave, with the Akhmediev breather providing the underlying mechanism. In addition, for 1D focusing case with small initial data, the appliction of the local Birkhoff transformation ensures that the error grows at most quadratically in time. Various numerical experiments confirm the sharp linear, quadratic, and exponential error growth rates.

In this paper, we show that the Cesàro operator on the Hardy space $H^p$, $0 < p < \infty$, is cyclic. Our techniques will involve semigroups.

We show that the formation of the product measure is the only natural way to assign to each tuple of probability measures on measurable spaces a probability measure on the product space. Here, naturality is meant in the sense of category theory and amounts to the condition that the assignment commutes with pushforward along measurable maps. In the standard Borel setting, we also prove an analogous result for relative products over a fixed base probability space. As a corollary, we conclude that the formation of relative products is a lax symmetric monoidal functor.

We study eigenvalues of discrete Schrödinger operators $H=\Delta+V$ on $\mathbb{Z}^2$, where $\Delta$ is the uncentered Laplacian, i.e., the un-normalized adjacency operator of $\mathbb{Z}^2$ and $V$ decays at infinity. By Weyl's theorem, the essential spectrum of $H$ is $[-4,4]$. We determine the sharp decay thresholds for existence of eigenvalues in three distinct spectral regimes. While it is natural to expect different behavior at the spectral edge $\lambda=\pm4$, and the bulk, there is a further distinction between the regular energies $0<|\lambda|<4$ and the interior critical energy $\lambda=0$ stemming from the reducibility of the corresponding Fermi surface in the latter case. For every $0<|\lambda|<4$, we construct potentials $V$ satisfying $|V(n)|\leq C|n|^{-1}$ for which $\lambda$ is an eigenvalue of $\Delta+V$, and prove absence of eigenvalues when $|V(n)|\leq C|n|^{-1-\varepsilon}$ for some $\varepsilon>0$. At $\lambda=0$, the critical power changes and we construct potentials $V$ satisfying $|V(n)|\leq C|n| ^{-2}$ for which $0$ is an eigenvalue of $\Delta+V$, as well as prove absence when $|V(n)|\leq C|n|^{-2-\varepsilon}$ for any $\varepsilon>0$. Finally, at each spectral edge, we show that, for every $K\geq3$, an eigenvalue can be created by potentials supported on exactly $K$ sites, whereas a potential supported on at most two sites cannot create an edge eigenvalue. The proofs combine Green-function expansions and moment cancellation, Hilbert-space-valued iterations, discrete Carleman estimates, and a uniform Green-kernel estimate.

We prove Llarull's scalar curvature rigidity theorem for spheres and scalar-mean rigidity for strictly convex Euclidean domains in all dimensions without the spin assumption.

We introduce the notion of the BKM coercivity constant and show that the positivity of this constant governs the exponential relative entropy decay of the semigroup. Indeed, we prove that the modified log-Sobolev constant is equivalent to the BKM coercivity constant up to a constant depending on the asymptotic conditional expectation. We also discover that the BKM coercivity constants for matrix amplifications are already attained with a qubit auxiliary system, which also coincides with the GNS spectral gap of the generating Lindbladian. As a corollary, we establish a criterion that the complete modified log-Sobolev inequality holds if and only if the GNS gap of the semigroup is strictly positive. All the discussion above applies to quantum Markov semigroups that admit a faithful asymptotic conditional expectation as long-time equilibration, with no detailed balance condition assumed. As an application, we show the finite-dimensional Chen-Kastoryano-Gilyén Gibbs samplers always satisfy the complete modified log-Sobolev inequality.

We study a continuous-time, multiasset trading game around index reconstitution, in which opportunistic traders have heterogeneous beliefs about the future index constitution and interact through transient cross-asset price impact and instantaneous execution costs. Opportunistic traders build inventory based on their individual beliefs about the future index constitution before a public announcement, adjust their positions to the revealed composition, and unwind them against the indexer's prescribed rebalancing trades at the implementation date when the new index composition takes effect. Under a no-price-manipulation condition, we construct a subgame-perfect Nash equilibrium. Its affine feedback policies are computed from a non-standard matrix Riccati equation and linear ODEs whose number and dimensions are independent of population size. Under mean-field scaling, we construct a mean-field equilibrium and obtain quantitative convergence and approximate-Nash bounds. Numerical illustrations show that greater competition among opportunists dampens price swings around implementation and that opportunistic trading lowers the indexer's implementation shortfall unless price resilience is very low. Holding the average belief fixed, greater belief dispersion can increase gross trading volume without changing net aggregate opportunist holdings or price impact.

The Erd\H os–Sós Conjecture was recently proved by GPT-6 Astra, using a very ingenious and surprising argument. In this note, we present a simplified version of this argument in an (arguably) more natural form. We also prove a related conjecture of Addario-Berry, Havet, Linhares Sales, Reed and Thomassé, and give a substantially simplified proof of an analogous result for eulerian digraphs, again proved by GPT-6 Astra, and announced by Mubayi and Verstraete. In addition, we determine the extremal (di)graphs for all three problems.

Elements of the Brauer group $\operatorname{Br}(S)$ of a variety $S$ have geometric incarnations as étale-projective $S$-bundles, yet producing minimalist constructions of such bundles, which often power arithmetic applications, remains a difficult problem. When $S$ is a K3 surface with Picard rank $1$, we use the birational geometry of moduli spaces of twisted sheaves on $S$ to construct geometric realizations of nontrivial elements of $\operatorname{Br}(S)$. We recover many known geometric constructions of Brauer classes on K3 surfaces while providing a common moduli-theoretic framework for them. As a by-product, we give a new proof of the period-index theorem for very general K3 surfaces.

Let $Z$ be any fixed closed connected PL $4$-manifold. We give a polynomial-time many-to-one reduction from the compressed word problem in Thompson's group $F$ to fixed-target recognition of $Z\#_{2}(S^2\times S^2)$. Once a finite triangulation of $Z$ has been fixed, the construction sends a straight-line program $A$ to a closed triangulated PL $4$-manifold $X_{A,Z}$ such that $\operatorname{val}(A)=1$ in $F$ if and only if $X_{A,Z}$ is PL-homeomorphic to $Z\#_2(S^2\times S^2)$. Since the compressed word problem in $F$ is PSPACE-complete, every such recognition problem is PSPACE-hard. In particular, fixed-target recognition of $\#_n(S^2\times S^2)$ is PSPACE-hard for every fixed $n\geq2$.

We show that logconcave probability measures along the Gaussian cooling path have thin-shell stability, generalizing the thin-shell theorem. This result leads to improved complexity for the fundamental problem of sampling an arbitrary logconcave distribution from a cold start. For (near-)isotropic logconcave distributions, the complexity is nearly $n^{2.5}$, improving the previous bound of $n^{2.75}$, and matching the complexity of the abstract Speedy walk.

The existence of a positive radial solution to the problem \begin{align*} -\Delta u + \lambda u = u^p + u^q, \quad u \in H^1(\mathbb{R}^N), ~~ N \ge 2, \end{align*} where $\lambda > 0$ and $1 < q < p < 2^* - 1$, has been known for a long time. For the pure-power nonlinearity, it is well known that this solution is unique. In contrast, the uniqueness problem for the double-power case is substantially more delicate and depends on the dimension and the exponents. It has been known that the uniqueness result is in general not true for the equation with double-power nonlinearities in dimension three and it is a long-standing open question to prove the uniqueness of positive radial solutions throughout the full subcritical range for $N \ge 4$ since the work of H. Berestycki, P.-L. Lions [2] (Arch. Ration. Mech. Anal. 1983). In this paper we provide the uniqueness results for all $\lambda > 0$ and all $1 < q < p < 2^* - 1$ when $N \ge 5$. In dimensions $N = 2,3,4$, with additional conditions on $p$ and $q$, we obtain the uniqueness results for all $\lambda > 0$. Our results are in sharp contrast to those of J. Dávila, M. del Pino and I. Guerra [6] (Proc. London Math. Soc. 2013), where some non-uniqueness results were obtained for $N=3$. At the same time, we give a positive answer and rigorous proof to the implication of their numerical simulation, clarify the open issues left by [6]. Finally, we completely resolve the uniqueness problem of the above equation with $L^2$-mass constraints in the mass-subcritical and mass-critical regimes for all $N \ge 2$.

We study an adapted Ricci flow of connections with metric torsion on surfaces with positive Euler characteristic. We first prove that there do not exist any nontrivial solitons of the flow on the $2$-sphere thus confirming a conjecture of Branding–Kröncke (J. Geom. Anal. 27.3 (2017), arXiv:1606.09121). We give an explicit family of torsion data for which the corresponding global solutions fail to converge on $\mathbb{S}^2$. Nevertheless, we provide several sufficient conditions for the convergence of the flow to a stationary point. We first prove that the normalized adapted Ricci flow always converges on $\mathbb{RP}^2$, which completely answers a question in the paper of Branding and Kröncke. Using this, we deduce that the flow converges on $\mathbb{S}^2$ whenever the initial metric and the torsion one-form are antipodally symmetric. We also prove a Łojasiewicz–Simon gradient inequality for the flow and use it to prove convergence to a stationary point provided the solution is close to an arbitrary stationary point.

Finite support indicates that the dependence is limited, but not how each finite context controls what can be observed. For a data symmetry G \leq Sym(A) and a finitely supported G-set X, we study the finite-dependence profile X^S = {x : Fix_G(S) fixes x}, indexed by finite contexts S \subseteq A. For relations on A^n this yields Boolean algebras B^{(n)}_{G,S} = Inv_n(Fix_G(S)), forming the invariance hierarchy. The hierarchy separates several nominal principles. Its meet law is the relation-algebra form of the Bojanczyk-Klin-Lasota least-support criterion, while level injectivity is fungibility. The independent join law governs composition across unions of contexts; it holds for locally oligomorphic automorphism groups of ultrahomogeneous structures in purely relational languages of arity at most two, but can fail for unary observations over ternary data. Freshness remains valid sort by sort, and its unrestricted some/any form characterizes full equality symmetry among closed groups. With least supports, orbit-finiteness implies finite context levels and a uniform support bound; under local oligomorphicity the converse also holds. Pointwise closure does not change the profile, and standard atomic-site presentations show that suitable ultrahomogeneous structures with the same age have equivalent action topoi, even though their material FSS universes may retain different external cardinal data. Contextual orientability gives a complementary pointed invariant whose threshold is the least support size of choice from unordered pairs. For countable omega-categorical structures with degenerate algebraic closure, the meet law is equivalent to weak elimination of imaginaries.

We solve the long-standing problem of determining the exact denominators of the coefficients of the logarithm $B(x)=x+\sum_{n\geq1}b_nx^{n+1}$, obtained from the universal formal group of complex cobordism by the modulus square construction: $$ b_n=\frac{C_n}{d_n},\qquad d_n=\frac{n+1}{2}\operatorname{lcm}(1,\ldots,2n+2), $$ where $C_n\in\Omega_{\mathrm U}^{-4n}$ is primitive and undecomposable. We prove that $C_n$ belongs to the coefficient ring $\Lambda$ of the universal two-valued law and to the subring $\Lambda_{\mathrm{St}}$ generated by quotient Stong manifolds, and give an explicit integral Stong manifold formula for it. The rational lift of $b_n$ to quaternionic cobordism modulo torsion has exact denominator $2d_n$. Consequently, $C_n$ has no integral quaternionic lift, whereas $2C_n$ does. Every Chern number of $C_n$ is divisible by $d_n$, and $c_{2n}(C_n)=d_n$. We introduce odd genera $\mathrm{Bc}_N$. For $N\geq n$, $\mathrm{Bc}_N$ detects the odd part of $d_n$. Its restriction to the Stong ring is integral exactly for $N\leq3$, while the universal odd genus $\mathrm{Bc}_{\infty}$ takes values there with only odd denominators. For every $n\geq5$, we construct undecomposable classes in $\Lambda^{-4n}$ with equal Ochanine genera and top Chern numbers but distinct $\mathrm{Bc}_3$ values. Finally, the oriented extension of $\mathrm{Bc}_3$ is integral on closed spin manifolds of real dimension below $24$ and on closed string manifolds of real dimension at most $24$. The spin bound is sharp: an Anderson-Brown-Peterson spin $24$-manifold has nonintegral $\mathrm{Bc}_3$ genus.

In 2006, Alexander Soifer conjectured that one cannot fully cover a square of side length $> n$ with $n^2 + O(1)$ unit squares. For small $n \in \{2, 3\}$, in 2009, Janusz Januszewski proved that it is impossible to fully cover a square of side length $> n$ with exactly $n^2 + 1$ unit squares. Recently in 2023, Baek and Lee proved that it is impossible to fully cover an equilateral triangle of side length $> n$ with exactly $n^2 + 1$ unit equilateral triangles whose sides are parallel to it. There were also some progress on a related problem: what is the largest square with side length $S(k)$ that can be fully covered by $k$ unit squares. However, there have been no improvement on the original conjecture. In this work, new tools have been developed that led to the proof of this conjecture for $n = 4$, with potential applications to related problems.

In this paper, we introduce the problem of private information retrieval (PIR) under arbitrary privacy requirements, in a graph-based storage system. This formulation is motivated by the server storage limitations, abundance of data (messages) and heterogeneous data privacy requirements. Under the arbitrary privacy requirement, each message has to be retrieved privately from a pre-specified subset of servers, where the subset always includes the servers storing it. Thus, each server is associated with a privacy set, which pre-specifies the message indices that should be privately retrieved from it. This setting is a generalization of the classical PIR setting, where the required message index needs to be kept private from all servers, i.e., there, the privacy set of each server comprises all message indices. Our setting is also a bridge between the newly formulated local PIR (LPIR) setting and the classical PIR setting, where in the former, the privacy set is exactly the set of stored message indices. In this paper, we derive general lower and upper bounds on the PIR capacity for general graphs, under certain privacy requirements, that capture the essence of both LPIR and classical PIR. Then, we focus on path and cyclic storage graphs under these and more fine-grained settings, for which we derive capacity results for certain cases, and establish lower and upper bounds for others. Their low degree allows for a more in-depth understanding of the new privacy formulation and admits more privacy requirement settings compared to other simple graphs. Finally, we introduce a new graph structure, the pyramid storage graph, to model server storage. Although this graph has never been investigated in the literature in any PIR context, it enjoys a nice symmetric structure for message storage and replication patterns.

Semigroups in which right divisibility is a partial order, that is, $\mathscr R$-trivial semigroups, have been studied by people in algebraic combinatorics in connection with Markov chains (especially left regular bands). Margolis and the author showed that the quiver of the algebra of an $\mathscr R$-trivial monoid is field independent. We prove here that the integral semigroup ring of an $\mathscr R$-trivial semigroup has a bound quiver presentation provided that it is unital. This then gives a uniform quiver presentation over any field via extension of scalars.

An interval edge coloring of a graph is a proper edge coloring by integers such that the colors on the edges incident with any vertex form an interval of integers. Not all graphs are interval colorable; a simple counterexample is $K_3$. The (interval coloring) deficiency of a graph $G$ is the minimum number of pendant edges whose addition to $G$ yields a graph with an interval edge coloring. In this paper, we introduce and study further measures of how far from being interval colorable a graph is. The local deficiency of a graph $G$ is the smallest number of pendant edges that needs to be added at every vertex of $G$ in order to obtain a graph with an interval edge coloring; we can think of the colors of these added edges as ”locally missing” at a vertex. We also study a weaker version of this notion, the weak local deficiency, which informally is the size of a largest set of consecutive integers ”locally missing” at a vertex in a proper edge coloring of $G$ minimizing this size. We compare weak local deficiency, local deficiency, and deficiency, and show that the difference can be arbitrarily large in both cases. Moreover, we give concrete examples of graphs whose weak local deficiency (and thus local deficiency) grows with the number of vertices as well as with the maximum degree. We also prove some constructive results on graphs with small weak local deficiency. In particular, all complete multipartite graphs have weak local deficiency at most $2$, and many complete multipartite graphs have weak local deficiency at most $1$. Moreover, bipartite graphs with maximum degree at most $6$, and Eulerian bipartite graphs with maximum degree at most $8$ both have weak local deficiency at most $1$. We conclude the paper by pointing to several open questions for further research.

We prove an unconditional Cauchy limit for a symmetrically truncated Stieltjes transform of the projected ordinates of the zeros of the Riemann zeta function. A suitable normalization of the logarithmic derivative of $\zeta$ on the critical line has the same limit provided that the sum of the distances to the critical line of the zeros lying to its right, counted with multiplicity up to height $2T$, is $o(T)$; we give an explicit bound on the comparison error. The proof rests on a convergence theorem for the Stieltjes transform of positive stationary point measures whose counting discrepancy satisfies an integrability criterion. This theorem is obtained by comparison with the transforms of periodic point measures, combined with truncation bounds and a transfer theorem. Over finite fields, a classical cotangent identity yields Cauchy limits for the logarithmic derivatives of zeta functions of varieties: the law is exactly Cauchy for nonconstant pure cohomological factors, and Poincaré duality gives explicit errors for full zeta functions, including Cauchy limits for smooth hypersurfaces of increasing degree, uniformly in the base field.

Establishing global curvature estimates for $k$-convex solutions of prescribed curvature equations is a longstanding problem in fully nonlinear partial differential equations and geometric analysis. We develop a new approach that combines fractional-linear transformations with the logarithmic concavity of hyperbolic polynomials. The key step is a coercive estimate for the quadratic forms arising from the third-order terms in the maximum-principle argument. Together with a complementary concavity inequality, this estimate yields global curvature bounds for closed, strictly star-shaped $k$-convex hypersurfaces in $\mathbb{R}^{n+1}$ satisfying $\sigma_k(\kappa)=f(X,\nu)>0$, throughout the range $3\leq k<n<2k$.

In this paper we prove the existence of monotone traveling wave solutions for an age-structured population model with a monotone and bistable nonlinearity. We formulate the problem in terms of a discrete-time recurrence $u_{n+1} = Q[u_n]$, with operator $Q$ encoding the role of the speed into its definition. For each speed, we prove the existence of a traveling wave by the general theory of monotone evolution systems, and from appropriate estimates on the speed, we deduce the existence of a traveling wave of the original problem. Also, the sign of the speed is established under an invasion condition, and the asymptotic behavior of the profile is determined by some Tauberian results.

Let $\mu\subseteq\lambda\subseteq(m^n)$. We characterize the image under the ordinary Robinson–Schensted–Knuth correspondence of matrices supported on the skew Ferrers diagram $\lambda/\mu$. The outer boundary determines an upper bound on the right key of the insertion tableau, while the inner boundary determines a lower bound on its left key; both bounds depend on the keys of the recording tableau. This yields tableau expansions of skew Ferrers Cauchy kernels using the standard basis polynomials of Lascoux and Schützenberger, indexed by intervals in Bruhat order. The proof first treats ordinary Ferrers diagrams. Using the supremum characterization of right keys from earlier work, we follow the $\lambda$-dependent bounds through single RSK insertions. When $\lambda$ has repeated parts, these weak column bounds need not form a semistandard tableau. Strictification determines a set $\operatorname{Comp}(\lambda)$ of admissible weak compositions and, for each $\alpha\in\operatorname{Comp}(\lambda)$, a composition $\alpha^\lambda$. Ordinary RSK then gives a weight-preserving bijective realization of the expansion \[ \prod_{(i,j)\in\lambda}\frac{1}{1-x_i y_j} = \sum_{\alpha\in\operatorname{Comp}(\lambda)} \hat K_\alpha(x)K_{\alpha^\lambda}(y), \] where $\hat K_\alpha$ and $K_\alpha$ denote Demazure atoms and key polynomials, respectively. We also give a direct admissibility criterion and a parking procedure for computing $\alpha^\lambda$. After translating conventions, these agree with the admissibility condition and half-bubble-sort construction of Feigin, Khoroshkin, and Makedonskyi. The staircase and truncated-staircase identities follow as special cases. Finally, we extend the weak-bound construction to an infinite alphabet, where strictification need not exist, and derive the infinite-variable Cauchy identity for the $m$-symmetric Schur functions.

We construct a simple rank-$7$ oriented matroid on $24$ elements with no simplicial tope, disproving the Las Vergnas simplex conjecture from 1980. We then show that the mutation graph on uniform oriented matroids is disconnected for infinitely many values of rank $r$ and ground-set size $n$, disproving the Cordovil–Las Vergnas conjecture from 1988.

Physically interpretable power take-off models support wave-energy development at every stage. For the seawater-pump class, few have been identified on a real machine and validated at the system level. This paper presents a block-modular grey-box model of a short-stroke seawater pump, identified from bench data alone. The pump decomposes into chamber, relief valve, piston seal and tip check valve, each structured by component physics, and the few parameters are identified block by block from terminal pressure, displacement, force and discharged-mass measurements, with a closed-cycle mass balance that decouples the degenerate seal and valve losses and closes to within 1% of the inflow. Over the sinusoidal campaign, the pressure-energised piston seal carries 84% of the lost mass and 97% of the lost energy, and the pump delivers about four-fifths of its input as high-pressure flow. Generalisation is demonstrated rather than assumed. With parameters fixed on ramps and sinusoids, the model reproduces 360 s sea-state records in free run to about 10% of range, captures 80% of several thousand peaks within 5 bar, transfers to a re-sprung valve on cracking strokes once its crack pressure is identified, follows thousand-cycle endurance records that drift below 0.1 bar, and reproduces the force and cycle-mean absorbed power, 0.1 to 5.7 kW, to a 4% median error. A practical-identifiability and uncertainty analysis confirms the terminal data resolve the retained parameters; thirty-seven repeated records set the experimental floor. The result is a compact, physically labelled model for design, energy assessment and model-based control of seawater-pump power take-offs.

For a large class of discrete- and continuous-time models satisfying a linear flow property, we prove convergence to the critical $2d$ stochastic heat flow under diffusive scaling of space and time. Convergence is in the sense of finite-dimensional distributions in the space of measure-valued stochastic flows. The class of discrete models we consider includes random walks in space-time random environments with finite-range jumps, and directed polymer models with finite-range spatial correlations. The class of continuum models we consider includes several distinct types of linear-multiplicative stochastic PDEs whose driving noise is Gaussian with a smooth and compactly supported covariance kernel.

The finite difference (FD) method is commonly used to approximate derivatives of smooth functions, with accuracy typically determined by stencil size and derivative order. However, certain centered stencils exhibit unexpectedly higher accuracy, a phenomenon known as superconvergence, which has been observed in practice but lacks rigorous explanation. We present a mathematical framework for superconvergence in centered FD approximations based on Taylor expansions of the truncation error and the resulting linear system for the FD coefficients. By analyzing symmetry properties of these coefficients and their interaction with the parity of the derivative order, we identify conditions under which higher-order error terms cancel. We show that superconvergence occurs for odd-order derivatives with even centered stencils and for even-order derivatives with odd centered stencils, while no superconvergence occurs for even derivatives with even centered stencils. Numerical experiments in MATLAB confirm the predicted convergence rates.

We establish a stochastic control representation for $L^p$-norms on Wiener space. For every $p\ge1$ and every non-negative universally measurable functional $\varphi$, we show that $$\|\varphi(W)\|_p = \sup_a \mathbb{E}\Big[ e^{-\frac12\int_0^T\|a_t\|^2\,dt} \varphi\Big( W+\sqrt{p-1}\int_0^\cdot a_t\,dt \Big) \Big],$$ where $W$ is a $d$-dimensional Brownian motion and the supremum is taken over all progressively measurable controls $a$ satisfying $\int_0^T\|a_t\|^2\,dt<\infty$ almost surely. This identity can be viewed as a multiplicative $L^p$-analogue of the Boué–Dupuis variational formula. We also provide an extension to strong solutions of stochastic differential equations with path-dependent coefficients. Building on the variational formula, we discuss finite-$p$ estimates for small-noise limits of stochastic differential equations with path-dependent coefficients. These include quantitative Freidlin-Wentzell bounds, a functional Gaussian approximation and concentration estimates, and moderate-deviation bounds. Moreover, we discuss consequences for functionals determined up to a stopping time, obtaining in particular probability bounds that identify the sharp exponential decay rate after optimization over controls. The proof of the variational formula combines probabilistic arguments with convex duality methods, viscosity theory for Hamilton-Jacobi-Bellman equations, and a dynamic programming principle.

We develop a theory of Sobolev, Besov and Triebel–Lizorkin spaces associated with weighted subcoercive operators on any real connected Lie group.

Spectral features can remain optimal under strongly uneven task preferences when they retain the directions most useful to the tasks. In a local-task model, expected probing gain depends on the task prior only through its covariance $\Lambda$: with $B$ recording dependence between views, $k$ spectral features span the leading eigenspace of $BB^\top$, while task-optimal features span that of $B\Lambda B^\top$. We derive alignment-dependent regret bounds, matching worst-case lower bounds for a flat leading spectrum, and bounds using the leading $2k$ directions with a spectral-tail term; a spectral gap makes the bound quadratic in small anisotropy. We also bound the imbalance from random preference patterns and from averaging independent tasks with an isotropic population covariance. With known $\Lambda$, changing one term of the spectral contrastive loss selects task-optimal features; with a labelled task bank, we propose a diagnostic and test correction without retraining. Correction reduces synthetic held-out regret from $0.858$ to $0.003$ with $2000$ tasks when less task-relevant directions dominate, and on CIFAR-100 lowers empirical regret by $0.031$–$0.053$ on held-out fine-label tasks using the same images. With a small task bank, however, correction can hurt.