Papers

Let $A$ be a bounded operator on a separable right quaternionic Hilbert space and let $\CA$ be the set of compact operators commuting with $A$. We establish the formula \[ \sigb(A)=\bigcap_{K\in\CA} \sS(A+K), \] a quaternionic analogue of the classical characterization of the complex Browder spectrum. The proof rests on a factorization lemma (Lemma~\ref{lem:fact}), a consequence of the Riesz–Schauder theorem and an ascent/descent argument, and on the characterization $\sigb(A)=\sS(A)\setminus\sd(A)$ in terms of S-eigenvalues of finite type, due to Arzini and Jaatit \cite{AJ26}, for which we give here an independent proof (Lemma~\ref{lem:A}). We also record two consequences of independent interest: the intersection defining $\sigb(A)$ may be restricted, without loss, to finite-rank operators commuting with $A$ (Corollary~\ref{cor:finite}); and, by a separate argument resting on the ascent/descent stability results of \cite{KD24}, the stronger pointwise statement $\sigb(A+K)=\sigb(A)$ holds for a fixed $K\in\CA$ and arbitrary $A\in\B(\V)$ (Theorem~\ref{thm:pointwise}), known in the complex case.

For all positive integers $n,k$, we prove that the layer-refined antichain polynomial of the product poset $[2]\times[n]\times[k]$ is real stable. Jacobi-polynomial interlacing further shows that its diagonal specialization, the ordinary antichain polynomial of the same poset, has only simple, strictly negative zeros. For the special family $[2]\times[m]\times[m+1]$, explicit reciprocal identities give palindromicity; reciprocal pairing of the simple negative zeros then shows that every coefficient in the gamma expansion is strictly positive. Thus we prove Conjecture 4.3 of Ding and Dong and resolve all parts of their Conjecture 4.5, while strengthening its stated gamma-positivity consequence. The enumerative input is an explicit first-crossing reflection for two lattice paths, specialized from work of Krattenthaler and Sulanke.

Let $(X _n )_n$ be a sequence of i.i.d. $R^d$-valued random variables uniformly distributed on a compact subset $M$ of $R^d$ . In this work, we study the asymptotic behavior of maximal packings of this compact subset $M$. Under some assumptions on $M$ (in particular, that it is a smooth submanifold of $R^d$ without boundary), we establish a precise extreme value law (of Gumbel type) for the minimal distances from the sample $(X_1 , \cdots, X_ n)$ to the centers of maximal packings. This result leads to explicit asymptotic confidence bounds for this maximal packing and thus to the support $M$. A distinctive feature of our contribution is the explicit derivation of the scaling sequences in the Gumbel convergence, depending only on the geometry of the support M. These formulas provide interpretable confidence bounds for $M$, which represent a novel complement to previous approaches such as those by Fasy et al. (2014). Our approach bridges geometric and combinatorial probability arguments and relies on analytic tools such as the Lambert $W$ function. In addition, we provide examples of main sample spaces in directional and circular statistics (circle, sphere, torus), along with simulations that illustrate and support the theoretical findings.

We study the least bending energy $m_L$ of a closed embedded wire of prescribed length $L$ confined to the closed unit disk. It is classical that the bending energy of any closed curve in the disk is at least its length, with equality only for multiply covered unit circles. Such a circle is the energy limit of a stack of disjoint embedded circles, but not of a single embedded loop, and the size of the resulting defect $m_L-L$ has remained open. We prove that \[ m_L \;=\; L+c_1\,L^{4/9}+O\bigl(L^{1/3}\bigr)\qquad\text{as }L\to\infty , \] with a sharp constant $c_1=5.223049\ldots$ given in closed form by two one-dimensional model problems. In particular $m_L-L\asymp L^{4/9}$, which improves the previously known upper bound of order $\sqrt{L}$ and provides the first lower bound beyond the trivial one. This paper was produced with substantial assistance from large language models; Section 7 sets out in detail what they contributed and what I verified.

Let $G=C_7\times C_7$, and let $S(G)$ denote the set of integer values of its group determinant. We determine $S(G)$ by classifying the values divisible by $7$; the coprime values are already known. Every nonzero divisible value has valuation at least $10$, and every multiple of $7^{12}$ occurs. At valuations $10$ and $11$, we give necessary and sufficient conditions on the cofactor in terms of two additive invariants of ideals in $\mathbb{Z}[\zeta_7]$. The first condition is a bounded signed sum of prime-ideal invariants. The second requires a prime ideal with nonzero invariant pair whose norm divides the cofactor. Neither cofactor set is a union of congruence classes modulo any positive integer. The least positive divisible value is $43\cdot7^{10}$, and the least positive value of valuation $11$ is $8\cdot7^{11}$. The proof combines integral reconstruction from character values with a calculation of the global-unit image modulo $7$. We conclude by identifying the additional local conditions and realization problems that arise at primes at least $11$.

We develop a Balancing Domain Decomposition by Constraints (BDDC) preconditioner for the interface systems arising from active-set semi-smooth Newton linearizations of elliptic optimal control problems with box-constrained controls, $L^1$-sparsity, and Moreau–Yosida regularized state constraints. The domain decomposition and BDDC construction are formulated directly at the infinite-dimensional level. Eliminating the interior variables yields a Schur complement for the state and adjoint traces assembled from local subdomain operators. We prove a global–local equivalence result, establish well-posedness of the local and partially assembled problems under verifiable conditions on the primal constraints. The BDDC operator admits a two-level additive Schwarz representation with independent local solves and a finite-dimensional coarse solve. We implement the preconditioner using conforming finite elements and test it for distributed and Neumann boundary control. For a fixed ratio of subdomain diameter $H$ to mesh size $h$, the GMRES iteration counts are nearly independent of mesh refinement and grow only moderately as this ratio increases. Enriching the coarse space substantially improves convergence and reduces sensitivity to the control regularization and sparsity weights. For the state-constrained problem, scaling the Moreau–Yosida penalty parameter proportionally to $h^2$ yields nearly constant GMRES iteration counts as the mesh is refined.

In this article, we study the second-order fluctuation of the solutions of one-dimensional polynomial stochastic partial differential equations (SPDEs) of the form \begin{equation*} (\partial_t - \Delta) \Phi_{\varepsilon} = -P(\Phi_{\varepsilon}) + \xi_{\varepsilon}, \end{equation*} where $P$ is a polynomial of degree greater than or equal to $2$, $\xi_\varepsilon$ is the white-noise after being convoluted (in space) by the heat kernel $K_\varepsilon = e^{\varepsilon \Delta}$. More precisely, taking advantage of the local solutions of pointwise well-posedness of limits $\Phi= \lim_{\varepsilon \to 0}\Phi_{\varepsilon}$, we characterise the limit of $\Phi^{err}=\lim_{\varepsilon \to 0} \varepsilon^{-1}(\Phi_{\varepsilon}-\Phi)$ as the solution of a more irregular stochastic partial differential equation. This is performed by applying the Da Prato–Debussche decomposition in the non-linear equation and characterising the limit of each of the terms. We also discuss possible characterisations of second-order fluctuations performed for higher-dimensional SPDEs in the weakly-coupled regime.

Wasserstein barycenters are a fundamental tool for the analysis of distribution-valued data, but their computation and statistical analysis become challenging in high-dimensional and multi-sample settings. The unregularized barycenter problem is computationally demanding and suffers severely from the curse of dimensionality, making regularization essential in many applications. While regularized optimal transport methods are widely used, approaches based on entropic penalization tend to overspread mass and may blur the geometric structure of the solution. Motivated by barycenter estimation, we propose a more general methodology based on sparse regularized multimarginal optimal transport (RMOT). Our formulation creates sparse transport plans and is designed to avoid this overspreading effect. We establish consistency, central limit theorems, and bootstrap validity for the optimal potentials and couplings for the empirical RMOT problem. Specializing to sparse regularized barycenters, our results yield a uniform weak limit and bootstrap consistency. Numerical simulations show that sparsity plays a fundamental role in preserving the structure of the solution and avoiding the overspreading behavior typical of entropic regularization.

A categorial approach to the non-metric geometry in Books V and VI of Euclid's Elements is presented. Specifically, we introduce a diagrammatic syntax that can be overlaid immediately on his diagrams, thus bridging intuitive presentation with fidelity to Euclid's arguments. This syntax makes complicated definitions like V.5, and indeed the arguments throughout books V and VI, including arguments about similar figures, intuitively clear. We show in an appendix that this syntax can be used to solve a puzzle regarding ancient mathematics. Finally, we offer evidence that this approach to Euclidean diagrams is rooted in the Aristotelian tradition itself, and that a similar syntax was utilized, in antiquity, for related questions of numeric and proportions. Thus the syntax is plausibly faithful to Euclid's own thought-world, and not an outside-imposition.

We exhibit a simple unital infinite-dimensional C$^*$-algebra $A$ which is not $K_1$-injective, i.e., the canonical map $ U(A)/U_0(A)\longrightarrow K_1(A)$ is not injective. This phenomenon is well known in the non-simple case. Indeed, a natural example is the generator $u$ of $\pi_4(U(2))\cong\mathbb{Z}/2$, and we use exactly this example as the seed of our construction. The construction proceeds by developing a higher-rank matrix trapping technique in homogeneous algebras generalizing the machinery of Villadsen in higher stable rank simple C$^*$-algebras. A hypothetical homotopy from the homomorphic image of $u$ to the identity in a certain homogeneous algebra is shown, through matrix trapping, to yield a spin bordism from the class of $u$ to that of the identity matrix $\mathrm{1}_2 \in M_2$, and this, in turn, is obstructed by the fact that the bordism class of $u$ is not trivial. We then fit our new arguments into a generalized version of Villadsen's higher stable rank construction to arrive at a simple example.

We give a complete characterization of the weak Lefschetz property (WLP) for monomial complete intersections over a field of positive characteristic. Richard Stanley observed that in characteristic zero WLP, and in fact the strong Lefschetz property (SLP), holds for all degree sequences, as a consequence of the hard Lefschetz theorem, and the same result was explained by Junzo Watanabe using the representation theory of $\mathfrak{sl}_2$. In positive characteristic, many partial results are known, most notably the classification for constant degree sequences due to Brenner–Kaid and Kustin–Vraciu. Our approach is based on a cohomological and representation-theoretic interpretation of WLP. Combined with an analysis of cohomology characters, this leads to a complete numerical criterion for WLP, expressed by simple inequalities involving the $p$-adic digits of the exponents. We also give a new proof of the known classification of SLP using Renaud's algorithm for multiplication in the Green–Han–Monsky ring.

The main aim of this paper is to introduce a Frobenius–Galois expansion of the substructural logic $\mathbf{FL}_{ew}$ and develop its algebraic and categorical semantics. The resulting logic, denoted by $\mathbf{FL}^{\mathbf{FGC}}_{ew}$, is obtained by adjoining a pair of unary connectives forming a Galois connection and satisfying suitable Frobenius-type compatibility conditions. Firstly, we prove that $\mathbf{FL}^{\mathbf{FGC}}_{ew}$ is algebraizable in the sense of Blok and Pigozzi and identify its equivalent algebraic semantics with the variety of Frobenius-adjoint residuated lattices, establishing the conservativity over $\mathbf{FL}_{ew}$ and relating its finite model property to residual finiteness of finitely generated free algebras. Secondly, for the distributive setting, we lift the Kalman construction to the Frobenius-adjoint algebraic framework. More precisely, we establish a categorical equivalence between Frobenius-adjoint residuated distributive lattices and Frobenius-adjoint $c$-differential residuated distributive lattices with the condition $\mathbf{CK}$. This equivalence yields a positive-cone representation of the former structures and, in turn, a logical counterpart of the categorical correspondence. Finally, for the distributive extension $\mathbf{FL}^{\mathbf{FGC},d}_{ew}$, we prove that derivability, validity over Frobenius-adjoint residuated distributive lattices, and validity over the corresponding positive cones determine the same consequence relation, and further show that this correspondence preserves equational and quasi-equational consequence, conservativity, and finite countermodels. These results provide a unified algebraic, categorical, and logical framework for Frobenius–Galois expansions of substructural logics.

In this paper we prove that simple Finsler manifolds are conformally stable. Given a simple Finsler manifold and a class of conformal factors, we characterize the singularity of their induced boundary distance functions which allows us to define an appropiate $H^2$ norm. We then obtain a stability estimate with respect to this Sobolev norm and the $L^2$ norm on the class of the conformal factors. To prove the main theorems, we adapt the classical integration by parts technique employed by Mukhometov \cite{Muhometov} to the Finsler setting.

We introduce Kleisli convolution representations for groups, rings, and algebras. We show that their representation categories are equivalent to the usual ones for groups and finite-dimensional algebras, but for rings only recover modules whose underlying Abelian groups are free of finite rank. We also apply the Kleisli convolution representation to provide a partial answer to the Margolis–Sakurai version of the modular isomorphism problem: If $G$ is a finite $p$-group with $D_3(G)=1$ and $\gcd(m,c_G!)=1$, then $\mathbb F_{p^m}G\cong\mathbb F_{p^m}H$ implies $G\cong H$, where $c_G$ is the number of geometric connected components of $\mathrm{Aut}(\overline{\mathbb F}_pG)$.

We study the oracle complexity of computing a point with small fixed-point residual $\|T(x)-x\| \leq \epsilon$, for a general norm $\|\cdot\|$ and a self-map $T$ of a compact convex set. We study this problem in the setting where $T$ is nonexpansive with respect to the same norm $\|\cdot\|$ and accessed via an unbiased stochastic oracle with bounded variance $\sigma^2$. We provide an algorithm that solves such instances for any norm with a weak Rademacher type $q > 1$, with high probability. The algorithm is based on a recursive anchoring technique. For type-$2$ spaces, such as $\ell_p$-spaces for $p \in [2, \infty]$, our algorithm attains stochastic oracle complexity $\tilde O(\sigma^2 \epsilon^{-3} + \epsilon^{-1})$. We further prove a near-matching lower bound (i.e., matching up to poly-log factors) for such $\ell_{\infty}$-norm instances in high dimensions. Our lower bound holds against any randomized algorithm that succeeds with constant probability. It further extends to settings with “sparse” noise, where variance measured with respect to any $\ell_p$ norm is of the same order, ruling out the possibility of improving oracle complexity as a function of $\varepsilon$ by measuring variance in a non-matching $\ell_p$ norm.

We investigates the transonic shock problem for steady relativistic Euler equations in hemispherical shells. We show the existence and uniqueness of spherically symmetric transonic shock solutions via the shooting method and the monotonicity between the exit pressure and shock position. Furthermore, without any restrictions on background transonic shocks, we prove the existence and stability of transonic shocks under three dimensional perturbations of the exit pressure. The key techniques involve the decoupling the hyperbolic and elliptic components in steady relativistic Euler equations using the deformation tensor and vorticity. To address the coordinate singularities, the “spherical projection coordinates" combining the spherical coordinates and the stereographic projection is employed. Subsequently, the Rankine-Hugoniot conditions are appropriately reformulated to determine the shock front and derive the boundary conditions on the shock front for a first-order nonlocal deformation-curl system.

The magnetic field strength is a central design object in the theory of magnetic confinement of plasma. Desirable confinement properties such as quasi-symmetry and omnigenity are characterized by special forms of the field strength when it is given in the particular coordinate system called “Boozer coordinates." We ask a basic realizability question: which functions of Boozer coordinates can arise as the magnitude of a magnetic field that is written in Boozer coordinates? Using exterior differential systems and the Cartan-Kähler theorem, we show that every positive analytic function is locally realizable. Thus there is no local analytic obstruction to prescribing the field strength in Boozer coordinates; though further restrictions might arise from global geometric or additional physical constraints.

Gravity-assist trajectory design requires searching numerous leg combinations, but applying nonlinear programming (NLP) for low-thrust trajectory optimization to all combinations is computationally prohibitive. Therefore, conventional frameworks first perform broad search using lightweight trajectory models such as Lambert-based calculation and simplified deep-space maneuver models, while pruning infeasible candidates. However, to maintain efficiency, conventional approaches handle only low-dimensional formulations with limited constraints such as up to a couple of impulse maneuvers per leg. Consequently, they cannot accommodate the constrained multivariable optimization required for low-thrust design, running the risk of prematurely pruning viable trajectories. To address this, this paper introduces a convex pruning approach capable of solving constrained multivariable problems directly within broad search while retaining the computational efficiency. We then augment the broad search stage with a robust local low-thrust optimization algorithm based on thrust regularization, which together enable global exploration and optimization in an automated fashion. We apply the proposed framework in a BepiColombo-inspired scenario involving nine gravity assists, successfully demonstrating its broad search capability to discover a far greater number of solutions (36,335) compared to a conventional approach (4,664) while identifying a wider feasible launch window by one year.

We give an explicit finite sum for the number of ideals of the Lie algebra of strictly lower triangular $n\times n$ matrices over $\mathbb F_q$, valid for every prime power $q$. The sum runs over integer compositions, with weights expressed using ordinary and Gaussian binomial coefficients. A contraction bijection transforms Gagnon's configuration sum into a weighted enumeration of nonnesting partitions, with antichains of intervals chosen independently in each block. We derive a Stieltjes continued fraction for the block weights. Lagrange inversion and the coefficient formula for Stieltjes–Rogers polynomials then yield the explicit sum.

We study the Vlasov equation with general power-law (Riesz-type) potentials with exponent \(\alpha\). We prove global well-posedness in every dimension \(d\ge4\), for both attractive and repulsive interactions, throughout the range \(0<\alpha<3\), for arbitrary nonnegative compactly supported bounded initial data. No smallness condition is imposed on the initial data. For the attractive problem, the virial identity gives finite-time breakdown for negative-energy solutions when \(\alpha\ge3\). The proof of global well-posedness combines the analysis of characteristics, estimates in Lagrangian coordinates, and a crucial application of a compensated integrability inequality due to Denis Serre.

Let $E$ be a non-CM elliptic curve defined over $\mathbb{Q}$. We present models of curves that have 2-adic image as large as possible, given the constraint that they have a rational 2-isogeny. These curves are known to be generic, and one of the aims of this article is to make that fact explicit. Our techniques make use of modular curves and their $j$-maps, as well as classification results on isogeny-graphs that have appeared in the literature recently.

Monge matrices arise throughout combinatorial optimization and algorithm design; the Monge polytope $\mathcal{M}_{pq}$ is the set of $p \times q$ Monge matrices lying inside the standard simplex on the set of matrix coordinates. We find a Stanley decomposition of the associated affine semigroup, and use it to obtain a volume formula for $\mathcal{M}_{pq}$ expressed as a sum over "Z-avoiding" Delannoy paths in a $p \times q$ grid. An efficient dynamic-programming implementation of this formula computes the volume in dimensions far beyond the reach of general-purpose exact-volume algorithms (e.g., the volume of $\mathcal{M}_{20,20}$, which has dimension 399, is computed in a fraction of a second). As a corollary of our Stanley decomposition, we also obtain a combinatorial closed form for the Ehrhart series of $\mathcal{M}_{pq}$.

We consider using randomized least squares to construct quadrature rules exact on a subspace of functions. Using new conditions that we call relative admissibility and reference weight concentration, we establish both that the quadrature weights from such a procedure concentrate close to their asymptotic values with prescribed and arbitrarily large probability, and this in turn provides useful finite-sample probabilistic bounds on the stability of the resulting quadrature rules for very general classes of possibly complex-valued functions. Our analysis both significantly generalizes the existing analysis of randomized least squares quadrature construction, and provides new bounds on stability for these rules. These results specialize to existence results for positive quadrature rules, when such rules can be theoretically expected. Because our procedures are formally algorithmic, our analysis is a substantive advance toward computationally constructive generalized Tchakaloff theorems.

We study the projected power method (PPM) for synchronizing \(n\) unknown permutations of \(m\) objects under a possibly sparse uniform corruption model. Each pair is observed with probability \(p\), and an observed measurement is uncorrupted with probability \(\pi_0\) and is otherwise an independent uniform permutation. Under \(\log m=o(np\pi_0^2)\), we prove exact one-step recovery (with high probability) of each prescribed block for an independent estimate with a fixed positive majority of correct blocks. When \(np\ge C_0\log n\) and \(m=o(np\pi_0^2)\), we prove that one high-probability event yields a block-error contraction simultaneously for every estimate whose optimally aligned error is at most \(0.5-\epsilon\). The contraction factor is \(O(m/(np\pi_0^2))\) and the error floor is \(O(e^{-cnp\pi_0}+e^{-cnp\pi_0^2}+{\log n}/{n})\). Consequently, one update maps every possibly data-dependent estimate in this basin to vanishing block error, and all subsequent iterates remain almost exact uniformly over the iteration index. The one-step and trajectory results extend to independent, non-identically distributed, permutation-valued corruptions with mean \(m^{-1} \mathbf{1}\mathbf{1}^{\top}\). Under the uniform model, a reference-block spectral initializer has aligned block error \(O_{\mathbb P}(m/(np\pi_0^2))\), yielding an end-to-end almost-exact recovery guarantee. Under a stronger all-block signal condition, PPM reaches exact recovery after finitely many iterations. The theory transfers exactly to partial permutations with common support; for varying supports, we establish deterministic and probabilistic co-visibility margins.

The twisted $C^*$-tensor product was exhaustively investigated by the authors in two previous papers. In this new context, generalising the usual tensor product and the Fermi one, we analyse the possibility of studying the set of symmetric states, that is those invariant under all finite permutations, in the setting of infinite twisted $C^*$-tensor products. As a preliminary result, we recognise that such an investigation can proceed only when the bicharacter, involved in the construction of such twisted tensor products, is hermitian. Otherwise, there is no natural action of the finitary symmetric group on the infinite twisted chain. Furthermore, even if the investigation of symmetric states is certainly meaningful for all twisted systems based on hermitian bicharacters, it is shown that it can be fruitfully carried out in three cases only. In these cases, we can provide the "genuine" version of the celebrated De Finetti Theorem already established by Hewitt and Savage for the general classical case, Stormer for the usual tensor product and Fidaleo for Fermi models. Very surprisingly, it emerges that one more twisted model can be treated exhaustively: it corresponds to the chain twisted by the so-called Klein four-group, and its Klein bicharacter unique up to equivalence.

Non-invasive pressure field estimation from velocity measurements is a longstanding engineering problem. We propose, analyze, and test a pressure-recovery method that computes a full stress field from velocity measurements, and leaves the pressure estimation as a cheap post-processing step. The method relies on a stress-velocity first order formulation of the Navier-Stokes equations, and we show that the formulation accounts for deviations from incompressibility in the measured velocity field by construction. In addition, we theoretically establish the convergence of the finite element (FE) approximation scheme, the stability of the stress recovery with respect to finite-resolution velocity measurements, and then validate this theory numerically. Our results show that the proposed estimator is robust in convective flow regimes and remains accurate at reduced spatial resolution, improving upon state-of-the-art pressure-recovery strategies.

We establish nonasymptotic Gaussian approximation bounds in higher-order Wasserstein distance $W_p$, $p\geq2$, for deterministic matrix-weighted sums of multivariate Poisson martingale differences generated by uniformly ergodic Markov chains. Under an $L^{(2+\eta)p}$-moment condition, $\eta>0$, we obtain the explicit bound $$O\left(p^3\|A\|_4^2+pd^{1/4}\|A\|_2^{1/2}\|A\|_4^2\right),$$ where $A\in\mathbb{R}^n$ collects the moment sizes of the weighted increments. When these sizes are of order $n^{-1/2}$, the bound achieves the first optimal $O(n^{-1/2})$ rate for fixed $p$ and $d$. Consequently, we also obtain the first optimal $O(n^{-1/2})$ $W_p$ Gaussian approximation rate for multivariate additive functionals of uniformly ergodic Markov chains. Our analysis develops two techniques for addressing the interplay between higher-order Wasserstein distance and temporal dependence. First, building on the Ornstein–Uhlenbeck relative-score approach of Fang and Koike (2023), we formulate the bound in terms of antisymmetric Stein couplings while retaining the conditional tensor structure. Second, we develop a refresh-then-maximal coupling that combines an independent first-step resampling, which preserves the desired Stein identity, with a subsequent maximal coupling that provides effective control of the coupling increment. We apply these bounds to nonlinear Markovian stochastic approximation by decomposing normalized errors into matrix-weighted Poisson martingale sums and remainders. This yields the first sharp $W_p$ Gaussian approximation rates for both last iterates and Polyak–Ruppert averages. In particular, the optimized rates are $O(n^{-1/3})$ for polynomial-stepsize last iterates and $O(n^{-1/4})$ for Polyak–Ruppert averages, with an $O(n^{-1/2})$ rate for harmonic-stepsize last iterates under an additional spectral condition.

Online gradient descent is usually studied through external regret, where the learner competes with fixed alternatives. Recent work shows that first-order methods control richer action-dependent deviations. We ask for a geometric characterization of the deviations with respect to which online gradient descent, mirror descent, and follow-the-regularized-leader (FTRL) achieve no regret. We identify exactness as the common principle. Exactness means that the relevant displacement field is generated by a scalar potential, or equivalently that the associated one-form is exact in the geometry used by the algorithm. This geometry depends on the algorithm. For gradient descent it is Euclidean geometry, for mirror descent it is the geometry induced by the regularizer, and for FTRL it is the cumulative dual state. Under mild regularity conditions, exactness yields sublinear regret, while nonzero circulation provides the complementary obstruction and leads to linear regret. This gives a unified geometric framework for understanding the deviation classes controlled by these algorithms and reveals that different first-order methods can control genuinely different classes of deviations. These deviation classes have direct consequences for learning, particularly in games. We study the equilibrium notions induced by exact-form deviations and introduce conservative correlated equilibrium, reflecting both the conservative geometry of the underlying displacement fields and the restricted family of deviations available to the players. We characterize its relation to correlated equilibrium, determine when the resulting equilibrium notions coincide and when they separate, and show how these relationships depend on the geometry and the learning algorithm. Overall, this work gives a unified geometric account of what first-order online learning algorithms are no-regret with respect to, beyond fixed comparators.

Quantum state tomography is a fundamental technique for estimating the state of a quantum system from measured data and plays a crucial role in evaluating the performance of quantum devices. However, standard estimation methods become computationally prohibitive as the system size increases due to the exponential growth of the density matrix, describing a quantum state, with the number of qubits. We propose a low-rank tensor-network framework for mixed-state quantum state tomography based on a block tensor train (Block-TT) factorization. Specifically, the density matrix is represented as the contraction of a Block-TT with its Hermitian transpose, yielding a TT analogue of the Burer-Monteiro factorization. This parameterization guarantees Hermiticity and positive semidefiniteness by construction while compressing the number of optimization variables from exponential to linear in the number of qubits. Building on this representation, we develop single-site and two-site density matrix renormalization group (DMRG) algorithms for estimating quantum states from compressed measurements. The resulting methods operate directly on the compressed parameterization, support adaptive rank refinement, and exploit efficient tensor-network contractions for expectation-value evaluation. The framework is applicable to a broad class of low-rank quantum states, including pure states, nearly pure states, and ground states that admit accurate tensor-network approximations. Numerical experiments demonstrate accurate state reconstruction from limited measurements together with substantial reductions in memory requirements and computational cost compared with conventional low-rank tomography methods.

For a finite group $G$ we write $\gamma_\infty(G)$ to denote the nilpotent residual of $G$, that is, the intersection of all terms of the lower central series. The following theorem is proved: Let $G$ be a finite group of odd order admitting an involutory automorphism $\varphi$ such that $G=[G,\varphi]$ and suppose that $\gamma_\infty(C_G(\varphi))$ has order $m$. Then the order of $\gamma_\infty(G')$ is bounded by a function depending only on $m$. This complements several earlier results on groups of odd order admitting involutory automorphisms.