Papers

We study the optimal score $T_{n,m}$ in a complete feedback card guessing game. The deck has $n$ types and $m$ cards of each type. Its predictable compensator $S_{n,m}$ is the sum of the conditional success probabilities. We prove that Var$(S_{n,m})\leq C((\log\log n)^2+m/\log n)$, uniformly in $m$ for large $n$. This bound yields a central limit theorem for every sequence $m=m_n\geq1$ as $n\to\infty$. The score variance is asymptotic to its mean. We also give uniform Kolmogorov bounds using the conditional Bernoulli moments. For $m,n\to\infty$, the excess mean is asymptotic to $(\log n)\log\min\{m,\log n\}+(\pi/\sqrt2)\sqrt{m\log n}$. The two contributions balance at $m\asymp(\log n)(\log\log n)^2$. The variance proof uses reversed occupancy hitting times. Discrete hypergeometric tail estimates control their fluctuations in the large-deviation, Gaussian, and saturation regimes. For fixed $n$, we also show that the compensator has a Brownian-bridge functional limit as $m\to\infty$.

In a smart grid, consumers can also play the role of producers and sellers of energy. Therefore, the control of a smart grid over time becomes a dynamic game with a large number of small players and one or several large players. This is much more complicated than the control of an energy system with a single producer. The framework of mean-field games with common noise allows the transformation of such a complicated model into a control problem for a typical prosumer (producer and consumer in one) optimizing his behavior against a flow of other such prosumers. In our model, we introduce 2 decision variables for such a controller: one describing the amount of electric energy bought from the grid (or sold to it), and the second one describing the charging rate for our battery. The cost function is designed to satisfy two goals: satisfying the demand of the consumer (a single home) and minimizing the payment for energy. Players are allowed to sell/buy energy from the grid. Common noise is hidden in price volatility and sun irradiance, which influence all the players. We compute the optimal controls for our model and simulate solutions on real data. The novelty of our construction, compared to other works on smart grid control, lies in combining the minimization of the cost of energy and sustaining some level of demand satisfaction. We analyze the impact of battery usage on achieving a balance between these two criteria. We also take into account that the smart grid is only part of the energy market, which makes consumer strategies dependent on energy consumption outside the grid.

Let $p$ be an odd prime and let $C_R$ be the van der Geer-van der Vlugt curve $y^p-y=xR(x)$ over an algebraically closed field of characteristic $p$, where $R(x)=\sum_{i=0}^{h}a_ix^{p^i}$ is an additive polynomial of degree $p^h$. We determine the image and the rank of every power of the Cartier operator $\Cart$ on $H^0(C_R,\Omg)$. For $1\le r\le h-1$ the image of $\Cart^r$ is spanned by an explicit set of monomial differentials that does not depend on $R$, and $\rank(\Cart^r)=p^{h-r-1}\binom{p+r}{r+2}$. The last iterate detects the linear coefficient: $\rank(\Cart^h)=0$ if $a_0=0$ and $\rank(\Cart^h)=\binom{(p-1)/2+h}{h+1}$ if $a_0\neq0$, so the nilpotency index of $\Cart$ is $h$ or $h+1$. Hence, for $h\ge2$, the loci $a_0=0$ and $a_0\ne0$ have the same $a$-number, which is the largest $a$-number of a one-point Artin-Schreier curve with pole order $p^h+1$, but they share no Ekedahl-Oort type. Over finite fields we deduce that a curve over $\F_{p^{2n}}$ that is geometrically isomorphic to $C_R$ can be $\F_{p^{2n}}$-maximal or $\F_{p^{2n}}$-minimal only if $n\ge h$, and only if $n\ge h+1$ when $a_0\neq0$. Explicit examples show that the second bound is attained for $(p,h)=(3,1)$, $(3,2)$ and $(5,2)$.

We consider the cone Yang-Mills functional recently introduced by Tseng and Zhou. This functional generalizes the Yang-Mills functional by coupling it to a closed 2-form which can represent a geometric structure of interest, e.g. a symplectic form or Ricci form. While the perspective of dimensional reduction is not always possible, we show that a cone Yang-Mills field is always locally the dimensional reduction of a pure Yang-Mills field. We then prove a monotonicity formula and use the dimensional reduction framework to prove an $\epsilon$-regularity theorem in two ways. Finally, we prove a compactness theorem for sequences of cone Yang-Mills connections with bounded action and $L^p$-bounded Higgs fields. Away from a singular set of codimension at least 4 and after suitable bundle identifications, the sequence subconverges in the $C^\infty_{\operatorname{loc}}$-topology.

We investigate a newsvendor problem where a decision-maker observes a $z$-dimensional vector of covariates, and wants to learn how much stock to order given a sample of size $n$ of censored demand data to minimize costs. Demand is censored when insufficient stock is available to meet it, and past stocking quantities are known. We propose a new model-agnostic loss function approach to estimate optimal order quantities using observational data, and characterize the optimal newsvendor policy as the policy that minimizes our proposed loss function. Our method requires only a single estimation step, can be scaled to datasets with hundreds of thousands of data points, and does not suffer from the same instabilities as inverse censoring probability weighting methods. We describe how this loss function can be implemented in linear empirical risk minimization, kernel weights optimization and deep neural network algorithms, and provide excess risk generalization bounds for their out-of-sample performance. Moreover, we show through numerical experiments that our method quickly converges to the optimal policy on synthetic data, and leads to improved performance on real world retail data.

We present a perceptual audio coder in which the entropy-coding model and the bit allocation co-adapt across blocks without model side information. Mantissas are coded by a hybrid of bit packing and arithmetic coding with per-band probability models learned online. Encoder and decoder accumulate identical histograms from decoded blocks and advance the same model state. The model enters bit allocation directly: per-band rate–distortion frontiers combine measured perceptual distortion with entropy-coded rate estimates under the current model, and a dynamic program attains the exact minimum of total objective perceptual distortion for each block's budget under the model's rate estimates, with reservoir carry-over. Under a strict per-block rate, entropy-coding-aware bit allocation lowers this distortion (band noise-to-mask ratio in linear scale) by 15-24% relative to the same solver without it, and co-adaptation removes a further 3-4% (up to 15% on oboe) that a backward-adaptive coder alone cannot realize in one pass; a MUSHRA test at 80 kbps per channel confirms the ranking.

Törnquist (arXiv:1310.3250) proved that if there exists a $\boldsymbol{\Sigma}^1_2$ maximal almost disjoint (MAD) family, then there exists a $\boldsymbol{\Pi}^1_1$ MAD family. Since then, several other "complexity reduction" proofs have been found for other combinatorial objects, such as maximal independent families or ultrafilters. All of these proofs have relied on $\boldsymbol{\Pi}^1_1$-uniformization, so they did not easily lift to higher projective levels. In this paper, we introduce a principle weaker than Uniformization which we call "Functional Projection", and show that this principle is sufficient for most complexity reduction proofs. On the other hand, we prove that in $L[U]$, the canonical inner model for a measurable cardinal, Functional Projection holds for $\boldsymbol{\Pi}^1_2$ relations. As an immediate corollary, we obtain that in $L[U]$ there are $\boldsymbol{\Pi}^1_2$ witnesses for many combinatorial families.

We prove that for $k=4$ the quantum uniformity measures are norms, whereas for every $k\geq5$ the triangle inequality fails. This gives a complete answer to an open question of Bu, Gu, and Jaffe. The $k=4$ case is proved using the Cauchy–Schwarz inequality for the quantum uniformity pairing $\langle\cdot,\cdot\rangle_{Q^4}$ introduced here, together with the triangle inequality in a suitable finite-dimensional Bochner $L^8$ space. For $k\geq5$, we construct systems of linear equations over a finite field and derive a contradiction from the structure theory of norming linear systems.

We propose a continuous learning formulation for $T$ non-forgetting tasks with the spherical parameter space $\mathbb{S}^{d-1}(\sqrt{d})$ in asymptotic regime $d\rightarrow \infty$. Each task $i$ has an optimum $\bm{\Theta}_i$, minimizing its corresponding loss function $R_i(\bm{w})$ on sphere. We reduce each task $i$ to an acceptable loss region modeled by a geodesic ball $B_{\bm{\Theta}_i}(r_i(d))$ on $\mathbb{S}^{d-1}(\sqrt{d})$ centered at $\bm{\Theta}_i$ with radius $r_i(d)$. To model task dependencies (similarities), task optima $\bm{\Theta}_i$ are distributed by a Neyman-Scott clustered point process $\Phi$. Cluster centers $\{\bm{Y}_j\}_{j=1}^{N_p}$ form a homogeneous Poisson point process on sphere, while intra-cluster task optima follow a von Mises-Fisher distribution with mean $\bm{Y}_j$ and concentration $\kappa$. Each task $t\in \{1, ..., T\}$ learning algorithm is considered to be a channel $Q_{t}(\mathrm{d}\bm{w}_t | \bm{w}_{t-1}, \mathcal{K}_{t-1}(\bm{\theta}_{1:t-1}), \bm{\theta}_t)$, where $\bm{W}_t$ is the learned parameter of task $t$ and $\mathcal{K}_{t-1}(\bm{\Theta}_{1:t-1})=\bigcap_{i=1}^{t-1}B_{\bm{\Theta}_i}(r_i(d))$. Since in continual learning, the learner does not need to reach $\bm{\Theta}_t$ exactly to successfully learn task $t$, to avoid forgetting as well as reliable learning, it only needs for the learned parameter vector $\bm{W}_{t}$ to enter $B_{\bm{\Theta}_t}(r_t(d)) \cap \mathcal{K}_{t-1}(\bm{\Theta}_{1:t-1})$. Using this model, we derive an upper bound on $T$ ensuring $\mathbb{P}[\mathcal{K}_{T}(\bm{\Theta}_{1:T}) \neq \emptyset] \rightarrow 1$ as $d\rightarrow \infty$ and $\kappa/d \rightarrow \alpha$, where $\alpha>0$ is a constant. Finally, for $T$ tasks sampled uniformly from $\Phi$, we derive an upper bound for total information theoretic capacity under the non-forgetting constraint for each task.

Let $Q \subseteq \mathbb{R}^n$ be a open, convex, and symmetric set and let $f:Q \rightarrow \mathbb{R}$ be a $\vartheta$-self-concordant barrier function, that is also symmetric. We show the associated spectral function $F(X)= \mathcal{O}^2 (f \circ \lambda)(X)$ is a $\mathcal{O}^2\vartheta$-self-concordant function, where $\mathcal{O} = 8\max\{1+3\vartheta, 12\}$, on the domain of all $n \times n$ symmetric matrices $X$, with eigenvalues $\lambda(X) \in Q$. This generalizes the relationship between the standard logarithmic barriers $-\sum_{i=1}^n\log x_i$ and $-\log \det X$ and gives a solution to a conjecture by L.~Tunçel, 1995.

Laurent phenomenon algebras generalise cluster algebras by allowing irreducible exchange polynomials in place of binomials. Lam and Pylyavskyy proved finite type for those arising from graphs and conjectured that the same holds for every Laurent phenomenon algebra admitting a linear seed. We prove this conjecture over an arbitrary unique factorisation domain. Our proof constructs a nonzero element of the upper algebra divisible by every cluster variable. Comparing noninitial cluster variables with the prime factors of this element in the initial Laurent ring bounds the number of cluster variable classes and yields a finite normalised mutation pattern.

We examine various applications of logic to $\mathrm{AW}^*$-algebras, particularly the monotone completeness conjecture. We develop a model theory for $\mathrm{AW}^*$-algebras equipped with a faithful normalized quasitrace. Using this language, we provide a separable coding which allows us to reduce the finite case of the monotone completeness conjecture to monotone sequential completeness of $\mathrm{II}_1$ $\mathrm{AW}^*$-factors. Combining an explicit recursive axiomatization with the completeness theorem for continuous logic, we show that this case of the conjecture, and Kaplansky's problem whether every $\mathrm{AW}^*$-factor of type $\mathrm{II}_1$ is a von Neumann algebra, are each equivalent in ZFC to a $\Pi^0_2$ sentence of arithmetic, so that their answers are the same in every $\omega$-model of ZFC. We also consider real $\mathrm{AW}^*$-algebras. Here the Baer-$*$-ring notion of an abelian projection must be distinguished from commutativity; following the real-operator-algebra terminology we call the former semi-abelian. Handelman's matrix theorem implies that a real $\mathrm{AW}^*$-algebra with no nonzero central semi-abelian projection has $\mathrm{AW}^*$ complexification. We answer a number of other open problems in real $\mathrm{AW}^*$-algebras.

We give a presentation by generators and relations of the group of birational transformations of the plane preserving a differential form with poles along a triple line. This group strictly contains the automorphism group of the affine plane.

Using the recent single-generation theorem for von Neumann $\mathrm{II}_1$ factors with separable predual, we prove the following results on the generators of a separable unital C*-algebra. Let $A$ be a separable unital C*-algebra, and suppose that $\partial \mathrm{T}(A)$ is nonempty, compact, and zero-dimensional. Then (1) If $A$ has strict comparison of positive elements by tracial states, then $A$ can be generated by two ordinary elements. (2) If $A$ has real rank zero and $A$ has strict comparison of projections by tracial states, then $A$ is singly generated. In this case, its single generators form a norm-dense $G_\delta$ subset of $A$. (3) If $A$ has strict comparison of positive elements by tracial states and there is a unitary $u\in A$ such that for every $\tau\in \partial \mathrm{T}(A)$, (a): the spectral distribution of $u$ with respect to $\tau$ is atomless and has full support on $\mathbb T$, and (b): for every $a\in A$, $$\lim_{t \to 0}\sup_{p\in D_\tau, 0<\tau(p)\leq t} \| p\bigl(\pi_\tau(a)-\mathbb E_{D_\tau}(\pi_\tau(a))\bigr)p\|=0, $$ where $M_\tau=\pi_\tau(A)''$, $D_\tau=\pi_\tau(u)''$, $\mathbb E_{D_\tau}: M_\tau \to D_\tau$ is the trace-preserving conditional expectation, and $\pi_\tau$ is the GNS representation associated to $\tau$, then $A$ is singly generated. In particular, Conclusion (3) above implies that the reduced group C*-algebra $\mathrm{C}_r^*(\mathbb F_n)$ is singly generated for every $n\in\{2,3,\ldots,\infty\}$.

The finite-dimensional $F$-algebras $D=F(e_{11}+e_{33})+Fe_{22}+Fe_{12}+Fe_{13}+Fe_{23} \subseteq UT_3$ and $D_0=Fe_{22}+Fe_{33}+Fe_{12}+Fe_{13}+Fe_{14}+Fe_{23}+Fe_{24}+Fe_{34} \subseteq UT_4$ occupy a distinguished position in the theory of central polynomials. Together with the infinite-dimensional Grassmann algebra, these algebras generate the only varieties of almost polynomial growth of proper central codimensions. We determine finite generating sets for their $T$-spaces of central polynomials and compute their central and proper central codimensions. We also prove that every group grading on $D_0$ is isomorphic to an elementary grading and determine the identities and codimensions associated with all its $\mathbb{Z}_2$-gradings.

A group of conformal automorphisms of a Riemann surface is said to act purely non-free, or to produce a gpnf-action, if each of its finite-order elements has fixed points. It is known that every finite group admits a gpnf-action on a suitable closed Riemann surface of genus at least two. If $G$ is an infinite countable group, and $X$ is an infinite-type orientable surface, without planar ends and whose space of ends is self-similar, then we observe that there is a Riemann surface structure on $X$ admitting a gpnf-action by $G$. We obtain upper bounds for those finite groups producing a gpnf-action on a closed Riemann surface of genus $g \geq 2$. If $G$ is a finite group, then its purely non-free genus $\sigma^{pnf}_{hyp}(G)$ is the minimum genus of at least two of a closed Riemann surface admitting a gpnf-action by $G$. This parameter is already known for cyclic, abelian non-cyclic groups, ${\mathcal A}_{4}$, ${\mathcal A}_{5}$ and ${\mathfrak S}_{4}$. For general $G$, we provide the value of $\sigma^{pnf}_{hyp}(G)$, correcting a previous statement on this value. In particular, we observe that every genus is the purely non-free genus of some group. As an application, we describe the minimal purely non-free actions for the family of metacyclic groups $G^{0}_{s}(n)=\langle x,y: x^{n}=y^{2}=1, yxy=x^{s}\rangle,$ where $n \geq 3$ and $2 \leq s \leq n-1$ is such that $s^{2} \equiv 1 \mod(n)$.

Bridge sampling estimates of marginal likelihoods are routinely reported with a delta-method Monte Carlo standard error (MCSE). Existing theory is asymptotic and cannot say whether a given run is long enough to trust the estimate or its MCSE. We give finite-sample answers. The optimal bridge estimator fits a logistic regression that separates posterior from proposal draws, and its accuracy is governed by their overlap, within a factor of two of the classification's Bayes error. Because the estimating function is a sum of bounded terms, no moment conditions are needed, and all bounds hold uniformly over targets, proposals and dimensions. A number of draws inversely proportional to the overlap suffices and is necessary in the worst case, paralleling the exponential-in-Kullback–Leibler requirement of importance sampling. A Berry–Esseen bound with an explicit constant holds at the optimal rate, and the MCSE is accurate once the effective overlap is large; it is estimated at no cost. For Markov chain draws, the required sample size grows in proportion to the sampler's relaxation time, unavoidably in the worst case. The estimated bridge, the fixed-point iteration and the fitted proposal matter only at second order, and we give free diagnostics for sampler failure and misestimated autocorrelation.

We prove the existence of Steiner minimal trees spanning a finite family of metric spaces in two cases of the Gromov–Hausdorff theory. In the compact Gromov–Hausdorff space $\mathcal{M}$, every finite family of nonempty compact metric spaces is spanned by a shortest Steiner minimal tree whose Steiner vertices are again nonempty compact metric spaces. In a fixed Gromov–Hausdorff cloud $\mathcal{C}$, every finite family of proper, isometrically cobounded spaces admits a finite-length Steiner minimal tree all of whose vertices remain in $\mathcal{C}$. The compact case uses compact replacement through distance profiles and the Hausdorff hyperspace, and the cloud case uses pointed ultralimits.

This paper concerns the Helmholtz scattering problem for open waveguide networks, extending the analysis from [arXiv:2302.04353, arXiv:2310.05816, arXiv:2401.04674] to plane-wave incident data. We first reduce the problem to a Fredholm second-kind integral equation on an infinite interface. This produces an integral representation of the solution to the original scattering problem, which is then used to obtain detailed asymptotic estimates of the solution at infinity. To leading order and away from the waveguides, the scattered field is itself a plane wave. Its amplitude and phase shift exhibit a jump across a critical polar angle depending on the direction of the incident plane-wave. Our estimates also imply that the scattered field is outgoing in a weak sense, allowing us to establish a uniqueness result. We conclude by calculating the waveguide-mode contributions to the scattered field in a perturbative regime.

Parametrized gates in a variational quantum eigensolver (VQE) live on a product of unitary groups, which makes Riemannian optimization a natural candidate for improving training. We test this candidate in the clean per-gate setting: each single-qubit gate is optimized directly on U (2) and compared with Adam on an unconstrained Euler-angle chart representing the same physical gate, initialized from the same unitaries and tuned with a separate learning-rate sweep. On a validated hydrogen-chain benchmark, we find no measurable practical advantage from the Riemannian machinery. The direct manifold optimizer and the Euler-coordinate baseline are interleaved in final error and success rate; the classical coordinate baseline is not slower in iterations on the successful cells; and the choices of Riemannian metric, retraction, and momentum vector transport do not change the conclusion. We also identify four failure modes that can manufacture an apparent positive result: a trivial product-state benchmark, a projected-matrix Adam baseline that discards most of its own step, a non-unitary inherited Cayley retraction that lets the reported energy pass below the variational minimum, and a learning-rate grid whose optimum sits at its boundary. Each failure mode produced a plausible multi-seed result in our own workflow before being caught. The practical output of this paper is therefore a controlled null result and a validation checklist for VQE optimizer comparisons.

The independence complex of the Kneser graph $\KG(n,k)$ is the simplicial complex of pairwise intersecting families of $n$-subsets of $[2n+k]$; equivalently, it is a Vietoris–Rips complex of the $n$-subsets of $[2n+k]$ under the symmetric-difference metric. Feng and Wang asked whether, for $n=3$, its reduced homology is nontrivial only in dimensions $6$ and $9$, and conjectured the rank in dimension $9$. We answer the question affirmatively over the rationals and determine all the associated Betti numbers. For every $k\ge0$, the only nonzero reduced Betti numbers using rationals of $\Ind(\KG(3,k))$ are \[ \beta_6=\binom{k+5}{7}+29\binom{k+6}{7} \qquad\text{and}\qquad \beta_9=\binom{k+6}{6}. \] The proof uses an inductive decomposition based on a simplicial retraction and Barmak's star clusters, Dowker duality together with Jonsson's computation of the homology of complexes of graphs with bounded covering number, and a relative Mayer–Vietoris spectral sequence which reduces the problem to the six-element sets.

The theory of matrix monotone functions of finite degree depends subtly on the matrix size. Motivated by this phenomenon, we introduce radial convexity of metrics with respect to matrix orders on the cone of positive definite matrices. We establish radial convexity of several natural metrics with respect to the Loewner order and then focus on the Frobenius norm. Using the Daleckii-Krein formula, we relate radial convexity to the positivity of the Hadamard square of the corresponding Loewner matrix. This leads us to conjecture that radial convexity of order $n$ is equivalent to matrix monotonicity of degree $n$, and we prove this equivalence in degree $2$. We further provide evidence in higher dimensions, obtain a characterization in degree $3$, and discuss extensions of our results to arbitrary unitarily invariant norms.

We present a method for enforcing exactly Dirichlet, Neumann, and Robin conditions on general triangular domains with arbitrarily curved boundaries. The algorithmic constructions developed here complements those in a previous work for general quadrilateral domains. Our method combines (i) the exact mapping between a general curved triangular domain and the standard triangle, (ii) TFC constrained expressions, and (iii) Barnhill transfinite interpolations, to provide explicit closed-form trial ansatze that mathematically satisfy the imposed boundary conditions exactly. Corner compatibility induced by the boundary conditions is pivotal, and enforcing exactly the imposed boundary conditions hinges on the exact enforcement of the induced compatibility constraints. The corner compatibility associated with the acute vertices of the standard triangle and the coupling between gradient components when the hypotenuse corresponds to a Neumann/Robin boundary pose particular challenges. Two strategies are critical to resolving these issues, by reformulating conditional corner constraints into equivalent unconditional forms and by introducing a separate TFC-style auxiliary edge construction together with an associated linear solve on the hypotenuse. We analyze the corner compatibility constraints and present formulations for the general form of functions that exactly satisfy the imposed Dirichlet, Neumann, or Robin conditions on curved triangular domains. The proposed method has been implemented using extreme learning machine type randomized neural networks for solving PDE problems. We present ample numerical experiments with linear/nonlinear PDE problems on a variety of domains with complex boundary geometries. Numerical results show that the current method has enforced the Dirichlet, Neumann, and Robin conditions on curved domain boundaries to the machine precision.

We characterize the central extensions, in the sense of the categorical theory of central extensions of Janelidze and Kelly, relative to the Birkhoff subcategory of $n$-nilpotent objects of a semi-abelian and algebraically coherent category. An extension $f \colon A \to B$ with kernel $K$ turns out to be central precisely when the iterated Huq commutator $[A,[A,\dots,[A,K]\dots]]$, with $n$ occurrences of $A$, vanishes. For $n=2$ this condition reads $[A,[K,A]] = 0$, that is, $[K,A]$ is a central subobject of $A$. In the special case of groups this description was obtained by Everaert and the first author. The present proof is categorical, and applies in particular to Lie algebras, to crossed modules, to cocommutative Hopf algebras over a field, and to any category of interest.

We investigate Eduard Hanslicks formalist aesthetics through the prism of Category Theory

Fourier-based spectral methods provide compact global approximations, permit simple differentiation and integration, and achieve spectral convergence for smooth periodic functions. Their use in finite-horizon optimal control problems (OCPs) is nevertheless challenged by the generally nonperiodic state and control trajectories: endpoint mismatch induces artificial discontinuities and can destroy spectral convergence. Leveraging Eckhoff-type endpoint-correction techniques, we develop a Fourier-based spectral framework for OCPs with possibly nonlinear dynamics and constraints. The correction improves convergence without enlarging the set of decision variables, yielding an efficient finite-dimensional numerical method. Furthermore, we formulate selected dynamics and obstacle-avoidance constraints through Sobolev residual bounds, which incorporate information about the residuals over the entire time horizon and improve constraint satisfaction. Numerical experiments on benchmark problems demonstrate competitive computational performance, lower residual errors, and improved solution accuracy.

Moment relaxations provide lower bounds for polynomial optimization problems, while the moment matrix carries more information than the bound itself. When the moment matrix satisfies the flat truncation condition, the bound is tight and global minimizers can be extracted from the matrix. In correlatively sparse relaxations, it is common that only some blocks are flat. We study whether such partial flatness still reveals part of a global minimizer. On a unicycle trajectory optimization problem, repeatedly extracting the atoms of flat cliques and fixing them recovers feasible, near-optimal trajectories. We then construct a counterexample whose relaxation is tight and has a rank-one flat clique, yet the unique extracted atom cannot be extended to a global minimizer. Finally, we give sufficient conditions for global extendability that involve only the boundary of a clique. Every extracted atom minimizes its clique objective given its boundary value, and when the bound is tight and the boundary moment block has rank at most one, the extracted atom of the clique is globally extendable.

Motivated by the opinion and social dynamics literature, we rigorously study dynamically co-evolving latent space network models, where agent opinions evolve based on a latent state space model influenced by and influencing an underlying dynamically evolving graph determining their interactions. In contrast to earlier work, see [18], agents are allowed to be heterogeneous, i.e., of different types, and their dynamic evolution can capture behaviors like attraction and repulsion between and within groups. In addition, the model encodes bidirectional feedback between the latent process and the underlying interaction network, persistence effects are included, and the interactions are localized, i.e., individual agents do not have global information. We produce weak and empirical limit theorems that encode the asymptotic joint distribution of agent opinion trajectories and their interaction network. In doing so, we describe and exploit the complex dependence structure of the model.

We study Yang–Baxter deformations of two-dimensional sigma models on compact symmetric spaces $\mathrm{SU}(N)/\mathrm{SO}(N)$, with particular emphasis on their ultraviolet behavior. We claim that the ultraviolet fixed point is described by the $\mathrm{SO}(N,\mathbb C)/\mathrm{SO}(N,\mathbb R)$ non-compact WZW model together with $N-1$ compact free bosons. We determine the leading correction to this fixed point and identify it with the perturbation generated by the corresponding screening currents. We also extend our construction to the broader class of symmetric spaces of maximal rank.

Let \(\Lambda\) be a \(\Z\)-graded Artin algebra which is Iwanaga–Gorenstein. We prove that \(\Lambda\) is CM-finite if and only if the category of gradable Gorenstein-projective modules has finite type, and if and only if the lengths of the indecomposable gradable Gorenstein-projective modules are bounded. Moreover, if \(\Lambda\) is CM-finite, then every Gorenstein-projective \(\Lambda\)-module is gradable. These results provide Gorenstein-projective analogues of the corresponding theorems of Gordon–Green for graded Artin algebras of finite representation type.