Papers

Large language model (LLM)-driven evolutionary search is an emerging algorithm-discovery paradigm that has already produced novel results in several scientific fields. Yet its application to wireless communications remains largely unexplored. To bridge this gap, we introduce The AI Telco Engineer (AITE), a framework to autonomously design algorithms for complex communication problems, while navigating performance-complexity tradeoffs. We showcase AITE on two challenging physical-layer problems: designing an equalizer for an orthogonal time-frequency space (OTFS) system, and constructing a receiver algorithm for an orthogonal frequency-division multiplexing (OFDM) system using a custom constellation and operating without pilots. For the first task, AITE develops algorithms that outperform the best-known solutions while reducing computational latency by a factor of 3.6 compared to the strongest baseline. For the second task, it discovers the first explicit, explainable algorithms that achieve performance parity with state-of-the-art neural receivers. These results demonstrate the strong potential of LLM-driven evolutionary search for the autonomous discovery of next-generation wireless communications algorithms.

Asymptotic equivalence between the quasi-Grammian and quasi-Wronskian representations of $N$-soliton solutions in the $J$-matrix formulation of the anti-self-dual Yang-Mills (ASDYM) equation is established up to a constant matrix factor. This formulation, known as the Yang equation, serves as the equation of motion of the four-dimensional Wess-Zumino-Witten (WZW$_4$) model and is equivalent to the ASDYM equation. To visualize the solitonic behavior, the action density of the WZW$_4$ model is evaluated for $\mathrm{G}=\mathrm{U}(2)$, demonstrating that the quasi-Grammian and quasi-Wronskian representations exhibit the same asymptotic soliton profiles, while the phase shift factors associated with $N$-soliton collisions are obtained explicitly. Hence, by virtue of the particle-like nature of solitons, the two representations describe the same class of ASDYM $N$-soliton solutions. These solitons can be regarded as a four-dimensional analogue of KP/KdV-type multi-solitons in fluid dynamics, suggesting a possible connection between the ASDYM equation and higher-dimensional Sato theory. Exact quasi-Grammian $N$-soliton solutions are also presented for $N\leq4$.

We introduce a moment-duality method for Korenblum's maximum principle in the Bergman space $A^2(\mathbb{D})$. Starting from an annular coefficient estimate of Wang, we show that admissibility of a constant~$c$ follows from the existence of a probability measure on $[c^2,1]$ whose ordinary and weighted moments lie on opposite sides of the Bergman moments $1/(k+1)$. This converts the norm comparison into a positive moment problem. We then give an explicit measure, consisting of eight atoms with rational data and Lebesgue measure on a terminal interval, for which the required inequalities admit a rigorous ball-arithmetic certificate. Consequently, \[ c_2\geq 0.4263, \] improving Wang's recent lower bound $c_2\geq0.3554$.

Cameron and Erdős asked if the number of sets free of arithmetic progressions of length $k$ is $2^{r_k(n)(1+o(1))}$, where $r_k(n)$ is the maximum cardinality of a $k$-AP-free subset of $\{1, \dots, n\}$. Balogh, Liu and Sharifzadeh made significant progress on this question showing that it is $2^{O(r_k(n))}$ for an infinite sequence of $n$. We improve their result in two ways. On the one hand, we prove that, for $k\geq 5$, the number of $k$-AP-free sets in $[n]$ is $2^{r_k(n)(1+o(1))}$ for an infinite sequence of $n$, solving the question of Cameron and Erdős for infinitely many values. On the other hand, we also prove that for $k \geq 3$ and all $n$ the number of $k$-AP-free sets in $[n]$ is $2^{O(r_k(n))}$. These results are in fact special cases of a general framework that we develop to count families of sets excluding certain arithmetic patterns, which applies as long as the corresponding extremal threshold satisfies certain Behrend-type lower bounds. As further examples, we get analogous results for solution sets to almost all systems of linear equations as well as counting versions of the multidimensional Szemerédi theorem.

We introduce and study spinal Hanoi towers groups, a family of groups acting on the $d$-adic tree that contains both the classical Hanoi towers group $\mathcal{H}^{(3)}$ and Skipper's generalizations as extreme cases. Each group is generated by $d$ automorphisms $a_1,\dots,a_d$, where $a_i$ has a unique non-trivial section, equal to $a_i$ itself, at the $i$-th coordinate, and root permutation $\sigma_i$ fixing $i$. The entire construction is thus encoded by the finite permutation group $P=\langle\sigma_1,\dots,\sigma_d\rangle\leq\mathrm{Sym}(d)$, and we develop a dictionary between the two: $G$ is fractal and level transitive if and only if $P$ is transitive; every group in the family is amenable and contracting, with explicit nucleus and a word problem solved by a length-halving recursion on syllables; and the abelianizations of $G$ and $P$ together control the first level stabilizer. The branch structure of the family is governed by the subgroup $J\leq\text{Aut}(\mathcal{T}_d)$, generated by the automorphisms with exactly two non-trivial sections, occupied by an element $h\in G$ and its inverse. We give a criterion for the containment $J\leq G$ that can be verified in an explicit finite quotient, and we prove that whenever it holds, a level transitive spinal Hanoi towers group is strongly fractal, regular branch over its commutator subgroup, has explicitly described rigid stabilizers at every level, and is just infinite. As an application, we show that the groups of type $(d,m)$, whose root permutations are $m$-cycles, satisfy $J\leq G$ for all $d\geq 4$ and are therefore just infinite, in contrast with the classical case of the Hanoi towers group on three pegs.

In this paper, we introduce skew CRT codes and the poly-skew metric with associated error model, and we provide a decoding algorithm for this family of codes under the poly-skew metric.

The skew-symmetric discretisation has recently been proposed as a new robust simulation method for weakly approximating stochastic differential equations (SDEs) with non-globally Lipschitz drift. This work develops a pathwise version of the scheme by representing the noise increment as a skew-normal distribution and coupling it with the driving Brownian increments, thereby enabling its use in the multilevel Monte Carlo (MLMC) framework. Under suitable conditions, we establish strong convergence of order 1/2 in $L^2$. Subsequently, the associated MLMC estimator is shown to have computational complexity ${O} \bigl(\varepsilon^{-2} (\log (1/\varepsilon))^2 \bigr)$ to achieve a mean-squared error $\varepsilon^2$. We then analytically compare the proposed scheme with the tamed Euler scheme, another benchmark for robust discretisation. Under a strong inward-drift regime with the current state being far from the stable region, we show that the probability of moving in the wrong direction tends to vanish in the skew-symmetric scheme, whereas the tamed Euler scheme makes such moves with a non-trivial probability. Furthermore, in the MLMC setting, employing a one-dimensional stochastic Ginzburg-Landau model, we specify the range of step sizes for which the asymptotic variance of the coupled level difference obtained via the pathwise skew-symmetric scheme is lower than that obtained via the tamed Euler scheme. Numerical experiments on several model examples support the theoretical rate of strong convergence and demonstrate the stability and effectiveness of the resulting MLMC in the superlinear drift setting.

We consider an insurance risk model with a random walk structure in which the underlying probability law is unknown. A decision maker observes a statistic $Q_n$ computed from $n$ historical observations and then needs to choose a capital buffer of the form $C_n=n f(c,Q_n)$, where $c=\log(1/\delta)/n$ balances the amount of data and the tolerated ruin probability $\delta$. The classical safe capital buffer is inversely proportional to the adjustment coefficient $\gamma$, the exponential rate at which the ruin probability decays; when the law is unknown, this coefficient must be inferred from the data. The profile $f$ couples the statistical cost of observing an atypical historical statistic with the future ruin exponent induced by the resulting decision. We study the joint ex ante probability (over both the historical sample and an independent future risk process) that the future maximum exceeds the data-dependent buffer. We show that the naive plug-in rule fails to achieve the prescribed logarithmic decay exponent $c$, illustrating the adverse impact of model uncertainty when making decisions under rare-event constraints. We then identify a profile $f^*=f^*(c,Q_n)$ with the following appealing properties: its lower semicontinuous majorants are safe, while regular continuous rules that fall strictly below $f^*$ are, under mild conditions, unsafe. We illustrate the potential applicability of our framework by developing three parametric examples. In nonparametric settings, we show that a single exponential envelope leads to degenerate capital buffers, whereas a two-level exponential envelope yields a nondegenerate capital buffer which we prove to be safe.

In this paper we obtain some sum-variants of our first $\zeta$-equivalent of the Fermat-Wiles theorem. For example, on the Riemann hypothesis, we give a new type of $\zeta$-equivalent based on a sum of isolated rectangle-shaped signals (pulses).

We develop a theory of Hardy spaces $H^p$ of discrete holomorphic functions on the upper half-lattice, within the classical framework of discrete holomorphicity on the square lattice. We prove Cauchy and Poisson reproducing formulas, establish a boundary norm identity, and obtain Paley–Wiener type characterizations for these spaces. In the Hilbert space case, we describe the associated reproducing kernel and Szegő projection, and we compare the discrete theory with the classical Hardy space on the upper half-plane through a family of discrete holomorphic approximants of classical $H^2$-functions. We also prove duality results for $H^p$, $1<p<\infty$, establish uniqueness and sampling results on horizontal lines, and introduce Bergman-type spaces, comparing two natural weighted scales.

Nonlinearly preconditioned inexact Newton methods form an effective class of solvers for large-scale nonlinear algebraic systems arising from the discretization of partial differential equations. A central challenge in nonlinear elimination (NE) preconditioning is the reliable identification of the slowly converging components to be eliminated. Existing selection strategies often rely on problem-specific physical information or user-tuned thresholds applied directly to the raw nonlinear residual, which may contain irregular oscillatory structures near stagnation regions, making the selected bad subset highly sensitive to threshold parameters. In this work, we propose an online-learning-enhanced NE preconditioner that identifies the bad subset from the dominant structure of the nonlinear residual rather than from the raw residual itself. Residual snapshots are collected online during the stagnation phase of the current Newton solve, and an unsupervised extraction model is trained to capture the principal nonlinear imbalance. We consider both a linear extractor based on principal component analysis and nonlinear extractors based on autoencoder neural networks. Moreover, we integrate the approach into a parallel domain decomposition framework, which trains a local extraction model independently on each subdomain. The learned residual reconstruction is then used to define the bad subset and guide the nonlinear elimination process. Numerical experiments on lid-driven cavity flows at Reynolds numbers up to 10,000 show that the proposed method produces more reliable and coherent bad subsets, is robust with respect to both NE and learning parameters, and outperforms the baseline NE preconditioner in terms of the convergence.

The spectral properties of the discrete Schrodinger operator on a two-dimensional lattice under uniform static fields are critically governed by the field direction: for angles parameterized by irrational frequencies, the operator is exponentially localized, and for a subset of such angles with asymptotically full Lebesgue measure, this localization persists under a given logarithmically decaying perturbation potential.

Over suitable base fields $k$ of characteristic not $2$, including algebraically closed ones, we construct universal abelian $\underline{\operatorname{K}}^{\operatorname{MW}}_2$-coverings for complements of closed embeddings $\mathbb{A}^1 \hookrightarrow \mathbb{A}^3$. Using these, we obtain a rectifiability invariant of such embeddings by lifting knot-theoretic ideas to algebraic geometry via motivic homotopy theory.

We investigate generalized topological recursion on compact spectral curves admitting exponential, and more generally essential, singularities as ramification points. Exploiting the global formulation of generalized topological recursion, we establish a contour deformation of the recursive residue formula that replaces contributions from these essential singularities by residues at meromorphic points. This provides a natural recursive framework for exponentially ramified spectral curves while remaining entirely within the generalized topological recursion formalism. Our formalism can also be viewed as a limiting case of the Bouchard-Eynard higher-order topological recursion, obtained when the order of a ramification point tends to infinity in a convergent manner, as occurs, for example, for exponential singularities while $dy$ remains regular and non-vanishing. We further illustrate the resulting formalism through several examples, including transcendental functions and the $x$-$y$ dual of the Mirzakhani curve.

A localized functional central limit theorem is established for kernel-weighted partial sum processes of piecewise locally stationary time series under geometric decay of the physical dependence measure. The localized process is shown to converge weakly to a centered Gaussian random distribution in $D'(0,1)$, and the limit extends naturally to an isonormal Gaussian process on $L^2([0,1])$. Weak convergence is further derived for processes indexed by totally bounded subsets of $L^2([0,1])$. As an application, the localized limit theory is used to construct tests for constant mean functions against linear, polynomial, and general alternatives in non-parametric regression with locally stationary errors. Simulation results and data examples illustrate the finite sample performance and practical applicability of the proposed methodology.

We develop an adjoint-sensitivity framework for positional influence in causal residual Transformers and separate unconditional analytic results from conditional boundary-shape conclusions. The principal unconditional theorem is the residual-to-depth-flow estimate for layer controls converging in $L^1$, complemented by a finite-token-to-Volterra attention estimate that explicitly controls the first cells near the causal endpoint. We define a normalized adjoint-energy influence density and derive its exact evolution along full-batch gradient flow. The adjoint admits an exact generator-term decomposition into residual transmission, nonlocal Volterra, and local channels, including all covariance cross terms. Causal masking can amplify early-position sensitivity and residual identity paths can transmit a right-localized terminal bias, but neither mechanism alone forces a U-shaped profile. We therefore state boundary advantages under independently checkable energy, correlation, and local-channel bounds; these conditions are sufficient rather than necessary. Finite-token influence balancing, positional reweighting, and task-aligned observability are presented as diagnostics or regularizers with explicit differentiation requirements, computational costs, and limitations. Controlled simulations illustrate that each intervention controls its designated surrogate, while observability balance or outer-loop reweighting need not monotonically reduce the influence-based Lost-in-the-Middle diagnostic.

Let $\mathcal{H}$ be an infinite dimensional separable Hilbert space and $\mathcal{N}$ a nest of projections on $\mathcal{H}$ with at least four projections. Let $\xi$ be a separating vector of $\mathcal{N}^{''}$ and $P_{\xi}$ the orthogonal projection from $\mathcal{H}$ onto the one-dimensional subspace of $\mathcal{H}$ generated by $\xi$. Let $\mathcal{L}$ be the lattice generated by $\mathcal{N}$ and $P_{\xi}$, and ${\rm{Alg}}\mathcal{L}$ the corresponding Kadison-Singer algebra. In this note, we show that every Lie automorphism $\psi$ on ${\rm{Alg}}\mathcal{L}$ can be decomposed as $\psi=\epsilon+\tau$ when $I_{-}^{\mathcal{N}}\vee P_{\xi}<I$, where $\epsilon$ is an automorphism and $\tau$ is a linear functional $\tau$ on ${\rm{Alg}}\mathcal{L}$ vanishing on each commutator. For the complementary case, where $I_{-}^{\mathcal{N}}<I$ with $I_{-}^{\mathcal{N}}\vee P_{\xi}=I$, we also give a construction of the Lie automorphism.

We study atomic splittable congestion games with $n\ge2$ players and nondecreasing $C^2$ resource costs $f$ with convex $x\mapsto xf(x)$. We characterize the largest resource cost class for which the Nash equilibrium is unique. This characterization combines a curvature inequality involving the first two derivatives and the number of players with strict increase of the scalar marginal costs $x\mapsto f(x)+xf'(x)$. The proof rests on first characterizing the largest resource cost classes for which the associated variational inequality operator is monotone, strictly monotone, or strongly monotone. We also draw a perhaps surprising connection to learning dynamics: the same curvature inequality characterizes universal local and global stability of Euclidean-regularized Frank–Wolfe dynamics on arbitrary compact convex strategy spaces. Both the uniqueness and stability characterizations assume closure under positive affine transformations. Both characterizations remain exact for network games; the necessity constructions use acyclic directed graphs whose arc costs can be chosen strictly positive. Finally, we show that on parallel-link networks, every interior equilibrium of the regularized Frank–Wolfe dynamics is locally exponentially stable even without the curvature condition.

For a finite commutative ring $R$, let $\GR$ denote its zero-divisor graph and $\WGR$ its weakly zero-divisor graph, the latter containing the former as a spanning subgraph. We introduce the weak zero-divisor difference graph $\DR:=\WGR-\GR$ and develop a complete structural theory for finite reduced rings $R\cong\mathbb F_{q_1}\times\cdots\times\mathbb F_{q_t}$. We show $\DR$ sits strictly between $\GR$ and $\WGR$ in a three-stage refinement that also contains Badawi's annihilator graph, and prove that distinct support classes $X_A,X_B$ are completely joined in $\DR$ if and only if $A\cap B\ne\emptyset$ – an exact criterion underlying every result that follows. Consequently $\DR$ is edgeless for $t\le2$ but connected with diameter $2$ and girth $3$ for every $t\ge3$, independently of the field orders. We establish a complete perfectness dichotomy – $\DR$ is perfect exactly when $t\in\{3,4\}$, with an elementary combinatorial proof at $t=4$, and never perfect for $t\ge5$ – and determine its clique number exactly at $t=3$ and $t=4$; for general $t$ we give two incomparable lower bounds and two upper bounds, sharp at $t=3$ but not beyond, together with a compression argument showing an extremal family may always be taken shifted without this alone resolving the problem. A reconstruction theorem shows $\DR$ recovers the multiset of field orders intrinsically, so $\DR\cong\mathcal D(S)$ forces $R\cong S$. We further give closed forms, valid for every $t\ge3$, for the degree sequence and minimum degree, the domination number, and the independence and vertex cover numbers. Finally, we briefly indicate, via the valuation structure of finite chain rings, why the reduced-ring hypothesis cannot simply be dropped.

Let $X$ be a subset of a real Hilbert space and let $v\colon X\to Y$, where $Y$ is a real Hilbert space. We prove that the following conditions are equivalent: whenever $A\subset X$, $\rho\geq0$, and $u\colon A\to Y$ is $1$-Lipschitz with $\left\lVert u(x)-v(x)\right\lVert\leq\rho$ for $x\in A$, there is a $1$-Lipschitz extension $\widetilde u\colon X\to Y$ with $\left\lVert \widetilde u(x)-v(x)\right \lVert\leq\rho$ for $x\in X$; and for every $1\leq k\leq\dim Y$, $$ \left\lVert v(x_0)-\sum_{i=1}^k t_i v(x_i)\right\lVert \leq \left\lVert x_0-\sum_{i=1}^k t_i x_i \right \lVert$$ whenever $x_0,\ldots,x_k\in X$, $t_1,\ldots,t_k\geq0$, and $\sum_{i=1}^k t_i=1$. Previous necessity results required $\dim Y\leq3$ or convexity of $X$. For finite-dimensional targets, an application gives an exact data processing characterisation for a finite branching hierarchy connecting Wasserstein and barycentric weak transport. If $Y$ is infinite-dimensional or $\dim\operatorname{Aff}X+1\leq\dim Y$, we also obtain a lifting theorem for convex Lipschitz functions and transfer convex Poincaré inequalities without increasing the constant.

We study deformations of unitary representations $\pi\colon G\to U(H)$ whose coefficients lie in the Hilbert-Schmidt ideal $HS(H)\subset B(H)$. Interesting applications arise for the left-regular representation of surface groups and, more generally, cocompact lattices in the automorphism group of a Fuchsian building of conformal dimension $<2$.

We prove that for every non-arithmetic set~$C$ and every arithmetic finite coloring of~$\mathbb{N}$, there is an infinite set $H \subseteq \mathbb{N}$ whose non-empty finite sums of distinct elements is monochromatic, and $C$ is not $H$-computable. We also study restrictions of Hindman's theorem to simple colorings.

Networked SEIR models describe epidemic spread within and between interacting subpopulations through contact-supported nonlinear transmission. Standard polynomial liftings based on complete ordered Kronecker tensors yield linear higher-dimensional representations, but their dimensions grow rapidly because they retain interactions absent from the transmission graph. This paper develops a graph-induced tensor lifting whose observables are selected from the effective transmission support. An exact edge-based quadratic representation separates linear compartmental transitions from nonlinear infection terms. A homogeneous hierarchy is then constructed recursively. The quadratic transmission field generates the next degree. The linear compartmental field saturates the resulting dictionary within that degree. The first edge-closure dynamics are linear up to an explicit cubic truncation residual, and higher-order truncations contain only next-degree terms. The first lifted dimension scales with the numbers of subpopulations and effective transmission channels. At fixed order, graph-induced dictionaries grow linearly with network size under uniformly bounded local connectivity, whereas complete polynomial liftings retain order-dependent polynomial growth. Uniform first edge-closure residual bounds depend on the transmission rate and the maximum weighted incoming transmission intensity. Numerical illustrations compare equal intensity per active channel with equal total incoming intensity. They confirm that dictionary dimensions depend only on graph support, whereas residual trajectories also reflect weight accumulation, weight distribution, and nonlinear propagation. These results provide a structured basis for reduced modeling and subsequent model-specific analysis and control.

A strong law of large numbers (SLLN) for sublinear expectation under a certain integrability condition is proved. It is shown that the proposed integrability condition is weaker than the majorant condition introduced by Cheng Hu (2018). The proof is based on the truncation method of N. Etemadi (1981).

We study an all-to-all coupled network of identical theta neurons with synaptic interaction mediated by a distributed time delay. Using the Watanabe-Strogatz reduction and passing to the thermodynamic limit under the assumption of uniformly distributed constants of motion, we derive a single delay differential equation for the complex order parameter. The delay is modeled by a family of delay kernels with prescribed mean delay, allowing discrete and distributed delays to be treated in a unified framework. The equilibria of the reduced system can be classified into two geometrically distinct families: type 1 equilibria on the unit circle and type 2 equilibria on the real axis. For both families, the local stability problem reduces to scalar characteristic equations involving the Laplace-Stieltjes transform of the delay kernel. We obtain stability criteria for admissible kernels and explicit Hopf bifurcation conditions for the Dirac kernel, with additional comparison to weak and strong Gamma kernels. The results show that the delay may either preserve stability, destabilize equilibria, or produce stability switching, depending on the equilibrium branch, parameter regime, and choice of kernel. Numerical simulations for the discrete-delay case support the analytical results and illustrate the corresponding phase portraits, basins of attraction, coexistence of attractors, and delay-induced periodic dynamics.

The master T-operator is a generating function for transfer matrices and a tau-function of the modified KP (mKP) hierarchy. It is conventionally introduced through a Schur-function expansion whose coefficients are fused transfer matrices satisfying the Cherednik–Bazhanov–Reshetikhin determinant formula. For rational inhomogeneous gl(M) spin chains, we give an alternative definition: we realize gl(M) on an auxiliary Fock space generated by finitely many families of Schwinger bosons, form a monodromy matrix from the resulting L-operator, and define the master T-operator as its trace over the Fock space. Howe duality then reproduces the Schur-function expansion. This realization relates two constructions of Baxter Q-operators. In the construction of [arXiv:1112.3310] (see also [arXiv:1010.4022]), the Q-operators are obtained from residues of the master T-operator with respect to selected Miwa variables. In the construction of [arXiv:1010.3699], they are defined using degenerate Yangian L-operators in oscillator form. When selected Miwa variables approach the inverses of boundary-twist eigenvalues, the defining trace develops poles. After a suitable rescaling of this L-operator, only the terms contributing to the highest-order pole survive in the normalized limit; before the trace is taken, they form the degenerate Yangian L-operator of the second construction. The trace then gives an explicit residue formula for the corresponding Q-operator. Independently, a Holstein–Primakoff-type large-occupation-number contraction of this L-operator on subspaces with fixed total occupation numbers yields the same degenerate Yangian L-operator.

We study free infinite divisibility in the two-parameter family of beta distributions $\{\beta_{p,q}:p,q>0\}$. Conditional positive definiteness of free cumulants yields a hierarchy of necessary Hankel conditions. We factor the first nontrivial determinant and obtain the explicit necessary inequality \[ 2s^3(s+1)+pq\bigl(s^3-7s^2-16s-12\bigr)\geq0, \qquad s=p+q. \] Its strict reverse defines an open two-dimensional non-freely-infinitely-divisible region not contained in the previously known exclusions. As a boundary consequence, we complete the classification of one boundary family: $\beta_{1/2,q}$ is freely infinitely divisible if and only if $q\geq3/2$. The $3\times3$ determinant is also obtained explicitly in the symmetric variables $s=p+q$ and $u=pq/s^2$. Finally, exact-rational $LDL^{\mathsf T}$ certificates show that each leading Hankel test from $3\times3$ through $12\times12$ strictly enlarges the exclusion supplied by all preceding leading tests. In particular, the $4\times4$ test already detects an open set with $p+q>3$, beyond the range accessible to the $2\times2$ determinant. The results are finite-order obstructions rather than a complete classification; two limiting arguments explain why no fixed member $H_N$ of the leading Hankel hierarchy can provide a uniform obstruction up to the small-parameter boundary.

Low-rank adaptation (LoRA) makes finetuning large language models cheaper by adding to each weight matrix a trainable low-rank update parameterized as the product of two matrices. These matrices are usually trained with Adam, which treats them as a single flat vector of parameters and ignores both the matrix and product structure of LoRA. Applying a matrix-aware optimizer such as Muon to each factor does not consistently improve over Adam, and neither do the product-aware Muon variants proposed in concurrent works. To realize consistent gains, we introduce PoLoRA, a Preconditioned Orthogonalized LoRA optimizer built from three ingredients: a product-aware spectral update direction, curvature preconditioning derived from controlling the per-sample loss change, and a magnitude rule that controls the sizes of both the factor and merged updates. We evaluate PoLoRA on instruction-tuning datasets for code and math across models from 1B to 8B parameters, and find that it reaches the final held-out loss achieved by tuned Adam in 1.2-1.7 times fewer steps, while adding at most 3% per-step overhead. Compared to Adam, PoLoRA is also less sensitive to the learning rate, and its optimal learning rate is stable across ranks.

One of the authors (Chen) had previously proved a result on the behavior of Betti-Whittaker periods under duality for cohomological cuspidal automorphic representations of ${\rm GL}_n/{\mathbb Q}$ under some regularity assumptions while using their relation to $L$-values as an anchor in his proof. In this article we prove a generalization of this result to ${\rm GL}_n$ over any number field $F$ without any regularity assumptions and without recourse to $L$-values, while using the outer-automorphism of ${\rm GL}_n$ as the main tool. Then, using results of Harder and one of the other authors (Raghuram), we give applications to new rationality results for the ratios of special values of general triple product $L$-functions and for general twisted Asai $L$-functions. We also give a new proof of a previous result of Bhagwat and Raghuram on the special values of $L$-functions for orthogonal groups. We present variations on period relations for the Betti-Shalika periods under duality, and the behavior of Betti-Whittaker periods under Galois automorphisms of $F$.

Stochastic nonconvex optimization is central to training deep networks and LLMs in modern machine learning. We give a black-box reduction from stochastic nonconvex optimization to ordinary static regret minimization in online convex optimization (OCO), addressing the open problem posed by Chen and Hazan (2024). Our reduction maintains a predictable gradient tracker, while a black-box online learner $\mathcal A$ selects a preconditioner that transforms this tracker into the update direction. Given a $\beta$-smooth function with initial gap at most $M$ and an unbiased gradient oracle with variance bounded by $\sigma^2$, we bound the expected average squared gradient norm by $O(\sigma\sqrt{M\beta/T}+\sqrt{M\beta}\mathrm{Reg}_T(\mathcal A)/T+\frac{M\beta}{T})$, where $\mathrm{Reg}_T(\mathcal A)$ is the static regret of $\mathcal A$ against the identity preconditioner. Thus, any OCO oracle with $O(\sqrt T)$ regret recovers the classical $O(T^{-1/2})$ convergence rate. We also prove that the additive variance term cannot be removed: every conversion making a constant number of stochastic-oracle queries per round, whose guarantee is stated through static regret against a comparator that the online learner can play, must contain an additive term of order $\sigma\sqrt{M\beta/T}$ for sufficiently large $T$. Moreover, for every online learner, our conversion bound is tight up to universal constants. We also extend the framework to nonsmooth nonconvex optimization, still relying only on ordinary static regret, and attain the optimal convergence rate for Goldstein-type stationarity. Finally, we conduct numerical experiments on nonconvex objectives to illustrate how the reduction exploits online-selected preconditioners while using the same stochastic-oracle budget as stochastic gradient descent.