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In this paper, we study the $S$ transforms of Jacobi processes in the frameworks of free and finite free probability theories. We begin by deriving a partial differential equation satisfied by the free $S$ transform of the free Jacobi process, and we provide a detailed analysis of its characteristic curves. We turn next our attention to the averaged characteristic polynomial of the Hermitian Jacobi process and to the dynamic of its roots, referred to as the \emph{frozen Jacobi process}. In particular, we prove, for a specific set of parameters, that the former aligns up to a Szegö variable transformation with the Hermite unitary polynomial. We also provide an expansion of the averaged characteristic polynomial of the Hermitian process in the basis of Jacobi polynomials. Finally, we establish the convergence of the frozen Jacobi process to the free Jacobi process in high dimensions by using the finite free S transform. In doing so, we prove a general result, interesting in its own, on the convergence of the finite differences of the finite free $S$ transform, which paves the way to obtain asymptotics of differential-difference equations satisfied by time-dependent finite free S-tran

We study selflessness in the general setting of reduced free products of $C^*$-algebras. Towards this end, we develop a suitable theory of rapid decay for filtrations in arbitrary $C^*$-probability spaces. We provide several natural examples and permanence properties of this phenomenon. By using this framework in combination with von Neumann algebraic techniques involving approximate forms of orthogonality, we are able to prove selflessness for general families of reduced free product $C^*$-algebras. As an instance of our results, we prove selflessness and thus strict comparison for the canonical $C^*$-algebras generated by Voiculescu's free semicircular systems. Our results also provide new examples of purely infinite reduced free products.

The free semigroup $\mathcal{F}$ over a finite alphabet $\mathcal{A}$ is the set of all finite words with letters from $\mathcal{A}$ equipped with the operation of concatenation. A subset $S$ of $\mathcal{F}$ is $k$-product-free if no element of $S$ can be obtained by concatenating $k$ words from $S$, and strongly $k$-product-free if no element of $S$ is a (non-trivial) concatenation of at most $k$ words from $S$. We prove that a $k$-product-free subset of $\mathcal{F}$ has upper Banach density at most $1/ρ(k)$, where $ρ(k) = \min\{\ell \colon \ell mid k - 1\}$. We also determine the structure of the extremal $k$-product-free subsets for all $k otin \{3, 5, 7, 13\}$; a special case of this proves a conjecture of Leader, Letzter, Narayanan, and Walters. We further determine the structure of all strongly $k$-product-free sets with maximum density. Finally, we prove that $k$-product-free subsets of the free group have upper Banach density at most $1/ρ(k)$, which confirms a conjecture of Ortega, Rué, and Serra.

In this paper we focus the attention on free boundary problems ruled by partial differential equations with nonstandard growth, presenting in particular some recent results. The interest in these problems stems from the diverse applications that motivate their study and from the challenging mathematical difficulties they pose.

The queue is conceptually one of the simplest data structures-a basic FIFO container. However, ensuring correctness in the presence of concurrency makes existing lock-free implementations significantly more complex than their original form. Coordination mechanisms introduced to prevent hazards such as ABA, use-after-free, and unsafe reclamation often dominate the design, overshadowing the queue itself. Many schemes compromise strict FIFO ordering, unbounded capacity, or lock-free progress to mask coordination overheads. Yet the true source of complexity lies in the pursuit of infinite protection against reclamation hazards--theoretically sound but impractical and costly. This pursuit not only drives unnecessary complexity but also creates a protection paradox where excessive protection reduces system resilience rather than improving it. While such costs may be tolerable in conventional workloads, the AI era has shifted the paradigm: training and inference pipelines involve hundreds to thousands of concurrent threads per node, and at this scale, protection and coordination overheads dominate, often far heavier than the basic queue operations themselves. This paper introduces Cyclic

Feed-forward 3D reconstruction models are efficient but rigid: once trained, they perform inference in a zero-shot manner and cannot adapt to the test scene. As a result, visually plausible reconstructions often contain errors, particularly under occlusions, specularities, and ambiguous cues. To address this, we introduce Free Geometry, a framework that enables feed-forward 3D reconstruction models to self-evolve at test time without any 3D ground truth. Our key insight is that, when the model receives more views, it produces more reliable and view-consistent reconstructions. Leveraging this property, given a testing sequence, we mask a subset of frames to construct a self-supervised task. Free Geometry enforces cross-view feature consistency between representations from full and partial observations, while maintaining the pairwise relations implied by the held-out frames. This self-supervision allows for fast recalibration via lightweight LoRA updates, taking less than 2 minutes per dataset on a single GPU. Our approach consistently improves state-of-the-art foundation models, including Depth Anything 3 and VGGT, across 4 benchmark datasets, yielding an average improvement of 3.73

In this paper, we extend the notion of microstate free entropy to the bi-free setting. In particular, using the bi-free analogue of random matrices, microstate bi-free entropy is defined. Properties essential to an entropy theory are developed, such as the behaviour of the entropy when transformations on the left variables or on the right variables are performed. In addition, the microstate bi-free entropy is demonstrated to be additive over bi-free collections provided additional regularity assumptions are included and is computed for all bi-free central limit distributions. Moreover, an orbital version of bi-free entropy is examined which provides a tighter upper bound for the subadditivity of microstate bi-free entropy and provides an alternate characterization of bi-freeness in certain settings.

Measurements of the mean free path of Lyman-continuum photons in the intergalactic medium during the epoch of reionization can help constrain the nature of the sources as well as sinks of hydrogen-ionizing radiation. A recent approach to this measurement has been to utilize composite spectra of multiple quasars at $z\sim 6$, and infer the mean free path after correcting the spectra for the presence of quasar proximity zones. This has revealed not only a steep drop in the mean free path from $z=5$ to $z=6$, but also potentially a mild tension with reionization simulations. We critically examine such direct measurements of the mean free path for biases due to quasar environment, incomplete reionization, and quasar proximity zones. Using cosmological radiative transfer simulations of reionization combined with one-dimensional radiative transfer calculations of quasar proximity zones, we find that the bias in the mean free path due to overdensities around quasars is minimal at $z\sim 6$. Patchiness of reionization at this redshift also does not affect the measurements significantly. Fitting our model to the data results in a mean free path of $λ_{\mathrm{mfp}}=1.49^{+0.47}_{-0.52}$~pMp

We survey free magmas and we explore the structure of their submagmas. By equipping the cyclic free magma with a second distributive operation we obtain a ringoid-like structure with some primitive arithmetical properties. A submagma is $k$-maximal when there are only $k-1$ submagmas between it and the free magma itself. These two tools, arithmetic and maximality, allow us to study the lattice of the submagmas of a free magma.

Optical pin beams (OPBs) are a promising candidate for realizing turbulence-resilient long-distance free-space optical communication links spanning hundreds of kilometers. In this work, we introduce a unified theoretical model to describe the propagation of OPBs and present comprehensive simulation results based on many realizations and link-budget analyses for constant turbulence strengths. For reference, we compare the performance of the OPBs to weakly diverging and focusing Gaussian beams. For a 100km long air-to-air link, 10km above sea level, our simulation results show that OPBs offer an improved link budget of up to 8.6dB and enhanced beam wander statistics of up to 3dB compared to the considered Gaussian beams. Additionally, we identified a quadratic relationship between the transmitter aperture diameter and the maximum achievable distances, which is crucial in deciding the suitability of OPBs for a given application scenario.

We prove existence and regularity of solutions to degenerate and singular elliptic free boundary problems, where the volume of the positivity set of the solution is prescribed.

In this paper, we extend the notion of non-microstate free entropy to the bi-free setting. Using a diagrammatic approach involving bi-non-crossing diagrams, bi-free difference quotients are constructed as analogues of the free partial derivations. Adjoints of bi-free difference quotients are discussed and used to define bi-free conjugate variables. Notions of bi-free Fisher information and non-microstate entropy are defined and properties of free entropy are extended to the bi-free setting.

Convex sets arising in a variety of applications are well-defined for every relevant dimension. Examples include the simplex and the spectraplex that correspond to probability distributions and to quantum states; combinatorial polytopes and their relaxations such as the cut polytope and the elliptope in integer programming; and unit balls of regularizers such as the $\ell_p$ and Schatten norms in inverse problems. Moreover, these sets are often specified using conic descriptions that can be obviously instantiated in any dimension. We develop a systematic framework to study such dimension-free descriptions of convex sets. We show that dimension-free descriptions arise from a recently-identified phenomenon in algebraic topology called representation stability, which relates invariants across dimensions in a sequence of group representations. Our framework yields structural results for dimension-free descriptions pertaining to the relations between the sets they describe across dimensions, extendability of a single set in a given dimension to a freely-described sequence, and continuous limits of such sequences. We also develop a procedure to obtain parametric families of freely-descri

Matrix convexity generalizes convexity to the dimension free setting and has connections to many mathematical and applied pursuits including operator theory, quantum information, noncommutative optimization, and linear control systems. In the setting of classical convex sets, extreme points are central objects which exhibit many important properties. For example, the Minkowski theorem shows that any element of a closed bounded convex set can be expressed as a convex combination of extreme points. Extreme points are also of great interest in the dimension free setting of matrix convex sets; however, here the situation requires more nuance. In the dimension free setting, there are many different types of extreme points. Of particular importance are free extreme points, a highly restricted type of extreme point that is closely connected to the dilation theoretic Arveson boundary. If free extreme points span a matrix convex set through matrix convex combinations, then they satisfy a strong notion of minimality in doing so. However, not all closed bounded matrix convex sets even have free extreme points. Thus, a major goal is to determine which matrix convex sets are spanned by their fr

The word problem of a finitely generated group is the formal language of words over the generators which are equal to the identity in the group. If this language happens to be context-free, then the group is called context-free. Finitely generated virtually free groups are context-free. In a seminal paper Muller and Schupp showed the converse: A context-free group is virtually free. Over the past decades a wide range of other characterizations of context-free groups have been found. The present notes survey most of these characterizations. Our aim is to show how the different characterizations of context-free groups are interconnected. Moreover, we present a self-contained access to the Muller-Schupp theorem without using Stallings' structure theorem or a separate accessibility result. We also give an introduction to some classical results linking groups with formal language theory.

Motivated by the study of the conjugacy problem for outer automorphism of free groups, we develop the algorithmic theory of the free-by-cyclic group produced by unipotent linearly growing automorphisms of f.g. free groups. We compute canonical splittings of these suspensions as well as their subgroups. We compute their automorphism groups. We show that this class of suspensions is effectively coherent. We solve the mixed Whitehead problem in these suspensions and show that their subgroups all satisfy the Minkowski property, i.e. that torsion in their outer automorphism group is faithfully represented in some computable finite quotients. An application of our results is a solution to the conjugacy problem for outer automorphisms of free groups whose polynomially growing part is unipotent linear.

We show that the space of traces of the free group $F_d$ on $2\leq d \leq \infty $ generators is a Poulsen simplex, i.e., every trace is a pointwise limit of extreme traces. This fails for many virtually free groups. The same result holds for free products of the form $C(X_1)*C(X_2)$ where $X_1$ and $X_2$ are compact metrizable spaces without isolated points. Using a similar strategy, we show that the space of traces of the free product of matrix algebras $M_n(\mathbb{C}) * M_n(\mathbb{C})$ is a Poulsen simplex as well, answering a question of Musat and R\ordam for $n \geq 4$. Similar results are shown for certain faces of the simplices above, such as the face of finite-dimensional traces or amenable traces.

We exhibit an orthonormal basis of cyclic gradients and a (non-orthogonal) basis of the homogeneous free divergence-free vector field on the full Fock space and determine the dimension of Voiculescu's free divergence-free vector field of degree k or less. Moreover, we also give a concrete formula for the orthogonal projection onto the space of cyclic gradients as well as the free Leray projection.

Let $E \subset Ω$ be a local almost-minimizer of the relative perimeter in the open set $Ω\subset \mathbb{R}^{n}$. We prove a free-boundary monotonicity inequality for $E$ at a point $x\in \partialΩ$, under a geometric property called ``visibility'', that $Ω$ is required to satisfy in a neighborhood of $x$. Incidentally, the visibility property is satisfied by a considerably large class of Lipschitz and possibly non-smooth domains. Then, we prove the existence of the density of the relative perimeter of $E$ at $x$, as well as the fact that any blow-up of $E$ at $x$ is necessarily a perimeter-minimizing cone within the tangent cone to $Ω$ at $x$.