Papers

This work considers reconfigurable intelligent surface (RIS) aided multi-way communication systems employing QPSK. The objectives of such system is to allow multiple users to exchange information, exploiting the full-duplex capabilities of RIS. The system under consideration comprises $K$ user terminals and an RIS with $N$ reflective elements. Each user terminal is equipped with one transmit and $M$ receive antennas. To mitigate inter-user interference caused by the full-duplex operation of the RIS, three successive interference cancellation (SIC) techniques are considered. We investigate three SIC ordering methods: $1)$ descending channel variances, $2)$ increasing mean square error, $3)$ optimization approach which maximizes the combined channel gain within the system. The bit error rate (BER) performance of these three SIC methods are evaluated using extensive Monte-Carlo simulations for different numbers of RIS elements, ranging from 4 to 64, and a varying number of transmitting users, ranging from 1 to 5. Additionally, an RIS phase shift optimization method for RIS with reflection elements of varying amplitudes is considered. In this case the previous SIC optimization technique is adapted to address phase-dependent amplitudes in RIS reflection elements. Simulation results indicate that in both ideal and varying amplitude scenarios, when the number of RIS elements is small, SIC with ordering based on increasing mean square error achieves the best BER performance. However, as the number of RIS elements increases, SIC ordering based on the optimization method achieves the best BER performance.

In this paper we introduce a geometrically constrained optimal transport problem on Riemannian manifolds endowed with a Borel foliation, in which mass is transported only within individual leaves. We characterize the existence of foliated transport plans through the equality of the transverse projections of the marginals and show that the corresponding quadratic transport cost decomposes into a measurable family of leafwise optimal transport problems. We also formulate the problem as a classical Kantorovich problem with an extended-valued leafwise cost. Finally, we introduce a foliated continuity equation and establish a Benamou-Brenier formula relating the static and dynamical foliated formulations.

We propose a data-driven framework for estimating the Bayesian Cramér-Rao bound (CRB) in high-dimensional imaging systems with complex, analytically intractable priors. Direct CRB computation is challenging in this setting due to the need to model the prior score and to form and invert the Bayesian Fisher information matrix in very high dimensions. To address these issues, we first reformulate the inverse problem in the latent space of a pre-trained variational autoencoder, thereby dramatically reducing the dimensionality of the bound estimation problem while preserving the spatial structure of the images. The Bayesian CRB is formed in this latent space and mapped back to the native parameter space using a change-of-variables formula. Second, to learn the latent prior score, we introduce a new sliced score matching objective defined in a Bochner space endowed with an $H^1(\Omega)$ Sobolev norm in space. This "Bochner-space sliced score matching" objective is consistent with standard sliced score matching, but penalizes errors in the spatial gradients of the score, suppressing high-frequency artifacts that otherwise contaminate the resulting CRB estimates. We validate the approach on a stylized quantitative photoacoustic computed tomography (qPACT) breast imaging problem with over one million unknown parameters, using a foundation-model autoencoder derived from Stable Diffusion. The proposed method yields stable, artifact-free Bayesian CRB estimates that reflect the highly non-Gaussian structure of the learned prior and reveal the substantial impact of the prior on the relative performance of competing qPACT design schemes.

In this paper, we study the ferromagnetic nearest-neighbor Ising model on rooted spherically symmetric trees with one or \(q\) children per vertex, according to a prescribed sequence of branching levels. Unbounded runs of one-child levels force uniqueness at every nonzero homogeneous field and every finite inverse temperature. We quantify this effect through the locations of sufficiently long nonbranching stretches. Conversely, uniqueness need not entail any uniform rate: for each prescribed vanishing sequence, we construct a density-one tree whose finite-volume boundary influence decays more slowly along a subsequence. The tree retains branching number \(q\), although its nonzero-field coexistence region disappears. For eventually bounded gaps, we prove an optimal uniform comparison with the periodic tree having the largest allowed gap and obtain an explicit interval of coexistence fields. Thus bounded gaps are equivalent, within this family, to coexistence at some finite temperature and nonzero field. For eventually periodic gaps, Gibbs uniqueness is equivalent to fixed-point uniqueness of the full-period return map. A negative Schwarzian derivative yields a complete classification: in the supercritical regime the map has three fixed points inside a closed coexistence interval, two at its endpoints, and one outside. The nonzero endpoints are automatically nondegenerate folds, with square-root splitting of the merging branches.

The hydrodynamical Hopf-Rinow problem asks whether a given reconfiguration of an ideal fluid can be attained by an Euler flow. On the flat two-dimensional torus, we give an equivalent criterion for the strong version of this problem (which requires that the fluid motion minimizes the total kinetic energy required) in terms of metric betweenness. Toward this, we prove that sufficiently small normal neighborhoods in the volume-preserving diffeomorphism group of the flat $n$-dimensional torus, $n \geq 2$, equipped with the kinetic energy metric, are geodesically convex; resolving the strong problem locally in the $H^s$ topology. We further establish local bi-Lipschitz equivalence of the intrinsic geodesic distance and the extrinsic $L^2$ distance. Finally, we prove that the squared geodesic distance from the identity admits a $C^{1,1}$ extension from a sufficiently small normal neighborhood to the ambient $L^2$ space.

We provide a modern overview of several forcing methods in bounded arithmetic and compare them with forcing in set theory. We specifically discuss the countable forcing framework of Atserias and Muller, the restricted ultrapowers of Bydzovsky and Muller, and forcing with random variables developed by Krajicek. We illustrate these forcing arguments by reviewing the Paris–Wilkie forcing and by giving a self-contained construction of the shallow PHP model constructed by Krajicek. We observe that, from a formal point of view, Krajicek's forcing with uncountable structures can be interpreted in the framework of Atserias and Muller after a countable collapse using an LS argument. We further observe that the countable restricted ultrapowers of Bydzovsky and Muller, used to obtain models of the theory $T_{PV}$, can also be interpreted in this framework. This part serves as a primer on modern forcing methods in bounded arithmetic. In the next part, we discuss the analogies and differences between forcing in arithmetic and set theory. We embed Krajicek's Boolean-valued models into generic extensions of models of set theory, observing that the set-theoretic forcing framework can be used to interpret forcing in arithmetics. In our interpretation, set theory offers a convenient method for formalizing forcing arguments with uncountable structures. We end with a question whether analogies with set theory can facilitate the discovery of new uncountable partial orders for arithmetic.

We prove the Aleksandrov-Bakelman-Pucci maximum principle for $L^n$-viscosity sub/super-solutions of fully nonlinear uniformly elliptic equations in $\Omega\subset\mathbb R^n$ with the drift coefficient $\gamma$ such that $\gamma(\ln(1+\gamma))^{\frac{n-1}{n}}\in L^n(\Omega)$. The approach is based on convex duality and Trudinger-Moser type inequalities.

The eigenvalues of a large normal random matrix form a two-dimensional Coulomb gas. At fixed weight, the spectral droplet grows with the matrix rank, and its semiclassical density increment is harmonic measure. We freeze an $N$-eigenvalue configuration and sample $M$ further eigenvalues using the projection onto polynomials of degree less than $N+M$ that vanish at the frozen points. Their conditional law is an exact polynomial ensemble in the potential of the frozen charges. For a regular analytic droplet, we prove that the spatial distribution of the new eigenvalues converges to harmonic measure on the old boundary when $M/\log N\to\infty$ and $M/N\to0$. This convergence holds conditionally on the frozen spectrum, outside a set of frozen configurations whose probability tends to zero. The equilibrium process formed from orthogonal polynomials of degrees $N,\ldots,N+M-1$ has the same limit. An exact finite-rank identity relates polynomial moments of the conditional mean to fluctuations of the frozen spectrum, with a remainder from polynomial modes of degree $N+M$ and above.

We study ramification invariants of finite separable extensions of valued fields that are pure on a primitive element. In the immediate unibranched case, we prove that the extended trace ideal is generated by minimal-polynomial derivatives of integral primitive elements and that the Dedekind different is its divisorial closure. For Galois extensions, we express this derivative ideal as a product of subgroup ramification ideals. These formulas extend prime-degree calculations to pure extensions of higher degree. We compute the differential module, traces, differents, ramification ideals, and character Swan ideals of a pure immediate elementary abelian extension of degree $p^2$. A totally ramified example of the same degree gives a comparison with the defectless case.

At face value, composition operators look like something first experienced in algebra. It's just composition of functions after all. However, there is a deep and technical setting in which these objects live, behaving like infinite-dimensional matrices acting on strange yet fascinating spaces. We take a novel approach to introducing a dense mathematical topic by mirroring that of a so-called scratch kitchen, where ingredients are not prepackaged, processed, or frozen but rather are fresh. While some mathematical maturity will certainly help in the same way that baking experience helps with a new recipe, nothing in one's background is assumed. Along the way, we make no reservations about making historical or philosophical comments to spark reader's interest as well. Bon appetit!

We prove the sharp horizontal isoperimetric inequality in $\mathbb H^n$, $n\ge2$, with equality precisely for left translates and Heisenberg dilations of Pansu balls, up to Haar-null sets.

For $n\ge2$, we prove a quantitative centre-rigidity estimate for bounded sets of finite Riemannian perimeter that are stationary and volume-constrained stable for normalized Riemannian perimeter with prescribed curvature $h_j$. Under confinement to a fixed compact set and cutoffs of vanishing full surface Dirichlet energy on the smooth regular boundary, the squared $L^2$ norms of the horizontal centre derivative, angle defect and weighted shape defect are bounded by $C(\varepsilon_j^2+\|h_j-2n\|_{C^2})|\Omega_j|$ when the parenthesized quantity is sufficiently small. The excess of normalized Riemannian perimeter over horizontal perimeter satisfies the same bound.

We study flows on finite volume homogeneous manifolds arising as directional restrictions of time changes of higher rank abelian unipotent actions. We establish Ratner-type rigidity results in this setting. For first row actions on finite volume quotients of $\operatorname{SL}_n(\mathbb{R})$, we show that natural quantitative assumptions on the cocycle force the flow to be algebraic and its ergodic invariant measure to be homogeneous. These results extend to higher dimensional subactions and to a general abstract setting. We also show that, on compact quotients of $\operatorname{SL}_3(\mathbb{R})$, the smooth time change is smoothly conjugate to a linear algebraic action by a leaf preserving diffeomorphism if and only if the cocycle of the directional flow has bounded deviations along one direction. We also obtain precise asymptotics for compact root increments. The principal thresholds in our results are sharp.

For $n\ge2$, we prove that the $C^2$ boundary of a bounded nonempty open set in $\mathbb H^n$ is volume-preserving stable if and only if it is a Pansu sphere up to left translation and Heisenberg dilation. The proof combines centre reconstruction with a volume-constrained test field derived from Pansu dilation. Consequently, every isoperimetric region admitting an open representative with $C^2$ boundary is a Pansu ball up to these symmetries.

For $n\ge2$, we establish second differentiability of horizontal perimeter for compact $C^2$ boundaries in $\mathbb H^n$ and the continuous extension of their affine second variation to $C^1$ vector fields along the boundary. For the unit Pansu sphere, the Hessian agrees with the closed Jacobi form across both characteristic poles, including variations which move them. We compute the full Friedrichs spectrum of its Jacobi operator. The operator has compact resolvent, Morse index one and nullity $2n+1$; its kernel consists exactly of normalized translation speeds. Consequently, the sphere is stable under volume-preserving ambient variations.

Graphoid automata evaluate graphs by composing relations between words of states. We examine the equations that make this evaluation independent of the chosen graph expression, and what changes when relations are replaced by matrices over a commutative semiring. We prove that the classical graphoid equations are equivalent to the special commutative Frobenius equations together with the required symmetry laws. The known cospan and hypergraph presentations then describe equality of structural and labelled expressions, with the bone law deleting precisely the isolated internal vertices. We use established partition representations to compare this equality with equality of evaluated matrices. For any nonempty finite state set $Q$ and commutative semiring $K$, we give a uniform criterion for two structural expressions to have the same matrix. The criterion determines which isolated-vertex counts are distinguished and when a vanishing scalar makes matrices from different boundary partitions equal. For the canonical structure, the bone law holds precisely when $|Q|\cdot1_K=1_K$. Preserving all relational compositions is a separate requirement, equivalent to additive idempotency for $|Q|\geq2$. We then examine the characteristic matrices of the known relational structures on $Q$ given by disjoint unions of Abelian groups. Their Frobenius equations hold over every commutative semiring, whereas specialness depends on the order of each group and the bone law depends on the number of groups. These conditions specify which equations remain valid when weights are assigned to a graphoid automaton.

The $k$-th power of a simple graph $G$, denoted $G^k$, is the graph with vertex set $V(G)$ where two vertices are adjacent if they are within distance $k$ in $G$. We investigate the zero forcing number of graph powers. Powers of graphs are much denser and generally not encompassed by existing results on zero forcing of graphs, hence their study requires a different approach. In contrast with the usual zero forcing behavior under edge deletion, we show that the zero forcing parameter (and variations of it) is monotone with respect to taking powers. We also determine the zero forcing number of powers of paths and cycles, together with upper bounds for powers of spiders and of grids. We then present spectral lower bounds on the zero forcing number of graph powers which uniquely use the spectrum of the base graph, as well as linear programming methods to compute these bounds. Finally, we study the sharpness of the derived bounds. To derive our results we use techniques ranging from graph theory, linear algebra and polynomial optimization.

Analog quantum computing with neutral atom computers is rapidly evolving into a promising way to perform combinatorial optimization tasks at scale. As technology matures, exploring real use cases leveraging such unique information processors becomes warranted; one example is logistics management where vehicles are assigned routes to perform a given task. Autonomous aerial vehicles (drones) are increasingly used for surveillance over a given territory. In this paper, we propose an innovative framework integrating classical machine learning forecasts and quantum annealing to find the best combination of tours for a given surveillance criterion. The risk maps produced by machine learning are used to find possible surveillance tours, while a quantum-assisted heuristic chooses the best combination, producing a full surveillance plan. By modelling our decision problem for drone paths as a Team Orienteering Problem (TOP) with additional constraints, we construct a new heuristic using quantum annealing to find good solutions. We apply this framework to wildfire prevention, where finding tours most likely to detect a starting fire has clear benefits. We study the effect of larger quantum resources on heuristic performance and compare those results to the use of classical resources.

We study correlation functions of topologically twisted $\mathcal N=2$ supersymmetric QCD with gauge group $\mathrm{SU}(N)$ on a compact four-manifold $X$. We derive the Coulomb branch integral for all $N$, including new couplings to background fluxes for the flavor symmetry. For $X$ of simple type with $b_2^+>1$, the correlators become sums over a finite number of strong-coupling singularities on the Coulomb branch, and can be expressed using a number of Coulomb branch couplings. The sum contains the $\mathcal N=1$ vacua, which become the supersymmetric vacua when the $\mathcal N=2$ theory is broken to $\mathcal N=1$ by an adjoint mass deformation, and whose contribution we attribute to the instanton branch. We focus on asymptotically free $\mathrm{SU}(N)$ QCD with $N_f<2N$ equal mass hypermultiplets, and conjecture that its instanton contribution to topological correlators agrees with the generating function of virtual Segre numbers of rank $N$, which are intersection numbers on the moduli space of semistable sheaves on an algebraic surface. We present a precise dictionary for all $N$ between universal series for Segre numbers and Coulomb branch couplings at the $\mathcal N=1$ vacua. The calculation of these couplings requires new higher-rank techniques, involving Siegel modular forms, an exact $\mathbb Z_N$ symmetry, charge conjugation symmetry, factorisation of the Seiberg–Witten curve, and the blowup formula. For pure $\mathrm{SU}(N)$, our result agrees with earlier work by Mariño and Moore, and the twisted correlators give the $\mathrm{SU}(N)$ Donaldson invariants of $X$. For $\mathrm{SU}(3)$ QCD, we determine closed-form expressions for all 13 Coulomb branch couplings for each $N_f=1,\dots, 5$, and find precise agreement with results of Göttsche and Kool for virtual rank 3 Segre numbers.

We prove that the general version—the one that includes the dimension of the tensor product—of Peskine and Szpiro's Dimension Inequality holds over noetherian local rings of depth at most one.

We present a general method to construct and solve time-dependent integrable chiral quantum field theories on finite intervals from autonomous scattering and reflection data. Spatial homogeneity forces the spectral labels of right- and left-moving particles to be affine with a common slope, so bulk and boundary couplings sample one spectral trajectory. A complete round trip shifts the spectral labels and gives boundary quantum Knizhnik–Zamolodchikov equations. The Yang–Baxter and reflection equations provide local consistency, while boundary-qKZ flatness provides global consistency. We reconstruct the local Hamiltonian by Cayley inversion of the bulk and boundary matching matrices. We apply the construction to rational $SU(N)$ scattering with fundamental boundary impurities and to an $O(3)$ model with a spin-$\tfrac12$ impurity. For the $SU(2)$ impurity, we construct explicit boundary-qKZ solutions as lattice sums of off-shell boundary Bethe vectors and recover the static open-chain Bethe equations when the drive rate vanishes

One of the primary objectives of this paper is to construct generating functions for Apostol-Dedekind sums. These exponential generating functions greatly facilitate the discovery of new formulas and relations involving these sums. Other primary objectives of this paper are to define Apostol-Dedekind sums of higher-order and other certain family of finite sums, and to construct their generating functions. These generating functions include the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Array polynomials, the Stirling numbers, the Carlitz polynomials with their partial derivative equations, and the Hardy sums etc. Moreover, by using these functions, many novel formulas for the Apostol-Dedekind sums of higher-order are derived.

We develop a higher-geometric framework for magnetic higher-form symmetries of pre-quantum field theories. The magnetic (topological) charges of a sigma-model with stacky target are encoded by the shape of the target, equivalently of its stack of fields, which is op-dual to the flat automorphism group controlling electric symmetries. Magnetic symmetry backgrounds are obtained by dualizing these charges against a chosen moduli stack of topological field theories, generalizing Pontryagin duality, and act on the theory by stacking. Cohesive modalities allow for targets with non-finite and non-discrete data. When the dualizing stack is a grouplike $E_\infty$-object, we prove a fibration theorem expressing the magnetic symmetries of an electrically gauged theory in terms of those of the ungauged theory and of the gauge group, computed by a homotopy-fixed-point spectral sequence with an explicit first page. For moduli stacks of higher $U(1)$-gerbes with connection, encoding bulk topological local Lagrangians, we prove a factorization theorem: the magnetic symmetry backgrounds of any connected target split, non-canonically, into Pontryagin duals of its integral homology groups, and are necessarily flat, as a consequence of dualizing discrete charges against a target with connections. We illustrate both theorems with pure $\mathrm{Pin}^\pm(2)$ gauge theory: $O(2)$ carries invertible magnetic $\mathbb{Z}_2$ symmetries in form degrees $d-3$, $d-4$ and $d-5$ that are absent for $\mathrm{Pin}^-(2)$, the first two distinguished by a single differential of the spectral sequence. Gauging and physical applications are developed in upcoming work.

We construct large families of mixed-state topological orders in (3+1)D, including intrinsically mixed-state topological orders. We realize these theories by relating the data parameterizing braided fusion 2-categories to the strong higher-form symmetries and their anomalies that characterize mixed-state topological order. We focus here on bosonic theories, which include theories which admit an emergent fermion in the spectrum of line operators. Our general framework is exemplified on the lattice, where we present concrete models and identify how the data of braided fusion 2-categories are physically encoded. We then formalize this data and show that every braided fusion 2-category arises from a general construction (decoherence followed by strong and weak gauging) as the category of topological operators of a (3+1)D mixed state, with intrinsically mixed-state topological orders distinguished by the presence of strong higher-form symmetries whose anomalies cannot be realized in any (3+1)D gapped ground state. Notably, we show that (3+1)D admits a new mechanism with no (2+1)D counterpart: intrinsically mixed-state order whose topological operators braid non-degenerately, but which carries a gravitational anomaly.

Let $\cP_{\ell,k}$ be the set of rooted binary phylogenetic networks with $\ell$ labelled leaves, $k$ reticulations and no parallel edges. We write $|\cP_{\ell,k}|=W_k(\ell)+D_k(\ell)$, where the weighted count $W_k(\ell)$ adds the inverse orders of the leaf-fixing automorphism groups and the defect $D_k(\ell)$ collects the remainder. The weighted count satisfies, for every $k$, a recursion over the source layers of the tree-component structure that involves neither a list of component graphs nor any distinction between symmetric and asymmetric configurations, and its exponential generating function is a Laurent polynomial in $\sqrt{1-2x}$. Automorphism groups of networks are $2$-groups, elementary abelian for $k\le5$ but not in general. For $k\le5$ the defect is the weighted count of networks with a distinguished involution, which obeys an extension of the same recursion. An exact symbolic evaluation of the two recursions yields $|\cP_{\ell,k}|$ in closed form for $k\le5$, the case $k=5$ being new; it reproduces the published counts for $k\le4$ and corrects a coefficient in a published generating function for $k=3$. For every $k$, uniformly over explicit ranges of $k$, we prove that the non-tree-child networks are a fraction $2k(k-1)/\ell$ of the tree-child networks to leading order, which gives the third term of the asymptotic expansion of $|\cP_{\ell,k}|$. We also prove that reticulation-visible networks exceed tree-child networks by the fraction $k(k-1)/\ell$, and that a uniformly random network in $\cP_{\ell,k}$ has a nontrivial automorphism with probability $k(k-1)/(4\ell^3)$ to leading order.

We prove that every recovering pair $(\mathcal{A},\mathcal{B})$ of nonempty families of subsets of an $n$-element set satisfies $$|\mathcal{A}||\mathcal{B}|\le\left(\frac{223949}{100000}\right)^n=(2.23949)^n.$$ A result of Mond, Souza and Versteegen, combined with the sharp $(9/4)^n$ bound of Fang and Huang for cancellative pairs, gives $(2.2499)^n$ for recovering pairs. We improve this to $(2.23949)^n$. The proof combines two entropy inequalities. One is derived directly from the recovering property, while the other applies the Fang-Huang bound to families obtained from the original pair by a construction of Mond, Souza and Versteegen. The two inequalities are then combined by a convex combination.

Probability-flow ordinary differential equations (PF-ODEs) are widely used as deterministic samplers for score-based diffusion models. Their usual justification is that the Fokker–Planck equation of a diffusion can be rewritten as a continuity equation driven by the score function of the forward diffusion. This identity does not, however, guarantee that the resulting velocity field generates a well-posed flow. We provide theoretical insights into the design of such deterministic samplers for generative models based on diffusions and reflected diffusions. We identify sufficient conditions for a regular Lagrangian PF-ODE flow to exist; reverse sampling and invertibility require two-sided divergence control. For learned scores, sampler stability is controlled by an unweighted velocity error, exposing a mismatch with density-weighted score matching and motivating architectural control of Jacobians, divergence, growth, and compression. Under the manifold hypothesis, positive-time regularization justifies an early-stopped PF-ODE while constants deteriorate near the data endpoint; an explicit sphere example shows that the exact deterministic flow becomes singular as the noise level vanishes even though the diffusion marginals remain well defined. These theoretical insights translate into concrete design principles for stable, invertible, and constraint-preserving diffusion samplers. We illustrate the practical relevance of these design principles using controlled numerical experiments.

We prove that NIP nilrings of bounded nilexponent are nilpotent. We also deduce that NIP $\omega$-categorical $n$-Engel Lie algebras of characteristic $p>n$ are nilpotent, yielding a case of the Lie algebra analogous of Wilson conjecture for $\omega$-categorical groups.

For decades, university mathematics assessed executable procedure, computing and reproducing standard proofs, as a proxy for the understanding it cared about, because procedure was cheap to mark and understanding was not. Generative AI now executes much of this procedure. It does not merely let students bypass the proxy; it exposes how much of what was certified was a proxy. The crisis is not simply cheating but a proxy problem, and resolving it forces a question the discipline has long avoided: what is the irreducible human core of mathematical competence, and how can it be built when some of the executable work can be offloaded? We set out two futures: displacement-and-re-centering, in which AI takes over some execution and education moves toward judgment, formulation and verification; and atrophy, in which premature offloading may erode capacities that depend on prior construction. At the heart of the optimistic future is a verification instability: some forms of oversight are cheap and bounded, while others approach the competence required to construct the work. The outcome is a design choice, not a technological destiny. Rather than add a model, we state the response as two principles: construct before you delegate, and let verification set the ceiling: how far a student can independently verify AI output limits what they may safely delegate. We illustrate with a sequence and two worked problems, and close on the question: whether the construct-before-delegate dependency is a hard constraint or an artefact of how mathematics has been taught.

Answering a question of Madill and Rampersad, we give an example a of a $2$-automatic sequence whose abelian complexity is not $2$-regular.