Papers

The Roman Domination Problem (RDP) on a simple, finite, undirected graph \(G=(V,E)\) asks for a labeling function \(f:V\rightarrow\{0,1,2\}\) such that every vertex assigned value \(0\) is adjacent to at least one vertex assigned value \(2\). The objective is to minimize the total weight \(\sum_{v\in V} f(v)\), and this minimum value is called the Roman domination number of \(G\), denoted by \(\gamma_R(G)\). Since the RDP is NP-complete for bipartite graphs, a natural direction is to study its complexity on restricted subclasses of bipartite graphs. The problem remains NP-complete even under strong structural restrictions, such as star-convex and comb-convex bipartite graphs. Therefore, identifying the borderline subclasses where the problem changes from NP-complete to polynomial-time solvable remains an important challenge. In this paper, we investigate the RDP on convex bipartite graphs and on their superclass, chordal bipartite graphs. First, we present a dynamic programming algorithm for convex bipartite graphs. The algorithm uses the interval ordering of one bipartition class and keeps a compact boundary state, which is sufficient to control the domination requirements of both processed and future vertices. This gives an \(O(n^3)\)-time algorithm for computing \(\gamma_R(G)\) on an \(n\) vertex convex bipartite graph. In contrast, we prove that the decision version of the RDP is NP-complete on chordal bipartite graphs by a polynomial reduction from Dominating Set on chordal bipartite graphs. Thus, our results show a clear separation between the tractability of convex bipartite graphs and the hardness of the larger chordal bipartite class.

We consider the maximal number of factors of a product of bushy trees that can be quasi-isometrically, or even coarsely embedded into various groups of interest, including mapping class groups, Torelli groups, Johnson kernels, surface braid groups, and Bestvina-Brady groups. We use this to quasi-isometrically distinguish groups from the above classes, and also to rule out coarse embeddings between them. All these are applications of general statements about coarse embeddings of products of bushy trees into hierarchically hyperbolic spaces.

Let $\kappa$ be a successor cardinal. We force a universe in which every model of PA of size $\kappa$ extends to a model M of the same size, where M has no splendid extensions. If there is an ineffable cardinal then this statement holds at some cardinal below it, in ZFC.

Respiratory syncytial virus (RSV) is a leading cause of bronchiolitis and other lower respiratory tract infections in infants. Increased viral circulation in the post-COVID era and heterogeneous prevention strategies across regions have made RSV control more challenging. We develop a stage-structured, age-stratified Susceptible-Infected-Recovered (SIR) compartmental model tailored to the Italian setting to investigate the population-level impact of infant prophylaxis with Nirsevimab, a long-acting monoclonal antibody. Scenario-based simulations over a multi-year horizon show that increasing infant protection coverage substantially reduces RSV incidence among infants and also yields indirect benefits in older age groups. In particular, extending coverage to infants born outside the epidemic season further lowers cumulative incidence, although infant-targeted prophylaxis alone does not reduce the control reproduction number below the epidemic threshold in the parameter range explored. These findings suggest that broader and more consistent infant Nirsevimab coverage may reduce RSV burden and support the evaluation of alternative implementation strategies in the Italian context.

A covariant reconstruction framework for Kantowski–Sachs (KS) geometries sourced by Chaplygin-type and polytropic fluids in teleparallel $F(T)$ gravity is developed using the coframe–spin-connection formalism and the teleparallel invariant approach. The matter sector is modelled by nonlinear equations of state, including the generalized Chaplygin gas $p=-A/\rho^{\alpha}$ and a polytropic law $p=K\rho^{\Gamma}$. The corresponding conservation laws determine the dependence of the fluid density on the anisotropic KS volume $V=A_2A_3^2$. These source scalings are then inserted into the symmetric part of the covariant teleparallel field equations and used to reconstruct the functional form of $F(T)$ directly from the KS dynamics. Power-law and exponential ansätze generate distinct invariant reconstruction branches. In the power-law sector, the Chaplygin fluid produces mixed constant-plus-power source terms, while the polytropic sector generates density powers controlled by the polytropic index. In the exponential sector, the natural reconstruction variable is the shifted invariant $X=T_0-T$, leading to shifted teleparallel de Sitter branches. The reconstructed models are interpreted as local anisotropic cosmological sectors and, for contracting angular KS scale factors, as local Kantowski–Sachs black-hole-interior reconstruction branches. The analysis is local and branch-dependent; leading-order viability is assessed through \(F_T>0\) and \(F_{TT}>0\), while a complete perturbative stability analysis is left for future work. The reconstruction is entirely driven by nonlinear matter conservation laws, thereby reversing the standard reconstruction strategy in which the gravitational Lagrangian is prescribed a priori.

We provide explicit upper and lower bounds for certain commonly occurring expressions that involve the Riemann zeta function $\zeta(s)$ and its derivatives on $s>1$. For example, this improves upon existing bounds for $(s-1)\zeta(s)$ and its derivatives, and for $(\zeta'/\zeta)(s)$.

We prove that the only planar, centrally symmetric, strictly convex body $K\subset\mathbb{R}^2$ with $C^1$ boundary that floats in equilibrium in every orientation for the perimetral densities $\sigma=\tfrac18$ or $\sigma=\tfrac38$ is a disk.

We consider the zero-temperature stochastic Ising process describing $\pm 1$ spin-flip dynamics on an infinite one-dimensional quasi-transitive graph $G=(V,E)$ with finite interaction range $K$. We prove that the zero-temperature limit of the Glauber dynamics for this Ising model exhibits a Type $\mathcal{I}$ behavior (infinite fluctuations of all vertices) if and only if the graph possesses the so-called shrink property. For graphs lacking this property, we introduce an algorithmic framework based on an auxiliary spatial automaton to distinguish, in finite time, between Type $\mathcal{F}$ behavior (almost sure local fixation) and Type $\mathcal{M}$ behavior (a mixed regime characterized by the presence of blinkers). We prove that the classification among these three regimes is algorithmically decidable. Furthermore, we provide a constructive example of a graph supporting blinkers of arbitrarily large size.

We study the geometric regularization of a positive closed current by the (twisted) Kähler-Ricci flow on a compact Kähler manifold. We conjecture that the local Arnold multiplicities linearly decrease to zero, while the flow produces complete Kähler metrics in the Zariski open subset of points that have small Lelong numbers. We prove this conjecture in complex dimension 1 and provide several partial results in higher dimension.

Working at the level of a sectorial operator $A$ on a Banach space $X$ and an unbounded operator $\partial$ defined on a subspace of $X$ in another Banach space $Y$, we show that a single reverse Riesz estimate $\|A^\alpha x\|_X \lesssim \|\partial x\|_Y$ for some $0 < \alpha < 1$, combined with the condition $0 \in \rho(A_0)$, where $A_0$ is the part of $A$ on the closure of the range of $A$, implies the Poincaré inequality $\|x - P(x)\|_X \lesssim \|\partial x\|_Y$, where $P$ is the Abel-ergodic projection onto the kernel of $A$. The condition $0 \in \rho(A_0)$ is the natural abstract substitute for a spectral gap, and is sharp already in the Hilbertian case. We also obtain a companion divergence inequality. The arguments are remarkably short, yet the principle is genuinely unifying: it covers commutative and noncommutative situations on the same footing and can be used with arbitrary Banach spaces. As a consequence, we recover Poincaré inequalities associated with hypercontractive semigroups and extend their validity to semigroups satisfying only a reduced spectral gap and a reverse Riesz estimate. We then illustrate the flexibility of the method across a wide spectrum of geometries, ranging from Riemannian manifolds, Lie groups, metric measure spaces, spin manifolds to genuinely noncommutative settings such as group von Neumann algebras, semigroups of Schur multipliers, $q$-Ornstein-Uhlenbeck semigroups and quantum tori, where we sometimes establish new inequalities and otherwise recover classical ones from a single principle.

In this paper, we consider a broad class of continuum two-scale Hamiltonians \begin{align*} H_\varepsilon:=T(-i\nabla_x+\mathbf A(x,\varepsilon x))+V(x,\varepsilon x),\qquad x\in \mathbb{R}^d \end{align*} where $T$ represents either a Dirac operator or a Schrödinger operator, and $x\mapsto \mathbf A(x,X)$ and $x\mapsto V(x,X)$ are $\mathbb L$-periodic with respect to some lattice $\mathbb L\subset\mathbb{R}^d$. No periodicity assumption is imposed on the second variable $X$. Let \begin{align*} \mathbb{R}^d\times \mathbb{R}^d\ni (k,X) \mapsto h(k,X):=T(-i\nabla_x+k+\mathbf A(x,X))+V(x,X) \end{align*} be a family of operators acting on $L^2(\mathbb{R}^d/\mathbb{L})$ with periodic boundary conditions. We assume only local spectral information near one point $(k_0,X_0)$: an isolated $J$-dimensional Bloch bands at energy $e_0$, possibly with internal crossings, and a Hermitian matrix-valued homogeneous function of degree $m\in\{1,2\}$. From these data we derive an effective Hamiltonian $\mathfrak{h}$. Then every sufficiently localized eigenpair $(\vec v,\mu)$ of $\mathfrak{h}$ gives an approximate eigenfunction $\Theta_\varepsilon$ such that for $\varepsilon$ small enough, \[ \|\Theta_\varepsilon\|_{L^2(\mathbb{R}^d)}=|\Omega|^{-1/2}+O(\varepsilon^{1/2}),\qquad \|(H_\varepsilon-e_0-\varepsilon^{m/2}\mu)\Theta_\varepsilon\|_{L^2(\mathbb{R}^d)} =O(\varepsilon^{m/2+1/4}). \]

We study the oscillatory arithmetic function $(-2)^{\Omega(n)}$, where $\Omega(n)$ counts the number of prime factors of $n$, with multiplicity. Sun conjectured a bound on its partial sums $W(x) = \sum_{n \leq x}^{} (-2)^{\Omega(n)}$ as $|W(x)| < x$ for all $x \geq 3078$. In this direction, we obtain new bounds for its logarithmically weighted average \begin{equation*} S(x)=\sum_{n \leq x} (-2)^{\Omega(n)} \log\biggl(\frac{x}{n}\biggr). \end{equation*} Using complex-analytic methods such as the log-weighted Perron's formula, we computed the bound $|S(x)| \leq 1.6x$

We construct a new family of random fields on the Heisenberg group $\mathbb{H}$, the sub-Riemannian analog of $\mathbb{R}^{n}$. These fields are generalized random eigenfunctions of the sub-Laplacian on $\mathbb{H}$, and can be viewed as the sub-Riemannian counterpart to the Berry random wave model in $\mathbb{R}^{n}$. The construction of such waves relies on the representation theory of $\mathbb{H}$, and differs from the Euclidean case because of the presence of infinite-dimensional unitary irreducible representations. This work represents a first step towards studying random waves and their geometry in sub-Riemannian spaces.

Randomized Kaczmarz is a natural fit for large sparse least-squares and tomographic reconstruction, and adaptive row selection can reduce iteration counts. However, deploying adaptive selection on a shared-memory machine means sampling from a residual that lock-free workers are concurrently modifying, often using stale data. We present the first systematic study of this regime: residual-weighted and greedy Kaczmarz under asynchronous execution, measured across 339 runs on a 96-core node with realized (not injected) delays. Four findings carry directly to practice. (i) Stability is governed by a boundary $\ell^*(T)$ between sampling aggressiveness and thread count; below it, more aggressive sampling is strictly better, so one should tune to just inside the cliff. (ii) Threshold-greedy selection (the standard accelerated rule) is unstable at high thread counts, diverging almost immediately. (iii) Under-relaxation buys back the cliff at a predictable cost, giving a usable safety knob. (iv) Consistent-snapshot reads admit a rare, scheduling-dependent divergence that live (inconsistent) reads never exhibited; live reads also cost less per step, making them the right default. We validate the implementation against published sequential results and outline the distributed two-level sampler these measurements motivate.

In this article we study complement ideals, and the dual concept of supplement ideals, in nearrings, both of which are generalizations of the concept of complement in a bounded modular lattice. We prove fundamental properties of complements and supplements in arbitrary nearrings. We then establish Galois connections between the ideal lattices of a nearring and of its matrix nearrings, yielding one-to-one correspondences between their respective complement and supplement ideals. We also define graphs associated with complement and supplement ideals of nearrings and study some of their combinatorial properties such as girth and clique number.

This book presents a part of the (complex) representation theory of $p$-adic reductive groups. Starting from a basis accessible to graduate students, it culminates with the theory of the "Bernstein center" and the Langlands classification of irreducible smooth representations. \vspace{0.5cm} \noindent The book consists of seven chapters, Chapters VI and VII constituting the core of the book. Chapter VI is devoted to the study of the category of smooth representations of a $p$-adic reductive group, establishing among other things the Bernstein decomposition theorem and the description of its center. Chapter VII deals with square-integrable and tempered representations, and the Langlands classification theorem is proved there. \vspace{0.5cm} \noindent The first four chapters are placed in a more general framework and tackle, in order, the study of algebras with idempotents, totally disconnected locally compact spaces and groups, smooth representations of the latter, and particular classes of representations (compact, unitary, square-integrable). Chapter V is a review of the structure results of $p$-adic reductive groups. Appendices give the elements of category theory and some results in algebra used in the text.

Determining absolute/convective instability boundaries conventionally requires repeated saddle searches in the complex-wavenumber plane and a subsequent scan of the physical parameter space to locate zero absolute growth. Such nested calculations become costly and sensitive to modal branch association for large non-normal eigenvalue problems. This work develops a direct continuation method for neutral stationary-saddle boundaries of frequency-affine generalised eigenvalue problems. The zero-group-velocity condition is expressed as an adjoint solvability residual and solved together with the direct and adjoint eigenproblems, complex gauge constraints and the neutral-growth condition. The resulting one-dimensional solution manifold in the combined state–parameter space is tracked by scaled pseudo-arclength continuation, allowing parameter folds to be crossed without switching the physical continuation variable. The formulation recovers the analytical Ginzburg–Landau boundary and, for a Gaussian-wake Orr–Sommerfeld problem, agrees with separately formulated finite-difference saddle corrections to approximately $10^{-8}$ in relative critical Reynolds number. Compared with nested complex-wavenumber and parameter-plane saddle scanning, the tested scans require $8.1$–$52.2$ times the wall time of the direct adjoint continuation. Extrapolation of the measured cost–accuracy trend to a boundary error of $E_H\sim10^{-6}$ suggests an estimated cost ratio of approximately $1.8\times10^{4}$ in favour of the direct continuation. Application to a coupled Oldroyd–B free-surface film reveals genuine folds of the neutral-saddle manifold and a re-entrant CI–AI–CI boundary geometry for the selected saddle family.

For a prime \(p\) and a positive integer \(m\), let \(\mathbb{F}_{p^m}\) be the finite field of cardinality \(p^m\), and let $ R_{u^2,v^2,p^m} =\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+v\mathbb{F}_{p^m} +uv\mathbb{F}_{p^m}, ~ u^2=v^2=0,\ uv=vu, $ be a finite non-chain ring. In this paper, we study skew polycyclic codes of length \(lj\) associated with \(f(x)^j\), where \(f(x)\) is a central polynomial of degree \(l\) in $R_{u^2, v^2, p^m}[x; \Theta],$ where $\Theta$ being an automorphism of \(R_{u^2,v^2,p^m}\). We describe these codes, characterize free skew polycyclic codes, and determine their ranks. Under suitable centrality assumptions, we decompose the quotient ring associated with \(x^{np^s}-\lambda\), where \(\gcd(n,p)=1\) and \(\Theta(\lambda)=\lambda\). This reduces the study of skew \((\lambda,\Theta)\)-constacyclic codes of length \(np^s\) to the study of left ideals of $\frac{R_{u^2,v^2,p^m}[x;\Theta]}{\langle f(x)^j\rangle}, $ where \(f(x)\) is a central irreducible divisor of degree \(l\) of \(x^{np^s}-\lambda\), for an invertible element \(\lambda\in R_{u^2,v^2,p^m}\) and \(j\in\mathbb{N}\). We then apply these results to skew \((\lambda,\Theta)\)-constacyclic codes of length \(p^s\) for different classes of units \(\lambda\). Several examples are presented to illustrate the theory and to obtain optimal codes. Finally, when \(\Theta\) is the identity automorphism, we study constacyclic codes of length \(np^s\) over \(R_{u^2,v^2,p^m}\), according as \(x^n-\alpha_0\) is irreducible or reducible over \(\mathbb{F}_{p^m}\). These results extend the work of \cite{CCDF18} and \cite{ZTG18} on constacyclic codes of length \(np^s\) over \(\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}\) to the finite non-chain ring \(R_{u^2,v^2,p^m}\).

In this article we show that the projected dynamical system obtained by restricting the state of an incrementally scattering passive system to a closed and convex subset K of the state space (a real Hilbert space), is also an incrementally scattering passive system. First we show that the projection of a maximal dissipative operator to the tangent cones of K is again maximal dissipative, hence, it determines a contraction semigroup.

For any $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{C}$, the $n$-th generalized central trinomial coefficient is defined as the constant term in the Laurent expansion of $(b+x+cx^{-1})^n$. In this paper, utilizing the constant term method and generating functions, we prove several congruences concerning generalized central trinomial coefficients. First, we establish a parametric congruence modulo $p^2$, from which several conjectural congruences of Z.-W. Sun follow as special cases. We also prove a Catalan-type congruence modulo $p^2$ and a congruence involving $\binom{3k}{k}T_{3k}(6,1)$ modulo $p$, thereby confirming another two conjectural congruences of Z.-W. Sun.

We study the quantum Wasserstein distances of De Palma and Trevisan associated with quadratic costs generated by self-adjoint observables in arbitrary finite dimension. We first show that any positive semidefinite cost operator is determined by the corresponding distance on pairs of pure states. For nonzero observable-generated quadratic costs, we then prove that the full isometry monoid consists exactly of unitary and antiunitary conjugations if and only if the distance is invariant under unitary conjugations on pure states. These conditions are equivalent to isotropy of the cost on the traceless subspace. They are also equivalent to a geometric characterization: two states attain the Wasserstein diameter exactly when their supports are orthogonal. It already suffices to impose this condition on pairs of pure states. We construct explicit mutually inverse maps between observable-generated quadratic costs and Hilbert–Schmidt frame-type operators formed from the traceless parts of their generators. This correspondence also expresses the transport variational problem directly in terms of the frame operator. For every such cost $C$ on a $d$-dimensional Hilbert space, the minimal number of self-adjoint generators equals the rank of its recovered frame operator $\Theta(C)$, and is therefore at most $d^2-1$. Nonzero isotropic costs require exactly $d^2-1$ generators, attaining the largest minimal generator number that any observable-generated quadratic cost can require. Their arbitrary generating families give tight frames, while their minimal realizations give scaled Hilbert–Schmidt orthonormal bases of the traceless subspace. These symmetry, frame, and diametral characterizations determine the same one-parameter family of quantum Wasserstein distances.

Background and Objective: Quantitative analysis of cell dynamics is central to modern biological research, providing critical insights into immune cell interactions, disease progression, and drug mechanisms. Automated cell tracking in time-lapse microscopy remains challenging due to noise, morphological variations, overlapping cells, and dynamic events such as divisions and fusions. Methods: We present ARGUS, a framework for Accelerated, Robust, General, and Unsupervised Cell Tracking Solutions. ARGUS combines adaptive cell detection, dense Farneback optical-flow prediction, frame-to-frame linear assignment, and a sequence-level tracklet-refinement step that reconnects trajectory fragments across short temporal gaps. Results: On publicly available Cell Tracking Challenge datasets, ARGUS achieved detection accuracy of 0.905-0.971 and tracking accuracy of 0.897-0.964, with runtimes within 1 minute (5-6 seconds for 3 frames). Conclusions: ARGUS is a modular, interpretable framework that can be adapted to different imaging modalities and biological applications without training data or GPU infrastructure. The implementation is publicly available at https://github.com/Gitinc/argus

In this paper, we construct braiding structures on the multi-interval Jones-Wassermann subfactor planar algebra associated with any unitary modular fusion category. Utilizing this construction, we provide a new proof of the self-duality of these subfactors. Furthermore, we demonstrate that these braidings induce a projective unitary representation of the balanced superelliptic mapping class group; consequently, these structures effectively encode the non-trivial higher-genus data of the underlying category. As an application of this correspondence, we derive a generalized Verlinde formula as 2-box Fourier duality of the planar algebra.

For a sequence $\mathbf a=(a_0,a_1,\dots)$ we define its dilated Hankel determinant $\ddot{H}_n(\mathbf a)=\det(a_{2i+j})_{0\le i,j\le n-1}$, the minor of the infinite Hankel matrix $(a_{i+j})$ formed from the even-indexed rows and the first $n$ columns. We prove that, for a broad class of sequences, $\ddot{H}_n$ admits a remarkably simple product evaluation. This mirrors the behaviour of the classical Hankel determinant $H_n$, but with two key distinctions: the class of sequences for which such formulas are known is far larger in the classical case; and, whereas $H_n$ enjoys a single universal evaluation – the Heilermann formula via the Jacobi continued fraction – no analogous general method exists for the dilated determinant, which is therefore considerably more challenging. Our evaluations instead rest on six methods developed here, four of general scope and two of a more specialised nature. The cases treated include the factorial numbers, the Catalan and central binomial coefficients; the Euler numbers and a one-parameter secant family; the involution numbers; the Springer numbers along with elliptic and derivative deformations; the reciprocal-sine function, whose evaluation rests on a new Catalan determinant proved by condensation; a Bessel analogue of the Euler numbers; and a multiplicative Bessel family. As an application, we settle a conjecture of Chapoton and the author on the roots of the Poupard and Kreweras polynomials.

Let $n\in\mathbb N\cap[2,\infty)$ and $\Omega\in L^1(\mathbb S^{n-1})$ with $\Omega\not\equiv 0$. In this article, we introduce a new family of lifted rough maximal operators $\{\mathcal{M}_\theta^\Omega\}_{\theta\in(0,\infty)}$ in the upper-half plane and establish their optimal weak-type estimates. Specifically, we prove that, for any $p \in (1, \infty)$, the estimate, with the positive equivalence constants independent of $f$, \[ \sup_{\theta,\lambda\in(0,\infty)}\lambda^p \underset{{\mathcal M}^\Omega_\theta(f)(x,t) > \lambda t^\frac{\gamma}{p}} {\int_{\mathbb R^n}\int_0^\infty} t^{\gamma-1}\,dt\,dx \sim \|f\|_{L^p(\mathbb{R}^n)}^p \] holds for all $f\in L^p(\mathbb R^n)$ if and only if $\gamma\in\mathbb R\setminus\{0\}$. For the endpoint case $p=1$ and $\Omega \in L(\log L)(\mathbb{S}^{n-1})$, we prove that the above estimate holds if and only if $\gamma \in (-\infty, -n) \cup (0, \infty)$. As applications, we obtain weak-type estimates for generalized Poisson integrals without any logarithmic integrability assumptions, which gives an affirmative answer to the question posed by Sjögren and Soria in page 228 of [Israel J. Math. 95 (1996)]. Moreover, although the operator $M^\ast_\Omega$, arising from the method of rotation of Calderón and Zygmund, is not of weak type $(1,1)$, we find that its lifted variant is weak type $(1,1)$. In addition, we establish a new characterization of Hardy spaces in terms of truncated rough singular integrals.

Robots are increasingly deployed in remote or hazardous areas for mission-critical control tasks. Due to their limited individual capabilities, they have to rely on other field sensors to obtain the state information of targets, and also a dedicated edge information hub (EIH) to enable information exchange, sensing data analysis and control command generation. Such configuration follows a sensing-communication-computing-control (SC3) closed loop. To optimize the whole closed-loop performance, this paper minimizes the linear quadratic regulator (LQR) control cost by designing the sensor-to-EIH bandwidth allocation. Specifically, we first model the distortion noise caused by limited communication data rate based on the mutual information theory. Next, under the control policy based on the Kalman filter and LQR controller, we formulate the control process as a partially observable Markov decision process (POMDP), and develop a deep reinforcement learning (DRL)-based sensor-to-EIH bandwidth allocation scheme. The proximal policy optimization (PPO) algorithm is utilized to train the DRL agent. Simulation results are provided to show the superiority of the proposed DRL-based scheme.

We study the $C^*$-algebra $C^*(\Gamma)$ of a locally finite undirected (Serre) graph $\Gamma$ and compute its K-theory. The algebra is defined intrinsically, as the graph-of-groups algebra with all groups trivial, and is shown to be independent of the choice of orientation. Its structure is accessed through a passage to directed graphs: for every locally finite $\Gamma$ there is a row-finite directed graph $E_\Gamma$ with $C^*(\Gamma)\cong C^*(E_\Gamma)$, and for essential $\Gamma$ this identifies $C^*(\Gamma)$ with the Cuntz–Krieger algebra of the Bass–Hashimoto (non-backtracking) matrix $T_\Gamma$. This makes the whole directed-graph K-theory machinery available and, unlike the directed case, produces K-groups that read off the geometry of $\Gamma$: for a finite essential graph of genus $g\ge 2$ one has $K_0(C^*(\Gamma))\cong\mathbb{Z}^{g}\oplus\mathbb{Z}/(g-1)\mathbb{Z}$ and $K_1(C^*(\Gamma))\cong H_1(\Gamma)\cong\mathbb{Z}^{g}$, and in general $K_\ast$ is governed by the genus, the number of ends and the number of dead-ends. We record the resulting classification by genus: the algebras are finite-dimensional or AF at genus $0$, AT precisely for finite graphs at genus $1$, and unital Kirchberg algebras for finite essential graphs of genus $\ge 2$.

We prove that for every Cayley graph $\mathcal{G}$ over any finite abelian group $G$, there is a weighted Cayley graph with $O(\log |G|)$ generators that is a spectral sparsifier for $\mathcal{G}$. This bound is optimal. Applying our bound to the group $G = \mathbb{F}_2^n$, yields, as a corollary, $O(n/\varepsilon^2)$-sized code sparsifiers for $\mathbb{F}_2$-linear codes, improving on the work of Khanna, Putterman and Sudan (SODA'24) who obtained a similar result with an additional $\mathrm{polylog}(n)$ loss. Our proof is strongly inspired by a recent work of Reis and Rothvoss for the construction of $\ell_1$-sparsifiers. Following their work, the abelian Cayley sparsification problem can be reduced to establishing a lower bound for the volume of a certain natural convex body. This volume bound follows from a short, elementary argument that relies on character symmetry.

In this paper, we consider the uplink of a multiuser Zak-OTFS system comprising users with heterogeneous delay-Doppler (DD) periods/frame sizes. Multiple access is achieved through time-frequency (TF) shifts that place the users in non-overlapping regions of the TF plane. Closed-form expressions for the effective DD domain channel between each user and the base station are derived for sinc and Gaussian pulse shaping filters. The inter-user interference (IUI) is shown to be negligible under the TF-shift-based multiple access, thereby decoupling the multiuser input-output relation (IOR) estimation problem into independent single-user estimation problems. For IOR estimation, a superimposed spread-pilot framework is employed. The spread-pilot sequence is obtained by applying FFT to a reshaped Zadoff-Chu sequence. To mitigate the pilot-data interference introduced by the superimposed spread-pilot, a DD dictionary-based IOR estimation scheme that iterates between IOR estimation and data detection is employed. Simulation results for a multiuser Zak-OTFS system demonstrate that the resulting IOR estimates achieve normalized mean-square error (NMSE) and bit error rate (BER) performances that closely match those of the corresponding single-user system. Furthermore, for sinc pulse shaping, the superimposed spread-pilot frame achieves higher spectral-efficiency compared to embedded pilot frame across a wide range of inter-user power ratios. For Gaussian pulse shaping, however, the embedded pilot frame achieves a higher spectral efficiency due to the combined effects of residual IUI and significant pilot-data interference in the case of superimposed spread-pilot. The robustness of the estimation framework to variations in channel power-delay profile and maximum Doppler shift is also demonstrated.

The structure of the standard Whittaker $(\mathfrak{g}, K)$-module is examined in the case when the group in question is a real split reductive linear Lie group. This module is an injective object in the category of Harish-Chandra $(\mathfrak{g}, K)$-modules which admit a fixed infinitesimal character. The global character of this module is determined. The main theorem of this paper is that it has a self-adjoint structure. Also obtained are the explicit socle filtrations of the standard Whittaker $(\mathfrak{g}, K)$-modules for the rank two split groups $SL(3,\mathbb{R})$, $Sp(2,\mathbb{R})$ and $G_{2}$(split).