Papers

Let $\Gamma$ be a discrete subgroup of a connected semisimple real algebraic group of higher rank. The growth indicator function $\psi_\Gamma$ records the directional exponential growth of the Cartan projections of elements of $\Gamma$ in the positive Weyl chamber $\mathfrak a^+$. We prove that if $\Gamma$ is a non-elementary relatively Borel Anosov group, then $\psi_\Gamma$ is strictly concave on non-collinear directions. We prove this by establishing the $\mathcal C^1$-smoothness of the Manhattan hypersurface, defined as the unit level set of the critical-exponent map $\phi\mapsto\delta^\phi(\Gamma)$. More generally, for a non-elementary $\theta$-transverse group, we prove local $\mathcal C^1$-regularity near every point of the $\theta$-Manhattan hypersurface that is positive on the $\theta$-limit cone and has a critical gap at infinity. In particular, the $\theta$-Manhattan hypersurface is globally $\mathcal C^1$ for relatively $\theta$-Anosov groups, and their $\theta$-growth indicator functions are strictly concave on non-collinear directions.

In this work, we consider the problem of computing a linear combination over a noiseless quantum many-to-one network. There are $k$ senders, Alice$_1$, $\ldots$, Alice$_k$, and a single receiver, Bob. Each Alice$_i$ has a data vector $W_i \in \mathbb{F}^{m_i}$, where $\mathbb{F}$ is a finite field. Bob wants to compute the linear combination $Y = V_1 W_1 + V_2 W_2 + \cdots + V_k W_k \in \mathbb{F}^m$, where $V_i$ is an $m \times m_i$ matrix over $\mathbb{F}$. The senders transmit quantum states to Bob through a noiseless many-to-one quantum network, but they are not allowed to communicate with each other. The senders share entanglement among themselves, while Bob does not share this entanglement. They encode their classical information $W_i$, $i=1,\ldots,k$, into their local subsystems and transmit them to Bob so that he can recover $Y$ through a quantum measurement and subsequent post-processing. The N-Sum Box protocol proposed by Allaix et al. (2025) considers this problem under certain constraints on the linear combination and the distribution of the data vectors among the senders. We present protocols that support the computation of a more general class of linear transformations by giving the senders access to more qudits and allowing them to judiciously precode their input symbols. The communication cost of our schemes is at most that of the best-known prior results in this area and is strictly lower in certain cases. Finally, we demonstrate that the communication cost is subadditive across instances. Specifically, we identify two linear functions for which the total cost of computing them individually is strictly larger than the cost of computing them jointly.

The purpose of this note is twofold. First, we prove quantitative estimates for the Ambrosio-Trevisan commutator which implies propagation of logarithmic Sobolev regularity. Second, we prove a similar estimate for the DiPerna-Lions commutator by relying on quantitative differentiation.

In this paper, we analyze the diffusion limit of Join-the-Shortest-Queue (JSQ) load balancing policies for a system with many parallel queues where each queue is being served by a server with a potentially different service rate. Prior asymptotic analyses of JSQ policies assumed servers to have service rates from a finite set which does not capture scenarios in modern data centers where service rates can vary at the level of individual servers. For systems with individually heterogeneous servers, tracking the empirical queue length distribution for each possible service rate becomes infeasible. To overcome this difficulty, we develop a framework based on measure-valued processes and provide a unified analysis of all JSQ-based load balancing policies under general tie-breaking rules. Our analysis identifies two key objects that distinguish the performance of different tie-breaking rules in the Halfin–Whitt regime, namely, the limiting fairness process which describes how idle servers are distributed across different service rates and the limiting routing measure which describes how arriving jobs are assigned across different service rates. In addition to characterizing the diffusion limits for different tie-breaking rules, we identify the tie-breaking rule which asymptotically minimizes the steady-state distributions of the diffusion-scaled total number of jobs and the diffusion-scaled number of waiting jobs. In proving these results, we develop crucial coupling-based sample-path comparisons which provide both policy-independent steady-state bounds and lower bounds to prove asymptotic optimality.

The monoid of all binary operations was first introduced by H. S. Kim and J. Neggers in 2008. Since then, different aspects and applications of this monoid were studied, while several questions about its semigroup-theoretic properties remain unanswered. We employ a transformation semigroup perspective to fully characterize principal left and right ideals, idempotent and regular elements of this monoid, as well as provide precise combinatorial enumerations of them. This approach gives a general framework for most of the existing results on ideals in the magma monoid. We also answer several open questions posed in the 2023 PhD dissertation of A. Rafieipour. Finally, we correct an error regarding the description of the center of the magma monoid from the 2011 paper of H. F. Fayomi.

Let $(X^{n+1},g_+)$ be a Poincaré–Einstein manifold with conformal infinity $(M^n,[h])$ of positive Yamabe constant $Y(M,[h])$. Then a relative comparison inequality \[ \frac{Y_1(X,M,[\bar g])}{Y_1(\mathbb{S}^{n+1}_+,\mathbb{S}^n,[g_{\mathbb{S}_+^{n+1}}])} \geq \left(\frac{Y(M,[h])}{Y(\mathbb{S}^n,[g_{\mathbb{S}^n}])}\right)^{\frac{n}{n+1}} \] holds for the type-I Escobar–Yamabe compactification $Y_1(X,M,[\bar g]$, with equality if and only if $\left(X, g_{+}\right)$ is isometric to the hyperbolic space $\left(\mathbb{H}^{n+1}, g_{\mathbb{H}^{n+1}}\right)$. This confirms a conjecture raised by Sun-Yung A. Chang.

For 2-complexes with trivial first Betti number, we study quantitative connections between filling inequalities for 1-cycles, the complexity of the second homology, and the size of the torsion part of the first homology. For 2-complexes whose fundamental groups have property $(\tau)$ with respect to all finite index normal subgroups, we prove a geometric lower bound on the logarithm of the size of the first homology of finite covers.

Let $G_q(d)$ be the unit-transfer graph on the nonnegative integer vectors whose $q$ coordinates sum to $d$, equivalently on the multiplicity profiles of size-$d$ multisets over $q$ symbols. The prime-checksum conjecture predicts that, for prime $q$ and all sufficiently large $d$, a largest independent set is a fiber of the natural cyclic checksum. We develop a finite-state weighted local-cover method for $G_q(d)$: translated induced subgraphs give local independence inequalities, while capped anchor profiles reduce the covering conditions for infinitely many degrees to a finite rational linear system. This method gives new proofs of the known cases $q=3$ and $q=4$ and determines $\alpha(G_q(d))$ exactly for $q=5$ and $q=7$ in every degree, thereby proving the next two odd-prime cases of the conjecture. In the complementary regime where $d$ is fixed and $q$ grows, the same method gives an explicit upper bound for $\alpha(G_q(5))$ for every $q\ge7$, determines $\alpha(G_q(d))$ exactly when $q$ is a power of two and $d\in\{6,8,10\}$, and yields an asymptotically sharp upper bound through three terms for every fixed $d\ge7$. The finite systems arising in these arguments are verified in exact arithmetic and supported by independently checkable certificates.

We solve [4, Question 4.2] left open by Dantas, Jung and Martínez-Cervantes by providing a complete characterization for the pairs $(L_p([0,1]),L_q([0,1]))$ having the weak maximizing property. More precisely, we prove that, for $1<p<\infty$ and $1\leq q<\infty$, the pair $(L_p([0,1]),L_q([0,1]))$ has the weak maximizing property if and only if $p=q=2$. A key step in the proof is a Gaussian construction that yields the failure of the compact perturbation property for the pair $(\ell_2,L_r([0,1]))$ whenever $r>2$.

We establish several results about Patterson–Sullivan measures for relatively Anosov groups. First, we prove that these measures are exact dimensional with respect to visual metrics induced by Gromov models in the Groves–Manning quasi-isometry class. Under the additional assumption that the group is relatively Morse, we show that the associated scalar Cartan metric is Gromov hyperbolic and that the corresponding boundary premetric is a visual metric to which the exact-dimensionality theorem applies. Second, we prove that their Manhattan manifolds are $C^1$-regular, from which we deduce that the growth indicator is $C^1$-regular and strictly concave on the interior of the limit cone. This extends the case of Anosov representations by Kim–Oh–Wang. Our methods are dynamical, and we exploit the fact due to Kim–Oh and Blayac–Canary–Zhu–Zimmer that Bowen–Margulis–Sullivan measures for relatively Anosov groups are finite and mixing.

Matrix models appear as fundamental descriptions of M-theory and D-brane dynamics, and via the gauge/gravity duality their gauge-invariant, or singlet, sector describes the purely gravitational degrees of freedom in the holographic dual. We present a new exact algorithm for computing observables of bosonic U(N) matrix models in the gauge-invariant singlet sector. This sector is spanned by an orthogonal basis of Schur polynomials (for a single matrix) and restricted Schur polynomials (for multiple matrices), which diagonalizes the free Hamiltonian and provides a natural truncation of the Hilbert space by excitation number. Matrix elements of the interaction Hamiltonian, or any gauge-invariant observable, are evaluated through a group-theoretic reduction to cosets and double cosets of suitable subgroups of the symmetric group, together with character sums on the symmetric group. The resulting entries are closed-form polynomials in the gauge-group rank N, assembled from group-theoretic data that are precomputed once and can be reused for any N and any coupling constants. We validate the one-matrix implementation against the exact mapping to N non-interacting fermions, demonstrating rapid convergence of the low-lying spectrum with the cutoff. The multi-matrix extension is outlined; its main bottleneck is the computation of restricted characters of the symmetric group, for which no algorithm comparable to the Murnaghan–Nakayama rule is currently known. The framework gives direct access to finite-N, finite-coupling dynamics of gauge-invariant states and opens a new computational window on the non-planar regime of holographic matrix models.

The cyclic probability, $\theta(G)$, of a finite group $G$, is defined as the probability that two randomly selected elements of $G$ generate a cyclic subgroup of $G$. In this paper, we study various upper and lower bounds of $\theta(G)$ and show that $\theta(G)$ can be used as a criterion for checking nilpotency and solvability of a finite group. We also compare $\theta(G)$ with other group invariants like commuting probability $cp(G)$ and normalized sum of element orders.

Generalized product codes (GPCs) combine excellent high-rate performance with low-complexity hardware implementations. We propose the refined dynamic reliability score decoder (\acs{RDRSD}), a hard-message-passing iterative error-and-erasure decoder that uses dynamic reliability scores. Its syndrome-domain implementation has complexity comparable to hard-decision \ac{iBDD}. Across various GPCs and decoding configurations, \acs{RDRSD} provides different complexity–performance trade-offs and achieves approximately \SI{1}{dB} coding gain over \ac{iBDD}. For GPCs with component codes of small error-correcting capability, we also analyze the error floor and propose a soft-aided post-processing step that significantly lowers it.

Motivated by the bifurcation formula for the Milnor number of pairs of holomorphic function germs, we introduce, via foliation theory, the Tjurina number of such pairs. We prove that this is an analytic invariant, derive an explicit formula for it, and establish several of its properties, including a bifurcation formula. As an application, we characterize semitame meromorphic function germs in terms of this invariant. We also investigate several properties of the Milnor number of a pair of holomorphic function germs. In particular, we establish a version of Teissier's Lemma for pairs, derive an upper bound for the Milnor number of a pair, and apply these results to generic pencils of algebraic curves.

This paper investigates the robustness of Timoshenko equations subject to external disturbances and infinite history memory within the integral input-to-state stability (iISS) framework. While polynomial iISS (PiISS) is a powerful tool for characterizing the influence of these two factors, the existing definitions of PiISS rely on graph norms of the state, which are inconsistent with the classical iISS notion for infinite-dimensional systems. In this paper, we provide a rigorous PiISS definition for a class of Timoshenko equations solely via the state norm, and derive sufficient conditions for PiISS under the Tolksdorf condition on the relaxation function, as well as exponential iISS (EiISS) under a stronger condition. The well-posedness is established by using the semigroup and elliptic equation theories. The PiISS and EiISS are assessed by constructing appropriate Lyapunov functionals and employing various a priori estimates of solutions, thereby fully characterizing the impact of disturbances and history memory on system stability.

We introduce illumination bodies and weighted illumination bodies in the class of $n$-dimensional $R$-ball convex bodies. These bodies may be regarded as dual counterparts of the recently introduced $R$-ball floating bodies. We prove that the illumination bodies are convex. We also show that a left derivative of volume gives rise to a surface area measure for ball-convex bodies, the relative surface area measure. In dimension $2$, we establish an isoperimetric inequality for this relative surface area.

In this paper, we investigate the global well-posedness of strong solution to a model recently derived by Constantin-Johnson \cite{CaJo} which describes the nonlinear wave propagation in the troposphere, especially for the morning glory cloud. Assuming that the initial velocity $v_0\in H^1$ and the thermodynamic forcing term $K\in L^2(0,T;L^2)$, we show that there exists a unique global strong solution to the initial boundary value problem of this system. Similar result was only known before for sufficiently small initial data in the existing literature.

Let \(n, m \ge 1\), and let \(H = C_{2^n}\wr C_2\) be the wreathed \(2\)-group and \(K = C_{2^m}=\langle z \rangle\) a cyclic group. We classify the Zappa–Szép products \(G = HK\) in which the base \(B \cong C_{2^n}\times C_{2^n}\) of \(H\) is normal in \(G\). When the matrix \(Z\) of the \(z\)-action on \(B\) commutes with the swap \(J\) of the two base generators – equivalently, \(Z\) is symmetric circulant – we classify these products by an explicit system of seven polynomial congruences in a tuple \((p,q,r,s,c)\). Dropping this hypothesis, we obtain a unified classification of all such products with \(B\) normal by five congruences on the entries of \(Z\) and the parameters \((r,s,c)\), of which the symmetric case is the specialisation \(JZ = ZJ\). Finally, we separately treat the classification for the modulus \(M = 2^m = 4\), since in this case the congruences degenerate.

Quadratic Sum-Of-Squares (QSOS) optimization problems appear in system identification and machine learning, but standard Schur-complement and second-order cone liftings enlarge conic dimensions and create computational bottlenecks for interior-point methods. This paper introduces a lifting-free regularization that preserves the original conic structure by adding a norm penalty to SOS variables, yielding closed-form primal updates and an unconstrained, concave dual with Lipschitz-continuous gradient. Accelerated first-order methods efficiently maximize this dual, and convergence analysis shows non-asymptotic recovery of the solution. Numerical experiments on constrained regression problems show the proposed method can be 40% faster than existing solvers such as SCS and handle larger problems than MOSEK, with memory scaling only in the number of equality constraints.

The Weyl calculus associates a function $a$ on phase space $\mathbb{R}^{2d}$ with the corresponding Weyl operator $L_a$ acting on $L^2(\mathbb{R}^d)$. At $p=2$, this correspondence is governed by an exact Hilbert–Schmidt identity. For $p\neq2$, two-sided $L^p$–Schatten estimates are known for Paley–Wiener type symbols, with constants depending on the Fourier-support scale. We study this quantitative dependence, improve the known upper bounds, and show that in large ranges of $p$ no support-independent global comparison can hold. Let $F_{\sigma}$ denote the symplectic Fourier transform, and let $u\in E'(\mathbb{R}^{2d})$ satisfy $\operatorname{supp}u\subset\overline{B(z_0,R)}$, where $R\geq1$. Then, for every $1\leq p\leq\infty$ and $\varepsilon>0$, we prove \[ \|L_{F_{\sigma} u}\|_{S_p}\lesssim_{d,p,\varepsilon}R^{(2d+1+\varepsilon)|1-2/p|}\, \|F_{\sigma} u\|_{L^p(\mathbb{R}^{2d})}, \] together with the reverse estimate with the same power of $R$. This sharpens the exponential dependence $\mathrm{e}^{cR^2}$ obtained by Luef and Samuelsen and Müller's polynomial dependence $R^{(5d+2)|1-2/p|}$. The main ingredient is a radial trace-class estimate based on the Hermite–Laguerre correspondence $\rho(\varphi_k)=P_k$, which reduces the relevant Weyl operators to finite-rank Hermite projections. We also show that dependence on $R$ is unavoidable. Compactly supported examples obtained by truncating Laguerre functions yield polynomial lower bounds for the best comparison constants. These examples refine Müller's operator-norm example and give nontrivial lower bounds for a larger range of Schatten exponents, which can cover the full range $1\le p\le\infty$ except for the Hilbert–Schmidt point $p=2$ as $d\to\infty$. Moreover, for every fixed $p>2$, the exponent in the reverse comparison estimate is asymptotically optimal as $d\to\infty$.

We prove norm inflation, in the sense of strong ill-posedness, for the two-dimensional Boussinesq system in supercritical Besov spaces. For the inviscid system, norm inflation holds in \(\dot B^\beta_{p,q}(\mathbb R^2)\times \dot B^\beta_{p,r}(\mathbb R^2)\) for \(\beta\neq0\), \(1<p\leq\infty\), \(1\leq q,r\leq\infty\), and \(-2<\beta-\frac{2}{p}<1\). For the fully dissipative system, the same conclusion holds in the range \(-2<\beta-\frac{2}{p}<-1\). In both cases, the results cover almost all supercritical Besov spaces satisfying the local integrability condition. Norm inflation occurs in the density component \(\rho\), while the velocity component \(u\) remains bounded. In the fully dissipative case, the inflation space is supercritical for \(u\), but subcritical for \(\rho\) with respect to its own scaling. This is not a contradiction: the density is transported by a velocity field in a supercritical regime, and this transport mechanism is precisely what produces norm inflation in \(\rho\).

Emerging sustainable materials increasingly rely on engineered hierarchy and microstructure to achieve control of their properties and mechanical behavior. Optimizing these materials with controllable microstructures requires efficient multiscale simulations. Data-driven surrogate models for the microscale can accelerate multiscale simulations, but require large amounts of data even for a fixed microstructure. When a range of microstructures is considered, as is the case in multiscale optimization, even more data is needed to train a surrogate. To overcome this challenge, we condition a hybrid physics-data surrogate on microstructural variables using a hypernetwork. This approach enables accurate predictions of multiscale mechanical behavior for a mycelium-woodchip composite material, even when trained on small datasets. The conditioned surrogate makes multiscale simulations of functionally graded structures tractable, and we validate it against a full FE^2 simulation. We optimize a graded multiscale disk, and reduce the peak stress by 43% compared to one with a random microstructure. Then, we go one step further, conditioning the network directly on manufacturing variables that can have a complex influence on the microstructure. This is a practical route to engineer the microscale for desired macroscale behavior. This contribution highlights the benefits of microarchitectured structures and demonstrates how conditioned surrogate models enable their multiscale optimization, which will accelerate the development and design of future sustainable materials and structures.

We study the $L_1(G)$-norms of Fejér kernels for the Walsh–Kaczmarz system. In the Walsh–Paley case the identity \[ \|K^w_{2^n}\|_1=1 \qquad (n\in\mathbb N) \] follows directly from the binary structure. Toledo's sharp result gives the global supremum \[ \sup_{n\in\mathbb P}\|K^w_n\|_1=\frac{17}{15}. \] The Walsh–Kaczmarz kernels behave differently. We first prove that \[ \sup_{n\in\mathbb N}\|K^\kappa_{2^n}\|_1=\lim_{n\to\infty}\|K^\kappa_{2^n}\|_1=\frac43, \] and we prove that the sequence $(\|K^\kappa_{2^n}\|_1:n\in\mathbb N)$ is non-decreasing and is strictly increasing from $n=2$ on. Then we use this result, an exact splitting formula from Skvortsov's decomposition, and the sharp Walsh–Paley estimates. This gives the exact global supremum \[ \sup_{n\in\mathbb P}\|K^\kappa_n\|_1=\frac{71}{50}. \]

Calegari introduced the laminations $\Lambda^\pm_u$ associated to a universal circle. We study the laminations $\Lambda^\pm_u$ for pseudo-Anosov orbit space universal circles of taut foliations, on atoroidal three-manifolds. We prove that $\Lambda^+_u$ and $\Lambda^-_u$ are completely determined by the stable and unstable prelaminations on the boundary of the orbit space. Then, using a result of Barthelmé, Bonatti, and Mann, we prove that for any action $\rho:\pi_1(M)\to \mathrm{Homeo}^+(S^1)$ coming from an orbit space of a pseudo-Anosov flow, there is a finite collection of lamination pairs such that $(\Lambda^+_u,\Lambda^-_u)$ lies in this collection for any minimal orbit space universal circle whose action is conjugate to $\rho$.

Fréchet means and $L^p$ centres of mass provide notions of average location in metric spaces. On finite-dimensional spheres, existence follows from compactness. On infinite-dimensional spheres, it is not known whether a centre of mass always exists. We show that this is not always the case, and give a simple assumption under which a centre of mass exists. We then show that finding the sample centre of mass of data $x_1, \ldots, x_n$ on the sphere is always an optimisation problem on a subsphere of manifold dimension at most $n$, regardless of the potentially infinite dimension of the sphere. We conclude with some statistical implications.

The finite-length performance of quantum low-density parity-check (LDPC) codes under iterative decoding is governed by small substructures of the Tanner graph, principally short cycles and absorbing sets. While the classical theory of these substructures for quasi-cyclic codes is well developed through discrete Fourier transform (DFT) methods, these tools do not directly address the two-block tensor structure $H_X = [\,\widetilde{H}_1 \mid I \otimes \widetilde{B}^T\,]$ of the lifted-product (quasi-cyclic generalised hypergraph product, QC-GHP) codes that dominate current quantum LDPC constructions. In this paper we develop a quantum-specific spectral framework that exploits this structure. At its core is a DFT block-diagonalisation of $H_X H_X^T$ that reduces moment-trace and cycle computations from an $(r_1\ell)\times(r_1\ell)$ matrix to a sum of $\ell$ small $r_1\times r_1$ Hermitian matrices, with the second block entering only as a scalar shift. From this result we derive a closed-form $4$-cycle count for generalised bicycle codes via additive energies, a joint Sidon characterisation of girth $6$ in the spirit of Fossorier's classical criterion, a Fourier expression for the number of $(3,3)$ elementary absorbing sets in column-weight-$3$ codes via the Wang-Dolecek-Wesel triangle bijection, and a lower bound on stopping-set sizes using the expander mixing lemma.

Let $p\ge 3$ be a prime and $b\ge 2$ an integer such that $p$ does not divide $b$. Then $1/p$ has a periodic digit expansion with respect to the basis $b$. The length $l$ of the period is the (multiplicative) order of $b$ mod $p$. If $l$ is even, then the mean value of the digits of a period is just $(b-1)l/2$. The case of an odd length $l$ is more interesting. If $l=(p-1)/2^m$ is odd, the mean value of the digits of a period was given previously. This mean value involves generalized Bernoulli numbers. However, it is not clear how this result can be generalized to an arbitrary odd length $l$. In the present note we settle this case.

Motivated by the Lester-Wigman vanishing area correlation conjecture for lattice points near the boundary of circles of growing radius, we investigate the dynamics of circle flows on tori (which are related to the motion of a charged particle in a magnetic field on a torus.) We show that an analogue of the vanishing correlation conjecture holds in this setting, i.e., we have "mixing for the area observable" despite the flow being essentially integrable. We also determine the probability density function of the areas, in global as well as local regimes.

The paper addresses the problem of finding the global minimum of a polynomial on a coordinate parallelepiped or on certain other sets with a simple structure. To solve the problem, the Variable Elimination Algorithm for polynomial optimization problems described in the author's previous works is used. It should be noted that the proposed method can be of practical significance only for problems of low dimensionality and for polynomials of not very high degree. This is due to the fact that, when implementing the VEA, a large number of alternative problems may arise concerning the determination of real roots of polynomials, and these polynomials may have very high degrees. Although, in theory, the Variable Elimination Algorithm is applicable to any polynomial optimization problems, there are often practically relevant problems that can be successfully solved using the proposed method.

Air and missile defense, naval force protection, and critical-infrastructure security require rapid allocation of scarce effectors against multiple incoming threats when neutralization is uncertain and observed only after a firing round. We formulate this defensive-allocation problem as a Markov decision process in which a commander assigns a fixed amount of effector capacity each round to heterogeneous threats under three operational objectives: minimizing expected threat-clearance time, maximizing the probability of clearance by a deadline, and maximizing effective assignments before a deadline. Rather than expensive full dynamic optimization, we study simple time-oblivious policies suited to fast implementation. Fair allocation, which ignores threat difficulty and spreads fire evenly, is highly effective: it is optimal for all three objectives under homogeneous threats or low-capacity engagements. Moreover, when effector capacity scales at least linearly with the number of threats, its threat-clearance time trails that of the optimal policy by at most a constant number of rounds. We further develop threat-difficulty-aware greedy policies for each objective, including a constant-factor guarantee for effective assignment maximization. Numerically, greedy policies are near-optimal across heterogeneous instances, while fair allocation remains a principled choice when threat difficulties are unknown.