Papers

We derive approximation rates for Sobolev functions defined on the complex unit sphere and complex projective space, representing quantum states, and on the special and projective unitary groups, representing quantum unitary operations. By using harmonic analysis on compact groups and compact homogeneous spaces, we characterize the corresponding Peter–Weyl decompositions and Sobolev spaces via irreducible representations and the Laplace–Beltrami operator. Combining spectral truncation with finite-depth quantum-circuit approximation, invoking the Solovay–Kitaev theorem, we establish quantitative rates that separate the spectral truncation error from the circuit approximation error.

We prove that each simplex in $\mathbb{R}^4$ contains a point with at least $14$ emanating normals to the boundary. Earlier, the authors proved that each generic simple polytope in $\mathbb{R}^4$ contains a point with at least $12$ emanating normals to the boundary. This problem belongs to a piecewise linear counterpart of a longstanding problem concerning normals to smooth convex bodies.

Let $R$ be a Noetherian regular local ring of mixed characteristic and dimension four, and let $r$ be a member of a regular system of parameters. We prove that every finitely generated projective module over $R[1/r]$ is free. We also prove that $\mathrm{Sp}_4(R[1/r])$ acts transitively on unimodular columns of length four and that every determinant orientation of a rank-two projective module is induced by a basis. The argument reduces freeness to two vanishing results: the degree-three Zariski cohomology of the naive degree-four Milnor–Witt sheaf and the alternating Witt group of $R[1/r]$. We establish these results using integral degree-two algebraic $K$-theory, supported hermitian $K$-theory, and local motivic cohomology. A finite-rank symplectic comparison relates the image of the completion obstruction to Milnor–Witt multiplication and yields the required symplectic frames.

We consider the question which functions $f$ may appear as limiting functions of the $N$-point correlation. We obtain sharp point-wise lower and upper bounds. The explicit examples constructed to show the sharpness of the bounds are based on van der Corput sequences.

The BROJA partial information decomposition provides an operationally motivated way to separate the information that two sources carry about a target into unique, redundant, and synergistic components. For continuous and high-dimensional variables, however, computing the BROJA decomposition requires solving a difficult optimization problem. Previous work showed that, when each source-target pair is jointly Gaussian, the problem can be reduced to finite-dimensional numerical optimization. Here, we derive a closed-form solution to this optimization problem, enabling direct computation of the BROJA decomposition in the general multivariate Gaussian setting. The resulting closed-form expression enables speedups exceeding two orders of magnitude for high-dimensional Gaussian BROJA PID. Code is available at https://github.com/brainsandmachines/Eigen_PID.

This paper deals with the problems of decision theoretic estimation of the scale parameter, $\sigma$, and the entropy (Shannon or Renyi) in general scale mixture models with unknown location. Employing Kubokawa's approach, under convexity of the loss $L(t)$ either in $t$ or in $lnt$, improved estimators of $\sigma$ are derived. If $L(t)$ satisfies an additional condition, the problem reduces to estimating $\sigma$ in the degenerate (scale mixture) model. As the entropy is a linear translation of $ln\sigma$, its estimation under a loss $L_0(u)$ amounts to estimating $\sigma$ under $L(t)=L_0(lnt)$, with the quadratic loss $L_0(u)=u^2$ and Brown's loss $L(t)=(lnt)^2$ being a typical case. Thus, in the case that $L_0(u)$ is convex improved estimators of the entropy are directly obtained from respective estimators of $\sigma$. The results are applied to scale mixtures of normal and exponential distributions, including the multivariate $t$ and multivariate Lomax distributions.

The DP color function $P_{DP}(\mathcal{H}, k)$ of a hypergraph $\mathcal{H}$ is defined as the minimum possible value of $P_{DP}(\mathcal{H}, \mathcal{F})$ taken over all $k$-fold covers $\mathcal{F}$ of $\mathcal{H}$. It serves as an extension of the classical chromatic polynomial. In this paper, we investigate the DP color function for two constructions: the join of a hypergraph with a complete graph and vertex-gluings of certain hypergraphs. Both constructions extend some results of Becker et al.

These notes summarize the content of a one-week course given by Luis Vega on the binormal flow model applied to describe the dynamics of three-dimensional vortex filaments in incompressible fluids following the Euler equations. This topic touches connections between the binormal flow, the Schrödinger map and the 1d cubic nonlinear Schrödinger equation, providing a rich casuistry linked to the physical and numerical phenomenology. This course was part of the $2^{\text{nd}}$ Advanced Summer School held at the Institut d'Etudes Scientifiques de Cargèse (IESC), at Cargèse (Corsica), from $20^{\text{th}}$ to $30^{\text{th}}$ June 2023. These notes have been written and completed by two of its attendees, Elena Danesi and Pablo Merino.

Recovery risk measures incorporate explicit liability-recovery requirements into asset-liability risk assessment. We study their specialization to optimized certainty equivalents (OCEs), allowing the underlying utility specification to vary across recovery levels and encompassing Recovery Average Value at Risk (RecAV@R) as a special case. For finite recovery specifications, we use the dual penalty representation to characterize redundant recovery regimes and derive position-independent screening conditions. For a subfamily of the Cressie-Read power-divergence family, including entropic and truncated-quadratic specifications, we obtain an exact characterization on atomless probability spaces: universal redundancy is equivalent to a two-dimensional convex-hull condition involving recovery levels and divergence-scale parameters. On such spaces, universal redundancy for RecAV@R reduces to pairwise dominance, whereas for power-divergence OCEs it can arise from the joint effect of several recovery regimes. For nonlinear OCE utilities, we develop a certified piecewise-linear approximation framework for portfolio optimization. In finite-scenario models, the approximating problems are finite-dimensional linear programs, while utility approximation errors transfer directly to the Recovery OCE, yielding explicit error bounds and convergence rates. Uniform approximation over the portfolio domain further implies convergence of optimal values and stability of optimal solutions. A numerical asset-liability study illustrates the effects of recovery-dependent risk preferences and the accuracy and convergence of the approximation.

This paper presents a finite-time stabilization method for the integrator chain via optimal control techniques by introducing a tailored cost function. We analyze key properties of the resulting value function to derive the corresponding Hamilton-Jacobi-Bellman equation. Numerical simulations illustrate the effectiveness of the approach.

We reprove and generalize a result of Derksen describing the Hilbert polynomial of the vanishing ideal of a transversal subspace arrangement in projective space. When the subspaces have all codimension 2, we describe explicitly the local cohomology modules of the corresponding defining ideal. This is achieved via a complex naturally associated to the subspace arrangement, whose basic properties are also investigated.

Compression schemes provide a general framework for deriving data-dependent risk bounds for learning algorithms. For learning rules based on preferent compression, existing results upper-bound the risk in terms of a sample-dependent complexity $\boldsymbol{k}$, defined as the cardinality of the compression augmented by the training examples on which the reconstructed hypothesis is inappropriate. These results, however, yield a margin of order $O(\sqrt{\ln N/N})$ above $\boldsymbol{k}/N$. In this paper, we establish that the logarithmic factor is not intrinsic. We prove that, for every fixed confidence level $1-\delta$, the risk is bounded with probability at least $1-\delta$ by $$ \frac{\boldsymbol{k}}{N} + O\left(\frac{1}{\sqrt{N}}\right), $$ uniformly over all possible values of $\boldsymbol{k}$. Thus, risk estimates for learning algorithms based on preferent compression attain the same law-of-large-numbers rate as the empirical estimate of the probability of a fixed event. We also show that the leading term $\boldsymbol{k}/N$ is necessary and investigate its relation to the empirical risk. In particular, we identify regimes in which the empirical risk retains the $O(1/\sqrt{N})$ convergence rate, as well as regimes in which it converges more slowly or ceases to be consistent, while the complexity-based estimator preserves the $O(1/\sqrt{N})$ rate.

We prove that the dimension of the deformation cone of any polytope combinatorially equivalent to a 2-cubical zonotope is bounded above by the number of generators of the zonotope. This bound is tight, and we show that it cannot be improved to an identity. We further extend our results to any 2-cubical polytope whose normal fan is combinatorially isomorphic to a pseudohyperplane arrangement, but we show that it cannot be extended to arbitrary 2-cubical polytopes.

For a finite group $G$ and a symmetric connection set $S$, we study the distance Laplacian ($D^L$) and distance signless Laplacian ($D^Q$) spectrum of the Cayley graph $\Cay(G,S)$ for $G$ a finite nilpotent group. Using the internal direct product decomposition over Sylow subgroups, we reduce every question to a single (possibly non-abelian) prime-power factor. In this setting, the required tools are furnished by ordinary representation theory rather than character theory. Based on this framework, we then derive a complete and explicit formula for every eigenvalue, with multiplicity, of the distance matrix, distance Laplacian, and distance signless Laplacian of $\Cay(G,S)$ for $G$ nilpotent: each eigenvalue is indexed by a tuple of irreducible characters of the Sylow factors. Finally, we use the explicit spectrum, together with Perron–Frobenius theory, to determine the eigenvalue-free interval of $D^L$ and $D^Q$.

Goal-oriented communication often struggles to sustain downstream task performance under variable and unpredictable bandwidth. We address this problem by proposing GO-OAE (Goal-Oriented Ordered AutoEncoding), which learns a progressively refinable quantized latent sequence, optimized directly for task utility rather than source reconstruction. Each component of the sequence is represented for transmission by its code index, and the learned ordering enables inference from any received prefix of the index sequence, with performance improving as more indices arrive. We prove that optimizing the goal-oriented distortion objective is equivalent to solving a weighted conditional information bottleneck formulation, and introduce an unbiased multi-prefix importance sampling (MPIS) estimator, to reduce GO-OAE training computational cost from O(BK) to O(Bm) with m << K, and reduces the estimator variance by a factor of 1/m. Experiments show that, on MiniImageNet, GO-OAE achieves over 96% of the lossless upper-bound classification accuracy using 324 bits per image (683X compression vs. raw RGB), and already attains over 94% of the upper bound with only 70% of the transmitted indices. It consistently outperforms reconstruction-optimized codecs and ordered autoencoders trained with uniform prefix sampling across all transmission budgets, demonstrating effective communication at extreme compression rates.

The open-loop minimum-energy control (MEC) problem asks for control inputs that minimize the control energy while ensuring the terminal state reaches a desired value. In this paper, we study the open-loop MEC problem for stochastic linear time-invariant systems. We consider a variety of stochastic constraints that enforce the terminal error to be small, and show that Stochastic MEC (SMEC) problem can be formulated as a QCQP. We obtain the optimal solution for SMEC and show that it generalizes the well-known Deterministic MEC problem by introducing novel stochastic variants of controllability and Gramian matrices. Finally, we show that there exists a fundamental lower bound on the optimal control energy which cannot be overcome by increasing the time horizon of the problem.

We produce the first examples of singular isoperimetric boundaries with non-zero constant mean curvature in Riemannian manifolds. The main technical ingredient is establishing that, for constant mean curvature hypersurfaces containing isolated singularities modelled on strictly stable, strictly minimising regular tangent cones, weak strict stability implies local unique volume-constrained minimisation.

We establish a vector-valued extension of a theorem of Marciszewski. Let $K$ be a compact Hausdorff space, let $\Gamma$ be a set, and let $X$ be a nonzero Banach space containing no isomorphic copy of $c_0$. We prove that if $C(K)$ embeds isomorphically into $c_0(\Gamma,X)$, then $C(K)$ is isomorphic to $c_0(\Lambda)$ for some set $\Lambda$. Moreover, when $\Gamma$ is infinite, the optimal distortion of embeddings of $C(K)$ into $c_0(\Gamma,X)$ equals that of embeddings into $c_0(\Gamma)$. The classification result answers a question posed in an earlier paper.

We study the recovery of a three-dimensional Riemannian manifold from the areas of minimal surfaces. We show that areas of minimal surfaces determine the metric on manifolds foliated by minimal surfaces, up to boundary-fixing isometry. This resolves the conjecture on boundary rigidity for least-area data in an exterior formulation, even without topological restrictions on the minimal surfaces. We also prove global rigidity for Einstein metrics on compact three-manifolds with boundary, assuming $\(\pi_1(M,\partial M)=0\)$ and a non-trapping condition. The Einstein result includes the

We propose a coupling-aware tri-hybrid precoding architecture for simultaneous wireless information and power transfer (SWIPT) using reconfigurable parasitic antennas. By integrating multiport network theory and information theory, we develop a third precoding layer that explicitly accounts for mutual coupling and reactive energy storage within the antenna domain. To efficiently optimize this architecture, we propose a low-complexity tri-hybrid solution comprising coordinate descent (CD) for parasitic reactive load optimization, a singular value decomposition (SVD)-based analog precoder, and a protective partial zero-forcing (PPZF) digital precoder. Numerical results demonstrate that the transmit array introduces impedance mismatch that reduces the real power accepted by the antenna array. Moreover, the proposed tri-hybrid architecture significantly outperforms hybrid precoding by constructively exploiting the coupling effects, yielding substantial gains in spectral efficiency (SE) and sum non-linear harvested energy (NLHE) at a fraction of the hardware cost.

We study the Yang–Mills flow on $\mathbb{S}^4$ restricted to $\mathrm{SO}(4)$-equivariant connections on the trivial $\mathrm{SU}(2)$ bundle. It was expected that antipodal instanton–anti-instanton configurations would lead to infinite-time singularities along the flow. We show that this does not occur. Moreover, for any smooth equivariant initial condition, the flow converges to the trivial flat connection.

A particle beam with constant intensity that propagates into a plasma triggers a bump-on-tail instability. It has been demonstrated that beam modulation can be used to significantly reduce this instability. However, finding the optimal beam profiles that accomplish this is a complex optimization problem that suffers from an extremely rugged optimization landscape and a relatively high-dimensional parameter space. In this paper, we evaluate a number of optimization algorithms and find that population based methods (in particular, CMA-ES and Differential Evolution) perform best. In particular, we investigate beam profiles where the beam is either completely turned ON or OFF. This configuration is easier to accomplish experimentally, but leads to a higher-dimensional and thus more challenging optimization problem. We obtain ON/OFF beam profiles that reduce the instability by more than 80% and thus perform significantly better than what has been reported in the literature.

We establish a spectral theorem for compact self-adjoint linear relations on a Hilbert space. The proof relies on the Arens decomposition of a closed linear relation into an operator part and a purely multivalued part, reducing the problem directly to the classical spectral theorem for compact self-adjoint operators. No finite-dimensionality condition is imposed on the multivalued part. We also prove a finiterank approximation theorem for compact bounded linear relations and a Fredholm alternative for compact self-adjoint linear relations. As an application, we employ the theory of boundary triples to show that the inverse of the Neumann Laplacian on a bounded domain is a compact self-adjoint linear relation. The abstract spectral theorem then recovers the classical eigenfunction expansion of solutions to the Neumann problem: an L2-convergent series in terms of the non-constant Neumann eigenfunctions together with an arbitrary additive constant. The framework extends naturally to general elliptic boundary value problems with finite-dimensional kernel.

We show that if a Banach space $X$ is smooth and almost locally uniformly rotund and if satisfies the following \[0\in W(T) \implies 0\in W(T^{-1})\] for all invertible bounded linear map on $X$ then Birkhoff-James orthogonality in $X$ is symmetric. This result, therefore, characterizes the inner product spaces out of all smooth and almost locally uniformly rotund spaces having three or more dimension via simultaneous zero inclusion property, introduced by Bračič et al. [J. Math. Anal. Appl., 449 (2017) 1413-1423]. Further, this gives answers to some open questions asked by E. Poon [J. Math. Anal. Appl., 534(2024), 128052].

Given a graph $G$, a set $W$ of vertices of $G$ is a resolving set if, for any pair of vertices $z_1$ and $z_2$ of $G$, there is a vertex $w\in W$ such that $d(w,z_1)\ne d(w,z_2)$, where $d(u,v)$ denotes the length of a shortest path between the vertices $u$ and $v$. The metric dimension of a graph $G$, $\beta(G)$, is the minimum cardinality of a resolving set of $G$. A resolving set $W$ is fault-tolerant if $W\setminus \{w\}$ is a resolving set for any $w\in W$. The fault-tolerant metric dimension of a graph $G$, denoted by $\beta'(G)$, is the minimum cardinality of a fault-tolerant resolving set of $G$. In this paper, we study the fault-tolerant metric dimension problem in maximal outerplanar graphs. In particular, we show that there are exactly two maximal outerplanar graphs with $\beta'(G)=3$. We also show that $4\le \beta'(G) \le 6$ for any maximal outerplanar graph $G$ of order at least 7 and metric dimension 2. Moreover, we establish the upper bound $\beta'(G) \le \left\lceil \frac{2n}{3} \right\rceil$ for maximal outerplanar graphs $G$ of order $n\ge 5$, by building in linear time a fault-tolerant resolving set of cardinality $\left\lceil \frac{2n}{3} \right\rceil$.

For an irreducible character $\chi$ of a finite group $G$, its codegree is defined by $cod(\chi)=|G:\ker\chi|/\chi(1)$. In this paper, we characterize fnite $p$-groups whose nonlinear irreducible characters all have the same codegree.

We study the expectation of the discrete total variation statistic $d=\sum_{i=1}^{n-1}|X_{i+1}-X_i|$ of counts $X_1,\dots,X_n$ over $n$ ordered bins with sample size $N$, under the uniform multinomial null. In the notation used throughout the series, a reference distribution $\mathbf r$ defines $d_{\mathbf r}(x)=\|x-N\mathbf r\|_V$; here $\mathbf r=\mathbf u$ is uniform and we write simply $d$. We obtain an exact single-sum formula, a closed form in central trinomial coefficients valid for every $n\ge2$, with one-term evaluations at $n=2,4$ and a complete D-finite characterization for $n=3$. The per-edge expectation $W_N=E|X_2-X_1|$ satisfies an exact three-term recurrence, from which we derive its complete $1/N$ expansion with the $n$-independent constant $E(d^\circ/\sqrt{Nn})\to 2/\sqrt\pi$ for the cyclic statistic $d^\circ=d^\circ_{\mathbf u}$, independent of the number of bins; the complementary sparse regime (fixed $N$, $n\to\infty$) is treated in a separate paper. Conditioning on a two-bin total reduces each edge to a symmetric random walk; as the subsequent parts of the series will show, this method carries the analysis to non-uniform reference laws.

We give a short autonomous recurrence that generates every odd prime, in increasing order, from the single initial value $8$. Its integer state determines both its output and its next update: the lowest set bit gives the next prime, while the highest set bit determines the sieve period. The same state also determines the following prime gap. A related recurrence represents the prime-counting function as the $2$-adic valuation of a single integer. Independently, a direct packing of a finite divisor sieve yields fixed arithmetic constructions for the $n$-th prime, the prime-counting function, and the twin-prime counting function. We express these constructions as straight-line programs using addition, truncated subtraction, multiplication, integer division, exponentiation, and remainder. Counting operations with reuse of computed values gives programs of lengths $166$, $132$, and $167$, respectively; one autonomous recurrence update uses $99$ operations. Explicit arithmetic subroutines remove all auxiliary operations, including bitwise AND. The programs unfold into fixed arithmetic terms without variable-length sums, products, or searches.

We prove finite-sample risk bounds for empirical quadratic and power Fréchet means in geodesic metric spaces with curvature bounded above for observations with arbitrary joint law. We compare the sample with exchangeable copies in which one observation is redrawn independently of the original sample and the others move to preserve the joint law. These displacements control the error, with no entropy or dimension assumptions. For quadratic Fréchet means, this yields a general “coupling bound” in any separable complete $CAT(\kappa)$ space, where for $\kappa>0$ the observations must lie in a closed convex set of circumradius below $\pi/(2\sqrt{\kappa})$. This bound is exact on the real line for nonnegatively correlated Gaussian observations and for comonotone functions of a common factor. We apply it in two settings. Under local dependence, the resulting risk bound grows linearly with the average size of the dependency neighborhoods. Under transport sensitivity and for bounded metric variances, summable sensitivities of the conditional laws to a change in one observation preserve the parametric mean squared error rate, while nonsummable polynomial decay gives explicit slower rates. We verify transport sensitivity for stationary reversible Markov chains with Wasserstein-contractive kernels and for bi-Lipschitz images of Gaussian sequences. In the more challenging setting of heavy-tailed, dependent data with possibly infinite variance, we consider power Fréchet means of order $\alpha\in(1,2)$. Under nonpositive curvature, we bound their $L^\alpha$ risk assuming only finite $\alpha$-th moments.

We provide a strengthened answer to the questions posed by Alonso, Martín, and Papini [Studia Math. 188 (2008), 135–150] concerning the comparison between the von Neumann–Jordan and James constants . The classical inequality $\CNJ(X)\leq J(X)$ between the von Neumann–Jordan and James constants was established by Takahashi and Kato [J. Math. Anal. Appl. 359 (2009), 602–609], Wang [Proc. Amer. Math. Soc. 138 (2010), 695–701], and Yang and Li [Appl. Math. Lett. 23 (2010), 277–281], while related sharp estimates were obtained by Kato and Takahashi [Rend. Circ. Mat. Palermo (2) Suppl. 82 (2010), 75–91]. We sharpen this inequality to $\CNJ(X)\leq\JB(X)$, where $\JB(X)$ is the Birkhoff–James orthogonal James constant introduced by Baronti and Papini [Constr. Math. Anal. 5 (2022), 37–45]. Our proof employs optimal distance ellipses and alternating contact directions.