Papers

Let $K=\mathbb{Q}(\sqrt{D})$ with $D>1$ squarefree, and let $\varepsilon_+$ be the totally positive fundamental unit of $\mathcal{O}_K$. B. M. Kim proved in 2000 that the octonary diagonal form \[ f=x_1^2+\cdots+x_4^2+\varepsilon_+(x_5^2+\cdots+x_8^2) \] is universal over $\mathcal{O}_K$ whenever $D=n^2-1$ is squarefree. We complete Kim's result to an if-and-only-if classification: $f$ is universal if and only if $D=n^2-1$ for some $n\ge2$, or $D=n^2-4$ for some odd $n\ge3$, in both cases subject to squarefreeness. The second family appears to be new in this context and contains $\mathbb{Q}(\sqrt{5})$ at $n=3$ as a degenerate boundary case, recovering Maaß's classical exceptional three-square phenomenon. Equivalently, $f$ is universal over $\mathcal{O}_K$ if and only if the Blomer–Kala invariant $M_D$ equals $1$; for the two stated families we have $M_D=1$, so the Blomer–Kala universal $8M_D$-variable construction specializes exactly to $f$. The converse leverages a continued-fraction involution $\tau(\gamma)=\varepsilon_+\gamma'$ together with a closed formula in convergent coordinates, a three-consecutive-square-values lemma for primitive quadratic polynomials of positive fundamental discriminant, and an even-root exclusion lemma derived from complete-quotient norm bounds.

We classify all left-invariant real affine connections in dimension three. Our approach reduces the three-dimensional problem to a two-dimensional one by decomposing each left-invariant affine connection into a two-dimensional part and an additional one-dimensional component. After characterizing all possible two-dimensional left-invariant affine connections, we return to the three-dimensional setting to obtain a simplified description of all three-dimensional left-invariant affine connections. We then explicitly solve the resulting simplified quadratic equations and perform a refined analysis up to isomorphism, leading to a complete classification. Furthermore, we determine several geometric and algebraic properties of these structures, including the Novikov, associative, radiant, and bi-symmetric conditions, as well as geodesic completeness.

We consider the Tanaka-Webster geometry of surfaces embedded in a 3-dimensional Lie group with a CR structure inherited by a contact form. We define the notions of Gauss and mean curvature and give specific examples.

We investigate indistinguishability of identical bosons and fermions undergoing arbitrary particle-number-preserving evolutions of their visible degrees of freedom. For the projective indistinguishability measure, defined by the projection of the visible state onto the symmetric/anti-symmetric subspace, we derive an equivalent expression in terms of the dynamically invariant internal state. We further generalize the textbook symmetrization/anti-symmetrization framework for bosons and fermions to arbitrary partial distinguishability by deriving an explicit reconstruction formula for the multiparticle visible state in terms of the indistinguishability function encoding the dynamical invariants. We give complete characterization of the class-functions of indistinguishability by projective measures on generalized symmetries. Finally, we reveal a strikingly counterintuitive effect: introducing additional particle label states can increase the multiparticle indistinguishability of identical particles. The effect originates from the cancellation of collective multiparticle phases.

Let $K/\mathbb Q_p$ be finite and let $f\in\mathcal O_K[X]$ be monic, of degree at least two, with $f'(X)\in\mathfrak m_K\mathcal O_K[X]$, equivalently $\bar f\in k[X^p]$. For a compatible inverse branch $f(t_{n+1})=t_n$ with $t_0\in\mathcal O_K$, put $K_n=K(t_n)$ and $K_\infty=\bigcup_nK_n$. We prove that $K_\infty/K$ is either unramified or deeply ramified. More precisely, once ramification appears, the ramification indices over the maximal unramified subfields tend to infinity and the finite-level differents are unbounded. In the Frobenius-type case $f(X)\equiv X^{p^a}\pmod{\mathfrak m_K}$ the unramified alternative is trivial, so $K_\infty=K$ or $K_\infty/K$ is deeply ramified. After completion, the non-unramified alternative gives perfectoid fields and examples show that APF property need not hold at the algebraic level.

We introduce a binary matroid framework for the enumeration of mod $2$ toric-colorable seeds of fixed Picard number. Working with binary matroids up to isomorphism, we organize them through their contraction structure and recursively enumerate weak pseudomanifold subcomplexes by a dynamic programming algorithm. The resulting method combines these structural reductions with a Gray-code traversal of the mod $2$ kernel of the ridge–facet incidence matrix. Using this framework, we complete the classification of mod $2$ toric-colorable seeds of dimension four and Picard number five, proving that there are exactly $198{,}846$ isomorphism classes. This case was computationally infeasible for the GPU-based algorithm previously used by Choi, Jang, and Vallée to treat Picard number four. We further verify that each of these seeds admits an integral characteristic map. As a validation of the method, we also reproduce that Picard number four classification: the weak pseudomanifold enumeration stage drops from over ten days on a GPU to ten minutes on a single CPU.

We study the ground-state entanglement entropy of free fermions on random graphs. We first establish a general criterion for the volume law on random graphs, based on the spectral characteristics of the Hamiltonian. We then apply this criterion to the Erdős–Rényi random graph, where each of the possible edges is present independently with some probability. Using random matrix theory and asymptotic freeness, we show that the ground-state entanglement entropy obeys an exact volume law in Erdős–Rényi graphs in the thermodynamic limit, with a universal coefficient that is independent of the edge probability of the graph. This coefficient is confirmed numerically to take the value approximately $0.3863$ nats, strictly below the Page value. The volume law therefore reflects the absence of geometric locality in the random graph.

It is well known that solutions of elliptic equations inherit analyticity from analytic coefficients, while much less is understood about the inheritance of ultra-analytic regularity, especially for nonlocal equations. This paper develops a systematic Fourier-analytic framework to study fractional and more general nonlocal pure-potential equations whose potentials satisfy ultra-analytic derivative bounds. We prove sharp quantitative logarithmic ultra-analytic estimates for normalized solutions, and show that both the logarithmic power and the leading constant involving the fractional exponent are optimal in natural periodic model examples. We also establish a general transfer principle for weighted ultra-analytic scales, which reveals why standard scales are not preserved, and singles out a natural family of invariant ultra-analytic spaces.

Given $n\in \mathbb{N}$, $p\in [1,\infty)$, and a weight $\gamma$ satisfying the local Muckenhoupt $A_p$ condition, we introduce a weakened version of the Ahlfors–David codimension-$\theta$ regularity condition for Ahlfors–David $d$-regular sets $E\subset\mathbb{R}^n$, where $d\in(0,n)$ and $\theta\in(0,p)$. Under this assumption, we provide a complete intrinsic description of the trace-space of the weighted Besov space $B^{\frac{\theta}{p}}_{p,1}(\mathbb{R}^n,\gamma)$ to $E$. In particular, our results cover the case of power-type weights $\gamma(x)=|x|^\alpha$ with $-n<\alpha<n(p-1)$, $\alpha\neq -(n-1)$, when $E=\mathbb{R}^{n-1}$. This extends earlier results obtained by Haroske and Schmeisser.

Let $A$ be a finite non-empty subset of an abelian group $G$, and let $r_A(d)=|\{(a,a')\in A^2:a-a'=d\}|$. Croot and Lev asked whether the pointwise half-threshold condition $r_A(d)\ge |A|/2$ for every $d\in A-A$ forces $A-A$ to be either a subgroup or a union of three cosets. We resolve this open problem in its sharp general form by identifying the essential obstruction: the statement is false in arbitrary abelian groups, but becomes true after excluding non-zero two-torsion. More precisely, if $G$ is two-torsion-free and the half-threshold condition holds, then either $A-A$ is a finite subgroup of $G$, or there are a finite subgroup $H\le G$ and elements $x,g\in G$ such that \[ A=(x+H)\cup(x+g+H). \] The two-torsion-free hypothesis is essential: for every $r\ge1$ we construct $A\subseteq\F_2^{2r+1}$ with $A-A=\F_2^{2r+1}\setminus\{t\}$ such that every non-zero represented difference has exactly $|A|/2$ representations, giving genuine counterexamples to the Croot–Lev conclusion. The proof of the positive result combines a Kneser quotient reduction with Lev's formulation of Kemperman's critical-pair theory.

We prove a Schur–Zassenhaus theorem for finite skew braces. More precisely, if \(B\) is a finite skew brace and \(I\) is an ideal of \(B\) such that \(|I|\) and \(|B/I|\) are coprime, then \(I\) admits a complement in \(B\).

A brief overview of results concerning the connection between the Hilbert-Polya conjecture and the Riemann hypothesis about the Riemann zeta function, some new results on p-adic quantum computing, quantum entanglement based on lattice spin models and algebraic entanglement models is given. Quantum computing uses both photons and electrons, so their known properties are (very briefly) presented.

We reconstruct Gaussian operator algebras from finite source-labelled observations, recovering coefficients, ordinary multiplication, and completion in independently specified Schatten norms. Gaussian duality and ordered contractions give a reconstruction theorem including the trace-class endpoint, while entire Mehler responses describe the resulting topology. In the Hilbert coefficient geometry, symmetric Rademacher systems realize the same algebra as a fluctuation limit: microscopic products converge at inverse site-number rate in the original norm scale, and finite decoding commutes with this limit. Relative Gram residuals determine target-dependent inverse costs. For bounded tuples, positive word observations close into completely positive maps, with contextual Schwarz defects controlling realization and multiplication. Positive mass and source innovations characterize compactness. Spectral invariants, Wick operators on closed manifolds, and Gaussian networks provide applications with explicit noise and tail bounds.

Parameter estimation in nonlinear dynamical systems from observational data is a fundamental inverse problem with applications in many disciplines. In practice, this is further complicated by the fact that observations are often noisy, sparse, and available only for a subset of the state variables. Furthermore, the initial condition (IC) may be unknown or inaccurate, causing further complications for chaotic systems with sensitive dependence on initial conditions. In this work, we develop a data-assimilation-augmented optimization framework for parameter estimation in ordinary differential equations using partial state observations. The method introduces a nudged system driven by the available observed component and estimates the unknown parameters by minimizing a cost functional, defined as a time-delayed mismatch between the observations and the corresponding observed component of the nudged solution over the admissible parameter space. Since the nudged system can be arbitrarily initialized, this approach eliminates the dependence on accurate IC. Using the Lorenz-63 system as a test case, we establish theoretical results showing synchronization of the nudged solution under parameter agreement, stability under parameter mismatch, and well-posedness of the data-to-parameter inverse map under suitable nondegeneracy conditions. Structural & practical identifiability, and Sobol sensitivity analyses are incorporated to assess which parameters can be reliably estimated from the observations. Numerical experiments in both chaotic and non-chaotic regimes show that this framework accurately recovers parameters from noisy partial observations. Comparisons with an on-the-fly parameter learning method and with Bayesian MCMC estimation demonstrate that the proposed method remains accurate under partial observations and higher noise levels while requiring substantially lower computational cost.

In this paper, we prove an approximation and interpolation theorem for maxfaces in the Lorentz–Minkowski $3$-space $\mathbb{L}^3$. Alarcón, Forstnerič, and López established approximation and interpolation theorems for conformal minimal surfaces using the Enneper–Weierstrass representation formula. We survey their methods and apply them to maxfaces. Furthermore, by incorporating singularity criteria based on the Weierstrass data of maxfaces into the approximation and interpolation theorem, we demonstrate the existence of a maxface with prescribed singularities at specified points, as well as the existence of a maxface whose singular set has a dense image in $\mathbb{L}^3$.

In this note, we give a Turán theorem for Cayley graphs $\Cay(\Z_p,S)$ over prime cyclic groups $\Z_p$. For a graph $F$ and a finite abelian group $G$, define the Cayley–Turán number by \[ \exCay(F,G) = \max\{|S|:S=-S\subseteq G\setminus\{0\},\ \Cay(G,S)\text{ is }F\text{-free}\}. \] Using a polynomial method, we prove that for every odd prime $p$ and every $1\le r\le p-1$, \[ \exCay(K_{r+1},\Z_p) = p-1-2\left\lfloor\frac{p}{r+1}\right\rfloor . \] The extremal construction is the complement of the short-difference interval \[ D_0=\{0,\pm1,\ldots,\pm\lfloor p/(r+1)\rfloor\}. \] We also discuss what changes for general finite abelian groups, showing why the exact prime-cyclic formula does not extend verbatim to composite cyclic groups.

Let $p$ be an odd prime number and let $\overline{\mathbb{F}}_p$ be a fixed algebraic closure of the finite field of order $p$. Let $K$ be a global function field of characteristic different from $p$ and let $G_{K}$ be the absolute Galois group of $K$. We prove that there are only finitely many isomorphism classes of continuous geometric semisimple representations $\rho:G_{K}\to \mathrm{GL}_{n}(\overline{\mathbb{F}}_{p})$ such that their Artin conductors are bounded. It is worth emphasizing that we do not need to assume that $p$ does not divide $n$.

The purpose of this paper is to give a natural divisor-theoretic formulation of the counting method introduced by Halburd for computing degree growth, in a form applicable to birational dynamical systems on varieties of arbitrary dimension. Instead of counting only preimages of special values, we follow time-indexed divisorial conditions through singularity patterns. These conditions are recorded on normalized finite-window orbit graphs, where the relevant multiplicities are realized as divisorial valuations of pullbacks of time-indexed divisors. This construction explains how the elementary computations appearing in singularity patterns can be interpreted as degree relations on a single normal variety. We then show that further relations arise from the failure of functoriality of pullbacks: when the center of a divisor enters the relevant indeterminacy locus, a degree-drop divisor appears. Under suitable finite-type assumptions, the two kinds of relations lead to closed linear difference systems governing degree sequences. Several examples, including higher-dimensional ones, demonstrate that the two mechanisms are complementary and that their combination determines the degree growth in cases where either mechanism alone is insufficient.

This letter addresses long-term fairness in uplink non-orthogonal multiple access integrated sensing and communication (NOMA-ISAC) systems. Existing resource allocation schemes that maximize instantaneous sum rate often favor strong users, leaving historically underserved users with poor long-term throughput. We propose PF-JUGPA, a proportional-fair scheduling based joint user grouping and power allocation method. PF-JUGPA first pre-selects users via a PF metric combining instantaneous rate proxies and historical averages, then performs fairness-aware grouping and power allocation by maximizing a weighted sum rate whose weights are inversely proportional to historical service rates. Simulation results show that PF-JUGPA significantly improves the Jain fairness index and weak-user average rates with only a modest sum-rate loss compared to sum-rate-oriented and round-robin baselines. The findings confirm that embedding long-term service history into both scheduling and resource allocation yields an effective throughput–fairness–sensing tradeoff in uplink NOMA-ISAC.

Let $M$ be a closed orientable 3-manifold and $k$ be a knot in $M$. Then the Dehn surgery of $M$ along $k$ with slope $\alpha$ is not surface fibered for all but a sparse set of slopes.

Recently, a family of unconventional integrators for higher order ODEs with polynomial vector fields was proposed, based on the polarization of vector fields. The simplest instance is the by now famous Kahan discretization for first order ODEs with quadratic vector fields. All these integrators possess remarkable conservation properties. In particular, for the first and the second order Hamiltonian ODEs, the discretization by polarization possesses an integral of motion and an invariant volume form. In this note, we extend our previously proposed algebraic approach to derivation of these integrals to discretizations of ODEs of an arbitrary order. For all orders $\ge 3$, these integrals are new.

In the category of digroups there is a natural pretorsion theory in which the torsion-free digroups are all groups, and torsion digroups form a category isomorphic to the category of non-empty sets. It is also possible to extend the theory of heaps from groups to digroups. The corresponding notion is that of a diheap.

Let $D$ be the ring of $S$-integers in a global field and $\widehat{D}$ its profinite completion. We propose a profinite version of the Bateman–Horn conjecture over $D$ and provide a first comparison with the classical one and its generalizations. Our approach is based on the new notion of procounting measure: a distribution on $\widehat{D}$ which should be seen as a profinite analogue of the counting function for a subset of $\mathbb{R}$. This allows us to deal with subsets of $\widehat{D}$ having Haar measure $0$ (corresponding to density zero in $\mathbb{R}$).

The celebrated Chern conjecture asserts that any closed minimal hypersurface in $\mathbb{S}^{n+1}$ with constant scalar curvature is isoparametric. In this paper, we resolve this conjecture in the affirmative for $M^4 \subset \mathbb S^5$ under the assumption that the Gauss-Kronecker curvature $K$ is constant. This result breaks the traditional reliance on consecutive trace conditions, demonstrating that the nonconsecutive spectral invariant set $\{H, S, K\}$ is sufficient to yield complete geometric rigidity. To overcome the analytical singular locus, we construct two novel weighted $3$-forms adapted to $S$ and $K$. Crucially, the global curvature estimates required to close our analysis are obtained unconditionally by proving the Euler characteristic $\chi(M)=0$. This local-to-global approach provides a new paradigm for higher-dimensional rigidity problems.

An $(n,R)$-covering sequence over a finite alphabet $\Sigma_q = \{0,1,\dots, q-1\}$ is a cyclic sequence whose consecutive length-$n$ windows form a covering code of radius $R$. Equivalently, every word in $\Sigma_q^n$ is within Hamming distance $R$ of at least one window. We give a deterministic and explicit construction of such sequences whose length, for every fixed alphabet size $q$, every fixed radius $R$, and every sufficiently large $n$, attains the sphere-covering lower bound up to a constant factor depending only on $q$ and $R$. Thus, in the fixed-radius regime, the construction removes the logarithmic factor in the general probabilistic upper bounds of [Chung and Cooper, Random Structures & Algorithms, 2004] and [Vu, Advances in Applied Mathematics, 2005]. It also complements the earlier explicit constructions of [Chee, Etzion, Ta, and Vu, Designs, Codes and Cryptography, 2025], which include constant factor bounds for the special binary radius-one families \(n=2^a-1\) and \(n=2^a\), where \(a\ge1\).

In this paper, we considered a heterogeneous Cattaneo-Vernotte equation with an exponential type of diffusion coefficient under the fundamental initial and boundary conditions stating that the solution vanishes at $+/-$ infinity. Owing to the Laplace transform method we obtain two forms of exact analytical solutions which are presented in terms of the ratio of modified Bessel functions. Using the theory of complete monotone functions, we show that the obtained solutions are probability density functions.

We establish four local-to-global propagation results for Dirichlet–to–Neumann maps. Our first two results are proved in the general setting of smooth compact Riemannian manifolds with boundary. The first shows that if two smooth Riemannian metrics coincide in a collar neighborhood of a connected boundary component \(\Gamma\), then equality of the corresponding local Dirichlet–to–Neumann maps on a nonempty open subset of \(\Gamma\) propagates to equality of the associated global Dirichlet–to–Neumann maps on all of \(\Gamma\). The proof combines unique continuation and self-adjointness arguments. The second replaces the geometric collar assumption by an exponential spectral assumption on the difference of the corresponding global Dirichlet–to–Neumann maps. The proof relies on the spectral unique continuation theory of Jerison–Lebeau, through the formulation of Le~Rousseau–Lebeau. Our third and fourth results establish local-to-global propagation principles under Ingham-type quasi–analytic spectral assumptions. Assuming that the boundary manifold is respectively a compact Riemannian symmetric space or a compact quasi–analytic Riemannian manifold, they rely on the propagation theorems of Ganguly–Thangavelu and of Bhowmik–Pradhan. As an application, we consider a class of conformally warped product metrics. In this setting, the local Borg–Marchenko theorem and Weyl–Titchmarsh theory relate the required Ingham-type spectral decay to a suitable quasi–analytic boundary closeness of the conformal factors, yielding new local-to-global uniqueness results for Dirichlet–to–Neumann maps.

We give explicit counterexamples to two questions. One is asked by Pogorelov and the other is by Toponogov. These questions concern the existence of closed asymptotic curves in a saddle surface, namely a complete immersed regular surface in $\mathbb{R}^3$ with nonpositive Gaussian/sectional curvature, and its geometric consequences under some topological conditions. We also modify the statements and prove a corrected version. In the appendix we include an example clarifying a conjecture of Milnor.

Circular operators have been studied extensively since the work of R. Gellar, who conjectured that every circular operator on a complex separable Hilbert space is strongly circular. In this short note, we show that circularity and strong circularity coincide for bounded operators that are finite or countably infinite direct sums of irreducible operators. This considerably narrows the search for potential counterexamples to Gellar's conjecture. As an application, we prove that every circular operator in the Cowen-Douglas class is strongly circular. In addition, we obtain several general results on circular operators that reveal the significance of the hyper-range and the Cauchy dual.

Generalized approximate message passing (GAMP) equipped with minimum mean-square error (MMSE) denoisers, commonly referred to as Bayes-GAMP, is a powerful framework for solving inverse problems described by generalized linear models (GLMs) with arbitrary component-wise nonlinearities in the observation process. However, despite its theoretical tractability and rigorously established asymptotic optimality, the range of practical observation models for which Bayes-GAMP admits a closed-form implementation remains severely limited, particularly in complex-valued settings. This limitation largely stems from the restrictive requirement that the corresponding output denoiser, given by a conditional expectation, admit a closed-form expression. To overcome this limitation, we propose a principled approach that enables the implementation of Bayes-GAMP for complex-valued models with virtually arbitrary nonlinear observation mappings. Specifically, within a score-matching framework, we train a neural network to emulate the output denoiser using training data generated from a characterization of the message dynamics based on state evolution (SE). Notably, the proposed approach requires neither explicit evaluation of the denoiser nor knowledge of an explicit functional form of the nonlinear mapping; it requires only access to forward evaluations of the mapping during offline training. We show that, under ideal training conditions, GAMP with the trained network replacing the analytically intractable denoiser asymptotically matches the performance of Bayes-GAMP with the exact denoiser.