Papers

We establish a countable-dimensional extension property for the family of continuous nowhere differentiable functions, which we denote by $\ND[0,1]$. More precisely, we prove that every finite-dimensional subspace $F\subset C[0,1]$ whose nonzero elements are nowhere differentiable can be extended to a countably infinite-dimensional subspace $G\subset C[0,1]$ with the same property.

Previously, An \cite{AL01} showed that the self-similar measure $\mu$ generated by a product-form Hadamard triple is a spectral measure. In this paper, we study its spectral eigenvalue problem. A set $A\subset\mathbb R$ is called a spectral eigenvalue set of $\mu$ if there exists a spectrum $\Lambda$ of $\mu$ such that $a\Lambda$ is a spectrum of $\mu$ for every $a\in A$. We introduce the Product-form Hadamard multiplier set $\mathcal{T}_*$, and prove that for any $s\in [0,\frac{\log \#\mathcal{D}}{\log N}]$, the spectral eigensubspace $$V^{(s)}(\mu_{N,\mathcal{D}},\mathcal{T}_*):=\{\Lambda :t \Lambda \text{ is a spectrum of }\mu \text{ for all }t \in\mathcal{T}_* \text{ and } \dim_{Be}(\Lambda)=s\}$$ has the cardinality of the continuum. This result allows us to show that for the four-digit self-similar measures, a real number $t$ is a spectral eigenvalue if and only if $t \in \left\{\frac{u}{v}:u,v\in 2\mathbb{Z}+1\right\}$. And for any subset $S$ of $\mathbb{R}$ is a spectral eigenvalue set if and only if $S \subset t^{-1} (2\mathbb{Z}+1)$ for some $t\in 2\mathbb{Z}+1$.

Inspired by work of Bugeaud-Corvaja-Zannier, we formulate a conjecture about unlikely intersections in powers of the multiplicative group over the ring of integers in a number field. Broadly speaking, if an intersection with a subgroup scheme is unlikely for dimension reasons, its “size" should not be too big compared to the “complexity" of the subgroup scheme. We first obtain some results on likely intersections that serve as a benchmark for the unlikely case and generalize work of Barroero-Capuano-Mérai-Ostafe-Sha. We then show that our conjecture in dimension $1$ follows from work of Corvaja-Zannier, we obtain some partial result in dimension $2$, and we present some open problems that are special cases of the conjecture.

We consider the Burgers equation with a scalar control acting through a fixed spatial profile, in a regime where the linearized system is not controllable. We identify natural conditions on the control profile leading to quadratic obstructions to small-time local null-controllability, quantified by negative Sobolev norms of the control of arbitrary integer order. We show that these conditions are related to iterated Lie brackets. The proof relies on repeated integrations by parts in the quadratic expansion combined with nonlinear remainder estimates. Finally, we construct smooth spatial profiles that produce obstructions at every negative integer order.

The Levi-Civita field $\mathcal{R}$ of formal Laurent series is a constructive, non-Archimedean ordered field that supports a full Lebesgue measure and integration theory, including a Dominated Convergence Theorem. This paper embeds that integration theory into the paraconsistent Chunk and Permeate framework, extending it from elementary calculus to genuine measure theory. The source chunk is modelled by $\mathcal{R}$ with its measure and integral, while the target chunk is the classical real line $\mathbb{R}$. A permeability relation exports the standard part of the internal integral, and it is shown that the Dominated Convergence Theorem permeates from the source chunk to the target chunk, yielding the classical Lebesgue Dominated Convergence Theorem without any choice principles and without the ultrafilters required by nonstandard measure theory. The construction is entirely explicit and demonstrates that paraconsistent logic can provide a rigorous foundation for deep analytical tools while keeping inconsistencies safely confined.

With the rapid development of integrated sensing and communication (ISAC) as a key enabler for future wireless networks, ensuring the security of both communication and sensing functions has become increasingly important. Current secure ISAC studies focus restrictively either on the communication or the sensing security, but not both. To bridge this gap, this paper investigates security for both, i.e., dual-security, in indoor orthogonal frequency division multiplexing (OFDM) based ISAC systems. Specifically, we consider a scenario in which a sensing user (SU) is authorized for sensing but may eavesdrop on communication data, while a communication user (CU) is authorized for communication but may perform unauthorized sensing. We chose this scenario as the pathological case where an authorized eavesdropper has more information and is more effective than an unauthorized one. To address this case, we propose the use of temporal artificial noise (AN) to prevent malicious CU sensing by enlarging its time-domain sensing error, and simultaneously degrade SU data eavesdropping by reducing its frequency-domain signal-to-interference-plus-noise ratio (SINR) with standard OFDM receiver processing. Meanwhile, our proposed scheme guarantees the sensing performance of the SU and the communication performance of the CU. We present numerical results that demonstrate AN can effectively provide dual protection for sensing and communication in OFDM-ISAC systems while guaranteeing the performance of legitimate users.

We develop a non-asymptotic approximation, sampling, and finite-iteration optimization theory for variational physics-informed approximation of uniformly monotone nonlinear multiscale elliptic equations. For boundary-compatible neural feature classes, the population error splits into approximation, empirical quadrature, and projected-gradient terms, with all non-approximation constants uniform in the microscopic scale \(\varepsilon\). Assuming a quantitative corrected \(H^1\)-estimate, a two-scale state class yields \[ \mathcal A_m^\varepsilon \le C\bigl(\varepsilon+\Phi_{0,m_0}^2+\Phi_{1,m_1}^2\bigr) \] in arbitrary dimension. We further introduce a convex primal-dual physics loss whose population value is a computable upper certificate for the state error. With additional flux-corrector regularity, a divergence-compatible two-scale flux class gives a certified state-flux bound combining \(O(\varepsilon)\) approximation, state and flux feature errors, empirical sampling error, and an \(O(K^{-1})\) optimization term. In contrast, for general periodic nonlinear fluxes satisfying a natural nondegeneracy condition, the empirical Rademacher complexities of strong-residual and squared-residual classes are bounded below by constant multiples of \((\varepsilon\sqrt N)^{-1}\) and \((\varepsilon^2\sqrt N)^{-1}\), respectively. These optimizer-independent lower bounds hold in every spatial dimension. Numerical experiments confirm the predicted \(\varepsilon\)- and \(N\)-scalings for nonlinear fluxes in \(d=1,2,3\), validate every computed primal-dual certificate, and show that corrector-enriched classes substantially reduce energy and \(H^1\) errors as the microscopic scale is refined.

Using geometric singular perturbation theory, we construct stationary, periodic, front-type far-from-equilibrium patterns in a cross-diffusion model for plant growth under the influence of toxicity. We show how existing techniques for the analysis of far-from-equilibrium patterns in one spatial dimension can be extended to include cross-diffusion terms and prove the existence of a one-parameter family of these patterns. For a general cross-diffusion model class, we show when front-type patterns may appear depending on the shape of the critical manifold, which is determined by the specific properties of the reaction terms.

We show that the velocity solution $u$ to the generalized Stokes equations with Dirichlet boundary conditions in dimension two and three is the superposition of two divergence-free velocity fields $\omega$ and $\thetaup$, $u=\omega+\thetaup$. Both $\omega$ and $\thetaup$ solve screened Poisson vector equations with right hand side respectively the rotational component in the Helmholtz decomposition of the external force, and the gradient of a harmonic scalar function $q$ called “solid pressure". Similarly, the fluid pressure $p$ is the superposition of the potential $\pi$ in the Helmholtz decomposition of the external force, and of $q$, $p=\pi+q$. It turns out that regardless the pressure $q$, $\omega+\thetaup$ is divergence-free and satisfies the generalized Stokes equations, except the normal component of the Dirichlet boundary condition. The role of $q$, which solves a boundary linear equation with a self-adjoint definite positive operator, is to constrain $\omega+ \thetaup$ to satisfy even the normal component of the Dirichlet boundary condition. The method is attractive for solving numerically the time dependent Navier-Stokes equations in two and three dimensions. It decouples the velocity from the pressure and solves the incompressibility constraint as a byproduct of unconstrained screened Poisson vector equations for $\omega$ and $\thetaup$, equipped with appropriate boundary conditions. In the two last sections we present two methods for solving numerically the generalized Stokes equations, and some results to demonstrate the efficiency of our approach.

We derive a rigorous sufficient condition for the absence of spin-glass order in the Ising spin glass with symmetric binary couplings on Migdal–Kadanoff (MK) hierarchical lattices with even branching number. The key observation is that a single exact renormalization-group step creates zero effective bonds with positive probability, thereby reducing the problem to bond percolation on the corresponding hierarchical lattice. When the induced dilution exceeds the percolation threshold, both spin-glass order and stiffness are absent at all temperatures, including zero temperature. As a consequence, our criterion gives a rigorous proof that the Ising spin glass on the square lattice does not exhibit a spin-glass phase within the MK approximation. More unexpectedly, by choosing sufficiently large scale factors and branching numbers, we construct MK hierarchical lattices whose fractal dimensions are arbitrarily close to three from below but still exhibit no spin-glass order. This sharply contrasts with numerical estimates obtained for MK hierarchical lattices with relatively small scale factors and branching numbers, which placed the lower critical dimension near \(2.52\).

Deodhar diagrams give a combinatorial way to compute point counts of open positroid varieties over finite fields. We introduce affine Deodhar diagrams, which extend this construction to affine patches of open positroid varieties. These diagrams are indexed by a bounded affine permutation together with a lattice path, and this extra flexibility makes them especially useful for recursive bijections. Motivated by connections with Dyck paths, open positroid varieties, and their cluster structure, we construct bijections between several classes of affine Deograms. These bijections give combinatorial proofs of point-count identities that were previously known from geometric isomorphisms.

A balanced $k$-partite $k$-graph is a $k$-uniform hypergraph such that every edge intersects each partition class in exactly one vertex, where each partition class has size $n$. Lo and Markström (2014) determined the minimum vertex-degree threshold for perfect matchings in balanced \(3\)-partite \(3\)-graphs. In this paper, we determine the minimum vertex-degree threshold for balanced \(4\)-partite \(4\)-graphs. The proof relies on a reduction framework for \(k\)-partite \(k\)-graphs, through which the existence of a perfect fractional matching is converted into a finite-dimensional optimization problem.

We study the singularities of some selected integrable lattice equations and show they all admit three types of singularities: one of finite and two of infinite extent. In particular, we show that all the equations we study possess a recently discovered, strip-like, singularity which is known under the moniker of ”taishi”. We study in detail the interaction of these taishi with the remaining two types of singularities and show that the rich behaviour first obtained in the case of the Korteweg-deVries equation is also present for other equations. Moreover, we find that in all cases this behaviour can be encoded in terms of simple symbolic dynamics which are just the dynamics governing a Box & Ball cellular automaton.

In this paper, we investigate divergent orbits for cone actions on products of certain homogeneous spaces. We introduce a notion of essential singularity for such actions, and estimate the Hausdorff dimension of the corresponding singular set. In particular, let $G/\Gamma=\mathrm{SL}(2,\mathbb{R})^s/\mathrm{SL}(2,\mathbb{Z})^s$, and let $C$ be a cone in the positive Weyl chamber with angular aperture $\epsilon>0$. Then the Hausdorff dimension of the set of points with essential divergent orbits under $C$ satisfies that when $\epsilon\in (0,\frac{1}{64})$, $$ 3s-\frac{1}{2}-4(s-1)\epsilon \leq \dim D^e(C, G/\Gamma)\leq 3s-\frac{1}{2}-\frac{1}{3}\epsilon. $$ This extends the previous result of An–Guan–Marnat–Shi \cite{AGMS} to higher-dimensional cone actions.

The minimal well-determined logic in the language with conjunction and implication is investigated. A calculus for this logic, in which the modus ponens rule is not postulated, is proposed. The main result consists in constructing a semantics for this logic: it is formed by the class of lower semilattices with a greatest element, where the implication is interpreted using a partial function defined via the partial order of the semilattice. This extension of the notion of interpreting logical connectives in a matrix allows for the correct determination of the truth of formulas in the language with conjunction and implication. Soundness and completeness theorems are proved. The proposed semantics creates an opportunity to investigate questions of finite model property for such systems and can also serve as a basis for studying other properties of both the minimal well-determined logic itself and its extensions. As an application of the obtained results, we prove that the set of tautologies of the minimal well-determined logic is decidable in polynomial time and present a corresponding decision algorithm.

In this paper, we investigate several geometric and function-theoretic properties of the domain $\mathbf{\Theta}_n$. We obtain new characterizations of its distinguished boundary and introduce the notion of a \textit{$\mathbf{\Theta}_n$-inner function}, together with several illustrative examples. We establish connections between $\mathbf{\Theta}n$-inner functions and $\Gamma_n$-inner functions, tetra-inner functions, and $\mathbf{\Theta}_{n+1}$-inner functions. Furthermore, we derive an explicit characterization of rational $\mathbf{\Theta}_n$-inner functions. As an application, for any finite collection of distinct interpolation nodes in $\mathbb{D}$ and prescribed target points in $\mathbf{\Theta}_n$, we obtain an explicit formula for the rational $\mathbf{\Theta}_n$-inner function satisfying the given interpolation data.

For a fixed $r$, let $f_r(n)$ denote the minimum number of complete $r$-partite $r$-uniform hypergraphs whose edge sets partition the complete $r$-uniform hypergraph on $n$ vertices. The Graham-Pollak theorem asserts that $f_2(n)=n-1$. Let $c_r$ be the smallest constant such that $f_r(n)\le c_r(1+o(1))\binom{n}{\lfloor r/2\rfloor}$. We give two constructions that improve upper bounds for $f_r(n)$. The first partitions $E(K_4)\times E(K_{17})$ into $44$ products of edge sets of complete bipartite graphs and yields $c_4\le11/12$, improving the previous bound of $c_4\le14/15$. The second gives an asymptotically improved exact cover for odd $r$. Combining these constructions, we prove that $c_r<1$ for every odd integer $r\ge65$, which improves upon the previous computer-assisted result of $113$. We further obtain the improved asymptotic bound $c_r\le \frac{r}{15}\left(\frac{11}{12}\right)^{r/4}+o(1)$ for general $r$. These results narrow the gap towards resolving the major open problem of determining whether $c_5<1$.

We investigate how the scaling of the interior-penalty parameter affects nonsymmetric interior-penalty Galerkin (NIPG) discretizations of the viscous rotating shallow-water equations in geopotential variables. The hyperbolic terms are approximated by a local Lax–Friedrichs flux, while viscosity acts on the momentum variables through a penalty law $\mu_e=\sigma h_e^{-\beta}$. The standard choice $\beta=1$ and the super-penalized choice $\beta=3$ are compared with a symmetric interior-penalty Galerkin reference. For the diffusion form, we establish consistency, continuity for $\beta\ge 1$, and an exact coercivity identity in the momentum DG seminorm. Manufactured-solution tests show that super-penalization can recover the expected momentum $L^2$ accuracy, whereas the coupled geopotential variable need not exhibit the same improvement. Rotating and topography-aware tests further show that the standard scaling generally gives the better accuracy-cost compromise for the explicit implementation considered here.

Assessing the asymptotic stability of linear self-adjoint homogeneous systems of differential-algebraic equations requires testing the Hurwitz stability of the associated Hermitian matrix polynomial $P(\lambda)$. Tests for known necessary and sufficient conditions rely on linearizations and eigensolvers, solving matrix equations and testing matrix inequalities, or generalized Bézoutians, and scale with either $O(d^2 n^3)$ or $O(d^3n^3)$ complexity, where $d$ and $n$ are the degree and size of $P(\lambda)$, respectively. We establish several novel sufficient conditions for stability, based on the numerical range of $P(\lambda)$. Based on the new results, we propose algorithms with $O(d n^3)$ asymptotic complexity. Our methods rely on very efficient core numerical linear algebra routines, such as the Cholesky decomposition of $n \times n$ matrices or the computation of the largest eigenvalue of $n \times n$ definite pencils. Therefore, a significant computational advantage can be expected in favor of the proposed approach even for moderate values of $d$ or $n$, and we verify this with numerical experiments.

We prove a Furstenberg-type criterion for a given orthogonal projection of a self-similar measure to be absolutely continuous, with quantified regularity. It requires exponential mixing of the rotational part at a rate that is sufficiently fast compared with an orbit relative analogue of its dimension. Using Ramanujan sets of irrational rotations in \(\mathrm{SO}(3)\) constructed by Lubotzky, Phillips and Sarnak (1986, 1987), we obtain explicit applications. In particular, we exhibit singular self-similar measures whose every line projection is absolutely continuous, measures of arbitrarily small Fourier dimension with smooth projections in all but a fully explicit exceptional set of directions, and a non-trivial example of a self-similar measure that is Salem with a $C^2 _0$ density.

Fuglede's conjecture asserts that a measurable set of positive and finite measure is spectral if and only if it tiles Euclidean space by translations. Counterexamples are known in every dimension $d\ge3$, whereas the one- and two-dimensional cases have remained unresolved. We construct two explicit $60$-point subsets of the rank-two finite Abelian group $\Z_{60}\times\Z_{12}$: one is a translational tile with no spectrum, and the other is spectral but does not tile. A finite-to-infinite transference principle lifts them to bounded subsets of $\R^2$ that are finite unions of unit squares. Consequently, both implications in Fuglede's conjecture fail in dimension two.

Topos causal models recast causal inference inside a topos: a causal world is a presheaf, an intervention is a sub-model named by a characteristic map into the subobject classifier $\Om$, and reasoning is Kripke-Joyal forcing in an intuitionistic internal language. We give the first axiom-free machine-checked account of this 1-topos core, in Cubical Agda over a previously verified probability monad and do-calculus; the framework is otherwise developed on paper, with central claims stated rather than proved. Three of our results go beyond faithful transcription. We exhibit a contextuality obstruction the programme does not treat: pairwise-consistent local causal data with no global model, detected by a degree-one holonomy class. We delimit the claim that interventions are modelled by the subobject classifier: an intervention and an observation name the same subobject, so $\Om$ fixes the target of a do-operation but not the operation itself, which is surgery on the kernels — where, on a confounder, the interventional and observational laws differ. And we settle the modal unit — inflationarity is derivable from $j\top = \top$ and naturality, not a fourth axiom. We also machine-check the classifier of sieves with its classification theorem, the pullback collating local mechanisms, and the Kripke-Joyal forcing clauses. The development assumes no axioms and typechecks under Agda's –safe flag, with the ordered field discharged at $\mathbb{Q}$; type-level sheafification and a directed do-calculus are future work.

Over the past 30 years, the special limiting procedure known as ultradiscretisation has become the tool of choice in the field of infinite dimensional integrable systems for constructing cellular automata that exhibit solitonic behaviour, as e.g. in the Korteweg-de Vries equation. A lesser known fact is that, in many cases, ultradiscretisation can also be used to construct cellular automata that retain the essential dynamical features (such as the existence of limit cycles etc.) of a dynamical system expressed in terms of ordinary differential equations (ODEs). In its standard application, the ultradiscretisation procedure relies on the prior construction of a `good' discretisation of the dynamical system at hand, that shares the essential dynamical features of the ODE and which is sign-free, making it amenable to the ultradiscrete limit. In this paper we show, on a simple but generic model, that even if one starts from a discrete system with different dynamical features than the ODE one obtains as its continuum limit, the ultradiscrete limit can be tweaked such that the dynamics of the resulting cellular automaton is closer to that of the continuum limit than to that of the discrete model itself.

Standard phylogenetic reconstruction techniques often yield unrooted phylogenetic networks; these are subsequently rooted to infer evolutionary history. A common problem in this process is to determine the structural classes to which the resulting network will belong. In this paper, we investigate unrooted networks in which the choice of any root results in a valid rooted phylogenetic network, a property we define as robustly orientable. We then establish a strict structural condition for this class, specifically, that an unrooted network is robustly orientable if and only if it contains no sink components. We also show that if an unrooted network is level-$2$ or less, or if it is tree-based, then it is robustly orientable. Furthermore, we define an unrooted network to be {\em robustly class $\mathcal C$} if the choice of any root results in a network belonging to class $\mathcal C$. We demonstrate that an unrooted network is robustly tree-child or robustly stack-free if and only if it is level-$1$ or less. Finally, we show that a phylogenetic network is robustly normal if and only if it is a phylogenetic tree.

We consider whether every quantum random access code (QRAC) with density-operator encodings and arbitrary decoding measurements obeys the conjectured bound $p\leq(1+\sqrt{m/n})/2$, where $n$ classical bits are encoded into $m$ qubits and $p$ is the worst-case success probability. We find that classical random access codes with private randomness, which form a subclass of this QRAC model, violate the bound. We embed these classical codes as QRACs with diagonal encoding states and commuting decoding measurements, and construct pure-state realizations with identical decoding statistics. The achievability theorem of Ambainis, Nayak, Ta-Shma, and Vazirani then yields violations for every fixed $p\in(1/2,1)$ at sufficiently large input length. The counterexamples span the full open interval between the conjectured and Nayak bounds at each fixed compression rate. A finite-blocklength analysis further yields order-optimal logarithmic qubit scaling for a recovery bias scaling as $\sqrt{\log_2 n/n}$ with a sufficiently large prefactor. These results identify the classical coding rate as the source of the separation and motivate restricted bounds based on quantitative spectral properties of decoding measurements.

For a 1-factorization $F=\{M_1,\ldots,M_d\}$ of the hypercube $Q_d$, let $G[F]$ have vertex set $F$, with $M_iM_j$ an edge exactly when $M_i\cup M_j$ is a Hamilton cycle. Behague proved that $Q_{k+\ell}$ has a 1-factorization $F$ with $G[F]\cong K_{k,\ell}$ for all positive $k,\ell$ except possibly $k=\ell=3$. We give an explicit 1-factorization of $Q_6$ for which $G[F]\cong K_{3,3}$, resolving the exceptional case. The construction is supplied as a finite certificate. Its correctness can be checked directly from the tables in the paper or by either of two independent, short, standard-library verifiers supplied with the certificate.

We introduce the conjugate-slope property of knots, a variant of Nie's Property (D), which can be used to study non-left-orderability of fundamental groups arising from Dehn filling. We show that this property holds for any knot in $S^3$ admitting a diagram whose negative crossings can be "controlled" in a precise combinatorial sense. The conjugate-slope property also places strong restrictions on the kinds of left-orderings that the knot group can support, and implies that the image of the meridian is generalised torsion in the fundamental group of many of the knot's Dehn fillings.

We determine the sharp $L^p$ ranges of the Riesz transform and the reverse Riesz inequality for a class of power-degenerate elliptic operators of Grushin type. When the degenerate variable has dimension at least two, the Riesz transform is bounded throughout the reflexive range. In dimension one, a capacity-driven transition occurs: weak degeneracy produces a sharp finite upper endpoint, whereas strong degeneracy separates the two half-lines and restores the full range. The reverse inequality displays a different sharp phenomenon. In the weakly degenerate one-dimensional regime it holds throughout the reflexive range except at a single Hardy-critical exponent, while in all remaining regimes it holds for every $1<p<\infty$.

Gupta and Iosevich introduced the edge complexity of a graph as the minimum Fourier ratio of its adjacency matrix over all vertex labelings and bounded it below by graph energy divided by the square root of twice the number of edges. We characterize equality for a fixed labeling: the Fourier transform of the adjacency matrix must have at most one nonzero entry in each row and column. This implies regularity, circulancy of every positive even power of an extremizing adjacency matrix, and a parity restriction on connected components, and it gives equality results for certain Laplacian spectral projectors. We construct equality cases from affine involutions on cyclic groups. Singer difference sets yield, for every prime power $q$, an equality-attaining $(q+1)$-regular graph that is not an abelian Cayley graph. We also establish Fourier-ratio estimates for weak, Cartesian, and strong graph products, including preservation of equality under weak products of coprime orders. We use Fourier-ratio recovery as a coding theorem to obtain entropy upper bounds for low-complexity adjacency matrices and complement them with a lower bound obtained by perturbing complete graphs. Finally, a concentration argument shows that if $Np_N/\log N\to\infty$ and $\limsup_{N\to\infty}p_N<1$, then $\operatorname{FR}_{\min}(G(N,p_N))$ is of order $N$ with probability tending to one.

In this paper, we classify the localizing ideals of the derived category D(U) of global representations over a field k of characteristic zero, for various infinite families U of finite groups. These families include elementary abelian p-groups, cyclic p-groups, and cyclic groups of prime order together with the trivial group. We deduce that the telescope conjecture holds for these D(U). In particular, via Pontryagin duality, our results for elementary abelian p-groups also establish the telescope conjecture and the corresponding classification for derived VI-modules. We also prove that the telescope conjecture holds for the derived category of FI-modules.