Papers

We study the k-delete recoverable robust 0-1 problem in which a decision-maker solves a combinatorial optimization problem subject to objective uncertainty. The model follows a two-stage robust setup. The decision-maker first commits to an initial plan and may then revoke up to k components of this decision after the uncertainty is revealed. The underlying uncertainty is modeled using a budgeted uncertainty set so that the decision-maker only hedges against a limited number of deviations in the uncertain parameters. We present four reformulations of the k-delete recoverable robust problem, which can be tackled using (i) general-purpose mixed-integer linear programming solvers, (ii) branch-and-cut methods, or (iii) column-and-constraint generation algorithms. For each formulation, we identify suitable solution methods and prove their correctness. Overall, we present eight approaches to solve the k-delete recoverable robust problem, which we assess and compare in an extensive computational study on instances of the assignment problem and the single-source capacitated facility location problem.

Let $G$ denote a finite, connected, simple graph, and let $D$ denote its diameter, and let $d_{G}(u, v)$ denote the distance between vertices $u$ and $v$ in $V(G)$. An antipodal labeling of $G$ is a mapping $f\colon V(G) \to \mathbb{N}_{0}$ such that, for each pair $(u, v)$ consisting of distinct vertices in $V(G)$, the relation $ |f(u) - f(v)| \geq D - d_{G}(u, v) $ holds. The span of $f$ is then defined so that $\operatorname{sp}(f) = \max\{ f(u) - f(v) : u, v \in V(G) \}$. The antipodal number of $G$, denoted with $\operatorname{an}(G)$, may then be defined as the minimum possible span among all antipodal labelings of $G$. Juan and Liu [Ars Combin., 2012] proved the upper bound $\operatorname{an}(C_{4k}) \leq 2k^2 - 1$, and conjectured that $\operatorname{an}(C_{4k}) = 2k^2 - 1$ for each positive integer $k$, and Juan and Liu verified this conjecture for $k \leq 5$. We succeed in proving Juan and Liu's conjecture, which seems to have remained open. The openness of Juan and Liu's conjecture has been noted, over the years, by many authors, including Rao et al. [Contrib. Discrete Math., 2015] and Saha et al. [Theory Comput. Syst., 2022].

This paper is concerned with solving the problem of finding a zero of the sum of a single-valued operator and a set-valued operator, neither of which is assumed to be monotone, in a real Hilbert space. Under the framework of pair monotonicity, we propose a generalized forward–backward algorithm and a generalized Tseng algorithm based on transformed and warped resolvents associated with an auxiliary linear operator. We establish weak, strong and linear convergence of the proposed algorithms under suitable assumptions involving pair strong monotonicity, pair cocoercivity, and pair Lipschitz continuity. Finally, numerical experiments are presented to validate the theoretical results and demonstrate the effectiveness of the proposed algorithms.

This is the third paper in a series on threshold cascades of coupled Ornstein-Uhlenbeck diffusions on graphs converging Benjamini-Schramm to a unimodular Galton-Watson tree. In arXiv:2608.15987 the reduction of the cascade to a finite-type Galton-Watson process (front decoupling) was proved unconditionally in the dissipative regime kappa < 1, where kappa = gamma L_sigma Delta_max / alpha_min is the single-step dissipation ratio. At kappa >= 1 the single-step echo bound resonates, and the reduction was left conjectural. We replace it by a multiscale (renormalisation-group) analysis. Coarse-graining into space-time blocks of depth L and duration T, we record each block's boundary-to-boundary gains in a 2x2 matrix with a trajectory channel and a stored-mass channel, fan-out absorbed. Three theorems result. The gain matrices compose under depth and time concatenation, so a renormalised per-level dissipation ratio kappa_eff exists and equals its infimum (Fekete). If the spectral radius of the gain matrix is below 1 at a single finite scale, a finite-horizon, finite-volume condition in principle certifiable numerically, then front decoupling holds at all scales with a quantitative total-variation bound; echoes may resonate locally, yet independence is recovered macroscopically. Finally we split influence steps into front, quiescent and saturated phases, price each step by the adapted band occupation of its source rather than by a global worst case, and deduce kappa_eff <= rho(T) for an explicit channel matrix T. The region rho(T) < 1 strictly contains kappa < 1: near criticality the effective condition degrades from gamma L_sigma Delta_max / alpha < 1 to gamma L_sigma / alpha < 1, and the degree disappears. We do not prove kappa_eff < 1 throughout kappa >= 1, and give a heuristic for why the resonance regime should be a genuine phase boundary.

We prove a strong-universality theorem at the exact fourth-moment threshold for finite families of non-Hermitian random matrices assembled from independent unordered-pair vector atoms. Within one atom, the matrix colors and the two endpoint orientations may have arbitrary joint real covariance, subject to reversal consistency across ordered type pairs and endpoint exchangeability in same-type blocks; the law may also depend on finitely many endpoint types. The matrices may be adjoined to an arbitrary deterministic tuple that converges jointly strongly with the type projections. For every fixed matrix amplification and fixed noncommutative \(*\)-polynomial, the resulting tuple converges strongly to an explicit covariance-matched free Gaussian family. Cross-type blocks are described by masked circular variables, whereas same-type blocks split into independent endpoint-symmetric and endpoint-antisymmetric semicircular sectors. Only a finite radial fourth moment is assumed off the diagonal, and a finite second moment suffices on the diagonal. The mode of convergence depends on the coupling: corners of one infinite array converge almost surely on a common event, while fixed-law nonnested triangular arrays converge in probability for each fixed test. As an application, we obtain exact-fourth-moment strong limits for finite-separable continuous left/right profiles, including separately weighted literal-transpose terms. For the nested almost-sure formulation, the fourth-moment threshold is sharp already on the Wigner subfamily; at this threshold arbitrary fresh rows admit no coupling-invariant almost-sure upgrade of our in-probability conclusion.

Although the Heil–Ramanathan–Topiwala conjecture is false in full generality, its failure makes the classification of positive geometric and arithmetic regimes more urgent. We settle the critical mixed-arithmetic regime for four time-frequency shifts. For \(z=(x,\omega)\) and \(w=(y,\eta)\), set \(\sigma(z,w)=x\eta-y\omega\). Let \(u,v\in\R^2\) satisfy \(\lvert\sigma(u,v)\rvert=1\), write \(\nu=\alpha u+\beta v\), and assume that \(0,u,v,\nu\) are distinct. If \(\dim_{\Q}\operatorname{span}_{\Q}\{1,\alpha,\beta\}=2\) then, for every nonzero \(f\in L^2(\R)\), the vectors \(f,\pi(u)f,\pi(v)f,\pi(\nu)f\) are linearly independent. This includes both four-point configurations in Chris Heil's Conjecture~9.2. The proof first converts a putative dependence into a scalar cocycle over an irrational rotation. On a positive-measure family of zero-free fibres, winding and continued-fraction returns force the periodic holonomy to be constant. The return multiplier is Laurent polynomial, however, so its ungauged holonomy is algebraic; the constant-holonomy law simultaneously makes it an irrational character. These conclusions are incompatible. The principal theorem and its complete formal dependency chain have been fully certified end-to-end in Lean~4, including the physical all-nonzero case, the complete three-point HRT theorem, the coefficient-reduction step, and the final four-point conclusion. The accompanying \href{https://www.dropbox.com/scl/fi/jd9xeqqera147wdc82r8k/3f260460-07d5-45f5-8d1c-ee1d2879c0af-aristotle-32-.tar.gz?rlkey=a24njadfds5ty9ez473lk03lv&e=1&dl=0} {Lean~4 certification archive} contains reproducible source, pinned build instructions, and transitive axiom audits.

Starting from the cyclic duality tensors introduced by Vernier, Miao, and Yamazaki (VMY), whose endpoint-traced matrix-product operators obey $\mathbb Z_N$ Tambara–Yamagami fusion and realize topological defect lines of the compactified-boson conformal field theory, we retain the virtual endpoint as an $N$-state dynamical degree of freedom and construct an exactly movable seam in the spin-$\tfrac12$ XXZ chain at $q=\mathrm e^{\mathrm i\pi M/N}$ with $\gcd(M,N)=1$. The local movement identity holds for any unitary $q$-Weyl pair; in the finite cyclic realization, all output phases are locally gauge equivalent and share the same four seam eigenvalues. On a ring, the output gauge becomes a directed twist on a single bond, while endpoint conjugacy reduces the $2N$ labels to two relative $\mathbb Z_2$ flux classes. An explicit antiunitary symmetry protects the class $\eta\equiv M-1\pmod 2$: for even $N$ it pairs distinct charge sectors isospectrally, whereas for odd $N$ it fixes one sector and squares to $-I$ there. Hence every many-body energy eigenspace in the protected class has even multiplicity. The source value $\eta_{\rm VMY}=1-M$ satisfies this condition for every coprime root, whereas the ungauged value $\eta=0$ does so only for odd $M$. Exact finite-size controls show that the opposite class can contain simple levels, so a global flux invisible to local gauge equivalence distinguishes the two classes spectrally.

Christoph, Müyesser and Wigderson recently asked whether an approximate form of the Erdős-Sós conjecture is robust under random edge deletions. We establish a density-sensitive transference theorem that converts global resilience for bounded-degree trees in sparse random graphs into resilient forest universality in arbitrary dense host graphs. More precisely, if $F$ is an $N$-vertex graph of edge density $\lambda$ bounded away from zero and $p\in[K/N,1]$, then, with probability $1-o(1)$ uniformly over the host and the percolation parameter, every subgraph obtained from $F_p$ by deleting at most an $\alpha$-fraction of its edges contains every bounded-degree forest on at most $((1-\alpha)\lambda-\xi)N$ vertices. Consequently, for every $c>0$, $L\ge1$, fixed $D$ and $\alpha<c$, every graph $F$ on at most $Ld$ vertices with average degree at least $d$ has the property that, after percolation at any rate $p\in[K/d,1]$ and any subsequent deletion of at most an $\alpha$-fraction of the surviving edges, the remaining graph is universal for all forests on at most $(1-c)d$ vertices and maximum degree at most $D$. This includes vertex-disjoint packings of any prescribed collection of bounded-degree trees of that total order. We further obtain forest-universality results for graphs whose connected components have vertex-cover number at most $Cd$, and for graphs that can be brought into this form by deleting sufficiently few edges on the $dn$-scale.

For n points in the unit square, the Heilbronn triangle problem asks for the largest possible minimum triangle area. We develop a variational stress theory for this max-min problem. At every positive-area local optimum, normalized Karush-Kuhn-Tucker multipliers assemble into a skew matrix B satisfying Bz = 2ib, where b is the outward square reaction. This equilibrium has the isotropic affine moment sum_i p_i b_i^T = Delta I_2; the identity also holds for every tight subfamily carrying weights inherited from the same multiplier, with Delta scaled by its multiplier mass. It follows that every positive stress component meets all four sides and that there are at most two such components. To handle nonunique multipliers, we introduce the intersection of the stress kernels over the whole KKT face and a canonical hybrid operator incorporating all tight determinant and boundary derivatives. A strictly convex selector removes every decomposable invisible motion, leaving a rank-one-free residual with a sharp dimension bound; the literal maximal two-dimensional residual is excluded. Two-terminal substresses satisfy an exact interface-flux law, and a five-internal-vertex rank-four block has a Pfaffian cofactor carrier. Finally, an analytic reduction followed by exact symbolic enumeration excludes every one-external completion of a specified one-silent five-cycle residual, for all orientation words. These results isolate the remaining degeneracy but do not solve the problem for arbitrary n.

Reconstructing the full-aperture channel of a fluid antenna system (FAS) from a few activated ports requires inferring unobserved responses from limited spatial samples. This paper proposes a covariance-domain FAS channel-reconstruction method based on nonnegative least squares (NNLS). A nonuniform port arrangement constructs a difference coarray with an enlarged aperture. Vectorizing the sample covariance maps the physical array to a virtual steering dictionary, and the resulting complex covariance-fitting model is recast as a strictly equivalent real-valued NNLS problem. A projected-gradient iteration initialized at zero estimates the nonnegative angular-occurrence coefficients, and the L_s strongest well-separated peaks yield the angles of arrival (AoAs). Simulations show that the proposed FAS-NNLS estimator exploits the enlarged virtual aperture and achieves lower AoA mean squared errors than uniform linear array (ULA) multiple signal classification (ULA-MUSIC) and ULA estimation of signal parameters via rotational invariance techniques (ULA-ESPRIT). Its non-line-of-sight (NLoS) path accuracy matches FAS covariance matching pursuit, while its line-of-sight (LoS) estimation is significantly improved.

Walsh-sequence-based receiver architectures offer an alternative approach for the reception and reconstruction of multiple simultaneous RF signals over wide bandwidths. While previous works have focused on the architecture and theoretical operation of Walsh-domain processing, limited experimental results have been reported so far on hardware operation under realistic conditions. This paper presents the experimental evaluation of a Walsh-sequence-based receiver prototype. A measurement campaign was conducted under various signal placement, interference, and receiver configuration scenarios. Reconstructed spectra, Walsh-lane power spectral densities (PSDs), and Error Vector Magnitude (EVM) measurements were used to investigate Walsh-domain signal representation, interference effects, and receiver performance. Finally, dense- spectrum experiments involving up to ten simultaneously active signals demonstrate the potential of the architecture for low power wideband spectrum sensing applications.

Traffic sensor placement in large urban networks must balance traffic monitoring needs with limited installation, maintenance, and data-management budgets. This study develops a framework for preserving network-wide traffic information while reconstructing flow and density at unobserved links and the macroscopic fundamental diagram (MFD). The framework combines normalized mutual information (NMI) with Inductive Graph Neural Network Kriging (IGNNK). NMI quantifies lag-aware dependence between joint flow-density states, while a greedy algorithm selects sensors by balancing weighted information coverage against redundancy. An IGNNK-based bidirectional graph-diffusion model reconstructs unobserved link states and generates the MFD. The framework was evaluated against a principal component analysis (PCA) benchmark using seven hours of dynamic traffic assignment simulation data for the Chicago network, with 4,805 directed links and 420 one-minute intervals split into training, validation, and independent testing subsets. Reconstruction performance for both PCA and NMI-IGNNK varied substantially over the sensor ratios of 0.5% to 30%. PCA generally yielded lower root mean squared error (RMSE) over the 0.5%-3% range, but its errors were nonmonotonic and reached their minimum at 2%. At 3%, normalized joint RMSE was similar for PCA and NMI-IGNNK (0.100 and 0.105, respectively). NMI-IGNNK remained applicable up to 30% coverage and generally improved as coverage increased, reducing testing-subset normalized joint RMSE from 0.212 at 0.5% to 0.043 at 30%, with consistent trends over the complete analysis period. The framework integrates lag-aware, information-theoretic sensor placement based on joint flow-density states with directed graph-based reconstruction, enabling a broad range of sensor budgets in large-scale networks without strict limits on the number of sensors.

The paper "Orthogonal completion of algebraic systems II" continues the line of "Orthogonal completion of algebraic systems" from the perspective of almost classical localizations of algebras and the conclusions of K.~I.~Beidar's doctoral dissertation. This allows one to apply the orthogonal completion method to describe non-degenerate orthogonally complete algebras bases on known classifications of strongly prime algebras.

Let $s(n)$ denote the number of ones in the binary expansion of an integer $n\in\mathbb{N}$, and let $\mu_t$ be the probability measure on $\mathbb{Z}$ defined by the asymptotic densities of the level sets of the function $\mathbb{N}\ni n\mapsto s(n+t)-s(n)\in\mathbb{Z}$. Let $P_t$ be the family of finitely supported measures defined by the convolution $\mu_t=\mu_1*P_t$. Recently, Tarlowski (2026) has shown that the family $P_t$ may be represented as a recursively grown binary tree $T_t$, and that the Cusick's conjecture - $\mu_t(\mathbb{N})>\frac12$, $t\in\mathbb{N}$, - follows from the asymmetry property of the family $T_t$, which was posed there as an open problem. Next, Cheng (2026) has provided the combinatorial description of the family $T_t$ in the language of principal subsequence ideals, and proved both conjectures. Both of these problems are directly related to the problem of determining the zeros of the function $\mathbb{N}\ni t \mapsto P_t(\mathbb{N})-\frac12\in[0,\tfrac12]$, a problem left open by Cheng (2026) as a saturation problem, and previously analyzed only numerically. In this paper we solve this problem completely. Writing an odd integer $t\ge3$ as $t=(1\,w\,1)_2$ with $w\in\{0,1\}^{\star}$, we show that $P_t(\mathbb{N})=\frac12$ if and only if $w$ is saturated in the following sense: in the block decomposition $w=1^{a_0}\,0\,1^{a_1}\,0\cdots0\,1^{a_k}$ with exactly $k$ zeros, every block of "1" satisfies $a_i\ge k$. Additionally, we show that the lower bound for $P_t(\mathbb{N})$ established by Cheng for $0$-initial words holds true for all non-saturated words.

Why the square? We present a geometry-axiom-free derivation of the Pythagorean theorem and the square at its core, establishing their algebraic origin from within the bare vector-space framework. Such concepts as the (right) angle, rotation, inner product, orthogonality etc also emerge as a logical construct rather than taken as given. They are necessitated by the square, and the ensuing theory, in turn, $\textit{canonically}$ stems from a $\textit{single}$ definitional primitive $-$ the ($\mathbb R^{\vcenter{\hbox{$\scriptscriptstyle+$}}}\!$-quantitative) invariant $\mathcal Q$-measure of a vector. This provides the core of an algebraic justification for Euclidean geometry. Equally important, these findings account (also canonically) for the complex modulus-squared $p = |\mathfrak a|^2$ $-$ the quantum Born rule $-$ and point out what is even admissible for being quantitatively interpreted. The linear structure and its endomorphisms are rigid in the sense that the $\textit{well-defined}$ interpretable turns out to be, up to gauge $\mathcal Q {\,\to\,} \mathrm{const} {\,\vcenter{\hbox{$\scriptstyle\times$}}\,} \mathcal Q$, the unique gauge-invariant measure $\mathcal Q=|\hspace{-0.18em}| {\cdot}{\cdot}{\cdot} |\hspace{-0.18em}|^2$; independently of the field $\mathbb R$ or $\mathbb C$.

We consider closed connected oriented three-manifolds equipped with positive cooriented contact structures. We prove that a symplectomorphism between their symplectizations forces the existence of an orientation-preserving diffeomorphism between the underlying three-manifolds under which the two contact structures are homotopic as oriented plane fields. Combining this formal rigidity with the classification of overtwisted contact structures, we show that two closed overtwisted contact three-manifolds have symplectomorphic symplectizations if and only if they are contactomorphic. Thus, we resolve the three-dimensional symplectization question when both contact structures are overtwisted.

Lobachevsky's discovery of Non-Euclidean geometry was a byproduct of his more ambitious plan of reforming the traditional Euclid-style foundations of geometry. This project aligned with d'Alembert's and Condillac's practically-oriented visions of mathematics and aimed at closing the perceived gap between the pure mathematics, on the one hand, and the natural sciences and engineering, on the other hand. Following d'Alembert Lobachevsky deliberately avoided the axiomatic development of his geometrical theory and combined instead some preliminary intuitive synthetic constructions with their rigorous analytic treatment. In addition to Non-Euclidean geometry, Lobachevsky's project brought about an early form of Dimension theory. The paper comprises a description of the local context of Lobachevsky's works, including some pointers to Russian geometry textbooks of the beginning of the 19th century. We summarise Lobachevsky's philosophical views which motivated his mathematical achievements and show how these motivations were misunderstood and misinterpreted in the beginning of the 20th century by Henri Poincaré and Ernst Cassirer. Finally we argue that Lobachevsky's epistemic views remain pertinent in the context of the ongoing discussion about the reunion of mathematics and physics. The Appendix comprises English translations of eight Lobachevsky's documents of various length and character where he expresses his general epistemic views on mathematics and, in particular, geometry.

In this paper we obtain new classes of the second generation of new $\zeta$-functionals and corresponding $\zeta$-equivalents of the Fermat-Wiles theorem. These are based on sets of elementary rectangular $\zeta$-signals generated by the Gramm's sequence.

In this work, we investigate implicit-explicit (IMEX) arbitrary high-order Deferred Correction (DeC) methods for the approximation of ordinary differential equations (ODEs). Such schemes are characterized by an iterative procedure that increases the order of accuracy by one at each iteration. More precisely, we study an efficient modification based on the introduction of interpolation processes between consecutive iterations, with the aim of systematically matching the accuracy achieved at each iteration with the order of the discretization employed. On the one hand, this modification leads to computational advantages, since the low-order iterations are performed on cheaper lower-order discretization structures; on the other hand, it endows the methods with a natural $p$-adaptive character, which is particularly appealing in the context of practical applications. We investigate this modification for two families of DeC schemes, providing numerical validation, efficiency assessments, and stability region plots. The numerical validation includes several examples involving stiff ODEs and partial differential equations (PDEs) with high-order spatial derivatives. The ability of the modified schemes to provide high-fidelity results at reduced computational cost, as well as the effectiveness of the adaptive strategy, is demonstrated through the numerical experiments.

By a result of Backelin, Veronese subrings of a standard $\mathbb{Z}$-graded algebra are eventually Koszul. Davis, Erman, and Martinova recently conjectured that the analogous statement holds for Veronese subrings of polynomial rings with positive nonstandard $\mathbb{Z}$-gradings. We disprove this conjecture. For the weighted polynomial ring $\mathbb{K}[x_1,x_2,x_3,x_4]$ with weights $(1,4,7,9)$, we prove that the $(9k+12)$-th Veronese subring is not nonstandard Koszul for every $k\geq1$. Our counterexamples arise from certain arrangements of lattice points, which we call cubic obstruction configurations. Each such configuration produces a minimal cubic generator in the defining ideal of the corresponding associated graded ring. This construction yields counterexamples in $n$ variables for all $n\geq4$. Moreover, we prove that the set of primitive four-variable weight vectors for which eventual nonstandard Koszulness fails has positive density. For a fixed three-variable grading, our cubic obstruction can occur at only finitely many Veronese indices, leaving that case open.

We use Khovanov-Floer theories and their spectral sequences to construct differentials on the corresponding skein-lasagna modules. Then we show that for 2-handlebodies, a spectral sequence of TQFTs coming from a Khovanov-Floer theory gives a spectral sequence of lasagna modules under the aforementioned differential. As a corollary, we recover the rank inequality between the Khovanov and Lee lasagna modules. We also prove certain properties like the connect-sum formula that these spectral sequences satisfy.

The linearity of the Stokes equations organises the hydrodynamic response of a rigid particle into a hierarchy of resistance operators, coupling successive truncations of the ambient-flow jet to moments of the surface traction. Since the work of Kelvin and Larmor, it has been known that this response does not resolve particle geometry faithfully: bodies with discrete rotational symmetry may be indistinguishable from bodies of revolution (Brenner's helicoidal symmetry) and a chiral body may respond isotropically, as in Kelvin's isotropic helicoid. We regard the resistance operators as elements of finite-dimensional O(3)-representation spaces and use character formulae to determine, at every level of the hierarchy, which point-group symmetries are hydrodynamically distinguishable and the dimension of each invariant space. This yields an explicit nested sequence of hydrodynamic symmetry-group sets, from the translation-force level to the quadratic-flow level. The framework reveals hydrodynamic classes that no shape can realise geometrically, gives helicoidal symmetry a level-dependent definition, and shows that polyhedral symmetry becomes visible in a strict order: tetrahedral symmetry in shear, octahedral symmetry through the stresslet, and icosahedral symmetry in quadratic flow. Projecting the resistance operators onto force- and torque-free motion provides symmetry-based parameter counts and a constructive route to the corresponding dynamical normal forms. We thereby complete the Jeffery-Bretherton-Ishimoto classification, characterise all hydrodynamic classes producing Jeffery dynamics, and identify chiral tetrahedral and octahedral normal forms that can generate irregular full-attitude dynamics.

Using an analogue of the Mandelstam-Giddings-Wolpert diagrams, we introduce a new canonical representative of the conformal class of a bounded domain of arbitrary connectivity in $\mathbb{C}$ as a flat conical surface with geodesic boundary (a "truncated light-cone diagram", simply LC-diagram in the sequel). The space of diagrams carries natural coordinates. We derive variational formulas for the determinant of the Dirichlet Laplacian on a LC-diagram with respect to these coordinates. Then, passing to the Schottky double of the LC-diagram, making use of the Burghelea-Friedlander-Kappeler formula and the known variational formulas for determinants of Laplacians on the moduli space of holomorphic differentials, we compute the Dirichlet-to-Neumann (DN) conformal invariant $\frac{{\rm det}'\Lambda}{|\Gamma|}$ (here $\Lambda$ is the DN operator on the boundary, $\Gamma$, of a multiply connected domain and $|\Gamma|$ is the length of the boundary). The resulting formula (which uses the periods of the Schottky double of the domain only) provides an elementary counterpart to the formulas of Guillarmou and Guillopé who had expressed the DN invariant through the Ruelle and Selberg zeta-functions. Our formula agrees with the recent result of Wentworth on the asymptotics of the DN invariant as all but one boundary components shrink; we have used this result to fix the undetermined constant of integration in our formula. As a corollary, we derive an explicit formula for the determinant of the Dirichlet Laplacian in a multiply connected domain.

We construct a subadditive weight on the type I$_\infty$ factor that preserves directed suprema but is not $\sigma$-weakly lower semi-continuous. This provides a counterexample to Haagerup's first problem posed in 1975.

This article presents the abridged core of A Mathematical Theory of Interpretation (MTI), which treats interpretation as observer-relative spectral measurement under an access structure. MTI makes interpretation a method-design problem: access, query, utility, and medium determine what an observer can select, identify, communicate, or refuse. On a learning-invariant Hilbert realization, Rational Entropy measures residual uncertainty across knowledge, utility, and medium. In the finite-effective regime, we classify its zero set. Pairwise confusability is equivalent to uniform atomic collapse, while a unique utility maximum can select one atom even when other zero-cost states remain non-atomic. This reverses the usual zero-error role of confusability: agreement in at least one observer direction excludes unresolved multi-atom readings, while the joint label preserves identification. The corresponding free-design capacity is the product of all but the smallest direction budget. A four-condition certificate characterizes sharp, decodable, medium-faithful, and order-independent readout on a finite commuting code sector and returns typed obstructions when those guarantees fail. Together, these results establish MTI as a theoretical basis for constructing interpretation methods with explicit access assumptions, guarantees, and failure modes.

We show that every closed, nonorientable surface can be minimally embedded in $\mathbb{S}^4$, providing in particular the first known examples of embedded, nonorientable minimal surfaces in $\mathbb{S}^4$ with negative Euler characteristic. All of the surfaces we obtain have area below $8\pi$, which has applications to the existence of nonorientable surfaces minimizing the Willmore functional with prescribed topology in $\mathbb{R}^n$ for $n\geq 4$. Moreover, the number of geometrically distinct embeddings of each nonorientable genus is shown to grow at least exponentially with respect to the genus. Among these surfaces, we identify a distinguished, highly symmetric family which converges in the large-genus limit to a union of four half-spheres, meeting along a great circle, whose poles are vertices of a regular tetrahedron. Rescalings of this family converge to a new singly-periodic nonorientable minimal surface in $\mathbb{R}^4$, which seems to provide the first example of a complete, embedded, nonorientable minimal surface in $\mathbb{R}^4$ without continuous symmetry group. The proofs further develop the equivariant eigenvalue optimization methodology from our earlier work, applied to carefully chosen families of symmetry groups acting on nonorientable surfaces. Interestingly, we also find natural pairs of surfaces and group actions for which there is no metric maximizing the first normalized eigenvalue of the Laplacian.

Plasticity under changing environments is central to both evolutionary biology and continual learning. Motivated by recent work on genotype–phenotype maps, we study a minimal deep-learning analogue where a network is trained alternately on two Boolean label sets, and ask which biological controls of plasticity survive the translation to gradient descent. Reinterpreting four proposed biological factors as quantities of training dynamics, we find the system reduces to two dimensionless controls: the task disagreement $r$, the fraction of disagreeing labels, and the reach $\eta T$, the product of learning rate and switching period. We derive two bounds on plasticity: $r$ alone fixes an extremal geometric floor on the utopia distance, while $r$ and $\eta T$ jointly bound forgetting. Across 9,720 trajectories, an ANOVA confirms that $r$, $\eta$, and $T$ dominate, while the effect of neutral-set size (emphasized in the biological setting) is negligible. The optimal reach itself follows an approximate inverse power law $\eta T^{*}\propto r^{-1.18}$, yielding a heuristic that sets the optimal reach $\eta T^*$ from the task disagreement alone. The analogy that survives is therefore dynamical rather than geometric, and our setting enables a view of plasticity through the lens of other driven systems in physics and engineering.

We say an $(n,n)$-function $F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ is a crooked function if for any nonzero $a \in \mathbb{F}_2^n$, the image of $D_aF(x)=F(x)+F(x+a)$ is an affine hyperplane. The only known examples of crooked functions are all quadratic almost perfect nonlinear (APN), or equivalently, for every known crooked function, $D_aF$ is affine for all $a \in \mathbb{F}_2^n$. The ortho-derivative $\pi_F \colon\mathbb{F}_2^n \to \mathbb{F}_2^n$ of a crooked function $F$ is the function such that $\pi_F(0)=0$, and for any nonzero $a$, the set $\{0,\pi_F(a)\}^\perp$ is the underlying vector space of $\mathrm{Im}(D_aF)$. We prove that for $n \geq 4$ and a crooked function $F$, if $k$ is a non-negative integer such that $F$ has $2^k-1$ quadratic component functions, $\pi_F$ has at least $2^n-2^{n-k}$ component functions of algebraic degree $n-2$. In particular, we resolve Gorodilova's conjecture that every component function of $\pi_F$ has algebraic degree $n-2$ when $F$ is quadratic APN. As a second main result, for $n \geq 4$, we associate to a crooked function $F$ a quadratic function $\varepsilon_F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ that satisfies a strong geometric-combinatorial condition regarding the sums of $F$ over $2$-dimensional linear subspaces. As a corollary to both of our main results, we prove that for any even $n \geq 4$, any quadratic APN $(n,n)$-function has at least $n$ semi-bent components. Furthermore, we obtain a congruence result on a problem on $m$-sequences introduced by Johansen, Helleseth, and Kholosha, and we determine the exact algebraic degrees of some Boolean functions associated to the bent and near-bent components of particular classes of plateaued vectorial functions.

We prove a Fourier-type toric spectral reciprocity formula over number fields, relating Tate integrals over a quadratic extension to a dual family of Hecke periods over the base field. The formula keeps the spectral parameter free and allows test functions adapted to arithmetic applications. We apply this reciprocity formula to canonical Hecke $L$-functions over CM fields and obtain explicit first moment formulas, including both central values and central derivatives. These formulas are uniform in the weight and the twisting conductor, and require no Heegner-type splitting hypothesis. Together with subconvexity bounds, they yield quantitative nonvanishing results for canonical Hecke $L$-functions and their derivatives. In weight one, this gives a rank-one analogue of the arithmetic applications of Masri–Yang, yielding quantitative Mordell–Weil rank-one results for quadratic twists of the associated CM abelian varieties.

We study the minimisers of the Hartree–Fock functional for flat bands in the chiral model of twisted bilayer graphene introduced by Tarnopolsky, Kruchkov, and Vishwanath. Our model is a continuous version of the discrete model considered by Becker, Lin and Stubbs in previous works and the first result is a direct analogue of main theorems of those papers: at half filling of the flat band of multiplicity two, there are exactly two minimisers, obtained by occupying either one of the two sub-bands. (A more general version of this result, stressing its topological origins, was presented in Appendix A.8 of the Lecture Notes by Tao-Zworski.) The new contribution is the strict coercivity of the Hessian of the Hartree–Fock functional at the two minimisers. It is a consequence of the difference between the Chern numbers of the component bands and can be optimistically considered as a soft version of a “spectral gap”.