Papers

For a smooth, connected, oriented surface $\Sigma$, topologically a sphere, immersed in the homogeneous space $E(\kappa,\tau)$ with $\tau\ne0$, we show that a one-sided Cauchy–Riemann-type inequality on the original Abresch–Rosenberg differential $\mathcal{Q}_{AR}$ – much weaker than requiring $\mathcal{Q}_{AR}$ to be holomorphic – already forces the mean curvature $H$ to be constant. The proof is uniform across every $E(\kappa,\tau)$, including the round sphere, and combines the Bers–Vekua similarity principle with the Poincaré-Hopf index formula for line fields; it is logically independent of the algebraic argument used, in the companion paper~\cite{AlencarRosenberg2026}, to characterize CMC immersions by holomorphy of $\mathcal{Q}_{AR}$ alone. We also show that the topological hypothesis is sharp: on every Berger sphere, including the round one, there exist compact non-CMC tori – the preimages under the Hopf-type submersion $\pi:E(\kappa,\tau)\to M^2(\kappa)$ of simple closed curves in the base with nonconstant geodesic curvature – satisfying the same inequality with a constant bound. As an elementary consequence of the classical Poincaré-Hopf theorem for line fields, we record that every closed surface on which $\det S<0$ everywhere – a condition satisfied, in particular, by every Hopf tube, where $\det S\equiv-\tau^2$ – is a torus.

This paper studies symmetry-induced reductions of scalar recurrence complexity for balanced banded Toeplitz determinants and permanents. Three mechanisms emerge: determinant state and spectral compression, exceptional permanent–determinant conversion, and symmetries of permanent state spaces. For symmetric determinants, straightening reduces the row-column states from $\binom{2m}{m}$ to the Catalan number $C_{m+1}$; primitive symplectic weight compression then leaves $(3^m+1)/2$ distinct autonomous modes. Skew symmetry has a parallel compound/Hodge explanation: the middle exterior representation splits into two Hodge halves of dimension $\binom{2m}{m}/2$, the one-step compound transfer exchanges the halves, and its two-step restriction has $3^{m-1}$ generic ternary modes. Thus the full skew bound is $2\cdot3^{m-1}$. Widom–Hankel arguments prove generic scalar minimality in both symmetry classes, and every even skew Toeplitz determinant admits an explicit half-size square factorization. For the zero-diagonal pentadiagonal support, a Pólya–Kasteleyn signing converts every permanent to a determinant. Among consecutive zero-diagonal two-sided bands, universal entrywise conversion—and separately Toeplitz diagonal-wise conversion—occurs only in the Hessenberg families and this pentadiagonal case. Paired renewal identities recover all restored-diagonal determinant and permanent layers. For permanents, transposition gives the open symmetric bound $(\binom{2m}{m}+2^m)/2$, while cyclic defect-sector pairing gives $(4^m+\binom{2m}{m})/2$. Skew-symmetry forces odd-size vanishing and corresponding even-subsequence bounds. In semibandwidth two the open symmetric bound is generically sharp; higher-semibandwidth permanent minimality is left separate from the symmetry reductions proved here.

We study the problem of finding tight explicit bounds on sums over zeros of $L$-functions near the $1$-line with a variety of zero-free regions and zero-density estimates. As an application to demonstrate the techniques, we apply them to the case of the Riemann zeta function and the prime number theorem to obtain new best estimates on the error terms in the prime number theorem. We show that for $x\geqslant21$ we have \begin{multline*} |\psi(x)-x| < 0.239\, x \exp\bigl(-0.1982767\,(1+\log\log\log(x)/(15\log\log(x)))\log(x)^{3/5}\log\log(x)^{-1/5}\bigr). \end{multline*} Combined with the zero-free half-plane $\Re(s)>7/8$ announced by OpenAI in October 2026, the same methods give $|\psi(x)-x| < 0.04934\,x^{3/4}\log(x)^2 + 0.02636425\,x^{7/8}$ for $x\geqslant 21$.

This paper provides a rigorous computer-assisted proof of the existence of generalized transmission eigenvalues arising in nonlinear optics in the context of high-order harmonic generation, a result conjectured in Cakoni, Hovsepyan, Lassas, and Vogelius, SIAM J. Math. Anal. 57 (2025), 1370-1405. The analysis is carried out for a one-dimensional nonlinear medium, where the problem reduces to a coupled system of nonlinear homogeneous ordinary differential equations subject to nonstandard boundary conditions. These eigenvalues correspond to probing frequencies $\omega$ for which there exists a nontrivial incident $\omega$-wave such that the second-harmonic field generated does not persist outside the compact support of the nonlinear medium, thereby rendering its nonlinear properties undetectable to an external observer. Building on earlier numerical evidence, we establish that, in the low-frequency regime, there exist generalized transmission eigenvalues whose associated eigenfunctions exhibit blow-up behavior as the frequency tends to zero. The proof combines analytical arguments with validated numerics, employing a Newton-Kantorovich framework together with interval arithmetic to rigorously control approximation errors. The algorithmic implementation underlying the computer-assisted proof is made available in the GitHub repository TransmissionEigenvalues.jl by Dominic Blanco (2026).

In this manuscript, we analyze a two-phase free boundary problem of Alt-Phillips-type driven by fully nonlinear elliptic equations with lower-order ingredients. More precisely, we consider equations of the type \[ F(x,D^{2}u)+\mathscr{H}(x,Du)=\mathscr{F}(x,u^{+},u^{-}) \quad \text{in } B_1 \subset \mathbb{R}^{n}, \] where the Hamiltonian term exhibits mixed linear and nonlinear gradient dependence, namely \[ \mathscr{H}(x,\xi):= \langle \mathcal{B}(x),\xi\rangle+\varrho(x)|\xi|^{\sigma}, \quad 0< \sigma\leq2, \,\,\, \sigma \neq 1, \] and the reaction is governed by a two-phase power-type nonlinearity of semilinear-type \[ \mathscr{F}(x,u^{+},u^{-}):=\mathfrak{g}(x)\big[(u^{+})^{m}-(u^{-})^{m}\big] \quad \text{for} \quad 0<m<1. \] Under suitable structural assumptions on the coefficients and the operator $F$, we develop a robust analytical framework to capture the fine properties of solutions near the free boundary. In particular, we establish improved growth estimates at higher-order singular nodal points, points where both phases meet and the solution exhibits critical degeneracy. The results are based upon a fine blow-up argument, stability and a Liouville-type theorem. We further prove quantitative non-degeneracy results, ensuring that solutions detach from zero at a controlled rate on each phase. As a consequence of our analysis, we derive a Liouville-type theorem for global solutions within this class. Our results extend and unify several previously known scenarios, and remain new even in the presence of gradient-dependent nonlinearities of order ($\sigma \neq 1$), thereby encompassing models with nonlinear drift and absorption effects arising in phase transition and reaction-diffusion phenomena.

We study the variance of the number of lattice points in smoothed narrow sectors, where the average is taken over the position of the sectors. There are three regimes corresponding to specific ranges for the open angle of the sectors and in each we obtain an asymptotic formula for the variance.

The Deck-of-cards-based Ordinal Regression (DOR) infers a value function from a ranking of reference alternatives in which the Decision Maker (DM) inserts blank cards between consecutive levels to express preference intensity. DOR, and its stochastic extension (SMAA-DOR), treat these answers as hard constraints defining a set of compatible value functions. We propose B-DOR, a probabilistic reformulation of DOR in which each pair of adjacent levels yields an ordinal observation, the declared direction and the number of cards, modelled through a cumulative-link likelihood that relates the number of blank cards to the latent value difference between alternatives. Two Bayesian inference algorithms are proposed: BAYES-DOR samples the whole posterior distribution by Hamiltonian Monte Carlo; FTRL-DOR tracks the maximum a posteriori estimate by constrained convex optimization. Moreover, through a multi-step elicitation process, elicitation can be spread over several short sessions reducing the cognitive burden on the DM. Both algorithms enjoy logarithmic regret bounds for prediction that hold for any sequence of DM responses and that guide the choice of the prior hyperparameters. A Monte Carlo study over 768 configurations shows that accuracy grows with the number of sessions, that blank cards add significant information over preference directions alone, that both algorithms maintain good performance under inconsistent answers, and that both outperform DOR and SMAA-DOR. An illustrative application to Italian regional healthcare performance demonstrates the practical applicability of the approach for building composite indicators.

We prove that the lattice covering density of the regular octahedron is $9/8$, settling a conjecture of Dougherty and Faber.

The classical theory of the wind-driven surface layer of a rotating ocean closes the momentum balance with a local flux-gradient law, in which the turbulent stress at a given depth is proportional to the shear at that depth. It predicts a surface current deflected forty-five degrees from the wind, which exceeds most direct measurements. This study asks what follows when locality is relaxed. Beginning from the exact integral relation between turbulent stress and mean shear, and requiring the memory kernel to carry no preferred vertical scale, we obtain a power-law kernel and therefore a stress law of fractional order. The resulting equation cannot be posed on the velocity, because the fractional derivative of a bounded profile vanishes at the surface, so the wind stress cannot be applied, while the alternative definition of the derivative leaves the surface current unbounded. Posed on the stress instead, the problem is solvable in closed form in Mittag-Leffler functions at every order between zero and one. The surface deflection then depends on the closure order alone and is smaller than forty-five degrees throughout, whereas the depth-integrated transport stays exactly normal to the wind, because that constraint follows from the momentum balance and not from the closure. The far field decays as a power law rather than exponentially, and its amplitude vanishes in the local limit, so that limit is singular. Under a suddenly applied stress the surface transient decays algebraically. Three independent algorithms agree closely, and no observational or model data are used.

An arrangement of $n$ lines in the plane defines cells, i.e., connected components of the complement of the arrangement. Answering a question of Volkmar Welker we show that, apart from two special cases, there is always a line $L$ in the arrangement such that one of the open halfplanes whose boundary is $L$ contains more than half of the bounded cells of the arrangement. We also consider some unusual discrepancy problems that are related to this question.

Let $(R,\mathfrak m)$ be a Noetherian local ring and let $\mathcal I^{(1)},\ldots,\mathcal I^{(s)}$ be graded $\mathfrak m$-primary families. We introduce weak asymptotic and strong homogeneous joint reductions and prove corresponding extensions of Rees's mixed multiplicity theorem relating joint reductions and mixed multiplicities. Actually, for weak asymptotic joint reductions, mixed multiplicities are realized as normalized limits of Hilbert–Samuel multiplicities, yielding an asymptotic Rees theorem. For multigraded admissible collections, a strong homogeneous joint reduction $\{x_{ij}\}$ of type $\mathbf d=(d_1,\ldots,d_s)$, with $x_{ij}\in I_{a_{ij}}^{(i)}$, satisfies the formula \[ e((x_{ij});R) = \left(\prod_{i,j}a_{ij}\right) e\!\left( \mathcal I^{(1)[d_1]},\ldots, \mathcal I^{(s)[d_s]} \right). \] We also prove a localized converse under formal equidimensionality, recovering the classical converse of Rees in the adic case. As an application, we derive a Böger-type theorem for graded families, whose adic specialization recovers Böger's classical theorem. In contrast, the converse to the weak asymptotic Rees theorem fails even for an adic family on a one-dimensional regular local ring.

We construct a one-parameter family $(X_c)_{0<c<1}$ of real reflexive Banach spaces, isomorphic to $\ell_2$ and satisfying $d_{\mathrm{BM}}(X_c,\ell_2)\le c^{-1}$, such that $(X_c,X_c)$ fails the weak maximizing property while $(X_c^*,X_c^*)$ has it. The failure of the WMP for $X_c$ is witnessed by a diagonal operator which does not attain its norm but admits a non-weakly null maximizing sequence. On the dual side, a separated-block estimate yields, after dualization, a reverse Pythagorean inequality and a quantitative form of the Opial property; together with property $(M)$, this gives the weak maximizing property for $X_c^*$. We also show that the compact perturbation property is self-dual for reflexive pairs. Since the weak maximizing property implies the compact perturbation property, each $(X_c,X_c)$ has the compact perturbation property. Consequently, the compact perturbation property does not imply the weak maximizing property even arbitrarily close to Hilbert space, and the weak maximizing property is not stable under duality in the reflexive setting.

We construct boundary observables that retain memory for a quantified time window in quantum chains whose basis configurations carry a conserved ordered product of matrix labels. Under strong-irreducibility and proximality hypotheses on the label matrices, projective contraction of random matrix products localizes a conserved charge in the root-mean-square over uniform basis configurations. A finite-time inequality then converts readout overlap and reset leakage into a correlation bound: for a conserved reference observable $H$, a readout $F$, accumulated squared leakage $\mathcal B$, and the normalized Hilbert-Schmidt inner product, $\langle F,\Phi(F)\rangle\ge 2\langle H,F\rangle^2/(\|H\|_2^2+\mathcal B)-\|F\|_2^2$. Here $\Phi$ is a sequence of conserving channels and partial resets. The bound holds for each such circuit, without averaging its gates. It yields an exponential memory window in the distance from the noisy region, with quantitative corrections for spatially distributed noise and imperfect conservation. For a thirteen-state elementary-matrix model, exact polynomial certificates give root-mean-square localization error $e^{-r/1400}$ between the charge and its restriction to the last $r$ sites, and limiting variance at least $1/19$ for a rational charge. A related normalized-Gram charge gives an eight-site certificate retaining more than half the specified readout correlation through two boundary-reset rounds. Uniform depolarization imposes an inverse-noise-rate lifetime ceiling. A two-qubit IBM experiment illustrates the general reset inequality; it does not realize the matrix-product chain. We provide proofs, exact certificate programs, and an independent recount of the archived experimental outcomes.

Quantum-enabled semantic communication networks (QESCs) leverage quantum technologies to transmit data meaning efficiently, yet face challenges from costly resources and noise. This letter introduces Quantum Markov Blankets (QMBs) to QESCs, a novel framework to isolate essential quantum information for semantic transmission. We prove QMBs' validity using quantum conditional mutual information, showing that they shield semantic content from irrelevant subsystems. An implementation strategy optimises QMB detection, reducing resource use. Simulations suggest that QMB-based QESCs cut qubit consumption by 50%-75% while enhancing fidelity compared with non-optimised quantum semantic schemes. Unlike classical approaches, QMBs offer inherent security by limiting an eavesdropper's access to classical data outside the blanket. We outline future directions, including real-time QMB adaptation. This work bridges quantum information theory and semantic communication, advancing resource-efficient and secure quantum networks.

We prove that learning an unknown mixed fermionic Gaussian state on $m$ modes to trace distance $\epsilon$ requires $\Omega(m^3/\epsilon^2)$ copies when measurements act on one copy at a time, even with arbitrary POVMs, fresh ancillas and classical adaptivity, but without quantum memory between copies. The bound holds for $0<\epsilon\le1/3600$ and separates this model from the known collective rate $\Theta(m^2/\epsilon^2)$. It follows from a uniform single-copy Fisher-information budget and a dimension-independent comparison between trace distance and Gaussian parameters. On covariance matrices of operator norm at most $1-c$, we prove a Frobenius-to-trace-norm continuity bound with constant $[2c(2-c)]^{-1/2}$; matchgate shadows then attain $O(m^3/(c\epsilon^2))$ copies. This also gives explicit error certificates for thermal free-fermion states. For a class of mixed passive bosonic Gaussian states with total mean photon number at most one, a reduction to bounded-block qudit tomography gives a single-copy lower bound $\Omega(m^3/(\epsilon^2\sqrt{\log(m/\epsilon)}))$, versus collective complexity $\Theta(m^2/\epsilon^2)$. These separations answer the mixed-state measurement-resource question posed by Chen et al. A two-mode example has optimal two-copy Fisher trace $5$, versus the separable ceiling $4$. An IBM replication gave a pre-registered witness lower bound $4.12$ at one-sided $95\%$ shot-noise confidence; its interpretation assumes the prescribed preparations and a common measurement channel and does not bound preparation systematics.

Resilience is a system's capability to prepare for, resist, absorb, and recover from adverse events. By definition, resilience therefore encompasses multiple criteria that can conflict and may prescribe different decisions. Importantly, a system must also maintain its effectiveness during routine operations and appropriately scale its preparedness for adverse events. Although resilience is inherently multicriteria, existing models often focus on a single criterion and do not explicitly analyze potential conflicts, the associated trade-offs, and their implications for decision-making. We formulate rebound, resistance, loss, and maximum performance degradation as separate resilience criteria in a general two-stage multicriteria model of network flows over time that integrates preparedness, effectiveness, and response. We show that, in general, such multicriteria optimization models are intractable and develop an enclosure-based heuristic for approximating the nondominated set. Our computational results show that different preparedness instruments and activation timings are associated with different resilience criteria and that the corresponding trade-offs are strongly instance-specific. Thus, resilience should be approached from an explicit multicriteria perspective rather than through a universal, preference-independent scalar index.

For $k, r, n$ natural numbers let $\rho_{r,k}(K_n)$ be the number of $r$-edge-colorings of $K_n$ that do not contain a rainbow copy of a $K_k$, that is, a copy of $K_k$ in which all edges receive different colors. When $k=3$, the quantity $\rho_{r,3}(K_n)$ represents the number of Gallai Colorings. It was proved by Balogh and Li and independently by Bastos, Benevides and Han, that most of the Gallai colorings are 2-colorings, for $n$ large. A natural analogue conjecture would be that when $k=4$, $r\ge 5$ and $n$ large, most rainbow-$K_4$-free $r$-edge-colorings are $5$-colorings. We show that this is not true in general and identify an exact threshold for $r$ where this ceases to be true. For the range where the conjecture is false, we determine the exponential growth of $\rho_{r,4}(K_n)$ for every fixed $r$. More precisely, for \(6\le r\le24\), we prove that \(\rho_{r,4}(K_n)=(\binom{r}{5}+o(1))5^{\binom{n}{2}}\); and for each \(r\ge25\), the proportion using at most five colors tends to zero, and \(\rho_{r,4}(K_n)=r^{(n^2/4)+o(n^2)}\). A bipartite construction, with all edges within the two parts assigned one common color, achieves the latter exponential growth rate. The lower bounds can be easily generalized for every $k$. Those results are related to other recent results about counting colorings that avoid rainbow cliques or given rainbow patterns in general. Our proof combines hypergraph containers with the graph removal lemma, structural estimates for color palettes and a refined count of colorings close to a fixed five-color palette.

This paper introduces the two-stage stochastic quadratic knapsack problem (TSSQKP) with uncertainty in profits and weights. To overcome the computational difficulties arising from the combined stochastic, binary, and quadratic structure, we propose a solution framework that integrates an adaptation of the progressive hedging algorithm with clustering-based scenario reduction. The enhanced progressive hedging algorithm (EPHA) incorporates a rounding procedure and a dynamic penalty update strategy driven by the detection of oscillation and stagnation patterns. In parallel, we propose a clustering framework based on opportunity-cost distances between scenarios, from which we derive the medoid lower bound (MLB), the cluster lower bound (CLB) and the cluster upper bound (CUB). To further improve computational efficiency, EPHA is also used to solve the cluster subproblems heuristically; in this case, the resulting upper bound is referred to as the estimated cluster upper bound (ECUB), and the corresponding gaps are interpreted as estimated optimality gaps. Computational experiments on 800 generated TSSQKP instances show that the proposed framework yields smaller gaps than CPLEX under comparable computational conditions. In the EPHA-based clustering framework, the average estimated gaps associated with the CLB, EPHA and MLB 1.65%, 1.88% and 2.41% respectively, compared with an average optimality gap of 35.86% for CPLEX. MLB is fastest to compute, with an average execution time of 420.88s, while EPHA provides a favourable compromise between solution quality and computational effort, reducing the average computational time associated with the cluster bounds by approximately 52.7%, while increasing the average estimated gap by only 0.23 percentage points.

We compute the effective six-wave interaction in spatial and temporal Hamiltonian Dysthe equations and compare it with the complete degree-six coefficient of the deep-water Euler Hamiltonian. On every compact subset $K$ of a generic quadratic-dispersion resonance patch, exact resonances persist under a small cubic correction to the dispersion, and \[ W_6\bigl(\Phi_{\lambda,\varepsilon}(z)\bigr) =\varepsilon\frac{4BD}{aL(z)}+O_K(\varepsilon^2) \] uniformly in the resonance point $z$ and the deformation parameter $\lambda$. Here $L$ is the spectral diameter and $B,D$ are the local cubic and mean-flow coefficients. Thus $BD\ne0$ gives nonzero scattering on relatively open resonance sets. At the Craig–Guyenne–Sulem temporal coefficients, the corresponding exact Euler resonances satisfy \[ \frac{\mu}{4}W_6^{\rm E}\bigl(1+\mu\Psi_\mu(z)\bigr) =-\frac{8}{L(z)}+O_K(\mu), \] where $\mu$ is the relative bandwidth. The leading term comes from nine contractions of quartic vertices; every other connected degree-six contribution is uniformly bounded. We classify the coefficient values for which the generic six-wave coefficient vanishes identically; in the temporal case, this also gives cancellation on every non-permutation four-wave resonance. Outside these sets, the Zakharov–Schulman condition restricts every regular $C^3$ quadratic leading symbol to $\operatorname{span}\{1,k,\omega(k)\}$, the span of the symbols of wave action, momentum, and linear energy. Consequently the physical Dysthe models cannot support an inverse-scattering hierarchy with infinitely many linearly independent regular $C^3$ quadratic leading symbols. The cancellation sets contain known nonlinear Schrödinger, mixed Chen–Lee–Liu, Calogero–Moser derivative nonlinear Schrödinger, and Hirota representatives.

Given any iterated function system (IFS) on the plane whose affine transformations share the same contracting matrix $T$, we characterize the set of extreme points of the attractor $A$ (along the convex hull $\operatorname{conv}(A)$) in terms of a directed graph on the edges of $\operatorname{conv}(A)$. This graph corresponds to a sofic system on the coding space associated to the IFS. When $T$ is proper orthogonal or purely real, we give conditions for $\operatorname{conv}(A)$ to be a finite polygon and, under additional symmetry of the transformations, we give conditions for $A$ itself to coincide with $\operatorname{conv}(A)$.

The critical dimension for free maps on m-manifolds is $q_m=m(m+3)/2$. We show that $Free^\infty(\mathbb{T}^m,\mathbb{R}^{q_m})\neq\emptyset$ for every $m\ge 1$. The main tool is a product construction: given free maps in critical dimension on $\mathbb{T}^a$ and on $\mathbb{T}^b$, together with a cross-free map on $\mathbb{T}^a\times \mathbb{T}^b$, whose mixed Hessian is invertible everywhere, we obtain a free map in critical dimension on $\mathbb{T}^{a+b}$, the osculating matrix of the product being block-triangular. Cross-free maps are additive in each argument, and in the torus setting none exists when one factor is one-dimensional. We construct explicit cross-free maps on $\mathbb{T}^2\times \mathbb{T}^2$ and on $\mathbb{T}^3\times\mathbb{T}^4$ using quaternionic multiplication. Combined with the low-dimensional cases $m\leq 5$, constructed explicitly by the author in a previous publication, these yield the result by induction.

Modern prediction-driven decision systems often rely on black-box predictors, but a point forecast alone does not provide the uncertainty scale required for robust downstream decision-making. We build a score-calibrated robustness framework that converts any fixed point predictor into a decision-relevant uncertainty representation through distribution-free conformal calibration. We use the conformal score, rather than a particular uncertainty set, as the primitive unit of robustness. The same score determines coverage-calibrated uncertainty sets for reliability-based robust optimization and normalizes target violations in a target-oriented formulation, Conformal Robust Satisficing. This formulation induces a conformal fragility measure that quantifies how rapidly performance deteriorates as the realized parameter departs from the forecast on the conformal score scale. For objective-uncertainty problems under standard convexity and duality conditions, we show that the reliability-based and target-oriented formulations parameterize the same score-calibrated robust decision frontier. This equivalence yields a data-driven mapping between reliability levels and acceptable targets and characterizes the marginal cost of robustness. Synthetic experiments validate the theoretical guarantees and illustrate the reliability-target correspondence. A real-data online-grocery case study demonstrates how the interface combines deep-learning demand forecasts with tractable inventory optimization, thereby improving reliability and reducing operational costs. Overall, our work shows that conformal scores endow fixed black-box predictors with an interpretable uncertainty scale for downstream decision-making while enabling reliability guarantees, acceptable-target selection, and fragility analysis within a unified framework.

The Asplund property plays an important role in Banach space theory due to its connections with differentiability properties of continuous convex functions, optimization problems, and the weak topology of Banach spaces. Motivated by the variety of nonequivalent definitions proposed in the literature for locally convex spaces, in this survey we provide a unified framework for studying the Asplund property beyond the Banach space setting. The main object of our study is the locally convex space of continuous functions $C_k(X)$ endowed with the compact-open topology, where $X$ is an arbitrary Tychonoff space. We completely characterize the Asplund property for $C_k(X)$ in terms of topological properties of the underlying space $X$. As an essential step, we revisit the proof of several classical results, including the Namioka–Phelps theorem. Our approach is independent of differentiability techniques and relies solely on topological methods. The exposition is self-contained, and all major results are provided with complete proofs, making the paper accessible to both specialists and newcomers.

Comparing probability measures in machine learning trades transport geometry against computational cost: Wasserstein distances encode the geometry of $\mathbb R^d$ but require solving a transport problem, while kernel discrepancies are cheap to evaluate yet depend delicately on their test class. We study the Toscani–Fourier family $\mathrm T_{s,p}$, the weighted $L^p$ norm of the difference of two characteristic functions, as a continuous Fourier-side discrepancy on $\mathbb R^d$. For $1\le p<\infty$ we show that $d/p<s<1+d/p$ is exactly the window in which $\mathrm T_{s,p}$ is finite on $\mathcal P_p(\mathbb R^d)$, both endpoints already failing for a pair of Dirac measures, and we establish the metric, embedding, and compactness structure of the resulting space, which we prove to be complete. Duality identifies $\mathrm T_{s,p}$ as an integral probability metric over a homogeneous Fourier–Lebesgue ball, with an explicit extremizer when $1<p<\infty$. We prove the global bound $\mathrm T_{s,p}\lesssim W_p^{\,s-d/p}$, whose exponent is sharp, show that no global converse of any form can hold, and recover topological equivalence with $W_p$ on bounded-support and uniform-tail classes, together with explicit reverse moduli on bounded-support classes that improve the imported energy-kernel exponent at $p=2$. There, $\mathrm T_{s,2}$ is a constant multiple of the classical energy distance, which yields an exact finite-sample identity for the mean of the empirical discrepancy; the numerical experiments are otherwise diagnostic.

For a finite family of real matrices, we bound the Gaussian width of the union of its image ellipsoids by its maximal Hilbert Schmidt norm, its maximal fixed input Gaussian width, and the square root of the product of its operator radius and decoupled Gaussian chaos supremum.

We study local bifurcations from the single-mode traveling wave family of the filamentation equation on the torus, within the real positive-frequency Sobolev space $X^s$. A Fourier null structure removes the apparent derivative loss and makes the traveling wave map real analytic for $s>3/2$. Its linearization splits into finitely many coupled two-mode blocks, a carrier block, and a diagonal high-frequency tail. For negative temporal frequency, we determine the critical values generated by both the finite coupled blocks and the tail. At every non resonant finite-block critical value, Lyapunov-Schmidt reduction yields a locally unique real-analytic branch, with vanishing linear correction to the bifurcation parameter. Moreover, each finite tail critical value produces a simple local branch. These values accumulate at a parameter where the linearization ceases to be Fredholm. The accumulation point is nevertheless a bifurcation point in the usual topological sense. For $\sigma=1$, a symmetry in the first Fourier mode produces an exact two-mode vertical branch, including the exceptional non-Fredholm case $k=2$.

The sets-as-trees interpretation of set theory in a dependent type theory with an impredicative universe of propositions validates Zermelo set theory, and it validates Replacement if the type theory has a choice or description operator, which turns a functional relation into a function. It has been natural to expect that without such an operator the strength of the type theory drops well below that of $\mathrm{ZF}$. We show that it does not. In the type theory of Lean with two predicative universes, from excluded middle as the only assumption and with no axiom (no choice, no propositional extensionality, no quotients), we prove the consistency of $\mathrm{ZF}$, stated outright for a first-order proof system. The proof is formalized. The mechanism is the large elimination of the accessibility predicate over a type as large as the type of sets: a recursion on accessibility whose recursive calls are guarded by propositions, and whose later calls are indexed by the value of an earlier call, computes as a term any ordinal that is specified by a proposition through a well-founded tree of a certain shape. We give a rule that produces such a tree for every ordinal, unless some $V_\rho$ is already a model of $\mathrm{ZF}$; the rule does not choose a cofinal map into a limit ordinal but takes all definable ones at once. In the first case the sets-as-trees satisfy Replacement for arbitrary propositional relations. Either way $\mathrm{ZF}$ has a model. Finally, the double negation of excluded middle suffices, and what remains of it is exactly that membership is not not well-founded in the stable reading of sets; this in turn implies the double negation of Markov's principle.

The determinant is traditionally introduced through permutation expansions, multilinearity, or recursive expansion formulas - approaches that, while rigorous, often obscure its conceptual significance. Instead, an approach is advocated wherein the determinant is defined via its natural characterization as a multiplicatice map on matrices, reflecting its role as a structural invariant of linear transformations under composition. This perspective defers combinatorial machinery until after the core properties are estaqblished, thereby lowering the cognitive barrier for students encountering the concept for the first time. The equivalence of this characterization with classical formulations is shown, and it is argued that this framework better aligns with the structural emphasis of m odern mathematical education.

In his seminal work from 1973, Delsarte conjectured that the Johnson schemes admit no nontrivial perfect codes. We prove this fundamental conjecture which is central in combinatorial coding theory. Our arguments combine lower bounds on the design strength of a perfect code with divisibility conditions for the size of a metric ball. We show that the ball size divides the greatest common divisor of specified binomial coefficients. An exact formula for this gcd implies squarefreeness and localizes a power of each prime divisor of the ball size to a short interval. Lloyd polynomials, explicit prime estimates, and bounds for linear forms in logarithms then exclude all sufficiently large radii. Additional arithmetic arguments handle the remaining radii.

The Albanese fibration of the Cartwright–Steger surface is a genus-$19$ Lefschetz fibration over a torus. We give an explicit formula for it: in the projective model of Borisov and Yeung, the map is represented by three homogeneous polynomials of degree $13$ with integer coefficients, and its target is the elliptic curve $Y^2Z=X^3-3888Z^3$. A recognition theorem shows that it suffices to verify one polynomial identity on the surface. A coefficient bound allows us to establish the identity over $Q$ from computations modulo primes. We also determine exact coordinates for the three nodes and their images on the elliptic curve.