Generative AI raises short-term productivity by completing tasks that learners would otherwise practice on their own. Whether this substitution erodes frontier skill, the skill behind top-tail non-AI-aided performance, is an open question of rising stakes. The sharper question is whether selection mechanisms can screen apart two coexisting types: substitute-users, who use AI in place of deliberate practice, and complement-users, who use it to accelerate skill development. In elite programming, the International Collegiate Programming Contest (ICPC) and the International Olympiad in Informatics (IOI) prohibit AI under proctoring and admit entrants through qualification rounds, whereas online Codeforces (CF) contests are unproctored and open to all. From CF histories we build an AI-prompt signature (more first-attempt acceptances, fewer attempts and retries) consistent with AI-assisted practice. Three patterns triangulate institutional screening. First, CF practice shifted toward this signature across cohorts over two AI rollouts. Second, in open CF contests a stronger signature predicts smaller rating gains for users with no ICPC/IOI affiliation, but not for those who qualified for the AI-prohibited contests. Third, inside the AI-prohibited ICPC environment, a shift toward AI-style practice predicts higher non-AI-aided scores for AI-era entrants. The same practice input carries opposite signs depending on whether the environment screens for it. The contrast points to two levers: how AI is integrated into training, since within the screened pool AI-style practice coincides with stronger non-AI-aided performance; and the design of AI-prohibited evaluation gates as a type-separating institution. Both extend beyond programming to credentialing systems (medical and legal boards, professional certification) that certify skill in a workforce increasingly shaped by AI.

First, this paper proves that Tennant's two claims (i.e. that his own logical system is paraconsistent, and that it overlaps minimal logic) are both false. Second, this proof is certified with Coq.

We show that negative Gaussian curvature regularizes the infrared sector of crystalline membranes. In a covariant formulation of embedded elasticity, the Green strain contains a symmetry-required linear coupling between the normal phonon and extrinsic curvature. Integrating out the in-plane phonons converts this coupling into a finite quadratic contribution to the inverse flexural response. Anomalous roughening is thereby replaced by curvature-controlled saturation, and the mechanism survives on minimal hyperbolic patches. Hyperbolic geometry preempts anomalous elasticity before the flat infrared regime is reached, implying the absence of a crumpling phase. The same Gaussian-order coupling admits sound propagation in the infrared.

Using edge weights is essential for modeling real-world systems where links possess relevant information, and preserving this information in low-dimensional representations is relevant for classification and prediction tasks. This paper systematically investigates how different random walk strategies - traditional unweighted, strength-based, and fully weight-aware - keeps edge weight information when generating node embeddings. Using network models, real-world graphs, and networks subjected to low-weight edge removal, we measured the correlation between original edge weights and the similarity of node pairs in the embedding space generated by random walk strategies. Our results consistently showed that weight-aware random walks significantly outperform other strategies, achieving correlations above 0.90 in network models. However, performance in real-world networks was more heterogeneous, influenced by factors like topology and weight distribution. Our analysis also revealed that removing weak edges via thresholding can initially improve correlation by reducing noise, but excessive pruning degrades representation quality. Our findings suggest that simply using a weight-aware random walk is generally the best approach for preserving node weight information in embeddings, but it is not a universal solution.

We provide multiple combinatorial expansion formulas - in terms of snake graphs, labelled posets, matrices, and $T$-walks - for elements in generalized cluster algebras associated to arcs on punctured orbifolds and illustrate their equivalence. This work generalizes and unifies existing work on combinatorial expansion formulas from surfaces and unpunctured orbifolds.

We study the stability of traveling wave solutions to the Burgers--Hilbert equation on $\mathbb{T}$ in the regime of small frequency $ω$ and large wave speed $c$. For $ω= 3$ and $c \approx 1.1$, we show that the linearized operator around these solutions has an eigenvalue with negative real part, indicating spectral instability. Our approach is computer-assisted: we reduce the problem to a finite-dimensional system and solve it rigorously using interval arithmetic. The Burgers--Hilbert equation arises as a quadratic approximation of the vortex patch problem for the two-dimensional Euler equations. In this setting, our results point to the instability of threefold symmetric V-states.

The integration of scandium aluminum nitride (ScAlN) into functional heterostructures has pushed semiconductor device physics into an extreme piezoelectric regime. Accurately determining the fundamental clamped dielectric permittivity (ε_33^S) of these alloys remains a major challenge, as standard density functional theory systematically overestimates this specific parameter due to well-known bandgap-underestimation limits. Furthermore, electrical capacitance-voltage (C-V) measurements of real-world devices do not yield this clamped baseline; instead, they capture an effective permittivity inflated by linear electromechanical coupling. Because epitaxial thin films are unconstrained in the out-of-plane direction, small-signal electrical measurements dynamically induce macroscopic lattice strain via the inverse piezoelectric effect. By applying these specific mechanical boundary conditions to the coupled equations of state, we utilize the analytical relation for operational permittivity: ε_eff=ε_33^S+e_33^2/C_33. Leveraging highly accurate first-principles predictions for the structural and piezoelectric tensors, we invert this model to extract the true, uninflated clamped dielectric baseline directly from experimental thin-film data. This approach circumvents the conventional limitations of pure first-principles dielectric calculations, flawlessly resolving the theoretical-experimental discrepancy and providing a critical framework for accurate device simulation in highly polar nitrides.

We consider an optimal transport problem between laws of random probability measures: given a base cost function, we build the associated OT cost between probability measures that in turn we use to define the OT cost between probability measures over probability measures. This setting admits a finer reformulation in terms of laws of random couplings, which retain more information than ordinary couplings. One of the main contributions of the paper is the characterization of the optimal ones in terms of Kantorovich potentials. Similarly, we also introduce the strict Monge problem, whose admissible competitors are more restrictive than in the usual Monge formulation. In this setting, we will give sufficient conditions under which the value of this problem is the same as the one considered above, in the spirit of the result by A. Pratelli. Then, for $p>1$, when the underlying cost is the distance to the power $p$ in a strictly convex Banach space, we will give sufficient conditions under which the optimal random coupling is unique and induced by a solution of the strict Monge problem, resembling the Brenier theorem.

Nikiforov's well-known spectral Turán inequality for walks states that, for every graph $G$ with clique number $ω(G)$, $λ^r(G)\le w_r(G)(1-1/ω(G))$, where $λ(G)$ is the largest eigenvalue of the adjacency matrix of $G$, and $w_r(G)$ is the number of walks with $r$ vertices in $G$. For $r=1$, this is Wilf's inequality; for $r=2$, it gives Nikiforov's spectral Turán theorem. Recently, Liu and Ning proved local versions of these two inequalities, strengthening both Wilf's inequality and Nikiforov's spectral Turán theorem. It is natural to ask whether Nikiforov's spectral Turán inequality for walks also admits a local strengthening. Motivated by this question, Kannan, Kumar, and Pragada conjectured the vertex-local bound $λ^r(G)\le \sum_{v\in V(G)} w_r(v)(1-1/c_G(v))$, where $w_r(v)$ denotes the number of walks with $r$ vertices starting at $v$, and $c_G(v)$ is the maximum order of a clique containing $v$. This conjecture is important because it gives the most natural local form of Nikiforov's spectral Turán inequality for walks. In this paper, we confirm this conjecture. More precisely, for $r\ge 2$, we prove the stronger edge-local inequality $$λ^r(G) \le \sum_{uv\in E(G)} \frac{c_G(uv)-1}{c_G(uv)} \bigl(w_{r-1}(u)+w_{r-1}(v)\bigr),$$ where $c_G(uv)$ is the maximum order of a clique containing the edge $uv$. Our result implies Nikiforov's spectral Turán inequality for walks and unifies several local spectral extremal results of Liu and Ning. We also determine all extremal graphs for both the edge-local and vertex-local inequalities. The main new ingredient is a Markov-chain estimate whose transition matrix is constructed from a Perron vector of $A(G)$; this estimate carries the local edge coefficient through walks of arbitrary length.

Energy transmission over long distances by waves is a key mechanism for many natural processes. This possibility arises when an inhomogeneous medium is arranged in such a manner that it enables a certain type of wave to propagate with virtually no reflection or scattering. By application of the Laplace cascade method for integrating second-order hyperbolic equations, a general algorithm for finding the parameters of inhomogeneous non-reflecting flows is proposed. The algorithm is applied to the problem of long linear surface waves propagation in a channel with variable cross-section. The general analysis of the problem is illustrated by a few representative solutions and compared with the results of previous studies. The results obtained may be of interest to mitigate the possible impact of waves on ships, marine engineering constructions, and human coastal activities.

We concern the structure of generating weighted IFSs of a self-similar measure on the real line. We provide various sufficient conditions for the existence of a minimal generating weighted IFS of a self-similar measure on the real line. Under the homogeneity, we show that `most' self-similar measures on the real line have a minimal generating weighted IFS, without separation conditions. The ingredients of our proofs are based on results of exponential polynomials (factorization theory and the distribution of zeros), logarithmic commensurability (with a dynamical system argument), and results on the structure of generating IFSs of a self-similar sets.

We show that for any smooth cubic in $\mathbb{P}^2$, there exists a dense $G_δ$ set of configurations of 9 distinct points such that blowing up $\mathbb{P}^2$ at these 9 points, the strict transform of the cubic is not linearizable and has nontorsion normal bundle. This answers a problem raised by Ogus in 1975.

For multi-strain models with increasing concave incidence and scalar, uncorrelated strain blocks, we prove a competitive-exclusion partition of parameter space, by constructing explicit global Lyapunov functions for the boundary and coexistence equilibria. We further extend the construction to models containing one irreducible rank-one infection block and several uncorrelated scalar strains. Our method combines Volterra entropy terms for the resident variables with Perron-weighted linear functionals for the invading variables, whose weights are obtained from transversal Metzler Jacobians on boundary faces. We have provided an algorithmic implementation in the Mathematica package E PID CRN (https:// github.com/florinav/EpidCRNmodels). The method organizes the dynamics via the minimal siphon lattice, recursively computes Perron eigenvectors of transversal Jacobians, and constructs candidate Lyapunov functions for all equilibria, producing a parameter partition into regions with a unique locally stable equilibrium and, for n = 2, explicit global stability certificates.

This paper develops a branch-level energy-localization framework for three-phase loads. The instantaneous terminal power of an admissible lumped equivalent is decomposed uniquely as Joule dissipation plus magnetic and electric stored-energy rates, branch by branch. Three formal results are established: a Branch-Level Localization Theorem (uniqueness given an admissible topology); a Topology-Indeterminacy Theorem (multiple admissible topologies reproduce identical terminal data with distinct localizations); and a Generalized Energetic Duality Theorem that organizes classical electrical dualities (Norton-Thevenin, series--parallel, L vs C, R vs G) as restrictions to Linear Time Invariant (LTI) sinusoidal regimes of a single time-domain principle in which constant-parameter equivalence is replaced by time-varying parameters. The framework is exercised on six test cases including the de Leon--Cohen open-phase paradox, switched-resistive loads, three-wire delta-versus-wye-virtual indeterminacy, fluctuating-phase loads, and a four-wire nonlinear load with hysteretic, linear, and switched branches. The framework is positioned as complementary to IEEE Std. 1459, CPC, instantaneous p-q, and Fryze-Buchholz-Depenbrock: each answers a different question, and the apparent paradoxes vanish once the question is posed precisely.

This paper presents a geometric time-domain method for identifying three-phase load equivalents from instantaneous voltage and current measurements at the point of common coupling. Measured waveforms are interpreted as trajectories in Euclidean signal spaces, and load-equivalent parameters are recovered from the geometry of those trajectories. The method extends a previously published single-phase geometric identification formulation to three- and four-wire systems and places special emphasis on the three-wire case, where no neutral voltage is measured and the terminal data must satisfy coupled Kirchhoff constraints. The main advance over the earlier analytical formulation is a sampled-data implementation based on local time windows, normalized matrix equations, harmonic-projection derivative and primitive coordinates, explicit geometric identifiability tests, passivity constraints, and energy/Kirchhoff residuals. The method does not force a model when the measured trajectory lacks enough information; instead, it reports low-rank or ill-conditioned windows as low-confidence evidence. Numerical simulations with clean data, measurement noise, window-length sweeps, and sensor delay show that the method accurately identifies informative three-phase trajectories and exposes structurally degenerate cases such as pure single-frequency excitation for higher-order three-wire models. For a given admissible topology the identified circuit closes the instantaneous terminal energy balance of the measured load over the analysis window.

Subnational studies of aid effectiveness often rely on repeated cross-sections or nighttime lights, making it difficult to separate local treatment effects from baseline differences and potentially favoring infrastructure-heavy projects. We address these limitations by studying World Bank and Chinese development projects in Africa with a balanced panel of 2,166 DHS clusters across 35 countries from 2002 to 2013. Geocoded AidData projects are linked to satellite-imputed International Wealth Index estimates, a household-centered measure of material living standards. We compare a conventional two-way fixed effects (TWFE) event-study with the switcher--stayer estimator of de Chaisemartin and D'Haultfoeuille (dCdH), which avoids contaminated comparisons under staggered treatment timing. Pre-treatment diagnostics show that project placement is frequently selective: clusters that later receive projects often begin from weaker relative positions before treatment onset. Consequently, TWFE often implies larger post-treatment gains than the preferred staggered-treatment design supports. Under dCdH, the evidence becomes more selective and sector-specific. For the World Bank, positive evidence is strongest in Health, while Education shows positive but less cleanly identified gains. For China, Water Supply and Sanitation and Other Social Infrastructure and Services show positive associations with local wealth, although residual selection concerns remain. By contrast, Chinese Energy Generation and Supply appears strongly positive under TWFE but falls close to zero under dCdH. Overall, the results do not support a donor-wide claim that either the World Bank or China uniformly improves local wealth. Instead, estimated effects are concentrated in a limited set of donor--sector panels and depend strongly on how treatment timing, selection, and outcome measurement are handled.

A b-coloring of a graph $G$ is a proper vertex coloring where each color class contains at least one vertex (a b-vertex) adjacent to a vertex in every other color class. The maximum number of colors in such a coloring is the b-chromatic number, ${\rm b}(G)$. A ${\rm b}^{\ast}$-coloring is a variation in which a b-vertex is adjacent to a b-vertex in every other color class. We employ the ${\rm b}^{\ast}$-coloring to prove that any $n$-vertex graph $G$ with independence number at most $t$ satisfies ${\rm b}(G) \leq [(t-1)n+tχ(G)]/(2t-1)$. This bound extends the bounds of Kouider and Zaker (2006) and Alkhateeb and Kohl (2011) and improves the bound in terms of the clique partition number. We show that this bound is sharp for all $t\geq 2$ and $χ(G)\geq 3$. Furthermore, we provide a refined bound based on the maximum number of vertex-disjoint independent sets of size $t$. Finally, we prove ${\rm b}^{\ast}(G) \leq [(t-2)n+(t-1)χ(G)]/(2t-3)$ for all $K_{1,t}$-free graphs $G$, a significant improvement over the analogous bound for ${\rm b}(G)$.

Given a frequency $λ$, we study the Riesz summability of $λ$-Dirichlet series $\sum_{n=1}^\infty a_n e^{-λ_n s}$ generating holomorphic functions of finite order. We present a new separation condition on the frequency $λ$ ensuring that, for any $k \geq 0$, each $λ$-Dirichlet series that is somewhere Riesz summable of some order and admits a holomorphic extension $f$ to the right half-plane $\mathbb{C}_0$ satisfying $f(s) = O(|s|^k)$ as $|s| \to \infty$ on $\mathbb{C}_0$, is in fact Riesz summable of order $k$ on $\mathbb{C}_0$. This extends Bohr's theorem, which corresponds to the case $k = 0$. Our work improves a recent result of Defant and Schoolmann, who showed the above property under Landau's condition (LC). Along the way, we also establish novel bounds on the coefficients of such $λ$-Dirichlet series and, under a mild condition on the frequency $λ$, show that they are optimal.

We study the convergence of SU(1,1) and SU(2) nonlinear Fourier transform with sparse square-summable data. The asymptotics of the associated polynomials orthogonal on the unit circle is obtained as a corollary.

We propose a non-Hermitian Floquet photonic lattice formed by a physical resonator coordinate and a synthetic frequency coordinate. A two-step modulation protocol realizes a chiral walk in this physical-synthetic plane, with a real synthetic flux controlling loop interference and imaginary gauge fields controlling non-reciprocal envelopes. We show that anomalous corner pairs at quasienergies zero and \(π/T\) exhibit three distinct layers of physics. A non-Bloch higher-order construction predicts whether the \(0/π\) corner pair exists under open boundaries. The imaginary gauge fields select where the right eigenmodes accumulate. The real flux controls the local interference matrix element that determines whether the doubled-period optical response is visible. As a result, the same topological coexistence sector can be bright, skin-dark, or flux-dark in a local optical measurement. We further show that the complex gauge can tune an exceptional point of the two-period corner propagator. At this point the anomalous response keeps its doubled-period sign alternation, but its envelope becomes algebraic because of a Jordan block. These results provide a photonic route to separate topological existence, skin-selected localization, optical visibility, and defective two-period dynamics in a non-Hermitian synthetic dimension.

The development of modern high-energy astrophysics has been deeply intertwined with advances in particle detector technology. Guido Barbiellini Amidei (1943-2024) played a pivotal role in bridging experimental particle physics and astrophysical observation. His scientific career spanned over four decades, from early electron-positron collider experiments at ADONE and LEP (DELPHI) to space based missions such as AGILE, Fermi, and PAMELA. This memorial paper reviews the evolution of high-energy astrophysics as a detector-driven science, highlighting key domains where Barbiellini left an indelible mark: gamma-ray astronomy, cosmic-ray physics, and antimatter studies. We discuss his personal contributions to silicon tracking, calorimetry, data analysis, and his leadership in international collaborations. The conceptual impact of his interdisciplinary approach is examined, and future perspectives in the observation of the high-energy universe are outlined, recognizing that the path forward is built on the foundations he helped lay

Gas-stellar kinematic misalignments trace how galaxies assemble and reorient their angular momentum. Although well documented locally, their continuous evolution remains largely unexplored. We aim to characterise the pathways linking aligned, misaligned, and counter-rotating phases, assessing how gas accretion channels and mergers drive reorientation. We analysed 44 central galaxies from the CIELO simulations from z=3.5 to z=0. We defined kinematic episodes using the intrinsic angle, psi, between the star-forming (SF) gas and stellar angular momentum vectors, interpreting these histories by tracking accreted gas origins and evaluating paired statistical contrasts. Nearly 80% of the simulated galaxies are aligned at z=0, yet 86% experience at least one non-aligned episode. Although 19% of non-aligned episodes last >2 Gyr, their durations do not differ significantly from aligned episodes. Abrupt changes in psi coincide with intervals where accreted gas dominates the pre-existing SF reservoir and is highly tilted relative to pre-existing stars. During transitions, the median mass ratio of accreted to pre-existing SF gas rises from 0.57 to 2.14, and the median angular offset increases from 21.2 to 64.6 deg. While relevant mergers cluster near these boundaries, they can drive either alignment or misalignment regardless of mass ratios or orbits. Instead, mergers triggering abrupt transitions typically encounter hosts that are already partially decoupled. In the simulated sample, gas-stellar misalignment is fundamentally driven by reservoir competition. Mergers act as conditional triggers, but a galaxy's ultimate kinematic fate depends strictly on how newly accreted gas couples to, replaces, or mixes with the pre-existing material as it joins the central SF reservoir.

We present two-epoch JWST NIRSpec and MIRI observations of the young protostar EC 53 (V371 Ser), a periodically variable source with well-characterized quiescent and burst phases. The spectra in both epochs show absorption in the CO overtone ($\sim$2.3 $μ$m) and fundamental ($\sim$4.6 $μ$m) bands and the H$_2$O stretching ($\sim$2.7 $μ$m) and bending ($\sim$6.0 $μ$m) modes. We also obtained high-resolution ($R\approx45{,}000$) IGRINS spectra during the burst to constrain the CO overtone line profiles. LTE slab modeling yields gas temperatures of $\sim$1800 K (CO overtone) and $\sim$1200 K (CO fundamental), consistent with the overtone tracing hotter gas at smaller radii. The H$_2$O stretching-mode absorption shows no compelling evidence for variability, and the current JWST CO overtone data do not provide a robust constraint on overtone variability. In contrast, the CO fundamental and H$_2$O bending-mode features weaken by a factor of $\sim$2 during the burst, which is most naturally explained by continuum changes rather than large variations in absorbing gas. To quantify continuum dilution, we introduce a ``relative veiling'' that treats the quiescent spectrum as an internal reference and measures the change in the continuum excess between the two epochs. This formalism yields burst-to-quiescent hot-continuum ratios of $2.9\pm0.2$ for the CO overtone and $1.71\pm0.11$ for the CO fundamental. Using a viscous-disk prescription, these imply representative accretion-rate ratios of $\sim$3.6 and $\sim$2.0, respectively. The differing ratios suggest that inner-disk regions traced at different temperatures, and thus radii, respond differently across the burst cycle, consistent with episodic mass buildup in the inner disk during quiescence followed by more efficient transport through the innermost disk onto the protostar during the burst.

A stabilisation of a $4$-manifold $X$ is the connected sum of $X$ with some number of copies of $S^2\times S^2$. If two smooth surfaces in a $4$-manifold are topologically isotopic, we investigate whether they must moreover be smoothly isotopic in some stabilisation of $X$. We prove this result holds whenever the surfaces are trivial in the $\mathbb{Z}/2$-homology of $X$. We also produce a large class of fundamental groups of the ambient $4$-manifold for which the result holds; this class includes free products of classical knot groups and, in particular, free groups.

The well-known Erdős-Hajnal conjecture states that for any graph $H$, there is a constant $c=c(H)>0$ such that every $n$-vertex graph $G$ with no induced copies of $H$ contains a clique or an independent set of size at least $n^{c}$. We prove that Erdős-Hajnal conjecture holds for two more graph classes-graphs with no induced copies of $E$-graph and graphs with no induced copies of Birds, where $E$-graph is the graph obtained from the five-vertex path by adding a pendent edge to the middle vertex of the path and Bird is the graph obtained from a bull by adding a pendent edge to one horn of the bull. Our results generalize the result of Nguyen, Scott and Seymour on the five-vertex path (Proceedings of London Mathematical Society 2026) and the result of Chudnovsky and Safra on the bull graph (Journal of Combinatorial Theory Series B 2008). The proof uses the iterative sparsification framework proposed by Nguyen, Scott and Seymour with our generalization. We first reduce, up to some technical condition, Erdős-Hajnal conjecture to a property called generlaized nice, which is a generalization of the ``nice'' property used in [T.~Nguyen, A.~Scott, and P.~Seymour. Induced subgraph density. VII. The five-vertex path. {\em Proceedings of the London Mathematical Society}, 132(3):e70133, 2026]. We ues Ramsey Theorem and a new idea for embedding graphs with no leaf vertices to prove that $E$-graph and Bird satisfy this technical condition. We then reduce the generalized nice property to a new property $(*)$. Finally, we show that $E$-graph and Bird graph satisfiy $(*)$. One key step in the proof is to prove, via defining appropriate equivalence relations, that certain auxiliary graph satisfies the Erdős-Hajnal conjecture.

This paper studies interdependent durations as equilibrium outcomes of a synchronization game, a continuous-time stopping game in which the incentive to stop increases when other players stop. We allow the payoffs to vary with both common shocks and observed and unobserved agent characteristics. The common shocks follow a spectrally negative Lévy process, a semiparametric process that includes Brownian motion as a special case but may also have jumps. We show that equilibrium outcomes can be represented as interdependent hitting times and use this to establish the game's nonparametric identification from data on stopping times and covariates. We develop maximum simulated likelihood and method of simulated moments estimators and evaluate their finite-sample and computational performance in Monte Carlo experiments. The results provide a tractable framework for identifying and estimating synchronization games from interdependent duration data.

White dwarfs (WDs) in binaries can form through either the stable mass-transfer process or common envelope evolution (CEE). Compared to CEE, the stable mass-transfer process can lead to a distinct mass-orbital period ($M_{\mathrm{WD}}-P_{\mathrm{orb}}$) relation. Thus, this relation of WDs contains the information about the evolution channels. We can study the relation in WD binary systems to determine whether their progenitors undergo a CEE. We use the stellar evolution code MESA as our primary computational tool and adopt the quasi-adiabatic criterion to ensure that our models satisfy the conditions for stable mass transfer. Our study considers different mass-transfer schemes, varying metallicities, and the relation for both low-mass and intermediate-mass progenitors. Previous studies have focused on the relation for low-mass progenitors, which cannot explain some long-period, high-mass WD binaries. Our results show that the relations for intermediate-mass progenitors whose cores remain non-degenerate prior to central helium burning can account for the formation channels of long-period and massive WD binaries.

In this note, we generalize a 1950 result of P. Erd\H os on upper bounds of $k$-th moment of Romanoff type representation functions. As an application, we give a conditional proof of a recent conjecture of Y.-G. Chen on Romanoff type sumsets under the assumption of the Hardy-Littlewood conjecture.

This paper studies two questions on valued fields: the metric dimension induced by an absolute value, and the uniform openness of multiplication. For nontrivial non-archimedean absolute values, we prove that the metric dimension equals the density character. In the archimedean case, the metric dimension is 2 for subfields of $\mathbb{R}$, while for non-real subfields of $\mathbb{C}$ it is either 2 or 3, according to whether the field contains a non-real element together with its complex conjugate. We also show that multiplication is uniformly open on every valued field. Finally, we prove that this property is genuinely metric, not purely topological, even on $\mathbb{R}$ with a suitable compatible choice of metric.

In this work, the exponential sum-connectivity Gourava index ($e^{SGO(G)}$) and the exponential product-connectivity Gourava index ($e^{PGO(G)}$) are computed and comparatively analyzed for benzenoid hydrocarbons. Our results demonstrate that these descriptors exhibit a strong mutual correlation and provide enhanced sensitivity in modeling the structural characteristics of molecular graphs. Regression analysis reveals that both indices are exceptionally reliable predictors of $π$-electronic energies, achieving correlation coefficients exceeding $0.999$. Notably, a comparative assessment indicates that the exponential product-connectivity variant offers a slightly superior fit, as its coefficients align more precisely with optimal least-squares results. These findings confirm that both exponential Gourava-based indices provide a robust framework for characterizing electronic properties, with the product-connectivity version showing particular promise for high-precision QSPR studies in benzenoid systems.