Understanding how nuclear size evolves with the number of protons and neutrons tests our models of strongly interacting matter. The nuclear charge (and proton) radii accessible through electromagnetic probes carry fundamental information on the saturation density and nuclear correlations. The radii of the neutron distribution are more difficult to measure, but they are important for our understanding of the isovector properties of nuclei that depend on the proton-to-neutron asymmetry, and on extended nucleonic matter in neutron stars. Interaction cross sections offer one of the few direct experimental windows into the neutron radii of nuclei far from stability, but translating these measurements into reliable structural information requires an integrated theoretical framework that links structure and reactions with a rigorous treatment of uncertainty. In this work, we compute interaction cross sections by using uncertainty-quantified proton and neutron distributions obtained in the self-consistent nuclear Density Functional Theory (DFT) with the Fayans energy density functional. The resulting densities are used in a modernized Glauber reaction framework, which features the refit of nucleon-nucleon profile functions. Applying this pipeline to the existing data on the calcium isotopic chain, we find no evidence for the dramatic neutron swelling reported earlier. While focusing here on the Ca chain, the methodology proposed in this work is applicable to interaction cross section measurements across the nuclear chart and is well-suited for new experiments currently planned at leading rare isotope facilities.

Sodium-ion batteries employing hard carbon electrodes are considered a drop-in technology for lithium-ion batteries. Electrode drying is a critical manufacturing step, as binder migration during pore emptying impacts the mechanical integrity and electrical performance of the electrode. Existing modeling approaches predominantly rely on the film shrinkage phase in a one dimensional way or neglect the capillary transport, resulting in a lack of physically consistent microstructure resolved predictions of binder migration. In this work, a spatially resolved pore scale continuum model is extended to explicitly describe capillary driven binder transport during pore emptying. The model is applied to hard carbon microstructures with varying mean particle diameters. The simulations reveal that smaller particle sizes lead to a more homogeneous binder distribution, whereas higher evaporation rates and increased surface tension promote stronger binder gradients. Variations in solvent viscosity show only a minor influence on binder migration, as long as no hydrophilic or hydrophobic behavior is present. Finally, the simulations demonstrate that an explicit description of capillary transport and microstructural effects is essential for accurately predicting binder migration and provides a basis for the targeted optimization of electrode drying processes.

The Fibonacci infinite word ${\bf f} = (f_i)_{i \geq 0} = 01001010\cdots$ is one of the most celebrated objects in combinatorics on words. There is a simple $5$-state automaton that, given $i$ in lsd-first Zeckendorf representation, computes its $i$'th term $f_i$, and a $2$-state automaton for msd-first. In this paper we consider the state complexity of the automaton generating the shifted sequence $(f_{i+c})_{i \geq 0}$, and show that it is $O(\log c)$ for both msd-first and lsd-first input. This is close to the information-theoretic minimum for an aperiodic sequence. The techniques involve a mixture of state complexity techniques and Diophantine approximation.

In this paper we introduce momentum-resolved metal Boltzmann-Bloch equations (MBBE) for the combined description of electronic intra- and interband processes in noble metals. This microscopic framework incorporates a full treatment of many-body electron-electron and electron-phonon interactions, relevant for relaxation and dephasing processes after optical excitation. For the example of gold, we calculate the linear optical response for near-infrared and visible energies. This provides insight into the interplay of microscopic processes hidden in phenomenological Drude-Lorentz models. The complex geometry of the Fermi surface is treated by an anisotropic electronic dispersion model, which is necessary to explain the temperature dependent spectrum over the whole frequency range of intra- and interband transitions.

Simultaneous wireless information and power transfer (SWIPT) critically depends on waveform design, which governs both reliable data delivery and efficient energy harvesting. Among waveform characteristics, the peak-to-average power ratio (PAPR) plays a pivotal role: low-PAPR signals improve power amplifier (PA) efficiency, while high-PAPR signals exploit rectifier nonlinearities to boost harvested energy. This duality makes PAPR a fundamental design challenge in SWIPT systems. To tackle this issue, we establish a unified analytical framework that characterizes the PAPR-dependent behaviors of both the PA and the rectifier, thereby revealing how waveform statistics determine end-to-end energy transfer efficiency. Building on this insight, we propose a frequency-domain resource allocation strategy for power-splitting SWIPT, where spectral segments are adaptively assigned to balance communication throughput with energy harvesting performance. Here, a key contribution is to extend SWIPT to MIMO-OFDM architectures. Despite concerns over excessive PAPR in large-scale antenna-subcarrier configurations, we demonstrate that appropriate waveform adaptation and resource optimization can transform MIMO-OFDM into an energy-efficient platform for joint data and power transfer. Finally, simulation results confirm significant improvements in PA efficiency, rectifier output, and overall energy transfer, thereby validating the practical benefits of the proposed approach.

The dynamic flow of cerebrospinal fluid (CSF) in brain ventricles exhibits flow features on several scales, both spatially and temporally. Most analysis of this complex flow and the accompanying transport has used instantaneous (Eulerian) flow variables. Such analysis makes understanding of unsteady transport challenging. Here, we analyze brain ventricular CSF flow both in a Eulerian sense and from the Lagrangian perspective -- a time-integrated view of the flow. With geometries generated from imaging data, we model CSF flow in adult human and embryonic zebrafish brain ventricles. In the human brain we model flow governed by cardiovascular pulsations, CSF secretion and motile cilia. The flow driven by cardiovascular pulsations is derived from a damped linear elastic model of brain ventricle deformations, as a result of applying displacement boundary conditions derived from experimental data. In the zebrafish brain we consider flow driven solely by motile cilia. The tissue and flow models are implemented and solved with finite element methods. We use the resulting velocity fields to compute finite-time Lyapunov exponent (FTLE) fields and use these fields to characterize Lagrangian coherent structures, which can be approximated by ridges in the FTLE fields. These coherent structures demonstrate prominent flow features in the brain ventricles congruent with findings in experimental research. In the human brain ventricles, we also investigate the role of inertia by comparing flow models governed by the Navier-Stokes and the Stokes equations. Comparisons show that solving the Stokes equations is adequate to compute integrated flow variables like stroke volumes, but that the Stokes approximation fails to resolve intricate features of flow and advective transport that are present in the solution to the Navier-Stokes equations, features that could be important to elucidating transport.

We present a minimal-component bidirectional digital interconnect that enables simultaneous transmission and reception of baseband logic signals over a single coaxial cable. The circuit consists of a passive resistive hybrid providing matched line termination and directional separation, a single CMOS logic gate as driver, and a commercial LVDS receiver used as a differential comparator. No active echo cancellation, calibration, or transformer coupling is required. An analytical treatment of the hybrid network is used to determine the system parameter that maximizes the received signal amplitude. SPICE simulations predict deterministic timing errors caused by incomplete separation of transmitted and received signals. Experimental measurements confirm the predicted deterministic jitter and show good agreement with the simulation results. For typical laboratory coaxial cables up to 6 m, the measured peak-to-peak edge timing error remains below 1 ns. A bidirectional transmission experiment with randomized data at 250 MBaud demonstrates a clearly open eye diagram and confirms reliable full-duplex operation. Due to its simplicity and compatibility with existing coaxial infrastructure, the proposed approach may be useful in laboratory and detector environments where cable routing or feedthrough density is constrained.

We prove a Sidorenko-type inequality for directed trees: for every oriented tree $T$ on $k$ vertices and every finite directed graph $G$, the homomorphism count hom$(T,G)$ is bounded above by the maximum of the two pure star counts hom$(S_{0,k-1},G)$ and hom$(S_{k-1,0},G)$. In other words, among all directed trees on $k$ vertices, the pure in- and out-stars maximise the homomorphism count into host digraphs. The proof is purely combinatorial, based on an iterative leaf-reallocation scheme combined with Hölder's inequality. We further investigate the corresponding homomorphism order on directed trees, discuss refinements via tail-truncation and pointwise bounds for rooted host graphs, and record several consequences, e.g. for random directed graph models and local weak limits, where the inequality reduces tree statistics to controlled pure in- and out-degree moments.

Although Hamiltonian Monte Carlo (HMC) scales as O(d^(1/4)) in dimension, there is a large constant factor determined by the curvature of the target density. This constant factor can be reduced in most cases through preconditioning, the state of the art for which uses diagonal or dense penalized maximum likelihood estimation of (co)variance based on a sample of warmup draws. These estimates converge slowly in the diagonal case and scale poorly when expanded to the dense case. We propose a more effective estimator based on minimizing the sample Fisher divergence from a linearly transformed density to a standard normal distribution. We present this estimator in three forms, (a) diagonal, (b) dense, and (c) low-rank plus diagonal. Using a collection of 114 models from posteriordb, we demonstrate that the diagonal minimizer of Fisher divergence outperforms the industry-standard variance-based diagonal estimators used by Stan and PyMC by a median factor of 1.3. The low-rank plus diagonal minimizer of the Fisher divergence outperforms Stan and PyMC's diagonal estimators by a median factor of 4.

Drilling investment is pivotal to operational planning in oil and gas (O\&G) exploration. Conventional deployment relies heavily on fragmented expert assessments of geological and economic factors, with limited integration ability of information. As the tool of portfolio show strong potential for mitigating uncertainty and selecting superior drilling plans, this study develops a multi-objective mean-variance portfolio model that accounts for geological-parameter uncertainty, enabling an effective risk-return trade-off and optimal selection. First, the probabilistic distribution of geological-parameters for prospect-list projects is obtained through expert-elicited priors. And considering the selection of the drilling projects as a portfolio, an optimization model is formulated jointly to describe the return and risk of short-term plan, under different constraints. Second, an improved OE-NSGA-II algorithm is proposed specifically for this model, in which (1) a directional crossover operator is designed to embed improving directions in objective space-derived from dominance and objective differences-into recombination, and (2) a structure-aware mutation operator is designed to prioritize high-utility bit flips via probabilistic sampling with feasibility repair, thus improving the search ability for superior Pareto solutions. Finally, using the case of 2023 exploration drilling deployment for verification, and then apply the validated method to the 2024 deployment to support decision-making. The results indicate that the proposed approach offers a reusable solution for drilling portfolio optimization in O\&G exploration.

The focus of this paper is the study of the moduli space of representations of fundamental groupoids of surfaces $Σ$ with boundaries with values in $G:=GL_n(\mathbb C)$. In absence of marked points on the boundary, this moduli space is realized in many equivalent ways: as the moduli space of linear local systems on $Σ$, as the moduli space of representations of the fundamental groupoid $Π_1 (Σ)$, as the space of monodromy data and as character variety. By adding marked points to the boundary of $Σ$ in order to capture irregular singularities, the Betti moduli space has been generalized in several ways by many authors. Although it is clear that these different approaches describe essentially the same spaces of mathematical objects, exactly how they fit together has not yet been established. Motivated by the broader programme of establishing an explicit and conceptually coherent relationship between the existing approaches to the study of the decorated Betti moduli space, in this paper, we develop a categorical framework that allows for a systematic definition of the \dfn{decorated Betti moduli spaces} space, in the presence of higher order poles, designed to specialize to the different points of view encountered in the literature.

Energy consumption is a significant concern for mobile network operators, and to enable further network energy improvements it is also an important target when developing the emerging 6G standard. In this paper we show that, despite the existence of many energy-saving features in 5G new radio (NR) networks, activating them in isolation yields only suboptimal savings and often compromises other network key performance indicators (KPIs) such as coverage or latency. We first introduce a compact taxonomy that distinguishes hardware capabilities from higher-layer features. Features fall into two classes: (i) signaling and scheduling mechanisms that create idle windows, and (ii) features that utilize those windows to save energy. We then present a feature orchestrator as a logical node to coordinate between features to maximize the gain. Using a 3GPP-aligned simulator with product-realistic parameters, we show that coordinating lean NR, scheduling, and advanced sleep modes significantly reduces gNodeB (gNB) energy consumption with negligible throughput loss, compared to the uncoordinated scenario. We conclude by outlining open issues in observability, system dynamics, coordination, and intelligent automation for energy performance management.

Reconfigurable intelligent surface (RIS)-aided rate-splitting multiple access (RSMA) systems have demonstrated remarkable potential in enhancing spectral efficiency. However, most existing works rely on ideal hardware, which is unrealistic.In practical deployments, RIS elements suffer from amplitude-phase coupling, where transceivers are subject to hardware impairments (HWI), and successive interference cancellation (SIC) in RSMA networks cannot achieve perfect interference elimination for decoded signals.To address these limitations, we investigate a robust beamforming design for RIS-aided RSMA systems under practical hardware imperfections. We first characterize the asymptotic signal-to-noise ratio (SNR) of practical RIS systems when the beamformer is designed based on ideal RIS model, thereby theoretically quantifying the resulting performance degradation. We then derive a closed-form expression for the distortion noise power induced by transceiver HWI, while also accounting for residual interference due to imperfect SIC. Building on these insights, we establish a comprehensive system model that jointly incorporates all hardware-induced impairments and formulate a multiuser sum rate maximization problem. To solve the resulting non-convex optimization problem, we develop an efficient block variable relaxation algorithm. Simulation results verify that the proposed scheme significantly outperforms conventional non-orthogonal multiple access (NOMA) approaches, and achieves superior robustness compared with benchmark schemes neglecting HWI, imperfect SIC, or amplitude-phase coupling.

Sparse functional data arise when measurements are observed infrequently and at irregular time points for each subject, often in the presence of measurement error. These characteristics introduce additional challenges for functional principal component analysis. In this paper, we propose a new approach for extracting functional principal components from such data by combining basis expansion with maximum likelihood estimation. Orthogonality of the estimated eigenfunctions is preserved throughout the optimization using modified Gram-Schmidt orthonormalization. An information criterion is proposed to select both the optimal number of basis functions and the rank of the covariance structure. Principal component scores are subsequently estimated via conditional expectation, enabling accurate reconstruction of the underlying functional trajectories across the full domain despite sparse observations. Simulation studies demonstrate the effectiveness of the proposed method and show that it performs favorably compared with existing approaches. Its practical utility is illustrated through applications to CD4 cell count data from the Multicenter AIDS Cohort Study and somatic cell count data from Irish research dairy cattle. Supplementary materials, including technical details, additional simulation results, and the R package mGSFPCA, are available online.

Understanding the impact of vegetation and small-scale antenna movements on signal propagation is important for the design and optimization of high-frequency wireless communication systems. This paper presents an experimental study analyzing signal propagation at 60 GHz and 80 GHz in the presence of vegetation, with a focus on forward scattering and microdiversity effects. A controlled measurement campaign was conducted in an indoor environment, where the influence of a potted plant placed in the line-of-sight (LOS) path between the transmitter and receiver was investigated. The study examines the effects of antenna micro-shifts on the channel impulse response (CIR), highlighting variations in received power due to small positional changes of the antennas. The results indicate that the 80 GHz band exhibits higher sensitivity to micro-movements compared to the 60 GHz band, leading to greater fluctuations in received power.

Controlling nonequilibrium responses in optically driven quantum materials is essential for advancing applications in energy conversion, ultrafast electronics, and quantum computation. Nonlinear optical spectroscopy serves as a powerful tool to investigate ultrafast electron and phonon dynamics in these systems; however, conventional nonlinear approaches often require intense laser pulses (> 10 GW/cm2) and typically encounter a strong background. Here, we introduce a phase-sensitive nonlinear spectroscopic technique that operates at low laser powers (~ 10 kW/cm2, pulse energies ~ 10 pJ) and enables real-time monitoring and active control of coherent phonons in a few-layer (three to five) thick 2H-MoTe2. Upon excitation with ultrashort (~ 10 fs) pump pulses, we achieve displacive excitation of coherent phonons, which periodically modulate the Kerr nonlinearity of the material, leading to cross-phase modulation (XPM) of a delayed probe pulse. This phase modulation induces spectral broadening and oscillations in the center of mass (COM) of the probe spectrum in time, enabling the detection of subtle nonlinear optical responses in a background-free manner. The nonlinear response can be selectively amplified or attenuated by adjusting the strength of the pump pulse, which controls the distribution of photoexcited carriers in the electronic bands. By combining two-color nondegenerate pump-probe measurements and time-dependent density-functional theory (TDDFT) calculations, we directly resolve the coupled nonequilibrium electronic and phonon dynamics. A dual-pump pulse scheme enables precise control of phonon oscillations, allowing selective activation or suppression of specific phonon modes and correspondingly the modulation of the Kerr nonlinearity.

Ergotropy, the maximum work extractable from a quantum system, is a central resource in quantum physics. Computing ergotropy is well established when the system state is fully known, but its estimation under partial information remains an open problem. Here we introduce a general certification framework that lower bounds ergotropy using only the expectation values of a limited set of arbitrary observables. The method naturally applies in the finite-statistics regime, yielding confidence-certified bounds that explicitly incorporate shot noise. We benchmark our approach on both synthetic data and experimental measurements from an IBM quantum processor. This establishes a robust and experimentally accessible tool for certifying extractable work in realistic quantum settings.

We review two types of fractional Kaehler potentials $K$ which reduce, along the inflationary path, to the form $N/(1-ϕ^{q_{\rm M}})^{p}$ with $q_{\rm M}=1$ or $2$ and $0.1\leq p\leq10$. Their coexistence, within a non-linear sigma model, with chaotic inflationary potentials of the form $ϕ^n$ (where $n=2$ or $4$) determines, independently from $q_{\rm M}$ and $n$, a class of $(p,N)$-inflationary attractors which leads to observables compatible with the ACT DR6. An implementation of these models in the context of supergravity can be also achieved by introducing two chiral superfields and a monomial superpotential, linear with respect to the inflaton-accompanying field, and supplementing the $K$'s above with a shift symmetry. Although inflation is attained for subplanckian inflaton values, the tensor-to-scalar ratio obtained for certain $N$ values can be possibly observable in the near future.

We investigate the encapsulation of water by a thin elastic film as a minimal model of elastocapillary self-folding with fluid transport. An equilateral triangular polydimethylsiloxane film is lifted quasi-statically from a water surface, while its side length and thickness are systematically varied. Depending on these parameters, the film exhibits three distinct morphologies: folding, recoiling, and liquid encapsulation. We show that the observed morphology is selected by the competition between surface energy, gravitational energy of the liquid, and bending energy of the film. In particular, encapsulation occurs in a narrow parameter region corresponding to the intersection of the elastocapillary, elastogravity, and capillary length scales. This result provides a simple physical criterion for liquid encapsulation by elastic films, based on the balance of bending, capillary, and gravitational energies.

We investigate the thermodynamic properties of the Kerr-Bertotti-Robinson black hole, an exact Petrov type D solution of Einstein-Maxwell theory describing a rotating black hole immersed in an external electromagnetic field. While the conserved angular momentum and electric charge can be computed straightforwardly, the conserved mass cannot be obtained through standard integrability methods due to the nontrivial asymptotically uniform external electromagnetic field. To overcome this difficulty, we adopt the Christodoulou-Ruffini mass relation as a thermodynamic definition of the conserved mass, and identify the associated generator, thereby fixing the ambiguity in defining this conserved mass and constructing the thermodynamic potentials. These thermodynamic quantities naturally satisfy the first law of black-hole thermodynamics as well as the Smarr formula.

Consider a piecewise affine Lipschitz map $ϕ: Ω\to \mathbb R$, where $Ω\subset \mathbb R^d$ is an open set, and assume that $x \mapsto x + t \nabla ϕ(x)$ is injective for almost every $t > 0$. In (J.-G. Liu, R.~L. Pego, \emph{Rigidly breaking potential flows and a countable Alexandrov theorem for polytopes}, Pure Appl. Anal., \textbf{7}(4), 2025) the authors conjecture that every such $ϕ$ must be locally convex. We prove the result assuming additionally $\nabla ϕ\in BV_{loc}(Ω)$, for a more general class of measure preserving maps.

Understanding the role of inertia in nanoscale heat transport is fundamental to the design of efficient nano-thermodynamics systems. In this work, we experimentally address the non-equilibrium dynamics of a Brownian gyrator, a paradigmatic model for nano-heat machines, that converts heat flow between two thermal baths into steady-state rotation. Using an optically levitated nanoparticle in a controlled vacuum environment, we study the transition from overdamped to underdamped dynamics of the gyrator. We demonstrate that, while the spatial signature of the non-equilibrium steady state vanishes as damping decreases, the rotational dynamics and energetics are optimized at a critical damping. Our findings reveal the importance of inertia for maximising the performance of nanoscale machines and provide fundamental insights into the design of efficient nano heat engines and processes.

These notes are based on lectures delivered by G. Schehr at the XVIth School on Fundamental Problems in Statistical Physics (FPSP), held in Oropa (Italy) from 30 June to 11 July 2025. After a brief introduction to extreme value statistics (EVS) for independent and identically distributed (IID) random variables, we discuss several paradigmatic examples of strongly correlated systems where classical extreme value theory no longer applies. In particular, we focus on time series generated by random walks and Brownian motion, as well as on eigenvalue statistics in random matrix theory. Emphasis is placed on applications of EVS to fundamental problems in statistical physics and disordered systems, including the Random Energy Model, stochastic search problems, as well as fluctuating interfaces, and directed polymers in random media within the Kardar-Parisi-Zhang universality class.

We study the performance of the Particle Transformer (ParT) for jet flavor tagging using ILD full simulation events (1M jets) as well as fast simulation samples (10M and 1M jets). We perform 3-category ($b/c/d$), 6-category ($b/c/d/u/s/g$), and 11-category trainings (including quark--antiquark separation), incorporating multivariate hadron particle identification information from $dE/dx$ and time-of-flight. For $b$/$c$ tagging, we observe a factor of 5--10 improvement over previous BDT-based taggers, and we obtain reasonable performance for strange tagging and quark/antiquark separation. The 10M-jet fast simulation study indicates that further gains are possible with higher training statistics.

We give an overview of the 2026 Computational Geometry Challenge targeting the problem of finding a Central Triangulation under Parallel Flip Operations in triangulations of point sets. A flip is the parallel exchange of a set of edges in a triangulation with opposing diagonals of the convex quadrilaterals containing them. The challenge objective was, given a set of triangulations of a fixed point set, to determine a central triangulation with respect to parallel flip distances. More precisely, this asks for a triangulation that minimizes the sum of flip distances to all elements of the input

A stochastic modeling methodology for 3-D foliage is presented, aimed at enhancing ray-tracing simulations. The model supports adjustable stochastic geometry, density, and shape to capture variability in foliage structures. The model is validated through experimental measurements of representative vegetation. The influence of foliage density and size on path loss and root mean square delay spread is analyzed to demonstrate the applicability of the model in the 80 GHz frequency band.

It is well known that every complex contact $3$-manifold, when regarded as a real manifold, gives rise to a fat $(4,6)$-distribution that admits two Reeb directions. Nonetheless, it was an open question whether the converse was true. This was not known even at the level of germs. The present work completely answers this question in the negative. We construct the first example of a fat distribution with two Reeb directions that does not support a complex contact structure anywhere, not even locally nor up to diffeomorphism. This result answers an open question by A. Bhowmick. In the second part of this work we prove a stronger result. By applying suitable $C^\infty$-perturbations to our construction, we show that the space of complex-contact germs has infinite codimension within the space of fat $(4,6)$-distribution germs with Reeb directions.

Quantum metrology enables parameter estimation beyond classical limits by exploiting nonclassical resources such as squeezing and entanglement. In distributed quantum sensing, Heisenberg scaling has been extended from $1/N^2$ to $1/(NM)^2$ through entanglement across both particles and spatial modes, where $N$ denotes the photon number and $M$ the number of spatially distributed modes. However, the overall sensitivity has remained limited to linear scaling with the number of measurement repetitions $R$. Here, we show that exploiting entanglement across temporal modes via time-domain multiplexing enables a scaling advantage with respect to $R$. As a result, the sensitivity can asymptotically approach simultaneous Heisenberg scaling in photons, spatial modes, and repetitions, yielding an overall sensitivity approaching $Δ^2 ϕ\propto 1/(NMR)^2$. Using the Bogoliubov transformation formalism, we prove the optimality of the protocol within the class of Gaussian states and show that the scaling is realizable via homodyne detection and maximum-likelihood estimation. We further show that the advantage persists under optical loss and propose an experimentally feasible loop-based photonic sensing scheme. Our results open a route to incorporating time-multiplexing techniques into quantum metrology.

Móricz and Nagy introduced the problem of maximizing the number of $r$-element subsets with rational sums in an $n$-element set of irrational numbers, and showed that it is equivalent to an extremal zero-sum problem. They determined the exact maximum in several cases. For the remaining range, they presented an explicit construction of an $n$-element set of irrational numbers containing exactly $m\binom{n-m}{r-1}$ such subsets, where $m=\lfloor n/r\rfloor$. They conjectured that this construction is always optimal for any $1<r<n$. In this paper, we confirm that conjecture. Our proof combines an order-theoretic antichain argument for zero-sum subsets with a sharp maximization of the resulting binomial expressions. As a consequence, we determine exactly the maximum number of $r$-term zero-sum subsequences in sequences of $n$ nonzero integers.

This article explores a new type of optimal covering of a complete graph by small cliques of different sizes, namely the minimum covering with minimum excess. In particular, the minimum size of a covering by triples and quadruples with minimum excess is determined. Moreover, some generalisation onto minimum coverings by triples, quadruples and quintuples with minimum excess is presented.