Exact Results Research Articles

Effects of group size and population size on the evolutionary stability of cooperation

Evolutionary game theory has classically been developed under the implicit assumption of an infinite population. Exact analytical results for finite populations are rare, and those that exist apply to situations in which strategy sets are discrete. We rigorously analyse a standard model for the evolution of cooperation (the multi-player continuous-strategy snowdrift game) and show that in many situations in which there is a cooperative evolutionarily stable strategy (ESS) if the population is infinite , there is no cooperative ESS if the population is finite (no matter how large). In these cases, contributing nothing is a globally convergently stable finite-population ESS, implying that apparent evolution of cooperation in such games is an artefact of the infinite population approximation. The key issue is that if the size of groups that play the game exceeds a critical proportion of the population then the infinite-population approximation predicts the wrong evolutionary outcome (in addition, the critical proportion itself depends on the population size). Our results are robust to the underlying selection process.

Vector bosons in the rotating frame of negative curvature wormholes

In this study, we investigate the relativistic dynamics of vector bosons within the context of rotating frames of negative curvature wormholes. We seek exact solutions for the fully-covariant vector boson equation, derived as an excited state of zitterbewegung. This equation encompasses a symmetric rank-two spinor, enabling the derivation of a non-perturbative second-order wave equation for the system under consideration. Our findings present exact results in two distinct scenarios. Notably, we demonstrate the adaptability of our results to massless vector bosons without compromising generality. The evolution of this system is shown to correlate with the angular frequency of the uniformly rotating reference frame and the curvature radius of the wormholes. Moreover, our results highlight that the interplay between the spin of the vector boson and the angular frequency of the rotating frame can give rise to real oscillation modes, particularly evident in excited states for massless vector bosons. Intriguingly, we note that the energy spectra obtained remain the same whether the wormhole is of hyperbolic or elliptic nature.

Coupled-cluster theory for trapped bosonic mixtures.

We develop a coupled-cluster theory for bosonic mixtures of binary species in external traps, providing a promising theoretical approach to demonstrate highly accurately the many-body physics of mixtures of Bose-Einstein condensates. The coupled-cluster wavefunction for the binary species is obtained when an exponential cluster operator eT, where T = T(1) + T(2) + T(12) and T(1) accounts for excitations in species-1, T(2) for excitations in species-2, and T(12) for combined excitations in both species, acts on the ground state configuration prepared by accumulating all bosons in a single orbital for each species. We have explicitly derived the working equations for bosonic mixtures by truncating the cluster operator up to the single and double excitations and using arbitrary sets of orthonormal orbitals for each of the species. Furthermore, the comparatively simplified version of the working equations are formulated using the Fock-like operators. Finally, using an exactly solvable many-body model for bosonic mixtures that exists in the literature allows us to implement and test the performance and accuracy of the coupled-cluster theory for situations with balanced as well as imbalanced boson numbers and for weak to moderately strong intra- and interspecies interaction strengths. The comparison between our computed results using coupled-cluster theory with the respective analytical exact results displays remarkable agreement exhibiting excellent success of the coupled-cluster theory for bosonic mixtures. All in all, the correlation exhaustive coupled-cluster theory shows encouraging results and could be a promising approach in paving the way for high-accuracy modeling of various bosonic mixture systems.

Boundary correlators and the Schwarzian mode

The effective low temperature dynamics of near-extremal black holes is governed by the quantum fluctuations of the Schwarzian mode of JT gravity. Utilizing as a proxy a planar charged black hole in asymptotically Anti-de-Sitter spacetime, we investigate the effects of these fluctuations on a probe scalar field. The corresponding holographic real-time boundary correlators are computed following a holographic renormalization procedure, using the dubbed gravitational Schwinger-Keldysh geometry (grSK) and known exact results of boundary correlators from the near-horizon region. This analysis gives rise to a retarded Green’s function that decays as a power law for late Lorentzian times. Its analytic structure indicates the presence of a branch cut in the complex frequency domain at finite temperature. These features are a non-perturbative hallmark that prevails as long as the planar transverse space is kept compact.

Food miles on the shelves: the share of local food products in the Hungarian retail sector

Share of national food products in retail is a frequent subject of policy debates. Local food is often associated with national security, sustainability, and support of local economy, contributing to value-added production, employment, rural development, and preservation of local food culture. Despite its importance, relevant academic literature about the proportion of national food in retail is basically non-existent. This paper presents a unique study that fills this gap and gives an account of the proportion of the main national food products in the Hungarian retail sector. The study presents a comprehensive picture of the food supply situation of the 10 largest retail chains in Hungary in the second half of 2020 for 16 key product categories (representing 67% of total food sales in value in the country), based on the experience of research that covered the physical examination of nearly 40 thousand individual food products. The study found that 70.85% of the analysed food products were supplied by domestic companies. Proportion of national products was higher for fresh food (meat, milk, perishable meat and milk products, eggs, fruits, and vegetables) and honey, while imports were dominant in higher value-added categories. Domestic chains had 15.37% points more local products in stock than international ones. The exact results can be used for benchmarking between companies and sectors, indicating comparative advantages and disadvantages, and provide solid basis for economic development plans.

Estimation of Energy Separation in Tunelling States in Symmetric-Hyperbolicus Double-Well Potential Problem using Filter Method

Double well potential (DWP) plays an important role in quantum physics but its analytical exact solutions are incomplete, except for some limited quasi-exact solvable (QES) and therefore numerical solutions are still required. This paper is aimed to obtain numerical solution of the recently proposed symnetric-hyperbolicus DWP and by using our newly developed filter method. The filter method is able to produce eigen-energies ​​and eigenfunctions those are in accordance with the exact analytical results. Next, we obtain the dependence of the ground state energy and the energy separation between the 2 lowest states on the DWP parameter k. For large k, the 2 lowest energy levels are below the potential barrier so the eigenfunctions penetrate the barrier, so the 2 lowest energy levels can be considered as the result of a single energy split. In this case, we observe that the energy separation is an exponential function k, as opposed to a linear function for smaller k in the non tunelling region. The exponential trend of the numerical results is in good agreement with the theoretical approach and allows one to estimate the 2 lowest energies for a given k. HIGHLIGHTS For small , the two lowest energy levels lie above the barrier potential, where both ground state and the energy separation is a linier function of For large , the two lowest energy levels lie below the barrier potential so that the eigenfunction penetrates the barrier. As a result, the two lowest energy levels can be thought of as splitting energy states. In this case, we observe that the energy separation is an exponential function . Since also depends on , it is possible to estimate the two lowest energy levels The exponential trend of the numerical results for large agrees well with the theoretical approach, shown by the linear relationship between and the classical actions in the classical forbidden regions, GRAPHICAL ABSTRACT

Fate of false vacuum in non-perturbative regimes

We use some exact results in scalar field theory to revise the analysis by Coleman and Callan about false vacuum decay and propose a simple non-perturbative formalism. We introduce an exact Green’s function which incorporates non-perturbative corrections in the strong coupling regimes of the theory. The solution of the scalar field theory involves the Jacobi elliptical function and has been used to calculate the effective potential for any arbitrary coupling values. We demonstrate the use of this formalism in a simple λ ϕ 4 theory, and show that the effective potential exhibits a false minimum at the origin. We then calculate the false vacuum decay rate in the thin wall approximation, and suggest simple analytic formulae that may be useful for the analysis for the first-order phase transition beyond the perturbative regime. In our methodology, we show that the standard results obtained in perturbation theory are reproduced by making the coupling values very small.

Nonequilibrium Response for Markov Jump Processes: Exact Results and Tight Bounds.

Generalizing response theory of open systems far from equilibrium is a central quest of nonequilibrium statistical physics. Using stochastic thermodynamics, we develop an algebraic method to study the static response of nonequilibrium steady state to arbitrary perturbations. This allows us to derive explicit expressions for the response of edge currents as well as traffic to perturbations in kinetic barriers and driving forces. We also show that these responses satisfy very simple bounds. For the response to energy perturbations, we straightforwardly recover results obtained using nontrivial graph-theoretical methods.

A possible coexistence of spontaneous magnetic order and thermal entanglement in a spin-1/2 Ising–Heisenberg diamond-decorated square lattice

The spin-1/2 Ising–Heisenberg model on a diamond-decorated square lattice is exactly solved by the decoration-iteration transformation, which establishes an exact mapping correspondence with the spin-1/2 Ising model on a square lattice with an effective temperature-dependent interaction between the nearest-neighbor spins. Exact results for the pair correlation function and spontaneous magnetization of the Heisenberg spins gained from the exact mapping equivalence allow a straightforward implementation of the concurrence, which may serve as a measure of the bipartite entanglement within decorating Heisenberg spin pairs. It is evidenced that the concurrence gradually diminishes upon increase of temperature above a disordered monomer-dimer ground state until it completely vanishes at a threshold temperature, while the concurrence is either always zero above a spontaneously ordered ferrimagnetic ground state or it shows a peculiar thermally assisted reentrance accompanied with a mutual coexistence of the spontaneous ferrimagnetic long-range order and the thermal entanglement.

An extension to Adhikari iterative method: A novel approach for obtaining complex eigensolutions in linear non-classically damped systems

This paper introduces an extended version of the Adhikari iterative method, adjusted for decoupling non-classically linear damped systems and refining the determination of complex eigenvalues. By integrating a self-adjoint theorem and leveraging spectral localization, the extension employs the complex eigenvectors from the Adhikari method to enhance the stability and accuracy of complex eigenvalue determination. This novel approach demonstrates a marked improvement in precision and stability. The study comprehensively assesses various dynamic characteristics, including damped and undamped eigenvalues, damping ratios, damping loss factors, and Frequency Response Function (FRF). Additionally, a novel methodology for computing FRF responses is proposed, involving the identification of eigensolutions within decoupled dynamical systems. The extension's effectiveness is assessed using the exact results from the state-space method as a benchmark, applying to three distinct non-classically damped models from previous research. The results are compared with those obtained using the original Adhikari method. The findings significantly reduce errors, affirming the extension's proficiency in approximating complex eigensolutions. These outcomes mark a notable advancement in the field, facilitating further research for developing more robust techniques for analysing complex dynamical systems.

Dynamics of packed swarms: Time-displaced correlators of two-dimensional incompressible flocks.

We analytically calculate the scaling exponents of a two-dimensional KPZ-like system: coherently moving incompressible polar active fluids. Using three different renormalization group approximation schemes, we obtain values for the roughness exponent χ and anisotropy exponent ζ that are extremely near the known exact results. This implies our prediction for the previously unknown dynamic exponent z is likely to be quantitatively accurate.

Time-dependent coupled cluster with orthogonal adaptive basis functions: General formalism and application to the vibrational problem.

We derive equations of motion for bivariational wave functions with orthogonal adaptive basis sets and specialize the formalism to the coupled cluster Ansatz. The equations are related to the biorthogonal case in a transparent way, and similarities and differences are analyzed. We show that the amplitude equations are identical in the orthogonal and biorthogonal formalisms, while the linear equations that determine the basis set time evolution differ by symmetrization. Applying the orthogonal framework to the nuclear dynamics problem, we introduce and implement the orthogonal time-dependent modal vibrational coupled cluster (oTDMVCC) method and benchmark it against exact reference results for four triatomic molecules as well as a reduced-dimensional (5D) trans-bithiophene model. We confirm numerically that the biorthogonal TDMVCC hierarchy converges to the exact solution, while oTDMVCC does not. The differences between TDMVCC and oTDMVCC are found to be small for three of the five cases, but we also identify one case where the formal deficiency of the oTDMVCC approach results in clear and visible errors relative to the exact result. For the remaining example, oTDMVCC exhibits rather modest but visible errors.

Capturing the electron-electron cusp with the coupling-constant averaged exchange-correlation hole: A case study for Hooke's atoms.

In density-functional theory, the exchange-correlation (XC) energy can be defined exactly through the coupling-constant (λ) averaged XC hole n̄xc(r,r'), representing the probability depletion of finding an electron at r' due to an electron at r. Accurate knowledge of n̄xc(r,r') has been crucial for developing XC energy density-functional approximations and understanding their performance for molecules and materials. However, there are very few systems for which accurate XC holes have been calculated since this requires evaluating the one- and two-particle reduced density matrices for a reference wave function over a range of λ while the electron density remains fixed at the physical (λ = 1) density. Although the coupled-cluster singles and doubles (CCSD) method can yield exact results for a two-electron system in the complete basis set limit, it cannot capture the electron-electron cusp using finite basis sets. Focusing on Hooke's atom as a two-electron model system for which certain analytic solutions are known, we examine the effect of this cusp error on the XC hole calculated using CCSD. The Lieb functional is calculated at a range of coupling constants to determine the λ-integrated XC hole. Our results indicate that, for Hooke's atoms, the error introduced by the description of the electron-electron cusp using Gaussian basis sets at the CCSD level is negligible compared to the basis set incompleteness error. The system-, angle-, and coupling-constant-averaged XC holes are also calculated and provide a benchmark against which the Perdew-Burke-Ernzerhof and local density approximation XC hole models are assessed.

Energy-performance aware virtual machines migration in cloud network by using prediction and fuzzy approaches

Energy wastage and service level access are challenges of cloud networks, while efficient migration of virtual machines is a solution for them. Of course, uncontrolled migration increases overhead in cloud comprising: violation in service and high energy consumption. Performance requirements is very important in cloud, generally, it in the form of service level agreements are formalized. The main goal of migration is energy and performance trade-off. Detecting overloaded and under-loaded physical machine as sub-problem of migration, which effects the performance and energy. In this paper, with definition a system modelling, in first for detect overloaded and under-loaded physical machines a predict detection method is proposed. The forecast of the future utilization of the processor by using predict physical machines utilization method is provided. Predict physical machine's utilization module in predict detection method has been called. Unlike the previous physical machine detection methods, the current load of physical machines along with the future load of them is considered, and will lead to more exact results. Then for selection of virtual machines on overloaded physical machines a fuzzy selection method is proposed. Unlike the previous virtual machine-selection methods, all affective variables are considered and will lead to better results. In overall, the predict detection-fuzzy selection algorithm as intelligent approach has proposed and simulated with PlanetLab workload, CloudSim toolkit and Matlab. Simulation results demonstrate that significantly outperforms benchmark algorithms in terms of the energy consumption, migration count, service level agreement violation, energy-service level agreement violation, and energy-service level agreement violation-migration.

The phase diagram of the baxter-Wu model in the magnetic field and temperature plane

Using the Fisher zeros of the partition function of a finite sized cluster with ‘spherical’ boundary conditions an approximate phase diagram in the magnetic field—temperature plane is obtained for the Baxter-Wu model. The phase diagram compares favourably with previous approximations as well as the known exact results when the magnetic field is zero. The accuracy of the approximated phase diagram, especially for small values of the magnetic field, is dependent on the fact, as shown previously, that for zero magnetic field the Fisher zeros for this cluster lie on loci giving the exact zero field critical temperature.

Quantum Many-Body Scars in Dual-Unitary Circuits.

Dual-unitary circuits are a class of quantum systems for which exact calculations of various quantities are possible, even for circuits that are nonintegrable. The array of known exact results paints a compelling picture of dual-unitary circuits as rapidly thermalizing systems. However, in this Letter, we present a method to construct dual-unitary circuits for which some simple initial states fail to thermalize, despite the circuits being "maximally chaotic," ergodic, and mixing. This is achieved by embedding quantum many-body scars in a circuit of arbitrary size and local Hilbert space dimension. We support our analytic results with numerical simulations showing the stark contrast in the rate of entanglement growth from an initial scar state compared to nonscar initial states. Our results are well suited to an experimental test, due to the compatibility of the circuit layout with the native structure of current digital quantum simulators.

Exact Results for the Distribution of Randomly Weighted Sums

Dependent random variables play a crucial role in various fields, from finance and statistics to engineering and environmental sciences. Often, interest lies in understanding the aggregate sum of a collection of dependent variables with random weights. In this paper, we provide a comprehensive study of the distribution of the aggregate sum with random weights. Expressions derived include those for the cumulative distribution function, probability density function, conditional expectation, moment generating function, characteristic function, cumulant generating function, moments, variance, skewness, kurtosis, cumulants, value at risk and the expected shortfall. Real data applications are discussed.

Exact solutions to the Erdős-Rothschild problem

Abstract Let $\boldsymbol {k} := (k_1,\ldots ,k_s)$ be a sequence of natural numbers. For a graph G, let $F(G;\boldsymbol {k})$ denote the number of colourings of the edges of G with colours $1,\dots ,s$ such that, for every $c \in \{1,\dots ,s\}$ , the edges of colour c contain no clique of order $k_c$ . Write $F(n;\boldsymbol {k})$ to denote the maximum of $F(G;\boldsymbol {k})$ over all graphs G on n vertices. There are currently very few known exact (or asymptotic) results for this problem, posed by Erdős and Rothschild in 1974. We prove some new exact results for $n \to \infty $ : (i) A sufficient condition on $\boldsymbol {k}$ which guarantees that every extremal graph is a complete multipartite graph, which systematically recovers all existing exact results. (ii) Addressing the original question of Erdős and Rothschild, in the case $\boldsymbol {k}=(3,\ldots ,3)$ of length $7$ , the unique extremal graph is the complete balanced $8$ -partite graph, with colourings coming from Hadamard matrices of order $8$ . (iii) In the case $\boldsymbol {k}=(k+1,k)$ , for which the sufficient condition in (i) does not hold, for $3 \leq k \leq 10$ , the unique extremal graph is complete k-partite with one part of size less than k and the other parts as equal in size as possible.

On the controllability results of semilinear delayed evolution systems involving fractional derivatives in Banach spaces

<abstract><p>The paper investigated the exact controllability of delayed fractional evolution systems of order $ \alpha\in (1, 2) $ in abstract spaces. At first, the exact controllability result is obtained when the nonlinear term $ f $ is locally Lipschitz continuous. Then, the certain compactness conditions and the measure of noncompactness conditions were applied to demonstrate the exact controllability of the concerned problem. The discussion was based on the fixed point theorems and the cosine family theory.</p></abstract>

Enhanced Brain Tumor Diagnosis Through Differential and Canonical Quadri –Partitioned Neutrosophic Set Classification Methods:A Comparative Study

An early cancer diagnosis is carried out for adequate management of diseases. Magnetic resonance imaging (MRI) is most commonly preferred method for cancer diagnosis. Due to the uncontrolled and rapid growth of cells, brain tumor is occurred. If not treated at a preliminary phase, it may lead to death. Thus, a noteworthy prerequisite for a successful treatment outcome is an early and precise diagnosis.Many conventional methods are discussed for performing efficient tumor detection. But, conventional classification methods not distinguish MRI as primary and metastases tumors in an accurate manner. Therefore, the performance comparison of deep learning-based classification (i.e., Differential Quadri-Partitioned Neutrosophic Interval-valued Polynomial Attention-based Deep CNN (DQNI-PADCNN) method and Canonical Quadri-Partitioned Neutrosophic Set based Otsuka–Ochiai Deep Recurrent Neural Network (CQNS-ODRNN) method) is introduced to provide exact image classification results. The brain MRI images are considered as an input. MRI image classification is carried out through CNN and RNN to find the brain tumor disease. Before the classification process, input images are de-noised. The noise-removed images are get segmented to identify the region of interested regions. Later, the images are classified into four classes such as glioma, meningioma, no tumor, and pituitary classes to detect the brain tumor. Both classification methods use Quadri-Partitioned Neutrosophic set for categorizing the images. Depending on CNNs and RNNs achievement in handling intricate tasks, an optimal multi-class brain tumor diagnosis is carried out. Experimental evaluation is implemented using MATLAB 2017 for brain tumor detection with the Brain Tumor MRI dataset. To the total number of MRI images, the various performance metrics are calculated in terms of sensitivity, specificity, accuracy, and time for the detection of brain tumors.