Difference equation

This paper is concerned with modification of the Adomian Decomposition Method for solving linear and non-linear Volterra and Volterra-Fredholm Integro-Differential equations. The Modified form of ADM was carried out by replacing the Adomian polynomials constructed in the conventional Adomian Decomposition Method with the constructed canonical polynomials. The modified Adomian Decomposition Method was applied to solve some existing example. The results obtained using the newly modified ADM proved superior when compared with the conventional ADM.

This article is concerned with the following rational difference equation y n+1 = (y n + y n-1 )/(p + y n y n-1 ) with the initial conditions; y -1 , y 0 are arbitrary positive real numbers, and p is positive constant. Locally asymptotical stability and global attractivity of the equilibrium point of the equation are investigated, and non-negative solution with prime period two cannot be found. Moreover, simulation is shown to support the results.

We investigate the periodic nature of solutions of the max difference equationxn+1=max⁡{xn,A}/(xnxn−1),n=0,1,…, whereAis a positive real parameter, and the initial conditionsx−1=Ar−1andx0=Ar0such thatr−1andr0are positive rational numbers. The results in this paper answer the Open Problem 6.2 posed by Grove and Ladas (2005).

In this paper, we investigate the asymptotic behavior and periodic nature of positive solutions of the difference equation where A ≥ 0 and 0 ≤ α ≤ 1. We prove that every positive solution of this difference equation approaches x = 1 or is eventually periodic with a period 2, 3 or 4.

We present a new stellar model which employs the Buchdahl metric potential for the temporal metric potential in the spherical symmetric configuration, following the Mazur-Mottola (MM) gravastar conjecture within the Einsteinian geometric framework. It is thought to be a promising alternative to the Black Holes (BH). Three regions make up the gravastar's structure: the interior, intermediate shell, and the exterior region. In our model, the interior core region is characterized by a pressure equal to the constant negative matter energy density. This gives rise to a constant repulsive force acting on the shell. This shell is modeled as being composed of an ultra-relativistic plasma fluid. In conformity with Zeldovich's stiff fluid conjecture, where the pressure is proportional to the energy density of the matter cancels the repulsive force exerted by the interior region. We have described the exterior region's geometry by the Schwarzschild solution with the spacetime being a vacuum. The specifications lead to a family of exact solutions for the gravastar, free of singularities possessing a physically valid features within the (3 + 1) dimensional spacetime paradigm. Moreover, our discussion has covered the junction and the energy conditions in detail, highlighting their role in the production of the thin shell. We conducted a comprehensive stability analysis of our gravastar model through the study of surface redshift and speed of sound. Thus, we have successfully formulated a stable gravastar model that overcomes the singularity problem of BHs, within the context of General Relativity (GR).

We investigate the deposition of energy due to the annihilations of neutrinos and antineutrinos on the rotation axis of rotating neutron and quark stars, respectively. The source of the neutrinos is assumed to be a neutrino-cooled accretion disc around the compact object. Under the assumption of the separability of the neutrino null geodesic equation of motion, we obtain the general relativistic expression of the energy deposition rate for arbitrary stationary and axisymmetric space-times. The neutrino trajectories are obtained by using a ray tracing algorithm, based on numerically solving the Hamilton-Jacobi equation for neutrinos by reversing the proper time evolution. We obtain the energy deposition rates for several classes of rotating neutron stars, described by different equations of state of the neutron matter, and for quark stars, described by the MIT bag model equation of state and in the colour-flavourlocked phase, respectively. The electron-positron energy deposition rate on the rotation axis of rotating neutron and quark stars is studied for two accretion disc models (isothermal disc and accretion disc in thermodynamical equilibrium). Rotation and general relativistic effects modify the total annihilation rate of the neutrino-antineutrino pairs on the rotation axis of compact stellar general relativistic objects, as measured by an observer at infinity. The differences in the equations of state for neutron and quark matter also have important effects on the spatial distribution of the energy deposition rate by neutrino-antineutrino annihilation.

Economic dynamics vary across business cycles, particularly in the relationship between output and unemployment. This study examines Okun's Law across 92 countries from 1980 to 2023, focusing on its validity and stability during expansion, recession, and recovery. The results confirm that Okun's Law broadly holds, but its strength is highly regime-and country-specific. A consistent pattern emerges: the Okun coefficient is steeper during recessions, flatter during expansions, and statistically weaker or insignificant during recoveries, consistent with jobless recoveries. Crosscountry evidence further reveals that high-income and OECD economies exhibit robust Okun relationships across regimes, while many low-income economies display weak or non-significant responses due to informality and structural labor market rigidities. Dynamic specifications confirm short-run stickiness in employment, with cumulative effects reinforcing the asymmetric adjustment of unemployment to output. By introducing a three-regime empirical design and incorporating post-pandemic labor market dynamics, this study provides the most comprehensive crosscountry assessment of Okun's Law to date. The findings underscore the importance of regime-contingent policy strategies: countercyclical support during recessions, targeted incentives during recoveries, and structural reforms in economies with weak output-employment linkages.

Gastrointestinal parasitism is one of the diseases that has the highest economic impact on the Argentinian beef production system, rendering it inefficient. In the region of the Humid Pampas, it has been estimated that 22 million dollars are lost annually because of the death of calves and 170 million dollars are lost in sub-clinic costs. A mathematical model with fuzzy parameters was constructed for the analysis of the free-living stages of gastrointestinal parasites, with the purpose of estimating the pool of L3 larvae available for migration to pasture and the levels of infection in pasture at any time of the year under different climatic conditions. The model is formulated in terms of a system of three difference equations. These equations describe the abundance of parasites in each of the successive stages of the population development. The model was calibrated and tested with data gathered through fieldwork carried out in Tandil (37 • 19 S, 59 • 08 05 W), province of Buenos Aires (Argentina) and the corresponding weather data. A comparison between model simulations and fieldwork data obtained in other locations achieved satisfactory results.

A conjecture of Lax [P. Lax, Differential equations, difference equations and matrix theory, Commun. Pure Appl. Math. 11 (1958) [175][176][177][178][179][180][181][182][183][184][185][186][187][188][189][190][191][192][193][194] says that every hyperbolic polynomial in two space variables is the determinant of a symmetric hyperbolic matrix. The conjecture has recently been proved by Lewis-Parillo-Ramana, based on previous work of Dubrovin and Helton-Vinnikov. In this note we prove related results for polynomials in several space variables which have rotational symmetries.

Consider MDAs (X.z) and (Y.i), and stopping times %(0, 0 < t -< 1. Denote ~(t) ~(t) S,(t)=ao + ~ X,i, T,(t)=bo + ~ Y,i, i=1 i=1 and let q0: ]R~IR be a function. If the common distribution converges and if St, T t denote the corresponding limiting processes then we give conditions such that the martingale transforms ~(t) ~o(S,,,_ 1) Y,i i=1 converge weakly to the stochastic integral t This result has important consequences for functional central limit theorems: (1) If the MDAs are connected by a difference equation of the form Xni=~O(Sn,i_ l) Yni, then weak convergence of T.(t) implies that of S.(t), and the limit satisfies the stochastic differential equation dS=q)(S)dT. This observation leads to functional limit theorems for diffusion approximations. E.g. we obtain easily a result of Lindvall, [4], on the diffusion approximation of branching processes. (2) If the MDA (X,i) arises from a likelihood ratio martingale then the limit satisfies t S,=I+5SdT, 0 84 H. Strasser which leads to the representation of the limiting likelihood ratios as exponential martingale: S~ = exp(T t -89 Tit ). This approximation by an exponential martingale has been proved previously by Swensen, [9], using a Taylor expansion of the log-likelihood ratio. (3) As a consequence we obtain a general functional central limit theo-2X2i) converges weakly to ([S,S]t), then X,i converges rem: If { weakly to (St), provided that the distribution of (St) is uniquely determined by that of (IS, Sit). This assertion embraces previous central limit theorems, dealing with cases where the increasing process (IS, S-lt) is deterministic.

We present a mathematically self-contained unification of Persistence-First (PF) and UGV Without Meta (UGV) into a theory that yields cosmic-scale, meta-free superintelligence from first principles under explicit assumptions. The core claims are: (i) a policy-and evaluator-independent order-equivalence between PF's persistence ratio and UGV's viability ratio via a fixed positive affine transform with constants determined solely by physical/communication/visibility assumptions; (ii) synergy/redundancy terms derived from the two axioms-rather than postulatedyielding a strategic exact potential for multi-agent interaction; (iii) evaluator coherence formalized in an information category of Blackwell-faithful morphisms, ensuring ratio-preserving aggregation under (co)limits; (iv) cosmological floors (thermodynamic and informational) that make the denominator uniformly positive under finite energy density, nonzero temperature, finite SNR/bandwidth, and a Doeblin-type visibility minorization; (v) stochastic goal-audit invariance via a Robbins-Siegmund construction with an explicit contraction-rate bound; (vi) adversarial safety expressed as an explicitly parameterized (practically estimable) threshold linking control-barrier geometry, audit strength, and SDPI floors, with an extension to convex collusion. A relational framing (dependent origination; process philosophy) situates "No-Meta" as governance internal to relations, not externally imposed. Together, these results provide sufficient conditions for a free, evaluator-plural, conflict-free, and cosmically consistent ascent to superintelligence without any meta-manager.

We give a deductively closed, assumption-minimized account showing that in a closed (no-meta) governance where all regulation is internally implemented (audits, evaluator reweighting, channel composition), and under finite dissipation, evil policies are non-persistent while good policies-including compassion and enlightenment endpoints-spread with strictly positive front speed. Previous assumptions (time-averaged Doeblin visibility, SDPI/LSI contraction, pathwise Cesàro limits, symmetry dominance, replicator-diffusion dynamics) are here derived from Persistence-First (PF), physics, and Blackwell-faithful evaluation, or constructed via entropic (Bregman-JKO) minimizing movements [10, 11, 12, 13]. The analysis connects Doeblin minorization to SDPI/LSI floors [4, 5, 6, 7, 9, 8], establishes a welldefined path objective with uniform integrability, proves a uniform path gap yielding ENPT/NSHS/NSA, and applies a semigroup KPP comparison [15, 16, 17, 18] to obtain an explicit front-speed lower bound 2 √ 𝐷 min 𝜆 min. Physical calibration rests on Landauer-type bounds [20, 21]. We also instantiate AC * /AC *+ internally, formalize an audit-head ⇒ MI-gain lemma, prove denominator monotonicity against gaming, make Δ𝐶 info,𝐻 explicit, include kernel-normalized (Abel-Toeplitz) averaging for heavy tails, and link the reaction rate 𝜆 to prior lemmas. Scientific posture. We refrain from stating logical impossibility as a matter of scientific discipline. This caveat does not constitute evidence that a successful attack exists; rather, under our stated constraints we know of no constructive attack, and the design raises both adversarial cost and detectability. Intuition. Under strictly positive floors for visibility, contraction, contact, and dissipation, the asymmetry is second-law-like: malign

We develop a theory-only framework for Unified Generative Viability (UGV): starting from a single axiom of causal fecundity (CF)-maximize the time-rate at which a world grows viable structure visible to a fixed evaluator-we derive, without any external rule stack (No-Meta), that two normative endpoints are first-class optima: Compassion (net hetero-creation) and Enlightenment (behavior invariant to self/other labels). The objective is a ratio with a fixed evaluator 𝐻 𝜁 : a conditional mutual information (CMI) term plus a dimensionless viable-mass increment in the numerator, divided by a denominator that is the least inflation of the larger of an information-cost and a constructive (cb-)SDPI floor tied to 𝐻 𝜁. The conditional DPI gives a three-line proof of anti-gaming monotonicity under world-side coarse-grainings (see Cover-Thomas [24], Csiszár-Körner [25]). Regularity follows from standard Borel measurability (Kallenberg [26]), Feller continuity for Markov kernels (Ethier-Kurtz [27]), and Uniform AC* bounds (strengthened to Uniform AC*+) that hold uniformly in time. Dinkelbach guarantees existence

We consider general, higher order difference equations of type in which the function f is non-increasing in each coordinate. We obtain sufficient conditions for the asymptotic stability of a unique fixed point relative to an invariant interval. We also discuss various applications of our main results.

Second order rational difference equations with quadratic terms in their numerators and linear terms in their denominators exhibit a rich variety of dynamic behaviors. It is demonstrated that depending on the parameters and initial values, there can be globally attracting fixed points, coexisting periodic solutions or chaotic trajectories.

We investigate the sampling theory associated with basic Sturm-Liouville eigenvalue problems. We derive two sampling theorems for integral transforms whose kernels are basic functions and the integral is of Jackson's type. The kernel in the first theorem is a solution of a basic difference equation and in the second one it is expressed in terms of basic Green's function of the basic Sturm-Liouville systems. Examples involving basic sine and cosine transforms are given.

A mon épouse Adjara et ma fille Samihra, A ma famille, mes amis et à tous ceux qui croient à l'effort et oeuvrent pour la justice, la paix et la dignité humaine. Tout d'abord, je remercie le Professeur Augustin BANYAGA qui me fait un grand honneur en acceptant de présider le jury de cette thèse. Je remercie également les Professeurs Saïd BELMEHDI, Jean-Pierre EZIN, Wolfram KOEPF et Côme GOUDJO pour avoir accepté de faire partie du jury. J'exprime ici ma reconnaissance aux professeurs M. Norbert HOUNKONNOU et André RONVEAUX pour les efforts fournis et les sacrifices énormes consentis pour la co-direction de cette thèse. Mes remerciements vont aussi à l'endroit du Professeur Jean-Pierre Ezin, Directeur de l'IMSP, pour sa constante sollicitude. Je remercie profondément le Service Allemand d'Echanges Universitaires (DAAD) qui, en m'octroyant une bourse doctorale, a rendu possible la réalisation de ce travail. Mes remerciements vont aussi à l'endroit de l'Université Nationale du Bénin et en particulier de l'Institut de Mathématiques et de Sciences Physiques pour l'hospitalité, le soutien financier et les sacrifices consentis tout au long de ma formation. Le séjour en Europe, de septembre 1997 à mars 1998, a été déterminant pour la finalisation de ce travail. Pour cela, je tiens a remercier le DAAD pour avoir financé ce séjour, le Centre "Konrad-Zuse-Zentrum für Informationstechnik Berlin" (ZIB) pour m'avoir acceuilli en son sein et aussi pour m'avoir offert d'excellentes conditions de travail. C'est l'occasion ici de remercier le Professeur Wolfram KOEPF pour son hospitalité pendant mon séjour à Berlin et aussi pour la formation qu'il m'a donnée en calcul symbolique. Harald B ÖING et le Professeur Winfried NEWN, tous de ZIB, m'ont aussi aidé notamment pour la programmation et je les en remercie. Il me plait ici de remercier très sincèrement le Professeur André RONVEAUX qui, malgré son admission à la retraite, a financé à ses propres frais mon séjour à Namur et aussi son séjour à Berlin. Ces rapprochements ont permis une évolution très significative du travail. Je remercie également Gérard LAGMAGO pour l'acheminement des copies de cette thèse à Namur,

The Operational Observatory of the Catalan Sea (OOCS), created in 2009 at CEAB-CSIC may be considered as a reference marine observatory because of its effectiveness and relatively low-cost functioning and maintenance. The number of time series obtained at the observation station of the meteorological conditions above the sea surface, along with physical and biogeochemical properties of the water layer over the continental shelf, supports its success. The strong fluctuations of atmospheric conditions registered in the last years altering the marine conditions make the simultaneous records of meteorological and marine observations essential for understanding present environmental fluctuations and for improving marine environmental predictions. Updated information regarding the observatory can be found at .

Experimental and simulated time series are necessarily discretized in time. However, many real and artificial systems are more naturally modeled as continuous-time systems. This paper reviews the major techniques employed to estimate a continuous vector field from a finite discrete time series. We compare the performance of various methods on experimental and artificial time series and explore the connection between continuous ͑differential͒ and discrete ͑difference equation͒ systems. As part of this process we propose improvements to existing techniques. Our results demonstrate that the continuous-time dynamics of many noisy data sets can be simulated more accurately by modeling the one-step prediction map than by modeling the vector field. We also show that radial basis models provide superior results to global polynomial models.

Combinatorial properties of 1D Fibonacci words is a well studied topic in Formal language theory. In the year 2000, Apostolico et.al. extended the concept of one dimensional Fibonacci words to two dimensional Fibonacci arrays and investigated the number of repetitions of some structures (squares, tandems). In this paper, we investigate the number of distinct Palindromic occurrences in any given Fibonacci array. We also investigate the number of palindromes in the conjugacy class of a Fibonacci array. Key-words: Fibonacci arrays, Palindromes, Parikh vector, Conjugacy.

Three different sets of shallow water equations, representing different levels of approximation are considered. The numerical solutions of these different equations for flow past bottom topography in several different flow regimes are compared. For several cases the full Euler solutions are computed as a reference, allowing the assessment of the relative accuracies of the different approximations. Further, the differences between the dispersive shallow water (DSW) solutions and those of the highly simplified, hyperbolic shallow water (SW) equations is studied as a guide to determining what level of approximation is required for a particular flow. First, the Green-Naghdi (GN) equations are derived as a vertically-integrated rational approximation of the Euler equations, and then the generalized Boussinesq (gB) equations are obtained under the further assumption of weak nonlinearity. A series of calculations, each assuming different values of a set of parameters—undisturbed upstream F...

Let ν be a rotation invariant Borel probability measure on the complex plane having moments of all orders. Given a positive integer q, it is proved that the space of ν-square integrable q-analytic functions is the closure of q-analytic polynomials, and in particular it is a Hilbert space. We establish a general formula for the corresponding polyanalytic reproducing kernel. New examples are given and all known examples, including those of the analytic case are covered. In particular, weighted Bergman and Fock type spaces of polyanalytic functions are introduced. Our results have a higher dimensional generalization for measure on C p which are in rotation invariant with respect to each coordinate.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Mazur and Mottola's gravastar model represents one of the few serious alternatives to the traditional understanding of the black hole. The gravastar is typically regarded as a theoretical alternative for the black hole. This paper investigates the creation of gravastar (gravitational vacuum star) within the realm of cylindrically symmetric space{time utilizing the Kuchowicz metric potential incorporating the role of Einstein's cosmological constant Ã. A stable gravastar comprises of three distinct regions, starting with an interior region marked by positive energy density and negative pressure ðp ¼ ÀÞ which is followed by an intermediate thin shell, where the interior negative pressure induces a outward repulsive force at each point on the shell. Ultrarelativistic sti® °uid makes up the thin shell governed by the equation of state (EoS) ðp ¼ Þ, which meets the Zel'dovich criteria, and then comes the region exterior to it which is total vacuum. In this scenario, the central singularity is eliminated and the event horizon is e®ectively substituted by the thin bounding shell. By employing the Kuchowicz metric potential, we have derived the remaining metric functions for the interior region and the shell regions yielding a nonsingular solution for both the regions. Additionally, we have investigated the impact of à on various characteristics of this shell region including its proper shell length, the energy content and entropy. Evidently, the in°uence of the nonzero cosmological constant in our stellar model opens up a new physically plausible scenario in the cylindrical space{time by the use of Kuchowicz metric function. The study of surface redshift had also helped in con¯rming the stability of our con¯guration.

The experiment was conducted to determine the crop coefficients and leaf area index of pigeonpea, using a digital weighing type lysimeters. The research was conducted under Department of Irrigation and Drainage Engineering, Dr. Panjabrao Deshmukh Krishi Vidyapeeth, Akola for kharif season of year 2024. The meteorological data was collected from the weather station installed at the experimental site to determine the reference crop evapotranspiration (ETo) using Penman

We investigate the higher order nonlinear discrete Volterra equations. We study solutions with prescribed asymptotic behavior. For example, we establish sufficient conditions for the existence of asymptotically polynomial, asymptotically periodic or asymptotically symmetric solutions. On the other hand, we are dealing with the problem of approximation of solutions. Among others, we present conditions under which any bounded solution is asymptotically periodic. Using our techniques, based on the iterated remainder operator, we can control the degree of approximation. In this paper we choose a positive non-increasing sequence u and use o(u n ) as a measure of approximation.

We establish some properties of iterations of the remainder operator which assigns to any convergent series the sequence of its remainders. Moreover, we introduce the spaces of multiple absolute summable sequences. We also present some tests for multiple absolute convergence of series. These tests extend the well-known classical tests for absolute convergence of series. For example we generalize the Raabe, Gauss, and Bertrand tests. Next we present some applications of our results to the study of asymptotic properties of solutions of difference equations. We use the spaces of multiple absolute summable sequences as the measure of approximation.

Parallel garbage collection systems consist of two processors, the mutator and the garbage collector, which operate on a conmwn memory of list nodes. The garbage collector repeatedly executes a two-stage cycle that maintains a list of nodes (the free list) for use by the mutator, which is dedicated to executing a user's list processing program. All list nodes contain two extra bits for marking purposes and are either white, black, or gray. During the mark stage, the garbage collector blackens all nodes that are accessible to the mutator. During the scan stage, white nodes are placed on the free list and black nodes are whitened. The color gray is used during the mark stage to guarantee that all accessible nodes are marked, it is shown that parallel garbage collection systems exhibit three fundamental types of performance (stable, alternating, and critical), and it is determined when each of the three will occur. Moreover, it can be determined when this parallel scheme is more efficient than sequential garbage collection by obtaining analytic formulas for a variety of performance measures. These results are obtained by expressing the performance of parallel garbage collection systems in terms of conditional difference equations (CDEs) in which the state of the system after a garbage collection cycle is described by a pair of difference equations and a Boolean function. The value of the Boolean function at the beginning of a cycle selects which of the two difference equations should be applied next.

We give an elementary proof of what we call Local Bézout Theorem. Given a system of n polynomials in n indeterminates with coefficients in a henselian local domain, (V, m, k), which residually defines an isolated point in k n of multiplicity r, we prove (under some additional hypothesis on V) that there are finitely many zeroes of the system above the residual zero (i.e., with coordinates in m), and the sum of their multiplicities is r. Our proof is based on techniques of computational algebra.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Let R be a local domain, v a valuation of its quotient field centred in R at its maximal ideal. We investigate the relationship between R h , the henselisation of R as local ring, and ṽ, the henselisation of the valuation v, by focussing on the recent result by de Felipe and Teissier referred to in the title. We give a new proof that simplifies the original one by using purely algebraic arguments. This proof is moreover constructive in the sense of Bishop and previous work of the authors, and allows us to obtain as a by-product a (slight) generalisation of the theorem by de Felipe and Teissier.

Monotonic, hump-shaped and zero-correlation productivity-diversity relationships have been found to date in many ecosystems. This diversity of responses has puzzled ecologists in their search for general principles on ecosystem functioning. Some state that the scale of observation is crucial in defining this relationship. We have developed a spatial model of tallgrass prairies where biomass and litter dynamics are defined by uncoupled difference equations. In this system, we periodically apply prescribed fire as a disturbance that propagates through neighboring cells. The model shows percolation thresholds at points where small-scale spatial heterogeneity and large-scale, global correlation coexist, resulting in power-law distributions in available areas for non-dominant species. These points maximize the biomass-diversity relationship. Our results suggest that spatial dependencies and the disturbance heterogeneity hypothesis are the cornerstone processes accounting for unimodality in productivity-diversity relationships.

The two-step regression process for whole stand survival modeling has been widely used for multiple species. However, estimating the parameters of a probability model for the discrete event of mortality and a whole stand survival model separately can lead to biased projections and poor estimates of the variability associated with future projections. This study derives a mixture probability density function for the combined equation used in the two-step survival regression method that can be used to estimate stand survival and projection uncertainty. Our new density function was used in the maximum likelihood framework to find optimal parameters for a whole stand survival function developed for loblolly pine (Pinus taeda) in the Western Gulf region of the US. The mixture probability density function results in accurate and unbiased projections of future stand density and provides a means to evaluate long-term model prediction uncertainty.

Motivated by the study of symmetry breaking operators for indefinite orthogonal groups, we give a Gegenbauer expansion of the two variable function |s-t| α in terms of the ultraspherical polynomials C λ ℓ (s) and C µ m (t). Generalization, specialization, and limits of the expansion are also discussed.

The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution.

Let P be a set of n points in the plane, not all on a line. We show that if n is large then there are at least n/2 ordinary lines, that is to say lines passing through exactly two points of P . This confirms, for large n, a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than n -C ordinary lines for some absolute constant C. We also solve, for large n, the "orchard-planting problem", which asks for the maximum number of lines through exactly 3 points of P . Underlying these results is a structure theorem which states that if P has at most Kn ordinary lines then all but O(K) points of P lie on a cubic curve, if n is sufficiently large depending on K.

Let P be a set of n points in the plane, not all on a line. We show that if n is large then there are at least n/2 ordinary lines, that is to say lines passing through exactly two points of P . This confirms, for large n, a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than n -C ordinary lines for some absolute constant C. We also solve, for large n, the "orchard-planting problem", which asks for the maximum number of lines through exactly 3 points of P . Underlying these results is a structure theorem which states that if P has at most Kn ordinary lines then all but O(K) points of P lie on a cubic curve, if n is sufficiently large depending on K.

This study investigates the 3D motion of a rigid body (RB) rotating around a fixed point, focusing on Lagrange's case while considering the effects of a gyrostatic moment (GM) and a Newtonian force field (NFF). It is notably that the center of mass is displaced very much from the principal dynamic symmetry axis. Utilizing the fundamental principle of angular momentum, the equations of motion (EOM) are formulated and solved applying the large parameter approach (LPA) to determine approximate solutions (AS) for the irrational frequencies' case. Euler's angles, which define the body's orientation at any moment, are explicitly calculated. Additionally, to assess the impact of applied moments on motion stability, we utilize advanced computational tools to generate graphical representations of the achieved solutions and the related Euler's angles. This work enhances the understanding of RB dynamics in complex motion scenarios, emphasizing the interaction between external forces and GMs in shaping stability and behavior. The study is poised to greatly influence the aerospace industry by advancing our understanding of rotational motion and the dynamics of celestial bodies, with direct applications in the design and operation of spaceships, spacecraft, and satellites.

Determinism and unpredictability are compatible since deterministic flows can produce, if sensitive to initial conditions, unpredictable behaviors. Within this perspective, the notion of scenario to chaos transition offers a new form of predictability for the behavior of sensitive to initial condition systems under the variation of a control parameter. In this paper I first shed light on the genesis of this notion, based on a dynamical systems approach and on considerations of structural stability. I then suggest a link to the figure of epigenetic landscape, partially inspired by a dynamical systems perspective, and offering a theoretical framework to apprehend developmental noise.

We study the connectedness property of the spectrum of forcing algebras over a noetherian ring. In particular we present for an integral base ring a geometric criterion for connectedness in terms of horizontal and vertical components of the forcing algebra. This criterion allows further simplifications when the base ring is local, or one-dimensional, or factorial. Besides, we discuss whether the connectedness of forcing algebras is a local property. Finally, we present a characterization of the integral closure of an ideal by means of the universal connectedness of the corresponding forcing morphism.

We show that, under some assumptions, the linear recurrence (or difference equation) of order one in a Banach space is nonstable in the Hyers-Ulam sense. Our results are also connected with the notion of shadowing in dynamical systems and computer sciences.

A systematic method to derive the nonlocal symmetries for partial differential and differentialdifference equations with two independent variables is presented and shown that the Korteweg-de Vries (KdV) and Burger's equations, Volterra and relativistic Toda (RT) lattice equations admit a sequence of nonlocal symmetries. An algorithm, exploiting the obtained nonlocal symmetries, is proposed to derive recursion operators involving nonlocal variables and illustrated it for the KdV and Burger's equations, Volterra and RT lattice equations and shown that the former three equations admit factorisable recursion operators while the RT lattice equation possesses (2 × 2) matrix factorisable recursion operator. The existence of nonlocal symmetries and the corresponding recursion operator of partial differential and differential-difference equations does not always determine their mathematical structures, for example, bi-Hamiltonian representation.

Six height‐age determination methods (Graves, Lenhart, Carmean, Newberry, ratio, and ISSA) were evaluated for their accuracy and sensitivity to sample size in determining height‐age pairs using stem analysis data from plantation-grown black spruce (Picea mariana[Mill.] B.S.P.) and jack pine (Pinus banksiana Lamb.) trees from Ontario, Canada. Twenty-three disks (sections) were used from 102 jack pine and 93 black spruce trees each for evaluation. The Graves, ratio, and Newberry methods were unbiased for determining height‐age pairs forboth black spruce and jack pine across the site productivity gradient and different crown classes. However, on the basis of the magnitude of height prediction bias, reconstructed tree profiles, and the amount of information required for height‐age determination, the Graves method withat least 13 stem sections is recommended for height‐age determination.

Thermal sampling peaks recorded after windowing polarization are studied for the segmental mode in poly͑⑀-caprolactone͒. Also, numerical decompositions of the global thermally stimulated depolarization current peak into pure Debye contributions are performed with direct signal analysis ͑DSA͒ and simulated annealing direct signal analysis procedures for Arrhenius and Vogel-Tammann-Fulcher ͑VTF͒ temperature dependences, respectively. It is found that the results between the experimental and the numerical procedures agree very well and the approximations made in the analysis of the experimental curves are thus validated, despite the unphysical values for the relaxation parameters found by both methods when using Arrhenius relaxation times. On the contrary, when VTF relaxation times are used for the numerical decomposition, agreement is found with the results of isothermal dielectric absorption as a function of frequency, together with reasonable values for reorientation energies, pre-exponential factors and VTF temperature. Thermal sampling and DSA also compare well when studying the departure from the zero-entropy line which indicates the onset of a cooperative character in the dynamics of molecular motion. Compensation is found whenever the primary relaxation is analyzed with Arrhenius or Eyring relaxation times and does not appear when VTF relaxation times are used in the numerical decomposition.