We prove limit theorems for the number of fixed points occurring in a random pattern-avoiding permutation distributed according to a one-parameter family of biased distributions. The bias parameter exponentially tilts the distribution towards favoring permutations with more or fewer fixed points than is typical under the uniform distribution. One case we study features a phase transition where the limiting distribution changes abruptly from negative binomial to Rayleigh to normal depending on the bias parameter. 17 pages, final version

This paper develops a practical computational framework for the Bayesian Cournot model with bilateral incomplete cost information, where each player is uncertain about the opponent’s marginal cost, drawn from a continuous compact interval [c*, c*] with 0<c*<c*<∞. The infinite dimensionality of the functional strategy spaces (mappings from types to production quantities) renders analytical closed-form solutions infeasible in this continuous-type setting. To overcome this challenge, we restrict the strategy spaces to finite-dimensional differentiable sub-manifolds—specifically, one-parameter families of oscillatory functions (cosine, sine, and mixed forms). After suitable affine Q-rescaling to map the oscillatory range into the production interval [0, Q], and with parameter ranges satisfying α, β>(π/2)/c*, these curves ensure near-exhaustivity: the joint production map (α, β)↦(xα(s), yβ(t)) covers [0, Q]2 densely for every fixed cost pair (s, t), thereby recovering (up to density and closure) the full ex-post payoff space. We introduce the ex-post payoff mapping Φ(s, t, x, y)=(es(x, y)(t), ft(x, y)(s)), which collects every realizable payoff pair once nature draws the types and players select their strategies. The image of Φ defines the general payoff space of the game, and its non-dominated points constitute the general ex-post Pareto frontier—all efficient realized outcomes across type-strategy realizations, without dependence on private probability measures over types. Using multi-objective genetic algorithms, we numerically approximate this frontier (and selected collusive compromises) within the restricted but representative sub-manifolds. The resulting frontiers are computationally accessible, robust to parameter variations, and validated through hypervolume convergence, sensitivity analysis, and comparisons with NSGA-II, PSO and scalarization methods. The findings are significant because they provide decision-makers in oligopolistic markets (e.g., electric vehicles) with viable, implementable production policies that explore efficient trade-offs under genuine cost uncertainty, without requiring explicit forecasts of the opponent’s type distribution—a limitation of traditional expected-utility approaches. By focusing on ex-post efficiency, the method reveals belief-independent compromise solutions that may guide tacit coordination or collusive outcomes in real-world strategic settings.

Let f t f_t be a one-parameter family of rational maps defined over a number field K K . We show that for all t t outside of a set of natural density zero, every K K -rational preperiodic point of f t f_t is the specialization of some K ( T ) K(T) -rational preperiodic point of f f . Assuming a weak form of the Uniform Boundedness Conjecture, we also calculate the average number of K K -rational preperiodic points of f f , giving some examples where this holds unconditionally. To illustrate the theory, we give new estimates on the average number of preperiodic points for the quadratic family f t ( z ) = z 2 + t f_t(z) = z^2 + t over the field of rational numbers.

Abstract NFTs or non-fungible tokens are digital assets stored on a blockchain. They can be traded or exchanged for money, cryptocurrencies or other NFTs. Examples include works of art and digital or other tokenised collectables. An important determinant of price for collectables is rarity within a collection. Many trading platforms offer to rank items in terms of rarity but rankings differ considerably and, often, little explanation is given of the methods used. This paper provides a mathematical framework for the analysis of a comprehensive class of collections. It examines individual and joint distributions of attributes over such collections, and shows how these can be combined to provide a rarity ranking for all items in the collection. There is, however, only a limited range of methods that give consistent results over different collections. These are identified as belonging to a one-parameter family of ranking functions. Each gives to every item of a collection a rarity score that is directly comparable between collections. Despite taking account of all possible combinations of attributes when ranking, the method is nonetheless computationally feasible.

We present a one-parameter family of smooth generalized pinball loss functions to overcome the challenges of non-differentiability, noise sensitivity, and resampling instability inherent in traditional loss functions such as hinge loss. These functions make the objective function in the formulation of the support vector machine (SVM) model twice continuously differentiable and improve model performance by reducing noise sensitivity and preserving the sparsity of the solution. Similarly, a novel twin-bounded support vector machine (TBSVM) model with a smooth generalized pinball loss function is obtained. Furthermore, we compare the performance of the TBSVM with the novel type of smooth loss function against other contemporary approaches, offering a comprehensive assessment of its strengths and limitations by conducting an evaluation with UCI datasets. The experimental results show that the proposed model has the best performance in the TBSVM with RBFSampler. Additionally, we prove that the generalized pinball loss function can be approximated by a novel smooth generalized pinball loss function in the uniform norm with arbitrary precision. We further show that the solutions of the proposed SVM and TBSVM models are unique and that they converge to the solutions of the models with non-smooth generalized pinball loss as the parameter approaches zero.

Abstract We propose an algebraic–probabilistic framework that turns finite-sample extrema into a simple exponential race. For one-parameter families whose cumulative distribution function (CDF) or survival function (SF) has the separable exponential form Hθ(x)=exp⁡{−ϕ(x)g−1(θ)}, the extremum of independent variables stays in the family with the parameter given by the pseudo-sum ⨁i=1nθi=g(∑i=1ng−1(θi)) induced by the generator g. The exponential-race view yields two general consequences: (i) the index of the extremum (arg-extreme) is independent of the extremum itself; (ii) its law is Boltzmann–Gibbs (softmax) with weights proportional to g−1(θi). The same mapping gives a closed-form description of the full ranking, clarifies links with existing constructions and extends immediately to exceedances and shortfalls. Under mild regularity, the separable exponential form is also necessary and sufficient for exact finite-n extremal stability. The framework unifies classical examples (Weibull, Pareto, Gumbel and Fréchet), with aggregation rules emerging as harmonic sums, ℓα norms or log-sum-exp, and it offers a boundary-behaviour perspective on the Fisher–Tippett–Gnedenko (FTG) trichotomy. The resulting theory is simple, closed-form and portable across reliability, insurance and finance, and it suggests natural Bayesian and dependence extensions.

The New Polynomial Exponential Distribution (NPED), introduced by Beghriche et al. (2022), provides a flexible one-parameter family capable of representing diverse hazard shapes and heavy-tailed behavior. Regression frameworks based on the NPED, however, have not yet been established. This paper introduces two methodological extensions: (i) a generalized linear model (NPED-GLM) in which the distribution parameter depends on covariates, and (ii) a Poisson–NPED count regression model suitable for overdispersed and heavy-tailed count data. Likelihood-based inference, asymptotic properties, and simulation studies are developed to investigate the performance of the estimators. Applications to engineering failure-count data and insurance claim frequencies illustrate the advantages of the proposed models relative to classical Poisson, negative binomial, and Poisson–Lindley regressions. These developments substantially broaden the applicability of the NPED in actuarial science, reliability engineering, and applied statistics.

We consider translators (i.e., initial condition of translating solitons) to mean curvature flow (MCF) in the hyperbolic 3 -space \mathbb{H}^{3} , providing existence and classification results. More specifically, we show the existence and uniqueness of two distinct one-parameter families of complete translators in \mathbb{H}^{3} , one containing catenoid-type translators, and the other parabolic cylindrical ones. We establish a tangency principle for translators in \mathbb{H}^{3} and apply it to prove that properly immersed translators to MCF in \mathbb{H}^{3} are not cylindrically bounded. As a further application of the tangency principle, we prove that any horoconvex translator which is complete or transversal to the z -axis is necessarily an open set of a horizontal horosphere. In addition, we classify all translators in \mathbb{H}^{3} which have constant mean curvature. We also consider rotators (i.e., initial condition of rotating solitons) to MCF in \mathbb{H}^{3} and, after classifying the rotators of constant mean curvature, we show that there exists a one-parameter family of complete rotators which are all helicoidal, bringing to the hyperbolic context a distinguished result by Halldorsson, set in \mathbb{R}^{3} .

Longitudinal oscillations of an inhomogeneous chain of linear oscillators coupled by springs are investigated. Both outer springs of the chain are rigidly fixed to immovable supports. The system is subjected to external periodic forces. The inhomogeneity of the chain (the perturbed system) is due to the different stiffness coefficients of the springs. These coefficients deviate slightly from a certain nominal value and depend on dimensionless deviation parameters. Zero values of these parameters correspond to a homogeneous (unperturbed) system. The resonant case is considered when the frequency of the external periodic force coincides with one of the eigenfrequencies of the unperturbed system. To construct an exact periodic solution of the perturbed system, the Lyapunov–Schmidt method is applied. As the problem is linear, this method allows to reduce it to a finite-dimensional algebraic problem of constructing a generalized Jordan chain for a degenerate linear operator. Necessary and sufficient conditions on the dimensionless deviation parameters are obtained, under which the length of such a chain is equal to 1 or 2. For each case, explicit exact formulas for the chain are derived, providing a complete description of the periodic solution. It is shown that for a generalized Jordan chain of length 1, the periodic solution of the perturbed system continuously transforms into a certain periodic solution of the unperturbed system as the small parameter $\varepsilon$ tends to zero. If the length of the generalized Jordan chain is $2$, the periodic solution of the perturbed system possesses a first-order pole at $\varepsilon=0$ and, reduces to a one-parameter family of periodic solutions of the unperturbed system. Numerical simulation was performed for a chain of eight oscillators. Plots of periodic solutions and phase trajectories of the perturbed system are constructed for various values of the small parameter.

This article considers a one-parameter family of piecewise-smooth vector fields that are invariant under reflection from the x-axis on a plane with Cartesian coordinates (x, y). The switching line passes through the origin O, transversally to the x-axis. For a zero value of the parameter, let the vector field of the family in the left half-neighborhood of the switching line coincide with a smooth vector field that has the O point as a rough stable node, and in its right half-neighborhood it coincides with a smooth vector field without singular points. Let this field also have a rough saddle S on the x-axis such that the open arc of the x-axis between the O and S points is an incoming separatrix of the saddle, and the two symmetric outgoing separatrices of the saddle do not contain any singular points and lead to the O point. The article demonstrates that if there is no singular point in the left semi-neighborhood of the switching line for the positive values of the parameter, then a unique, stable, periodic trajectory arises from each of the two symmetrical contours formed by the separatrices. Under certain additional conditions, the emerging periodic trajectory is unique and hyperbolic.

The main object of our study is the Dirichlet kernel. The properties of this trigonometric polynomial — the sum of cosines of multiple arcs — are of undoubted interest in the theory of trigonometric series. For example, the results on the asymptotic behavior of the Lebesgue constants, which are the integral norms of the Dirichlet kernels, are well known. These results are constantly being developed and generalized as applied to various systems of functions in both one-dimensional and multidimensional situations. In this paper, we find the leading term of the asymptotics for the value of the global minimum of the Dirichlet kernel as its number tends to infinity. The leading term is the product of the said number by a negative constant, which coincides with the value of the global minimum of the sinc-function (cardinal sine). The proof uses the connection between Dirichlet kernels and Chebyshev polynomials of the second kind. As can be seen from the authors’ previous works, the result undergoes quantitative changes in the transition to lacunary sums of cosines. Our interest in such constructions is caused by the problem posed several years ago by L. E. Rossovskii and A. A. Tovsultanov on calculating the spectral radius for a special one-parameter family of functional operators. The question reduces to studying the behavior of “long” products of sines with lacunae in the arguments. It is shown that the revealed asymptotic property of Dirichlet kernels turns out to be useful in a similar “non-lacunary” problem.

Embankments are extensively applied in civil engineering and exist naturally, characterized by a constant slope ensuring stable material stacking. Despite their ubiquity, research on embankment surface geometric modeling remains limited. Three novel construction methods for embankment surfaces are proposed: the envelope of a one-parameter family of cones, the envelope of constant-slope tangent planes, and the geometric limit. Subsequently, the properties of embankment surfaces and their geometric interpretation are elaborated. Specific examples are provided to illustrate that these methods offer convenient and feasible alternatives for engineering geometric design related to these surfaces.

We extend our earlier work on the linearized perturbations of static black holes in Einstein-Maxwell-dilaton (EMD) theories to the case where the black holes are solutions in an enlarged theory including also an axion. We study the perturbations in a three-parameter family of such (Einstein-Maxwell-dilaton-axion) EMDA theories. The systems of equations describing the linearized perturbations can always be separated, but they can only be decoupled when the three parameters are restricted to a one-parameter family of EMDA theories, characterized by a parameter b that determines the coupling of the axion to the ε μ ν ρ σ F μ ν F ρ σ term. In the specific case when b = 1 , the theory is related to an N = 2 supergravity. In this one case we find that the perturbations in the axial and the polar sectors are related by a remarkable transformation, which generalizes one found by Chandrasekhar for the perturbations of Reissner-Nordström in Einstein-Maxwell theory. This transformation is of a form found in supersymmetric quantum mechanical models. The existence of such mappings between the axial and polar perturbations appears to correlate with those cases where there is an underlying supergravity supporting the solution, even though the black hole backgrounds are nonextremal and therefore not supersymmetric. We prove the mode stability of the static black hole solutions in the supersymmetric EMDA theory. For other values of the parameter b in the EMDA theories that allow decoupling of the modes, we find that one of the radial potentials can be negative outside the horizon if b is sufficiently large, raising the possibility of there being perturbative mode instabilities in such a case.

The projective transformations of n-dimensional space give rise to a one-parameter family of inhomogeneous first-order differential operator representations of s l n + 1 . By restricting to the subalgebra o n + 1 , we obtain new explicit representations for the simple Lie algebras of type B or type D. Applying these operators on the space of exponential-polynomial functions or partially swapping differential operators and multiplication operators, we realize more general explicit representations for o n + 1 . All of these representation spaces are infinite-dimensional. We obtain the composition series of these o n + 1 -modules under certain conditions.

In the paper we consider a continuum class functions defined by terms of the $Q_s$-representation of real numbers on the segment $[0;1]$, which generalizes the classical $s$-adic representation. The dependence of the $n$-th digit of the $Q_s$-representation of the function value is specified by a finite function $\varphi_n(a_n,\alpha_n)$ of two variables, whose arguments are the corresponding $Q_s$-digits $\alpha_n(x)$ and $a_n(a)$ of the input $x$ and the parameter $a$, respectively. We prove continuity of each function in this class at every $Q_s$-unary number, i.e., at points possessing a unique $Q_s$-representation. Necessary and sufficient conditions for continuity on the entire domain are established. Conditions involving the digits of the parameter $a$ and the sequence of defining functions $(\varphi_n)$, under which the function $f_a$ admits finite or continuum cardinality level sets are obtained. For particular cases ($s=2$), we study integral and differential properties, as well as the fractal properties of the sets of values. Using the self-similarity properties of the function graph and the established connection between the functions under consideration and the inversor of digits of the $Q_2$-representation of numbers, we compute the Lebesgue integral of these functions. Furthermore, we identify a subclass of functions that are piecewise singular or singular on intervals; that is, continuous non-constant functions whose derivative is zero almost everywhere in the sense of Lebesgue measure.

Abstract We use exponential asymptotic analysis to identify the relevance of Stokes’ phenomenon to integrability in discrete systems. We study Stokes’ phenomenon in two discrete problems with the same (leading-order) continuous limit, a finite-difference discretisation of the first continuous Painlevé equation and the first discrete Painlevé equation, as well as a family of differential equation associated with each discrete problem. This analysis reveals two important observations. Firstly, the orderly behaviour that characterises Stokes’ phenomenon in discrete equations emerges naturally from corresponding continuous differential equations as the order of the latter increases, although this is not apparent at low orders. Secondly, Stokes’ phenomenon vanishes in the continuum limit of the integrable discrete equation, but not the non-integrable discrete equation. This means that subdominant exponentials do not appear in the integrable equation, and therefore do not cause moveable singularities to form in the solution. The results are clarified further by consideration of one-parameter family of difference equations that interpolates between the two considered in detail.

We study [Formula: see text]-positive linear maps on matrix algebras and address two problems, (i) characterizations of [Formula: see text]-positivity and (ii) generation of nondecomposable [Formula: see text]-positive maps. On the characterization side, we derive optimization-based conditions equivalent to [Formula: see text]-positivity that (a) reduce to a simple check when [Formula: see text], (b) reveal a direct link to the spectral norm of certain order-3 tensors (aligning with known NP-hardness barriers for [Formula: see text]), and (c) recast [Formula: see text]-positivity as a novel optimization problem over separable states, thereby connecting it explicitly to separability testing. On the generation side, we introduce a Lie-semigroup-based method that, starting from a single [Formula: see text]-positive map, produces one-parameter families that remain [Formula: see text]-positive and nondecomposable for small enough times. We illustrate this by generating such families for [Formula: see text] and [Formula: see text]. We also formulate a semidefinite program (SDP) to test an equivalent form of the positive partial transpose (PPT) square conjecture (and do not find any violation of the latter). Our results provide practical computational tools for certifying [Formula: see text]-positivity and a systematic way to sample [Formula: see text]-positive nondecomposable maps.

Abstract We extend well-known results on the Newtonian limit of Lorentzian metrics to orthonormal frames. Concretely, we prove that, given a one-parameter family of Lorentzian metrics that in the Newtonian limit converges to a Galilei structure, any family of orthonormal frames for these metrics converges pointwise to a Galilei frame, assuming that the two obvious necessary conditions are satisfied: the spatial frame must not rotate indefinitely as the limit is approached, and the frame’s boost velocity with respect to some fixed reference observer needs to converge.

Abstract A Gordian unlink is a finite number of unknots that are not topologically linked, each with prescribed length and thickness, and that cannot be disentangled into the trivial link by an isotopy preserving length and thickness throughout. In this note, we provide the first examples of Gordian unlinks. As a consequence, we identify the existence of isotopy classes of unknots that differ from those in classical knot theory. More generally, we present a one-parameter family of Gordian unlinks with thickness ranging in $[1,2)$ and absolute curvature bounded by 1, concluding that thinner normal tubes lead to different rope geometries than those previously considered. Knots or links in the one-parameter model introduced here are called thin knots or links. When the thickness is equal to 2, we obtain the standard model for geometric knots, also called thick knots.

This work reveals a fundamental link between general covariance and Birkhoff's theorem. We extend Birkhoff's theorem from general relativity to a broad class of generally covariant gravity theories formulated in the Hamiltonian framework. Conversely, we show that each one-parameter family of static, spherically symmetric spacetimes determines a class of covariant theories, each of which has that family of spacetimes as its entire vacuum solution space. Our systematic and model-independent framework applies to a wide range of spacetimes, including observationally inferred, quantum-gravity-inspired, and regular black holes. It provides a universal tool for probing their dynamical origins and enables the reconstruction of the underlying covariant theories from observational data, including gravitational-wave and black-hole-shadow measurements.