A New Approach of Constrained Interpolation Based on Cubic Hermite Splines

Suppose we have a constrained set of data and wish to approximate it using a suitable function. It is natural to require the approximant to preserve the constraints. In this work, we state the problem in an interpolating setting and propose a parameter-based method and use the well-known cubic Hermite splines to interpolate the data with a constrained spline to provide with a C 1 interpolant. Then, more smoothing constraints are added to obtain C 2 continuity. Additionally, a minimization criterion is presented as a theoretical support to the proposed study; this is performed using linear programming. The proposed methods are demonstrated with illustrious examples.

Characterisation of homogeneous fractional Sobolev spaces

Our aim is to characterize the homogeneous fractional Sobolev–Slobodeckiĭ spaces $$\mathcal {D}^{s,p} (\mathbb {R}^n)$$ D s , p ( R n ) and their embeddings, for $$s \in (0,1]$$ s ∈ ( 0 , 1 ] and $$p\ge 1$$ p ≥ 1 . They are defined as the completion of the set of smooth and compactly supported test functions with respect to the Gagliardo–Slobodeckiĭ seminorms. For $$s\,p < n$$ s p < n or $$s = p = n = 1$$ s = p = n = 1 we show that $$\mathcal {D}^{s,p}(\mathbb {R}^n)$$ D s , p ( R n ) is isomorphic to a suitable function space, whereas for $$s\,p \ge n$$ s p ≥ n it is isomorphic to a space of equivalence classes of functions, differing by an additive constant. As one of our main tools, we present a Morrey–Campanato inequality where the Gagliardo–Slobodeckiĭ seminorm controls from above a suitable Campanato seminorm.

Quasipolynomial Computation of Nested Fixpoints

It is well-known that the winning region of a parity game with n nodes and k priorities can be computed as a k-nested fixpoint of a suitable function; straightforward computation of this nested fixpoint requires $$\mathcal {O}(n^{\frac{k}{2}})$$ O ( n k 2 ) iterations of the function. Calude et al.’s recent quasipolynomial-time parity game solving algorithm essentially shows how to compute the same fixpoint in only quasipolynomially many iterations by reducing parity games to quasipolynomially sized safety games. Universal graphs have been used to modularize this transformation of parity games to equivalent safety games that are obtained by combining the original game with a universal graph. We show that this approach naturally generalizes to the computation of solutions of systems of any fixpoint equations over finite lattices; hence, the solution of fixpoint equation systems can be computed by quasipolynomially many iterations of the equations. We present applications to modal fixpoint logics and games beyond relational semantics. For instance, the model checking problems for the energy $$\mu $$ μ -calculus, finite latticed $$\mu $$ μ -calculi, and the graded and the (two-valued) probabilistic $$\mu $$ μ -calculus – with numbers coded in binary – can be solved via nested fixpoints of functions that differ substantially from the function for parity games but still can be computed in quasipolynomial time; our result hence implies that model checking for these $$\mu $$ μ -calculi is in $$\textsc {QP}$$ QP . Moreover, we improve the exponent in known exponential bounds on satisfiability checking.

Regularized vortex approximation for 2D Euler equations with transport noise

We study a mean field approximation for the 2D Euler vorticity equation driven by a transport noise. We prove that the Euler equations can be approximated by interacting point vortices driven by a regularized Biot–Savart kernel and the same common noise. The approximation happens by sending the number of particles [Formula: see text] to infinity and the regularization [Formula: see text] in the Biot–Savart kernel to [Formula: see text], as a suitable function of [Formula: see text].

Failure rate properties of parallel systems

We study failure rate monotonicity and generalised convex transform stochastic ordering properties of random variables, with an emphasis on applications. We are especially interested in the effect of a tail-weight iteration procedure to define distributions, which is equivalent to the characterisation of moments of the residual lifetime at a given instant. For the monotonicity properties, we are mainly concerned with hereditary properties with respect to the iteration procedure providing counterexamples showing either that the hereditary property does not hold or that inverse implications are not true. For the stochastic ordering, we introduce a new criterion, based on the analysis of the sign variation of a suitable function. This criterion is then applied to prove ageing properties of parallel systems formed with components that have exponentially distributed lifetimes.

Spectral temperature of π− mesons in proton–carbon interactions at 4.2 GeV/c in the framework of UrQMD model

Transverse momentum [Formula: see text] spectra of [Formula: see text] mesons calculated using ultra-relativistic quantum molecular dynamic (UrQMD) model (Latest version 3.3-p2) simulations have been compared with [Formula: see text] spectra of [Formula: see text] mesons, obtained experimentally in interactions of protons beam with carbon nuclei (propane as target) at momentum of 4.2 GeV/c. Spectral temperatures of negative pions obtained in experimental and UrQMD model simulated interactions of protons beam with carbon nuclei have been calculated by fitting both spectra with four different fitting functions, i.e. Hagedorn thermodynamic, Boltzmann distribution, Gaussian and exponential functions. These functions are used commonly for describing hadron spectra and their spectral temperatures. Hagedorn thermodynamic function has been recommended as the most suitable function to extract the temperature of negative pions at above momentum among these four functions.

A Generalized Cubic Exponential B-Spline Scheme with Shape Control

In this paper, a generalized cubic exponential B-spline scheme is presented, which can generate different kinds of curves, including the conics. Such a scheme is obtained by generalizing the cubic exponential B-spline scheme based on an iteration from the generation of exponential polynomials and a suitable function with two parameters μ and ν. By changing the values of μ and ν, the sensitivity of the shape of the subdivision curve to the initial control value v0 can be changed and different kinds of curves can then be obtained by adjusting the value of v0. For this new scheme, we show that, with any admissible choice of μ and ν, it owns the same smoothness order and support as the cubic exponential B-spline scheme. Besides, based on a different iteration and another suitable function, we construct a similar nonstationary scheme to generate more curves with different shapes and show the role of iterations and suitably chosen functions in the construction and analysis of such schemes. Several examples are given to illustrate the performance of our new schemes.

Functions and the Aesthetics of Technical Artefacts

In this paper, it is examined to what extent functions, as analysed in the philosophy of technical artefacts, can serve a role in explaining the aesthetic appreciation of these objects. The main conclusion is that, despite first appearances, so-called ‘Functional Beauty’ accounts cannot derive strength from analyses of artefact functions; on the contrary, these analyses constrain the possibilities for developing a suitable, function-based account of aesthetic appreciation. The paper follows a conceptual-engineering approach. After presenting desiderata for an account of aesthetic appreciation, relevant insights are reviewed that are drawn from philosophical work on artefact functions. Combining the desiderata and insights, three major issues are identified that complicate the relation between function ascriptions to and aesthetic appreciation of technical artefacts. In closing, options are offered for resolving these complications or avoiding them altogether.

Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle

In this paper, we study optimal lower and upper bounds for functionals involving the first Dirichlet eigenvalue {\lambda_{F}(p,\Omega)} of the anisotropic p-Laplacian, {1<p<+\infty} . Our aim is to enhance, by means of the {\mathcal{P}} -function method, how it is possible to get several sharp estimates for {\lambda_{F}(p,\Omega)} in terms of several geometric quantities associated to the domain. The {\mathcal{P}} -function method is based on a maximum principle for a suitable function involving the eigenfunction and its gradient.

Estimates of Solutions to Nonlinear Evolution Equations

Consider the equation u’(t) = A (t, u (t)), u(0)= U0 ; u' := du/dt (1). Under some assumptions on the nonlinear operator A(t,u) it is proved that problem (1) has a unique global solution and this solution satisfies the following estimate ||u (t)|| < µ (t) -1 for every t belongs to R+ = [0,infinity). Here µ(t) > 0, µ belongs to C1 (R+), is a suitable function and the norm ||u || is the norm in a Banach space X with the property ||u (t) ||’ <= ||u’ (t) ||.