Solutions to homogeneous Monge–Ampère equations of homothetic functions and their applications to production models in economics

In economics, homothetic functions are production functions whose marginal technical rate of substitution is homogeneous of degree zero. Mathematically, a homothetic function is a function of the form f(x)=F(h(_x1,…,_xn))$f(\mathbf{x})=F(h(x_1,\dots ,x_n))$, where F is a monotonically increasing function and h   is a homogeneous function of any degree d≠0$d\ne 0$. In this paper, we classify homothetic functions satisfying the homogeneous Monge–Ampère equation. Several applications to production models in economics will also be given.     

Regularity in the obstacle problem for parabolic non-divergence operators of Hörmander type

In this paper we continue the study initiated in [15] concerning the obstacle problem for a class of parabolic non-divergence operators structured on a set of vector fields X={_X1,…,_Xq}$X=\{X_1,\dots ,X_q\}$ in Rⁿ${\mathbf{R}}^n$ with C^∞$C^{\infty }$-coefficients satisfying Hörmander?s finite rank condition, i.e., the rank of Lie[_X1,…,_Xq]$\mathrm{Lie}[X_1,\dots ,X_q]$ equals n   at every point in Rⁿ${\mathbf{R}}^n$. In [15] we proved, under appropriate assumptions on the operator and the obstacle, the existence and uniqueness of strong solutions to a general obstacle problem. The main result of this paper is that we establish further regularity, in the interior as well as at the initial state, of strong solutions. Compared to [15] we in this paper assume, in addition, that there exists a homogeneous Lie group G=(Rⁿ,°,_δλ)$\mathbf{G}=({\mathbf{R}}^n,°,{\delta }_{\lambda })$ such that _X1,…,_Xq$X_1,\dots ,X_q$ are left translation invariant on G and such that _X1,…,_Xq$X_1,\dots ,X_q$ are _δλ${\delta }_{\lambda }$-homogeneous of degree one.     

On fractal properties of non-normal numbers with respect to Rényi f-expansions generated by piecewise linear functions

The paper is devoted to the study of fractal properties of subsets of the set of non-normal numbers with respect to Rényi f  -expansions generated by continuous increasing piecewise linear functions defined on [0,+∞)$[0,+\infty )$. All such expansions are expansions for real numbers generated by infinite linear IFS f={_f0,_f1,…,_fn,…}$f=\{f_0,f_1,\dots ,f_n,\dots \}$ with the following list of ratios _Q∞=(_q0,_q1,…,_qn,…)$Q_{\infty }=(q_0,q_1,\dots ,q_n,\dots )$.     

Exact finite-difference schemes for two-dimensional linear systems with constant coefficients

An exact finite-difference scheme for a system of two linear differential equations with constant coefficients, (d/dt)x(t)=Ax(t)$(\mathrm{d}/\mathrm{d}t)\mathbf{x}(t)=A\mathbf{x}(t)$, is proposed. The scheme is different from what was proposed by Mickens [Nonstandard Finite Difference Models of Differential Equations, World Scientific, New Jersey, 1994, p. 147], in which the derivatives of the two equations are formed differently. Our exact scheme is in the form of (1/φ(h))(_xk+1-_xk)=A[θ_xk+1+(1-θ)_xk]$(1/\phi (h))({\mathbf{x}}_{k+1}-{\mathbf{x}}_k)=A[\theta {\mathbf{x}}_{k+1}+(1-\theta ){\mathbf{x}}_k]$; both derivatives are in the same form of (_xk+1-_xk)/φ(h)$(x_{k+1}-x_k)/\phi (h)$.     

S-Noetherian properties on amalgamated algebras along an ideal

Let A and B   be commutative rings with identity, f:A→B$f:A\to B$ a ring homomorphism and J an ideal of B  . Then the subring A?^fJ:={(a,f(a)+j)|a∈A and j∈J}$A{?}^{f}J:=\{(a,f(a)+j)|a\in A\text{and}j\in J\}$ of A×B$A\times B$ is called the amalgamation of A with B along with J with respect to f. In this paper, we investigate a general concept of the Noetherian property, called the S  -Noetherian property which was introduced by Anderson and Dumitrescu, on the ring A?^fJ$A{?}^{f}J$ for a multiplicative subset S   of A?^fJ$A{?}^{f}J$. As particular cases of the amalgamation, we also devote to study the transfers of the S  -Noetherian property to the constructions D+(_X1,…,_Xn)E[_X1,…,_Xn]$D+(X_1,\dots ,X_n)E[X_1,\dots ,X_n]$ and D+(_X1,…,_Xn)E?_X1,…,_Xn?$D+(X_1,\dots ,X_n)E?X_1,\dots ,X_n?$ and Nagata?s idealization.     

On characterized subgroups of compact abelian groups

Let X be a compact abelian group. A subgroup H of X   is called characterized if there exists a sequence u=(_un)$\mathbf{u}=(u_n)$ of characters of X   such that H=_su(X)$H=s_{\mathbf{u}}(X)$, where _su(X):={x∈X:(_un,x)→0 in T}$s_{\mathbf{u}}(X):=\{x\in X:(u_n,x)\to 0\text{in}\mathbb{T}\}$. Every characterized subgroup is an _Fσδ$F_{\sigma \delta }$-subgroup of X  . We show that every _Gδ$G_{\delta }$-subgroup of X is characterized. On the other hand, X   has non-characterized _Fσ$F_{\sigma }$-subgroups.     

On the convergence of hybrid polynomial approximation to higher derivatives of rational curves

In this paper, we derive the bounds on the magnitude of l  th (l=2,3)$(l=2,3)$ order derivatives of rational Bézier curves, estimate the error, in the _L∞$L_{\infty }$ norm sense, for the hybrid polynomial approximation of the l  th (l=1,2,3)$(l=1,2,3)$ order derivatives of rational Bézier curves. We then prove that when the hybrid polynomial approximation converges to a given rational Bézier curve, the l  th (l=1,2,3)$(l=1,2,3)$ derivatives of the hybrid polynomial approximation curve also uniformly converge to the corresponding derivatives of the rational curve. These results are useful for designing simpler algorithms for computing tangent vector, curvature vector and torsion vector of rational Bézier curves.     

Kolmogorov's problem on the class of multiply monotone functions

In this paper we give necessary and sufficient conditions for the system of positive numbers _Mk1,_Mk2,…,_Mkd$M_{k_1},M_{k_2},\dots ,M_{k_d}$, 0≤_k1<…<_kd≤r$0\le k_1<\dots <k_d\le r$, to guarantee the existence of an r  -monotone function defined on the negative half-line _R−${\mathbb{R}}_{-}$ and such that _{‖x(ki)‖∞}=_Mki${\Vert x^{(k_i)}\Vert }_{\infty }=M_{k_i}$, i=1,2,…,d$i=1,2,\dots ,d$. We also discuss some applications of the obtained results and connections with other problems.     

From intervals to domains: Towards a general description of validated uncertainty,with potential applications to geospatial and meteorological data

When physical quantities _xi$x_i$ are numbers, then the corresponding measurement accuracy can be usually represented in interval terms, and interval computations can be used to estimate the resulting uncertainty in y=f(_x1,…,_xn)$y=f(x_1,\dots ,x_n)$.     

Inequalities with exponential weights

Let R=(-∞,∞)$\mathbb{R}=(-\infty ,\infty )$ and let Q∈C²:R→R⁺=[0,∞)$Q\in C^2:\mathbb{R}\to {\mathbb{R}}^{+}=[0,\infty )$ be an even function. Then in this paper we consider the infinite–finite range inequality, an estimate for the Christoffel function, and the Markov–Bernstein inequality with the exponential weights _wρ(x)=|x|^ρe^-Q(x),x∈R$w_{\rho }(x)=|x{|}^{\rho }{\mathrm{e}}^{-Q(x)},x\in \mathbb{R}$.     

Asymptotics of solutions to the generalized Ostrovsky equation

We consider the Cauchy problem for the generalized Ostrovsky equation

_utx=u+_(f(u))xx,$u_{tx}=u+{(f(u))}_{xx},$

where f(u)=|u|^ρ−1u$f(u)={|u|}^{\rho -1}u$ if ρ   is not an integer and f(u)=u^ρ$f(u)=u^{\rho }$ if ρ   is an integer. We obtain the L^∞${\mathbf{L}}^{\infty }$ time decay estimates and the large time asymptotics of small solutions under suitable conditions on the initial data and the order of the nonlinearity.     

Reduced order models for random functions. Application to stochastic problems

A method is developed for constructing reduced order models for arbitrary random functions. The reduced order models are simple random functions, that is, functions with a finite range (_x1,…,_xm)$({\mathit{x}}_1\text{,}\dots \text{,}{\mathit{x}}_m)$. The construction of the reduced order models involves two steps. First, a range (_x1,…,_xm)$({\mathit{x}}_1\text{,}\dots \text{,}{\mathit{x}}_m)$ is selected based on somewhat heuristic arguments. Second, the probabilities (_p1,…,_pm)$(p_1\text{,}\dots \text{,}{p}_m)$ of (_x1,…,_xm)$({\mathit{x}}_1\text{,}\dots \text{,}{\mathit{x}}_m)$ are obtained from the solution of an optimization problem. Reduced order models are applied to calculate the distributions of the modal frequencies of a linear dynamic system with random stiffness matrix and statistics of the hydraulic head in a soil deposit with random heterogeneous conductivity. The performance of reduced order models in both applications is remarkable.     

Frequency-dependent interpolation rules using first derivatives for oscillatory functions

We construct frequency-dependent rules to interpolate oscillatory functions y(x)$y(x)$ with frequency ω$\omega$ of the form,

y(x)=_f1(x)cos(ωx)+_f2(x)sin(ωx),$y(x)=f_1(x)\cos (\omega x)+f_2(x)\sin (\omega x)\text{,}$

On oriented graphs whose skew spectral radii do not exceed 2

Let S(G^σ)$S(G^{\sigma })$ be the skew adjacency matrix of the oriented graph G^σ$G^{\sigma }$ of order n   and _λ1,_λ2,…,_λn${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ be all eigenvalues of S(G^σ)$S(G^{\sigma })$. The skew spectral radius _ρs(G^σ)${\rho }_s(G^{\sigma })$ of G^σ$G^{\sigma }$ is defined as max{|_λ1|,|_λ2|,…,|_λn|}$\max \{|{\lambda }_1|,|{\lambda }_2|,\dots ,|{\lambda }_n|\}$. In this paper, we investigate oriented graphs whose skew spectral radii do not exceed 2.     

Generic residual intersections and intersection numbers of movable components

Given a sequence x of elements of a commutative equidimensional noetherian ring R  , cycles _zi(x,R)$z_i(\mathbf{x},R)$ (i∈N$i\in \mathbb{N}$) in the cycle group of polynomial rings over R are defined by generic residual intersections. The study of these cycles gives new insight into the theory for excess intersections in projective space developed by Stückrad and Vogel, in particular concerning the contribution to the intersection cycle of embedded components not defined over the base field.