A novel method for mathematical modeling of a form-cutting-tool of the optimum design

A method that pertains to design of the optimal form-cutting-tool for machining of a given sculptured surface on a multi-axis NC   machine is discussed in the paper. The results reported in the paper are based in much on the author’s previous work in the field of R$\mathbb{R}$-mapping of a sculptured surface onto the machining surface of cutting tools. Mathematical foundations of a novel method of experimental modeling of the interaction of the form-cutting-tool and the work are disclosed. The last method is of critical importance for the experimental determination of the rate of conformity functions, those essential for consequent use of R$\mathbb{R}$-mapping of surfaces. The presented geometric criteria is helpful for designers since the maximal rate of conformity of the generating surface of the form-cutting-tool to the sculptured surface is a prerequisite for the development of extremely efficient machining operations, and in solving design problems. The results of the research reported in this paper can be considered as a portion of the DG/K-method of surface generation on a multi-axis NC machine earlier developed ¹ by the author. The method is based on the extensive use of reliable results worked out in classical differential geometry of surfaces. Topics covered in the paper enables one designing the form-cutting-tool for optimal machining of a given sculptured surface on a multi-axis NC machine. The usefulness of the approach is verified from two simple examples that are clear and easy for understanding.     

Finite Dimensional de Branges Spaces on Riemann Surfaces

We study certain finite dimensional reproducing kernel indefinite inner product spaces of multiplicative half order differentials on a compact real Riemann surface; these spaces are analogues of the spaces introduced by L. de Branges when the Riemann sphere is replaced by a compact real Riemann surface of a higher genus. In de Branges theory an important role is played by resolvent-like difference quotient operators Rα; here we introduce generalized difference quotient operators Ryα for any non-constant meromorphic function y on the Riemann surface. The spaces we study are invariant under generalized difference quotient operators and can be characterized as finite dimensional indefinite inner product spaces invariant under two operators Ry1αi and Ry2α2, where y1 and y2 generate the field of meromorphic functions on the Riemann surface, which satisfy a supplementary identity, analogous to the de Branges identity for difference quotients. Just as the classical de Branges spaces and difference quotient operators appear in the operator model theory for a single nonselfadjoint (or nonunitary) operator, the spaces we consider and generalized difference quotient operators appear in the model theory for commuting nonselfadjoint operators with finite nonhermitian ranks.     

Relations between the half turns of the hyperbolic plane

Let R_a denote the half turn about the point a of the hyperbolic plane H. If the points a, b, c, d lie on the same line and the pair (c, d) is obtained from the pair (a, b) by a translation, then we have R_aR_b = R_cR_d. We study the group G whose generating set is {R_a:a∈H} and whose defining relations are the ones mentioned above together with the relations R²_a = 1. We show that G can be made into a Lie group, G has two connected components, and its identity component G₀ is the universal covering group of PSL₂(R). In particular, it follows that all relations between the half turns in PSL₂(R) follow from the abovementioned relations and a single additional relation of length five.     

Non-linear response of an electrode–electrolyte interface impedance with the frequency

In this paper we analyze the frequency response of a distributed parameters non-linear circuit. This circuit is a fractal model of an electrode–electrolyte interface. The results show that the series equivalent resistance R_eq decreases as the number of fractal levels rises. Its frequency behavior is similar to the results described by other authors that used a constant phase angle element. At low frequency range, the series equivalent reactance X_eq decreases with frequency and R_eq shows a flat response in this range. The model also explains how interface geometry modifies the values of some electrochemical parameters. As roughness increases the Argand diagram shows smaller semicircles. The model behavior is like a single dispersion system with its central frequency increasing with the electrode roughness.     

Morphic group rings

An element a in a ring R is called left morphic if there exists b∈R such that l_R(a)=Rb and l_R(b)=Ra, where l_R(a) denotes the left annihilator of a in R. The ring R is called left morphic if every element of R is left morphic. Left morphic rings have been studied by Nicholson and Sánchez Campos. In this paper, the question of when a group ring is left morphic is discussed in great detail and various morphic group rings are identified.     

Constraint programming for computing non-stationary (R, S) inventory policies

This paper proposes a constraint programming model for computing the finite horizon single-item inventory problem with stochastic demands in discrete time periods with service-level constraints under the non-stationary version of the “periodic review, order-up-to-level” policy (i.e., non-stationary (R, S) or, simply (R_n, S_n)). It is observed that the modeling process is more natural and the required number of variables is smaller compared to the MIP formulation of the same problem. The computational tests show that the CP approach is more tractable than the conventional MIP formulation. Two different domain reduction methods are proposed to improve the computational performance of solution algorithms. The numerical experiments confirmed the effectiveness of these methods.     

Order-constrained uniform approximations to the exponential based on restricted rationals

LetR _n/m(z∶γ)=P _n(z∶γ)/(1?γz) ^m be a rational approximation to exp (z),z ∈C, of ordern for all real positiveγ. In this paper we show there exists exactly one value ofγ in each of min(n+1,m) interpolation intervals such that the uniform error overR ^? is at a local minimum.     

Numerical estimates for a Grötzsch ring constant

Numerical lower and upper estimates are obtained for the constant λ_n defined by λn =lim _a→0(modR _G,n (a)+loga) associated with the Grötzsch extremal ringR _G,n (a) in euclideann-space, for 3≤n≤22. Improved lower estimates in terms ofn are provided for λ_n, the modulus ofR _G,n (a) is compared with its counterpart in the plane, and bounds for modR _G,n (a) are obtained that are of the correct order asa tends either to 0 or to 1. The ratio of the latter bounds is bounded by constants depending only onn.     

An infinite color analogue of Rado's theorem

Let R be a subring of the complex numbers and a be a cardinal. A system L of linear homogeneous equations with coefficients in R is called a-regular over R if, for every a-coloring of the nonzero elements of R, there is a monochromatic solution to L in distinct variables. In 1943, Rado classified those finite systems of linear homogeneous equations that are a-regular over R for all positive integers a. For every infinite cardinal a, we classify those finite systems of linear homogeneous equations that are a-regular over R. As a corollary, for every positive integer s, we have ℵ₀²>ℵs if and only if the equation x₀+sx₁=x₂+?+xs+2 is ℵ₀-regular over R. This generalizes the case s=1 due to Erd?s.     

On oscillation of solutions of a nonautonomous quasilinear second-order equation

Sufficient conditions are obtained for the initial values of nontrivial oscillating (for t=ω) solutions of the nonautonomous quasilinear equation $$y'' \pm \lambda (t)y = F(t,y,y'),$$ wheret ∈ Δ=[a, ω[,-∞ <a < ω ≤+ ∞, λ(t) > 0, λ(t) ∈ C _Δ ⁽¹⁾ , |F((t,x,y))|≤L(t)(|x|+|y|)^1+α, L(t) ≥-0, α ∈ [0,+∞[, F: Δ × R² →R,F ∈C _Δ×R ²,R is the set of real numbers, and R² is the two-dimensional real Euclidean space.     

The average quality of greedy-algorithms for the Subset-Sum-Maximization Problem

This paper deals with the quality of approximative solutions for the Subset-Sum-Maximization-Problem maximize $$\sum\limits_{i = l}^n {a_i x_i } $$ subject to $$\sum\limits_{i = l}^n {a_i x_i } \leqslant b$$ wherea _l,...,a_n,bεR⁺ andx _l,...x_nε{0,1}. produced by certain heuristics of a Greedy-type. Every heuristic under consideration realizes a feasible solution (x ₁, ..., x_n) whose objective value is less or equal the optimal value, which is of course not greater thanb. We use the gap between capacityb and realized value as an upper bound for the error made by the heuristic and as a criterion for quality. Under the stochastic model:a ₁, ..., a_n, b independent,a ₁...,a_n uniformly distributed on [0, 1], b uniformly distributed on [0,n] we derive the gap-distributions and the expected size of the gaps. The analyzed algorithms include four algorithms which can be done in linear time and four heuristics which require sorting, which means that they are done inO(nlnn) time.     

On the monotonicity properties of additive representation functions, II

For a set A of positive integers and any positive integer n, let R₁(A,n), R₂(A,n) and R₃(A,n) denote the number of solutions of a+a^′=n with the additional restriction a,a^′∈A; a,a^′∈A,a<a^′ and a,a^′∈A,a≤a^′ respectively. In this paper, we specially focus on the monotonicity of R₃(A,n). Moreover, we show that there does not exist any set A⊂N such that R₂(A,n) or R₃(A,n) is eventually strictly increasing.     

The fundamental constituents of iteration digraphs of finite commutative rings

For a finite commutative ring R and a positive integer k ? 2, we construct an iteration digraph G(R, k) whose vertex set is R and for which there is a directed edge from a ∈ R to b ∈ R if b = a ^k . Let R = R ₁ ⊕ … ⊕ R _s , where s > 1 and R _i is a finite commutative local ring for i ∈ {1, …, s}. Let N be a subset of {R ₁, …, R _s } (it is possible that N is the empty set \(\not 0\) ). We define the fundamental constituents G _N ^* (R, k) of G(R, k) induced by the vertices which are of the form {(a ₁, …, a _s ) ∈ R: a _i ∈ D(R _i ) if R _i ∈ N, otherwise a _i ∈ U(R _i ), i = 1, …, s}, where U(R) denotes the unit group of R and D(R) denotes the zero-divisor set of R. We investigate the structure of G* _N (R, k) and state some conditions for the trees attached to cycle vertices in distinct fundamental constituents to be isomorphic.     

Partitions of natural numbers with the same representation functions

For a set A of nonnegative integers the representation functions R₂(A,n), R₃(A,n) are defined as the number of solutions of the equation n=a+a^′,a,a^′∈A with a<a^′, a?a^′, respectively. Let D(0)=0 and let D(a) denote the number of ones in the binary representation of a. Let A₀ be the set of all nonnegative integers a with even D(a) and A₁ be the set of all nonnegative integers a with odd D(a). In this paper we show that (a) if R₂(A,n)=R₂(N?A,n) for all n?2N−1, then R₂(A,n)=R₂(N?A,n)?1 for all n?12N²−10N−2 except for A=A₀ or A=A₁; (b) if R₃(A,n)=R₃(N?A,n) for all n?2N−1, then R₃(A,n)=R₃(N?A,n)?1 for all n?12N²+2N. Several problems are posed in this paper.     

An upper bound for Cubicity in terms of Boxicity

An axis-parallel b-dimensional box is a Cartesian product R₁×R₂×?×R_b where each R_i (for 1≤i≤b) is a closed interval of the form [a_i,b_i] on the real line. The boxicity of any graph G, is the minimum positive integer b such that G can be represented as the intersection graph of axis-parallel b-dimensional boxes. A b-dimensional cube is a Cartesian product R₁×R₂×?×R_b, where each R_i (for 1≤i≤b) is a closed interval of the form [a_i,a_i+1] on the real line. When the boxes are restricted to be axis-parallel cubes in b-dimension, the minimum dimension b required to represent the graph is called the cubicity of the graph (denoted by ). In this paper we prove that , where n is the number of vertices in the graph. We also show that this upper bound is tight.Some immediate consequences of the above result are listed below:

1.

Planar graphs have cubicity at most 3⌈log₂n⌉.

2.

Outer planar graphs have cubicity at most 2⌈log₂n⌉.

3.

Any graph of treewidth tw has cubicity at most (tw+2)⌈log₂n⌉. Thus, chordal graphs have cubicity at most (ω+1)⌈log₂n⌉ and circular arc graphs have cubicity at most (2ω+1)⌈log₂n⌉, where ω is the clique number.

The above upper bounds are tight, but for small constant factors.     

On nonparametric change point estimator based on empirical characteristic functions

We propose a nonparametric change point estimator in the distributions of a sequence of independent observations in terms of the test statistics given by Huˇskov′a and Meintanis(2006) that are based on weighted empirical characteristic functions. The weight function ω(t; a) under consideration includes the two weight functions from Huˇskov′a and Meintanis(2006) plus the weight function used by Matteson and James(2014),where a is a tuning parameter. Under the local alternative hypothesis, we establish the consistency, convergence rate, and asymptotic distribution of this change point estimator which is the maxima of a two-side Brownian motion with a drift. Since the performance of the change point estimator depends on a in use, we thus propose an algorithm for choosing an appropriate value of a, denoted by a_s which is also justified. Our simulation study shows that the change point estimate obtained by using a_s has a satisfactory performance. We also apply our method to a real dataset.     

The hardness of approximating the boxicity, cubicity and threshold dimension of a graph

A k-dimensional box is the Cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line. The boxicity of a graph G, denoted as , is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the Cartesian product R₁×R₂×?×R_k where each R_i is a closed interval on the real line of the form [a_i,a_i+1]. The cubicity of G, denoted as , is the minimum integer k such that G can be represented as the intersection graph of a collection of k-cubes. The threshold dimension of a graph G(V,E) is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. In this paper we will show that there exists no polynomial-time algorithm for approximating the threshold dimension of a graph on n vertices with a factor of O(n^0.5−?) for any ?>0 unless NP=ZPP. From this result we will show that there exists no polynomial-time algorithm for approximating the boxicity and the cubicity of a graph on n vertices with factor O(n^0.5−?) for any ?>0 unless NP=ZPP. In fact all these hardness results hold even for a highly structured class of graphs, namely the split graphs. We will also show that it is NP-complete to determine whether a given split graph has boxicity at most 3.     

Semiprime Goldie centralizers

LetG be a finite group of automorphisms acting on a ringR, andR ^G={fixed points ofG}. We show that under certain conditions onR andG, whenR ^Gis semiprime Goldie then so isR. In particular, ifa∈R is invertible anda ⁿ∈Z(R), thenR ^G,withG generated by the inner automorphism determined bya, is the centralizer ofa—C _R(a). The above result withR ^Greplaced byC _R(a) is shown without the assumption thata is invertible.     

A new relation between molar sound velocity and molar volume in carboxy acid solutions

A new relationship between molar sound velocity ‘R ’ and molar volume (M/ρ) in carboxy acid solutions has been established. Using the composite ultrasonic interferometer, both monocarboxylic and dicarboxylic acid solutions in common solvents like methyl alcohol, carbontetrachloride have been studied and it has been established that a linear relationshipR =a ₀ +a ₁ · M/ρ holds good in all solutions of homologous series of acids in a common solvent at a particular temperature. The constantsa ₀ anda ₁ for a number of carboxy acid solutions have been evaluated.     

Points entiers de certains schémas de matrices

Let R be an Euclidean domain, M _n (R) the R-module of matrices with n rows and n columns, V an intersection of n?1 affine hyperplanes of M _n (R). Assume V is not empty. Then, for all a in R, there exists a matrix in V with determinant equal to a.