text:
Journal of Applied Mathematics
Journal of Applied Mathematics/
AbstractIntroductionReferencesCopyrightRelated Articles
About Nodal Systems for Lagrange Interpolation on the Circle
E. Berriochoa,1A. Cachafeiro,2and J. M. García Amor2
Academic Editor: Nicola Guglielmi
Accepted12 Oct 2011
We study the convergence of the Laurent polynomials of Lagrange interpolation on the unit circle for continuous functions satisfying a condition about their modulus of continuity. The novelty of the result is that now the nodal systems are more general than those constituted by the n roots of complex unimodular numbers and the class of functions is different from the usually studied. Moreover, some consequences for the Lagrange interpolation on [−1,1] and the Lagrange trigonometric interpolation are obtained.
The aim of this paper is to study the Lagrange interpolation problem on the unit circle 𝕋∶={𝑧∶|𝑧|=1} for nodal systems more general than those constituted by the 𝑛 roots of complex unimodular numbers. This last case has been studied in [1], where there is posed as an open problem its extension to more general nodal systems. Recently a similar problem has been solved in [2] for the Hermite interpolation problem. Now we follow the ideas in [2] to obtain some results for the Lagrange case. Moreover, in [1] there is obtained a result about convergence of the interpolants for continuous functions satisfying a condition related to their modulus of continuity. In the present paper our aim is to obtain a similar result for the new nodal systems and with a different condition on the modulus of continuity for the functions.
The Lagrange interpolation problem on the real line has been widely studied for a long time and many results about convergence are known (see [3–6]). If we only assume the continuity of the function, it is well known that the behavior is rather irregular. Faber has proved that for each nodal system there exists a continuous function such that the sequence of Lagrange interpolation polynomials is not uniformly convergent. Bernstein has also proved the existence of a continuous function such that the sequence of Lagrange interpolation polynomials is unbounded on a prefixed point. In the case of the nodal systems constituted by the zeros of the Tchebychef polynomials of the first kind, many results are known. Although these last nodal systems are good for interpolation, Grünwald in [7] and Marcinkiewicz in [8] have proved the existence of a continuous function such that the sequence of Lagrange interpolation polynomials, corresponding to the Tchebychef nodal system, is divergent. After this result a natural problem was to obtain an analogous result for an arbitrary nodal system. This result was obtained by Erdös and Vértesi in [9], where they prove that for each nodal system on [−1,1] there exists a continuous function such that the sequence of Lagrange interpolation polynomials diverges for almost every point in [−1,1]. Thus, to obtain better properties about the convergence of the sequence of Lagrange interpolation polynomials, it is needed to impose some restriction on the function, such as a condition on its modulus of continuity. In the case of Jacobi abscissas, Szegő has obtained important results about convergence by imposing some conditions to the modulus of continuity of the function (see [10]). For example, in the case of the Tchebychef abscissas of first kind, he obtained the uniform convergence to the function on [−1,1], under the assumption that its modulus of continuity is 𝑜(|log𝛿|−1). Szegő has also obtained uniform convergence of the sequence of Lagrange interpolation polynomials for more general nodal systems, under the assumptions that the nodes are the zeros of the orthogonal polynomials with respect to a weight function 𝑤(𝑥) such that √𝑤(𝑥)1−𝑥2≥𝜇>0,𝑥∈(−1,1) and the modulus of continuity of the functions is 𝑜(𝛿1/2) with 𝛿→0.
In the present paper we improve some results about convergence of the Lagrange interpolation polynomials in [−1,1], by using the Szegő transformation and the results concerning the unit circle. The organization of the paper is the following. In Section 2 we obtain our main result concerning the uniform convergence of the Laurent polynomial of Lagrange interpolation for nodal systems described in terms of some properties and for continuous functions with modulus of continuity 𝑜(𝛿𝑝) when 𝛿→0 and 𝑝≥1/2. Section 3 is devoted to obtain some consequences of the preceding results concerning the Lagrange interpolation on [−1,1]. Finally, in the last section, we obtain some improvements concerning the Lagrange trigonometric interpolation.
2. Lagrange Interpolation in the Space of Laurent Polynomials
Let {𝑧𝑗}𝑛𝑗=1 be a set of complex numbers such that |𝑧𝑗|=1 for all 𝑗=1,…,𝑛 and 𝑧𝑖≠𝑧𝑗 for 𝑖≠𝑗. Let {𝑢𝑗}𝑛𝑗=1 be a set of arbitrary complex numbers, and let 𝑝(𝑛) and 𝑞(𝑛) be two nondecreasing sequences of nonnegative integers such that 𝑝(𝑛)+𝑞(𝑛)=𝑛−1,𝑛≥2 with lim𝑛→∞𝑝(𝑛)=lim𝑛→∞𝑞(𝑛)=∞.
We recall that the Lagrange interpolation problem in the space of Laurent polynomials consists in determining the unique Laurent polynomial 𝐿−𝑝(𝑛),𝑞(𝑛)(𝑧)∈Λ−𝑝(𝑛),𝑞(𝑛)=span{𝑧𝑘∶−𝑝(𝑛)≤𝑘≤𝑞(𝑛)} such that𝐿−𝑝(𝑛),𝑞(𝑛)𝑧𝑗=𝑢𝑗,for𝑗=1,…,𝑛.(2.1)
If we denote by 𝑊𝑛∏(𝑧)=𝑛𝑗=1(𝑧−𝑧𝑗) the nodal polynomial, then 𝐿−𝑝(𝑛),𝑞(𝑛)(𝑧) can be written as follows: 𝐿−𝑝(𝑛),𝑞(𝑛)(𝑧)=𝑛𝑗=1𝑙𝑗,𝑛−1(𝑧)𝑢𝑗,(2.2) where 𝑙𝑗,𝑛−1(𝑧) are the fundamental polynomials of Lagrange interpolation given by 𝑙𝑗,𝑛−1(𝑧𝑧)=𝑗𝑝(𝑛)𝑊𝑛(𝑧)𝑊′𝑛𝑧𝑗𝑧−𝑧𝑗𝑧𝑝(𝑛),for𝑗=1,…,𝑛,(2.3) and they are characterized by satisfying 𝑙𝑗,𝑛−1(𝑧𝑘)=𝛿𝑗,𝑘, for all 𝑗,𝑘.
We are also going to consider the Lagrange interpolation polynomial for a function 𝐹 defined on 𝕋, that we are going to denote by 𝐿−𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧) and which is characterized by fulfilling the conditions 𝐿−𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧𝑗)=𝐹(𝑧𝑗) for 𝑗=1,…,𝑛.
When the nodal system is constituted by the 𝑛-roots of a complex number with modulus 1, and the function 𝐹 is continuous on 𝕋, and its modulus of continuity satisfies 𝜆(𝐹,𝛿)=𝒪(𝛿𝑝), 𝑝>1/2, the following result about convergence is known (see [1]).
Theorem 2.1. Let 𝐹 be a continuous function on 𝕋, let 𝑝(𝑛) and 𝑞(𝑛) be two nondecreasing sequences of nonnegative integers such that 𝑝(𝑛)+𝑞(𝑛)=𝑛−1 and lim𝑛→∞𝑝(𝑛)/(𝑛−1)=𝑟 with 0<𝑟<1, and assume that the modulus of continuity of 𝐹 is 𝜆(𝐹,𝛿)=𝒪(𝛿𝑝) for some 𝑝>1/2, if 𝛿→0.
Let 𝐿−𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧) be the Laurent polynomial of Lagrange interpolation for the function 𝐹 with nodal system {𝑧𝑗}𝑛𝑗=1 being the 𝑛-roots of complex numbers 𝜏𝑛 with |𝜏𝑛|=1.
Then lim𝑛→∞𝐿−𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.
Proof. See [1].
The main tools to prove the preceding result are the explicit expression of the Laurent polynomial of Lagrange interpolation and some properties concerning the nodal system. In [2] the Hermite interpolation problem was studied for general nodal systems satisfying certain properties. Following similar ideas we prove, in the next theorem, a result about the convergence of the Lagrange interpolants for a different class of functions and more general nodal systems.
Theorem 2.2. Let 𝐹 be a continuous function on 𝕋, with modulus of continuity 𝜆(𝐹,𝛿)=𝑜(𝛿1/2), if 𝛿→0. Let 𝑝(𝑛) and 𝑞(𝑛) be two nondecreasing sequences of nonnegative integers such that 𝑝(𝑛)+𝑞(𝑛)=𝑛−1 and lim𝑛→∞𝑝(𝑛)/(𝑛−1)=𝑟 with 0<𝑟<1.
Let {𝑧𝑗}𝑛𝑗=1 be a set of complex numbers such that |𝑧𝑗|=1 for all 𝑗=1,…,𝑛 and 𝑧𝑖≠𝑧𝑗 for 𝑖≠𝑗 and let 𝑊𝑛(𝑧)=Π𝑛𝑗=1(𝑧−𝑧𝑗) be the nodal polynomial. Assume that there exist positive constants 𝐵 and 𝐿 such that for every 𝑧∈𝕋 and 𝑛 large enough the following relations hold:
(i)𝐵≤|𝑊′𝑛(𝑧)|/𝑛, (ii)|𝑊𝑛(𝑧)|2/𝑛2∑𝑛𝑗=1(1/|𝑧−𝑧𝑗|2)≤𝐿.
If 𝐿−𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧)∈Λ−𝑝(𝑛),𝑞(𝑛) is the Laurent polynomial of Lagrange interpolation related to the nodal system and the function 𝐹, then lim𝑛→∞𝐿−𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.
Proof. First we prove that there exists a positive constant 𝐶 such that ∑𝑛𝑗=1|𝑙𝑗,𝑛−1√(𝑧)|≤𝐶𝑛 for every 𝑧∈𝕋 and 𝑛 large enough. Indeed, taking into account (2.3) and applying the hypothesis we get 𝑛𝑗=1||𝑙𝑗,𝑛−1(||=𝑧)𝑛𝑗=1||𝑊𝑛(𝑧)𝑧𝑗𝑝(𝑛)||||𝑊𝑛𝑧𝑗𝑧−𝑧𝑗𝑧𝑝(𝑛)||=𝑛𝑗=1||𝑊𝑛||(𝑧)||𝑊𝑛𝑧𝑗𝑧−𝑧𝑗||≤1𝐵𝑛𝑛𝑗=1||𝑊𝑛||(𝑧)||𝑧−𝑧𝑗||≤1𝐵𝑛𝑛𝑗=1||𝑊𝑛||(𝑧)2||𝑧−𝑧𝑗||21/2𝑛𝑗=111/2≤√𝐿𝐵√𝑛.(2.4) Let us consider the Laurent polynomial of best uniform approximation to 𝐹, 𝑇−𝑝(𝑛),𝑞(𝑛)(𝑧)∈Λ−𝑝(𝑛),𝑞(𝑛). If 𝐸−𝑝(𝑛),𝑞(𝑛)(𝐹)=max𝑧∈𝕋|𝐹(𝑧)−𝑇−𝑝(𝑛),𝑞(𝑛)(𝑧)|, then it holds that 𝐸−𝑝(𝑛),𝑞(𝑛)𝜋(𝐹)≤2𝜆𝐹,𝑠(𝑛),(2.5) where 𝑠(𝑛)=min(𝑝(𝑛),𝑞(𝑛)) (see [1]). Since lim𝑛→∞𝜋/𝑠(𝑛)=0, then by hypothesis 𝜆(𝐹,𝜋/𝑠(𝑛))=𝑜((𝜋/𝑠(𝑛))1/2).
If we write𝐹(𝑧)−𝐿−𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧)=𝐹(𝑧)−𝑇−𝑝(𝑛),𝑞(𝑛)(𝑧)−𝐿−𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧)+𝑇−𝑝(𝑛),𝑞(𝑛)(𝑧)=𝐹(𝑧)−𝑇−𝑝(𝑛),𝑞(𝑛)(𝑧)−𝐿−𝑝(𝑛),𝑞(𝑛)𝐹−𝑇−𝑝(𝑛),𝑞(𝑛);𝑧=𝐹(𝑧)−𝑇−𝑝(𝑛),𝑞(𝑛)(𝑧)−𝑛𝑗=1𝑙𝑗,𝑛−1𝐹𝑧(𝑧)𝑗−𝑇−𝑝(𝑛),𝑞(𝑛)𝑧𝑗,(2.6) then we have ||𝐹(𝑧)−𝐿−𝑝(𝑛),𝑞(𝑛)||≤||(𝐹;𝑧)𝐹(𝑧)−𝑇−𝑝(𝑛),𝑞(𝑛)||+(𝑧)𝑛𝑗=1||𝑙𝑗,𝑛−1||||𝐹𝑧(𝑧)𝑗−𝑇−𝑝(𝑛),𝑞(𝑛)𝑧𝑗||≤𝐸−𝑝(𝑛),𝑞(𝑛)(𝐹)1+𝑛𝑗=1||𝑙𝑗,𝑛−1||√(𝑧)≤2𝜆(𝐹,𝜋/𝑠(𝑛))1+𝐶𝑛=2𝜆(𝐹,𝜋/𝑠(𝑛))(𝜋/𝑠(𝑛))1/2√𝜋(𝑠(𝑛)/(𝑛−1))1/2√1+𝐶𝑛√,𝑛−1(2.7) and it is easy to prove that the last expression tends to zero because lim𝑛→∞𝜆(𝐹,𝜋/𝑠(𝑛))/(𝜋/𝑠(𝑛))1/2=0 and lim𝑛→∞𝑠(𝑛)=1𝑛−12lim𝑛→∞𝑝(𝑛)𝑛−1+lim𝑛→∞𝑞(𝑛)𝑛−1−lim𝑛→∞||||𝑝(𝑛)−𝑞(𝑛)=⎧⎪⎨⎪⎩1𝑛−11−𝑟,if𝑟∈2,1,1𝑟,if𝑟∈0,2.(2.8)
Remark 2.3. (i) Since 𝜆(𝐹,𝛿)=𝑜(𝛿𝑝) for 𝑝>1/2 implies 𝜆(𝐹,𝛿)=𝑜(𝛿1/2), then the preceding result is also valid for functions with modulus of continuity 𝑜(𝛿𝑝), with 𝑝>1/2, if 𝛿→0. Hence, in the sequel and for simplicity, we establish all the results with the condition 𝜆(𝐹,𝛿)=𝑜(𝛿1/2).
(ii) Since it is clear that the nodal systems in Theorem 2.1 satisfy the hypothesis of Theorem 2.2, we have that the result given in Theorem 2.1 is also valid for functions with modulus of continuity 𝑜(𝛿1/2), if 𝛿→0.
Next we recall a sufficient condition given in [2] in order that the nodal system satisfies the conditions imposed in the previous theorem. We use the so-called para-orthogonal polynomials (see [11–13]) and the class of measures satisfying the Szegő condition, (see [10, 13–15]). Notice that the nodal systems in Theorem 2.1 are constituted by the 𝑛 roots of complex unimodular numbers, and indeed they are the 𝑛 roots of the para-orthogonal polynomials with respect to the Lebesgue measure on [0,2𝜋].
Theorem 2.4. Let 𝜈 be a measure on [0,2𝜋] in the Szegő class with Szegő function having analytic extension up to |𝑧|>1. Let {𝜙𝑛(𝑧)} be the monic orthogonal polynomial sequence with respect to 𝜈, MOPS(𝜈), and 𝜔𝑛(𝑧,𝜏)=𝜙𝑛(𝑧)+𝜏𝜙∗𝑛(𝑧), with |𝜏|=1 being the para-orthogonal polynomials. Then there exist positive constants 𝐴,𝐵1,𝐵2, and 𝐿 such that for every 𝑧∈𝕋 and 𝑛 large enough the following relations hold: (i)|𝜔𝑛(𝑧,𝜏)|≤𝐴, (ii)𝐵1≤|𝜔′𝑛(𝑧,𝜏)|/𝑛≤𝐵2, (iii)|𝜔𝑛(𝑧,𝜏)|2/𝑛2∑𝑛𝑗=1(1/|𝑧−𝑧𝑗|2)≤𝐿, where one assumes that 𝑧1,…,𝑧𝑛 are the zeros of 𝜔𝑛(𝑧,𝜏).
Taking into account the preceding results, we are in conditions to prove the following corollary.
Corollary 2.5. Let 𝐹 be a continuous function on 𝕋, with modulus of continuity 𝜆(𝐹,𝛿)=𝑜(𝛿1/2), if 𝛿→0. Let 𝑝(𝑛) and 𝑞(𝑛) be two nondecreasing sequences of nonnegative integers such that 𝑝(𝑛)+𝑞(𝑛)=𝑛−1 and lim𝑛→∞𝑝(𝑛)/(𝑛−1)=𝑟 with 0<𝑟<1.
Let 𝜈 be a measure on [0,2𝜋] in the Szegő class with Szegő function having analytic extension up to |𝑧|>1. Let {𝜙𝑛(𝑧)} be the MOPS(𝜈) and let 𝜔𝑛(𝑧,𝜏)=𝜙𝑛(𝑧)+𝜏𝜙∗𝑛(𝑧), with |𝜏|=1, be the para-orthogonal polynomials.
If 𝐿−𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧)∈Λ−𝑝(𝑛),𝑞(𝑛) is the Laurent polynomial of Lagrange interpolation related to the function 𝐹 and with nodal system the zeros of the para-orthogonal polynomials 𝜔𝑛(𝑧,𝜏), then lim𝑛→∞𝐿−𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.
Proof. Taking into account that the zeros of 𝜔𝑛(𝑧,𝜏) belong to 𝕋 (see [11]), the result is immediate from Theorems 2.2 and 2.4.
Remark 2.6. Notice that the preceding result is valid for the Bernstein-Szegő measures (see [13]).
3. Lagrange Interpolation on [−1,1]
In this section we present some consequences of Theorem 2.2 concerning the Lagrange interpolation problems on [−1,1]. Let us recall that the Lagrange interpolation polynomial related to a nodal system {𝑥𝑗}𝑛𝑗=1⊂[−1,1] and satisfying the conditions {𝑢𝑗}𝑛𝑗=1 is given by 𝑙𝑛−1∑(𝑥)=𝑛𝑗=1(𝑝𝑛(𝑥)/(𝑝𝑛(𝑥𝑗)(𝑥−𝑥𝑗)))𝑢𝑗, where 𝑝𝑛(𝑥)=Π𝑛𝑗=1(𝑥−𝑥𝑗).
Theorem 3.1. Let 𝑝𝑛∏(𝑥)=𝑛𝑗=1(𝑥−𝑥𝑗) be a nodal system in [−1,1] such that 𝑊2𝑛(𝑧)=2𝑛𝑧𝑛𝑝𝑛((𝑧+1/𝑧)/2) satisfies the following inequalities ||𝑊𝐵≤2𝑛||(𝑧),||𝑊2𝑛2𝑛||(𝑧)2𝑛𝑗=11||𝑧−𝑧𝑗||2+1||𝑧−𝑧𝑗||2≤𝐿(2𝑛)2,(3.1) with (𝑧𝑗+(1/𝑧𝑗))/2=𝑥𝑗 for 𝑗=1,…,𝑛 and for some positive constants 𝐵 and 𝐿, 𝑛 large enough and every 𝑧∈𝕋.
Let 𝑓 be a continuous function on [−1,1] such that 𝜆(𝑓,𝛿)=𝑜(𝛿1/2), if 𝛿→0.
If 𝑙𝑛−1(𝑓,𝑥) is the Lagrange interpolation polynomial such that 𝑙𝑛−1(𝑓,𝑥𝑗)=𝑓(𝑥𝑗) for 𝑗=1,…,𝑛, then 𝑙𝑛−1(𝑓,𝑥) converges to 𝑓(𝑥) uniformly on [−1,1].
Proof. It is easy to see that the polynomial 𝑊2𝑛(𝑧) has the following expression 𝑊2𝑛(𝑧)=Π𝑛𝑗=1(𝑧−𝑧𝑗)(𝑧−𝑧𝑗), with (𝑧𝑗+𝑧𝑗)/2=𝑥𝑗.
Let us define a continuous function on 𝕋 by 𝐹(𝑧)=𝐹(𝑧)=𝑓(𝑥), with 𝑥=(𝑧+(1/𝑧))/2 and 𝑧∈𝕋. It is clear that𝜆(𝐹,𝛿)=sup𝑧1,𝑧2∈𝕋;|𝑧1−𝑧2|<𝛿||𝐹𝑧1𝑧−𝐹2||≤sup𝑥1,𝑥2∈[−1,1];|𝑥1−𝑥2|<𝛿||𝑓𝑥1𝑥−𝑓2||=𝜆(𝑓,𝛿).(3.2)
If we take 𝑊2𝑛(𝑧) as nodal system on 𝕋, we can consider the following Lagrange interpolation problem: find the Laurent polynomial of Lagrange interpolation 𝐿−𝑛,𝑛−1(𝐹;𝑧)∈Λ−𝑛,𝑛−1 satisfying the interpolation conditions𝐿−𝑛,𝑛−1𝐹;𝑧𝑗=𝐿−𝑛,𝑛−1𝐹;𝑧𝑗𝑥=𝑓𝑗,𝑗=1,…,𝑛.(3.3) By applying Theorem 2.2 we have that lim𝑛→∞𝐿−𝑛,𝑛−1(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.
On the other hand, for 𝑥=(𝑧+(1/𝑧))/2 and 𝑧∈𝕋 it holds𝐿−𝑛,𝑛−1(𝐹;𝑧)=𝑛𝑗=1𝑊2𝑛(𝑧)𝑧𝑛𝑗𝑧𝑛𝑊2𝑛𝑧𝑗𝑧−𝑧𝑗𝐹𝑧𝑗+𝑛𝑗=1𝑊2𝑛(𝑧)𝑧𝑗𝑛𝑧𝑛𝑊2𝑛𝑧𝑗𝑧−𝑧𝑗𝐹𝑧𝑗=𝑛𝑗=1𝑝𝑛(𝑥)𝑝𝑛𝑥𝑗𝑥−𝑥𝑗𝑓𝑥𝑗=𝑙𝑛−1(𝑓;𝑥).(3.4) Hence lim𝑛→∞𝑙𝑛−1(𝑓,𝑥)=𝑓(𝑥) uniformly on [−1,1].
As a consequence we obtain, in the next corollary, a result that was proved by Szegő in [10] under weaker conditions. Although our result is not new, we give the proof because the way in which it is obtained is different from Szegő's proof.
Corollary 3.2. Let 𝑓 be a continuous function on [−1,1] such that 𝜆(𝑓,𝛿)=o(𝛿1/2), if 𝛿→0. Let 𝑑𝜇(𝑥)=𝑤(𝑥)𝑑𝑥 be a finite positive Borel measure on [−1,1] satisfying the Szegő condition ∫1−1√(log𝑤(𝑥)/1−𝑥2)𝑑𝑥>−∞, and let {𝑃𝑛(𝑥)} be the MOPS(𝜇). Assume that the function √𝑤(𝑥)1−𝑥2 is positive on [−1,1] and it is analytic in an open set containing [−1,1].
If 𝑙𝑛−1(𝑓,𝑥) is the Lagrange interpolation polynomial satisfying the interpolation conditions 𝑙𝑛−1(𝑓,𝑥𝑗)=𝑓(𝑥𝑗),𝑗=1,…,𝑛, where {𝑥𝑗}𝑛𝑗=1 are the zeros of the orthogonal polynomial 𝑃𝑛(𝑥), thenlim𝑛→∞𝑙𝑛−1(𝑓,𝑥)=𝑓(𝑥)(3.5) uniformly on [−1,1].
Proof. By using the Szegő transformation (see [10]), the measure 𝑑𝜇(𝑥) becomes into the measure 𝑑𝜈(𝜃)=(1/2)𝑤(cos𝜃)|sin𝜃|𝑑𝜃, which is in the Szegő class with Szegő function having analytic extension up to |𝑧|>1 (see [14]). If we denote by {𝜙𝑛(𝑧)} the MOPS(𝜈) and by {𝑃𝑛(𝑧)} the MOPS(𝜇), then both sequences are related by𝑃𝑛1(𝑥)=2𝑛1+𝜙2𝑛(𝜙0)2𝑛(𝑧)+𝜙∗2𝑛(𝑧)𝑧𝑛=12𝑛1+𝜙2𝑛(𝜔0)2𝑛(𝑧,1)𝑧𝑛.(3.6)
The zeros of 𝑃𝑛(𝑥), 𝑥1,…,𝑥𝑛, are simple and belong to (−1,1) and they are related to the zeros of 𝜔2𝑛(𝑧,1), 𝑧1,…,𝑧𝑛,𝑧𝑛+1=𝑧𝑛,…,𝑧2𝑛=𝑧1, by 𝑥𝑗=(𝑧𝑗+𝑧𝑗)/2,𝑗=1,…,𝑛. By applying Theorem 2.4 we get that the system 𝜔2𝑛(𝑧,1) satisfies the hypothesis of Theorem 3.1. Then we have that 𝑙𝑛−1(𝑓,𝑥) converges to 𝑓(𝑥) uniformly on [−1,1].
Analogous results can be obtained for other nodal systems related to those given in Corollary 3.2. Let 𝑑𝜇1(𝑥)=𝑤(𝑥)𝑑𝑥 be a finite positive Borel measure on [−1,1]. Let us consider the measures𝑑𝜇2(𝑥)=1−𝑥2𝑑𝜇1(𝑥),𝑑𝜇3(𝑥)=(1−𝑥)𝑑𝜇1(𝑥),𝑑𝜇4(𝑥)=(1+𝑥)𝑑𝜇1(𝑥),(3.7) and let us denote the MOPS with respect to these measures by {𝑃𝑛(𝑥,𝜇𝑖)}4𝑖=1. Let us consider the Szegő transformed measure of 𝑑𝜇1(𝑥), 𝑑𝜈(𝜃)=(1/2)𝑤(cos𝜃)|sin𝜃|𝑑𝜃 with MOPS(𝜈), {𝜙𝑛(𝑧)}. Taking into account the relation between the measures, we can relate the orthogonal sequences as follows (see [16]):𝑃𝑛𝑥,𝜇1=12𝑛1+𝜙2𝑛𝑧(0)𝑛𝑤2𝑛√(𝑧,1),1−𝑥2𝑃𝑛𝑥,𝜇2=12𝑛+1𝚤1−𝜙2𝑛+2𝑧(0)𝑛+1𝑤2𝑛+2√(𝑧,−1),1−𝑥𝑃𝑛𝑥,𝜇3=12𝑛+1/2𝚤1−𝜙2𝑛+1𝑧(0)𝑛+(1/2)𝑤2𝑛+1√(𝑧,−1),1+𝑥𝑃𝑛𝑥,𝜇4=12𝑛+1/21+𝜙2𝑛+1𝑧(0)𝑛+(1/2)𝑤2𝑛+1(𝑧,1).(3.8)
We denote by ±1,𝑥1,…,𝑥𝑛 the zeros of √1−𝑥2𝑃𝑛(𝑥,𝜇2), by 1,𝑦1,…,𝑦𝑛 the zeros of √1−𝑥𝑃𝑛(𝑥,𝜇3), and by −1,𝑣1,…,𝑣𝑛 the zeros of √1+𝑥𝑃𝑛(𝑥,𝜇4).
If we denote by ±1,𝑧1,…,𝑧𝑛,𝑧1,…,𝑧𝑛 the zeros of 𝜔2𝑛+2(𝑧,−1), by 1,𝑤1,…,𝑤𝑛,𝑤1,…,𝑤𝑛 the zeros of 𝑤2𝑛+1(𝑧,−1), and by −1,𝑢1,…,𝑢𝑛,𝑢1,…,𝑢𝑛 the zeros of 𝑤2𝑛+1(𝑧,1), then the following relations hold: ℜ(𝑧𝑖)=𝑥𝑖,ℜ(𝑤𝑖)=𝑦𝑖, and ℜ(𝑢𝑖)=𝑣𝑖,𝑖=1,…,𝑛. By taking nodal systems related to the zeros of 𝑃𝑛(𝑥,𝜇𝑖), 𝑖=2,3,4, we obtain the next result.
Theorem 3.3. Let 𝑓 be a continuous function on [−1,1] such that 𝜆(𝑓,𝛿)=o(𝛿1/2), if 𝛿→0. Let 𝜇1 be a finite positive Borel measure on [−1,1], 𝑑𝜇1(𝑥)=𝑤(𝑥)d𝑥, satisfying the Szegő condition. Assume that the function √𝑤(𝑥)1−𝑥2 is positive in [−1,1] and it is analytic in an open set containing [−1,1]. Let 𝑑𝜇𝑖(𝑥),𝑖=2,3,4, be the measures given in (3.7). Let us consider the Lagrange interpolation polynomials for the function 𝑓 with the following nodal systems: (i)the zeros of 𝑃𝑛(𝑥,𝜇2) joint with ±1,(ii)the zeros of 𝑃𝑛(𝑥,𝜇3) joint with 1,(iii)the zeros of 𝑃𝑛(𝑥,𝜇4) joint with −1.Then the corresponding Lagrange interpolation polynomials uniformly converge to 𝑓(𝑥) on [−1,1].
Proof. By the Szegő transformation the measure 𝑑𝜇1(𝑥) becomes into the measure 𝑑𝜈(𝜃)=(1/2)𝑤(cos𝜃)|sin𝜃|𝑑𝜃, which is in the Szegő class with Szegő function having analytic extension up to |𝑧|>1. We denote by {𝜙𝑛(𝑧)} the MOPS(𝜈). If we define a continuous function 𝐹 on 𝕋 by 𝐹(𝑧)=𝐹(𝑧)=𝑓(𝑥) with 𝑥=(𝑧+1/𝑧)/2 and 𝑧∈𝕋, then it is clear that 𝜆(𝐹,𝛿)≤𝜆(𝑓,𝛿).
(i) We consider the para-orthogonal polynomial 𝜔2𝑛+2(𝑧,−1), whose zeros are ±1,𝑧1,…,𝑧𝑛, 𝑧1,…,𝑧𝑛∈𝕋 and they are related to the zeros of 𝑃𝑛(𝑥,𝜇2) by 𝑥𝑗=(𝑧𝑗+𝑧𝑗)/2;𝑗=1,…,𝑛.
Let us consider the following Lagrange interpolation problem: find the Laurent polynomial of Lagrange interpolation 𝐿−(𝑛+1),𝑛(𝐹;𝑧) satisfying 𝐿−(𝑛+1),𝑛𝐹;𝑧𝑗=𝐿−(𝑛+1),𝑛𝐹;𝑧𝑗𝑧=𝐹𝑗𝐿,𝑗=1,…,𝑛,−(𝑛+1),𝑛(𝐹;1)=𝐹(1),𝐿−(𝑛+1),𝑛(𝐹;−1)=𝐹(−1).(3.9) By applying Corollary 2.5 we have that lim𝑛→∞𝐿−(𝑛+1),𝑛(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋. If we take 𝑙𝑛+1𝐿(𝑓,𝑥)=−(𝑛+1),𝑛(𝐹;𝑧)+𝐿−(𝑛+1),𝑛(𝐹;1/𝑧)2(3.10) for 𝑥=(𝑧+(1/𝑧))/2, then 𝑙𝑛+1(𝑓,𝑥) fulfills 𝑙𝑛+1(𝑓,𝑥𝑗)=𝑓(𝑥𝑗),𝑗=1,…,𝑛, and 𝑙𝑛+1(𝑓,±1)=𝑓(±1). Therefore, 𝑙𝑛+1(𝑓,𝑥) is the Lagrange interpolation polynomial for the function 𝑓 and the nodal system given in (i) and lim𝑛→∞𝑙𝑛+1(𝑓,𝑥)=𝑓(𝑥) uniformly on [−1,1].
(ii) We consider the para-orthogonal polynomials 𝜔2𝑛+1(𝑧,−1) whose zeros, 1,𝑤1,…,𝑤𝑛,𝑤1,…,𝑤𝑛, are related to the zeros of 𝑃𝑛(𝑥,𝜇3) by 𝑦𝑗=(𝑤𝑗+𝑤𝑗)/2,𝑗=1,…,𝑛.
We pose the problem of finding the Laurent polynomial of Lagrange interpolation 𝐿−𝑛,𝑛(𝐹;𝑧) satisfying 𝐿−𝑛,𝑛𝐹;𝑤𝑗=𝐿−𝑛,𝑛𝐹;𝑤𝑗𝑤=𝐹𝑗𝐿,𝑗=1,…,𝑛,−𝑛,𝑛(𝐹;1)=𝐹(1).(3.11) Since lim𝑛→∞𝐿−𝑛,𝑛(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋, if we define 𝑙𝑛(𝑓,𝑥)=(𝐿−𝑛,𝑛(𝐹;𝑧)+𝐿−𝑛,𝑛(𝐹;1/𝑧))/2 for 𝑥=(𝑧+(1/𝑧))/2 and 𝑧∈𝕋, then 𝑙𝑛(𝑓,𝑥) fulfills 𝑙𝑛(𝑓,𝑥𝑗)=𝑓(𝑥𝑗),𝑗=1,…,𝑛, and 𝑙𝑛(𝑓,1)=𝑓(1). Therefore, lim𝑛→∞𝑙𝑛(𝑓,𝑥)=𝑓(𝑥) uniformly on [−1,1].
(iii) It is obtained proceeding the same way as in the previous items.
Remark 3.4. (i) In particular, the preceding result is valid for the following nodal systems: the zeros of the Tchebychef polynomials of the second kind joint with ±1, the zeros of the Tchebychef polynomials of the third kind joint with 1, and the zeros of the Tchebychef polynomials of the fourth kind joint with −1.
(ii) Moreover it is also valid for the polynomial modifications, by positive polynomials, of the Bernstein measures corresponding to the Tchebychef measures mentioned before.
4. Trigonometric Interpolation
Next we obtain some consequences of Theorem 2.2, which are related to the Lagrange trigonometric interpolation. Now the nodal points are in [0,2𝜋] and they are obtained as follows. Let 𝑑𝜇(𝑥)=𝑤(𝑥)𝑑𝑥 be a positive finite Borel measure on [−1,1] satisfying the Szegő condition. Assume that the function √𝑤(𝑥)1−𝑥2>0 for all 𝑥∈[−1,1] and it is analytic in an open set containing [−1,1]. If {𝑃𝑛(𝑥)} is the MOPS(𝜇) and {𝑥𝑗}𝑛𝑗=1 are the zeros of 𝑃𝑛(𝑥), we consider the following nodal system on [0,2𝜋],{𝜃𝑗}2𝑛𝑗=1, such that 𝜃𝑗=arccos𝑥𝑗,𝑗=1,…,𝑛 with 0<𝜃𝑗<𝜋 and 𝜃𝑛+𝑗=2𝜋−𝜃𝑛−𝑗+1 for 𝑗=1,…,𝑛; that is, the points are symmetric with respect to 𝜋.
Theorem 4.1. Let 𝑓 be a real continuous function on [0,2𝜋], with modulus of continuity 𝜆(𝑓,𝛿)=𝑜(𝛿1/2), if 𝛿→0.
Let the nodal system be {𝜃𝑗}2𝑛𝑗=1 with 𝜃𝑗=arccos𝑥𝑗,𝑗=1,…,𝑛 with 0<𝜃𝑗<𝜋 and 𝜃𝑛+𝑗=2𝜋−𝜃𝑛−𝑗+1 for 𝑗=1,…,𝑛, where {𝑥𝑗}𝑛𝑗=1 are the zeros of the orthogonal polynomial 𝑃𝑛(𝑥) with respect to the measure 𝑑𝜇. One also assumes that 𝑑𝜇(𝑥)=𝑤(𝑥)𝑑𝑥 is a positive finite Borel measure on [−1,1] satisfying the Szegő condition and such that √𝑤(𝑥)1−𝑥2>0 for all 𝑥∈[−1,1] and it is analytic in an open set containing [−1,1].
Then there is a Lagrange interpolation trigonometric polynomial of degree ≤𝑛, 𝜏𝑛(𝜃), such that 𝜏𝑛(𝜃𝑗)=𝑓(𝜃𝑗) for 𝑗=1,…,2𝑛 and it satisfies that lim𝑛→∞𝜏𝑛(𝜃)=𝑓(𝜃) uniformly on [0,2𝜋].
Proof. Proceeding like in Corollary 3.2 we obtain that the transformed measure of 𝑑𝜇(𝑥) by the Szegő transformation, 𝑑𝜈(𝜃), satisfies the hypothesis of Theorem 2.4 and the para-orthogonal polynomials satisfy the bound condition of Theorem 2.4.
Let us define 𝐹 by 𝐹(𝑒𝚤𝜃)=𝑓(𝜃) for 𝜃∈[0,2𝜋]. Let 𝐿−𝑛,𝑛−1(𝐹,𝑧) be the Lagrange interpolation polynomial such that 𝐿−𝑛,𝑛−1(𝐹,𝑧𝑗)=𝑓(𝜃𝑗), where 𝑧𝑗=𝑒𝚤𝜃𝑗,𝑗=1,…,2𝑛.
Since 𝐹 is continuous on 𝕋 and 𝜆(𝐹,𝛿)=𝑜(𝛿1/2), we can apply Corollary 3.2 and therefore lim𝑛→∞𝐿−𝑛,𝑛−1(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.
If we take 𝜏𝑛(𝜃)=ℜ(𝐿−𝑛,𝑛−1(𝐹;𝑒𝚤𝜃)), then 𝜏𝑛(𝜃) satisfies the interpolation conditions, 𝜏𝑛(𝜃𝑗)=𝑓(𝜃𝑗), and lim𝑛→∞𝜏𝑛(𝜃)=𝑓(𝜃).
In the next result we denote the integer part of 𝑥 by [𝑥] and we consider another type of nodal system on [0,2𝜋].
Theorem 4.2. Let 𝜈 be a measure on [0,2𝜋] in the Szegő class with Szegő function having analytic extension up to |𝑧|>1. Let {𝑧𝑗}𝑛𝑗=1 be the zeros of the para-orthogonal polynomials 𝜔𝑛(𝑧)=𝜙𝑛(𝑧)+𝜏𝜙∗𝑛(𝑧), with |𝜏|=1, and let 𝜃𝑗∈[0,2𝜋] such that 𝑒𝚤𝜃𝑗=𝑧𝑗,𝑗=1,…,𝑛.
If 𝑓 is a continuous function on [0,2𝜋] with 𝜆(𝑓,𝛿)=𝑜(𝛿1/2), if 𝛿→0, then there is a Lagrange interpolation trigonometric polynomial of degree ≤[𝑛/2], 𝜏[𝑛/2](𝜃), such that 𝜏[𝑛/2](𝜃𝑗)=𝑓(𝜃𝑗) for 𝑗=1,…,𝑛 and it satisfies that lim𝑛→∞𝜏[𝑛/2](𝜃)=𝑓(𝜃) uniformly on [0,2𝜋].
Proof. Let 𝐹 be a continuous function defined by 𝐹(𝑒𝚤𝜃)=𝑓(𝜃). Since 𝜆(𝐹,𝛿)≤𝜆(𝑓,𝛿), then 𝜆(𝐹,𝛿)=𝑜(𝛿1/2). By applying Corollary 2.5 we obtain for 𝑝(𝑛)+𝑞(𝑛)=𝑛−1, with lim𝑛→∞𝑝(𝑛)/(𝑛−1)=𝑟 and 0<𝑟<1, there exists 𝐿−𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧) such that 𝐿−𝑝(𝑛),𝑞(𝑛)(𝐹;𝑒𝚤𝜃𝑗)=𝑓(𝜃𝑗) and lim𝑛→∞𝐿−𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧)=𝐹(𝑧) uniformly on 𝕋.
We distinguish the two following cases.
(i)If 𝑛 is even, we take 𝑝(𝑛)=𝑛/2 and 𝑞(𝑛)=𝑛/2−1. Then [𝑛/2]=𝑛/2.(ii)If 𝑛 is odd, we take 𝑝(𝑛)=(𝑛−1)/2 and 𝑞(𝑛)=(𝑛−1)/2. Then [𝑛/2]=(𝑛−1)/2.In any case the real part of 𝐿−𝑝(𝑛),𝑞(𝑛)(𝐹;𝑧) is a trigonometric polynomial of degree [𝑛/2], that satisfies the interpolation conditions and the convergence property.
The research was supported by Ministerio de Ciencia e Innovación under grant number MTM2011-22713.
L. Daruis and P. González-Vera, "Some results about interpolation with nodes on the unit circle," Indian Journal of Pure and Applied Mathematics, vol. 31, no. 10, pp. 1273–1296, 2000.
View at: Google Scholar | Zentralblatt MATH
E. Berriochoa, A. Cachafeiro, and E. Martĺnez, "About measures and nodal systems for which the Hermite interpolants uniformly converge to continuous functions on the circle and interval," Applied Mathematics and Computation, vol. 218, no. 9, pp. 4813–4824, 2012.
J.-P. Berrut and L. N. Trefethen, "Barycentric Lagrange interpolation," SIAM Review, vol. 46, no. 3, pp. 501–517, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
T. J. Rivlin, The Chebyshev Polynomials, Wiley-Interscience, New York, NY, USA, 1974.
J. Szabados and P. Vértesi, Interpolation of Functions, World Scientific Publishing, Teaneck, NJ, USA, 1990.
P. Turán, "On some open problems of approximation theory," Journal of Approximation Theory, vol. 29, no. 1, pp. 23–85, 1980.
G. Grünwald, "On the theory of interpolation," Acta Mathematica, vol. 75, pp. 219–245, 1943.
J. Marcinkiewicz, "Sur la divergence des polynmes d'interpolation," Acta Scientiarum Mathematicarum, vol. 8, pp. 131–135, 1937.
P. Erdös and P. Vértesi, "On the almost everywhere divergence of Lagrange interpolation," in Approximation and Function Spaces (Gdańsk, 1979), pp. 270–278, North-Holland, Amsterdam, The Netherlands, 1981.
G. Szegő, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, vol. 23, American Mathematical Society, Providence, RI, USA, 4th edition, 1975.
W. B. Jones, O. Njåstad, and W. J. Thron, "Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle," The Bulletin of the London Mathematical Society, vol. 21, no. 2, pp. 113–152, 1989.
L. Golinskii, "Quadrature formula and zeros of para-orthogonal polynomials on the unit circle," Acta Mathematica Hungarica, vol. 96, no. 3, pp. 169–186, 2002.
B. Simon, Orthogonal Polynomials on the Unit Circle, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, USA, 2005.
P. Nevai and V. Totik, "Orthogonal polynomials and their zeros," Acta Scientiarum Mathematicarum, vol. 53, no. 1-2, pp. 99–104, 1989.
L. Ya. Geronimus, Orthogonal Polynomials: Estimates, Asymptotic Formulas, and Series of Polynomials Orthogonal on the Unit Circle and on an Interval, Authorized Translation from the Russian, Consultants Bureau, New York, NY, USA, 1961.
J. M. García Amor, Ortogonalidad Bernstein-Chebyshev en la recta real, Doctoral Dissertation, Universidad de Vigo, 2003.
Copyright © 2012 E. Berriochoa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A 630,000 sq ft logistics project on green belt land, brought forward by Eddie Stobart Logistics and developer Liberty Properties, has been backed by Warrington Council planners.
The scheme, designed by architect RGP and supported by planner GVA How, will add to Eddie Stobart's existing facilities just off Junction 9 of the M56, which currently covers 145,000 sq ft of logistics space.
It features a warehouse of around 600,000 sq ft along with circa 30,000 sq ft of office and ancillary space.
According to Eddie Stobart, the development is set to add 480 permanent jobs at the site, as well as supporting 250 off-site employment opportunities. The development features around 465 car parking spaces and more than 120 trailer parking bays.
Although situated on the green belt, Warrington Council planners have supported the application ahead of next week's development management committee.
There have been objections from local councillors, including Cllrs Sharon Harris, Judith Wheeler; and Gerry Palmer. All three argued the land's green belt status should be protected, while also highlighting the additional traffic impact of the development.
However, the planners' report said Stobart had demonstrated 'very special circumstances' to allow the development to go ahead; these include the creation of more than 650 jobs, and Stobart's local spend of more than £2m per year on businesses within the borough.
In total, the development is expected to add around £18m to area's GVA, and the construction phase is also expected to create around 240 jobs. The project is set to complete in early 2020.
The scheme is recommended for approval at the meeting on 12 November subject to a Section 106 agreement; this is to include contributions of £2.1m towards public transport, public footpath, and road improvements.
This should not be built on Green Belt Land. There are plenty of other places this could be sited. No exceptions. As usual, council planners don't have a clue.
Green belt land should never be built on!
They want to start paying drivers wages correctly first!!!
Another one with very special circumstances, that turn out to be not so special after all. Same in St Helens with the build at Florida Farm North, 4 strands to the VSC all of which have now been shown to be a farce – just hoodwinking the council such is their desperation. No difference here, should be adopting a brownfield first policy. Too much money involved.
HS2 is cutting straight through the green belt in Warrington, which is being re-assessed and some parts potentially being re-assigned anyway (grass must be greener on the other side). Surely everybody knows that when it suits councils and government it happens no matter what, always has and always will. If it creates jobs and the £2.1m is spent on roads etc then what real harm has it done. Sorry but the same people complaining about this will no doubt complain there aren't enough jobs too.
An absolute disgrace if the Council back this one, the traffic in that area is already dreadful and whilst I'm sure it will help employment it's green belt land.


Find out what areas are affected by power cuts. Powerco is urging drivers of large loads to take more safety precautions around power lines Electricity . Find out …... Power lines, extremely low frequency magnetic fields and health effects. Current research has mainly concerned magnetic fields, not only from high voltage power lines from the electricity supply in general.
How To Find Out About Someone - Our site provides best quality background check information, you will get access to the personal information, police records, contact information and more. We must understand, however, that background checks are not only done on the borrowers.
Information on powerline safety. For those working in close proximity to overhead and underground powerlines, info on requirements and exclusion zones. For those working in close proximity to overhead and underground powerlines, info on requirements and exclusion zones.
Crews are currently investigating the cause of the out 12 Jan 2019 MOUNT HOPE, COBAR, TILPA areas - We currently have over 2700 customers without supply.


text:
Global Outreach Conference Sermons
Category Ministry Update
by Sarah Boghosian November 12, 2013
The Global Outreach Conference 2013 may have just ended, but you can listen to two of the wonderful preachers from the conference any time you want.
On Sunday, November 3, Dr. Giotis Kantartzis brought us the powerful message "In Athens" based on Acts 17:16-34. Dr. Kantartzis pastors the First Greek Evangelical Church, an historic Presbyterian church in Athens, Greece. He holds a M.Div. from Gordon Conwell Theological Seminary; a MST from Boston University, School of Theology; and a Ph.D. from the Divinity School of the University of Thessaloniki. Through the 2012 Easter Sacrificial Offering, Tenth partnered with his church in its outreach to immigrants.
Sunday morning and evening, November 10, Rev. Norman MacKay preached the series, "How shall we sing the Lord's song in a strange land?". In the morning he addressed "Part 1: Jesus as Founder" based on Hebrews 12:1-3. In the evening he addressed "Part 2: Jesus as Visionary" expounding upon Hebrews 10:1-10. Rev. MacKay served with WEC International as a pioneer missionary in the Central Asian Republics of the former USSR. He recently was appointed as a church planter with the Free Church of Scotland, and Tenth supports this work as part of our UK Partnership.
Whether you've heard all three sermons already or you're hearing about them for the first time, I encourage you make time to listen to these messages from two of God's faithful servants.
© 2023 Tenth Presbyterian Church.
Permissions: You are permitted and encouraged to reproduce and distribute this material in its entirety or in unaltered excerpts, as long as you do not charge a fee. For Internet posting, please use only unaltered excerpts (not the content in its entirety) and provide a hyperlink to this page, or embed the entire material hosted on Tenth channels. You may not re-upload the material in its entirety. Any exceptions to the above must be approved by Tenth Presbyterian Church.
Please include the following statement on any distributed copy: By Sarah Boghosian. © 2023 Tenth Presbyterian Church. Website: tenth.org
Sarah Boghosian
Sarah Boghosian is Website & Social Media Manager for Tenth.org. Read more
Show Hospitality By John P
Scripture: Hebrews 13:1-16
Back to the Future By Chris Andino
Series: 1 Peter
Scripture: 1 Peter 1:10-12
And His Courts with Praise By Jonathan Cruse
Scripture: Exodus 27:1-19
Browse By Index:
Year Index
Ministry Index

redpajama_set_name: RedPajamaCommonCrawl


Yes, it can be stressful to determine what you'll need and what you can definitely live without. And for those of us who enjoy elaborate hair and makeup routines, choosing just a few essential tools to bring with us is especially difficult.
The key to eliminating pre-vacation stress is to plan it all out. Choose the exact outfits you will bring, right down to the shoes, and then plan a makeup look and hairstyle to complement it. The more overlap you can achieve between the shoes, eyeshadow and lipstick colors, and hairstyle components, the fewer things you will need to bring with you.
Also, think about what events you plan to attend or the activities you plan to do. If you're going to the beach or to a festival, you'll need to pack sunscreen formulated for both your skin and your hair. On the other hand, if you'll primarily be inside, you probably won't need these things (or at least not such a large bottle).
What Amenities Will Be Available at Your Accommodation?
Another factor to consider is whether your hotel, rental, B&B or other accommodation will have any supplies available for your use. For example, it's unlikely that a hotel would have curling irons and flat irons in each room, but most do have a hair dryer at your disposal. Of course, this won't be your top rated hair dryer complete with temperature settings and multiple attachment heads. However, it will get the job done.
However, at an upscale B&B or a rental vacation home, you might have more tools to choose from. Find out ahead of time whether you need to pack some of your styling equipment. Knowing that your top-of-the-line curling iron won't get damaged in transit will be a load off your mind.
Yes, we all have multiple mascaras, foundations, cleansers, toners, concealers and highlighters, not to mention a variety of palettes for everything from eyeshadow to rouge to lip liner and lipstick. But will you use all of these? No. Will you even have room in your bags for all of them? Probably not. Remember, you can let go of some anxiety ("But what if I want/need it when I get there?") by meticulously planning your entire look from head to toe for each day of your vacation.
For all of your non-colorful beauty products, like your foundation and mascara, bring just one bottle or variety. For the colorful products, bring either your favorites or your most comprehensive palette. This will help avoid a mish-mash of products that you never end up using while on your vacation.
Finally, remember that you can always go to a convenience store to pick up anything you might have forgotten. Relax and enjoy your time off, opting for simple styles if need be to keep things fun and lighthearted.


These materials were built for a German auto racer based in Atlanta. He wanted his vehicle competing in a battle of tug-of-war with a Mustang… and his winning, of course. We also ended up constructing a clean and elegant logo to use alongside his other brand materials.
This was the final logo Antonio selected. It features a simple filled outline of an inverted V-oriented engine, and is a solid color throughout except for the outline of the pulley on the right, which is a linear gradient that fades from solid at 100% opacity to solid at 0% opacity on both sides. The logotype is set in Proxima Nova.


We celebrate daily and weekend Masses on our online service and we invite you to become a part of it. Hear the Gospel, reflect on the daily homily and worship God in the Eucharist. The Sacrifice of the Mass is the source and summit of our Catholic faith and is the center of our mission. We welcome you to share these videos with friends, family and parish members.
You may click on the play icon in the middle of the player to the right to see the current broadcast. If no broadcasts are active, click the button called "previous broadcasts" to view the archive of our recent live broadcasts right in this player.
Father Vogel's homilies are also available as a podcast. See below to listen to this week's homily, or follow the links to the right to subscribe to YouTube and watch the homily from the original webcast recording.
Fr. Kucera discusses how to intentionally create the silence and the time for prayer and deepening our relationship with God during the penitential season of Lent.
Fr. Kucera discusses the need for the church to work together as one body to further the mission of the church.
Subscribe to the podcast for automatic downloads whenever a new homily is uploaded!


text:
Europe Global terrorism World
Islamic State claims responsibility for machete attack in Belgium (Update 2)
By Reuters News Service August 7, 2016 November 7, 2016 084
Belgian police officers stand guard outside the main police station after a machete-wielding man injured two female police officers before being shot in Charleroi
By Foo Yun Chee
Islamic State claimed responsibility on Sunday for an attack by a machete-wielding man in Belgium that left two female police officers seriously injured, the group's Amaq news agency said on Twitter.
Shortly before the claim, Belgian prosecutors had identified the assailant as a 33-year-old Algerian, K.B., who had lived in Belgium since 2012, saying that he may have been inspired by terrorism.
The attacker, who shouted "Allahu Akbar" (God is greatest) during the assault on Saturday, was shot by police and died of his injuries. He was carrying a rucksack but no explosives or other weapons were found.
"There are indications that the attack may have been inspired by a terrorist motive," the prosecutors said in a statement. "He is known to police for criminal acts, not for terrorism."
Prosecutors searched two houses in Charleroi after the attack but did not provide details.
Early on Sunday Belgian police arrested a man with a machete in the eastern city of Liege, broadcaster VRT reported.
It said Liege police had cordoned off an area before detaining the man. It said he was of Turkish origin, did not use the machete and was not previously known to the police.
Belgium is increasing security at police stations after Saturday's attack in Charleroi, Prime Minister Charles Michel told a press conference on Sunday.
Islamist bombers killed 32 people in suicide attacks at Brussels airport and a metro station in March. Many of those who carried out attacks in Paris in November, which killed 130 people, were based in Belgium.
Belgium and its capital Brussels, which houses European Union institutions and the headquarters of NATO, are currently on a security alert level of three out of a maximum four, denoting a "possible and probable" threat.
attackBelgiumPolice
Bouncer slightly injured in Napa club scuffle
Golden start for Australia on day of records
Reuters News Service
Reuters News Service January 20, 2022
Reuters News Service January 19, 2022 January 19, 2022
Betting Omicron has peaked, Johnson drops Covid rules in England
UK sees highest inflation since 1992, pressuring BoE and households
Blinken tells Ukraine he will keep working to avert Russian attack
Greece imposes rolling fines to push Covid vaccinations in older people


Bruce McAvaney moved to television in 1978, when he joined Adelaide station ADS-7 to read sport news and produce the weekly Racetrack program. His career received a boost when colleague Sandy Roberts covered the 1980 Moscow Olympics for Seven, and Bruce was chosen to host the Adelaide end of the telecast for the station.
From 1981 until 1983, McAveney was the chief sports presenter for Seven News in Adelaide. He was also the lead commentator for Seven's telecasts of the South Australian National Football League competition, calling the 1983 SANFL Grand Final with former player Robert Oatey. He also hosted the league's Magarey Medal telecasts.
In late 1983 he moved to Melbourne and joined Ten Melbourne to read sport news. The following year he was the secondary host and commentated track and field events at the 1984 Los Angeles Olympics for the Ten Network.
Between 1985 and 1988, Bruce McAvaney also called the Melbourne Cup and hosted various major sporting telecasts for Ten, including the 1986 Edinburgh Commonwealth Games, the 1987 World Athletics Championships in Rome and the 1988 Grand Prix athletics in Berlin. Bruce McAvaney went on to co-host Ten's telecast of the 1988 Seoul Olympics, a role which won him significant acclaim.
In 1989, Bruce McAvaney negotiated a two-year premature end to his contract with Ten, and returned to the Seven Network on the condition that he could cover the 1992 Olympics.
Since his return to Seven, McAvaney has hosted and called a broad range of the network's sports coverage, including the Melbourne Cup, World Athletics Championships, Motor Racing, the Australian Open Tennis, Australian Masters Golf and all Summer Olympic Games from Barcelona 1992 toBeijing 2008. His extensive history covering Olympic Games has led to the nickname "Mr Olympics".
Bruce McAvaney was awarded a Medal of the Order of Australia (OAM) in June 2002 for service to sports broadcasting, and to the community through charitable and sporting organisations. He was also inducted into the Sport Australia Hall of Fame in that year.
Info:Bruce McAvaney is an Australian sports commentator & television personality.


text:
Introduction | Revolution | Romantic Satire | Censorship | Irish Uprising of 1798 | The Storming of the Bastille | The Reign of Terror
Romantic Politics
The Governess
Working-Class Children
William Blake and the Child
Wordsworth and the Child
Animal Rights and Welfare
Nature's Economy
"Nutting"
The Voice of the Earth
Feminine Sensibility & Virtue
The Female Body
British East India Company
The Trial of Warren Hastings
Sir William Jones
Pantisocracy
Southey and Ireland
Oriental Tales
A Woman's Power
Ideas on Breastfeeding
The Mother's Duty
The "Dangerously Good" Mother
The "Naturally Bad" Mother
A Priestley Polemicist
A Faithful Disbelief
Mad Shelley
Escaping the Infernal Abode
Romanticism and Revolution
Romantic Satire
Censorship and the Print Culture
Irish Uprising of 1798
The Storming of the Bastille
The Reign of Terror
Somerset Case
The Zong Case
Abolishing the Slave Trade
Ending Slavery
"The Rime of the Ancient Mariner"
Blake's Plates for Stedman
Wordsworth's "To Toussaint L'Ouverture"
Urban Poetry
The Consumer-Based Economy
New Forms of Art
Radical Political Groups
The Masses
Censorship and Print Culture During the Romantic Era
Because of the passage of a series of increasingly stringent laws, radical speech in the Regency era became equated with vice and subject to laws enforcing morality. While vice has always been a prime target for moral reform, in the late eighteenth and early nineteenth centuries, the call for societal reform and the suppression of vice took on heated interest by government parties. A proclamation issued by George III on June 1, 1787 called for "the Encouragement of Piety and Virtue, and for [the] preventing and punishing of Vice, Profaneness, and Immorality," yet this proclamation included provisions for acting against the publication and dissemination of print material found to contain "licentious" or indecent content (Donelan 1). The term "vice," as it was known in the eighteenth and nineteenth centuries, included the traditional definition of moral degeneracy as well as a more vague implication of social deviance in general. In the wake of the French Revolution, Donelan suggests that "social deviance" also came to include radical political views (and speech) as well as moral ones. A succession of public societies formed over the next three decades in order to help enforce the campaign against vice, starting with the Proclamation Society (1787), the Society for the Suppression of Vice (1802), and the Constitutional Society (1830). All three societies acted against "those who published and distributed...obscenity," including "seditious and blasphemous publications" (Donelan 1).
During the early eighteenth century, the government's increasing fear of internal rebellion prompted the passage of several additional laws aimed at curtailing the citizens' ability to form organized rebellions. One such series of laws is the Six Acts of 1819, which, although focused on suppressing sedition in general, also included restrictions on the press, including the Criminal Libel Act. The emphasis on policing print culture contained in these laws created a stringent atmosphere of censorship for Romantic poets, particularly for proponents of political reform such as Byron, Shelley, Thomas Moore, and Leigh Hunt. As Donelan says, "For Lord Byron and his liberal friends, the war on vice had become an attack on their right to address emerging problems of popular culture through literature in a time of political reaction" (2).
Shannon Heath
Donelan, Charles. Romanticism and Male Fantasy in Byron's Don Juan: A Marketable Vice. London: MacMillan
Gilmartin, Kevin. Print Politics: The press and radical opposition in early nineteenth-century England. Cambridge
University Press, 1996.
Haslett, Moyra. Byron's Don Juan and the Don Juan Legend. Oxford: Clarendon Press, 1997.
Stauffer, Andrew. Anger, Revolution, and Romanticism. Cambridge University Press, 2005.
This site was made with the support of the University of Tennessee Department of English


When I first mentioned to friends and colleagues that I was going on a Caribbean cruise with Seabourn, I heard over and over about their spectacular caviar and champagne beach day. It sounded pretty incredible, but the actual event was even more spectacular.
Isla Catalina (Catalina Island) in the Dominican Republic was being rented out by Seabourn for the afternoon, but they needed some time to get it ready. That gave me some time to spend with Mom and to catch up on a bit of social media.
We had breakfast overlooking the Caribbean Sea in the Collanade. I was more than satisfied with a Feta Cheese Egg White Omelet and a small Chocolate Croissant. I knew there would be plenty of food later on.
I climbed off the tender and was greeted with Caribbean music and dancers, as well as towels and rum punch.
The beach and the turquoise water couldn't have looked any better if I designed it myself. An abundance of chairs lined the sand and the water was crystal clear and warm.
I was escorted to a lounge chair and relaxed in front of the beautiful beach, enjoying the warm sunshine as the waiters came by with cocktails. Some of the guests also enjoyed snorkeling and watersports.
It wasn't long before I heard the boat sirens and headed down to the water. There, tuxedo-clad waiters stepped off the boats and into the water. Some had silver trays filled with caviar while others popped open bottles of champagne and poured it to the guests in the water!
That was just part of the excitement. After they finished with the champagne and caviar, we were told lunch was ready.
The lunch spread was like nothing I had ever seen. Lobster tails, ribs, burgers, chicken, potatoes and corn on the cob were being barbecued and placed with an array of salads, fruit and desserts. And, as was Seabourn's way, the drinks were continually flowing and complimentary.
I reveled in the warm sun for another hour or so before heading back to the boat.
When I got back to the ship, I showered and changed and took Mom up to the Observation Deck for High Tea. In addition to the selection of freshly brewed teas, there were warm scones with clotted cream and jam, as well as finger sandwiches.
I enjoyed a beautiful sunset over the Dominican Republic as we sailed away, and then joined Mom in our Penthouse Suite to dress for dinner.
My courses were adjusted to accommodate my seafood allergy, with no shrimp in the Chef's Cocktail, lamb shank and quail were substituted for the lobster course. Although it seems like an enormous amount of food, all of the portions were small.
At Restaurant 2, guests were given the option to pay a bit extra for a premium wine pairing, but we were happy with the selection of wines of the day. We started off with a 2012 Possessioni Bianco, which I loved. With dessert was a wonderful 2009 Chateau-Saint Armannd Sauternes, full of fruit, but not too overly sweet.
After dinner, we visited the Seabourn Sojourn's casino for the first time during the trip. Mom headed to the show with some friends she had made while onboard and I retired for the night. It had been our last stop on the Sojourn, but we had a bit more time onboard before departing in Fort Lauderdale.


text:
The Big Road Trip, Part 4: The Field of Dreams
We left G-Fest XXIII, and the Crowne Plaza O'Hare in Rosemont, IL, at 2:20 pm, and worked our way through a maze of highways and roadwork to I-90 west. After some pleading and arm twisting by my wonderful daughter, Melody, I agreed to make another stop at Mitsuwa Market in Arlington Heights, to get her another Bubble Tea. She is addicted.
The highway was not very scenic, and the on and off again rain was disappointing, but after passing Rockford and entering western Illinois, the sun came out and the drive became gorgeous! We stopped at a scenic rest area near Galena and grabbed a few pics. We also visited with a nice couple from Iowa who had been traveling all over the country on a motorcycle. We told them about our road trip and asked for recommendations for places to see in Memphis, where the wife used to live.
We continued west and stopped in Galena after seeing signs for President U.S. Grant's home. Grant had lived in Galena just prior to the Civil War, and returned to a hero's welcome after, receiving the house as a gift from the town. It was just a few years later that he was elected President, and he spent very little time in Galena. We arrived too late in the day to visit the house, but we walked around the grounds and took some pictures.
We soon reached the Mississippi River, and we crossed over into Iowa for the first time. Dubuque was a really nice town, and we found Iowa to be more scenic than we imagined. We continued west towards Davenport, heading towards the legendary Field of Dreams movie site. The GPS took us on an unexpected drive that ended up being more direct than what I had mapped out, but ended up being on dirt roads that stretched forever through corn fields and farm land.
After cresting a small rise, the road went back to asphalt, and there it was: The Field of Dreams!
There were a couple of guys on motorcycles pulling out of the long driveway to the house as we pulled up, and there was one lone car in the gravel parking lot. We pulled in and it became clear that the site was closed for the evening. We parked and quickly took some pictures. It was so quiet! You could actually hear the corn swaying in the slight breeze that was blowing. While Melody really had no reference, since she hadn't seen the movie, I was in awe to be there. FIELD OF DREAMS is an incredibly impactful movie, and it was just so cool to be there.
We hopped back in the Jeep and headed out. We didn't disturb anything while there, since I was nervous about being there after their posted hours, and back at the main road, I checked the GPS to see if there was a place we could stop to eat. We had been driving for 200 miles, and while we vowed to avoid fast food, we wanted to get down the road as far as we could and that meant a quick stop. We found a Hardees right in Dyersville, and stopped to eat.
We ate quickly and I figured out how far I wanted to drive, knowing how far we had to go tomorrow without wearing ourselves out tonight. I made a reservation at the Sleep Inn in Charles City, Iowa, figuring another 100 miles or so would allow us to stop by 10 pm or so.
We walked out of the Hardees and saw a sight that haunted us for the remainder of the trip. It was a lapse of judgment that would never be repeated, but never forgotten. And it was only through God's grace that it wasn't worse than it was, or could have been, which is either a statement on the town and people of the area we were in, in the heart of America, or God was really looking out for us. Or it was a combination of both. And we agreed then and there and that we would never tell a soul what it was that happened. So there. That's the end of it.
We jumped back on the highway, watched an amazing sunset, and just kept thanking God for the incredible trip we were on. US 20 turned into I-380, and we turned northwest near Waterloo onto US 218 all the way to Charles City. The Sleep Inn was very nice, as was the staff, and we quickly settled in to sleep. Our mileage for the day was 311.9, and we were at 1112.1 miles for the trip.
The Big Road Trip, Part 3: G-Fest XXIII
G-Fest XXIII, Chicago (Rosemont), Illinois. G-Fest is the largest regular gathering of Godzilla and Japanese monster fans in the world. It's a family-oriented convention featuring presentations by fans and actors and crew from Japanese Godzilla films, including topics of interest, contests and gaming, new and classic kaiju movies, a dealers room, and lots of fun.
After arriving on Friday and attending the opening ceremonies, introducing all of the guests, daughter Melody and I crashed for the night. We were excited to be at G-Fest, and we were looking forward to the beginning of our big road trip.
The Crowne Plaza Hotel O'Hare, in Rosemont, is a wonderful venue. The rooms are very nice, and the views from the higher floors are actually kind of cool, as planes on final approach to O'Hare airport fly right by the windows (and they are appropriately soundproofed, too. During G-Fest, the hotel airs rare TV shows and movies with a Japanese monster (kaiju)/sci-fi theme on the in-house TV channel, so during our downtime, we enjoyed watching some of the programs we wouldn't have a chance to see otherwise. The hotel is fairly large, able to handle the needs of the convention (except for the long wait for the elevators), and we like that the convention has a home that we've become comfortable with.
I woke up early on Saturday morning. I went down to the main level and got some breakfast stuff for us, returning to our room to eat. I allowed Melody to sleep in, and I went down in time for one of the early sessions with Akira Takarada, star of the original GOJIRA (Godzilla). Takarada is every bit the big star, and hearing his stories of his time on the sets of movies and working with other actors and directors in Japan, is just so interesting. I particularly enjoyed his stories about Nick Adams, an American actor who starred with him in INVASION OF THE ASTRO MONSTER (aka, MONSTER ZERO) in the 60s. He was interviewed by JD Lees, the publisher of G-FAN Magazine, and the organizer of the convention, and was translated by Robert Scott Field, an actor who played professional baseball in Japan and learned the language and lives there now, and is a popular guest at G-Fest every year.
After an hour with Takarada, the next session was with Bin Furuya, the suit-actor who played the original Ultraman, and acted in many other kaiju movies and TV shows. ULTRAMAN was one of my favorite TV shows when I was a kid growing up in the Maryland suburbs of Washington, DC, back in 70s. I watched the show on Channel 20, which had a wonderful lineup of shows for kids each weekday afternoon, hosted by Captain 20 (Dick Dyszel). I loved hearing Furuya's stories about the show, and the show he actually acted on (instead of being a suit-actor), ULTRASEVEN. August Ragone, author of the wonderful book, "Eiji Tsuburaya: Master of Monsters," interviewed Furuya, and Field again translated.
Melody joined me during the session with Furuya, and after it ended, we went to the Dealers Room, which has so much great stuff for sale. Melody got a few t-shirts, and we enjoyed browsing through the merchandise. We then walked around the convention to see the sites, including the wonderful model and art gallery.
We then went to another session, this time with the wonderful actress, Hiroko Sakurai, who starred in ULTRAMAN, and also its predecessor, ULTRA Q. This was her first trip to the US, so it was great to see her excitement and humbleness on display as she met with her fans.
We lunched on some leftover Giordano's, then went to see a session with Tony Isabella, prolific writer and comic book historian, as well as the founder of the First Church of Godzilla (see Facebook). He shared a nice history and slide show about monster kaiju in the comics, and I loved hearing about his love of monster movies. I've long wanted to see him at a G-Fest, so this was pretty cool.
We went on a drive to one of our favorite places when we come to G-Fest, Mitsuwa Market, in Arlington Heights, where we got some goodies and some Japanese carry-out for dinner. Melody got some Bubble Tea for the first time, and thus began a new addiction...
Back at the Crowne Plaza, we ate, then went to the Main Event of every G-Fest: The costume parade. We got some good seats since we arrived early, and we watched the Final G-Pardy, a trivia game in the same format as the Jeopardy TV game show. Following that, there were raffle drawings for some neat prizes, with proceeds going to "G-Fans Helping G-Fans," which helps needy folks in the G-Fan family.
This year's costumes were very good, and the parade lasted more than an hour. There are simple costumes, and some that are very elaborate, and all are inspired by Japanese Kaiju movies and TV shows. The participants vary in age from young toddlers to the most "experienced" folks in fandom. It's a fun party, and one of the highlights of every G-Fest.
Following the festivities, we headed back to our room to watch movies on the in-house channel and rest. Sunday morning, we got room service for breakfast, which was really great, probably our best meal of the trip so far (and definitely our most expensive!).
We checked out after getting ready, then we spent the rest of the morning wandering around the convention and making some last purchases at the dealers room, including meeting "Big In Japan" author Timothy Price, and artist Alan Barnes. We shared with Timothy that we were taking a big cross-country road trip, and we chatted about our mutual love of roller coasters.
We ate lunch at the hotel's restaurant (the Front Desk gave us a voucher for a free meal when the hotel staff forgot to pick up our room service request for a Saturday breakfast late Friday night), then we hit the road.
It was another fantastic G-Fest! I feel like we missed out on a lot of stuff this time, given how late we arrived on Friday, and then hitting the road on Sunday afternoon. But this was one of the best G-Fests with so many wonderful guests, and the sessions were as enjoyable as ever. But we were also excited to be hitting the road to continue our trip.
The Big Road Trip, Part 2
The biggest problem with traveling on the road with a teenage daughter is her propensity to crave sleep above the adventure of travel. I'm the type of person that wants to be on the road at sun-up, but... Let's just say it became a challenge for both of us as the trip wore on.
Day 2 of our trip would take us to our first and only destination: Chicago, IL, for G-Fest XXIII. G-Fest is the largest regular gathering of Godzilla and Japanese monster fans in the world. It's a family-oriented convention featuring presentations by fans and actors and crew from Japanese Godzilla films, including topics of interest, contests and gaming, new and classic kaiju movies, a dealers room, and lots of fun. My daughter and I have been going for about 5 years now, and I went to several of the first G-Fests back in the 90s. It's held each year in mid-July at the Crowne Plaza Chicago O'Hare, in Rosemont, IL.
Making the trip to Chicago each year means trying to vary the route a little bit so that is doesn't get old, because it tends to get old. We've flown, taken the train, and driven the turnpikes. This time, I decided to take the "southern" route across Ohio and Indiana, and then drive north into Chicagoland. We left our hotel at 7:35 a.m. and headed west on I-70. It wasn't long before we hit the mother of all traffic jams. There was an accident just east of Columbus, OH, and traffic came to a standstill.
The cell phone is a masterful tool, and has changed the face of travel. With this little device, we can check our route and traffic issues on the spot, find restaurants and pit stops along the way, including making reservations at hotels, and find every roadside attraction in existence, as well as stay in touch with our friends and family through online social media. Oh, yeah... we can also call and talk to people with it!
Upon hitting this traffic jam, we found out that there was an accident ahead that had completely shut down the interstate. We quickly found a detour around the incident, and we were able to continue on with a minimal delay. Just east of Indianapolis, we stopped for gas, ran into a Wal-mart for a few needed supplies, and then had lunch at a Cracker Barrel. We were back on the road at 1 p.m. and headed towards Chicago.
Unfortunately, it was rush hour on a Friday afternoon. Traffic was really bad. We criss-crossed our way through the Chicago suburbs, sticking with highways instead of surface roads, and hitting pockets of stop and go traffic all the way to Rosemont. We arrived at about 4:30. Given the itinerary of G-Fest and not wanting to miss any of the afternoon sessions, we decided to go eat dinner first. As is customary, that meant a stop at the nearby Giordano's, which is some of the best stuffed deep dish pizza I've ever had (yes, even better than Uno's and Lou Malnati's). We weren't disappointed.
After a delicious meal, we headed over to the Crowne Plaza, checked in, and made our way to the intro session for G-Fest XXIII! It was fantastic seeing so many big stars from some of my favorite Japanese monster shows and movies, and I received personal greetings from Akira Takarada, Bin Furuya, and Hiroko Sakurai, as they passed us after the session.
We were really excited to be back at G-Fest, and, exhausted from the 463 mile drive, we decided to crash for the night back in our room.
Posted by Eric Freed at 10:23 AM 1 comment:
It was July 14, 2016. We were planning to hit the road at around lunch time, but my boss found it necessary to keep me around the office well past noon. By the time I finished taking care of the things that my boss needed me to do, it was 3 p.m. I quickly said my goodbyes, wrapped up the office chores, and headed towards home. I picked up my daughter from the church day camp, and we rushed home to load up our Jeep. The only good thing all day to that point was that we had packed everything the day before, so it was just a matter of moving it all from our living room and into the back of the Jeep.
At 4:30 p.m., I locked the front door and we pulled out of our neighborhood. I felt the stress drip away off of my body, and I felt really good for the first time all day. Daughter Melody said a prayer asking for God's safe hands around us as we traveled, and for good health and happiness along the way. We were beginning our cross-country adventure, our biggest road trip ever.
We headed out Route 32, dodging traffic and hoping any delays would be short. Heading west on Interstate 70, we hit a fairly major jam through Frederick, and then stop-and-go traffic all the way to Hagerstown. Things cleared out and we were at speed to Hancock, where we stopped for dinner. The Park N' Dine was a favorite of my family's since long before I was born. Hancock was the halfway point for us when we traveled from home to my grandparent's house in little Hopwood, PA, where my father had grown up from the age of 10 to when he left home after high school to get a Federal Government job at 18. When we visited my grandparents, we almost always stopped on the way to their place at the Park N' Dine to eat.
We arrived at 6:30, 95 miles from home. We sat in the front part of the restaurant, which is the original diner section, instead of the main dining room towards the back. The owners had added on to the original building over the years to expand with its growing popularity, eventually taking over the attached gas station. Even with the expansion, we always had a long wait for a table when we stopped here to eat over the years. Its popularity has obviously diminished over time. The place was not at all busy tonite. I ordered my favorite dish, the hot turkey sandwich, loaded with gravy and mashed potatoes with dressing. Melody got chicken tenders and broccoli. It was delicious! Our young server took great care of us, though it took a while to get our food. It was 7:30 by the time we pulled out and returned to the road, jumping onto I-68 west.
Our next stop was at the Hill Top Fruit Market in Grantsville, MD. We spent ten minutes of mad rushing around the little store loading up on snacks for the trip. They have miniatures of just about every kind of candy currently being made, and Melody got a bag of her favorites. I got some fruit and nuts, and a few sugar-free items, and we returned to the road. It was 8:30 p.m.
We continued on US 40, the National Road, through the mountains towards Uniontown, PA. It was rough going due to it being a 2-lane road, and getting stuck behind cars that had difficulty maintaining the speed limit because of the changing topography kept our speeds down. We were not making good time. It was almost dark as we crested the Summit of Chestnut Ridge and headed down the mountain towards Hopwood. There were heavy, dark clouds all along the horizon, and we ended up hitting rain before we got to the base of the mountain. As it was already dusk, it got dark very quickly, and the rains came down hard. I could barely see. Even though I was very familiar with the area around Uniontown, the road we planned to take, Route 43, was new and I didn't know exactly where the exit ramp was located. With visibility practically at zero, I missed the exit, so we took the Route 51 exit and stopped at a Sheetz to wait out the storm. It was a good pit stop for us, and I was able to get a cup of coffee.
I had hoped to get somewhere into mid-Ohio by the time we stopped for the night, but our late departure, the off and on heavy traffic, and now the weather, combined to knocked us off our schedule. It was time to begin making our plans for the night. The weather app on my iPhone told us that we would likely exit the immediate storm very soon, but there were bunches more west of us, where we were headed. I used the Choice Hotels app to see where we might want to stay along our planned route, and I picked out a Quality Inn in Cambridge, Ohio. It was 9:30 p.m., and the GPS estimated our arrival at around midnight. It was well short of our plan for Day One of our trip, but that would be my limit for today.
Back on the road, we pulled out of the big storm a few miles down Route 43, but the rain continued in spurts for the remainder of our drive. We didn't make another stop, having gassed up in Keysers Ridge, MD, just before crossing into PA earlier, and we arrived at our Destination at 11:50 p.m. It was clean and comfortable, and sleep took over quickly. We were 315 miles from home.


The Ice Run slot machine invites you to take an advantageous trip to the north. The slot developed by Playtech has 5 reels and 25 paylines. Each spin of the reels can bring the sums with the coefficients of up to 4,000. There are such prize features, as free spins and a bonus round.
The winning combinations are formed from 2, 3, 4, and 5 identical symbols on the lines on consecutive reels from left to right. If there are several combinations on one line at once, only the highest paid is the winning one. If the prize combinations were formed on different lines during one spin, then the winnings for them are summed up and accrued to the player's account. The balance of the account is shown in the information block in the lower left corner. There is no risk game or progressive jackpot in the game.
The northern inhabitant symbol is a wild symbol It replaces all the icons in the winning combinations except for the scatter and the icon that launches the prize round. The wild symbol appears only on 3 middle reels and does not bring the wins by itself. The Eskimo girl symbol has the largest coefficients. When 2, 3, 4, or 5 such icons appear on the active line, players get the winnings with multipliers of 5, 100, 400, and 4,000. The seal symbol has multipliers of 3, 30, 100, and 500. The symbols of the bonfire and fish multiply the linear bet by 2, 20, 50, and 200. The remaining 6 icons are card symbols that have coefficients from 7 to 100.
Under certain conditions, the wild symbol triggers repeated spins without taking the credits from the player's account. If this icon appears in the center of the middle reel, it expands to the entire row vertically. Then two respins start. If the wild symbol appears anywhere on the second, third, and fourth reel, it also expands. After that, players receive one respin.
The yurt symbol is a scatter. It is paid regardless of active lines. When 3, 4, or 5 such symbols appear anywhere on the playing field, players receive a win in the amount of the total bet multiplied by 10, 25, and 100, respectively. A combination of at least three scatter symbols anywhere in the playing field triggers free spins. There are 10 free spins. During free spins, all the wins are multiplied at once by 3.
If the symbol of northern wolves simultaneously appears on the first and fifth reels, the prize round begins. During the game, three igloos appear on the screen. You need to enter one of them. Every igloo has its winnings.
The theoretical percentage of the return is 93.83%. Free respins can bring you large payouts. You can win even more during free spins. In order to get the maximum for this round, you can gradually raise the bet until the combination of three scatters appears. To win more often during standard spins, it is recommended to use all the paylines.


I finished my last round of saw renovations, and have ended up with a serious collection of panel saws. Some were dropped some where chopped, but most made it through in decent condition.
In all I did a major restoration on 3 sets of saws for a total of 9. Some of the ones pictured above were purchased in pretty good shape, and didn't need any serious restoration work. Sanding all those saws was messy work and left my fingers stained black. Luckily I put down a sheet of particle board to protect my bench. I guess gloves are in order next time. After 9 saws, the rust remover bath looks like used motor oil.
I like to think that after all the neglect these saws are happy to be back on the job and free of decades of rust. Look, they have even taken up synchronized swimming!
The last set of saws came from a really cheap lot purchase. Any buy from ebay can be a crap shoot. It is hard to tell quality from the pictures and sellers rarely take a picture down the length of the saw blade to show bends. Still, for 15 bucks for the three, I am willing to take my chances.
Unfortunately the top most saw, in the left hand picture, was beyond saving. I couldn't get past the heavy pitting and bends in the blade. I disassembled it and will keep the parts for potential future repairs of other saws. If I could find another medium sized backsaw or two and a good miter box/saw my collection would be complete… yeah right!
I have been collecting old panel saws for a while. Some are in decent shape and only need a sharpen, some have bad rust and hard bends in the saw plate. In this endeavor, no saw will go untreated. I tried to straighten when possible, but a shortening happens. Check out these work horses of american history.
They have seen better days, and the one in the middle has some serious bend to it. Still, all indications are that the age of these saws is pre-WW2. That was a golden era of saws where you probably can't go wrong with anything you find. I started by removing the handles and ran into my first serious snag.
Someone had lost all the split nuts and decided to replace them with wood screws. Based on the screws it was probably a number of decades ago. They were screwed or pounded in, then had the tips bent over. After a lot of messing around with pliers and cutters I eventually grabbed a hack saw and sawed the screw off between the plate and handle.
Next the saw plates went into a shallow tub with Evapo-Rust. I found that one of those under-bed storage containers was perfect. Once again this stuff is amazing. It works pretty well over night, but there appear to be no ill effects from leaving it longer.
After their bath they look almost black. I tried a few different techniques and found spraying down the blade with WD-40, then scrubbing with a sanding sponge did the best. A 220 grit 3M sandblaster sanding sponge did a fantastic job. It holds together really well under wet conditions, even with the saw teeth raking it, it doesn't get loaded, and it cut through the grime quickly. Each side took less than a minute to clean. I have never used sanding sponges, but I can see why they are popular. This poor plate had to get cut down a bit, but I saved the cutoff for use as scratch stock and tools.
The handles were mostly in decent shape. Lots of scratches, and small breaks from getting dropped. The finish was stripped off with a corse sanding sponge. Again these sanding sponges surprise me. They do a really great job of adapting to all the handle contours. Once smoothed and cleaned up I rubbed a coat of boiled linseed oil on each one and let it dry. Everything went back together quickly. The tall saw had to get a new set of hardware. Luckily my local Ace had brass plated binding post hardware. Not exactly authentic, but it is a huge improvement over the wood screws the last guy used.
Sharpening is the next step. I have everything I need, but my table is covered in saw rebuilding stuff. I will keep the refurb train going until they are all done, then convert over to sharpening. In the mean time I need to come up with a good measure for saw sharpness. Maybe some time trials are in order.
I have been doing a lot of old tool cleanup lately. Maybe half of it ends up on my blog because I doubt anyone wants to see yet another number 4 cleanup. My technique hasn't really changed since I first posted about it. Maybe there is a better way however. Maybe the power of ultrasonics can speed up the cleaning of old tools a thousand fold! Or not as we shall see.
My subject is a Number 9 1/2 block plane, a pretty common model. I picked it up from an antique store for a reasonable price. All the parts were present, but they were very full of dirt and rust. Specifically the adjustable shoe was completely stuck.
Now previously I would just soak it overnight in the ole Evapo Rust. This time I put it in an ultrasonic cleaner with Evapo Rust. The results were…. CRAP! After nearly an hour of going in an out of the cleaner I had what appeared to be a wet plane with ever so slightly less junk. A bit of google searching seems to indicate that rust removal is a big thing with ultrasonics. My cleaner is an old all stainless unit I got from a surplus place years ago. Time for a shaming!
That should teach it a lesson. Oh well this block plane and all its parts completely filled the basket. I would never be able to clean anything bigger. So much for time saving technology. A new large unit would probably due the trick, but those are expensive. I should just be patient and let em soak overnight.
There we are, no rushing this process I guess. I'll leave the flattening and sharpening for a later date. While I was working on my big secret woodworking project I might as well soak something else. I got an mixup of heirloom twist bits and brace accessories over christmas. They look rough now, but a trip through the Evapo Rust cures all!
After the rust bath a soaking in WD-40 and a quick scrub with a scotch brite pad does a good job of getting into the twist areas. After the block plane, the twist bits and another small project the tub of rust remover was starting to look like the black lagoon!
The liquid is completely opaque. I hope that in a few days the sediment will fall out to the bottom and leave a clearer liquid at the top. If not, I will pour it through a coffee filter to keep the junk from contaminating my main supply of Evapo Rust.
As part of a big haul of heirloom (my great grandfather) tools I got a hand grinder. It will hold a 6 inch wheel, clamps to a surface, and gets powered by tacos and coffee. Gorgeous!
It was pretty dirty when I started in on it. A little brushing with a brass brush got rust and dirt all over my newly cleaned work top. So much for a pristine work surface.
I didn't really bother with any rust conversions. It wasn't very rusty, just mostly dirty, and I was worried about getting wet stuff into a nook and not being able to get it back out again.
A flat piece of wood with a nice cleat attached served as a good base to clamp to. I started with a single layer of oak, but later doubled up because it warped too much under heavy grinding.
Time to accessorize! I picked up a grinding set from lee valley. The veritas set came with a soft 150 grit cool grinding wheel, Adjustable tool rest, blade holding jig, and angle setting guide. I have to say it was pricey but feels really well put together. Link to the product.
I had to cut off the loop that would hold the original tool rest. It was a simple right angle bit of metal with a screw to hold it in place. With this upgrade I don't think it will be missed. To test it I pulled out an old Ohio Tools chisel. It was part of a small lot purchase from ebay. I really wanted the mortise chisel but wasn't sure what to do with this guy.
Still, no reason to get rid of it. Ohio Tools made good stuff, and it was a socket chisel. Instead of making another bench chisel I could use a skew chisel. I don't own any, and this guy requires a lot of grinding no matter what. Lets get it clamped up and going. The blade holder jig has pins that help hold the blade either square or at a 30 degree angle. Handy for grinding a skew!
I was able to keep a finger on the back of the blade and control pressure while feeling for heat. The slow speed and a close finger meant that heat damage was not an issue. The whole process took a while. 150 grit leaves a great surface that will only need a bit of work at the stone, but moves slowly when trying to make a radical change. I might use my power grinder to rough in something like this next time.
A somewhat shiny new skew chisel for me all by hand, what a great feeling. It is so satisfying to take old misfit tools and give them a good home. Well, more of a work home. No rest for the wicked!
As I mentioned earlier the single board was not stiff enough under heavy grinding. The gearing ratio means it takes a good bit of force on the handle to keep the wheel spinning quickly. I glued up another layer and put everything back on. A quick test showed that this was what the doctor ordered. Very stiff and very ready for its next job. The whole thing sits on a shelf and gets clamped into my front vice when needed.
I posted last week about my haul from the fall tractor and engine show. It was a saw, drill, side rabbet plane, scissors, rasp, chisel, and wooden molding plane. All in decent but not great condition. I went through and reconditioned all of them in the past week except the side rabbet plane. It was in great shape and didn't really need anything.
These wiss scissors were probably not that old, but had a little rust starting. I just gave them a scuff with a scotch brite pad, then did a little edge work to get them back in good cutting condition. I used Paul Seller's method of sharpening. It may not be different from everyone else's method, but credit due, that is where I learned it.
This was another one that looked a little rough, but only really needed sharpening. I re-flattened the iron, sharpened it up, and stuck it back in. Depth setting is a little tricky, but I managed to get a nice looking round over on this piece of maple. I might have to become a fan of wooden planes now!
I didn't have any mortise chisels so this was a nice find for only a few bucks. It had seen a lot of use, but would still work with some love. I removed the handle and gave it a good sanding. Boiled linseed oil is my finish of choice for tools. It is cheap, easy to apply and does a decent job of soaking in and protecting wood.
I hit the iron with some brass brushing near the socket to clean the junk off. The rest got flattening and a 35 degree edge for doing heavy mortise chopping. I did a few test hits with it and was happy with the results.
Another cheap buy to help fill my collection. This one is huge and aggressive compared to my other rasps. I didn't bother with trying to do anything to the surface, I just made a handle for it. I bought some brass tubing a while back for just this purpose and put it to good use. I turned the shoulder long expecting to trim off any excess when I pressed the brass down. Once the brass was flush with the end I couldn't get it down any further. Oops, next time I will cut the shoulder to length. I drilled out for the tang and pounded it home. The handle is a little shorter than it should be for a rasp this big, but it beats the heck out of no handle.
This is a lovely Millers Falls drill. I would label it a medium size, but these are not my area of expertise. A little scrubbing for rust and oil for movement was most of what this guy needed. Just for grins I decided to try and repaint the center hub. I used a foam brush to apply primer and red paint to all the insides.
It is a great looking drill and should give me a lot of years of service.
This was purchased mostly for practice. I want to start sharpening saws, and am not willing to experiment on my expensive and still sharp lie-neilson back saws. Big cheap panel saw to the rescue!
I started by pulling the handle off and scrubbing off as much of the rust and junk from the saw panel as I could. Here is a close up of what I had to start with. Unfortunately, I don't think the logo is going to survive the cleaning in tact.
After cleaning (forgot a shot of that when I was finished) I moved on to sharpening. I have never sharpened a saw before, so this was very new territory. After watching some how-to vids and mustering up some courage I jumped in. You can see a new metal shine starting on some of the right most teeth. It is kind of hard to shoot. If they look really oddly sized it is because of the set (teeth are bent left and right to cut a wider swath) in this camera angle.
I took a 2×4 and tried to rip it before sharpening and afterwards. The before wandered a bit and was tough work. The afterwards took about half the time and went straighter. I had a lot of trouble sharpening in places because the saw was much taller than my vice. It would rattle and vibrate as I sharpened. I need to make a clamp fixture that goes up and supports just under the teeth. I have seen plenty made in the past, I just need to get around to doing it. Till then, this saw is in good working order. The handle was in really good shape, so I didn't touch it.
I used Monica's grandfather's brace to drill the holes in her yarn swift. It worked well, and I felt it needed a cleanup. I have been picking up a few things from ebay recently, and am falling into an old tool refurbishment tear. There are probably more to come. First, the before shot.
Very dirty looking, but still in working order. I started by taking the chuck out and the head (mushroom shaped thing you push against) apart. The ratchet portion is riveted together and the center spinner handle is not removable. A few trips through the Krud Kutter (or as I call it now The Green Machine) got all the rust converted on the ratchet portion and the head area. I didn't want to dunk the center handle, so some naval jelly was brushed onto the rest of the tool.
With all the rust converted I tried doing a bit of scrubbing with a scotch brite pad. It helped clean up the darker areas, but wasn't getting as far as I wanted. On a whim I decided to try the brass brush on my bench grinder. WOW! What a result. It takes the grime and converted rust off and leaves a really clean looking surface.
Before I started cleaning I spent some time trying to identify the brace. I couldnt see any markings on it, and went as far as posting it on reddit to beg for help. Low and behold once I got the rust removed and cleaned off, the maker's marks appeared.
I did some research and based on the model number and jaw type, it has to be newer than 1949. We know that her grandfather built a house in the early 50s, so my guess is that it was purchased around then. Another interesting history fact is that the handles appeared to have been painted red at some point, probably originally. The underside of the head was left red, and some tinges of red were left on the center handle. They either wore away or were removed and varnished sometime in the past. I sanded off a majority of it and recoated them with boiled linseed oil.
With the wood preserved I re-greased the bearings at the head, oiled everything up and hit it with a protectant spray. The final results are gorgeous. It has some of the original scratches and a hint of the red is left over in the handle. You wouldn't ever mistake it for new, but it surely does look gorgeous. Now that I am keeping it oiled and protected the tool should last another 60 years at least.


Northgate Chrysler Dodge Jeep Ram wants to provide Cincinnati, Ohio with a simple car-buying experience. We accomplish this through a Live Market Pricing program, which gives you the best prices on our entire selection of new and used vehicles. This removes questionable pricing tactics, such as marking vehicles up in order to mark them down. Don't settle for haggling or lengthy negotiations. Instead, use our Live Market Pricing which searches 20,000 vehicle databases each hour. With the resulting real-time prices, you can be assured we only offer fair prices. Use the money you save on memorable outings with the whole family, like going to the Great American Ball Park or the Cincinnati Zoo & Botanical Garden.
Take a look at our model lineup page to see all of the quality vehicles we offer. Perhaps you're in the market for a high-tech and swift mid-size sedan, such as the Chrysler 200. Thirsty for adventure and exploration? Stay safe and comfortable with the Jeep Cherokee. Don't forget to check our constantly changing specials and new vehicle incentives to ensure you are receiving the best deal.
When it's time to find a great vehicle, our sales and finance teams will be by your side. Our sales team is dedicated to helping you find the ideal vehicle, not just simply making a sale. If you'd like to speak to one of our sales representative, give us a call at (877) 652-7019 or just drop by our Cincinnati, Ohio dealership. Financing your purchase can seem daunting, but we proactively work to make sure your experience at Northgate Chrysler Dodge Jeep Ram is as smooth as possible. Take a look at our financing center to explore the best purchasing options for you specific situation, whether that be buying or leasing. We'll work with you to create the best plan for your lifestyle.


In case you're wondering if you cannot make calls through Skype, WhatsApp, Viber, FaceTime in countries in the Middle East like the UAE and Qatar, it is because the calling features of these apps are blocked or disabled by the telecom providers in the country. You can still use these apps in order to 'chat', but the VoIP (Voice Over Internet Protocol) service, which includes video calling and voice calling, is banned.
Although it is not unusual for a country to regulate the usage of internet for their citizens, The United Arab Emirates and Qatar now officially joins China, Egypt, Jordan, and Belize (to name a few) to have blocked Skype services in their regions.
Skype, the Microsoft acquired company in 2011, has just been added to the list of banned VoIPs in the UAE and Qatar. The two main Internet Service Providers in the United Arab Emirates, Etisalat and Du, reasoned out that the services provided by this application are classified as unlicensed Voice over Internet Protocol Services and should be disallowed per the regulations.
Despite the economical benefits (this allows you to communicate with someone who may as well be on the other side of the world for free) gained in using VoIP services, the UAE currently has rigid provisions in place for these which focuses on keeping the privacy and security related concerns of its internet users protected and safe.
Unfortunately just this weekend, most consumers only learned about this news due to the unpleasant experiences they had in using the application. One person mentioned that he could not hear anything from the earpiece while using the app despite having really good internet connection. While another said that his call would not connect at all when he tried calling a friend.
If you also visit the support page of Skype, it validates this direction and mentions that there isn't much that they can do about it. As well, users are advised that they should be contacting their internet service providers for the explanation and that they may ask to have the block on the these services lifted.
For people looking to still utilize VoIPs however, Etisalat and Du offer their own VoIP based applications namely, BOTIM and C'Me which currently go for 50 AED and 100 AED depending on the package that suits them.
Apparently, VoIP services may actually still be allowed to operate in the country as long as certain qualifications are satisfied; the Telecommunications Regulatory Authority (TRA) of the United Arab Emirates has issued a statement that their stand on VoIPs has not changed.