Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

Label_Title

Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

Lacunary Interpolation (0,2;3) Problem and Some Comparison from Quartic Splines

Ambrish Kumar Pandey, Q S Ahmad*, Kulbhushan Singh

Department of Mathematics, Integral University, Lucknow

Abstract: This paper discuss a method for lacunary interpolation and their comparison with quartic splines in Generalised form as an local methods for solving lacunary interpolation problems using piecewise polynomials with certain specific properties.

Keywords: lacunary interpolation, quartic splines, interpolation, piecewise polynomial

Received July 21, 2013; Revised September 07, 2013; Accepted November 24, 2013

Citation: Ambrish Kumar Pandey, Q S Ahmad, Kulbhushan Singh, Lacunary Interpolation (0,2;3) Problem and Some Comparison from Quartic Splines. American Journal of Applied Mathematics and Statistics, 1 (6). 117-120. January 2013. doi:10.12691/ajams-1-6-2

1. Introduction

During the last century, splines theory has received considerable attention. Lacunary interpolation was initiated in 1957 [1]. Several researchers have studied the use of splines to solve such interpolation problems [2,3,4,5]. All of these methods are global and require the solution of a large system of equations. The most appropriate method solving lacunary interpolation problems using piecewise polynomials with certain continuity properties.

Spline functions are a good tool for the numerical approximation of functions on the one hand and they also suggest new, challenging and rewarding problems on the other. Piecewise linear functions, as well as step functions, have been an important theoretical and practical tools for approximation of such functions. Lacunary interpolation by spline appears whenever observation gives scattered or irregular information about a function and it's derivatives. Also, the data in the problem of lacunary interpolation are values of the functions and of it's derivatives but without hermite condition in which consecutive derivatives are used at each nodes.

2. Construction

Let S(x) ε S⁽⁵⁾n,6 denote the class of sixtic splines S(x) on [0,1] such that

•  S(x) ε C⁵ [0, 1]

•  S(x) is a polynomial of degree six on each subinterval

[picture] | (1)

It can be verified that if P(x) is a sixtic on [0, 1] then:

Where:

[picture] | (2)

For later references we have:

and

and

and

Further, a sixtic P(x) on [1, 2] can be written as:

Where:

[picture] | (3)

It is easy to verify that a sixtic Q(x) on [0, 1] can be expressed in the following form:

[picture] | (4)

Where:

[picture] | (5)

for later references we have

Also a sixtic Q(x) on [1, 2] can be written as:

[picture] | (6)

Where:

[picture] | (7)

3. The Approximation of the Spline Functions

Descriptions of the method: Let (Sn, C⁵) be the class of spline functions with respect to the set of knots xi. The spline functions will denoted by Si(x), where i = 0, 1,..., n. We shall prove the following:

Theorem 1 (Existence and Uniqueness)

For every odd integer n and for every set of 5n+9/2 real numbers

there exists a unique S(x) ε S⁽⁵⁾n,6 such that:

Let fεC⁵[0,1] and n an odd integer. then the unique sixtic spline Sn(x) satisfying conditions of Theorem 3.1, with fv = f(v/n), v = 0,1,…,n;

[picture] | (8)

we have:

Proof of Theorem 1

For a given S(x) ε S⁽⁵⁾n,6 set h = n⁻¹, Mv = S⁽⁵⁾ (vh+), v = 0,1,…..,n-1, Nv = S⁽⁵⁾ (vh-), v = 0,1,….,n. Since, S⁽⁵⁾ is linear in each internal (vh, (v+1)h), it is completely determined by the (2n) constants . Also, if S(x) satisfies the requirements of Theorem 1 that for 2vh ≤ x ≤ (2v+1)h, v = 0,1,…, n-1/2 , it must have the following form:

[picture] | (9)

and for (2v+1)h ≤ x ≤ (2v+2)h , v=0,1,…., n-3/2, S(x) has the form:

[picture] | (10)

We shall show that it is possible to determine the (2n) parameters , such that the function S(x) given by Eq. 9 and 10 will also satisfy (5) in Theorem 1 and S' (x), S"(x) and S⁽⁴⁾ will be continuous on [0,1]. S(x) is continuous because of the interpolating condition Eq. 8 in Theorem 1, S'(x) and S⁽⁴⁾(x) are continuous on [0,1] except at the points (2vh) and (2v+1)h, respectively, v = 0, 1,… , n-1/2.

From Eq. 10 we see that Eq. 8 in Theorem 1 is equivalent to:

[picture] | (11)

[picture] | (12)

Simple calculations show that S"((2v+2)h-) = S"((2v+2)h+) and S⁽⁵⁾((2v+2)h+) are equivalent to:

[picture] | (13)

[picture] | (14)

Similarly and are equivalent to:

[picture] | (15)

[picture] | (16)

Thus, the theorem will be established if we show that the system of linear Eq. 12-16 has a unique solution. This end will be achieved by showing that the homogeneous system corresponding to Eq. 12-16 has only zero solution.

The following is the homogeneous system of equations for v= 0,1,…, n-3/2:

[picture] | (17)

[picture] | (18)

[picture] | (19)

[picture] | (20)

[picture] | (21)

[picture] | (22)

Form Eq. 19 and 20 we have for v = 0, 1,…, n-3/2:

and

[picture] | (23)

Putting the values and Mn-1 = -13/7 Nn from Eq. 19 in 20 we have:

[picture] | (24)

Also, from Eq. 14 and 20 we have:

[picture] | (25)

and M₀ = -N₁ from Eq. 21.

Using Eq. 19 we obtain 17Mn-3+5Nn-2 and using Eq. 21 with 25 and 24 we have Mn-3 = Nn-2 = Mn-2 = Nn-1 = 0. Also obtain the system:

By the same manner we get M₀ = M₁ = … = Mn-1 = 0 and N₁=N₂= N₃ = … = Nn = 0, to solution of the homogeneous system for n = 4p and n = 4p+2.

References

[1]   | J. BALAZS AND P. TURAN, Notes on interpolation. II, III, IV, Acta Math. Acud. Sci.Hungar, 8 (1957), 201-215; 9 (1958), 195-214, 243-258.
[2]   | TH. FAWZY, Note on lacunary interpolation with splines. 1. (0, 3)-Lacunary interpolation,Ann. Univ. Sci. Budapest. Eiifciis Sect. Math. 28 (1985).
[3]   | TH. FAWZY, Note on lacunary interpolation by splines. II, Ann. Univ. Sci. Budapest. Sect.Comput. 5, in press.
[4]   | TH. FAWZY, Notes on lacunary interpolation by splines. III. (0, 2) Case, Acra Math. Acud.Sci. Hungar., in press.
[5]   | TH. FAWZY, Lacunary interpolation by g-splines: (0, 2, 3) Case, Studia Sci. Math.Hungar., in press.