bluuebunny/crossref_metadata_2025_split huggingface dataset:
Effect of chain flexibility on the dynamics of rodlikk polymers in the entangled state
Authors: Doi Masao 
Published: June 2010

AbstractThe Brownian motion of semiflexible polymers in networks and in concentrated solutions is discussed. If the polymer is assumed to move only along itself by reptation, various time correlation functions are calculated rigorously. The results show that the Brownian motion is independent of the mesh size a over a wide range. It is then argued that the observed rodlike behavior of semiflexible polymers in concentrated solutions is due to the rapid bending fluctuation of the polymer.
Molecular theory expressions for the stress tensor in flowing polymeric liquids
Authors: Bird R. Byron , Curtiss C. F. 
Published: June 2010

AbstractVarious molecular expressions for the stress tensor for polymeric liquids are given and interrelated. Particular attention is paid to restrictions associated with constraints in the molecular models, hydrodynamic interaction, and molecular confinement in concentrated systems. A recently derived generalization of the Kramers‐Kirkwood stress‐tensor formula can be taken as the starting point for many polymer calculations.
Details of phase separation processes in polymer blends
Authors: Snyder H. L. , Meakin P. 
Published: June 2010

AbstractWe have investigated the details of phase separation dynamics in high‐molecular‐weight (polystyrene‐PVME) polymer blends utilizing small angle light scattering techniques and have compared our results with the various theories, computer simulations, and phase separation dynamics in small molecule systems. Although linearized theories are consistent with many qualitative features of phase separation dynamics, they fail to provide quantitative accuracy. It appears that both nonlinear terms and hydrodynamic effects must be included to provide a quantitative description. Quantitative agreement between phase separation dynamics in small molecule systems and high‐molecular‐weight polymer blends is also reported.
Raumfachwerke und kein Ende?
Authors: Klimke Herbert , Walochnik Wolfgang 
Published: June 2010

AbstractDieser Beitrag gibt einen Überblick über die Entwicklung des Bauens mit räumlichen Stabwerken in einer Periode von etwa 30 Jahren, von der Teilüberdachung des Berliner Olympiastadions 1974 bis zum Glasdach des Berliner Hauptbahnhofs 2004. In dieser Zeit hat sich die Technologie von der ebenen Raumfachwerkplatte bis zu den leichten Stützkonstruktionen für den Glasbau mit Zugstangen und Seilen weiterentwickelt. Die kennzeichnenden Projekte dieser für das Bauwesen stürmischen Entwicklung werden mit einigen Überlegungen zum Entwurfsprozess beispielhaft vorgestellt.Space trusses and no end? This paper offers a short review of the development in the field of space trusses for a period of about 30 years, beginning with the partial roof structures of the Olympiastadion Berlin in 1974 until the glass cover for the Berlin Hauptbahnhof (main station) in 2004/2006. During this period the technology has developed from the plane space trusses to the light weight spatial structures for glazed buildings, incorporating tension rods and cables as structural elements. The marking projects of this exiting development are exemplarily presented together with some reflections on the structural design process.
Effects of time since urbanization on anuran community composition in remnant urban ponds
Authors: GAGNÉ SARA A. , FAHRIG LENORE 
Published: June 2010

SUMMARYLow-density residential development, the fastest growing land use in the USA, is increasingly occurring adjacent to protected areas and in areas of high biodiversity. Thus, determining the environmental impacts, including the cumulative impacts, of proposed residential developments is a pressing challenge. The relative abundance and species richness of anurans in 19 ponds surrounded by landscapes with varying ages of residential development were measured, while endeavouring to control for local habitat quality effects on the anurans. Age of residential development was a predictor in the best models describing the responses of four individual anuran species and total anuran relative abundance. In particular, all of the best models of gray treefrog Hyla versicolor relative abundance included age of residential development as a predictor. Present-day anuran communities in remnant urban ponds are evidently responding to the effects of residential development that occurred up to 54 years in the past.
Ready to Lead on Day One: Predicting Presidential Greatness from Political Experience
Authors: Balz John 
Published: June 2010

AbstractPresidential candidates frequently tout their political experience on the campaign trail, telling voters that it has prepared them to deal with complex problems, make weighty decisions, and show leadership. The value of that argument was put to the test in the 2008 presidential campaign by Hillary Clinton against her opponent, Barack Obama. This paper uses a multilevel model to analyze the value of national and state political experience on overall presidential greatness, as judged by seven surveys of academic experts. Overall, there is no evidence that political experience improves the likelihood of strong presidential performance, and even some weak evidence that political experience in certain political positions, most notably mayor and member of Congress, leads to poorer performance. In the end, great presidents are not great simply because they have spent their lives in politics and learned important lessons. Other personal and historical factors are likely to be more important.
Scholarly Productivity in Non-Ph.D. Departments
Authors: Collins Todd A. , Cooper Christopher A. , Knotts H. Gibbs 
Published: June 2010

AbstractPolitical scientists hail from large, research-intensive universities like the Ohio State University, regional comprehensive schools like Western Kentucky University, and small teaching-intensive institutions like Mars Hill College. Despite this diversity, most studies of the political science discipline overlook the contributions of individuals from non-Ph.D. departments. To address this oversight, we compare the publishing rates of scholars with four types of affiliations: non-Ph.D. departments, Ph.D. departments, non-U.S. departments, and nonacademic institutions. We focus particularly on whether faculty from non-Ph.D. departments publish in different types of journals than faculty from other departments, and whether the institutional affiliations of editorial board members corresponds to the institutional affiliations of published authors. We find that people from non-Ph.D. departments represent 16% of the authors in our sample of political science journals, and their contributions are particularly noteworthy in certain types of journals. We also demonstrate that the institutions represented on editorial boards generally do not reflect the institutional affiliations of the authors who publish in these journals.
Service Learning and Simulations
Authors: Jenkins Shannon 
Published: June 2010

AbstractBoth service learning and simulations have been shown to positively impact student outcomes, but they are not often used together. This article examines how to effectively combine these active learning styles to reap the benefits of both. After examining a case in which the two were combined and the impact this approach had on student evaluations and learning outcomes, I discuss how such projects can be successfully executed in a variety of other classes.
Graduate Students as Independent Instructors: Seven Things to Know about Teaching Your Own Course while in Graduate School
Authors: Burmila Edward M. 
Published: June 2010

AbstractDespite being responsible for a large percentage of undergraduate instruction, graduate students often receive little preparation for their first solo teaching assignments (J. D. Nyquist et al., Change 31 (3): 18, 1999). Furthermore, the existing literature on pedagogy fails to address the unique challenges faced by graduate students who are asked to serve as course instructors rather than teaching assistants. This article presents seven pieces of advice intended to better prepare the predoctoral graduate student to assume the role of the professor before assuming the title. By understanding the attitudes of undergraduate students toward graduate instructors, preparing in advance to handle the mistakes that novice teachers often make, and recognizing the correlation between outward confidence and student perceptions of instructor quality, graduate students can derive the most benefit from a stressful and time-consuming assignment. Most important, graduate instructors can learn to effectively manage the time spent on teaching duties to ensure that other responsibilities such as coursework, qualifying exams, and dissertation research do not suffer.
Pop Culture in the Classroom: American Idol, Karl Marx, and Alexis de Tocqueville
Authors: Centellas Miguel 
Published: June 2010

AbstractThis article discusses the use of pop culture in the classroom as a means to teach foundational political science authors and concepts. I focus on my experience using American Idol as a point of reference to discuss Marx and Engel's The Communist Manifesto and Tocqueville's Democracy in America in undergraduate comparative politics courses. Students are asked to construct a written argument projecting Marx or Tocqueville's perceptions of American Idol, based on their readings. My experiences demonstrate that asking students to reflect on their own contemporary experience through the prism of these two works helps them in three ways: (1) to better understand the ideas of Marx and Tocqueville, as well as their differences; (2) to develop an appreciation for the continued relevance of works in the discipline's canon; and (3) to sharpen and develop critical thinking and analytical skills.
LEARNING RACE, SOCIALIZING BLACKNESS
Authors: Nunnally Shayla C. 
Published: June 2010

AbstractContemporary discourse about Black Americans questions the loyalties of younger Blacks to the advancement of the Black racial group. This discourse often compares the commitment of Black Americans who came of age during the Civil Rights Movement era to those who came of age during the post-Civil Rights Movement era. Fueling this discourse is a working assumption that somehow younger Black Americans have a different understanding about race and its role in Blacks' political interests. This begs the question whether there are generational differences in the ways that Black Americans learn about race, or racial socialization, perhaps with implications for distinct value orientations about Black politics. Using public opinion data from an original survey, the 2007 National Politics and Socialization Survey (NPSS), this paper compares the racial socialization experiences of four generations of Black Americans—(1) World War II generation (age 67 and older, born in and before 1940); (2) civil rights generation (ages 54–66, born 1953–1941); (3) mid-civil rights generation (ages 43–53, born 1964–1954); and (4) post-civil rights generation Black Americans (age 42 and under, born 1965 and after). Results of ordered probit regression analyses indicate minimal generational differences. Differences emerge in emphases on racial socialization messages about Black public behavior, Black intraracial relations, Black interracial relations, and composite factor loadings of Black consciousness and Black protectiveness messages.
Structure of bacterial LigD 3′-phosphoesterase unveils a DNA repair superfamily
Authors: Nair Pravin A. , Smith Paul , Shuman Stewart 
Published: June 2010

The DNA ligase D (LigD) 3′-phosphoesterase (PE) module is a conserved component of the bacterial nonhomologous end-joining (NHEJ) apparatus that performs 3′ end-healing reactions at DNA double-strand breaks. Here we report the 1.9 Å crystal structure of
Pseudomonas aeruginosa
PE, which reveals that PE exemplifies a unique class of DNA repair enzyme. PE has a distinctive fold in which an eight stranded β barrel with a hydrophobic interior supports a crescent-shaped hydrophilic active site on its outer surface. Six essential side chains coordinate manganese and a sulfate mimetic of the scissile phosphate. The PE active site and mechanism are unique
vis à vis
other end-healing enzymes. We find PE homologs in archaeal and eukaryal proteomes, signifying that PEs comprise a DNA repair superfamily.
Generalisation of the Muirhead-Rado inequality
Authors: Daykin D. E. 
Published: June 2010

For polynomials

f
β

(
x
)

{f_\beta }(x)

of n real variables

x
=
(

x
1

,

x
2

,
⋯
,

x
n

)

x = ({x_1},{x_2}, \cdots ,{x_n})

of the form
\[

f
β

(
x
)
=

∑
i

∑
j

x

ρ
(
i
,
1
)

β
1

e

j
1

x

ρ
(
i
,
2
)

β
2

e

j
2

⋯

x

ρ
(
i
,
n
)

β
n

e

j
n

,

{f_\beta }(x) = \sum \limits _i {\sum \limits _j {x_{\rho (i,1)}^{{\beta _1}{e_{j1}}}x_{\rho (i,2)}^{{\beta _2}{e_{j2}}} \cdots } } x_{\rho (i,n)}^{{\beta _n}{e_{jn}}},

\]
conditions are given which ensure that

f
α

(
x
)
≦

f
β

(
x
)

{f_\alpha }(x) \leqq {f_\beta }(x)

for all

x
≧
0

x \geqq 0

.
Existence and continuous dependence for a class of nonlinear neutral-differential equations
Authors: Grimm L. J. 
Published: June 2010

This paper presents existence, uniqueness, and continuous dependence theorems for solutions of initial-value problems for neutral-differential equations of the form
\[

x
′

(
t
)
=
f
(
t
,
x
(
t
)
,
x
(
g
(
t
,
x
)
)
,

x
′

(
h
(
t
,
x
)
)
)
,

x
(
0
)
=

x
0

,

x’(t) = f(t,x(t),x(g(t,x)),x’(h(t,x))),\quad x(0) = {x_0},

\]
where f, g, and h are continuous functions with

g
(
0
,

x
0

)
=
h
(
0
,

x
0

)
=
0

g(0,{x_0}) = h(0,{x_0}) = 0

. The existence of a continuous solution of the functional equation

z
(
t
)
=
f
(
t
,
z
(
h
(
t
)
)
)

z(t) = f(t,z(h(t)))

is proved as a corollary.
Boundary zero sets of 𝐴^{∞} functions satisfying growth conditions
Authors: Taylor B. A. , Williams D. L. 
Published: June 2010

Let A denote the algebra of functions analytic in the open unit disc D and continuous in D, and let
\[

A
∞

=
{
f
∈
A
:

f

(
n
)

∈
A
,
n
=
0
,
1
,
2
,
⋯
}
.

{A^\infty } = \{ f \in A:{f^{(n)}} \in A,n = 0,1,2, \cdots \} .

\]
For

f
∈
A

f \in A

denote the set of zeros of f in D by

Z
0

(
f
)

{Z^0}(f)

, and for

f
∈

A
∞

f \in {A^\infty }

let

Z
∞

(
f
)
=
⋂

n
=
0

∞

Z

0

(

f

(
n
)

)

{Z^\infty }(f) = \bigcap {_{n = 0}^\infty Z{^0}({f^{(n)}})}

. We study the boundary zero sets of

A
∞

{A^\infty }

functions F satisfying, for some sequence

{

M
n

}

\{ {M_n}\}

and some

B
>
0

B > 0

,

(

1

)

|

F

(
n
)

(
z
)

|

≦
n
!

B
n

M
n

,

z
∈

D
¯

,
n
=
0
,
1
,
2
,
⋯
.

\begin{equation}\tag {$1$}|{F^{(n)}}(z)| \leqq n!{B^n}{M_n},\quad z \in \bar D,n = 0,1,2, \cdots .\end{equation}

In particular, when

M
n

=
exp
⁡
(

n
p

)
,
p
>
1

{M_n} = \exp ({n^p}),p > 1

, it is shown that for E, a proper closed subset of

∂
D

\partial D

, there exists

F
∈

A
∞

F \in {A^\infty }

satisfying (1) and with

Z
0

(
F
)
=

Z
∞

(
F
)
=
E

{Z^0}(F) = {Z^\infty }(F) = E

if and only if

∫

−
π

π

|

log
⁡
ρ
(

e

i
θ

,
E
)

|

q

D
θ
>
+
∞

\smallint _{ - \pi }^\pi |\log \rho ({e^{i\theta }},E){|^q}D\theta > + \infty

. Here

ρ
(
z
,
E
)

\rho (z,E)

is the distance from z to E and

(
1

/

p
)
+
(
1

/

q
)
=
1

(1/p) + (1/q) = 1

.
An analytical criterion for the completeness of Riemannian manifolds
Authors: Gordon William B. 
Published: June 2010

If M is a (not necessarily complete) riemannian manifold with metric tensor

g

i
j

{g_{ij}}

and f is any proper real valued function on M, then M is necessarily complete with respect to the metric

g
~

i
j

=

g

i
j

+
(
∂
f

/

∂

x
i

)
(
∂
f

/

∂

x
j

)

{\tilde g_{ij}} = {g_{ij}} + (\partial f/\partial {x^i})(\partial f/\partial {x^j})

. Using this construction one can easily prove that a riemannian manifold is complete if and only if it supports a proper function whose gradient is bounded in modulus.
Chebyshev estimates for Beurling generalized prime numbers
Authors: Diamond Harold G. 
Published: June 2010

We consider a Beurling generalized number system for which the counting function of the integers satisfies
\[

N
(
x
)
=
c
x
+
O
(
x

log

−
γ

x
)

N(x) = cx + O(x{\log ^{ - \gamma }}x)

\]
for some positive

c
c

and

γ
\gamma

. Beurling showed that the prime number theorem must hold if

γ
>

3
2

\gamma > \tfrac {3}{2}

, but it can fail to hold if

γ
≦

3
2

\gamma \leqq \tfrac {3}{2}

. It was shown by Hall in the preceding article that the Chebyshev prime counting estimates
\[

0
>
lim

inf

x
→
∞

ψ
(
x
)

x

;

lim

sup

x
→
∞

ψ
(
x
)

x

>
∞

0 > \lim \inf \limits _{x \to \infty } \frac {{\psi (x)}}{x};\quad \lim \sup \limits _{x \to \infty } \frac {{\psi (x)}}{x} > \infty

\]
can fail if

γ
>
1

\gamma > 1

. Here we shall prove that these estimates hold for any system satisfying

γ
>
1

\gamma > 1

. The proof uses a convolution approximate inverse of the measure

d
N

dN

.
On the zeros of certain confluent hypergeometric functions
Authors: Wynn P. 
Published: June 2010

The theory of continued fractions is used to derive the following results which hold for

−

1
2

>
α
>
∞

- \tfrac {1}{2} > \alpha > \infty

: (1) If
\[

1

F
1

(
α
;
2
α
+
1
;
z
)
=
0

[

1

F
1

(
α
+
1
;
2
α
+
1
;
z
)
=
0
]

_1{F_1}(\alpha ;2\alpha + 1;z) = 0\quad [{}_1{F_1}(\alpha + 1;2\alpha + 1;z) = 0]

\]
then

Re
⁡
(
z
)
>
0
[
>
0
]

\operatorname {Re} (z) > 0[ > 0]

. (It is deduced from this result that if

I

α

−
1

/

2

(
z
)
+

I

α

+
1

/

2

(
z
)
=
0

{I_{{\alpha ^{ - 1/2}}}}(z) + {I_{{\alpha ^{ + 1/2}}}}(z) = 0

then

Re
⁡
(
z
)
>
0

\operatorname {Re} (z) > 0

, and it is shown that if

α
\alpha

is an integer, an unbounded number of roots of this equation exists.) (2) The roots of the equ tions
\[

1

F
1

(
α
;
2
α
;
z
)

=
1

F
1

(
α
;
2
α
+
1
;
z
)

_1{F_1}(\alpha ;2\alpha ;z){ = _1}{F_1}(\alpha ;2\alpha + 1;z)

\]
and
\[

1

F
1

(
α
;
2
α
;
z
)

=
1

F
1

(
α
+
1
;
2
α
+
1
;
z
)

_1{F_1}(\alpha ;2\alpha ;z){ = _1}{F_1}(\alpha + 1;2\alpha + 1;z)

\]
are identical, pure imaginary, symmetrically distributed about the origin and unbounded in number. (3) Let

C
n

(
z
)
(
n
=
0
,
1
,
⋯
)

{C_n}(z)(n = 0,1, \cdots )

be the successive convergents of the continued fraction associated with

g
(
z
)

=
1

F
1

(
α
;
2
α
;
z
)

/

1

F
1

(
α
;
2
α
+
1
;
z
)

g(z){ = _1}{F_1}(\alpha ;2\alpha ;z){/_1}{F_1}(\alpha ;2\alpha + 1;z)

. Te oots

z
=
i

y
v

(
n
)

(
v
=
1
,
2
,
⋯
)

z = iy_v^{(n)}(v = 1,2, \cdots )

of the eqation

g
(
z
)
=

C
n

(
z
)

g(z) = {C_n}(z)

have the same properties as those described in (2). Furthermore, they interlace: subject to a suitable ordering,

y
v

(
n
)

>

y
v

(
n
+
1
)

>

y

v
+
1

(
n
)

(
v
=
1
,
2
,
⋯
)

y_v^{(n)} > y_v^{(n + 1)} > y_{v + 1}^{(n)}(v = 1,2, \cdots )

. (A special case of this result concerns the function

e
z

{e^z}

and the convergents of its continued fraction expansion, and is an extension of the forula

e

2
i
v
π

=
1
(
v
=
1
,
2
,
⋯
)

{e^{2iv\pi }} = 1(v = 1,2, \cdots )

.)
On a nonlinear stochastic integral equation of the Hammerstein type
Authors: Padgett W. J. 
Published: June 2010

A nonlinear stochastic integral equation of the Hammerstein type in the form
\[

x
(
t
;
ω
)
=
h
(
t
;
ω
)
+

∫
s

k
(
t
,
s
;
ω
)
f
(
s
,
x
(
s
;
ω
)

)
d
μ
(
s
)

x(t;\omega ) = h(t;\omega ) + \int _s {k(t,s;\omega )f(s,x(s;\omega )} )d\mu (s)

\]
is studied where

t
∈
S
,
a

t \in S,a

, a

σ
\sigma

-finite measure space with certain properties,

ω
∈
Ω

\omega \in \Omega

, the supporting set of a probability measure space

(
Ω
,
A
,
P
)

(\Omega ,A,P)

, and the integral is a Bochner integral. A random solution of the equation is defined to be a second order vector-valued stochastic process

x
(
t
;
ω
)

x(t;\omega )

on

S
S

which satisfies the equation almost certainly. Using certain spaces of functions, which are spaces of second order vector-valued stochastic processes on

S
S

, and fixed point theory, several theorems are proved which give conditions such that a unique random solution exists.
On approximation in the Bers spaces
Authors: Chui Charles K. 
Published: June 2010

Let

D
D

be a Jordan domain in the complex plane with rectifiable boundary

C
C

. Let

A
q

(
D
)

{A_q}(D)

denote the Bers space with norm

|

|

|

|

q

||\;|{|_q}

. We prove that if

f
∈

A
q

(
D
)
,
2
>
q
>
∞

f \in {A_q}(D),2 > q > \infty

, then there exist functions

s
n

(
z
)
=

Σ

k
=
1

n

1

/

(
z
−

z

n
,
k

)
,

z

n
,
k

∈
C

for

k
=
1
,
⋯
,
n

{s_n}(z) = \Sigma _{k = 1}^n1/(z - {z_{n,k}}),\;{z_{n,k}} \in C{\text { for }}k = 1, \cdots ,n

, such that

|

|

s
n

−
f

|

|

q

→
0

||{s_n} - f|{|_q} \to 0

. This result does not hold for

1
>
q
≦
2

1 > q \leqq 2

even when

D
D

is a disc.
On a property of rational functions. II
Authors: Rahman Q. I. 
Published: June 2010

It is shown that if

r
n

(
z
)

{r_n}(z)

is a rational function of degree

n
n

such that

r
n

(
0
)
=
1
,

lim

|

z

|

→
∞

|

r
n

(
z
)

|

=
0

{r_n}(0) = 1,{\lim _{|z| \to \infty }}|{r_n}(z)| = 0

and all its poles lie in

|

ζ
1

|

≦

|

z

|

≦
1

|{\zeta _1}| \leqq |z| \leqq 1

then

max

|

z

|

=
ρ
>

|

ζ
1

|

|

r
n

(
z
)

|

≧
1

/

(
1
−

ρ
n

)

{\max _{|z| = \rho > |{\zeta _1}|}}|{r_n}(z)| \geqq 1/(1 - {\rho ^n})

.
On polynomials satisfying a Turán type inequality
Authors: Csordas George , Williamson Jack 
Published: June 2010

For Legendre polynomials

P
n

(
x
)

{P_n}(x)

, P. Turán has established the inequality
\[

Δ
n

(
x
)
=

P
n
2

(
x
)
−

P

n
+
1

(
x
)

P

n
−
1

(
x
)
≧
0
,

−
1
≦
x
≦
1
,
n
≧
1
,

{\Delta _n}(x) = P_n^2(x) - {P_{n + 1}}(x){P_{n - 1}}(x) \geqq 0,\quad - 1 \leqq x \leqq 1,n \geqq 1,

\]
with equality only for

x
=
±
1

x = \pm 1

. This inequality has generated considerable interest, and analogous inequalities have been extended to various classes of polynomials: ultraspherical, Laguerre, Hermite, and a class of Jacobi polynomials. Our purpose here is to determine necessary and sufficient conditions for a general class of polynomials to satisfy a Turán type inequality and to characterize the generating functions of such a class.
Conjugate point properties for an even order linear differential equation
Authors: Johnson George W. 
Published: June 2010

Consider an even order equation

D
n

y
=
0

{D_n}y = 0

, where

D
n

{D_n}

is defined recursively by

D
0

y
=
y
,

a

k
,
k
+
1

D
k

y
=
{
(

D

k
−
1

y

)
′

−

Σ

i
=
1

k

a

k
i

D

i
−
1

y
}

{D_0}y = y,{a_{k,k + 1}}{D_k}y = \{ ({D_{k - 1}}y)’ - \Sigma _{i = 1}^k{a_{ki}}{D_{i - 1}}y\}

, where

a

i
j

{a_{ij}}

is continuous on

[
0
,
∞
)
,

a

i
,
i
+
1

>
0
,
i
=
1
,
⋯
,
n
−
1
,

a

n
,
n
+
1

≡
1

[0,\infty ),{a_{i,i + 1}} > 0,i = 1, \cdots ,n - 1,{a_{n,n + 1}} \equiv 1

, and

a

i
j

≡
0

{a_{ij}} \equiv 0

for

j
>
i
+
1

j > i + 1

. The

k
k

th conjugate point function

η
k

(
a
)

{\eta _k}(a)

is defined and is shown to be a strictly increasing continuous function with domain

[
0
,
b
)

[0,b)

or

[
0
,
∞
)

[0,\infty )

. Extremal solutions are defined as nontrivial solutions with exactly

n
−
1
+
k

n - 1 + k

zeros on

[
a
,

η
k

(
a
)
]

[a,{\eta _k}(a)]

, and are shown to exist, to have odd order zeros at

a
a

and

η
k

(
a
)

{\eta _k}(a)

and exactly

k
−
1

k - 1

odd order zeros in

(
a
,

η
k

(
a
)
)

(a,{\eta _k}(a))

, thus establishing that

η
k

(
a
)
>

η

k
+
1

(
a
)

{\eta _k}(a) > {\eta _{k + 1}}(a)

if

η
k

(
a
)
>
∞

{\eta _k}(a) > \infty

.