bluuebunny/crossref_metadata_2025_split huggingface dataset:
PANCREATIC TRAUMA: UNIVERSITIES OF MELBOURNE HPB GROUP
Authors: Fleming William R. , Collier Neil A. , Banting Simon W. 
Published: July 2002

Background: Pancreatic trauma is uncommon, but carries high morbidity and mortality rates, especially when diagnosis is delayed or inappropriate surgery is attempted. Although the retroperitoneal position of the pancreas confers it some immunity to injury, the force required to do so often results in severe associated injuries to other organs, which may be life threatening. Diagnosis may be difficult and surgery can be a considerable technical challenge. Methods: All patients with pancreatic trauma who attended one of three Melbourne teaching hospitals from 1977 to 1998 were identified. Injuries were graded and the method of diagnosis and treatment studied. The incidence and causation of postoperative morbidity and mortality was identified. Results: Thirty‐eight patients (26 men and 12 women) were studied. Blunt trauma was responsible in 30 patients, stab wounds in five, gunshot wounds in two and iatrogenic injury in one. Injuries to other organs occurred in 30 patients. Surgical procedures were undertaken in 34 patients, resulting in the death of five and complications in 25. Conclusion: Complications and death are related to the associated injuries, as much as to the pancreatic injury itself. In this study, we review the experience of the management of pancreatic trauma in three large teaching hospitals in Melbourne over a 21‐year period, and suggest a strategy for dealing with these difficult patients. Adherence to the basic concepts of control of bleeding from associated vascular injury, minimization of contamination, accurate pancreatic assessment, judicious resection and adequate drainage can diminish the risk. By approaching the problem in a systematic way and adopting a generally conservative management plan, complications and deaths can be minimized in these complex cases.
TESTICULAR TORSION: TIME IS THE ENEMY
Authors: Dunne Patrick J. , O’Loughlin Barry S. 
Published: July 2002

Background: The acute scrotum is a diagnostic dilemma, and testicular torsion is of primary interest because of its fertility problems for the patient and medico‐legal issues for the surgeon. The present study aimed to correlate operative findings of patients with suspected testicular torsion with certain clinical variables and investigations to see if diagnosis and outcome could be improved. Methods: A total of 99 patients underwent scrotal exploration for suspected testicular torsion at the Royal Brisbane Hospital between 1990 and 1995. Colour Doppler ultrasound, white blood count and urine microscopy results were documented, along with the patient’s age and duration of testicular pain. Results: Fifty‐six patients were found to have torsion, and the testicular loss rate was 23%. Patients who experienced testicular pain for longer than 12 h had a testicular loss rate of 67%. A negative urine microscopy was suggestive of testicular torsion, but was not diagnostic. The white blood count did not aid in the diagnosis. Colour Doppler ultrasound of the scrotum was used on nine occasions with three false negative results and a sensitivity of only 57%. Conclusions: Time is the enemy when managing the acute scrotum. No investigation substantially improves clinical diagnosis enough to warrant any delay in definitive surgical intervention.
Characteristic functional structure of infinitesimal symmetry mappings of classical dynamical systems. I. Velocity-dependent mappings of second-order differential equations
Authors: Katzin Gerald H. , Levine Jack 
Published: July 2002

The primary purpose of this paper is to show that infinitesimal velocity-dependent symmetry mappings [(a) x̄i =xi +δxi, δxi  ≡ ξi(ẋ,x,t)δa with associated change in path parameter (b) t̄=t+δt, δt ≡ ξ0(ẋ,x,t)] of classical (including relativistic) particle systems (c) Ei(ẍ,ẋ,x,t) =0 are expressible in a form with a characteristic functional structure which is the same for all dynamical systems (c) and is manifestly dependent upon constants of motion of the system. In this characteristic form the symmetry mappings are determined by (d) ξi =Zi(ẋ,x,t) +ẋiξ0,ξ0 arbitrary; the functions Zi appearing in (d) have the form (e) Zi =BAgiA(C1,...,Cr; t), 0≤r≤2n, A=1,...,2n, where the BA are arbitrary constants of motion and the C’s appearing in the functions giA are specified constants of motion.
A procedure is given to determine the giA. For Lagrangian systems it follows that velocity-dependent Noether mappings are a subclass of the above-mentioned general symmetry mappings of the form (a)–(e). An analysis of velocity-dependent Noether mapping theory is included in order to compare for Lagrangian systems the procedure for obtaining the characteristic form (e) of the general mappings with the procedure for obtaining the well-known formula (f) ZiN =Hij(ẋ,x,t)∂Z/∂ẋ j (Z ≡  constant of motion), characteristic of velocity-dependent Noether mappings. It is shown how any given velocity-dependent symmetry mapping function Zi(ẋ,x,t) (including Noether mappings) can be expressed in the form (e). A collection of variational formulas and identities is derived in order to develop from first principles the velocity-dependent symmetry mapping theory. Throughout, comparisons are made between velocity-dependent and velocity-independent symmetry theory.
Ethanol metabolism by macrophages: possible role in organ damage
Authors: Wickramasinghe S. N. 
Published: July 2002

AbstractMacrophages and hepatocytes oxidize ethanol to acetate in vitro at comparable rates but by different biochemical pathways. Ethanol metabolism by macrophages is largely ADH‐independent and mainly based on cytochrome P450 and on the extracellular release of superoxide anion radicals. There is also evidence that during ethanol metabolism, macrophages release more acetaldehyde extracellularly than hepatocytes; the high concentrations of acetaldehyde around macrophages may damage surrounding tissue cells. Some of this acetaldehyde forms unstable cytotoxic complexes with serum albumin and with erythrocytes. The superoxide anion radicals released by macrophages may not only oxidize ethanol to acetaldehyde but also react with and damage cells in their immediate vicinity. After exposure to ethanol, macrophage‐depleted rodents show markedly reduced levels of cytotoxic acetaldehyde‐albumin complexes in the blood and reduced levels of hydroxyethyl radicals in the bile compared to control animals, indicating that the generation of such potentially pathogenic molecules is, to a large extent, dependent on macrophage activity. Macrophage‐depleted animals also show less early liver damage than control animals. The reduction in ethanol‐induced liver damage in macrophage‐depleted mice and rats may be due to a reduction or elimination of the generation of various Kupffer‐cell‐derived hepatotoxic substances, including acetaldehyde and reactive oxygen radicals, in such animals. These data suggest that ethanol metabolism by tissue macrophages may play an important role in mediating ethanol‐related tissue damage.
Using an interdisciplinary approach to training to develop the quality of communication with adults with profound learning disabilities by care staff
Authors: Dobson Susan , Upadhyaya Shripati , Stanley Brian 
Published: July 2002

AbstractThe study examines the effects of training on care staff's communication to service users with profound and severe learning disabilities. An interdisciplinary therapeutic training programme aimed to improve care staff's cognitions about disability, to alter their communicative practice and to provide them with practical solutions to the communication difficulties of the service users. Participants were each video recorded with a single service user before and during training. The recordings formed the basis of the communication workshops. Both baseline and final recordings were evaluated using criteria based on verbal and non‐verbal interactions. The criteria assessed alterations in the care staff's language use, verbal responses, and interpretation and praise of service users' communication. The use of posture, position, eye contact and gaze monitoring during interactions was also evaluated. The pre‐ and post‐training analysis demonstrated alterations and improvements in care staff's use of verbal communication as well as gaze monitoring and position during interactions. A follow‐up study 6 months later showed many of the post‐training positive changes had been maintained or had continued to develop. It is suggested that the success and implementation of the communication training in the workplace was due to the collaborative therapeutic nature of the training package. The interdisciplinary and integrated training programme had facilitated a willingness to change and led to the development of a positive view of each other's practice. The recommendations from each of the therapist's training programme formed an integral part of the care staff's new care plans for each service user. This contributed to the continuing development the care staff's therapeutic role.
Statistical extensions of some classical Tauberian theorems
Authors: Fridy J. , Khan M. 
Published: July 2002

Hardy’s well-known Tauberian theorem for Cesàro means says that if the sequence

x
x

satisfies

lim
C
x
=
L

\lim Cx = L

and

Δ

x
k

=
O
(
1

/

k
)

\Delta x_k = O (1/k)

, then

lim
x
=
L

\lim x = L

. In this paper it is shown that the hypothesis

lim
C
x
=
L

\lim Cx = L

can be replaced by the weaker assumption of the statistical limit: st-lim

C
x
=
L

Cx = L

, i.e., for every

ϵ
>
0

\epsilon >0

,

lim

n

−
1

|

{
k
≤
n
:

|

(
C
x

)
k

−
L

|

≥
ϵ
}

|

=
0

\lim n^{-1} | \{ k \leq n: | ( Cx)_k - L | \geq \epsilon \} | = 0

. Similarly, the “one-sided” Tauberian theorem of Landau and Schmidt’s Tauberian theorem for the Abel method are extended by replacing

lim
C
x

\lim Cx

and

lim
A
x

\lim Ax

with st-lim

C
x

Cx

and st-lim

A
x

Ax

, respectively. The Hardy-Littlewood Tauberian theorem for Borel summability is also extended by replacing

lim
t

(
B
x

)
t

=
L

\lim _t (Bx)_t=L

, where

t
t

is a continuous parameter, with

lim
n

(
B
x

)
n

=
L

\lim _n (Bx)_n =L

, and further replacing it by

(

B

∗

)

(B^{*})

-st-lim

B

∗

x
=
L

B^{*} x =L

, where

B

∗

B^{*}

is the Borel matrix method.
On algebraic polynomials with random coefficients
Authors: Farahmand K. 
Published: July 2002

The expected number of real zeros and maxima of the curve representing algebraic polynomial of the form

a
0

(

n
−
1

0

)

1

/

2

+

a
1

(

n
−
1

1

)

1

/

2

x
+

a
2

(

n
−
1

2

)

1

/

2

x
2

+
⋯
+

a

n
−
1

(

n
−
1

n
−
1

)

1

/

2

x

n
−
1

a_0\binom {n-1}{0}^{1/2}+a_1\binom {n-1}{1}^{1/2}x +a_2\binom {n-1}{2}^{1/2}x^2+\cdots +a_{n-1}\binom {n-1}{n-1}^{1/2}x^{n-1}

where

a

j

,
j
=
0
,
1
,
2
,
…
,
n
−
1

a_{j}, j= 0, 1, 2, \ldots , n-1

, are independent standard normal random variables, are known. In this paper we provide the asymptotic value for the expected number of maxima which occur below a given level. We also show that most of the zero crossings of the curve representing the polynomial are perpendicular to the

x
x

axis. The results show a significant difference in mathematical behaviour between our polynomial and the random algebraic polynomial of the form

a

0

+

a

1

x
+

a

2

x

2

+
⋯
+

a

n
−
1

x

n
−
1

a_{0}+a_{1}x +a_{2}x^{2}+\cdots +a_{n- 1}x^{n-1}

which was previously the most studied.
An addition theorem for the color number
Authors: Aarts Jan , Fokkink Robbert 
Published: July 2002

There is a close relation between the color number of a continuous map

f
:
X
→
X

f\colon X \to X

without fixed points and the topological dimension. If

f
f

is an involution, the color number is also related to the co-index. An addition theorem for the color number is established thus underscoring the interrelations between color number, dimension and co-index.
Primary decomposition: Compatibility, independence and linear growth
Authors: Yao Yongwei 
Published: July 2002

For finitely generated modules

N
⊊
M

N \subsetneq M

over a Noetherian ring

R
R

, we study the following properties about primary decomposition: (1) The Compatibility property, which says that if

Ass
⁡
(
M

/

N
)
=
{

P
1

,

P
2

,
…
,

P
s

}

\operatorname {Ass} (M/N)=\{ P_1, P_2, \dots , P_s\}

and

Q
i

Q_i

is a

P
i

P_i

-primary component of

N
⊊
M

N \subsetneq M

for each

i
=
1
,
2
,
…
,
s

i=1,2,\dots ,s

, then

N
=

Q
1

∩

Q
2

∩
⋯
∩

Q
s

N =Q_1 \cap Q_2 \cap \cdots \cap Q_s

; (2) For a given subset

X
=
{

P
1

,

P
2

,
…
,

P
r

}
⊆
Ass
⁡
(
M

/

N
)

X=\{ P_1, P_2, \dots , P_r \} \subseteq \operatorname {Ass}(M/N)

,

X
X

is an open subset of

Ass
⁡
(
M

/

N
)

\operatorname {Ass}(M/N)

if and only if the intersections

Q
1

∩

Q
2

∩
⋯
∩

Q
r

=

Q
1
′

∩

Q
2
′

∩
⋯
∩

Q
r
′

Q_1 \cap Q_2\cap \cdots \cap Q_r= Q_1’ \cap Q_2’ \cap \cdots \cap Q_r’

for all possible

P
i

P_i

-primary components

Q
i

Q_i

and

Q
i
′

Q_i’

of

N
⊊
M

N\subsetneq M

; (3) A new proof of the ‘Linear Growth’ property, which says that for any fixed ideals

I
1

,

I
2

,
…
,

I
t

I_1, I_2, \dots , I_t

of

R
R

there exists a

k
∈

N

k \in \mathbb N

such that for any

n
1

,

n
2

,
…
,

n
t

∈

N

n_1, n_2, \dots , n_t \in \mathbb N

there exists a primary decomposition of

I
1

n
1

I
2

n
2

⋯

I
t

n
t

M
⊂
M

I_1^{n_1}I_2^{n_2}\cdots I_t^{n_t}M \subset M

such that every

P
P

-primary component

Q
Q

of that primary decomposition contains

P

k
(

n
1

+

n
2

+
⋯
+

n
t

)

M

P^{k(n_1+n_2+\cdots +n_t)}M

.
The identical equation in 𝜓-products
Authors: Sitaramaiah V. , Subbarao M. 
Published: July 2002

In Bull. Amer. Math. Soc. 36 (1930), 762–772, R. Vaidyanatha- swamy established a remarkable identity valid for any multiplicative arithmetic function and involving Dirichlet convolution. D. H. Lehmer (Trans. Amer. Math. Soc. 33 (1931), 945–952) introduced a very general class of arithmetical convolutions, called

ψ
\psi

-products, which include the well-known Dirichlet products, Eckford Cohen’s unitary convolutions, and in fact Narkiewicz’s so-called regular

A
A

-convolutions. In this paper, we establish an identical equation valid for multiplicative arithmetic functions and Lehmer’s

ψ
\psi

-convolutions which yields, as special cases, all known identical equations valid for the Dirichlet and unitary convolutions, besides establishing identical equations for several new convolutions.
On nonlinear 𝑛-widths
Authors: Dung Dinh , Thanh Vu 
Published: July 2002

For characterization of best nonlinear approximation, DeVore, Howard, and Micchelli have recently suggested the nonlinear

n
n

-width

δ
n

(
W
,
X
)

\delta _n(W,X)

of a subset

W
W

in a normed linear space

X
X

. We proved by a topological method that for

δ
n

(
W
,
X
)

\delta _n(W,X)

and the well-known Aleksandrov

n
n

-width

a
n

(
W
,
X
)

a_n(W,X)

in a Banach space

X
X

the following inequalities hold:

δ

2
n
+
1

(
W
,
X
)
≤

a
n

(
W
,
X
)
≤

δ
n

(
W
,
X
)

\delta _{2n+1}(W,X)\le a_n(W,X)\le \delta _n(W,X)

. Let

K

p
,
θ

α

K_{p,\theta }^{\alpha }

be the unit ball of Besov space

B

p
,
θ

α

,

α
>
0
,

1
≤
p
,
θ
≤
∞

B_{p,\theta }^{\alpha },\quad \alpha >0,\quad 1\le p,\theta \le \infty

, of multivariate periodic functions. Then for approximation in

L
q

,

1
≤
q
≤
∞

L_q,\quad 1\le q\le \infty

, with some restriction on

p
,
q

p,q

and

α
\alpha

, we established the asymptotic degree of these

n
n

-widths:

a
n

(

K

p
,
θ

α

,

L
q

)
≈

δ
n

(

K

p
,
θ

α

,

L
q

)
≈

n

−
α

a_n(K_{p,\theta }^{\alpha },L_q)\approx \delta _n(K_{p,\theta }^{\alpha }, L_q)\approx n^{-\alpha }

.
Mean theoretic approach to the grand Furuta inequality
Authors: Fujii Masatoshi , Kamei Eizaburo 
Published: July 2002

Very recently, Furuta obtained the grand Furuta inequality which is a parameteric formula interpolating the Furuta inequality and the Ando-Hiai inequality as follows : If

A
≥
B
≥
0

A \ge B \ge 0

and

A
A

is invertible, then for each

t
∈
[
0
,
1
]

t \in [0,1]

,

F

p
,
t

(
A
,
B
,
r
,
s
)
=

A

−
r

/

2

{

A

r

/

2

(

A

−
t

/

2

B

p

A

−
t

/

2

)

s

A

r

/

2

}

1
−
t
+
r

(
p
−
t
)
s
+
r

A

−
r

/

2

\begin{equation*}F_{p,t}(A,B,r,s) = A^{-r/2}\{A^{r/2}(A^{-t/2}B^{p}A^{-t/2})^{s}A ^{r/2}\}^{\frac {1-t+r}{(p-t)s+r}}A^{-r/2} \end{equation*}

is a decreasing function of both

r
r

and

s
s

for all

r
≥
t
,

p
≥
1

r \ge t, ~p \ge 1

and

s
≥
1

s \ge 1

. In this note, we employ a mean theoretic approach to the grand Furuta inequality. Consequently we propose a basic inequality, by which we present a simple proof of the grand Furuta inequality.
Paracompact subspaces in the box product topology
Authors: Nyikos Peter , Piatkiewicz Leszek 
Published: July 2002

In 1975 E. K. van Douwen showed that if

(

X
n

)

n
∈
ω

( X_n )_{ n \in \omega }

is a family of Hausdorff spaces such that all finite subproducts

∏

n
>
m

X
n

\prod _{ n > m } X_n

are paracompact, then for each element

x
x

of the box product

◻

n
∈
ω

X
n

\square _{n \in \omega } X_n

the

σ
\sigma

-product

σ
(
x
)
=
{
y
∈

◻

n
∈
ω

X
n

:
{
n
∈
ω
:
x
(
n
)
≠
y
(
n
)
}
is finite
}

\sigma ( x ) = \{ y \in \square _{n \in \omega } X_n : \{ n \in \omega : x (n) \neq y (n) \} \text { is finite} \}

is paracompact. He asked whether this result remains true if one considers uncountable families of spaces. In this paper we prove in particular the following result: Theorem Let

κ
\kappa

be an infinite cardinal number, and let

(

X

α

)

α
∈
κ

( X_{\alpha } )_{\alpha \in \kappa }

be a family of compact Hausdorff spaces. Let

x
∈

◻

=

◻

α
∈
κ

X
α

x \in \square = \square _{\alpha \in \kappa } X_\alpha

be a fixed point. Given a family

R

\mathcal {R}

of open subsets of

◻

\square

which covers

σ
(
x
)

\sigma ( x )

, there exists an open locally finite in

◻

\square

refinement

S

\mathcal {S}

of

R

\mathcal {R}

which covers

σ
(
x
)

\sigma ( x )

. We also prove a slightly weaker version of this theorem for Hausdorff spaces with “all finite subproducts are paracompact" property. As a corollary we get an affirmative answer to van Douwen’s question.
A counterexample to a 𝐵𝑃-analogue of the chromatic splitting conjecture
Authors: Devinatz Ethan 
Published: July 2002

We prove that, if

n
≥
2

n\ge 2

, the

E
(
n
−
1

)
∗

E(n-1)_\ast

-localization of the

K
(
n

)
∗

K(n)_\ast

-localization map

B

P
p

→

L

K
(
n
)

B
P

BP_p\to L_{K(n)}BP

is not a split monomorphism in the stable category by exhibiting spectra

Z
Z

for which the map

π
∗

(

L

n
−
1

(
B

P
p

)
∧
Z
)
→

π
∗

(

L

n
−
1

(

L

K
(
n
)

B
P
)
∧
Z
)

\pi _\ast (L_{n-1}(BP_p)\wedge Z)\to \pi _\ast (L_{n-1}(L_{K(n)}BP)\wedge Z)

is not injective. If

p
≥
max
{

1
2

(

n
2

−
2
n
+
2
)
,
n
+
1
}

p\ge \max \{\frac {1}{2}(n^2-2n +2), n+1\}

and

n
≥
3

n\geq 3

, we show that

Z
Z

may be taken to be a two-cell complex in the sense of

E
(
n
−
1

)
∗

E(n-1)_\ast

-local homotopy theory. The question of whether the map

L

n
−
1

(
B

P
p

)
→

L

n
−
1

L

K
(
n
)

B
P

L_{n-1}(BP_p)\to L_{n-1}L_{K(n)}BP

splits was asked by Hovey and is in some sense a

B
P

BP

-analogue of Hopkins’ chromatic splitting conjecture.
A pointwise spectrum and representation of operators
Authors: Bertoglio N. , Martínez Servet , Martín Jaime 
Published: July 2002

For a self-adjoint operator

A
:
H
→
H

A:H\to H

commuting with an increasing family of projections

P

=
(

P

t

)

{\mathcal {P}}=(P_{t})

we study the multifunction

t
→

Γ

T

(
t
)
=
⋂
{

σ

I

:
I

t\to \Gamma ^{\mathcal {T}}(t)=\bigcap \{\sigma _{I}:I

an open set of the topology

T

{\mathcal {T}}

containing

t
}

t\}

, where

σ

I

\sigma _{I}

is the spectrum of

A
A

on

P

I

H

P_{I}H

. Let

m

P

m_{\mathcal {P}}

be the measure of maximal spectral type. We study the condition that

Γ

T

\Gamma ^{\mathcal {T}}

is essentially a singleton,

m

P

{
t
:

Γ

T

(
t
)

m_{\mathcal {P}}\{t:\Gamma ^{\mathcal {T}}(t)

is not a singleton

}
=
0

\}=0

. We show that if

T

{\mathcal {T}}

is the density topology and if

m

P

m_{\mathcal {P}}

satisfies the density theorem, in particular if it is absolutely continuous with respect to the Lebesgue measure, then this condition is equivalent to the fact that

A
A

is a Borel function of

P

{\mathcal {P}}

. If

T

{\mathcal {T}}

is the usual topology then the condition is equivalent to the fact that

A
A

is approched in norm by step functions

∑

n
∈

N

Γ

T

(

α

n

)
⟨

P

I

n

f
,
f
⟩

\sum \limits _{n\in \mathbb {N}}\Gamma ^{\mathcal {T}} (\alpha _{n})\langle P_{I_{n}} f,f\rangle

, where the set of intervals

{

I

n

:
n
∈

N

}

\{I_{n}:n\in \mathbb {N}\}

covers the set where

Γ

T

\Gamma ^{\mathcal {T}}

is a singleton.
A new characterization of 𝑃𝑟𝑜𝑗¹𝒳=0 for countable spectra of (LB)-spaces
Authors: Wengenroth Jochen 
Published: July 2002

The derived projective limit functor Proj

1

^1

is a very useful tool for investigating surjectivity problems in various parts of analysis (e.g. solvability of partial differential equations). We provide a new characterization for vanishing Proj

1

^1

on projective spectra of (LB)-spaces which improves a classical result of V. P. Palamodov and V. S. Retakh.
Norm of convolution by operator-valued functions on free groups
Authors: Buchholz Artur 
Published: July 2002

We present a connection between the Leinert sets and the non-crossing two-partitions and we use this connection to give a simple proof of the free Khintchine inequality in discrete non-commutative

L
p

L_p

-spaces. Moreover we extend the inequality of Haagerup-Pisier,
\[

‖

∑

g
∈
S

λ
(
g
)
⊗

a
g

‖

C
λ
∗

(

F
n

)
⊗
A

≤
2
max

{

‖

∑

g
∈
S

a
g
∗

a
g

‖

1
2

,

‖

∑

g
∈
S

a
g

a
g
∗

‖

1
2

}

,

\left \| \sum _{g\in S} \lambda (g)\otimes a_g\right \|_{C_\lambda ^*(F_n)\otimes A} \le 2\max \left \{\left \| \sum _{g\in S} a_g^*a_g\right \|^{\frac 12}, \left \|\sum _{g\in S} a_g a_g^*\right \|^{\frac 12}\right \},

\]
where

λ
\lambda

is the left regular representation of the group

F
n

F_n

,

a
g

a_g

are elements of the

C
∗

C^*

-algebra

A
A

, and

S
S

is the set of the words with length one, to the set

S
S

of the words with arbitrary fixed length.
A simple proof of a curious congruence by Sun
Authors: Shan Zun , Wang Edward 
Published: July 2002

In this note, we give a simple and elementary proof of the following curious congruence which was established by Zhi-Wei Sun:
\[

∑

k
=
1

(
p
−
1
)

/

2

1

k
⋅

2
k

≡

∑

k
=
1

[
3
p

/

4
]

(
−
1

)

k
−
1

k

(

m
o
d

p
)
.

\sum ^{(p-1)/2}_{k=1}\frac {1}{k\cdot 2^k}\equiv \sum ^{[3p/4]}_{k=1} \frac {(-1)^{k-1}}{k}\quad (\mathrm {mod}\,p).

\]
An extension of the work of V. Guillemin on complex powers and zeta functions of elliptic pseudodifferential operators
Authors: Bucicovschi Bogdan 
Published: July 2002

The purpose of this note is to extend the methods and results of Guillemin on elliptic self-adjoint pseudodifferential operators of order one, from operators defined on smooth functions on a closed manifold to operators defined on smooth sections in a vector bundle. The case of bundles of Hilbert modules of finite type over a finite von Neumann algebra will also be treated.
Periodic solutions of a periodic delay predator-prey system
Authors: Yongkun Li 
Published: July 2002

The existence of a positive periodic solution for

{

d

H
(
t
)

d

t

=
r
(
t
)
H
(
t
)

[
1
−

H
(
t
−
τ
(
t
)
)

K
(
t
)

]

−
α
(
t
)
H
(
t
)
P
(
t
)
,

d

P
(
t
)

d

t

=
−
b
(
t
)
P
(
t
)
+
β
(
t
)
P
(
t
)
H
(
t
−
σ
(
t
)
)

\begin{equation*} \begin {cases} \frac {\mathrm {d}H(t)}{\mathrm {d}t}=r(t)H(t) \left [1-\frac {H(t-\tau (t))}{K(t)}\right ] -\alpha (t)H(t) P(t),\ \frac {\mathrm {d}P(t)}{\mathrm {d}t}=-b(t)P(t)+\beta (t)P(t)H(t-\sigma (t)) \end{cases} \end{equation*}

is established, where

r
r

,

K
K

,

α
\alpha

,

b
b

,

β
\beta

are positive periodic continuous functions with period

ω
>
0

\omega >0

, and

τ
\tau

,

σ
\sigma

are periodic continuous functions with period

ω
\omega

.