# Rocky Mountain Journal of Mathematics

Published by Rocky Mountain Mathematics Consortium

Print ISSN: 0035-7596

Published by Rocky Mountain Mathematics Consortium

Print ISSN: 0035-7596

Publications

A $\mathbf{Q}$-Cartier divisor $D$ on a projective variety $M$ is {\it almost nup}, if $(D , C) > 0$ for every very general curve $C$ on $M$. An algebraic variety $X$ is of {\it almost general type}, if there exists a projective variety $M$ with only terminal singularities such that the canonical divisor $K_M$ is almost nup and such that $M$ is birationally equivalent to $X$. We prove that a complex algebraic variety is of almost general type if and only if it is neither uniruled nor covered by any family of varieties being birationally equivalent to minimal varieties with numerically trivial canonical divisors, under the minimal model conjecture. Furthermore we prove that, for a projective variety $X$ with only terminal singularities, $X$ is of almost general type if and only if the canonical divisor $K_X$ is almost nup, under the minimal model conjecture. Comment: 11 pages, LaTeX2e; The published version

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- 30 Reads
- 1 Citation

In this article, we derive some identities for multilateral basic hypergeometric series associated to the root system A_n. First, we apply Ismail's argument to an A_n q-binomial theorem of Milne and derive a new A_n generalization of Ramanujan's 1-psi-1 summation theorem. From this new A_n 1-psi-1 summation and from an A_n 1-psi-1 summation of Gustafson we deduce two lemmas for deriving simple A_n generalizations of bilateral basic hypergeometric series identities. These lemmas are closely related to the Macdonald identities for A_n. As samples for possible applications of these lemmas, we provide several A_n extensions of Bailey's 2-psi-2 transformations, and several A_n extensions of a particular 2-psi-2 summation.

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- 136 Reads
- 10 Citations

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy both conditions below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is diagonal and the matrix representing $A$ is irreducible tridiagonal. We call such a pair a {\it Leonard pair} on $V$. Referring to the above Leonard pair, we investigate 24 bases for $V$ on which the action of $A$ and $A^*$ takes an attractive form. With respect to each of these bases, the matrices representing $A$ and $A^*$ are either diagonal, lower bidiagonal, upper bidiagonal, or tridiagonal.

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- 16 Reads
- 86 Citations

A congruence relation satisfied by Igusa's cusp form of weight 35 is
presented. As a tool to confirm the congruence relation, a Sturm-type theorem
for the case of odd-weight Siegel modular forms of degree 2 is included.

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- 85 Reads
- 13 Citations

Let $B$ be a smooth projective surface, and $\mathcal{L}$ an ample line
bundle on $B$. The aim of this parer is to study the families of elliptic
Calabi--Yau threefolds sitting in the bundle $\mathbb{P}(\mathcal{L}^a \oplus
\mathcal{L}^b \oplus \mathcal{O}_B)$ as anticanonical divisors. I will show
that the number of such families is finite.

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- 30 Reads

Suppose $m(\alpha)$ denotes the Mahler measure of the non-zero algebraic
number $\alpha$. For each positive real number $t$, the author studied a
version $m_t(\alpha)$ of the Mahler measure that has the triangle inequality.
The construction of $m_t$ is generic, and may be applied to a broader class of
functions defined on any Abelian group $G$. We prove analogs of known results
with an abstract function on $G$ in place of the Mahler measure. In the
process, we resolve an earlier open problem stated by the author regarding
$m_t(\alpha)$.

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- 17 Reads
- 6 Citations

We give a conjectural description for the kernel of the map assigning to each finite $\mathbb Z_p$-free $G\times\mathbb Z_p$-set its rational permutation module where G is a finite p-group. We prove that this conjecture is true when G is an elementary abelian p-group or a cyclic p-group.

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- 16 Reads
- 2 Citations

We study the reduced Beurling spectra $sp_{\Cal {A},V} (F)$ of functions
$F \in L^1_{loc} (\jj,X)$ relative to certain function spaces $\Cal{A}\st
L^{\infty}(\jj,X)$ and $V\st L^1 (\r)$ and compare them with other spectra
including the weak Laplace spectrum. Here $\jj$ is $\r_+$ or $\r$ and $X$ is a
Banach space. If $F$ belongs to the space $ \f'_{ar}(\jj,X)$ of absolutely
regular distributions and has uniformly continuous indefinite integral with
$0\not\in sp_{\A,\f(\r)} (F)$ (for example if F is slowly oscillating and $\A$
is $\{0\}$ or $C_0 (\jj,X)$), then $F$ is ergodic. If $F\in \f'_{ar}(\r,X)$ and
$M_h F (\cdot)= \int_0^h F(\cdot+s)\,ds$ is bounded for all $h > 0$ (for
example if $F$ is ergodic) and if $sp_{C_0(\r,X),\f} (F)=\emptyset$, then
${F}*\psi \in C_0(\r,X)$ for all $\psi\in \f(\r)$.
We show that tauberian theorems for Laplace transforms follow from results
about reduced spectra. Our results are more general than previous ones and we
demonstrate this through examples

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- 26 Reads

It is well known that a countable group admits a left-invariant total order if and only if it acts faithfully on R by orientation preserving homeomorphisms. Such group actions are special cases of group actions on simply connected 1-manifolds, or equivalently, actions on oriented order trees. We characterize a class of left-invariant partial orders on groups which yield such actions, and show conversely that groups acting on oriented order trees by order preserving homeomorphism admit such partial orders as long as there is an action with a point whose stabilizer is left-orderable.

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- 14 Reads
- 1 Citation

We consider a cocompact discrete reflection group $W$ of a CAT(0) space $X$. Then $W$ becomes a Coxeter group. In this paper, we study an analogy between the Davis-Moussong complex $\Sigma(W,S)$ and the CAT(0) space $X$, and show several analogous results about the limit set of a parabolic subgroup of the Coxeter group $W$.

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- 12 Reads
- 2 Citations

Let $A$ be a stably finite simple unital $C^*$-algebra and suppose $\alpha $ is an action of a finite group $G$ with the tracial Rokhlin property. Suppose further $A$ has real rank zero and the order on projections over $A$ is determined by traces. Then the crossed product $C^*$-algebra $C^*(G,A, \alpha)$ also has real rank zero and order on projections over $A$ is determined by traces. Moreover, if $A$ also has stable rank one, then $C^*(G,A, \alpha)$ also has stable rank one. Comment: 10 pages, 0 figures. Minor corrections made and typos corrected. To appear in Rocky Mountain Journal of Mathematics

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- 98 Reads
- 16 Citations

Morita equivalence of twisted inverse semigroup actions and discrete twisted partial actions are introduced. Morita equivalent actions have Morita equivalent crossed products.

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- 19 Reads
- 8 Citations

We define inverse semigroup actions on topological groupoids by partial
equivalences. From such actions, we construct saturated Fell bundles over
inverse semigroups and non-Hausdorff \'etale groupoids. We interpret these as
actions on C*-algebras by Hilbert bimodules and describe the section algebras
of these Fell bundles.
Our constructions give saturated Fell bundles over non-Hausdorff \'etale
groupoids that model actions on locally Hausdorff spaces. We show that these
Fell bundles are usually not Morita equivalent to an action by automorphisms.
That is, the Packer-Raeburn Stabilisation Trick does not generalise to
non-Hausdorff groupoids.

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- 108 Reads
- 25 Citations

A result of Graber, Harris, and Starr shows that a rationally connected variety defined over the function field of a curve over the complex numbers always has a rational point. Similarly, a separably rationally connected variety over a finite field or the function field of a curve over any algebraically closed field will have a rational point. Here we show that rationally connected varieties over the maximally unramified extension of the p-adics usually, in a precise sense, have rational points. This result is in the spirit of Ax and Kochen's result saying that the p-adics are usually $C_{2}$ fields. The method of proof utilizes a construction from mathematical logic called the ultraproduct. The ultraproduct is used to lift the de Jong, Starr result in the equicharacteristic case to the mixed characteristic case. Comment: 16 pages

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- 25 Reads
- 3 Citations

In this short note, we give a proof of the Riemann hypothesis for Goss
$v$-adic zeta function $\zeta_{v}(s)$, when $v$ is a prime of
$\mathbb{F}_{q}[t]$ of degree one.

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- 80 Reads

The dimension of any module over an algebra of affiliated operators ${\mathcal U}$ of a finite von Neumann algebra ${\mathcal A}$ is defined using a trace on ${\mathcal A}.$ All zero-dimensional ${\mathcal U}$-modules constitute the torsion class of torsion theory $(\mathrm{{\bf T}},\mathrm{{\bf P}})$. We show that every finitely generated ${\mathcal U}$-module splits as the direct sum of torsion and torsion-free part. Moreover, we prove that the theory $(\mathrm{{\bf T}},\mathrm{{\bf P}})$ coincides with the theory of bounded and unbounded modules and also with the Lambek and Goldie torsion theories. Lastly, we use the introduced torsion theories to give the necessary and sufficient conditions for ${\mathcal U}$ to be semisimple. Comment: To appear in Rocky Mountain Journal of Mathematics

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- 27 Reads

It is shown that if the processes $B$ and $f(B)$ are both Brownian motions
then $f$ must be an affine function. As a by-product of the proof, it is shown
that the only functions which are solutions to both the Laplace equation and
the eikonal equation are affine.

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- 27 Reads
- 2 Citations

We describe the C∗-Algebra generated by an irreducible Toeplitz operator T , with continuous symbol on the unit circle T, and finitely many composition operators on the Hardy space H2 induced by certain linear fractional self-maps of the unit disc, modulo the ideal of compact operators K(H2). For specific automorphisminduced composition operators and certain types of irreducible Toeplitz operators, we show that the above C∗-Al- gebra is not isomorphic to that generated by the shift and composition operators.

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- 132 Reads
- 5 Citations

The conditions on a Banach space, $E$, under which the algebra,
$\mathcal{K}(E)$, of compact operators on $E$ is right flat or homologically
unital are investigated. These homological properties are related to
factorization in the algebra and it is shown that, for $\mathcal{K}(E)$, they
are closely associated with the approximation property for $E$. The class of
spaces, $E$, such that $\mathcal{K}(E)$ is known to be right flat and
homologically unital is extended to include spaces which do not have the
bounded compact approximation property.

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- 12 Reads

Bounded orbit injection equivalence is an equivalence relation defined on
minimal free Cantor systems which is a candidate to generalize flip Kakutani
equivalence to actions of the Abelian free groups on more than one generator.
This paper characterizes bounded orbit injection equivalence in terms of a mild
strengthening of Rieffel-Morita equivalence of the associated
C*-crossed-product algebras. Moreover, we construct an ordered group which is
an invariant for bounded orbit injection equivalence, and does not agrees with
the K\_0 group of the associated C*-crossed-product in general. This new
invariant allows us to find sufficient conditions to strengthen bounded orbit
injection equivalence to orbit equivalence and strong orbit equivalence.

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- 43 Reads
- 7 Citations

Mark Kac gave one of the first results analyzing random polynomial zeros. He considered the case of independent standard normal coefficients and was able to show that the expected number of real zeros for a degree n polynomial is on the order of (2/pi)log(n), as n goes to infinity. Several years later, Sambandham considered two cases with some dependence assumed among the coefficients. The first case looked at coefficients with an exponentially decaying covariance function, while the second assumed a constant covariance. He showed that the expectation of the number of real zeros for an exponentially decaying covariance matches the independent case, while having a constant covariance reduces the expected number of zeros in half. In this paper we will apply techniques similar to Sambandham's and extend his results to a wider class of covariance functions. Under certain restrictions on the spectral density, we will show that the order of the expected number of real zeros remains the same as in the independent case. Comment: 14 pages, final edited version incorporating referee's suggestion to substantially shorten several arguments. To appear in the Rocky Mountain Journal of Mathematics

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- 16 Reads
- 7 Citations

We discuss existence of explicit search bounds for zeros of polynomials with coefficients in a number field. Our main result is a theorem about the existence of polynomial zeros of small height over the field of algebraic numbers outside of unions of subspaces. All bounds on the height are explicit. Comment: 10 pages, revised version: to appear in Rocky Mountain Journal of Mathematics; minor editorial revisions, removed section 5 for brevity of exposition

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- 16 Reads
- 2 Citations

We find the limiting proportion of periodic points in towers of finite fields
for polynomial maps associated to algebraic groups, namely pure power maps z^d
and Chebyshev polynomials.

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- 44 Reads
- 6 Citations

It was proved over a century ago that an algebraic curve C in the real projective plane, of degree n, has at most connected components. If C is nonslngular, then each of its commponents is a topological circle. A circle in the projectlve plane either separates it into a disk (the interior of the circle) and a Möbius band (the circle's exterior), or does not separate it. In the former case, the circle is an oval. If C is nonsingular, then all its components are ovals if n is even, and all except one are ovals if n is odd. An oval is included in another if it lies in the other's interior. The topological type of (a nonsingular) C is completely determined by (1) the parity of n, (2) how many ovals it has, and (3) the partial ordering of its ovals by inclusion. We present an algorithm which, given a homogeneous polinomial f(x,y,z) of degree n with integer coefficients, checks whether tlte curve defined hy f = 0 is nonsingular and if so, computes its topological type. The algorithm's maximum computing time is O(n27L(d)3), where d is the sum of the absolute values of the integer coofficients of f, and L(d) is the length of d.

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- 20 Reads
- 78 Citations

Let I = (x(upsilon 1), ... , x(upsilon q)) be a square-free monomial ideal of a polynomial ring K[x(1), ... , x(n)] over an arbitrary field K, and let A be the incidence matrix with column vectors upsilon(1), ... , upsilon(q). We will establish some connections between algebraic properties of certain graded algebras associated to I and combinatorial optimization properties of certain polyhedra and clutters associated to A and I, respectively. Sonic applications to Rees algebras and combinatorial optimization are presented.

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- 53 Reads
- 45 Citations

In this paper, by analogy with the case of C*-algebras, we define the notion of induced representation of a locally C*-algebra, and then we prove a imprimitivity theorem for induced representations of locally C*-algebras.

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- 49 Reads
- 5 Citations

We describe a method for associating a $C^{*}$-correspondence to a Mauldin-Williams graph and show that the Cuntz-Pimsner algebra of this $C^{*}$-correspondence is isomorphic to the $C^{*}$-algebra of the underlying graph. In addition, we analyze certain ideals of these $C^{*}$-algebras. We also investigate Mauldin-Williams graphs and fractal $C^{*}$-algebras in the context of the Rieffel metric. This generalizes the work of Pinzari, Watatani and Yonetani. Our main result here is a {}``no go'' theorem showing that such algebras must come from the commutative setting.

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- 41 Reads
- 11 Citations

Let X be a Hilbert bimodule over a C*-algebra A and $O_X= A \rtimes_X \Z$. Using a finite section method we construct a sequence of completely positive contractions factoring through matrix algebras over A which act on $s_{\xi} s_{\eta}^*$ as Schur multipliers converging to the identity. This shows immediately that for a finitely generated X the algebra $O_X$ inherits any standard approximation property such as nuclearity, exactness, CBAP or OAP from A. We generalise this to certain general Pimsner algebras by proving semi-splitness of the Toeplitz extension under certain conditions and discuss some examples.

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- 33 Reads
- 5 Citations

We discuss a problem of Dixmier on continuous fields of postliminal
C*-algebras and the greatest liminal ideals of the fibers.

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- 15 Reads
- 1 Citation

Given a correspondence X over a C*-algebra A, we construct a C*-algebra and a Hilbert C*-bimodule over it whose crossed product is isomorphic to the augmented Cuntz-Pimsner C*-algebra of X. This construction enables us to establish a condition for two augmented Cuntz-Pimsner C*-algebras to be Morita equivalent.

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- 91 Reads
- 10 Citations

We compute the monoid of isomorphism classes of finitely generated projective
modules of a Leavitt path algebra over an arbitrary directed graph. Our result
generalizes the result of Ara, Moreno, and Pardo in which they computed this
monoid of a Leavitt path algebra over a countable row-finite directed graph.

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- 38 Reads
- 12 Citations

We expand on some invariants used for classifying nonselfadjoint operator
algebras. Specifically to nonselfadjoint operator algebras which have a
conditional expectation onto a commutative diagonal we construct an
edge-colored directed graph which can be used as an operator algebra invariant.

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- 7 Reads
- 1 Citation

The higher rank graphs of Kumjian and Pask are discrete Conduche fibrations
over the monoid of k-tuples of natural numbers for some k in which every
morphism in the base has a finite preimage under the the fibration. We examine
the generalization of this construction to discrete Conduche fibrations with
the same finiteness condition and a lifting property for completions of cospans
to commutative squares, over any category satisfying a strong version of the
right Ore condition, including all categories with pullbacks and right Ore
categories in which all morphisms are monic.

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- 64 Reads
- 4 Citations

Let $A$ be a local Artinian Gorenstein ring with algebraically closed residue
field $A/\fM=k$ of characteristic 0, and let $P_A(z) := \sum_{p=0}^{\infty}
(\tor_p^A(k,k))z^p $ {be} its Poincar\'e series. We prove that $P_A(z)$ is
rational if either $\dim_k({\fM^2/\fM^3}) \leq 4 $ and $ \dim_k(A) \leq 16,$ or
there exist $m\leq 4$ and $c$ such that the Hilbert function $H_A(n)$ of $A$ is
equal to $ m$ for $n\in [2,c]$ and equal to 1 for $n > c$. The results are
obtained thanks to a decomposition of the apolar ideal $\Ann(F)$ when $F=G+H$
and $G$ and $H$ belong to polynomial rings in different variables.

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- 23 Reads
- 4 Citations

This work is motivated by Radulescu's result on the comparison of C*-tensor
norms on C*(F_n) x C*(F_n). For unital C*-algebras A and B, there are natural
inclusions of A and B into their unital free product, their maximal tensor
product and their minimal tensor product. These inclusions define three
operator system structures on the internal sum A+B, the first of which we
identify as the coproduct of A and B in the category of operator systems.
Partly using ideas from quantum entanglement theory, we prove various
interrelations between these three operator systems. As an application, the
present results yield a significant improvement over Radulescu's bound on
C*(F_n) x C*(F_n). At the same time, this tight comparison is so general that
it cannot be regarded as evidence for a positive answer to the QWEP conjecture.

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- 28 Reads
- 20 Citations

For a free partial action of a group in a set we realize the associated
partial skew group ring as an algebra of functions with finite support over an
equivalence relation and we use this result to characterize the ideals in the
partial skew group ring. This generalizes, to the purely algebraic setting, the
known characterization of partial C*-crossed products as groupoid C*-algebras.
For completeness we include a new proof of the C* result for free partial
actions.

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- 69 Reads
- 17 Citations

Let $\mathcal{R}$ be a commutative ring with identity, $I(X,\mathcal{R})$ be
the incidence algebra of a locally finite pre-ordered set $X$. In this note, we
characterise the derivations of $I(X,\mathcal{R})$ and prove that every Jordan
derivation of $I(X,\mathcal{R})$ is a derivation provided that $\mathcal{R}$ is
$2$-torsion free.

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- 168 Reads
- 26 Citations

With each Fell bundle over a discrete group G we associate a partial action
of G on the spectrum of the unit fiber. We discuss the ideal structure of the
corresponding full and reduced cross-sectional C*-algebras in terms of the
dynamics of this partial action.

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- 30 Reads
- 13 Citations

A C*-algebra is n-homogeneous (where n is finite) if every its nonzero
irreducible representation acts on an n-dimensional Hilbert space. An
elementary proof of Fell's characterization of n-homogeneous C*-algebras (by
means of their spectra) is presented. A spectral theorem and a functional
calculus for finite systems of elements which generate n-homogeneous
C*-algebras are proposed.

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- 26 Reads
- 4 Citations

Consider an ideal I in K[x,y,z] corresponding to a point configuration in P2
where all but one of the points lies on a single line. In this paper we study
the symbolic generic initial system obtained by taking the reverse
lexicographic generic initial ideals of the corresponding uniform fat point
ideals. We describe the limiting shape of this system of ideals and, in proving
this result, demonstrate that infinitely many of the uniform fat point ideals
are componentwise linear.

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- 19 Reads
- 3 Citations

In this article we formulate a version of the analytic Novikov conjecture for
semigroups rather than groups, and show that the descent argument from coarse
geometry generalises e?ectively to this new situation.

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- 5 Reads

We propose a generalisation of analytic in a domain function of bounded index, which
was introduced by J. G. Krishna and S. M. Shah \cite{krishna}. In fact, analytic in the unit ball
function of bounded index by Krishna and Shah is an entire function. Our
approach allows us to explore properties of analytic in the unit ball functions.
We proved the necessary and sufficient conditions of bounded $L$-index in direction for
analytic functions. As a result, they are applied to study partial differential equations and get sufficient
conditions of bounded $L$-index in direction for analytic solutions. Finally, we estimated
growth for these functions.

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- 48 Reads
- 29 Citations

All the rings are assumed to be commutative Noetherian and all the modules are finitely generated. Let A be a ring of dimension n ≥ 2, and let L be a projective A-module of rank 1. In [3], Bhatwadekar and Sridharan defined an abelian group, called the Euler class group of A with respect to L which is denoted by E(A,L). To the pair (P, χ), where P is a projective Amodule of rank n with determinant L and χ : L ∼→∧nP an isomorphism, called an L-orientation of P, they attached an element of E(A,L) which is denoted by e(P, χ). One of the main result in [3] is that P has a unimodular element if and only if e(P, χ) is zero in E(A,L).

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- 37 Reads

With the launch of second line anti-retroviral therapy for HIV infected
individuals, there has been an increased expectation on surviving period of
people with HIV. We consider previously well-known models in HIV epidemiology
where the parameter for incubation period is used as one of the important
components to explain the dynamics of the variables. Such models are extended
here to explain the dynamics with respect to a given therapy that prolongs life
of an HIV infected individual. A deconvolution method is demonstrated for
estimation of parameters in the situations when no-therapy and multiple
therapies are given to the infected population. The models and deconvolution
method are extended in order to study the impact of therapy in age-structured
populations. A generalization for a situation when n-types of therapies are
available is given. Models are demonstrated using hypothetical data and
sensitivity of the parameters are also computed.

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- 12 Reads
- 3 Citations

A necessary and sufficient condition for existence of a Banach space with a finite dimensional decomposition but without the π-property in terms of norms of compositions of projections is found. 2000 Mathematics Subject Classification. Primary 46B15; Secondary 46B07, 46B28. The problem of existence of Banach spaces with the π-property but without a finite dimensional decomposition is one of the well-known open problems in Banach space theory. It was first studied by W. B. Johnson [3]. P. G. Casazza and N. J. Kalton [2] found important connections of this problem with other problems of Banach space theory. See in this connection the survey [1]. Recall the definitions. A separable Banach space X has the π-property if there is a sequence Tn: X → X of finite dimensional projections such that If in addition the projections satisfy (∀x ∈ X) ( lim n→∞ ||x − Tnx| | = 0).

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- 63 Reads
- 2 Citations

For a real polynomial $p = \sum_{i=0}^{n} c_ix^i$ with no negative coefficients and $n\geq 6$, let $\beta (p) = \inf_{i=1}^{n-1} c_i^2/c_{i+1}c_{i-1}$ (so $\beta (p) \geq 1$ entails that $p$ is log concave). If $\beta(p) > 1.45...$, then all roots of $p$ are in the left half plane, and moreover, there is a function $\beta_0 (\theta)$ (for $\pi/2 \leq \theta \leq \pi$) \st $\beta \geq \beta_0(\theta)$ entails all roots of $p$ have arguments in the sector $| \arg z| \geq \theta$ with the smallest possible $\theta$; we determine exactly what this function (and its inverse) is (it turns out to be piecewise smooth, and quite tractible). This is a one-parameter extension of Kurtz's theorem (which asserts that $\beta \geq 4$ entails all roots are real). We also prove a version of Kurtz's theorem with real (not necessarily nonnegative) coefficients.

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- 64 Reads
- 5 Citations

It has been noted in several papers that an arithmetic-geometric mean
inequality incorporating variance would be useful in economics and finance.
There have been previous partial results in this direction; nevertheless, the
question of finding sharp bounds for the geometric mean in terms of the
arithmetic mean and variance has remained unanswered. In this paper we prove
such an inequality; it implies that for all positive sequences of given length
with fixed arithmetic mean and nonzero variance, the geometric mean is extremal
when all terms in the sequence except one are equal.

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- 973 Reads
- 4 Citations

Inspired by Rearick (1968), we introduce two new operators, LOG and EXP. The LOG operates on generalized Fibonacci polynomials giving generalized Lucas polynomials. The EXP is the inverse of LOG. In particular, LOG takes a convolution product of generalized Fibonacci polynomials to a sum of generalized Lucas polynomials and EXP takes the sum to the convolution product. We use this structure to produce a theory of logarithms and exponentials within arithmetic functions giving another proof of the fact that the group of multiplicative functions under convolution product is isomorphic to the group of additive functions under addition. The hyperbolic trigonometric functions are constructed from the EXP operator, again, in the usual way. The usual hyperbolic trigonometric identities hold. We exhibit new structure and identities in the isobaric ring. Given a monic polynomial, its infinite companion matrix can be embedded in the group of weighted isobaric polynomials. The derivative of the monic polynomial and its companion matrix give us the different matrix and the infinite different matrix. The determinant of the different matrix is the discriminate of the monic polynomial up to sign. In fact, the LOG operating on the infinite companion matrix is the infinite different matrix. We prove that an arithmetic function is locally representable if an only if it is a multiplicative function. An arithmetic function is both locally and globally representable if it is trivially globally represented.

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- 392 Reads
- 5 Citations

We study arithmetic progression in the $x$-coordinate of rational points on genus two curves. As we know, there are two models for the curve $C$ of genus two: $C: y^2=f_{5}(x)$ or $C: y^2=f_{6}(x)$, where $f_{5}, f_{6}\in\Q[x]$, $\operatorname{deg}f_{5}=5, \operatorname{deg}f_{6}=6$ and the polynomials $f_{5}, f_{6}$ do not have multiple roots. First we prove that there exists an infinite family of curves of the form $y^2=f(x)$, where $f\in\Q[x]$ and $\operatorname{deg}f=5$ each containing 11 points in arithmetic progression. We also present an example of $F\in\Q[x]$ with $\operatorname{deg}F=5$ such that on the curve $y^2=F(x)$ twelve points lie in arithmetic progression. Next, we show that there exist infinitely many curves of the form $y^2=g(x)$ where $g\in\Q[x]$ and $\operatorname{deg}g=6$, each containing 16 points in arithmetic progression. Moreover, we present two examples of curves in this form with 18 points in arithmetic progression.

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- 42 Reads
- 17 Citations

In an isomorphic copy of the ring of symmetric polynomials we study some families of polynomials which are indexed by rational weight vectors. These families include well known symmetric polynomials, such as the elementary, homogeneous, and power sum symmetric polynomials. We investigate properties of these families and focus on constructing their rational roots under a product induced by convolution. A direct application of the latter is to the description of the roots of certain multiplicative arithmetic functions (the core functions) under the convolution product.

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- 192 Reads
- 15 Citations

Top-cited authors

- Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan

- 596 Citations