This paper surveys the close relationships among the topics included in the title. Emphasis is given to the family of positive Perron-Caratheodory continued fractions (PPC-fractions) which play a central role in the theory of trigonometric moment problems and Szego polynomials orthogonal on the unit circle. An important application of PPC-fractions is frequency analysis of discrete time signals using Wiener-Levinson digital filters with illustrations given from computational experiments.

It is known that a necessary condition for $T:=\sum Q_k(x) D^k$ to be hyperbolicity preserving is that $Q_k(x)$ and $Q_{k-1}(x)$ have interlacing zeros. We characterize all quadratic linear operators; as a consequence, we find several classes of the $P_n$-multiplier sequence.

It is shown how solving a discrete convolution problem gives a unique insight into the famous Faa di Bruno formula for the $n$th derivative of a composite function.

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.

Let $0\lt \alpha \lt 1$. We show that Bernoulli polynomials appear in the difference $\sum _{n \leq x}\Delta ^j(n+\alpha )-\int _1^x \Delta ^j(t)\,dt$ for $j=1, \ldots , 4$. As a corollary of this fact, we get better approximations of $\int _1^x \Delta ^j(t)\,dt$ by using zeros of Bernoulli polynomials. For $j=1,2$, we give some interpretation of this fact by means of Dirichlet series with the coefficients $\Delta ^j(n+\alpha )$.

In this note, we prove several non-vanishing results of Carlitz-Fermat quotients.

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

The method of Riccati equations is used to study some properties of third order linear ordinary differential equations. Some criteria of asymptotic behavior and non stability of solution of this equation are obtained. Two oscillatory criteria are proved.

In this paper we consider the class of complex-valued harmonic univalent functions that map the unit disc onto the complex plane, half-plane or a strip slit along finitely many horizontal half-lines.

We show how to integrate operator valued functions with respect to a spectral or orthogonally scattered measure. Such measures typically have a variation which has either the value zero or infinity on any set and cannot therefore be treated by the approaches of Bartle or Dobrakov. Bilinear integrals of this type arise from trace class operators between Banach function spaces and in the connection between stationary-state scattering theory and time-dependent scattering theory in Hilbert space.

In this paper, we compute terms of the matrix $A_{(k)}^{\infty }$, which contains Fibonacci type numbers and polynomials, with the help of determinants and permanents of various Hessenberg matrices. In addition, we show that determinants of these Hessenberg matrices can be obtained by using combinations. The results that we obtain are important, since the matrix $A_{(k)}^{\infty } $ is a general form of Fibonacci type numbers and polynomials, such as $k$ sequences of the generalized order-$k$ Fibonacci and Pell numbers, generalized bivariate Fibonacci $p$-polynomials, bivariate Fibonacci and Pell $p$-polynomials, second kind Chebyshev polynomials and bivariate Jacobsthal polynomials, etc.

In this paper, a method is given for calculating the HOMFLY polynomials of two bridge knots by using a representation of the braid group $\mathbb {B}_4$ into a group of $3\times 3$ matrices. Also, examples will be given of a 2-bridge knot and a 3-bridge knot that have the same Jones polynomial, but different HOMFLY polynomials.

Let $\mathscr {L}_{n,t}$ be the set of all $n$-vertex connected graphs with clique number $t$ $(2\leq t\leq n)$. For $n$-vertex connected graphs with given clique number, lexicographic ordering by spectral moments ($S$-order) is discussed in this paper. The first $\sum _{i=1}^{\lfloor ({n-t-1})/{3}\rfloor }(n-t-3i)+1$ graphs with $3\le t\le n-4$, and the last few graphs, in the $S$-order, among $\mathscr {L}_{n,t}$ are characterized. In addition, all graphs in $\mathscr {L}_{n,n}\bigcup \mathscr {L}_{n,n-1}$ have an $S$-order; for the cases $t=n-2$ and $t=n-3$, the first three and the first seven graphs in the set $\mathscr {L}_{n,t}$ are characterized, respectively.

Consider an ideal $I \subseteq K[x,y,z]$ corresponding to a point configuration in $\mathbb {P}^2$ where all but one of the points lies on a single line. In this paper, we study the symbolic generic initial system $\{\rm{gin\,}(I^{(m)})\}_m$ obtained by taking the reverse lexicographic generic initial ideals of the uniform fat point ideals $I^{(m)}$. We describe the limiting shape of $\{\rm{gin\,}(I^{(m)})\}_m$ and, in proving this result, demonstrate that infinitely many of the ideals $I^{(m)}$ are componentwise linear.

The number of singleton blocks in all partitions of a set $\{a_1,\ldots ,a_n\}$ is known to be equal to the number adjacencies, that is, pairs of consecutively numbered elements $(a_i,a_{i+1})$ in a block. We give a generalization of this relation by introducing the $d$-adjacency which is a pair of elements $(a_i,a_j)$ satisfying $j-i=d>0$. It is proved that the number of $d$-adjacencies in all partitions is independent of $d$. Then we show that the number of $d$-adjacencies in noncrossing partitions is a function of $d$ by means of an exact formula.

In this paper, we establish an extension of a noncommutative Bennett inequality with a parameter $1\leq r\leq 2$ and use it together with some noncommutative techniques to establish a Rosenthal inequality. We also present a noncommutative Hoeffding inequality as follows: Let $(\mathfrak {M}, \tau )$ be a noncommutative probability space, $\mathfrak {N}$ be a von Neumann subalgebra of $\mathfrak {M}$ with the corresponding conditional expectation $\mathcal {E}_{\mathfrak {N}}$ and let subalgebras $\mathfrak {N}\subseteq \mathfrak {A}_j\subseteq \mathfrak {M}\,\,(j=1, \cdots , n)$ be successively independent over $\mathfrak {N}$. Let $x_j\in \mathfrak {A}_j$ be self-adjoint such that $a_j\leq x_j\leq b_j$ for some real numbers $a_j\lt b_j$ and $\mathcal {E}_{\mathfrak {N}}(x_j)=\mu $ for some $\mu \geq 0$ and all $1\leq j\leq n$. Then for any $t>o$ it holds that \[ {\rm Prob}\bigg (\bigg |\sum _{j=1}^n x_j-n\mu \bigg |\geq t\bigg )\leq 2 \exp \bigg \{\frac {-2t^2}{\sum _{j=1}^n(b_j-a_j)^2}\bigg \}. \]

Turing's method uses explicit bounds on $|\int _{t_{1}}^{t_{2}} S(t)\, dt|$, where $\pi S(t)$ is the argument of the Riemann zeta-function. This article improves the bound on $|\int _{t_{1}}^{t_{2}} S(t)\, dt|$ given in \cite {TrudgianTuring}.

In this paper, a local $M$-estimation for the conditional variance function in heteroscedastic regression models under stationary $\alpha $-mixing dependent samples is developed. The local $M$-estimator is based on the local linear smoothing technique and the $M$-estimation technique, and it is shown to be not only asymptotically equivalent to the local linear estimator but also robust. The weak consistency as well as the asymptotic normality of the local $M$-estimator for the conditional variance function are obtained under mild conditions.