Sequences, q-multinomial Identities, Integer Partitions with Kinds, and Generalized Galois Numbers

The line spectral frequency (LSF) of a causal finite length sequence is a frequency at which the spectrum of the sequence annihilates or the magnitude spectrum has a spectral null. A causal finite-length sequence with (L + 1) samples having exactly L-LSFs, is referred as an Annihilating (AH) sequence. Using some spectral properties of finite-length sequences, and some model parameters, we develop spectral decomposition structures, which are used to translate any finite-length sequence to an equivalent set of AH-sequences defined by LSFs and some complex constants. This alternate representation format of any finite-length sequence is referred as its LSF-Model. For a finite-length sequence, one can obtain multiple LSF-Models by varying the model parameters. The LSF-Model, in time domain can be used to synthesize any arbitrary causal finite-length sequence in terms of its characteristic AH-sequences. In the frequency domain, the LSF-Model can be used to obtain the spectral samples of the sequence as a linear combination of spectra of its characteristic AH-sequences. We also summarize the utility of the LSF-Model in practical discrete signal processing systems.

On a novel critically-sampled discrete-time real Gabor transform

Combinatorial proof is an important topic both for combinatorics education and proof education researchers, but relatively little has been studied about the teaching and learning of combinatorial proof. In this paper, we focus on one specific phenomenon that emerged during interviews with mathematicians and students who were experienced provers as they discussed and engaged in combinatorial proof. In particular, participants used a wide variety of cognitive models to interpret multiplication by a constant when reasoning about binomial identities, some of which seemed to be more (or less) effective in helping produce a combinatorial proof. We present these cognitive models and describe episodes that illustrate implications of these cognitive models for our participants’ work on proving binomial identities. Our findings both inform research on combinatorial proof and highlight the importance of understanding subtleties of the familiar operation of multiplication.

A class of binary sequences, constrained with respect to the length of zero runs, is considered. For such sequences, termed (d, k)-sequences, new combinatorial and computational results are established. Explicit expressions for enumerating (d, k)-sequences of finite length are obtained. Efficient computational procedures for calculating the capacity of a (d, k)-code are given. A simple method for constructing a near-optimal (d, k)-code is proposed. Illustrative numerical examples demonstrate further the theoretical results.

Combinatorial proofs and generalizations of conjectures related to Euler’s partition theorem

Combinatorics For each integer partition mu, let e (F) over tilde (mu)(q; t) be the coefficient of x(1) ... x(n) in the modified Macdonald polynomial (H) over tilde (mu). The polynomial (F) over tilde (mu)(q; t) can be regarded as the Hilbert series of a certain doubly-graded S(n)-module M(mu), or as a q, t-analogue of n! based on permutation statistics inv(mu) and maj(mu) that generalize the classical inversion and major index statistics. This paper uses the combinatorial definition of (F) over tilde (mu) to prove some recursions characterizing these polynomials, and other related ones, when mu is a two-column shape. Our result provides a complement to recent work of Garsia and Haglund, who proved a different recursion for two-column shapes by representation-theoretical methods. For all mu, we show that e (F) over tilde (mu)(q, t) is divisible by certain q-factorials and t-factorials depending on mu. We use our recursion and related tools to explain some of these factors bijectively. Finally, we present fermionic formulas that express e (F) over tilde ((2n)) (q, t) as a sum of q, t-analogues of n!2(n) indexed by perfect matchings.

Chapter 40 Non-linear dynamical systems: Instability and chaos in economics

This paper presents a direct method of reducing convolution processing time using hardware computing and implementations of discrete linear convolution of two finite length sequences (NXN). This implementation method is realized by simplifying the convolution building blocks. The purpose of this research is to prove the feasibility of an application specific integrated circuit (ASIC) that performs a convolution on an acquired image in real time. The proposed implementation uses a modified hierarchical design approach, which efficiently and accurately speeds up computation; reduces power, hardware resources, and area significantly. The efficiency of the proposed convolution circuit is tested by embedding it in a top level FPGA. Simulation and comparison to different design approaches show that the circuit uses only 5 mw that saves almost 35% of area and is four times faster than what is implemented in. In addition, the presented circuit uses less power consumption and has a delay of 20 ns from input to output using 32 nm process library. It also provides the necessary modularity, expandability, and regularity to form different convolutions for any number of bits.

Properties of the magnitude terms of orthogonal scaling functions

In this paper, we consider determinants of some families of Toeplitz–Hessenberg matrices having various translates of the Catalan numbers for the nonzero entries. These determinant formulas may also be rewritten as identities involving sums of products of Catalan numbers and multinomial coefficients. Combinatorial proofs may be given for several of the identities that are obtained.

We introduce a new combinatorial optimization problem in this paper, called the Minimum Common Integer Partition (MCIP) problem, which was inspired by computational biology applications including ortholog assignment and DNA fingerprint assembly. A partition of a positive integer n is a multiset of positive integers that add up to exactly n, and an integer partition of a multiset S of integers is defined as the multiset union of partitions of integers in S. Given a sequence of multisets S 1, ⋯, S k of integers, where k ≥ 2, we say that a multiset is a common integer partition if it is an integer partition of every multiset S i , 1≤ i≤ k. The MCIP problem is thus defined as to find a common integer partition of S 1, ⋯, S k with the minimum cardinality. It is easy to see that the MCIP problem is NP-hard since it generalizes the well-known Set Partition problem. We can in fact show that it is APX-hard. We will also present a $\frac{5}{4}$ -approximation algorithm for the MCIP problem when k = 2, and a $\frac{3k(k-1)}{3k-2}$ -approximation algorithm for k ≥ 3.

The number of ways to distribute r identical objects into n identical boxes can usually be obtained by a method of generating function or by a recursive formula. In this paper, for another approach, it is shown that we can obtain this number by using generalization of Polya’s theorem. From this, we can also find the number of partitions of integer n as a sum of k positive integers. Computing for the results is discussed.

Our main results are asymptotic zero-one laws satisfied by the diagrams of unimodal sequences of positive integers. These diagrams consist of columns of squares in the plane; the upper boundary is called the shape. For various types of unimodal sequences, we show that, as the number of squares tends to infinity, 100% of shapes are near a certain curve---that is, there is a single limit shape. Similar phenomena have been well-studied for integer partitions, but several technical difficulties arise in the extension of such asymptotic statistical laws to unimodal sequences. We develop a widely applicable method for obtaining these limit shapes, based in part on a method of Petrov. We also mention a few notable corollaries---for example, we obtain a limit shape for so-called "overpartitions'' by a simple DeSalvo-Pak-type transfer. To aid in the proof of these limit shapes, we prove an asymptotic formula for the number of partitions of the integer n into distinct parts where the largest part is at most t times the square root of n for fixed t. Our method follows a probabilistic approach of Romik, who gave a simpler proof of Szekeres' asymptotic formula for distinct parts partitions when instead the number of parts is bounded by t times the square root of n. The probabilistic approach is equivalent to a circle method/saddle-point method calculation, but it makes some of the steps more intuitive and even predicts the shape of the asymptotic formula, to some degree. Finally, motivated by certain problems concerning Rogers-Ramanujan-type identities, we give combinatorial proofs of three families of inequalities among certain types of integer partitions.

Recently it was shown that the Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. In this paper, we introduce a similar representation for the Padovan numbers. As a corollary, we derive an infinite family of double inequalities.

We deal with a sequence of integer-valued random variables $\{Z_N\}_{N=1}^{\infty}$ which is related to restricted partitions of positive integers. We observe that $Z_N=X_1+ \ldots + X_N$ for independent and bounded random variables $X_j$'s, so $Z_N$ has finite mean ${\bf E}Z_N$ and variance ${\bf Var}Z_N$. We want to find the limit distribution of ${\hat Z}_N=\left(Z_N-{\bf E}Z_N\right)/{\sqrt{{\bf Var}Z_N}}$ as $N \to \infty.$ While in many cases the limit distribution is normal, the main results established in this paper are that ${\hat Z}_N \overset{d}{\to} Z_{*},$ where $Z_{*}$ is a bounded random variable. We find explicitly the range of values of $Z_*$ and derive some properties of its distribution. The main tools used are moment generating functions, cumulant generating functions, moments and cumulants of the random variables involved. Useful related topics are also discussed.