Related papers: Row-column factorial designs with multiple levels
The normalized factorial moments $F_q$ are continued to noninteger values of the order $q$, satisfying the condition that the statistical fluctuations remain filtered out. That is, for Poisson distribution $F_q = 1$ for all $q$. The…
If $x$ is a non-empty string then the repetition $xx$ is called a tandem repeat. Similarly, a tandem in a two dimensional array $X$ is a configuration consisting of a same primitive block $W$ that touch each other with one side or corner.…
A de Bruijn array code is a set of $r \times s$ binary doubly-periodic arrays such that each binary $n \times m$ matrix is contained exactly once as a window in one of the arrays. Such a set of arrays can be viewed as a two-dimensional…
An $r$-matrix is a matrix with symbols in $\{0,1,\ldots,r-1\}$. A matrix is simple if it has no repeated columns. Let ${\cal F}$ be a finite set of $r$-matrices. Let $\hbox{forb}(m,r,{\cal F})$ denote the maximum number of columns possible…
A commutative semigroup of abstract factorials is defined in the context of the ring of integers. We study such factorials for their own sake, whether they are or are not connected to sets of integers. Given a subset X of the positive…
We combine several folklore observations to provide a working framework for iterating constructions which contradict the axiom of choice. We use this to define a model in which any kind of structural failure must fail with a proper class of…
Let $\mathbb{F}_q$ be the finite field with $q$ elements, where $q$ is a prime power and $n$ be a positive integer. In this paper, we explore the factorization of $f(x^{n})$ over $\mathbb{F}_q$, where $f(x)$ is an irreducible polynomial…
It is well understood that different neural network architectures are suited to different tasks, but is there always a single best architecture for a given task? We compare the expressive power of transformers, RNNs, and transformers with…
Let A be any set of positive integers and n a positive integer. A composition of n with parts in A is an ordered collection of one or more elements in A whose sum is n. We derive generating functions for the number of compositions of n with…
For the quantum integer [n]_q = 1+q+q^2+... + q^{n-1} there is a natural polynomial multiplication such that [mn]_q = [m]_q \otimes_q [n]_q. This multiplication is given by the functional equation f_{mn}(q) = f_m(q) f_n(q^m), defined on a…
This report introduces a novel class of reasoning architectures, termed Quantum Circuit Reasoning Models (QCRM), which extend the concept of Variational Quantum Circuits (VQC) from energy minimization and classification tasks to structured…
A $q$-ary $t$-covering array is an $m \times n$ matrix with entries from $\{0, 1, ..., q-1\}$ with the property that for any $t$ column positions, all $q^t$ possible vectors of length $t$ occur at least once. One wishes to minimize $m$ for…
The focus of this thesis is the study and construction of covering arrays, relying on maximal period sequences and other tools from finite fields. A covering array of strength $t$, denoted $\mathrm{CA}(N; t, k,v)$, is an $N\times k$ array…
In this paper we study saturated fractions of factorial designs under the perspective of Algebraic Statistics. We define a criterion to check whether a fraction is saturated or not with respect to a given model. The proposed criterion is…
Factorial designs are frequently used in different fields of science, e.g. psychological, medical or biometric studies. Standard approaches, as the ANOVA $F$-test, make different assumptions on the distribution of the error terms, the…
We study a family of subcodes of the $m$-dimensional product code $\mathscr{C}^{\otimes m}$ ('subproduct codes') that have a recursive Plotkin-like structure, and which include Reed-Muller (RM) codes and Dual Berman codes as special cases.…
We extend a factorization due to Krein to arbitrary analytic functions from the upper half-plane to itself. The factorization represents every such function as a product of fractional linear factors times a function which, generally, has…
A sparsity pattern in $\mathbb{R}^{n \times m}$, for $m\geq n$, is a vector subspace of matrices admitting a basis consisting of canonical basis vectors in $\mathbb{R}^{n \times m}$. We represent a sparsity pattern by a matrix with…
We introduce two related notions of pattern enforcement in $(0,1)$-matrices: $Q$-forcing and strongly $Q$-forcing, which formalize distinct ways a fixed pattern $Q$ must appear within a larger matrix. A matrix is $Q$-forcing if every…
The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the…