Related papers: Prenex normalization and the hierarchical classifi…

`Categorification' is the process of replacing equations by isomorphisms. We describe some of the ways a thoroughgoing emphasis on categorification can simplify and unify mathematics. We begin with elementary arithmetic, where the category…

The scalar difference equation $x_{n+1}=f_{n}(x_{n},x_{n-1},...,x_{n-k})$ may exhibit symmetries in its form that allow for reduction of order through substitution or a change of variables. Such form symmetries can be defined generally…

Motivated by bifurcation of branches of homoclinic orbits of dynamical systems, we consider families of first-order equations on the real line and introduce a generalisation of previous index theorems by Pejsachowicz, and by Hu and…

The notion of a randomization of a first order structure was introduced by Keisler in the paper Randomizing a Model, Advances in Math. 1999. The idea was to form a new structure whose elements are random elements of the original first order…

The systematic method for the conversion of first class constraints to the equivalent set of Abelian one based on the Dirac equivalence transformation is developed. The representation for the corresponding matrix performing this…

Integral operators of Abel type of order a > 0 arise naturally in a large spectrum of physical processes. Their inversion requires care since the resulting inverse problem is ill-posed. The purpose of this work is to devise and analyse a…

Regularization is one of the crucial ingredients of deep learning, yet the term regularization has various definitions, and regularization methods are often studied separately from each other. In our work we present a systematic, unifying…

We develop algebraic models of simple type theories, laying out a framework that extends universal algebra to incorporate both algebraic sorting and variable binding. Examples of simple type theories include the unityped and simply-typed…

This paper considers a formalisation of classical logic using general introduction rules and general elimination rules. It proposes a definition of `maximal formula', `segment' and `maximal segment' suitable to the system, and gives…

Various feature descriptions are being employed in logic programming languages and constrained-based grammar formalisms. The common notational primitive of these descriptions are functional attributes called features. The descriptions…

We establish why li(x) outperforms x/log x as an estimate for the prime counting function pi(x). The result follows from subdividing the natural numbers into the intervals s_k :={p_k^2,..., p_{k+1}^2-1}, k>=1, each being fully sieved by the…

We carry out the complete group classification of the class of (1+1)-dimensional linear Schr\"odinger equations with complex-valued potentials. After introducing the notion of uniformly semi-normalized classes of differential equations, we…

We introduce a new method for building classification models when we have prior knowledge of how the classes can be arranged in a hierarchy, based on how easily they can be distinguished. The new method uses a Bayesian form of the…

We develop normalisation by evaluation (NBE) for dependent types based on presheaf categories. Our construction is formulated in the metalanguage of type theory using quotient inductive types. We use a typed presentation hence there are no…

Formalized $1$-category theory forms a core component of various libraries of mathematical proofs. However, more sophisticated results in fields from algebraic topology to theoretical physics, where objects have "higher structure," rely on…

Let H be a non semi-simple Ariki-Koike algebra. According to [18] and [14], there is a generalisation of Lusztig's a-function which induces a natural order (parametrised by a tuple m) on Specht modules. In some cases, Geck and Jacon have…

First-order logic is the basis for many knowledge representation formalisms and methods. Providing technological support for learning to write first-order formulas for natural language specifications requires methods to test formulas for…

In this article, we develop a new approach to the Poincar\'e--Dulac normal form theory for a system of differential equations near a singular point. Using the continuous averaging method, we construct a normalization flow that moves a…

The way of finding all the constraints in the Hamiltonian formulation of singular (in particular, gauge) theories is called the Dirac procedure. The constraints are naturally classified according to the correspondig stages of this…

In this paper, we establish lower bounds for the oracle complexity of the first-order methods minimizing regularized convex functions. We consider the composite representation of the objective. The smooth part has H\"older continuous…