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Sébastien Tremblay

Publications and source records attributed to Sébastien Tremblay.

At least 19 recordsLinked to original sources

Dimensional Distribution Emotion State: Leveraging Valence and Arousal as a Common Embedding Space for Visual Emotion Analysis

Museums are important sites for the dissemination of culture and art. They are institutions rooted in history and tradition; their exhibitions are often designed to highlight these aspects. Recently, a new approach is being explored in the field: emotion-based exhibitions. These exhibitions are designed specifically to elicit emotions in the visitors, in order to maximize engagement, and as a way to democratize access to art and attract a wider, more diverse audience. To do so, the emotional content of the artworks must first be extracted, however, manually annotating the artworks by experts is a prohibitively labor-intensive process, and risks introducing the personal bias of curators. To assist the museum curators in their design of these exhibitions, we wish to develop a tool that can predict the emotional response evoked by a work of art. In this article, we leverage a continuous bi-dimensional emotion space to enhance emotion representations and the training process of deep learning models. Drawing inspiration from existing categorical and dimensional emotion representations, we introduce a new representation, Dimensional Distribution Emotion State (DDES), along with a pipeline for multi-dataset training. We show that DDES provides multiple advantages compared to widely used representations while exhibiting similar baseline performance.

cs.CV↗

Binary Representation in Multicomplex and Clifford Algebras

Using a binary representation for basis elements of an algebra combined with a framework of multiplier and index functions, a connection has been established between the structure of a large class of algebras and the XOR componentwise operation. This result covers both multicomplex and Clifford algebras of any dimension. Necessary conditions for the existence of a diagonal basis are derived directly from this result, with an algorithm-like method to obtain it through a change-of-coordinates matrix in the commutative case. The same framework is then used to define an algebra of conjugate operations and study its properties.

math-ph↗

Multicomplex Ideals, Modules and Hilbert Spaces

In this article we study some algebraic aspects of multicomplex numbers $\mathbb M_n$. For $n\geq 2$ a canonical representation is defined in terms of the multiplication of $n-1$ idempotent elements. This representation facilitates computations in this algebra and makes it possible to introduce a generalized conjugacy $Λ_n$, i.e. a composition of the $n$ multicomplex conjugates $Λ_n:=\dagger_1\cdots \dagger_n$, as well as a multicomplex norm. The ideals of the ring of multicomplex numbers are then studied in details, free $\mathbb M_n$-modules and their linear operators are considered and, finally, we develop Hilbert spaces on the multicomplex algebra.

math-ph↗

Supersymmetric generalized power functions

Complex-valued functions defined on a finite interval $[a,b]$ generalizing power functions of the type $(x-x_0)^n$ for $n\geq 0$ are studied. These functions called $Φ$-generalized powers, $Φ$ being a given nonzero complex-valued function on the interval, were considered to contruct a general solution representation of the Sturm-Liouville equation in terms of the spectral parameter \cite{kravchenko2008, kravporter2010}. The $Φ$-generalized powers can be considered as a natural basis functions for the one-dimensional supersymmetric quantum mechanics systems taking $Φ=ψ_0^2$, where the function $ψ_0(x)$ is the ground state wave function of one of the supersymmetric scalar Hamiltonians. Several properties are obtained such as $Φ$-symmetric conjugate and antisymmetry of the $Φ$-generalized powers, a supersymmetric binomial identity for these functions, a supersymmetric Pythagorean elliptic (hyperbolic) identity involving four $Φ$-trigonometric ($Φ$-hyperbolic) functions as well as a supersymmetric Taylor series expressed in terms of the $Φ$-derivatives. We show that the first $n$ $Φ$-generalized powers are a fundamental set of solutions associated with a nonconstant coefficients homogeneous linear ordinary differential equations of order $n+1$. Finally, we present a general solution representation of the stationary Schrödinger equation in terms of geometric series where the Volterra compositions of the first type is considered.

math-ph↗

On a three-dimensional Riccati differential equation and its symmetries

A three-dimensional Riccati differential equation of complex quaternion-valued functions is studied. Many properties similar to those of the ordinary differential Riccati equation such that linearization and Picard theorem are obtained. Lie point symmetries of the quaternionic Riccati equation are calculated as well as the form of the associated three-dimensional potential of the Schrödinger equation. Using symmetry reductions and relations between the three-dimensional Riccati and the Schrödinger equation, examples are given to obtain solutions of both equations.

math-ph↗

On two-dimensional supersymmetric quantum mechanics, pseudoanalytic functions and transmutation operators

Pseudoanalytic function theory is considered to study a two-dimensional supersymmetric quantum mechanics system. Hamiltonian components of the superhamiltonian are factorized in terms of one Vekua and one Bers derivative operators. We show that imaginary and real solutions of a Vekua equation and its Bers derivative are ground state solutions for the superhamiltonian. The two-dimensional Darboux and pseudo-Darboux transformations correspond to Bers derivatives in the complex plane. Results on the completeness of the ground states are obtained. Finally, superpotential is studied in the separable case in terms of transmutation operators. We show how Hamiltonian components of the superhamiltonian are related to the Laplacian operator using these transmutation operators.

math-ph↗

Wave polynomials, transmutations and Cauchy's problem for the Klein-Gordon equation

We prove a completeness result for a class of polynomial solutions of the wave equation called wave polynomials and construct generalized wave polynomials, solutions of the Klein-Gordon equation with a variable coefficient. Using the transmutation (transformation) operators and their recently discovered mapping properties we prove the completeness of the generalized wave polynomials and use them for an explicit construction of the solution of the Cauchy problem for the Klein-Gordon equation. Based on this result we develop a numerical method for solving the Cauchy problem and test its performance.

math.AP↗

Spatial pseudoanalytic functions arising from the factorization of linear second order elliptic operators

Biquaternionic Vekua-type equations arising from the factorization of linear second order elliptic operators are studied. Some concepts from classical pseudoanalytic function theory are generalized onto the considered spatial case. The derivative and antiderivative of a spatial pseudoanalytic function are introduced and their applications to the second order elliptic equations are considered.

math.CV↗

Complete systems of recursive integrals and Taylor series for solutions of Sturm-Liouville equations

Consider an arbitrary complex-valued, twice continuously differentiable, nonvanishing function $ϕ$ defined on a finite segment $[a,b]\subset \mathbb{R}$. Let us introduce an infinite system of functions constructed in the following way. Each subsequent function is a primitive of the preceding one multiplied or divided by $ϕ$ alternately. The obtained system of functions is a generalization of the system of powers ${(x-x_{0}%)^{k}}_{k=0}^{\infty}$. We study its completeness as well as the completeness of its subsets in different functional spaces. This system of recursive integrals results to be closely related to so-called $L$-bases arising in the theory of transmutation operators for linear ordinary differential equations. Besides the results on the completeness of the system of recursive integrals we show a deep analogy between the expansions in terms of the recursive integrals and Taylor expansions. We prove a generalization of the Taylor theorem with the Lagrange form of the remainder term and find an explicit formula for transforming a generalized Taylor expansion of a function in terms of the recursive integrals into a usual Taylor expansion. As a direct corollary of the formula we obtain the following new result concerning solutions of the Sturm-Liouville equation. Given a regular nonvanishing complex valued solution $y_{0}$ of the equation $y^{\prime\prime}+q(x)y=0$, $x\in(a,b)$, assume that it is $n$ times differentiable at a point $x_{0}% \in\lbrack a,b]$. We present explicit formulas for calculating the first $n$ derivatives at $x_{0}$ for any solution of the equation $u^{\prime\prime}+q(x)u=λu$. That is, an explicit map transforming the Taylor expansion of $y_{0}$ into the Taylor expansion of $u$ is constructed.

math.CA↗

Explicit solutions of generalized Cauchy-Riemann systems using the transplant operator

In [8] it was shown that the transplant operator transforms solutions of one Vekua equation into solutions of another Vekua equation, related to the first via a Schrödinger equation. In this paper we demonstrate a fundamental property of this operator: it transforms formal powers of the first Vekua equation into formal powers of the same order for the second Vekua equation. This property allows us to obtain positive formal powers and a generating sequence of a "complicated" Vekua equation from positive formal powers and a generating sequence of "simpler" Vekua equation. Similar results is obtained regarding construction of Cauchy kernels. Elliptic and hyperbolic pseudoanalytic function theory are considered and examples are given to illustrate the procedure.

math.CV↗

Symmetries of differential-difference dynamical systems in a two-dimensional lattice

Classification of differential-difference equation of the form $\ddot{u}_{nm}=F_{nm}\big(t, \{u_{pq}\}|_{(p,q)\in Γ}\big)$ are considered according to their Lie point symmetry groups. The set $Γ$ represents the point $(n,m)$ and its six nearest neighbors in a two-dimensional triangular lattice. It is shown that the symmetry group can be at most 12-dimensional for abelian symmetry algebras and 13-dimensional for nonsolvable symmetry algebras.

math-ph↗

Zakharov-Shabat system and hyperbolic pseudoanalytic function theory

In [1] a hyperbolic analogue of pseudoanalytic function theory was developed. In the present contribution we show that one of the central objects of the inverse problem method the Zakharov-Shabat system is closely related to a hyperbolic Vekua equation for which among other results a generating sequence and hence a complete system of formal powers can be constructed explicitly.

math-ph↗

Solvable Lie algebras with triangular nilradicals

All finite-dimensional indecomposable solvable Lie algebras $L(n,f)$, having the triangular algebra T(n) as their nilradical, are constructed. The number of nonnilpotent elements $f$ in $L(n,f)$ satisfies $1\leq f\leq n-1$ and the dimension of the Lie algebra is $\dim L(n,f)=f+{1/2}n(n-1)$.

math.RA↗

Bicomplex quantum mechanics: I. The generalized Schrödinger equation

We introduce the set of bicomplex numbers $\mathbb{T}$ which is a commutative ring with zero divisors defined by $\mathbb{T}=\{w_0+w_1 \bold{i_1}+w_2\bold{i_2}+w_3 \bold{j}| w_0,w_1,w_2,w_3 \in \mathbb{R}\}$ where $\bold{i^{\text 2}_1}=-1, \bold{i^{\text 2}_2}=-1, \bold{j}^2=1,\ \bold{i_1}\bold{i_2}=\bold{j}=\bold{i_2}\bold{i_1}$. We present the conjugates and the moduli associated with the bicomplex numbers. Then we study the bicomplex Schrödinger equation and found the continuity equations. The discrete symmetries of the system of equations describing the bicomplex Schrödinger equation are obtained. Finally, we study the bicomplex Born formulas under the discrete symetries. We obtain the standard Born's formula for the class of bicomplex wave functions having a null hyperbolic angle.

quant-ph↗

Fine grading of $sl(p^2,\mathbb{C})$ generated by tensor product of generalized Pauli matrices and its symmetries

Study of the normalizer of the MAD-group corresponding to a finegrading offers the most important tool for describing symmetries in the system of non-linear equations connected with contraction of a Lie algebra. One fine grading that is always present in any Lie algebra $sl(n,\mathbb{C})$ is the Pauli grading. The MAD-group corresponding to it is generated by generalized Pauli matrices. For such MAD-group, we already know its normalizer; its quotient group is isomorphic to the Lie group $Sl(2,\mathbb{Z}_n)\times v\mathbb{Z}_2$. In this paper, we deal with a more complicated situation, namely that the fine grading of $sl(p^2, \mathbb{C})$ is given by a tensor product of the Pauli matrices of the same order $p$, $p$ being a prime. We describe the normalizer of the corresponding MAD-group and we show that its quotient group is isomorphic to $Sp(4,\mathbb{Z}_p)\times\mathbb{Z}_2$.

quant-ph↗

Lie symmetries of multidimensional difference equations

A method is presented for calculating the Lie point symmetries of a scalar difference equation on a two-dimensional lattice. The symmetry transformations act on the equations and on the lattice. They take solutions into solutions and can be used to perform symmetry reduction. The method generalizes one presented in a recent publication for the case of ordinary difference equations. In turn, it can easily be generalized to difference systems involving an arbitrary number of dependent and independent variables.

math-ph↗

Integrable lattice equations and their growth properties

In this paper we investigate the integrability of two-dimensional partial difference equations using the newly developed techniques of study of the degree of the iterates. We show that while for generic, nonintegrable equations, the degree grows exponentially fast, for integrable lattice equations the degree growth is polynomial. The growth criterion is used in order to obtain the integrable deautonomisations of the equations examined. In the case of linearisable lattice equations we show that the degree growth is slower than in the case of equations integrable through Inverse Scattering Transform techniques.

math-ph↗

Integrable systems without the Painlevé property

We examine whether the Painlevé property is a necessary condition for the integrability of nonlinear ordinary differential equations. We show that for a large class of linearisable systems this is not the case. In the discrete domain, we investigate whether the singularity confinement property is satisfied for the discrete analogues of the non-Painlevé continuous linearisable systems. We find that while these discrete systems are themselves linearisable, they possess nonconfined singularities.

math-ph↗