Lotfi Hermi | University of Arizona (original) (raw)

Papers by Lotfi Hermi

Research paper thumbnail of An isoperimetric inequality of Saint-Venant-type for a wedge-like membrane

Why care? Consider right isoceles triangle with equal sides of unit length (Payne-Weinberger) α =... more Why care? Consider right isoceles triangle with equal sides of unit length (Payne-Weinberger) α = 1, λ 1 ≥ 45.0734 α = 2, λ 1 ≥ 47.6325 α = 4, λ 1 ≥ 45.9094 Faber-Krahn (among all domains): λ 1 ≥ 36.3368 Faber-Krahn (among all triangles): λ 1 ≥ 4π 2 √ 3 A ≈ 45.5858 Exact value: λ 1 = 49.350625 An isoperimetric inequality of Saint-Venant-type for a wedge-lik

Research paper thumbnail of Green’s Functions and Euler’s Formula for <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow></mrow><annotation encoding="application/x-tex"></annotation></semantics></math></span><span class="katex-html" aria-hidden="true"></span></span>\zeta (2n)$$

Research paper thumbnail of Trace Formulas Applied to the Riemann ζ-Function

Integrability, Supersymmetry and Coherent States, 2019

We use a spectral theory perspective to reconsider properties of the Riemann zeta function. In pa... more We use a spectral theory perspective to reconsider properties of the Riemann zeta function. In particular, new integral representations are derived and used to present its value at odd positive integers.

Research paper thumbnail of On Rayleigh-type formulas for a non-local boundary value problem associated with an integral operator commuting with the Laplacian

Applied and Computational Harmonic Analysis, 2018

In this article we prove the existence, uniqueness, and simplicity of a negative eigenvalue for a... more In this article we prove the existence, uniqueness, and simplicity of a negative eigenvalue for a class of integral operators whose kernel is of the form |x − y| ρ , 0 < ρ ≤ 1, x, y ∈ [−a, a]. We also provide two different ways of producing recursive formulas for the Rayleigh functions (i.e., recursion formulas for power sums) of the eigenvalues of this integral operator when ρ = 1, providing means of approximating this negative eigenvalue. These methods offer recursive procedures for dealing with the eigenvalues of a one-dimensional Laplacian with non-local boundary conditions which commutes with an integral operator having a harmonic kernel. The problem emerged in recent work by one of the authors [48]. We also discuss extensions in higher dimensions and links with distance matrices.

Research paper thumbnail of Abstract. We review upper and lower bound isoperimetric properties of the fundamental eigenfunction of the Dirichlet Laplacian and announce new reverse H¨older type inequalities for norms of this function in the case of a wedgelike membrane

We review upper and lower bound isoperimetric properties of the fundamental eigenfunction of the ... more We review upper and lower bound isoperimetric properties of the fundamental eigenfunction of the Dirichlet Laplacian and announce new reverse H¨older type inequalities for norms of this function in the case of a wedgelike membrane.

Research paper thumbnail of Laplacian and Bilaplacian Based Features for Shape Classification

Finite difference schemes of various eigenvalue problems are used to generate size, rotation, and... more Finite difference schemes of various eigenvalue problems are used to generate size, rotation, and translation invariant sets of features for shape recognition and classification of binary images. These feature sets are based on the eigenvalues of the Dirichlet Laplacian, the clamped plate problem, and the buckling problem. The stability and effectiveness of these features is demonstrated by using them in the classification of 6 types of computer generated and hand-drawn shapes. The classification was done using 4 to 20 features fed to simple feed-forward neural networks trained using the backpropagation algorithm. All features performed very well and correct classification rates of up to 99.7% were achieved on the computer generated shapes and 97.2% on the hand-drawn shapes.

Research paper thumbnail of Max-to-mean ratio estimates for the fundamental eigenfunction of the Dirichlet Laplacian

Contemporary Mathematics, 2011

We review upper and lower bound isoperimetric properties of the fundamental eigenfunction of the ... more We review upper and lower bound isoperimetric properties of the fundamental eigenfunction of the Dirichlet Laplacian and announce new reverse Hölder type inequalities for norms of this function in the case of wedgelike membrane.

Research paper thumbnail of Feature generation using the Laplacian operator with neumann boundary condition

Proceedings 2007 IEEE SoutheastCon, 2007

... Engineering Dept University of West Florida mkhabou@uwf.edu Mohamed BH Rhouma Mathematics and... more ... Engineering Dept University of West Florida mkhabou@uwf.edu Mohamed BH Rhouma Mathematics and Statistics Dept Sultan Qaboos University rhouma@squ.edu.om Lotfi Hermi Mathematics Dept University of Arizona hermi@math.arizona.edu Abstract ...

Research paper thumbnail of Two new Weyl-type bounds for the Dirichlet Laplacian

Transactions of the American Mathematical Society, 2008

Research paper thumbnail of Shape Recognition Based on Eigenvalues of the Laplacian

Advances in Imaging and Electron Physics, 2011

Shape Recognition Based on Eigenvalues of the Laplacian Mohamed Ben Haj Rhouma, Mohamed Ali Khabo... more Shape Recognition Based on Eigenvalues of the Laplacian Mohamed Ben Haj Rhouma, Mohamed Ali Khabou, and Lotfi Hermi* Contents 1. Introduction 186 2. Related Work 187 3. Eigenvalues of the Dirichlet and Neumann Laplacians 189 3.1. ... 186 M. Ben Haj Rhouma etal. ...

Research paper thumbnail of Quadratic Interpolation and Rayleigh–Ritz Methods for Bifurcation Coefficients

SIAM Journal on Mathematical Analysis, 2010

In this article we study the estimation of bifurcation coefficients in nonlinear branching proble... more In this article we study the estimation of bifurcation coefficients in nonlinear branching problems by means of Rayleigh-Ritz approximation to the eigenvectors of the corresponding linearized problem. It is essential that the approximations converge in a norm of sufficient strength to render the nonlinearities continuous. Quadratic interpolation between Hilbert spaces is used to seek sharp rate of convergence results for bifurcation coefficients. Examples from ordinary and partial differential problems are presented.

Research paper thumbnail of On Extending the Inequalities of Payne, Pólya, and Weinberger Using Spherical Harmonics

Rocky Mountain Journal of Mathematics, 2008

Using spherical harmonics, rearrangement techniques, the Sobolev inequality, and Chiti's reverse ... more Using spherical harmonics, rearrangement techniques, the Sobolev inequality, and Chiti's reverse Hölder inequality, we obtain extensions of a classical result of Payne, Pólya, and Weinberger bounding the gap between consecutive eigenvalues of the Dirichlet Laplacian in terms of moments of the preceding ones. The extensions yield domain-dependent inequalities.

Research paper thumbnail of Shape recognition using eigenvalues of the Dirichlet Laplacian

Pattern Recognition, 2007

The eigenvalues of the Dirichlet Laplacian are used to generate three different sets of features ... more The eigenvalues of the Dirichlet Laplacian are used to generate three different sets of features for shape recognition and classification in binary images. The generated features are rotation-, translation-, and size-invariant. The features are also shown to be tolerant of noise and boundary deformation. These features are used to classify hand-drawn, synthetic, and natural shapes with correct classification rates ranging from 88.9% to 99.2%. The classification was done using few features (only two features in some cases) and simple feedforward neural networks or minimum Euclidian distance.

Research paper thumbnail of A unified approach to universal inequalities for eigenvalues of elliptic operators

Pacific Journal of Mathematics, 2004

We present an abstract approach to universal inequalities for the discrete spectrum of a self-adj... more We present an abstract approach to universal inequalities for the discrete spectrum of a self-adjoint operator, based on commutator algebra, the Rayleigh-Ritz principle, and one set of "auxiliary" operators. The new proof unifies classical inequalities of Payne-Pólya-Weinberger, Hile-Protter, and H.C. Yang and provides a Yang type strengthening of Hook's bounds for various elliptic operators with Dirichlet boundary conditions. The proof avoids the introduction of the "free parameters" of many previous authors and relies on earlier works of Ashbaugh and Benguria, and, especially, Harrell (alone and with Michel), in addition to those of the other authors listed above. The Yang type inequality is proved to be stronger under general conditions on the operator and the auxiliary operators. This approach provides an alternative route to recent results obtained by Harrell and Stubbe.

Research paper thumbnail of Isoperimetric Inequalities for a Wedge-Like Membrane

Annales Henri Poincaré, 2013

For a wedge-like membrane, Payne and Weinberger proved in 1960 an isoperimetric inequality for th... more For a wedge-like membrane, Payne and Weinberger proved in 1960 an isoperimetric inequality for the fundamental eigenvalue which in some cases improves the classical isoperimetric inequality of Faber-Krahn. In this work, we introduce "relative torsional rigidity" for this type of membrane and prove new isoperimetric inequalities in the spirit of Saint-Venant, Pólya-Szegő, Payne, Payne-Rayner, Chiti, and Talenti, which link the eigenvalue problem with the boundary value problem in a fundamental way.

Research paper thumbnail of Performance Comparison of Laplacian-Based Features

... Mohamed A. Khabou 1 , Mohamed BH Rhouma 2 , and Lotfi Hermi 3 ... The problem in the two-dime... more ... Mohamed A. Khabou 1 , Mohamed BH Rhouma 2 , and Lotfi Hermi 3 ... The problem in the two-dimensional case remained open until 1992, when Gordon et al [3] constructed Bilby and Hawk (Figure 1), the first pair of regions in the plane that ...

Research paper thumbnail of Bounds for Eigenvalues of the Schrodinger Operator

math.umass.edu

0.1. Bounds for Eigenvalues of the Schrödinger Operator [4]. Another re-search topic, related wit... more 0.1. Bounds for Eigenvalues of the Schrödinger Operator [4]. Another re-search topic, related with integrable systems, is a joint work with, and a problem suggested by Dr. Lotfi Hermi. Consider the the Schrödinger operator: ... N ∑ i=1 λ 1/2 i + 1 4π ... N ∑ i=1 λ 3/2 i − 3 2π

Research paper thumbnail of Universal Inequalities for Higher-Order Elliptic Operators

preprint, 2003

Abstract. Two new abstract explicit formulas relating gaps of eigenval-ues of a symmetric operato... more Abstract. Two new abstract explicit formulas relating gaps of eigenval-ues of a symmetric operator to two sets of symmetric and skew-symmetric operators and their commutators are developed. The formulas prove to be equivalent to earlier work by Hook and Harrell & ...

Research paper thumbnail of On Harrell-Stubbe Type Inequalities for the Discrete Spectrum of a Self-Adjoint Operator

preprint

We produce a new proof and extend results by Harrell and Stubbe for the discrete spectrum of a se... more We produce a new proof and extend results by Harrell and Stubbe for the discrete spectrum of a self-adjoint operator. An abstract approach–based on com-mutator algebra, the Rayleigh-Ritz principle, and an “optimal” usage of the Cauchy-Schwarz inequality–is ...

Research paper thumbnail of An isoperimetric inequality of Saint-Venant-type for a wedge-like membrane

Why care? Consider right isoceles triangle with equal sides of unit length (Payne-Weinberger) α =... more Why care? Consider right isoceles triangle with equal sides of unit length (Payne-Weinberger) α = 1, λ 1 ≥ 45.0734 α = 2, λ 1 ≥ 47.6325 α = 4, λ 1 ≥ 45.9094 Faber-Krahn (among all domains): λ 1 ≥ 36.3368 Faber-Krahn (among all triangles): λ 1 ≥ 4π 2 √ 3 A ≈ 45.5858 Exact value: λ 1 = 49.350625 An isoperimetric inequality of Saint-Venant-type for a wedge-lik

Research paper thumbnail of Green’s Functions and Euler’s Formula for <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow></mrow><annotation encoding="application/x-tex"></annotation></semantics></math></span><span class="katex-html" aria-hidden="true"></span></span>\zeta (2n)$$

Research paper thumbnail of Trace Formulas Applied to the Riemann ζ-Function

Integrability, Supersymmetry and Coherent States, 2019

We use a spectral theory perspective to reconsider properties of the Riemann zeta function. In pa... more We use a spectral theory perspective to reconsider properties of the Riemann zeta function. In particular, new integral representations are derived and used to present its value at odd positive integers.

Research paper thumbnail of On Rayleigh-type formulas for a non-local boundary value problem associated with an integral operator commuting with the Laplacian

Applied and Computational Harmonic Analysis, 2018

In this article we prove the existence, uniqueness, and simplicity of a negative eigenvalue for a... more In this article we prove the existence, uniqueness, and simplicity of a negative eigenvalue for a class of integral operators whose kernel is of the form |x − y| ρ , 0 < ρ ≤ 1, x, y ∈ [−a, a]. We also provide two different ways of producing recursive formulas for the Rayleigh functions (i.e., recursion formulas for power sums) of the eigenvalues of this integral operator when ρ = 1, providing means of approximating this negative eigenvalue. These methods offer recursive procedures for dealing with the eigenvalues of a one-dimensional Laplacian with non-local boundary conditions which commutes with an integral operator having a harmonic kernel. The problem emerged in recent work by one of the authors [48]. We also discuss extensions in higher dimensions and links with distance matrices.

Research paper thumbnail of Abstract. We review upper and lower bound isoperimetric properties of the fundamental eigenfunction of the Dirichlet Laplacian and announce new reverse H¨older type inequalities for norms of this function in the case of a wedgelike membrane

We review upper and lower bound isoperimetric properties of the fundamental eigenfunction of the ... more We review upper and lower bound isoperimetric properties of the fundamental eigenfunction of the Dirichlet Laplacian and announce new reverse H¨older type inequalities for norms of this function in the case of a wedgelike membrane.

Research paper thumbnail of Laplacian and Bilaplacian Based Features for Shape Classification

Finite difference schemes of various eigenvalue problems are used to generate size, rotation, and... more Finite difference schemes of various eigenvalue problems are used to generate size, rotation, and translation invariant sets of features for shape recognition and classification of binary images. These feature sets are based on the eigenvalues of the Dirichlet Laplacian, the clamped plate problem, and the buckling problem. The stability and effectiveness of these features is demonstrated by using them in the classification of 6 types of computer generated and hand-drawn shapes. The classification was done using 4 to 20 features fed to simple feed-forward neural networks trained using the backpropagation algorithm. All features performed very well and correct classification rates of up to 99.7% were achieved on the computer generated shapes and 97.2% on the hand-drawn shapes.

Research paper thumbnail of Max-to-mean ratio estimates for the fundamental eigenfunction of the Dirichlet Laplacian

Contemporary Mathematics, 2011

We review upper and lower bound isoperimetric properties of the fundamental eigenfunction of the ... more We review upper and lower bound isoperimetric properties of the fundamental eigenfunction of the Dirichlet Laplacian and announce new reverse Hölder type inequalities for norms of this function in the case of wedgelike membrane.

Research paper thumbnail of Feature generation using the Laplacian operator with neumann boundary condition

Proceedings 2007 IEEE SoutheastCon, 2007

... Engineering Dept University of West Florida mkhabou@uwf.edu Mohamed BH Rhouma Mathematics and... more ... Engineering Dept University of West Florida mkhabou@uwf.edu Mohamed BH Rhouma Mathematics and Statistics Dept Sultan Qaboos University rhouma@squ.edu.om Lotfi Hermi Mathematics Dept University of Arizona hermi@math.arizona.edu Abstract ...

Research paper thumbnail of Two new Weyl-type bounds for the Dirichlet Laplacian

Transactions of the American Mathematical Society, 2008

Research paper thumbnail of Shape Recognition Based on Eigenvalues of the Laplacian

Advances in Imaging and Electron Physics, 2011

Shape Recognition Based on Eigenvalues of the Laplacian Mohamed Ben Haj Rhouma, Mohamed Ali Khabo... more Shape Recognition Based on Eigenvalues of the Laplacian Mohamed Ben Haj Rhouma, Mohamed Ali Khabou, and Lotfi Hermi* Contents 1. Introduction 186 2. Related Work 187 3. Eigenvalues of the Dirichlet and Neumann Laplacians 189 3.1. ... 186 M. Ben Haj Rhouma etal. ...

Research paper thumbnail of Quadratic Interpolation and Rayleigh–Ritz Methods for Bifurcation Coefficients

SIAM Journal on Mathematical Analysis, 2010

In this article we study the estimation of bifurcation coefficients in nonlinear branching proble... more In this article we study the estimation of bifurcation coefficients in nonlinear branching problems by means of Rayleigh-Ritz approximation to the eigenvectors of the corresponding linearized problem. It is essential that the approximations converge in a norm of sufficient strength to render the nonlinearities continuous. Quadratic interpolation between Hilbert spaces is used to seek sharp rate of convergence results for bifurcation coefficients. Examples from ordinary and partial differential problems are presented.

Research paper thumbnail of On Extending the Inequalities of Payne, Pólya, and Weinberger Using Spherical Harmonics

Rocky Mountain Journal of Mathematics, 2008

Using spherical harmonics, rearrangement techniques, the Sobolev inequality, and Chiti's reverse ... more Using spherical harmonics, rearrangement techniques, the Sobolev inequality, and Chiti's reverse Hölder inequality, we obtain extensions of a classical result of Payne, Pólya, and Weinberger bounding the gap between consecutive eigenvalues of the Dirichlet Laplacian in terms of moments of the preceding ones. The extensions yield domain-dependent inequalities.

Research paper thumbnail of Shape recognition using eigenvalues of the Dirichlet Laplacian

Pattern Recognition, 2007

The eigenvalues of the Dirichlet Laplacian are used to generate three different sets of features ... more The eigenvalues of the Dirichlet Laplacian are used to generate three different sets of features for shape recognition and classification in binary images. The generated features are rotation-, translation-, and size-invariant. The features are also shown to be tolerant of noise and boundary deformation. These features are used to classify hand-drawn, synthetic, and natural shapes with correct classification rates ranging from 88.9% to 99.2%. The classification was done using few features (only two features in some cases) and simple feedforward neural networks or minimum Euclidian distance.

Research paper thumbnail of A unified approach to universal inequalities for eigenvalues of elliptic operators

Pacific Journal of Mathematics, 2004

We present an abstract approach to universal inequalities for the discrete spectrum of a self-adj... more We present an abstract approach to universal inequalities for the discrete spectrum of a self-adjoint operator, based on commutator algebra, the Rayleigh-Ritz principle, and one set of "auxiliary" operators. The new proof unifies classical inequalities of Payne-Pólya-Weinberger, Hile-Protter, and H.C. Yang and provides a Yang type strengthening of Hook's bounds for various elliptic operators with Dirichlet boundary conditions. The proof avoids the introduction of the "free parameters" of many previous authors and relies on earlier works of Ashbaugh and Benguria, and, especially, Harrell (alone and with Michel), in addition to those of the other authors listed above. The Yang type inequality is proved to be stronger under general conditions on the operator and the auxiliary operators. This approach provides an alternative route to recent results obtained by Harrell and Stubbe.

Research paper thumbnail of Isoperimetric Inequalities for a Wedge-Like Membrane

Annales Henri Poincaré, 2013

For a wedge-like membrane, Payne and Weinberger proved in 1960 an isoperimetric inequality for th... more For a wedge-like membrane, Payne and Weinberger proved in 1960 an isoperimetric inequality for the fundamental eigenvalue which in some cases improves the classical isoperimetric inequality of Faber-Krahn. In this work, we introduce "relative torsional rigidity" for this type of membrane and prove new isoperimetric inequalities in the spirit of Saint-Venant, Pólya-Szegő, Payne, Payne-Rayner, Chiti, and Talenti, which link the eigenvalue problem with the boundary value problem in a fundamental way.

Research paper thumbnail of Performance Comparison of Laplacian-Based Features

... Mohamed A. Khabou 1 , Mohamed BH Rhouma 2 , and Lotfi Hermi 3 ... The problem in the two-dime... more ... Mohamed A. Khabou 1 , Mohamed BH Rhouma 2 , and Lotfi Hermi 3 ... The problem in the two-dimensional case remained open until 1992, when Gordon et al [3] constructed Bilby and Hawk (Figure 1), the first pair of regions in the plane that ...

Research paper thumbnail of Bounds for Eigenvalues of the Schrodinger Operator

math.umass.edu

0.1. Bounds for Eigenvalues of the Schrödinger Operator [4]. Another re-search topic, related wit... more 0.1. Bounds for Eigenvalues of the Schrödinger Operator [4]. Another re-search topic, related with integrable systems, is a joint work with, and a problem suggested by Dr. Lotfi Hermi. Consider the the Schrödinger operator: ... N ∑ i=1 λ 1/2 i + 1 4π ... N ∑ i=1 λ 3/2 i − 3 2π

Research paper thumbnail of Universal Inequalities for Higher-Order Elliptic Operators

preprint, 2003

Abstract. Two new abstract explicit formulas relating gaps of eigenval-ues of a symmetric operato... more Abstract. Two new abstract explicit formulas relating gaps of eigenval-ues of a symmetric operator to two sets of symmetric and skew-symmetric operators and their commutators are developed. The formulas prove to be equivalent to earlier work by Hook and Harrell & ...

Research paper thumbnail of On Harrell-Stubbe Type Inequalities for the Discrete Spectrum of a Self-Adjoint Operator

preprint

We produce a new proof and extend results by Harrell and Stubbe for the discrete spectrum of a se... more We produce a new proof and extend results by Harrell and Stubbe for the discrete spectrum of a self-adjoint operator. An abstract approach–based on com-mutator algebra, the Rayleigh-Ritz principle, and an “optimal” usage of the Cauchy-Schwarz inequality–is ...