text:
## Alessandro CALAMAI

49 pubblicazioni classificate nel seguente modo:

 Nr. doc. Classificazioni 47 1 Contributo su Rivista 2 2 Contributo in Volume
Anno
Risorse
2021
Global persistence of the unit eigenvectors of perturbed eigenvalue problems in hilbert spaces: The odd multiplicity case
MATHEMATICS
Autore/i: Benevieri, P.; Calamai, A.; Furi, M.; Pera, M. P.
Classificazione: 1 Contributo su Rivista
Abstract: We study the persistence of eigenvalues and eigenvectors of perturbed eigenvalue problems in Hilbert spaces. We assume that the unperturbed problem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation result. The approach is topological, based on a notion of degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/290094

2021
Boundary value problems associated with singular strongly nonlinear equations with functional terms
Autore/i: Biagi, Stefano; Calamai, Alessandro; Marcelli, Cristina; Papalini, Francesca
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/285330

2020
Existence results for boundary value problems associated with singular strongly nonlinear equations
JOURNAL OF FIXED POINT THEORY AND ITS APPLICATIONS
Autore/i: Biagi, S.; Calamai, A.; Papalini, F.
Classificazione: 1 Contributo su Rivista
Abstract: We consider a strongly nonlinear differential equation of the following general type: (Φ(a(t,x(t))x′(t)))′=f(t,x(t),x′(t)),a.e. on[0,T],where f is a Carathédory function, Φ is a strictly increasing homeomorphism (the Φ -Laplacian operator), and the function a is continuous and non-negative. We assume that a(t, x) is bounded from below by a non-negative function h(t), independent of x and such that 1 / h∈ Lp(0 , T) for some p> 1 , and we require a weak growth condition of Wintner–Nagumo type. Under these assumptions, we prove existence results for the Dirichlet problem associated with the above equation, as well as for different boundary conditions. Our approach combines fixed point techniques and the upper/lower solution method.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/284544

2020
Global continuation in euclidean spaces of the perturbed unit eigenvectors corresponding to a simple eigenvalue
TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS
Autore/i: Benevieri, P.; Calamai, A.; Furi, M.; Pera, M. P.
Classificazione: 1 Contributo su Rivista
Abstract: In the Euclidean space Rk, we consider the perturbed eigenvalue problem Lx + εN(x) = λx, ||x|| = 1, where ε, λ are real parameters, L is a linear endomorphism of Rk, and N: Sk−1 → Rk is a continuous map defined on the unit sphere of Rk . We prove a global continuation result for the solutions (x, ε, λ) of this problem. Namely, under the assumption that x_* is one of the two unit eigenvectors of L corresponding to a simple eigenvalue λ_*, we show that, in the set of all the solutions, the connected component containing (x_*, 0, λ_*) is either unbounded or meets a solution (x*, 0, λ*) having x* ≠ x_*. Our result is inspired by a paper of R. Chiappinelli concerning the local persistence property of eigenvalues and eigenvectors of a perturbed self-adjoint operator in a real Hilbert space.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/276779

2020
Nonzero Positive Solutions of Elliptic Systems with Gradient Dependence and Functional BCs
Autore/i: Biagi, S.; Calamai, A.; Infante, G.
Classificazione: 1 Contributo su Rivista
Abstract: We discuss, by topological methods, the solvability of systems of second-order elliptic differential equations subject to functional boundary conditions under the presence of gradient terms in the nonlinearities. We prove the existence of nonnegative solutions and provide a non-existence result. We present some examples to illustrate the applicability of the existence and non-existence results.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/284545

2020
Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces
ZEITSCHRIFT FUR ANALYSIS UND IHRE ANWENDUNGEN
Autore/i: Benevieri, P.; Calamai, A.; Furi, M.; Pera, M. P.
Classificazione: 1 Contributo su Rivista
Abstract: We consider the nonlinear eigenvalue problem where are real parameters, L,C : G H are bounded linear operators between separable real Hilbert spaces, and N : S H is a continuous map defined on the unit sphere of G. We prove a global persistence result regarding the set of the solutions (x,) SR R of this problem. Namely, if the operators N and C are compact, under suitable assumptions on a solution p= (x, 0, ) of the unperturbed problem, we prove that the connected component of containing pis either unbounded or meets a triple p= (x, 0,) with p6= p. When C is the identity and G = H is finite dimensional, the assumptions on (x, 0,) mean that xis an eigenvector of L whose corresponding eigenvalue,is simple. Therefore, we extend a previous result obtained by the authors in the finite dimensional setting. Our work is inspired by a paper of R. Chiappinelli concerning the local persistence property of the unit eigenvectors of perturbed self-adjoint operators in a real Hilbert space.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/290093

2019
Sustainable Engineering for Resilient Built and Natural Environments
The First Outstanding 50 Years of “Università Politecnica delle Marche”
Autore/i: Alici, Antonello; Bocci, Maurizio; Bonvini, Paolo; Brocchini, Maurizio; Calamai, Alessandro; Canestrari, Francesco; Capozucca, Roberto; Carbonari, Alessandro; Carbonari, Sandro; Cardone, Fabrizio; Clementi, Francesco; Clini, Paolo; Cocchi, Giammichele; Corvaro, Sara; Darvini, Giovanna; Davì, Fabrizio; Dezi, Luigino; Di Giuseppe, Elisa; D’Orazio, Marco; Ferretti, Maddalena; Ferrotti, Gilda; Gara, Fabrizio; Giretti, Alberto; Graziani, Andrea; Lancioni, Giovanni; Lemma, Massimo; Lenci, Stefano; Lorenzoni, Carlo; Malinverni, Eva Savina; Mancinelli, Alessandro; Mariano, Fabio; Mentrasti, Lando; Mondaini, Gianluigi; Montecchiari, Piero; Munafò, Placido; Naticchia, Berardo; Postacchini, Matteo; Quagliarini, Enrico; Quattrini, Ramona; Ragni, Laura; Serpilli, Michele; Soldini, Luciano; Virgili, Amedeo; Zampini, Giovanni
Editore: Springer
Classificazione: 2 Contributo in Volume
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/272866

2019
Heteroclinic solutions for a class of boundary value problems associated with singular equations
NONLINEAR ANALYSIS
Autore/i: Biagi, Stefano; Calamai, Alessandro; Papalini, Francesca
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/264799

2019
Branches of Forced Oscillations Induced by a Delayed Periodic Force
Autore/i: Calamai, Alessandro; Pera, Maria Patrizia; Spadini, Marco
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/259831

2019
New Multimedia Technologies as Tools for a Modern Approach to Scientific Communication and Teaching of Mathematical Sciences
The First Outstanding 50 Years of “Università Politecnica delle Marche”
Autore/i: Alessio, F. G.; Brambilla, M. C.; Calamai, A.; de Fabritiis, C.; Demeio, L.; Franca, M.; Marcelli, C.; Marietti, M.; Montecchiari, P.; Papalini, F.; Petrini, M.; Telloni, A. I.
Editore: Springer
Classificazione: 2 Contributo in Volume
Abstract: This paper describes various technological tools and e-learning resources used by the authors in their work as university teachers and communicators of mathematics to a general public. As pointed out by recent developments in mathematics education research, a technology enhanced teaching may have significant potentialities at tertiary level: e-learning is an essential support to overcome some logistic obstacles such as the large number of students per teacher, the small number of hours of lesson available, the heterogeneity of the freshmen’s mathematical background. Moreover, a fine design of digital environments can foster an engaging, inclusive, flexible and meaningful learning of mathematical topics. In this paper we give a perspective on some educational experiments carried out with engineering students of Università Politecnica delle Marche through specific functionalities of the Moodle platform (quiz, forum, workshop) and the dynamic geometry software Geogebra. We also report shortly on our approach to the use of new technological devices in the dissemination of scientific knowledge to a broader audience.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/272790

2018
Boundary value problems for singular second order equations
FIXED POINT THEORY AND APPLICATIONS
Autore/i: Calamai, Alessandro; Marcelli, Cristina; Papalini, Francesca
Classificazione: 1 Contributo su Rivista
Abstract: We investigate strongly nonlinear differential equations of the type (Phi(k(t)u′(t)))′=f(t,u(t),u′(t)),a.e.on[0,T],where Phi is a strictly increasing homeomorphism and the nonnegative functionkmay vanish on a set of measure zero. By using the upper and lower solutions method,we prove existence results for the Dirichlet problem associated with the above equation, as well as for different boundary conditions involving the function k. Our existence results require a weak form of a Wintner–Nagumo growth condition.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/259810

2018
Global continuation of the eigenvalues of a perturbed linear operator
ANNALI DI MATEMATICA PURA ED APPLICATA
Autore/i: Benevieri, Pierluigi; Calamai, Alessandro; Furi, Massimo; Pera, Maria Patrizia
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/252490

2018
Multiplicity of forced oscillations for scalar retarded functional differential equations
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
Autore/i: Calamai, Alessandro; Pera, Maria Patrizia; Spadini, Marco
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/252491

2017
On general properties of n-th order retarded functional differential equations
RENDICONTI DELL'ISTITUTO DI MATEMATICA DELL'UNIVERSITÀ DI TRIESTE
Autore/i: Benevieri, Pierluigi; Calamai, Alessandro; Furi, Massimo; Pera, Maria Patrizia
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/252517

2017
Multiplicity of forced oscillations for the spherical pendulum acted on by a retarded periodic force
NONLINEAR ANALYSIS
Autore/i: Calamai, Alessandro; Pera, Maria Patrizia; Spadini, Marco
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/244836

2017
On the persistence of the eigenvalues of a perturbed fredholm operator of index zero under nonsmooth perturbations
ZEITSCHRIFT FUR ANALYSIS UND IHRE ANWENDUNGEN
Autore/i: Benevieri, Pierluigi; CALAMAI, Alessandro; Furi, Massimo; Pera, Maria Patrizia
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/246998

2017
On the position of chaotic trajectories
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
Autore/i: CALAMAI, Alessandro; Diblík, J.; FRANCA, Matteo; Pospíšil, M.
Classificazione: 1 Contributo su Rivista
Abstract: The main purpose of this paper is to locate trajectories of a perturbed system, which is known to behave chaotically. The unperturbed system is assumed to have the origin as a hyperbolic fixed point, and to admit a trajectory homoclinic to the origin. This homocline is assumed to lie in a prescribed region having the origin in its border. Using a Mel'nikov type approach, we introduce natural conditions ensuring that all the chaotic trajectories of the perturbed system, given by classical results, lie in the same region. The applicability of our results is illustrated in two examples. In the first one, we find positive radial solutions for a class of P.D.E.'s, obtaining new results in the case of critical equations ruled by Laplacian with Hardy potentials. In the other one, we show that under certain conditions one of two weakly coupled pendula moves in one direction only.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/231642

2017
Multiplicity of periodic solutions for systems of weakly coupled parametrized second order differential equations
NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS
Autore/i: Calamai, Alessandro; Sfecci, Andrea
Classificazione: 1 Contributo su Rivista
Abstract: We prove a multiplicity result of periodic solutions for a system of second order differential equations having asymmetric nonlinearities. The proof is based on a recent generalization of the Poincaré-Birkhoff fixed point theorem provided by Fonda and Urena.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/246994

2016
Nontrivial solutions of boundary value problems for second-order functional differential equations
ANNALI DI MATEMATICA PURA ED APPLICATA
Autore/i: Calamai, Alessandro; Infante, G.
Classificazione: 1 Contributo su Rivista
Abstract: In this paper we present a theory for the existence of multiple nontrivial solutions for a class of perturbed Hammerstein integral equations. Our methodology, rather than to work directly in cones, is to utilize the theory of fixed point index on affine cones. This approach is fairly general and covers a class of nonlocal boundary value problems for functional differential equations. Some examples are given in order to illustrate our theoretical results.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/223115

2015
Periodic perturbations of constrained motion problems on a class of implicitly defined manifolds
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/202514

2015
On the degree for oriented quasi-Fredholm maps: its uniqueness and its effective extension of the Leray–Schauder degree
TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS
Autore/i: P., Benevieri; CALAMAI, Alessandro; M., Furi
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/202515

2014
Global continuation of forced oscillations of retarded motion equations on manifolds
JOURNAL OF FIXED POINT THEORY AND ITS APPLICATIONS
Autore/i: P., Benevieri; Calamai, Alessandro; M., Furi; M. P., Pera
Classificazione: 1 Contributo su Rivista
Abstract: We investigate T-periodic parametrized retarded functional motion equations on (possibly) noncompact manifolds; that is, constrained second order retarded functional differential equations. For such equations we prove a global continuation result for T-periodic solutions. The approach is topological and is based on the degree theory for tangent vector fields as well as on the fixed point index theory. Our main theorem is a generalization to the case of retarded equations of an analogous result obtained by the last two authors for second order differential equations on manifolds. As corollaries we derive a Rabinowitz-type global bifurcation result and a Mawhin-type continua-tion principle. Finally, we deduce the existence of forced oscillations for the retarded spherical pendulum under general assumptions.
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/202516

2014
Periodic solutions of semi-explicit differential-algebraic equations with time-dependent constraints
BOUNDARY VALUE PROBLEMS
Autore/i: L., Bisconti; Calamai, Alessandro; M., Spadini
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/202513

2013
Mel’nikov methods and homoclinic orbits in discontinuous systems
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
Autore/i: Calamai, Alessandro; Franca, Matteo
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/112676

2013
A continuation result for forced oscillations of constrained motion problems with infinite delay
Autore/i: P. BENEVIERI; A. CALAMAI; M . FURI; M. P. PERA
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54504

2013
On general properties of retarded functional differential equations on manifolds
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
Autore/i: P., Benevieri; Calamai, Alessandro; M., Furi; M. P., Pera
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/66064

2013
Global continuation of periodic solutions for retarded functional differential equations on manifolds
BOUNDARY VALUE PROBLEMS
Autore/i: BENEVIERI P; A. CALAMAI; FURI M; PERA M. P
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/112677

2012
On the existence of forced oscillations of retarded functional motion equations on a class of topologically nontrivial manifolds
RENDICONTI DELL'ISTITUTO DI MATEMATICA DELL'UNIVERSITÀ DI TRIESTE
Autore/i: P. BENEVIERI; A. CALAMAI; M. FURI; M.P. PERA
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/66065

2012
Branches of forced oscillations for a class of constrained ODEs: a topological approach
NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/64419

2011
Branches of harmonic solutions for a class of periodic differential-algebraic equations
COMMUNICATIONS IN APPLIED ANALYSIS
Autore/i: A. CALAMAI
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54505

2011
Heteroclinic solutions of boundary value problems on the real line involving singular Phi-Laplacian operators
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
Autore/i: A. CALAMAI
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54583

2011
On the existence of forced oscillations for the spherical pendulum acted on by a retarded periodic force
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
Autore/i: Benevieri, P; Calamai, Alessandro; Furi, M; Pera, M. P.
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54503

2010
A Borsuk-type theorem for some classes of perturbed Fredholm maps
TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS
Autore/i: A. CALAMAI; P. BENEVIERI
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54507

2010
Yet another spectrum for nonlinear operators in Banach spaces
NONLINEAR FUNCTIONAL ANALYSIS AND APPLICATIONS
Autore/i: J. APPELL; A. CALAMAI; A. SCHMIED
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54508

2009
A new spectrum for continuous nonlinear operators in Banach spaces
NONLINEAR FUNCTIONAL ANALYSIS AND APPLICATIONS
Autore/i: A. CALAMAI; M. FURI; A. VIGNOLI
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54502

2009
Fast forced oscillations for constrained motion problems with delay
COMMUNICATIONS IN APPLIED ANALYSIS
Autore/i: P. BENEVIERI; A. CALAMAI; M. FURI; M.P. PERA
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54513

2009
An overview on spectral theory for nonlinear operators
COMMUNICATIONS IN APPLIED ANALYSIS
Autore/i: A. CALAMAI; M. FURI; A. VIGNOLI
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54514

2009
Retarded functional differential equations on manifolds and applications to motion problems for forced constrained systems
Autore/i: P. BENEVIERI; A. CALAMAI; M. FURI; M.P. PERA
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54510

2009
Delay differential equations on manifolds and applications to motion problems for forced constrained systems
ZEITSCHRIFT FUR ANALYSIS UND IHRE ANWENDUNGEN
Autore/i: P. BENEVIERI; A. CALAMAI; M. FURI; M.P. PERA
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54511

2009
A general approach for front-propagation in functional reaction-diffusion equations
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
Autore/i: CALAMAI A; C. MARCELLI ; PAPALINI F
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/52227

2009
On forced fast oscillations for delay differential equations on compact manifolds.
JOURNAL OF DIFFERENTIAL EQUATIONS
Autore/i: P. BENEVIERI; A. CALAMAI; M. FURI; M.P. PERA
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54512

2008
Bifurcation results for a class of perturbed Fredholm maps
FIXED POINT THEORY AND APPLICATIONS
Autore/i: P. BENEVIERI; A. CALAMAI
Classificazione: 1 Contributo su Rivista
Abstract: Art. ID 752657
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54509

2007
Saddle type solutions to a class of semilinear elliptic equations
Autore/i: F. ALESSIO; A. CALAMAI; P. MONTECCHIARI
Classificazione: 1 Contributo su Rivista
Abstract: We consider a class of semilinear elliptic equations of the form $$-\Delta u(x,y)+W'(u(x,y))=0,\quad (x,y)\in\R^{2}$$ where $W:\R\to\R$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-1)^{2}$. We show, via variational methods, that for any $j\geq 2$, the equation has a solution $v_{j}\in C^{2}(\R^{2})$ with $|v_{j}(x,y)|\leq 1$ for any $(x,y)\in\R^{2}$ satisfying the following symmetric and asymptotic conditions: setting $\tilde v_{j}(\rho,\theta)=v_{j}(\rho\cos( \theta),\rho\sin(\theta))$, there results $\tilde v_{j}(\rho,\frac{\pi}{2}+\theta)=-\tilde v_{j}(\rho,\frac{\pi}{2}-\theta)$, $\tilde v_{j}(\rho,\theta+\frac{\pi}{j})=-\tilde v_{j}(\rho,\theta)$, $\forall (\rho,\theta)\in \R^{+}\times\R$ and $\tilde v_{j}(\rho,\theta)\to 1$ as $\rho\to+\infty$ for any $\theta\in [\frac{\pi}2-\frac{\pi}{2j},\frac{\pi}2)$.}
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/34140

2007
Forced oscillations for delay motion equations on manifolds
ELECTRONIC JOURNAL OF DIFFERENTIAL EQUATIONS
Autore/i: P. BENEVIERI; A. CALAMAI; M. FURI; M.P. PERA
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54501

2007
Global branches of periodic solutions for forced delay differential equations on compact manifolds
JOURNAL OF DIFFERENTIAL EQUATIONS
Autore/i: P. BENEVIERI; A. CALAMAI; M. FURI; M.P. PERA
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54499

2006
A degree theory for a class of perturbed Fredholm maps II
FIXED POINT THEORY AND APPLICATIONS
Autore/i: P. BENEVIERI; A. CALAMAI; M. FURI
Classificazione: 1 Contributo su Rivista
Abstract: Art. ID 27154
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54498

2005
A degree theory for a class of perturbed Fredholm maps
FIXED POINT THEORY AND APPLICATIONS
Autore/i: P. BENEVIERI; A. CALAMAI; M. FURI
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54497

2004
On existence and uniqueness of solutions for ordinary differential equations with nonlinear boundary conditions
BOLLETTINO DELL'UNIONE MATEMATICA ITALIANA. B
Autore/i: A. CALAMAI
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54495

2004
The invariance of domain theorem for compact perturbations of nonlinear Fredholm maps of index zero.
NONLINEAR FUNCTIONAL ANALYSIS AND APPLICATIONS
Autore/i: A. CALAMAI
Classificazione: 1 Contributo su Rivista
Scheda della pubblicazione: https://iris.univpm.it/handle/11566/54496

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text:
# B About the surface area of a prolate ellipsoid

Tags:
1. Feb 18, 2017

### Tahmeed

Is there any limit for which we can approximately write the surface area of a prolate ellipsoid to be 4piA*B comparing with the spherical 4piR*R??

Last edited: Feb 18, 2017
2. Feb 18, 2017

### haruspex

Look at eqn 9 at http://mathworld.wolfram.com/ProlateSpheroid.html.
You could substitute for a in terms of c and e using eqn 8, then make an approximation for small e.

3. Feb 18, 2017

### Tahmeed

I don't know why but the link isnt working for me. However does it bring 4piAC as a result?

4. Feb 18, 2017

### haruspex

It can't be exactly that, of course, so it depends how good an approximation you want.
The exact equations at that link are
Area = $2\pi a^2+2\pi ac\sin^{-1}(e)/e$, eqn (9), and $a^2=c^2(1-e^2)$, eqn 8
where the radii are a, a, c.

format: plain


text:
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Singular boundary value problems for superlinear second order ordinary and delay differential equations. (English) Zbl 0863.34022

The authors study the existence of solutions for the following singular boundary value problems for nonlinear second order ordinary and delay differential equations:

$\left\{\begin{array}{cc}{y}^{\text{'}\text{'}}\left(t\right)+q\left(t\right)f\left(t,y\left(t\right)\right)=0,\hfill & 0
$\left\{\begin{array}{cc}\frac{1}{p\left(t\right)}{\left(p\left(t\right){y}^{\text{'}}\left(t\right)\right)}^{\text{'}}+q\left(t\right)f\left(t,y\left(t\right)\right)=0,\hfill & 0
$\left\{\begin{array}{cc}{y}^{\text{'}\text{'}}\left(t\right)+q\left(t\right)f\left(t,y\left(t-r\right)\right)=0,\hfill & t\in \left(0;1\right)\setminus \left\{r\right\},\hfill \\ y\left(t\right)=\mu \left(t\right),\hfill & t\in \left[-r;0\right],\hfill \\ y\left(1\right)=0,\hfill & 0

.

##### MSC:
 34B15 Nonlinear boundary value problems for ODE 34K10 Boundary value problems for functional-differential equations


text:
nykkyo Posts: 132, Reputation: 1 Junior Member #1 Apr 25, 2013, 10:06 AM
Why does light converge
On the observer and bathe him in light ?because light is emitted in all diections from the aource?
 ebaines Posts: 12,129, Reputation: 1307 Expert #2 Apr 25, 2013, 10:20 AM
If you're asking how it is that an observer on earth can see light emitted from a distant source - such as stars, planets, nebulae etc - it's because most objects emit light in all directions, so any observer would be able to see them regardless of location. But some sources actually emit light in a highly-directional way, so your ability to see it depends on your position relative to the source. An example of this would be pulsars, which are neutron stars that emit a directional beam of electromagnetic radiation, much like a spot light rotating. That beam can only be observed on Earth when it is pointing in our direction, so to us it appears as a pulsating light source that flashes in time with the rotation of the star.
 nykkyo Posts: 132, Reputation: 1 Junior Member #3 Apr 25, 2013, 04:51 PM
I mean why do we see points of light when light is emitted in all directions. It implies light rays from long distances are separate. In that case our understanding how light is emitted is suspect.
 ebaines Posts: 12,129, Reputation: 1307 Expert #4 Apr 26, 2013, 06:04 AM
Consider how the particles of light (photons) travel from the source to your eye. Photons are emitted from all parts of the star's surface, and if you are relatively close to the star (like we are to the sun) it appears as a round disc. If you moved further away the apparent size of the disc gets smaller. Get far enough away and the apparent size of the disc becomes too small to be distinguished by your eyes, yet there are enough photons hitting your eye so that you can still see that there is astar, but you can''t see the disc. Seeing light coming fom a source of negligible apparent size makes the object appear as a pin point of light.

I hope this makes it clear - again think of star light as a series of photon particles eminating in all directions from the star, some of which travel on a path that intersects with your eyes with the result that you see the star.
 nykkyo Posts: 132, Reputation: 1 Junior Member #5 Aug 25, 2013, 11:56 PM
Originally Posted by ebaines
Consider how the particles of light (photons) travel from the source to your eye. Photons are emitted from all parts of the stars surfac, and if you are relatively close to the star (like we are to the sun) it appears as a round disc. If you moved further away the apparent size of the disc gets smaller. Get far enough away and the apparanet size of the disc becomes too small to be distinguished by your eyes, yet there are enough photons hitting your eye so that you can still see that there is astar, but you can''t see the disc. Seeing light coming fom a source of negligible apparent size makes the object appear as a pin point of light.

I hope this makes it clear - again think of star light as a series of photon particles eminating in all directions from the star, some of which travel on a path that intersects with your eyes with the result that you see the star.
I understand the inverse square law; but why are some photons funneled to our aperture?
 ebaines Posts: 12,129, Reputation: 1307 Expert #6 Aug 26, 2013, 05:52 AM
Originally Posted by nykkyo
I understand the inverse square law; but why are some photons funneled to our aperture?
If you understand the inverse square law then you understand that as you get further away from a lighrt-emitting object the "density" of photons that passes through any given surface area (such as the surface area of a telescope's mirror) gets smaller as well, but it never goes all the way to 0. So some photons are going to hit the telescope no matter how far away you are. It's not that these photons are "channeled" to the scope, but rather that some numberof photons inevitably hit by chance.

Let's run some numbers - imagine that a light source emits N photons per second. Consider a spherical shell 1 meter in radius cenered on the source - N photons must pass through the surface of that shell every second. So if you stand 1 meter away with a telescope that has surface area of 1 m^2 the proportion of those photons that will hit the telescope is the ratio of the telescope's area to the surface area of the 1 meter shell:

$
n = N \left( \frac {1m^2}{4 \pi 1^2} \right) = 0.0796 N
$

If you stand temn meters away the ratioj gets smaller: n= 0.000796 N

At 100 meters: n= 7.96E-6 N

At 100 meters n = 7.96E-8 N

and so on. Note that the ratio never gets to zero.
 nykkyo Posts: 132, Reputation: 1 Junior Member #7 Aug 26, 2013, 07:15 AM
Thec inverse suar law decrease4s intensity (photon density) but does not decrease coverage of aprteure.
 ebaines Posts: 12,129, Reputation: 1307 Expert #8 Aug 26, 2013, 07:24 AM
Nykko - I don't understand the point of your last post - are you answering your own question from post #5?
 nykkyo Posts: 132, Reputation: 1 Junior Member #9 Aug 26, 2013, 07:42 AM
Originally Posted by ebaines
nykko - I don't understand the point of your last post - are you answering your own question from post #5?
Phton density is the same in all directions; but the farther away from the source the lower number of phtons hit an area (like an aperture). The divegence of the rays is what cause the density to decrease. Since the rays are radially continuous the area is wholly illuminated. If the rays are radially quantized some rays may miss the aperture.
 Tuttyd Posts: 53, Reputation: 4 Junior Member #10 Sep 6, 2013, 11:37 PM
Originally Posted by nykkyo
I understand the inverse square law; but why are some photons funneled to our aperture?
I think I get what your are saying.

Photons don't need to be "funneled" in any way in order for us to see them. Photons are force carriers- charged particles interacting result in charged particles being exchanged. Photons carrier just enough energy to activate the light sensitive parts of the eyes.

 Question Tools Search this Question Search this Question: Advanced Search


text:
# Points of information for lowly members

I'm looking at this page, which includes an R expression (for the gam function in the mgcv package). The R expression includes two elements (functions?) which do not appear on the help page for that function ('te' and 'bs').

R, of course, almost glories in its perverse unsearchability - a Google search on "te" brings up 4.6bn results!

The obvious resort would be to comment on the answer, asking for clarification.

However, as I understand it, those with fewer than 50 points in the bank are barred from doing this, and needs must remain in ignorance.

Is this a bug or a feature?

• In the meantime are you familiar with RSeek? With it you can quickly find the mgcv manual where gam and te are documented. – whuber Apr 25 '13 at 19:04


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# Past Analytic Topology in Mathematics and Computer Science

16 January 2008
15:00
Rolf Suabedissen
Abstract
• Analytic Topology in Mathematics and Computer Science
10 January 2008
10:30
Jacob Vosmaer
Abstract
We show that the profinite completion (a universal algebraic construction) and the MacNeille completion (an order-theoretic construction) of a modal algebra $A$ coincide, precisely when the congruences of finite index of $A$ correspond to principal order filters. Examples of such modal algebras are the free K4-algebra and the free PDL-algebra on finitely many generators.
• Analytic Topology in Mathematics and Computer Science
21 November 2007
15:00
Rolf Suabedissen
Abstract
• Analytic Topology in Mathematics and Computer Science
7 November 2007
15:00
Martin Escardo
Abstract
• Analytic Topology in Mathematics and Computer Science
17 October 2007
16:00
Abstract
• Analytic Topology in Mathematics and Computer Science
10 October 2007
16:00
Robin Knight
Abstract
• Analytic Topology in Mathematics and Computer Science
1 October 2007
16:00
Abstract
In this talk we discusss some notions of small sets in Polish groups. We give some examples and applications of these notions in analysis and group theory. Moreover, we introduce a new notion of smallness which we call Haar meager sets. This notion coincides with the meager sets in locally compact groups. However, it is strictly stronger in the setting of nonlocally compact groups. We argue that this is the right topological analogue of Christian's Haar null sets. The speaker gratefully acknowledges the support of the LMS under a Scheme 2 Grant.
• Analytic Topology in Mathematics and Computer Science
30 August 2007
14:00
Gavin Hitchcock
Abstract
• Analytic Topology in Mathematics and Computer Science
20 July 2007
15:45
Paul Gartside
Abstract
• Analytic Topology in Mathematics and Computer Science
19 July 2007
14:00
Paul Gartside
Abstract
• Analytic Topology in Mathematics and Computer Science


text:
#### How can classical multidimensional scaling go wrong?

Given a matrix $D$ describing the pairwise dissimilarities of a data set, a common task is to embed the data points into Euclidean space. The classical multidimensional scaling (cMDS) algorithm is a widespread method to do this. However, theoretical analysis of the robustness of the algorithm and an in-depth analysis of its performance on non-Euclidean metrics is lacking. In this paper, we derive a formula, based on the eigenvalues of a matrix obtained from $D$, for the Frobenius norm of the difference between $D$ and the metric $D_{\text{cmds}}$ returned by cMDS. This error analysis leads us to the conclusion that when the derived matrix has a significant number of negative eigenvalues, then $\|D-D_{\text{cmds}}\|_F$, after initially decreasing, willeventually increase as we increase the dimension. Hence, counterintuitively, the quality of the embedding degrades as we increase the dimension. We empirically verify that the Frobenius norm increases as we increase the dimension for a variety of non-Euclidean metrics. We also show on several benchmark datasets that this degradation in the embedding results in the classification accuracy of both simple (e.g., 1-nearest neighbor) and complex (e.g., multi-layer neural nets) classifiers decreasing as we increase the embedding dimension.Finally, our analysis leads us to a new efficiently computable algorithm that returns a matrix $D_l$ that is at least as close to the original distances as $D_t$ (the Euclidean metric closest in $\ell_2$ distance). While $D_l$ is not metric, when given as input to cMDS instead of $D$, it empirically results in solutions whose distance to $D$ does not increase when we increase the dimension and the classification accuracy degrades less than the cMDS solution.


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# 10-327/Homework Assignment 5 Solutions

### Solution

An assignment without a solution is like a nightmare to me. I like every question accompanied with a clean solution aside.(Might not be the case for research question because they are just simply too hard.) I would like to share this happiness of understanding and acquiring knowledge with everybody because I don't think this class is a battle. I certainly don't like the idea that we should keep information/answers as something like business secrets. This learning process should be enjoyable which should be full of discussions instead of things like "you have to think on your own/ I can't tell you the answer". I know maybe other people might not agree with me but I believe a positive learning environment is crucial to truly understanding something well although we should not neglect independent thinking at the same time. That is why I share whatever I have with you. If I am wrong feel free to criticize me, and I am pretty sure a lot of people don't agree with me. But that is OK because there is just no absolute right or wrong and everybody is doing what they think is right. Just like you can't say if Axiom of choice is right or not. If you believe then it is right. If you don't believe it then it is wrong. Or maybe, for the entire subject--math, if you believe it then it is right. If you don't believe it then it is completely wrong.-Kai

Too late for marks, but just in time for the holidays are the extra problems (1,6,8,12 pp 170-171) - just sketches:

1. (a) is trivial. For (b), suppose X is compact Hausdorff relative to topologies T1,T2 which are comparable. The identity function Id is continuous in one direction and so takes compact sets to compact sets. The rest is easy: fix any open set U and consider Id(X-U) ...

6. Also trivial. As in 1(b), use (i) compact subsets of T2 spaces are closed and (ii) closed subsets of compact spaces are compact.

8. If f is continuous, fix any point (x,y) not on the graph. Separate y and f(x) by open sets U, V, respectively. By continuity, find W open in X st f(W) is a subset of V. Then W x U is open disjoint from the graph. Conversely, if the graph is closed, apply the hint ...

12. Fix an open cover Uα for X. For each y in Y, its preimage under p is compact, so can be covered by some finite collection of the Uα, which also gives an open set. Do this for every y in Y. But then apply the hint to notice that we get a covering of Y by open neighbourhoods; apply compactness and the hint to conclude we only care about the preimages of finitely many of these nbds in Y, which are already inside finitely many finite unions of open sets.

Bcd 22:51, 19 December 2010 (EST)


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I Injective immersion that is not a smooth embedding

cianfa72

Hi, I'm aware of a typical example of injective immersion that is not a topological embedding: figure 8

$\beta: (-\pi, \pi) \to \mathbb R^2$, with $\beta(t)=(\sin 2t,\sin t)$

As explained here an-injective-immersion-that-is-not-a-topological-embedding the image of $\beta$ is compact in $\mathbb R^2$ subspace topology while the domain open interval is not, thus $\beta$ is not a smooth embedding.

Consider it from the point of view of "homeomorphism onto its image" definition, I was trying to find out an instance of image subset open in the subspace topology that actually is not open in the domain topology or the other way around.

Can you help me ? Thanks.

Related Differential Geometry News on Phys.org

WWGD

Gold Member
Hi, I'm aware of a typical example of injective immersion that is not a topological embedding: figure 8

$\beta: (-\pi, \pi) \to \mathbb R^2$, with $\beta(t)=(\sin 2t,\sin t)$

As explained here an-injective-immersion-that-is-not-a-topological-embedding the image of $\beta$ is compact in $\mathbb R^2$ subspace topology while the domain open interval is not, thus $\beta$ is not a smooth embedding.

Consider it from the point of view of "homeomorphism onto its image" definition, I was trying to find out an instance of image subset open in the subspace topology that actually is not open in the domain topology or the other way around.

Can you help me ? Thanks.
You can show the complement in the image is open, to show the image itself is closed. Take any point x in the complement. It is at a non-zero distance from the image. Consider a ball centered at x with distance d/2 from the image, to show complement is open, so image itself is closed.

cianfa72

Thinking again about it, I believe we can find an example of that subset looking at the inverse map $\beta^{-1}$.
Take an open subset $(-\varepsilon, \varepsilon)$ belonging to the $\mathbb R^1$ subspace topology induced on $(-\pi,\pi)$. Its preimage under $\beta^{-1}$ includes the Figure 8 central point and is not an open subset of Figure 8 endowed with $\mathbb R^2$ subspace topology (it cannot be obtained through the intersection of an $\mathbb R^2$ open subset with the Figure 8 itself) therefore $\beta$ is not an homomorphism with the image (in the subspace topology)

Is that right ?

Last edited:

cianfa72

Thinking again about it, I believe we can find an example of that subset looking at the inverse map $\beta^{-1}$.
Take an open subset $(-\varepsilon, \varepsilon)$ belonging to the $\mathbb R^1$ subspace topology induced on $(-\pi,\pi)$. Its preimage under $\beta^{-1}$ includes the Figure 8 central point and is not an open subset of Figure 8 endowed with $\mathbb R^2$ subspace topology (it cannot be obtained through the intersection of an $\mathbb R^2$ open subset with the Figure 8 itself) therefore $\beta$ is not an homomorphism with the image (in the subspace topology)

Is that right ?
Help ! I'm a beginner...just to check I got it correctly, can you confirm that ? Thanks

WWGD

Gold Member
Thinking again about it, I believe we can find an example of that subset looking at the inverse map $\beta^{-1}$.
Take an open subset $(-\varepsilon, \varepsilon)$ belonging to the $\mathbb R^1$ subspace topology induced on $(-\pi,\pi)$. Its preimage under $\beta^{-1}$ includes the Figure 8 central point and is not an open subset of Figure 8 endowed with $\mathbb R^2$ subspace topology (it cannot be obtained through the intersection of an $\mathbb R^2$ open subset with the Figure 8 itself) therefore $\beta$ is not an homomorphism with the image (in the subspace topology)

Is that right ?
Careful, homeomorphism, not homomorphism. But be careful, a map may be open --taking open sets to open sets -- and not be a homeomorphism.

WWGD

Gold Member
Thinking again about it, I believe we can find an example of that subset looking at the inverse map $\beta^{-1}$.
Take an open subset $(-\varepsilon, \varepsilon)$ belonging to the $\mathbb R^1$ subspace topology induced on $(-\pi,\pi)$. Its preimage under $\beta^{-1}$ includes the Figure 8 central point and is not an open subset of Figure 8 endowed with $\mathbb R^2$ subspace topology (it cannot be obtained through the intersection of an $\mathbb R^2$ open subset with the Figure 8 itself) therefore $\beta$ is not an homomorphism with the image (in the subspace topology)

Is that right ?
Just remember that a map may be open but not a homeomorphism. I think this map is a local diffeomorphism, so it sends open to open. .

cianfa72

Just remember that a map may be open but not a homeomorphism. I think this map is a local diffeomorphism, so it sends open to open. .
ok, but just for the very fact exists an open set that $\beta$ does not send in an open set, it suffices to say it is not homeomorphism, do you ?

WWGD

Gold Member
ok, but just for the very fact exists an open set that $\beta$ does not send in an open set, it suffices to say it is not homeomorphism, do you ?
, Yes, that is enough to guarantee a maps not a diffeomorphism

EDIT: But as you said, there is an open interval in #(-\epsilon, \epsilon)# that hits the center part of the figure 8, whose image is not open.

Last edited:

"Injective immersion that is not a smooth embedding"

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3k views

### How to prove this identity $\pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;$?

How to prove this identity? $$\pi=\sum_{k=-\infty}^{\infty}\left(\dfrac{\sin(k)}{k}\right)^{2}\;$$ I found the above interesting identity in the book $\bf \pi$ Unleashed. Does anyone knows how to ...
### Prove: $\int_0^1 \frac{\ln x }{x-1} d x=\sum_1^\infty \frac{1}{n^2}$
I'd like your help with proving that $$\int_0^1 \frac{\ln x }{x-1}d x=\sum_{n=1}^\infty \frac{1}{n^2}.$$ I tried to use Fourier series, or to use a power series and integrate it twice but it didn't ...


text:
CoCalc Shared FilesUse-Python3-based-SageMath-in-a-CoCalc-project.md
Authors: Frédéric Chapoton, Samuel Lelièvre
Views : 17
Description: Use Python3-based SageMath in CoCalc

# Python3-based SageMath in CoCalc

SageMath is being ported to Python 3.

This is an ongoing effort, and some things might not work in the Python3-based SageMath.

If you already want to try it however, there are several ways to go:

• just copy a folder to a CoCalc project of yours,
• or build from source in CoCalc,
• or build from source elsewhere.

Here we document the first two options.

For the third option, see the "Installation from source" section of the Sage installation manual:

## Get started by copying a folder to your project

In CoCalc, just copy this folder to your project:

In a CoCalc terminal, you can then use Python3-based SageMath by typing:

$sage/sage  which will give you the famous Sage prompt sage:  at which you can type any Sage command. To add a "Sage Py3" Jupyter kernel to your project, we follow instructions adapted from the "Custom Jupyter Kernel" page on the CoCalc wiki: Here, just run the following in a CoCalc terminal $ cd
$OLD_KERNEL=/ext/jupyter/kernels/sage-develop$ NEW_KERNEL=/home/user/.local/share/jupyter/kernels/sage_py3
$cp -arv$OLD_KERNEL $NEW_KERNEL$ cd $NEW_KERNEL$ open kernel.json

and then replace the content of kernel.json by:

{
"display_name": "Sage Py3",
"argv": [
"/home/user/sage/sage",
"--python",
"-m",
"sage.repl.ipython_kernel",
"--matplotlib=inline",
"-f",
"{connection_file}"
]
}

Now if you open a Jupyter notebook worksheet (file ending in .ipynb), you can select the "Sage Py3" kernel from the "Kernel > Change kernel" menu.

## Or build from source

Instead of copying a ready-to-go Sage folder as above, you might prefer to build from source. Here is how to do that.

In a CoCalc project with internet access and some extra disk space, open a terminal.

If your project has many cores (say 4) and you want to build in parallel, first run the following command (replacing 4 by the number of cores available):

$export MAKE='make -j4'  Or you could edit your .bashrc, adding the following line to it. export MAKE='make -j4'  and then either close the terminal and open it again, or run source /home/user/.bashrc  Once you have set MAKE as above (or skipped that), run: $ cd
$mkdir bin$ git clone https://github.com/sagemath/sage.git
$cd sage$ make configure
$./configure --with-python=3$ make build
$ln -s /home/user/sage/sage /home/user/bin  You can then try: $ sage --version
$sage --python --version  $ sage -q   # start Sage in quiet mode (no banner)
sage: print(version())      # SageMath version
sage: print(sys.version)    # Python version

and try your favourite Sage commands, see if they work!

You can also install a "Sage Py3" Jupyter kernel, by following the instructions at the end of the previous section.


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• 论文 •

### 收入不平等、信贷约束与家庭消费

• 出版日期:2016-08-15 发布日期:2016-08-11

### Income Inequality，Credit Constraints and Household Consumption

Hang Bin & Xiu Lei

• Online:2016-08-15 Published:2016-08-11

Abstract:

The paper, based on social status seeking, makes an analysis of the impacts of income inequality upon urban family consumption. We think that, in the economic circumstance with less-developed financial market and low-level social security, the increase of income gap will force the families with middle or low income to reduce the non-positional consumption and increase the savings. An empirical study, based on the micro-tracking data of CFPS , has indicated that: 1) Such show-off consumption of the families with high social status may exert a substantial demonstration effect upon the families with middle and low social status. In addition, the higher the social status of the families, the stronger their comparative motives are. 2) Income inequality may greatly reduce consumption of the families with middle and low social status. 3) The status seeking may largely stimulate positional consumption, while the increase in income difference is more likely to restrain non-positional consumption. 4) The inhibit effect on consumption of income gap is related to credit constraints.


text:
# Simulation of a function of normal, uniform, and exponential random variables [closed]

I need to solve this please Let $A$ be a normal random variable with $\mu = 70$ and $\sigma= 15$, $B$ a normal random variable with $\mu = 20$ and $\sigma= 3$, $C$ an exponential random variable with $\lambda= 4$, and $D$ a uniform random variable on the interval $[0,1]$. Consider the model $T = 4A^2 - (B/C) + 10D$

Use Monte Carlo simulation method with number of trials (or iteration) equals 1000 to estimate 1. $P(-22.325 < T < 256.458).$ 2. the mean of $T$.

## closed as off-topic by Did, Namaste, BruceET, Jean Marie, ShaileshDec 28 '17 at 1:49

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Namaste, BruceET, Jean Marie, Shailesh
If this question can be reworded to fit the rules in the help center, please edit the question.

• Welcome to Math Stack Exchange. Can you tell what you have attempted to solve your problem, where you are blocked, etc.? We don't want to serve a solution on a tray without some effort from the asker. – Jean Marie Dec 27 '17 at 23:20
• To your question, I reply by questions: what simulation language do you use ? How do you simulate an exponential variable with $\lambda = 4$ (do you know that is done using the logarithm of a certain uniform random variable ?) etc. – Jean Marie Dec 27 '17 at 23:25
• No answer. I vote to close this question. – Jean Marie Dec 28 '17 at 1:20
• this is an assignment, with any simulator based on Monti Carlo methods – Ayman Dec 28 '17 at 22:35
• You come back a day after you have asked your question : it's not serious... – Jean Marie Dec 28 '17 at 23:09

## 1 Answer

I will try to get you started on this by showing a simulation in R statistical software. I used a million trials rather than the requested 1000. I am assuming $\lambda = 4$ is the exponential rate, not the exponential mean.

Because you have shown noting to reveal the level, methods, or computer language of your course, I have no way of knowing whether this is helpful.

 set.seed(1227);  m = 10^6
a = rnorm(m, 70, 15);  b = rnorm(m, 20, 3)
c = rexp(m, rate=4);   d = runif(m, 0, 1)
t = 4*a^2 - b/c + 10*d
mean((t > -22.325) & (t < 256.485))
## 0.000154
mean(t)
## 13434.11

An exponential distribution puts much of its probability near 0. What difficulties do you suppose that might present?

• this is an assignment, with any simulator based on Monti Carlo methods – Ayman Dec 28 '17 at 22:35
• So is my Answer helpful? // @JeanMarie asked about metnods: For example, are you supposed to simulate exponential RVs vis uniform, or is it OK to use a function such as R's rexp for that? Similarly for rnorm? Unless you show some attempt to solve the problem, it is difficult for us to give useful answers. – BruceET Dec 28 '17 at 23:36


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# Understanding Generalized Barycentric Coordinates

## Recommended Posts

Hey guys, so I'm trying to create a quad renderer that renderers quads using 4 half space functions instead of rendering 2 triangles. I was previously following ryg's (https://fgiesen.wordpress.com/2013/02/10/optimizing-the-basic-rasterizer/) tutorial on triangle rasterization and he uses barycentric coordinates to interpolate values as well as see if the point is inside all of the edges. This works great for triangles but I know some people previously made quad rasterizers and I found this paper (http://geometry.caltech.edu/pubs/MHBD02.pdf) about generalized barycentric coordinates for n-sided polygons. Now this is my first ever attempt at understanding a mathematical paper so I'm probably not understanding a lot of basic principles here.

They use a couple variables, q1->qn are the vertices, Q is the polygon, p is a point inside the polygon and alpha1->alphan are the coefficients for each vertice (a percent saying how much percent of each vertex weight applies to that point), and w1->wn which is the actual weight. They have 3 properties that they are trying to gain in their generalized barycentric coords: affine combination, smoothness and convex combination.

For affine combination, the first summation is a bit confusing. They're saying if you sum all coefficients multiplied by the vertices then you get the point p but I'm not entirely sure what that is achieving. They're also saying all the coefficients add up to 1 which makes sense since the coefficients are percents so they have to reach a total of 100% or 1. After they state the properties, they then say you can rewrite the affine summations as the summation of each wj(qj - p) which equals to 0. Now I thought I'd do a example with a triangle and see if this works.

My vertices would be (0,0), (0.5, 1), (1, 0) with p being (0.5, 0.5) and my weights being 5, 4 and 3. Applying the formula I would get:

5(0.5 - 0.5, 1 - 0.5) + 4(0 - 0.5, 0 - 0.5) + 3(1 - 0.5, 0 - 0.5)

= (-0.5, -1)

First off, I don't get how the vector value is supposed to equal to 0 unless its the vector 0. Second, I don't get why this example doesn't work. Is there something wrong with the vertices I chose or am I missing some step in between? Thanks in advance.

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The affine combination part means each vertex has a coefficient, you multiply all vertices with all the coefficients and sum the result. The coefficients themselves sum to 1. It's just a standard affine combination of points. The goal of this is to generate a point from a combination of the polygon vertices, as a way of mapping coefficients of a polygon to the space around the polygon.

I skimmed the paper and I'm pretty skeptical; if you have more edges than 3 for a 2D shape, I don't think you can guarantee a 1 to 1 mapping of coefficients to points in the polygon. This property is pretty much the entire reason barycentric coordinates are interesting, and you lose that once you have too many degrees of freedom. Maybe someone else will come along that actually reads it thoroughly, but to me it seems to be pretty lame. Additionally, modern hardware has built in functionality for dealing with triangles making the entire paper really useless for modern purposes. I hate to give you such a lame suggestion but: you are probably better off studying a better paper.

I hope this helps!

Edited by Randy Gaul

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Hmm, im going to read up on affine math, I've never actually learned it, the most I know about it is from ryg's tutorials on barycentric coordinates but thats really rough. Thanks for the affine combination term, will help me figure all of this out.

The paper has over 200 references so doesn't that make it more useful? :P Just to clarify, this is for a software renderer and in my case I'm only rendering cube's (or 6 quads) so I wanted to see if a quad renderer would be faster than rendering 2 triangles for each quad. I was hoping SIMD would work out better with 4 half space functions vs 3. I came across this paper because I didn't know how to interpolate with 4 points vs 3 (its not as well documented or used probably).

I'll read up more on affine math but I know that interpolating with 4 points has to be possible because I've seen mentions of people who created software renderers that only render quads (not as 2 triangles but as 4 halfspace functions).

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You can think of an affine combination as a weighted average of points. I recommend Essential Math by Van Verth, 2nd Ed.

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Your example doesn't work because of 2 things:

1. You're using the normal Cartesian coordinates of the triangle points instead of converting them to barycentric coordinates.
2. You're basically pre-selecting the barycentric coordinates. You need to use the formulas to compute all the $$w_j$$ values and then you'll see $$\sum_{j=0}^n w_j(\mathbf{q}_j-\mathbf{p}) = 0$$, where $$\mathbf{q}_j, \mathbf{p}$$ are in Cartesian coordinates.

In order to get the proper interpolation you're looking for, You have to use the formulas in that paper to convert the point to the barycentric coordinates (the $$\alpha_j$$ vector) and then dot that with the values you want those points to have in order to get the interpolated value of the given point.

I will agree with Randy Gaul that you do lose coordinate uniqueness if the polytope isn't a simplex. However, I disagree with him that this paper is not interesting. True, barycentric coordinates with a simplex will guarantee a unique coordinate value for any point, but there are a lot of uses beyond just finding points in a simplex, although you have to be sufficiently interested in computational geometry to appreciate them. In a game programming setting, there might be only a couple of relevant uses beyond this one.

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Oh okay, that makes a lot more sense. I read up on affine spaces and I think i can properly visualize them now. I read a bit of essential math but the matrix example confused me so I might go back with a better understanding now and see how else it is applied in graphics. From my pdf, it explained affine spaces as 2 sets and a operator overload. We have a set E of points, a set E' of vectors which is a vector space, and we overload + in order to satisfy certain addition axioms. The set E' is basically all the possible vectors that you can apply to displace any point E and still be left with a point in E (this way of looking at it made it a lot easier to visualize for me).

I also now understand why the weights sum to 1 and that's so that affine combinations behave similarly to linear combinations (this was actually quite interesting figuring out why this is the case and I wouldn't imagine that the weights would have to be constrained like that just because we shift the origin and remove its significance).

I'm going to back now and look over the paper so I might still update this post with questions, but I feel like if we just define an affine space with 4 points instead of 3 in 2D, we would basically have a quad in affine space the same way we have a triangle in affine space so we could interpolate from the barycenters and what not the same way we can interpolate a triangle. Are there any properties that get lost if we do just this? You guys are saying that you lose coordinate uniqueness once you go further than a triangle but if you did that, you would have 4 weights instead of 3. Wouldn't that guarantee that each coordinate is unique especially since its also a valid affine space and every coordinate is unique in a affine space?

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Not quite.

Say we have 2 points A and B to define a line AB.  We can create a combination of those two points:

c0 * A + c1 * B = P

Where P is a point on the line, and c0 + c1 == 1. c0 and c1 map to the point P given A and B.

Similarly, if we take any point P, we can directly compute c0 and c1:

n = norm( B - A )

divisor = 1.0 / length( B - A )

c0 = dot( P - A, n ) * divisor

c1 = dot( B - P, n ) * divisor

Since we have only two points A and B that define this line AB, there is a one to one mapping of coefficients to any point P on the line. Note that AB defines a one dimensional space where c0 and c1 are barycentric coordinates of the segment AB. Now say we have three points to define this line: A B and C. Imagine an affine combination of these three points that maps to P.

c0 * A + c1 * B + c2 * C = P

c0, c1 and c2 still need to sum to 1. This means we can choose any of these three coefficients and set it equal to zero. When we do this we result in the original equation above. This shows that only two points are necessary to construct a one to one mapping in this one dimensional space.

In 2D we can define a one to one mapping of three coefficients in an affine combination to a point P, and these coefficients would be called barycentric coordinates. Similar to the one dimensional example space above, we can write down equations to compute barycentric coordinates given P and 3 points, or vice versa.

If we added another point (4 instead of 3) then four (four choose three, binomial theorem) different combinations of all four points can map to a single point P. Pick any three from the set of four, and you can calculate an affine combination of those three points that maps to P.

The property you lose by adding too many degrees of freedom is one to one mapping.

Edited by Randy Gaul

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That can't be right for points inside the quad. Lets say I define a triangle from a,b,c. If I make another triangle from a,b,d the points inside both triangles don't have to overlap. So theres no way to define for triangle abc the coordinates in abd without using negative weights which at that point are outside the triangle. If I define a quad from a,b,c,d the points inside the quad should be unique but the points outside might not since like you said, your adding to many degrees of freedom. So I guess wouldn't the points inside the quad be defined uniquely? ie: the weights must all be positive, you cannot have negative weights.

EDIT: I guess this assumes both triangles are also on the same plane in order to be apart of the same affine space and that abc and abd don't overlap. I'll try to do a bit of math to figure this out but it seems correct.

Edited by D.V.D

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Yeah, 3 unique points would represent a 2-space (plane).

If you have 4 unique points you can represent 3-space, though the same rule applies: more than 4 points for a 3-space mapping and you will not have a one to one mapping.

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Randy Gaul is right. You lose the one-to-one mapping because you can find other sets of weights that satisfy all the equations.

However, if I understand the application right (which maybe I don't), I don't think you need to care about that for a rasterizer. Losing the one-to-one mapping in this case means that you could have two points be rendered as the same color, but does that matter? I wouldn't think so, but then I might be missing the point of the application.

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Sorry, I've been stuck with exams, just finished my first one and Im not done till end of april.

Yeah I'm not too sure if it matters either. The triangle rasterizer that I had before would look at what changed in the half plane equations as you stepped through x and y independantly and only update the small parts of the math that actually change. I was hoping to do the same for this. As for whether the one to one will even matter, I'm not entirely sure. I feel like it could screw up the weights and how they are distributed if you got 4 of them. I guess I'll just test it afterwards or maybe Ill grab a bit of time to make the quad rasterizer quickly.

Basically, my concern is that because of no one to one mapping, you will have a point where you could represent it as maybe 50%w0, 20%w1, 20%w2 and 10%w3 but might also be represented as 10%w0, 20%w1, 20%w2, and 50%w3 (i just pulled these numbers out of no where but I think it illustrates the point). If this is the case, then when i interpolate the z at each point, I might be getting completely wrong answers depending on which of the above cases happens to be the result we get for that pixel.

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Yeah I'm not sure if computing the same pixel multiple times matters for your application. So long as your algorithm of choice generates a consistent color for a single pixel you'll be fine.

Ryg's triangle rasterizer is fast because you can use SIMD to compute 4 pixels at once. Barycentric coordinates are just used for point in triangle testing, in which you can reject pixels outside of your triangle (if I'm not mistaken). Additionally, the problem of rasterization triangles is embarrassingly parallel to the point where each thread can be running SIMD code simultaneously -- the only difficult threading consideration would be false sharing.

For a quad you can do the exact same thing, except instead of barycentric coordinates you can do four plane tests, one for each edge of the quad. Normals don't need to be normalized since you're only sign testing, and you should be able to come up with a robust and fast rasterizer. It may even be faster in some specific cases where the quads fill up their own bounding AABB better than triangles do.

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Yeah I'm not sure if computing the same pixel multiple times matters for your application. So long as your algorithm of choice generates a consistent color for a single pixel you'll be fine.

Ryg's triangle rasterizer is fast because you can use SIMD to compute 4 pixels at once. Barycentric coordinates are just used for point in triangle testing, in which you can reject pixels outside of your triangle (if I'm not mistaken). Additionally, the problem of rasterization triangles is embarrassingly parallel to the point where each thread can be running SIMD code simultaneously -- the only difficult threading consideration would be false sharing.

For a quad you can do the exact same thing, except instead of barycentric coordinates you can do four plane tests, one for each edge of the quad. Normals don't need to be normalized since you're only sign testing, and you should be able to come up with a robust and fast rasterizer. It may even be faster in some specific cases where the quads fill up their own bounding AABB better than triangles do.

Okay I found the time and made a quick one. I didn't really go for implementing what was in the paper yet, I wanted to see if just extending ryg's code would work. Here's what I got:

void RenderQuad(ScreenBuffer* Screen, Vector2i P[4], float32 z[4], uint32 Color)
{
//SortPoints4(P);
// Get bounding box
int32 MinX = Min4(P[0].x, P[1].x, P[2].x, P[3].x);
int32 MaxX = Max4(P[0].x, P[1].x, P[2].x, P[3].x);
int32 MinY = Min4(P[0].y, P[1].y, P[2].y, P[3].y);
int32 MaxY = Max4(P[0].y, P[1].y, P[2].y, P[3].y);
MinX = Max2(0, MinX);
MinY = Max2(0, MinY);
MaxX = Min2(Screen->Width, MaxX-1);
MaxY = Min2(Screen->Height, MaxY-1);
Vector2i Pos = { MinX, MinY };
// Increment values
int32 A01 = P[0].y - P[1].y; int32 B01 = P[1].x - P[0].x;
int32 A12 = P[1].y - P[2].y; int32 B12 = P[2].x - P[1].x;
int32 A23 = P[2].y - P[3].y; int32 B23 = P[3].x - P[2].x;
int32 A30 = P[3].y - P[0].y; int32 B30 = P[0].x - P[3].x;
// Calculate row starting positions
int32 w0_row = (P[2].x - P[1].x)*(Pos.y - P[1].y) - (P[2].y - P[1].y)*(Pos.x - P[1].x);
int32 w1_row = (P[3].x - P[2].x)*(Pos.y - P[2].y) - (P[3].y - P[2].y)*(Pos.x - P[2].x);
int32 w2_row = (P[0].x - P[3].x)*(Pos.y - P[3].y) - (P[0].y - P[3].y)*(Pos.x - P[3].x);
int32 w3_row = (P[1].x - P[0].x)*(Pos.y - P[0].y) - (P[1].y - P[0].y)*(Pos.x - P[0].x);
// Rasterize
for (Pos.y = MinY; Pos.y < MaxY; ++Pos.y)
{
int32 w0 = w0_row;
int32 w1 = w1_row;
int32 w2 = w2_row;
int32 w3 = w3_row;

for (Pos.x = MinX; Pos.x < MaxX; ++Pos.x)
{
if ((w0 | w1 | w2 | w3) >= 0)
{
float32 DepthValue = (w0 + w1 + w2 + w3) / (w0*z[0] + w1*z[1] + w2*z[2] + w3*z[3]);
Screen->BackBuffer[Pos.y*Screen->Width + Pos.x] = Color;
Screen->DepthBuffer[Pos.y*Screen->Width + Pos.x] = DepthValue;
}
// One step to the right
w0 += A12;
w1 += A23;
w2 += A30;
w3 += A01;
}
// One row step
w0_row += B12;
w1_row += B23;
w2_row += B30;
w3_row += B01;
}
}

So far, this works properly at rendering one of the faces of a quad while keeping proper depth interpolation (ie: the triangles don't get drawn overlapping when they shouldn't). I didn't do extensive testing to see if the depth is exactly the same as drawing the quad with 2 triangles (exams maybe later). I'm wondering if the paper is any different now, it looked more complicated to be honest but I still wanna figure it out, maybe after exams. I do have a question on sorting the triangles points, ryg's blog says the vertices have to be sorted counter clockwise so I made this function to do so:

void SortPoints4(Vector2i* P)
{
float32 angles[4] = { atan2(P[0].y, P[0].x), atan2(P[1].y, P[1].x), atan2(P[2].y, P[2].x), atan2(P[3].y, P[3].x) };
for (int32 i = 1; i < 4; i++)
{
float32 tmp = angles[i];
Vector2i tmp2 = P[i];
int j = i;
for (; j && tmp < angles[j - 1]; --j)
{
angles[j] = angles[j - 1];
P[j] = P[j - 1];
}
angles[j] = tmp;
P[j] = tmp2;
}
}

This doesn't work properly even though it should sort the points counter clock wise. It works by finding the angles from the origin which is bottom left of the screen, and then it makes it go from lowest to highest. Unfortunately, this makes the quad disappear sometimes depending on its orientation to the camera. Is there a easy way of doing this? Or is there a way of making it invisible all the time or only when the normal of the quad is facing the camera?

Not sure if this helps, but here is a really nice tutorial on point arithmetic (it starts about chapter 7):

http://www.flipcode.com/archives/3D_Geometry_Primer_Chapter_1-Introduction.shtml

Thanks for the link, will look over it!!

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There's a lot of ways to sort 3 verts. You can use a series of if-statements to enumerate all the possibilities (once in screen space), or you can cross two edges and see if the resulting vector points towards or away from the camera.

I found some really old code of mine for doing backface culling (read with caution):

bool Camera::IsBackFace( const Point4& a, const Point4& b, const Point4& c )
{
Vec4 A, B, C;
A.Set( a.x, a.y, a.z, a.w );
B.Set( b.x, b.y, b.z, b.w );
C.Set( c.x, c.y, c.z, c.w );
Vec4 v1( B - A );
Vec4 v2( C - A );
Vec4 N( Vec4::CrossProduct( v1, v2 ) );
Point4 pos( position.x, position.y, position.z, position.w );
Vec4 view( pos - a );
return Vec4::DotProduct( N, view ) < 0.0f;
}

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It looks like your code is looking at which way your object is facing (towards camera or away). I know that is one of the ways to check clockwise vs counter clockwise ordering but my method which sorts points based on their angle from the origin or bottom left part of the screen should also work yet it doesn't. For most of the random rotations my quad goes through, it goes invisible even though the vertices are ordered properly. Is there anything wrong with my code? Its just a sort that I found somewhere online.

I've seen other people do similar software rasterizing as I am but they don't order their vertices, they just transform them to camera space, figure out which ones are together via index buffer and render. For some reason when I define my quad as

Vector3f Cube[8] = { { 1.0f, 1.0f, 1.0f }, { 1.0f, 0.0f, 1.0f }, { 0.0f, 1.0f, 1.0f }, { 0.0f, 0.0f, 1.0f },
{ 1.0f, 1.0f, 0.0f }, { 1.0f, 0.0f, 0.0f }, { 0.0f, 1.0f, 0.0f }, { 0.0f, 0.0f, 0.0f } };

and render each face, some of my faces appear correctly under arbitrary rotations but others disappear sometimes. Is there a way to somehow arrange the vertices before hand so I don't need to sort each face after? If it helps, I will only ever be rendering cubes. If some of my faces work properly under arbitrary rotations, then I think there has to be some way to get this to work.

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