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## Nos 50 dernières publications

• 24 March 2023

Quelques aspects de l’élaboration de l’analyse fonctionnelle jusqu’à Banach. Rôle des opérations dans une approche générale

Frédéric Jaeck

• 24 March 2023

La these se divise en deux parties. Dans la premiere, nous montrons comment les notions de multiplicite et d'ensemble frontiere sont intimement liees aux classes a#m#,#n. Plus precisement nous donnons une condition suffisante en termes de multiplicite pour qu'une contraction appartienne a une classe donnee. Nous conjecturons que ces criteres caracterisent en particulier les classes a#1#,#n. Ces resultats permettent deja de construire avec une tres grande efficacite de multiple exemples d'operateurs dans une classe donnee et de tester certaines conjectures classiques. La deuxieme partie met en place le cadre general permettant d'etudier l'algebre duale engendree par une paire commutative de contractions. En particulier nous caracterisons les paires continues (ie. Possedant un calcul fonctionnel h#(t#2) faible#* continu en termes de mesure spectrale et de bande de mesures. Ensuite, nous donnons des resultats de factorisation fonctionnelle dans l#1() ou est dans une large classe de mesures incluant les mesures spectrales du dessus. Enfin nous mettons en place un processus d'approximation qui est une variante de la technique de s. Brown, basee sur une hypothese de type e#r##,#. On demontre alors qu'une paire de contractions a calcul isometrique avec cette propriete possede un sous-espace invariant non trivial.

Frédéric Jaeck

• 23 March 2023

Une élégante balade mathématique. Avec beaucoup d'humour, le physicien David Acheson nous présente sa vision des mathématiques : un univers magique, immédiatement accessible et avant tout ludique, où le plaisir de raisonner domine. Les multiples énigmes, observations troublantes, constructions géométriques et manipulations de chiffres en tout genre proposés sont les jalons d'un voyage extravagant à la Lewis Carroll, qui n'oublie pas les applications les plus remarquables de la discipline. Du nombre Pi au pendule chaotique, de Pythagore à Andrew Wiles qui démontra le théorème énoncé par Pierre de Fermat 350 ans plus tôt, embarquez pour une fascinante balade au pays des mathématiques ! De savoureuses illustrations achèveront de convaincre le profane comme l'initié qu'il s'agit là d'un des livres les plus réjouissants jamais écrit sur le sujet.

Frédéric Jaeck

• 23 March 2023

Une élégante balade mathématique. Avec beaucoup d'humour, le physicien David Acheson nous présente sa vision des mathématiques : un univers magique, immédiatement accessible et avant tout ludique, où le plaisir de raisonner domine. Les multiples énigmes, observations troublantes, constructions géométriques et manipulations de chiffres en tout genre proposés sont les jalons d'un voyage extravagant à la Lewis Carroll, qui n'oublie pas les applications les plus remarquables de la discipline. Du nombre Pi au pendule chaotique, de Pythagore à Andrew Wiles qui démontra le théorème énoncé par Pierre de Fermat 350 ans plus tôt, embarquez pour une fascinante balade au pays des mathématiques ! De savoureuses illustrations achèveront de convaincre le profane comme l'initié qu'il s'agit là d'un des livres les plus réjouissants jamais écrit sur le sujet.

Frédéric Jaeck, Isabelle Chalendar

• 23 March 2023

Une élégante balade mathématique. Avec beaucoup d'humour, le physicien David Acheson nous présente sa vision des mathématiques : un univers magique, immédiatement accessible et avant tout ludique, où le plaisir de raisonner domine. Les multiples énigmes, observations troublantes, constructions géométriques et manipulations de chiffres en tout genre proposés sont les jalons d'un voyage extravagant à la Lewis Carroll, qui n'oublie pas les applications les plus remarquables de la discipline. Du nombre Pi au pendule chaotique, de Pythagore à Andrew Wiles qui démontra le théorème énoncé par Pierre de Fermat 350 ans plus tôt, embarquez pour une fascinante balade au pays des mathématiques ! De savoureuses illustrations achèveront de convaincre le profane comme l'initié qu'il s'agit là d'un des livres les plus réjouissants jamais écrit sur le sujet.

Frédéric Jaeck

• 17 March 2023

For a fixed finite group G, we study the fields of definition of geometrically irreducible components of Hurwitz moduli schemes of marked branched G-covers of the projective line. The main focus is on determining whether components obtained by "gluing" two other components, both defined over a number field K, are also defined over K. The article presents a list of situations in which a positive answer is obtained. As an application, when G is a semi-direct product of symmetric groups or the Mathieu group M 23 , components defined over Q of small dimension (6 and 4, respectively) are shown to exist.

Béranger Seguin

• 13 March 2023

The net impact of aircraft contrails on global climate change is a matter of controversy today. Among the many parameters potentially influencing this issue, the role played by the aircraft wake has received only little attention so far. Yet the interaction between the jets causing these contrails and the aircraft wake leads to large modifications in the contrail's altitude (ranging in hundreds of meters). This change in altitude heavily influences the net impact of contrails to global climate change, since the related change in temperature affects the ice content and the radiative properties of these contrails. This vortex entrainment strongly depends on the relative positioning of the jet with respect to the tip vortices. Furthermore, after the formation of the tip vortices and the turbulent diffusion of the jet, the jet dispersion is also driven by buoyant forces associated with atmospheric stratification. Here we focus on the vortex entrainment and buoyancy effects by running a large number of two-dimensional simulations, scaled by the Brunt-Väisälä frequency and jet to tip vortex spacing, of the flow from the aftermath of the jet turbulent diffusion, vortex roll-up and initial ice formation up to the vortex destabilization stage. The very near wake dynamics are not simulated and instead replaced by an analytical description for the vortex, the jet and the ice plume. Ice water content is determined from the offset to ice saturation, given prescribed ambient conditions. The jet lateral spacing is considered in the range from fuselage to wing tip. The potential radiative impact of the early wake is calculated using the total extinction induced by the ice plume. The results are indicative of the impact of older contrail cirrus clouds, the largest proportion in the whole contrail radiative impact. The parametric mapping (stratification, jet spacing) highlights the important role played by the jet position on the opacity of early contrails, for regular stratification levels. In particular, a jet located closer to the wing tip results in contrails located at lower altitudes and reduced optical thickness, suggesting that jet positioning could be an interesting mean of contrail mitigation. Eventually the results also explain several real life observations.

Pierre Saulgeot, Vincent Brion, Nicolas Bonne, Emmanuel Dormy, Laurent Jacquin

• 9 March 2023

The net impact of aircraft contrails on global climate change is a matter of controversy today. Among the many parameters potentially influencing this issue, the role played by the aircraft wake has received only little attention so far. Yet the interaction between the jets causing these contrails and the aircraft wake leads to large modifications in the contrail's altitude (ranging in hundreds of meters). This change in altitude heavily influences the net impact of contrails to global climate change, since the related change in temperature affects the ice content and the radiative properties of these contrails. This vortex entrainment strongly depends on the relative positioning of the jet with respect to the tip vortices. Furthermore, after the formation of the tip vortices and the turbulent diffusion of the jet, the jet dispersion is also driven by buoyant forces associated with atmospheric stratification. Here we focus on the vortex entrainment and buoyancy effects by running a large number of two-dimensional simulations, scaled by the Brunt-Väisälä frequency and jet to tip vortex spacing, of the flow from the aftermath of the jet turbulent diffusion, vortex roll-up and initial ice formation up to the vortex destabilization stage. The very near wake dynamics are not simulated and instead replaced by an analytical description for the vortex, the jet and the ice plume. Ice water content is determined from the offset to ice saturation, given prescribed ambient conditions. The jet lateral spacing is considered in the range from fuselage to wing tip. The potential radiative impact of the early wake is calculated using the total extinction induced by the ice plume. The results are indicative of the impact of older contrail cirrus clouds, the largest proportion in the whole contrail radiative impact. The parametric mapping (stratification, jet spacing) highlights the important role played by the jet position on the opacity of early contrails, for regular stratification levels. In particular, a jet located closer to the wing tip results in contrails located at lower altitudes and reduced optical thickness, suggesting that jet positioning could be an interesting mean of contrail mitigation. Eventually the results also explain several real life observations.

Guillaume Peureux, Bénédicte Louvat, Louvat Bénédicte, Sandrine Berrégard, Sylvain Garnier, Dominique Moncond'huy, Françoise Poulet, Pierre Pasquier, Roussillon Marine, Guillaume Peureux

• 4 March 2023

The net impact of aircraft contrails on global climate change is a matter of controversy today. Among the many parameters potentially influencing this issue, the role played by the aircraft wake has received only little attention so far. Yet the interaction between the jets causing these contrails and the aircraft wake leads to large modifications in the contrail's altitude (ranging in hundreds of meters). This change in altitude heavily influences the net impact of contrails to global climate change, since the related change in temperature affects the ice content and the radiative properties of these contrails. This vortex entrainment strongly depends on the relative positioning of the jet with respect to the tip vortices. Furthermore, after the formation of the tip vortices and the turbulent diffusion of the jet, the jet dispersion is also driven by buoyant forces associated with atmospheric stratification. Here we focus on the vortex entrainment and buoyancy effects by running a large number of two-dimensional simulations, scaled by the Brunt-Väisälä frequency and jet to tip vortex spacing, of the flow from the aftermath of the jet turbulent diffusion, vortex roll-up and initial ice formation up to the vortex destabilization stage. The very near wake dynamics are not simulated and instead replaced by an analytical description for the vortex, the jet and the ice plume. Ice water content is determined from the offset to ice saturation, given prescribed ambient conditions. The jet lateral spacing is considered in the range from fuselage to wing tip. The potential radiative impact of the early wake is calculated using the total extinction induced by the ice plume. The results are indicative of the impact of older contrail cirrus clouds, the largest proportion in the whole contrail radiative impact. The parametric mapping (stratification, jet spacing) highlights the important role played by the jet position on the opacity of early contrails, for regular stratification levels. In particular, a jet located closer to the wing tip results in contrails located at lower altitudes and reduced optical thickness, suggesting that jet positioning could be an interesting mean of contrail mitigation. Eventually the results also explain several real life observations.

Rossana Tazzioli, Laurent Mazliak, Frédéric Jaeck, Emma Sallent del Colombo

• 4 March 2023

We study the dynamics of random walks hopping on homogeneous hyper-cubic lattices and multiplying at a fertile site. In one and two dimensions, the total number $\mathcal{N}(t)$ of walkers grows exponentially at a Malthusian rate depending on the dimensionality and the multiplication rate $\mu$ at the fertile site. When $d>d_c=2$, the number of walkers may remain finite forever for any $\mu$; it surely remains finite when $\mu\leq \mu_d$. We determine $\mu_d$ and show that $\langle\mathcal{N}(t)\rangle$ grows exponentially if $\mu>\mu_d$. The distribution of the total number of walkers remains broad when $d\leq 2$, and also when $d>2$ and $\mu>\mu_d$. We compute $\langle \mathcal{N}^m\rangle$ explicitly for small $m$, and show how to determine higher moments. In the critical regime, $\langle \mathcal{N}\rangle$ grows as $\sqrt{t}$ for $d=3$, $t/\ln t$ for $d=4$, and $t$ for $d>4$. Higher moments grow anomalously, $\langle \mathcal{N}^m\rangle\sim \langle \mathcal{N}\rangle^{2m-1}$, in the critical regime; the growth is normal, $\langle \mathcal{N}^m\rangle\sim \langle \mathcal{N}\rangle^{m}$, in the exponential phase. The distribution of the number of walkers in the critical regime is asymptotically stationary and universal, viz. it is independent of the spatial dimension. Interactions between walkers may drastically change the behavior. For random walks with exclusion, if $d>2$, there is again a critical multiplication rate, above which $\langle\mathcal{N}(t)\rangle$ grows linearly (not exponentially) in time; when $d\leq d_c=2$, the leading behavior is independent on $\mu$ and $\langle\mathcal{N}(t)\rangle$ exhibits a sub-linear growth.

Michel Bauer, P. L. Krapivsky, Kirone Mallick

• 2 March 2023

We study the dynamics of random walks hopping on homogeneous hyper-cubic lattices and multiplying at a fertile site. In one and two dimensions, the total number $\mathcal{N}(t)$ of walkers grows exponentially at a Malthusian rate depending on the dimensionality and the multiplication rate $\mu$ at the fertile site. When $d>d_c=2$, the number of walkers may remain finite forever for any $\mu$; it surely remains finite when $\mu\leq \mu_d$. We determine $\mu_d$ and show that $\langle\mathcal{N}(t)\rangle$ grows exponentially if $\mu>\mu_d$. The distribution of the total number of walkers remains broad when $d\leq 2$, and also when $d>2$ and $\mu>\mu_d$. We compute $\langle \mathcal{N}^m\rangle$ explicitly for small $m$, and show how to determine higher moments. In the critical regime, $\langle \mathcal{N}\rangle$ grows as $\sqrt{t}$ for $d=3$, $t/\ln t$ for $d=4$, and $t$ for $d>4$. Higher moments grow anomalously, $\langle \mathcal{N}^m\rangle\sim \langle \mathcal{N}\rangle^{2m-1}$, in the critical regime; the growth is normal, $\langle \mathcal{N}^m\rangle\sim \langle \mathcal{N}\rangle^{m}$, in the exponential phase. The distribution of the number of walkers in the critical regime is asymptotically stationary and universal, viz. it is independent of the spatial dimension. Interactions between walkers may drastically change the behavior. For random walks with exclusion, if $d>2$, there is again a critical multiplication rate, above which $\langle\mathcal{N}(t)\rangle$ grows linearly (not exponentially) in time; when $d\leq d_c=2$, the leading behavior is independent on $\mu$ and $\langle\mathcal{N}(t)\rangle$ exhibits a sub-linear growth.

Tristan Ozuch

• 2 March 2023

Given a diffeomorphism which is homotopic to the identity from the [Formula: see text]-torus to itself, we construct an isotopy whose norm is controlled by that of the diffeomorphism in question.

Dmitri Burago, Jinpeng Lu, Tristan Ozuch

• 2 March 2023

We introduce a new functional inspired by Perelman's $\lambda$-functional adapted to the asymptotically locally Euclidean (ALE) setting and denoted $\lambda_{\operatorname{ALE}}$. Its expression includes a boundary term which turns out to be the ADM-mass. We prove that $\lambda_{\operatorname{ALE}}$ is defined and analytic on convenient neighborhoods of Ricci-flat ALE metrics and we show that it is monotonic along the Ricci flow. This for example lets us establish that small perturbations of integrable and stable Ricci-flat ALE metrics with nonnegative scalar curvature have nonnegative mass. We then introduce a general scheme of proof for a Lojasiewicz-Simon inequality on non-compact manifolds and prove that it applies to $\lambda_{\operatorname{ALE}}$ around Ricci-flat metrics. We moreover obtain an optimal weighted Lojasiewicz exponent for metrics with integrable Ricci-flat deformations.

Alix Deruelle, Tristan Ozuch

• 2 March 2023

In this second article, we prove that any desingularization in the Gromov-Hausdorff sense of an Einstein orbifold by smooth Einstein metrics is the result of a gluingperturbation procedure that we develop. This builds on our first paper where we proved that a Gromov-Hausdorff convergence implied a much stronger convergence in suitable weighted Hölder spaces, in which the analysis of the present paper takes place.

Tristan Ozuch

• 2 March 2023

In this second article, we prove that any desingularization in the Gromov-Hausdorff sense of an Einstein orbifold by smooth Einstein metrics is the result of a gluingperturbation procedure that we develop. This builds on our first paper where we proved that a Gromov-Hausdorff convergence implied a much stronger convergence in suitable weighted Hölder spaces, in which the analysis of the present paper takes place.

Tristan Ozuch

• 27 February 2023

Lectures grothendieckiennes rassemble les textes qui font suite à un séminaire qui s’est tenu au département de mathématiques de l’École Normale Supérieure de 2017 à 2018. Le livre présente une pensée complexe à l’œuvre, celle de l’un des mathématiciens les plus influents et énigmatiques du 20e siècle : Alexander Grothendieck. Les auteurs, Pierre Cartier, Olivia Caramello, Alain Connes, Laurent Laforgue, Colin McLarty, Gilles Pisier, Jean-Jacques Szczeciniarz et Fernando Zalamea, dévoilent à leur façon les conséquences mathématiques ou philosophiques que l’on peut tirer d’une œuvre monumentale qui a transformé le paysage mathématique du 20e siècle et qui a probablement ouvert une nouvelle ère mathématique que nous avons seulement commencé à explorer. Préface de Peter Scholze.

Frédéric Jaeck, P. Cartier, A. Connes, C. Maclarty, J.-J. Szczeciniarz, O. Caramello, L. Lafforgue, G. Pisier, F. Zalamea

• 24 February 2023

Let $H(q,p)$ be a Hamiltonian on $T^*T^n$. We show that the sequence $H_{k}(q,p)=H(kq,p)$ converges for the $\gamma$ topology defined by the author, to $\overline{H}(p)$. This is extended to the case where only some of the variables are homogenized, that is the sequence $H(kx,y,q,p)$ where the limit is of the type ${\overline H}(y,q,p)$ and thus yields an effective Hamiltonian''. We give here the proof of the convergence, and the first properties of the homogenization operator, and give some immediate consequences for solutions of Hamilton-Jacobi equations, construction of quasi-states, etc. We also prove that the function $\overline H$ coincides with Mather's $\alpha$ function which gives a new proof of its symplectic invariance proved by P. Bernard.

Claude Viterbo

• 24 February 2023

Soit $X$ une vari\'et\'e projective et lisse, d\'efinie sur un corps de nombres. Sous l'hypoth\ese $H^2(X,\mathcal O_X)=0,$ Colliot-Th\'el\ene et Raskind ont d\'emontr\'e que le sous-groupe de torsion $CH^2(X)_{tors}$ du groupe de Chow en codimension $2$ est fini. Dans cette note, on donne des bornes uniformes pour le groupe fini $CH^2(X)_{tors}$ quand $X$ varie en famille. Let $X$ be a smooth projective variety defined over a number field. Assuming $H^2(X,\mathcal O_X)=0,$ Colliot-Th\'el\ene and Raskind proved that the torsion subgroup $CH^2(X)_{tors}$ in the Chow group of cycles of codimension $2$ is finite. In this note, we give uniform bounds for the finite group $CH^2(X)_{tors}$ when $X$ varies in a family.

François Charles, Alena Pirutka

• 23 February 2023

Let $\mathscr{M}_{\mathbb R}$ be the moduli space of smooth real binary quintics. We show that each connected component of $\mathscr{M}_{\mathbb R}$ is isomorphic to an arithmetic quotient of an open subset of the real hyperbolic plane. Our main result is that the induced metric on $\mathscr{M}_{\mathbb R}$ extends to a complete hyperbolic metric on the moduli space of stable real binary quintics, making it isometric to the hyperbolic triangle of angles $\pi/3$, $\pi/5$ and $\pi/10$.

Olivier de Gaay Fortman

• 23 February 2023

We construct non-arithmetic lattices in $PO(n,1)$ for each integer $n$ greater than one. To do so, we provide a construction of glueing together complete real hyperbolic orbifolds along totally geodesic subspaces. The hyperbolic orbifolds that we glue are those that arise from anti-unitary involutions on a unitary Shimura variety. This construction generalizes a method of Allcock-Carlson-Toledo, who glued hyperbolic quotient spaces in dimensions two, three and four.

Olivier de Gaay Fortman

• 23 February 2023

We prove the real integral Hodge conjecture for several classes of real abelian threefolds. For instance, we prove the property for real abelian threefolds $A$ whose real locus $A(\mathbb R)$ is connected, and for real abelian threefolds $A$ which are a product $A = B \times E$ of an abelian surface $B$ and an elliptic curve $E$ with connected real locus $E(\mathbb R)$. Moreover, we show that every real abelian threefold satisfies the real integral Hodge conjecture modulo torsion, and reduce the general case to the Jacobian case.

Olivier de Gaay Fortman

• 23 February 2023

We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, any product of Jacobians of smooth projective curves over the complex numbers satisfies the integral Hodge conjecture for one-cycles. The main ingredient is a lift of the Fourier transform to integral Chow groups. Similarly, we prove the integral Tate conjecture for one-cycles on the Jacobian of a proper curve of compact type over the separable closure of a finitely generated field. Furthermore, abelian varieties satisfying such a conjecture are dense in their moduli space.

Thorsten Beckmann, Olivier de Gaay Fortman

• 23 February 2023

This thesis intends to make a contribution to the theories of algebraic cycles and moduli spaces over the real numbers. In the study of the subvarieties of a projective algebraic variety, smooth over the field of real numbers, the cycle class map between the Chow ring and the equivariant cohomology ring plays an important role. The image of the cycle class map remains difficult to describe in general; we study this group in detail in the case of real abelian varieties. To do so, we construct integral Fourier transforms on Chow rings of abelian varieties over any field. They allow us to prove the integral Hodge conjecture for one-cycles on complex Jacobian varieties, and the real integral Hodge conjecture modulo torsion for real abelian threefolds. For the theory of real algebraic cycles, and for several other purposes in real algebraic geometry, it is useful to have moduli spaces of real varieties to our disposal. Insight in the topology of a real moduli space provides insight in the geometry of a real variety that defines a point in it, and the other way around. In the moduli space of real abelian varieties, as well as in the Torelli locus contained in it, we prove density of the set of moduli points attached to abelian varieties containing an abelian subvariety of fixed dimension. Moreover, we provide the moduli space of stable real binary quintics with a hyperbolic orbifold structure, compatible with the period map on the locus of smooth quintics. This identifies the moduli space of stable real binary quintics with a non-arithmetic ball quotient.

Olivier de Gaay Fortman

• 23 February 2023

Abstract For fixed k < g and a family of polarized abelian varieties of dimension g over ${{\mathbb{R}}}$, we give a criterion for the density in the parameter space of those abelian varieties over ${{\mathbb{R}}}$ containing a k-dimensional abelian subvariety over ${{\mathbb{R}}}$. As application, we prove density of such a set in the moduli space of polarized real abelian varieties of dimension g and density of real algebraic curves mapping non-trivially to real k-dimensional abelian varieties in the moduli space of real algebraic curves as well as in the space of real plane curves. This extends to the real setting results by Colombo and Pirola [10]. We then consider the real locus of an algebraic stack over $\mathbb{R}$, attaching a topological space to it. For a real moduli stack, this defines a real moduli space. We show that for ${{\mathcal{M}}}_g$ and ${{\mathcal{A}}}_g$, the real moduli spaces that arise in this way coincide with the moduli spaces of Gross–Harris [13] and Seppälä–Silhol [23].

Olivier de Gaay Fortman

• 22 February 2023

We address the problem of deterministic sequential estimation for a nonsmooth dynamics governed by a variational inequality. An example of such dynamics is the Skorokhod problem with a reflective boundary condition. For smooth dynamics, Mortensen introduced in 1968 a nonlinear estimator based on likelihood maximisation. Then, starting with Hijab in 1980, several authors established a connection between Mortensen's approach and the vanishing noise limit of the robust form of the so-called Zakai equation. In this paper, we investigate to what extent these methods can be developed for dynamics governed by a variational inequality. On the one hand, we address this problem by relaxing the inequality constraint by penalization: this yields an approximate Mortensen estimator relying on an approximating smooth dynamics. We verify that the equivalence between the deterministic and stochastic approaches holds through a vanishing noise limit. On the other hand, inspired by the smooth dynamics approach, we study the vanishing viscosity limit of the Hamilton-Jacobi equation satisfied by the Hopf-Cole transform of the solution of the robust Zakai equation. In contrast to the case of smooth dynamics, the zero-noise limit of the robust form of the Zakai equation cannot be understood in our case from the Bellman equation on the value function arising in Mortensen's procedure. This unveils a violation of equivalence for dynamics governed by a variational inequality between the Mortensen approach and the low noise stochastic approach for nonsmooth dynamics.

Louis-Pierre Chaintron, Álvaro Mateos González, Laurent Mertz, Philippe Moireau

• 22 February 2023

Kudla and Millson constructed a $q$-form $\varphi_{KM}$ on an orthogonal symmetric space using Howe's differential operators. It is a crucial ingredient in their theory of theta lifting. This form can be seen as a Thom form of a real oriented vector bundle. In \cite{mq} Mathai and Quillen constructed a {\em canonical} Thom form and we show how to recover the Kudla-Millson form via their construction. A similar result was obtained by Garcia for signature $(2,q)$ in case the symmetric space is hermitian and we extend it to an arbitrary signature.

Romain Branchereau

• 22 February 2023

We consider the Kudla-Millson theta series associated to a quadratic space of signature $(N,N)$. By combining a see-saw' argument with the Siegel-Weil formula, we show that its (regularized) integral along a torus attached to a totally real field of degree $N$ is the diagonal restriction of an Eisenstein series. It allows us to express the Fourier coefficients of the diagonal restriction as intersection numbers, which generalizes a result of Darmon-Pozzi-Vonk to totally real fields.

Romain Branchereau

• 21 February 2023

We consider the Kudla-Millson theta series associated to a quadratic space of signature $(N,N)$. By combining a see-saw' argument with the Siegel-Weil formula, we show that its (regularized) integral along a torus attached to a totally real field of degree $N$ is the diagonal restriction of an Eisenstein series. It allows us to express the Fourier coefficients of the diagonal restriction as intersection numbers, which generalizes a result of Darmon-Pozzi-Vonk to totally real fields.

Nathanael Berestycki, Raphaël Cerf

• 16 February 2023

We consider the Kudla-Millson theta series associated to a quadratic space of signature $(N,N)$. By combining a see-saw' argument with the Siegel-Weil formula, we show that its (regularized) integral along a torus attached to a totally real field of degree $N$ is the diagonal restriction of an Eisenstein series. It allows us to express the Fourier coefficients of the diagonal restriction as intersection numbers, which generalizes a result of Darmon-Pozzi-Vonk to totally real fields.

Raphaël Cerf, Joseba Dalmau

• 12 February 2023

We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs with uniformly bounded angles and Z-invariant weights. Specifically, we show that in the massive scaling limit, i.e., as the mesh size $\delta$ tends to zero at the same rate as the inverse temperature goes to the critical one, the two-point spin correlations in the full plane behave as $\delta^{-\frac{1}{4}}\mathbb{E}\left[\sigma_{u_{1}}\sigma_{u_{2}}\right]\ \to\ C_{\sigma}^{2}\cdot\Xi\left(|u_{1}-u_{2}|,m\right)\quad\text{as}\quad\delta\to0,$ where the universal constant $C_{\sigma}$ and the function $\Xi(|u_{1}-u_{2}|,m)$ are independent of the lattice. The mass $m$ is defined by the relation $k'-1\sim 4m\delta$, where $k'$ is the Baxter elliptic parameter. This includes $m$ of both signs as well as the critical case when $\Xi(r,0)=r^{-1/4}.$ These results, together with techniques developed to obtain them, are sufficient to extend to isoradial graphs the convergence of multi-point spin correlations in finite planar domains on the square grid, which was established in a joint work of the first two authors and C. Hongler at criticality, and by S.C. Park in the sub-critical massive regime. We also give a simple proof of the fact that the infinite-volume magnetization in the Z-invariant model is independent of the site and of the lattice. As compared to techniques already existing in the literature, we streamline the analysis of discrete (massive) holomorphic spinors near their ramification points, relying only upon discrete analogues of the kernel $z^{-1/2}$ for $m=0$ and of $z^{-1/2}e^{\pm 2m|z|}$ for $m\ne 0$. Enabling the generalization to isoradial graphs and providing a solid ground for further generalizations, our approach also considerably simplifies the proofs in the square lattice setup.

Dmitry Chelkak, Konstantin Izyurov, Rémy Mahfouf

• 11 February 2023

We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs with uniformly bounded angles and Z-invariant weights. Specifically, we show that in the massive scaling limit, i.e., as the mesh size $\delta$ tends to zero at the same rate as the inverse temperature goes to the critical one, the two-point spin correlations in the full plane behave as $\delta^{-\frac{1}{4}}\mathbb{E}\left[\sigma_{u_{1}}\sigma_{u_{2}}\right]\ \to\ C_{\sigma}^{2}\cdot\Xi\left(|u_{1}-u_{2}|,m\right)\quad\text{as}\quad\delta\to0,$ where the universal constant $C_{\sigma}$ and the function $\Xi(|u_{1}-u_{2}|,m)$ are independent of the lattice. The mass $m$ is defined by the relation $k'-1\sim 4m\delta$, where $k'$ is the Baxter elliptic parameter. This includes $m$ of both signs as well as the critical case when $\Xi(r,0)=r^{-1/4}.$ These results, together with techniques developed to obtain them, are sufficient to extend to isoradial graphs the convergence of multi-point spin correlations in finite planar domains on the square grid, which was established in a joint work of the first two authors and C. Hongler at criticality, and by S.C. Park in the sub-critical massive regime. We also give a simple proof of the fact that the infinite-volume magnetization in the Z-invariant model is independent of the site and of the lattice. As compared to techniques already existing in the literature, we streamline the analysis of discrete (massive) holomorphic spinors near their ramification points, relying only upon discrete analogues of the kernel $z^{-1/2}$ for $m=0$ and of $z^{-1/2}e^{\pm 2m|z|}$ for $m\ne 0$. Enabling the generalization to isoradial graphs and providing a solid ground for further generalizations, our approach also considerably simplifies the proofs in the square lattice setup.

Nicolas Curien, Emmanuelle Fauchart, Gilbert Laffond, Jean Lainé, Jacques Lesourne, François Moreau

• 10 February 2023

We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs with uniformly bounded angles and Z-invariant weights. Specifically, we show that in the massive scaling limit, i.e., as the mesh size $\delta$ tends to zero at the same rate as the inverse temperature goes to the critical one, the two-point spin correlations in the full plane behave as $\delta^{-\frac{1}{4}}\mathbb{E}\left[\sigma_{u_{1}}\sigma_{u_{2}}\right]\ \to\ C_{\sigma}^{2}\cdot\Xi\left(|u_{1}-u_{2}|,m\right)\quad\text{as}\quad\delta\to0,$ where the universal constant $C_{\sigma}$ and the function $\Xi(|u_{1}-u_{2}|,m)$ are independent of the lattice. The mass $m$ is defined by the relation $k'-1\sim 4m\delta$, where $k'$ is the Baxter elliptic parameter. This includes $m$ of both signs as well as the critical case when $\Xi(r,0)=r^{-1/4}.$ These results, together with techniques developed to obtain them, are sufficient to extend to isoradial graphs the convergence of multi-point spin correlations in finite planar domains on the square grid, which was established in a joint work of the first two authors and C. Hongler at criticality, and by S.C. Park in the sub-critical massive regime. We also give a simple proof of the fact that the infinite-volume magnetization in the Z-invariant model is independent of the site and of the lattice. As compared to techniques already existing in the literature, we streamline the analysis of discrete (massive) holomorphic spinors near their ramification points, relying only upon discrete analogues of the kernel $z^{-1/2}$ for $m=0$ and of $z^{-1/2}e^{\pm 2m|z|}$ for $m\ne 0$. Enabling the generalization to isoradial graphs and providing a solid ground for further generalizations, our approach also considerably simplifies the proofs in the square lattice setup.

Nicolas Curien, François Moreau

• 10 February 2023

We consider the problem of defining and computing real analogs of polynomial Hurwitz numbers, in other words, the problem of counting properly normalized real polynomials with fixed ramification profiles over real branch points. We show that, provided the polynomials are counted with an appropriate sign, their number does not depend on the order of the branch points on the real line. We study generating series for the invariants thus obtained, determine necessary and sufficient conditions for the vanishing and nonvanishing of these generating series, and obtain a logarithmic asymptotic for the invariants as the degree of the polynomials tends to infinity.

Ilia Itenberg, Dimitri Zvonkine

• 10 February 2023

From a topological viewpoint, a rational curve in the real projective plane is generically a smoothly immersed circle and a finite collection of isolated points. We give an isotopy classification of generic rational quintics in R P 2 \mathbb {RP}^2 in the spirit of Hilbert’s 16th problem.

Ilia Itenberg, Grigory Mikhalkin, Johannes Rau

• 10 February 2023

We establish the enumerativity of (original and modified) Welschinger invariants for every real divisor on any real algebraic Del Pezzo surface and give an algebro-geometric proof of the invariance of that count both up to variation of the point constraints on a given surface and variation of the complex structure of the surface itself.

Ilia Itenberg, Viatcheslav Kharlamov, Eugenii Shustin

• 10 February 2023

We show that the maximal number of (real) lines in a (real) nonsingular spatial quartic surface is 64 (respectively, 56). We also give a complete projective classification of all quartics containing more than 52 lines: all such quartics are projectively rigid. Any value not exceeding 52 can appear as the number of lines of an appropriate quartic.

Alex Degtyarev, Ilia Itenberg, Ali Sinan Sertöz

• 10 February 2023

Given a tropical variety X and two non-negative integers p and q we define homology group $H_{p,q}(X)$. We show that if X is a smooth tropical variety that can be represented as the tropical limit of a 1-parameter family of complex projective varieties, then $\dim H_{p,q}(X)$ coincides with the Hodge number $h^{p,q}$ of a general member of the family.

Ilia Itenberg, Ludmil Katzarkov, Grigory Mikhalkin, Ilia Zharkov

• 10 February 2023

Soit k un corps ultramétrique complet. Nous démontrons un substitut au théorème de la fibre réduite (de Bosch, Lütkebohmert et Raynaud) valable pour tout morphisme Y Ñ X plat et à fibres géométriquement réduites entre espaces k-affinoïdes au sens de Berkovich, sans supposer que X et Y sont stricts ni que la dimension relative de Y sur X est constante. Nous ne faisons pas appel au théorème de la fibre réduite original, ni aux techniques ou au langage de la géométrie formelle. Notre énoncé est formulé en termes de réduction graduée à la Temkin ; notre preuve repose sur un théorème de finitude de Grauert et Remmert et sur la théorie de la réduction (graduée) des germes d'espaces analytiques, due à Temkin. Abstract (Reduction of affinoid spaces in family).-Let k be a nonarchimedean complete field. We prove a substitute for the reduced fiber theorem (of Bosch, Lütkebohmert and Raynaud) that holds for every morphism Y Ñ X flat and with geometrically reduced fibers between k-affinoid spaces in the sense of Berkovich, without assuming that X and Y are strict, nor that the relative dimension of Y over X is constant. We do not use the original reduced fiber theorem, nor the language or the techniques of formal geometry. Our statement is formulated in terms of Temkin's graded reduction; our proof rests on a finiteness result of Grauert and Remmert and on Temkin's theory of (graded) reduction of germs of analytic spaces. Lors de la rédaction de cet article, l'auteur a bénéficié du soutien de l'ANR à travers les projets Valuations, combinatoire et théorie des modèles (ANR-13-BS01-0006), et Définissabilité en géométrie non archimédienne (ANR-15-CE40-0008), ainsi que de celui de l'IUF dont il était membre junior d'octobre 2012 à octobre 2017. Il a aussi profité en mars 2019 de l'hospitalité de l'université hébraïque de Jérusalem, avec le soutien du projet ERC Consolidator 770922 (BirNonArchGeom) de Michael Temkin.

Antoine Ducros

• 10 February 2023

Nous développons dans cet article des techniques d'aplatissement des faisceaux cohérents en géométrie de Berkovich, en nous inspirant de la stratégie générale que Raynaud et Gruson ont mise en oeuvre pour traiter le problème analogue en théorie des schémas. Nous donnons ensuite quelques applications à l'étude des morphismes entre espaces analytiques compacts, et obtenons notamment une description de l'image d'un tel morphisme. Abstract (Blow-ups flattening in non-archimedean analytic geometry) We develop in this article flattening techniques for coherent sheaves in the realm of Berkovich spaces; we are inspired by the general strategy that Raynaud and Gruson have used for dealing with the analogous problem in scheme theory. We then give some applications to the study of morphisms between compact analytic spaces; among other things, we get a description of the image of such a morphism.

Antoine Ducros

• 10 February 2023

We explain how non-archimedean integrals considered by Chambert-Loir and Ducros naturally arise in asymptotics of families of complex integrals. To perform this analysis we work over a non-standard model of the field of complex numbers, which is endowed at the same time with an archimedean and a non-archimedean norm. Our main result states the existence of a natural morphism between bicomplexes of archimedean and non-archimedean forms which is compatible with integration.

Antoine Ducros, Ehud Hrushovski, François Loeser

• 31 January 2023

We prove that the word problem is undecidable in functionally recursive groups, and that the order problem is undecidable in automata groups, even under the assumption that they are contracting.

Laurent Bartholdi, Ivan Mitrofanov

• 31 January 2023

We prove that the semigroup generated by a reversible Mealy automaton contains a free subsemigroup of rank two if and only if it contains an element of infinite order.

Dominik Francoeur, Ivan Mitrofanov

• 31 January 2023

Let A ∼ = k〈X 〉/I be an associative algebra. A finite word over alphabet X is I-reducible if its image in A is a k-linear combination of length-lexicographically lesser words. An obstruction is a subword-minimal Ireducible word. If the number of obstructions is finite then I has a finite Gröbner basis, and the word problem for the algebra is decidable. A cogrowth function is the number of obstructions of length ≤ n. We show that the cogrowth function of a finitely presented algebra is either bounded or at least logarithmical. We also show that an uniformly recurrent word has at least logarithmical cogrowth. Résumé. Soit A ∼ = k〈X 〉/I une algèbre associative. Un mot fini sur l'alphabet X est I-réductible si son image dans A est une combinaison linéaire k de mots de longueur lexicographiquement moindre. Une obstruction dans un mot minimal I-réductible. Si le nombre d'obstructions est fini, alors I a une base finie Gröbner, et le mot problème pour l'algèbre est décidable. Une fonction co-croissance est le nombre d'obstructions de longueur ≤ n. Nous montrons que la fonction de co-croissance d'une algèbre finement présentée est soit bornée, soit au moins logarithmique. Nous montrons également qu'un mot uniformément récurrent a au moins une co-croissance logarithmique.

Alexei Ya. Kanel-Belov, Igor Melnikov, Ivan Mitrofanov

• 31 January 2023

We prove that for a compact metric space the property of having finite covering dimension is equivalent to the existence of a total order with finite snake number.

Ivan Mitrofanov

• 30 January 2023

We investigate the existence of sufficient local conditions under which poset representations decompose as direct sums of indecomposables from a given class. In our work, the indexing poset is the product of two totally ordered sets, corresponding to the setting of 2-parameter persistence in topological data analysis. Our indecomposables of interest belong to the so-called interval modules, which by definition are indicator representations of intervals in the poset. While the whole class of interval modules does not admit such a local characterization, we show that the subclass of rectangle modules does admit one and that it is, in some precise sense, the largest subclass to do so.

Magnus Bakke Botnan, Vadim Lebovici, Steve Oudot

• 30 January 2023

We introduce a general definition of hybrid transforms for constructible functions. These are integral transforms combining Lebesgue integration and Euler calculus. Lebesgue integration gives access to well-studied kernels and to regularity results, while Euler calculus conveys topological information and allows for compatibility with operations on constructible functions. We conduct a systematic study of such transforms and introduce two new ones: the Euler-Fourier and Euler-Laplace transforms. We show that the first has a left inverse and that the second provides a satisfactory generalization of Govc and Hepworth's persistent magnitude to constructible sheaves, in particular to multi-parameter persistent modules. Finally, we prove index-theoretic formulae expressing a wide class of hybrid transforms as generalized Euler integral transforms. This yields expectation formulae for transforms of constructible functions associated to (sub)level-sets persistence of random Gaussian filtrations.

• 29 January 2023

Let $G$ be a reductive group over a finite field with a maximal unipotent subgroup $U$, we consider certain sheaves on $G/U$ defined by Kazhdan and Laumon and show that their cohomology produces the cohomology of the Deligne-Lusztig varieties. We then use this comparison to give a new proof of a result of Dudas.

Arnaud Eteve

• 27 January 2023

Let $G$ be a reductive group over a finite field with a maximal unipotent subgroup $U$, we consider certain sheaves on $G/U$ defined by Kazhdan and Laumon and show that their cohomology produces the cohomology of the Deligne-Lusztig varieties. We then use this comparison to give a new proof of a result of Dudas.

Joshua Frisch, Forte Shinko

• 27 January 2023

We study topological realizations of countable Borel equivalence relations, including realizations by continuous actions of countable groups, with additional desirable properties. Some examples include minimal realizations on any perfect Polish space, realizations as $K_\sigma$ relations, and realizations by continuous actions on the Baire space. We also consider questions related to realizations of specific important equivalence relations, like Turing and arithmetical equivalence. We focus in particular on the problem of realization by continuous actions on compact spaces and more specifically subshifts. This leads to the study of properties of subshifts, including universality of minimal subshifts, and a characterization of amenability of a countable group in terms of subshifts. Moreover we consider a natural universal space for actions and equivalence relations and study the descriptive and topological properties in this universal space of various properties, like, e.g., compressibility, amenability or hyperfiniteness.

Joshua Frisch, Alexander Kechris, Forte Shinko, Zoltán Vidnyánszky

• 27 January 2023

Given a countable group $G$ and a $G$-flow $X$, a measure $\mu\in P(X)$ is called characteristic if it is $\mathrm{Aut}(X, G)$-invariant. Frisch and Tamuz asked about the existence of a minimal $G$-flow, for any group $G$, which does not admit a characteristic measure. We construct for every countable group $G$ such a minimal flow. Along the way, we are motivated to consider a family of questions we refer to as minimal subdynamics: Given a countable group $G$ and a collection of infinite subgroups $\{\Delta_i: i\in I\}$, when is there a faithful $G$-flow for which every $\Delta_i$ acts minimally?

Joshua Frisch, Brandon Seward, Andy Zucker

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# I need help understanding proofs of Gibbs' Inequality using Jensen's Inequality.

I've recently begun to study the basics of Information Theory using this pdf: https://web.stanford.edu/~montanar/RESEARCH/BOOK/partA.pdf, and have just read about Kullback-Leiber Divergence; there Gibbs' inequality is proven (although it isn't called by that name) using the convexity of the function: $-\log_2 x$ and Jensen's Inequality; some other proofs use the same idea with other convex functions like $\ x\log_2 x$ .

All well so far except for one thing: The functions that are used in the proofs are not $-\log_2 x$ or $\ x\log_2 x$ , but their composition with other functions like: $-\log_2 \left(\frac{q(x)}{p(x)}\right)$ or $\ \left(\frac{q(x)}{p(x)}\right)\log_2 \left(\frac{q(x)}{p(x)}\right)$ with $q(x)$ and $p(x)$ being probability distributions with $x \in \mathcal I$ such that $\ \forall x \in \mathcal I, \ q(x),p(x) \neq 0$ and $\sum \limits_{x \in \mathcal I} q(x) = 1$ and $\sum \limits_{x \in \mathcal I} p(x) = 1$.

The thing is: How do i know if the new composite functions are convex and so Jensen's Inequality applies? depending on $q(x)$ and $p(x)$, $f(x)=q(x)/p(x)$ may or may not be convex, and may or may not be non-decreasing; and i know examples of compositions of convex functions that are not convex.

I understood another proof that didn't use Jensen's Inequality at all, it's this kind of proofs that i have trouble with.

All help would be appreciated, regarding the question and also regarding style (i'm new so...)

Thank you all.

• Jensen says that for convex $f$ and random variable $Y$, $\mathbb{E}f(Y)\ge f(\mathbb{E}Y)$
• $f(x)=-\log x$ is convex
• We can let $Y=\frac{p(X)}{q(X)}$. As $X$ is a random variable, $Y$ is also just a random variable.
• Jensen thus says that $\mathbb{E}[-\log Y] \ge -\log\mathbb{E}Y$
• Taking expectations with respect to $q$, this becomes

$$\mathbb{E}\left[-\log \frac{p(X)}{q(X)}\right]=\mathbb{E}\left[\log \frac{q(X)}{p(X)}\right]=D_{KL}(q||p) \ge -\log\mathbb{E}_q\left(\frac{p(X)}{q(X)}\right)=-\log\mathbb{E}_p(1)=0$$

and thus, Gibbs' inequality is proven.

• $Y$ can be any function? Isn't there a problem if the composite function: $f \circ Y (x)$ isn't convex? – IDUTDP Dec 20 '16 at 20:56
• have edited answer to address this - $Y$ is just a random variable – πr8 Dec 20 '16 at 20:58
• Well a random variable is a function =P. It seems my problem comes from a misunderstanding of Jensen's Inequality, i'll have to review the topic. Thank you for your answer :) – IDUTDP Dec 20 '16 at 20:58
• you're correct! though in the context of jensen, you typically have one "variable" ($Y$) and one "function" ($f$), so i thought that might be unhelpful/confounding in trying to explain the solution. – πr8 Dec 20 '16 at 21:00


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# dissection of rectangle into triangles of the same area

Given $m \times n$ rectangle with area $A$, and $m,n \in \mathbb{N}$. Let $S_k(m,n)$ be the number of way to dissect this rectangle into $k$ non-overlapping triangles whose area is $\frac{A}{k}$.

It is known that when $k$ is odd $S_k(n,n)=0$ (For example see Paul Monsky, "On Dividing A Square Into Triangles", American Mathematical Monthly, Vol 77 no 2, 1970)

Is it true that $S_k(m,n) < \infty$?
Is there any known bound for $S_k(m,n)$ (even when $m=n$ and $k$ even)?

Edit: There are several possible interpretations of the problem, one may assume that the vertices of the triangles are lattice points. Or perhaps (harder?), when we are given an arbitrary rectangle with a fixed dimension (not necessarily has an integer dimension).

-
Just to check, are there any conditions on the triangles? Do the vertices need to be lattice points? Then again, judging from your $S_k(m,n) < \infty$, I guess that the answer is no? –  Calvin Lin Jan 3 '13 at 7:49
it's entirely an open question, so it can be lattice point or not. –  Ajat Adriansyah Jan 3 '13 at 8:14
@AjatAdriansyah, open question for you means unknown answer or no more conditions needed on the triangles? –  Sigur Jan 3 '13 at 10:40
To my knowledge it is open, and I would like to present it in a more flexible open-ended fashion. For example one can consider the variant of the problem where the dimension of rectangle is not specified. –  Ajat Adriansyah Jan 3 '13 at 12:41
It seems when the triangle’s vertices are nor necessarily the lattice points then $S_k(n,m)$ does not depends on $n$ and $m$, because an affine transformation can transform the dissected rectangle into a square, preserving all triangles and their relative areas. –  Alex Ravsky May 1 '13 at 0:22


text:
# General Thoughts: Physics (1 Viewer)

#### spatula232

##### Active Member
Atar cannot be created nor destroyed but can be transformed. It usually transforms into something called a mystery mark
I'm sig-ing this

#### Mr_Kap

##### Well-Known Member
RIP for that copper question.

I drew a straight line with negative gradient and then from C to D it was FAIRLY straight (but not exactly).

lol. at least my explanation was alright, maybe 2/5 lol.

#### Drsoccerball

##### Well-Known Member
RIP for that copper question.

I drew a straight line with negative gradient and then from C to D it was FAIRLY straight (but not exactly).

lol. at least my explanation was alright, maybe 2/5 lol.
Exactly this is why I used Extension 2 since I saw Integrand do it once and it actually worked

#### PhysicsMaths

##### Active Member
Exactly this is why I used Extension 2 since I saw Integrand do it once and it actually worked
They can't award you marks for doing this.

#### Drsoccerball

##### Well-Known Member
They can't award you marks for doing this.
I didn't think they would... It was for me not them.

#### Mr_Kap

##### Well-Known Member
Biggest regret: Dropping 4u. Should've kept it just for physics.

was this the mechanics topic where you learnt this..because that is the ONLY 4u topic i didn;t learn.

or was it the 4u integration topic ---> because i already know 4u integration so if i could have used it i'm such a dickhead for not using it.

#### Drsoccerball

##### Well-Known Member
Biggest regret: Dropping 4u. Should've kept it just for physics.

was this the mechanics topic where you learnt this..because that is the ONLY 4u topic i didn;t learn.

or was it the 4u integration topic ---> because i already know 4u integration so if i could have used it i'm such a dickhead for not using it.
Mechanics helped me alot for physics... Mainly resistance questions...

#### SlimiJay

##### Member
no one has the paper??

#### IamKirby

##### Member
I didn't think they would... It was for me not them.
Hey man, I did 4u but I don't know how it could be used here. Can you explain? Thanks

Sent from my iPhone using Tapatalk

#### apurba

##### Member
Okay haha thought so, thanks

Sent from my iPhone using Tapatalk
Okay this was my reasoning for my graph:

I said as eddy currents are induced on the copper plate, causing it to slow down.
but then, the trolley passes through the magnet, and the direction of the eddy currents are then switched, causing it to experience a force in the opposite direction (i.e. accelerating it to the right slightly).

Where did I go wrong?

#### Drsoccerball

##### Well-Known Member
Hey man, I did 4u but I don't know how it could be used here. Can you explain? Thanks

Sent from my iPhone using Tapatalk
acceleration = -k

$\bg_white v \frac{dv}{dx} = -k$

$\bg_white \frac{dx}{dv}= -\frac{v}{k}$

$\bg_white [x]_0^x =[ \frac{-v^2}{2k}]_v^W$

Where W is the final speed even though k is changing it gave me a rough estimate of what the graph looked like.

#### Thjv

##### Member
acceleration = -k

$\bg_white v \frac{dv}{dx} = -k$

$\bg_white \frac{dx}{dv}= -\frac{v}{k}$

$\bg_white [x]_0^x =[ \frac{-v^2}{2k}]_v^W$

Where W is the final speed even though k is changing it gave me a rough estimate of what the graph looked like.
4u motion..

#### Drsoccerball

##### Well-Known Member
Okay this was my reasoning for my graph:

I said as eddy currents are induced on the copper plate, causing it to slow down.
but then, the trolley passes through the magnet, and the direction of the eddy currents are then switched, causing it to experience a force in the opposite direction (i.e. accelerating it to the right slightly).

Where did I go wrong?

As it leaves it is also slowed down remember how a similar experiment where a magnet dropped through a metal induces an m field to always slow down the magnet whether it enters or leaves.

#### Mr_Kap

##### Well-Known Member
As it leaves it is also slowed down remember how a similar experiment where a magnet dropped through a metal induces an m field to always slow down the magnet whether it enters or leaves.
if i explained all the shit correctly with CORRECT REASONING (i did) but drew the graph that i did (negative straight line from A to C, and then a fairly straight line from C to D), can i get 2/3 out of 5???

#### InteGrand

##### Well-Known Member
acceleration = -k

$\bg_white v \frac{dv}{dx} = -k$

$\bg_white \frac{dx}{dv}= -\frac{v}{k}$

$\bg_white [x]_0^x =[ \frac{-v^2}{2k}]_v^W$

Where W is the final speed even though k is changing it gave me a rough estimate of what the graph looked like.
$\bg_white This is basically a derivation of the well-known formula v^2 = u^2 + 2a\Delta s for \textsl{constant} a.$

#### Drsoccerball

##### Well-Known Member
$\bg_white This is basically a derivation of the well-known formula v^2 = u^2 + 2a\Delta s for \textsl{constant} a.$
Agahahahaahah how many things did I derive in the exam...

#### Drsoccerball

##### Well-Known Member
if i explained all the shit correctly with CORRECT REASONING (i did) but drew the graph that i did (negative straight line from A to C, and then a fairly straight line from C to D), can i get 2/3 out of 5???
I think so. Also the question would've been more amazing if there was friction from the floor aswell... But since V didnt change there was no friction...

#### Thjv

##### Member
I think so. Also the question would've been more amazing if there was friction from the floor aswell... But since V didnt change there was no friction...
Pls no... this is hsc physics


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# A gumball machine contains 7 blue, 5 green, and 4 red

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A gumball machine contains 7 blue, 5 green, and 4 red [#permalink]

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A gumball machine contains 7 blue, 5 green, and 4 red gumballs - identical besides colour. If the machine disperses 3 gumballs at random, what is the probability that it dispenses one of each colour.

Use the comibatorics method only!
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22 Oct 2012, 01:27
I tried to solve using combinatorics but I was wrong - answer below.

I thought there is 16 balls in total and if we select three from 16, we have 16!/3!13! ways to choose three balls.

BGR is one of each and there is 6 ways of organising

we have 1C7x 1C5 x 1C4 x 6 / number of ways to choose three balls

so we have 7 x 5 x 4 x 6 / 16 x 15 x 14 x 3 x 2

Simplifying the sixes 7 x 5 x 4 / 16 x 15 x 14

simplifying 7 and 14 1 x 5 x 4 / 16 x 15 x 2

Simplifying 5 and 15 1 x 1 x 4 / 16 x 3 x 2

Simplifying 4, 16 1 x 1 x 1 / 4 x 3 x 2

However

Using simple probabilities

6x 7/16 x 5/15 x 4/14 = 1/4
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Total No. of ways to select 3 out of 16 is = 16C3

No. Of ways to select one blue ball = 7C1
No. Of ways to select one green ball = 5C1
No. Of ways to select one red ball = 4C1

So. total no. of ways for favourable outcome = 7*5*4 = 140

So Probability = 140/560 = 1/4

When using simple probability, after choosing the first ball, there are only 15 balls to choose from and one less of the color that has already been chosen and after selecting the second, there are only 14 balls to choose from and one less of the color that has already been chosen and so on. Hence order has to be taken into account.

However while using combinations, the order does not matter as we are selecting one ball from one color only each time.
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jordanshl wrote:
I tried to solve using combinatorics but I was wrong - answer below.

I thought there is 16 balls in total and if we select three from 16, we have 16!/3!13! ways to choose three balls.

BGR is one of each and there is 6 ways of organising

we have 1C7x 1C5 x 1C4 x 6 / number of ways to choose three balls

so we have 7 x 5 x 4 x 6 / 16 x 15 x 14 x 3 x 2

Simplifying the sixes 7 x 5 x 4 / 16 x 15 x 14

simplifying 7 and 14 1 x 5 x 4 / 16 x 15 x 2

Simplifying 5 and 15 1 x 1 x 4 / 16 x 3 x 2

Simplifying 4, 16 1 x 1 x 1 / 4 x 3 x 2

However

Using simple probabilities

6x 7/16 x 5/15 x 4/14 = 1/4

You did a computational mistake, the 3! from the denominator of 16C3 should go up to the numerator.
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jordanshl wrote:
A gumball machine contains 7 blue, 5 green, and 4 red gumballs - identical besides colour. If the machine disperses 3 gumballs at random, what is the probability that it dispenses one of each colour.

Use the comibatorics method only!

We need to find the probability of BGR (a marble of each color).

Probability approach:

$$P(BGR)=\frac{7}{16}*\frac{5}{15}*\frac{4}{14}*3!=\frac{1}{4}$$, we are multiplying by 3! since BGR scenario can occur in several different ways: BGR, BRG, RBG, ... (# of permutations of 3 distinct letters BGR is 3!).

Combination approach:

$$P(BGR)=\frac{C^1_7*C^1_5*C^1_4}{C^3_{16}}=\frac{1}{4}$$.

Hope it's clear.

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Re: A gumball machine contains... [#permalink]

### Show Tags

09 Jul 2013, 08:05
Bunuel wrote:
jordanshl wrote:
A gumball machine contains 7 blue, 5 green, and 4 red gumballs - identical besides colour. If the machine disperses 3 gumballs at random, what is the probability that it dispenses one of each colour.

Use the comibatorics method only!

We need to find the probability of BGR (a marble of each color).

Probability approach:

$$P(BGR)=\frac{7}{16}*\frac{5}{15}*\frac{4}{14}*3!=\frac{1}{4}$$, we are multiplying by 3! since BGR scenario can occur in several different ways: BGR, BRG, RBG, ... (# of permutations of 3 distinct letters BGR is 3!).

Combination approach:

$$P(BGR)=\frac{C^1_7*C^1_5*C^1_4}{C^3_{16}}=\frac{1}{4}$$.

Hope it's clear.

Is there a way to solve this using $$\frac{16!}{7!*5!*4!}$$ as the denominator?
Math Expert
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Posts: 34902
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Re: A gumball machine contains... [#permalink]

### Show Tags

10 Jul 2013, 00:14
1
KUDOS
Expert's post
elementbrdr wrote:
Bunuel wrote:
jordanshl wrote:
A gumball machine contains 7 blue, 5 green, and 4 red gumballs - identical besides colour. If the machine disperses 3 gumballs at random, what is the probability that it dispenses one of each colour.

Use the comibatorics method only!

We need to find the probability of BGR (a marble of each color).

Probability approach:

$$P(BGR)=\frac{7}{16}*\frac{5}{15}*\frac{4}{14}*3!=\frac{1}{4}$$, we are multiplying by 3! since BGR scenario can occur in several different ways: BGR, BRG, RBG, ... (# of permutations of 3 distinct letters BGR is 3!).

Combination approach:

$$P(BGR)=\frac{C^1_7*C^1_5*C^1_4}{C^3_{16}}=\frac{1}{4}$$.

Hope it's clear.

Is there a way to solve this using $$\frac{16!}{7!*5!*4!}$$ as the denominator?

Yes.

Total # of outcomes: $$\frac{16!}{7!*5!*4!}$$ (the number of arrangements of the mables).

Favorable outcomes: we need the first three marbles to be BGR in any combination, so 3!. The remaining 13 marbles (6 blue, 4 green, and 3 red) can be arranged in 13!/(6!4!3!).

P = (Favorable)/(Total) = $$\frac{(3!\frac{13!}{6!4!3!})}{(\frac{16!}{7!*5!*4!})}=\frac{1}{4}$$.

Hope it's clear.
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Re: A gumball machine contains 7 blue, 5 green, and 4 red [#permalink]

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01 Sep 2016, 03:10
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Re: A gumball machine contains 7 blue, 5 green, and 4 red   [#permalink] 01 Sep 2016, 03:10
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text:
# Knightmare

Time Limit: 2000/1000 MS (Java/Others)????Memory Limit: 65536/65536 K (Java/Others)
Total Submission(s): 346????Accepted Submission(s): 144

Problem Description
A knight jumps around an infinite chessboard. The chessboard is an unexplored territory. In the spirit of explorers, whoever stands on a square for the first time claims the ownership of this square. The knight initially owns the square he stands, and jumps?N?times before he gets bored.
Recall that a knight can jump in 8 directions. Each direction consists of two squares forward and then one squaure sidways.

After?N?jumps, how many squares can possibly be claimed as territory of the knight? As?N?can be really large, this becomes a nightmare to the knight who is not very good at math. Can you help to answer this question?
?

Input
The first line of the input gives the number of test cases,?T.?T?test cases follow.
Each test case contains only one number?N, indicating how many times the knight jumps.
1T105
0N109
?

Output
For each test case, output one line containing “Case #x: y”, where?x?is the test case number (starting from 1) and?y?is the number of squares that can possibly be claimed by the knight.
?

Sample Input
3
0
1
7
?

Sample Output
Case #1: 1
Case #2: 9
Case #3: 649
?

（在大部分情况下，a,b,c三个连续结果，有如下规律：(c-b)-(b-a)==28 ）。

#include
#include
#include
#include
#include
using namespace std;
int main() {
/*  const int com[16] = { -2, -1, -2, 1, -1, 2, -1, -2, 1, 2, 1, -2, 2, 1, 2, -1};
set<> >se1;
se1.insert ( pair ( 0, 0 ) );
int T = 20;
while ( T-- ) {
set<> >se2;
set<> >::iterator it;
for ( it = se1.begin(); it != se1.end(); it++ )
for ( int i = 0; i < 8; i++ )
se2.insert ( pair ( ( *it ).first + com[i * 2], ( *it ).second + com[i * 2 + 1] ) );
for ( it = se2.begin(); it != se2.end(); it++ )
se1.insert ( ( *it ) );
cout << se1.size() << endl;
}*/
int T;
unsigned long long n;
cin >> T;
const int com[6] = {1, 9, 41,109,205,325};
for ( int i = 1; i <= T; i++ ) {
scanf ( "%llu", &n );
if ( n <= 5 ) printf ( "Case #%d: %d\n", i, com[n] );
else printf ( "Case #%d: %llu\n", i, 205 + ( n - 4 ) * 120 + ( n * ( n + 1 ) / 2 - 5*n + 10 ) * 28 );
}
return 0;
}

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text:
# American Institute of Mathematical Sciences

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Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $L^2$-supercritical case
February  2021, 41(2): 747-775. doi: 10.3934/dcds.2020299

## Large time behavior of exchange-driven growth

 1 WMG, University of Warwick, UK, Mathematics Institute, University of Warwick, UK 2 Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60 (room 2.023), D-53115 Bonn, Germany

*Corresponding author: E.esenturk.1@warwick.ac.uk

Received  April 2019 Revised  April 2020 Published  August 2020

Exchange-driven growth (EDG) is a model in which pairs of clusters interact by exchanging single unit with a rate given by a kernel $K(j, k)$. Despite EDG model's common use in the applied sciences, its rigorous mathematical treatment is very recent. In this article we study the large time behaviour of EDG equations. We show two sets of results depending on the properties of the kernel $(i)$ $K(j, k) = b_{j}a_{k}$ and $(ii)$ $K(j, k) = ja_{k}+b_{j}+\varepsilon\beta_{j}\alpha_{k}$. For type I kernels, under the detailed balance assumption, we show that the system admits unique equilibrium up to a critical mass $\rho_{s}$ above which there is no equilibrium. We prove that if the system has an initial mass below $\rho_{s}$ then the solutions converge to a unique equilibrium distribution strongly where if the initial mass is above $\rho_{s}$ then the solutions converge to cricital equilibrium distribution in a weak sense. For type II kernels, we do not make any assumption of detailed balance and equilibrium is shown as a consequence of contraction properties of solutions. We provide two separate results depending on the monotonicity of the kernel or smallness of the total mass. For the first case we prove exponential convergence in the number of clusters norm and for the second we prove exponential convergence in the total mass norm.

Citation: Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299
##### References:

show all references

##### References:
 [1] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [2] Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426 [3] Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318 [4] Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465 [5] Puneet Pasricha, Anubha Goel. Pricing power exchange options with hawkes jump diffusion processes. Journal of Industrial & Management Optimization, 2021, 17 (1) : 133-149. doi: 10.3934/jimo.2019103 [6] Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457 [7] Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302 [8] João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138 [9] Sören Bartels, Jakob Keck. Adaptive time stepping in elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 71-88. doi: 10.3934/dcdss.2020323 [10] Andreu Ferré Moragues. Properties of multicorrelation sequences and large returns under some ergodicity assumptions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020386 [11] Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321 [12] Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252 [13] Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240 [14] Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460 [15] Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119 [16] Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046 [17] Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336 [18] Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339 [19] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [20] Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112

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text:
Moving Average of Something Other than the Close Price

I am trying to calculate a simple average of something other than the close price. So, for example, the Low Price * .95. Or the High Price * 1.05. Does anyone have any suggestions? All I can find is the MAVG function and it seems to only work off of the close price.

6 responses

There's a section in the Help & API Docs titled History (https://www.quantopian.com/help#ide-history) that discusses how to get historical low/high etc prices for securities. You can define the number of days you want to include in your history look up by setting "bar_count" as a parameter to the history function. This will return a pandas DataFrame to which you can apply the standard pandas transforms. e.g. calculating a 5-day moving average (from the docs where price_history.mean() gives you a 5-day rolling moving average):

def initialize(context):
set_universe(universe.DollarVolumeUniverse(90.0, 90.1))

def handle_data(context, data):
price_history = history(bar_count=5, frequency='1d', field='price')
log.info(price_history.mean())

In your case, you can change the field parameter from 'price' (which gives you closing prices) to 'low' or 'high'.

Hi Sekuri-

Thanks so much for your reply. I really appreciate it. This looks like a way to calculate an average off of low or high price. How would I manipulate those prices first before running and Mean of them? Is there a way to look through them and modify their values?

Not sure I exactly understand what you're after, but do you mean something like this?

lows = 0.95*history(bar_count=5, frequency='1d', field='low')
lows_mavg = lows.mean()
highs = 1.05* history(bar_count=5, frequency='1d', field='high')
highs_mavg = highs.mean()

Thanks again for the quick reply. I'm anxious to get this working.

No, not quite. I want to manipulate each value individually. So, I would like to take the (High-Low/High) for each day, then calculate an average for the last 10 days from that number.

like this?

        lows = history(bar_count=11, frequency='1d', field='low') #bar_count =11 to get last 10 complete days (the last - 11th - bar is today's info before closing and will always be grabbed)
highs = history(bar_count=11, frequency='1d', field='high')
mavg = (highs[0:-1]-lows[0:-1]).divide(highs[0:-1]) # [0:-1] to account for last 10 complete days - ignore today's incomplete info
print mavg.mean()

Hi Sekuri,

Thank you very much. I was trying to do it in a loop, your method of using highs[0:-1] is much better. I really appreciate your help!


text:
### Session

Moderator: Felix Xinnan Yu

Abstract:

### Chat is not available.

Thu 22 July 6:00 - 6:20 PDT

(Oral)
##### Improved, Deterministic Smoothing for L_1 Certified Robustness

Alexander Levine · Soheil Feizi

Randomized smoothing is a general technique for computing sample-dependent robustness guarantees against adversarial attacks for deep classifiers. Prior works on randomized smoothing against L1 adversarial attacks use additive smoothing noise and provide probabilistic robustness guarantees. In this work, we propose a non-additive and deterministic smoothing method, Deterministic Smoothing with Splitting Noise (DSSN). To develop DSSN, we first develop SSN, a randomized method which involves generating each noisy smoothing sample by first randomly splitting the input space and then returning a representation of the center of the subdivision occupied by the input sample. In contrast to uniform additive smoothing, the SSN certification does not require the random noise components used to be independent. Thus, smoothing can be done effectively in just one dimension and can therefore be efficiently derandomized for quantized data (e.g., images). To the best of our knowledge, this is the first work to provide deterministic "randomized smoothing" for a norm-based adversarial threat model while allowing for an arbitrary classifier (i.e., a deep model) to be used as a base classifier and without requiring an exponential number of smoothing samples. On CIFAR-10 and ImageNet datasets, we provide substantially larger L1 robustness certificates compared to prior works, establishing a new state-of-the-art. The determinism of our method also leads to significantly faster certificate computation. Code is available at: https://github.com/alevine0/smoothingSplittingNoise.

[Paper]
[ ]
Thu 22 July 6:20 - 6:25 PDT

(Spotlight)
##### Mixed Nash Equilibria in the Adversarial Examples Game

Laurent Meunier · Meyer Scetbon · Rafael Pinot · Jamal Atif · Yann Chevaleyre

This paper tackles the problem of adversarial examples from a game theoretic point of view. We study the open question of the existence of mixed Nash equilibria in the zero-sum game formed by the attacker and the classifier. While previous works usually allow only one player to use randomized strategies, we show the necessity of considering randomization for both the classifier and the attacker. We demonstrate that this game has no duality gap, meaning that it always admits approximate Nash equilibria. We also provide the first optimization algorithms to learn a mixture of classifiers that approximately realizes the value of this game, \emph{i.e.} procedures to build an optimally robust randomized classifier.

[Paper]
[ ]
Thu 22 July 6:25 - 6:30 PDT

(Spotlight)
##### Learning to Generate Noise for Multi-Attack Robustness

Divyam Madaan · Jinwoo Shin · Sung Ju Hwang

Adversarial learning has emerged as one of the successful techniques to circumvent the susceptibility of existing methods against adversarial perturbations. However, the majority of existing defense methods are tailored to defend against a single category of adversarial perturbation (e.g. $\ell_\infty$-attack). In safety-critical applications, this makes these methods extraneous as the attacker can adopt diverse adversaries to deceive the system. Moreover, training on multiple perturbations simultaneously significantly increases the computational overhead during training. To address these challenges, we propose a novel meta-learning framework that explicitly learns to generate noise to improve the model's robustness against multiple types of attacks. Its key component is \emph{Meta Noise Generator (MNG)} that outputs optimal noise to stochastically perturb a given sample, such that it helps lower the error on diverse adversarial perturbations. By utilizing samples generated by MNG, we train a model by enforcing the label consistency across multiple perturbations. We validate the robustness of models trained by our scheme on various datasets and against a wide variety of perturbations, demonstrating that it significantly outperforms the baselines across multiple perturbations with a marginal computational cost.

[Paper]
[ ]
Thu 22 July 6:30 - 6:35 PDT

(Spotlight)
##### Query Complexity of Adversarial Attacks

Grzegorz Gluch · Rüdiger Urbanke

There are two main attack models considered in the adversarial robustness literature: black-box and white-box. We consider these threat models as two ends of a fine-grained spectrum, indexed by the number of queries the adversary can ask. Using this point of view we investigate how many queries the adversary needs to make to design an attack that is comparable to the best possible attack in the white-box model. We give a lower bound on that number of queries in terms of entropy of decision boundaries of the classifier. Using this result we analyze two classical learning algorithms on two synthetic tasks for which we prove meaningful security guarantees. The obtained bounds suggest that some learning algorithms are inherently more robust against query-bounded adversaries than others.

[Paper]
[ ]
Thu 22 July 6:35 - 6:40 PDT

(Spotlight)
##### Training Adversarially Robust Sparse Networks via Bayesian Connectivity Sampling

Ozan Özdenizci · Robert Legenstein

Deep neural networks have been shown to be susceptible to adversarial attacks. This lack of adversarial robustness is even more pronounced when models are compressed in order to meet hardware limitations. Hence, if adversarial robustness is an issue, training of sparsely connected networks necessitates considering adversarially robust sparse learning. Motivated by the efficient and stable computational function of the brain in the presence of a highly dynamic synaptic connectivity structure, we propose an intrinsically sparse rewiring approach to train neural networks with state-of-the-art robust learning objectives under high sparsity. Importantly, in contrast to previously proposed pruning techniques, our approach satisfies global connectivity constraints throughout robust optimization, i.e., it does not require dense pre-training followed by pruning. Based on a Bayesian posterior sampling principle, a network rewiring process simultaneously learns the sparse connectivity structure and the robustness-accuracy trade-off based on the adversarial learning objective. Although our networks are sparsely connected throughout the whole training process, our experimental benchmark evaluations show that their performance is superior to recently proposed robustness-aware network pruning methods which start from densely connected networks.

[Paper]
[ ]
Thu 22 July 6:40 - 6:45 PDT

(Spotlight)
##### Efficient Training of Robust Decision Trees Against Adversarial Examples

Daniël Vos · Sicco Verwer

Current state-of-the-art algorithms for training robust decision trees have high runtime costs and require hours to run. We present GROOT, an efficient algorithm for training robust decision trees and random forests that runs in a matter of seconds to minutes. Where before the worst-case Gini impurity was computed iteratively, we find that we can solve this function analytically to improve time complexity from O(n) to O(1) in terms of n samples. Our results on both single trees and ensembles on 14 structured datasets as well as on MNIST and Fashion-MNIST demonstrate that GROOT runs several orders of magnitude faster than the state-of-the-art works and also shows better performance in terms of adversarial accuracy on structured data.

[Paper]
[ ]
Thu 22 July 6:45 - 6:50 PDT

(Spotlight)
##### Scalable Certified Segmentation via Randomized Smoothing

Marc Fischer · Maximilian Baader · Martin Vechev

We present a new certification method for image and point cloud segmentation based on randomized smoothing. The method leverages a novel scalable algorithm for prediction and certification that correctly accounts for multiple testing, necessary for ensuring statistical guarantees. The key to our approach is reliance on established multiple-testing correction mechanisms as well as the ability to abstain from classifying single pixels or points while still robustly segmenting the overall input. Our experimental evaluation on synthetic data and challenging datasets, such as Pascal Context, Cityscapes, and ShapeNet, shows that our algorithm can achieve, for the first time, competitive accuracy and certification guarantees on real-world segmentation tasks. We provide an implementation at https://github.com/eth-sri/segmentation-smoothing.

[Paper]
[ ]
Thu 22 July 6:50 - 6:55 PDT

(Q&A)


text:
# How do I render from one render target to another?

I have two render targets:

1. a fake backbuffer; a special render target where I do all my rendering.

2. a light render target; where I render my light fx.

I'm sure I'm rendering correctly on both.

The problem arises when I overlay the light render target onto the fake backbuffer by drawing a quad covering it:

DxEngine.DrawSprite(0.0f, 0.0f, 0.0f,
(float)DxEngine.GetWidth(), (float)DxEngine.GetHeight(),
0xFFFFFFFF, LightSurface->GetTexture());

Regardless of what's in the light target, nothing is rendered onto the other target. I tried clearing the light target with full-white or full-black, but still get nothing.

Fake backbuffer created with

Direct3dDev->CreateTexture(Width, Height, 1, D3DUSAGE_RENDERTARGET,
D3DFMT_X8R8G8B8, D3DPOOL_DEFAULT, &Texture, nullptr);

Light render target created with

Direct3dDev->CreateTexture(Width, Height, 1, D3DUSAGE_RENDERTARGET,
D3DFMT_A8R8G8B8, D3DPOOL_DEFAULT, &Texture, nullptr);

I also tried to create both with D3DFMT_A8R8G8B8, again without difference. Both targets have the same width and height.

Only the fixed pipeline is used

DirectX setup for rendering :

Direct3dDev->SetSamplerState(0, D3DSAMP_MINFILTER, D3DTEXF_LINEAR );
Direct3dDev->SetSamplerState(0, D3DSAMP_MAGFILTER, D3DTEXF_LINEAR );
Direct3dDev->SetSamplerState(0, D3DSAMP_MIPFILTER, D3DTEXF_NONE );
Direct3dDev->SetRenderState(D3DRS_CULLMODE, D3DCULL_NONE);
Direct3dDev->SetRenderState(D3DRS_LIGHTING, false);
Direct3dDev->SetRenderState(D3DRS_ZENABLE, D3DZB_TRUE);
Direct3dDev->SetRenderState(D3DRS_ZWRITEENABLE,D3DZB_TRUE);
Direct3dDev->SetRenderState(D3DRS_ZFUNC,D3DCMP_LESSEQUAL);

Direct3dDev->SetRenderState(D3DRS_ALPHABLENDENABLE,  true );
Direct3dDev->SetRenderState(D3DRS_ALPHAREF, 0x00000000ul);
Direct3dDev->SetRenderState(D3DRS_ALPHATESTENABLE, true);
Direct3dDev->SetRenderState(D3DRS_ALPHAFUNC,D3DCMP_GREATER);
Direct3dDev->SetRenderState(D3DRS_SRCBLEND,  D3DBLEND_SRCALPHA );
Direct3dDev->SetRenderState(D3DRS_DESTBLEND, D3DBLEND_INVSRCALPHA );

Direct3dDev->SetTextureStageState(0, D3DTSS_COLORARG1, D3DTA_TEXTURE);
Direct3dDev->SetTextureStageState(0, D3DTSS_COLORARG2, D3DTA_DIFFUSE);
Direct3dDev->SetTextureStageState(0, D3DTSS_COLOROP, D3DTOP_MODULATE);
Direct3dDev->SetTextureStageState(0, D3DTSS_ALPHAARG1, D3DTA_TEXTURE);
Direct3dDev->SetTextureStageState(0, D3DTSS_ALPHAARG2, D3DTA_DIFFUSE);
Direct3dDev->SetTextureStageState(0, D3DTSS_ALPHAOP, D3DTOP_MODULATE);

Direct3dDev->SetTextureStageState(0, D3DTSS_RESULTARG, D3DTA_CURRENT);

Direct3dDev->SetTextureStageState(0, D3DTSS_TEXCOORDINDEX, D3DTSS_TCI_PASSTHRU);

//ensure the first stage is not used for now
Direct3dDev->SetTextureStageState(1, D3DTSS_COLOROP, D3DTOP_DISABLE);

How can I do this right?

• Should i add some more details to attract an answer?? – Chaotikmind Aug 17 '14 at 16:53
• Yes, specifically how you set up your render targets, and how you set them to be rendered to. Are you trying to create a deferred renderer and want to write to multiple render targets at the same time MRT (msdn.microsoft.com/en-us/library/windows/desktop/… ) – Roy T. Aug 19 '14 at 18:19
• Ok, i'll add that, and no i don't think we could call what i'm doing deferred rendering, i just want to write to a single target at a time, btw i'm only rendering quads, no real 3d here. – Chaotikmind Aug 19 '14 at 19:32
• What's this: DxEngine.DrawSprite? – Laurent Couvidou Aug 21 '14 at 12:50
• You have the depth tests enabled so it might be that all your pixels are discarded by that. Or maybe a problem with alpha. This is the kind of issue that can be easily solved using PIX / Visual Studio Graphics Debugger. Learn how to use it, you won't regret it. – Laurent Couvidou Aug 21 '14 at 12:59


text:
number of p element subsets whose sum is divisible by p

caffeinemachine

Well-known member
MHB Math Scholar
Let $S=\{ 1, 2, \ldots , 2p\}$, where $p$ is an odd prime. Find the number of $p$-element subsets of $S$ the sum of whose elements is divisible by $p$.

Attempt.

Let $\mathcal{K}$ be the set of all the $p$ element subsets of $S$. Let $\sigma(K)$ denote the sum of the elements of a member $K$ of $\mathcal{K}$. Define a relation $R$ on $\mathcal{K}$ as: $ARB$ if $\sigma(A) \equiv \sigma(B) (\mod p)$, for $A,B \in \mathcal{K}$. Its clear that $R$ is an equivalence relation. Choose $K_0,K_1, \ldots , K_{p-1} \in \mathcal{K}$ such that $\sigma(K_i) \equiv i (\mod p)$. Let $[K_i]$ denote the equivalence class of $K_i$ under $R$.

Claim: $|[K_i]|=|[K_j]|$ whenever $2p>i,j>0$
Proof. Let $x \in \{ 1, \ldots , p-1\}$ be such that $ix \equiv j (\mod p)$ (Such an $x$ exists). If $x$ is odd put $y=x$ else put $y=p+x$. So in any case $y$ is odd and $0<y<2p$.
Now let $K_i = \{ a_1, \ldots , a_p \}$. Consider a set $K^{*}=\{ ya_1, \ldots , ya_p\}$, where each element of $K^{*}$ is reduced mod $2p$ if necessary. One can easily show that $ya_i$'s are all distinct, thus $K^{*} \in \mathcal{K}$. Also $\sigma(K^{*}) \equiv iy \equiv ix \equiv j (\mod p))$, thus $K^{*} \in [K_j]$. SO from each element of $[K_i]$ we can produce one element of $[K_j]$. Also, since $gcd(y,2p)=1$, it follows that two distinct elements of $[K_i]$ don't match to a same element of $[K_j]$. The claim follows.
To arrive at the required answer I just need to show that $|[K_0]|=|[K_1]|+2$. But here I am stuck.

Opalg

MHB Oldtimer
Staff member
Unable to get anywhere with this, I did an internet search and found that this is a problem (an exceptionally difficult one) from the 1995 International Olympiad. There is a solution in a book by Ross Honsberger (Mathematical Chestnuts from Around the World, pp.220-223; if you search for the book, you will find it freely available online in DjVu format). The solution given there consists of a direct proof, using a generating function, that $|[K_0]| = \frac1p\Bigl\{{2p\choose p} + 2(p-1)\Bigr\}.$ It does not make use of your very elegant proof that all the other sets $[K_i]$ have the same size, and it gives no clue as to why $[K_0]$ should contain two extra elements.

caffeinemachine

Well-known member
MHB Math Scholar
Unable to get anywhere with this, I did an internet search and found that this is a problem (an exceptionally difficult one) from the 1995 International Olympiad. There is a solution in a book by Ross Honsberger (Mathematical Chestnuts from Around the World, pp.220-223; if you search for the book, you will find it freely available online in DjVu format). The solution given there consists of a direct proof, using a generating function, that $|[K_0]| = \frac1p\Bigl\{{2p\choose p} + 2(p-1)\Bigr\}.$ It does not make use of your very elegant proof that all the other sets $[K_i]$ have the same size, and it gives no clue as to why $[K_0]$ should contain two extra elements.
Thank You Opalg. Although the proof given in Honsberger uses a totally different approach, I myself tried a generating function approach and failed. So I am happy to see a proof using generating functions.

It does not make use of your very elegant proof that ...
*rubs eyes* Thank you so much for the compliment.


text:
# GATE1995-1.4

1.2k views

In the following Pascal program segment, what is the value of X after the execution of the program segment?

X := -10; Y := 20;
If X > Y then if X < 0 then X := abs(X) else X := 2*X;
1. $10$
2. $-20$
3. $-10$
4. None

edited

Answer of $X$ remains unchanged. As the if condition becomes false.

X := -10

Answer is C . This is classic example of $if-else$ issue. Always $else$ matches for nesting to the closest $if$  in $C$ Programming & Pascal .
https://en.wikipedia.org/wiki/Dangling_else

if (x>y)
{
if (x<0)
x=abs(x)
else
x=2*x
}

edited
0
else matches to closest if in all languages?
3
For sure in C & Pascal ! We don't get any other languages in GATE !
2
But here x < y i.e.  -10 < 20
0
Yes you seem correct. Updated.
2
When having confusion, always try to match else with last unmatched if condition.

This is usually the solution the compiler follows for dangling else problem during compilation of a program.

Reference: Ullman for compilers
0

always try to match else with last unmatched if condition.

means?

0
Means

if(a>b) then if(b>c) then t=x else v=y will be

if(a>b) then

if(b>c) then

t=x

else

v=y
X is remains unchanged. As the if condition is becomes false.
X = -10

Hope this helps:)

## Related questions

1
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text:
×

# Prove this

$$x,y,z$$ are integers such that $$(x-y)^2+(y-z)^2+(z-x)^2=xyz$$.
Prove that $$x^3+y^3+z^3$$ is divisible by $$x+y+z+6$$.

Note by Abhishek Alva
7 months, 2 weeks ago

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$$x^{3}+y^{3}+z^{3}-3xyz=1/2×(x+y+z)((x-y)^{2}+(y-z)^{2}+(z-x)^{2})$$ Using this you can solve in 1 minute · 7 months, 2 weeks ago


text:
# Revision history [back]

### simon_two_descent error

Hi. I'm having some difficulty with simon_two_descent, which is used to give bounds on the rank of elliptic curves. I'm running Sage 5.0 on Ubuntu 11.04 64bit. In Sage 4.7.2 I get a different error trying to do the same thing, for which I can provide details if needed.

My objective is to compute the rank of an EllipticCurve over a specific number field. I am quite new to Sage so I am likely doing something wrong. Can someone please tell me how to fix this problem? Thank you. My code is as follows:

----------------------------------------------------------------------
| Sage Version 5.0, Release Date: 2012-05-14                         |
| Type notebook() for the GUI, and license() for information.        |
----------------------------------------------------------------------
sage: S = CyclotomicField(43).subfields(3)
sage: poly = S[0][0].polynomial()
sage: K = NumberField(poly, 'a')
sage: E = EllipticCurve(K, '37')
sage: E
Elliptic Curve defined by y^2 + y = x^3 + (-1)*x over Number Field in a with defining polynomial x^3 + x^2 - 14*x + 8
sage: G = E.simon_two_descent()
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)

/home/blatm/sage/sage-5.0-linux-64bit-ubuntu_10.04.3_lts-x86_64-Linux/<ipython console> in <module>()

/home/blatm/sage/sage-5.0-linux-64bit-ubuntu_10.04.3_lts-x86_64-Linux/local/lib/python2.7/site-packages/sage/schemes/elliptic_curves/ell_number_field.pyc in simon_two_descent(self, verbose, lim1, lim3, limtriv, maxprob, limbigprime)
281                               verbose=verbose, lim1=lim1, lim3=lim3, limtriv=limtriv,
282                               maxprob=maxprob, limbigprime=limbigprime)
--> 283         prob_rank = Integer(t[0])
284         two_selmer_rank = Integer(t[1])
285         prob_gens = [self(P) for P in t[2]]

/home/blatm/sage/sage-5.0-linux-64bit-ubuntu_10.04.3_lts-x86_64-Linux/local/lib/python2.7/site-packages/sage/rings/integer.so in sage.rings.integer.Integer.__init__ (sage/rings/integer.c:6865)()

TypeError: unable to convert x (=f) to an integer
sage: