text:
for Journals by Title or ISSN for Articles by Keywords help
 Subjects -> MATHEMATICS (Total: 966 journals)     - APPLIED MATHEMATICS (82 journals)    - GEOMETRY AND TOPOLOGY (20 journals)    - MATHEMATICS (711 journals)    - MATHEMATICS (GENERAL) (43 journals)    - NUMERICAL ANALYSIS (22 journals)    - PROBABILITIES AND MATH STATISTICS (88 journals) MATHEMATICS (711 journals)                  1 2 3 4 | Last

1 2 3 4 | Last

 Annali di Matematica Pura ed ApplicataJournal Prestige (SJR): 1.197 Citation Impact (citeScore): 1Number of Followers: 1      Hybrid journal (It can contain Open Access articles) ISSN (Print) 1618-1891 - ISSN (Online) 0373-3114 Published by Springer-Verlag  [2350 journals]
• On a nonlinear model for arctic gyres
• Authors: Jifeng Chu
Pages: 651 - 659
Abstract: We prove some results on the existence and uniqueness of solutions to a recently derived nonlinear model for the ocean flow in arctic gyres. We derive an equivalent formulation of the problem in the form of an integral equation on a semi-infinite interval and then use fixed point techniques. We also obtain some stability result.
PubDate: 2018-06-01
DOI: 10.1007/s10231-017-0696-6
Issue No: Vol. 197, No. 3 (2018)

• Finite groups of units of finite characteristic rings
• Authors: Ilaria Del Corso; Roberto Dvornicich
Pages: 661 - 671
Abstract: Fuchs (Abelian groups, Pergamon, Oxford, 1960, Problem 72) asked the following question: which groups can be the group of units of a commutative ring' In the following years, some partial answers have been given to this question in particular cases. The aim of the present paper is to address Fuchs’ question when A is a finite characteristic ring. The result is a pretty good description of the groups which can occur as group of units in this case, equipped with examples showing that there are obstacles to a “short” complete classification. As a by-product, we are able to classify all possible cardinalities of the group of units of a finite characteristic ring, so to answer Ditor’s question (Ditor in Am Math Mon 78(5):522–523, 1971).
PubDate: 2018-06-01
DOI: 10.1007/s10231-017-0697-5
Issue No: Vol. 197, No. 3 (2018)

• Functional capacities on the Grushin space $${\mathbb {G}}^n_\alpha$$ G
α n
• Authors: Yu Liu; Jie Xiao
Pages: 673 - 702
Abstract: Functional capacities on the Grushin space $${\mathbb {G}}^n_\alpha$$ are introduced, developed, and subsequently applied to the theory of Sobolev embeddings.
PubDate: 2018-06-01
DOI: 10.1007/s10231-017-0699-3
Issue No: Vol. 197, No. 3 (2018)

• Isoparametric hypersurfaces in a Randers sphere of constant flag curvature
• Authors: Ming Xu
Pages: 703 - 720
Abstract: In this paper, I study the isoparametric hypersurfaces in a Randers sphere $$(S^n,F)$$ of constant flag curvature, with the navigation datum (h, W). I prove that an isoparametric hypersurface M for the standard round sphere $$(S^n,h)$$ which is tangent to W remains isoparametric for $$(S^n,F)$$ after the navigation process. This observation provides a special class of isoparametric hypersurfaces in $$(S^n,F)$$ , which can be equivalently described as the regular level sets of isoparametric functions f satisfying $$-f$$ is transnormal. I provide a classification for these special isoparametric hypersurfaces M, together with their ambient metric F on $$S^n$$ , except the case that M is of the OT-FKM type with the multiplicities $$(m_1,m_2)=(8,7)$$ . I also give a complete classification for all homogeneous hypersurfaces in $$(S^n,F)$$ . They all belong to these special isoparametric hypersurfaces. Because of the extra W, the number of distinct principal curvature can only be 1, 2 or 4, i.e. there are less homogeneous hypersurfaces for $$(S^n,F)$$ than those for $$(S^n,h)$$ .
PubDate: 2018-06-01
DOI: 10.1007/s10231-017-0701-0
Issue No: Vol. 197, No. 3 (2018)

• From the Gross–Pitaevskii equation to the Euler Korteweg system,

• Authors: Corentin Audiard; Boris Haspot
Pages: 721 - 760
Abstract: We consider the Euler equations with a capillary tensor in the case of the so-called quantum hydrodynamics, which is formally equivalent to the Gross–Pitaevskii equation. Our main result is the existence and uniqueness of global solutions without vortices for small data in dimension at least 3. The absence of vortices means that the density remains bounded away from 0. Previous results include existence of global solutions without uniqueness (Antonelli and Marcati in Commun Math Phys 287(2):657–686, 2009) and lower bounds on the first occurrence of vortices (Béthuel et al. in J Anal Math 110:297–338, 2010). Our proof uses in a crucial way some deep results on the scattering of the Gross–Pitaevskii equation due to Gustafson et al. (Commun Contemp Math 11(4):657–707, 2009). The optimality of our assumptions is discussed, in particular we show that for slightly less regular initial data the density does not even remain bounded. We also sketch in the appendix the key arguments for the scattering of solutions of the Gross–Pitaevskii equation.
PubDate: 2018-06-01
DOI: 10.1007/s10231-017-0702-z
Issue No: Vol. 197, No. 3 (2018)

• Extension of modular functions and measures
• Authors: Hans Weber
Pages: 761 - 774
Abstract: We prove extension theorems for group-valued modular functions defined on orthomodular lattices or modular complemented lattices and for modular measures defined on (pseudo-)D-lattices generalizing results of Riečan (Mat Časopis Sloven Akad Vied 19:44–49, 1969; 20:239–244, 1970; Comment Math Univ Carolin 20:309–316, 1979), Avallone and De Simone (Ital J Pure Appl Math 9:109–122, 2001) and Avallone et al. (Bollettino UMI 9-B(8):423–444, 2006; Math Slovaca 66:421–438, 2016). As basic tool, we first prove an extension theorem for lattice uniformities. This also yields as immediate consequence a result on the extension of modular functions on arbitrary lattices similar to that of Fox and Morales (Fundam Math 78:99–106, 1973) and Kranz (Fundam Math 91:171–178, 1976).
PubDate: 2018-06-01
DOI: 10.1007/s10231-017-0703-y
Issue No: Vol. 197, No. 3 (2018)

• Quasistatic evolution of perfectly plastic shallow shells: a rigorous
variational derivation
• Authors: G. B. Maggiani; M. G. Mora
Pages: 775 - 815
Abstract: In this paper, we rigorously deduce a quasistatic evolution model for shallow shells by means of $$\Gamma$$ -convergence. The starting point of the analysis is the three-dimensional model of Prandtl–Reuss elasto-plasticity. We study the asymptotic behaviour of the solutions, as the thickness of the shell tends to zero. As in the case of plates, the limiting model is genuinely three-dimensional, limiting displacements are of Kirchhoff–Love type, and the stretching and bending components of the stress are coupled in the flow rule and in the stress constraint. However, in contrast with the case of plates, the equilibrium equations are not decoupled, because of the presence of curvature terms. An equivalent formulation of the limiting problem in rate form is also discussed.
PubDate: 2018-06-01
DOI: 10.1007/s10231-017-0704-x
Issue No: Vol. 197, No. 3 (2018)

• Energy estimates for the Cauchy problem of Klein–Gordon-type equations
with non-effective and very fast oscillating time-dependent potential
• Authors: Fumihiko Hirosawa; Wanderley Nunes do Nascimento
Pages: 817 - 841
Abstract: The aim of this paper is to prove some energy estimates for Klein–Gordon equations with time-dependent potential. If the potential is “non-effective” and has “very slow oscillations” in the time-dependent coefficient, then energy estimates are proved in Ebert et al. (in: Dubatovskaya and Rogosin (eds) AMADE 2012, Cambridge Scientific Publishers, Cambridge, 2014). In contrast, the main goal of the present paper is to generalize the previous results to potentials with “very fast oscillations” in the time-dependent coefficient; consequently, the positivity of the potential is not required anymore.
PubDate: 2018-06-01
DOI: 10.1007/s10231-017-0705-9
Issue No: Vol. 197, No. 3 (2018)

• Unique strong and $${ \mathbb {V} }$$ V -attractor of a three-dimensional
globally modified two-phase flow model
• Authors: T. Tachim Medjo
Pages: 843 - 868
Abstract: In this article, we study a globally modified Allen–Cahn–Navier–Stokes system in a three-dimensional domain. The model consists of the globally modified Navier–Stokes equations proposed in Caraballo et al. (Adv Nonlinear Stud 6(3):411–436, 2006) for the velocity, coupled with an Allen–Cahn model for the order (phase) parameter. We prove the existence and uniqueness of strong solutions. Using the flattening property, we also prove the existence of global $${ \mathbb {V} }$$ -attractors for the model. Using a limiting argument, we derive the existence of bounded entire weak solutions for the three-dimensional coupled Allen–Cahn–Navier–Stokes system with time-independent forcing.
PubDate: 2018-06-01
DOI: 10.1007/s10231-017-0706-8
Issue No: Vol. 197, No. 3 (2018)

• Parabolic solutions for the planar N -centre problem: multiplicity and
scattering
• Authors: Alberto Boscaggin; Walter Dambrosio; Duccio Papini
Pages: 869 - 882
Abstract: For the planar N-centre problem \begin{aligned} \ddot{x} = - \sum _{i=1}^N \frac{m_i (x-c_i)}{\vert x - c_i \vert ^{\alpha +2}}, \qquad x \in \mathbb {R}^2 {\setminus } \{ c_1,\ldots ,c_N \}, \end{aligned} where $$m_i > 0$$ for $$i=1,\ldots ,N$$ and $$\alpha \in [1,2)$$ , we prove the existence of entire parabolic trajectories, having prescribed asymptotic directions for $$t \rightarrow \pm \,\infty$$ and prescribed topological characterization with respect to the set of the centres.
PubDate: 2018-06-01
DOI: 10.1007/s10231-017-0707-7
Issue No: Vol. 197, No. 3 (2018)

• Multiple solutions for a problem with discontinuous nonlinearity
• Authors: Claudianor O. Alves; Jefferson A. dos Santos; Rodrigo C. M. Nemer
Pages: 883 - 903
Abstract: In this work, we use the Lusternik–Schnirelmann category to estimate the number of nontrivial solutions for a problem with discontinuous nonlinearity and subcritical growth.
PubDate: 2018-06-01
DOI: 10.1007/s10231-017-0708-6
Issue No: Vol. 197, No. 3 (2018)

• On square functions with independent increments and Sobolev spaces on the
line
• Authors: Julià Cufí; Artur Nicolau; Andreas Seeger; Joan Verdera
Pages: 905 - 940
Abstract: We prove a characterization of some $$L^p$$ -Sobolev spaces involving the quadratic symmetrization of the Calderón commutator kernel, which is related to a square function with differences of difference quotients. An endpoint weak type estimate is established for functions in homogeneous Hardy–Sobolev spaces $$\dot{H}^1_\alpha$$ . We also use a local version of this square function to characterize pointwise differentiability for functions in the Zygmund class.
PubDate: 2018-06-01
DOI: 10.1007/s10231-017-0709-5
Issue No: Vol. 197, No. 3 (2018)

• Qualitative properties of singular solutions to nonlocal problems
• Authors: Luigi Montoro; Fabio Punzo; Berardino Sciunzi
Pages: 941 - 964
Abstract: We consider positive weak solutions to $$(-\Delta )^s u=f(x,u)$$ in $$\Omega {\setminus } \Gamma$$ under zero Dirichlet boundary condition. The domain $$\Omega$$ is bounded or is the whole space, and the solution has a singularity on the singular set $$\Gamma$$ . Under suitable assumptions on f we prove symmetry and monotonicity properties of the solutions when the singular set $$\Gamma$$ has zero s-capacity.
PubDate: 2018-06-01
DOI: 10.1007/s10231-017-0710-z
Issue No: Vol. 197, No. 3 (2018)

• Hardy spaces for semigroups with Gaussian bounds
• Authors: Jacek Dziubański; Marcin Preisner
Pages: 965 - 987
Abstract: Let $$T_t=e^{-tL}$$ be a semigroup of self-adjoint linear operators acting on $$L^2(X,\mu )$$ , where $$(X,d,\mu )$$ is a space of homogeneous type. We assume that $$T_t$$ has an integral kernel $$T_t(x,y)$$ which satisfies the upper and lower Gaussian bounds: \begin{aligned} \frac{C_1}{\mu (B(x,\sqrt{t}))} \exp \left( {-\,c_1d(x,y)^2/ t} \right) \le T_t(x,y)\le \frac{C_2}{\mu (B(x,\sqrt{t}))} \exp \left( {-\,c_2 d(x,y)^2/ t} \right) . \end{aligned} By definition, f belongs to $$H^1(L)$$ if $$\Vert f\Vert _{{H^1(L)}}=\Vert \sup _{t>0} T_tf(x) \Vert _{L^1(X,\mu )} <\infty$$ . We prove that there is a function $$\omega (x)$$ , $$0<c\le \omega (x)\le C$$ , such that $$H^1(L)$$ admits an atomic decomposition with atoms satisfying: $${\mathrm{supp}}\, a\subset B$$ , $$\Vert a\Vert _{L^\infty } \le \mu (B)^{-1}$$ , and the weighted cancelation condition $$\int a(x)\omega (x)\hbox {d}\mu (x)=0$$ .
PubDate: 2018-06-01
DOI: 10.1007/s10231-017-0711-y
Issue No: Vol. 197, No. 3 (2018)

• Erratum to: A $$\varvec{\mathbb {Q}}$$ Q -factorial complete toric variety
is a quotient of a poly weighted space
• Authors: Michele Rossi; Lea Terracini
Pages: 989 - 998
Abstract: After the publication of [2], we realized that Proposition 3.1, in that paper, contains an error, whose consequences are rather pervasive along the whole section 3 and for some aspects of examples 5.1 and 5.2. Here we give a complete account of needed corrections.
PubDate: 2018-06-01
DOI: 10.1007/s10231-017-0698-4
Issue No: Vol. 197, No. 3 (2018)

• Uniform boundary estimates in homogenization of higher-order elliptic
systems
• Authors: Weisheng Niu; Yao Xu
Abstract: This paper focuses on uniform boundary estimates in homogenization of a family of higher-order elliptic operators $$\mathcal {L}_\varepsilon$$ , with rapidly oscillating periodic coefficients. We derive uniform boundary $$C^{m-1,\lambda } (0\!<\!\lambda \!<\!1)$$ and $$W^{m,p}$$ estimates in $$C^1$$ domains, as well as uniform boundary $$C^{m-1,1}$$ estimate in $$C^{1,\theta } (0\!<\!\theta \!<\!1)$$ domains without the symmetry assumption on the operator. The proof, motivated by the works “Armstrong and Smart in Ann Sci Éc Norm Supér (4) 49(2):423–481 (2016) and Shen in Anal PDE 8(7):1565–1601 (2015),” is based on a suboptimal convergence rate in $$H^{m-1}(\Omega )$$ . Compared to “Kenig et al. in Arch Ration Mech Anal 203(3):1009–1036 (2012) and Shen (2015),” the convergence rate obtained here does not require the symmetry assumption on the operator, nor additional assumptions on the regularity of $$u_0$$ (the solution to the homogenized problem), and thus might be of some independent interests even for second-order elliptic systems.
PubDate: 2018-06-02
DOI: 10.1007/s10231-018-0764-6

• Centralizers on a super-reflexive Schatten ideal
• Authors: Jesús Suárez de la Fuente
Abstract: We give a simple proof that there is no strictly singular bicentralizer on a super-reflexive Schatten ideal. This result applies, in particular, to the p-Schatten class for $$1<p<\infty$$ .
PubDate: 2018-05-29
DOI: 10.1007/s10231-018-0760-x

• Peak functions in $$\mathbb {C}$$ C -convex domains
• Authors: Peter Pflug; Włodzimierz Zwonek
Abstract: In this paper, we show the existence of different types of peak functions in classes of $$\mathbb {C}$$ -convex domains. As one of the tools used in this context is a result on preserving the regularity of $$\mathbb {C}$$ -convex domains under projection.
PubDate: 2018-05-29
DOI: 10.1007/s10231-018-0759-3

• Local behavior of solutions to fractional Hardy–Hénon equations
with isolated singularity
• Authors: Yimei Li; Jiguang Bao
Abstract: In this paper, we study the local behaviors of positive solutions of \begin{aligned} (-\Delta )^\sigma u= x ^{\tau }u^p \end{aligned} with an isolated singularity at the origin, where $$(-\Delta )^\sigma$$ is the fractional Laplacian, $$0<\sigma <1$$ , $$\tau >-2\sigma$$ and $$p>1$$ . Our first results provide a blowup rate estimate near an isolated singularity, and show that the solution is asymptotically radially symmetric.
PubDate: 2018-05-28
DOI: 10.1007/s10231-018-0761-9

• Transverse Kähler structures on central foliations of complex
manifolds
• Authors: Hiroaki Ishida; Hisashi Kasuya
Abstract: For a compact complex manifold, we introduce holomorphic foliations associated with certain abelian subgroups of the automorphism group. If there exists a transverse Kähler structure on such a foliation, then we obtain a nice differential graded algebra which is quasi-isomorphic to the de Rham complex and a nice differential bi-graded algebra which is quasi-isomorphic to the Dolbeault complex like the formality of compact Kähler manifolds. Moreover, under certain additional condition, we can develop Morgan’s theory of mixed Hodge structures as similar to the study on smooth algebraic varieties.
PubDate: 2018-05-26
DOI: 10.1007/s10231-018-0762-8

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# Definition:Orbit (Physics)

The orbit of body $A$ around body $B$ is the path taken by $A$ as it travels around $B$ under the influence of the force acting between $A$ and $B$.
The period of an orbit of body $A$ around body $B$ is the length of time it takes for $A$ to travel once around $B$ and return to its original position.

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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 19th 2016
• (edited Feb 19th 2016)

I notice that there seemed to be no-where a really explicit semantics explanation of the claim that “univalence implies the existence of quotient types”. I checked with Peter Lumsdaine, and he kindly provided some text which, up to some minor reformatting, I have put into quotient type – Properties – from univalence.

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeMar 29th 2016

What does ’u.m.p.’ mean in

(For the subobject classification u.m.p…. ?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMar 29th 2016

I forget, but the lead-in of the sentence just means to say “To see how the axioms for a subobject classifier are satisfied: ….”. I have edited accordingly.

• CommentRowNumber4.
• CommentAuthorZhen Lin
• CommentTimeMar 29th 2016

“Universal mapping property”, I believe.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMar 29th 2016
• (edited Mar 29th 2016)

That must be it.

If anyone feels energetic, it would be great if there were more on the nLab on the classical construction of quotients from subobject classifiers. Such, or pointer to such, is presently missing at elementary topos and quotient object etc.

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeMar 29th 2016

I’ve been meaning to complete and release some notes on the “down-to-earth” way to construct colimits in a topos, including quotients. At some point I can add a description to quotient object. Of course there is also the argument due to Paré.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMar 29th 2016
• (edited Mar 29th 2016)

That would be great if you could add something.

What’s the precise citation to Paré?

• CommentRowNumber8.
• CommentAuthorTodd_Trimble
• CommentTimeMar 29th 2016

I mean the proof that $P: E^{op} \to E$ is monadic, which is the standard approach (see, e.g., Mac Lane-Moerdijk); I believe this result is usually credited to Paré.

• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeMar 29th 2016

Okay, I’ve created a first installment at quotient object, indicating a Galois connection between quotient objects and binary relations.

I may be adding more later.

• CommentRowNumber10.
• CommentAuthorThomas Holder
• CommentTimeMar 29th 2016
• (edited Mar 29th 2016)

The reference for Paré is:

Robert Paré, Colimits in Topoi , Bull. AMS 80 (1974) pp.556-661. (pdf)

This has the by now standard proof but the result itself is attributed to Mikkelsen, a student of Anders Kock, in Johnstone’s 1977 book.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeMar 30th 2016

Thomas, thanks a lot for the reference! That’s what I was looking for. I have added a pointer at quotient type.

Todd, thanks for your additions to quotient object. What I think would be particularly useful to eventually add to the nLab is discussion of how quotients may be constructed using a subobject classifier. If you would find time to get to that, that would be a big service, I think.

• CommentRowNumber12.
• CommentAuthorTodd_Trimble
• CommentTimeMar 30th 2016

Urs, as I said, this was a first installment. A full write-up will take some time. I’ve been thinking though that it should probably have its own article (i.e., it won’t all fit comfortably at quotient object or at elementary object), although maybe I can sketch just the construction at quotient object.

• CommentRowNumber13.
• CommentAuthorTodd_Trimble
• CommentTimeMar 30th 2016

Okay, you can now have another look at quotient object. I have added a subsection “In toposes”.

Obviously this is just in outline form for now.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeMar 30th 2016
• (edited Mar 30th 2016)

Thanks, Todd!!

I have added some more hyperlinks and then I have cross-linked these two sections:

• CommentRowNumber15.
• CommentAuthorRodMcGuire
• CommentTimeMar 31st 2016

Off topic but related. I came across generalized kernel while looking for information about quotients in enriched categories.

It contains the unknown iTeX ulcorner f urcorner which has been present since Mike Shulman started that page in 2012.

Is there some other notational convention, say a hat, that could be used there?

• CommentRowNumber16.
• CommentAuthorMike Shulman
• CommentTimeAug 2nd 2018

Reference Hofmann’s thesis for “higher inductive” quotients (predating general HITs), and mention Maietti’s result that “effectiveness” for quotients of “Type-valued equivalence relations” implies LEM.

1. Fixed broken links to home page Ahrens

Anonymous

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# Risk free Yield Curve Building in Excel using Negative Overnight Index Swap (OIS) Rates

Long are the dates when all-purpose risk free yield curves were produced out of deposits, futures and swaps. Since the credit crisis of 2007/08, banks realized they should use different curves for different purposes. In the intervening years, quants have proven mathematically that uncollateralized contingent flows should be discounted using a curve that incorporates the default risk associated with these flows, whereas collateralized flows should be discounted using a curve that reflects the risk of the collateral account. Also the bank's own default risk, as reflected in its cost of funding, should be taken into account.

Without entering into details in the present article, the current derivatives pricing practice by most major banks today is to discount expected values of collateralized future flows using the curve implied by the overnight index related to the denomination currency of the posted collateral. The future flows themselves are being projected using a different curve, the so called "forecasting curve", which is obtained by the current prices of market traded instruments, the flows of which are related to the flows of the referred derivative.

It turns out, the proper generation of the discounting yield curve is of paramount importance since it is involved in the pricing and risk management of diverse types of financial instruments.

With the current article I want to show you how to create a yield curve in Excel using the open source QuantLib analytics library, when the input market data are OIS (Overnight Index Swap) rates. As an example I will build a curve out of negative EONIA (European Overnight Index Average) swap rates, despite the fact that such swaps will likely seize to exist after January 1st 2020 due to new European regulation that intends to replace the EONIA index with the new ESTER index. The example will still remain valid, since one may replace the EONIA index with any other overnight index.

Please note that currently QuantLib provides direct support for the five overnight indices shown below along with their respective currencies:

EONIA for EUR

FedFunds and SOFR for USD

SONIA for GBP

AONIA for AUD

Nzocr for NZD

Nevertheless one may still build a custom overnight index with user-specified conventions.

### The Bootstrapping Process

We assume we are given the current market values (prices) Vi of an increasing-maturity series of Overnight Index Swaps Si, i = 1, …, n.

For each i, Vi is just a known number, whereas Si is a random variable that represents the stream of cash flows received under the respective swap. They are related through the equation:

Vi = PV(Si)

where the function PV takes as input a random variable that represents a stream of future cash flows and returns a single number being interpreted as the arbitrage-free Present Value of that cash flow stream.

It is a mathematical fact that the PV(.) function can be written in terms of the expectation (probability average) function E(.), where the dot . is a placeholder of the random variable being expected as input to the corresponding function.

I remind you here that the expectation E(X) of any random variable X, the realization of which may assume a value out of a discrete set of numbers {x1, x2, …, xk} with respective probabilities {p1, p2, …, pk}, is defined as:

E(X) = p1x1+ p2x2+…+pkxk

If X assumes infinite possible values taken out of a continuous interval [a,b], with a probability density p(.), then its expectation E(X) is defined through an integral as:
$E(X)=\int_{a}^{b}p(x)dx$

The exact equation for the PV function on the random variable X is:

$PV\left(X\right)=N_{0}E\left(\frac{X}{N_{T}}\right)$

where T is the latest time occurring in X and Nt is an arbitrarily chosen numeraire stochastic process, i.e. for each time t, Nt is a random variable that represents the price of some arbitrarily chosen tradable asset as of time t. Omitting the intermediate steps, the conclusion is that the price V0 of the first swap S0 depends only on the discount factor as of that swap's maturity. This stems from the fact that the expectation E[Fi] of the compounded rate Fi on which the present value of the ith floating payment depends is given by the – almost – geometric average of the daily forward rates.

As explained in my article about Overnight Index Swaps, E[Fi] is given by:

$E\left[F_{i}\right]=\frac{\prod_{j=1}^{n}\left (1\left[I_{j}\right]\tau_{j}\right)-1}{{T}'_{i}-{T}'_{i-1}}$

where j enumerates the daily time intervals within the ith accrual period and the forecasted daily index E[I j] is given by:

$\frac{\frac{P(T_{j-1})}{P(T_{j})}-1}{\tau_{j}}$

So, cancellations occur and the product takes the form:

$E\left[F_{i}\right]=\frac{\prod_{j=1}^{n}\left&space;(1\left[I_{j}\right]\tau_{j}\right)-1}{{T}'_{i}-{T}'_{i-1}}=\frac{\frac{P(t_{0})}{P(t_{n})}-1}{{T}'_{i}-{T}'_{i-1}}$

The unknown discount factors P(tn) needs to be calculated out of one single equation involving the known market price of the first swap.

One then proceeds iteratively solving for the unknown discount factors involved in the remaining swaps, in a process conventionally referred as bootstrapping.

### Creating the Yield Curve

I will use the wizard to generate the correct formula that creates an object of type Yield Curve out of OIS market rates. After I select the Yield Curve type in Type Selector I must check the Use OIS flag inside the input parameters screen and finally hit the Go button, as this video demonstrates:

This is the result:

### Understanding the formula

As you see, cell A1 contains the formula =ds(A2:B7;A9;A10:B12), which takes three input arguments and returns the text &YldCrv_A1:1.1

The first input argument contains the overnight index and the currency. They are shown as %Eonia and %EUR respectively because the wizard is aware of my location in Germany and accordingly used my local settings for creating the default index and currency. Had I been located in the US, these would have been set as %FedFunds and %USD respectively.

The last argument A10:B12 contains a table consisting of two columns.

The first column bears the title #Maturity and contains the lengths of the supplied swaps, expressed as time intervals.

The second column bears the title #Rate and contains the respective swap rates in fractional units, so that 0.01 means a rate of 1%.

The wizard sets the swap rates to 1% because it has no knowledge of actual market data. It is my responsibility to enter rates that are actually traded in the market.

On the global-rate.com web site I can see the following recent historical EONIA fixings:

Due to the unfortunate fact that I have no access to current prices of actually traded Eonia OIS, for the sake of demonstration I will overwrite the cells B11 and B12 with the values -0.349% and -0.330%, which are at least conformant to the actual historical fixings.

I decide also to insert two more rows with maturities of 18 months and 2 years with respective slightly increasing rates so that I can generate a curve up to 2 years.

After these manual edits, my spreadsheet looks like in the image below. Due to cell A1 being selected, the wizard displays the contents of the referred object, i.e. the contents of the created OIS.

You may notice the three entries:

OIS Settlement Days= 2

OIS Payment Lag= %2D{TARGET|F}

OIS Payment Frequency= Annual.

These are the default conventions affecting the floating legs of the swaps that are usually associated with the chosen EONIA index. I could overwrite these settings by supplying the respective key-value pairs as part of the spreadsheet range that constitutes the first input argument to my ds() spreadsheet formula.

### Creating a Yield Curve out of Forward Starting OIS that carry a Spread

QuantLib supports a yield curve construction out of OIS that start in the future and the floating leg of which pays the compounded index plus a spread. The following video shows how I may instruct the wizard to generate the appropriate spreadsheet formula, which this time references a table consisting of four rather than two columns:

This is the result:

### Using the Yield Curve Object and Constructing a Zero Rate Curve

With regard to how a Yield Curve object may be used in general, I would refer you to the respective section of the Yield Curve out of Deposit Rates article.

As an example here, I may use the function Zero Rate made available by all objects of type Yield Curve to set up a table with increasing maturities and associated calculated zero rates. As usually, the easiest way of creating the zero rate spreadsheet formula is by first selecting the cell containing the Yield Curve object and then choosing through the wizard the Zero Rate function.

The video below demonstrates how I can create such a table with monthly (30 day) maturity increments and also construct the associated zero rate chart in only 90 seconds.

Watching the video, you will notice that the default input of the Zero Rate function within the wizard consists of one maturity date, but I decide to add one more date. The only reason I do that is to inform the wizard that my intention is to supply more than one input maturities. Then the wizard generates the spreadsheet formula with a syntax designed to accept an array of input maturities rather than a single maturity. The resulting formula can be then manually edited by me much easier than if it were handling a single maturity input.

The resulting zero rate table and the associated chart are shown below:


text:
# Help with a proof

1. Mar 6, 2005

### stunner5000pt

Let f = 0 if x=1/n for all n belonging to positive integers
and 1 otherwise

i need to prove that the integral of f over 0 to 1 exists and that the integral is 1.

first of all i cant understand the function. (great start) If x = 1, then x is in the form 1/n then f(1) = 0.

If i picked a partition [a,1] where a<1 then the upper sum would be U(f,P) = 0 (1-a) = 0 (right?)

and lower sum would vary because A could be either of the form (1/n) of not of that form

if a was of the form 1/n then L(f,P) = 0(1-a) = 0
if a was not of the form 1,n then L(f,P) = 1(1-a) = 1-a

stuck right here what should ido
i'd realy \like to know if i am in right in terms of the fucntions value

any help would be greatly appreciated

2. Mar 6, 2005

### Hurkyl

Staff Emeritus
Just to be clear, you're saying the function is defined on the interval [0, 1] by:

$$f(x) := \left\{ \begin{array}{ll} 0 \quad & x = 1/n \, \mathrf{for some} \, n \in \mathbb{N} \\ 1 \quad & \mathrf{otherwise} \end{array}$$

Since you're having trouble understanding the function, let's start with that first.

What is f(1/2)? f(1/6)? f(3/4)? $f(\pi/4)$? f(√2 / √8)?

Last edited: Mar 6, 2005
3. Mar 6, 2005

### stunner5000pt

all values of x that are fractions of the form 1/2 are zero so f(1/2) = f(1/6) = 0 because no matter what the fraction you can lawys write it with a natural in the denom right? however the function at the square roots would be 1 becasue they are irrational

as for 3/4 and n/4 they are also 1 because 1/0.75 is not a natural and 1/(n/4) = 4/n only if n is a multiple of 4

Last edited: Mar 6, 2005
4. Mar 6, 2005

### Hurkyl

Staff Emeritus
This is almost correct. Note that f(n/4) = 0 if n = 1, or 1/2, or ...

Now, I actually wrote $f(\pi/4)$, but the default font renders pi as π, so it was confusing... but I'm happy about the mistake because seeing you write the above made it more clear that you're understanding how it works.

Now, it's actually the case that f(√2 / √8) = 1! Can you figure out why?

Last edited: Mar 6, 2005
5. Mar 6, 2005

### stunner5000pt

becasue the division of two irrationals gives an irrational number?
i would think you suppose that two rationals divided would give a rational so then two irarationals divided should give an irrational.

but isnt root2/ root8 = root 2 / 2 root 2 = 1/2??

6. Mar 6, 2005

### AKG

You can easily show the set of discontinuities to be countable, as they can very obviously be indexed by the naturals. Since it is a countable set of points, it has measure 0, so f is integrable. Alternatively, the upper sum is:

$$U(f, P) = \sum _{S \in P} M_S(f)v(S)$$

Where v is the volume function, S is a sub-rectangle (in this case, sub-interval) of P, and $M_S(f) = \sup \{f(x) : x \in S \}$, i.e. the maximum value that f take on in the interval S. clearly, $M_S(f) = 1$ for all S, so:

$$U(f, P) = \sum _{S \in P}v(S) = v([0, 1]) = 1$$

Therefore, the upper integral, UI(f) = 1, since:

$$UI(f) = \inf _P \{U(f, P)\} = \inf _P \{1\} = 1$$

Now, if you are allowed to use the argument I gave initially to prove that f is integrable, then the above is sufficient to show that the integral is 1. On the other hand, if you have to show integrability by showing that the lower integral equals the upper integral, then it remains to show that the lower integral is 1:

$$L(f, P) = \sum _{S \in P} m_S(f)v(S)$$

Suppose the lower integral were not 1, suppose it is 1 - e. Choose a partition P so that it starts with the sub-interval $S_1 = [0, e/2]$. Now, on the interval [e/2, 1], there are finitely many points of the form 1/n. So, partition [e/2, 1] by placing small intervals around the points of the form 1/n. For such a partition:

$$L(f, P) = \sum _{S \in P} m_S(f)v(S) = 0(e/2) + \sum _{S \in (P - \{S_1\})} m_S(f)v(S)$$

Since we can choose the rectangles surrounding the finitely many points of the form 1/n in [e/2, 1] to be arbitrarily small, i.e. we can take "better" and "better" partitions with smaller and smaller intervals surrounding these points, the rectangles on which $m_S(f) = 0$ become smaller and smaller, and since there are finitely many of them, we know that if we take the supremum of L(f, P) over all partitions P of this form, we must get a value of 1 - e/2. This contradicts the assumption that the supremum of L(f, P) over all partitions P (of any form) is 1 - e (note the the supremum over these L(f, P) is exactly, by definition, the lower integral), so the supremum must be 1, as required.

7. Mar 6, 2005

### Hurkyl

Staff Emeritus
Er, bleh, typo. I was trying to contradict when you said " however the function at the square roots would be 1 becasue they are irrational" and I said 1 myself! Yes, f(√2 / √8) = 0.

Anyways, I think you understand the function now. Do you know where it is discontinuous?

8. Mar 6, 2005

### Hurkyl

Staff Emeritus
PS. don't do the entire problem for him. :tongue2: One learns more by getting hints and filling in the missing steps than reading how someone else solved the problem.

9. Mar 6, 2005

### stunner5000pt

what is a volume function?? (note i ma in a first year intro to real analysis course)

p.s. i probably know this as the width of the interval

so the function is 1 whenever x is not of the form a/n and n is not an integer multiple of a. i understand the function now
as for hte proof now

what i posted in post #1 is erroneous in some way??

p.s. coirrected

yes the upper sum is 1. So i cannot compute the lower sum because that would give a zer owouldnt it ?/

and bascially your proof states that the function is 1 for the most part but the discontuities are finite because the set of 1/n is finite so they are removable .

Last edited: Mar 6, 2005
10. Mar 6, 2005

### AKG

The volume function in 1 dimension is just the length of the interval. Good point Hurkyl, my bad.

11. Mar 6, 2005

### Hurkyl

Staff Emeritus

Right, so you see that already...

Also, [a, 1] is not a partition. A partition of the interval [0, 1] requires a collection of subintervals that cover the whole thing. For instance, {[0, 1/3], [1/3, π/4], [π/4, 1]} is a partition, though you would usually write it as the sequence (0, 1/3, π/4, 1).

12. Mar 6, 2005

### stunner5000pt

i dont undersatnd the step
$$L(f, P) = \sum _{S \in P} m_S(f)v(S) = 0(e/2) + \sum _{S \in (P - \{S_1\})} m_S(f)v(S)$$

where does 0(e/2 ) comes from??

13. Mar 6, 2005

### AKG

Well, the sum is over all subintervals S in P. Any partition has finitely many sub-intervals, so we can write P = {S1, S2, ..., Sm} (so it has m sub-rectangles). The sum over all S in P is the same as the "sum" over just S1 plus the sum over all the remaining sub-rectangles S2, S3, ..., Sm. The sum over just S1 is 0 x e/2. 0 is the minimum value of f on that interval, and e/2 is the "volume" (or length) of that interval.


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# What is a Non-Interactive Zero Knowledge Proof?

I understand the concept of a Zero Knowledge Proof thanks to the easy to understand analogy of Alibaba's cave. However, this seems to require interaction between the verifier and the other party.

I have not found an explanation of non-interactive zero knowledge proofs (NIZK). The wikipedia article is way too complex for someone without advanced training to understand.

https://en.wikipedia.org/wiki/Non-interactive_zero-knowledge_proof

Can someone explain the concept of non-interactive zero-knowledge in a simple way?

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## migrated from security.stackexchange.comFeb 7 '14 at 8:45

This question came from our site for information security professionals.

Isn't this more of an algorithms question? How is this really related to security? – Daisetsu Feb 7 '14 at 1:50

A non-interactive ZK proof is when you play with yourself. Or, more accurately, with an impartial version of yourself.

In a normal ZK proof, the prover first issues a bunch of commitments, then the verifier issues challenges that the prover complies with; this proves anything only as long as the verifier is assumed to issue challenges normally without any prior understanding with the prover.

In a non-interactive ZK proof, the verifier is replaced by a hash function (or something similar) which is computed over the whole set of commitments: the hash function result is the challenge. If the hash function is really a random oracle then the prover cannot guess its output before trying it, i.e. before having produced his commitments, and that's where the security comes from.

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I wonder how many crypto schemes are involved when playing with yourself. – Steve Feb 6 '14 at 17:07
"Zero-Knowledge" has a precise definition. This MAC protocol is not ZK (though it uses non-interactive ZK-proofs internally). – Thomas Pornin Feb 6 '14 at 18:01
@Steve: Schnorr Signature uses a non-interactive ZK proof. – David 天宇 Wong Dec 23 '14 at 13:04
How would the random oracle concept meet with the definition of zero-knowledge stating that there should exist a simulator who can reproduce the proof in probabilistic polinomial time? How can a simulator find the correct hashes if he does not know the original information? – Onheiron Feb 22 at 19:55
From now on, when I play with myself I will say "I'm making a non-interactive ZK proof" – cygnusv Jul 1 at 9:56

In very simplified terms, a NI-ZK proof works in 2 stages: First, you run the protocol with a simulator (who is just a verifier, but the random choices are done differently), and then you can give the transcript of the protocol to anyone and convince them that the proof is real.

The most important ways to achieve this are:

• In the random oracle model (assuming that you have access to ideal hash functions), you can use the Fiat Shamir heuristic: You replace the verifier with a simulator, and whenever the verifier would have to choose randomly, you use the hash function over the entire protocol so far (most importantly, the commitments of the prover) to emulate a random unpredictable choice. Since the hash function is deterministic, you can't change the value without changing the previous interactions. When you're done with the protocol, you can just take the transcript and give it to someone. However, there is a tradeoff in the security level: If the prover is able to cheat with a certain probability, then he could just try different commitments if he doesn't like the result of the hash function. So in order to get e.g. an error of less than $1/2^{80}$, you would need an error probability of less than $1/2^{160}$ with the hash function replacing true random choices. (And this is why it's called a heuristic, there is no proof for this)
• With a common reference string you can achieve NI-ZK. The common reference string is a random string of symbols, which is drawn from some probability distribution. The assumption is, that these random values are available to all parties, but no party has any influence on their actual choice. And if you simulate the ZK proof with the random choices according to the common reference string, you should get the same result as anyone else validating the protocol transcript. I believe you don't have to double the security parameter in this case (like for Fiat Shamir), but on the other hand a common reference string is harder to realize than a hash function.
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Intuitively you want to think of NIZK as more of "static" proofs, where once you(the verifier) have a proof transcript (handed to you by the prover), you should be able to use that transcript to convince yourself about the (in)validity of the proof statement, Without needing any further assistance or queries to the verify.

In interactive proof systems, there are more rounds involving more queries and answers between the prover and verifier. NIZK proofs are almost always 1 round.

Now, on a high level overview one might actually wonder what is the use of this, so as it turns out, many protocols (like Undeniable signatures) that use ZKIP to prove the (in)validity of the signatures, are vulnerable to man-in-the-middle attacks, if they use an interactive version, the NIZK is particularly interesting there.

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text:
Past Mathematical Geoscience Seminar

20 June 2014
14:15
Tarje Nissen-Meyer
Abstract
Seismology currently undergoes rapid and exciting advances fueled by a simultaneous surge in recorded data (in both quality and quantity), realistic wave-propagation algorithms, and supercomputing capabilities. This enables us to sample parameter spaces of relevance for imaging the Earth's interior 3D structure with fully numerical techniques. Seismic imaging is the prime approach to illuminate and understand global processes such as mantle convection, plate tectonics, geodynamo, the vigorous interior of the Sun, and delivers crucial constraints on our grasp of volcanism, the carbon cycle and seismicity. At local scales, seismic Earth models are inevitable for hydrocarbon exploration, monitoring of flow processes, and natural hazard assessment. \\ \\ With a slight focus on global-scale applications, I will present the underlying physical model of realistic wave propagation, its numerical discretization and link such forward modeling to updating Earth models by means of inverse modeling. The associated computational burden to solve high-resolution statistical inverse problems with precise numerical techniques is however entirely out of reach for decades to come. Consequently, seismologists need to take approximations in resolution, physics, data and/or inverse methodology. I will scan a number of such end-member approximations, and focus on our own approach to simultaneously treat wave physics realistically across the frequency band while maximizing data usage and allow for uncertainty quantification. This approach is motivated by decisive approximations on the model space for typical Earth structures and linearized inverse theory.
• Mathematical Geoscience Seminar
6 June 2014
14:15
Teresa Kyrke-Smith
Abstract
Ice sheets are among the key controls on global climate and sea-level change. A detailed understanding of ice sheet dynamics is crucial so to make accurate predictions of their mass balance into the future. Ice streams are the dominant negative component in this balance, accounting for up to 90$\%$ of the Antarctic ice flux into ice shelves and ultimately into the sea. Despite their importance, our understanding of ice-stream dynamics is far from complete. A range of observations associate ice streams with meltwater. Meltwater lubricates the ice at its bed, allowing it to slide with less internal deformation. It is believed that ice streams may appear due to a localization feedback between ice flow, basal melting and water pressure in the underlying sediments. I will present a model of subglacial water flow below ice sheets, and particularly below ice streams. This hydrologic model is coupled to a model for ice flow. I show that under some conditions this coupled system gives rise to ice streams by instability of the internal dynamics.
• Mathematical Geoscience Seminar
16 May 2014
14:15
Roger Cropp
Abstract
The classical separate treatments of competition and predation, and an inability to provide a sensible theoretical basis for mutualism, attests to the inability of traditional models to provide a synthesising framework to study trophic interactions, a fundamental component of ecology. Recent approaches to food web modelling have focused on consumer-resource interactions. We develop this approach to explicitly represent finite resources for each population and construct a rigorous unifying theoretical framework with Lotka-Volterra Conservative Normal (LVCN) systems. We show that mixotrophy, a ubiquitous trophic interaction in marine plankton, provides the key to developing a synthesis of the various ways of making a living. The LVCN framework also facilitates an explicit redefinition of facultative mutualism, illuminating the over-simplification of the traditional definition. We demonstrate a continuum between trophic interactions and show that populations can continuously and smoothly evolve through most population interactions without losing stable coexistence. This provides a theoretical basis consistent with the evolution of trophic interactions from autotrophy through mixotrophy/mutualism to heterotrophy.
• Mathematical Geoscience Seminar
9 May 2014
14:15
Andrew Nicholas
Abstract
Numerical models provide valuable tools for integrating understanding of riverine processes and morphology. Moreover, they have considerable potential for use in investigating river responses to environmental change and catchment management, and for aiding the interpretation of alluvial deposits and landforms. For this potential to be realised fully, such models must be capable of representing diverse river styles, and the spatial and temporal transitions between styles that can be driven by environmental forcing. However, while numerical modelling of rivers has advanced significantly over the past few decades, this has been accomplished largely by developing separate approaches to modelling different styles of river (e.g., meanders and braided networks). In addition, there has been considerable debate about what should constitute the ‘basic ingredients’ of river models, and the degree to which the environmental processes governing river evolution can be simplified in such models. This seminar aims to examine these unresolved issues, with particular reference to the simulation of large rivers and their floodplains.
• Mathematical Geoscience Seminar
14 March 2014
14:15
Heather Viles
Abstract
Microbial biofilms grow on most rock and stone surfaces and may play critical roles in weathering. With climate change and improving air quality in many cities in Europe biofilms are growing rapidly on many historic stone buildings and posing practical problems for heritage conservation. With many new field and lab techniques available it is now possible to identify the microbes present and start to clarify their roles. We now need help modelling microbial biofilm growth and impacts in order to provide better advice for conservators.
• Mathematical Geoscience Seminar
28 February 2014
14:15
Christine Lane
Abstract
• Mathematical Geoscience Seminar
14 February 2014
14:15
Abstract
Hazardous geophysical mass flows, such as snow avalanches, debris-flows and pyroclastic flows, often spontaneously develop large particle rich levees that channelize the flow and enhance their run-out. Measurements of the surface velocity near an advancing flow front have been made at the United States Geological Survey (USGS) debris-flow flume, where 10m^3 of water saturated sand and gravel are allowed to flow down an 80m chute onto a run-out pad. In the run-out phase the flow front is approximately invariant in shape and advances at almost constant speed. By tracking the motion of surface tracers and using a simple kinematic model, it was possible to infer bulk motion as incoming material is sheared towards the front, over-run and shouldered to the side. At the heart of the levee formation process is a subtle segregation-mobility feedback effect. Simple models for particle segregation and the depth-averaged motion of granular avalanches are described and one of the first attempts is made to couple these two types of models together. This process proves to be non-trivial, yielding considerable complexity as well as pathologies that require additional physics to be included.
• Mathematical Geoscience Seminar
31 January 2014
14:15
Mauro Werder
Abstract
Jakobshavn Isbrae and many other fast flowing outlet glaciers of present and past ice sheets lie in deep troughs which often have several overdeepened sections. To make their fast flow possible their bed needs to be slippery which in turn means high basal water pressures. I will present a model of subglacial water flow and its application to Jakobshavn. I find that, somewhat surprisingly, the reason for Jakobshavn's fast flow might be the pressure dependence of the melting point of ice. The model itself describes the unusual fluid dynamics occurring underneath the ice; it has an interesting mathematical structure that presents computational challenges.
• Mathematical Geoscience Seminar
6 December 2013
14:15
Daniel Goldberg
Abstract
Stick-slip behavior is a distinguishing characteristic of the flow of Whillans Ice Stream. Distinct from stick-slip on northern hemisphere glaciers, which is generally attributed to supraglacial melt, the behavior is thought be be controlled by fast processes at the bed and by tidally-induced stress. Modelling approaches to studying this phenomenon typically consider ice to be an elastically-deforming solid (e.g. Winberry et al, 2008; Sergienko et al, 2009). However, there remains a question of whether irreversible, i.e. viscous, deformation is important to the stick-slip process; and furthermore whether the details of stick-slip oscillations are important to ice stream evolution on longer time scales (years to decades). To address this question I use two viscoelastic models of varying complexity. The first is a modification to the simple block-and-slider models traditionally used to examine earthquake processes on a very simplistic fashion. Results show that the role of viscosity in stick-slip depends on the dominant stress balance. These results are then considered in the context of a continuum description of a viscoelastic ice stream with a rate-weakening base capable of exhibiting stick-slip behavior. With the continuum model we examine the spatial and temporal aspects of stick-slip, their dependence on viscous effects, and how this behavior impacts the mean flow. Different models for the evolution of basal shear stress are examined in the experiments, with qualitatively similar results. A surprising outcome is that tidal effects, while greatly affecting the spectrum of the stick-slip cycle, may have relatively little effect on the mean flow.
• Mathematical Geoscience Seminar


text:
# Position of a Point with Respect to the Hyperbola

We will learn how to find the position of a point with respect to the hyperbola.

The point P (x$$_{1}$$, y$$_{1}$$) lies outside, on or inside the hyperbola $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1 according as $$\frac{x_{1}^{2}}{a^{2}}$$ - $$\frac{y_{1}^{2}}{b^{2}}$$ – 1 < 0, = or > 0.

Let P (x$$_{1}$$, y$$_{1}$$) be any point on the plane of the hyperbola $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1 ………………….. (i)

From the point P (x$$_{1}$$, y$$_{1}$$) draw PM perpendicular to XX' (i.e., x-axis) and meet the hyperbola at Q.

According to the above graph we see that the point Q and P have the same abscissa. Therefore, the co-ordinates of Q are (x$$_{1}$$, y$$_{2}$$).

Since the point Q (x$$_{1}$$, y$$_{2}$$) lies on the hyperbola $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1.

Therefore,

$$\frac{x_{1}^{2}}{a^{2}}$$ - $$\frac{y_{2}^{2}}{b^{2}}$$ = 1

$$\frac{y_{2}^{2}}{b^{2}}$$ = $$\frac{x_{1}^{2}}{a^{2}}$$ - 1 ………………….. (i)

Now, point P lies outside, on or inside the hyperbola according as

PM <, = or > QM

i.e., according as y$$_{1}$$ <, = or > y$$_{2}$$

i.e., according as $$\frac{y_{1}^{2}}{b^{2}}$$ <, = or > $$\frac{y_{2}^{2}}{b^{2}}$$

i.e., according as $$\frac{y_{1}^{2}}{b^{2}}$$ <, = or > $$\frac{x_{1}^{2}}{a^{2}}$$ - 1, [Using (i)]

i.e., according as $$\frac{x_{1}^{2}}{a^{2}}$$ - $$\frac{y_{1}^{2}}{b^{2}}$$ <, = or > 1

i.e., according as $$\frac{x_{1}^{2}}{a^{2}}$$ - $$\frac{y_{1}^{2}}{b^{2}}$$ - 1 <, = or > 0

Therefore, the point

(i) P (x$$_{1}$$, y$$_{1}$$) lies outside the hyperbola $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1 if PM < QM

i.e., $$\frac{x_{1}^{2}}{a^{2}}$$ - $$\frac{y_{1}^{2}}{b^{2}}$$ - 1 < 0.

(ii) P (x$$_{1}$$, y$$_{1}$$) lies on the hyperbola $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1 if PM = QM

i.e., $$\frac{x_{1}^{2}}{a^{2}}$$ - $$\frac{y_{1}^{2}}{b^{2}}$$ - 1 = 0.

(ii) P (x$$_{1}$$, y$$_{1}$$) lies inside the hyperbola $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1 if PM < QM

i.e., $$\frac{x_{1}^{2}}{a^{2}}$$ - $$\frac{y_{1}^{2}}{b^{2}}$$ - 1 > 0.

Hence, the point P(x$$_{1}$$, y$$_{1}$$) lies outside, on or inside the hyperbola $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1 according as x$$\frac{x_{1}^{2}}{a^{2}}$$ - $$\frac{y_{1}^{2}}{b^{2}}$$  - 1 <, = or > 0.

Note:

Suppose E$$_{1}$$ = $$\frac{x_{1}^{2}}{a^{2}}$$ - $$\frac{y_{1}^{2}}{b^{2}}$$ - 1, then the point P(x$$_{1}$$, y$$_{1}$$) lies outside, on or inside the hyperbola $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1 according as E$$_{1}$$ <, = or > 0.

Solved examples to find the position of the point (x$$_{1}$$, y$$_{1}$$) with respect to an hyperbola $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1:

1. Determine the position of the point (2, - 3) with respect to the hyperbola $$\frac{x^{2}}{9}$$ - $$\frac{y^{2}}{25}$$ = 1.

Solution:

We know that the point (x$$_{1}$$, y$$_{1}$$) lies outside, on or inside the hyperbola $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1 according as

$$\frac{x_{1}^{2}}{a^{2}}$$ - $$\frac{y_{1}^{2}}{b^{2}}$$ – 1 < , = or > 0.

For the given problem we have,

$$\frac{x_{1}^{2}}{a^{2}}$$ - $$\frac{y_{1}^{2}}{b^{2}}$$ - 1 = $$\frac{2^{2}}{9}$$ - $$\frac{(-3)^{2}}{25}$$ – 1 = $$\frac{4}{9}$$ - $$\frac{9}{25}$$ - 1 = - $$\frac{206}{225}$$ < 0.

Therefore, the point (2, - 3) lies outside the hyperbola $$\frac{x^{2}}{9}$$ - $$\frac{y^{2}}{25}$$ = 1.

2. Determine the position of the point (3, - 4) with respect to the hyperbola $$\frac{x^{2}}{9}$$ - $$\frac{y^{2}}{16}$$ = 1.

Solution:

We know that the point (x$$_{1}$$, y$$_{1}$$) lies outside, on or inside the hyperbola $$\frac{x^{2}}{a^{2}}$$ - $$\frac{y^{2}}{b^{2}}$$ = 1 according as

$$\frac{x_{1}^{2}}{a^{2}}$$ - $$\frac{y_{1}^{2}}{b^{2}}$$ - 1 <, = or > 0.

For the given problem we have,

$$\frac{x_{1}^{2}}{a^{2}}$$ - $$\frac{y_{1}^{2}}{b^{2}}$$ - 1 = $$\frac{3^{2}}{9}$$ - $$\frac{(-4)^{2}}{16}$$ - 1 = $$\frac{9}{9}$$ - $$\frac{16}{16}$$ - 1 = 1 - 1 - 1 = -1 < 0.

Therefore, the point (3, - 4) lies outside the hyperbola $$\frac{x^{2}}{9}$$ - $$\frac{y^{2}}{16}$$ = 1.

The Hyperbola


text:
Changeset 1549

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Timestamp:
Oct 22, 2011, 2:04:00 PM (7 years ago)
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Update parlensort doc

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Legend:

Unmodified
 r1302 \maketitle \section{The Idea} \section{Length-Group Bitstreams} Given streams $S$ and $E$ that mark the starts and ends of Names. ends of Names of length 1, $E_1$, and the stream of ends of Names of greater than length 1, $E_{>1}$, as follows. Here  $n(S)$ is the bitstream next'' operation, shifting the stream forward one position. \begin{eqnarray*} \end{eqnarray*} \section{Parallel Hashing} Given length groups identified as in the previous section, it is possible to perform parallel hash value calculation for names in each group as follows. The first step is to create hash streams. Let $B_0, B_1, \ldots, B_7$ be the {\em basis} bitstreams determined by character stream transposition.   That is, the $k$th basis bitstream $B_k$ consists of the $k$th bit of each character in the source data stream. Then we may compute hash streams by some bitwise logical combination of basis bit streams. For example, the following two hash streams may be useful. \begin{eqnarray*} h_a & = & B_2 \oplus B_4 \oplus B_6 \\ h_b & = & B_3 \oplus B_5 \oplus B_7 \\ \end{eqnarray*} \section{Follow-up} lengths?   This avoids allocation of length-based position arrays, the memory operations of insertion of positions into those arrays, bounds-checking on those arrays and maintaining an auxiliary array of lengths of the each array. arrays and maintaining an auxiliary array of lengths of each array. A further benefit may be from the programming side: the length sorting can


text:
I was talking with @looselycoupled the other day about how we generate passwords for use on websites. We both agree that every single domain should have its own password (to prevent one crack ruling all your Internets). However, we’ve both evolved on the method over time, and I’ve written a simple script that allows me to generate passwords using methodologies discussed in this post.

In particular I use the generator to create passwords for pwSafe, the tool I currently use for password management (due to its use of the open source database format created by Bruce Schneier). It is my hope that this script can be embedded directly into pwSafe, or at least allow me to write directly to the database; but for now I just copy and paste with the pbcopy utility.

The point of this post, though, is not the generator (though I hope it can be useful). The point of the post is that because I do not believe in security through obscurity or obfuscation I want to expose my techniques to criticism and testing publicly. I believe that an approach to security must be ongoing, living, and continuous. While the foundation of security revolves around some shared secret (a key, a password) that must not be made public, the way that secret is used should be open and constantly inspected for flaws.

So although this post isn’t really about anything and is quite simple, it does put my money where my mouth is, philosophically speaking. If you’re reading this, I hope you read with a critical eye and report any flaws you spot in the comments of the snippet on GitHub Gist.

## Previous Approaches

My original approach was pretty simple: keep three versions of the same password in a variety of strengths:

password
p4ssw0rd
p4$$w0rd  In order to generate domain specific passwords, just append a few characters from the site. So for example, for Facebook, Gmail, and PayPal you might use passwordfb, p4ssw0rdgmail, and p4$$w0rdpp respectively, in order of increasing “security”. The benefit of this approach is simple, memorable passwords that meet a variety of security features required by websites.

Of course, the flaws are numerous here. All passwords should have the same level of security, using a variety of characters. Additionally the use of a discoverable pattern could make a targeted attack easy. The real problem for me, however, was that on sites that required a routine password change (which should be done anyway). This method does not lend flexibility. However, it does highlight the trade-off between ease of use and security, however, which led my to my next approach.

The next approach did the same thing, but instead of adding letters and symbols, I used a long, correct sentence inspired by the xkcd Password Strength comic. A characteristic of the domain is simply fit into the base sentence:

base: "Silly banana, happy elephant."

This allowed me to change the password by moving the domain phrase around or performing other manipulations, while also having longer, more secure passwords that I could easily remember. Still, having a template seemed like a bad idea.

After a while, however, I finally started using pwSafe and started generating random passwords for each site. This is clearly the best way to do things but in all honesty - the most important thing to secure is your email and any social auth accounts. To that end, I set up 2 factor authentication using Google Authenticator as much as I could, and text messaging where I couldn’t.

I don’t think I’m paranoid, this just seems like good Internet hygiene. The appendix includes a list of the sites that I use 2 factor authentication and please let me know if there are any others I should include.

The issue, of course, is that I’m paranoid that I’ll lose or misplace my pwSafe master key, have the database corrupted, or otherwise be locked out of my accounts. I want a methodology that creates passwords that are as secure as possible that I can also reproduce on my own. The password should be domain specific, unique, long, and contain a variety of characters.

## Reproducible Methodology

So here’s the idea:

2. Create a password string by concatenating the master, the base url of the site, and some salt or incrementing value.
4. Encode the hash and use as password.

Now all you have to remember is the master password and the salt or incrementing value on a per-domain basis.

Issues:

1. What encoding scheme do you choose? Some sites require special characters and numbers, others cannot accept them. The domain of the characters is now also reduced to 16 chars for hex encoding, or 64 chars for base64 (even if you use one with special characters like hexbin).
2. How do you verify the master password (ensuring there was no typo).
3. How do you remember the salt? Perhaps something device specific like a UUID?

I’m sure there are more issues here as well.

However, I’m hoping this is good enough when combined with 2FA, this password is just a first step, and gives me the ability to reproduce the sequence of characters for crucial sites in case the pwSafe database is corrupted. For most sites, I’ll simply use a randomly generated password and rely on the “forgot password” link.

## Conclusion

So in this post, I’ve presented a simple, reproducible method for generating somewhat secure passwords that I intend to use in combination with two factor authentication. It is my hope that these passwords are long enough, unique to the domain, but able to be generated on demand.

This is a list of the sites that I have set up 2 factor authentication with, for reference.


text:
Hi, I’m having an issue update fom 12.0.0 to 13.0.12.

When running the updater I get:

I can download the file manually - I’ve tried also an approach found here:

• Download the required artifact manually (which worked fine from the browser
• Put it in the \$DATA_DIRECTORY/updater-{instanceid}/

Strangly I get the same message - even with the function commented out? Did I do something wrong here?
(I’ve just added /* at the start and */ at the end of the function)

/*

/*
}


text:
# HW5

Make sure you check the syllabus for the due date.

Hand written digit recognition. Training data, labels.  Testing data, labels.

### PROBLEM 1 [50 points]

Run an SVM implementation by training over the training set and testing over the testing set. Try several kernels, including the polynomial and the RBF ones. Report the results. Use one of these packages: SVMlight, SGDSVM, osu SVMLIBSVM or other software  (here, here).  If you choose an SVM packge that does not provide multi-class implementations, you should write a wrapper that will make the code run for each class versus the others, separately.
This problem have been well researched, and there is a comprehensive archive of results here; take a look at the several SVM papers listed before you start working on it. Data is in the IDX format, described on the same page; you will have to write your own code to read the data, or to find some code that does just that.

### PROBLEM 2 [50 points]

Same as problem 1. Instead of using a SVM package, implement your own using SMO optimization (do not have implement the quadratic solver). Compare with Problem 1 results

### PROBLEM 3 [Extra Credit]

Same problem, but dont use SMO. Instead use a built in  (or existing library) quadratic solver from Matlab, Python, Java, C etc.

### PROBLEM 4 [Extra Credit]

Same data as in problem 1. Use a neural network to make the classification. You can perhaps adapt the NN code from hw2.

The repository page points to many NN approaches; you should do a bit of research first. Report findings and be ready to give a short demo of your code.

### PROBLEM 5 [20 points]

Explain why 0$\le$ $\alpha$$\le \mathrm{C/m}\frac{}{}$ is a constraint in the dual optimization with slack variables. (HINT: read Chris Burges tutorial first) Distinguish three cases, and explain them in terms of the classification and constraints: a) 0$=$ $\alpha$; b) 0$<$ $\alpha$$<\mathrm{C/m}\frac{}{}$; c) $\alpha$$=\mathrm{C/m. This has been discussed in class and in SVM notes; a detailed rigurous explanation is expected.}\frac{}{}$

### PROBLEM 6 [20 points]

Consider the following 6 points in 2D, for two classes:

class 0:   (1,1)   (2,2)    (2,0)

class 1:   (0,0)   (1,0)    (0,1)

a) Plot these 6 points, construct the optimal hyperplane by inspection and intuition (give the W,b) and calculate the margin.

b) Which points are support vectors ?

c) Construct the hyperplane by solving the dual optimization problem using the Lagrangian. Compare with part (a).

### PROBLEM 5 [Extra Credit]

Prove of the harmonic functions property discussed in class based on this paper. Specifically, prove that to minimize the energy function

f must be harmonic, i.e. for all unlabeled datapoints j, it must satisfy


text:
This can be related to the "regular" regularization tradeoff in the following way. SVMs are usually formulated like $$\min_{w} \mathrm{regularization}(w) + C \, \mathrm{loss}(w; X, y) ,$$ whereas ridge regression / LASSO / etc are formulated like: $$\min_{w} \mathrm{loss}(w; X, y) + \lambda \, \mathrm{regularization}(w) .$$ The two are of course equivalent with $C = \tfrac{1}{\lambda}$. I think it's more intuitive to see in the latter case, though, that as $\lambda \to \infty$ your solution is determined entirely by the regularization term, so that your bias is very high and variance very low; as $\lambda \to 0$, you take away all the regularization bias but also lose its variance reduction.
• Thanks. Just wondering, why isn't $\lambda$ used in the SVM cost function? It seems confusing to have a new parameter, $C$. – sir_thursday Oct 16 '16 at 12:58


text:
Last edited by Arashishakar
Friday, April 17, 2020 | History

3 edition of Almost invariant subspaces and high gain feedback found in the catalog.

# Almost invariant subspaces and high gain feedback

Published by Centrum voor Wiskunde en Informatica in Amsterdam, The Netherlands .
Written in English

Subjects:
• Invariant subspaces.,
• Feedback (Electronics)

• Edition Notes

The Physical Object ID Numbers Statement H. L. Trentelman. Series CWI tract -- 29. Pagination iv, 239 p. : Number of Pages 239 Open Library OL14278827M ISBN 10 9061963087

The fundamental subspaces are four vector spaces defined by a given m × n m \times n m × n matrix A A A (and its transpose): the column space and nullspace (or kernel) of A A A, the column space of A T A^T A T (((also called the row space of A), A), A), and the nullspace of . Almost invariant subspaces; an approach to high gain feedback design, part I: almost controlled invariant subspaces. IEEE Trans Aut. Control. vAC Google Scholar; Willems, Almost invariant subspaces: an approach to high gain feedback design, part II: almost conditionally invariant subspaces. IEEE Trans Aut. Control. vAC Cited by: You can write a book review and share your experiences. Other readers will always be interested in your opinion of the books you've read. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them., Free ebooks since Don't show me this again. Welcome! This is one of over 2, courses on OCW. Find materials for this course in the pages linked along the left. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.. No enrollment or registration.

Presents Results from a Very Active Area of ResearchExploring an active area of mathematics that studies the complexity of equivalence relations and classification problems, Invariant Descriptive Set Theory presents an introduction to the basic concepts, methods, and results of this theory. It brings together techniques from various areas of mathem.

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### Almost invariant subspaces and high gain feedback by H. L. Trentelman Download PDF EPUB FB2

Almost invariant subspaces and high gain feedback. Amsterdam, The Netherlands: Centrum voor Wiskunde en Informatica, © (OCoLC) Online version: Trentelman, H.L. Almost invariant subspaces and high gain feedback.

Amsterdam, The Netherlands: Centrum voor Wiskunde en Informatica, © (OCoLC) Material Type: Internet resource. Abstract. This talk was devoted to an exposition of the theory of ‘almost invariant subspaces’ which was developed in [1,2]. This theory provides a geometric approach to the synthesis of high gain feedback control synthesis and is therefore intimately related to singular : Jan C.

Willems. The book contains 11 lectures and begins with a discussion of analytic functions. This is followed by lectures covering invariant subspaces, individual theorems, invariant subspaces in Lp, invariant subspaces in the line, and analytic vector functions. Almost invariant subspaces: High gain feedback.

or -singularly perturbed feedback. Conference Paper in Proceedings of the IEEE Conference on Decision and Control January with 13 Reads. ×Close. The Infona portal uses cookies, i.e.

strings of text saved by a browser on the user's device. The portal can access those files and use them to remember the user's data, such as their chosen settings (screen view, interface language, etc.), or their login data.

Abstract. In this paper we will solve, for finite dimensional linear time invariant systems, the problem of the existence of a dynamic state feedback control law such that in the closed loop system the exogenous variables are noninteracting to any arbitrary degree of by: 5.

This paper is concerned with a generalization of the almost disturbance decoupling problem by state feedback.

Apart from approximate decoupling from the external disturbances to a first to-be-controlled output, we require a second output to be uniformly bounded with respect to the accuracy of decoupling. The problem is studied using the geometric approach to linear by: 6.

J.C. Willems, Almost invariant subspaces: an approach to high gain feedback design -Part II: Almost conditionally invariant subspaces, IEEE Trans. Automat. Control 27 (Oct. ) to appear. (6) C. Commalilt and J.M.

Dian, Structure at infinity of linear l11ultivariable systems: a geometric approach, Presented to the 20th IEEE Conference Cited by: [23] [ [25. [26~ [ J.C. Willems, Almost invariant subspaces: An approach to high gain feedback design - part 1: Almost controlled invafiant subspaces, IEEE Trans.

Automat. Control 26 () J.C. WiUems, Almost i,wariant subspaces: An approach to high gain feedback desiD~, - part If: Almost conditionally invariant subspaces Cited by: We recall the pole placement flexibilities and constraints that both exist when using a particular almost invariant subspace as a support for the construction of specific (including high gain.

Harry Trentelman is a full professor in Systems and Control at the Johann Bernoulli Institute for Mathematics and Computer Science of the University of to he served as an assistant professor and as an associate professor at the Mathematics Department of the Eindhoven University of Technology, the Netherlands.

He obtained his PhD degree in Mathematics from the. Invariant subspaces. Eigenvalues and eigenvectors. A list of eigenvectors correpsonding to distinct eigenvalues is linearly indepenedent. The number of disti.

Almost invariant subspaces and high gain feedback () Pagina-navigatie: Main; Save publication. Save as MODS; Export to Mendeley; Save as EndNoteCited by: A situation of great interest is when we have T-invariant subspaces W 1;;W t and V = W 1 W t.

For if = 1 [[ t, where i is a basis for W i, we see that [T] = [T W 1] 1 1 t[T Wt] t: There are two important examples of T{invariant subspaces that arise in our study of Jordan and rational canonical forms - Kerpt(T) and T{cyclic subspaces.

T File Size: KB. Invariant subspaces and quadratic matrix equations suppose V = R(M) is A-invariant, where M ∈ Rn×k is rank k, so AM = MX for some X ∈ Rk×k conformally partition as A11 A12 A21 A22 M1 M2 = M1 M2 X A11M1 +A12M2 = M1X, A21M1 +A22M2 = M2X eliminate X from ﬁrst equation (assuming M1 is nonsingular): X = M−1 1 A11M1 +M −1 1 A12M2.

invariant subspaces. (i) =)(iii) is immediate. (iii) =)(i): There is an invariant subspace Wof V that is maximal with respect to being a direct sum of simple invariant subspaces. We must show W= V. If not, since V is assumed to be generated by its simple invariant subspaces, there exists a simple invariant subspace SˆV that is not contained in Size: KB.

without using almost invariant subspaces. In solvable cases a high-gain feedback is explicitly given which includes the one proposed in [14] in special cases.

The LP case for arbitrary p is treated directly in [17]. In this note we address the almost disturbance decoupling problem for.

Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics is the first book to provide a systematic construction of exact solutions via linear invariant subspaces for nonlinear differential operators.

Acting as a guide to nonlinear evolution equations and models from physics and mechanics, the Cited by: Almost (A;B)-invariant subspaces are of interest to study subspaces invariant under high gain state feedback. Thus, in general, almost (A;B)-invariant subspaces cannot be made invariant under state feedback, so there is no friend, but they can be made almost-invariant in the sense that for every x 2 V and any " > 0 there exists aFile Size: KB.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Almost invariant subspaces for WOT closure of an algebra of operators.

Ask Question Feedback on Q2 Community Roadmap. Autofilters for Hot Network Questions. namely, we introduce the Almost Self-Bounded Controlled-Invariant subspaces.

We recall the pole placement exibilities and constraints that both exist when using a particular almost invariant subspace as a support for the construction of speci c (including high gain) feedbacks.

A subspace ℳ ⊂ C / n is called invariant for the transformation A, or A invariant, if Ax ∈ ℳ for every vector x ∈ ℳ. In other words, ℳ is invariant for A means that the image of ℳ under A is contained in ℳ; Aℳ ⊂ ℳ. Trivial examples of invariant subspaces are {0}.

In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself.

Many variants of the problem have been solved, by restricting the class of bounded operators considered or by specifying a particular class of Banach spaces. CHAPTER 5 Eigenvalues, Eigenvectors, and Invariant Subspaces 5.A Invariant Subspaces In this chapter we develop the tools that will help us understand the structure of operators.

Recall that an operator is a linear map from a vector space to. Thanks for contributing an answer to Mathematics Stack Exchange. Please be sure to answer the question.

Provide details and share your research. But avoid Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience. Use MathJax to format equations. Invariant Subspaces Recall the range of a linear transformation T: V!Wis the set range(T) = fw2Wjw= T(v) for some v2Vg Sometimes we say range(T) is the image of V by Tto communicate the same idea.

We can also generalize this notion by considering the image of a particular subspace U of V. We usually denote the image of a subspace as followsFile Size: KB. Invariant Subspaces (Dover Books on Mathematics) Paperback – J by Heydar Radjavi (Author) › Visit Amazon's Heydar Radjavi Page.

Find all the books, read about the author, and more. See search results for this author. Are you an author. Learn about Author Central Author: Heydar Radjavi, Peter Rosenthal. In the eighties further contributions are due to Willems (with the theory of almost controlled and almost conditioned invariant subspaces to deal with high-gain feedback problems), Anderson, Akashi, Bhattacharyya, Commault, Dion, Kucera, Imai, Malabre, Molinari, Pearson, Silverman and Schumacher, who contributed with a complete study of system.

[19] J.C. WILLEMS, "Almost invariant subspaces: An approach to high gain feedback design—Part l: Almost controlled invariant subspaces, Part Il: Almost conditionally invariant subspaces," IEEE.

Conditions for solvability (via high-gain feedback) are given in terms of plant transmission zeros. Simplifications in the geometry of almost invariant subspaces for plants that satisfy the conditions are exploited to obtain explicit control law constructions.

Cyclic subspaces for linear operators Let V be a nite dimensional vector space and T: V!V be a linear operator. One way to create T-invariant subspaces is as follows. Choose a non-zero vector v 2V, and let k2N v is the smallest T-invariant subspaces that contains T.

Abstract: We show that any bounded operator $T$ on a separable, reflexive, infinite-dimensional Banach space $X$ admits a rank one perturbation which has an invariant Author: Alexey I.

Popov, Adi Tcaciuc. (the almost controlled invariant by Willems), invisible at the output. Unfortunately, we don’t like to manage distributions as inputs, because of saturation. Willems, “Almost invariant subspaces: an approach to high gain feedback design - Part I: Almost controlled invariant subspaces”, IEEE Autom.

Cont., 26, –, T-inarianvt subspaces f0g= W 0 W W n= V with dimW k= kfor each kif and only if there is a basis for V for which [T] is upper triangular. Proof. Suppose f0g= W 0 W W n= V is a ag of T-inarianvt subspaces with dimW k= kfor each k. We construct a basis for Vinductively. Let 0 6= v 1 2W 1.

Then, fv 1gis a basis for W 1 as dimW 1 = 1 File Size: KB. Homework Statement Let T be a linear operator on a vector space V and let W be a T-Invariant subspace of V.

Prove that W is g(T)-invariant for any polynomial g(t). Homework Equations Cayley-Hamilton Theorem. The Attempt at a Solution Im not sure how to begin.

Ok so g(t) is the. Invariant and controlled invariant subspaces In this chapter we introduce two important concepts: invariant subspace and controlled invariant subspace, which will be used later on to solve many control problems.

Invariant subspaces Consider an n-dimensional linear system () x˙ = File Size: KB. If is a 1-dim invariant subspace, then the same conclusion holds. If is invariant for, then we regard as an operator from to. If is invariant for, then we regard as an operator from to.

If are invariant for, then so are and. For, if, then there are, such that. By continuity of. By invariance, for all and for all. Thus. Purchase Introduction to Operator Theory and Invariant Subspaces, Volume 42 - 1st Edition. Print Book & E-Book. ISBNBook Edition: 1. Invariant and reducing subspaces of composition operators 23 Cϕg = Cϕf.χϕ−1(E) = Cϕf.χE because ϕ −1(E)=E Hence Cϕg ∈ Lp[0,1] Thus M is a non-trivial proper invariant subspace of Cϕ 2.

Reducing subspaces of composition operators on 2: Theorem Let ϕ: N → N be an injection which is not a surjection. Then Cϕ has a reducing subspace if and only if there exists two points in N.

So by Definition KLT, we see that T\left (z\right) ∈K\kern pt \left ({T}^{k}\right).Thus K\kern pt \left ({T}^{k}\right) is an invariant subspace of V relative to T (Definition IS).

Two interesting special cases of Theorem KPIS occur when choose k = 0 and k = than give an example of this theorem, we will refer you back to Example KPNLT where we work with null.

MEASURABLE CHOICE AND THE INVARIANT SUBSPACE PROBLEM BY EDWARD A. AZOFF AND FRANK GILFEATHER1 Communicated by P. R. Halmos, February 8, In [1], J. Dyer, A. Pedersen and P. Porcelli announced that an affir mative answer to the invariant subspace problem would imply that every reductive operator is normal.Finite-dimensional vector spaces admit cyclic decompositions, which are direct sum decompositions into invariant subspaces relative to linear operators acting on them.

If I understand your question correctly, I would like to show here how cyclic d.A–invariant if A(H0) ⊆ H0. (Throughout this paper, all subspaces will be assumed to be closed.) The subspace H0 is said to be A–hyperinvariant if it is S–invariant whenever S ∈ B(H) commutes with A.

Recall that the invariant subspace prob-lem asks whether, for .


text:
# Pure Python script that saves an HTML page with all images

Here is a pure Python script that saves an HTML page without CSS but with all images on it and replaces all hrefs with a path of an image on the hard drive.

I know that there are great libraries like BeautifulSoup and others but I would like to try myself with pure Python.

Actually there is no practical usage of this script, it was just test task from one of companies where I applied.

Once again what it does:

This script can pre-launched from command line, which takes 2 arguments:

1. Address of web page to save (this one is required)
2. Name of folder where images from page should be saved (this one is optional) Script saves html content of page(without CSS) and also it looks for all images on page and save them too, replacing their href attribute to actual path of image on hard drive.

How can I improve it?

import random
import string
import sys
import urllib2
import os
import re
from urlparse import urlparse

page_to_open = urllib2.urlopen(url_name)
base_dir = os.path.dirname(os.path.realpath(__file__))
dir_to_save = os.path.join(base_dir, dir_name)
new_file_name = '%s.html' % ''.join(random.choice(string.ascii_uppercase + string.ascii_lowercase) for _ in range(10))
if not os.path.exists(dir_to_save):
os.makedirs(dir_to_save)

images_on_page = re.findall('img .*?src="(.*?)"', target_page)
internal_images = [img for img in images_on_page if img.startswith('/')]
external_images = [img for img in images_on_page if not img.startswith('/')]

for image in internal_images:
image_url = '%s%s' % (page_to_open.geturl()[:-1], image)
new_image_name = urlparse(image_url).path.split('/')[-1]
with open(os.path.join(dir_to_save, new_image_name), 'w') as new_image:
target_page = re.sub(image, new_image.name, target_page)

for image_url in external_images:
new_image_name = urlparse(image_url).path.split('/')[-1]
with open(os.path.join(dir_to_save, new_image_name), 'w') as new_image:
target_page = re.sub(image_url, new_image.name, target_page)

with open(os.path.join(base_dir, new_file_name), 'w') as new_file:
new_file.write(target_page)

if __name__ == '__main__':
target_url = sys.argv[1]
if len(sys.argv) > 2:
dir_name = sys.argv[2]
else:

Your code is pretty good! I do have a few tips regarding style, and such.

• As mentioned in the comments, even though it looks like you aren't, you shouldn't be trying to parse HTML with Regexes.
• Where are your comments? While good code can be pretty readable, comments are still a valuable asset. You should probably flesh out your page_loader function with a docstring, and any unclear blocks of code with an inline comment.
• If you're only going to use part of a module, e.g, one function or variable, you should do from ... import ....

That's about all I can really think of right now. If there's anything else you want me to comment on, just mention it below, and I'll see if I can cover it. Hope this helps!

You have some very long lines. Making them shorter will improve readability on smaller displays and when your IDE has right and left panes open.

EDIT: This is the line I was talking about (sorry for saying multiple lines)

new_file_name = '%s.html' % ''.join(random.choice(string.ascii_uppercase + string.ascii_lowercase) for _ in range(10))

I usually break the lines on the + sign, whenever possible. This is how I'd do it:

new_file_name = '%s.html' % ''.join(random.choice(string.ascii_uppercase +
string.ascii_lowercase) for _ in range(10))

This is one of the recommendations from Python's style guide, which I prefer, and it's applicable when you have parenthesis. If you don't, you're advised to put some. You can also use the backslash to separate the lines, like this:

new_file_name = '%s.html' % ''.join(random.choice(string.ascii_uppercase + \
string.ascii_lowercase) for _ in range(10))

The key point is that, whatever means you choose, you need to clearly separate the continuation's indentation from the indentations of what's above and below.


text:
So far we have run a sequential tasks in a number of configurations.

This example introduces two new concepts: running multi-core MPI tasks and specifying input data for the task, in this case a simple python MPI script.

## 5.4.1. Preparation¶

Download the file mpi_tasks.py with the following command:

curl -O https://raw.githubusercontent.com/radical-cybertools/radical.pilot/master/examples/docs/mpi_tasks.py

Open the file mpi_tasks.py with your favorite editor. The example might work right out of the box on your local machine, this depends whether you have a local MPI installation. However, if you want to try it out with different resources, like remote HPC clusters, look for the sections marked:

# ----- CHANGE THIS -- CHANGE THIS -- CHANGE THIS -- CHANGE THIS ------

and change the code below according to the instructions in the comments.

This example makes use of an application that we first download to our own environment and then have staged as input to the MPI tasks.

Download the file helloworld_mpi.py with the following command:

curl -O https://raw.githubusercontent.com/radical-cybertools/radical.pilot/master/examples/helloworld_mpi.py

## 5.4.2. Execution¶

** This assumes you have installed RADICAL-Pilot either globally or in a Python virtualenv. You also need access to a MongoDB server.**

Set the RADICAL_PILOT_DBURL environment variable in your shell to the MongoDB server you want to use, for example:

export RADICAL_PILOT_DBURL=mongodb://<user>:<pass>@<mongodb_server>:27017/

If RADICAL-Pilot is installed and the MongoDB URL is set, you should be good to run your program:

python mpi_tasks.py

The output should look something like this:

Initializing Pilot Manager ...
Submitting Compute Pilot to Pilot Manager ...
Initializing Unit Manager ...
Registering Compute Pilot with Unit Manager ...
Submit Compute Units to Unit Manager ...
Waiting for CUs to complete ...
...
Waiting for CUs to complete ...
All CUs completed successfully!
Closed session, exiting now ...

## 5.4.3. Logging and Debugging¶

Since working with distributed systems is inherently complex and much of the complexity is hidden within RADICAL-Pilot, it is necessary to do a lot of internal logging. By default, logging output is disabled, but if something goes wrong or if you’re just curious, you can enable the logging output by setting the environment variable RADICAL_PILOT_VERBOSE to a value between CRITICAL (print only critical messages) and DEBUG (print all messages).

Give it a try with the above example:

RADICAL_PILOT_VERBOSE=DEBUG python simple_bot.py


text:
# Overview¶

PyLops is an open-source Python library focused on providing a backend-agnostic, idiomatic, matrix-free library of linear operators and related computations. It is inspired by the iconic MATLAB Spot – A Linear-Operator Toolbox project.

Linear operators and inverse problems are at the core of many of the most used algorithms in signal processing, image processing, and remote sensing. For small-scale problems, matrices can be explicitly computed and manipulated with Python numerical scientific libraries such as NumPy and SciPy.

Large-scale problems often feature matrices that are prohibitive in size—but whose operations can be described by simple functions. PyLops operators exploit this to represent a linear operator not as array of numbers, but by functions which describe matrix-vector products in forward and adjoint modes. Moreover, many iterative methods (e.g. cg, lsqr) are designed to not rely on the elements of the matrix, only these matrix-vector products. PyLops offers such solvers for many different types of problems, in particular least-squares and sparsity-promoting inversions.

Get started by installing PyLops and following our quick tour.

## Terminology¶

A common terminology is used within the entire documentation of PyLops. Every linear operator and its application to a model will be referred to as forward model (or operation)

$\mathbf{y} = \mathbf{A} \mathbf{x}$

while its application to a data is referred to as adjoint model (or operation)

$\mathbf{x} = \mathbf{A}^H \mathbf{y}$

where $$\mathbf{x}$$ is called model and $$\mathbf{y}$$ is called data. The operator $$\mathbf{A}:\mathbb{F}^m \to \mathbb{F}^n$$ effectively maps a vector of size $$m$$ in the model space to a vector of size $$n$$ in the data space, conversely the adjoint operator $$\mathbf{A}^H:\mathbb{F}^n \to \mathbb{F}^m$$ maps a vector of size $$n$$ in the data space to a vector of size $$m$$ in the model space. As linear operators mimics the effect a matrix on a vector we can also loosely refer to $$m$$ as the number of columns and $$n$$ as the number of rows of the operator.

Ultimately, solving an inverse problems accounts to removing the effect of $$\mathbf{A}$$ from the data $$\mathbf{y}$$ to retrieve the model $$\mathbf{x}$$.

For a more detailed description of the concepts of linear operators, adjoints and inverse problems in general, you can head over to one of Jon Claerbout’s books such as Basic Earth Imaging.

## Implementation¶

PyLops is build on top of the scipy class scipy.sparse.linalg.LinearOperator.

This class allows in fact for the creation of objects (or interfaces) for matrix-vector and matrix-matrix products that can ultimately be used to solve any inverse problem of the form $$\mathbf{y}=\mathbf{A}\mathbf{x}$$.

As explained in the scipy LinearOperator official documentation, to construct a scipy.sparse.linalg.LinearOperator, a user is required to pass appropriate callables to the constructor of this class, or subclass it. More specifically one of the methods _matvec and _matmat must be implemented for the forward operator and one of the methods _rmatvec or _adjoint may be implemented to apply the Hermitian adjoint. The attributes/properties shape (pair of integers) and dtype (may be None) must also be provided during __init__ of this class.

Any linear operator developed within the PyLops library follows this philosophy. As explained more in details in Implementing new operators section, a linear operator is created by subclassing the scipy.sparse.linalg.LinearOperator class and _matvec and _rmatvec are implemented.

## History¶

PyLops was initially written by Equinor It is a flexible and scalable python library for large-scale optimization with linear operators that can be tailored to our needs, and as contribution to the free software community. Since June 2021, PyLops is a NUMFOCUS Affiliated Project.