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# Scottish Book Problem 172

The problem is formulated using old terminology and I want to understand what it actually says.

The problem reads: "A space $$E$$ of type (B) has the property (a) if the weak closure of an arbitrary set of linear functionals is weakly closed. [A sequence of linear functionals $$f_n(x)$$ converges weakly to $$f(x)$$ if $$f_n(x) \to f(x)$$ for every $$x$$.] The space $$E$$ of type (B) has the property (b) if every sequence of linear functionals weakly convergent converges weakly as a sequence of elements in the conjugate space $$\bar{E}$$. Question: Does every separable space of type (B) which has property (a) also possess property (b)?"

I know that "space of type (B)" means "Banach space" in modern terminology.

Q1: Does "linear functional" include the assumption of continuity?

Q2: What is the conjugate space? Does it mean the dual space of continuous linear functionals on $$E$$?

If the answers to Q1 and Q2 are yes, then this would mean Banach spaces with property (b) are exactly what nowadays are called Grothendieck spaces. [A Grothendieck space is a Banach space $$E$$ such that every sequence in the dual space $$E^*$$ that converges weak* - that is, with respect to $$\sigma(E^*,E)$$ - also converges weakly - that is, with respect to $$\sigma(E^*,E^{**})$$.]

I am also unsure how to understand the condition "the weak closure of an arbitrary set of linear functionals is weakly closed", as this seems like a tautology. Is the point here that we consider a sequential closure?

• Thank you. So the problem is: Assume that $E$ is a Banach space such that the weak* sequential closure of every subset of $E^*$ is weak* sequentially closed. Is $E$ a Grothendieck space? – Hannes Thiel Oct 8 at 7:47
• The terminology "weak* sequential closure" is counterintuitive. Tthe weak* sequential closure of a set $X\subset E^*$, lets denote it by $c(X)$, is the set of all limits of weak* convergent sequences in $X$. It is not necessarily true that $c(c(X))=c(X)$. Thus, the weak* sequential closure need not be weak* sequentially closed. I found a paper of Godun (Weak* derivatives of sets of linear functionals), which shows that a separable Banach space $E$ is quasireflexive iff every subspace $F\subseteq E^*$ satisfies $c(c(F))=c(F)$. Thus, separable Banach spaces with (a) are quasireflexive. – Hannes Thiel Oct 8 at 12:50
• In the same paper, it is also shown that a Banach space $E$ is Grothendieck if and only if $c(F)=F$ for every norm-closed subspace $F\subset E^*$. Further, for every (not necessarily norm-closed) subspace $F\subset E^*$ with norm-closure $\overline{F}$, we have $c(F)=c(\overline{F})$. – Hannes Thiel Oct 8 at 13:03

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# Research Publications, Xue-Mei Li

The list of my publications is given below by topics, A thesis and a dissertation are also appended at the end. These topics are:

(1) Stochastic Averaging and Homogenization, Perturbations to Conservation Laws; (2) Hypoelliptic Operators;

(3) Estimates for Fundamental Solutions of Parabolic Equations; (4) Strictly Local Martingales, Martingales, Semi-martingales;

(5) Existence of Global Smooth Stochastic Flow; (6) Complete SDEs Lacking Strong Completeness; (7) Other Stochastic Flows;

(8) Moment Estimates, Moment Stability, Long Time Asymptotics of Stochastic Dynamics, Explosion;

(9) Interplay between Stochastic Processes and their Underlying Spaces; (10) Differential Forms and Harmonic Maps;

(11) The Geometry of Diffusions; (12) Books on Diffusion Operators, Semi-elliptic Riemannian Geometry, Geometry of Filtering;

(13) Malliavin Calculus; (14) Generalised KPP Equation; (15) Spectral Gap and Poincare Inequality on Spaces of Continuous Paths;

(16) L2 Hodge-de Rham Theory in Infinite Dimensional Spaces; (17) Sobolev Calculus on Path and Loop Spaces;

## Stochastic Averaging and Homogenization, Perturbations to Conservation Laws

Problems from mathematical physics or from geometry lead to the study of singular perturbation problem. This includes dynamical descriptions for Brownian motions, convergence of manifolds, and in-homogeneously scaled Riemannian manifolds. This singular perturbation problem involves multi-scale analysis, averaging and stochastic differential equations. We study and view singular perturbation problems as perturbations to conservations laws'. Our models do not always have conventional conservation laws, e.g. real valued energy functions. Typically they are manifold valued, representing orbits. The models are motivated by Physics and Geometry. In particular we study dynamical descriptions for the convergence of a family of Riemannian manifolds to lower dimensional spaces, modelling Brownian motions as geodesics with fast changing directions, and perturbation to integrable stochastic systems which has connection to evolution of energy of weakly interactive particle systems. The analysis involving multi-scale analysis, separation of slow and fast, reduction. The stochastic parallel transport processes along the slow variables will typically also converge, in the Wasserstein distance, with rate of convergence. The state space are typically not compact. The singular operators, themselves not necessarily hypoelliptic. The fast variables are typically hypoelliptic, or more generally Fredholm operators with zero Fredholm index allowing non-unique invariant probability measures.
• Perturbation of Conservation Laws and Averaging on Manifolds (2017), article. Xue-Mei Li, Submitted to: Abelsymposium 2016.
• Homogenisation on Homogeneous Spaces (with an appendix by D. Rumynin) (2016) Article arXiv:1505.06772. To appear: Journal of the Mathematical Society of Japan. Xue-Mei Li
• Limits of Random Differential Equations on Manifolds. Article. Probability Theory Related Fields. Vol 166:659-712, No 3-4 (2016). 10.1007/s00440-015-0669-x arXiv:1501.04793, Xue-Mei Li
• Random Perturbation to the geodesic equation. The Annals of Probability , 44(1), 544-566, (2016), Xue-Mei Li. arxiv:1402.5861 article
• Effective Diffusions with Intertwined Structures, Xue-Mei Li (2012) Article Arxiv:1204.3250
• An averaging principle for Integrable stochastic Hamiltonian systems Nonlinearity 21 (2008) pp.803--822, Xue-Mei Li.

### Fredholm, Hypoelliptic Operators

In the following article we study aspects of hypoelliptic operators. Hypoelliptic operators are also studied in the following articles: Homogenisation on Homogeneous Spaces (with an appendix by D. Rumynin) (2016), Limits of Random Differential Equations on Manifolds; and Perturbation of Conservation Laws and Averaging on Manifolds (2017)Perturbation of conservation laws and averaging on manifolds (2017), arxiv 1705.08857, submitted to Abelsymposium 2016'.
• On Hypoelliptic bridge. Aticle Electronic Communications in Probability, Vol 24, no 20, 1-12 (2016), Xue-Mei Li
• Perturbation of Conservation Laws and Averaging on Manifolds (2017), article. Xue-Mei Li, submitted to Abelsymposium 2016.

### Strictly Local Martingales, Martingales, Semi-martingales

In the first and the second article we study local martingales which are not true martingales (and name them as strictly local martingales). In the third we give more examples of strict local martingale. In the fourth we study manifold valued martingales. In the fifth we show a class of hypoelliptic bridge processes are semi-martingale and has L1 integrability property.

### Brownian, Generalised and hypoelliptic Bridges

• On the Semi-Classical Brownian Bridge Measure' arxiv:1607.06498 Article Electronic Communications in Probability 2017, Vol. 22, paper no. 38, 1-15
• Generalised Brownian bridges: examples. arxiv:1612.08716 (2016), Xue-Mei Li. To appear: Markov Processes and Related Fields.

### Existence of Global Smooth Stochastic Flows (strong p-completeness)

• Strong p-completeness of stochastic differential equations and the existence of smooth flows on non-compact manifolds.  Probab. Theory Relat. Fields. 100, 4, 485-511. 1994, Xue-Mei Li.
• For this problem we assume pathwise uniqueness and local existence. An SDE is complete (or conservative) if for every initial point x, its solution F_t(x, omega) exists for almost surely all omega. An SDE is said to be strongly complete (or has a smooth global solution flow) if there exists a set omega of full measure on which F_t(x, omega) is jointly continuous. If so, the regularity of the solution will be automatically prescribed by the regularity of its vector field. If the adjoint SDE (the SDE with its drift vector field reversed) is also strongly complete then F_t(., omega) are diffeomorphisms. If the driving vector fields of and SDE are smooth, the solution is unique. These however do not imply that the solutions are continuous w.r.t. its initial data. The random solution flow needs to start with a random initial data, e.g. restart from a random variable. The technical problem is that the life times and exit times from relatively compact open sets may not be continuous with respect to the its initial value. For example the life time of the ODE (d/dt) z(t)=z(t)^2 on the complex plane is not continuous on any open set containing a segment of the real line. The solution to this equation is global if and only if the initial point is not on the real axis.

For a compact smooth manifold this is automatic (solved by lifting the SDE to the space of diffeomorphism group). For Lipschitz continuous vector fields in an Euclidean space this is solved by fixed point method. Neither method seems to extend non-compact non-linear manifolds.

Under what conditions is a complete SDE driven by smooth vector fields strongly complete? We solved this open question which was open since the 70's when the first counter example was given. This leads also to improvements to the known results on Rn, in a non-trivial way.

The strong completeness problem is roughly equivalent to the problem that for a full set of omega, a connected relatively compact open set pushed by the solution remains connected for all time.

The existence of a derivative of F_x(., -) in probability is much easier, which will solve an SDE and is called the derivative flow. We prove that the order of integration f(F_t(x, omega), where f is a smooth compactly supported function, (with respect to omega) and differentiation (in probability with respect to the initial point) can be exchanged provided the following holds: the solution with initial point from a smooth compact curve remains a connected smooth curve (We say the SDE is strongly 1-completeness).

We also introduced the concept of strong p-compeletenss, examples of SDEs which are strong p-complete not strongly (p-1) complete are also given. The main existence theorem in the following article uses the moments of the derivative flow (Thm. 4.1). We found that both the growth of the driving vector fields and their derivatives are needed, and they compensate each other. Application to SDE in Euclidean space is in section 6. Application in differentiation hear semi-group is in section 9.
• Strong completeness for a class of stochastic differential equations with irregular coefficients , X. Chen and Xue-Mei Li. Electronic J. of Probability. 19 (2014) no. 91 1-34. article. also in arXiv:1402.5079
• In this article, we separate the problems caused by the lack of regularity in the driving force from the problem that the derivatives' of the driving forces are not bounded, allowing also the ellipticity of the Markov generators decaying at infinity.

### Complete SDEs Lacking Strong Completeness

• Lack of strong completeness for stochastic flows, Xue-Mei Li and Michael Scheutzow. Annals of Probability, vol 39(4), 1407-1421 (2011).
• We give such an example on R2 with one single real valued Brownian motion and bounded smooth driving vector fields, by proving that solutions starting from two nearby points, with the same one dimensional driving noise, becomes progressively independent.

### Other Stochastic Flows

• Reflected Brownian Motion: selection, approximation, and Linearization. M. Arnaudon and Xue-Mei Li. arxiv:1602.00897 Article Electronic Journal of Probability 2017, Vol. 22, paper no. 31, 1-55
• We construct a family of approximate SDEs whose solutions stay in the interior whose damped parallel translations approximate that of a reflected Brownian motion, the latter solves the heat equation for differential 1-forms with absolute boundary conditions. This allows us to easily pass the differentiation formula from manifolds without boundary to manifolds with boundary. The difficulty, besides the non-linearity, is the convergence of continuous stochastic processes with stochastic processes with jumps. On the real line, the approximation selects the Skorohod solution versus the Tanaka solution. The damped parallel transports along the approximated paths, which are sample continuous, converge to a tangent process' along the reflected Brownian motion, with jumps upon its hitting the boundary, at the ends of the excursions of the process to the interior.
• Barycentres of probability measures transported by stochastic flows. M. Arnaudon and Xue-Mei Li. (2005) Annals of Probability, vol 33, No.4, 1509-1543.
• Given a subset of a manifold, push it with the stochastic flow of an SDE. WE study the problem how does the centre of the mass evolve.

### Interplay between Stochastic Processes and their Underlying Spaces

• On extensions of Myers' theorem. Xue-Mei Li.   Bull. London Math. Soc. 27, 392-396. 1995.
• In the article above we prove that the weighted measure exp(2h)dx is finite, hence the existence of a finite invariant probability measure for the Laplace-Beltrami operator with drift the gradient of a function h, under stochastic positivity condition on Ricci-Hess (h). This leads to a theorem on the finiteness of the fundamental group.
• Bounded and L^2 Harmonic Forms on Universal Covers.    Geom.  And Funct. Anal. Vol. 8 , 283-303 (1998). K. D. Elworthy, Xue-Mei Li and and S. Rosenberg.
• In the article above, we further explore the concept of stochastic positivity of a function. The function is question in this article are the Weitzenbock curvatures on differential forms.
• On compactness of manifolds and existence of certain type functional inequalities. Xue-Mei Li and F.-Y. Wang. Infinite Dimensional Analysis, Quantum Probability and Related Topics. Volume 6, pages 29-38, (2003).
• Curvature and topology: spectral positivity.  Methods and Applications of Global Analysis, Ed. Yu Gliklikh. In Voronezh Series on New Developments in Global Analysis.p. 45-60. 1993. K. D. Elworthy, Xue-Mei Li and Steven Rosenberg.

### The Geometry of Diffusions

Please also see the two books The Geometry of Filtering' and On the Geometry of Diffusion Operators and Stochastic Flows'.

### Books on the Geometry of Diffusion Operators and Semi-elliptic Geometry

• On the Geometry of Diffusion Operators and Stochastic Flows. Lecture Notes in Mathematics, volume 1720, Springer-Verlag (1999). K. D. Elworthy, Y. LeJan, and Xue-Mei Li. Introduction
• This is original research, not published elsewhere, on diffusion processes and their associated Riemannian and sub-Riemannian geometry. The main theorem of the book is given in the beginning of the book (page 8), the rest of book is on applications of the theorem.

We describe the main theorem. Consider a sub-elliptic operator of the form sum(X_i)^2+X_0 where X_1, ..., X_p are smooth vector fields. This a gives rise to a sub-bundle, E, of the tangent bundle. The main theorem is: a semi-elliptic second order differential operator without zero order term (diffusion operator) in Hormander form determines a metric linear connection on the sub-bundle of the tangent bundle with respect to its intrinsic metric. Furthermore every metric connection on a sub-bundle is determined this way. This connection is furthermore determined by the following property: if X is the bundle map from the trivial bundle over the manifold to the tangent bundle, the covariant derivative of the vector field, X(e), vanishes in the directions the vector fields are not trivial.

This connection has in general a torsion. For example the vector fields determined by an isometric embedding of a Riemannian manifold determines the Levi-Civita connection. We compute and study this connection: its torsion, its curvatures, Weiztenbock formulas, the symmetry property of its Ricci curvature in associated with properties of its torsion. We study also its semi-groups. A Hormander form diffusion operator of constant rank leads to an semi-elliptic stochastic differential equation (SDE) and thus probabilistic representations for various problems in analysis, especially analysis on path spaces. In particular we have Bismut type formula, Logarithmic Sobolev and Clark-Ocone formula, and decomposition of noise.
• The Geometry of Filtering, Birkhauser, (2010). K. D. Elworthy, Yves Le Jan and Xue-Mei Li.
• This is original research, not published elsewhere, on intertwined diffusion and differential operators on a principal bundle. In this book we study second order differential operators of constant rank, A and B, without zero order term, intertwined by a map from one manifold to another. The intertwining property determines a splitting of the operator B, into the sum of two Hormander type operators, which are called the horizontal operator and the vertical operator. The vertical operator can be thought of being independent' of the operator A, and hence leading to applications to stochastic filtering and to analysis on path spaces for probability measures determined by degenerate diffusion operators. This method are also useful even when the states of the operators are Euclidean spaces.

### Malliavin Calculus

Following the elementary proof for Bismut formula in my Ph.D. thesis, we explore a range of Bismut type formulae, along with their applications. See also the following articles Generalised Brownian bridges: examples', Xue-Mei Li, arxiv:1612.08716 (2016) and On Hypoelliptic bridge' Aticle  Xue-Mei Li, Electronic Communications in Probability. Vol 24, no 20, 1-12 (2016); First Order Feynman-Kac Formula. Xue-Mei Li and J. Thompson. (2016) arxiv:1608.03856 Article

### Spectral Gap and Poincare Inequality on Path Spaces

We are concerned with the differential operator d*d where the adjont is taken with respect to the probability distribution of the continuous paths of a Brownian motion or a Brownian bridge in a finite dimensional manifold. On such inequalities path spaces, for measure generated by semi-elliptic (degenerate) diffusion operators, this is also discussed in our books. Here we focus on the Brownian bridge measure on the space of continuous loops over a finite dimensional manifolds.
• A Concrete Estimate For The Weak Poincare Inequality On Loop Space. X. Chen, Xue-Mei Li, and B. Wu. Probab. Theory Relat. Fields (2011) 151:559-590
• A spectral gap for the Brownian bridge measure on hyperbolic spaces X. Chen, Xue-Mei Li, and B. Wu, Progress in analysis and its applications, 398-404, World Sci. Publ., Hackensack, NJ, 2010.
• A Poincare Inequality on Loop spaces, X. Chen, Xue-Mei Li, and B. Wu. J. of Funct. Anal. vol 259 (6), pp 1421-1442 (2010)
• ### Toward an L2 Hodge-de Rham Theory on Infinite Dimensional Spaces

The well known DeRham theorem relates DeRham cohomogolies, a differential geometric object, with singer cohomologies, a topological concept, with far reaching consequences. Toward this aim we first study the existence of an L2 Hodge decomposition theorem on path spaces, manifolds modelled on infinite dimensional Banach spaces. The difficulty is that there is no natural L2 tensor spaces of the tangent spaces.

### Toward a Sobolev Calculus on Path and Loop Spaces

The aim is to establish Sobolev calculus in infinite dimensional spaces where the derivatives are in the sense of Malliavin calculus. The idea is to establish a chart that pulls the corresponding calculus from Wiener space. One technique is using stochastic flows from an SDE whose measure would provide the basis of the analysis, and whose vector fields would induce a good covariant derivative on the finite dimensional manifold, and a stochastic filtering technique keeping all concepts intrinsic.
• Itô maps and analysis on path spaces. K. D. Elworthy and Xue-Mei Li. Math. Zeitschrift. (2007) 257: pp 643--706.
• The stochastic differential equation approach to analysis on path space. New trends in stochastic analysis and related topics, 207-226, (2012)
• Intertwining and the Markov uniqueness problem on path spaces K. D. Elworthy and Xue-Mei Li. In Stochastic Partial Differential Equations and Applications - VII, eds G. Da Prato, L. Tubaro. (2005).
• Gross-Sobolev spaces on path manifolds: uniqueness and intertwining by Itô maps. K. D. Elworthy and Xue-Mei Li. C. R. Acad. Sci. Paris, Ser. I 337 (2003) 741-744.
• Here we attempt to prove Markov uniqueness for the operator d*d. We only managed to prove that the closure of BC2 functions are the same as the closure of smooth compactly supported functions. The gap would be filled if we know the closure of BC1 functions are the same as well.

### Thesis and Dissertation

• Stochastic Flows on Noncompact manifolds (1992) In chapter 2, d+d* is proven to be essentially self-adjoint (on differential forms of all degrees), d* being the L2 adjoint of d w.r.t the measure exp(2h)dx, h a smooth function, dx the Riemmannian volume measure, followed by a Hodge decomposition theorem, commutation of d with the corresponding semi-group, a discussion on recurrent property (chapter 3) and invariant measure. In chapter 4, an over determined heat equation is considered together with a criterion for non-explosion, non inplosion, and C_0 property. In Chapter 5, the open problem for the existence of a global solution flow of an SDE on non-compact manifold is solved. In chapter 6, we define \delta P_t phi(v)=E df (TF_t(v)), where TF_t(v) is the derivative flow of the Brownian flow with drift nabla h, phi a differential 1-form. The questions weather dP_tf=\delta P_t df and \delta P_t df=e^{1/2t]\Delta}df, where f is a smooth function, are studied. differentiation under the expectation is also studied. These are important for the next chapter. In chapter 7, moment stability of Brownian system are studied together with the topological properties of the manifolds, relating to vanishing of harmonic differential forms and cohomologies. In chapter 8, A Bismut type formula are extended to differential forms of degree greater than one, employing the gradient Brownian system. As a by product, a simple elementary proof for Bismut’s formula (for functions ) is given.
• Behaviour at infinity of stochastic solutions and diffusion semi-groups (1989), M.Sc. dissertation.


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# Excel – How to prevent Excel from modifying the file on exit

microsoft excelversion control

Whenever I open an Excel file and then close it without changing anything, the Excel would modify the file without asking a permission or giving any kind of warning. It modifies just 4 bytes at offset 0x046C, and I suspect this might be just the hard-coded last-modified time or something like that.

The problem is that this really screws up the versioning control system, since the file's MD5 sum changes.

Is there any way to stop Excel from doing this? As far as I can see there are no links of formulas or any other dynamic elements in the file.

• I had the same problem, and after several hours reading through several forums and pages I found the root cause of the problem and solution, so I’ll be copy&pasting my answer on the forums I read.

There seems to be an issue with excel files, in particular Excel 2003 xls files, whenever they get opened in Excel, even if they don't get changed and don't get saved, Excel automatically updates some of the file's metadata, such as the "Document Properties and Personal Information" and "Last Accessed Statistics", therefore the file every time it gets opened changes a little bit, and this makes that the MD5 changes also.

Solution:

One way to avoid this problem is to remove document properties and personal information.

To remove the hidden data and personal information from Office documents, in Excel 2007 and for Excel 2013/2010.

Other way to avoid this would be to use .xlsx files. I have been trying to replicate this behavior in .xlsx files, but it seems to only happen on .xls (2003) files.


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1.

DFG is a straight line. What is the value of x?

A. 12
B. 14
C. 16
D. 18
E. 20

2. If the hypotenuse of an isosceles right triangle is 7\sqrt { 2 } , what is the area of the triangle?

(A) 14
(B) 18
(C) 24.5
(D) 28
(E) 49

3.  If p is the perimeter of a triangle with one side of 6 and another side of 9, what is the range of possible values for p?

(A) 3 < p < 15
(B) 15 < p < 24
(C) 18 < p < 30
(D) 18 < p < 42
(E) 21 < p < 42

4. A triangle has sides of 8, m, and n.

 Quantity A Quantity B |m-n| 8

5.

 Quantity A Quantity B The clockwise angle made by the hour hand and the minute hand at 12:15pm The counterclockwise angle made by the hour hand and the minute hand at 12:45pm

6. A 10 foot ladder leans against a vertical wall and forms a 60 degree angle with the floor. Assuming the ground below the ladder is perfectly horizontal, how far above the ground is the top of the ladder?

(A) 5 feet
(B) 5\sqrt { 3 } feet
(C) 7.5 feet
(D) 10 feet
(E) 10\sqrt { 3 } feet

7.  In the figure, if line CE bisects ∠ACB, then x =

(A) 45
(B) 50
(C) 55
(D) 65
(E) 70

8. In the triangle, what is the value of x ?

(A) 25
(B) 55
(C) 60
(D) 77
(E) 85

9. The following are the measures of the sides of five different triangles. Which one of them represents a right triangle?

(A) \sqrt { 3 },\sqrt { 4 },\sqrt { 5 }
(B) 1, 5, 4
(C) 7, 3, 4
(D) \sqrt { 3 },\sqrt { 7 },\sqrt { 4 }
(E) 4, 8, 10

10. In the figure, ABC and ADC are right triangles. Which of the following could be the lengths of AD and DC, respectively?

(I) 3 and 4
(II) 4 and 6
(III) 1 and 24
(IV) 1 and 26
(A) I and II only
(B) II and III only
(C) III and IV only
(D) IV and I only
(E) I, II and III only

11.

 Quantity A Quantity B X Y

12.

 Quantity A Quantity B \frac { x }{ y } 1

13.

Which of the following statements individually provide(s) sufficient additional information to determine the area of triangle ABC above?
Indicate all such statements.

(A) DBC is an equilateral triangle.
(B) ABD is an isosceles triangle.
(C) The length of BC is equal to the length of AD.
(D) The length of BC is 10.
(E) The length of AD is 10.

14. In the figure below, AB=BC=CD. If the area of triangle CDE is 42, what is the area of triangle ADG?

15. What are the lengths of sides NO and OP in triangle NOP below?

1 20
2 C
3 C
4 |m-n|<8
5 B
6 B
7 D
8 A
9 D
10 C
11 D
12 A
13 A and D
14 378
15 NO=30 and OP=10\sqrt { 34 }


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Error in FUN(left, right) : non-numeric argument to binary operator in RStudio
1
0
Entering edit mode
27 days ago
Dimaris • 0

Hello

I'm having a problem that I get the message

Error in FUN(left, right) : non-numeric argument to binary operator in R

This is the code that I'm using for my data

library(vegan)
logespecies2mm <- log(especies2mm+1)

I really appreciate all the help, I have the script if you want more detail but the main goal is that I want to do a RDA graph analysis in two sampling times to answer a question in correlation sorting (sand particles grain size) with the abundance of two main groups bivalves and gastropods. Also I been trying to transform the data without using log, instead use hellinger transformation but in my RStudio it says hellinger object error not found in R. Really appreciate all the help

rstudio • 282 views
0
Entering edit mode

You really think people can troubleshoot this without knowing anything at all about what's in the especies2mm object?

0
Entering edit mode
26 days ago
ATpoint 49k

Your data.frame is not of type double (numeric) or integer so you cannot apply a mathematical / log transformation function:

> log2(data.frame(A="A")+1)
Error in FUN(left, right) : non-numeric argument to binary operator

https://stackoverflow.com/questions/2288485/how-to-convert-a-data-frame-column-to-numeric-type


text:
1. ## dimension

Hi, i need some help in understanding the following question:

Let V be the vector space of all 2x2 matrices over a field F. Show that V has dimension 4 over F.

2. ## Re: dimension

Try to find a basis for V.

3. ## Re: dimension

Originally Posted by alexmahone
Try to find a basis for V.
ok, to find a basis for V, i must show that it is linearly independent and that it spans the vector space.

but where and how do i start to do this?

4. ## Re: dimension

Originally Posted by ibnashraf
ok, to find a basis for V, i must show that it is linearly independent and that it spans the vector space.

but where and how do i start to do this?
How about four 2x2 matrices, each with one 1 (at each position) and three 0's?


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# [tex-live] lua path

Reinhard Kotucha reinhard.kotucha at web.de
Tue Dec 10 02:31:08 CET 2013

On 2013-12-09 at 20:29:20 +0000, Robin Fairbairns wrote:

> Philip Taylor <P.Taylor at Rhul.Ac.Uk> wrote:
>
> > Since, in the opinion of the TeX Live cadre, most questions
> > are stupid, if one is to judge by the responses that Norbert
> > et al are wont to give, then the simplest thing would be
> > to modify TeXdoc to simply display the text "If you need to
> > ask, you are too stupid to use TeX", regardless of any
> > parameter(s) passed.
> >
> > >> Why is anyone asking texdoc about perl? IMNSHO, it's perfectly okay for
> > >> texdoc to give a stupid answer to a stupid question.
> > >
> > > +1
> >
> > Maxwell, Adam R wrote:
> >
> > > Where does that end, though? How about texdoc {python,tcl,r,ruby,sh}?
> > > Why is anyone asking texdoc about perl? IMNSHO, it's perfectly okay
> > > for texdoc to give a stupid answer to a stupid question.
> >
> > -\infty.  There are no stupid questions, there are only stupid
> > answers, of which yours (IMENSHO) is just about as stupid as an
> > answer can get.
>
> gasp ... robin agrees with philip (on a matter other than music!)
>
> a text for texdoc perl, explaining tex live's perl (and why an actual
> user shouldn't want it) surely doesn't need more than a couple of
> paragraphs.  (if that.)
>
> i appreciate how little most tex experts like to think of the average
> user (who has no control of his tex system, and doesn't know how to
> install hir own).  i don't think tex live should go that way.

I prefer not to mention Perl at all.  There is no need to bother
innocent users with the details and experienced users will find it out
anyway.  I fear that documentation can be understood as advertising.
TeX Live doesn't provide Perl for general use.  Period.

Phil successfully called the hidden Perl by specifying an absolute path.
But it doesn't work this way.  All you can achieve this way is to get
a help or usage message, retrieve Perl's version number, or maybe you
can use the -e option in order to print "Hello World!" to screen.

If the hidden Perl is invoked this way it's not able to find any
of its modules.  As I said before, this is on purpose.  TeX Live is
not a Perl distribution.

The biggest problem is that if the *deliberatly* hidden Perl is
mentioned in the documentation, it leads to wrong expectations.

Assume that it simply doesn't exist.

Regards,
Reinhard

--
----------------------------------------------------------------------------
Reinhard Kotucha                                      Phone: +49-511-3373112
Marschnerstr. 25
D-30167 Hannover                              mailto:reinhard.kotucha at web.de
----------------------------------------------------------------------------
Microsoft isn't the answer. Microsoft is the question, and the answer is NO.
----------------------------------------------------------------------------


text:
The ratio of the whole surface areas of a hemisphere and a sphere of same radius is

Free Practice With Testbook Mock Tests

Options:

1. 4 ∶ 3

2. 2 ∶ 3

3. 3 ∶ 4

4. None of the above

Correct Answer: Option 3 (Solution Below)

This question was previously asked in

Official Paper 7: Tripura TET 2017 Paper 1

Solution:

Concept

Total surface area of sphere = 4πr2

Total surface area of hemisphere = 3πr2

Calculation

Ratio of total surface area of a hemisphere to surface area of sphere = 3πr2 : 4πr2

⇒ 3 : 4


text:
# something that looks sort of symmetrical but also not

Given the set $S_0$ of finite binary strings whose digit sum is congruent to 0 mod 2 and the set $S_1$ of finite binary strings whose digit sum is congruent to 1 mod 2,

what are the implications of the fact that $F: \{s_1 \in S_1 : s_1 \mbox{ends in 1} \} \to S_0$ that removes the trailing 1 from $s_1$ is onto $S_0$ but $“F^{-1}” : \{s_0 \in S_0 \} \to S_1$ that appends a 1 to the end of $s_0$ is not onto $S_1$?

-
Didn't you mean mod 2? – tomasz Jun 19 '12 at 21:56
Also, what do you mean by "what are the implications"? – tomasz Jun 19 '12 at 21:59
When you write or edit a post, the result is shown below. you don't need to save the edit each time to see what happens... – tomasz Jun 19 '12 at 22:03
@tomasz I'm on my mobile browser... And yes I did want to say mod 2 :) – idonutunderstand Jun 19 '12 at 22:06
I just saw a silver notification indicating there was a suggested edit but it said "rejected" at the bottom? – idonutunderstand Jun 19 '12 at 22:12

In a sense, this shows that you can add or subtract a single element from an infinite set and still have a bijection between the domain and range. This is not true for finite sets. If we allow $0 \in \mathbb N$, consider $f(x)=x+1$ on $\mathbb N$. $f$ is not onto, but $f^{-1}$ is. This is equivalent to your example, but may be less surprising. One's view of the implications can range from "trivial" to "the base of all the theory of infinite sets".
Should your $f^{-1}$ be in quotes or something since it doesn't map 0 to anything? – idonutunderstand Jun 19 '12 at 22:38
@AbstractionOfMe: $f^-1$ has a domain of $\mathbb N \setminus \{0\}$, it is true. It is still a fine function. This is similar to the fact that your $F$ does not have a domain of all of $S_1$, which you made explicit. It maps "half of $S_1$" onto $S_0$. – Ross Millikan Jun 19 '12 at 22:48


text:
過去の記録

2010年01月08日(金)

談話会・数理科学講演会

16:30-17:30   数理科学研究科棟(駒場) 123号室
お茶&Coffee&お菓子: 16:00～16:30 (コモンルーム)

[ 講演概要 ]

2010年01月07日(木)

作用素環セミナー

16:30-18:00   数理科学研究科棟(駒場) 128号室
Luc Rey-Bellet 氏 (Univ. Massachusetts)
Large deviations, Billiards, and Non-equilibrium Statistical Mechanics

GCOEセミナー

16:30-18:00   数理科学研究科棟(駒場) 128号室
LucRey-Bellet 氏 (Univ. Massachusetts)
Large deviations, Billiards, and Non-equilibrium Statistical Mechanics
[ 講演概要 ]
Large deviations have applications in many aspects of statistical mechanics. New applications for the steady states of non-equilibrium statistical mechanics have emerged during the past ten years and these applications deal with the fluctuations of the entropy production. After discussing some general aspects of entropy production we turn to concrete examples, in particular billiards with and without external forces and discuss large deviations theorems for such systems.

2010年01月05日(火)

トポロジー火曜セミナー

16:30-18:30   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

The volume growth of hyperkaehler manifolds of type $A_{\\infty}$
[ 講演概要 ]

複素解析幾何セミナー

16:30-18:00   数理科学研究科棟(駒場) 126号室

Alan Huckleberry 氏 (Ruhr-Universität Bochum)
Hyperbolicity of cycle spaces and automorphism groups of flag domains

応用解析セミナー

16:00-17:30   数理科学研究科棟(駒場) 002号室
Hatem Zaag 氏 (CNRS / パリ北大学)
A Liouville theorem for a semilinear heat equation with no gradient structure
[ 講演概要 ]
We prove a Liouville Theorem for entire solutions of a vector
valued semilinear heat equation with no gradient structure. Classical tools such as the maximum principle or energy techniques break down and have to be replaced by a new approach. These tools involve a very good understanding of the dynamical system formulation of the equation in the selfsimilar setting. Using the Liouville Theorem, we derive uniform estimates for blow-up solutions of the same equation.

2009年12月16日(水)

統計数学セミナー

15:00-16:10   数理科学研究科棟(駒場) 128号室
Stefano Maria Iacus 氏 (Department of Economics, Business and Statistics, University of Milan, Italy)
ecent results on volatility change point analysis for discretely sampled stochastic differential equations
[ 講演概要 ]
In this seminar we review recent advances on change point analysis for the volatility component of stochastic differential equations under different discrete sampling schemes. We consider both ergodic and high frequency and non ergodic cases. Results have been obtained by means of least squares, CUSUM tests and quasi-maximum likelihood approach. We show an application to the recent financial crisis and finally present a Monte Carlo study to compare the three methods under different setups.

Join work with Prof. Nakahiro Yoshida.
[ 講演参考URL ]
http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/12.html

2009年12月15日(火)

Lie群論・表現論セミナー

17:00-18:00   数理科学研究科棟(駒場) 056号室
トポロジー火曜セミナーと合同です。この週に砂田利一氏の集中講義が行われます

Open Problems in Discrete Geometric Analysis
[ 講演概要 ]
Discrete geometric analysis is a hybrid field of several traditional disciplines: graph theory, geometry, theory of discrete groups, and probability. This field concerns solely analysis on graphs, a synonym of "1-dimensional cell complex". In this talk, I shall discuss several open problems related to the discrete Laplacian, a "protagonist" in discrete geometric analysis. Topics dealt with are 1. Ramanujan graphs, 2. Spectra of covering graphs, 3. Zeta functions of finitely generated groups.
[ 講演参考URL ]

トポロジー火曜セミナー

17:00-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:40 - 17:00 コモンルーム

Open Problems in Discrete Geometric Analysis
[ 講演概要 ]
Discrete geometric analysis is a hybrid field of several traditional disciplines: graph theory, geometry, theory of discrete groups, and probability. This field concerns solely analysis on graphs, a synonym of "1-dimensional cell complex". In this talk, I shall discuss several open problems related to the discrete Laplacian, a "protagonist" in discrete geometric analysis. Topics dealt with are 1. Ramanujan graphs, 2. Spectra of covering graphs, 3. Zeta functions of finitely generated groups.

2009年12月14日(月)

統計数学セミナー

14:00-15:10   数理科学研究科棟(駒場) 128号室
L. VOSTRIKOVA 氏 (LAREMA, Departement de Mathematiques, Universite d’Angers, FRANCE)
On the stability of contingent claimes in incomplet models under statistical estimations.
[ 講演概要 ]
In exponential semi-martingale setting for risky asset we estimate the difference of prices of options when initial physical measure P and corresponding martingale measure Q change to tilde{P} and tilde{Q} respectively. Then, we estimate L1 distance of option’s prices for corresponding parametric models with known and estimated parameters. The results are applied to exponential Levy models with special choise of martingale measure as Esscher measure, minimal entropy measure and f^q -minimal martingale measure. We illustrate our results by considering GMY and CGMY models.
[ 講演参考URL ]
http://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2009/11.html

代数幾何学セミナー

14:40-16:10   数理科学研究科棟(駒場) 126号室
いつもと時間帯が異なります。ご注意ください。
Sergey Galkin 氏 (IPMU)
Invariants of Fano varieties via quantum D-module
[ 講演概要 ]
We will introduce and compute Apery characteristic
class and Frobenius genera - invariants of Fano variety derived from
it's Gromov-Witten invariants. Then we will show how to compute them
and relate with other invariants.

2009年12月11日(金)

GCOE社会数理講演シリーズ

16:20-17:50   数理科学研究科棟(駒場) 117号室

2009年12月10日(木)

講演会

10:40-12:10   数理科学研究科棟(駒場) 128号室

作用素環セミナー

16:30-18:00   数理科学研究科棟(駒場) 128号室

Symmetric norms and spaces of operators modelled on a semifinite von Neumann algebra


text:
# Translating custom \section format to titlesec

I have some custom \section formatting with a horizontal rule, and I'm trying to DRY out my code by using titlesec to define default formatting options. But I can't exactly reproduce what happens when not using titlesec. Here's an example:

\documentclass{article}
\usepackage[explicit]{titlesec}
\usepackage{xhfill}

\begin{document}

% Plain xrfill:
\section*{\large\normalfont First Section \xrfill[0.6ex]{0.4pt}}
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.

% Via titlesec:
\titleformat{\section}[runin]{\large}{\thetitle}{1em}{#1 \xrfill[0.6ex]{0.4pt}}
\section*{Second Section (\texttt{titlesec})}
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.

% Titlesec with a \\ at the end:
\titleformat{\section}[runin]{\large}{\thetitle}{1em}{#1 \xrfill[0.6ex]{0.4pt}\\}
\section*{Third Section (line break)}
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.

\end{document}

Note that using titlesec doesn't put the following text in a new paragraph. I tried to solve this with \\ but that doesn't work either. What can I do to make this work the same way as it would without titlesec?

-

Use the [display] style instead of the [runin] style:

\titleformat{\section}[display]{\large}{\thetitle}{1em}{#1\space\xrfill[0.6ex]{0.4pt}}
\section*{Second Section (\texttt{titlesec})}
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor
incididunt ut labore et dolore magna aliqua.

-
Great, thanks! But the titlesec documentation seems to claim that runin mimics the default behavior. So why wouldn't this work with runin? – jtbandes Dec 25 '12 at 5:54
The [runin] style is equivalent to the standard classes' \paragraph style. The default style (which in your example would also work) is [hang], which is like the standard \section command (see p.3 of the docs). Because of the rule, I think that the [display] is more appropriate, though. – Alan Munn Dec 25 '12 at 6:04


text:
# Resistors 5

Solution
The effective resistance between points $A$ and $B$ is $\frac{10}{7}\Omega$. To arrive at this result, the easiest method was to label the current passing through each resistor $I_1$ to $I_{18}$ and then create equations by considering each node. The equations are shown in the diagram below. Solution of these equations was then completed using Mathematica (largely because I'm lazy).

Equations A through K come from consideration of the nodes labelled in the diagram (foolishly I've not been consistent with my labelling in relation to the original problem, but hopefully its still easy enough to follow).

The latter equations come from considering the voltage around each of the small triangles (by applying Kirchoff's second law (the potential differences around any closed loop add to zero)).

The solution is over specified, we have 19 equations in 18 variables, so one equation can be legitimately ignored or used as a check of the final answer.


text:
Mathematician:Reinhold Remmert

Jump to: navigation, search

Mathematician

German mathematician whose work has mainly been in developing the theory of complex spaces.

German

History

• Born: June 22, 1930, Osnabrück, Germany

Publications

• Classical Topics in Complex Function Theory


text:
# PH¶

## Description¶

Plugin for the Quantum Espresso ph.x executable.

## Supported codes¶

• tested from ph.x v5.0 onwards.

## Inputs¶

• parent_calculation, can either be a PW calculation to get the ground state on which to compute the phonons, or a PH calculation in case of restarts.

Note: There are no direct links between calculations. The use_parent_calculation will set a link to the RemoteFolder attached to that calculation. Alternatively, the method use_parent_folder can be used to set this link directly.

• qpoints, class KpointsData Reciprocal space points on which to build the dynamical matrices. Can either be a mesh or a list of points. Note: up to QE 5.1 only either an explicit list of 1 qpoint (1 point only) can be provided, or a mesh (containing gamma).

• parameters, class ParameterData Input parameters of ph.x, as a nested dictionary, mapping the input of QE. Example:

{"INPUTPH":{"ethr-ph":1e-16},
}

A full list of variables and their meaning is found in the ph.x documentation.

Following keywords are already taken care of by AiiDA:

'INPUTPH', 'outdir': scratch directory
'INPUTPH', 'prefix': file prefix
'INPUTPH', 'iverbosity': file prefix
'INPUTPH', 'fildyn': file prefix
'INPUTPH', 'ldisp': logic displacement
'INPUTPH', 'nq1': q-mesh on b1
'INPUTPH', 'nq2': q-mesh on b2
'INPUTPH', 'nq3': q-mesh on b3
'INPUTPH', 'qplot': flag for list of qpoints

• settings, class ParameterData (optional) An optional dictionary that activates non-default operations. Possible values are:

• ‘PARENT_CALC_OUT_SUBFOLDER’: string. The subfolder of the parent scratch to be copied in the new scratch.
• ‘PREPARE_FOR_D3’: boolean. If True, more files are created in preparation of the calculation of a D3 calculation.
• ‘NAMELISTS’: list of strings. Specify all the list of Namelists to be printed in the input file.
• ‘PARENT_FOLDER_SYMLINK’: boolean # If True, create a symlnk to the scratch of the parent folder, otherwise the folder is copied (default: False)
• ‘CMDLINE’: list of strings. parameters to be put after the executable and before the input file. Example: [“-npool”,”4”] will produce ph.x -npool 4 < aiida.in
• ‘ADDITIONAL_RETRIEVE_LIST’: list of strings. Extra files to be retrieved. By default, dynamical matrices, text output and main xml files are retrieved.

## Outputs¶

There are several output nodes that can be created by the plugin, according to the calculation details. All output nodes can be accessed with the calculation.out method.

• output_parameters ParameterData Contains small properties. Example: dielectric constant, warnings (possible error messages generated in the run). calculation.out.output_parameters can also be accessed by the calculation.res shortcut. Furthermore, various dynamical_matrix_* keys are created, each is a dictionary containing the keys q_point and frequencies.

## Errors¶

Errors of the parsing are reported in the log of the calculation (accessible with the verdi calculation logshow command). Moreover, they are stored in the ParameterData under the key warnings, and are accessible with Calculation.res.warnings.


text:
# rational exponents? Solve please

• Jun 2nd 2012, 06:36 AM
zbest1966
rational exponents? Solve please
(-3a1/4)(9a)-3/2
• Jun 2nd 2012, 06:53 AM
Reckoner
Re: rational exponents? Solve please
Quote:

Originally Posted by zbest1966
(-3a1/4)(9a)-3/2

Here are some properties you should know:

$a^ma^n = a^{m+n}$

$(ab)^n = a^nb^n$

$a^{m/n} = \sqrt[n]{a^m}$
• Jun 2nd 2012, 07:03 AM
zbest1966
Re: rational exponents? Solve please
I understand the properties but how do you solve it.
• Jun 2nd 2012, 07:25 AM
Reckoner
Re: rational exponents? Solve please
Quote:

Originally Posted by zbest1966
I understand the properties but how do you solve it.

If you understood the properties, I suspect you would not be having difficulties with the problem. By the way, we have nothing to "solve" here, this is not an equation. Rather, we are "simplifying" the expression.

$\left(-3a^{1/4}\right)(9a)^{-3/2}$

$=-3a^{1/4}\cdot9^{-3/2}a^{-3/2}$

$=-\frac{3a^{1/4}}{9^{3/2}a^{3/2}}$

$=-\frac{3a^{1/4}}{27a^{3/2}}$

Now, use the properties $\frac{a^m}{a^n} = a^{m-n}$ and $a^{-n} = \frac1{a^n}$ to finish it:

$=-\frac{a^{-5/4}}{9}=-\frac1{9a^{5/4}}$

As a final step, you may want to convert $a^{5/4}$ to radical form, depending on the problem's instructions.
• Jun 2nd 2012, 07:51 AM
HallsofIvy
Re: rational exponents? Solve please
Quote:

Originally Posted by zbest1966
I understand the properties but how do you solve it.

You could actually apply the properties you say you understand or you could ask someone else to solve it for you. I see you chose the second solution method.


text:
# A quaternion valued elliptic complex

This blog entry delivers an other example of an elliptic complex which can be used in discrete Atiyah-Singer or Atiyah-Bott type setups as examples. We had seen that when deforming an elliptic complex with an integrable Lax deformation, we get complex elliptic complexes. We had wondered in that blog entry whether a complex can lead to quaternion-valued fields. The discussion which follows here is all in a setting where the underlying geometry is a finite abstract simplicial complex. It almost verbatim would apply also to elliptic complexes over Riemannian manifolds.

## Quantum dynamics

Quantum mechanics studies the unitary evolution $U(t)=exp(i D t)$, where D is selfadjoint operator on a Hilbert space. Given a state $\psi(0)=u+iv$ encoding the initial situation, one gets the future $\psi(t) = exp(i D t) \psi = (\cos(D t) + i \sin(D t)) (u+iv)$ so that $u(t) = \cos(D t) u - \sin(D t) v$ solves the wave equation $u_{tt} = - D^2 u$ with initial condition $u(0)=u$ and initial velocity $u'(0)=D v$. The analogy is perfect if $D^2$ is a Laplacian. Since the Schrödinger equation $i \psi'= -D\psi$ for a complex wave $\psi=u+iv$ is equivalent to the wave equation for a real valued wave, quantum mechanics is also called “wave mechanics”. It is quite amazing that on a fundamental level, the mathematics of quantum mechanics is simpler than classical mechanics, where one has non-linear dynamical systems. It is also remarkable also that on a fundamental level, complex valued fields appear naturally. The unitary dynamics is at the core of quantum mechanics. When discussing the system itself, one can avoid discussing the role of measurements. There are still many mathematical problems: like the relation between spectral properties of D and the dynamical transport properties of the wave. Even before that there is the problem to determine the nature of the spectrum for concrete Hamiltonians in the continuum. In applications like chemistry, one wants to make quantitative predictions about chemical properties of matter, in cryptological applications one wants to use the system secure message transmission and in computer science one aims to use the dynamics for making computations. Still, the underlying dynamical system could not be simpler: the trajectory of simple path in an unitary group. But this evolution is not the most general one as there is a symmetry present. But any unitary conjugation of D produces a system which is equivalent. We just watch it from a different point of view. Principles of relativity have historically been very successful. So much that they appear to be a cliché (but that should note make them less important). It is also entrenched in quantum mechanics: unitarily equivalent operators produce equivalent physics as chosing a particular coordinate system is just chosing a particular basis in the Hilbert space.

## Quaternion valued waves

By results of Hurwitz and Frobenius, the list of associative complete division algebras contains only two elements: the complex numbers and the quaternions. This is related to the fact that spheres in Euclidean spaces Rn are connected Lie groups only if n=2 or n=4. Both completeness as well as associativity are important in a dynamical quantum frame work. This excludes the other two division algebras: the real line is not complete and the octonion algebra is not associative. As complex-valued field naturally enter quantum dynamics, it is natural to ask whether quaternion-valued fields are relevant. It was Rudolf Fueter, a student of David Hilbert, who has investigated the Dirac equation for quaternion valued fields and saw it as a Cauchy-Riemann type equation. All three, Adolf Hurwitz, Ferdinand Frobenius and Rudolf Fueter were Zurich-based for part of their careers.

Do quaternion-valued waves appear naturally in quantum dynamics? If one would insist also on commutativity, the complex plane would be the unique division algebra, but in a quantum frame work, we really do not want to insist on commutativity. So, we have to look at quaternion-valued fields too. But they have to appear naturally in the sense that we don’t want to place the structure in artificially. It should emerged by itself! Now there appears a superficial difficulty with quaternion-valued matrices like that one can not define naturally a determinant for such matrices. But this is no problem as we can look at complex $n \times n$ matrices as real $2n \times 2n$ matrices, we can also look at quaternion-valued $n \times n$ matrices as complex-valued $2n \times 2n$ matrices. This is known as under the name of Cayley-Dickson construction and allows us to deal with complex matrices.

## From geometry to dynamics

Given a geometric space like a simplicial complex or CW complex G or a ring element generated by such objects, there are natural incidence matrices d and so a Dirac operator D and a Laplacian L=D2. More generally we have elliptic complexes defined over G. What matters is the spectrum of D as well as how the eigenfunctions are linked to geometry. One way how D links to geometry is that D determines the geometry: it is possible to measure distances on all dimension level using the Connes formula. Given two simplices x,y, their distance is the supremum over all $|f(x)-f(y)|$, where f runs over all functions satisfying $|Df|\leq 1$. This formula links the algebra with distances in the geometry. Now, if we let the operator evolve freely in its symmetry group, we can get complex valued or even quaternion valued operators. What quantum dynamics should we chose? Given any unitary path V(t) we can look at D(t) = V(t)* D(0) V(t). Now taking V(t)=exp(i tD) or any V(t)=exp(i t f(D)) does not work as V(t) commutes with D so that there is no motion. So, if we want to move, the deformation of the operator D has to be done with something not obtained directly from the functional calculus of D. What works is a Lax deformation, or equivalently introducing a symplectic structure. A functional H(L)=tr(h(L)) can be seen as a Hamiltonian function. Its gradient is h'(L). The analogue of a symplectic matrix J changing the classically the gradient flow $u'=\nabla h(u)$ on the cotangent bundle into a Hamiltonian system $u'=J \nabla h(u)$ is replacing d/dt L = h'(L) with d/dt L = [B(L),h'(L)].

## A tumbling rock analogy

As a motivation for the following lets look at the analogy of a rock moving through space without the influence of any external force. Lets for simplicity assume the center of mass is fixed. This can be achieved by translating into a non-accelerating coordinate system of the rock. In order not to be distracted by particularities inherent to three dimensional space (like cross product), it is helpful to look at a more general Lie group setup and look for example at a rock tumbling through n dimensional space. Its motion is described by a geodesic in the orthogonal group SO(n) which is a Lie group. The motion is best described when describing an element in its Lie algebra so(n), which is the tangent space. We get now a Hamiltonian system L’=[B,L] =J dH(L), where B(L) depends on the geometry of the rigid body moving. There is a moment of inertia mass tensor M, so that B=M(L). Geometrically, the inertia tensor defines a Riemannian structure on the tangent bundle TSO(n) and the free body motion is a geodesic with respect to this structure. The Hamiltonian system takes place on the cotangent bundle of the tangent bundle of the rotation group. This is Vladimir Arnold’s picture of the free body motion. The tumbling block picture is only an analogy. but it explains something important: we do not have to justify why the rock tumbles in space. It does because it can and because it is a very improbable situation if it does not make use of the symmetry group (here SO(n)) it is placed in. A rotating coordinate system has effects on the rock. There are forces appearing like the centrifugal or Coriolis force. [By the way, I learned these basic forces in a mechanics course given by Juerg Froehlich. It was one of the best courses at ETH. Froehlich covered lots of ground up to Arnold’s theorem telling that a completely integrable Hamiltonian system must be a translation on a torus. The course early on also covered all basic forces in mechanics. In the course, these forces were introduced in a way that they can be understood in arbitrary dimensions using matrix computations. They are usually explained only in $R^3$, where one has a cross product. I had been very impressed by the power of mathematics being able to cover physics situations which are not realized in reality as we live in a three dimensional world, where a cross product exists. This course was given to Sophomores and as custom, we were examined orally. Fröhlich set up the exams with “Pflicht” und “Kuehr”, meaning that some questions were given by him but that there was a free component in which the student could present anything. I had been reading Arnold’s book “Methods of classical mechanics”, which is a treasure trove containing appendices, where many have later become research areas. One of the appendices dealt with the motion of a free body and its infinite dimensional limit, the Euler equations of a fluid. As just mentioned, it deals with cotangent bundles of tangent bundles. I must have done a rather poor job explaining this in first few minutes of the allocated 15 minutes for that component. Froehlich drew the Eifel tower on the blackboard (he had been working near Paris at IHES for many years), painted a rotating top on its tip and asked me: “How does it move?”. After the exam, he told me: “Herr Knill, the Bourbaki times are over”. He was right. A time where mathematics has been very concrete covering basic examples has been in full swing and remained so for some time. Of course, Bourbaki has come back in recent years, much stronger and almost with a vengence. In a talk of 2015 [PDF] it was painted as oscillations between Descartian and Baconian times (terms coined by Freeman Dyson) but it could now also be different: the concrete non-Bourbaki mathematics has drifted more and more to computer science, applied mathematics or statistics departments.

## Deformation of a Quaternion complex

Back to the quantum case, we want a Lax system D’=[B,D], in order to have an isospectral deformation. Given a real differential complex $d: E \to F$ we can form $D=d+d^*$ and deform it with $D'=[B,D]$, where $B=d-d^*+I b$, where $I=oi+pj+qk$ is a space quaternion. This gives a deformation in a complex plane spanned by 1 and the normalized quaternion I/|I|. Since I2 =-|I|^2, the deformation stays in that plane. We can also form a quaternion-valued complex by taking three. Let $D_1,D_2,D_3$ be real elliptic complexes which commute, define the quaternion valued complex $D=i D_1 + j D_2 + k D_3$. We initially have the Pythagorean relation $L=D^2 = -L_1^2 - L_2^2 - L_3^2$. Now, as suggested in the particles and primes note, one could think of deforming each of the complexes $D_1,D_2,D_3$ isospectrally. This works and we indeed get an isospectral deformation of D, but only if the complex parameter $\beta$ in the deformation $D(t)=d(t)+d(t)^* + b(t), B(t)=d(t)-d(t)^* + i \beta b(t)$ is zero. In that case, we evolve each of the three space components independently. But if we turn on the $i \beta b(t)$ part, the real parts of the three complex planes which intersect start to interact and D is no more isospectral. In the “particles and primes” paper, we were led to such a deformation by the analogy of neutrino oscillations. Neutrini in that paper were the real primes in a complex plane and if the three complex planes are interpreted as different generations of particles the association is natural. But we stay in mathematics and want an isospectral deformation of the quaternion complex. This can be achieved as follows. Choose a space quaternion $I=\alpha i + \beta j + \beta k$ and deform the quaternion valued operator with the dynamical system D’ = [B,D], where if $D=d+d^*+b$, then $B=d-d^* + I b$. The guiding quaternion $I$ could be made time dependent so that we have an $I(t)$. This would already have been possible in the complex case, where the coupling $\beta(t)$ appearing in B could be made time dependent. For the experiments done so far, we just took a constant I=i so far. A time dependent coupling constant is not just a time change. It guides the coupling between the kinetic part of the deformed D, which corresponds to a new exterior derivative and the potential part of the deformed D, which corresponds to a “dark matter” part which has no geometric interpretation except that it converges to the square root of the Laplacian which drives the wave equation in the d’Alembert setup in the limit of large time. But we have already without this a lot of choice as there are many commuting isospectral deformations. As in Hamiltonian dynamics, one could make the Hamiltonian time dependent and still get a symplectic isospectral deformation.

## Actual implementation

When playing with that system on the computer, quaternions are best implemented as complex $2 \times 2$ matrices using Pauli matrices $\sigma_i$. [Pauli is an other “Zürich connection” joining Fueter or Hurwitz even so timely later. I’m personally convinced that this is not an accident. Mathematicians and physicists influenced each other much more at that time when they were geographically close. Today of course, science has become global.] The real number 1 is the identity matrix, the standard imaginary square root of -1 is $i \sigma_3$, the quaternion $j$ is $i \sigma_2$ and the quaternion $k$ is $i \sigma_1$. In other words, the explicit translation from a quaternion a+ib+jc+kd to a complex $2 \times 2$ matrix $\left[ \begin{array}{cc} a+ib & c+id \\ -c+id & a-ib \end{array} \right]$. The complex parameter I is now just an arbitrary element in the Lie algebra of SU(2). As mentioned before, it could be made time dependent. In the following we just take I=i. To illustrate the story, lets take the smallest simplicial complex which leads to a deformation. It is the Whitney complex of the complete graph $K_2$. global Dirac operator is $\left[ \begin{array}{ccc}0 & 0 & -1 \\0 & 0 & 1 \\-1 & 1 & 0 \\ \end{array} \right]$. To evolve the elliptic complex into the quaternion domain, take three copies of this and glue it together to get the new complex which is now quaternion valued:

$D=\left[ \begin{array}{cccccc} 0 & 0 & 0 & 0 & -i & -1-i \\ 0 & 0 & 0 & 0 & 1-i & i \\ 0 & 0 & 0 & 0 & i & 1+i \\ 0 & 0 & 0 & 0 & -1+i & -i \\ -i & -1-i & i & 1+i & 0 & 0 \\ 1-i & i & -1+i & -i & 0 & 0 \\ \end{array} \right]$

Its Laplacian $L=D^2$ is just the $-3$ times the original Hodge Laplacian, where each entry 1 is replaced with a $2 \times 2$ identity matrix. It is evident that all the Betti numbers have just doubled. No new topology has emerged by building the quaternion complex and since the deformation is isospectral, this does not happen also when deforming this complex. While topology has not changed, it is the metric which has changed. The isospectral deformation of calculus produced a deformation of distance. It is evident from the isospectral nature that the deformed complex is an elliptic complex. What however is interesting dynamically that we get a natural quaternion-valued wave evolution in the limit. Again, this had not to be constructed. It emerged just naturally by allowing the deformation to happen in the quaternion space. [One question is then why a complex chooses to evolve in the quaternion space or why it does not stay in the complex plane. If we look at the “particle and prime” allegory then both should happen. In that picture, the complex case describes Leptons, while the quaternion case describes Baryons. But this was a combinatorical cartoon.] Anyway, here is the deformed operator at time t=0.1:

$\left[\begin{array}{cccccc}0.\,+0.92i&0.&0.\,-0.92i&0.&-0.14-0.65i&-0.78-0.51i\\0.&0.\,+0.92i&0.&0.\,-0.92i&0.51\,-0.78i&0.14\,+0.65i\\0.\,-0.92i&0.&0.\,+0.92i&0.&0.14\,+0.65i&0.78\,+0.51i\\0.&0.\,-0.92i&0.&0.\,+0.92i&-0.51+0.78i&-0.14-0.65i\\0.14\,-0.65i&-0.51-0.78i&-0.14+0.65i&0.51\,+0.78i&0.\,-1.84i&0.\\0.78\,-0.51i&-0.14+0.65i&-0.78+0.51i&0.14\,-0.65i&0.&0.\,-1.84i\\\end{array}\right]$

We made the evolution with a Runge-Kutta integration so that $D^2 - \overline{D}^2$ has entries smaller than 10-12. Also the maximal difference between eigenvalues of $D$ and $\overline{D}$ is of this order.

So, we have an isospectral deformation D(t) but there is one problem which would have been hard to catch if not tried out: the deformed entries in the off diagonal part are not quaternions. The matrix D(t) is a complex 6 x 6 matrix isospsectral to D(0) but it is not a quaternion 3 x 3 matrix. The matrix entry $D_{33}$ for example which originally at t=0 was a space quaternion has developed a real diagonal entry which is not a multiply of the identity matrix as the diagonal parts in that matrix have opposite signs.

What was the problem? It is a bit more than a programming error: if we write $i b$ where $b$ is a quaternion written in matrix form, we can not just multiply the matrix by i because the quaternion i is implemented by a 2×2 diagonal matrix with i, -i in the diagonal. So, as we have decided to implement quaternions using 2×2 matrices, we also have to implement the scalar multiplication with i as such. The isospectral deformation was fine and produced a deformation of the elliptic complex. It was just that if we take the quaternion 2×2 matrix $\left[ \begin{array}{cc} a+i b & c+i d \\i d-c & a-i b \\ \end{array} \right]$ and multiply it with i, then it is no more a quaternion 2×2 matrix. Multiplication by i is a multiplication with a diagonal matrix, where each diagonal is $\left[ \begin{array}{cc} i &0 \\0 & -i \\ \end{array} \right]$ as this preserves the quaternion structure. Now, the Lax differential equation is a differential equation in a module over the quaternion algebra and the resulting deformation will be a quaternion valued complex.

Here is the deformation up to t=0.1, but where things are done correctly. We have $D=d+d^* + b$, where now both d and b are quaternion valued. As the “dark energy” b is not real after the deformation, we take $B=d-d^* + i (b+b^*)$, but where now i the quaternion i. It could be replaced by any unit length quaternion I as such a quaternion still satisfies $I^2=-1$.

$\left[ \begin{array}{cccccc} 0\,+0.36i&0.38\,+0.33i&0\,-0.36i&-0.38-0.33i&-0.19-0.85i& -0.86-0.86i\\ -0.38+0.33i&0\,-0.36i&0.38\,-0.33i&0\,+0.36i&0.86\,-0.86i &-0.19+0.85i\\ 0\,-0.36i&-0.38-0.33i&0\,+0.36i&0.3 \,+0.33i&0.19\,+0.85i &0.86\,+0.86i\\ 0.38\,-0.33i&0\,+0.36i&-0.38+0.33i&0\,-0.36i&-0.86+0.86i& 0.19\,-0.85i\\ 0.19\,-0.85i&-0.86-0.86i&-0.19+0.85i&0.86\,+0.86i&0\,-0.72i &-0.67-0.77i\\ 0.86\,-0.86i&0.19\,+0.85i&-0.86+0.86i&-0.19-0.85i&0.67\,-0.77 i&0\,+0.72i\\ \end{array} \right]$


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# Tag Info

26

The $GF$ in $GF(p^n)$ is not a function — it just stands for "Galois field (of $p^n$ elements)". As for what a Galois field is, it's a finite set of things (which we might represent e.g. with the numbers from $0$ to $p^n-1$), with some mathematical operations (specifically, addition and multiplication, and their inverses) defined on them that let us ...

18

Elliptic curves are not the only curves that have groups structure, or uses in cryptography. But they hit the sweet spot between security and efficiency better than pretty much all others. For example, conic sections (quadratic equations) do have a well-defined geometric addition law: given $P$ and $Q$, trace a line through them, and trace a parallel line ...

11

Discrete logarithms in $\mathbb{F}_{p}$ share the same asymptotic complexity as integer factorization for general numbers: $L_p[1/3,1.923]$ for general integers, $L_p[1/3,1.587]$ for special integers. Discrete logarithms in $\mathbb{F}_{p^n}$ have the same asymptotic complexity as factoring special integers, i.e. $L_{p^n}[1/3, 1.587]$, via the Function ...

10

The quoted recommendations do little to prevent fields that are subject to the recent developments. Take the $\mathbb{F}_{2^{6120}}$ example: it clearly passes the field size criterion, but also the subgroup rule, as the group order $2^{6120} - 1$ has one $1536$-bit prime factor. Not all binary fields are affected equally, however. Both Göloğlu et al and ...

10

It depends. If the order $m$ of $g$'s group is known and $a$ has an inverse modulo $m$ (which is the case if and only if $a$ is coprime to $m$), then it is easy: Calculate the inverse $b:=a^{-1}\bmod m$ (for instance, using the Euclidean algorithm), and compute the power $(g^a)^b$. By Lagrange's theorem, this equals $g$. However, there are cases for which ...

9

I think there are some gaps and some misunderstandings in what you say. A finite field or Galois field $GF(p^n)$ is a collection of $p^n$ $n$-dimensional vectors. Here, $p$ is a prime, and each coordinate in a vector is an integer in the range $[0,p-1]$; that is, an element of $GF(p)$. Thus, $$\mathbf A = (a_0, a_1, \ldots, a_{n-1}), ~~ a_i \in GF(p)$$ is ...

7

There is no reason in Shamir's scheme for the finite field $\mathbb F$ to have a prime number $p$ of elements; the field can have $p^m$ elements for suitable prime $p$ and integer $m \geq 1$. So, using $F_{2^8}$, the field with $2^8$ elements is perfectly all right. However, choosing $m = 1$ has the advantage that calculations in $\mathbb F_p$ can be done ...

7

For $p = 2q+1$, one can note that elements of $\mathbb{G}_q$ are exactly the non-zero quadratic residues modulo $p$: Since $p$ is prime, $\mathbb{Z}_p$ is a field. Hence, the polynomial $X^q-1$, being of degree $q$, cannot have more than $q$ roots in $\mathbb{Z}_p$. So $\mathbb{G}_q$ contains all the $q$ values of order $1$ or $q$. If $x$ is a non-zero ...

7

Let $n = \lceil \log q \rceil$ (with "$\log$" being the base-2 logarithm, so $n$ is the size, in bits, of $q$). If $q$ is a prime integer (i.e. $\mathbb{F}_q$ is the field of integers modulo $q$), then classical implementations will have cost $O(n)$ for addition and subtraction, $O(n^2)$ for multiplications and divisions. The cost of multiplications can be ...

7

What is Rijndael's finite field? Rijndaels finite field is $F=\mathrm{GF}(2^8)$ with minimal polynomial $f(x)=x^8 + x^4 + x^3 + x + 1$. Formally, we have $F=\mathbb F_2[x] / (f)$ but don't worry about that. So what does this mean? Well, elements of $F$ should be thought of as polynomials over $\mathbb{F}_2$, with the added fact that the minimal polynomial ...

6

Antoine Joux very kindly sent me the following on the topic: People worry that [logarithms over fields with composite exponent] might be easier, this is why they use prime exponent. For some factorization of the exponent, viewing the finite field as a tower of extensions $(p^{n_1})^{n_2}$ indeed makes things easier. See "The function field sieve ...

6

Well, if $q$ is a prime (and not $p^n$ with $n>1$), then addition, subtraction and multiplication can be performed by doing the traditional operations modulo $q$, that is: $a +_{\mathbb{F}_q}b \equiv (a+b) \bmod q$ $a -_{\mathbb{F}_q}b \equiv (a-b) \bmod q$ $a \times_{\mathbb{F}_q}b \equiv (a\times b) \bmod q$ As such, addition and subtraction can be ...

6

For security of ECC, the choice of the irreducible polynomial is unimportant, because all finite fields with the same cardinal are isomorphic to each other (and the isomorphisms are easy to compute); this is why we can say "the finite field $GF(2^{163})$" even though there are many irreducible polynomials of degree $163$ over $GF(2)$. So the algebraic ...

6

Antoine Joux announced the computation of discrete logarithm over $\mathbb{F}_{2^{257 \times 24}}$, which is now pretty close to what was being used in pairing-based cryptography. According to Joux, "a direct consequence of this record is that supersingular curves (of genus 1 or 2) defined over GF(2^257) cannot be used securely for pairing-based ...

6

In GF(28), 7 × 11 = 49. The discrete logarithm trick works just fine. Your mistake is in assuming that Galois field multiplication works the same way as normal integer multiplication. In prime-order fields this actually is more or less the case, except that you need to reduce the result modulo the order of the field, but in fields of non-prime order ...

6

The process is pretty simple. As you say, each party multiplies their two shares. They then use Shamir secret sharing to share the resulting value with the other parties. Once they have received a "subshare" from each other party, each party simply runs Lagrangian interpolation on the subshares they received (plus their own subshare). The result is a share ...

6

Yes, there is an algorithm for efficiently computing square roots in $GF(2^n)$. I don't know if this is the most efficient known, but the existence of an efficient algorithm can be shown by observing that squaring within $GF(2^n)$ is a bitwise linear operation, hence it is equivalent to taking the bit representation of the value, and multiplying it by an ...

6

No, RSA encryption and signature is performed in (the multiplicative semigroup of) the factor ring $\mathbb Z/n\mathbb Z$ which is not a field since the non-zero elements $kp+n\mathbb Z$ (for $0<k<q$) and $kq+n\mathbb Z$ (for $0<k<p$) do not have multiplicative inverses. (However, one easily observes that all other non-zero elements are ...

6

GF$(2^8)$ or $\mathbb F_{2^8}$ can also be viewed as the vector space $\mathbb F_2^8$ of $8$-bit vectors (or bytes) over GF$(2)$ or $\mathbb F_2$. Suppose $\{\beta_0, \beta_1, \cdots, \beta_7\}$ is a basis of $\mathbb F_2^8$ over $\mathbb F_2$, that is, the sum $$a_0\beta_0 \oplus a_1\beta_1 \oplus \cdots \oplus a_7\beta_7, ~ a_i \in \mathbb F_2$$ equals ...

6

It does not; the equation holds for any element $g$. The fact that $g$ is a generator means only that every element of the group can be obtained a key. This is not at all necessary for the protocol.

5

$((g \mod p)^{(p-1)/q})^{142363323} = (t \mod p)^{(p-1)/q}$ Equivalently, $(g \mod p)^{362274084216648467976382636880} = (t \mod p)$ That is, $362274084216648467976382636880 = 142363323 \mod \frac {p-1}q$

5

Göloğlu, Granger, McGuire, and Zumbrägel broke the DLP in the Galois finite field with $2^n$ elements for $n=6120$, and Antoine Joux did it for $n=6168$, with modest computing power. These recent announcements directly imply that cryptosystems (if any) based on the DLP in a field with $2^n$ elements are no longer secure, including for $n$ big enough that ...

5

For the second case, mapping numbers from $\mathbb{Z}_q$ to $\mathbb{G}_q$ and back when: $p=aq+1$ with an $a$ such that, e.g., |p|=1024 and |q|=160 It appears an efficient subgroup encoding/decoding scheme does not exist. Although it has not been proven that one cannot exist, notable cryptographers have conjectured it in the literature. For example, ...

5

Actually, the choice of irreducible polynomial is unimportant in AES; for any polynomial representation of $GF(2^8)$, you can modify the affine tranformation (and the MixCollumn) operation to come up with a block cipher that is equivalent to AES (meaning that any break to that can be translated to a break on the original AES). The key observation here is ...

5

Elliptic curves have a number of nice features that make them good for cryptography. One could write a whole book on the topic (as some have), so I'll highlight a few points. The points on an elliptic curve over a finite field forms a group. The same is not true for the ideas you mentioned. Discrete log on many of these EC groups is hard. In fact, there ...

5

Actually, the answer is "such an $n$ exists iff neither $i, j$ are 0". We have $x^{2^q-1} = 1$ for $x \ne 0$ (because multiplication in $GF(2^q)$ over the nonzero elements forms a group of order $2^q-1$), hence if $i, j \ne 0$, $n=2^q-1$ is one answer (if not necessarily the minimal one). If we add the simple observation that if one of $i, j$ is 0, then ...

5

To complete poncho's answer, if you know some Galois theory. The map $\sigma: x\mapsto x^2$ from $\mathbf{F}_{2^n}$ to itself is simply the Frobenius automorphism (relative to $\mathbf{F}_2$). It generates the Galois group $\mathrm{Gal}\left(\mathbf{F}_{2^n}/\mathbf{F}_2\right)$, which is cyclic of order $n$, and so its inverse (which is, by definition, the ...

4

To complete @Samuel's answer, there are a few shortcuts that can be used when n is composite; however, they only contribute small constant factors, hence they do not change the asymptotic behavior: If n can be divided by r, then one can first solve the discrete logarithm in the subfield GF(2r). In a sieve-based algorithm, this can provide up to half the ...

4

The Handbook on Applied Cryptography (link to the pdf version is on Alfred's webpage) has some of the known techniques to do finite field arithematic. If you are doing arithmetic to implement Elliptic Curve Cryptography (note the comment made by Paulo), then there are methods that depends on whether you are doing it in Jacobian or Projective plane (inverse ...

4

Probably the easiest solution for the case a=2 is to map $m\in\{1\ldots q\}$ to $(m/p)m$ where $(m/p)$ is the Legendre symbol. The inverse can be obtained by mapping a quadratic residue $x\in Z/(pZ)^*$ either to x or -x depending on which of the two residue classes contains an integer in $\{1\ldots q\}$. This is of course a well know solution, but I can't ...

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# How to calculate univariate conditional distribution of a trivariate gaussian [duplicate]

I am trying to find the conditional distribution of a trivariate gaussian. So here is a hypothetical trivariate gaussian: $$\mathcal{N}(\mu_{ABC},\Sigma_{ABC}),\;\mu_{ABC}=\begin{bmatrix}\mu_A \\ \mu_B \\ \mu_C \end{bmatrix},\;\Sigma_{ABC}=\begin{bmatrix} \sigma^2_{AA} & \sigma^2_{AB} & \sigma^2_{AC}\\ \sigma^2_{BA} & \sigma^2_{BB} & \sigma^2_{BC} \\ \sigma^2_{CA} & \sigma^2_{CB} & \sigma^2_{CC} \end{bmatrix}$$

I would like to find the conditional distribution parameters as follows:

$$\mu_{A}|B,C$$ $$\Sigma_{A}|B,C$$

To get the resulting distribution $$\mathcal{N}(\mu_{A},\Sigma_{A}|B,C)$$