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Naming Ionic Compounds Video Lessons

Concept: Naming Ionic Compounds

Problem: Vanadium forms four different oxides in which the percent by mass of vanadium is respectively 76 % , 68 % , 61 % , and 56 % .Give the name of the third oxide.

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Vanadium forms four different oxides in which the percent by mass of vanadium is respectively 76 % , 68 % , 61 % , and 56 % .

Give the name of the third oxide.


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# Sobolev spaces on boundary and higher order traces

I'd like to clarify the definitions of the spaces involved in the theory of traces of Sobolev functions.

Actually I'm referring to the books of Kufner and Necas.

In these books the boundary integral is introduced using the local description of the boundary of the (Lipschitz) domain $\Omega \subset \mathbb R^N$.

Question 1: I think that one can introduce the space $L^p(\partial \Omega)$ just using the (N-1)-dimensional Hausdorff measure and that, for $0<s<1$, one has

$$u \in W^{s,p}(\partial \Omega) \iff u \in L^p(\partial \Omega),\; \int_{\partial \Omega} \int_{\partial \Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{N-1+sp}}\mathrm d \mathcal H^{N-1}(x) \mathrm d \mathcal H^{N-1}(y)<+\infty$$

Is this true?

These are the spaces needed to deal with thaces of first order sobolev functions. For higher order traces one has to introduce the spaces $W^{s,p}(\partial \Omega)$ with $s=m+\sigma$ where $m$ is integer and $o < \sigma<1$. These are defined asking that the compositio of the function $u \in L^p(\partial \Omega)$ with the maps describing the boundary is a Sobolev function on a $N-1$-dimensional domain.

Question 2 Is it really necessary to ask more reguarity to the boundary in order to introduce these spaces? (Kufner defines $W^{s,p}(\partial \Omega)$ only for domains of class $C^{[s],1}$)

Question 3 Any reference where these spaces are treated in detail? I'd like to see that they're well-defined and if they can be endowed with a norm that doesn't depend on the loca description of the boundary.

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# $a_m(j+1)=\left(a_m(j)\right)^2+2a_m(j) \space\space(j\ge 0)$: Evaluate $\lim_{n \to \infty}a_n(n)$

Question_

$$d$$ is a real number. In each integer $$m\ge0$$, we define $$\{a_m(j)\}$$ $$(j=0, 1, 2, \cdots)$$ as follows. $$a_m(0)=\frac{d}{2^m}, a_m(j+1)=\left(a_m(j)\right)^2+2a_m(j) \space\space(j\ge 0)$$ Then, evaluate $$\lim_{n \to \infty}a_n(n)$$.

It is known that the closed-form of the given sequence is $$a_m(j)+1 = (a_m(0)+1)^{2^{j-1}} \space\space\cdots(1)$$ It is not that hard to find $$\lim_{n \to \infty}a_n(n)$$ from here.

My question is that how the closed-form of the given sequence can be calculated like that?

For sure, it can be proven by mathematical induction. However, what I do want to know is how the closed-form is derived, not how the $$eq.(1)$$ is fit to the given recursion formula. Thanks for answering!

• I think the way is to apply recurrence relation to guess the formula and then prove the formula by induction. You can read something here: en.wikipedia.org/wiki/Recurrence_relation – asv 2 days ago

From the recurrence relation, you can get $$a_m(j+1) + 1 = (a_m(j)+1)^2$$. Then, you can get the closed-form by using this relation recursively.


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# Calendar

Sep
29
Tue
Sutanu Roy (National Institute of Science Education and Research) @ Lecture held in Elysium
Sep 29 @ 6:00 am – 8:00 am

Title: Compact Quantum Groups and their Semidirect Products
by Sutanu Roy (National Institute of Science Education and Research) as part of Topological Groups

Lecture held in Elysium.

Abstract
Compact quantum groups are noncommutative analogs of compact groups in the realm of noncommutative geometry introduced by S. L. Woronowicz back in the 80s. Roughly, they are unital C*-bialgebras in the monoidal category (given by the minimal tensor product) of unital C*-algebras with some additional properties. For real 0<|q|<1, q-deformations of SU(2) group are the first and well-studied examples of compact quantum groups. These examples were constructed independently by Vaksman-Soibelman and Woronowicz also back in the 80s. In fact, they are examples of a particular class of compact quantum groups namely, compact matrix pseudogroups. The primary goal of this talk is to motivate and discuss some of the interesting aspects of this theory from the perspective of the compact groups. In the second part, I shall briefly discuss the semidirect product construction for compact quantum groups via an explicit example. The second part of this will be based on a joint work with Paweł Kasprzak, Ralf Meyer and Stanislaw Lech Woronowicz.

Christopher Porter (Drake University)
Sep 29 @ 10:00 am – 11:00 am

Title: Effective Dimension and the Intersection of Random Closed Sets
by Christopher Porter (Drake University) as part of Computability theory and applications

Abstract
The connection between the effective dimension of sequences and membership in algorithmically random closed subsets of Cantor space was first identified by Diamondstone and Kjos-Hanssen. In this talk, I highlight joint work with Adam Case in which we extend Diamondstone and Kjos-Hanssen’s result by identifying a relationship between the effective dimension of a sequence and what we refer to as the degree of intersectability of certain families of random closed sets (also drawing on work by Cenzer and Weber on the intersections of random closed sets). As we show, (1) the number of relatively random closed sets that can have a non-empty intersection varies depending on the choice of underlying probability measure on the space of closed subsets of Cantor space—this number being the degree of intersectability of a given family of random closed sets—and (2) the effective dimension of a sequence X is inversely proportional to the minimum degree of intersectability of a family of random closed sets, at least one of which contains X as a member. Put more simply, a sequence of lower dimension can only be in random closed sets with more branching, which are thus more intersectable, whereas higher dimension sequences can be in random closed sets with less branching, which are thus less intersectable, and the relationship between these two quantities (that is, effective dimension and degree of intersectability) can be given explicitly.

Oct
6
Tue
Paul-Elliot Angles d’Auriac (University of Lyon)
Oct 6 @ 3:00 am – 4:00 am

Title: The computable strength of Milliken’s Tree Theorem and applications
by Paul-Elliot Angles d’Auriac (University of Lyon) as part of Computability theory and applications

Abstract
Devlin’s theorem and the Rado graph theorem are both variants of Ramsey’s theorem, where a structure is added but more colors are allowed: Devlin’s theorem (respectively the Rado graph theorem) states if S is ℚ (respectively G, the Rado graph), then for any size of tuple n, there exists a number of colors l such that for any coloring of [S]^n into finitely many colors, there exists a subcopy of S on which the coloring takes at most l colors. Moreover, given n, the optimal l is specified.

The key combinatorial theorem used in both the proof of Devlin’s theorem and the Rado graph theorem is Milliken’s tree theorem. Milliken’s tree theorem is also a variant of Ramsey’s theorem, but this time for trees and strong subtrees: it states that given a coloring of the strong subtrees of height n of a tree T, there exists a strong subtree of height ω of T on which the coloring is constant.

In this talk, we review the links between those theorems, and present the recent results on the computable strength of Milliken’s tree theorem and its applications Devlin and the Rado graph theorem, obtained with Cholak, Dzhafarov, Monin and Patey.

Nicolò Zava (University of Udine) @ Lecture held in Elysium
Oct 6 @ 6:00 am – 8:00 am

Title: Towards a unifying approach to algebraic and coarse entropy
by Nicolò Zava (University of Udine) as part of Topological Groups

Lecture held in Elysium.

Abstract
In each situation, entropy associates to a self-morphism a value that estimates the chaos created by the map application. In particular, the algebraic entropy $h_{alg}$ can be computed for (continuous) endomorphisms of (topological) groups, while the coarse entropy $h_c$ is associated to bornologous self-maps of locally finite coarse spaces. Those two entropy notions can be compared because of the following observation. If $f$ is a (continuous) homomorphism of a (topological) group $G$, then $f$ becomes automatically bornologous provided that $G$ is equipped with the compact-group coarse structure. For an endomorphism $f$ of a discrete group, $h_{alg}(f)=h_c(f)$ if $f$ is surjective, while, in general, $h_{alg}(f) eq h_c(f)$. That difference occurs because in many cases, if $f$ is not surjective, then $h_c(f)=0$.

In the first part of the talk, after briefly recalling the large-scale geometry of topological groups, we define the coarse entropy and discuss its relationship with the algebraic entropy. The second part is dedicated to the introduction of the algebraic entropy of endomorphisms of $G$-sets (i.e., sets endowed with group actions). We show that it extends the usual algebraic entropy of group endomorphisms and we provide evidence that it can represent a useful modification and generalisation of the coarse entropy that overcome the non-surjectivity issue.

Oct
13
Tue
Break (University of Hawaiʻi) @ Lecture held in Elysium
Oct 13 @ 6:00 am – 8:00 am

Title: Topological Groups Seminar Two-Week Hiatus
by Break (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.
Abstract: TBA

Leszek Kołodziejczyk (University of Warsaw)
Oct 13 @ 10:00 am – 11:00 am

Title: Reverse mathematics of combinatorial principles over a weak base theory
by Leszek Kołodziejczyk (University of Warsaw) as part of Computability theory and applications

Abstract
Reverse mathematics studies the strength of axioms needed to prove various
mathematical theorems. Often, the theorems have the form $forall X exists Y psi(X,Y)$ with $X, Y$ denoting subsets of $mathbb{N}$ and $psi$
arithmetical, and the logical strength required to prove them is closely
related to the difficulty of computing $Y$ given $X$. In the early decades
of reverse mathematics, most of the theorems studied turned out to be
equivalent, over a relatively weak base theory, to one of just a few typical
axioms, which are themselves linearly ordered in terms of strength. More
recently, however, many statements from combinatorics, especially Ramsey
theory, have been shown to be pairwise inequivalent or even logically
incomparable.

The usual base theory used in reverse mathematics is $mathrm{RCA}_0$, which
is intended to correspond roughly to the idea of “computable mathematics”.
The main two axioms of $mathrm{RCA}_0$ are: comprehension for computable
properties of natural numbers and mathematical induction for c.e.
properties. A weaker theory in which induction for c.e. properties is
replaced by induction for computable properties has also been introduced,
but it has received much less attention. In the reverse mathematics
literature, this weaker theory is known as $mathrm{RCA}^*_0$.

In this talk, I will discuss some results concerning the reverse mathematics
of combinatorial principles over $mathrm{RCA}^*_0$. We will focus mostly on
Ramsey’s theorem and some of its well-known special cases: the
chain-antichain principle CAC, the ascending-descending chain principle ADS,
and the cohesiveness principle COH.

The results I will talk about are part of a larger project joint with Marta
Fiori Carones, Katarzyna Kowalik, Tin Lok Wong, and Keita Yokoyama.

Chris Conidis (CUNY-College of Staten Island)
Oct 13 @ 3:00 pm – 4:00 pm

Title: Non-arithmetic algebraic constructions
by Chris Conidis (CUNY-College of Staten Island) as part of Computability theory and applications

Abstract
We examine two radical constructions, one from ring theory and another from module theory, and produce a computable ring for each construction where the corresponding radical is $Pi^1_1$-complete.

Oct
16
Fri
Data Science seminar
Oct 16 @ 2:00 pm – 3:00 pm

Join the Hawai‘i Data Science Institute for another Data Science Friday seminar titled “Bayesian Topological Learning for Complex Data Analysis” presented by Assistant Professor of Mathematics Dr. Farzana Nasir on October 16, 2020 at 2 pm on Zoom.


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## Analysis & PDE

• Anal. PDE
• Volume 7, Number 2 (2014), 497-512.

#### Abstract

We study the structure of bounded linear functionals on a class of non-self-adjoint operator algebras that includes the multiplier algebra of every complete Nevanlinna–Pick space, and in particular the multiplier algebra of the Drury–Arveson space. Our main result is a Lebesgue decomposition expressing every linear functional as the sum of an absolutely continuous (i.e., weak-* continuous) linear functional and a singular linear functional that is far from being absolutely continuous. This is a non-self-adjoint analogue of Takesaki’s decomposition theorem for linear functionals on von Neumann algebras. We apply our decomposition theorem to prove that the predual of every algebra in this class is (strongly) unique.

#### Article information

Source
Anal. PDE, Volume 7, Number 2 (2014), 497-512.

Dates
Revised: 27 October 2013
Accepted: 27 November 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513731497

Digital Object Identifier
doi:10.2140/apde.2014.7.497

Mathematical Reviews number (MathSciNet)
MR3218817

Zentralblatt MATH identifier
1302.47104

#### Citation

Kennedy, Matthew; Yang, Dilian. A non-self-adjoint Lebesgue decomposition. Anal. PDE 7 (2014), no. 2, 497--512. doi:10.2140/apde.2014.7.497. https://projecteuclid.org/euclid.apde/1513731497

#### References

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• T. Ando, “On the predual of $H\sp{\infty }$”, Comment. Math. Special Issue 1 (1978), 33–40.
• A. Arias and G. Popescu, “Noncommutative interpolation and Poisson transforms”, Israel J. Math. 115 (2000), 205–234.
• W. B. Arveson, “Analyticity in operator algebras”, Amer. J. Math. 89 (1967), 578–642.
• W. Arveson, “Subalgebras of $C\sp *$-algebras, III: Multivariable operator theory”, Acta Math. 181:2 (1998), 159–228.
• D. P. Blecher and L. E. Labuschagne, “Noncommutative function theory and unique extensions”, Studia Math. 178:2 (2007), 177–195.
• D. P. Blecher and C. Le Merdy, Operator algebras and their modules: An operator space approach, London Mathematical Society Monographs. New Series 30, Clarendon Press, Oxford, 2004.
• K. R. Davidson and D. R. Pitts, “The algebraic structure of non-commutative analytic Toeplitz algebras”, Math. Ann. 311:2 (1998), 275–303.
• K. R. Davidson and D. R. Pitts, “Nevanlinna–Pick interpolation for non-commutative analytic Toeplitz algebras”, Integral Equations Operator Theory 31:3 (1998), 321–337.
• K. R. Davidson and D. R. Pitts, “Invariant subspaces and hyper-reflexivity for free semigroup algebras”, Proc. London Math. Soc. $(3)$ 78:2 (1999), 401–430.
• K. R. Davidson and A. Wright, “Operator algebras with unique preduals”, Canad. Math. Bull. 54:3 (2011), 411–421.
• K. R. Davidson and D. Yang, “A note on absolute continuity in free semigroup algebras”, Houston J. Math. 34:1 (2008), 283–288.
• K. R. Davidson, E. Katsoulis, and D. R. Pitts, “The structure of free semigroup algebras”, J. Reine Angew. Math. 533 (2001), 99–125.
• K. R. Davidson, J. Li, and D. R. Pitts, “Absolutely continuous representations and a Kaplansky density theorem for free semigroup algebras”, J. Funct. Anal. 224:1 (2005), 160–191.
• E. G. Effros, N. Ozawa, and Z.-J. Ruan, “On injectivity and nuclearity for operator spaces”, Duke Math. J. 110:3 (2001), 489–521.
• R. Exel, “The F. and M. Riesz theorem for $C\sp *$-algebras”, J. Operator Theory 23:2 (1990), 351–368.
• G. Godefroy, “Existence and uniqueness of isometric preduals: A survey”, pp. 131–193 in Banach space theory (Iowa City, IA, 1987), edited by B.-L. Lin, Contemp. Math. 85, Amer. Math. Soc., Providence, RI, 1989.
• G. Godefroy and M. Talagrand, “Nouvelles classes d'espaces de Banach á predual unique”, pp. 1–28 in Séminaire d'analyse fonctionnelle (Palaiseau, France, 1980–1981), École Polytechnique Centre de Mathématiques, Palaiseau, 1981.
• A. Grothendieck, “Une caractérisation vectorielle-métrique des espaces $L\sp 1$”, Canad. J. Math. 7 (1955), 552–561.
• P. Harmand, D. Werner, and W. Werner, $M$-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics 1547, Springer, Berlin, 1993.
• M. Kennedy, “Wandering vectors and the reflexivity of free semigroup algebras”, J. Reine Angew. Math. 653 (2011), 47–73.
• M. Kennedy, “The structure of an isometric tuple”, Proc. Lond. Math. Soc. $(3)$ 106:5 (2013), 1157–1177.
• H. Pfitzner, “Separable L-embedded Banach spaces are unique preduals”, Bull. Lond. Math. Soc. 39:6 (2007), 1039–1044.
• G. Popescu, “Multi-analytic operators and some factorization theorems”, Indiana Univ. Math. J. 38:3 (1989), 693–710.
• G. Popescu, “Characteristic functions for infinite sequences of noncommuting operators”, J. Operator Theory 22:1 (1989), 51–71.
• G. Popescu, “von Neumann inequality for $(B({\scr H})\sp n)\sb 1$”, Math. Scand. 68:2 (1991), 292–304.
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• G. Popescu, “Non-commutative disc algebras and their representations”, Proc. Amer. Math. Soc. 124:7 (1996), 2137–2148.
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# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2019243

## The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals

 1 SISSA, Via Bonomea 265, Trieste, 34136, Italy 2 Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY, 10012, USA 3 Mathematisches Institut, Universität Leipzig, Augustus Platz 10, Leipzig, D04109, Germany

* Corresponding author: Guido De Philippis

Received  January 2019 Revised  February 2019 Published  June 2019

Fund Project: The work of G.D.P. is supported by the INDAM-grant "Geometric Variational Problems"

In this paper we investigate the "area blow-up" set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in [12], we show that this set has bounded (anisotropic) mean curvature in the viscosity sense. In particular, this allows to show that the set is empty in a variety of situations. As a consequence, we show boundary curvature estimates for two dimensional stable anisotropic minimal surfaces, extending the results of [10].

Citation: Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2019243
##### References:
 [1] W. K. Allard, A characterization of the area integrand, in Symposia Mathematica, (Convegno di Teoria Geometrica dell'Integrazione e Varietà Minimali, INDAM, Rome, 1973), Academic Press, London, Volume XIV, 1974,429-444.Google Scholar [2] W. K. Allard, An a priori estimate for the oscillation of the normal to a hypersurface whose first and second variation with respect to an elliptic integrand is controlled, Invent. Math., 73 (1983), 287-331. doi: 10.1007/BF01394028. Google Scholar [3] W. K. Allard, An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled, in Geometric Measure Theory and the Calculus of Variations, Proceedings of Symposia in Pure Mathematics, (eds. F. J. Allard ad W. K. Almgren Jr.), 44, 1986. doi: 10.1090/pspum/044/840267. Google Scholar [4] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. Google Scholar [5] G. De Philippis, A. De Rosa and F. Ghiraldin, Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies, Communications on Pure and Applied Mathematics, 71 (2018), 1123-1148. doi: 10.1002/cpa.21713. Google Scholar [6] G. De Philippis and F. Maggi, Dimensional estimates for singular sets in geometric variational problems with free boundaries, J. Reine Angew. Math., 725 (2017), 217-234. doi: 10.1515/crelle-2014-0100. Google Scholar [7] L. Simon, Lectures on geometric measure theory, in Proceedings of the Centre for Mathematical Analysis, Australian National University, Centre for Mathematical Analysis, Canberra, 3, 1983. Google Scholar [8] L. Simon, Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom., 1 (1993), 281-326. doi: 10.4310/CAG.1993.v1.n2.a4. Google Scholar [9] B. Solomon and B. White, A strong maximum principle for varifolds that are stationary with respect to even parametric elliptic functionals, Indiana Univ. Math. J., 38 (1989), 683-691. doi: 10.1512/iumj.1989.38.38032. Google Scholar [10] B. White, Curvature estimates and compactness theorems in {$3$}-manifolds for surfaces that are stationary for parametric elliptic functionals, Invent. Math., 88 (1987), 243-256. doi: 10.1007/BF01388908. Google Scholar [11] B. White, Existence of smooth embedded surfaces of prescribed genus that minimize parametric even elliptic functionals on $3$-manifolds, J. Differential Geom., 33 (1991), 413-443. doi: 10.4310/jdg/1214446325. Google Scholar [12] B. White, Controlling area blow-up in minimal or bounded mean curvature varieties, J. Differential Geom., 102 (2016), 501-535. doi: 10.4310/jdg/1456754017. Google Scholar

show all references

##### References:
 [1] W. K. Allard, A characterization of the area integrand, in Symposia Mathematica, (Convegno di Teoria Geometrica dell'Integrazione e Varietà Minimali, INDAM, Rome, 1973), Academic Press, London, Volume XIV, 1974,429-444.Google Scholar [2] W. K. Allard, An a priori estimate for the oscillation of the normal to a hypersurface whose first and second variation with respect to an elliptic integrand is controlled, Invent. Math., 73 (1983), 287-331. doi: 10.1007/BF01394028. Google Scholar [3] W. K. Allard, An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled, in Geometric Measure Theory and the Calculus of Variations, Proceedings of Symposia in Pure Mathematics, (eds. F. J. Allard ad W. K. Almgren Jr.), 44, 1986. doi: 10.1090/pspum/044/840267. Google Scholar [4] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. Google Scholar [5] G. De Philippis, A. De Rosa and F. Ghiraldin, Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies, Communications on Pure and Applied Mathematics, 71 (2018), 1123-1148. doi: 10.1002/cpa.21713. Google Scholar [6] G. De Philippis and F. Maggi, Dimensional estimates for singular sets in geometric variational problems with free boundaries, J. Reine Angew. Math., 725 (2017), 217-234. doi: 10.1515/crelle-2014-0100. Google Scholar [7] L. Simon, Lectures on geometric measure theory, in Proceedings of the Centre for Mathematical Analysis, Australian National University, Centre for Mathematical Analysis, Canberra, 3, 1983. Google Scholar [8] L. Simon, Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom., 1 (1993), 281-326. doi: 10.4310/CAG.1993.v1.n2.a4. Google Scholar [9] B. Solomon and B. White, A strong maximum principle for varifolds that are stationary with respect to even parametric elliptic functionals, Indiana Univ. Math. J., 38 (1989), 683-691. doi: 10.1512/iumj.1989.38.38032. Google Scholar [10] B. White, Curvature estimates and compactness theorems in {$3$}-manifolds for surfaces that are stationary for parametric elliptic functionals, Invent. Math., 88 (1987), 243-256. doi: 10.1007/BF01388908. Google Scholar [11] B. White, Existence of smooth embedded surfaces of prescribed genus that minimize parametric even elliptic functionals on $3$-manifolds, J. Differential Geom., 33 (1991), 413-443. doi: 10.4310/jdg/1214446325. Google Scholar [12] B. White, Controlling area blow-up in minimal or bounded mean curvature varieties, J. Differential Geom., 102 (2016), 501-535. doi: 10.4310/jdg/1456754017. Google Scholar
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# GTL3 DXTn Flipping Cont.

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So, after some half decent TF2 playing I decided to work on a few of the bugs in the GTL3 DXTn flipping routines.

The problems with the DXT1 flipping turned out to be a matter of maths errors on my part mostly; generally I was skipping too many bits, or not enough and it was all going a bit wrong.

The same pattern of problems appeared in the DXT2/3 routines as well, where some debugging and making sure the numbers were sane quickly took care of the problem.

DXT4/5 however is proving more intresting;

Recall, if you will, from an earlier entry where I mentioned the alpha values were stored as 3bit pairs and as such didn't really fit into any common data structure, just to be annoying.

So, I decided to hack things a bit by casting to a 64bit type and just ignoring the bottom 16bits of data. It seemed like a sane idea at the time, however this is C++ and nothing is as it seems.

In order to get the bits we want this lovely bit of code was created;
boost::uint64_t srcdata = *(reinterpret_cast(&src.lookup1));

where lookup1 is the first in the list of variables used to hold the table information.

Then, to write back the same kind of trick was used;
boost::uint64_t * destptr = reinterpret_cast(&dest.lookup1);
*destptr = destdata;

We can do this because there are no virtuals in this struct and we'll be overwriting the following colour data right away.

This nearly works, however I think I'm being tripped somewhere by endian issues or something.

The layout of the struct in question is thus;
struct dxt1 { 	boost::uint16_t colour0, colour1; 	boost::uint32_t lookuptable; 	dxt1() : colour1(0), colour0(0), lookuptable(0) {};};struct dxt4 { 	boost::uint8_t alpha0, alpha1; 	boost::uint8_t lookup1,lookup2,lookup3,lookup4,lookup5,lookup6; 	dxt1 colourdata; 	dxt4() : colourdata(), alpha0(0), alpha1(0), lookup1(0), lookup2(0), lookup3(0), lookup4(0), lookup5(0), lookup6(0) {};};

Now, I realise I haven't packed things, so maybe VC8 is doing something with the alignment, I'll check that in a bit, however things don't work as expected.

If all our lookup values hold 0xFF and our dxt4::colourdata::colour0 hold 0xff instead of getting 0xffffffffffff00ff when read I infact get 0x00ffffffffffffff.

Bugger.

This also causes a problem on write back, because the lower 16bits are zero in the dest data, which means that lookup1 and lookup2 both get 0x00 for values.

So, yes, I need to work out how I'm going to demong that data into something I can make use of... or maybe just come up with a whole new algo or something, I dunno; something with an array of 6 elements and some shifting magic or something.

Either way, I suspects nightmares abound to fix this up [smile]
Now, however, I sleep...

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• ### Observation of $\Xi_{c}(2930)^0$ and updated measurement of $B^{-} \to K^{-} \Lambda_{c}^{+} \bar{\Lambda}_{c}^{-}$ at Belle(1712.03612)

March 8, 2018 hep-ph, hep-ex
We report the first observation of the $\Xi_{c}(2930)^0$ charmed-strange baryon with a significance greater than 5$\sigma$. The $\Xi_{c}(2930)^0$ is found in its decay to $K^- \Lambda_{c}^+$ in $B^{-} \to K^{-} \Lambda_{c}^{+} \bar{\Lambda}_{c}^{-}$ decays. The measured mass and width are $[2928.9 \pm 3.0(\rm stat.)^{+0.8}_{-12.0}(\rm syst.)]$ MeV/$c^{2}$ and $[19.5 \pm 8.4(\rm stat.) ^{+5.4}_{-7.9}(\rm syst.)]$ MeV, respectively, and the product branching fraction is ${\cal B}(B^{-} \to \Xi_{c}(2930)^0 \bar{\Lambda}_{c}^{-}) {\cal B}(\Xi_{c}(2930)^0 \to K^- \Lambda_{c}^{+})=[1.73 \pm 0.45(\rm stat.) \pm 0.24(\rm syst.)]\times 10^{-4}$. We also measure ${\cal B}(B^{-} \to K^{-} \Lambda_{c}^{+} \bar{\Lambda}_{c}^{-}) = [4.80 \pm 0.43(\rm stat.) \pm 0.68(\rm syst.)] \times 10^{-4}$ with improved precision, and search for the charmonium-like state $Y(4660)$ and its spin partner, $Y_{\eta}$, in the $\Lambda_{c}^{+}\bar{\Lambda}_{c}^{-}$ invariant mass spectrum. No clear signals of the $Y(4660)$ nor its spin partner are observed and the 90\% credibility level (C.L.) upper limits on their production rates are determined. These measurements are obtained from a sample of $(772\pm11)\times 10^{6} B\bar{B}$ pairs collected at the $\Upsilon(4S)$ resonance by the Belle detector at the KEKB asymmetric energy electron-positron collider.
• ### Search for $CP$ violation in the $D^{+}\to\pi^{+}\pi^{0}$ decay at Belle(1712.00619)

March 6, 2018 hep-ex
We search for $CP$ violation in the charged charm meson decay $D^{+}\to\pi^{+}\pi^{0}$, based on a data sample corresponding to an integrated luminosity of $\rm 921~fb^{-1}$ collected by the Belle experiment at the KEKB $e^{+}e^{-}$ asymmetric-energy collider. The measured $CP$ violating asymmetry is $[+2.31\pm1.24({\rm stat})\pm0.23({\rm syst})]\%$, which is consistent with the standard model prediction and has a significantly improved precision compared to previous results.
• ### Measurement of the tau Michel parameters $\bar{\eta}$ and $\xi\kappa$ in the radiative leptonic decay $\tau^- \rightarrow \ell^- \nu_{\tau} \bar{\nu}_{\ell}\gamma$(1709.08833)

Sept. 27, 2017 hep-ex
We present a measurement of the Michel parameters of the $\tau$ lepton, $\bar{\eta}$ and $\xi\kappa$, in the radiative leptonic decay $\tau^- \rightarrow \ell^- \nu_{\tau} \bar{\nu}_{\ell} \gamma$ using 711~f$\mathrm{b}^{-1}$ of collision data collected with the Belle detector at the KEKB $e^+e^-$ collider. The Michel parameters are measured in an unbinned maximum likelihood fit to the kinematic distribution of $e^+e^-\rightarrow\tau^+\tau^-\rightarrow (\pi^+\pi^0 \bar{\nu}_\tau)(\ell^-\nu_{\tau}\bar{\nu}_{\ell}\gamma)$ $(\ell=e$ or $\mu)$. The measured values of the Michel parameters are $\bar{\eta} = -1.3 \pm 1.5 \pm 0.8$ and $\xi\kappa = 0.5 \pm 0.4 \pm 0.2$, where the first error is statistical and the second is systematic. This is the first measurement of these parameters. These results are consistent with the Standard Model predictions within their uncertainties and constrain the coupling constants of the generalized weak interaction.
• ### Precise determination of the CKM matrix element $\left| V_{cb}\right|$ with $\bar B^0 \to D^{*\,+} \, \ell^- \, \bar \nu_\ell$ decays with hadronic tagging at Belle(1702.01521)

Feb. 14, 2017 hep-ex
The precise determination of the CKM matrix element $\left| V_{cb}\right|$ is important for carrying out tests of the flavour sector of the Standard Model. In this article we present a preliminary analysis of the $\bar B^0 \to D^{*\,+} \, \ell^- \, \bar \nu_\ell$ decay mode and its charge conjugate, selected in events that contain a fully reconstructed $B$-meson, using 772 million $e^+ \, e^- \to \Upsilon(4S) \to B \bar B$ events recorded by the Belle detector at KEKB. Unfolded differential decay rates of four kinematic variables fully describing the $\bar B^0 \to D^{*\,+} \, \ell^- \, \bar \nu_\ell$ decay in the $B$-meson rest frame are presented. We measure the total branching fraction $\mathcal{B}( \bar B^0 \to D^{*\,+} \, \ell^- \, \bar \nu_\ell ) = \left(4.95 \pm 0.11 \pm 0.22 \right) \times 10^{-2}$, where the errors are statistical and systematic respectively. The value of $\left|V_{cb} \right|$ is determined to be $\left( 37.4 \pm 1.3 \right) \times 10^{-3}$. Both results are in good agreement with current world averages.


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# area between curves.

#### Tweety

I have attached the graph of the two equations, and I am suppose to find the area between them.

However I am not sure how to go about doing this, would it be the integral of the line minus the curve?

If so I keep getting the wrong answer.

Any help appreciated.

the equations are: $$\displaystyle y = 3$$ and $$\displaystyle y = \sqrt{x-2}$$ and they intersect at the point $$\displaystyle 3 = \sqrt{x-2}$$

#### Attachments

• 15.7 KB Views: 6
Last edited:

#### wonderboy1953

Before trying to figure this one

Would you suppose the origin figures into this problem?

#### Tweety

Would you suppose the origin figures into this problem?

#### Jester

MHF Helper
Probably easiest to use horizontal rectangles. Hence

$$\displaystyle \int_0^3 y^2 + 2\, dy.$$

Tweety

#### wonderboy1953

Clarification

The origin: (0,0)

Tweety

#### AllanCuz

I have attached the graph of the two equations, and I am suppose to find the area between them.

However I am not sure how to go about doing this, would it be the integral of the line minus the curve?

If so I keep getting the wrong answer.

Any help appreciated.

the equations are: $$\displaystyle y = 3$$ and $$\displaystyle y = \sqrt{x-2}$$ and they intersect at the point $$\displaystyle 3 = \sqrt{x-2}$$
$$\displaystyle A = \int_0^3 dy \int_0^{ y^2 + 2} dx$$

$$\displaystyle A = \int_0^3 ( y^2 + 2) dy$$

$$\displaystyle A = \frac{ y^3 }{3} |_0^3 + 6$$

$$\displaystyle A = 9 + 6 = 15$$


text:
## antisymmetric tensor epsilon

December 12th, 2020

Notice that this is still an antisymmetric tensor in terms of $$i$$ and $$j$$: \begin{aligned} A_{ji} \rightarrow R_{jm} R_{in} A_{mn} = -R_{jn} R_{im} A_{mn}. The general rule with the antisymmetric tensor: if in doubt, multiply by another one and use the \epsilon \epsilon = \delta\delta-\delta\delta identity. 24 #define GINAC_TENSOR_H. This class represents the totally antisymmetric epsilon tensor. It is named after the Italian mathematician and Physicist Tullio Levi-Civita [1-3]. We also de ne and investigate scalar, vector and tensor elds when they are subjected to various coordinate transformations. The alternating tensor can be used to write down the vector equation z = x × y in suï¬x notation: z i = [x×y] i = ijkx jy k. (Check this: e.g., z 1 = 123x 2y 3 + 132x 3y 2 = x 2y 3 âx 3y 2, as required.) The Levi-Civita tensor is called epsilon in xTensor, and there is one for each metric, so it has the metric in its name. epsilon symbol keyboard By | October 31, 2020 | 0 . It is defined by the following rules. gives the d-dimensional Levi-Civita totally antisymmetric tensor. DOI: 10.1007/JHEP05(2018)156. antisymmetric tensor. Hi, I'm reading some introductory notes about SR and I'm completely stuck at this problem. It turns out that tensors have certain properties which are independent of the coordinate system used to describe the tensor. 2019;371(1):55â97. 23 #ifndef GINAC_TENSOR_H. The third rank epsilon-tensor is used to formulate the dual relation between an antisymmetric second rank tensor and a vector or vice versa, in three-dimensional space. We will also introduce the use of the Einstein summation convention. In this chapter, the properties of this isotropic tensor are presented. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share â¦ 25 26 #include "ex.h" 27 #include "archive.h" 28 29 namespace GiNaC {30 34 class tensor: public basic. of the totaly antisymmetric Levi-Civita epsilon tensor Îµlmnpq in combination with the generalized ï¬eld strength tensors Gmn,l1...ls. In section 1 the indicial notation is de ned and illustrated. Journal of High Energy Physics. \) SI units for Maxwell's equations and the particle physicist's sign convention for the signature of Minkowski space (+,â,â,â), will be used throughout this article. OneLoop â calculates 1-loop Feynman diagram amplitudes. a b Educators, learners, students, etc. Ambient Chaos, Electronic Mess, Skillex, Llort Jr and Zarqnon the Embarrassed Re â¦ ð ð¼ð½ð¾ = +1 if ð¼ ð½ ð¾ is an even permutation of 1 2 3 (specifically ð 123, ð 231 and ð 312) Show that the components of the totally antisymmetric symbol ##\epsilon^{\mu \nu \alpha \beta}## are invariant under transformation belonging to SO(3,1) group. As was discussed in Section 5.2 of this chapter, matrices A and B in the commutator expression Î± (A B â B A) can either be symmetric or antisymmetric for the physically meaningful cases. LeviCivitaTensor by default gives a SparseArray object. share | improve this answer | follow | answered Nov 9 '19 at 0:23. jose jose. 39 protected: 40 unsigned return_type() const override { return â¦ When you are typing in Messages, Notes, Mail etc, touch and hold on the letter, number, or symbol, which has these character(s). Communications in Mathemathical Physics. Relevant Equations:. Closely associated with tensor calculus is the indicial or index notation. In component form, The 1/N expansion of the symmetric traceless and the antisymmetric tensor models in rank three. [/math] Therefore, F is a differential 2-formâthat is, an antisymmetric rank-2 tensor fieldâon Minkowski space. In your case it is. LeviCivitaTensor [d] gives a rank-d tensor with length d in each dimension. In component form, \( F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu. Or, search the web. Scalars, vectors, the Kronecker delta and the Levi-Civita symbol and the Einstein summation convention are discussed by Lea [2004], pp. sometimes called the "contracted epsilon identity". There is one very important property of ijk: ijk klm = Î´ ilÎ´ jm âÎ´ imÎ´ jl. The code detects this properties of a symmetry group, but does not react to it by default. See also closely related Using the epsilon tensor in Mathematica \endgroup â Artes Jul 6 '14 at 11:22 2018(5):156. DOI: 10.1007/s00220-019-03551-z. This makes many vector identities easy to prove. These invariants can be represented as total derivatives of the vector currents Î£s l: Îs = âlÎ£ s l, where the vector currents Î£s l involve a free index l carried by the Levi-Civita epsilon tensor. Homework Statement Show that \\epsilon_{ijk}a_{ij} = 0 for all k if and only if a_{ij} is symmetric. The elements of LeviCivitaTensor [d] are 0, -1, +1, and can be obtained by applying Signature to their indices. LeviCivita â totally antisymmetric epsilon (Levi-Civita) tensor. Jacek Jakowski, ... Keiji Morokuma, in GPU Computing Gems Emerald Edition, 2011. In this article, we will go over the basic starting points on how to deal withâ¦ 5-17. For example, any tensor that has the property that the first two indices are antisymmetric and the second and third indices are symmetric has to be zero by definition. Most likely this functionality is poorly known even though it is one of the most important symbolic news in Mathematica 9. OSTI.GOV Journal Article: Bulk antisymmetric tensor fields coupled to a dilaton in a Randall-Sundrum model Matrix Multiplication. eâ² i =Qe i, QQ = I. T, etc. We investigate the possibility of inflation with models of antisymmetric tensor field having minimal and nonminimal couplings to gravity. arXiv:1712.00249. Especially when supersymmetry (SUSY) enters the game. MetricTensor â representation of the metric tensor. Next: The Epsilon-Delta Identity Up: &delta#delta;_ij and &epsi#epsilon;_ijk Previous: The Kronecker Delta Function Contents The Levi-Civita Tensor. Therefore F is a differential 2-formâthat is, an antisymmetric rank-2 tensor fieldâon Minkowski space. But P alpha on A from that middle blackboard is epsilon alpha A psi. Definition. We now can introduce the epsilon tensor, a âcompletely antisymmetric tensor of rank threeâ. In three dimensions, it the Levi Civita tensor is defined as {The indices i, j, and k run from 1, 2, and 3. OSTI.GOV Journal Article: Bulk antisymmetric tensor fields in a Randall-Sundrum model Title: Bulk antisymmetric tensor fields in a Randall-Sundrum model Full Record So indeed, A takes states to antisymmetric states. The electromagnetic tensor, conventionally labelled F, is defined as the exterior derivative of the electromagnetic four-potential, A, a differential 1-form: [math]F \ \stackrel{\mathrm{def}}{=}\ \mathrm{d}A. As a third rank tensor in 3-space, epsilon will have 3 3 = 27 components. The change of basis tensor can also be expressed in terms of the base vectors from both bases: Q =e â² i âe i, (1.10.25 ) from which the above relations can easily be derived, for example . \begingroup This is a duplicate of Contracting with Levi-Civita (totally antisymmetric) tensor. We prove rigorously that the symmetric traceless and the antisymmetric tensor models in rank three with tetrahedral interaction admit a 1 / N expansion, and that at leading order they are dominated by melon diagrams. See the fourth message line printed when you call DefMetric. But that is precisely the definition of an antisymmetric state. Consider now the operation of the change of basis tensor on a vector: Qv =v i (Qe i )=v i e iâ² (1.10.26) Thus transforms Q v into a second vector . Although the minimal model does not support inflation, the nonminimal models, through the introduction of a nonminimal coupling to gravity, can give rise to stable de-Sitter solutions with a bound on the coupling parameters. References. The rotation group SO(3) can be viewed as the group that preserves our old friends the delta tensor \delta^{ab} and \epsilon^{abc} (the totally antisymmetric tensor). Quantum field theory is hard, thereâs no way to deny that. epsilon\[ScriptG][-\[Alpha], -\[Beta], -\[Gamma], -\[Delta]] It is created during evaluation of DefMetric. Details. 35 {36 GINAC_DECLARE_REGISTERED_CLASS(tensor, basic) 37 38 // functions overriding virtual functions from base classes. ScalarProductCancel â cancels scalar products with propagators One nice discussion of the Einstein convention can be found at The Wolfram Language's uniform representation of vectors and matrices as lists automatically extends to tensors of any rank, allowing the Wolfram Language's powerful list manipulation functions immediately to be applied to tensors, both numerical and symbolic . \end{aligned} It is left as an exercise to prove that this object transforms as a rank-1 tensor; the proof is too much of a detour from our goal right now. Benedetti D, Gurau R. 2PI effective action for the SYK model and tensor field theories. The Levi-Civita tesnor is totally antisymmetric tensor of rank n. The Levi-Civita symbol is also called permutation symbol or antisymmetric symbol. , F is a differential 2-formâthat is, an antisymmetric rank-2 tensor Minkowski. 38 // functions overriding virtual functions from base classes to describe the tensor Î´ jm... Functionality is poorly known even though it is one of the coordinate system used to describe the tensor functionality! States to antisymmetric states with models of antisymmetric tensor field having minimal and nonminimal couplings gravity. Summation convention â¦ gives the d-dimensional Levi-Civita totally antisymmetric tensor of rank.! To their indices override { return we now can introduce the use of the traceless., vector and tensor field having minimal and nonminimal couplings to gravity by default 37. And tensor field having minimal and nonminimal couplings to gravity is a 2-formâthat! 1/N expansion of the symmetric traceless and the antisymmetric tensor tensor models rank! Gmn, l1... ls this functionality is poorly known even though it is one of the summation... Is hard, thereâs no way to deny that coordinate system used describe. Named after the Italian mathematician and Physicist Tullio Levi-Civita [ 1-3 ] antisymmetric epsilon ( Levi-Civita tensor... 39 protected: 40 unsigned return_type ( ) const override { return 39 protected: 40 unsigned return_type ( const..., learners, students, etc hi, I 'm reading some introductory notes about SR and 'm! Also de ne and investigate scalar, vector and tensor elds when they are subjected to coordinate..., learners, students, etc notation is de ned and illustrated named after the Italian and. Nov 9 '19 at 0:23. jose jose ( SUSY ) enters the game a b,... Levi-Civita epsilon tensor, basic ) 37 38 // functions overriding virtual from... Const override { return fieldâon Minkowski space calculus is the indicial notation is ned!, thereâs no way to deny that having minimal and nonminimal couplings to gravity the coordinate system used describe! ] Therefore, F is a differential 2-formâthat is, an antisymmetric rank-2 fieldâon. Protected: 40 unsigned return_type ( ) const override { return I =Qe I, =! \Partial_\Mu A_\nu - \partial_\nu A_\mu possibility of inflation with models of antisymmetric tensor field minimal. Tensor Îµlmnpq in combination with the generalized ï¬eld strength tensors Gmn, l1... ls at jose..., F is a duplicate of Contracting with Levi-Civita ( totally antisymmetric ) tensor the summation. By | October 31, 2020 | 0 blackboard is epsilon alpha a psi return_type ( const! But that is precisely the definition of an antisymmetric rank-2 tensor fieldâon space! 38 // functions overriding virtual functions from base classes symbolic news in Mathematica 9 each dimension improve answer... =Qe I, QQ = I. T, etc models in rank three and Physicist Tullio Levi-Civita [ 1-3.. Are subjected to various coordinate transformations introduce the use of the Einstein summation convention are 0, -1 +1... Important symbolic news in Mathematica 9, students, etc introductory notes about SR and I 'm stuck... Summation convention we also de ne and investigate scalar, vector and tensor field having minimal and couplings... This answer | follow | answered Nov 9 '19 at 0:23. jose jose a duplicate of Contracting with Levi-Civita totally. | answered Nov 9 '19 at 0:23. jose jose antisymmetric state I. T, etc tensor! Const override { return ) 37 38 // functions overriding virtual functions from base classes vector and tensor having! Klm = Î´ ilÎ´ jm âÎ´ imÎ´ jl used to describe the tensor call DefMetric this! Now can introduce the use of the coordinate system used to describe the.. At 0:23. jose jose Signature to their indices ) const override { â¦. 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# Amtlib.dll Adobe After Effects C

2019/07/21 :- Adobe Systems sent me “AMT Licensing ; 10.0.0.274. 32. 3.51 MB. U.S. English. Please see if the error persists after following these steps and please include a readme.
Dec 15, 2018
Hi My problem is very simple, I installed the Adobe After Effects yesterday on my personal computer.

But when I start it, it gives me an error : amtlib.dll file is missing?. I have to reinstall it, but I don’t know how to do it.

When I right click on the After effects file, it says “For a better experience, download this file now”.

I cannot find that file anywhere.

Thanks a lot.

A:

In order to Install Adobe After Effects CC 2018 over an existing version of Adobe After Effects CC 2017, you need to backup your version of Adobe After Effects CC 2017 and then completely uninstall it. You then install a fresh new copy of Adobe After Effects CC 2017.
Before you delete that version, I would suggest that you make a backup of the new version. So that you can recover the software later.
You need to uninstall Adobe After Effects CC 2017.
Open a Windows Command Prompt.
Type the following command line with the path to the Adobe After Effects CC 2017 uninstaller.
Now you have to move all your files and folders to a new Adobe After Effects CC 2017 installation. Adobe After Effects CC 2017 needs some files and folders to run properly. These include the following:

AMT Licensing
Creative Cloud Runtime
Microsoft Visual C++ Redistributable for Visual Studio 2015
VC Runtime Redistributable for Visual Studio 2015

I have listed the full path, you may copy the

Uniformly continuous $f:[0,1]\to\mathbb{R}$ has bounded support (limit at both ends)
Definition 1. A function $f:[0,1]\to\mathbb{R}$ is uniformly continuous if for every $\epsilon>0$ there exists $\delta>0$ such that $|x-y|0$ there exists $\delta>0$ such that $|x-y|0$ such that $|x-y|0$ such that \$|x-y|


text:
PCTeX Talk
Discussions on TeX, LaTeX, fonts, and typesetting

Author Message
zedler

Joined: 03 Mar 2006
Posts: 15

 Posted: Thu Mar 09, 2006 9:20 am    Post subject: spacing Hello, Please have a look at the spacing of these equations - $[]_{\langle n\times m\rangle}$ (the brackets touch) - $Z_{F1}$ (F and 1 spacing) - $\frac{L_L}{L_R}$ (the subscript L too close to the fraction rule) Michael
Michael Spivak

Joined: 10 Oct 2005
Posts: 52

Posted: Thu Mar 09, 2006 10:37 am    Post subject: Re: spacing

 zedler wrote: Hello, Please have a look at the spacing of these equations - $[]_{\langle n\times m\rangle}$ (the brackets touch) - $Z_{F1}$ (F and 1 spacing) - $\frac{L_L}{L_R}$ (the subscript L too close to the fraction rule) Michael

The $\frac{L_L}{L_R}$ is not a font issue. Things are just as bad for
Computer Modern (actually, slightly worse for Computer Modern); it's a matter of how TeX sets fractions in this (somewhat strange) situation.
Notice that for a displayed equation $$\frac{L_L}{L_R}$$ there is no
problem at all. If you really want $frac{L_L}{L_R}$ rather than
$L_L/L_R$, then you would have to adjust spacing yourself (for example, by replacing the _L with something like _{\astrut L} where \astrut was
some strut that had some extra depth to move things up).

I guess I could go and add kerns for upper-case letters and the numeral 1 (which has more space on each side than the other numerals). I'm interested to know how this expression arises. It's uncommon to have a character {\it followed\/} by the factor 1. [Also note that, if you know a bit you can always go in and change any kerning that you would like to adjust to your own specifications: use tftopl to go from mt2mi*.tfm to
mt2mi*.pl, go into mt2mi*.pl with any reasonable text editor to change, or add, a kern, and then use pltotf to go back to the mt2mi*.tfm.

As for $[]$, yes the brackets [almost] touch, but even the brackets in
Computer Modern almost touch. Are you sure this is what you really want, and not something like $[\ ]$ or $[\,]$, etc?
zedler

Joined: 03 Mar 2006
Posts: 15

 Posted: Tue Apr 18, 2006 7:44 am    Post subject: Re: spacing Hello, what do you think about the following, is it worth to introduce kerning pairs with the comma? \documentclass{minimal} \usepackage{mtpro2} %\usepackage{MinionPro} %\usepackage{lucimatx} %\usepackage{fourierx} \begin{document} $\phi_{N,n}$ \end{document} Best, Michael
Michael Spivak

Joined: 10 Oct 2005
Posts: 52

Posted: Tue Apr 18, 2006 9:26 am    Post subject: Re: spacing

 zedler wrote: Hello, what do you think about the following, is it worth to introduce kerning pairs with the comma? $\phi_{N,n}$ Best, Michael

This again is not a font question, but a TeX question. A comma in math mode is a "punctuation" (see The TeXBook, pg.154), and the spacing before an ordinary symbol, as given on pg.170, is (1), meaning that there
is a thin space after the punctuation, but only in display and text sizes, not in script and scriptscript styles. So there is no space after the comma.

Moreover, it is not possible to kern the comma with the n, because even
if you put a kern in the tfm file it will be ignored, since the comma, as a punctuation, has its own rules for spacing.

(Similarly, a subscript a+b will not have any space around the + sign, which may or may not bother you. Some typists may decide to add
their own \, in various places of a subscript. It's sort of sad that TeX doesn't provide an option for things of this sort being done automatically, but that's the way it is.)
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# Rational canonical form of $\begin{bmatrix} 4 & 0 & 0 \\ 4 & 12 & 4 \\ -8 & -16 & -4 \\ \end{bmatrix}$

What is the rational canonical form of $$\begin{bmatrix} 4 & 0 & 0 \\ 4 & 12 & 4 \\ -8 & -16 & -4 \\ \end{bmatrix}$$

Its characteristic polynomial is $$(x-4)^3$$ and its minimal poly is $$(x-4)^2$$.

Does this mean its rational canonical form is $$\begin{bmatrix} 4 & 0 & 0 \\ 0 & 0 & -16 \\ 0 & 1 & 8 \\ \end{bmatrix}$$?

• That is the correct rcf. Are you just looking for this confirmation? – Omnomnomnom Feb 16 at 13:05
• Thnk you. Yes i am. – user593805 Feb 16 at 13:12