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DescTools (version 0.99.44)

# VIF: Variance Inflation Factors

## Description

Calculates variance-inflation and generalized variance-inflation factors for linear and generalized linear models. It's a measure describing how much the variance of an estimated coefficient is increased because of collinearity.

## Usage

VIF(mod)

## Arguments

mod

an object that responds to coef, vcov, and model.matrix, such as an lm or glm object.

## Value

A vector of vifs, or a matrix containing one row for each term in the model, and columns for the GVIF, df, and $$GVIF^{1/(2\times df)}$$.

## Details

If all terms in an unweighted linear model have 1 df, then the usual variance-inflation factors are calculated.

The vif are defined as $$vif_j=\frac{1}{1-R_j^2}$$

where $$R_j^2$$ equals the coefficient of determination for regressing the explanatory variable j in question on the other terms in the model. This is one of the well-known collinearity diagnostics.

If any terms in an unweighted linear model have more than 1 df, then generalized variance-inflation factors (Fox and Monette, 1992) are calculated. These are interpretable as the inflation in size of the confidence ellipse or ellipsoid for the coefficients of the term in comparison with what would be obtained for orthogonal data.

The generalized vifs are invariant with respect to the coding of the terms in the model (as long as the subspace of the columns of the model matrix pertaining to each term is invariant). To adjust for the dimension of the confidence ellipsoid, the function also prints $$GVIF^{1/(2\times df)}$$ where $$df$$ is the degrees of freedom associated with the term.

Through a further generalization, the implementation here is applicable as well to other sorts of models, in particular weighted linear models and generalized linear models.

Values of vif up to 5 are usually interpreted as uncritical, values above 5 denote a considerable multicollinearity.

## References

Fox, J. and Monette, G. (1992) Generalized collinearity diagnostics. JASA, 87, 178--183.

Fox, J. (2008) Applied Regression Analysis and Generalized Linear Models, Second Edition. Sage.

Fox, J. and Weisberg, S. (2011) An R Companion to Applied Regression, Second Edition, Sage.

## Examples

Run this code
# NOT RUN {
VIF(lm(Fertility ~ Agriculture + Education, data=swiss))
VIF(lm(Fertility ~ ., data=swiss))
# }

Run the code above in your browser using DataCamp Workspace


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Unlimited Plugins, WordPress themes, videos & courses! Unlimited asset downloads! From \$16.50/m

# Should You Start Using CSSLint?

Getting our code reviewed by a pro is a great way of improving code quality but what happens if you don't have access to a rockstar programmer? You do the next best thing and grab a 'lint' for that language.

Today, I'd like to talk a little about CSSLint, a recently released code analysis tool for, you guessed it, CSS. Join me after the jump!

## What Exactly is a Lint?

Let's hit Wikipedia for a definition:

Lint is a tool that flags suspicious usage in software written in any computer language.

Basically, a lint looks at our code and checks for bad programming practices. Undefined variables, inefficient constructs, things like that.

You're probably wondering why you'd ever need such a tool. Let's face it: not all of us possess supreme knowledge of what we're working with or have the luxury of getting our code peer reviewed. In these cases, sticking our code in a lint is the next best option. And unlike clean up tools, lint spits out tidbits about your code and how to improve it.

## Introducing CSSLint

Created by Nicholas Zakas and Nicole Sullivan, CSSLint is an open source tool that checks your syntax against a set of predefined rules to root out possible inefficiencies and make sure that your presentation works as expected with little surprises.

After heading over to the CSSLint site, you can merely paste in your source CSS and choose which rules you'd like enforced. Hit the lint button and CSSLint will start eroding your smugness.

If you're a Node junkie like me, there's a version for that as well. Just type in sudo npm install -g csslint and you're good to go!

## The CSS Lint Rules

Let's take a quick look at the rules that CSSLint enforces.

• Parsing errors should be fixed
• Remove empty rules
• Use correct properties for a display
• Don't use too many floats
• Don't use too many web fonts
• Don't use too may font-size declarations
• Don't use IDs in selectors
• Heading styles should only be defined once
• Zero values don't need units
• Vendor prefixed properties should also have the standard
• CSS gradients require all browser prefixes
• Avoid selectors that look like regular expressions
• Beware of broken box models
• Don't use @import
• Don't use !important
• Include all compatible vendor prefixes
• Avoid duplicate properties

If you're anything like me, a few of the rules must have had you flummoxed.

## Do the Rules make Sense?

Quite frankly, yes, no and everything in between.

After lurking at a number of discussion boards and IRC rooms, I found out that many developers seem to be in uproar over the rules and maybe right so. Let me attempt to explain why most of the rules make sense but others do not.

### The Controversial Rules

#### Don't use IDs in selectors

IDs shouldn't be used in selectors because these rules are too tightly coupled with the HTML and have no possibility of reuse. It's much preferred to use classes in selectors and then apply a class to an element in the page.

This one struck a nerve with a lot of developers since we're quite accustomed to assigning IDs for the main structural components of a layout like the header and footer.

CSSLint argues that the styling for such elements, by definition, can't be directly reused. If you want dual sidebars on your page, a class lets you reuse styling while an ID will not.

Keep in mind that just because a class can be reused doesn't mean it has to be. Unique classes are perfectly acceptable and a swell way to reuse styling if the need arises.

Heading elements (h1-h6) should be defined as top-level styles and not scoped to particular areas of the page.

Most developers, including me, have gotten used to writing context sensitive headers. As in, we define separate styling for headers depending on, say, which page it is being displayed on. An argument in favor of this approach is that it moves all cruft from the markup to the stylesheet. You can merely define a tag and let the CSS cascade accordingly.

CSSLint argues that such an approach compromises the predictability of your design. If someone else were to pick up your design and tried putting in a heading somewhere, he/she would need to be aware of the context and placement of the element. With headings defined unconditionally, he or she can use a heading anywhere confident of its presentation regardless of its parents.

#### Heading styles should only be defined once

Heading elements (h1-h6) should have exactly one rule on a site.

An extension of the rule above to improve predictability of presentation. Right or wrong, keep in mind that this basically excludes including some sort of reset code within your stylesheet. Every reset sheet works on your headings as well and thus CSSLint will mark it as an error.

### The Questionable Rules

Adjoining classes look like .foo.bar. While technically allowed in CSS, these aren't handled properly by Internet Explorer 6 and earlier.

With this rule enabled, CSSLint flags rules like .nav.red, with the official reason being that Internet Explorer 6 and below don't play well with the selector. I respect the developers but I have to disagree here. Just because it doesn't work with Internet 'Dev-buster' Explorer 6 is not a great reason to stop working with a useful feature.

#### Beware of broken box models

Borders and padding add space outside of an element's content. Setting width or height along with borders and padding is usually a mistake because you won't get the visual result you're looking for. CSS Lint warns when a rule uses width or height in addition to padding and/or border.

The box model maybe broken but almost every front end developer I know is intimately aware of the shortcomings and how to overcome the disparities with implementation. Are we really ready to give up a layer of control because some older browsers have a different implementation?

#### Don't use too many web fonts

Using web fonts comes with performance implications as font files can be quite large and some browsers block rendering while downloading them. For this reason, CSS Lint will warn you when there are more than five web fonts in a style sheet.

I don't foresee a situation where I'd be using more than five fonts in a page but I feel that dipping into this territory is a bit questionable. If anything, this is a design flaw than a development flaw. If a developer is referencing five separate fonts in his stylesheet, chances are, it ain't by accident.

#### Don't use too many floats i.e. abstract the functionality away

CSS Lint simply checks to see if you've used float more than 10 times, and if so, displays a warning. Using this many floats usually means you need some sort of abstraction to achieve the layout.

While I concur with the creators' argument that having more than ten instances of float is a bad idea, I also feel that this will affect the markup once past a given size.

For example, in a situation where you'd want to float two elements, traditionally you'd use:

... and the styling, by traditional methods.

The CSSLint method would be abstracting the float like so:

... and styling like so:

While I agree that this is a viable approach, I feel that the markup will get significantly crowded once you try to abstract more away. I'd rather see a class containing most of its styling in one place than clutter the markup with 10+ classes. Again, I feel that this is a subjective topic.

### The Obvious Rules

• Remove empty rules
• Avoid duplicate properties
• Zero values don't need units
• Vendor prefixed properties should also have the standard
• Parsing errors should be fixed
• Include all compatible vendor prefixes
• ... rest of the rules

All of the rules above adhere to the current standard practices. Sure, some of the rules have little real world significance, like zero values not needing units, or will be caught by a decent IDE, like parsing errors, but nevertheless these are good rules to have in a CSS test suite.

## A Few Concerns

CSSLint is going to help a lot of developers down the road but...

CSSLint is, no doubt, written by developers with great credentials and is definitely going to help a lot of developers and designers down the road.

What I find a little irksome is that a lot of the controversial rules come from Object Oriented CSS, a CSS framework intended to let developers write maintainable CSS. While I have nothing against the framework, and its paradigm, you'd have to agree that it's a way of doing things, not the way to do things.

As opposed to JSLint where I feel like all the rules make sense, with CSSLint, it feels like I'm being told that one style of writing CSS is right and the others are wrong. It'd be like someone asking me to give up the Beatles because Rolling Stones is their preferred band.

## It's Only a Tool

Tools are just that -- tools.

Of course, we, as a group, tend to be rather clingy when it comes to our code. We don't enjoy hearing that our code could have *gasp* potential issues or be written in an entirely different way.

The main thing to make a note of here is that CSSLint is, ultimately, a tool. It merely lets you know that some of the areas may have errors.

CSSLint doesn't have to be the iron fist around which you base your entire ego on. There's no reason to bend over backwards to avoid a warning if you know precisely what you're doing.

## So, Should you Start Using CSSLint?

Yes.

In CSS, as in integral calculus, there are many solutions to a given problem. There necessarily isn't a 'best' way to do things -- you may favor readability while I may favor efficiency. What's important is that you realize that each and everyone of us have our unique way of doing things.

If you don't have the same perspectives towards solving a problem, you may disagree with another approach and may even find it questionable.

Having said that, there's never a good reason for not learning new things. Looking at problems from the perspective of another developer is a great way to see if you can learn something new.

## Wrapping Up

What do you think about CSSLint? Find it useful? Confusing? Has it helped with your real world problems? Let us know in the comments.

Be excellent to each other and thank you so much for reading!

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# p-sylow subgroup

• Nov 5th 2008, 06:05 AM
deniselim17
p-sylow subgroup
Let $o(G)=pq$, where $p>q$ are primes.
Since $p$ divides $o(G)$, $G$ contains an element $b$ of order p.
Let $S=$.
Since $S$ is a $p$-sylow subgroup of $G$, the number of conjugates is $1+kp$ for some $k \geqslant 0$.
But $1+kp=[G:N(S)]$ which divides $o(G)=pq$.
Since $(1+kp,p)=1$, then $1+kp$ divides $q$.
Since $q, then $k=0$ and $S$ is normal in $G$.

I don't understand why S is normal in G. If it is because [G:N(S)]=1, why?


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# Derivative Calculator with step-by-step Explanations

The following variables and constants are reserved:

You can enter expressions the same way you see them in your math textbook. Implicit multiplication (5x = 5*x) is supported. If you are entering the derivative from a mobile phone, you can also use ** instead of ^ for exponents. The interface is specifically optimized for mobile phones and small screens.

## Supported differentiation rules

• Sum rule
• Elementary power rule
• Constant factor rule
• Product rule (Leibniz rule)
• Quotient rule
• Chain rule
• Inverse function rule
• Logarithmic derivatives
• Exponential functions
• Trigonometric functions


text:
# What happens to acid when put in water?

It is conceived to increase the concentration of the hydronium ion, ${H}_{3} {O}^{+}$, the characteristic cation of water.
The lower hydrogen halides, $H - X$, $\left(X \ne F\right)$, are all room temperature gases. They are all quite soluble in water, and in aqueous solution they are conceived to protonate the water molecule:
$H X + {H}_{2} O \rightarrow {H}_{3} {O}^{+} + {X}^{-}$
The $\text{hydronium}$ or $\text{acidium ion}$ is a cluster of 3-4 water molecules with an extra ${H}^{+}$ added. We use representations such as ${H}^{+}$ or ${H}_{3} {O}^{+}$ for simplicity.


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## Introduction

Electronic excitations underlie numerous biological and physical processes of interest, including photosynthesis1,2, DNA damage3,4, photodynamic therapy5, photopharmacology6,7, and solar energy conversion8,9. Excited-state dynamics simulations are a powerful tool to uncover the mechanism of photoresponse in these systems, particularly when combined with ab initio electronic-structure calculations, since the important molecular motions do not need to be known a priori but are revealed by the dynamical trajectories10,11. Given its reasonable accuracy within current approximations, linear-response time-dependent density-functional theory (LR-TDDFT)12,13 has become a work-horse method in the field; however, its computational expense (formally scaling with system size as N3) and the need to solve the electronic-structure problem at every simulation timestep limits the system size and timescale amenable to study. Thus, a cheaper approach to electronic excitations and excited-state dynamics is highly desirable but poses a significant challenge.

One might question whether the expense of LR-TDDFT is needed in the first place. For the ground state (GS), the first Hohenberg–Kohn (HK1) Theorem14, which provides the foundation of DFT, establishes the existence of a bijection from the ground-state electron density to the external potential of a many-body system. As a consequence, there exists a formal map from the ground-state electron density not only to the ground-state energy but also to every property of the system. The map from density to ground-state energy is encoded by an unknown universal density functional; however, practical approaches make use of the Kohn-Sham (KS) theory15, which expresses the functional via a fictitious non-interacting system that shares the density of the true system, allowing an exact treatment of kinetic energy and Coulomb contributions to the functional and approximations to the remaining small exchange and correlation terms. While KS theory provides a practical scheme for solving simultaneously for the ground-state density and energy of a system, functionals that map from the ground-state density to excited-state energies are currently unknown, although, their formal existence is established by the HK1 theorem. As an alternative, LR-TDDFT provides an indirect means to the excited-state energies from knowledge of the ground-state density and its response to an external driving field12,13. Given the expense of LR-TDDFT, a direct map from density to excited-state energies would be preferable.

Theoretical support for a direct route to excited-state energies comes from a generalization of the HK1 theorem to an excited-state density-energy bijection. While a universally general HK1 theorem for excited states with arbitrary external potentials has been disproved16, a special excited-state HK1 theorem for Coulombic external potentials (i.e., molecular systems) has been argued to exist17 and has motivated recent orbital-optimized DFT approaches to electronic excited states18,19.

Alongside theoretical developments in excited-state DFT, there has been significant recent progress in using data-driven approaches to machine-learn ground-state density functionals20,21,22,23,24,25,26,27,28. Related to the current work, there has also been progress in developing machine-learning models to directly predict electronic excited-state energies, gradients, and non-adiabatic couplings given a particular molecular geometry, and the SchNarc approach by Westermayr, Marquetand et al. has been demonstrated to predict non-adiabatic dynamics in agreement with ab initio simulations using a reasonable amount of training data29,30,31,32,33,34,35. In the current work, we explore the proposition that a multistate density functional can be machine learned via an excited-state HK map. This new development establishes a framework that is potentially more powerful than directly learning excited-state energies, since densities and density functionals can yield any desired property.

In order to construct an ML excited-state density functional, we work in the framework introduced in ref. 22, which uses a data-driven ML model with a physically motivated representation of the molecule. The fundamental object in this approach is a potential-to-density map, n[v](r), which is machine-learned by computing the external potential and corresponding density at a desired level of theory for a range of nuclear geometries of the system and then inputting these functions as training data. For ground-state densities, this map is known as the machine-learned Hohenberg–Kohn (ML-HK) map; once learned, it can be used to compute any related quantities, including ground-state energies, and other observables at the same level of theory22,23. It is worth noting that the level of theory is not restricted to DFT but can also be performed, for example, using coupled-cluster theory23. In this work, we generalize the potential-to-density map to excited states in a multistate Hohenberg–Kohn framework (ML-MSHK), allowing us to learn excited-state densities and energies simultaneously with comparable accuracy to the ground state.

In order to test our ML-MSHK model, we consider the excited-state proton transfer (ESPT) reaction in malonaldehyde (MA), a small organic molecule exhibiting a non-trivial internal reaction that is sensitive to electronic excitation. ESPT is at the heart of photoacidity and is a key step in the photocycle of many photoactive proteins, such as green fluorescent protein (GFP)36,37. Intramolecular proton transfer, as in MA, serves as a useful model to study ESPT, since the reaction rate is not limited by the diffusion of the proton donor and acceptor together, and can therefore be probed by ultrafast spectroscopy38,39.

The structural simplicity of MA makes it an especially appealing model of ESPT. In particular, one conformation of malonaldehyde is a ring structure with an internal hydrogen bond that allows for a proton-transfer reaction between the two oxygen atoms (see molecular graphic in Fig. 1a). Excitation of this molecule from the S0 ground state to the S2 singlet excited state (ππ*) leads to a substantial reduction of the proton-transfer barrier from several kcal/mol in the ground state to an essentially barrierless reaction in the excited state. However, the presence of a lower-lying singlet nπ* state in MA significantly complicates its ESPT reaction, and ultrafast non-adiabatic transitions from S2 to S1 are believed to compete with the ESPT reaction40,41,42. The barrier to proton transfer on S1 is even higher than the ground-state40,43, and furthermore, a three-state intersection is predicted to be energetically accessible41, meaning the electronic excitation can be efficiently quenched to the ground state. As a result, the ESPT reaction in MA is largely hindered by competing processes and has yet to be directly observed experimentally42.

One approach to suppressing the competing processes in MA and, thereby, controlling the ESPT reaction is to raise the nπ* state energy via functional group modifications. For example, the nπ* and ππ* energetic ordering is reversed in methyl salicylate (MS)44, which shares the chelate ring of MA yet exhibits efficient ESPT, as the non-adiabatic transitions to the nπ* state are removed38,45,46. In this work, using excited-state dynamics with our ML-MSHK method, we seek to explore whether mechanical restrictions applied to MA can instead be used to promote the ESPT reaction. Recognizing that the S2/S1 transition in MA is brought about by torsional motions41, we propose that a restraint of planarity on the molecule will allow the ESPT reaction to proceed unhindered. This restraint can be viewed as biomimetic of a similar mechanism in GFP whose protein environment enforces planarity of the 4-(p-hydroxybenzylidene)imidazolidin-5-one chromophore, preventing nonadiabatic transitions and enhancing its fluorescence quantum yield47.

## Results

### Theory

There is a growing recognition of the importance of using physics-based principles in choosing descriptors and models for machine-learning molecular properties48,49. The blessing, and perhaps curse, of ML models is their great flexibility to predict almost any quantity given sufficient data. This can lead to a loss of physical insight, and concerns of overfitting must always be addressed. By using physical principles, fundamental constraints are automatically introduced into the model that mitigate these problems. In the context of learning electronic structure, a very natural physical framework relies on the connections between the electron density, n(r), external potential, v(r), and energy, E, of a many-body quantum system. These connections are formalized by the HK1 theorem, which proves the existence of a bijection from the ground-state electron density to the external potential of a many-body system: n0(r) ↔ v(r)14. The electron density, which gives the probability for finding an electron at a certain location in space, is a particularly convenient quantity to work with since, unlike the many-body wavefunction, it is a three-dimensional scalar field regardless of the number of electrons and is, therefore, straightforward to represent numerically. For molecular systems, the external potential is simply the Coulombic scalar potential arising from the nuclear charges: $$v({{{{{{{\bf{r}}}}}}}})=-\mathop{\sum }\nolimits_{a}^{N}{Z}_{a}/|{{{{{{{\bf{r}}}}}}}}-{{{{{{{{\bf{r}}}}}}}}}_{a}|$$. Since the external potential uniquely defines the molecular electronic Hamiltonian (for a given spin state and number of electrons), the electronic energy must also be uniquely determined from knowledge of the external potential or, per HK1, the density14. These maps from external potential to density and energy are realized as functionals, e.g., n[v](r), which inspired the framework for a previous approach that machine-learned density functionals for ground-state energy predictions (ML-HK)22. Part of the success of the ML-HK approach can be attributed to the uniqueness of the external potential and density as molecular descriptors, a property not shared by some other choices50.

We now demonstrate how these ideas can be extended to electronic excited states. As mentioned in the introduction, an excited-state extension of HK1 has been argued to exist for Coulombic external potentials17, meaning that the map from external potential to each state’s density is encoded as a functional, nj[v](r), where j = 0, 1, 2, . . . . labels each excited state. Furthermore, a property of the (unknown) exact universal functional is that each extremal density of the energy functional corresponds to an electronic eigenstate with the functional returning the exact (excited-state) eigenvalue, Ej = E[nj]51. This motivates a machine-learning approach that leverages an excited-state Hohenberg–Kohn mapping.

### Learning excited states with ML-MSHK

The first step of our ML model is to learn multiple electronic state densities. We start by expanding the densities in an orthonormal basis set, ϕl(r), as $${n}_{j}^{{{{{{{{\rm{ML}}}}}}}}}[v]=\mathop{\sum }\nolimits_{l=1}^{L}{u}_{j}^{(l)}[v]{\phi }_{l}({{{{{{{\bf{r}}}}}}}})$$, and we learn the set of basis coefficients $$\{{u}_{j}^{(l)}\}$$ by training on a set of input potentials corresponding to different geometries of a system. We allow for the learning of ground- and excited-state densities with a unique map to each state j. Various ML models, including artificial neural networks and kernel methods, have been used to learn total energies from electronic-structure calculations22,23,30,52,53,54,55,56,57,58,59. However, learning excited-state HK maps is unique to our approach, and we follow the ground-state ML-HK functional’s successful use of the kernel ridge regression (KRR) method60. In principle, the functionals could instead be learned with neural networks, but a comparison of different methods is beyond the scope of this manuscript.

With a set of learned densities, a second KRR model is used to learn ground- or excited-state energies from a set of input training densities. We consider two types of energy functionals, depending on the nature of the input used. The first relies on the proven existence of a single functional that maps ground or excited-state densities to their respective energy eigenvalues51. We thus learn a single map from density to energy and use multiple states in the training of this map:

$${E}^{{{{{{{{\rm{ML-MSHK}}}}}}}}}[{n}_{j}]=\mathop{\sum }\limits_{i=1}^{M}\mathop{\sum}\limits_{k}{\alpha }_{i,k}K({{{{{{{{\bf{u}}}}}}}}}_{i,j}[v],{{{{{{{{\bf{u}}}}}}}}}_{i,k}[{v}_{i}]),$$
(1)

where K(ui,j[v], ui,k[vi]) is the kernel, k runs over the states used in the training of the model, and {αi,k} are the coefficients learned in the energy functional model. Since energy predictions from this model rely on training with multiple electron densities for a given molecular geometry, we call the combination of the two types of maps the multistate Hohenberg–Kohn approach (ML-MSHK). Learning Eq. (1) comes at the cost of retaining multiple densities for each training example, increasing storage needs; however, the advantage is the resulting energy functional is not state specific. We note that while this work focuses on the planar S2 state, we also train on densities from the S1 excited state to demonstrate that including additional densities does not degrade the performance of the ML-MSHK model.

The second energy functional we consider is of the external potential itself, without using the density as an intermediate descriptor. Following the naming of a similar approach for ground-state energies22, we call this the multistate Kohn-Sham map (ML-MSKS):

$${E}_{j}^{{{{{{{{\rm{ML-MSKS}}}}}}}}}[v]=\mathop{\sum }\limits_{i=1}^{M}{\gamma }_{i,j}K\left[{v}_{i},v\right],$$
(2)

where {γi,j} are the coefficients learned in the ML-MSKS energy functional model. Such functionals must also exist since the external potential uniquely defines the molecular electronic Hamiltonian and, therefore, also its eigenstates14. Eq. (2) has the advantage that it uses only external potentials rather than densities, reducing computational storage requirements compared to Eq. (1). However, a disadvantage is that the energy maps in this model are state specific, which we expect will introduce errors near electronic crossings30.

In all cases, models are trained against ab initio densities and energies following the procedure described in section “Methods”. A schematic of the different maps used in this study is shown in Fig. 1a.

### Excited-state energy predictions

In order to test the performance of the excited-state machine-learned density functionals, we start by considering learning curves for training S2 excited-state energies of MA, shown in Fig. 1b, in which the lowest three densities (n0, n1, n2) were used for the ML-MSHK model. The smallest training set considered contains 2000 molecular conformations of MA generated from a ground-state AIMD trajectory, propagated as described in section “Training and test set generation”. A training set similar to this was previously found to yield a ML-HK potential that accurately described the proton-transfer reaction on the ground state22. From Fig. 1b, however, we see somewhat large out-of-sample mean absolute errors (MAEs) of 0.5 and 0.8 kcal/mol for the ML-MSHK and ML-MSKS models, respectively, with this training set. Furthermore, for the ML-MSKS model, the error is seen to vary strongly with geometry, as revealed by the orange curve in Fig. 1c, which shows energy prediction errors along a minimal-energy pathway for the ESPT reaction as a function of the proton-transfer coordinate, r = rHO1 − rHO2, where rHOi is the distance of the proton from oxygen atom i. Using the same 2000 training geometries, the ML-MSHK error (dashed blue curve) is more uniform than the ML-MSKS predictions, demonstrating the advantage of the former.

In order to use the ML-MSHK model for excited-state dynamics, we must further reduce its prediction errors; however, we found essentially no improvement upon adding more samples from the ground-state trajectory. As we will see below, the remaining source of error in models trained only with ground-state samples is that the S2-initiated ESPT involves nuclear responses also in modes orthogonal to the direct proton transfer mode, and these are not adequately sampled by the ground-state trajectory. Thus, we extended the training set by including geometries extracted from 30 S2 excited-state AIMD trajectories, in a manner described in Supplementary Note 1, yielding the learning curves shown in Fig. 1b.

After including geometries extracted from excited-state trajectories in the training (ground and excited-state samples), both ML models perform significantly better than the ML model constructed using only ground-state samples in the training. Interestingly, the ML-MSKS map (Fig. 1b, orange curve) always performs worse than the MSHK map (blue curve). Similar behavior was noted in the ground-state ML-HK study of ref. 22. The rest of this work pertains to the ML-HK models since learning the densities as an intermediate step yields the models that outperform the direct potential-to-energy approach, and the inclusion of density learning has other benefits such as the capability of density-based delta-learning to add corrections from high-level wavefunction theory23. Encouragingly, the out-of-sample error for ML-MSHK converges below 0.2 kcal/mol once the sample size reaches M = 5000. This error is comparable to that found in the previous ground-state study of MA22, suggesting that ML-MSHK will be suitable for molecular dynamics simulations (confirmed below). Having found convergence at M = 5000, our final training set generation follows the protocol described in section “Training and test set generation”. For the adiabatic excited-state dynamics considered in this work, it is possible to train state-specific ML-HK models to return the S2 energies using only ground-state or S2 densities as input (see Supplementary Fig. 1). While we do not include any non-adiabatic transitions in this study, we note that the state-specific models are worse than ML-MSHK in the vicinity of electronic crossings (see Supplementary Note 5). Given the excellent performance of the ML-MSHK model and its ability to capture multiple electronic states simultaneously, we focus exclusively on this model for the remainder of this work.

### Excited-state density predictions

Equally important to energy predictions is the question of how well the model reproduces electron densities. This is explored in Fig. 2, which shows electron densities for the CS-symmetric ground-state optimized geometry of MA (similar results for the proton transfer transition state are shown in Supplementary Fig. 4). In order to highlight the changes in bonding structure, excited-state densities are displayed as density differences with respect to the ground state. For all states considered, there is no discernible difference between the ML-MSHK predicted densities (left) and ab initio TD-PBE0 densities (right), even though the ground-state optimized structure was not included in the training set. Indeed, the out-of-sample integrated MAE in the total electron densities is very small (0.012 e).

Knowledge of the density differences also provides insight into the nature of the electronic excitations. From Fig. 2, S1 clearly corresponds to an nπ* transition with a depletion of density from the proton-accepting oxygen’s px orbital and an accumulation on the backbone π* orbital. S2, on the other hand, is a ππ* transition, as identified by a depletion of density on the central conjugated carbon’s pz orbital.

### Excited-state MD using ML potentials

Having demonstrated that our ML-MSHK model provides a quantitative prediction of excited-state electron densities and total energies within our test set of geometries, we next consider its ability to generate a functional for use in excited-state MD. We thus initiated non-equilibrium MD trajectories of MA on its S2 state following vertical excitation from the ground state. 1000 ML-MSHK and 50 AIMD trajectories were run and compared. It should be noted that the computational effort associated with propagating dynamics on the ML-MSHK surface is negligible compared to AIMD dynamics. Thus, for the cost of 50 excited-state AIMD trajectories used to generate the training and test sets, we gain the ability to perform excited-state dynamics with 1000 ML-MSHK trajectories using the resulting machine-learned functional to converge the excited-state non-equilibrium averaged properties.

We consider three metrics to assess the quality of the machine-learned functional. The first concerns how well ML-MSHK is able to interpolate the energy between geometries explicitly included in the training. This is accomplished by evaluating ML-MSHK energies along an AIMD excited-state trajectory from which some geometries were included in the training. The result is shown in Fig. 3a, where we see almost perfect agreement between ML-MSHK and the AIMD TD-PBE0 energies. Note that only 32 geometries out of 481 from this trajectory were used in the training set of the ML-MSHK model, highlighting the fidelity to which ML-MSHK is able to interpolate the functional between training points. Second, we assess how well ML-MSHK propagates the dynamics starting from a geometry included in the training set, shown in panel b, where again we see almost perfect agreement with AIMD energies evaluated on the ML trajectory snapshots. Finally, we check how well the machine-learned functional does on arbitrary geometries by evaluating ab initio TD-PBE0 energies along an ML-MSHK trajectory where no geometry was included in the training set, shown in panel c. Here, slightly larger deviations between ML-MSHK and TD-PBE0 energies are seen; however, the difference is within the expected MAE bounds from the learning curve and does not grow with time, showing that ML-MSHK faithfully reproduces the excited-state energy functional, at least in the configurational space sampled during non-equilibrium dynamics. The error could be reduced by further increasing the training set; however, we found this unnecessary since this small error does not influence the dynamical predictions as we show below.

### ESPT in planar MA

Having seen that ML-MSHK provides an accurate description of the S2 excited-state functional of MA, we next consider the predictions of excited-state dynamics of the ESPT reaction using this functional. To that end, we compute the following non-equilibrium response function:

$$S(t)=\overline{\left({r}_{-}(0){r}_{-}(t)\right)/\left|{r}_{-}(0){r}_{-}(t)\right|},$$
(3)

where t = 0 corresponds to the instance of vertical excitation from an equilibrium configuration on the S0 state to the S2 state, and the overbar indicates a non-equilibrium average over initial conditions. S(t) thus measures the memory of which reaction basin the proton is in (reactant or product). A value of S = 1 means the proton is in the reactant basin, a value of S = − 1 means the proton is in the product basin, and a value of S = 0 means memory has been lost. Since MA has a symmetric proton donor and acceptor, S = 0 indicates the reaction is complete.

Figure 4a shows the ESPT reaction’s non-equilibrium response function, S(t), computed for 50 AIMD trajectories (red) and 1,000 ML-MSHK trajectories (blue). Shaded areas represent the 95% confidence intervals computed with the bootstrap method61, as implemented in SciPy v. 1.7.162. The results show that the AIMD and ML-MSHK agree within statistical certainty, highlighting the predictive power of ML-MSHK. Furthermore, an interesting feature of S(t) becomes clear in the ML-MSHK results: apart from a ~10% fraction of trajectories that undergo almost immediate proton transfer, there is a waiting period of ~40 fs before the remainder of ESPT reaction proceeds, which is then rapidly completed by 60 fs. This behavior is hinted at in the AIMD results but becomes much more apparent in the ML-MSHK predictions due to averaging over many more trajectories.

The observed 40-fs waiting period arises from important responses of the heavy atoms following excitation to S2. This can be seen for example in the average oxygen–oxygen distance, $$\overline{{d}_{{{{{{{{\rm{OO}}}}}}}}}}$$, plotted in Fig. 4b, which exhibits a decrease from ~2.57 Å to ~2.42 Å in the first 30 fs following excitation. This timescale is comparable to the waiting period of the ESPT reaction, suggesting that the oxygen–oxygen motion gates the reaction, which does not proceed rapidly until a distance of dOO ≤ 2.45 Å is reached.

The response of dOO following excitation to S2 can be understood from changes in the electron density shown in Fig. 2. The reduction in π-bonding character lessens the rigidity of the conjugated system, allowing the chelating ring to contract and the oxygen atoms to become closer. Bringing together the proton donor and acceptor oxygen atoms then reduces the barrier to ESPT, thus explaining the gating mechanism. This is explored explicitly in Fig. 5, which plots the potential energy profile of the minimum energy pathway of ESPT on S2 for a series of fixed dOO distances. Encouragingly, quantitative agreement between ML-MSHK (blue curves) and TD-PBE0 (dashed red curves) is seen, despite none of the minimum energy pathway geometries being included explicitly in the training set. For the initial average dOO distance (~2.57 Å), a proton-transfer barrier of >2 kcal/mol is seen, explaining why only a small fraction of excited trajectories undergo ESPT in the first 20 fs. The S2 potential is downhill for motions that reduce dOO to a value of 2.45 Å (following the minimum of the potential energy profile as a function of dOO), which also brings about a reduction of the barrier to proton transfer, having a value of 0.3 kcal/mol at dOO = 2.45 Å. The barrier disappears completely for dOO = 2.35 Å; however, this oxygen–oxygen distance is disfavored due to an overall increase in the potential, explaining why the initial reduction in dOO is seen to reverse at t = 30 fs in Fig. 4b.

## Discussion

In this paper, we developed a multistate Hohenberg–Kohn machine-learning framework for the accurate prediction of excited-state densities and energies. We demonstrated that excited-state energies, which are expressible as a single functional of multiple state densities, can be learned as accurately as the ground-state energy, even for the moderately sized molecule malonaldehyde, which exhibits a non-trivial excited-state proton transfer reaction. The resulting machine-learned functional is faithful to the underlying excited-state AIMD trajectories on which the model was based, allowing accurate non-equilibrium excited-state molecular dynamics trajectories to be performed with a 10-fold computational saving compared to AIMD. Furthermore, our study yielded new insight into the excited-state proton transfer reaction in malonaldehyde. This was aided by the low-cost of ML-MSHK energy predictions that allowed us to converge the non-equilibrium dynamics with 1,000 excited-state trajectories, revealing the 40-fs waiting period following photoexcitation, after which ESPT promptly completed.

The observed gating mechanism of proton transfer in planar MA by heavy atom motion has been seen previously in other intramolecular ESPT reactions which are not precluded by ultrafast non-adiabatic transitions. In particular, the lack of isotope effect for ESPT in methyl salicylate was explained by a delocalization of the reaction coordinate to modes other than the donor OH stretch38. Later studies in related ESPT molecules found the reaction path was dominated by anharmonic low-frequency backbone modes that modulated the donor-acceptor distance63,64,65. Thus, by enforcing planarity in MA, we have suppressed nonadiabatic transitions that compete with ESPT and find the mechanism of proton transfer is indeed multidimensional in nature and follows the concensus picture that has emerged for intramolecular chelate ring structures. That we correctly captured the multidimensional nature of ESPT from a limited training of excited-state trajectories is a testament to the power of our ML-MSHK method.

This work can be extended in a number of directions. As an example, we removed the restraint of planarity on malonaldehyde (see Supplementary Note 5). Models using additional non-planar training geometries show that the multistate density functional outperforms the state-specific functionals near S2/S1 state crossings, which is vital for running future non-adiabatic MD trajectories. In addition, the LR-TDDFT training data could be replaced with high-level wavefunction-based electronic structure for input to train the ML-MSHK density and energy maps. A similar approach was recently shown to be successful for the ground-state ML-HK method to learn coupled-cluster energies23. Based on the promising results presented in this work, we are optimistic that ML-MSHK will find practical use in excited-state non-adiabatic simulations with the costly electronic-structure steps done only once to build the training set.

## Methods

### Machine-learning model

In order to predict the excited-state energies of a molecule using electron densities via the ML-MSHK map introduced in section “Learning excited states with ML-MSHK”, we start by using a basis expansion of the densities:

$${n}_{j}^{{{{{{{{\rm{ML}}}}}}}}}[v]({{{{{{{\bf{r}}}}}}}})=\mathop{\sum }\limits_{l=1}^{L}{u}_{j}^{(l)}[v]{\phi }_{l}({{{{{{{\bf{r}}}}}}}}),$$
(4)

where j indexes an electronic state (j = 0 is the ground state, j = 1 is the first excited state, etc). Following the ground-state ML-HK method22, we choose a Fourier basis. In Eq. (4), l indexes a basis function, of which there are L. In this work 50 functions are used in each dimension (125,000 = 50 × 50 × 50 in total). The coefficients of the basis set expansion are learned using KRR. In particular, $${u}_{j}^{(l)}$$ are represented as a kernel expansion of the form

$${u}_{j}^{(l)}[v]=\mathop{\sum }\limits_{i=1}^{M}{\beta }_{i,j}^{(l)}\kappa [{v}_{i},v].$$
(5)

Here, $${\beta }_{i,j}^{(l)}$$ parameterizes the kernel model, and the kernel functional κ has a Gaussian form

$$\kappa [{v}_{i},v]=\exp \left[-\frac{1}{{\sigma }^{2}}\int\ {{{{{{{\rm{d}}}}}}}}{{{{{{{\bf{r}}}}}}}}{\left(v({{{{{{{\bf{r}}}}}}}})-{v}_{i}({{{{{{{\bf{r}}}}}}}})\right)}^{2}\right],$$
(6)

where σ is a kernel width hyperparameter. The external potentials, vi(r), i = 1, . . . , M, are unique to each of the M training geometries, and are paired with M training densities, ni,j(r), for each state j. Like the density, we use a basis representation of the external potential; however, care must be taken to avoid the Coulomb singularities in v(r). We therefore use a Gaussian representation of the potential:

$$v({{{{{{{\bf{r}}}}}}}})=\mathop{\sum }\limits_{a}^{N}{Z}_{a}{e}^{-|{{{{{{{{\bf{R}}}}}}}}}_{a}-{{{{{{{\bf{r}}}}}}}}{|}^{2}/2{\sigma }_{{{{{{{{\rm{pot}}}}}}}}}^{2}},$$
(7)

where Ra is the position of the ath nucleus, Za is its corresponding nuclear charge, and σpot is a width parameter22,66. The potential from Eq. (7) is then formed on a 3D grid surrounding the molecule and stored as a vector, vi, for each sample i to be used in Eq. (6). 60 × 50 × 30 grid points were used with a spacing of 0.2 Å, commensurate with the shape of the molecule. We used previously optimized values of σpot = 0.2 Å and a grid spacing for MA from ref. 22.

As a final step, we train a multistate total energy functional:

$${E}^{{{{{{{{\rm{ML-MSHK}}}}}}}}}[{n}_{j}]=\mathop{\sum }\limits_{i=1}^{M}\mathop{\sum}\limits_{k}{\alpha }_{i,k}\kappa [{{{{{{{{\bf{u}}}}}}}}}_{i,k},{{{{{{{{\bf{u}}}}}}}}}_{j}],$$
(8)

where k sums over the states of the densities used in the training and κ is another Gaussian kernel.

### Reference ab initio electronic structure

Reference calculations on MA’s excited states were performed at the LR-TDDFT level using the PBE067,68 approximate exchange and correlation functional. This level of theory, which we abbreviate as TD-PBE0, was chosen as a compromise between accuracy and efficiency in this first demonstration of our ML-MSHK method. In particular, PBE0 provides qualitatively correct ground-state thermochemical properties of MA, yielding proton transfer barrier height of 2.0 kcal/mol compared to the predicted barrier from high-level theory of 4.1 kcal/mol69,70. In addition, TD-PBE0 provides reasonable spectroscopic quantities for MA, with a S0-S2 vertical excitation of 5.3 eV compared to the experimental value of 4.7 eV71. Electronic structure calculations were performed in CPMD v. 4.3.072 using a plane-wave basis with a kinetic energy cutoff of 90 Rydberg. Core electrons were replaced with Troullier-Martins norm-conserving pseudopotentials73.

### AIMD excited-state molecular dynamics

Ab initio excited-state Born-Oppenheimer MD simulations of a gas-phase malonaldehyde molecule were performed in CPMD using the same TD-PBE0 level of theory discussed above. 50 independent non-equilibrium trajectories were initiated on the S2 state following vertical excitations spaced every 100 fs from an AIMD ground-state trajectory sampled at 300 K taken from ref. 22. Excited-state dynamics were propagated in the microcanonical ensemble with a 0.25 fs timestep for 120 fs, which takes 5027 min for one simulation with all cores (24) of dual Intel Xeon E5 2650 v4 (2.2 GHz) CPUs. Planarity was maintained with the following added restraining potential: $${V}^{rest.}=\mathop{\sum }\nolimits_{a}^{N}\frac{1}{2}{k}_{a}{({z}_{a})}^{2}$$, where ka is the strength of each restraint and is 9 kcal/(mol bohr2) for hydrogen and 40 kcal/(mol bohr2) for heavy atoms. za is the z-coordinate of each atom. The planarity restraints were chosen to be stiffer for the heavy atoms, since backbone torsions have been identified as bringing about the S2/S1 electronic crossing in MA41.

### Training and test set generation

It has been shown that exploiting molecular point group symmetries helps to increase the effective dataset size significantly without performing additional quantum chemical calculations23. We thus generated training sets for malonaldehyde that included reflection about the mirror symmetry plane perpendicular to the molecular axis, $${\sigma }_{v{\prime} }$$, of an idealized planar C2v structure, since this symmetry operation ensures the equivalence of the two oxygen atoms as proton donors/acceptors. To achieve a symmetrization of the training sets while avoiding unnecessary duplication of samples, we combined symmeterization with a clustering of molecular geometries. For example, the training set for MD runs was formed by starting with 2000 structures taken from a previous ground-state AIMD trajectory22 and 14,400 structures taken every timestep from 30 independent ab initio excited-state trajectories described in section “AIMD excited-state molecular dynamics”. Next, all geometries were aligned to a reference geometry representing the planar proton-transfer transition state with C2v symmetry (see Supplementary Note 4). Then, following alignment, any molecular structures with a negative proton-transfer coordinate, r, were reflected in the $${\sigma }_{v{\prime} }$$ plane, i.e., to ensure r > 0. The training set was then clustered with K-Means74 to make a set of 2500 samples, using a metric related to the L2 deviation of the external potential, $$\sqrt{\int\ {{{{{{{\rm{d}}}}}}}}{{{{{{{\bf{r}}}}}}}}{\left({v}_{i}({{{{{{{\bf{r}}}}}}}})-{v}_{j}({{{{{{{\bf{r}}}}}}}})\right)}^{2}}$$, where vi represents the potential from Eq. (7). Finally, the training set was doubled in size by applying the reflection operator in the $${\sigma }_{v{\prime} }$$ plane to yield the production training set of 5000 geometries. The test set contains 240 aligned snapshots extracted every 10 fs from 20 independent ab initio excited-state trajectories described in section “AIMD excited-state molecular dynamics”.

The electronic-structure calculations for excited-state energies and densities take 40 min with 8 cores of an Intel Xeon E5 2650 v4 (2.2 GHz) CPU for one geometry and the training takes 10 min using the same number of cores with hyperparameters given.

### ML excited-state molecular dynamics

To generate initial conditions to perform excited-state dynamics on the ML S2 potential, a ground-state AIMD trajectory was run in CPMD using the PBE exchange and correlation functional67 with the same kinetic energy cutoff and pseudopotentials discussed above. This trajectory was run in the canonical ensemble at 300 K using massive Nosé-Hoover chain thermostats with a timestep of 0.5 fs75. 1000 independent non-equilibrium trajectories were initiated on the S2 state following vertical excitations spaced every 100 fs from the ground-state trajectory. Excited-state dynamics were propagated in the microcanonical ensemble with a 0.25 fs timestep using the atomistic simulation environment v.3.19.076. Atomic forces were evaluated numerically using central differences with a step length of dx = 0.001 Å. Dynamics were propagated for 60 fs, which was sufficient to observe the ESPT reaction. The propagation of one 60-fs dynamics simulation takes 20 min with 2 cores of an Intel Xeon E5 2650 v4 (2.2 GHz) CPU.


text:
Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# KöMaL Problems in Mathematics, November 2012

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## Problems with sign 'K'

Deadline expired on January 10, 2013.

K. 349. Kate is younger than her husband, but the age of each of them is a two-digit number that consists of the same two digits. The sum of their ages equals 11 times the difference of their ages. How old is Kate?

(6 pont)

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K. 350. Steve lives far away from his new workplace, so he decided to go to work by car. On the first day, he travelled at an average speed of 70 km/h (practically uniformly), and arrived 1 minute late. On the second day, he started out at the same time but travelled at an average speed of 75 km/h (practically uniformly) and arrived 1 minute early. How far does he live from his workplace?

(6 pont)

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K. 351. Find such numbers of the form TARTAR, divisible by 49 such that the number of the form RATRAT obtained by rearranging the digits is also divisible by 49.

(6 pont)

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K. 352. Make a 5×5 table, and fill it out with the numbers 1 to 25, left to right, top to bottom. Interchange the rows an arbitrary number of times, and then interchange the columns an arbitrary number of times. Add 7 to each entry of the mixed'' table obtained in this way. Finally, add the numbers along one of the diagonals. Prove that the result is always 100.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

(6 pont)

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K. 353. Where are those points P(x;y) in the rectangular coordinate plane for which xy-2y=3x-6?

(6 pont)

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K. 354. The sum of the squares of a certain positive integer and its two neighbours can be expressed as the sum of five consecutive integers. How many three-digit numbers have this property?

(6 pont)

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## Problems with sign 'C'

Deadline expired on December 10, 2012.

C. 1140. Ann, Beth, Connie and Dora shared a room in a class trip. The girls were given 1 bottle of orange juice, 2 bottles of apple juice, 2 bottles of peach juice and 3 bottles of mineral water to take with them on the forest walk. In how many different ways can they distribute the drinks among themselves if everyone is to get two bottles?

Suggested by A. Balga, Budapest

(5 pont)

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C. 1141. The last digit of n3+3n2+3n is 4 (n is a positive integer). What is the last digit of 4n2+5n+6?

(5 pont)

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C. 1142. The points P, A, B and Q, in this order, lie on the same line such that PA=BQ, and a triangle ABC with interior angles of 36o, 72o, 72o can be constructed out of the three line segments. Let R be the intersection of the circle of radius CP centred at C and the extension of the line AC beyond C. Find the angles subtended by the line segment PQ at the points C and R.

(5 pont)

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C. 1143. The first few terms of a number sequence are the even numbers etc. From a certain term onwards, the sequence continues as an arithmetic progression with a common difference of d=3. Which term is that if the sum of the first 50 terms of the sequence is 2985?

(5 pont)

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C. 1144. Find the largest possible value of S=x2y+2x+3+2y+xy2, given that x2+y2=2.

(5 pont)

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## Problems with sign 'B'

Deadline expired on December 10, 2012.

B. 4482. In Elastica, only natural numbers less than ten thousand are known. These numbers are written down in a flexible way. The rule for Elastican numbers is as follows: each number is represented in the lowest possible base such that the resulting number have at most four digits. Unfortunately, the numbers represented in this way are not always possible to decode uniquely: For example, the four-digit Elastican number 1101 may mean 13 as well as 37. Is there a three-digit Elastican number that represents more than one natural number?

Based on the idea of T. Lakatos, Balassagyarmat

(3 pont)

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B. 4483. On a white sheet of squared paper, 40 small squares are coloured red. Prove that it is possible to select 10 red squares such that no two of them have a point in common.

(3 pont)

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B. 4484. Prove that the equation has no solution of positive integers x, y, z where z>1.

(5 pont)

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B. 4485. What is the largest possible fraction of a triangle that can be covered by a square with all its vertices on the perimeter of the triangle?

Suggested by I. Légrádi, Sopron

(4 pont)

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B. 4486. Let a and b be positive integers. How many non-negative integers n are there such that ? ([x] denotes the greatest integer not greater than x.)

Suggested by R. F. Stöckli, Buenos Aires

(5 pont)

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B. 4487. Is there a plane that does not pass through any vertex of a regular octahedron but intersects all the faces?

Suggested by V. Vígh, Szeged

(5 pont)

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B. 4488. Show that 168 cannot be expressed as the sum of the squares of two rational numbers.

(4 pont)

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B. 4489. Given the points P1, P2, P3 and the distances e1, e2, e3, construct a circle such that the length of the tangent drawn from Pi is ei.

(5 pont)

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B. 4490. P and Q are interior points on side AC of a non-isosceles triangle ABC such that . The interior angle bisectors drawn from A and C intersect the line segment BP at points K and L, and the line segment BQ at points M and N, respectively. Prove that the lines AC, KN and LM are concurrent.

Suggested by Sz. Miklós, Herceghalom

(6 pont)

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B. 4491. The names of 100 convicts are placed in 100 numbered drawers in some order. Then the convicts are brought one by one from their cells in a random order, and each of them is allowed to pull out 50 drawers one by one. If he finds his own name in a drawer then he is led to a separate room. Otherwise all 100 convicts are executed immediately. Finally, if each of them succeeds, they are all set free. Show that, knowing the rules, the prisoners can follow a strategy that results in a larger than 30% probability of going free.

(6 pont)

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## Problems with sign 'A'

Deadline expired on December 10, 2012.

A. 572. Two circles k1 and k2 with centres O1 and O2, respectively, intersect perpendicularly at P and Q. Their external homothety center is H. The line t is tangent to k1 at T1 and tangent to k2 at T2. Let X be a point in the interior of the two circles such that HX=HP=HQ, and let X' be the reflection of X about t. Let the circle XX'T2 and the shorter arc PQ of k1 meet at U1, and let the circle XX'T1 and the shorter arc PQ of k2 meet at U2. Finally, let the lines O1U1 and O2U2 meet at V. Show that VU1=VU2.

(5 pont)

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A. 573. Let D={0,1,2,...,9} be the set of decimal digits, and let RD×D be a set of ordered pairs of digits. An infinite sequence (a1,a2,a3,...) of digits is said to be compatible with R if (aj,aj+1)R for all positive integer j. Determine the smallest positive integer K with the property that if an arbitrary set RD×D is compatible with at least K distinct digit sequences then R is compatible with infinitely many digit sequences.

Based on the 5th problem of CIIM 2012, Guanajuato, Mexico

(5 pont)

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A. 574. Let n2 and let be a polynomial with real coefficients. Prove that if for some positive integer k the polynomial (x-1)k+1 divides p(x) then .

CIIM 2012, Guanajuato, Mexico

(5 pont)

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text:
#### Problem 21E

21. The altitude of a triangle is increasing at a rate of $1 \mathrm{~cm} / \mathrm{min}$ while the area of the triangle is increasing at a rate of $2 \mathrm{~cm}^{2} / \mathrm{min}$. At what rate is the base of the triangle changing when the altitude is $10 \mathrm{~cm}$ and the area is $100 \mathrm{~cm}^{2}$ ?


text:
# Does there exist a field which has infinitely many subfields?

Does there exist a field which has infinitely many subfields? Does there exist an enormous supply of such fields?

I don't know how to begin.

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$\mathbb Q(\pi)$. – Andrés Caicedo Jul 17 '13 at 5:00
A good one @Andres! The only "shortcoming" being that all the intermediate fields are isomorphic to each other (Lüroth). Not that the example in my answer would be any better in that respect :-) – Jyrki Lahtonen Jul 17 '13 at 7:27

The complex numbers $\mathbb{C}$ is an example of such a field. It has infinitely many subfields, since you can adjoin family of irrational numbers (pick your favourite ones!) to $\mathbb{Q}$. My favourites would be roots of $\sqrt[n]{2}$ for each $n\in\mathbb{N}$. So in this case, the infinite family of subfields would be $\{\mathbb{Q}(\sqrt[n]{2})\}_{n=1}^{\infty}$

-

Recall the notion of prime subfield $K_0$ of a field $K$: this is the unique smallest subfield of $K$. If $1+\ldots + 1 = 0$ for some $n$, then the least such $n$ is a prime number $p$ and the prime subfield is $\mathbb{F}_p$. Otherwise the prime subfield is the rational numbers $\mathbb{Q}$. Now here is the full answer to your question:

A field $K$ has finitely many subfields iff $K/K_0$ has finite degree.

Proof: $K/K_0$ has finite degree, then $K$ is either a finite field or a number field. The first case is obvious; the second case can be done e.g. by replacing $K$ with its Galois closure over $\mathbb{Q}$ (showing that a larger field has only finitely many subfields will be sufficient!) and using the Galois correspondence.

Now suppose $K/K_0$ has infinite degree.

Case 1: If $K/K_0$ is algebraic, then choose $x_1 \in K \setminus K_0$. Then $K_0(x_1)$ has finite degree over $K_0$ so is proper in $K_1$. Repeating this argument generates an infinite ascending chain of finite degree subfields of $K$.

Case 2: If $K/K_0$ is not algebraic, let $t \in K$ be transcendental over $K_0$. Then $\{K_0(t^n)\}_{n=1}^{\infty}$ are infinitely many subfields of $K$.

Afterthought: Let $L/K$ be any field extension, and consider the lattice $\operatorname{Sub}(L/K)$ of subextensions, i.e., fields $F$ with $K \subset F \subset L$. Then the OP's original question asks about when for a field $K$, $\operatorname{Sub}(K/K_0)$ is infinite, and my answer characterizes when this happens. But upon further reflection, in each of Cases 1 and 2 I am showing the failure of a different weaker finiteness property. Namely, if $K/K_0$ is infinite degree algebraic, then $\operatorname{Sub}(K/K_0)$ contains an infinite ascending chain of subfields, so it is not Noetherian (this is just the definition of a Noetherian partially ordered set: it doesn't contain infinite ascending chains). If $K/K_0$ is transcendental, then $\operatorname{Sub}(K/K_0)$ contains an infinite descending chain -- if $t$ is transcendental over $K_0$, take $F_n = K_0(t^{2^n})$ -- so it is not Artinian (again, by definition).

In fact this part of the answer doesn't use any special properties of $K_0$. In fact it shows that if $L/K$ is infinite degree algebraic then $\operatorname{Sub}(L/K)$ is not Noetherian, and if $L/K$ is transcendental then $\operatorname{Sub}(L/K)$ is not Artinian. (The converse works when $K_0$ is perfect but not in general; see Jyrki's answer and my comments on it below.)

You can check that it doesn't work the other way around.

Claim: Let $K_0$ be any prime field (i.e., $\mathbb{F}_p$ or $\mathbb{Q}$). Then:
a) There is an infinite algebraic field extension $K/K_0$ such that $\operatorname{Sub}(K/K_0)$ is Artinian.
b) There is a transcendental field extension $K/K_0$ such that $\operatorname{Sub}(K/K_0)$ is Noetherian.

I leave the verifications to you (if you care to do them), with the following hints: a) take a $\mathbb{Z}_p$-extension. b) Use Luroth's Theorem.

Maybe we should ask for characterizations of field extensions $L/K$ such that $\operatorname{Sub}(L/K)$ is Noetherian, or Artinian?

Note that Jyrki's answer also fits into this framework: he shows that $L/K$ having finite degree need not imply that $\operatorname{Sub}(L/K)$ is finite (although it is evidently Noetherian and Artinian, i.e., of finite length). As I like to mention, I got a PhD in arithmetic geometry / number theory before I learned about this example: up until not so very long ago I was sure that basic field theory showed that such examples could not exist.

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I like complete answers :) Thanks for teaching me this theorem that I didn't know (+1) – Prism Jul 17 '13 at 5:07
@Prism: I didn't know the "theorem" until about 10 minutes ago either. – Pete L. Clark Jul 17 '13 at 5:12
I checked your field theory notes, and quick search showed that it wasn't there. I then concluded that you had some secret theorems you hid from us :) Didn't know you figured that out instantaneously... – Prism Jul 17 '13 at 5:14

Let $K$ be any field and let $X$ be any infinite set. For each choice of subset $Y\subseteq X$, we get a different subfield $K(Y)\subseteq K(X)$.

Let $K$ be any field and let $t$ be an indeterminate. For each choice of integer $n\geq 1$, we get a different subfield $K(t^n)\subseteq K(t)$.

In case you're not familiar with it, here is the definition of $K(X)$: it is the field of fractions of $K[X]$, the ring of polynomials in the indeterminates $X$ over $K$. That is, the elements of $K[X]$ are polynomials where the coefficients are elements of $K$, and the "variables" or "indeterminates" are the elements of $X$. Thus for example, if $K=\mathbb{Q}$ and $X=\{s,t\}$, then $K[X]=\mathbb{Q}[s,t]$ consists of polynomials such as $\frac{1}{2}+5s+t^2+3st$, and $K(X)=\mathbb{Q}(s,t)$ consists of rational functions such as $$\frac{\frac{1}{2}+5s+t^2+3st}{7-3s^3+4st^2}.$$

-
This is surely the most conceptually economical answer, and also the most elementary. $\mathbb{C}$ is more natural and more mathematically important, but proving that adjoining algebraically independent irrational numbers gives infinitely many distinct subfields is a bit more work than simply observing that $K(Y) \neq K(Y')$ for distinct $Y, Y' \subseteq X$. – Peter LeFanu Lumsdaine Jul 17 '13 at 14:11
You should probably make the assumption that $X$ is an infinite algebraically independent set. – tomasz Jul 28 '13 at 14:21
@tomasz: Any set can be used as a set of indeterminates; there would be nothing wrong with using the set $X=\{2,5\}\subset\mathbb{Q}$ as a set of indeterminates over $\mathbb{Q}$, making a field $\mathbb{Q}(X)$ of transcendence degree $2$ over $\mathbb{Q}$ (though notationally it would certainly be confusing, since of course $\mathbb{Q}(2,5)=\mathbb{Q}$). It is this sense in which I intend the notation $K(X)$. – Zev Chonoles Jul 28 '13 at 15:17

Adding an example of infinitely many intermediate fields of a finite extension $L/K$.

Let be $F$ be an algebraic closure of $\mathbb{F}_2$, so $F$ is infinite. Let $x$ and $y$ be algebraically independent trascendental elements over $F$. Let $L=F(x,y)$ and $K=F(x^2,y^2)$. We easily see that $[L:K]=4$. A basis for this extension consists of $\{1,x,y,xy\}$. Let $\alpha\in F$ be arbitrary. Consider the intermediate space $L_\alpha$ spanned over $K$ by $\{1,x+\alpha y\}$. We have $$L_\alpha\cap L_{\alpha'}=K$$ whenever $\alpha\neq\alpha'$ (linear algebra), so the spaces $L_\alpha$ are distinct. The key is that $L_\alpha$ is also a subfield. This is because $$(x+\alpha y)^2=x^2+2\alpha xy+\alpha^2y^2=x^2+\alpha^2y^2\in K.$$

This does not contradict the argument in Pete L. Clark's answer, because the extension $L/K$ is purely inseparable, and thus cannot be contained in a Galois extension. And that was the key in his argument in the case of number fields.

-
A similar construction is available in characteristic $p>2$ as well. – Jyrki Lahtonen Jul 17 '13 at 7:24


text:
# Linear transformations and norm

I am studying Linear Algebra II, and I came across several questions in which, for a certain linear transformation ($T\colon\mathbf{V}\to\mathbf{V}$) I was told that: $$||T(a)|| \leq ||a||.$$

I am not completely certain how to use this information. For instance, consider the following question (please forgive my translation, it's the first time I write math in English):

For a linear transformation $T\colon\mathbf{V}\to\mathbf{V}$in a unitary space [i.e., complex inner product space], such that

• $|c|=1$ for every eigenvalue $c$ of $T$;
• $||T(a)|| \leq ||a||$ for every vector $a$ in $\mathbf{V}$; prove that T is a unitary operator.

How does the fact that $||T(a)|| \leq ||a||$ help me?

Thanks.

-
Isn't "unitary space" bordering on ancient terminology? The modern term is "inner product space" –  kahen Feb 10 '11 at 21:16
Hila, apparently $V$ is finite dimensional, correct? Have you seen that every operator on $V$ can be unitarily triangularized? –  Jonas Meyer Feb 10 '11 at 21:24
@kahen @Arturo Forgot to add - we use the term "unitary space" for "inner product space over the complex numbers", and "euclidean space" for "inner product space over the real numbers", and just "inner product space" when we don't care about the field. –  Hila Feb 10 '11 at 22:37
@Hila: Ehr: the page lists three equivalent definitions: $UU^* = U^*U = I$; the range of $U$ is dense; and "$U$ respects the inner product". I am going to guess you mean the first, but you really could have saved some trouble by simply typing it... –  Arturo Magidin Feb 11 '11 at 1:01
@Arturo: Dense range and respect of inner product combine to give just one characterization (and in this case dense range is redundant). –  Jonas Meyer Feb 11 '11 at 1:28

Every linear transformation on a finite dimensional complex inner product space is unitarily triangularizable. See for example Hogben's Handbook of linear algebra. This means you can find an orthonormal basis $e_1,\ldots e_n$ such that $Te_k=\sum_{i\leq k}a_{ik}e_i$. The eigenvalues of a triangular matrix are the diagonal entries, so $|a_{kk}|=1$ for each $k$. Thus $1\geq\|Te_k\|^2=\sum_{i\leq k}|a_{ik}|^2\geq|a_{kk}|^2=1$, forcing $a_{ik}=0$ for $i\lt k$. So the basis actually diagonalizes $T$, and it is now straightforward to show that $T$ is unitary regardless of which definition you use.
Given that $T$ can be triangularizable, it is easy to see that after diagonalizing and computing $T^\dagger T$ you get identity. But it seems that the condition $\parallel Tv\parallel\leq \parallel v\parallel$ is not being used here. –  Marcos Villagra Feb 11 '11 at 1:53
@Marcos Villagra: In order to show that $T$ is actually diagonalized, the condition $\|Tv\|\leq\|v\|$ must be used. (Otherwise the result would be false, e.g. consider a linear transformation on a 2-dimensional space that has matrix $\begin{bmatrix}1 & 1\\0&1\end{bmatrix}$ with respect to some orthonormal basis.) The condition is used above at "$1\geq\|Te_k\|^2$". –  Jonas Meyer Feb 11 '11 at 1:58
my mistake was that I thought that the eigenvectors of the triangularized matrix automatically diagonalized $T$ –  Marcos Villagra Feb 11 '11 at 2:13
@Marcos: I see. There is no knowledge a priori that the $e_k$'s are eigenvectors. In order to have an orthonormal basis of eigenvectors an operator must be normal, and that is not given. But in this case, they do turn out to be eigenvectors. –  Jonas Meyer Feb 11 '11 at 2:23


text:
# Penrose Tiling #1

Penrose tiles are any of a set of plane figures which can be combined to tile the plane aperiodically (without translational symmetry). They were described by the British mathematician Roger Penrose in the 1970s.

The code described in this post is available under the GPL licence on github. Some examples are given in the next blog post.

One pair of Penrose tiles, known as P3 are the following rhombuses:

To ensure an aperiodic tiling, the tiles must be arranged so that the coloured arcs match up. For the purposes of constructing such a tiling, each rhombus may be considered to consist of a pair of mirror-image Robinson triangles:

The larger triangle on the left, referred to here as BL has sides in the ratio $1:1:\phi$. The smaller one on the right, BS has sides in the ratio $1:1:\psi$ where $\phi = (\sqrt{5}+1)/2$ and $\psi = 1/\phi = (\sqrt{5}-1)/2$ are the Golden ratio and its inverse respectively. To build a tiling one starts with an arrangement of such triangles and repeatedly applies the process of "inflation", replacing each triangle with the following arrangement of smaller triangles in the indicated orientation:

That is, a BL triangle, ABC, is replaced with two scaled-down BL triangles (EDA and CEB) and a BS triangle (DEB); in the lefthand above figure, D divides AC in the ratio $1:(\phi-1)\equiv 1:\psi$ and E divides AB in the ratio $\psi:(1-\psi) \equiv \psi:\psi^2$. Each BS triangle, A'B'C', is replaced with a BL triangle (C'D'B') and a BS triangle (D'C'A'); in the righthand figure, D' divides A'B' in the ratio $(1-\psi):\psi\equiv \psi^2:\psi$.

The code defines a class, Penrose, which can be used to create SVG images of Penrose tilings. The base class, RobinsonTriangle defines a generic Robinson triangle with its vertices given as complex numbers, $x + iy$ (note that the orientation of the SVG canvas (increasing $y$-index down the page) results in the $y$-axis appearing flipped in the resulting image). Methods are defined for producing the SVG paths for the rhombus associated with a Robinson triangle and for drawing the matching arcs if requested. We also define two more useful methods for returning the centre of the rhombus and for returning a "conjugate" triangle from the existing one with its $y$-coordinates inverted.

class RobinsonTriangle:
"""
A class representing a Robinson triangle and the rhombus formed from it.

"""

def __init__(self, A, B, C):
"""
Initialize the triangle with the ordered vertices. A and C are the
vertices at the equal base angles; B is at the vertex angle.

"""

self.A, self.B, self.C = A, B, C

def centre(self):
"""
Return the position of the centre of the rhombus formed from two
triangles joined by their bases.

"""

return (self.A + self.C) / 2

def path(self):
"""
Return the SVG "d" path element specifier for the rhombus formed
by this triangle and its mirror image joined along their bases.

"""

AB, BC = self.B - self.A, self.C - self.B
xy = lambda v: (v.real, v.imag)
return 'm{},{} l{},{} l{},{} l{},{}z'.format(*xy(self.A) + xy(AB)
+ xy(BC) + xy(-AB))

def get_arc_d(self, U, V, W):
"""
Return the SVG "d" path element specifier for the circular arc between
sides UV and UW, joined at half-distance along these sides.

"""

start = (U + V) / 2
end = (U + W) / 2

# ensure we draw the arc for the angular component < 180 deg
cross = lambda u, v: u.real*v.imag - u.imag*v.real
US, UE = start - U, end - U
if cross(US, UE) > 0:
start, end = end, start
r = abs((V - U) / 2)
return 'M {} {} A {} {} 0 0 0 {} {}'.format(start.real, start.imag,
r, r, end.real, end.imag)

def arcs(self):
"""
Return the SVG "d" path element specifiers for the two circular arcs

"""

D = self.A + self.C - self.B
arc1_d = self.get_arc_d(self.A, self.B, D)
arc2_d = self.get_arc_d(self.C, self.B, D)
return arc1_d, arc2_d

def conjugate(self):
"""
Return the vertices of the reflection of this triangle about the
x-axis. Since the vertices are stored as complex numbers, we simply
need the complex conjugate values of their values.

"""

return self.__class__(self.A.conjugate(), self.B.conjugate(),
self.C.conjugate())

From this base class we derive a class for each of the two types of P3 triangle, BtileL and BtileS:

class BtileL(RobinsonTriangle):
"""
A class representing a "B_L" Penrose tile in the P3 tiling scheme as
a "large" Robinson triangle (sides in ratio 1:1:phi).

"""

def inflate(self):
"""
"Inflate" this tile, returning the three resulting Robinson triangles
in a list.

"""

# D and E divide sides AC and AB respectively
D = psi2 * self.A + psi * self.C
E = psi2 * self.A + psi * self.B
# Take care to order the vertices here so as to get the right
# orientation for the resulting triangles.
return [BtileL(D, E, self.A),
BtileS(E, D, self.B),
BtileL(self.C, D, self.B)]

class BtileS(RobinsonTriangle):
"""
A class representing a "B_S" Penrose tile in the P3 tiling scheme as
a "small" Robinson triangle (sides in ratio 1:1:psi).

"""

def inflate(self):
"""
"Inflate" this tile, returning the two resulting Robinson triangles
in a list.

"""
D = psi * self.A + psi2 * self.B
return [BtileS(D, self.C, self.A),
BtileL(self.C, D, self.B)]

The method inflate simply returns a list of new triangles with vertices determined by the "inflation" process described above. Note that the orientation of these triangles is important: the vertices are given in the order A, B, C using the notation of the above figures.

Finally, the main Penrose class is responsible for generating the tiling out of the P3 rhombuses. The appearance can be altered by passing key,value pairs to the config dictionary on initialization.

class PenroseP3:
""" A class representing the P3 Penrose tiling. """

def __init__(self, scale=200, ngen=4, config={}):
"""
Initialise the PenroseP3 instance with a scale determining the size
of the final image and the number of generations, ngen, to inflate
the initial triangles. Further configuration is provided through the
key, value pairs of the optional config dictionary.

"""

self.scale = scale
self.ngen = ngen

# Default configuration
self.config = {'stroke-colour': '#fff',
'base-stroke-width': 0.05,
'margin': 1.05,
'tile-opacity': 0.6,
'random-tile-colours': False,
'Stile-colour': '#08f',
'Ltile-colour': '#0035f3',
'Aarc-colour': '#f00',
'Carc-colour': '#00f',
'draw-tiles': True,
'draw-arcs': False,
'reflect-x': True
}
self.config.update(config)

self.elements = []

def set_initial_tiles(self, tiles):
self.elements = tiles

def inflate(self):
""" "Inflate" each triangle in the tiling ensemble."""
new_elements = []
for element in self.elements:
new_elements.extend(element.inflate())
self.elements = new_elements

def remove_dupes(self):
"""
Remove triangles giving rise to identical rhombuses from the
ensemble.

"""

# Triangles give rise to identical rhombuses if these rhombuses have
# the same centre.
selements = sorted(self.elements, key=lambda e: (e.centre().real,
e.centre().imag))
self.elements = [selements[0]]
for i, element in enumerate(selements[1:], start=1):
if abs(element.centre() - selements[i-1].centre()) > TOL:
self.elements.append(element)

""" Extend the tiling by reflection about the x-axis. """

self.elements.extend([e.conjugate() for e in self.elements])

def make_tiling(self):
""" Make the Penrose tiling by inflating ngen times. """

for gen in range(self.ngen):
self.inflate()
self.remove_dupes()
if self.config['reflect-x']:
self.remove_dupes()

def get_tile_colour(self, e):
if self.config['random-tile-colours']:
return '#' + hex(random.randint(0,0xfff))[2:]
if isinstance(e, BtileL):
return self.config['Ltile-colour']
return self.config['Stile-colour']

def make_svg(self):
""" Make and return the SVG for the tiling as a str. """

xmin = ymin = -self.scale * self.config['margin']
width =  height = 2*self.scale * self.config['margin']
viewbox ='{} {} {} {}'.format(xmin, ymin, width, height)
svg = ['<?xml version="1.0" encoding="utf-8"?>',
'<svg width="100%" height="100%" viewBox="{}"'
' preserveAspectRatio="xMidYMid meet" version="1.1"'
' baseProfile="full" xmlns="http://www.w3.org/2000/svg">'
.format(viewbox)]
# The tiles' stroke widths scale with ngen
stroke_width = str(psi**self.ngen * self.scale *
self.config['base-stroke-width'])
svg.append('<g style="stroke:{}; stroke-width: {};">'
.format(self.config['stroke-colour'], stroke_width))
for e in self.elements:
if self.config['draw-tiles']:
svg.append('<path fill="{}" opacity="{}" d="{}"/>'
.format(self.get_tile_colour(e),
self.config['tile-opacity'], e.path()))
if self.config['draw-arcs']:
arc1_d, arc2_d = e.arcs()
svg.append('<path fill="none" stroke="{}" d="{}"/>'
.format(self.config['Aarc-colour'], arc1_d))
svg.append('<path fill="none" stroke="{}" d="{}"/>'
.format(self.config['Carc-colour'], arc2_d))
svg.append('</g>\n</svg>')
return '\n'.join(svg)

def write_svg(self, filename):
""" Make and write the SVG for the tiling to filename. """
svg = self.make_svg()
with open(filename, 'w') as fo:
fo.write(svg)

The method responsible for generating the tiling is make_tiling and it simply inflates the initial configuration of triangles ngen times. Since the rhombuses are generated from Robinson triangles, it is possible for the inflation of adjacent triangles to produce the same rhombus: the method remove_dupes removes such duplicate rhombuses from the ensemble: two rhombuses are the same if they have the same centre (this is established using the RobinsonTriangle centre method).

Finally, a careful choice of initial conditions (such that the initial triangles all have edges along the $x$-axis) enables the tiling area to be doubled simply by reflection in the $x$-axis (equivalent to taking the complex conjugate of all triangles' vertex coordinates). This functionality is implemented with the method add_conjugate_elements.

Current rating: 4.5


text:
WordPress.org

How to add blank lines in a post/page (24 posts)

1. talia
Member
Posted 5 years ago #

the system seems to automatically close up lines between paragraphs but I need to have some larger spaces between some paragraphs to break up the text.

how do i insert a double line space instead of a single line???

2. godsassass1n
Member
Posted 5 years ago #

i would suggest adding some page breaks to get the desired spacing. a simple
. that is if your talking about what i think you are.
add as many as needed. hope this helps or maybe not....

3. tiopourtu
Member
Posted 5 years ago #

Hi, I have the same problem too. But when I add
tag. It works as long as I don't edit the post. Because when I edited after I published, the
tag is disappeared.

4. godsassass1n
Member
Posted 5 years ago #

are you editing your post in html or visual? i'm gonna go ahead and say that you are in visual try editing your text in html just like you would in visual except add that tag anywhere you want a space. also there are no spaces in that tag i just wrote it spaced out so it doesn't get recognized as html in my post. let me know how it worked out for you

5. csleh
Member
Posted 5 years ago #

There are plugins to remove wordpress formatting. Or perhaps turning off the wsiwyg editor under users/your profile.

How about using a small heading like an
 to separate the different bits? Easy to do and good for viewer scanning.

6. talia
Member
Posted 5 years ago #

hi thanks for the replies
i have already tried
 several times in both html and visual format

I have also tried

 thinking it would insert a para but no luck

[spaces used here but not in formatting]

thanks i have been using the h4 setting but sometimes it's just a matter of wanting extra space between text rather than needing a sub-heading!

compared to blogger the formatting is a pain in the ***
i'm also having trouble with images not floating right left next to text and ended up manually inserting formatting in html

is that a wp 2.5.1 fault or would it be the theme?

7. Sergio Urra
Member
Posted 5 years ago #

have you tried
I'm using it, and it works. I use also a return to make separated paragraphs in the visual mode.

8. talia
Member
Posted 5 years ago #

have i tried what? if you mean the
 then yes several times. i tried it in html mode before i came to the forum

i have used the return to separate paragraphs but i need a DOUBLE line space
using the return key multiple times still only returns a single line space

unfortunately it's not wysisyg. it will look fine with the (br) or the double return key but as soon as you save it wp performs some voodoo and the darn thing reformats itself

9. talia
Member
Posted 5 years ago #

if i change

p
{
line-height:1.8em;
}

in the css to 2.8em will that achieve what I want and make bigger lines between paragraphs?

10. Sergio Urra
Member
Posted 5 years ago #

Well, to have two blank lines I use this:

It works for me.

After the edit it looks OK in the visual mode. When I come back to the HTML editor it loks like:

One line here....

Other line here....

I supose that you can write directly the code tags in two lines.

11. Sergio Urra
Member
Posted 5 years ago #

in the css to 2.8em will that achieve what I want and make bigger lines between paragraphs?

Yest, but the result will be proportional to the separation of the lines, so it's not a good solution.

12. talia
Member
Posted 5 years ago #

ah ok thanks sergiourra didn't realise that was how line spacing worked.

so in addition to putting in the br i also have to use the code?

i will try that now thanks!

13. talia
Member
Posted 5 years ago #

BRILLIANT sergiourra!!!

phew so now i now to use the code before and after and it works fine.

my next challenge is with the images not floating beside the text. is there a code fix for that too?

thanks

14. Sergio Urra
Member
Posted 5 years ago #

Select the image and click the alignment button (left, right or center)

You can edit the margins of the image in the style.css file.

15. vdubplate
Member
Posted 5 years ago #

This is so retarded. I can't enter more than one line of space either and I have tried what sergiourra suggested. I have wasted so much time on this.

16. DIYer
Member
Posted 5 years ago #

Thanks Sergo, I got it to work eventually.
For two blank lines you have to enter :

I wonder why normal
does not work, it is so time consuming to find tags that work from one application vs another!

17. DIYer
Member
Posted 5 years ago #

I dont know why codes did not appear in the post first time, hope it appears now (sigh)

18. arnaudh
Member
Posted 5 years ago #

I wasn't having this problem until I installed Safari 4 beta. I can add blank lines on my Safari 3 machine. But my Safari 4 beta one will refuse to do so.

19. gravityandlevity
Member
Posted 5 years ago #

I tried using sergiourra/DIYer's suggestion, and it left funny gray marks on the blank lines (my theme is Vigilance).

I got around it by cheating with LaTex code: I made a blank space with LaTex that was not interpreted as empty space and deleted. Just put this line on its own where you want there to be a blank space:
$latex \hspace{10mm}$

Of course, 10mm is completely arbitrary. For really big spacings you can also use the \vspace command instead of a bunch of \hspace's.

20. francissimisim
Member
Posted 4 years ago #

Hello everyone try using this code to create a br space between your posts. It works like a charm

21. Arash
Member
Posted 4 years ago #

I wished wordpress would finally fix this. I wish I could contribute to fix this.

People have been requesting this since WP 1.5 ;)

22. Arash
Member
Posted 4 years ago #

There is an excellent plugin that allows you to insert
 and

23. newcasl
Member
Posted 4 years ago #

Thanks francissimisim, just what I needed.

24. tracelemental
Member
Posted 4 years ago #

THANK U everyone ... fixed my issue as well.

I had installed tinymce but deactivated because of another issue with the visual editor not working. I got that sorted out by going into the database where the value for visual was stuck on false.

Then I had same issue with trying to format my posts.

Yaaah all fixed, and reinstalled tinymce which has built feature for it anyway.

This topic has been closed to new replies.


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Monday, December 7 • 19:00 - 23:59

The completion of low rank matrices from few entries is a task with many practical applications. We consider here two aspects of this problem: detectability, i.e. the ability to estimate the rank $r$ reliably from the fewest possible random entries, and performance in achieving small reconstruction error. We propose a spectral algorithm for these two tasks called MaCBetH (for Matrix Completion with the Bethe Hessian). The rank is estimated as the number of negative eigenvalues of the Bethe Hessian matrix, and the corresponding eigenvectors are used as initial condition for the minimization of the discrepancy between the estimated matrix and the revealed entries. We analyze the performance in a random matrix setting using results from the statistical mechanics of the Hopfield neural network, and show in particular that MaCBetH efficiently detects the rank $r$ of a large $n\times m$ matrix from $C(r)r\sqrt{nm}$ entries, where $C(r)$ is a constant close to $1$. We also evaluate the corresponding root-mean-square error empirically and show that MaCBetH compares favorably to other existing approaches.


text:
On Ajtai's Lower Bound Technique for -way Branching Programs and the Hamming Distance Problem Jakob Pagter May 2000

### Abstract:

In this report we study the proof employed by Miklos Ajtai [Determinism versus Non-Determinism for Linear Time RAMs with Memory Restrictions, 31st Symposium on Theory of Computation (STOC), 1999] when proving a non-trivial lower bound in a general model of computation for the Hamming distance problem: given elements decide whether any two of them have small'' Hamming distance. Specifically, Ajtai was able to show that any -way branching program deciding this problem using time must use space . We generalize Ajtai's original proof allowing us to prove a time-space trade-off for deciding the Hamming distance problem in the -way branching program model for time between and , for some suitable . In particular we prove that if space is , then time is

Available as PostScript, PDF, DVI.