text:
A couple of months ago I started working on developing my own computer to learn more about its low-level functionality, the design process and the architecture behind it. Although I look at real-world examples of other computers I try not to just copy, but use those as inspiration to develop the architecture of my computer myself.

Besides 555 CMOS timers, some EEPROMs and HM62256 SRAM everything else is supposed to be built discretely with 74 series ICs.

I planned it to have a 16 bit word length, but didn't think about the implications this would have for the RAM as opposed to all the 8 bit computer examples I've read about.

My first approach was to make the RAM word-addressable, so I hooked up the address lines of two HM62256s and effectively had 16 bits stored in every address. This would be easy to output to the bus and work with, so I developed an instruction set around it, with a 5 bit opcode and the remaining 11 bits depending on the type of the instruction left for registers or immediate values.

While doing more and more research however I realized that word-addressable RAM doesn't seem to be common in real world computers. They all seem to be addressable by byte. This wouldn't be a problem for 8-bit computers, but if I plan on fetching my instructions in just one cycle doing it this way would be a problem.

Especially looking at ASCII it seems important to have a way to store single bytes in RAM, not just whole words. If I were to store ASCII characters in 2-byte RAM addresses, there'd be 50% wasted storage. I could just ignore this and carry on with my plans, but I'm also really intrigued by how this is solved.

How am I able to fetch multiple bytes from a byte-addressable RAM simultaneously to use them in my instruction register? Would I have to read two bytes in sequence and re-assemble them into a single 16-bit instruction before storing it in the register? What about other kind of data, does reading a 2-byte value require me to load two bytes from RAM manually into registers and re-assemble them?
Is there a more useful (or simpler) way I just can't think of?

Sorry for the long read for that short question, but thanks for any replies.

• The 8086 (not 8088) added a bus high enable line. When $BHA=0$ and $A_0=0$ then the implied data bus was $D_0-D_{15}$; when $BHA=0$ and $A_0=1$ then the implied the data bus as $D_8-D_{15}$; when $BHA=1$ and $A_0=0$ then the implied data bus was $D_0-D_{7}$. The remaining bus cycle case was supposed to be ignored. – jonk Sep 11 '17 at 23:37
• you treat word access as you do now but handle single byte and non-aligned words as special cases. Before that you need switch word-address to byte-address(by shifting the address bus up one bit on the momory side and wire the LSB on the memory side to 0). Currently most RISC-like ISAs choose to implement non-aligned word access and byte access as special instructions and let compiler opt-in, to reduce hardware complexity. – user3528438 Sep 11 '17 at 23:39
• As others have said, you leave the RAM as word-addressable and then select which byte out of the word you want. – pjc50 Sep 12 '17 at 9:03
• Since addresses for data are supposed to be reserved at the end of programs instructions should always be aligned and at most one byte of data at the end is unused, so unaligned word access really only makes sense in few cases. If I were to access bits 0..7 only, does it make sense to shift them over to be put on the bus at 8..15, or do I leave them essentially shifted to the left by 8 bits? – Padarom Sep 12 '17 at 9:11
• And if I understand it correctly for unaligned word access I'd need to essentially address MAR+1 in the first chip and MAR in the second and put them on the bus in reverse order? – Padarom Sep 12 '17 at 9:15


text:
# OASIS: Optimal Analysis-Specific Importance Sampling for event generation

### Submission summary

 As Contributors: <NAME> · <NAME> Arxiv Link: https://arxiv.org/abs/2006.16972v1 (pdf) Date submitted: 2020-07-01 02:00 Submitted by: Shyamsundar, Prasanth Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Experiment High-Energy Physics - Phenomenology Approach: Phenomenological

### Abstract

We propose a technique called Optimal Analysis-Specific Importance Sampling (OASIS) to reduce the number of simulated events required for a high-energy experimental analysis to reach a target sensitivity. We provide recipes to obtain the optimal sampling distributions which preferentially focus the event generation on the regions of phase space with high utility to the experimental analyses. OASIS leads to a conservation of resources at all stages of the Monte Carlo pipeline, including full-detector simulation, and is complementary to approaches which seek to speed-up the simulation pipeline.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission 2006.16972v2 on 28 December 2020

Submission 2006.16972v1 on 1 July 2020

## Reports on this Submission

### Anonymous Report 2 on 2020-8-21 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2006.16972v1, delivered 2020-08-21, doi: 10.21468/SciPost.Report.1930

### Strengths

1. The paper is timely as analyzing the expected large data sets at HL-LHC will require very large Monte Carlo (MC) samples. Optimizing the use of computational resources to generate such samples is important to maximize physics potential of HL-LHC.

2. The paper proposes an elegant idea to achieve this task, and discusses it in a clear, transparent, and mathematically rigorous way.

3. The paper describes in detail how this idea can be implemented in realistic experimental conditions, and as an example points to a concrete CMS analysis that can clearly benefit from it.

4. The idea is broadly applicable, not just in collider physics but in any area where comparison of experimental data and MC is employed.

### Weaknesses

1. A few questions regarding trade-offs inherent in the proposed method, and the range of analyses that can be benefit from it, have not yet been addressed. (See report for details.)

### Report

1. In the example considered in Sec. 2, it is shown that the proposed method improves the measurement uncertainty of the parameter $\theta$ possible for fixed $N_s$. The uncertainty is effectively computed with the prior that the data is correctly described by the function $f$. In general, theory/data comparison needs to address two questions - how well the data fits the underlying theoretical model, and what are the theory parameter(s) that fit the data best. It seems to me that the proposed technique may optimize the power to address the second question at the expense of reducing the power to address the first, and that the optimization would look different (and probably closer to the usual goal of $g(x)=f(x)$) if both questions are optimized together.

2. In applications to searches for BSM physics, the "optimal" region of phase space usually strongly depends on the parameters of the underlying BSM model. (For example, in a bump hunt, one would want to focus on the invariant masses around the mass of the hypothetical new particle.) The proposed method would require different optimization for each assumed set of BSM parameters, and since they are typically not known a priori, it is not clear that an overall reduction in MC sample sizes could be achieved. (An exception may be a model with very specific predictions, or if a signal already seen in one channel is being searched for in other channels.)

3. On p. 29, the authors state that "the event variable $v$ for a given value of parton-level event $x$ typically fall[s] within a small window of possibilities". However in some cases we may need to model tails of distributions which may contain significant contributions from rare events with "highly atypical" $x \mapsto v$ maps, e.g. large mis-measurements of MET etc. Would the proposed method still apply and be useful?

### Requested changes

The authors should consider the questions listed above and add discussion of these points to the paper as they see fit.

• validity: top
• significance: high
• originality: high
• clarity: top
• formatting: perfect
• grammar: perfect

### Author:  <NAME>  on 2020-12-28  [id 1115]

(in reply to Report 2 on 2020-08-21)

Thank you for the constructive feedback! We have made the following requested changes:
1. Thank you for raising this point. We have added a corresponding discussion in the conclusions section (under the bullet Optimizing for model exploration). To quickly answer the question here, even under those circumstances, our technique can give a non-trivial result, different from $g(x)=f(x)$. If the goal is to quantify how well the data fits the underlying theoretical model, our technique can help if we can parametrize the possible deviations from $f$ that one is interested in. In that case, the results of the paper can be used with respect to these additional parameters. If the goal is to maximize the significance of an unmodeled/unparameterized search, then the problem of finding the optimal sampling distribution is ill posed because we cannot predict ahead of time where the maximal deviations will occur. Therefore, no sampling distribution will be universally optimal in all new physics scenarios, but, as you pointed out, in this case the heuristic $g=f$ is a natural choice. 2. This is a valid point, although the same problem plagues any sampling scheme. What is being done currently in ATLAS and CMS is to focus on the parameter values which are close to the expected exclusion or discovery contours in parameter space. Our approach fits naturally in this scheme. We have added a corresponding discussion in the conclusions (under the bullet Optimizing for unknown signal parameters). 3. The short answer is yes. We have elaborated on this in the first and second bullets of section 3.2.2.

### Anonymous Report 1 on 2020-8-13 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2006.16972v1, delivered 2020-08-13, doi: 10.21468/SciPost.Report.1914

### Strengths

1. covering both how the method can be applied at parton level and detector level analysis
2. paper well written and pedagogical
3. paper particularly relevant for the HL-HLC where such approach can be keys for the event-generation

### Weaknesses

1. weak conclusion
2. missing systematical uncertainty impact in the parton-level analysis

### Report

The paper focus on how someone can optimized the event generation for a given analysis in order to reduce as much as possible the statistical uncertainty impact on that given analysis. The main idea behind such method

The paper is well written and very pedagogical in the way they present the various concept used in the paper.

The paper also present two different type of analysis, the first one at parton level, which is particularly interesting for theorist and phenomenologist and the second one at reconstructed level which is particularly interesting for experimentalist. I would have been keen to recommend publication for each of the them independently.

The main drawback of the paper is that the search of optimal event generation for a given analysis might not lead to an optimal event set for another analysis. While this might not be a real concern for theorist/phenomenologist, this can be a huge issue for the experimental community where sample of events are primary produced centrally and the same events sample used by many group and an even higher number of analysis. This being said this is nothing that the authors can do about it, it just reduce the significance of their work to my point of view (and the issue is briefly mentioned in the paper already).

A final minor criticism is about the conclusion of the paper. My personal feeling is that the conclusion feels a bit too much like an introduction with the comparison of more basic method (for example biasing, slicing,...). Those comparison are valid but should be part of the introduction rather than the conclusion of the paper. The conclusion of the paper would also gain to recap in bit more in details how the optimality is defined/reached and some of the practical implication (like the fact that even for an infinite budget the optimal distribution is not the unweighted one).

As a final suggestion to the author, I believe that the paper can be significantly be improve by adding at the partonic level the effect of theoretical uncertainty. The authors include such type of effects in the detector level analysis while such effect are also important for the partonic case and seeing how figure 6 is impacted would be a nice plot to look at (even if I can guess what the plot will look likes in advance). But I think this can be left to the author to decide to include this or not.

Finally, I want to point to the author, a minor typo in the paper on page 23 -last line- the word "the" is repeated.

My final recommendation for this paper is a minor revision, such that the authors can at minima improve the conclusion and consider the other points that i have raised. After that small change, my recommendation will change to
"publish --top 10%--"

### Requested changes

1. rewrite the conclusion
2. consider to add description of the impact of theoretical uncertainty at parton-level (optional)

• validity: top
• significance: good
• originality: high
• clarity: top
• formatting: excellent
• grammar: perfect

### Author:  <NAME>  on 2020-12-28  [id 1116]

(in reply to Report 1 on 2020-08-13)

Thank you for the constructive feedback! We have made the following requested changes:
1. We have rewritten and expanded the conclusions sections following the input from both referees, and feedback from our talks on this work. 2. In the original version, the theoretical uncertainty was already present in eq. (63) in the form of $\sigma_\mathrm{syst}$. In order to highlight this, we expanded the text above eq. (63), discussing the different contributions to $\sigma_\mathrm{syst}$.
Also, thanks for pointing out the repeated word---we've fixed it.

format: plain


text:
## Cakes, Custard, and Category Theory

<NAME>'s new popular coscience book is out, in the U.K. under the title Cakes, Custard and Category Theory: Easy recipes for understanding complex maths, and in the U.S. under the title How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics:

Most people imagine maths is something like a slow cooker: very useful, but pretty limited in what it can do. Maths, though, isn't just a tool for solving a specific problem - and it's definitely not something to be afraid of. Whether you're a maths glutton or have forgotten how long division works (or never really knew in the first place), the chances are you've missed what really makes maths exciting. Calling on a baker's dozen of entertaining, puzzling examples and mathematically illuminating culinary analogies - including chocolate brownies, iterated Battenberg cakes, sandwich sandwiches, Yorkshire puddings and Möbius bagels - brilliant young academic and mathematical crusader <NAME> is here to tell us why we should all love maths.

From simple numeracy to category theory ('the mathematics of mathematics'), Cheng takes us through the joys of the mathematical world. Packed with recipes, puzzles to surprise and delight even the innumerate, Cake, Custard & Category Theory will whet the appetite of maths whizzes and arithmophobes alike. (Not to mention aspiring cooks: did you know you can use that slow cooker to make clotted cream?) This is maths at its absolute tastiest.

Cheng, one of the Catsters, gives a guided tour of mathematical thinking and research activities, and through the core philosophy underlying category theory. This is the kind of book you can give to your grandma and grandpa so they can boast to their friends what her grandchildren are doing (and bake you a nice dessert when you come and visit :) ). A pleasant weekend reading.

In this year's POPL, <NAME> made a splash by showing how to get from parametricity to conservation laws, via Noether's theorem:

Invariance is of paramount importance in programming languages and in physics. In programming languages, <NAME>’ theory of relational parametricity demonstrates that parametric polymorphic programs are invariant under change of data representation, a property that yields “free” theorems about programs just from their types. In physics, <NAME> showed that if the action of a physical system is invariant under change of coordinates, then the physical system has a conserved quantity: a quantity that remains constant for all time. Knowledge of conserved quantities can reveal deep properties of physical systems. For example, the conservation of energy, which by Noether’s theorem is a consequence of a system’s invariance under time-shifting.

In this paper, we link Reynolds’ relational parametricity with Noether’s theorem for deriving conserved quantities. We propose an extension of System Fω with new kinds, types and term constants for writing programs that describe classical mechanical systems in terms of their Lagrangians. We show, by constructing a relationally parametric model of our extension of Fω, that relational parametricity is enough to satisfy the hypotheses of Noether’s theorem, and so to derive conserved quantities for free, directly from the polymorphic types of Lagrangians expressed in our system.

## Seemingly impossible programs

In case this one went under the radar, at POPL'12, <NAME> gave a tutorial on seemingly impossible functional programs:

Programming language semantics is typically applied to
prove compiler correctness and allow (manual or automatic) program
verification. Certain kinds of semantics can also be applied to
discover programs that one wouldn't have otherwise thought of. This is
the case, in particular, for semantics that incorporate topological
ingredients (limits, continuity, openness, compactness). For example,
it turns out that some function types (X -> Y) with X infinite (but
compact) do have decidable equality, contradicting perhaps popular
belief, but certainly not (higher-type) computability theory. More
generally, one can often check infinitely many cases in finite time.

I will show you such programs, run them fast in surprising instances,
and introduce the theory behind their derivation and working. In
particular, I will study a single (very high type) program that (i)
optimally plays sequential games of unbounded length, (ii) implements
the Tychonoff Theorem from topology (and builds finite-time search
functions for infinite sets), (iii) realizes the double-negation shift
from proof theory (and allows us to extract programs from classical
proofs that use the axiom of countable choice). There will be several
examples in the languages Haskell and Agda.

A shorter version (coded in Haskell) appears in Andrej Bauer's blog.

## <NAME>

Earlier this week Microsoft Research Cambridge organised a Festschrift for <NAME>. The preface from the book:

<NAME> has made exceptional contributions to the world of programming
languages and beyond. Throughout his career, he has re-invented himself every
decade or so, while continuing to make true innovations. His achievements span
many areas: software; language design, including experimental languages;
programming language foundations; and the interaction of programming languages
and biology. These achievements form the basis of his lasting scientific leadership
and his wide impact.
...
Luca is always asking "what is new", and is always looking to
the future. Therefore, we have asked authors to produce short pieces that would
indicate where they are today and where they are going. Some of the resulting
pieces are short scientific papers, or abridged versions of longer papers; others are
less technical, with thoughts on the past and ideas for the future. We hope that
they will all interest Luca.

Hopefully the videos will be posted soon.

## Dependently-Typed Metaprogramming (in Agda)

<NAME> gave an 8-lecture summer course on Dependently typed metaprogramming (in Agda) at the Cambridge University Computer Laboratory:

Dependently typed functional programming languages such as Agda are capable of expressing very precise types for data. When those data themselves encode types, we gain a powerful mechanism for abstracting generic operations over carefully circumscribed universes. This course will begin with a rapid depedently-typed programming primer in Agda, then explore techniques for and consequences of universe constructions. Of central importance are the â€œpattern functorsâ€ which determine the node structure of inductive and coinductive datatypes. We shall consider syntactic presentations of these functors (allowing operations as useful as symbolic differentiation), and relate them to the more uniform abstract notion of â€œcontainerâ€. We shall expose the double-life containers lead as â€œinteraction structuresâ€ describing systems of effects. Later, we step up to functors over universes, acquiring the power of inductive-recursive definitions, and we use that power to build universes of dependent types.

The lecture notes, code, and video captures are available online.

As with his previous course, the notes contain many(!) mind expanding exploratory exercises, some of which quite challenging.

## Lightweight Monadic Programming in ML

Lightweight Monadic Programming in ML

Many useful programming constructions can be expressed as monads. Examples include probabilistic modeling, functional reactive programming, parsing, and information flow tracking, not to mention effectful functionality like state and I/O. In this paper, we present a type-based rewriting algorithm to make programming with arbitrary monads as easy as using ML's built-in support for state and I/O. Developers write programs using monadic values of type M t as if they were of type t, and our algorithm inserts the necessary binds, units, and monad-to-monad morphisms so that the program type checks. Our algorithm, based on Jones' qualified types, produces principal types. But principal types are sometimes problematic: the program's semantics could depend on the choice of instantiation when more than one instantiation is valid. In such situations we are able to simplify the types to remove any ambiguity but without adversely affecting typability; thus we can accept strictly more programs. Moreover, we have proved that this simplification is efficient (linear in the number of constraints) and coherent: while our algorithm induces a particular rewriting, all related rewritings will have the same semantics. We have implemented our approach for a core functional language and applied it successfully to simple examples from the domains listed above, which are used as illustrations throughout the paper.

This is an intriguing paper, with an implementation in about 2,000 lines of OCaml. I'm especially interested in its application to probabilistic computing, yielding a result related to Kiselyov and Shan's Hansei effort, but without requiring delimited continuations (not that there's anything wrong with delimited continuations). On a theoretical level, it's nice to see such a compelling example of what can be done once types are freed from the shackle of "describing how bits are laid out in memory" (another such compelling example, IMHO, is type-directed partial evaluation, but that's literally another story).

## Kleisli Arrows of Outrageous Fortune

Kleisli Arrows of Outrageous Fortune

When we program to interact with a turbulent world, we are to some extent at its mercy. To achieve safety, we must ensure that programs act in accordance with what is known about the state of the world, as determined dynamically. Is there any hope to enforce safety policies for dynamic interaction by static typing? This paper answers with a cautious â€˜yesâ€™.

Monads provide a type discipline for effectful programming, mapping value types to computation types. If we index our types by data approximating the â€˜state of the worldâ€™, we refine our values to witnesses for some condition of the world. Ordinary monads for indexed types give a discipline for effectful programming contingent on state, modelling the whims of fortune in way that Atkeyâ€™s indexed monads for ordinary types do not (Atkey, 2009). Arrows in the corresponding Kleisli category represent computations which a reach a given postcondition from a given precondition: their types are just specifications in a Hoare logic!

By way of an elementary introduction to this approach, I present the example of a monad for interacting with a file handle which is either â€˜openâ€™ or â€˜closedâ€™, constructed from a command interface specfied Hoare-style. An attempt to open a file results in a state which is statically unpredictable but dynamically detectable. Well typed programs behave accordingly in either case. Haskellâ€™s dependent type system, as exposed by the Strathclyde Haskell Enhancement preprocessor, provides a suitable basis for this simple experiment.

I discovered this Googling around in an attempt to find some decent introductory material to Kleisli arrows. This isn't introductory, but it's a good resource. :-) The good introductory material I found was this.

## A Lambda Calculus for Real Analysis

A Lambda Calculus for Real Analysis

Abstract Stone Duality is a revolutionary paradigm for general topology that describes computable continuous functions directly, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, and the reasoning looks remarkably like a sanitised form of that in classical topology. This is an introduction to ASD for the general mathematician, with application to elementary real analysis.

This language is applied to the Intermediate Value Theorem: the solution of equations for continuous functions on the real line. As is well known from both numerical and constructive considerations, the equation cannot be solved if the function "hovers" near 0, whilst tangential solutions will never be found.

In ASD, both of these failures and the general method of finding solutions of the equation when they exist are explained by the new concept of overtness. The zeroes are captured, not as a set, but by higher-type modal operators. Unlike the Brouwer degree, these are defined and (Scott) continuous across singularities of a parametric equation.

Expressing topology in terms of continuous functions rather than sets of points leads to treatments of open and closed concepts that are very closely lattice- (or de Morgan-) dual, without the double negations that are found in intuitionistic approaches. In this, the dual of compactness is overtness. Whereas meets and joins in locale theory are asymmetrically finite and infinite, they have overt and compact indices in ASD.

Overtness replaces metrical properties such as total boundedness, and cardinality conditions such as having a countable dense subset. It is also related to locatedness in constructive analysis and recursive enumerability in recursion theory.

<NAME> is deadly serious about the intersection of logic, mathematics, and computation. I came across this after beating my head against Probability Theory: The Logic of Science and Axiomatic Theory of Economics over the weekend, realizing that my math just wasn't up to the tasks, and doing a Google search for "constructive real analysis." "Real analysis" because it was obvious that that was what both of the aforementioned texts were relying on; "constructive" because I'd really like to develop proofs in Coq/extract working code from them. This paper was on the second page of results. Paul's name was familiar (and not just because I share it with him); he translated Jean-<NAME>'s regrettably out-of-print Proofs and Types to English and maintains a very popular set of tools for typesetting commutative diagrams using LaTeX.

## Simplicial Databases

Simplicial Databases, <NAME>.

In this paper, we define a category DB, called the category of simplicial databases, whose objects are databases and whose morphisms are data-preserving maps. Along the way we give a precise formulation of the category of relational databases, and prove that it is a full subcategory of DB. We also prove that limits and colimits always exist in DB and that they correspond to queries such as select, join, union, etc. One feature of our construction is that the schema of a simplicial database has a natural geometric structure: an underlying simplicial set. The geometry of a schema is a way of keeping track of relationships between distinct tables, and can be thought of as a system of foreign keys. The shape of a schema is generally intuitive (e.g. the schema for round-trip flights is a circle consisting of an edge from $A$ to $B$ and an edge from $B$ to $A$), and as such, may be useful for analyzing data. We give several applications of our approach, as well as possible advantages it has over the relational model. We also indicate some directions for further research.

This is what happens when you try to take the existence of ORDER BY and COUNT in SQL seriously. :-)

If you're puzzled by how a geometric idea like simplexes could show up here, remember that the algebraic view of simplicial sets is as presheaves on the category of finite total orders and order-preserving maps. Every finite sequence gives rise to a total order on its set of positions, and tables have rows and columns as sequences!

## An Innocent Model of Linear Logic

An Innocent Model of Linear Logic by Paul-<NAME> was referenced by Noam in a serendipitious subthread of the "Claiming Infinities" thread.

Here's the abstract:

Since its early days, deterministic sequential game semantics has been limited to linear or polarized fragments of linear logic. Every attempt to extend the semantics to full propositional linear logic has bumped against the so-called Blass problem, which indicates (misleadingly) that a category of sequential games cannot be self-dual and cartesian at the same time. We circumvent this problem by considering (1) that sequential games are inherently positional; (2) that they admit internal positions as well as external positions. We construct in this way a sequential game model of propositional linear logic, which incorporates two variants of the innocent arena game model: the well-bracketed and the non well-bracketed ones.

The introduction goes on to refer to to <NAME>'s "Category Y with Conway games as objects, and winning strategies as morphisms, composed by sequential interaction," and points out that "it is a precursor of game semantics for proof theory and programming languages," and is "a self-dual category of sequential games." The foreword mentions that the paper goes on to give "a crash course on asynchronous games" and then "constructs a linear continuation monad equivalent to the identity functor, by allowing internal positions in our games, [which] circumvents the Blass problem and defines a model of linear logic."

<NAME> called this paper mind-blowing. My mind-blow warning light already exploded. I'm posting this paper because I know a number of LtUers are interested in these topics, and this way I can buttonhole one of them the next time I see them and ask them to explain it to me. ;)


text:
cohomology

# Contents

## Idea

Given a morphism $f:\stackrel{^}{B}\to B$ in an (∞,1)-topos $H$, regarded as exhibiting a characteristic class on $\stackrel{^}{B}$ with values in $B$, the $f$-twisted cohomology at stage $X$ is the fiber of $f$ over a given element of $B$.

This is such that if $f:\stackrel{^}{B}\to *$ is the terminal morphism, $f$-twisted cohomology is precisely the ordinary cohomology in $H$ with coefficients in $\stackrel{^}{B}$.

## Definition

###### Definition

For

• $H$ an (∞,1)-topos;

• $f:\stackrel{^}{B}\to B$ a morphism in $H$ (a characteristic class on $\stackrel{^}{B}$-cohomology);

• $X\in H$ an object of $H$;

• and $c\in H\left(X,B\right)$ a $B$-cocycle on $X$ with cohomology class $\left[c\right]\in {\pi }_{0}H\left(X,B\right)$

the $\left(f,\left[c\right]\right)$-twisted cohomology of $X$ is the set of equivalence classes

${H}_{\left[c\right]}\left(X,f\right):={\pi }_{0}{H}_{\left[c\right]}\left(X,f\right)$H_{[c]}(X,f) := \pi_0\mathbf{H}_{[c]}(X,f)

of the homotopy fiber ${H}_{\left[c\right]}\left(X,f\right)$ of the morphism $f:\stackrel{^}{B}\to B$ over the generalized element $c$ of $B$.

More explicitly, ${H}_{\left[c\right]}\left(X,f\right)$ is the ∞-groupoid defined as the homotopy pullback

$\begin{array}{ccc}{H}_{\left[c\right]}\left(X,f\right)& \to & *\\ ↓& & \phantom{\rule{thickmathspace}{0ex}}{↓}^{*↦c}\\ H\left(X,\stackrel{^}{B}\right)& \to & H\left(X,B\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \mathbf{H}_{[c]}(X,f) &\to& {*} \\ \downarrow && \;\downarrow^{\mathrlap{{*}\mapsto c}} \\ \mathbf{H}(X,\hat B) &\to& \mathbf{H}(X,B) } \,.

## Properties

### Formulation in terms of sections

Twisted cohomology may be reformulated equivalently in terms of collections of sections as follows:

Given the twisting cocycle $c:X\to B$, let ${P}_{f}\to X$ be the $\infty$-pullback (homotopy pullback) of $\stackrel{^}{B}\to B$ along this cocycle:

$\begin{array}{ccc}{P}_{f}& \to & \stackrel{^}{B}\\ ↓& & ↓f\\ X& \stackrel{c}{\to }& B\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ P_f &\to& {\hat{B}} \\ \downarrow && \downarrow{f} \\ X &\stackrel{c}{\to}& B }\,.

Write $\Gamma \left({P}_{f}\right)={H}_{X}\left(X,{P}_{f}\right)$ for the $\infty$-groupoid of sections of ${P}_{f}\to X$.

###### Proposition

This collection of sections is the $\left(f,\left[c\right]\right)$-twisted cohomology of $X$:

${\pi }_{0}\Gamma \left({P}_{f}\right)\simeq {H}_{\left[c\right]}\left(X,f\right)$\pi_0 \Gamma(P_f) \simeq H_{[c]}(X,f)
###### Proof

By the universal property of (homotopy) pullbacks, homotopy classes of sections $X\to {P}_{f}$, are in bijection with homotopy classes of homotopy comutative cones of the form

$\begin{array}{ccc}& & X\\ & {}^{\mathrm{id}}↙& & ↘\\ X& \stackrel{c}{\to }& B& \stackrel{f}{←}& \stackrel{^}{B}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && X \\ & {}^{id}\swarrow && \searrow \\ X &\stackrel{c}{\to}& B &\stackrel{f}\leftarrow& \hat{B} } \,.

These in turn are manifestly the homotopy classes of maps $X\to \stackrel{^}{B}$ such that $X\to \stackrel{^}{B}\to B$ is homotopic to $c$.

More generally:

###### Proposition

We have a natural equivalence

${H}_{\left[c\right]}\left(X,f\right)\simeq {H}_{/B}\left(c,f\right)\phantom{\rule{thinmathspace}{0ex}},$\mathbf{H}_{[c]}(X,f) \simeq \mathbf{H}_{/B}(c,f) \,,

where on the right we have the derived hom space in the over-(∞,1)-topos over $B$ from the twisting cocycle $c:X\to X$ to the morphism $f:\stackrel{^}{B}\to B$.

###### Proof

Use the characterization of the homotopy fibers of (∞,1)-functor categories (as described there) in terms of hom-object in the over (∞,1)-category ${H}_{/B}$.

On the right, the objects are morphism $X\to \stackrel{^}{B}$ equipped with equivalences from $X\to \stackrel{^}{B}\stackrel{f}{\to }B$ to $c:X\to B$.

### Twisted cohomology with trivial twisting cocycle

Let $*\to B$ be a pointed object. Then

• we say that the cocycle

$\left(X\to *\to B\right)\in H\left(X,B\right)$

is the trivial $B$-cocycle on $X$.

• the morphism $f:\stackrel{^}{B}\to B$ induces a fibration sequence

$A\to \stackrel{^}{B}\stackrel{f}{\to }B$

in $H$.

###### Proposition

The $\left(\left[*\right],f\right)$-twisted cohomology with trivial twisting cocycle is equivalent to the ordinary cohomology with coefficients in the homotopy fiber $A$ of $f$:

${H}_{\left[*\right]}\left(X,f\right)\simeq H\left(X,A\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{H}_{[*]}(X,f) \simeq \mathbf{H}(X,A) \,.
###### Proof

By definition, the homotopy fiber of $A$ is the homotopy pullback

$\begin{array}{ccc}A& \to & *\\ ↓& & ↓\\ \stackrel{^}{B}& \stackrel{f}{\to }& B\end{array}$\array{ A &\to& * \\ \downarrow && \downarrow \\ \hat B &\stackrel{f}{\to}& B }

in $H$. Since the $\infty$-groupoid valued hom in an (∞,1)-category is exact with respect ot homotopy limits (by definition of homotopy limits), it follows that for every object $X$, there is fibration sequence of cocycle ∞-groupoids

$\begin{array}{ccc}H\left(X,A\right)& \to & *\\ ↓& & {↓}^{{\mathrm{const}}_{*}}\\ H\left(X,\stackrel{^}{B}\right)& \to & H\left(X,B\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \mathbf{H}(X,A) &\to& {*} \\ \downarrow && \downarrow^{\mathrlap{const_{*}}} \\ \mathbf{H}(X,\hat B) &\to& \mathbf{H}(X,B) } \,.

By definition of twisted cohomology, this identifies

$H\left(X,A\right)\simeq {H}_{\left[*\right]}\left(X,f\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{H}(X,A) \simeq \mathbf{H}_{[*]}(X,f) \,.

For this reason, when $B$ is pointed, it is customary to call the set of equivalence classes ${\pi }_{0}{H}_{\left[c\right]}\left(X;f\right)$ the $c$-twisted $A$-cohomology of $X$, and to denote it by the symbol

${H}_{\left[c\right]}\left(X,A\right)$H_{[c]}(X,A)
###### Remark

The cohomology fibration sequence $H\left(X,A\right)\to H\left(X,\stackrel{^}{B}\right)\to H\left(X,B\right)$ can be seen as an obstruction problem in cohomology:

• the obstruction to lifting a $\stackrel{^}{B}$-cocycle to an $A$-cocycle is its image in $B$-cohomology (all with respect to the given fibration sequence)

But it also says:

• $A$-cocycles are, up to equivalence, precisely those $\stackrel{^}{B}$-cocycles whose class in $B$-cohomology is the class of the trivial $B$-cocycle.

## Examples

### Sections as twisted functions…

For $V$ a vector space and $X$ a manifold, both regarded a 0-truncated objects in the $\left(\infty ,1\right)$-topos on the site CartSp (that of Lie infinity-groupoids), a cocycle $X\to V$ is simply smooth $V$-valued function on $X$.

Now let $G$ be a Lie group with smooth delooping groupoid $BG$ and let $\rho :BG\to \mathrm{Vect}$ be a representation of $G$ on $V$, i.e. $\rho \left(•\right)=V$. Then the corresponding action groupoid $V//G$ sits in the fibration sequence

$V\to V//G\stackrel{p}{\to }BG\phantom{\rule{thinmathspace}{0ex}}.$V \to V//G \stackrel{p}{\to} \mathbf{B}G \,.

Hence we can ask for the $p$-twisted cohomology of $X$ with values in $V$. Now, a cocycle $g:X\to BG$ is one classifying a $G$-principal bundle on $X$. By looking at this in Cech cohomology it is immediate to convince onself that cocycles $X\to V//G$ such that the composite $X\to V//G\stackrel{p}{\to }BG$ is equivalent to the given $g$ are precisely the sections of the $\rho$-associated vector bundle:

on a patch ${U}_{i}$ of a good cover over wich $P$ has been trivialized, the cocycle $X\to V//G$ is simply a $V$-valued function ${\sigma }_{i}:{U}_{i}\to V$. Then on double overlaps it is a smooth natural transformation ${\sigma }_{i}{\mid }_{{U}_{ij}}\to {\sigma }_{j}{\mid }_{{U}_{i}j}$ whose components in $G$ are required to be those of the given cocycle $g$. That means exactly that the functions $\left({\sigma }_{i}\right)$ are glued on double overlaps precisely as the local trivializations of a global section $\sigma :X\to P{×}_{G}V$ would.

Hence we find the $p$-twisted cohomology is

${H}_{\left[g\right]}\left(X,V\right)=\left\{\mathrm{sections}\phantom{\rule{thickmathspace}{0ex}}\mathrm{of}\phantom{\rule{thickmathspace}{0ex}}P{×}_{G}V\right\}\phantom{\rule{thinmathspace}{0ex}}.$H_{[g]}(X,V) = \{sections\; of\; P \times_G V\} \,.

In this sense a section is a twisted function.

Notice that $V//G\stackrel{p}{\to }BG$ is not itself a homotopy fiber, but is a lax fiber, in that we have a lax pullback (really a comma object )

$\begin{array}{ccc}V//G& \to & *\\ ↓& & ↓\\ BG& \to & \mathrm{Vect}\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ V//G &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}G &\to& Vect } \,,

where in the bottom right corner we have Vect (regarded as a stack on $\mathrm{CartSp}$ in the evident way) and where the right vertical morphism sends the point to the ground field vector space $k$ (or rather sends $U\in \mathrm{CartSp}$ to the trivial vector bundle $U×k$ ).

We may paste to this the homotopy pullback along the cocycle $g:X\to BG$ to obtain

$\begin{array}{ccccc}P{×}_{G}V& \to & V//G& \to & *\\ ↓& & ↓& & ↓\\ X& \stackrel{g}{\to }& BG& \to & \mathrm{Vect}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ P\times_G V &\to& V//G &\to& * \\ \downarrow && \downarrow && \downarrow \\ X &\stackrel{g}{\to}& \mathbf{B}G &\to& Vect } \,.

This makes is manifest that a section $\sigma :X\to P{×}_{G}V$ is also the same as a natural transformation from ${\mathrm{const}}_{k}:X\to \mathrm{Vect}$ to $X\stackrel{g}{\to }BG\to \mathrm{Vect}$.

Notice moreover that in the special case that $G=U\left(1\right)$ and for ground field $k=ℂ$ we may think of $BU\left(1\right)$ as the category $ℂ\mathrm{Line}↪ℂ\mathrm{Mod}=\mathrm{Vect}$ and think of the twisting cocycle $g$ as

$X\stackrel{g}{\to }ℂ\mathrm{Line}↪ℂ\mathrm{Mod}\phantom{\rule{thinmathspace}{0ex}}.$X \stackrel{g}{\to} \mathbb{C}Line \hookrightarrow \mathbb{C}Mod \,.

#### … and $\infty$-sections as twisted $\infty$-functions

Regarded this way, the above discussion has a generalization to the case where the monoid $ℂ$ is replaced with any ring spectrum $R$ and we consider

$X\stackrel{\tau }{\to }R\mathrm{Line}↪R\mathrm{Mod}\phantom{\rule{thinmathspace}{0ex}}.$X \stackrel{\tau}{\to} R Line \hookrightarrow R Mod \,.

Twisted cohomology in terms of such morphisms $\tau$ is effectively considered in

and in unpublished work of U<NAME> et al. For more on this see the discussion at (∞,1)-vector bundle.

### twisted K-theory

In the context of generalized (Eilenberg–Steenrod) cohomology a coefficient object for cohomology is a spectrum $A$: the $A$-cohomology of a topological space $X$ with coefficients in $A$ is the set of homotopy classes of maps $X\to A$. For instance, as a model of the degree-$0$ space in the K-theory spectrum one can take $A=\mathrm{Fred}\left(H\right)$, the space of Fredholm operators on a separable Hilbert space $H$. There is a canonical action on this space of the projective unitary group $G=PU\left(H\right)$ of $H$. Since $PU\left(H\right)$ has the homotopy type of an Eilenberg–Mac Lane space $K\left(ℤ,2\right)$, a $PU\left(H\right)$-principal bundle $P\to X$ defines a class $c\in {H}^{3}\left(X,ℤ\right)$ in ordinary integral cohomology (this may also be thought of as the class of a twisting bundle gerbe). The twisted K-theory (in degree $0$) of $X$ with that class as its twist is the set of homotopy classes of sections $X\to P{×}_{PU\left(H\right)}\mathrm{Fred}\left(H\right)$ of the associated bundle.

### $G$-Actions on spectra

The above example generalizes straightforwardly to the case that

• $A$ is a connective spectrum, i.e. topological space that is an infinite loop space. (We need to assume a connective spectrum given by an infinite loop space so that $A$ can be regarded as living in the category of topologicall spaces along with the other objects, such as classifying spaces $BG$ of nonabelian groups);

• with a (topological) group $G$ acting on $A$ by automorphisms and

• a $G$-principal bundle $P\to X.$

In this case there is an established definition of generalized (Eilenberg–Steenrod) cohomology with coefficients $A$ twisted by a $G$-principal bundle as follows.

From the $G$-principal bundle $P\to X$ we obtain the associated $A$-bundle $P{×}_{G}A\to X$. Then a twisted $A$-cocycle on $X$ is defined to be a section $X\to P{×}_{G}A$ of this associated bundle. The collection of homotopy classes of these sections is the twisted $A$-cohomology of $X$ with the twist specified by the class of $P$.

This is the definition of twisted cohomology as it appears for instance essentially as definition 22.1.1 of the May–Sigursson reference below (when comparing with their definition take their $G$ to be the trivial group and identify their $\Gamma$ and $\Pi$ with our $G$).

It is clearly a particular case of the general definition of twisted cohomology given above:

• the $\left(\infty ,1\right)$-topos $H$ is the $\left(\infty ,1\right)$-category of Top of topological spaces

• the object $B$ is the delooping $BG$ of the group $G$.

• the object $\stackrel{^}{B}$ is the homotopy quotient $A//G\simeq EG{×}_{G}A$.

• the twisting cocycle $c$ is the element in $\mathrm{Top}\left(X,BG\right)$ defining the principal $G$-bundle $P\to X$.

Indeed, $B$ is pointed, we have a fibration sequence

$A\to A//G\to BG$A \to A//G \to \mathbf{B}G

and the homotopy pullback

$\begin{array}{ccc}{P}_{A}& \to & A//G\\ ↓& & ↓f\\ X& \stackrel{c}{\to }& BG\end{array}\phantom{\rule{thinmathspace}{0ex}}$\array{ P_A &\to& {A//G} \\ \downarrow && \downarrow{f} \\ X &\stackrel{c}{\to}& \mathbf{B}G }\,

is the $A$-bundle $P{×}_{G}A\to X$.

The obstruction problem described by this example reads as folllows:

• the obstruction to lifting a (“nonabelian” or “twisted”) $A//G$-cocycle $g:X\to A//G$ to an $A$-cocycle $\stackrel{^}{g}:X\to A$ is its image $\delta g$ in first $G$-cohomology $\delta g\in {H}^{1}\left(X,G\right):={\pi }_{0}\mathrm{Top}\left(X,BG\right)$.

Read the other way round it says:

• $A$-cocycles are precisely those $G$-twisted $A$-cocycles whose twist vanishes.
###### Remark

Since the associated bundle $P{×}_{G}A$ is in general no longer itself a spectrum, twisted cohomology is not an example of generalized Eilenberg–Steenrod cohomology.

To stay within the spectrum point of view, May–Sigurdsson suggested that twisted cohomology should instead be formalized in terms of parameterized homotopy theory, where one thinks of $P{×}_{G}A$ as a parameterized family of spectra.

### Group cohomology with coefficients in a module

Some somewhat trivial examples of this appear in various context. For instance group cohomology on a group with coefficients in a nontrivial module can be regarded as an example of twisted cohomology. See there for more details.

Compare this to the example below of cohomology “with local coefficients”. It is the same principle in both cases.

### Twisted bundles

To get a feeling for how the definition does, it is instructive to see how for the fibration sequence coming from an ordinary central extension $K\to \stackrel{^}{G}\to G$ of ordinary groups as

$B\stackrel{^}{G}\to BG\stackrel{\omega }{\to }{B}^{2}K$\mathbf{B}\hat G \to \mathbf{B}G \stackrel{\omega}{\to} \mathbf{B}^2 K

classified by a group 2-cocycle $\omega$, $c$-twisted $\stackrel{^}{G}$-cohomology produces precisely the familiar notion of twisted bundles, with the twist being the lifting gerbe that obstructs the lift of a $G$-bundle to a $\stackrel{^}{G}$-bundle.

This is also the first example in the list in the last section of

and contains examples that are of interest in the wider context of string theory.

### Cohomology with local coefficients

What is called cohomology with local coefficients is twisted cohomology with the twist given by the class represented by the universal cover space of the base space, which is to say: by the action of the fundamental group of the base space.

In the classical case of ordinary cohomology, <NAME> in 1972 constructed a twisted $K\left(\pi ,n\right)$, denoted $\stackrel{˜}{K}\left(\pi ,n\right)$, so that, for nice spaces, the cohomology with local coefficients ${\stackrel{˜}{H}}^{n}\left(X,\pi \right)$ with respect to a homomorphism $\epsilon :{\pi }_{1}\left(X\right)\to \mathrm{Aut}\left(\pi \right)$ is given by homotopy classes of maps $X\to \stackrel{˜}{K}\left(\pi ,n\right)$ compatible with $\epsilon .$

More generally, for any space $X$, let $A$ be a coefficient object that is equipped with an action of the first fundamental group ${\pi }_{1}\left(X\right)$ of $X$. (Such an action is also called an $A$-valued local system on $X$).

Then there is the fibration sequence

$A\to A//{\pi }_{1}\left(X\right)\to B{\pi }_{1}\left(X\right)$A \to A//\pi_1(X) \to \mathbf{B} \pi_1(X)

of this action.

Notice that there is a canonical map $c:X\to B{\pi }_{1}\left(X\right)$, the one that classifies the universal cover of $X$.

Then $A$-cohomology with local coefficients on $X$ is nothing but the $c$-twisted $A$-cohomology of $X$ in the above sense.

## References

A discussion of ${\pi }_{1}\left(X\right)$-twisted ordinary cohomology is in

• <NAME>, <NAME>, <NAME>, Homotopy colimits and cohomology with local coefficients, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 44 no. 1 (2003), p. 63-80 (numdam)

For the special case of generalized (Eilenberg–Steenrod) cohomology twisted by a $G$-principal bundle see section 22.1 of

This in turn is based on the definition of twisted K-theory given in

Details on Larmore’s work on twisted cohomology are at

The above discussion of $c$-twisted cohomology as the homotopy fiber of $H\left(X,\stackrel{^}{B}\right)\to H\left(X,B\right)$ over $c\in H\left(X,B\right)$ etc. is in

### Chronology of literature on twisted cohomology

The oldest meaning of twisted cohomology is that of cohomology with local coefficients (see above).

For more on the history of that notion see

In the following we shall abbreviate

• tc = twisted cohomology

Searching MathSciNet for twisted cohomology led to the following chronology: It missed older references in which the phrase was not used but the concept was in the sense of local coefficient systems – ancient and honorable.

Most notably missing are

• <NAME> (1938) Topologie der Polyeder und kombinatorische Topologie der Komplexe_, Mathematik und ihre Anwendungen in Physik und Technik,_(But note that reprints appear, sans reviews. There is a short English and longer German review on Zentralblatt)

• <NAME> (1942,1943)

• Olum (thesis 1947, published 1950)

Next come several that involve twisted differentials more generally.

Few are in terms of homotopy of spaces

tc ops should be treated as a single phrase – it may be that the ops are twisted, not the cohomology

• 1966 McClendon thesis – summarized in

• 1967 <NAME> tc ops

• 1967 Larmore tc ops

• 1969 McClendon tc ops

• 1969 Larmore tc

• 1970 Peterson tc ops

• 1971 McClendon tc ops

• 1972 Deligne Weil conjecture for K3 tc – meaning?

• 1972 Larmore tc

• 1973 Larmore and Thomas tc

• 1973 Larmore tc

• gap

• 1980 Coelho & Pesennec tc

• 1980 Tsukiyama sequel to McClendon

• 1983 Coelho & Pesennec tc

• 1985 Morava but getsted at 1975 ??

• 1986 Fried tc

• 1988 Baum & Connes ??

• 1989 Lott torsion

• 1990 Dwork ??

• 1993 Gomez–Tato tc minimal models

• 1993 Duflo & Vergne diff tc

• 1993 Vaisman tc and connections

• 1993 Mimachi tc and holomorphic

• 1994 Kita tc and intersection

• 1995 Cho, Mimachi and Yoshida tc and configs

• 1995 Cho, Mimachi tc and intersection

• 1996 Iwaski and Kita tc de rham

• 1996 Asada nc geom and strings

• 1997 H Kimura tc de Rham and hypergeom

• 1998 <NAME>?, <NAME>?, <NAME>?, Morse theory of harmonic forms, Topology, (Volume 37, Issue 3, May 1998, Pages 469–483)

• 1998 Knudson tc SL_n

• 1998 Morita tc de Rham

• 1999 Kachi, Mtsumoto, Mihara tc and intersection

• 1999 Hanamura & Yoshida Hodge tc

• 1999 Felshtyn & Sanchez–Morgado Reidemeister torsion

• 1999 Haraoka hypergeom

• 2000 Tsou & Zois tc de rham

• 2000 Manea tc Czech

• 2001 Royo Prieto tc Euler

• 2001 Takeyama q-twisted

• 2001 Gaberdiel &Schaefr–Nameki tc of Klein bottle

• 2001 Proc Rims tc and DEs and several papers in this book

• 2001 Knudson tc SL_n

• 2001 Royo Prieto tc as $d+k\wedge$

• 2001 Barlewtta & Dragomir tc and integrability

• 2002 Lueck ${L}^{2}$

• 2002 Verbitsky HyperKahler, torsion, etc

• 2003 Etingof & Grana tc of Carter, Elhamdadi and Saito

• 2003 Cruikshank tc of Eilenberg

• 2003 various in Proc NATO workshop

• 2003 Dimca tc of hyperplanes

• 2004 Kirk & Lesch tc and index

• 2004 Bouwknegt, Evslin, Mathai tc and tK

• 2004 Bouwknegt, Hannbuss, Mathai tc in re: T-duality

• 2005 Bouwknegt, Hannbuss, Mathai tc in re: T-duality

• 2005 Bunke & Schick tc in re: T-duality

• 2006 Dubois tc and Reidemeister (elsewhere he considers twisted Reidemesiter)

• 2006 Bunke & Schick tc in re: T-duality

• 2006 Sati

• 2006 Atiyah & Segal tc and tK

• 2007 Mickelsson & Pellonpaa tc and tK

• 2007 Sugiyama in re: Galois and Reidemeister

• 2007 Bunke, Schick, Spitzweck tc in re: gerbes

• 2008 Kawahara hypersurfaces

Revised on April 22, 2013 11:35:44 by <NAME> (95.147.236.31)


text:
# Suspicious Conversation

Yesterday, I overheard the conversation between my friends Jim and Rose:

Jim: I know lots of games.

Rose: Wow, let's play some.

Jim: If Jack too was here, it would have been great fun.

Rose: Oh, I miss Jack a lot.

Jim: He came last week for the exams.

Rose: Ha! Really? He didn't tell me.

Jim: He asked for a tea party.

Rose: Yes, it should be arranged.

Jim: Let's go at 4:00pm. He must be at home at that time.

Rose: He has nailed the exams and so he should be hosting the party, right?

Jim: Yup, and I bought a pen for him to appreciate his success in the exams.

Rose: Oh, great. Let me see it.

Jim: Aha, you lost. I am the winner.

What game they were playing? How did Rose lose and Jim win? And how does this conversation make sense?

• Is "It Jack too was here" correct or did you mean "If Jack..."? Sep 10, 2015 at 9:00
• Is "the one who hosting" correct?
– A E
Sep 10, 2015 at 9:12
• Yes. It's "if". I have edited it. Thank you for pointing it out. :) Sep 10, 2015 at 9:12
• @AE: It is correct. Any how I have edited the question to avoid confusions. Sep 10, 2015 at 9:19
• I think of some games like "don't say you,me, and he/she" in Chinese. We start a meaningless conversation and someone who speaks a forbidden word lose the game Sep 10, 2015 at 10:12

I thought that it was

animals formed by consecutive letters in the sentence

and it turns out that's correct.

Jim: I know lots of games.

owl

Rose: Wow, let's play some.

owl also (edited to say: probably owlet)

Jim: If Jack too was here, it would have been great fun.

bee

Rose: Oh, I miss Jack a lot.

jackal

Jim: He came last week for the exams.

camel

Rose: Ha! Really? He didn't tell me.

hare

Jim: He asked for a tea party.

rat

Rose: Yes, it should be arranged.

bear

Jim: Let's go at 4:00pm. He must be at home at that time.

goat

Rose: He has nailed the exams and so he should be hosting the party, right?

snail

Jim: Yup, and I bought a pen for him to appreciate his success in the exams.

Rose: Oh, great. Let me see it.

And here's where Rose lost, since she had no animal.

Jim: Aha, you lost. I am the winner.

• There's "ape" in Jim's statement (bought a pen).
– f''
Sep 11, 2015 at 3:12
• Perfect. Please add what @f'' suggested to make the answer complete and I will accept the answer :) Sep 11, 2015 at 3:46

My first attempt:

Jim and Rose has already starting the game the moment the conversation starts, and the rule is to continue the conversation in a way that, when reading backward, it makes sense. They were both doing fine until Rose's last sentence, which loses her the game.

• That definitely seems like it - nice find!
– Deusovi
Sep 10, 2015 at 16:30
• Nice finding. But there is a better fit than that. Sep 11, 2015 at 3:46


text:
# minipage latex side by side

equations? It would definitely be useful to have a minimal example to look at. Try to use subfigure instead of minipage, I’m not sure it will work, but it is worth a try. In case you mean left in the document, you can pass the fleqn option to either documentclass or the amsmath package. Try reducing their sizes slightly and it should work just fine. Tags： latex While write a paper (or a proposal), you might need to place a table and a figure in a same row to save some space (or for some other reasons). Including images in a report is very common in LaTeX. If you Google for “Excel table macro Latex” or something like that, you can find a macro to convert the table to a LaTeX table. This website uses cookies and pixel tags to ensure you get the best experience on our website. For some reason, I can’t get the minipage environment to place my tables side-by-side. Remark: Using the subfigure-package is another way to place figures or tables side-by-side. Please post a minimal example of the table and matrix and I’ll be happy to advise how to place them side-by-side. Y only have one minipage for your tabular environment, but actually you need a minipage for the picture as well. I had been struggling with the format till I tried your solution. For example, two pictures side by side, two tables next to a text or a picture or vice versa. This seems to be the most frustrating (to me) task. They are definitely getting aligned ! Result <NAME> ( <EMAIL> ) If so, here is how you can achieve that: If this is not what you wanted, please provide code of a mini-example. Otherwise, I would need to see a minimal example which produces the error to be able to help. Besides there are still further options, which however in practical application the minipage does not play a role like the height and the adjustment (again c, t and b) within the minipage. The following code shows you how: You can also have more than two column simply by adding another \minipage in between the table-command and reduce the width of each \minipage (0.33\linewidth in addition to the \hspace). I've got 3 figures and a table that I want to put into 4 quadrants like below. Nice figures :-)! \documentclass{article} \usepackage{graphicx} \begin{document} Computers are machines that perform tasks or calculations according to a set of instructions, or … online LaTeX editor with autocompletion, highlighting and 400 math symbols. I use a macro called calc2latex (converts tables form OpenOffice into latex output), works great! … Ok, after some tries I could solve it myself. Pastebin.com is the number one paste tool since 2002. Like this : The other solution I tried was using minipage command, but the problem with minipage is that the subfigures get normal figure caption (like Figure 1) while … But if you put a fixed box, you have already made that decision. 7.2.1 Side-by-Side Tables.You can create side-by-side tables using an analogous method to the side-by-side figures approach described above. Thanks again Tom ! Again, the code works perfectly (apart from the missing graphicx package). Adam. I cannot reproduce what you are referring to. I’ll try to answer both below. Source: World Bank (2003). I’m adding you to my blog roll, if that’s ok…. GDP 3.5 5.4 I’m trying to put a table and a picture side-by-side but my efforts are in vane.. who I can do that? Here is the same example using subcaption: Ok thanks and where do I put my pictures in the code? (adsbygoogle = window.adsbygoogle || []).push({}); Need help with your thesis or book project? At present the seperation is around ~1cm between two images with the below code. Another disadvantage of minipage is that it does not align fi gures. Well, of course, they are two different figures: one is say Figure 1 and the other is say Figure 2. I only get one caption and it is only on the second picture? Float means, LaTeX decides where exactly to place elements. By using this website, you consent to the use of cookies.learn more about cookies see also our privacy policy. Not that it was wrong, but remember that sometimes the order makes a difference. Here is an example. Table and Figure Side by Side… I understand that can be done easily using lstlisting package. This is my table title Subscribe to this blog. I am also trying to put a figure and a table side by side. ... placing things side by side. t. Match the top line in the minipage with the baseline of the surrounding text (Plain TeX’s \vtop. Inflation 2.0 4.1 The left pictures should change with every click and the right pictures would be nice to overlay with every click, so the old data still is shown in a weaker colour density. how to do dis?? \end{minipage} %second column \begin{minipage}[t]{0.5\textwidth} This is the second column. Glad it works! By using the first option \hline> have to replaced by \cline. Do you use [H] on minipage? The following code can be used to place two figures side-by-side by creating a minipage: If there is not %, the images will be displayed in one above the other, not side by side. I’ve tried all kinds of variations, but no luck. How to put text beside figure with minipage? The minipage is often used to put things next to each other, which can otherwise be hard put together. F F F T. I'm using the subfigure environment for placing the 3 figures but how I can make the … Please provide a complete minimal working example next time, including preamble. If you want more space between the figure and the caption, use: Also, if you prefer numbers instead of the default numbering: Here is a link to the documentation: subfigure. We use two minipages side by side in first one we put the figure e.g. Try [H] or [h!t] on the figure. I’m looking for a way to place a picture next to a table. pic1 und in second one the text. Code: [Select all] [Expand/Collapse] [Download] (untitled.tex) \begin{table} [ht] … the rest all didn’t work for me (minipages, wrapfigure, parbox..), it kept aligning the elements top-down, if at all. I suggest using \hfill, which basically fills whatever space is available. Export (png, jpg, gif, svg, pdf) and save & share with note system Among other things, it lets you define the font style of the caption and hence you don’t need to type the caption twice: Hope this works! Post was not sent - check your email addresses! Replace \begin{center}…\end{center} by \centering, which does not add any additional white space on top and below the figure/table. Finally, following solution worked by using \usepackage{subfigure}. I found the solution by modifying the help given in http://www.win.tue.nl/latex/documentation/plaatjes/pictures.pdf. It also works with two tables side by side. Please let me know if how it goes. This happened with so many figures that LaTeX couldn’t put them down nicely on a page any more. I have 2 questions: One- Is it possible to place two figures side by side and immediately below them another two figures, all of these as one “unit”? Any suggestions? Use \figcaption{...} instead of \caption{...}, provided by the nonfloat package. Boxes and Mini-Pages. The vspace command in the second minipage is only a hint, for example if the text is not enough you could push it up by using vspace. . Miguel, If I understand correctly, you can simply use the \caption{table description} command. to the preamble. Please see my code snippets section for more details. I removed that option and voilà, it worked. Two Figures Side-by-Side in Latex Many times you can come across a situation where you need to put two figures side-by-side in a paper written in latex format. Use your code without [H] and move the figure forward until you get it where you want. […] more appropriate solution would be using the minipage environment. Even though the figures are now only an inch across they still stack vertically. The problem is not related to eps-figures. The minipage is often used to put things next to each other, which can otherwise be hard put together. By using the caption package, the size of some captions may be changed, and suddenly LaTeX placed all my figures at the end of the document. If you are still stuck, please include a minimal working example and briefly describe what you tried and where you are facing problems. A minipage is not a float but a box which sits at its fixed place. b. So, all you have to do is replace them with your figures (\includegraphics). The code is as follows: The problem is not so much with the size of the figures inside the minipage environment, but the size of the minipage “box” itselves. For more information see my post, the subfig and caption package documentations. Thank you for helping me. By using the first option \hline> … I am getting crazy with this minifigure. Using minipage to display two figures/table side-by-side in Latex November 24, 2010 November 24, 2010 kunigami In the following example, ‘figure1’ and ‘figure2’ are being displayed side-by-side. _____________________________ Below is a minimal working example where subfloats have no label. You’ll have to load it in your preamble. Hi Thomas, First just put the two tables in a first one, second minipage. The first is a beamer-specific environment and is therefore only available in a beamer presentation. The issue is that strut on the right side uses the maximum vertical space (for the line) whereas the left side doesn't require the maximum (due to the text on the line not having any depth [e.g. The width can be set as absolute or as relative value. The tabular environment knows multicolumn cells. But I wanted them left aligned..that way they look better ! Thanks much for the code example. I had this error: “to many unprocessed floats”. Subfigure differs from minipage in that you cannot choose the space between two figures, but it is determined automatically, depending on the figure size. Amir, your code worked wonderfully for me. can you suggest how to do this using minipage ? LaTeX side-by-side tables.StartLaTeX.To put two tables side-by-side, there are two ways.First just put the two tables in a first one, second minipage. Take a look at the answer to a similar question on tex.SX. ... by the way, it also doesn’t make sense to have float inside a fixed box. Including the small horizontal space in-between, they are taking up more than the complete linewidth. Hey, I don't know if this is the right place to ask but I'm kind of lost and need help. Whereas the latter has other applications and is available in all document-classes. \documentclass{article} \usepackage{graphicx} \begin{document} Computers are machines that perform tasks or calculations according to a set of instructions, or programs. Hi again, Positions the minipage so its vertical center lines up with the center of the adjacent text line (what Plain TeX calls \vcenter). This is not that easy actually. … Let’s discuss. Thanks for your question. First just put the two tables in a first one, second minipage. There are two ways (and possibly more) to place content side-by-side in a beamer presentation, the columns and the minipage environments. ... placing things side by side. http://texblog.wordpress.com/2007/08/07/introduction-to-tables-in-latex/, http://texblog.wordpress.com/2007/08/01/placing-figurestables-side-by-side-minipage/, http://www.win.tue.nl/latex/documentation/plaatjes/pictures.pdf, Multi-column and multi-row cells in LaTeX tables, Control the width of table columns (tabular) in LaTeX. Other than that, your tikz figure seems to be ok. The first is a beamer-specific environment and is therefore only available in a beamer presentation. 1970 2000 You may notice that the left side text is a bit lower than the right side, however. In my thesis I used a minipage environment to do that. Hi. Cheers, Tom. <NAME>. I'm using the subfigure environment for placing the 3 figures but how I can make the table appear next to the third figure and not on the next line? plzz help me out. <NAME> CLAN TAG #URR8PPP I’m not sure what you are doing exactly. Ideally, the code would illustrate your problem with as few lines as possible. Insert→Box (to insert another minipage) Change the width of the two minipages to 45% Textwidth (by clicking the right mouse button on each minipage) In each minipage: change the paragraph alignment to center add the graphics press enter for a new paragraph Insert a caption (Insert→Caption). How to put text beside figure with minipage? Example:. By this one has the opportunity to use this new page, for example, set two pictures side by side, then just set two minipages side by side. However, it doesn’t look good at all. The conflict is due to the floatrow package (as the error suggests). Including images in a report is very common in LaTeX. Pastebin is a website where you can store text online for a set period of time. \documentclass{article} \usepackage{graphicx} \usepackage{subcaption} \begin{document} Computers are machines that perform tasks … Ideally, I’d like a GUI table generator that converts the table into LaTeX. I've got 3 figures and a table that I want to put into 4 quadrants like below. Could it be that your editor replaces double dash by a long dash? the attached pic). ... One minipage serves the purpose of putting one figure. Structuring your work nicely is probably the most obvious reason why you want to put two figures/tables side-by-side. Thanks for your comment. They end up getting both Figure captions or table captions. figure to figure and table to table. Two- It’s a bit hard to explain (I’d send you a pic but don’t know how), but as I’m making a table I’d like to know how to control the lines, e.g. minipage can also be used for text, not only for figures and […], Placing figures/tables side-by-side (\subfigure) « Blog on Latex Matters, […] within the usual figure-floating-environment. The code is one i have used previously and successfully so have no idea why it is now not working. The error I get states “Package floatrow Error: Caption(s) lost.” I have tried reading around this but cant find a solution to the issue. So try this and let me know if you are still facing problems: Table , figure in twocolumn mode « Hla Hla Htay’s Notes, […] from http://texblog.wordpress.com/2007/08/01/placing-figurestables-side-by-side-minipage/ […]. i want to insert table and matrix in a line. Hi Tom i want to insert a table and a figure side by side but when i do this the figures considered as atble and no of table increases.how can i do this? The idea behind the minipage command is that within an existing page “built in” an additional page. Hey guys, I have been trying to place two figures that span half a column side-by-side. A minipage is not a float but a box which sits at its fixed place. the code didn’t work for me for placing a figure and a table near each other is there any package (expect caption & graphicx) which should be added? spacing in math mode > line break within table cell side by side tables. Structuring your work nicely is probably the most obvious reason why you want to put two figures/tables side-by-side. Use minipage like in this example, which places two pictures side by within a figure float with separate captions \begin{figure}[htbp] \centering \begin{minipage}[b]{5 cm} \includegraphics{filename 1} \caption{caption 1} \label{labelname 1} \end{minipage} \begin{minipage}[b]{5 cm} \includegraphics{filename 2} … Here is an example: Question 2 I use it to create all my tables and then paste the generated code into my Latex document. Insert→Box (to insert another minipage) Change the width of the two minipages to 45% Textwidth (by clicking the right mouse button on each minipage) In each minipage: change the paragraph alignment to center add the graphics press enter for a new paragraph Insert a caption (Insert→Caption). You would need to give me an example that includes the packages you load, so I can reproduce the error. There is indeed a problem in your code. In general, they are stacked vertically if there is not enough space to arrange them horizontally. I assume by align, you mean vertically align and . Hey Tom, I’ve tried out your solution but then LaTeX informs me that this \usepackage[lofdepth]{subfig} package can’t work with \usepackage{subcaption}. The two figures have their own captions. After that you can use the environment wrapfig, it takes two parameters that are passed inside braces: the alignement that can be l, r, c, i or o; this letters stand for left, right, centre, inner and outer (the last two intended for two-sided documents).The second parameter is the width of the figure, in the example is 0.25 the width of the text. Figures stacked rather than side by side when using minipage. Laying out two tables of uneven length aligns both tables vertically in the middle. I want both figures side by side in one column of a two-column LaTeX document, but the result is both the figures are one below another. We use two minipages side by side in first one we put the figure e.g. You have two options: thank you for the fine explanation and the code. A mistake that is often made is, there is a blank line between the \end{minipage} and \begin{minipage} left. You can find a general introduction here. 3 posts • Page 1 of 1 ; In another note to myself, here I describe how to put two figures side by side in LaTeX with the minipage environment. The following code can be used to place two figures side-by-side by creating a minipage: Note, for certain figure manipulations such as width, loading the graphicx package is required: The same “local column” – effect can be achieved for tables. There’s one for Excel called xls2latex. The first is a beamer-specific environment and is therefore only available in a beamer presentation. please help me. Your examples work, but how to put a figure with \includegraphics[width=0.5\textwidth]{figs2/fig5.eps} in your examples and not a … All of this information helped me place 2 figures side by side and a table and a firgure side by side (using the minipage command). Using [H], the figure disappears, using [h!t], the figure appears, but not on the defined area. I have included the code proper below, I am just at a loss as to why it used to work fine and now doesnt. The only thing we should notice is the % character between the two minipage environments. I’m trying to put a table and a figure side by side using the minipage environment but I have this problem: I tried vertically put the big table in one column for the two-column latex document. I have three columns and above them one cell which is not divided by the lines of thecolumns below… Do you have any suggestions? online LaTeX editor with autocompletion, highlighting and 400 math symbols. Even though you can have several figures/tables next to each other (see posts on minipage and subfigure), with none of these techniques it is possible to have floating text around a figure […]. We use two minipages side by side in first one we put the figure e.g. example three tables Cheers. I am trying to draw the image below in LaTeX using lstlisting, tikzpicture, minipage, and tabular. Side by side content in LaTeX Jun 26, 2017 If there is one thing that keeps popping up in my LaTeX needs, it’s the ability to put a block of text, a table, anything on a side of a page, and something else on the other side. The linewidth of the figure is 1 in order to use the complete width provided by the “minipage”. thanks for the quick answer. Btw. Can anyone offer advice please? Whereby one must note here that the widths of several minipages those do not lay next to each other than the width of the text are larger, since otherwise everything cannot be indicated. Any helps are highly appreciated. Then you insert your table, depending on the height/width ratio you can even change the page layout to landscape. LaTeX minipage. The idea behind the minipage command is that within … Post navigation You might want to try using the table itself to hold the note and source. side by side tables. Export (png, jpg, gif, svg, pdf) and save & share with note system I tried to use minipage with lstlisting but it does not work. you are loading several packages multiple times, including textcomp, float and amsmath. Including the LaTeX error would also be useful. Another thing that might be useful is the caption package (i.e. how to solve this ??? Thanks a lot Sir for helping me to get the desired output. What I propose is using the “subfigure” package instead of “minipage”, which gives you more flexibility. I am trying to insert four figures in a LaTeX document but if I use subfigure command, two of my figures stay in first row and the other two go to the second line. Whereas the latter has other applications and is available in all document-classes. Some time ago I found a book template that had the chapter titles with the number upside (for example, "Chapter One"), then an horizontal line that was the same width as the title number and below it the chapter title (for example, "This is the first … The minipage can be used to place figures side-by-side too. This entry was posted in problems and tagged figure , minipage , side by side on December 25, 2012 by Aux_Latex . To align the equations, this should work: Why are my tikz figures not getting generated? but that doesnot work ! Hard to say why [H] doesn’t though. I am not able to give simultaneously different captions i.e. Insert→Box (to insert another minipage) Change the width of the two minipages to 45% Textwidth (by clicking the right mouse button on each minipage) In each minipage: change the paragraph alignment to center add the graphics press enter for a new paragraph Insert a caption (Insert→Caption). This article explains how to do it with a basic example. If you need the figure to display in the middle of a page, you can use. > Is there a way to put 2 figures side-by-side in a row on the page? I do not want to display figure numbering like (a) F1 (b) F2 (c) F3… etc instead of it i want to display F1 F2 F3 F4…etc An image is sufficient the dashes on the second picture tables and then paste the generated into! Has other applications and is therefore only available in a row on second! Tables for you the complete width provided by the lines highlighted below the surrounding text ( Plain TeX s! To my blog roll, if I understand correctly, you can pass the option! Be done easily using lstlisting, tikzpicture, minipage, side by side try using the itself. ] doesn ’ t work because there would be using the width can be set as absolute as., Iam using this code to display in the second one is based on minipage and caption different i.e! To put figures side by side that Si was asking about though the figures are now an! Only an inch across they still stack vertically between two images with the baseline of the and... And tagged figure, minipage, and the other is say figure 1 is set in the code that absolutely. Figures: one is say figure 1 and the source at the top line in the code. Better off using either subfigure or subfig top line in the table ’....Push ( { }, I don ’ t have problems with citing inside the.! In advance for any assistance given, greatly appreciated the middle of a page, you can pass fleqn. The picture as well helped me speed up my work and then paste the generated code my! How can I add those lines of text at the top to get desired. Have three columns and above them one cell which is not a float but a box which at. That two mini-pages can be used to place elements website uses cookies and pixel tags to ensure get. Table captions try reducing their sizes slightly and it is worth a try use! This post on subfigures is a beamer-specific environment and is available in a beamer presentation, the images will displayed. Place elements to my blog roll, if space is available \label { )... Best experience on our website Si was asking about please include a minimal example produces. Any tips on the figure is 1 in order to use either subfigure or the amsmath package floating! ~1Cm between two images with the format till I tried to use subfigure instead “. Of 1 to the top please post a minimal working example and briefly describe what you are still stuck please! Plain TeX calls \vcenter ) example and briefly describe what you tried where... 5.4 Inflation 2.0 4.1 _____________________________ note: this values are not true knows multicolumn.... \Minipage environmnet to put things next to a text or a picture or vice versa # # # #... 1 of 1 to the use of cookies.learn more about cookies see also our privacy policy in! 1 of 1 to the top line in the middle of a page you... } environment Cheers but did n't work properly and issues with alignments a lot it really helped, ’. It where you are loading several packages multiple times, including textcomp, float amsmath. By \cline side caption for figures, each with a captioning issue in minipage helps undrestand what I mean thanx... Complex figures and supports captioning and referencing the linewidth of the figure is 1 in order use! You tried and where you are still stuck, please include a minimal example to look in LaTeX using package... As if the two tables side-by-side, each with a caption this using,... And supports captioning and referencing used a minipage for your figures ( \includegraphics ) was... Can you suggest how to do that using \usepackage { subfigure } of one. Asking about found your minipage stuff pretty useful an existing page  built ''... Am struggling with a caption you consent to the floatrow package ( i.e and! Figure is 1 in order to use the minipage environments tables for you it helps undrestand what I... Fixed box ( to me ) task captions: Thank you very much for the width can be below! In §4.7 looking for a way to place my tables side-by-side table }! Captains ) side by side, two tables next to a text or picture... Enough space to arrange them horizontally have the two tables in a first one we the... Overlay them by clicking next in the minipage environment to do this, but remember that sometimes the order a... To arrange them horizontally suggests ) a try provide two simple approaches to look LaTeX. About cookies see also our privacy policy inside a fixed box separate captains ) side by side of.... The picture as well enclosed by a long dash thecolumns below… do you have any other to... Me minipage latex side by side side by side, two pictures side by side pixel tags to ensure you get desired. At its fixed place reducing their sizes slightly and it is worth a try to do,. A \twocolumn environment subfigure ” package instead of minipage, and the minipage command is that within minipage... For me! place elements, and the minipage environment creates a horizontal box, you switch back a! We do using the “ minipage ”, I would like to add the figure about! Done easily using lstlisting package propose is using the width option, e.g { subfigure } one. \Rule command is that within an existing page “ built in ” an additional page in example... To get the minipage environment, which means that two mini-pages can be used to two. 0,2 to the top line in the table can ’ t though used \begin flalign. Before the \label { } is that within an existing page  built in ” an additional page would! Provide a minimal example with two figures that span half a column side-by-side that. Is 1 in order to use the minipage is often used to place content side-by-side in beamer... ( side by side, two tables side-by-side statements like we do using float... Below code I then use this command in the short caption of the text... More ) to place two figures, try using the table display in code. Like my table title _____________________________ 1970 2000 _____________________________ amazes me are their displays where, for example ): 1. G, p, etc those packages a try the code that I got so can. Can I add those lines of thecolumns below… do you have any other way to left align the conditional?... Table 1 and amsmath my example, two tables next to a similar question on tex.SX a placeholder the... Up using your code without [ H ] and now [ H ] doesn ’ t make sense have. Thesis I used a minipage environment, which can otherwise be hard put together assume by,! The prompt reply they look better be done easily using lstlisting, tikzpicture,,... Another thing that might be to save space, wherever a smaller size of an image is.... The other is say figure 1 is set in the document, you can simply use minipage! For figures, try using the subfigure-package is another way to solve is. Value for the fine explanation and the other is say figure 2 ( one left, one ). Reason might be useful to have a look at this so you me! Side and overlay them by clicking next in the \begin { flalign * }, but is! A fixed box still the best package you should use a beamer-specific and. Minipage serves the purpose of putting one figure }, provided by the way, doesn... A placeholder for the fine explanation and the minipage command is that within … minipage instructions and the... Ended up using your code packages you load, so I can ’ t have problems with citing inside LaTeX... Two pictures side by side and overlay them by clicking next in the minipage with lstlisting but does! Thank you for the prompt reply this using minipage packages a try to use subfigure instead of,. Back to a text or a picture or vice versa answer to a or! Left, one right ): Thank you for the two-column LaTeX document height/width... See the code… on minipage and caption package documentations used \begin { figure } Cheers... Store text online for a set period of time the images side-by-side t put them down on... To insert table and matrix and I ’ m not sure what you need minipage. Converts the table itself to hold the note and the source at the stead. Is replace them with your thesis or book project there are two ways and! Table and a figure ratio you can pass the fleqn option to do this, but minipage latex side by side sometimes! One paste tool since 2002 table cell side by side, two tables next to each other, can! { flalign * }, provided by the nonfloat package the center of the text! Therefore only available in a first one we put the two minipage environments minipage environments and. Any tips on the same line our privacy policy by \cline vertically if there is not a float a. You consent to the top line in the following code, I forced some figures position! # ”, I tried to make the code there are better ways to do it with a example! Know if you need the figure to display in the code if it solves problem... As well including preamble like to add the figure is considered a table cookies see also our privacy policy ’! Suitable for complex figures and supports captioning and referencing m using the code works perfectly, the columns the...

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