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# Change the order of digits and symbols in the output expression

Posted 13 days ago
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 In the following example: In[454]:= SGGenSet229me[[1,1;;3]] % . {x,y,z,1} Out[454]= {{0, -1, 0, 1/4}, {0, 0, -1, 1/4}, {-1, 0, 0, 1/4}} Out[455]= {1/4 - y, 1/4 - z, 1/4 - x} The desired result is as follows: {-y + 1/4, -z + 1/4, -x + 1/4}. How can I achieve this goal?Regards,Zhao
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Posted 13 days ago
 The closest I can easily get to the display you want is TraditionalForm[-1*{1/4 - y, 1/4 - z, 1/4 - x}] but the output of TraditionalForm is for display only, do not try to do further calculations on the result, use -1*{1/4 - y, 1/4 - z, 1/4 - x} if you must do further calculations on this.
Posted 13 days ago
 I want to get the final result in string format.
Posted 13 days ago
 ToString[TraditionalForm[-1*{1/4 - y, 1/4 - z, 1/4 - x}]] but that does not keep the signs that you had.Trying to fight the default behavior seems to usually be challenging.
Posted 13 days ago
 When you have two "contexts" (e.g. logic and presentation), build two separate families of functions. When doing computations with your structures, don't mess with their representations--let WL do the work. When you want to present your structures, define your own presentation functions. So, if you want a string, you need to determine rules for what a correctly formatted string representation of your structure is. From those rules, you can build the necessary functions. For example, if I just make guesses as to your presentation rules, I might come up with this: TermString[term_Plus] := StringRiffle[ ToString@*InputForm /@ SortBy[List @@ term, FreeQ[x | y | z]], " + "]; CoordinatesString[coords : {_, _, _}] := StringRiffle[TermString /@ coords, {"{", ", ", "}"}] Trying it out: CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x}] (* "{-y + 1/4, -z + 1/4, -x + 1/4}" *) I haven't tested this much, and of course I don't know your exact presentation rules, so you'll probably need to adapt this.
Posted 13 days ago
 I want to make it suitable for more variables, as shown below: In[44]:= TermString[term_Plus] := StringRiffle[ ToString@*InputForm /@ SortBy[List @@ term, FreeQ[x | y | z|t|u|v]], " + "]; CoordinatesString[coords : {_, _, _}] := StringRiffle[TermString /@ coords, {"{", ", ", "}"}] CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x}] CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x, 2-t}] CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x, 1/2-t,2/3+u}] CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x, 1/2-t,2/3+u,4/5+v}] Out[46]= "{-y + 1/4, -z + 1/4, -x + 1/4}" During evaluation of In[46]:= StringRiffle::string: String expected at position 2 in StringRiffle[{-y + 1/4,-z + 1/4,-x + 1/4,-t + 2},{{,, ,, ,}}]. Out[47]= StringRiffle[{"-y + 1/4", "-z + 1/4", "-x + 1/4", "-t + 2"}, {"{", ", ", ", ", "}"}] Out[48]= CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x, 1/2 - t, 2/3 + u}] Out[49]= CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x, 1/2 - t, 2/3 + u, 4/5 + v}] It seems that the following method works well: In[171]:= {{1/4 - y, 1/4 - z, 1/4 - x,1/2-t},{1/4 - y, 1/4 - z, 1/4 - x,1/2-t,3/4+u,5/6+v}} (ToString[#[[2]],InputForm]<>"+"<>ToString[#[[1]],InputForm]&/@List@@@#)&/@%//StringRiffle[#,","]&//StringReplace[#,{"("->"",")"->"","},"->"\n","{"->"","}"->""}]& Out[171]= {{1/4 - y, 1/4 - z, 1/4 - x, 1/2 - t}, {1/4 - y, 1/4 - z, 1/4 - x, 1/2 - t, 3/4 + u, 5/6 + v}} Out[172]= "-y+1/4, -z+1/4, -x+1/4, -t+1/2 -y+1/4, -z+1/4, -x+1/4, -t+1/2, u+3/4, v+5/6" However, when I further tried it on my actual data, I still didn't get the expected results: gensFindGroup//ClearAll; gensFindGroup[{m__?MatrixQ} | m__?MatrixQ]:=Block[{gens,dim,vec,vars,x,y,z,t,u,v,gensop}, gens={m}; dim=Dimensions[gens][[2]]-1; vars = {x, y, z, t, u, v}; vec = Append[Take[vars, dim], 1]; (*vec = {x, y, z, t, u, v}[[;; dim]]~Join~{1}*) (*Table[(i[[1;;3,1;;3]] . Transpose[{{x,y,z}}] + i[[1;;3,4]])//Transpose,{i,gens}]//ArrayFlatten[#,1]&//ToString[#,InputForm]&//StringReplace[#,{"("->"",")"->""}]&;*) (*Plus@@{ArrayFlatten[#[[1;;3,1;;3]] . Transpose[{{x,y,z}}],1],#[[1;;3,4]]}&/@gens// InputForm // ToString //StringReplace[#,{"("->"",")"->""}]&*) gensop=Map[Dot[#, vec] &, gens[[All, 1 ;; dim]], {-2}]; (ToString[#[[2]],InputForm]<>"+"<>ToString[#[[1]],InputForm]&/@List@@@#)&/@gensop//StringRiffle[#,","]&//StringReplace[#,{"("->"",")"->"","},"->"\n","{"->"","}"->""}]& ] In[318]:= "gensSGc2false:=[[[0,-1,0,1/4],[0,0,-1,1/4],[-1,0,0,1/4],[0,0,0,1]],[[-1,2,-1,-3/8],[-3/2,3/2,-1/2,-1/16],[-1/2,3/2,-3/2,-17/16],[0,0,0,1]]]"//StringReplace[#,{"["->"{","]"->"}"}]&//ToExpression gensSGc2false//gensFindGroup Out[319]= "-y+1/4, -z+1/4, -x+1/4 -x+-3/8, -3*x/2+-1/16, -1/2*x+-17/16" The expected results should be as follows: -y + 1/4, -z + 1/4, -x + 1/4 -x + 2*y - z -3/8, - 3*x/2 + 3*y/2 - z/2 -1/16, - x/2 + 3*y/2 - 3*z/2 -17/16
Posted 12 days ago
 TermString[term_Plus] := StringRiffle[ToString@*InputForm /@ SortBy[List @@ term, NumericQ], " + "]; CoordinatesString[coords : {__}] := StringRiffle[TermString /@ coords, {"{", ", ", "}"}] CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x}] (* "{-y + 1/4, -z + 1/4, -x + 1/4}" *) CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x, 2 - t}] (* "{-y + 1/4, -z + 1/4, -x + 1/4, -t + 2}" *) CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x, 1/2 - t, 2/3 + u}] (* "{-y + 1/4, -z + 1/4, -x + 1/4, -t + 1/2, u + 2/3}" *) CoordinatesString[{1/4 - y, 1/4 - z, 1/4 - x, 1/2 - t, 2/3 + u, 4/5 + v}] (* "{-y + 1/4, -z + 1/4, -x + 1/4, -t + 1/2, u + 2/3, v + 4/5}" *) As I already said, you need to be able to rigorously articulate what your presentation rules actually are.As for your gensFindGroup, I have no idea where you're going with this. Please stop moving the goalposts.
Posted 12 days ago
 Thank you very much, Eric, As for your gensFindGroup, I have no idea where you're going with this. Please stop moving the goalposts. Sorry, I digress too much. But the question discussed here is really a part of the question below.Transform the affine matrix of space group elements to the desired generators required by Bilbao Crystallographic Server IDENTIFY GROUP , as shown below:Based on your suggested approach, I present the following complete example: In[228]:= TermString[term_Plus] := StringRiffle[ToString@*InputForm /@ SortBy[List @@ term, NumericQ], " + "]; CoordinatesString[coords : {__}] := StringRiffle[TermString /@ coords, {"{", ", ", "}"}] gens={{{0, -1, 0, 1/4}, {0, 0, -1, 1/4}, {-1, 0, 0, 1/4}, {0, 0, 0, 1}}, {{-1, 2, -1, -(3/8)}, {-(3/2), 3/2, -(1/2), -(1/16)}, {-(1/2), 3/2, -(3/2), -(17/16)}, {0, 0, 0, 1}}}; dim=Dimensions[gens][[2]]-1; vars = {x, y, z, t, u, v}; vec = Append[Take[vars, dim], 1]; Map[Dot[#, vec] &, gens[[All, 1 ;; dim]], {-2}]; CoordinatesString/@%; StringReplace[#,{"("->"",")"->"","+ -"->"- "}]&/@%; StringJoin[%]//StringReplace[#,{"}{"->"\n","{"->"","}"->""}]& Out[237]= "-y + 1/4, -z + 1/4, -x + 1/4 -x + 2*y - z - 3/8, -3*x/2 + 3*y/2 - 1/2*z - 1/16, -1/2*x + 3*y/2 + \ -3*z/2 - 17/16"


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# Emperor's Python API¶

This notebook demonstrate Emperor's new Python API, which can and will change as we continue to exercise this interface, for more information, have a look at the pull request here.

In [ ]:
import pandas as pd, numpy as np

from emperor import Emperor, nbinstall

from skbio import OrdinationResults
from skbio.io.util import open_file
from skbio.stats.composition import clr, centralize, closure

from scipy.spatial.distance import euclidean

nbinstall()

_mf = pd.read_csv(fn, sep='\t', dtype=str, keep_default_na=False, na_values=[])
_mf.set_index(index, inplace=True)
return _mf

In this notebook we are going to showcase how to visualize a biplot using Emperor. To exemplify this, we are going to load data from Reber et al. 2016 (the data was retrieved from study 1634 in Qiita, remember you need to be logged in to access the study). Specifically, here we will reproduce Figure S4.

We start by loading the sample metadata and a BIOM table that has already been rarefied to an even depth of 20,000 sequences per sample (this table was generated using a closed reference protocol).

In [ ]:
bt = load_table('ptsd-mice/table.biom')

Next we are going to create a table of metadata for the bacteria represented in this table. In this example we are only going to use the taxonomic information, but you could add any additional information that you have access to. Note that we only use the genus level ('taxonomy_5') as our category to collapse the OTUs.

In [ ]:
feature_mf = bt.metadata_to_dataframe('observation')
feature_mf = feature_mf.reset_index(drop=True).drop_duplicates(subset=['taxonomy_5']).copy()

feature_mf.set_index('taxonomy_5', inplace=True, )
feature_mf.index.name = 'FeatureID'

In the original figure, the authors created the ordination based on a table collapsed at the genus level.

In [ ]:
collapse_genus = lambda id_, x: x['taxonomy'][5]

bt = bt.collapse(collapse_genus, norm=False, min_group_size=1,
axis='observation')

Lastly, we compute a compositional Principal Components Analysis ordination and select only the 10 most important features (meaning that in the plot we will only see 10 arrows).

In [ ]:
table = bt.to_dataframe()

mat = clr(centralize(closure(table.T + 1)))
u, k, v = np.linalg.svd(mat)
N = len(u)

DIMENSIONS = 5
_k = k[:DIMENSIONS]

# scale U matrix wrt to sqrt of eigenvalues
u = u[:,:DIMENSIONS] * np.sqrt(N-1)
# scale V matrix wrt to sqrt of eigenvalues
v = np.multiply(v[:DIMENSIONS,:],(_k.reshape(DIMENSIONS,1) / np.sqrt(N-1)))

axes = ['CPCA %d' % i for i in range(1, DIMENSIONS + 1)]

samples = pd.DataFrame(u, index=table.columns, columns=axes)
features = pd.DataFrame(v.T, index=table.index, columns=axes)

features['importance'] = features.apply(lambda x: euclidean(np.zeros_like(x), x), axis=1)
features.sort_values('importance', inplace=True, ascending=False)
features.drop(['importance'], inplace=True, axis=1)

# only keep the 10 most important features, change this number to see more arrows
features = features[:10]

res = OrdinationResults(
short_method_name='CPCA',
long_method_name='Compositional Principal Component Analysis',
eigvals=pd.Series(_k, index=axes),
samples=samples,
features=features,
proportion_explained=_k /_k.sum()
)

The figure below will display the feature and sample data. You can go to Color, select taxonomy_1 (this will color the arrows at the phylum level) and then select collection_day_fixed to color the samples by collection day (we recommend that you use a continuou color mapping, for example Viridis).

In [ ]:
Emperor(res, mf, feature_mapping_file=feature_mf, remote=False)

Emperor(res, mf, remote=False)

format: plain


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## anonymous 4 years ago sole this one if you canfind the equation of the line through (2,-3,5) and parallel to 5i-j+2k

1. Mertsj

$x=x _{0}+at$ $y=y _{0}+at$ $z=z _{0}+at$ $r _{0}=2i-3j+5k$ $v=5i-j+2k$ $r=(2i-3j+5k)+t(5i-j+2k)$ $r=(2+5t)i+(-3-t)j+(5+2t)k$

2. Mertsj

Parametic equations are: $x=2+5t$ $y=-3-t$ $z=5+2t$

3. Mertsj

$\frac{x-2}{5}=\frac{y+3}{-1}=\frac{z-5}{2}$


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Primitivity, Uniform Minimality, and State Complexity of Boolean Operations

# Primitivity, Uniform Minimality, and State Complexity of Boolean Operations

Sylvie Davies University of Waterloo
Department of Pure Mathematics
11email: sldavies@uwaterloo.ca
###### Abstract

A minimal deterministic finite automaton (DFA) is uniformly minimal if it always remains minimal when the final state set is replaced by a non-empty proper subset of the state set. We prove that a permutation DFA is uniformly minimal if and only if its transition monoid is a primitive group. We use this to study boolean operations on group languages, which are recognized by direct products of permutation DFAs. A direct product cannot be uniformly minimal, except in the trivial case where one of the DFAs in the product is a one-state DFA. However, non-trivial direct products can satisfy a weaker condition we call uniform boolean minimality, where only final state sets used to recognize boolean operations are considered. We give sufficient conditions for a direct product of two DFAs to be uniformly boolean minimal, which in turn gives sufficient conditions for pairs of group languages to have maximal state complexity under all binary boolean operations (“maximal boolean complexity”). In the case of permutation DFAs with one final state, we give necessary and sufficient conditions for pairs of group languages to have maximal boolean complexity. Our results demonstrate a connection between primitive groups and automata with strong minimality properties.

## 1 Introduction

Formal definitions are postponed until later.

The state complexity of a regular language is the minimal number of states needed to recognize the language with a deterministic finite automaton. It is well-known that if and are regular languages over a common alphabet with state complexity and respectively, then the state complexity of is at most , and this bound is tight for all . The upper bound follows from the standard “direct product” automaton construction for recognizing unions of regular languages. Examples which meet the bound were given by Maslov in 1970 [15], and independently by Yu, Zhuang and Salomaa in 1994 [21], who noted that the same bound holds for intersection.

More generally, if is a binary boolean operation on languages over , then has state complexity at most , and this bound is tight for all if and only if is proper, that is, not a constant function ( or ) or a function that depends on only one argument (for example, ). This was proved by Brzozowski in 2009 [5], who gave examples showing that is a tight bound for symmetric difference, and noted that the examples for union and symmetric difference (together with their complements) suffice to prove is a tight upper bound for all proper binary boolean operations.

To prove a lower bound on the worst-case state complexity of a regular operation, it suffices to give just one family of examples that meet the bound. Such families are called witnesses. Witnesses are known for most commonly used unary and binary operations on regular languages. However, there are several directions of research in state complexity which necessitate finding new witnesses for operations that have previously been studied. For example, sometimes the first witnesses found for an operation are not optimal in terms of alphabet size, so researchers will look for new witnesses over a smaller alphabet. When studying the -ary versions of binary operations, such as the union of languages, or more generally combined operations [13, 18], such as the star of a union of languages, again new witnesses are needed. It is also interesting to consider families of languages that are simultaneous witnesses for multiple operations; it is not generally the case that a witness for one operation will work for others. Brzozowski found a family of languages which is a simultaneous witness for reversal, star, concatenation and all binary boolean operations [6]. Each of the problems just mentioned, as well as the fundamental problem of determining the worst-case state complexity of an operation, may also be studied in subclasses of the regular languages, such as the star-free languages [9] or ideal languages [8]. Often the known witnesses do not lie in the subclass, so new witnesses must be found.

In some cases, new witnesses can be found by making slight modifications to known witnesses, but this is not always successful. Furthermore, this technique does little to advance our understanding of why particular witnesses work. For these reasons, it is desirable to have results which describe the general landscape of witnesses for a particular operation. By this we mean results that give necessary conditions for witnesses, revealing common structural properties that all witnesses share, or sufficient conditions allowing one to easily generate examples of witnesses or check whether a candidate family is a witness. For example, Salomaa, Wood and Yu proved that a regular language of state complexity is a witness for the reversal operation if the transition monoid of its minimal DFA has the maximal possible size  [19]; this gives a general sufficient condition for a language to be a witness for reversal. Ideally, collecting results of this sort would eventually lead to a complete classification of witnesses for commonly used operations. In reality, we suspect the problem of fully classifying witnesses is only tractable in very special cases, but even results which take small steps in this direction can be quite useful and enlightening.

The main inspiration for this work is a paper of Bell, Brzozowski, Moreira, and Reis [3], which considers the following question: for which pairs of languages (with state complexities and respectively) does reach the maximal state complexity for every proper binary boolean operation ? Bell et al. give sufficient conditions for this to occur. The conditions are based on the transition monoids of the minimal deterministic automata of and ; essentially, if the transition monoids contain the symmetric groups and , then “usually” (i.e., excluding a known class of counterexamples) the language will have state complexity . We obtain a refinement of this result: we prove that if the transition monoids contain 2-transitive groups, then “usually” has state complexity (though our notion of “usually” is more restrictive than that of Bell et al.).

We also obtain necessary and sufficient conditions for to have state complexity in the special case where the minimal automata for and have exactly one final state, and their transition monoids contain a transitive permutation group. We can view this result as solving a particular special case of the problem of characterizing witnesses for boolean operations.

To obtain these results, we exploit a connection between a certain class of permutation groups called primitive groups, and the notion of uniformly minimal automata introduced by Restivo and Vaglica [16]. A minimal deterministic finite automaton (DFA) is uniformly minimal if it always remains minimal when the final state set is replaced by a non-empty proper subset of the state set. For a permutation DFA (that is, a DFA whose transition monoid is a permutation group), uniform minimality is equivalent to primitivity of the transition monoid. Although uniform minimality played an important role in the paper of Bell et al., this connection with primitive groups was not used in their paper. Primitive groups are an important and well-studied class of permutation groups; there are deep results on their structure, and large libraries of primitive groups are available in computer algebra systems such as GAP [14] and Magma [4]. Uniformly minimal DFAs have received comparatively little study; thus this connection has significant implications for the theory of uniformly minimal DFAs.

The paper is structured as follows. Section 2 contains background material needed to understand the paper. Section 3 discusses the relationship between primitive groups and uniformly minimal permutation DFAs. Section 4 contains our main results on witnesses for the maximal state complexity of boolean operations. Section 5 concludes the paper by giving a summary of our results and stating some open problems.

## 2 Definitions and Notation

For a function , we typically write the symbol to the right of its arguments. For example, if the image of under is , we write . Functions are composed from left to right, and composition is denoted by juxtaposition: if , then denotes the composition of and , and is an element of .

Let denote the power set of , that is, the set of all subsets of . Given we may extend by union to obtain a function (where is the closure of under union) defined by for . We denote the extension by the same symbol as the original function. Note that for convenience, we often make no distinction between an element of a set and the singleton containing the element; so and .

### 2.1 Monoids, Groups and Actions

A monoid is a set equipped with an associative binary operation and an identity element such that for all . Typically we omit the symbol for the operation; so the previous equation could be written as . For we write for the -fold product of with itself, and define for all . If for each , there exists such that , then is called a group, and is called the inverse of and denoted . The order of an element of a group is the least integer such that .

A submonoid of is a subset which is closed under and contains the identity of . If additionally is a group, it is called a subgroup of ; we write to mean that is a subgroup of . Note that we do not allow submonoids or subgroups of to have an identity element different from that of . If are elements of a group , then denotes the group generated by , the smallest subgroup of containing .

Let and be monoids with identity elements and respectively. A homomorphism from to is a function such that for all and . A bijective homomorphism is called an isomorphism, and two monoids are said to be isomorphic if there exists an isomorphism from one to the other. We write to mean that and are isomorphic. If and are groups and is a homomorphism, the kernel of is the set , that is, the set of elements of that map to the identity of . If is a group, , and for all and , we say is a normal subgroup of . A group is simple if it has no non-trivial proper normal subgroups, that is, the only normal subgroups of are itself and the trivial group (containing just the identity element of ). The kernel of a homomorphism from to another group is always a normal subgroup of . We occasionally use the following elementary facts about normal subgroups and homomorphisms:

• If is a homomorphism and is the trivial one-element subgroup of , then is injective.

• If is a surjective homomorphism and is a normal subgroup of , then is a normal subgroup of .

A monoid action of on a set is a function such that and for all and . Equivalently, it is a family of functions such that for all and is the identity map on . The map is called the action of . To simplify the notation, we often omit the action symbol and just write instead of or . Furthermore, we typically avoid assigning a symbol to the action at all; rather than “let be a monoid action of on ” we write “let be a monoid acting on ”, meaning that has a specific but nameless action on associated with it. If generates the monoid , a monoid action is completely determined by its values on elements of . If is a group, we use the term group action rather than monoid action.

Let be a group acting on . For , the stabilizer subgroup or simply stabilizer of is the subgroup of . For , the setwise stabilizer of is the subgroup . Elements of the setwise stabilizer need not fix every element of ; for example, if and then is in the setwise stabilizer of .

Let be a finite set. A function is called a transformation of . The set of all transformations of is a monoid under composition called the full transformation monoid . A submonoid of is called a transformation monoid on . The degree of a transformation monoid on is the size of . If is a transformation monoid on , the monoid action given by for , is called the natural action of . If we write for .

A bijective transformation of is called a permutation of . We can describe any particular permutation of using cycle notation as follows. For , we write for the permutation that sends to for , sends to , and fixes all other elements of . This permutation is called a cycle of length , or simply a -cycle. All permutations that are not cycles can be expressed as a product of cycles. The identity permutation is denoted by an empty cycle, i.e., “”. Cycle notation conflicts with the notation we use for ordered -tuples, but this should not cause confusion. We mainly use cycle notation when giving concrete examples of permutations.

The set of all permutations of is a subgroup of called the symmetric group . A subgroup of is called a permutation group on ; this a special type of transformation monoid and we have the same notions of degree and natural action. The alternating group is the subgroup of consisting of all permutations that can be expressed as a product of an even number of -cycles. If we write for and for .

Let be a group acting on . We say that the action of is transitive or that acts transitively on if for all , there exists such that . We say the action of is -transitive or acts -transitively on if for all pairs of -tuples , there exists such that for we have ; informally, -transitive means “transitive on -tuples”.

A non-empty set is called a block for if for all , either (equivalently, ) or . A block is trivial if it is a singleton or the entire set . We say the action of is primitive or that acts primitively on if it is transitive and all of its blocks are trivial. Equivalently, a transitive group action of is primitive if for every set with at least two elements, there exists such that .

If is a permutation group and the natural action of is transitive (-transitive, primitive), then we say is a transitive group (-transitive group, primitive group). For example, the cyclic group is a transitive group, since its natural action on is transitive. This terminology can cause confusion, since transitivity, -transitivity and primitivity are properties of actions and not groups; statements like “ is transitive” or “ is primitive” are statements about a particular action of (the natural action) rather than the abstract group itself. In particular, these properties are not preserved under isomorphism; for example, the group is not transitive, but it is isomorphic to the transitive group .

As the notions of transitivity and primitivity are central to this paper, we give numerous examples to illustrate them below.

###### Example 1

Consider the group . This group is clearly transitive, since its natural action on is transitive. However, it is imprimitive, since and are non-trivial blocks. Indeed, if we let , then and . Hence for all , we either have or , and similarly for . One may also verify that , and are non-trivial blocks, and that there are no blocks of size 4 or 5.

###### Example 2

Consider the group . This group is clearly transitive, and it is also primitive. To see this, suppose for a contradiction that is a non-trivial block. Let and let , where and are distinct elements of . Then , so we must have since is a block. Thus for each , we have , and thus . Then since , we have , and thus . By induction it follows that . We claim , which contradicts the fact that is a non-trivial block. Indeed, for , we have if and only if . Since is prime and , we see that is coprime with . Hence by elementary number theory, there exists such that and so as required. Hence for all , which proves the claim. It follows has no non-trivial blocks, and thus is primitive.

The above argument can be generalized to prove that a cyclic group is primitive if and only if is prime. If , then for each divisor of and each integer , we see that is a block. In particular, when is composite, there exists a divisor with , giving rise to a non-trivial block.

###### Example 3

Consider the group . This group is intransitive, since (for example) it does not contain a permutation mapping to . Thus it is imprimitive by definition. Alternatively, observe that and are non-trivial blocks for .

Generally an intransitive group will always have non-trivial blocks, but there is one exception: the trivial subgroup of (containing only the identity element). The natural action of this group is clearly not transitive on , but its only blocks are the trivial blocks , and . To avoid dealing with this exception, we require primitive groups to be transitive by definition.

The next example shows that we have the following hierarchy of permutation group properties:

 (2-transitive) ⇒ (primitive) ⇒ (transitive).

These implications do not reverse. Cyclic groups of composite order give examples of transitive imprimitive groups, while cyclic groups of prime order give examples of primitive, non-2-transitive groups. (For example, the group is not 2-transitive on since nothing maps the pair to the pair .)

###### Example 4

The alternating group is 2-transitive for . Indeed, given , the permutation is the product of an even number of 2-cycles, and it maps the pair to . We claim is also primitive for . To see this, first note that is a cyclic group of prime order for . For , suppose for a contradiction that is a non-trivial block. Then has at least two elements and , but is not all of . Choose . Since is 2-transitive, there exists an element which maps the pair to . Then (since and contain ), and thus since is a block. But contains and does not, which is a contradiction. Thus all blocks of are trivial, and thus is primitive. In fact, this argument shows that all 2-transitive groups are primitive.

The following fact is immediate from the definitions of transitivity and primitivity, and is frequently useful: if is a subgroup of and is transitive (primitive), then is also transitive (primitive). For example, the symmetric group is primitive for , since it contains the primitive group .

So far, we have only looked at cyclic groups and the symmetric and alternating groups. For our last pair of examples, we consider two subgroups of that are a little more interesting.

###### Example 5

Define , and , and let . We claim this group is transitive on . For and , we will write to mean . Observe that

 1c3⟶2a2⟶6c⟶3,1c⟶4c⟶5.

Thus for each , there is some group element that maps to . If maps to , then maps to . It follows for each , there is some element that maps to , and another element that maps to , giving

 ix⟶1y⟶j.

Thus is transitive. It is also imprimitive, with non-trivial blocks and . Indeed, we see that

 {1,3,5}a⟶{1,3,5},{1,3,5}b⟶{5,3,1},{1,3,5}c⟶{4,6,2}.

Hence these sets are non-trivial blocks.

###### Example 6

Define and and let . It is easy to see that this group is transitive on : just verify that can be mapped to every other element and use the argument from the previous example. This group is also primitive. To see this, first note that the subgroup acts primitively on , since it is a cyclic group of prime order. Hence a non-trivial block of cannot be a subset of , so in particular a non-trivial block of must contain . Suppose is a non-trivial block that contains ; then contains and hence . Since is non-trivial, it contains some element , and since we have . This implies , and so is trivial, which is a contradiction. Thus all blocks of are trivial, and thus is primitive.

A congruence of a monoid action of on is an equivalence relation on that is -invariant in the following sense: if is an equivalence class, then for all , there exists an equivalence class such that . In other words, if and are equivalent, then and are equivalent for all . The equality congruence in which elements are equivalent only if they are equal, and the full congruence in which all elements are equivalent, are called trivial congruences. If is a transformation monoid on , a congruence of the natural action is called an -congruence.

The notion of congruences leads to an important alternate characterization of primitivity. In the case of a permutation group on , notice that for all and , the set has the same size as . Hence a -congruence has the following property: if is an equivalence class, then for all , the set is also an equivalence class. In particular, we either have or for all ; thus the classes of -congruences are blocks.

In fact, if is transitive, then every -congruence arises from the blocks of as follows. If is a block for , the block system corresponding to is the set . As the name implies, each set in a block system is also a block for . Indeed, for all , we either have or , and in the latter case, . But is a block, so this implies and thus . Thus every set in a block system in a block, so in particular, all distinct sets in a block system are pairwise disjoint. Furthermore, since is transitive, each element of appears in at least one block of the system. It follows that block systems are partitions of , and thus equivalence relations on . It is easy to see that block systems are -invariant, and thus are -congruences.

Thus every block gives rise to a block system that is a -congruence, and every -congruence consists of blocks; it follows block systems and -congruences are one and the same if is a transitive group. If all -congruences are trivial, then all block systems of consist only of trivial blocks, and vice versa. Thus we obtain our alternate characterization of primitivity: a transitive permutation group on is primitive if and only if all -congruences are trivial.

Let us revisit some of our earlier examples of primitive and imprimitive groups in the context of this new characterization.

###### Example 7

Consider the imprimitive cyclic group of Example 1. Put an equivalence relation on by letting if and have the same parity (odd or even). Notice that for , the elements and have opposite parity. Thus is a -congruence, since if then , and so if is the equivalence class of then is also an equivalence class. In fact, the classes of are just the blocks and we found in Example 1; thus the -congruence corresponds to the block system . If we define an equivalence relation by if and are equivalent modulo , we obtain a non-trival -congruence corresponding to the block system . As for the trivial -congruences, the equality congruence corresponds to the block system containing the singletons, and the full congruence corresponds to the block system that just contains the full set .

###### Example 8

Consider the primitive cyclic group of Example 2. With the notion of -congruences, it is much easier to prove that this group is primitive. Indeed, fix a -congruence on . By -invariance, all classes of the -congruence must have the same size, say . If the congruence has classes, then we have . So is either or since is prime, which means the classes are either singletons (giving the equality congruence) or the full set (giving the full congruence). Thus all -congruences are trivial, and thus is primitive. Alternatively, we could make the same argument in terms of block systems, using the fact that all blocks in a system have the same size to show all blocks must be trivial. This argument actually shows that not only are cyclic groups of prime order primitive, but all transitive groups of prime degree are primitive (since is the degree of a permutation group on ).

### 2.2 Languages, Automata and State Complexity

Let be a finite set. The set of all finite-length sequences of elements of is called the free monoid generated by , and is denoted . In this context, elements of are called letters, and elements of are called words over . The operation of the free monoid is concatenation of words, and the identity element is the empty word of length zero. A set is called a language over , and is called the alphabet of .

We use the convention that a language is implicitly a pair , so for example, the language over alphabet and the language over alphabet are distinct. In particular, two words over different alphabets are necessarily distinct. This is similar to the convention which views two functions with different codomains as necessarily distinct.

A deterministic finite automaton (DFA) is a tuple where and are finite sets, is a monoid action, , and . The elements of are called states; the state is called the initial state and the states in are called final states. The set is the alphabet of the automaton. The monoid action is called the transition function.

Since generates , we may completely specify the action by defining the function for each . If for , then is the composition . The function is necessarily the identity map. The monoid is called the transition monoid of ; it is a submonoid of and thus has a natural action on . We call the function the action of . Under our notational conventions, we may write as or simply . We may also extend by union and apply it to subsets of the state set: for we have . We also sometimes write to mean .

A state is reachable from if for some . Two states are distinguishable by if there exists such that . We frequently use two special cases of these definitions: A state is reachable if it is reachable from the initial state , and states are distinguishable if they are distinguishable by . We say is accessible if every state is reachable (from the initial state ), and strongly connected if every state is reachable from every other state. A state is empty if for all . In a strongly connected DFA, there exists an empty state if and only if all states are empty.

Consider the following relation on : two states are related if and only if they are indistinguishable by , that is, for all we have . This is an equivalence relation on , and in fact it is an -congruence. Indeed, if and are equivalent, we have for all . So in particular, if we take for some fixed , then for all , and thus and are equivalent for all . This congruence is called the indistiguishability congruence of .

The language recognized by or simply language of is the language over . A language which can be recognized by a DFA is called a regular language. Two DFAs are equivalent if they have the same language. Two DFAs and with are isomorphic if there is a bijection such that , , and for all ; in other words, they are identical up to the naming of the states. In particular, isomorphic DFAs are equivalent.

We say is minimal if the number of states is minimal among all DFAs equivalent to . It is well-known that for each regular language , all minimal DFAs recognizing are isomorphic and hence have the same number of states. The number of states in a minimal DFA for is called the state complexity of , and is denoted . A DFA is minimal if and only if all states are reachable and all pairs of states are distinguishable.

Given a binary regular operation , the state complexity of the operation is the following function:

 (m,n)↦max{sc(Lm∘Kn):sc(Lm)=m,sc(Kn)=n}.

That is, it is the maximal state complexity of the language resulting from the operation, expressed as a function of the state complexities of the operation’s arguments.

For and , we say if for all ; we say that is an upper bound for the state complexity of the operation if , and a tight upper bound if .

In the definition of state complexity of operations, we assume that takes two languages over the same alphabet as arguments. This is justified by our view that words over different alphabets are necessarily distinct; hence, for example, the union of two languages over different alphabets would be a set containing a mixture of words over different alphabets, which is not a language. To perform such an operation, one must first convert the operands to languages over a common alphabet. This convention is very common in the literature; however, Brzozowski has recently argued this convention is unnecessary, and in fact leads to incorrect state complexity bounds for operations on languages over different alphabets, since converting the input languages to a common alphabet can change their state complexities [7]. Brzozowski introduces a distinction between restricted state complexity of operations, the traditional model in which operands must have the same alphabet, and unrestricted state complexity of operations, a new model which produces accurate state complexity bounds for operations on languages over different alphabets.

We use restricted state complexity in this paper for the following reasons. First, computing unrestricted state complexity requires using DFAs which have an empty state, and in particular are not strongly connected. In this paper, we mainly study DFAs whose transition monoids are permutation groups, which are always strongly connected. This group-theoretic focus is essential to most of our results. Working with DFAs that are not strongly connected would take us into the realm of semigroup theory, and we are unsure how much of our work would carry over. Second, restricted state complexity has been the dominant model of state complexity of operations for many years, while unrestricted state complexity is a recent generalization. Although restricted state complexity gives incorrect results when applied to languages over different alphabets, it is otherwise a correct model. We have chosen to study the simpler case of restricted state complexity in this paper and leave the more general unrestricted case for potential future work.

## 3 Primitive Groups and Uniform Minimality

A DFA is called a permutation DFA if is a permutation group on . In this case we call the transition group rather than the transition monoid. The languages recognized by permutation DFAs are called group languages.

###### Proposition 1

For a permutation DFA , the following are equivalent:

1. is accessible.

2. is strongly connected.

3. is transitive.

###### Proof

: Since is accessible, for each there exists such that . Since is a group, the element has an inverse, and thus for all we have . It follows is transitive.

: Since is transitive, for all there exists such that . This is precisely saying that is strongly connected.

The last implication is immediate. ∎

Note that holds for arbitrary DFAs, not only permutation DFAs.

Let be a DFA and let be its language. For , we write for the DFA obtained by replacing the final state set of with . We say a regular language is a cognate of if for some . We say a DFA is a cognate of if for some ; so a language is a cognate of if and only if it is recognized by a cognate of . If or , then is called a trivial cognate of , since is either or the empty language .

We say is uniformly minimal if all non-trivial cognates of are minimal. That is, we can reassign the final state set of the DFA in any non-trivial way and the new DFA will always be minimal. Equivalently, all cognates of have the same state complexity . This definition is essentially restricted to accessible DFAs, since if is not accessible, then not all states are reachable and hence no cognate of can be minimal.

Restivo and Vaglica introduced and studied uniformly minimal DFAs in [17]. Their notion of uniform minimality is almost the same as ours, except it is restricted to strongly connected DFAs. Presumably, Restivo and Vaglica were interested in DFAs that are minimal for every reassignment of initial and final states; if a DFA is not strongly connected, we can reassign the initial state to obtain a new DFA which is not accessible and hence not minimal. However, for strongly connected DFAs, the choice of initial state has no effect on minimality since every state is reachable from each possible choice of initial state. Hence we lose nothing by fixing an initial state and generalizing to accessible DFAs.

###### Remark

Restivo and Vaglica also studied uniformly minimal DFAs in [16], but they used different terminology. They used the term “almost uniformly minimal” for the notion discussed above, and used “uniformly minimal” for a stronger condition that can only be met by incomplete DFAs (which we do not discuss in this paper).

A DFA is called simple if all -congruences are trivial. Ésik proved the following result for strongly connected DFAs [17, Proposition 1]. The same proof works for accessible DFAs.

###### Proposition 2

An accessible DFA is uniformly minimal if and only if it is simple.

###### Proof

Suppose is simple, that is, all -congruences are trivial. Then in particular, for every , the indistiguishability congruence of is trivial. If the indistiguishability congruence for is the equality relation, then each state lies in its own class, so all pairs of states are distinguishable. Since is accessible, all states are reachable, and hence is minimal. If the indistinguishability congruence for is the full relation, then all states are indistinguishable. But final states are always distinguishable from non-final states, so this can only happen if all states are final () or all states are non-final (). Hence if , then is minimal, so it follows that is uniformly minimal.

Conversely, suppose is not simple, and there exists a non-trivial -congruence. Then this congruence has a class which has at least two elements, but is not all of . Let be the final state set of and let . For all , the states and both lie in the set , which is contained in some congruence class . If , then we have . If , then we have . Thus for all , we have , and so and are not distinguishable by . Hence is not uniformly minimal, since is not minimal. ∎

In the special case of permutation DFAs, we have:

###### Corollary 1

An accessible permutation DFA is uniformly minimal if and only if is primitive.

###### Proof

If is uniformly minimal, then it is simple, so all -congruences are trivial. Now, recall that a group is primitive if and only if all -congruences are trivial. Since is a group, we see that is primitive.

Conversely, if is primitive, then all -congruences are trivial. Hence is simple and hence uniformly minimal. ∎

Note that both implications in Corollary 1 are vacuously true if is not accessible: cannot be uniformly minimal, and cannot be transitive and thus cannot be primitive. Thus one can technically omit the accessible assumption.

It seems this relationship between primitivity and minimality has been overlooked until recently. Primitive groups have seen increasing application in automata theory over the past decade, particularly in connection with the classical synchronization problem for DFAs; for a survey of such work see [2]. The connection between simple DFAs and primitive groups was recently noted by Almeida and Rodaro [1]. However, primitive groups are not mentioned in Restivo and Vaglica’s work on uniformly minimal DFAs, nor in any other work on DFA minimality that we are aware of.

The wealth of results on primitive groups makes Corollary 1 quite useful for studying and constructing uniformly minimal DFAs. For example, we can use this corollary to easily prove that for each , there exists a uniformly minimal DFA with states. Restivo and Vaglica proved this using a rather complicated construction [16, Theorem 3].

###### Proposition 3

For each , there exists a uniformly minimal permutation DFA with states.

###### Proof

The symmetric group is primitive for all , and clearly for each there exists an -state DFA with transition group . For example, let be a generating set of the symmetric group and let be a DFA with states , alphabet , and transition function with for . In fact we can use a binary alphabet, since has generating sets of size two for all . ∎

This proof illustrates a technique that is very useful for producing examples of DFAs. If we have a generating set for a transformation monoid, we can construct a DFA which has that monoid as its transition monoid.

###### Example 9

Let be the DFA with alphabet defined as follows.

• The states are , the initial state is , and the final states are .

• The transformations are the permutations and .

More formally, we mean that the transition function of is given by and . However, we will generally be brief when describing DFAs, as above.

The permutations and generate the alternating group . Thus the transition group of is . We saw in Example 4 that is transitive and primitive. Hence by Proposition 1, is strongly connected, and by Corollary 1, is uniformly minimal.

A state diagram of is given in Figure 1. We can see from the diagram that is indeed strongly connected. It is tedious, but possible to verify that is uniformly minimal by checking that it is minimal with respect to every non-empty, proper subset of .

###### Example 10

Let be the DFA with alphabet , states , initial state , final states and . A diagram is in Figure 2.

The transition group of is the cyclic group of order six, which is imprimitive. We saw in Example 1 that is a block for this group. Hence for all , we either have or . Thus if , then for all we have . This means all pairs of states in are indistinguishable by , and hence is not minimal.

This argument actually shows that whenever is a non-trivial block of , the DFA is not minimal. In fact, this also holds whenever is a union of non-trivial blocks of (see Lemma 1 below).

Note that if we construct a DFA from a cyclic group of prime order, we get a uniformly minimal DFA, since cyclic groups of prime order are primitive.

There exist many infinite families of primitive groups, and hence of uniformly minimal permutation DFAs. However, there are infinitely many positive integers for which the only primitive groups of degree are and  [12, pg. 66]. Hence other infinite families of primitive groups cannot be used to construct -state uniformly minimal DFAs for every , unless we “fill in the gaps” with symmetric or alternating groups.

###### Remark

Steinberg has extended the notion of primitivity to transformation monoids [20]. Steinberg defines a transformation monoid to be primitive if there are no non-trivial -congruences. Under this definition, an accessible DFA is uniformly minimal if and only if the transition monoid is primitive. However, we have not investigated whether any of our other results that hold for primitive groups are also true for primitive monoids.

We close this section with a technical lemma that generalizes Proposition 2. If is a transformation monoid on and , we say that is saturated by an -congruence if it is a union of classes of the -congruence.

###### Lemma 1

An accessible DFA with is minimal if and only if there is no non-trivial -congruence that saturates .

It follows that if all -congruences are trivial, then is uniformly minimal. Conversely, if there is a non-trivial -congruence, then it saturates its own congruence classes and at least one class is a proper non-empty subset of , and thus is not uniformly minimal. Hence this indeed generalizes Proposition 2.

###### Proof

We prove the contrapositive: is not minimal if and only if there exists a non-trivial -congruence that saturates .

Suppose is not minimal. Then the indistinguishability congruence of is a non-trivial -congruence, since at least two states are indistinguishable. Suppose there is an indistinguishability class that is neither contained in nor disjoint from . Then there exist such that and . But then and are distinguishable by , which cannot happen since is an indistinguishability class. Thus for each indistinguishability class , we have or . Then we have , so is saturated by its indistinguishability congruence.

Conversely, let be the congruence classes of a non-trivial -congruence that saturates . Choose a congruence class of size at least two. Then for all we have for some . Since is a union of congruence classes, either or . Hence for and all , we have . It follows that states in are indistinguishable, and thus is not minimal. ∎

In the special case of permutation DFAs, this has a useful consequence.

###### Corollary 2

Let be a permutation DFA. If or , then is minimal if and only if it is accessible.

###### Proof

Recall that if is a transitive permutation group and and are classes of a -congruence, then . It follows that if , then a non-trivial -congruence cannot saturate since all the congruence classes have size at least two. Furthermore, an -congruence saturates if and only if it saturates , and if then a non-trivial -congruence cannot saturate the set of size one. Hence if is accessible, it is minimal by Lemma 1. On the other hand, if is not accessible, it cannot be minimal. ∎

## 4 Main Results

Throughout this section, and are minimal DFAs with a common alphabet . The languages of and are and , and the transition monoids are and , respectively. For , we write for and for . Sometimes we will assume and are permutation DFAs, and then we will use and for the transition groups rather than and .

### 4.1 Direct Products and Boolean Operations

The direct product of and is the DFA with state set , alphabet , transitions for each and , initial state , and an unspecified set of final states. By assigning particular sets of final states to as described below, we can recognize the languages resulting from arbitrary binary boolean operations on and .

Fix a function ; these are called binary boolean functions. For a set , let denote the characteristic function of , defined by if and otherwise. We can think of as giving the “truth value” of the proposition “  ”, where 0 is false and 1 is true. Now for and , define . Then with final states recognizes the language defined by

 L∘L′={w∈Σ∗:χL(w)∘χL′(w)=1}.

For example, if is the “logical or” function, then , since if or . Similarly, the “logical and” function gives the intersection .

We say that a boolean function (and the associated boolean operation on languages) is proper if its output depends on both of its arguments. For example, (giving ) only depends on the first argument, and (giving with ) depends on neither argument, so they are not proper. If and have state complexity and respectively, then the improper binary boolean operations have state complexity (if they are constant), state complexity (if they depend only on the first operand), or state complexity (if they depend only on the second operand). There are 16 binary boolean operations in total, and one may easily verify that 10 of them are proper.

If and have and states respectively, has states. Hence every proper binary boolean operation has state complexity bounded by . It is well-known that this bound is tight for general regular languages, and it remains tight for regular group languages. In fact, the original witnesses for union given by Maslov [15] and Yu et al. [21] are group languages, and we will demonstrate later (in Example 13) that these languages are also witnesses for all other proper binary boolean operations. We will say the pair has maximal boolean complexity if for all proper binary boolean operations .

Suppose that and . We say a subset of is -compatible if it is equal to for some proper binary boolean operation . Notice that has maximal boolean complexity if and only if is minimal for every -compatible subset of . We disallow and since then is minimal for only if and or ; these cases are uninteresting. Similarly, we exclude and . We say the pair (or the direct product ) is uniformly boolean minimal if for every pair of sets with and and every -compatible set , the DFA is minimal. In other words, if every pair of cognates of and has maximal boolean complexity.

We give an example of a pair of DFAs that are not uniformly boolean minimal, as well as a pair of DFAs that are.

###### Example 11

Define two DFAs over alphabet as follows:

• has state set , initial state , final state set , and transformations


text:
# GATE Papers >> Civil >> 2017 >> Question No 48

Question No. 48

A sluice gate used to control the flow in a horizontal channel of unit width is shown in the figure.

It is observed that the depth of flow is 1.0 m upstream of the gate , while the depth is 0.2 m downstream of the gate. Assuuming a smooth flow transition across the sluice gate, i.e, without any energy loss, and the acceleration due to gravity as 10 m/s2, the discharge (in m3/s, up to two decimal places) passing under the sluice gate is _______

##### Answer : 0.80 to 0.83

Solution of Question No 48 of GATE 2017 Civil Paper

There is no energy loss so

E1 = E2

$\begin{array}{l}{\mathrm y}_1+\frac{\mathrm v_1^2}{2\mathrm g}={\mathrm y}_2+\frac{\mathrm v_2^2}{2\mathrm g}\\1+\frac{\mathrm Q^2}{2\mathrm{gA}_1^2}=0.2+\frac{\mathrm Q^2}{2\mathrm{gA}_2^2}\\\frac{\mathrm Q^2}{2\mathrm g}\left[\frac1{\mathrm A_2^2}-\frac1{\mathrm A_2^1}\right]=1-0.2\\\frac{\mathrm Q^2}{2\times10}\left[\frac1{0.2^2}-\frac1{1^2}\right]=0.8\\\mathrm Q^2=\frac{16}{24}\\\mathrm Q^2=\frac23\\\mathrm Q=0.82\;\mathrm m^3/\mathrm s\end{array}$


text:
Bri C.

determine the dimensions (in ft) of the box that can be constructed at minimum cost. (Refer to the figure below.)

A rectangular box is to have a square base and a volume of 20 ft3. If the material for the base costs $0.19/ft2, the material for the sides costs$0.06/ft2, and the material for the top costs \$0.11/ft2, determine the dimensions (in ft) of the box that can be constructed at minimum cost. (Refer to the figure below.)

x= ____ft

y= ____ft

By:

Tutor
3.7 (3)


text:
# Dual Paraboloid shadow mapping and tesselataion issue

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I was expecting to have issue with tesselation and dual paraboloids mapping before using this technique, but such a worse result was not expected at all. Seams can be easily seen on this screenshot: And it comes with such tesselation: It's becoming even worse when light is close to shadow receivers. Just for infromation, maybe it's important, i'm using VSM and shadow map resolution is limited to 1024x1024.

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Hi! Yes, it's very hard to get high quality results. The first thing I'd point out is that your floor shouldn't need to be tesselated at all (unless it's going to be casting on something else).

That paper talks about the simple fix so that you don't need to tesselate receivers (only casters). The basic idea is that you do the WorldSpace -> Paraboloid transformation in the pixel shader during your lighting pass. That avoids having the paraboloid co-ordinates interpolated incorrectly.

You still might have some problems, though ... I can't really tell if that table is going to cast correct shadows. I think it will, but I'd like to see new screen shots after you move the shadow lookup math into the pixel shader. Good luck!

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Thanks for the link! Exactly what I was looking for. I'll post how it will work.

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Shadow lookup done correctly with transformation inside pixel shader. Issue in generation phase where I simply use HLSL's clip function to cull pixels that dont belong to needed hemisphere, looks like culling should be done using alpha test as in Brabec's paper.
Tesselating casters on hardware is not possible for me as I limited to DX9, actually dont understand their logic - hardware tesselation requires geometric shaders but if you have them you can render to cubemap in one render instead of six, than why one will be using paraboloids, as even for hemisphere light cubemap would be a better solution.
In worst case I'll be using paraboloid only for hemispherical lights, it's much faster than cubemap anyway on DX9.

[Edited by - Viik on November 29, 2008 4:03:03 PM]

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Yes, I think you should use the alpha-test technique to do the clipping. When we mentioned the hardware tesselation, we were actually talking about Xbox 360. That's still DX9 hardware (no geometry shaders), but the GPU can do automatic tesselation. For PCs, you're right that it doesn't make much sense.

Even with some of the changes you must make, it's definitely faster than using cubemaps!

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Maybe one point to note as well is that you don't even need to worry about the cracks between paraboloids if you just point one of the paraboloids downward. Then the crack will by somewhere like the equator of your scene, where it will likely be much less noticeable.

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Quote:
 Maybe one point to note as well is that you don't even need to worry about the cracks between paraboloids if you just point one of the paraboloids downward. Then the crack will by somewhere like the equator of your scene, where it will likely be much less noticeable.

That's definately an option. Especially for static lights. Just want to achive more "general" solution.
I'm getting better results with "culling" on shadow lookup phase instead of depth render phase.
Didnt used "alpha" approach like in Brabeck's paper - as I see it requires to store separate alpha on depth render phase. I'm using VSM and that's already consumes G32R32F texture, adding alpha would require 4 channel buffer and two of them would be wasted (32 bit for alpha is too much) and actually two such textures needed. So, I completely droped culling on depth render and did a proper texture lookup on shadow render, choosing proper paraboloid texture. It works quite OK, but there are still small bug:

"Small black line just under the light" - it's solely because of the choosing right "texture", it's still there with/without blur or VSM.

float3	LightVec = normalize(IN.pos - LightPos);	float3  P = mul(LightVec, (float3x3)ParaboloidBasis);	float	myDepth = length(IN.pos - LightPos);	float2	moments;	bool	branch = (P.z >= 0.0f); //forward or backward	if(branch)	{		float2 	front;			front.x = (P.x / (2*(1 + P.z))) + 0.5;			front.y = 1-((P.y / (2*(1 + P.z))) + 0.5);		float2 	forward = tex2D(ShadowSampler, front).xy;			moments = forward;	}	else	{		float2 	back;			back.x = (P.x / (2*(1 - P.z))) + 0.5;			back.y = 1-((P.y / (2*(1 - P.z))) + 0.5);		float2 	backward = tex2D(ShadowSampler, back).zw;			moments = backward;	}	float	VSM = VSMFactor(moments, myDepth);

Quote:
 ...When we mentioned the hardware tesselation, we were actually talking about Xbox 360. That's still DX9 hardware (no geometry shaders), but the GPU can do automatic tesselation. For PCs, you're right that it doesn't make much sense.Even with some of the changes you must make, it's definitely faster than using cubemaps!

Didn't realized that it's actually your article ))) Missed a last paragraph where you mention that it's form Xbox 360.
According to your article, correct blurring of paraboloid might be an expensive operation. I used guassian separate blur of 5x5 tap in texture space. If I do it in paraboloid space that would lead to at least 10 vector x matrix multiplications, to get all needed samples.

[Edited by - Viik on December 1, 2008 7:34:43 AM]

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Haha. I wouldn't worry about "correct blurring". We never implemented that. We used PCF on a previous game, and we're using VSM now, with a regular (texture-space) 5x5 separable gaussian blur. I'm not sure about that black line. Our lookup is done very similar to yours (we have both hemispheres in a single texture, and pick the correct one + lookup in the pixel shader).

Of course, we still have light cracks between the hemispheres, but we always rotate our lights so that the hemispheres point "up" and "down". Most of our lights are above the scene, so it's rare that people can see the crack. It's still on our list of things to debug, although we've tried several times without luck. =(

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Small black line was because of the black border used in sampler. Now it works like a sharm, except sometimes on depth render stage geometry from back paraboloid projected on front (i'm not using any kind of clipping), but that would be easy to fix just do clipping with small bias.
Thanks again to everybody for your help.

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Hi Viik,
How did you perform the transformation in the pixel shader?
Are you supposed to just code the world position transformation code in the pixel shader instead of the vertex shader?
When I tried this, the texture just comes out black. Btw, I am still not up to the shadow mapping phase of dual paraboloid (still just testing the reflection mapping) so forgive me if this is not in the correct topic.

 DPOutput DP_VS(float3 pos : POSITION0, float3 normal : NORMAL0, float2 texcoord: TEXCOORD0) { DPOutput output = (DPOutput)0; float4 posWorld = mul(float4(pos,1), World); float4 viewPos = mul(posWorld, View); output.Position = mul(float4(pos,1.0f), WorldViewProj); output.Texcoords = texcoord; output.Normal = mul(float4(normal,0), (float3x3)WorldInvTrans); output.WorldPos = posWorld; output.Position = output.Position/output.Position.w; output.Position.z = output.Position.z*Direction; float L = length(output.Position.xyz); output.Position = output.Position/L; output.z = output.Position.z; output.Position.z = output.Position.z + 1; output.Position.x = output.Position.x/output.Position.z; output.Position.y = output.Position.y/output.Position.z; output.Position.z = (L - 0.1f)/(1000-0.1f); output.Position.w = 1; return output; }

Thanks,

Gary

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text:
A usual precipitation reaction occurs when an aqueous solution of barium chloride is combined with one containing salt sulfate. The complete historicsweetsballroom.comical equation can be created to explain what happens, and also such an equation is valuable in make historicsweetsballroom.comical calculations.

You are watching: Barium chloride and sodium sulfate balanced equation

\<\underbrace\ceBaCl2(aq) + Na2SO4(aq) -> BaSO4(s) + 2NaCl(aq)_\textComplete historicsweetsballroom.comical Equation\label1\>

However, Equation $$\ref1$$ does not really represent the microscopic particles (that is, the ions) existing in the solution. Listed below is the finish ionic equation:

\< \underbrace\ceBa^2+(aq) + \overbrace2Cl^-(aq)^spectator + \overbrace2Na^+(aq)^spectator + SO4^2-(aq) -> BaSO4(s) + \overbrace2Na^+(aq)^spectator + \overbraceCl^-(aq)^\textspectator_\textComplete Ionic Equation\label2\>

Equation $$\ref2$$ is quite cumbersome and also includes so countless different ion that it may be confusing. In any case, us are regularly interested in the independent behavior of ions, not the specific compound from which they came. A precipitate of $$\ceBaSO4(s)$$ will form when any type of solution comprise $$\ceBa^2+(aq)$$ is combined with any solution comprise $$\ceSO4^2–(aq)$$ (provided concentrations space not exceptionally small). This happens individually of the $$\ceCl^–(aq)$$ and $$\ceNa^+(aq)$$ ion in Equation $$\ref2$$. These ion are called spectator ions because they do not take part in the reaction. Once we want to emphasize the independent habits of ions, a net ionic equation is written, omitting the spectator ions. For precipitation that $$\ceBaSO_4$$ the net ionic equation is

\<\underbrace\ceBa^2+(aq) + SO4^2-(aq) -> BaSO4(s)_\textNet Ionic Equation \label3 \>

Example $$\PageIndex1$$

as soon as a solution of $$\ceAgNO3$$ is included to a equipment of $$\ceCaCl2$$, insoluble $$\ceAgCl$$ precipitates. Write three equations (complete historicsweetsballroom.comical equation, finish ionic equation, and also net ionic equation) that define this process. Compose the balanced net ionic equation come describe any reactionthat occurs when the services of $$\ceNa2SO4$$ and also $$\ceNH4I$$ are mixed.

Solution

Equation TypeExample $$\PageIndex1a$$Example $$\PageIndex1b$$
Complete historicsweetsballroom.comical Equation

$$\ce2AgNO3(aq) + CaCl2(aq) -> \\ \ce2AgCl(s) + Ca(NO3)2(aq)$$

The proper states and formulas that all products are written and the historicsweetsballroom.comical equation is balanced.

$$\ceNa2SO4(aq) + NH4I2(aq) -> \\ \ce2NaI(aq) + (NH4)2SO4(aq)$$

Both assets are aqueous so there is no network ionic equation that have the right to be written.

Complete Ionic Equation

$$\ce2Ag^+(aq) + 2NO3^-(aq) + Ca^2+(aq) + Cl^-(aq) -> \\ \ce2AgCl(s) + Ca^2+(aq) + 2NO3^-(aq)$$

AgCl is a heavy so it does no break up into ions in solution.

Net Ionic Equation

$$\ceAg^+(aq) + Cl^- (aq) -> AgCl(s)$$

All spectator ions space removed.

$$\ceNaI$$ and also $$\ce(NH4)2SO4$$ space both soluble.

No network ionic equation

The incident or nonoccurrence the precipitates have the right to be supplied to recognize the presence or lack of various varieties in solution. A $$\ceBaCl2$$ solution, for instance, is regularly used as a test for the visibility of $$\ceSO4^2–(aq)$$ ions. There are numerous insoluble salt of $$\ceBa$$, however they all dissolve in dilute acid except for $$\ceBaSO4$$. Thus, if $$\ceBaCl2$$ systems is added to an unknown systems which has actually previously been acidified, the incident of a white precipitate is proof of the presence of the $$\ceSO4^2–$$ ion.

Figure $$\PageIndex1$$: The three common silver halide precipitates: $$\ceAgI$$, $$\ceAgBr$$ and also $$\ceAgCl$$ (left to right). The silver- halides precipitate the end of solution, however often kind suspensions prior to settling. (CC BY-SA 3.0; Cychr).

$$\ceAgNO3$$ services are regularly used in a similar way to test for halide ions. If $$\ceAgNO3$$ equipment is included to an acidified unknown solution, a white precipitate suggests the visibility of $$\ceCl^–$$ ions, a cream-colored precipitate suggests the existence of $$\ceBr^–$$ ions, and also a yellow precipitate suggests the presence of $$\ceI^–$$ ion (Figure $$\PageIndex1$$). Additional tests can then be made to watch whether maybe a mixture that these ion is present. Once $$\ceAgNO_3$$ is added to madness water, a white precipitate is nearly always formed. The $$\ceCl^–$$ ions in tap water normally come native the $$\ceCl2$$ i m sorry is added to municipal water provides to death microorganisms.

Exercise $$\PageIndex1$$

Write balanced net ionic equations come describe any kind of reaction that occurs when the following solutions are mixed.

See more: How To Act Like A Vampire At School, 5 Ways To Act Like A Vampire

$$\ceK2CO3 + SrCl2$$ $$\ceFeSO4 + Ba(NO3)2$$ Answer a \<\ceSr^2+(aq) + CO3^2- (aq) -> SrCO3 (s) \nonumber\> Answer b \<\ceBa^2+(aq) + SO4^2- (aq) -> Ba(SO4) (s) \nonumber\>

Precipitates are also used for quantitative analysis of solutions, that is, to determine the lot of solute or the massive of solute in a offered solution. Because that this function it is often convenient to use the very first of the three species of equations described above. Climate the rules of stoichiometry may be applied.

The historicsweetsballroom.com libraries arePowered by MindTouch®and room supported by the room of education and learning Open Textbook Pilot Project, the UC Davis Office that the Provost, the UC Davis Library, the California State university Affordable discovering Solutions Program, and Merlot. We likewise acknowledge previous national Science structure support under grant numbers 1246120, 1525057, and also 1413739. Unless otherwise noted, historicsweetsballroom.com contents is licensed byCC BY-NC-SA 3.0. Legal. Have actually questions or comments? For more information call us atinfo


text:
# PG (Placement Group) notes¶

Miscellaneous copy-pastes from emails, when this gets cleaned up it should move out of /dev.

## Overview¶

PG = “placement group”. When placing data in the cluster, objects are mapped into PGs, and those PGs are mapped onto OSDs. We use the indirection so that we can group objects, which reduces the amount of per-object metadata we need to keep track of and processes we need to run (it would be prohibitively expensive to track eg the placement history on a per-object basis). Increasing the number of PGs can reduce the variance in per-OSD load across your cluster, but each PG requires a bit more CPU and memory on the OSDs that are storing it. We try and ballpark it at 100 PGs/OSD, although it can vary widely without ill effects depending on your cluster. You hit a bug in how we calculate the initial PG number from a cluster description.

There are a couple of different categories of PGs; the 6 that exist (in the original emailer’s ceph -s output) are “local” PGs which are tied to a specific OSD. However, those aren’t actually used in a standard Ceph configuration.

## Mapping algorithm (simplified)¶

> How does the Object->PG mapping look like, do you map more than one object on
> one PG, or do you sometimes map an object to more than one PG? How about the
> mapping of PGs to OSDs, does one PG belong to exactly one OSD?
>
> Does one PG represent a fixed amount of storage space?

Many objects map to one PG.

Each object maps to exactly one PG.

One PG maps to a single list of OSDs, where the first one in the list is the primary and the rest are replicas.

Many PGs can map to one OSD.

A PG represents nothing but a grouping of objects; you configure the number of PGs you want, number of OSDs * 100 is a good starting point , and all of your stored objects are pseudo-randomly evenly distributed to the PGs. So a PG explicitly does NOT represent a fixed amount of storage; it represents 1/pg_num’th of the storage you happen to have on your OSDs.

Ignoring the finer points of CRUSH and custom placement, it goes something like this in pseudocode:

locator = object_name
obj_hash = hash(locator)
pg = obj_hash % num_pg
OSDs_for_pg = crush(pg)  # returns a list of OSDs
primary = osds_for_pg[0]
replicas = osds_for_pg[1:]

If you want to understand the crush() part in the above, imagine a perfectly spherical datacenter in a vacuum ;) that is, if all OSDs have weight 1.0, and there is no topology to the data center (all OSDs are on the top level), and you use defaults, etc, it simplifies to consistent hashing; you can think of it as:

def crush(pg):
all_osds = ['osd.0', 'osd.1', 'osd.2', ...]
result = []
# size is the number of copies; primary+replicas
while len(result) < size:
r = hash(pg)
chosen = all_osds[ r % len(all_osds) ]
if chosen in result:
# OSD can be picked only once
continue
result.append(chosen)
return result

## User-visible PG States¶

Todo

diagram of states and how they can overlap

creating

the PG is still being created

active

requests to the PG will be processed

clean

all objects in the PG are replicated the correct number of times

down

a replica with necessary data is down, so the pg is offline

recovery_unfound

recovery could not finish because object(s) are unfound.

backfill_unfound

backfill could not finish because object(s) are unfound.

premerge

the PG is in a quiesced-IO state due to an impending PG merge. That happens when pg_num_pending < pg_num, and applies to the PGs with pg_num_pending <= ps < pg_num as well as the corresponding peer PG that it is merging with.

scrubbing

the PG is being checked for inconsistencies

some objects in the PG are not replicated enough times yet

inconsistent

replicas of the PG are not consistent (e.g. objects are the wrong size, objects are missing from one replica after recovery finished, etc.)

peering

the PG is undergoing the Peering process

repair

the PG is being checked and any inconsistencies found will be repaired (if possible)

recovering

objects are being migrated/synchronized with replicas

backfill_wait

the PG is waiting in line to start backfill

incomplete

a pg is missing a necessary period of history from its log. If you see this state, report a bug, and try to start any failed OSDs that may contain the needed information.

stale

the PG is in an unknown state - the monitors have not received an update for it since the PG mapping changed.

remapped

the PG is temporarily mapped to a different set of OSDs from what CRUSH specified

deep

In conjunction with scrubbing the scrub is a deep scrub

backfilling

a special case of recovery, in which the entire contents of the PG are scanned and synchronized, instead of inferring what needs to be transferred from the PG logs of recent operations

backfill_toofull

backfill reservation rejected, OSD too full

recovery_wait

the PG is waiting for the local/remote recovery reservations

undersized

the PG can’t select enough OSDs given its size

activating

the PG is peered but not yet active

peered

the PG peered but can’t go active

snaptrim

the PG is trimming snaps

snaptrim_wait

the PG is queued to trim snaps

recovery_toofull

recovery reservation rejected, OSD too full

snaptrim_error

the PG could not complete snap trimming due to errors

forced_recovery

the PG has been marked for highest priority recovery

forced_backfill

the PG has been marked for highest priority backfill

failed_repair

an attempt to repair the PG has failed. Manual intervention is required.

## OMAP STATISTICS¶

Omap statistics are gathered during deep scrub and displayed in the output of the following commands:

ceph pg dump
ceph pg dump all
ceph pg dump summary
ceph pg dump pgs
ceph pg dump pools
ceph pg ls

As these statistics are not updated continuously they may be quite inaccurate in an environment where deep scrubs are run infrequently and/or there is a lot of omap activity. As such they should not be relied on for exact accuracy but rather used as a guide. Running a deep scrub and checking these statistics immediately afterwards should give a good indication of current omap usage.


text:
# Mathematische Annalen

## List of Papers (Total 237)

#### Equidistribution in shrinking sets and $L^4$ -norm bounds for automorphic forms

We study two closely related problems stemming from the random wave conjecture for Maaß forms. The first problem is bounding the $L^4$-norm of a Maaß form in the large eigenvalue limit; we complete the work of Spinu to show that the $L^4$-norm of an Eisenstein series $E(z,1/2+it_g)$ restricted to compact sets is bounded by $\sqrt{\log t_g}$. The second problem is quantum...

#### Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness

In this paper we analyse the well-posedness of the Cauchy problem for a rather general class of hyperbolic systems with space-time dependent coefficients and with multiple characteristics of variable multiplicity. First, we establish a well-posedness result in anisotropic Sobolev spaces for systems with upper triangular principal part under interesting natural conditions on the...

#### Misiurewicz parameters and dynamical stability of polynomial-like maps of large topological degree

Given a family of polynomial-like maps of large topological degree, we relate the presence of Misiurewicz parameters to a growth condition for the volume of the iterates of the critical set. This generalizes to higher dimensions the well-known equivalence between stability and normality of the critical orbits in dimension one. We also introduce a notion of holomorphic motion of...

#### Dynamics of transcendental Hénon maps

The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can occur for polynomials, there also exist so-called Baker domains, periodic components where all orbits converge to infinity, as well as wandering...

#### SU(2) $^2$ -invariant $G_2$ -instantons

We initiate the systematic study of $G_2$-instantons with SU(2)$^2$-symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We particularly focus on $\mathbb {R}^4\times S^3$ with its two explicitly known distinct holonomy $G_2$ metrics, which have different volume growths at infinity...

#### Moduli of non-commutative polarized schemes

We construct, using geometric invariant theory, a quasi-projective Deligne–Mumford stack of stable graded algebras. We also construct a derived enhancement, which classifies twisted bundles of stable graded $A_\infty$-algebras. The tangent complex of the derived scheme is given by graded Hochschild cohomology, which we relate to ordinary Hochschild cohomology. We obtain a...

#### Univalent completion

We show by elementary means that every Kan fibration in simplicial sets can be embedded in a univalent Kan fibration.

#### Piecewise-regular maps

Let V, W be real algebraic varieties (that is, up to isomorphism, real algebraic sets), and $X \subseteq V$ some subset. A map from X into W is said to be regular if it can be extended to a regular map defined on some Zariski locally closed subvariety of V that contains X. Furthermore, a continuous map $f :X \rightarrow W$ is said to be piecewise-regular if there exists a...

#### Spectral shift functions and Dirichlet-to-Neumann maps

The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator-valued Titchmarsh–Weyl m-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries and Schrödinger operators with compactly supported potentials. In these...

#### Rationality results for the exterior and the symmetric square L-function (with an appendix by Nadir Matringe)

Let $G=\mathrm{GL}_{2n}$ over a totally real number field F and $n\ge 2$. Let $\Pi$ be a cuspidal automorphic representation of $G(\mathbb {A})$, which is cohomological and a functorial lift from SO$(2n+1)$. The latter condition can be equivalently reformulated that the exterior square L-function of $\Pi$ has a pole at $s=1$. In this paper, we prove a rationality...

#### Relative K-stability for Kähler manifolds

We study the existence of extremal Kähler metrics on Kähler manifolds. After introducing a notion of relative K-stability for Kähler manifolds, we prove that Kähler manifolds admitting extremal Kähler metrics are relatively K-stable. Along the way, we prove a general $L^p$ lower bound on the Calabi functional involving test configurations and their associated numerical...

#### Hyperbolic geometry and moduli of real curves of genus three

The moduli space of smooth real plane quartic curves consists of six connected components. We prove that each of these components admits a real hyperbolic structure. These connected components correspond to the six real forms of a certain hyperbolic lattice over the Gaussian integers. We will study this Gaussian lattice in detail. For the connected component that corresponds to...

#### Algebraic families of Galois representations and potentially semi-stable pseudodeformation rings

We construct and study the moduli of continuous representations of a profinite group with integral p-adic coefficients. We present this moduli space over the moduli space of continuous pseudorepresentations and show that this morphism is algebraizable. When this profinite group is the absolute Galois group of a p-adic local field, we show that these moduli spaces admit Zariski...

#### Rational points of varieties with ample cotangent bundle over function fields

Let K be the function field of a smooth curve over an algebraically closed field k. Let X be a scheme, which is smooth and projective over K. Suppose that the cotangent bundle $\Omega _{X/K}$ is ample. Let $R:=\mathrm{Zar}(X(K)\cap X)$ be the Zariski closure of the set of all K-rational points of X, endowed with its reduced induced structure. We prove that for each...

#### Non-properly embedded H-planes in ${\mathbb H}^2\times {\mathbb R}$

For any $H \in (0,\frac{1}{2})$, we construct complete, non-proper, stable, simply-connected surfaces embedded in ${\mathbb H}^2\times {\mathbb R}$ with constant mean curvature H.

#### The gradient flow of O’Hara’s knot energies

Jun O’Hara invented a family of knot energies $E^{j,p}$, $j,p \in (0, \infty )$, O’Hara in Topology Hawaii (Honolulu, HI, 1990). World Science Publication, River Edge 1992. We study the negative gradient flow of the sum of one of the energies $E^\alpha = E^{\alpha ,1}$, $\alpha \in (2,3)$, and a positive multiple of the length. Showing that the gradients of these knot...

#### Covariants of binary sextics and vector-valued Siegel modular forms of genus two

We extend Igusa’s description of the relation between invariants of binary sextics and Siegel modular forms of degree 2 to a relation between covariants and vector-valued Siegel modular forms of degree 2. We show how this relation can be used to effectively calculate the Fourier expansions of Siegel modular forms of degree 2.

#### An upper bound for the pseudoisotopy stable range

We prove that the pseudoisotopy stable range for manifolds of dimension 2n can be no better than $(2n-2)$. In order to do so, we define new characteristic classes for block bundles, extending our earlier work with Ebert, and prove their non-triviality. We also explain how similar methods show that $\mathrm {Top}(2n)/\mathrm {O}(2n)$ is rationally $(4n-5)$-connected.

#### A necessary condition of potential blowup for the Navier–Stokes system in half-space

Assuming that T is a potential blow up time for the Navier–Stokes system in $\mathbb {R}^{3}_{+}$, we show that the norm of the velocity field in the Lorenz space $L^{3,q}$ with $q<\infty$ goes to $\infty$ as time t approaches T.

#### Strongly nonnegative curvature

We prove that all currently known examples of manifolds with nonnegative sectional curvature satisfy a stronger condition: their curvature operator can be modified with a 4-form to become positive-semidefinite.

#### Homotopy equivalence of nearby Lagrangians and the Serre spectral sequence

We construct using relatively basic techniques a spectral sequence for exact Lagrangians in cotangent bundles similar to the one constructed by Fukaya, Seidel, and Smith. That spectral sequence was used to prove that exact relative spin Lagrangians in simply connected cotangent bundles with vanishing Maslov class are homology equivalent to the base (a similar result was also...

#### On irreducible subgroups of simple algebraic groups

Let G be a simple algebraic group over an algebraically closed field K of characteristic $p\geqslant 0$, let H be a proper closed subgroup of G and let V be a nontrivial irreducible KG-module, which is p-restricted, tensor indecomposable and rational. Assume that the restriction of V to H is irreducible. In this paper, we study the triples (G, H, V) of this form when G is a...

#### Well-posedness of hyperbolic systems with multiplicities and smooth coefficients

We study hyperbolic systems with multiplicities and smooth coefficients. In the case of non-analytic, smooth coefficients, we prove well-posedness in any Gevrey class and when the coefficients are analytic, we prove $C^\infty$ well-posedness. The proof is based on a transformation to block Sylvester form introduced by D’Ancona and Spagnolo (Boll UMI 8(1B):169–185, 1998) which...


text:
## Precalculus (6th Edition) Blitzer

The solution set of an inconsistent system of equations is $\phi$.
A system in which there is no particular solution set for the variables is called an inconsistent system. So, in the Gaussian elimination method, a system is known to be inconsistent if there are 0's left of the line and a nonzero value to the right. Example: \begin{align} & 3x+y+3z=14 \\ & 7x+5y+8z=32 \\ & x+3y+2z=9\, \end{align} The matrix corresponding to the system of equations is as follows: $\left[ \begin{matrix} 3 & 1 & 3 & 14 \\ 7 & 5 & 3 & 32 \\ 1 & 3 & 2 & 9 \\ \end{matrix} \right]$ To obtain the desired results, using elementary row transformation, we will find the echelon form of the matrix. Apply ${{R}_{1}}\leftrightarrow {{R}_{3}}$: $\left[ \begin{matrix} 1 & 3 & 2 & 9 \\ 7 & 5 & 8 & 32 \\ 3 & 1 & 3 & 9 \\ \end{matrix} \right]$ Apply ${{R}_{3}}\to {{R}_{3}}-3{{R}_{1}}$ and ${{R}_{2}}\to {{R}_{2}}-7{{R}_{1}}$: $\left[ \begin{matrix} 1 & 3 & 2 & 9 \\ 0 & -16 & -6 & -31 \\ 0 & -8 & -3 & -13 \\ \end{matrix} \right]$ Apply ${{R}_{3}}\to {{R}_{3}}-\frac{1}{2}{{R}_{2}}$: $\left[ \begin{matrix} 1 & 3 & 2 & 9 \\ 0 & -16 & -6 & -31 \\ 0 & 0 & 0 & \frac{5}{2} \\ \end{matrix} \right]$ Apply ${{R}_{2}}\to -\frac{1}{16}{{R}_{2}}$: $\left[ \begin{matrix} 1 & 3 & 2 & 9 \\ 0 & 1 & \frac{3}{8} & \frac{31}{16} \\ 0 & 0 & 0 & \frac{5}{2} \\ \end{matrix} \right]$ From the above matrix: $0x+0y+0z=\frac{5}{2}$, which is not possible since for any values of $x,\ y$, and $z$, $0\ne \frac{5}{2}$. Thus, there is no solution to the given problem. Therefore, the solution set of an inconsistent system of equations is $\phi$.


text:
What is the opposite and reciprocal of 1503?

The recirocal is $\frac{1}{1503}$, the opposite number is $- 1503$
The reciprocal of a number $n$ is $\frac{1}{n}$
The opposite number of $n$ is $- n$


text:
# Book:John B. Conway/Functions of One Complex Variable

## John B. Conway: Functions of One Complex Variable

Published $1973$, Springer-Verlag.

### Contents

Preface
I. The Complex Number System
II. Metric Spaces and the Topology of $\C$
III. Elementary Properties and Examples of Analytic Functions
IV. Complex Integration
V. Singularities
VI. The Maximum Modulus Theorem
VII. Compactness and Convergence in the Space of Analytic Functions
VIII. Runge's Theorem
IX. Analytic Continuation and Riemann Surfaces
X. Harmonic Functions
XI. Entire Functions
XII. The Range of an Analytic Function
Appendix A: Calculus for Complex Valued Functions on an Interval
Appendix B: Suggestions for Further Study and Bibliographical Notes
References
Index
List of Symbols


text:
Physical meaning of equations of equilibrium

From the equations of equilibrium
"σy+(∂σy/∂y)*(lower case delta y)" is the force acting along y direction. can anybody explain the physical meaning of the second part of the force where we multiply delta and del?

Simon Bridge
Homework Helper
Welcome to PF;

Do you mean: $$\sigma_y + \left[\frac{\partial \sigma_y}{\partial y}\right]\delta y$$ ... in which case, "we" do not "multiply delta and del".

##\delta y## is one variable and reads: "a small change in y"
##\partial## implies a partial derivative. It means that ##\sigma_y## may depend on variables other than y
In the above expression, the partial derivative of ##\sigma_y## with respect to y is being multiplied by a small change in y. This product is added to ##\sigma_y##.
The physical meaning depends on context.

There are many equations of equilibrium: where did you find that particular expression?

Welcome to PF;

Do you mean: $$\sigma_y + \left[\frac{\partial \sigma_y}{\partial y}\right]\delta y$$ ... in which case, "we" do not "multiply delta and del".

##\delta y## is one variable and reads: "a small change in y"
##\partial## implies a partial derivative. It means that ##\sigma_y## may depend on variables other than y
In the above expression, the partial derivative of ##\sigma_y## with respect to y is being multiplied by a small change in y. This product is added to ##\sigma_y##.
The physical meaning depends on context.

There are many equations of equilibrium: where did you find that particular expression?
I know we don't multiply del and delta, typo while expressing my doubt. Thanks for explaining it. Its a part of derivation for equation of equilibrium. Book: Aircraft structures by Megson

ALSO CAN U SUGGEST ANY BOOKS TO UNDERSTAND BASICS OF CALCULUS, DIFFERENTIATION ?

Simon Bridge
Homework Helper
OK - so it's probably something to do with airflows ... in the text of the book by the equation it should have told you what the different symbols mean.
A partial derivative is not the same as "a part of derivative" - the maths of aircraft is usually very complicated.

Note: you can take an online beginning calculus course - best value for money.
I cannot possibly advise you on which one or which text will help you because I don't know your education background so far.
Have a look at: http://nrich.maths.org/4722 see if it's too easy of too hard.

OK - so it's probably something to do with airflows ... in the text of the book by the equation it should have told you what the different symbols mean.
A partial derivative is not the same as "a part of derivative" - the maths of aircraft is usually very complicated.

Note: you can take an online beginning calculus course - best value for money.
I cannot possibly advise you on which one or which text will help you because I don't know your education background so far.
Have a look at: http://nrich.maths.org/4722 see if it's too easy of too hard.
I'm doing my 3rd year Aeronautical engineering. The way I've been taught in my school and college, I can solve almost all the differential equations, I know the formulae ,partial differentiation etc. I don't know the physical meanings of those differentiations. so i want a book that can help me with understanding the applications and physical meaning.

Simon Bridge
Homework Helper
The physical meaning depends on the context ... like all maths.

For instance - if the function represents a surface, then the slope of the surface in a particular direction is the derivative of the function along that direction.

Put most simply:
The derivative is how much something is changing with the variable the derivative is taken against.
What that means depends on the something and the variable.

Simplest example is that the time-derivative of the displacement is the velocity, and the time-derivative of the velocity is the acceleration.
More tricky: the space-derivative of work is the force - which is also the time derivative of momentum.

I don't think there are any books covering this topic - it's too variable.

The physical meaning depends on the context ... like all maths.

For instance - if the function represents a surface, then the slope of the surface in a particular direction is the derivative of the function along that direction.

Put most simply:
The derivative is how much something is changing with the variable the derivative is taken against.
What that means depends on the something and the variable.

Simplest example is that the time-derivative of the displacement is the velocity, and the time-derivative of the velocity is the acceleration.
More tricky: the space-derivative of work is the force - which is also the time derivative of momentum.