This paper presents a comprehensive study aimed at understanding the dynamics of single-vehicle and multi-vehicle crashes through a binary classification approach. By harnessing high-resolution, multi-source data, including high-resolution traffic profile data captured by weigh-in-motion stations, weather conditions, roadway attributes, and pavement properties, we delved into distinctive characteristics of the two crash types. Particularly, a meticulous data fusion approach was applied to integrate the diverse data sources, enabling a holistic investigation of influential factors. Framing it as a classification task, key factors differentiating between single-vehicle and multi-vehicle crashes were identified. The results of the study provide valuable insights into the underlying mechanisms of the two distinct crash types, supporting the development of targeted safety measures.
Citation: Hao Zhen, Oscar Lares, Jeffrey Cooper Fortson, Jidong J. Yang, Wei Li, Eric Conklin. Unraveling the dynamics of single-vehicle versus multi-vehicle crashes: a comparative analysis through binary classification[J]. Applied Computing and Intelligence, 2024, 4(2): 349-369. doi: 10.3934/aci.2024020
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This paper presents a comprehensive study aimed at understanding the dynamics of single-vehicle and multi-vehicle crashes through a binary classification approach. By harnessing high-resolution, multi-source data, including high-resolution traffic profile data captured by weigh-in-motion stations, weather conditions, roadway attributes, and pavement properties, we delved into distinctive characteristics of the two crash types. Particularly, a meticulous data fusion approach was applied to integrate the diverse data sources, enabling a holistic investigation of influential factors. Framing it as a classification task, key factors differentiating between single-vehicle and multi-vehicle crashes were identified. The results of the study provide valuable insights into the underlying mechanisms of the two distinct crash types, supporting the development of targeted safety measures.
Let
The classical results of Lefschetz [13] say that the rational map associated to
There have been several improvements in this direction, by work of several authors, for instance [15] showed that for
Now, by adjunction, the canonical sheaf of
We succeed in this paper to find (respectively: conjecture) simple results for general such hypersurfaces.
Our work was motivated by a theorem obtained by the first author in a joint work with Schreyer [4] on canonical surfaces: if we have a polarization of type
The connection of the above result with the Lefschetz theorems is, as we already said, provided by adjunction, we have the following folklore result, a proof of which can be found for instance in [6] (a referee pointed out that the proof in the case of a principal polarization appears in 2.10 of [12], and that of course Green's proof works in general)
Lemma 1.1. Let
Let
If
(θ2,…,θd,∂θ1∂z1,…,∂θ1∂zg). |
Hence first of all the canonical map is an embedding if
This is our main result:
Theorem 1.2. Let
Then the canonical map
The first observation is: the hypothesis that we take a general such pair, and not any pair, is necessary in view of the cited result of [4].
The second observation is that the above result extends to more general situations, using a result on openness of birationality (this will be pursued elsewhere). This allows another proof of the theorem, obtained studying pull-backs of Theta divisors of hyperelliptic curves (observe that for Jacobians the Gauss map of the Theta divisor is a rational map, see [7] for a study of its degree).
Here we use the following nice result by Olivier Debarre [8]:
Theorem 1.3. Let
1. The branch divisor
2. The ramification divisor of
3. the local monodromies of the covering at the general points of
4. the Galois group of
The next question to which the previous result paves the way is: when is
An elementary application of the Severi double point formula [19] (see also [10], [2]) as an embedding obstruction, yields a necessary condition (observe that a similar argument was used by van de Ven in [20], in order to study the embeddings of Abelian varieties).
Theorem 1.4. Let
If the canonical map
d≥g+1. |
With some optimism (hoping for a simple result), but relying on the highly non-trivial positive result of the second author [6] concerning polarizations of type
Conjecture 1. Assume that
Then the canonical map
We end the paper discussing the conjecture.
We give first a quick outline of the strategy of the proof.
The first step 2.1 reduces to the case where the Pfaffian
Step 2.2 considers the particular case where
Step 2.3 shows that if
Step 2.4, the Key Step, shows that if
Step 2.5 finishes the proof, showing that, in each of the two possible cases corresponding to the subgroups of
We shall proceed by induction, basing on the following concept.
We shall say that a polarization type
Lemma 2.1. Assume that the polarization type
Proof. We let
There exists an étale covering
By induction, we may assume without loss of generality that all numbers
Then we have
Since by assumption
To contradict the second alternative, it suffices to show that the canonical system
Since then we would have as projection of the canonical map a rational map
F(x)=(x0,x1)≠F(gx)=(x0,ζx1), |
thereby separating the points of a general fibre.
Now, if there were only one non-zero eigenspace, the one for the eigenvalue
H0(X,OX(KX))=H0(X,π∗(OX′(KX′))=H0(X′,π∗π∗(OX′(KX′))= |
=⊕p1H0(X′,OX′(KX′+iη)), |
(here
But this is a contradiction, since the dimension
Here, we shall consider a similar situation, assuming that
We define
We consider the Gauss map of
f=Ψ∘ϕ, ϕ:X→Y, Ψ:Y→P. |
The essential features are that:
(i)
(ii)
Either the theorem is true, or, by contradiction, we have a factorization of
In this case
That the canonical map of
We prove here a result which might be known (but we could not find it in [1]; a referee points out that, under the stronger assumption that
Lemma 2.2. The general divisor in a linear system
Proof. In this case of a polarization of type
We consider, as before, the inverse image
The curve
The double covering
Hence
Assume now that
Hence the hyperelliptic curves in
We shall use the Grauert-Remmert [11] extension of Riemann's theorem, stating that finite coverings
Remark 1. Given a connected unramified covering
(1) Then the group of covering trasformations
G:=Aut(X→Y)≅NH/H, |
where
(2) The monodromy group
core(H)=∩γ∈ΓHγ=∩γ∈Γ(γ−1Hγ), |
which is also called the normal core of
(3) The two actions of the two above groups on the fibre over
We have in fact an antihomomorphism
Γ→Mon(X→Y) |
with kernel
(4) Factorizations of the covering
We consider the composition of finite coverings
To simplify our notation, we consider the corresponding composition of unramified covering spaces of Zariski open sets, and the corresponding fundamental groups
1→K1→H1→Γ1. |
Then the monodromy group of the Gauss map of
We shall now divide all the above groups by the normal subgroup
1→K→H→Γ, Γ=Mon(X0→P0). |
Since
● we have a surjection of the monodromy group
●
●
●
●
●
(I)
(II)
We consider now the case where there is a nontrivial factorization of the Gauss map
Define
1→K→ˆH→Γ, |
and set:
H′:=ˆH∩core(H), H″:=ˆH/H′, H″⊂SN. |
Obviously we have
● in case (I), where
(Ia)
(Ib)
● in case (II), where
(IIa)
(IIb)
(IIc)
We first consider cases b) and c) where the index of
Lemma 2.3. Cases (b) and (c), where the degree
Proof. Observe in fact that
X→Y→P, X→Σ→P, |
hence we have that the degree of
(2p)N=m(N2pm). |
Consider now the respective ramification divisors
Since
Since the branch locus is known to be irreducible, and reduced, and
From the equality
To show that these special cases cannot occur, we can use several arguments.
For the case
For the case
Z:=Σ×PY, |
so that there is a morphism of
If
Remark 2. Indeed, we know ([4]) that the monodromy group of
1→K→K1→H→Γ, |
where
At any rate, if
While, if
Excluded cases (b) and (c), we are left with case (a), where
H″⊂SN−1⇒ˆH⊂H, |
equivalently
Here, we can soon dispense of the case
For the case where
r(θi)=ζiθi. |
If
r(s(θi))=s(r−1(θi))=s(ζ−1θi)=ζ−1s(θi), |
hence we may assume without loss of generality that
s(θi)=θ−i, −i∈Z/p. |
It is then clear that
In the special case
ι(z)=−z+η. |
If for a general deformation of
Theorem 3.1. Let
Assume moreover that
If
d≥n+2=g+1. |
Proof. Assume the contrary,
Observe that the pull back of the hyperplane class of
The Severi double point formula yields see ([10], also [5])
m2=cn(Φ∗TP2n−TX), |
where
By virtue of the exact sequence
0→TX→TA|X→OX(X)→0, |
we obtain
m2=[(1+X)2n+2]n=(2n+2n)Xn+1⇔d(n+1)!=m=(2n+2n). |
To have a quick proof, let us also apply the double point formula to the section of
In view of the exact sequence
0→TY→TA|Y→OY(X)n−d+2→0, |
we obtain
m2=[(1+X)n+d+1Xn−d+1]n=(n+d+1d−1)Xn+1 |
equivalently,
d(n+1)!=m=(n+d+1d−1). |
Since, for
(n+1)(n+1)!=(2n+2n)⇔(n+2)!=(2n+2n+1). |
We have equality for
(n+3)(n+2)=n2+5n+6>2(2n+3)=4n+6. |
We are done with showing the desired assertion since we must have
Recall Conjecture 1:
Conjecture 2. Assume that
Then the canonical map
The first observation is that we can assume
The second remark is that we have a partial result which is similar to lemma 2.1
Lemma 4.1. Assume that the polarization type
Proof. As in lemma 2.1 we reduce to the following situation: we have
Since by assumption
Recalling that
H0(X,OX(KX))=⊕p1H0(X′,OX′(KX′+iη)), |
(here
Under our strong assumption
Already in the case of surfaces (
Were our conjecture too optimistic, then the question would arise about the exact range of validity for the statement of embedding of a general pair
the first author would like to thank Edoardo Sernesi and Michael Lönne for interesting conversations. Thanks to the second referee for useful suggestions on how to improve the exposition.
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