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a364e41d09231a3579e4c66fb48947101804f9a6 | subsection | 41 | 45 | Derivation of optimal transverse control | Then, from Hamilton's equations, we have a two point boundary value problem\frac{\mathrm {d}}{\mathrm {d}t} \tilde{\eta }_{k_2} = \frac{\partial H_{k_2}}{\partial p_{k_2}} & = - (\kappa \tilde{k}_2^2 + \tilde{k}_2^4) \tilde{\eta }_{k_2} + \tilde{\zeta }_{k_2}, \\
- \frac{\mathrm {d}}{\mathrm {d}t} p_{k_2} = \frac{\par... | {
"cite_spans": []
} | 1805.08236 | Optimal control of thin liquid films and transverse mode effects | [
"Ruben J. Tomlin",
"Susana N. Gomes",
"Grigorios A. Pavliotis",
"Demetrios T. Papageorgiou"
] | [
"physics.flu-dyn"
] | 2,018 | en | Physics | [
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338f1284c832126f06ffb0d60709a2ec3c302675 | subsection | 42 | 45 | Derivation of optimal transverse control | To solve this two point boundary value problem, we make the ansatz thatp_{-k_2}(t) = - \gamma r_{k_2}(t)\tilde{\eta }_{k_2}(t) + q_{k_2}(t).Taking the time derivative of this and equating with the complex conjugate of (), after manipulations we arrive at& \gamma \left[ - \frac{\mathrm {d}}{\mathrm {d}t}r_{k_2} - r_{k_2... | {
"cite_spans": []
} | 1805.08236 | Optimal control of thin liquid films and transverse mode effects | [
"Ruben J. Tomlin",
"Susana N. Gomes",
"Grigorios A. Pavliotis",
"Demetrios T. Papageorgiou"
] | [
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1da40a497e78b47999e98b9545be7dcb9b6afaf7 | subsection | 43 | 45 | Estimate for proof of existence of optimal control | Here we give a derivation of inequality (REF ) used in section . Multiplying (REF ) by \eta and taking the spatial average gives the energy equation\frac{1}{2} \frac{\mathrm {d}}{\mathrm {d}t} \Vert \eta \Vert _{L_0^2}^2 = (1-\kappa ) \Vert \eta _x \Vert _{L_0^2}^2 - \kappa \Vert \eta _y \Vert _{L_0^2}^2 - \Vert \eta \... | {
"cite_spans": []
} | 1805.08236 | Optimal control of thin liquid films and transverse mode effects | [
"Ruben J. Tomlin",
"Susana N. Gomes",
"Grigorios A. Pavliotis",
"Demetrios T. Papageorgiou"
] | [
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de72e4de7111045081638ba40276dd678a41ea55 | subsection | 44 | 45 | Estimate for proof of existence of optimal control | Then, using Gronwall's inequality, we may compute& \frac{\mathrm {d}}{\mathrm {d}t} \left( \Vert \eta \Vert _{L_0^2}^2 e^{-C_2 t} \right) \le - \Vert \eta \Vert _{H_0^2}^2 e^{-C_2 t} + \Vert \zeta \Vert _{L_0^2}^2 e^{-C_2 t}, \\
\Rightarrow \quad & \Vert \eta (t) \Vert _{L_0^2}^2 e^{-C_2 t} \le \Vert v \Vert _{L_0^2}^2... | {
"cite_spans": []
} | 1805.08236 | Optimal control of thin liquid films and transverse mode effects | [
"Ruben J. Tomlin",
"Susana N. Gomes",
"Grigorios A. Pavliotis",
"Demetrios T. Papageorgiou"
] | [
"physics.flu-dyn"
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a8dd7c0beb4e4841b7c3a19b889a8f9f7272e79f | abstract | 0 | 65 | Abstract | We provide a tool how one can view a polynomial on the affine plane of
bidegree $(a,b)$ - by which we mean that its Newton polygon lies in the
triangle spanned by $(a,0)$, $(0,b)$ and the origin - as a curve in a
Hirzebruch surface having nice geometric properties. As an application, we
study maximal $A_k$-singularitie... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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292c93855f1cda04600e2fa0f433b5a1d0978e19 | subsection | 1 | 65 | Introduction | We study algebraic curves (not necessarily reduced) on the affine plane \mathbb {A}^2(\mathbb {C}) that have a singularity of type A_k, which means that there is an analytical local isomorphism such that the curve is given by y^2-x^{k+1}=0 in a neighbourhood of the singular point (c.f. Definition REF ). We ask:Question... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1017/s0305004100074144",
"end": 671,
"openalex_id": "https://openalex.org/W2016460738",
"raw": "C. T. C. Wall. Highly singular quintic curves. Mathematical Proceedings of the Cambridge Philosophical Society, 119(2):257–277, 1996.",
... | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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d88866b1c0b0b5302d58b19b0e5127f0e649f60a | subsection | 2 | 65 | Introduction | Initially, we hoped to improve this bound, but the best we get with our results is N(3,11)=17 yielding \alpha \ge \frac{12}{11}\simeq 1.09.In fact, using N(3,b) it is not possible to obtain a better lower bound than Orevkov's \alpha \ge \frac{7}{6}:
A result in knot theory by Feller about the existence of algebraic co... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.4310/cag.2016.v24.n5.a4",
"end": 559,
"openalex_id": "https://openalex.org/W1537346968",
"raw": "Peter Feller. Optimal cobordisms between torus knots. Communications in Analysis and Geometry, 24(5):993–1025, 2016.",
"source_ref_id"... | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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81f6bdeecaeb100839f68324a6379a849d137a0a | subsection | 3 | 65 | Introduction | It would be interesting to have a family of curves of bidegree (3,b) with increasing b that have maximal A_k-singularity.Moreover, in Remark REF we observe a connection to Weierstrass points on \mathbb {P}^1\times \mathbb {P}^1, recently introduced in .I thank Peter Feller for introducing knot theory and its connection... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 254,
"openalex_id": "",
"raw": "P. Aleksander Maugesten and T. Karoline Moe. Special Weierstrass points on algebraic curves in \\mathbb {P}^1\\times \\mathbb {P}^1. ArXiv e-prints, January 2018.",
"source_ref_id": "fdb0c15a3... | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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03c7f4dd88c624fc8f6bbd82620e6a4ac2f26aa8 | subsection | 4 | 65 | Preliminaries | In Section REF we recall what a Hirzebruch surface is and fix our notation.
Then, we introduce singularities of type A_k in Section REF and observe what happens when blowing up such a singularity.
We continue to provide some easy bounds in Section REF .
To conclude the preliminaries, we introduce in Section REF the not... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0a302f1dc57625ff67f0da1efe68d8b114544b46 | subsection | 5 | 65 | Hirzebruch surfaces | Let m\ge 0 be an integer.
The m-th Hirzebruch surface \mathbb {F}_m is defined to be the quotient of \left(\mathbb {A}^2\setminus \lbrace (0,0)\rbrace \right)^2 modulo the following equivalence relation on it: The two points \left((x_0,x_1),(y_0,y_1)\right) and \left((x_0^{\prime },x_1^{\prime }),(y_0^{\prime },y_1^{\p... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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95a3be911ced4326e18a1c90e64e7f24bbad05e2 | subsection | 6 | 65 | Hirzebruch surfaces | Hence we can embedd \mathbb {A}^2 into \mathbb {F}_m for example with \iota _m:\mathbb {A}^2\hookrightarrow \mathbb {F}_m, (x,y)\mapsto [x:1;y:1], as the following picture illustrates:[scale=0.9]
(-3-0.2+0,2) to (-1+0.2+0,2);
(-3+0,0-0.2) to (-3+0,2+0.2);
a) at (-2+0,2);
) [above=-0.2cm of a] x=0;
b) at (-3+0,0.5);
)... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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51776c52550d4b07e0a30490d131e989ee65147a | subsection | 7 | 65 | Singularities of type | Definition 2.2
Let C be a curve on a smooth surface.
A point s\in C is called singularity of type A_k for some integer k\ge 1 if there are local analytic coordinates in which C around s is given by the equation y^2-x^{k+1}=0.
We sometimes abuse the notation and say that a smooth point has a “singularity” of type A_0.F... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.5860/choice.43-1004",
"end": 1986,
"openalex_id": "https://openalex.org/W643171281",
"raw": "C. T. C. Wall. Singular Points of Plane Curves. London Mathematical Society Student Texts. Cambridge University Press, 2004.",
"source_ref... | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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6d4ceef7ab7e664d5f54640caa44f6b9dcd5e82b | subsection | 8 | 65 | Singularities of type | Let C\subset X be a curve reduced at s and let \tilde{C}\subset Y be its strict transform.The following are equivalent:
m_s(C)=2
\tilde{C}\cdot E=2
C has an A_k-singularity at s for some k\ge 1.
If REF holds, then the following statements hold:
\tilde{C}\cap E contains two distinct points if and only if k=1.... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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d5e46d65d1a7ac7d77d805d657487df9ae2f0198 | subsection | 9 | 65 | Singularities of type | For i=1,\ldots ,n, the exceptional divisor of the i-th blow-up is denoted by E_i.]If k=1 or k=2 we are done with applying Lemma REF once.
If k\ge 3 let n={\frac{k}{2}}\ge 2.
By applying Lemma REF n times, we get a sequence of n blow-ups as described in this lemma. Figure REF depicts the situation. | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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ca6936130f4be2765f5c32f3dd7afe1df1d6d940 | subsection | 10 | 65 | Baby bounds | As a warm-up, we give bounds for N(1,b), N(2,b), and N(3,3) in this section and remark that an irreducible curve of genus g has at most an A_{2g}-singularity (c.f. Lemma REF ).Example 2.8
Let us prove that N(1,b)=0 for all integers b.
Let F be a (reduced) polynomial of bidegree (1,b), so F=\lambda x+ G(y), where G\in ... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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9b41e12b45b090473377b010b431549192905271 | subsection | 11 | 65 | Baby bounds | This gives an A_3-singularity, as in Example REF .Corollary 2.12
N(3,3)=3.The upper bound comes from Lemma REF and the existence of such a singularity comes from Example REF . | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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6d14ec2ba318f932c0d9675d9a08f26482326c1e | subsection | 12 | 65 | Links and cofiberedness | Recall Corollary REF and Figure REF .
Instead of blowing up the singular point n times, we will do one blow-up at a time in the following way.Definition 2.13
Let m be an integer, let p\in \mathbb {F}_m be a point and let f be the fiber containing it.
A birational map \pi :\mathbb {F}_m\dashrightarrow \mathbb {F}_{m\pm... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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47065f78246d3808cd49ef859534f8894e3338c5 | subsection | 13 | 65 | Polynomial in | In this section we study polynomials F in \mathbb {A}^2 of bidegree (a,am-r) for some a,m\ge 1 and 0\le r<a and divisors C\sim aS_+ in \mathbb {F}_m.
We obtain a correspondence between such polynomials and divisors in Lemma REF , which is the main statement of this section.
As an application, we find an upper bound for... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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68c311daa6390f6420afa7976733c64c4d4ddb48 | subsection | 14 | 65 | Polynomial in | Let C\subset \mathbb {F}_m be a divisor with C\sim aS_+. Then, there exists a polynomial F (unique up to multiplication with a constant) of bidegree (a,am) such that C is its (a,am)-divisor.
Moreover, if C is irreducible, then so is F.If REF and / or REF hold, let G=\sum _{i=0}^ax_0^i\,x_1^{a-i}\,G_{m(a-i)}(y_0,y_1) be... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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-0.018568411469459534,
... | |
614c3b443b026fc399eee265ec1088d906367027 | subsection | 15 | 65 | Polynomial in | So we have(C-C^{\prime })\mid _{\iota _m(\mathbb {A}^2)}=0and because \mathbb {F}_m\setminus \iota _m(\mathbb {A}^2)=S_-\cup f holds there are some \alpha ,\beta \ge 0 such that C-C^{\prime }=\alpha S_-+\beta f\sim 0, since C\sim aS_+\sim C^{\prime }.
Hence \alpha =\beta =0 and so C=C^{\prime }, and REF is proved.Let u... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.010374382138252258,
-0.005572416353970766,
-0.02854480780661106,
-0.072559654712677,
0.01277727261185646,
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0.04180265963077545,
0.04079573228955269,
-0.00015995428839232773,
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-0.002374284202232957,
-0.017880553379654884,
0... | |
0048c1dc231fc542ce1e940b292813ac090e5fdb | subsection | 16 | 65 | Polynomial in | As G_{m(a-i)} is of degree m(a-i), this implies thaty_1^{m(a-i)-N(i)}=y_1^{r-{\frac{ir}{a}}}needs to divide G_{m(a-i)}(y_0,y_1), which is REF .For the converse direction, we assume REF and find G_{m(a-i)}=0 or G_{m(a-i)}=y_1^{r-{\frac{ir}{a}}}P_{N(i)}(y_0,y_1), where the P_{N(i)} are homogeneous polynomials of degree N... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.01969723030924797,
-0.0022866942454129457,
-0.005130281671881676,
-0.07225877791643143,
0.018202010542154312,
0.002433546120300889,
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0.022565610706806183,
0.04394114762544632,
-0.0023515380453318357,
-0.03725843131542206,
0.0008567951736040413,
-0.03120126575231552,
... | |
7a8ee0c17b262f000415db37628b60143d5d5c07 | subsection | 17 | 65 | Polynomial in | For a=3, we have m_p(C)\le C\cdot f=3S_+\cdot f=3, where f is the fiber going through p, and thus we have REF or REF .Note that for r=2 and any a\ge 3 we have \left(2-{\frac{2i}{a}}\right)_{i=0}^2=(2,2,1).Assuming REF yields y_1^2\mid G_{ma}, y_1^2\mid G_{m(a-1)} and y_1\mid G_{m(a-2)}.
On the affine chart \lbrace [x:1... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.04400799423456192,
-0.01228378526866436,
-0.005539147648960352,
-0.06695806980133057,
0.013176457956433296,
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0.017029445618391037,
-0.04068145528435707,
-0.006008373107761145,
0.023072153329849243,
0.... | |
2bb76ab16bd9091854d2fcb1fd5bf75d2d27810e | subsection | 18 | 65 | Polynomial in | Then, there exists an automorphism \alpha \in \operatorname{Aut}(\mathbb {F}_m) such that \alpha (s)=[0:1;0:1] and \alpha (t)=[0:1;1:0].Applying an automorphism of the form [x_0:x_1;y_0:y_1]\rightarrow [x_0:x_1;ay_0+by_1:cy_0+dy_1] with \left(\begin{}
a & b\\
c & d\\
\end{}\right)\in \operatorname{GL}_2(\mathbb {C}) we... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.009079796262085438,
0.004051573108881712,
-0.037265315651893616,
-0.029650496318936348,
-0.0015241086948662996,
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0.038089364767074585,
0.026430601254105568,
0.004829841665923595,
-0.007843723520636559,
-0.0034583343658596277,
-0.011468011885881... | |
60045f8090289a0067be33e02a7e5dcc98eaa52f | subsection | 19 | 65 | Polynomial in | Hence we have C\sim aS_+\subset \mathbb {F}_m irreducible and we can compute its arithmetic genusg(C) &= \frac{1}{2}C\cdot (C+K_{\mathbb {F}_m})+1\\
& = \frac{1}{2}aS_+\cdot \left((a-2)S_-+\left((a-1)m-2\right)f\right)+1\\
& = \frac{1}{2}a\left((a-1)m-2\right)+1.Lemma REF yieldsk\le 2g(C)=a\left((a-1)m-2\right)+2=(am-2... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.037709612399339676,
0.023782072588801384,
-0.036519747227430344,
-0.03115009143948555,
0.011982564814388752,
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0.02031925693154335,
-0.027595747262239456,
0.012981747277081013,
-0.014507217332720757,
0.0... | |
458712b281472ba2261de24070d28892e170976e | subsection | 20 | 65 | Polynomial in | Recall that C\sim 2\,S_+=2\,(S_-+m\,f).
We can apply Lemma REF .In case REF , we can write C_1\sim aS_-+bf with 0\le a\le 2 and 0\le b\le 2m.
If a=0 (respectively a=1), then C_1 is a fiber (respectively a section, since C_1 is irreducible and contains thusly no fibers) and therefore smooth.
So let us assume that a=2.
T... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.011095666326582432,
0.023168116807937622,
-0.051128584891557693,
-0.03794198855757713,
0.0101112499833107,
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0.01839102804660797,
-0.020665105432271957,
0.01015703659504652,
0.004578679334372282,
0.0198... | |
e009ba3cd67efb2d4f5a98912bdc4873b2238b86 | subsection | 21 | 65 | Polynomial in | If a=0 (respectively a=1), C_1 is a fiber (respectively a section) and therefore smooth.If a>1, then 0\le C_1\cdot S_-=-am+b and hence b\ge am.
So a=3 is impossible, because that would give b=3m and hence C=C_1 would be irreducible.
The only remaining possibility is a=2 and therefore C_1\sim 2S_-+bf with 2m\le b\le 3m.... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.010418959893286228,
0.03511631861329079,
-0.051438845694065094,
-0.04384200647473335,
0.0119673116132617,
0.011707982048392296,
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0.03478071466088295,
0.013630073517560959,
0.010678289458155632,
-0.013958049938082695,
0.025704167783260345,
0.033377282321453094,
0.01... | |
19b67cb56845afc485259043e4649ecea0df37ff | subsection | 22 | 65 | Polynomial in | Then, we have 0\le C_2\cdot S_-=-m+d and hence b\ge 2m and d\ge m, which implies with b+d\le 3m that b=2m and d=m.
Hence C_1\sim 2S_+, C_2\sim S_+ and so we have C_1\cdot C_2=2m and C=C_1+C_2.
So only in the latter case the equality C_1\cdot C_2=2m can occur. Assuming n=I_p(C_1,C_2)=2m, the point p is the only point i... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.014740033075213432,
0.006412524729967117,
-0.05697648599743843,
-0.037048447877168655,
-0.008529688231647015,
0.037231553345918655,
-0.009582547470927238,
0.04284680634737015,
0.015449568629264832,
-0.009124782867729664,
-0.026825029402971268,
0.015510604716837406,
0.01239780243486166,
... | |
b868133ef6f7056fbb7100f0d16595232b144271 | subsection | 23 | 65 | To Be ... | In this section the goal is to give a lower bound for N(3,b) where b\le 12, namely the existence of a polynomial of bidegree (3,b) with a certain singularity of type A_k is shown.In what follows we will not give the specific equation of a polynomial, but rather prove that a polynomial with certain properties exists.In ... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.04226521775126457,
-0.02253636345267296,
-0.02571007050573826,
-0.04479807987809181,
-0.036741748452186584,
0.020400214940309525,
0.0504130981862545,
-0.008010555990040302,
-0.01707392744719982,
-0.0188743956387043,
-0.007301049306988716,
0.05358680337667465,
-0.007598584517836571,
0.01... | |
1ddb7c63fb79f0d28129b6a813540aadb0b00077 | subsection | 24 | 65 | The recipe | We start by introducing some definitions that simplify the statements that follow.
Now is a good time to go back to take a look at Figures REF and REF .Definition 4.1
Let m\ge 0 be an integer and let C\subset \mathbb {F}_m be an effective divisor and p\in \mathbb {F}_m a point.
We say that p is a transversal point of ... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.00808725506067276,
0.03222694620490074,
-0.03643842414021492,
-0.011451858095824718,
0.00498586893081665,
-0.005615301430225372,
0.008743390440940857,
0.023392003029584885,
0.03878830373287201,
0.04666193202137947,
-0.00797281228005886,
-0.016754349693655968,
0.019531482830643654,
0.014... | |
dc2d580c1fa6521330f2d971bc5522f26be64000 | subsection | 25 | 65 | The recipe | We say that \mathcal {C} is of type -1, 0 or k\ge 1 in the following cases:If C\cap f=\lbrace p,s,t\rbrace for a point t distinct from p and s, \mathcal {C} is of type -1,
if C\cap f=\lbrace p,s\rbrace , and f and C are tangent at s, then \mathcal {C} is of type 0,
if C\cap f=\lbrace p,s\rbrace , and s is an A_k-si... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.018141690641641617,
0.015837742015719414,
-0.03970116004347801,
-0.01823323778808117,
0.024702604860067368,
0.026838716119527817,
0.041989851742982864,
0.05379949510097504,
0.017989110201597214,
0.02952411398291588,
-0.014693396165966988,
0.007903613150119781,
0.024900957942008972,
0.00... | |
99934ea7448ab866481596a82d3efb4c2d9bbbfc | subsection | 26 | 65 | The recipe | Consider a p-link \pi :\mathbb {F}_m\dashrightarrow \mathbb {F}_{m^{\prime }}.
Let D^{\prime }:=\pi _*(D). Then, D is an a-section if and only if D^{\prime } is an a-section.
If this holds, then D is irreducible if and only if D^{\prime } is irreducible.Let s^{\prime }\in \mathbb {F}_{m^{\prime }} be the inverse point ... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.02707591839134693,
0.0001526263658888638,
-0.012286422774195671,
-0.011981170624494553,
0.002020391635596752,
0.03507354110479355,
0.02171873301267624,
0.04316273704171181,
0.059493761509656906,
0.05024460330605507,
-0.05891377851366997,
0.0030887762550264597,
0.0051167989149689674,
0.0... | |
2b51e8d6e2e15aea13bfffaf361128058222305e | subsection | 27 | 65 | The recipe | To prove REF , recall that D and f intersect transversally at p, hence \tilde{D} and \tilde{f} do not meet on E, the strict transform of \rho .
Moreover, \tilde{D} intersects E transversally at a point \hat{p}\in X (not lying on the strict transform \tilde{f}). Therefore, \sigma _*(D)=D^{\prime } and \sigma _*(E)=f^{\p... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.029243700206279755,
0.007246057502925396,
-0.015110261738300323,
-0.01909387670457363,
-0.01167611125856638,
0.0369361974298954,
0.03208259865641594,
0.018269680440425873,
0.047986529767513275,
0.042491890490055084,
-0.010203242301940918,
-0.01491947565227747,
0.022421186789870262,
0.01... | |
7c4f02904202106bb305d195e70816f028a6fdc6 | subsection | 28 | 65 | The recipe | Let \mathcal {C}=\mathcal {C}_0 be an a-configuration (C_0=C,\bullet ,\bullet ,p_0=p)_{m}.
For i=1,\ldots ,n, let \pi _i:\mathbb {F}_{m_{i-1}}\dashrightarrow \mathbb {F}_{m_i} be a p_{i-1}-link with inverse point s_i.
Let \mathcal {C}_i=(C_i,\bullet ,s_i,p_i)_{m_i} be the direct image of \mathcal {C}_{i-1}.
We call the... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.03660128638148308,
0.036692868918180466,
-0.015324309468269348,
0.01191297173500061,
0.017995378002524376,
0.015301413834095001,
0.009806642308831215,
0.05253612622618675,
0.014126143418252468,
0.05253612622618675,
-0.04081394523382187,
0.007997945882380009,
0.00984480045735836,
0.01354... | |
bd64c352e60d237b50d39f8b7a6491ca518d809f | subsection | 29 | 65 | The recipe | So we have \tilde{C}^{\prime }\cap E=\tilde{C}\cap \tilde{f}.By REF of Lemma REF , we know that m_{s^{\prime }}(C^{\prime })=2, so part REF of Lemma REF is satisfied (note that C^{\prime } is a 3-section and is therefore reduced) and C^{\prime } has a singularity of type A_K at s^{\prime } for some K\ge 1.
Hence, part ... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.022745909169316292,
0.01638437621295452,
-0.024195179343223572,
-0.01571313664317131,
0.01341718714684248,
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0.03533167392015457,
0.05287546664476395,
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0.041037220507860184,
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-0.015308866277337074,
-0.01836758852005005,
0.... | |
ba8e32870763a53f05d32af799e3f20ced14266b | subsection | 30 | 65 | The recipe | (0,5);
(0,5) ..controls (-2,2) and (-2,0).. (2,3);(-2,0) ..controls (0,2.5) and (1,2).. (2,1);(-2.2,0.5) nodeS^{\prime };
(-2,5.5) nodeC^{\prime };
(0.3,-1.5) nodef^{\prime };
(0.3,1.3) nodep^{\prime };
(0.3,4.5) nodes^{\prime };
(-0.4,5.7) nodeA_1;k=0:
[xscale=0.5,yscale=0.3,baseline=(b.base)]
(1.1,-2) – (1.1, 7);
(-2... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.021150803193449974,
0.007511892355978489,
-0.05344163998961449,
0.01661848835647106,
0.026675039902329445,
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0.01025493536144495,
0.059759411960840225,
0.017091557383537292,
0.02559155598282814,
-0.05573068931698799,
0.013772438280284405,
0.004166067112237215,
0.002... | |
cf15b9b867f0584777a10da2cffacb5df3a7ebe7 | subsection | 31 | 65 | The recipe | (2,1);
(-2.2,0.5) nodeS^{\prime };
(-2,5) nodeC^{\prime };
(0.3,-1.5) nodef^{\prime };
(0.3,1.3) nodep^{\prime };
(0.3,4.5) nodes^{\prime };
(-0.7,5.7) nodeA_{k+2};Lemma 4.12
Let m\ge 0 and a\ge 1 be two integers, and let \mathcal {C}=(C,S,\bullet ,p)_m be a tangent a-configuration.
Let 1\le n\le I_p(S,C) be an intege... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.019226958975195885,
0.04834206774830818,
-0.03955260291695595,
0.01158195361495018,
0.022858718410134315,
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0.021439585834741592,
0.04834206774830818,
-0.0314040333032608,
-0.011429359205067158,
0.013733542524278164,
-0.00675... | |
49fe49771c5c028c41b59acecd3c5953c43af893 | subsection | 32 | 65 | The recipe | Hence, \tilde{C} and \tilde{S} intersect at most in \hat{p}, givingI_{\hat{p}}(\tilde{C},\tilde{S})=\tilde{C}\cdot \tilde{S}=I_p(S,C)-1,and so also I_{p^{\prime }}(C^{\prime },S^{\prime })=I_{p}(S,C), as the intersection multiplicity is a local property.To summarize the lemmas of this section, we equip a 3-configuratio... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.0021799239329993725,
0.026548152789473534,
-0.035855263471603394,
-0.005805501248687506,
0.018431132659316063,
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0.0270821675658226,
0.028806272894144058,
-0.03216293454170227,
0.0031163564417511225,
0.005179941654205322,
-0.... | |
dcdef611815b23578332fcdf18f92dfcbd56eac0 | subsection | 33 | 65 | The recipe | Let \mathcal {C}=(C,S,\bullet ,p)_m be a tangent a-configuration and let S^2\le n\le I_p(S,C) be an integer.
Assume that C^2=2an-a^2S^2. Let \pi :\mathbb {F}_{m}\dashrightarrow \mathbb {F}_{m^{\prime }} be a transversal \mathcal {C}-chain of n links and let \pi _*(\mathcal {C})=(C^{\prime },S^{\prime },\bullet ,p^{\pri... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.03816676139831543,
0.054061952978372574,
-0.03798370808362961,
-0.014934157021343708,
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0.016062989830970764,
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0.04146173223853111,
0.004622113890945911,
0.029853058978915215,
-0.030036112293601036,
-0.001041119685396552,
0.02515467256307602,
0... | |
d69a851b20472523362450db524f9236519ccb80 | subsection | 34 | 65 | The recipe | Let \pi :\mathbb {F}_{m}\dashrightarrow \mathbb {F}_{m^{\prime }} be an s-link with inverse point p^{\prime }.
Then, (\pi _*(C),\bullet ,\bullet ,p^{\prime }) is a 3-configuration, which we will call the direct image of \mathcal {C} and denote it by \pi _*(\mathcal {C}).
Moreover, \pi _*(\mathcal {C}) is of type k-2\ge... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.07805738598108292,
0.013701237738132477,
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0.023755932226777077,
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0.05379795655608177,
-0.035824619233608246,
0.03484813868999481,
0.0201093889772892,
0.024549... | |
5a930f884e76b8ebf0d3635843f7fa682d47de89 | subsection | 35 | 65 | The recipe | We say that the composition \pi =\pi _n\circ \cdots \circ \pi _1:\mathbb {F}_{m_0}\dashrightarrow \mathbb {F}_{m_n} is a singular \mathcal {C}-chain of n links and we say that \mathcal {C}_n is the direct product \pi _*(\mathcal {C}) of \mathcal {C}.
The singular \mathcal {C}-chain of n links is the inverse of a transv... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.02349521592259407,
0.026775391772389412,
-0.02557011879980564,
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0.04213882237672806,
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0.021206721663475037,
0.015615164302289486,
0.... | |
b2f05a05fbda552d1fc0f2fc34abe7939d7286d8 | subsection | 36 | 65 | The recipe | Then, we will contract the (-1)-curve and get a situation in the projective plane \mathbb {P}^2.
Finally, we will show that this situation does not exist in certain cases.The following lemma has some assumptions that are specific to the cases that we want to study.
To arrive in some \mathbb {F}_{l}, where l is odd, we ... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.06678520888090134,
0.025288641452789307,
-0.014689388684928417,
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0.0021499923896044493,
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0.016299499198794365,
-0.03568186238408089,
0.011392860673367977,
-0.02512076310813427,
0... | |
ef9cd753ef188c63f97a5fdb5af8813e28ca6512 | subsection | 37 | 65 | The ingredients | In this section we will show the existence of some tangent 3-configurations \mathcal {C}=(C,S,\bullet ,p)_m satisfying the assumptions of Lemma REF with m=0 (that is, a configuration in \mathbb {F}_0=\mathbb {P}^1\times \mathbb {P}^1) or m=1 (that is, a configuration in \mathbb {F}_1 obtained by the blow-up at one poin... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.012251624837517738,
0.018751241266727448,
-0.048548780381679535,
-0.024716852232813835,
0.01684407703578472,
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0.018858041614294052,
-0.0035930979065597057,
0.012694086879491806,
0.008429666981101036,
0... | |
493c4993bd3ee5f20937ddfa40cc6ee4d8c02155 | subsection | 38 | 65 | The ingredients | We prove that \mathcal {C}=(C,S,\bullet ,p)_0 is a tangent 3-configuration.First, we show that p is the unique intersection point of C and S.
We can parametrize S by [y_0:y_1]\mapsto ([y_0^2:y_1^2],[y_0:y_1]).
Inserting this parametrization into F, we findF(y_0^2,y_1^2,y_0,y_1)=(y_0+y_1)^7,and so C and S intersect only... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.005296111572533846,
0.041208937764167786,
-0.04334569722414017,
-0.020406054332852364,
0.029594121500849724,
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0.03409657999873161,
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-0.011034836992621422,
0.051312755793333054,
... | |
c4274692e361081dfa7c8ba486fdf4f01ddbb7ef | subsection | 39 | 65 | The ingredients | Then there exists an irreducible polynomial of bidegree (3,3m) with at least an A_{4m+1}-singularity.It can be proved by writing m=2a-1 for an integer a\ge 2 and considering the curves C=V(F) and S=V(G) in \mathbb {F}_0, whereG = & x_0y_1^a-x_1y_0^a,\\
F = & x_0^3\sum _{i=0}^{a-1}{{4a-1}{i}}y_0^{a-1-i}y_1^i + x_0^2x_1\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1273,
"openalex_id": "",
"raw": "P. Aleksander Maugesten and T. Karoline Moe. Special Weierstrass points on algebraic curves in \\mathbb {P}^1\\times \\mathbb {P}^1. ArXiv e-prints, January 2018.",
"source_ref_id": "fdb0c15a... | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.032066963613033295,
0.02015245519578457,
-0.016643699258565903,
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0.005877163726836443,
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0.023676464334130287,
0.04195249825716019,
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0.004843606613576412,
0.026407189667224884,
... | |
7e3999c0a5260f08d6acd77d3e88dca85b4d2414 | subsection | 40 | 65 | The ingredients | For instance, the curve C of Remark REF , which has degree (3,a-1), contains p as a smooth (1,a)-Weierstrass point, with hyperosculating curve S of degree (1,a).Looking ahead, the examples of tangent 3-configurations we obtain on \mathbb {F}_1 (after blowing up a situation in \mathbb {P}^2) can be transformed to a tang... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.024768706411123276,
0.016481949016451836,
-0.018755847588181496,
-0.020861875265836716,
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0.042822547256946564,
0.020907657220959663,
-0.014688774012029171,
0.032231368124485016,
... | |
80cdb8ac62dee831f96b6024a50941f65eda0431 | subsection | 41 | 65 | The ingredients | With REF we have \tilde{C}\cdot \tilde{L}=C\cdot L-m_q(C)=\deg C -m_q(C)=3, hence \tilde{C} is a 3-section.
Similarly, we find with REF that \tilde{S} is a section.
To see that \hat{p} is a transversal point of \tilde{C} we insert REF and REF into
REF and findI_p(S,C)=3\left(m_q(S)+1\right)+m_q(C)\ge 3.Since p is disti... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.04907381907105446,
0.0459914468228817,
-0.022461649030447006,
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0.01536608673632145,
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0.0012569825630635023,
0.027192022651433945,
-0.019486090168356895,
0.0021038721315562725,
0.02139350026845932,
0.01... | |
bfe7e825ced5fe8f6e627712738c894148e2cce6 | subsection | 42 | 65 | The ingredients | If we want to achieve a case with r=1 or r=2, we will also add to the situation in \mathbb {P}^2 a line T (going through q) and describe the intersection with C at a point t\in T.
This corresponds then to a situation such as REF or REF in Lemma REF , using the fact that the line T is a fiber after a blow-up.The example... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.05042431131005287,
0.026646746322512627,
-0.03858131170272827,
-0.04663943499326706,
0.003891706932336092,
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0.009057756513357162,
0.0047616176307201385,
0.025181632488965988,
0.017703451216220856,
0.0340... | |
37f7cc426ac0f4595be4c12120ef28b65af8c1b7 | subsection | 43 | 65 | The ingredients | Therefore, C^{\prime } intersects a fiber T^{\prime } at only one point t^{\prime }, giving I_{t^{\prime }}(T^{\prime },C^{\prime })=3.
So we are in case REF of Lemma REF .
Finally, REF of Lemma REF asserts that there exists a polynomial of bidegree (3,3m^{\prime }-1)=(3,5) with a singularity of type A_7.
This polynomi... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.028596211224794388,
0.011101250536739826,
-0.0503256693482399,
-0.05530025064945221,
0.02002039924263954,
0.005089026875793934,
0.005455253645777702,
-0.0010786524508148432,
0.012947644107043743,
0.004371832590550184,
-0.022431394085288048,
0.01149036642163992,
0.027024488896131516,
-0.... | |
743c7fcbad83100f9b4b4241e6a88cb8d62a22bd | subsection | 44 | 65 | The ingredients | By Lemma REF , \mathcal {C}^{\prime } is of type 2n+2=10.The largest power of y in the polynomial of C is y^2 with unique tangent direction x=0, which corresponds to T.
Hence, m_t(C)=2 and T is the unique tangent direction to C at t, and so there is a fiber T^{\prime }\subset \mathbb {F}_{m^{\prime }} containing a poin... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.010722510516643524,
0.008181161247193813,
-0.0525364875793457,
-0.05314702168107033,
0.03168673813343048,
-0.0022666091099381447,
-0.0021693052258342505,
-0.002075816970318556,
0.02208608202636242,
0.019750788807868958,
-0.013767551630735397,
0.008967224508523941,
0.008967224508523941,
... | |
82d66347965620d475f7a742e078cbd9bc68a802 | subsection | 45 | 65 | The ingredients | Hence, we can apply Lemma REF to a transversal \mathcal {C}-chain of 6 links and get a 3-configuration \mathcal {C}^{\prime }=(C^{\prime },S^{\prime },s^{\prime },p^{\prime })_{m^{\prime }} where m^{\prime }=n-\tilde{S}^2=6-3=3 and C^{\prime }\sim 3S_+\subset \mathbb {F}_3.Note that we did not give a point “s” in the l... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.013415314257144928,
0.024175036698579788,
-0.04371042177081108,
-0.030875062569975853,
0.014041056856513023,
0.019718527793884277,
0.03629307821393013,
0.02379348687827587,
0.0075546992011368275,
0.0004027741961181164,
-0.025319688022136688,
0.025106020271778107,
0.001695037935860455,
0... | |
1b2603b464b9ddc9068beff1d75cc2a003a07c1b | subsection | 46 | 65 | The ingredients | (-3,1)
.. controls (-2.6,0) and (-1,-1) .. (0.3,0.3);(-1.2,0.5) nodeS;
(-0.3,-0.6) nodeq;
(0.3,3.5) nodeL;
(0.3,2.2) nodep;
(0.3,5.2) nodes;
(-1.1,5.7) nodeC;
(-3,5.3) nodet;
(-2,3) nodeT;We want to prove that the assumptions of Lemma REF are satisfied.
The line going through p and q is L.
For REF note that p\in C is s... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.04477040469646454,
0.033600691705942154,
-0.03982643410563469,
-0.030823523178696632,
0.03698822855949402,
0.004490010906010866,
0.016907161101698875,
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0.027435988187789917,
0.007198513485491276,
-0.002973630791530013,
0.014999764040112495,
0.02432311698794365,
0.00... | |
3323e789ce4d736c0c90ec714470efad857862a7 | subsection | 47 | 65 | The ingredients | So assume that it is of degree 3.
Then, C=C_1+C_2, where C_2 is irreducible and of degree 1.
Having I_p(S,C_1)\le 6, we achieve 7=I_p(S,C)=I_p(S,C_1)+I_p(S,C_2) only if I_p(S,C_2)\ge 1. Hence, p\in C_2\,\cap \, C_1 and so p is a singular point of C. This is a contradiction to REF and therefore, C is irreducible.We have... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.018403733149170876,
0.018190091475844383,
-0.027071496471762657,
-0.019502464681863785,
0.0018693676684051752,
-0.009278167970478535,
0.030993353575468063,
0.030184565111994743,
0.013688349165022373,
0.018403733149170876,
0.0037997758481651545,
-0.00490613654255867,
0.0032923761755228043,... | |
77dce99fd8cd3d1098e0d24e17d2ca860b254f0e | subsection | 48 | 65 | The ingredients | This polynomial is irreducible by Corollary REF .Lemma 4.23 There exists an irreducible polynomial of bidegree (3,11) with a singularity of type A_{17}.Let \omega =i\sqrt{3} and letF = & {z}^{3}+
\frac{3}{8}\left(\omega +3\right)\left(y-x\right)z^2+
\frac{9}{8}\left( {\frac{\omega }{2}}+{1} \right) {x}^{2}z+
\frac{3}{6... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.019088713452219963,
-0.004238121677190065,
-0.03222651034593582,
-0.06701648980379105,
0.017364472150802612,
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0.01690671034157276,
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0.04046624153852463,
-0.0023326834198087454,
-0.04086297005414963,
0.02981562353670597,
-0.012771585024893284,
... | |
8fd03610a9a7a2d4555063471568d1bbc5d260a6 | subsection | 49 | 65 | The ingredients | We insert x=0 into C and see that it is not the zero polynomial. Therefore, C does not contain L and so does not contain any line meeting q.We have shown that the assumptions of Lemma REF are satisfied. The lemma implies that \mathcal {C}=(\tilde{C},\tilde{S},\hat{s},\hat{p})_1 is a tangent 3-configuration that satisfi... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.018123187124729156,
0.010609997436404228,
-0.0587935745716095,
-0.03642943874001503,
0.01992330141365528,
0.005545268300920725,
0.021326780319213867,
0.017665531486272812,
-0.008794627152383327,
-0.005713075399398804,
0.007192830555140972,
0.01630781777203083,
0.004908363334834576,
-0.0... | |
e4de4a5585571184faa9d0cbf4437c8b2cd2ee84 | subsection | 50 | 65 | The ingredients | This polynomial is irreducible by Corollary REF .Lemma 4.24 There exists an irreducible polynomial of bidegree (3,12) with a singularity of type A_{18}.Let \omega =i\sqrt{3} and letF= &{z}^{3}+ \frac{9}{2}\left( -1+\omega \right) {x}^{3}-9\,y{x}^{2}+9\,z{
y}^{2}+ 3\left( -\omega +3 \right) xyz-\\
& 6\,y{z}^{2}-3\,x{z}^... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.027460765093564987,
-0.00644565187394619,
-0.022578852251172066,
-0.07585273683071136,
0.014660998247563839,
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0.026926806196570396,
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-0.003137011080980301,
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0.036919474601745605,
-0.015164445154368877,
... | |
e012b44fe75e5612681bf2852693cc7ce2133363 | subsection | 51 | 65 | The ingredients | Hence, C does not contain L.We have proven that the assumptions of Lemma REF are satisfied and get a 3-configuration \mathcal {C}=(\tilde{C},\tilde{S},\hat{s},\hat{p})_{1} that satisfies \tilde{C}^2=6n-9\tilde{S}^2 and is equipped with [9,5,\bullet ,9;1].
Note that the line L is tangent to C at s, asF(0,1,z)=z(z-3)^2.H... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.034967198967933655,
0.008741799741983414,
-0.05138286575675011,
-0.037377677857875824,
0.017361551523208618,
0.007429767400026321,
0.014546782709658146,
0.005785912275314331,
-0.003823583945631981,
0.006377089768648148,
-0.008566353470087051,
0.03090905211865902,
-0.0024467124603688717,
... | |
6bc2114f38eae411787a9b4e5a87558ebe00ebfb | subsection | 52 | 65 | ... Or Not To Be | In this chapter we find an upper bound for N(3,b) for small b.
We use the “recipe” of Section REF but in converse direction. First, we verify in Section REF that we are allowed to go backwards, and then determine in Section REF that the configurations we get do not occur. | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.024550028145313263,
0.02702181413769722,
-0.011039121076464653,
0.008086709305644035,
-0.005752244032919407,
0.045926403254270554,
0.03845001384615898,
0.020384609699249268,
0.004634599667042494,
0.015990322455763817,
-0.025389214977622032,
0.03616132214665413,
0.017622921615839005,
0.0... | |
0124227d62c4e45d69641adf4fe54f762f057c18 | subsection | 53 | 65 | The non-ingredients | In this section, we give an upper bound for N(3,9) and N(3,12).
We assume that there is a curve with a larger A_k-singularity and find a contradiction to the configuration in \mathbb {P}^2 that we obtain with the “recipe” from Section REF .Lemma 5.6
There is an upper bound N(3,9)\le 13. In particular, there is no poly... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.0045594642870128155,
0.012423109263181686,
-0.033057067543268204,
-0.04154263064265251,
-0.022022783756256104,
0.004807468503713608,
-0.00398333091288805,
-0.0049219317734241486,
0.017551075667142868,
0.030859369784593582,
-0.008287159726023674,
0.0128657016903162,
0.0054484643042087555,
... | |
90ee2b52c83ad11e4d0966c3300eb79bfd4f6edb | subsection | 54 | 65 | The non-ingredients | Then, the singularity of C is a node.
[Figure: The situation of Lemma]First of all, let us prove that there exists a point q, distinct from p, on S\cap L, where L is the line going through p and s.
If S\cap L=\lbrace p\rbrace were true, then S and L would be tangent at p, and since C and S are tangent at p, also L and... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.025857096537947655,
0.030998006463050842,
-0.06657249480485916,
-0.03365236520767212,
0.01343196164816618,
-0.03777119517326355,
0.012417508289217949,
0.016078690066933632,
0.013729432597756386,
-0.005556606221944094,
-0.005896028596907854,
-0.0013348058564588428,
0.004065437242388725,
... | |
88b78e9cbcb4e30d0f655474aa6a2973903f5d4f | subsection | 55 | 65 | The non-ingredients | By Lemma REF , we know that k\le 20.
We assume that C_0 has an A_k-singularity at a point s_0, where k\in \lbrace 19,20\rbrace .
So we consider the 3-configuration \mathcal {C}_0=(C_0,S_-,s_0,p_0)_4, which is disjoint because C_0\sim 3S_+ is irreducible.
Let \pi :\mathbb {F}_{4}\dashrightarrow \mathbb {F}_{m^{\prime }}... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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27612d3fe5f237b4397205d668bc04c05b4144c7 | subsection | 56 | 65 | The non-ingredients | We consider the linear map\varphi :\mathbb {C}[x,y,z]_3&\rightarrow \mathbb {C}[u,v]_9\\
G&\mapsto G(u^3,u^2v,v^3)and notice that both source and target are of dimension 10.
The equation of the curve being F, we have \ker \varphi =\mathbb {C}\cdot F and so \operatorname{Im}\varphi has dimension 9.
One can check that y^... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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8c32625b73038858da51e6fe6917b7bf0337f715 | subsection | 57 | 65 | The non-ingredients | Therefore, we can assume that \lambda =1.One concludes the proof by checking that not all partial derivatives of G can be zero at the same point.
So G is not singular.Lemma 5.10
Let C\subset \mathbb {P}^2 be an irreducible cubic with a node.
Let D\subset \mathbb {P}^2 be an irreducible cubic that intersects C in only ... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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102d0c385550729fb194089577135ddc5863889b | subsection | 58 | 65 | The non-ingredients | So \lambda is a ninth root of 1.We look at\begin{split} H=&z^3+z^2 9\lambda (x+\lambda ^7y)\\
&+z\left(9\lambda ^2 x^2(-\lambda ^6+4) + 3xy(28\lambda ^3-1)+9\lambda y^2(4\lambda ^6-1)\right)\\
&+9\lambda xy\left(x\,(-4\lambda ^6+14\lambda ^3-1)+\lambda y(-\lambda ^6+14\lambda ^3-4)\right)\\
&+84\lambda ^3 y^3(\lambda ^... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
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a51ad43a7704f8894f326f4eb4d7cd0a4a1cb45a | subsection | 59 | 65 | The non-ingredients | (For instance, if \lambda =1 we find that G=(3x+3y+z)^3.)
Therefore, \lambda is not a third root of 1 and so \lambda ^6+\lambda ^3+1=0.We insert s into the differentials of G and find, using \lambda ^6+\lambda ^3+1=0, that they are not zero, a contradiction to s being a singular point of D:\frac{dG}{dx}\left(-\lambda ^... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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87793fbad3d79fff5188969084633debfb71f916 | subsection | 60 | 65 | Let's Tie the Knot | In Section REF we present a result of knot theory and explain how it is used to obtain an upper bound for N(3,b).
With this, we can finally finish the proof of Theorem REF in Section REF , by marrying the lower bound from Section with the obtained upper bound.
Therefore, despite the Shakespearean section titles and th... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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dd96d7beda9772d32901e65af81fd815230dd1c5 | subsection | 61 | 65 | Detour to knot theory | A knot is a smooth and oriented embedding of S^1 into S^3.
A link is the disjoint union of finitely many knots.For any positive r\in \mathbb {R}, let S_r^3 denote the sphere of dimension 3 with radius r embedded in \mathbb {C}^2 as S_r^3=\lbrace (x,y)\in \mathbb {C}^2\mid |x|^2+|y^2|=r^2\rbrace .Definition 6.1 Let C\su... | {
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} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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d65642295b829be2b0329e96af14f29f5020266a | subsection | 62 | 65 | Detour to knot theory | Then, V(G) is smooth (for general s,t), and V(G)\pitchfork S_r^3 is isotopic to C\pitchfork S_r^3, that is K.
The polynomial G was chosen in such a way that x^a and y^b appear with non-zero coefficients. Hence, the link of infinity of G is T_{a,b}.
This means that V(G) gives rise to an algebraic cobordism between K, th... | {
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"arxiv_id": "",
"doi": "10.4310/cag.2016.v24.n5.a4",
"end": 834,
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"raw": "Peter Feller. Optimal cobordisms between torus knots. Communications in Analysis and Geometry, 24(5):993–1025, 2016.",
"source_ref_id"... | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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9625b3033ec8b47e7c8dc45acb2818b9cd95498b | subsection | 63 | 65 | Detour to knot theory | If b is no multiple of 3, we have with Lemma REF that\frac{2(N(3,b)+1)}{3b}\le \frac{2(5b-1)}{9b}=\frac{10}{9}-\frac{2}{9b}<\frac{10}{9}<\frac{7}{6}.If b is a multiple of 3, we write b=3m-3 for some m\ge 2, and hence b+1=3m-2 is not a multiple of 3, so Remark REF givesN(3,b)\le N(3,b+1)\le 5m-5.We find that \frac{2(N(3... | {
"cite_spans": []
} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
] | [
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a5ad6c5c80dae6f58c4c3af74db74543bacc88d5 | subsection | 64 | 65 | Proof of Theorem | [Proof of Theorem REF ]
Let us look at the souvenirs collected on our journey.
In Remark REF we have found lower bounds (LB), and the upper bounds (UB) from Remarks REF and REF combined give the following values for N(3,b):
[Table: NO_CAPTION]Remark 6.7
Theorem REF shows that in the cases studied, the upper bound obta... | {
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} | 1808.05133 | Plane curves of fixed bidegree and their $A_k$-singularities | [
"Julia Schneider"
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f4dcc3ae4669c3f731c5632745b75ac0ff0327e3 | abstract | 0 | 55 | Abstract | This paper considers the problem of predicting the number of events that have
occurred in the past, but which are not yet observed due to a delay. Such
delayed events are relevant in predicting the future cost of warranties,
pricing maintenance contracts, determining the number of unreported claims in
insurance and in ... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
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78a85f9752569369a8713bee3476d54dbc5510bc | subsection | 1 | 55 | Introduction | In many domains within operational research analysts are interested in building a stochastic model for the occurrence of events. However, the events of interest are often observed or reported with some delay. Analysts should account for these unobserved events since ignoring them will bias the decisions based on the st... | {
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"Jonas Crevecoeur",
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"Roel Verbelen"
] | [
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ddf2fcff0326a5ce764d6fe861f4103e7150adf2 | subsection | 2 | 55 | Introduction | A machine failure (`the observed event') is often the result of previous defects (`the hidden event') which remained unobserved. These defects can be detected by on site inspections and timely repairs will prevent expensive failures or breakdowns of the machine. However, the profitability of these inspections depends l... | {
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"raw": "Christer, A. (1973). Innovatory decision making. In White, D. and Brow, K., editors, The Role and Effectiveness of Theories of Decision in Practice, pages 369–377. Hodder and Stoughton, London.",
... | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
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32e6839ef8e2606db804c890261379779c6be2f1 | subsection | 3 | 55 | Introduction | This weekday pattern relates to calendar day effects in the reporting process which are difficult to model using classical techniques designed for aggregated data (see ).
provide a method to incorporate this weekday pattern for reporting delays of less than one week. extend this weekday pattern to reporting delays bey... | {
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"Jonas Crevecoeur",
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a3153735d12301d128df8e92b48521b0fa720903 | subsection | 4 | 55 | A granular model for the occurrence of events subject to delay | Denote by N_t the number of events occurring on date t, where t = 1 is the date of the first event. These events remain hidden until their observation at date s after a delay s-t. Let N_{t, s} be the number of events that occurred on date t and are observed on date s. Since all events will be observed at some point in ... | {
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"raw": "Jewell, W. S. (1990). Predicting IBNYR events and delays. ASTIN Bulletin, 20, 93–111. https://doi.org/10.2143/AST.19.1.2014914.",
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af6af5685f44dc3c0bca69ae945acc17a9e14948 | subsection | 5 | 55 | A granular model for the occurrence of events subject to delay | Let {\chi } denote the available data, consisting of all events that are observed on the evaluation date \tau{\chi } = \lbrace N_{t, s} \mid t \leqslant s \leqslant \tau \rbrace .The loglikelihood of the observed data is\ell ({\lambda }, {p} ; {\chi } ) = \sum _{t = 1}^\tau \sum _{s = t}^\tau \Big [N_{t, s} \cdot \log ... | {
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} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
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e4ee3d5b47779a84c30a73f46ecb83ef4b61945c | subsection | 6 | 55 | A time change strategy to model observation delay | We are interested in structuring the observation probabilities p_{t, s} based on covariates corresponding to the occurrence date t and the reporting date s of the event. The probabilistic nature of the data enforces the constraintsp_{t, s} \geqslant 0, \quad \forall t, s \quad \text{and} \quad \sum _{s \geqslant t} p_{... | {
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852d5722afb8251eb87e3e7ce7ed2a7eab24e879 | subsection | 7 | 55 | A time change strategy to model observation delay | We set\log (\alpha _{t, s}) = {x}_{t, s}^{^{\prime }} \cdot {\gamma },for a vector {x}_{t, s} of covariates related to observing on date s an event that occurred on date t and the corresponding parameter vector {\gamma }. In contrast with classical regression methods, the reporting probabilities p_{t, s} not only depen... | {
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} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
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00f6f36d6b8910fed703b399451369b6928c690b | subsection | 8 | 55 | Calibration | Our approach divides the observation delay model into two components. The time change transformation \varphi _t defined in (\ref {eq:transformation}) captures the heterogeneity in the observation process. This transformation is expressed by the daily observation exposures, which require the calibration of the regressio... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
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2f5f6b0344116c51b0c183ff169823734e002a34 | subsection | 9 | 55 | Predicting the number of hidden events | At the evaluation date \tau we predict the number of events from past occurrence dates t that will be observed on future dates s. Hence our focus is onN_{t, s}, \quad \text{for } t \leqslant \tau \text{ and } s > \tau .We aggregate these future daily observation counts to find the total number of hidden eventsN^{\mathr... | {
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"raw": "Bonetti, M., Cirillo, P., Musile Tanzi, P., and Trinchero, E. (2016). An analysis of the number of medical malpractice claims and their a... | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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9c583d91210be647ab61de17d822029cb2575915 | subsection | 10 | 55 | Data characteristics | We illustrate our approach with the analysis of a liability insurance data set from the Netherlands. The same data is studied in , and with focus on calculating reserves in discrete time, model reserves in continuous time and who propose a model for the number of hidden claim counts at a daily level. The data registers... | {
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{
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"raw": "Pigeon, M., Antonio, K., and Denuit, M. (2013). Individual loss reserving with the multivariate skew normal distribution. ASTIN Bulletin, 43, 399–428. https://doi.org... | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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759138b5f95208f3b065659ff34001c8a2abb157 | subsection | 11 | 55 | Occurred accidents | Figure REF shows the daily number of accidents that occurred between July, 1996 and August, 2009 and initiated a claim reported to the insurance company before August 31, 2009. Since only claims reported before August 31, 2009 are observed, we see a decrease in observed event counts for the most recent dates which have... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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c3a2fcebb7608a194fe5f38190932c2c08103ccf | subsection | 12 | 55 | Reported claims | Figure REF shows the daily number of claims reported between July 1996 and August 2009. Again the red line shows the moving average of the number of reported claims, calculated over the latest 30 days. The seasonality in event counts observed in Figure REF leads to a similar seasonal pattern in reported claim counts, t... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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babe36cd64a481c5121f45847b6382ed0d58de50 | subsection | 13 | 55 | Reporting delay | Figure REF illustrates the empirical reporting delay distribution in days over the first three weeks after the occurrence of the insured event. The empirical probability of reporting peaks the day after the claim occurred and strongly decreases afterwards. The increase in reporting after exactly fourteen days is most l... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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c922822d12fc93615c1ba0d66d32eb531b11608a | subsection | 14 | 55 | The number of hidden events | The evaluation date refers to the date on which the insurer computes the reserve. In practice this date is often the last day of a quarter or the financial year. Figure REF uses a rolling evaluation date to illustrate the daily number of IBNR claims. For each evaluation date we show the number of claims corresponding t... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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9850e83d20ea0d0a0fadbcf26d23cc0f3c4f17a4 | subsection | 15 | 55 | Model specification | We opt for computational efficiency and model the time-changed reporting delay \tilde{U} with an exponential distribution. The reporting exposures include six effects and are structured as\alpha _{t, s} &=
\alpha ^{\texttt {occ.\,dom}}_t
\cdot \alpha ^{\texttt {occ.\,month}}_t
\cdot \alpha ^{\texttt {rep.\,holiday}}_{s... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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740905a2e7e445108b9d550b056d2003fa587f21 | subsection | 16 | 55 | Model specification | Finally, \alpha _{s-t}^{\texttt {delay}} partitions the time elapsed since the accident occurred in 23 bins according to the strategy specified in online Appendix . These bins adapt the tail of the distribution as well as increase the probability of reporting after 14, 30 and 365 days. | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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b704260f616e03e20e1f64c4c9d2321ab46945d8 | subsection | 17 | 55 | Parameter estimates | We estimate the model parameters by maximizing the loglikelihood in (\ref {eq:loglikelihoodExp}) using 8 years of data i.e. all accidents that occurred and were reported between July 1, 1996 and September 5, 2004. The resulting training data set contains 274187 reported claims, for which we model the reporting process ... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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487c595a25d6eb639f269fe285513ef453353d55 | subsection | 18 | 55 | Occurrence day of month | Figure REF shows the effect of the day of the month on which the accident occurred. Reporting exposure is lower for accidents that occur on the first or fifteenth of the month, which implies that accidents from these days have a longer reporting delay. This is most likely the result of data quality issues. Insureds who... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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467f05ec6cc22ef23290b60a804ca76d566a33e5 | subsection | 19 | 55 | Month | Two month effects are included in the reporting exposure structure. Figure REF shows the effect for \exp ({\gamma }^{\texttt {occ.\,month}}) which considers the month in which the accident occurs. These parameters indicate that reporting is slower for accidents that occurred around the beginning of the year (January, F... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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b3e328fde3e44c032528285533aad033cc0f0ef6 | subsection | 20 | 55 | Holiday | Figure REF shows the effect of holidays on reporting exposure. Hardly any claim gets reported on national holidays and the reporting probability is reduced by more than 50\% on unofficial holidays (Good Friday and New Year's Eve). These estimates are of the same magnitude as the effects found in the empirical analysis ... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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dc887dfad2cc9bf39f678df0d2b7ad69e2296ab1 | subsection | 21 | 55 | Reporting day of the week | We include the day of the week effect in the reporting exposure specification (REF ) through an interaction between the time elapsed after the accident occurred s-t and the day of the week on which the claim is reported. Figure REF shows a grouping of the estimated coefficients based on the time elapsed since the occur... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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24c55d0c384166fe6fa5348def5dc33ee9c9e680 | subsection | 22 | 55 | Delay | Figure REF shows the evolution of the reporting exposure component \exp ({\gamma }^{\texttt {delay}}) in (REF ) as a function of the time elapsed since the accident occurred. This effect scales the reporting probability at specific delays such that the time-changed reporting delay \tilde{U} better resembles an exponent... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
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] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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35b4c032878f45655383ee67f584e6177c5d3407 | subsection | 23 | 55 | Out-of-time predictions | We predict the number of hidden events, i.e. the IBNR claim count, following the strategy outlined in Section REF . Because the non-parametric occurrence estimators are unreliable for recent event dates for which few events are observed, we propose a pragmatic approach to get around this drawbacks. Insurance companies ... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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a1d37245cf068d97fffc4006beeacc95ee91df80 | subsection | 24 | 55 | Future observation of hidden events | Our daily model splits the total IBNR point estimate of 2012.7 claims by future reporting date. Figure REF shows the estimated number of daily reported claims in September and October, 2004 for accidents that occurred before August 31, 2004. The dashed line in Figure REF indicates the computation date. We do not make p... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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036e529fd8f3046080debb853d60c3bf76982deb | subsection | 25 | 55 | Evolution of the number of hidden events | The primary focus of our granular model is estimating the total IBNR count. The top panel of Figure REF plots the predicted number of unreported claims on each evaluation date between September, 2003 and August, 2004. Each point estimate is an out-of-time IBNR estimate obtained from the granular model calibrated on the... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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2fa371a175a171e47b356ba9ed701c9b85be8796 | subsection | 26 | 55 | Benchmark with a model for aggregate data | We benchmark our granular approach to Mack's chain ladder method on aggregated data, which is the industry standard in claims reserving. This method discretizes time and aggregates the observed events into a two dimensional table based on the occurrence period and the discretized reporting delay. A Poisson generalized ... | {
"cite_spans": [
{
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"openalex_id": "https://openalex.org/W2134548066",
"raw": "Mack, T. (1993). Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin, 23, 213–225. https://d... | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
] | 2,018 | en | Quantitative Finance | [
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53b1cfe1fcd18e7c21b793aacd6917fb52e38ff3 | subsection | 27 | 55 | Investigated scenarios | We further evaluate our approach with portfolios simulated along four different scenarios. Each scenario generates data from an insurance portfolio from January 1, 1998 onwards. Figure REF outlines the structure of these data sets. The insurer observes the claims that are reported before the computation date (the gray ... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
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] | [
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4a1a0d48e0f6f4461518eeb475172f1975541538 | subsection | 28 | 55 | Scenario 1: Baseline scenario | This is the basic scenario from which the other three scenarios will slightly deviate. The occurrence of insured events follows a Poisson distribution with an average of 100 claims on each occurrence date. For these occurrences the reporting delay is simulated along the model specification outlined in Section , i.e. th... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
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] | [
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88b0fbd6936db49ef591a7cd78eb1c42002dd1ad | subsection | 29 | 55 | Scenario 2: Volatile occurrences | In this scenario external causes, such as the weather, have a large effect on the number of accidents that occur on a given date. The environment can be in two states, a good state with an average of 100 accidents per day and a bad state in which there are on average 400 accidents. The transitions between these states ... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
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"q-fin.RM"
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37384bc4f25fa261fe433ef179490079d593f451 | subsection | 30 | 55 | Scenario 3: Low claim frequency | This scenario illustrates the effect of a strong reduction in the number of occurred accidents. The occurrence process is modeled by a Poisson distribution with a daily average of two claims. The reporting model from the baseline scenario is used. This scenario is visualized in the bottom row of Figure REF . We observe... | {
"cite_spans": []
} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
"Jonas Crevecoeur",
"Katrien Antonio",
"Roel Verbelen"
] | [
"q-fin.RM"
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