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64931e1f3c6080059a6a3658846c2401c5778de5 | subsection | 31 | 55 | Scenario 4: Online reporting | In this scenario the insurer introduces an online tool for claim reporting. This online tool is launched at January 1, 2003 and increases the number of reports in the weekend and on holidays. The new reporting exposures become\alpha _{t, s} =
{\left\lbrace \begin{array}{ll}
0.10 \cdot (0.20)^{\mathbb {1}_{s \in \texttt... | {
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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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ab42add24242065c3dc407591625b3785e817ead | subsection | 32 | 55 | Calibrated models: granular versus aggregate | We compare the accuracy of the predictions of the hidden event counts using three models, namely the exact granular model from which we simulated the data, an approximate granular model and a model for yearly aggregated data. The historical information (gray area in Figure REF ) is used to predict the number of IBNR cl... | {
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1f46d14db58c1fd516761c487f46ea3b067889f8 | subsection | 33 | 55 | Exact granular model | We use our knowledge of the shape of the distribution and reporting exposure structure behind the various scenarios and calibrate the exact same model for reporting delay on the historical data. Hence we estimate the variance parameter in the lognormal distribution for the smoothed reporting delay \tilde{U} and the par... | {
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eaffda589dc54087975b20c1e0c18a74e2361edf | subsection | 34 | 55 | Approximate granular model | This model considers the more realistic situation where the insurer wants to fit the model of Section , but is unaware of the exact underlying distribution. Motivated by computational benefits the insurer chooses an exponential distribution for the smoothed reporting delay \tilde{U}, and structures the reporting exposu... | {
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1efde09eb3e56512af0ba5dd7bff3a5e5175b0d5 | subsection | 35 | 55 | A model for aggregated data: the chain ladder | The chain ladder method described in Section REF is the industry standard for predicting the number of unreported claims. We aggregate the simulated data by calendar year and benchmark our granular approach to the chain ladder method on this aggregated data. | {
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1e5020f0472b595bf04ac51bb8e7df39f5f104f7 | subsection | 36 | 55 | Results and discussion | We evaluate the performance of the reserving models by predicting the total number of IBNR claims at the evaluation date, which corresponds to the hatched area in Figure REF . This prediction is compared with the actual number of unreported claims as observed in the simulated data set. We simulate 1000 data sets and ca... | {
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d4771a66bcf191eec224b3e487ed5ceaadb2cb79 | subsection | 37 | 55 | Impact of evaluation date | We observe in all four scenarios an increase in unreported claims on New Year's Eve (see the last column in Figure REF ). This is the result of multiple holidays at the end of the year, which prevents clients from reporting their claim. We compare the average percentage error in Table REF on December 31, 2003 and Augus... | {
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32c4e495f69e8c3c25ece37b787dda1fc4e4c6da | subsection | 38 | 55 | Baseline | The top row of Figure REF visualizes a single data set from the baseline scenario. Both the occurrence and reporting process are stable. This leads to a yearly periodical pattern in IBNR counts, which is easy to predict. Since all three models perform well (see Figure REF ), there is no reason to replace the chain ladd... | {
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"Jonas Crevecoeur",
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4e8d24904f849b31503b4e36bf768666cf05594b | subsection | 39 | 55 | Volatile occurrences | The range of IBNR values encountered throughout a year is much wider in this scenario compared to the other three scenarios. Table REF and Figure REF show that the performance of the granular models is in line with their performance in the baseline scenario. The occurrence process has little effect on the prediction ac... | {
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fe91a319fd470162cf17c35a801dd7dab02ac703 | subsection | 40 | 55 | Low claim frequency | The occurrence frequency is reduced from an average of hundred daily claims to only two claims. The third row of Figure REF visualizes a data set from this scenario. Since on average only two accidents occur per day, our predictions for the intensities \lambda _t in the occurrence process are less reliable. As seen in ... | {
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273214e83a87c7ee6e9704c4d711054f36ea2169 | subsection | 41 | 55 | Online reporting | On January 1, 2003 the insurer introduces an online tool to report claims, which creates a breakpoint in the reporting process. The granular model performs well on both evaluation dates, since we estimate different exposure parameters after the breakpoint. Both evaluation dates correspond with around one year of post b... | {
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867517a6dda302ae524261692c68dd473c20c09d | subsection | 42 | 55 | Conclusion | We propose a new method to model the number of events that occurred in the past, but which are not yet registered due to an observation delay. Our approach provides an elegant and flexible framework for modeling the observation delay subject to calendar day covariates by introducing the concept of observation exposure.... | {
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e8a522fd42c4cac6cf6240bfcfa11c473c4a89fe | subsection | 43 | 55 | Maximum likelihood estimation of observation exposure parameters | We model a parameter vector {\gamma } which structures the observation exposures.\ell ({\gamma } ; {\chi } ) &= \sum _{t=1}^\tau \sum _{s=t}^\tau N_{t, s} \cdot \log (p_{t, s} ) - \sum _{t=1}^\tau N_t^{\mathrm {R}}(\tau ) \cdot \log (p_t^{\mathrm {R}}(\tau ) )\\
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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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404160cf38dcefacb7fc9e7e82c5e60541f1a756 | subsection | 44 | 55 | Maximum likelihood estimation of observation exposure parameters | The components of the score vector {S} are\frac{\partial \ell ({\gamma }, {\xi } ; {\chi })}{\partial \gamma _i} = &
\sum _{t=1}^\tau \sum _{s=t}^\tau \frac{N_{t, s}}{p_{t,s}} \cdot \left[
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} | 10.1016/j.ejor.2019.02.044 | 1801.02935 | Modeling the number of hidden events subject to observation delay | [
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82c7b9adc1184631b596b56d4519a8b90f13b87f | subsection | 45 | 55 | Maximum likelihood estimation of observation exposure parameters | The Hessian {H} is given by& \frac{\partial \ell ({\gamma } ; {\chi })}{\partial \gamma _i \partial \gamma _j} = \\
& \qquad \sum _{t=1}^{\tau } \sum _{s=t}^\tau \frac{N_{t, s}}{p_{t, s}} \cdot \Bigg [
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e444400a473d6c4184b6806aaa038a1fd844a006 | subsection | 46 | 55 | Maximum likelihood estimation of observation exposure parameters | Together with the observation parameters, the simulation study of Section REF estimates the variance parameter \sigma in the lognormal time-changed distribution. The Newton-Raphson algorithm in (\ref {eq:iterativeNR}) can easily be extended to this case, where the distribution function of F_{\tilde{U}} depends on param... | {
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c72045a6b907ae58f97dd4e788541538304e50c9 | subsection | 47 | 55 | Simulation procedure | We outline the algorithm that was used to generate data sets from the four scenarios specified in Section REF . This algorithm combines a model for the occurrence of events with a model for the observation delay as described in Section . We divide the algorithm in three steps. | {
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4882eeb2f6597df7a35f65c97d29418e6ef3ddad | subsection | 48 | 55 | Step 1. Occurrence | We first generate the number of occurred events. The number of daily events follows a Poisson distributionN_t \sim \text{Poisson}(\lambda _t),where the intensity \lambda _t is obtained from the occurrence process specification for the scenarios in Section REF . | {
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0727dec2e4aedb0ec8b53a5b96df5169155a9461 | subsection | 49 | 55 | Step 2. Observation | We now simulate the observation date for each occurred event. Combining equation (\ref {eq:transformation}) and (\ref {eq:formulaP}), we can write the probability that an event from date t is observed on date s asp_{t, s} = P\left(\tilde{U} \in \left[ \sum _{v=t}^{s-1} \alpha _{t, v}, \sum _{v=t}^{s} \alpha _{t, v} \ri... | {
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c1775214b5b234398558a68452337e022b356728 | subsection | 50 | 55 | Step 3. Truncation | With steps 1 and 2 we have simulated an observation date for each occurred event. We split this data set into observed and hidden events. We use the data set with observed events to calibrate the model and to predict the number of hidden events. The hidden events are kept only for evaluating the prediction accuracy. | {
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1abdc0a102f939cde5bf4dd358716d5b1924cc35 | subsection | 51 | 55 | A standard distribution for the time changed observation delay | Modeling the time-changed observation delay with an exponential distribution has significant computational benefits. Therefore, this section puts focus on the use of the exponential distribution as a standard distribution for modeling the time-changed observation delay \tilde{U}. Since the exponential distribution is l... | {
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28fc6c0beaf0c6f2cc62d9bf7b8b6a07861ec57b | subsection | 52 | 55 | Binning observation delay | Our binning strategy maximizes the loglikelihood in (REF ) when the observation exposures depend only on the time elapsed since the event occurred, i.e.\alpha _{t, s} = \exp ({\gamma }^{\texttt {delay}} \cdot x_{s-t}^{\texttt {delay}}) = \exp (\gamma {^\texttt {s-t}}),where we estimate for each delay s-t a separate par... | {
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5a4498a50d01e26579fc7fc28f4c4a99c107c9e0 | subsection | 53 | 55 | Binning observation delay | This figure shows in red the estimated delay parameters using approximation (REF ). The top panel shows the estimates for delays up to 31 days, whereas the parameters for larger delays (up to 400 days) are shown in the bottom panel. Based on this knowledge observation delay is grouped in 23 bins, separated by vertical ... | {
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82a19d53fa99c043a497ad858d30a19f2f9f50ea | subsection | 54 | 55 | A link with the Kaplan-Meier estimator | We show that under the binning strategy of Appendix REF the time changed model has the same flexibility as the Kaplan-Meier estimator and is as such suitable for modelling a wide range of portfolios.The Kaplan-Meier estimator for the survival function of the observation delay random variable is\widehat{P(\texttt {delay... | {
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d08c82f66bd96a462c4e7f6ac5defb8e0c8ab231 | abstract | 0 | 3 | Abstract | This document presents the syntax, classification rules, realizability
semantics, and soundness theorem for Cedille, an extrinsic (i.e., Curry-style)
type theory extending the Calculus of Constructions, and designed for deriving
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469102fed93e49cffe1c1c3a5d0576a93612251e | subsection | 1 | 3 | Introduction | The type theory of Cedille is called the Calculus of Dependent Lambda
Eliminations (CDLE). This document presents the version of CDLE as of
June 1, 2018. We have made many changes from the first paper on CDLE ,
mostly in the form of dropping constructs we discovered (to our
surprise) could be derived . I have also omit... | {
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b641604668bc48d690ddad77dba9074ab591b59a | subsection | 2 | 3 | Classification Rules | The classification rules are given in
Figures REF , REF , and . For
brevity, we take these figures as implicitly specifying the syntax of
kinds \kappa , types T, and annotated terms t; these may use term
variables x and type variables X, which we assume come from
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2e1554b4cfe4bf9334c9d80832cb49969d25bdfc | abstract | 0 | 33 | Abstract | In this paper, we introduce a parameterized discrete curvature
($\alpha$-curvature) for piecewise linear metrics on polyhedral surfaces, which
is a generalization of the classical discrete curvature. A discrete
uniformization theorem is established for the parameterized discrete curvature,
which generalizes the discret... | {
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} | 1806.04516 | Parameterized discrete uniformization theorems and curvature flows for
polyhedral surfaces, I | [
"Xu Xu"
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a0a8c67b02d7b2af37b505b065404bdc30700b33 | subsection | 1 | 33 | Introduction | In this paper, we study the combinatorial \alpha -curvature of piecewise linear metrics on surfaces.
Combinatorial \alpha -curvature was introduced by Ge and the author ,
for Thurston's circle packing metrics as a generalization of the classical
combinatorial curvature K.
The classical combinatorial curvature K, defin... | {
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c23b659f245f156aba88f90081dc5be3d9e22099 | subsection | 2 | 33 | Introduction | Combinatorial Yamabe flow with surgery for polyhedral metrics were defined in , ,
where the long time existence and convergence of the combinatorial
Yamabe flow with surgery are proved.
Following Luo's approach, Ge introduced the combinatorial Calabi flow for piecewise linear metrics on surfaces.
Recently, Zhu and the ... | {
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ed722654a4660c831d30fee9ef7c591e135c6f46 | subsection | 3 | 33 | Introduction | Note that
the combinatorial curvature K is independent of the geometric triangulations of (S, V) with a PL metric d.Definition 1.1 Suppose (S, V, \mathcal {T}) is a triangulated surface with a PL metric d and
w: V\rightarrow (0, +\infty ) is a conformal factor of d on (S, V, \mathcal {T}).
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b84752103a29ddd73d24d974ece58ece153fdc34 | subsection | 4 | 33 | Introduction | Following Luo's approach, Ge introduced the combinatorial Calabi flow to find the constant curvature PL metric, which is
a negative gradient flow of the combinatorial Calabi energy.To study the combinatorial \alpha -Yamabe problem, we introduce the combinatorial \alpha -Yamabe flow and combinatorial
\alpha -Calabi flow... | {
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a74958661891ffe4dc23c976419e20cebce3490a | subsection | 5 | 33 | Introduction | The combinatorial \alpha -curvature R_\alpha evolves according to\frac{dR_{\alpha , i}}{dt}=(\Delta ^\mathcal {T}_{\alpha }R_{\alpha })_i+\alpha R_{\alpha , i}(R_{\alpha , i}-R_{\alpha , av})along the combinatorial \alpha -Yamabe flow (REF ),
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168412f5b0c398a8b96d9c32ec8e54c0419788d1 | subsection | 6 | 33 | Introduction | Note that the weight in the \alpha -Laplace operator is
\omega _{ij}=\frac{\cot \theta _{k}^{ij}+\cot \theta _l^{ij}}{w_i^\alpha }.
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d85980e05f9eb3467858976cee848785ee5dab2d | subsection | 7 | 33 | Introduction | Then there exists a PL metric in the conformal class \mathcal {D}(d_0) with constant \alpha -curvature if and only if
one of the following two conditions is satisfied:(1)
The combinatorial \alpha -Yamabe flow with surgery
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e0ac35c671952c2d5e46ea21425bb365513867e6 | subsection | 8 | 33 | Rigidity of | Suppose (S, V, \mathcal {T}) is a triangulated surface with a PL metric d and w: V\rightarrow (0, +\infty ) is a positive function defined on V.
Set h: \mathbb {R}^n_{>0}\rightarrow \mathbb {R}^n be the homeomorphism defined by u_i=h(w_i)=\ln w_i.
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ff403e4be810cfc3a364f8001d062827a8d01c5b | subsection | 9 | 33 | Rigidity of | Luo studied Bobenko-Pinkall-Springborn's extension and obtained a general extension method for similar problems
without involving Milnor's Lobachevsky function, which has lots of applications (see , , , for example). Here we take Luo's approach.Lemma 2.2 ()
Let l_1, l_2, l_3 and \theta _1, \theta _2, \theta _3 be the ... | {
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ef6c3b7498148bca3c4a55dc1386253a0bed8133 | subsection | 10 | 33 | Rigidity of | If w=\sum _{i=1}^na_i(x)dx_i is a continuous closed 1-form on A so that
F(x)=\int _a^x w is locally convex on A and each a_i can be extended continuous to X by constant functions to a
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bdfc828b6601a56ab7b1b53c2e93f16b9ba75ef4 | subsection | 11 | 33 | Rigidity of | By direct calculations, we have\operatorname{Hess}_u F= L-\alpha \left(
\begin{array}{ccc}
\overline{R}_1w_i^{\alpha } & & \\
& \ddots & \\
& & \overline{R}_Nw_N^{\alpha } \\
\end{array}
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f949d3d6d733fd30887c7aeb0da47b82bcba9a87 | subsection | 12 | 33 | Rigidity of | And the second term \int _{u_0}^u\sum _{i=1}^N(2\pi -\overline{R}_iw_i^\alpha )du_i in(REF )
can be naturally defined on \mathbb {R}^N, then we have the following extension \widetilde{F}(u) defined on \mathbb {R}^N
of the Ricci energy function F(u)\widetilde{F}(u)=-\sum _{\triangle ijk\in F}\widetilde{F}_{ijk}+\int _{u... | {
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4a3753dea50d137114c19cca34de5418ec318a40 | subsection | 13 | 33 | Rigidity of | It follows that\nabla \widetilde{F}(\overline{u}_A)=\nabla \widetilde{F}(\overline{u}_B)=0.Set\begin{aligned}f(t)=&\widetilde{F}((1-t)\overline{u}_A+t\overline{u}_B)\\
=&\sum _{\triangle ijk\in F}f_{ijk}(t)+\int _{u_0}^{(1-t)\overline{u}_A+t\overline{u}_B}\sum _{i=1}^N(2\pi -\overline{R}_iw_i^\alpha )du_i,
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596a797ed8ec08dee953ec08aa506ef91f17eadb | subsection | 14 | 33 | Rigidity of | \end{aligned}By Lemma REF , we have \overline{u}_A-\overline{u}_B=c\textbf {1} for some constant c\in \mathbb {R}, which
implies that \overline{w}_A=e^{c}\overline{w}_B. So there exists at most one conformal factor
with combinatorial \alpha -curvature \overline{R} up to scaling.
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901d6fa44204dae5cccd14533bc3630838401e3f | subsection | 15 | 33 | Rigidity of | (2)
If \alpha \overline{F}\le 0 and \alpha \overline{F}\lnot \equiv 0, then there exists at most one conformal factor u^*\in \mathcal {U}(d) such that \mathbf {A}^{-1}(p, w(u^*))\in \mathcal {D}(d) has combinatorial \alpha -curvature \overline{F}.Define the energy functionW_\alpha (u)=W(u)-\int _0^u\sum _{i=1}^N \overl... | {
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4534ae6cfc007a5e6e1322da3423b20daa16a72b | subsection | 16 | 33 | Combinatorial Yamabe flow of | By direct calculations, we have the following properties of combinatorial \alpha -Yamabe flow.Lemma 2.5
If \alpha =0, \sum _{i=1}^N u_i is invariant along the normalized combinatorial \alpha -Yamabe flow (REF ).
If \alpha \ne 0, ||w||_\alpha ^\alpha =\sum _{i=1}^Nw_i^\alpha is invariant
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dd3b88a940c2fa9aa51d3855560253c93ed08298 | subsection | 17 | 33 | Combinatorial Yamabe flow of | By direct calculations, we have\begin{aligned}\frac{\partial \Gamma _i}{\partial u_j}|_{u=u^*}
=&-\frac{1}{w_i^\alpha }\frac{\partial K_i}{\partial u_j}+\alpha R_{\alpha , av}(\delta _{ij}-\frac{w_j^\alpha }{||w||_\alpha ^\alpha })\\
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ab45025dbcd8449a26aacc16e738b485063b9ce0 | subsection | 18 | 33 | Combinatorial Yamabe flow of | Specially, if \alpha R_{\alpha , av}\le 0, we have
u^* is a local attractor of the normalized combinatorial \alpha -Yamabe flow (REF ).
Then the conclusion follows from the Lyapunov Stability Theorem (, Chapter 5).
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725e813a948dfa87d4b1ed39dd94ccb0b6a45f82 | subsection | 19 | 33 | Combinatorial Calabi flow of | Similar to the combinatorial \alpha -Yamabe flow, we have the following properties of combinatorial \alpha -Calabi flow.Lemma 2.6
If \alpha =0, \sum _{i=1}^N u_i is invariant along the combinatorial \alpha -Calabi flow (REF ).
If \alpha \ne 0, ||w||_\alpha ^\alpha =\sum _{i=1}^Nw_i^\alpha is invariant
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baf49a4506324589bd4ba69b4c1f53289d6e7ffe | subsection | 20 | 33 | Combinatorial Calabi flow of | \end{aligned}If \alpha \chi (S)\le 0, then we have D\Gamma |_{u=u^*} has N-1 negative eigenvalue and a zero eigenvalue with
1-dimensional kernel orthogonal to the space \lbrace w\in \mathbb {R}^N|\sum _{i=1}^Nw_i^\alpha =N\rbrace ,
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b5b400cc34e6cb95871353e5144158af644e32f8 | subsection | 21 | 33 | Body | Theorem REF and Theorem REF
gives the long time existence and convergence of the combinatorial \alpha -Yamabe flow (REF ) and combinatorial
\alpha -Calabi flow (REF ) for initial PL metrics with small initial energy.
However, for general initial PL metrics, the combinatorial \alpha -Yamabe flow and combinatorial
\alph... | {
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274bf83b40add84abac07334bb1f11e830453458 | subsection | 22 | 33 | Gu-Luo-Sun-Wu's work on discrete uniformization theorem | Definition 3.1 ( Definition 1.1)
Two PL metrics d, d^{\prime } on (S, V) are discrete conformal if
there exist sequences of PL metrics d_1=d, \cdots , d_m=d^{\prime }
on (S, V) and triangulations \mathcal {T}_1, \cdots , \mathcal {T}_m of
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e56072c976aa6585750cd167015f3f74227c6ba5 | subsection | 23 | 33 | Gu-Luo-Sun-Wu's work on discrete uniformization theorem | In the proof of Theorem REF , Gu-Luo-Sun-Wu proved the following result.Theorem 3.2 ()
There is a C^1-diffeomorphism \mathbf {A}: T_{PL}(S, V)\rightarrow T_D(S, V) between T_{PL}(S, V) and T_D(S-V).
Furthermore, the space \mathcal {D}(d)\subset T_{PL}(S, V) of all equivalence classes of PL metrics
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8e9f33510b0beae9b8e61987fbe2b82011acfe30 | subsection | 24 | 33 | Gu-Luo-Sun-Wu's work on discrete uniformization theorem | Then we can extend the Euclidean discrete \alpha -Laplace operator to be defined on \mathbb {R}^n_{>0}, which is the space of
the conformal factors for the discrete conformal class \mathcal {D}(d).Definition 3.2
Suppose (S, V) is a marked surface with a PL metric d_0,
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3dadd5cb7ac7fc483bfa4c0b526b9c64ae8afe68 | subsection | 25 | 33 | Combinatorial | By Gu-Luo-Sun-Wu's discrete conformal theory ,
the normalized combinatorial \alpha -Yamabe flow with surgery takes the following form.Definition 3.4 Suppose (S, V) is a marked surface with a PL metric d_0. The combinatorial \alpha -Yamabe flow with surgery is defined to be\begin{aligned}\left\lbrace
\begin{array}{ll}
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ee2e5a0ceaedba90f1621a92b76c175a03f22623 | subsection | 26 | 33 | Combinatorial | For any n\in \mathbb {N}, there exists \xi _n\in (n, n+1) such
thatu_i(n+1)-u_i(n)=u_i^{\prime }(\xi _n)=\mathbf {F}_{\alpha , av}-\mathbf {F}_{\alpha , i}(u(\xi _n)).Set n\rightarrow +\infty , then we have\mathbf {F}_{\alpha , i}(u^*)=\lim _{n\rightarrow +\infty }\mathbf {F}_{\alpha , i}(u(\xi _n))=\mathbf {F}_{\alpha... | {
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981cf39028f5ca7502acf29716c2d634a5908ea0 | subsection | 27 | 33 | Combinatorial | Then the solution of the combinatorial \alpha -Yamabe flow with surgery (REF ) exists for all time
and \lim _{t\rightarrow +\infty }W_{\alpha }(u(t)) exists. Furthermore,\begin{aligned}0=&\lim _{n\rightarrow +\infty }(W_\alpha (u(n+1)-W_\alpha (u(n))))
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0844ae00cbf237b75c4f78f8e291d4c30b7420c3 | subsection | 28 | 33 | Combinatorial | \end{aligned}along the combinatorial \alpha -Ricci flow.Note that the surgery ensures that the weight\omega _{ij}=\frac{1}{w_i^\alpha }\frac{\partial \mathbf {F}_i}{\partial u_j}=\frac{\cot \theta _{k}^{ij}+\cot \theta _{l}^{ij}}{w_i^\alpha }\ge 0along the combinatorial \alpha -Yamabe flow with surgery (REF ).
This mot... | {
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6ffa3f87723ac4497a8156825080025211789a07 | subsection | 29 | 33 | Combinatorial | \alpha \in \mathbb {R} is a constant such that \alpha \mathbf {F}_{\alpha ,i}(u(0))<0 for all i\in V,
then the normalized \alpha -Yamabe flow with surgery (REF ) exists for all time and converges
exponentially fast to a constant \alpha -curvature PL metric.Note that the combinatorial \alpha -curvature \mathbf {F}_{i,\a... | {
"cite_spans": []
} | 1806.04516 | Parameterized discrete uniformization theorems and curvature flows for
polyhedral surfaces, I | [
"Xu Xu"
] | [
"math.GT",
"math.DG"
] | 2,018 | en | Mathematics | [
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bb88182f62ca24231025e6040b0541ee29f766f0 | subsection | 30 | 33 | Combinatorial | For \alpha =0, \alpha -curvature \mathbf {F}_\alpha is the classical discrete curvature \mathbf {F}. The existence of constant curvature PL metric
is ensured by Theorem REF .
If \chi (S)=0, the constant \alpha -curvature metric is a zero \alpha -curvature metric for all \alpha \in \mathbb {R}.
Specially, it is a PL met... | {
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b2cc41482d6bfd345bbf943b2517f60d52e2c3fa | subsection | 31 | 33 | Combinatorial | It is straightway to check that if the combinatorial \alpha -Calabi flow with surgery (REF )
converges, the limit metric is a constant \alpha -curvature PL metric.We have the following result for combinatorial \alpha -Calabi flow with surgery (REF ).Theorem 3.7 Suppose (S, V) is a closed connected marked surface with a... | {
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} | 1806.04516 | Parameterized discrete uniformization theorems and curvature flows for
polyhedral surfaces, I | [
"Xu Xu"
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5ef653df6b5b1c3e7ce330c36cb25dcf46b3d50a | subsection | 32 | 33 | Combinatorial | By the properness of W_\alpha , u(t) is bounded
along the combinatorial \alpha -Calabi flow with surgery (REF ),
which implies the long-time existence of combinatorial \alpha -Calabi flow with surgery.As W_\alpha (u(t)) is bounded along the combinatorial \alpha -Calabi flow with surgery
and \frac{dW_\alpha (u(t))}{dt}\... | {
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fe8b46634562844d812e27bd5979b1679c7e6bbc | abstract | 0 | 9 | Abstract | In a recent paper by Harrison et al., the concept of program completion is
extended to a large class of programs in the input language of the ASP grounder
gringo. We would like to automate the process of generating and simplifying
completion formulas for programs in that language, because examining the output
produced ... | {
"cite_spans": []
} | 1810.00453 | anthem: Transforming gringo Programs into First-Order Theories
(Preliminary Report) | [
"Vladimir Lifschitz",
"Patrick Lühne",
"Torsten Schaub"
] | [
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418f7b2413d56c336c914324f0e7d9a17f0ac78c | subsection | 1 | 9 | Introduction | Harrison et al. extended the concept of program
completion to a large class of nondisjunctive programs in the input
language of the ASP grounder gringo . They argued that it
would be useful to automate the process of generating and simplifying
completion formulas for (tightTightness is a syntactic condition
that guar... | {
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(Preliminary Report) | [
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660da17bdcfb6301220ab8d6a06d6aaaf62bbcce | subsection | 2 | 9 | Introduction | Differences between atoms in gringo programs and atomic parts of
formulas are related mostly to arithmetic expressions (see
Section below).The GitHub repository of anthem
contains the source code, installation steps, usage instructions, as well as
multiple examples to experiment with.https://github.com/potassco/anthem | {
"cite_spans": []
} | 1810.00453 | anthem: Transforming gringo Programs into First-Order Theories
(Preliminary Report) | [
"Vladimir Lifschitz",
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06738312a7f649945d56b06583871ff650bcdd6c | subsection | 3 | 9 | Examples | Example 1.
Given the input file
[language=anthem]
s(X) :- p(X).
s(X) :- q(X).external p(1).
external q(1).
anthem generates the formula
[language=FOL]
forall V1 (s(V1) <-> (p(V1) or q(V1)))
The first two lines of the input file express the conditions=p\cup qin the language of logic programming. The last two lines tel... | {
"cite_spans": []
} | 1810.00453 | anthem: Transforming gringo Programs into First-Order Theories
(Preliminary Report) | [
"Vladimir Lifschitz",
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55fe6b7de2dcd5ab75b7de28b1e0303b8a7ea315 | subsection | 4 | 9 | Examples | These simplifications will be implemented in a future version of anthem. | {
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(Preliminary Report) | [
"Vladimir Lifschitz",
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1f37405affdace4852f3dc9127a69812e749ce23 | subsection | 5 | 9 | Arithmetic Expressions in Formulas | In the output of anthem, an integer variable can be recognized by its
first character—the letter N. For instance, the formula\texttt {\textbf {exists} N p(N)}is stronger than\texttt {\textbf {exists} X p(X)}—it expresses that the set p contains at least one integer (and not
only ground terms formed using symbolic const... | {
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07308144d816ac806fd70bbcdc5a65e79a5c41c6 | subsection | 6 | 9 | Examples Involving Arithmetic Expressions | Example 5.
The program
[language=anthem]
letter(a). letter(b). letter(c).
p(1..3, Y) :- letter(Y).
:- p(X1, Y), p(X2, Y), X1 != X2.
q(X) :- p(X, ).
:- X = 1..3, not q(X).show p/2.
encodes the set of permutations of the letters a, b, c. anthem
transforms it into
[language=FOL]
forall N1, V1 (p(N1, V1) -> (N1 in 1..3 an... | {
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} | 1810.00453 | anthem: Transforming gringo Programs into First-Order Theories
(Preliminary Report) | [
"Vladimir Lifschitz",
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c91e74d786de3ac70ddbec9a279e1d4a37a24337 | subsection | 7 | 9 | Implementation | The implementation of anthem takes advantage of gringo's library
functionality for accessing the abstract syntax tree (AST) of a nonground
program. The AST obtained from gringo is taken by anthem and
turned into the AST of the collection of formulas representing the rules
of the program . That tree is then turned into ... | {
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023685fb31e7987653468ed2e7d4b6783b379fa3 | subsection | 8 | 9 | Future Work | Future work on anthem will proceed in two main directions. First, we
would like to support aggregates and conditional literals. When this is
accomplished, we will be able to replace, for instance, the first four rules
of Example 4 by a single rule
[language=anthem]
1 color(V, C) : color(C) 1 :- vertex(V).
With aggrega... | {
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670d9f99454769cc116a371416624a8cf3d5ef72 | abstract | 0 | 100 | Abstract | For online resource allocation problems, we propose a new demand arrival
model where the sequence of arrivals contains both an adversarial component and
a stochastic one. Our model requires no demand forecasting; however, due to the
presence of the stochastic component, we can partially predict future demand as
the seq... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
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e15179a42f05009783d672e9d2a76ff52ceb515d | subsection | 1 | 100 | Introduction | E-commerce platforms host markets for perishable resources in various industry sectors ranging from airlines to hotels to internet advertising.
In these markets, demand realizes sequentially, and the firms need to make online (irrevocable) decisions regarding how (and at what price) to allocate resources to arriving de... | {
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211ebb4adf8ed029a434252bd4b7ad8836ef8311 | subsection | 2 | 100 | Introduction | Therefore parameter p determines the level oflearnability predictability of demand.From a practical point of view, our demand model requires no forecast for the number of customers from each fare class of each type prior to arrival; instead, it assumes a rather mild “regularity” in the arrival pattern: a fraction p of ... | {
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f0408e3ec7afe051b3df5f045109cf0ed7ac8e56 | subsection | 3 | 100 | Introduction | In Section ?, we discuss how to estimate the parameter p, and comment on how robust our algorithm is to overestimating p.For the above problem, we design two online algorithms (a non-adaptive and an adaptive oneWe call an algorithm “adaptive” if it makes decisions based on the sequence of arrivals it has observed so fa... | {
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"raw": "Cheapair (2016). What the airlines never tell you about airfares. https://www.cheapair.com/blog/travel-tips/what-the-airlines-never-tell-you-about-airfares/.",
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d9ea0da60b933d6d4f6a40b9a6bbf5eab8fdfb2c | subsection | 4 | 100 | Introduction | This highlights the need to design new algorithms when departing from traditional approachesarrival models.We alsoconsider a classic stopping time problem known as study the classic secretary problem under our partially predictable arrival model.
The secretary problem, a stopping time problem,
corresponds to the setti... | {
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865926c74c15340b3d2031d5d77ecf6937c13689 | subsection | 5 | 100 | Literature Review | Online allocation problems have broad applications in revenue management, internet advertising, scheduling appointments (through web applications) in health care, just to name a few. Thus it has been studied in various forms in operations research and management, as well as computer science.
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9069c556db1d52448c96a19fb78d7ff9279714c1 | subsection | 6 | 100 | Literature Review | However, when 0 < p < 1, we show, in Subsection REF , that for a certain class of instances our algorithms perform better than that of .Nonstationary stochastic models:
Motivated by advanced service reservation and scheduling, and studied online allocation problems where demand arrival follows a known nonhomogeneous Po... | {
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"raw": "Agrawal, S., Z. Wang, and Y. Ye (2014). A Dynamic Near-Optimal Algorithm for Online Linear Programming. Operations Research 62(4), 876–890.",
... | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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5a7054757c4b11b892dc2a72f72fe8ea5f036dad | subsection | 7 | 100 | Literature Review | Parameter \lambda plays a similar role as parameter p in our model, in that it controls the adversary's power.
However, the underlying arrival processes in these two models differ considerably and cannot be directly compared.
In particular, we do not assume any prior knowledge of the stochastic component; instead we pa... | {
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c3e5d9418310c73e3aadefc76eed05587204b666 | subsection | 8 | 100 | Model and Preliminaries | A firm is endowed with b (identical) units of a product to sell over n \ge 3 periods, where n \ge b. In each period, at most one customer arrives demanding one unit of the product; customers belong to twoclasses types depending ontheir willingness-to-pay: class-1 and class-2 customers are willing to pay \$ 1 and \$ a r... | {
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c1ed7bcfa39e82443eb9ce3108af73d7042fe4c7 | subsection | 9 | 100 | Model and Preliminaries | Each customer joins the predictablestochastic group independently and with the same probability p.
Other customers are in the unpredictableadversarial group denoted by {\color {black}\mathcal {A}}.
Customers in the predictablestochastic group are permuted uniformly at random among themselves. Formally, a permutation \s... | {
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3400090c1c891d66ab1304c0dc1c94ba4d073333 | subsection | 10 | 100 | Model and Preliminaries | This idea is formalized later in Subsection REF along with further analysis of our model.Having described the arrival process, we now define the competitive ratio of an online algorithm under the proposed partiallylearnable predictable model as follows:An online algorithm is c-competitive in the proposed partiallylearn... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
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2832d4d8ac780f3e0d121b48a43a70bf1a03b5bc | subsection | 11 | 100 | Notational Conventions | Throughout the paper, we use uppercase letters for random variables and lowercase ones for realizations. We have already used this convention in defining \vec{V} vs. \vec{v}.
We normalize the time horizon to 1, and represent time steps by \lambda = 1/n, 2/n, \dots , 1.
First, we introduce notations related to the rando... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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c585d26abe0cfcbd3e82eaff4596dc2ae67df3e5 | subsection | 12 | 100 | Notational Conventions | Looking at the top row that shows sequence {\color {black}\vec{v}_I}, we have: n_1=4, \eta _1(5/8) = 3, \tilde{o}_1(5/8) = 0.5\times 3 + 0.5\times 4 \times (5/8) = 2.75, and \tilde{o}^{{\color {black}\mathcal {S}}}_1(5/8) = 0.5\times 4 \times (5/8)=1.25 that are all deterministic quantities.
Similarly, forclass type-2 ... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
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8a0fb5fa858774994e33a89e3761c83a61227d00 | subsection | 13 | 100 | Estimating Future Demand | At time \lambda < 1, upon observing o_j(\lambda ), j=1,2 (but not n_j and \eta _j(\lambda )), we wish to estimate future demand, or equivalently the total demand n_j.
To make such an estimation, we establish the following concentration result:Define constants \alpha \triangleq 10 + 2 \sqrt{6}, \bar{\epsilon }\triangleq... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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a90d7c824d55c4371ac293a7327860157cf0ab1a | subsection | 14 | 100 | Estimating Future Demand | Roughly, a total of p n_jclass type-j customers belong to the predictablestochastic group, and a \lambda fraction of them arrive by time \lambda , because these customers are spread almost uniformly over the entire time horizon.
As a result, there are approximately p n_j \lambda class type-j customers from {{{\color {b... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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ecc493cdc865f64b80fe2b0f4dd09ce51e574976 | subsection | 15 | 100 | Estimating Future Demand | Combining these with our deterministic approximations leads us to compute upper bounds on the total number of customers as established in Lemma REF .Finally, based on Lemma REF , fixing \epsilon \in [\frac{1}{n}, \bar{\epsilon }], we partition the sample space of arriving sequences into two subsets, \mathcal {E} and it... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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345e1389db852b3567b7657c9ea40d14a3a7210d | subsection | 16 | 100 | A Non-Adaptive Algorithm | In this section, we designpresent and analyze aour first onlinenon-adaptive algorithm for the resource allocation problem and the demand model described in Section .
First, in Section REF , we describe the algorithm.
Then, in Section REF , we present the analysis of its competitive ratio. | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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8bef95c7322dd1b72e759f24de693072d19ee2fe | subsection | 17 | 100 | The Algorithm | Our first algorithm is a non-adaptive online algorithm that uses predetermined dynamic thresholds to accept or reject customers.
This algorithm combines some ideas from the primal algorithm of and the threshold algorithm of to capturegenerate maximal revenue from both the predictablestochastic and unpredictableadvers... | {
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f2ccad4fce02bbd2a6a2520f83a44b9b8ae714d1 | subsection | 18 | 100 | The Algorithm | For a certain range of c, we show that ALG_{2,c} attains a competitive ratio of c (up to an error term); however, if c becomes too large (for example if c=1),
then ALG_{2,c} no longer guarantees a c fraction of the optimum offline solution.
We call this algorithm adaptive because it makes decisions based on the sequenc... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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e3e1ee50c09626dcc293bb3a465e9585af49d7db | subsection | 19 | 100 | The Algorithm | Recall that we defined \Delta to be \alpha \sqrt{b \log n}, where constant \alpha itself is defined in Lemma REF .Under event \mathcal {E}, ifsuppose n_1\ge \frac{k}{p^2} \log n and b > \left({\frac{1}{{\color {black}\bar{\epsilon }}} \frac{n\sqrt{\log n}}{(1-c)^2ap^{3/2}}}\right)^\frac{2}{3}, where
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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0b87c8d3038a5614c769bfa14d58ee56796c2093 | subsection | 20 | 100 | The Algorithm | The decision of whether to accept the customer is based on the following two observations:Observation 1 If u_1(\lambda ) \ge n_1, thenOPT(\vec{v}) \le \min \lbrace n_1, b\rbrace + a (b-n_1)^+ = (1-a) \min \lbrace n_1, b\rbrace +ab \le \min \lbrace u_1(\lambda ) , b\rbrace (1-a)+ab.Observation 2
If we accept the curren... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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4f6e0cd2673af8ce997931944638929631346f93 | subsection | 21 | 100 | The Algorithm | Repeat for time \lambda = 1/n, 2/n, \dots , 1:
Calculate functions u_1 (\lambda ) and u_{1,2} (\lambda ) (to construct upper bounds for n_1 and n_1+n_2):
u_1 (\lambda ) & \triangleq {\left\lbrace \begin{array}{ll}
b &\text{ if }\lambda <\delta \text{ (not enough data {to learn}{\color {black}observed})}.\\
\min \left... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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e8a4b7249b4b41572baa1398387dcd49d37143c4 | subsection | 22 | 100 | The Algorithm | We identify the sufficient condition for c-competitiveness by solving the following mathematical program whose feasibility region contains all such instances. Tthe factor-revealing mathematical program is presented in (REF ). We will explain the construction of this program in the analysis of the competitive ratio (in ... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
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ae9ff61f58024deef4f266c38666d8e9a8754f1c | subsection | 23 | 100 | The Algorithm | First, we solve (REF ) numerically for the regime where b= \kappa n (where 0< \kappa \le 1 is a constant), and show that if b/n > 0.5, then Algorithm REF achieves a better competitive ratio than Algorithm REF .
[Figure: Solution of (), c^*, vs. p for a=0.50 and 0.70]In Figure REF , we fix a=0.5,0.7, and plot c^* for p ... | {
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} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
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bf5df8796a901768890cc68e301b9305bc452c61 | subsection | 24 | 100 | Competitive Analysis | ::The theorem was stated twice! We can't just copy the whole appendix here...
::I restructured the proof; defined v_A in Lemma 4.7 instead of \bar{v}; fixed numerous typos. One question: we don't use \theta b integer anymore, right? I changed all of them to \lfloor \theta b \rfloor .In this subsection, we analyze the ... | {
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b8882dad02e3f0be5addd62632315c8b0b059c0e | subsection | 25 | 100 | Competitive Analysis | In particular,
recalling that we defined constant \bar{\epsilon }= 1/24 in Lemma REF , if \frac{1}{a(1-p)p}\sqrt{\frac{\log n}{b}} \ge \bar{\epsilon },
then O\left(\frac{1}{a(1-p)p}\sqrt{\frac{\log n}{b}}\right) becomes O(1) and Theorem REF becomes trivial.
Therefore, without loss of generality, we assume \frac{1}{a(1-... | {
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0.003209067974239588,
0.007635368499904871,
0.020345719531178474,
-0.042095646262168884,
0.027809379622340202,
-0.007188922725617886,
... | |
bb16dda783581da8dd45d9306ceb3967eac36765 | subsection | 26 | 100 | Competitive Analysis | In the main part of the proof we show that, for any realization \vec{v} belonging to event \mathcal {E},\frac{{ALG_1(\vec{v})}}{OPT({\color {black}\vec{v}_I})} \ge p+\frac{1-p}{2-a} - O(\epsilon ).Fixing a realization \vec{v} that belongs to event \mathcal {E}, we define q_1(\lambda ), q_{2,e}(\lambda ), and q_{2,f}(\l... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.05313192307949066,
-0.033478301018476486,
-0.015442132949829102,
0.016266120597720146,
0.005416953936219215,
0.009971772320568562,
0.002143892925232649,
0.01087968423962593,
0.02536049857735634,
0.04428169131278992,
-0.019790954887866974,
0.00862134899944067,
-0.01876860111951828,
0.019... | |
6437db8cc51dfaf57571f6c64b16e16a825b26b1 | subsection | 27 | 100 | Competitive Analysis | Note that because the algorithm accepts all type-1 customers, this implies q_1(\lambda , \vec{v}) \ge q_{1}(\lambda , \vec{v}_{M}), which proves our claim (i.e., q_{2,e}(\lambda , \vec{v}) \le q_{2,e}(\lambda , \vec{v}_{M})). Thus, without loss of generality, we assume n_1 \le b.
Further, note that because of condition... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.062035344541072845,
-0.0024277849588543177,
-0.010893564671278,
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0.022504214197397232,
-0.015333379618823528,
0.012609988451004028,
-0.032589152455329895,
0.013... | |
32521dd1f0c8b35addae7be350b1d75aa73236a3 | subsection | 28 | 100 | Competitive Analysis | Therefore, the ratio between ALG_1(\vec{v}) and OPT(\vec{v}) can be expressed as:\frac{ALG_1(\vec{v})}{OPT(\vec{v})} = \frac{n_1 + a \left[q_{2,e}(1) +q_{2,f}(1)\right]}{n_1 + a \min \lbrace n_2,(b-n_1)\rbrace }.The only “mistake” that the algorithm may make is to reject too many classtype-2 customers. The following le... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.05690944194793701,
-0.013395842164754868,
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0.01039779745042324,
-0.018552783876657486,
... | |
6c80c9225f349bddfed1ccc783ebed4853c7697d | subsection | 29 | 100 | Competitive Analysis | With the same realization of the predictablestochastic group and random permutation, we claim that:q_{2,e}(\lambda , \vec{v}) \ge q_{2,e}(\lambda , \vec{v}_A), ~~ \lambda \in \lbrace 0, 1/n, \ldots , 1\rbrace .Otherwise, we consider an alternative adversarial instance \vec{v}_{I,A} that keeps the values of arbitrary b-... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.0626886710524559,
0.012719329446554184,
-0.014031701721251011,
-0.01474892906844616,
0.013879100792109966,
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0.0463297963142395,
0.019044658169150352,
0.029360515996813774,
-0.028750110417604446,
-0.... | |
3d8fe367c712f52c512b59b7ed98dedb212a7777 | subsection | 30 | 100 | Competitive Analysis | Because o_1(\lambda ,{\vec{v}}_A) + q_{2,e}(\lambda , {\vec{v}}_A) \le {\lfloor \lambda pb \rfloor }, and o_1(\lambda ,{\vec{v}}) = o_1(\lambda ,{\vec{v}}_A), we can conclude that q_{2,e}(\lambda , {\vec{v}}_A) \le q_{2,e}(\lambda , \vec{v}) (REF ) holds in the last case as well. This concludes the induction.
Thus, wit... | {
"cite_spans": []
} | 1810.00447 | Online Resource Allocation under Partially Predictable Demand | [
"Dawsen Hwang",
"Patrick Jaillet",
"Vahideh Manshadi"
] | [
"cs.DS"
] | 2,018 | en | Computer Science | [
-0.03941824287176132,
0.020029470324516296,
-0.03337736427783966,
-0.012219044379889965,
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0.008222303353250027,
0.010434240102767944,
0.03719104826450348,
0.002284396905452013,
0.0066625... |
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