diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzczy" "b/data_all_eng_slimpj/shuffled/split2/finalzczy"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzczy"
@@ -0,0 +1,5 @@
+{"text":"\n\n\\subsection{Benchmark Applications}\n\\label{subsec:applications}\n\\input{applications}\n\n\\subsection{Metrics Collected by Benchmarking}\n\\label{subsec:metrics}\n\\input{metrics}\n\n\\subsubsection*{Adding New Benchmarks and Metrics}\nExtensibility is a consideration in the design of \\texttt{DeFog}. Therefore, new benchmarks can be included to the portfolio of applications. \nThe benchmarks will need to follow a specific directory structure. The template scripts are provided within the software for setting up and running the containers for the new application. Similarly, new application metrics can be obtained by including appropriate functions in a \\texttt{DeFog} script. This is beyond the scope of this paper (detailed steps are provided in the software repository).\n\n\n\n\n\\subsection{Motivation}\nFog benchmarking as a research area is still in its early stages of development. The dependencies between cloud-edge services of an application are not fully known given that there are only a few open source Fog applications available. \nThe following five general observations regarding Fog benchmarking have been considered while developing \\texttt{DeFog}:\n\n(i) \\textit{Fog benchmarking is complex as dependencies between cloud and edge services of a Fog application need to be considered}: In cloud benchmarking, the dependencies of the application are mostly in the cloud. However, the dependencies between cloud and edge services of a Fog application will need to be considered for Fog benchmarking. In other words, the bespoke requirements and dependencies of an application will need to be considered ideally for benchmarking the interactions in a Fog system, which is considered in \\texttt{DeFog}. This makes Fog benchmarking more complex than cloud benchmarking. \n\n(ii) \\textit{Fog benchmarks are not readily available}: There is a limited understanding of the real workloads that can benefit the most from using Fog computing. Consequently, there are no open source Fog benchmarks readily available. While the portfolio of benchmarks for traditional computing platforms, such as high-performance clusters or even the cloud are diverse and comprehensive, Fog benchmarks are not yet developed. \\texttt{DeFog} attempts to create an early repository of six benchmark applications that can be expanded upon by the community as our understanding of Fog applications evolves and as applications become available. \n\n(iii) \\textit{Fog benchmarking should generate rapid results}: Ideal benchmarking should generate results quickly and in the context of the Fog it is essential given that the edge is a transient environment. Resources located on the edge (routers or gateways made accessible for general purpose computing) are anticipated to have intermittent profiles when compared to dedicated resources in a data center. \\texttt{DeFog} is therefore designed to execute in a lightweight manner and under one minute.  \n\n(iv) \\textit{Metrics captured during Fog benchmarking must be generalised to a wide variety of workloads}: Many benchmarking solutions are workload specific and therefore generate workload specific metrics on a target platform. Hence, they cannot provide insight into the suitability of the target platform for a different class of workloads. The aim of \\texttt{DeFog} is to evolve over time by adding more workloads so that a wide range of metrics can be captured for diverse Fog platforms and application benchmarks. \n\n(v) \\textit{Benchmarking needs to be consistent}: To ensure that benchmarking is consistent each execution of the benchmark must be on the same application build version and package dependencies. This is ensured in \\texttt{DeFog} so that cloud-edge services are consistently benchmarked. In addition, to minimise the impact of any noise within the Fog platform (due to temporary network congestion or spikes in workloads), the benchmark needs to be executed multiple times for a range of input data using identical containers to obtain an averaged values of metrics. \n\n\\subsection{Benchmarking Methodology}\n\\label{subsec:benchmarkingmethodology}\n\nThe aim of \\texttt{DeFog} is to automate the deployment of benchmark applications on the target platform, the transfer of assets required for these applications to run on the target platform, the generation of metrics relevant to the target platform and benchmark, and finally gather results. The proposed benchmarking methodology used by \\texttt{DeFog} accounts for these and is a sequence of six steps, which is as follows: (1) Build and run the benchmark application container image for the target platforms, (2) transfer the required asset to the cloud resource on the target platform, (3) transfer the required asset to the edge resource on the target platforms, (4) execute the benchmark application, (5) gather the values for the catalogue of metrics, and (6) return the metrics to the user. The third step is required for the edge-only or cloud-edge (Fog) deployment modes. The second to fifth step is repeated multiple times for each application for consistency. The metrics are measured by an observing system. The methodology in detail is as follows:\n\n\\textit{Step 1 - Build and run a container}: In this first step, a container image is built and tagged for a target platform (cloud and\/or edge) to ensure that a consistent execution environment is available for all future runs of the benchmark. This research employs Docker containers and it is run in the detached mode to allow for concurrent benchmarking tasks to be executed on other application containers. \n\n\\textit{Step 2 - Transfer assets to the cloud}: An asset that is transferred to the cloud is the input data required by the application's service that runs on the cloud. Each benchmark application has its corresponding asset and a single benchmark application may have multiple assets. For example, there are currently six benchmark applications in \\texttt{DeFog} and each application has its own assets. In addition, each application may have multiple assets corresponding to similar or different sizes or types of input. This ensures that \\texttt{DeFog} can be used for varying inputs of the application to identify performance gain under different operating conditions for the application. Some of the communication related metrics discussed in a subsequent section are also measured during this step.   \n\n\\textit{Step 3 - Transfer assets to the edge}: The asset transferred in this case is the input data required by the application's service that will be hosted on the edge resource. Currently, the data required by the application benchmark has to be manually partitioned and provided as assets. Automation within this step is desirable, but is not within the scope of this article. Some of the communication related metrics discussed in a subsequent section are measured during this step.  \n\n\\textit{Step 4 - Execute the benchmark application}: Given the running container (from Step 1) and the assets (Step 2 and Step 3) for each application the number of times the benchmark needs to be executed is determined. If there are $N$ assets for an application, then the benchmark is executed $N$ times. The benchmark applications are further considered in Section~\\ref{subsec:applications}. Additional benchmarking tools, namely UnixBench\\footnote{\\url{https:\/\/github.com\/kdlucas\/byte-unixbench\/tree\/master\/UnixBench}} and Sysbench\\footnote{\\url{https:\/\/github.com\/akopytov\/sysbench}} are included within \\texttt{DeFog}. They provide a large number of target platform related metrics, including CPU performance, concurrency and I\/O read write.\n\n\\textit{Step 5 - Gather the values for a catalogue of identified metrics}: A combination of communication and computation related metrics obtained from DeFog. They can be categorised as: (i) target platform metrics, which are attributes that capture the characteristics of the specific target platform under consideration, (ii) Fog application performance metrics, which are attributes that provide insight into the performance of the application on the target platform, and (iii) metrics obtained from external tools, which are attributes obtained by using third party tools. The metrics will be discussed further in Section~\\ref{subsec:metrics}.\n\n\\textit{Step 6 - Provide the metrics to the user}:\nThe target audience that can benefit from \\texttt{DeFog}, include (i) vendors of new edge hardware who want to demonstrate the benefit of using the Fog via benchmarks, (ii) Internet Service Providers (ISPs) who want to deploy micro data centres at the edge of the network and want to tabulate performance of Fog applications for their customers, (iii) system software administrators who want to investigate the impact on Fog applications when a system level change, such as update to operating system or libraries, is required on the edge resource, and (iv) network administrators who want to quantify the performance of Fog applications when a new networking protocol is introduced. When the benchmarks have completed execution the metrics are provided to the observing system in the form of both comma separated and verbose text files. \n\n\n\\begin{figure}[]\n\t\\centering\n\t\\includegraphics[width=0.5\\textwidth]{deploymentmodes.png}\n\t\\caption{\\texttt{DeFog} deployment modes, namely cloud-only, edge-only and cloud-edge (Fog).}\n\\label{fig:deploymentmodes}\n\\end{figure}\n\n\\subsection{Deployment Modes}\nThe \\texttt{DeFog} benchmarking suite works across three distinct deployment modes, namely a cloud-only, edge-only, and cloud-edge (Fog), as shown in Figure~\\ref{fig:deploymentmodes} and considered below. The underlying principle is that the relative performance of the benchmark applications should be compared on different target platforms to quantify any benefit of leveraging the edge for a cloud application. \n\nThe resources available for the target platform as shown in Figure~\\ref{fig:deploymentmodes} are from: (i) the cloud resource layer in which a large amount of computational and storage resources are available, (ii) the edge resource layer, which comprises either dedicated micro clouds or traffic routing nodes, such as gateways, routers and base stations, that are augmented with computing and\/or storage capabilities, and (iii) the user devices layer, which comprises devices, such as smartphones, wearables, and sensors. A computer system is utilised to observe the benchmarking process and collect the metrics. \n\n\\textbf{\\textit{(i) Cloud-only deployment}}: \nThe cloud-only deployment is typical of conventional cloud applications in which all requests originating from an end user-device are serviced by a cloud resource. Figure~\\ref{fig:cloudonlydeploymentmode} shows the two tier cloud-only deployment mode comprising the cloud resources and user devices. In \\texttt{DeFog}, the application is built and deployed on the cloud resource using Docker containers. The application is then executed in the container and the benchmark metrics are generated. The method adopted is based on a container-based cloud benchmarking approach previously reported~\\cite{cloudcontainerbenchmarking-01}. The user device uses the secure shell to interact with the cloud container during the benchmarking process.\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.5\\textwidth]{cloudonlydeploymentmode.png}\n\t\\caption{The \\texttt{DeFog} cloud-only deployment mode.}\n\\label{fig:cloudonlydeploymentmode}\n\\end{figure}\n\nA \\texttt{DeFog} user specifies which applications in the repository need to be benchmarked and the accompanying asset is transferred to the cloud (in this research we used Amazon Web Services (AWS) Elastic Compute Cloud (EC2)\\footnote{\\url{https:\/\/aws.amazon.com\/ec2\/}}). The output data from the application is uploaded to an AWS Simple Secure Storage (S3)\\footnote{\\url{https:\/\/aws.amazon.com\/s3\/}} bucket. The output is also transferred to the observing system along with the metrics generated during the benchmarking process. \n\n\\textbf{\\textit{(ii) Edge-only deployment}}:\nThe edge-only deployment mode assumes that all services of an application can be entirely run on an edge resource as shown in Figure~\\ref{fig:edgeonlydeploymentmode}. This is practical if it is assumed that dedicated micro clouds or modular data centres are located at the edge of the network and the application provider replicates the application on multiple geographic locations closer to the end user. Similar to the cloud-only deployment mode, the application is deployed on the the edge resource using Docker containers. The application is then run within the container and metrics are generated. In this research resource constrained single board computers, such as Odroid XU4\\footnote{\\url{https:\/\/magazine.odroid.com\/odroid-xu4}} and Raspberry Pi 3\\footnote{\\url{https:\/\/www.raspberrypi.org\/products\/raspberry-pi-3-model-b\/}} are used as edge resources. The outputs and the metrics are stored in an S3 bucket as well as sent to the observing system.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.5\\textwidth]{edgeonlydeploymentmode.png}\n\t\\caption{The \\texttt{DeFog} edge-only deployment mode.}\n\\label{fig:edgeonlydeploymentmode}\n\\end{figure}\n\n\n\\textbf{\\textit{(iii) Cloud-edge (Fog) deployment}}:\nIn the Fog deployment mode, services of an application may be distributed across the cloud and edge as shown in Figure~\\ref{fig:fogdeploymentmode}. Communication latency or bandwidth sensitive services of the application are offloaded to the edge so that the overall QoS of the application can be maximised. \n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.5\\textwidth]{fogdeploymentmode.png}\n\t\\caption{The \\texttt{DeFog} cloud-edge (Fog) deployment mode.}\n\\label{fig:fogdeploymentmode}\n\\end{figure}\n\nConsider for example a deep learning application for object detection. This application will require the training of a model, which is a computationally intensive task, and cannot be easily carried out on the edge. If large micro cloud like resources are assumed at the edge, then it may be possible. However this will not be the case if there are only smaller form factor and resource constrained embedded devices available on the edge. Therefore, the training service of the application is ideally suited for the cloud. The detection service could be offloaded to the edge. The assets required by this service are a trained model and associated weights which will need to be offloaded from the cloud to the edge.  \n\\section{Introduction}\n\\label{sec:introduction}\n\\input{introduction}\n\n\\section{DeFog Benchmarking}\n\\label{sec:defogbenchmarking}\n\\input{defog}\n\n\\section{Benchmark Applications and Metrics Collected by \\texttt{DeFog}}\n\\label{sec:applicationsandmetrics}\n\\input{applicationsandmetrics}\n\n\\section{Experimental Studies}\n\\label{sec:experimentalstudies}\n\\input{experiments}\n\n\\section{Related Work}\n\\label{sec:relatedwork}\n\\input{relatedwork}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\\input{conclusions.tex}\n\n\\balance\n\\bibliographystyle{ACM-Reference-Format}\n\n\\subsection{Implementation and Setup}\n\\label{sec:implementation}\n\\input{implementation.tex}\n\n\n\\subsection{Results}\n\\label{sec:results}\n\\input{results}\n\n\n\n\n\n\n\\subsubsection{Application latencies for different deployment modes}\nFigure~\\ref{fig:latencies} show the communication and computation round trip times for the three deployment modes, namely the cloud-only, edge-only, and cloud-edge (Fog) deployments. The standard deviation of the executions is highlighted in the observed results. In these executions, the edge resources are exclusively used by the benchmark application. The communication latency (Figure~\\ref{fig:communicationlatency}) is consistently lower for all applications running on the edge when compared to the cloud. The computation latency (Figure~\\ref{fig:computationlatency}) is significantly larger for applications, such as YOLO and PocketSphinx, on the edge compared to the cloud. This is because the edge resources employed in this research are hardware limited compared to a cloud resource. The computational cost of these applications exceeds the gains in communication latencies on the edge. There is a slight increase in the latency times for the cloud-edge deployment modes when compared to the edge only deployment, which is due to the time needed to transfer assets from the cloud to the edge. Aeneas and FogLAMP show comparable and sometimes lower computational latency on the edge when compared to the cloud. The lack of performance gain for the former two applications may be due to the specific manner in which the application is partitioned. \n\nFigure~\\ref{fig:realfdlatencies} shows the computation and communication latencies for different combination of services of the RealFD application (FD, MD, GSC) on the three deployment modes. For this application, the results obtained using the Odroid XU4 are presented. Communication latency in this figure is the single trip time taken (not round trip) and the computation latency is the time taken to process a single video frame. There are two potential deployment options across the cloud and edge - FD on the cloud, and MD and GSC on the edge, or alternatively FD and MD on the cloud, and GSC on the edge. There is a difference in the performance of the two Fog deployments and \\texttt{DeFog} highlights this variation for the benchmark. \n\n\n\\subsubsection{Impact of stressing edge resources on application latencies}\nFor this experiment Gaussian workloads are simulated on the edge nodes in the cloud-edge deployment mode for YOLO, PocketSphinx, Aeneas, and RealFD and the edge-only deployment mode for FogLAMP. The motivation is to simulate a real world multi-tenant distributed system where there are competing workloads residing on the same edge under variable network conditions. The \\texttt{stress}\\footnote{\\url{https:\/\/people.seas.harvard.edu\/~apw\/stress\/}} package is used to stress the edge resource by simulating computations in the background. The \\texttt{stress-ng}\\footnote{\\url{https:\/\/kernel.ubuntu.com\/git\/cking\/stress-ng.git\/}} package is used to stress the network. These packages use stressors to subject the computing cores and network to various levels of stress. \n\nIn this paper, we explicitly define minimal stress when one CPU core of the edge resource is stressed. The network is stressed by transferring a file of size 256MB at roughly 21740 bytes per second. For \\textit{low stress}, two CPU cores are stressed. For \\textit{medium} and \\textit{high stress}, three CPU cores and four CPU cores are stressed respectively. For \\textit{very high stress} all CPU cores are stressed and the RAM memory is stressed using two stressor processes.  \n\nThe communication latency of benchmark applications when network bandwidth is stressed is shown in Figure~\\ref{fig:stresscommunicationlatencies} (for the RealFD application on Odroid XU4 is shown in Figure~\\ref{fig:stressrealfdcommunication}). Applications that transfer larger amounts of data, such as Aeneas, are affected. Figure~\\ref{fig:stresscomputationlatencies} shows the trend in the computation latencies of the applications when the computing cores are stressed (Figure~\\ref{fig:stressrealfdcomputation}). As expected there is a significant increase in the computation latencies. Computationally less intensive applications, such as Aeneas and FogLAMP have less effect with stressed CPU cores. It is immediately inferred that applications demonstrate different trends when edge resources are stressed. For RealFD it is noted that different combination of services across the cloud and the edge have different performance. \nFor the iPokeMon application as shown in Figure~\\ref{fig:jmeterstress}, the edge node is subject to similar stress as above. There is an improvement in the response latency of the cloud-edge deployment by over 7 times when compared to the cloud-only deployment when the stress as defined in this paper is very high.   \n\n\n\n\n\\begin{figure*}[ht]\n\\begin{center}\n\t\\subfloat[Communication latency on Odroid XU4]\n\t{\\label{fig:userscommunicationodroid}\n\t\\includegraphics[width=0.236\\textwidth]\n\t{users-communication-odroid.pdf}}\n\\hfill\n\t\\subfloat[Communication latency on Raspberry Pi 3]\n\t{\\label{fig:userscommunicationraspberry}\n\t\\includegraphics[width=0.236\\textwidth]\n\t{users-communication-raspberry.pdf}}\n\\hfill\n\t\\subfloat[Computation latency on Odroid XU4]\n\t{\\label{fig:userscomputationodroid}\n\t\\includegraphics[width=0.236\\textwidth]\n\t{users-computation-odroid.pdf}}\n\\hfill\n\t\\subfloat[Computation latency on Raspberry Pi 3]\n\t{\\label{fig:userscomputationraspberry}\n\t\\includegraphics[width=0.236\\textwidth]\n\t{users-computation-raspberry.pdf}}\n\\end{center}\n\\caption{Impact of concurrent users on latency of benchmark applications.}\n\\label{fig:userslatencies}\n\\end{figure*}\n\n\\begin{figure*}[ht]\n\\begin{center}\n\t\\subfloat[On Odroid XU4]\n\t{\\label{fig:usersbytestransferodroid}\n\t\\includegraphics[width=0.48\\textwidth]\n\t{users-bytestransfer-odroid.pdf}}\n\\hfill\n\t\\subfloat[On Raspberry Pi 3]\n\t{\\label{fig:usersbytestransferraspberry}\n\t\\includegraphics[width=0.48\\textwidth]\n\t{users-bytestransfer-raspberry.pdf}}\n\\end{center}\n\\caption{Impact of concurrent users on average bytes transferred.}\n\\label{fig:usersbytestransfer}\n\\end{figure*}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.48\\textwidth]{users-rtf.pdf}\n\t\\caption{Impact of concurrent users on real time factor for PocketSphinx.}\n\\label{fig:usersrtf}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.48\\textwidth]{jmeter-users.pdf}\n\t\\caption{Impact of concurrent users on average response latency per request for iPokeMon.}\n\\label{fig:jmeterusers}\n\\end{figure}\n\n\n\\subsubsection{Impact of concurrent users on application latencies}\nFor this experiment, concurrent users (2, 5, 10, and 50) of the application benchmark are considered as shown in Figure~\\ref{fig:userslatencies}. The individual requests from the application are simulated for concurrent users using JMeter. It is noted that the communication latency as shown in Figure~\\ref{fig:userscommunicationodroid} and Figure~\\ref{fig:userscommunicationraspberry} increases with the number of users. This is because the time in flight and time to transfer the results also increases because with the increasing number of users the rate at which data movement occurs decreases (this is highlighted in Figure~\\ref{fig:usersbytestransfer}). Similar increases are noted for computation latency. Although the rate at which the computation latency increases is different. For example, FogLAMP has a much lower rate of increase in computation latency when compared to PocketSphinx. This is because of the nature of the computational intensity underpinning the benchmarks. The figures highlight the need for \\texttt{DeFog} - to differentiate workloads that may not have a significant increase in computation or communication latencies even when there are multiple users versus those that may have the computation or communication latencies significantly affected.  \n\nApplications that have a larger execution time for a single user, are impacted the most by concurrent users, which results in larger computation latencies. For example, consider PocketSphinx as shown in Figure~\\ref{fig:usersrtf}. Given the current decomposition of services for the PocketSphinx application it is evident that the cloud-edge deployment is not advantageous over the cloud-only deployment. \n\nHowever, consider the impact of concurrent users on iPokeMon as shown in Figure~\\ref{fig:jmeterusers}. The cloud-edge deployment clearly has significant advantages over the cloud-only deployment. Latency spike observed for the edge nodes when there are 250 concurrent users was due to a single communication request that increased the average response time of the requests.\n\n\n\\subsubsection{Summary}\nThe experimental results have demonstrated that the communication and computation latencies vary across the different deployments, namely cloud-only, edge-only and cloud-edge. Although the communication latency of the application may improve by using the cloud-edge deployment, there will be no overall gain if the computation latency is high on resource constrained edge nodes. Applications are noted to behave differently when the edge node is stressed, both its system and network resources. The rate at which performance degrades varies across applications. Similarly, when concurrent users request service from the same edge node, the rate at which the communication and computation latencies increase vary; some workloads exhibit significant increase in latencies where as others have a negligible increase. These observations highlight the use of \\texttt{DeFog} \nto identify any performance gain in using the Fog over the cloud-only deployment (\\textbf{\\textit{Q1}}), which services of an application would benefit from moving to the edge (\\textbf{\\textit{Q2}}), and which deployment platforms are best suited for an application when there are multiple hardware choices (\\textbf{\\textit{Q3}}). ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nUltracold and interacting dilute alkali-metal vapors trapped by one-dimensional double-well potentials \\cite{oliver}\nprovides the possibility of studying the formation of macroscopic coherent states \\cite{smerzi,stringari,anglin,mahmud,anna}\nand macroscopic Schr\\\"odinger-cat states \\cite{cirac,dalvit,huang,carr,brand,main}.\nThe two-site Bose-Hubbard (BH) Hamiltonian \\cite{twomode:milburn} efficiently describes the microscopic\ndynamics of such systems. When the boson-boson interaction is repulsive and the number of bosons is even, the crossover from a delocalized atomic coherent\nstate to a (fully incoherent) localized Fock state (the so called twin Fock state with the particles equally shared between the two wells) takes place by increasing the interatomic coupling strength\n\\cite{stringari,anglin,mahmud,anna,main}. For attractively interacting bosons, the two-spatial mode BH model predicts the formation of a macroscopic Schr\\\"odinger-cat state\n\\cite{cirac,dalvit,huang,carr,brand} when the interatomic attraction becomes sufficiently large. Finally, when the attraction between\nthe bosons is sufficiently strong the collapse should take place \\cite{sb,io-e-boris}.\n\nMotivated by the concrete possibility to isolate single atomic ions \\cite{heidelberg,wineland1,bergquist,cheinet} and manipulate quantum gases at single-atom level \\cite{cheinet, diedrich, monroe,haroche-raimond,nobel} (note that D. J. Wineland was awarded in 2012 with the physics Nobel prize for his activity in this sector), we focus on the behavior of few trapped bosonic atoms at zero temperature.\n\nThe aim of the present work, then, is to study the ground state of a system consisting of a low number $N$ of bosons confined in a symmetric double-well trap and characterize it from the quantum correlations point of view. To do this we use the two-site Bose-Hubbard model. We diagonalize the underlying Hamiltonian by analytically finding the eigenvector and the eigenvalue of its lowest energetic state for $N=2$ - this case has already been discussed in \\cite{main} - and $N=3,4$ bosons. Hence, we provide analytical formulas for the parameters that describe the correlation properties of the ground state of the system. These parameters are: the Fisher information $F$ \\cite{braunstein,pezze} which is related to the fluctuation of the number of bosons in a given well and achieves its maximum in correspondence to the Schr\\\"odinger-cat state; the coherence visibility $\\alpha$ \\cite{stringari,anglin,anna} which measures the coherence related to the single-particle tunneling across the central barrier and attains its maximum value in correspondence to the atomic coherent state; the entanglement entropy $S$ \\cite{bwae} which quantifies the amount of the genuine quantum correlations of the ground state from the bi-partition perspective. In particular, we calculate $F$ and $\\alpha$ following two paths: on one hand by taking the average, with respect to the ground state, of the left-right population imbalance variance and the left-well hopping operator, respectively, and on the other hand by applying the Hellmann-Feynman theorem \\cite{cohen}. For both the calculations (that, as we shall comment, provide the same results) we use the analytically determined ground-state eigenvectors and eigenvalues.\n\nWe study the ground state and the parameters $F$, $\\alpha$, $S$ by widely exploring the atom-atom interaction range, from strong attractions to strong repulsions. In this latter regime, we comment about the of $N$ even-$N$ odd  difference: when $N$ is even (the ratio of the number of bosons to the number of wells is a positive integer) the ground state is a separable Fock state  with $N\/2$ particles in the left well and $N\/2$ particles in the right well (this is, as commented at the beginning, the twin Fock state), while when $N$ is odd (the total number of bosons is not commensurate with the number of wells) the ground state is given by a symmetric combination of two separable Fock states. When the boson-boson repulsion becomes sufficiently large, the quantities $F$, $\\alpha$, $S$, tend to zero for an even number of particles; they remain, instead, finite when $N$ is odd.\n\n\n\n\n\\section{The system}\n\nWe analyze a finite number $N$ of identical interacting bosonic atoms at zero temperature confined by a trapping potential $V_{trap}(\\bf{r})$. We suppose that this potential is given by the superposition of an isotropic harmonic confinement in the radial plane ($x-y$) and a double-well potential $V_{DW}(z)$ in the axial ($z$) direction, i.e.\n\\begin{equation}\n\\label{trapping}\nV_{trap}({\\bf r}) = \\frac{m\\omega_{\\bot}^2}{2}(x^2+y^2)+ V_{DW}(z)\n\\;,\\end{equation}\nwhere $m$ is the mass of the bosons and $\\omega_{\\bot}$ the trapping frequency in the radial plane. We assume that the double-well is symmetric in the $z$ direction and that the system is quasi one-dimensional due to a strong transverse radial harmonic confinement.\n\n\n\nIn the second quantization language, the Hamiltonian that controls the microscopic dynamics of the system is\n\\begin{eqnarray}\n\\label{system:ham0}\n\\hat{H} &=& \\int d^{3}{\\bf r}\\hat{\\Psi}^\\dagger({\\bf r})(-\\frac{\\hbar^2}{2m}\\nabla^2+V_{trap}({\\bf r}))\\hat{\\Psi}({\\bf r}) \\nonumber\\\\\n&+&\\frac{1}{2}\\int d^{3} {\\bf r}d^{3} {\\bf r'}\\hat{\\Psi}^\\dagger({\\bf r})\\hat{\\Psi}^\\dagger({\\bf r'}) V({\\bf r}-{\\bf r'})\n\\hat{\\Psi}({\\bf r'})\\hat{\\Psi}({\\bf r})\n\\;.\\end{eqnarray}\nThe field operator $\\hat{\\Psi}({\\bf r})$ ($\\hat{\\Psi}^\\dagger({\\bf r})$) destroys (creates) a boson in the position ${\\bf r}$. $\\hat{\\Psi}({\\bf r})$ and $\\hat{\\Psi}^\\dagger({\\bf r})$ satisfy the usual bosonic commutation rules: $[\\hat{\\Psi}({\\bf r}),\\hat{\\Psi}^\\dagger({\\bf r'})]=\\delta^{(3)}({\\bf r}-{\\bf r'})$, and $[\\hat{\\Psi}({\\bf r}),\\hat{\\Psi}({\\bf r'})]=0=[\\hat{\\Psi}({\\bf r})^\\dagger,\\hat{\\Psi}^\\dagger({\\bf r'})]$.\nWe assume that the bosons interact between each other via short-range interactions, so that the atom-atom interaction potential $V({\\bf r}-{\\bf r'})$  can be described (in the dilute regime and for ultra-low temperatures) by a contact potential given by\n\\begin{equation}\n\\label{contact}\nV({\\bf r}-{\\bf r'})=g\\delta^{(3)}({\\bf r}-{\\bf r'})\n\\;,\\end{equation}\nwhere the coupling constant $g$ is equal to $\\displaystyle{\\frac{4\\pi\\hbar a_s}{m}}$ with $a_s$ the s-wave scattering length.\nTherefore the Hamiltonian (\\ref{system:ham0}) becomes\n\\begin{eqnarray}\n\\label{system:ham1}\n\\hat{H} &=& \\int d^{3}{\\bf r}\\hat{\\Psi}^\\dagger({\\bf r})(-\\frac{\\hbar^2}{2m}\\nabla^2+V_{trap}({\\bf r}))\\hat{\\Psi}({\\bf r}) \\nonumber\\\\\n&+&\\frac{g}{2}\\int d^{3} {\\bf r}d^{3} {\\bf r'}\\hat{\\Psi}^\\dagger({\\bf r})\\hat{\\Psi}^\\dagger({\\bf r'})\n\\hat{\\Psi}({\\bf r'})\\hat{\\Psi}({\\bf r})\n\\;.\\end{eqnarray}\nUnder the hypothesis that only the lowest energetic doublet of the potential $V_{DW}(z)$ is populated, we expand the field operator $\\hat{\\Psi}({\\bf r})$ according the two-spatial mode decomposition:\n\\begin{equation}\n\\label{expansion}\n\\hat{\\Psi}({\\bf r}) = \\Phi_L({\\bf r})\\hat{a}_L + \\Phi_R({\\bf r})\\hat{a}_R\n\\;,\\end{equation}\nwhere $\\hat{a}_k$ ($\\hat{a}^\\dagger_k$) - $k=L,R$, with $L (R)$ denoting the left (right) well - destroys (creates) a boson in the $k$th well. The single-particle operators $\\hat{a}_k$ and $\\hat{a}^\\dagger_k$  satisfy the\nbosonic commutation rules:\n\\begin{eqnarray}\n\\label{spcommutation}\n&&[\\hat{a}_k,\\hat{a}^\\dagger_j]=\\delta_{k,j} \\nonumber\\\\\n&&[\\hat{a}_k,\\hat{a}_j]=0=[\\hat{a}^\\dagger_k,\\hat{a}^\\dagger_j]\n\\;.\\end{eqnarray}\nDue to the form  of the trapping potential given by Eq. (\\ref{trapping}), the single-particle wave function  $\\Phi_k({\\bf r})$ ($k=L,R$) can be written according to the factorization\n\\begin{equation}\n\\label{system:dec}\n\\Phi_k({\\bf r}) = w(x)w(y)\\phi_k(z)\n\\;,\\end{equation}\nwhere $w(x)$ and $w(y)$ are the ground-state wave functions of the harmonic oscillator potentials $m\\omega_{\\bot}^{2} x^2\/2$ and $m\\omega_{\\bot}^{2} y^{2}\/2$, respectively.\nThe single-particle wave functions $\\phi_L(z)$ and $\\phi_R(z)$ are tightly localized in the left and right well, respectively, and satisfy the orthonormalization conditions $(k,l = L,R)$\n\\begin{eqnarray}\n\\label{system:rel1}\n&&\\int_{-\\infty}^{+\\infty}dz|\\phi_k(z)|^2 = 1 \\nonumber\\\\\n&&\\int_{-\\infty}^{+\\infty}dz \\phi^{*}_{k}(z)\\phi_l(z) =\\delta_{k,l}\n\\;,\\end{eqnarray}\n(with $\\phi^{*}_{k}(z)$ the complex conjugate of $\\phi_{k}(z)$) so that\n\\begin{eqnarray}\n\\label{system:rel2}\n&&\\int d^3 {\\bf r}|\\Phi_k({\\bf r})|^2 =1\\nonumber\\\\\n&&\\int d^3 {\\bf r} \\Phi^{*}_{k}({\\bf r})\\Phi_l({\\bf r})=\\delta_{k,l}\n\\;.\\end{eqnarray}\nWe use the expansion (\\ref{expansion}) and its Hermitian conjugate at the right-hand side of Eq. (\\ref{system:ham1}); by exploiting the orthonormalization conditions\n(\\ref{system:rel2}) and the fact that $V_{DW}(z)$ is symmetric, the well known two-site Bose-Hubbard Hamiltonian \\cite{main,twomode:milburn} is achieved\n\\begin{equation}\n\\label{ham:bh}\n\\hat{H} = -J(\\hat{a}_L^\\dagger\\hat{a}_R + \\hat{a}_R^\\dagger\\hat{a}_L) +\\frac{U}{2}\\big(\\hat{n}_L(\\hat{n}_L-1)+\\hat{n}_R(\\hat{n}_R-1)\\big)\n\\;.\\end{equation}\nHere $\\hat{n}_k = \\hat{a}^\\dagger_k\\hat{a}_k$ is the operator counting the number of bosons in the $k$th well. Note that the Hamiltonian (\\ref{ham:bh}) commutes with the total number operator $\\hat{N}=\\hat{n}_L+\\hat{n}_R$. The amplitude $U$ measures the strength of the boson-boson interaction in the same well (on-site or intra-well interaction)\n\\begin{equation}\n\\label{u}\nU =\\frac{g}{2\\pi a_\\perp^2}\\int_{-\\infty}^{+\\infty}dz |\\phi_k(z)|^4\n\\end{equation}\nwith $a_\\bot =\\displaystyle{\\sqrt{\\frac{\\hbar}{m\\omega_{\\bot}}}}$. The sign of $U$ is controlled by that of $a_s$ which can be experimentally tuned via the Feshbach resonance technique, so that when $a_s$ is positive (negative) the bosons are repulsively (attractively) interacting. $J$ is the tunnel matrix element between the two wells:\n\\begin{equation}\n\\label{j}\nJ =-\\int_{-\\infty}^{+\\infty} dz\\phi^{*}_{L}(z)\\,\\big(-\\frac{\\hbar^2}{2m}\\frac{d^2}{dz^2}+V_{DW}(z)\\big)\\,\\phi_R(z)\n\\;.\\end{equation}\n\nTo capture the main properties of the system, we focus on the eigenproblem\n\\begin{equation}\n\\label{eigenproblem}\n\\hat {H} |E_j\\rangle = E_j |E_j \\rangle\n\\end{equation}\nfor a fixed number $N$ of bosons. In this case the Hamiltonian $\\hat {H}$ can be represented by\na $(N+1)\\times(N+1)$ matrix in the Fock basis $|i,N-i\\rangle=|i\\rangle_L \\otimes |N-i\\rangle_R$ (with $\\otimes$ denoting the tensor product) with $i=0,...,N$. For each eigenvalue $E_j$, with\n$j=0,1,...,N$, the corresponding eigenstate $|E_j\\rangle$ will be of the form\n\\begin{equation}\n\\label{eigenstate}\n|E_j\\rangle=\\sum_{i=0}^{N}\\,c_{i}^{(j)} \\, |i,N-i\\rangle \\;,\n\\end{equation}\nwhere $|c_i^{(j)}|^2$ is the probability to have $i$ ($N-i$) bosons in the left (right) well when the system is in the $j$th eigenstate of the two-site BH Hamiltonian.\nNote that since the left-right symmetry of the Hamiltonian (\\ref{ham:bh}), for any eigenstate one has that\n\\begin{equation}\n\\label{lrs}\n\\langle \\hat{n}_L \\rangle=\\langle \\hat{n}_R \\rangle\n\\;,\\end{equation}\nwhere the average $\\langle...\\rangle$ is taken with respect to the given eigenstate.\nWe are analyzing the system at zero temperature. Then, the only two-site BH Hamiltonian eigenstate to be occupied is the lowest one, so that in the following we shall denote the corresponding eigenvector and eigenvalue simply by $|E\\rangle$ and $E$, respectively. The expansion coefficients with respect to the basis $|i,N-i\\rangle$ shall be, then, denoted by $c_i$. As discussed in \\cite{main}, the ground state of the Hamiltonian (\\ref{ham:bh}) features different behaviors depending on the interplay between the on-site interaction $U$ and the hopping amplitude $J$. Following the same path followed in \\cite{main}, we study the ground state in terms of the dimensionless parameter $\\xi =U\/J$. Let us start with some limit cases.\n\\begin{itemize}\n\\item $\\xi=0$. The ground state is the atomic coherent state \\cite{arecchi}\n\\begin{equation}\n\\label{ACS}\n|ACS\\rangle = \\frac{1}{\\sqrt{N!}}(\\frac{1}{\\sqrt{2}}\\big(\\hat{a}_L^\\dagger+\\hat{a}_R^\\dagger)\\big)^N|0,0\\rangle\n\\;,\\end{equation}\n(the energy associated to this state is $-NJ$) where $|0,0\\rangle=|0\\rangle_L\\otimes|0\\rangle_R$ is the tensor product between the vacuum of the operator $\\hat{a}_L$ and the vacuum of $\\hat{a}_R$, i.e. we have no particles in the left well and no particles in the right well.\n\\item $U>0$ : $\\xi\\rightarrow+\\infty$. In the case of a strong repulsive interaction and with an even number $N$ of bosons, as well known, the ground state tends to the twin Fock state\n\\begin{equation}\n\\label{fock}\n|FOCK\\rangle = |\\frac{N}{2},\\frac{N}{2}\\rangle\n\\;.\\end{equation}\n\nIf $N$, instead, is odd, when  $\\xi\\rightarrow+\\infty$ the ground state tends to\n\\begin{equation}\n\\label{pseudo:fock}\n|pseudoFOCK\\rangle = \\frac{1}{\\sqrt{2}}\\Big(\\Big|\\frac{N-1}{2},\\frac{N+1}{2}\\Big\\rangle + \\Big|\\frac{N+1}{2},\\frac{N-1}{2}\\Big\\rangle\\Big)\n\\;.\\end{equation}\nTo understand this, let us consider the extreme case of complete absence of hopping, that is $J=0$. In this case the eigenvalues of the Hamiltonian (\\ref{ham:bh}) are given by those of the intra-well term: $E=\\displaystyle{\\frac{U}{2}(2i^2-2Ni+N^2-N)}$. We are here considering the state with $i$ $(N-i)$ bosons in the left (right) well. Requiring that $\\partial E\/\\partial i=0$ provides $i=N\/2$. Since $N$ is odd and $i$ must be an integer, the values of $i$ which minimize $E$ are those integer closest to $N\/2$, i.e. $i=(N-1)\/2$ and $i=(N+1)\/2$ that correspond to the two separable Fock states\n\\begin{eqnarray}\n&&|\\varphi\\rangle_1 = |\\frac{N-1}{2},\\frac{N+1}{2}\\rangle \\nonumber\\\\\n&&|\\varphi\\rangle_2 = |\\frac{N+1}{2},\\frac{N-1}{2}\\rangle\n\\;.\\end{eqnarray}\nThese states, although having the (same) minimum energy, do not satisfy the condition (\\ref{lrs}). Nevertheless, it is easy to prove that the state\n\\begin{equation}\n|\\varphi\\rangle_3 = \\frac{1}{\\sqrt{2}}(|\\varphi\\rangle_1+|\\varphi\\rangle_2)\n\\end{equation}\nhas the same energy of $|\\varphi\\rangle_l$ ($l=1,2$) and satisfies the condition (\\ref{lrs}).\n\\item $U<0$ : $\\xi\\rightarrow-\\infty$. In the case of a strong attractive interaction, the ground state tends to the macroscopic superposition state\n\\begin{equation}\n\\label{cat}\n|CAT\\rangle=\\frac{1}{\\sqrt{2}}(|N,0\\rangle+|0,N\\rangle)\n\\;.\\end{equation}\n\\end{itemize}\nThis state, frequently called NOON state, is the boson-version of the Schr\\\"odinger cat state \\cite{cirac,dalvit,huang,carr,brand}.\n\nAt this point it is worth to observe that apart the issue of the possible collapse (related to attractive interactions), the realization of the cat state is not trivial due to the very tiny separation (in the presence of finite couplings) between the two lowest levels that makes the cat state very fragile, see, for example, \\cite{huang}.\n\n\n\\section{Analysis parameters}\nIn this section we introduce the parameters that we use to characterize the correlations of the ground state of the two-site BH Hamiltonian (\\ref{ham:bh}). These parameters are the Fisher information, the coherence visibility, and the entanglement entropy.\n\nIn the meanwhile, it is useful to remind the well-known properties:\n\\begin{equation}\n\\label{ort}\n\\langle j,N-j|i,N-i\\rangle=\\delta_{i,j}\n\\;,\\end{equation}\nand\n\\begin{eqnarray}\n\\label{spaction}\n&&\\hat{a}^\\dagger_L |i,N-i\\rangle = \\sqrt{i+1}|i+1,N-i\\rangle\\nonumber\\\\\n&&\\hat{a}_L |i,N-i\\rangle = \\sqrt{i}|i-1,N-i\\rangle\\nonumber\\\\\n&&\\hat{a}^\\dagger_R |i,N-i\\rangle = \\sqrt{N-i+1}|i,N-i+1\\rangle\\nonumber\\\\\n&&\\hat{a}_R |i,N-i\\rangle = \\sqrt{N-i}|i,N-i-1\\rangle\\nonumber\\\\\n&&\\hat{n}_L|i,N-i\\rangle = i|i,N-i\\rangle. \\nonumber\\\\\n&&\\hat{n}_R|i,N-i\\rangle = (N-i)|i,N-i\\rangle\n\\;.\\end{eqnarray}\n\\begin{itemize}\n\\item {\\it Fisher Information.}\\\\\nThe quantum Fisher information $F_{QFI}$ is the quantity \\cite{braunstein,pezze}\n\\begin{equation}\n\\label{qfi}\nF_{QFI} =(\\Delta\\hat{n}_{L,R})^2 = \\langle(\\hat{n}_L-\\hat{n}_R)^2\\rangle - (\\langle\\hat{n}_L-\\hat{n}_R\\rangle)^2\n\\;,\\end{equation}\nwhere the expectation values are taken with respect to the ground state $|E\\rangle$.\nBy using  the orthonormality condition (\\ref{ort}) and rules (\\ref{spaction}) in Eq. (\\ref{qfi}), we can express $F_{QFI}$ in terms of the expansion coefficients $c_i$ as follows:\n\\begin{equation}\nF_{QFI} = \\sum_{i=0}^N (2i-N)^2|c_i|^2\n\\;.\\end{equation}\nIt is convenient to normalize $F_{QFI}$ at its maximum value $N^2$ by defining the Fisher information $F$ as\n\\begin{equation}\n\\label{fi}\nF=\\frac{F_{QFI}}{N^2}\n\\;,\\end{equation}\nso that we have a quantity varying in the range $[0,1]$. In terms of the coefficients $c_i$, $F$ is\n\\begin{equation}\n\\label{fi2}\nF=\\frac{1}{N^2}\\sum_{i=0}^N (2i-N)^2|c_i|^2\n\\;.\\end{equation}\nThis $F$ will be equal to $1$ for the NOON state (\\ref{cat}).\n\n\\item {\\it Coherence visibility.}\\\\\nIn ultracold atom physics, it is customary to investigate the coherence properties in terms of the momentum distribution $n(p)$ which is the Fourier transform of the one-body\ndensity matrix $\\rho_1(x,x')$ \\cite{stringari,anglin,anna}:\n\\begin{equation}\n\\label{np}\nn(p)=\\int dx dx'\\exp\\big(-ip(x-x')\\big)\\,\\rho_1(x,x') \\;,\n\\end{equation}\nwhere\n\\begin{equation}\n\\rho_1(x,x')=\\langle \\hat{\\Psi}(x)^{\\dagger}\\hat{\\Psi}(x')\\rangle \\;\n\\end{equation}\nwith the operators $\\hat{\\Psi}(x)$ and $\\hat{\\Psi}^{\\dagger}(x)$ -\nsatisfying the standard bosonic commutation rules - annihilating and creating,\nrespectively, a boson at the point $x$, and the average $\\langle...\\rangle$ being the ground-state average.\nFollowing Refs. \\cite{stringari,anglin,anna}, it is possible to show that the momentum distribution $n(p)$ can be written as\n\\begin{equation}\n\\label{npexp}\nn(p)= n_0(p) \\bigg(1 +\\alpha \\cos\\big(pd \\big) \\bigg) \\;.\n\\end{equation}\nHere $n_0(p)$ is the momentum distribution in the fully incoherent regime\n($n_0(p)$ depends on the shape of the double-well potential $V_{DW}(z)$), and\n$d$ is the distance between the two minima of $V_{DW}(z)$. $\\alpha$ is\na real quantity which measures the visibility of the interference fringes. This visibility is given by\n\\begin{equation}\n\\label{visibility}\n\\alpha=\\frac{2\\,|\\langle \\hat{a}^{\\dagger}_L\\hat{a}_R\\rangle|}{N} \\;,\n\\end{equation}\nwhere the expectation value is taken with respect to the ground state. The quantity $\\alpha$ characterizes the degree of coherence, between the two wells, related to the left-right (and back) tunneling.\n\nWe can express the  coherence visibility (\\ref{visibility}) in terms of the coefficients $c_i$ by using in Eq. (\\ref{visibility}) the rules (\\ref{spaction}) and the orthonormalization condition (\\ref{ort}), so that one has\n\\begin{equation}\n\\label{visibility2}\n\\alpha = \\frac{2}{N}|\\sum_{i=0}^N c_{i} c^{*}_{i+1} \\sqrt{(i+1)(N-i)}|\n\\;,\\end{equation}\nwhere $c^{*}_{i+i}$ is the complex conjugate of $c_{i+1}$. $\\alpha$ is maximum, that is $1$, for the atomic coherent state (\\ref{ACS}).\n\n\\item {\\it Entanglement entropy.}\\\\\nFinally, it is interesting to analyze the genuine quantum correlations pertaining to the ground state $|E\\rangle$. In particular, we study the quantum entanglement\nof $|E\\rangle$ from the perspective of the bi-partition. In this framework, the two partitions are given by the left well and right one.  When the system is in $|E\\rangle$, the\ndensity matrix $\\hat{\\rho}$ is\n\\begin{equation}\n\\label{dm}\n\\hat{\\rho} =|E\\rangle\\langle E|\n\\;.\\end{equation}\nAn excellent measure of the entanglement between the two wells\nis provided by the entanglement entropy $S$ \\cite{bwae}. This quantity is the von Neumann entropy of the reduced density matrix $\\hat{\\rho}_{L(R)}$\ndefined by\n\\begin{equation}\n\\hat{\\rho}_{L(R)} =Tr_{R(L)} \\hat{\\rho}\n\\;,\\end{equation}\nthat is the matrix obtained by partial tracing the total density matrix (\\ref{dm}) over the degrees of freedom of the right (left) well (note that $\\hat{\\rho}_{L}=\\hat{\\rho}_{R}$). By using the definition of trace of a matrix and the orthonormalization condition (\\ref{ort}), it is possible to show that the entanglement entropy\n\\begin{equation}\n\\label{ee0}\nS=-Tr \\hat{\\rho}_{L(R)} \\log_{2}\\hat{\\rho}_{L(R)}\n\\end{equation}\nis given by\n\\begin{equation}\n\\label{ee}\nS=-\\sum_{i=0}^{N}|c_{i}|^2\\log_{2}|c_{i}|^2\n\\;.\\end{equation}\nFor a given number of bosons $N$, the theoretical maximum value of $S$ is $\\log_2(N+1)$ that would correspond to the situation in which the quantities $|c_{i}|^2$ are all equal: $|c_i|^2=1\/(N+1)$ whatever $i$.\n\\end{itemize}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n~~~$State (N=2)$~~~ & ~~~$F$~~~ &~~~$\\alpha$~~~ & ~~~$S$~~~\\\\\n\\hline\n$|ACS\\rangle$\n& $1\/2$\n& $1$ & $3\/2$  \\\\\n$|FOCK\\rangle$\n&$0$ & $0$ & $0$ \\\\\n$|CAT\\rangle$\n&$1$ & $0$ & $1$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\nTable 1. {\\small The Fisher information $F$, the coherence visibility $\\alpha$, and the entanglement entropy $S$, for the atomic coherent state (\\ref{ACS}), the twin Fock state (\\ref{fock}), and the NOON state (\\ref{cat}) with $N=2$ bosons.}\\\\\n\n\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n~~~$State (N=3)$~~~ & ~~~$F$~~~ &~~~$\\alpha$~~~ & ~~~$S$~~~\\\\\n\\hline\n$|ACS\\rangle$\n& $1\/3$\n& $1$ & $1.81128$  \\\\\n$|pseudoFOCK\\rangle$\n&$1\/9$ & $2\/3$ & $1$ \\\\\n$|CAT\\rangle$\n&$1$ & $0$ & $1$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\nTable 2. {\\small The Fisher information $F$, the coherence visibility $\\alpha$, and the entanglement entropy $S$, for the atomic coherent state (\\ref{ACS}), the state (\\ref{pseudo:fock}), and the NOON state (\\ref{cat}) with $N=3$ bosons.}\\\\\n\n\n\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n~~~$State (N=4)$~~~ & ~~~$F$~~~ &~~~$\\alpha$~~~ & ~~~$S$~~~\\\\\n\\hline\n$|ACS\\rangle$\n& $1\/4$\n& $1$ & $2.03064$  \\\\\n$|FOCK\\rangle$\n&$0$ & $0$ & $0$ \\\\\n$|CAT\\rangle$\n&$1$ & $0$ & $1$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\nTable 3. {\\small The Fisher information $F$, the coherence visibility $\\alpha$, and the entanglement entropy $S$, for the atomic coherent state (\\ref{ACS}), the twin Fock state (\\ref{fock}), and the NOON state (\\ref{cat}) with $N=4$ bosons.}\\\\\n\n\n\n\n\n\\section{Analysis}\nIn this section, we determine the ground state of the two-site Bose-Hubbard Hamiltonian when $N=1$, $N=2$, $N=3$, and $N=4$. We calculate the Fisher information (\\ref{fi}), the coherence visibility (\\ref{visibility}), and the entanglement entropy (\\ref{ee}) for a system with $N=1$, $N=2$, $N=3$, and $N=4$ bosons. We analyze the structure of the ground state and $F$, $\\alpha$, $S$  in terms of the scaled on-site interaction $\\xi=U\/J$.\n\nAs first, we represent the Hamiltonian $\\hat{H}$ with respect to the Fock basis $|i,N-i\\rangle$. We start from the right-hand side of Eq. (\\ref{ham:bh}) and use the rules (\\ref{spaction}) and the orthonormalization condition (\\ref{ort}). Note that, here, we measure the energies in units of $J$. We shall denote by the symbols $\\hat{\\tilde H}$ and $\\tilde E$ the dimensionless energetic quantities, i.e. $\\hat{\\tilde H}=\\hat{H}\/J$ and $\\tilde E=E\/J$.\n\nWhen $N=1$, the Hamiltonian (\\ref{ham:bh}) consists of the only hopping term. In this case, the two-site Bose-Hubbard Hamiltonian, given by Eq. (\\ref{ham:bh}), in the Fock basis $|i,N-i\\rangle$ is\n\\[\\hat{\\tilde H}= \\left(\\begin{array}{cc}\n  0  & -1 \\\\\n  -1 & 0 \\end{array}\n   \\right).\\]\nThe eigenvector $|E\\rangle$ associated to the ground state is\n\\begin{equation}\n\\label{sgs1}\n|E\\rangle=\\frac{1}{\\sqrt{2}}\\big(|0,1\\rangle+|1,0\\rangle\\big)\n\\;,\\end{equation}\nand the related eigenvalue is\n\\begin{equation}\n\\label{e1}\n\\tilde E=-1\n\\;.\\end{equation}\nThen, it is easy to see that the state $|E\\rangle$ is the atomic coherent state (\\ref{ACS}) with $N=1$ that coincides with the state NOON, Eq. (\\ref{cat}), and with the state (\\ref{pseudo:fock}) with $N=1$. In this case, by using Eqs. (\\ref{fi2}), (\\ref{visibility2}), and (\\ref{ee}), we immediately see that $F=\\alpha=S=1$.\n\nAt this point, let us focus on a number of bosons larger than one. We therefore consider the cases $N=2$, $N=3$, and $N=4$. The matrices corresponding to the two-site Bose-Hubbard Hamiltonian (\\ref{ham:bh}) with $N=2$, $N=3$, and $N=4$ are, respectively\n\n\n\n\\[\\hat{\\tilde H}= \\left(\\begin{array}{ccc}\n  \\xi & -\\sqrt{2} & 0 \\\\\n -\\sqrt{2} & 0 & -\\sqrt{2} \\\\\n  0 & -\\sqrt{2} & \\xi \\end{array} \\right),\\]\n\n\n\n\\[\\hat{\\tilde H}= \\left(\\begin{array}{cccc}\n 3\\xi & -\\sqrt{3} & 0 & 0 \\\\\n  -\\sqrt{3} & \\xi & -2 & 0 \\\\\n  0 & -2 & \\xi & -\\sqrt{3} \\\\\n  0 & 0 & -\\sqrt{3} & 3\\xi\n\\end{array} \\right),\\]\n\n\n\n\\[\\hat{\\tilde H}= \\left(\\begin{array}{ccccc}\n 6\\xi & -2 & 0 & 0 & 0 \\\\\n -2 & 3\\xi & -\\sqrt{6} & 0 & 0 \\\\\n  0 & -\\sqrt{6} & 2\\xi & -\\sqrt{6} & 0 \\\\\n  0 & 0 & -\\sqrt{6} & 3\\xi & -2 \\\\\n  0 & 0 & 0 & -2 & 6\\xi\n\\end{array} \\right),\\]\n\n\n\n\nwhere $\\xi=U\/J$.\n\nThe ground-state energy $\\tilde E$ pertaining to the case $N=2$ and the corresponding eigenvector $|E\\rangle$ are, respectively\n\\begin{equation}\n\\label{egs2}\n\\tilde E=\\frac{1}{2} (\\xi - \\sqrt{16 + \\xi^2})\n\\;,\\end{equation}\n\\begin{equation}\n\\label{sgs2}\n|E\\rangle =  A_2\\Big(|0,2\\rangle + \\frac{\\xi + \\sqrt{16 + \\xi^2}}{2 \\sqrt{2}}|1,1\\rangle+ |2,0\\rangle \\Big)\n\\;,\\end{equation}\nso that\n\\begin{eqnarray}\n\\label{c2}\n&&c_0=c_2=A_2\\nonumber\\\\\n&&c_1=\\frac{A_2(\\xi + \\sqrt{16 + \\xi^2})}{2 \\sqrt{2}}\n\\;.\\end{eqnarray}\n\nFor $N=3$, we get\n\\begin{equation}\n\\label{egs3}\n\\tilde E = -1 + 2 \\xi - \\sqrt{4 + 2 \\xi + \\xi^2}\n\\;,\\end{equation}\n\\begin{eqnarray}\n\\label{sgs3}\n&&|E\\rangle = A_3 \\Big( |0,3\\rangle + \\frac{1 + \\xi + \\sqrt{4 + 2 \\xi + \\xi^2}}{\\sqrt{3}}|1,2\\rangle\\nonumber\\\\\n&+& \\frac{1 + \\xi + \\sqrt{4 + 2 \\xi + \\xi^2}}{\\sqrt{3}}|2,1\\rangle + |3,0\\rangle \\Big)\n\\;,\\end{eqnarray}\nso that\n\\begin{eqnarray}\n\\label{c3}\n&&c_0=c_3=A_3\\nonumber\\\\\n&&c_1=c_2=\\frac{A_3(1 + \\xi + \\sqrt{4 + 2 \\xi + \\xi^2})}{\\sqrt{3}}\n\\;.\\end{eqnarray}\n\nWhen $N=4$, for the energy of the ground state we obtain the following result:\n\n\\begin{eqnarray}\n\\label{egs4}\n&&\\tilde E=\\frac{1}{3}\\big(11\\xi-2\\sqrt{k_4}\\cos \\frac{\\theta}{3}\\big) \\nonumber\\\\\n&&\\theta=\\theta_1=\\arctan \\frac{b_4}{a_4} \\nonumber\\\\\n&&\\theta=\\theta_2=\\arctan \\frac{b_4}{a_4} +\\pi\n\\;,\\end{eqnarray}\nwhere $k_4=13 \\xi^2+48$. $\\theta=\\theta_1$ (the second row of Eq. (\\ref{egs4})) when $\\xi \\le-\\bar \\xi$ and $0<\\xi \\le \\bar \\xi$, and $\\theta=\\theta_2$ (the third row of Eq. (\\ref{egs4})) when $-\\bar \\xi<\\xi<0$ and $\\xi>\\bar \\xi$ with $\\bar\\xi=12\\sqrt{2\/35}$. Moreover $a_4=288\\xi-35\\xi^3$, $b_4=6\\sqrt{3}\\sqrt{9\\xi^6+412\\xi^4+64 \\xi^2+1024}$. Note that when $\\xi \\rightarrow 0^{+}(^{-})$, the energy in Eq. (\\ref{egs4}) gives back $-4$ (in units of $J$) for $\\theta=\\theta_1$ ($\\theta_2$), i.e. the energy of to the atomic coherent state (\\ref{ACS}).\nThe eigenvector pertaining to the energy in Eq. (\\ref{egs4}) is\n\\begin{eqnarray}\n\\label{sgs4}\n&&|E\\rangle = A_4 \\Big( |0,4\\rangle + (3\\xi-\\frac{\\tilde E}{2})|1,3\\rangle\\nonumber\\\\\n&+& \\big(\\frac{18\\xi^2-9\\tilde E\\xi+\\tilde E^2-4}{2\\sqrt{6}}\\big)|2,2\\rangle \\nonumber\\\\\n&+&(3\\xi-\\frac{\\tilde E}{2})|3,1\\rangle +|0,4\\rangle\\Big)\n\\;,\\end{eqnarray}\nso that\n\\begin{eqnarray}\n\\label{c4}\n&&c_0=c_4=A_4\\nonumber\\\\\n&&c_1=c_3=A_4(3\\xi-\\frac{\\tilde E}{2})\\nonumber\\\\\n&&c_2=\\frac{A_4(18\\xi^2-9\\tilde E\\xi+\\tilde E^2-4)}{2\\sqrt{6}}\\;.\\nonumber\\\\\n\\end{eqnarray}\n\nThe factors $A_2$, $A_3$, and $A_4$ are normalization factors given by the following formulas:\n\\begin{equation}\n\\label{a2}\nA_2 = \\frac{2}{\\sqrt{16+\\xi^2+\\xi \\sqrt{\\xi^2+16}}}\n\\;,\\end{equation}\n\\begin{equation}\n\\label{a3}\nA_3 = \\frac{1}{\\sqrt{2 + \\frac{2}{3} \\big(1 + \\xi + \\sqrt{4 + \\xi (2 + \\xi)}\\big)^2}}\n\\;,\\end{equation}\n\\begin{eqnarray}\n\\label{a4}\n&&A_4=2\\sqrt{\\frac{6}{d_4}}\\nonumber\\\\\n&&d_4=48+12(\\tilde E-6\\xi)^2+\\nonumber\\\\\n&&(18\\xi^2-9\\tilde E \\xi+\\tilde E^2-4)^2\n\\;\\end{eqnarray}\nwith $\\tilde E$ given by Eq. (\\ref{egs4}). Note that Eqs. (\\ref{egs2}) and (\\ref{sgs2}) are the same that we found in \\cite{main}.\nWe observe that in the limit $\\xi\\rightarrow -\\infty$, the states (\\ref{sgs2}),  (\\ref{sgs3}) and (\\ref{sgs4})  becomes, as expected,\n\\begin{equation}\n\\label{cat2}\n|E\\rangle = \\frac{1}{\\sqrt{2}}(|0,2\\rangle+|2,0\\rangle)\n\\;,\\end{equation}\ni.e. the boson-version of the Schr\\\"odinger cat state (\\ref{cat}) with $N=2$,\n\\begin{equation}\n\\label{cat3}\n|E\\rangle = \\frac{1}{\\sqrt{2}}(|0,3\\rangle +|3,0\\rangle)\n\\;,\\end{equation}\nwhich is the Schr\\\"odinger cat state (\\ref{cat}) with $N=3$, and similarly\n\\begin{equation}\n\\label{cat4}\n|E\\rangle = \\frac{1}{\\sqrt{2}}(|0,4\\rangle +|4,0\\rangle)\n\\;,\\end{equation}\n\nWhen $\\xi\\rightarrow 0$, from the state (\\ref{sgs2}) ($N=2$) we retrieve\n\\begin{equation}\n\\label{acs2}\n|E\\rangle = \\frac{1}{2}(|0,2\\rangle+\\sqrt{2}|1,1\\rangle+|2,0\\rangle)\n\\end{equation}\nand from the state (\\ref{sgs3}) ($N=3$) one gets\n\\begin{eqnarray}\n\\label{acs3}\n|E\\rangle &=& \\frac{1}{2}\\Big(\\frac{1}{\\sqrt{2}}|0,3\\rangle + \\sqrt{\\frac{3}{2}}|1,2\\rangle \\nonumber\\\\\n &+&\\sqrt{\\frac{3}{2}}|2,1\\rangle + \\frac{1}{\\sqrt{2}}|3,0\\rangle\\Big)\n\\;.\\end{eqnarray}\nWhen $N=4$ and $\\xi \\rightarrow 0$, the state (\\ref{sgs4}) gives\n\\begin{eqnarray}\n\\label{acs4}\n|E\\rangle &=& \\frac{1}{4}\\Big(|0,4\\rangle + 2|1,3\\rangle +\\sqrt{6}|2,2\\rangle \\nonumber\\\\\n &+&2|3,1\\rangle +|4,0\\rangle\\Big)\n\\;.\\end{eqnarray}\n\nThese last three states represent the forms assumed by the atomic coherent state $|ACS\\rangle$ (\\ref{ACS}) when $N=2$, $N=3$, and $N=4$, respectively.\n\nIn the limit $\\xi\\rightarrow +\\infty$, the state (\\ref{sgs2}) ($N=2$) becomes $|1,1\\rangle$ which is the twin Fock state (\\ref{fock}) with $N=2$. Instead for $N=3$, in the deep repulsive regime, $\\xi\\rightarrow +\\infty$,  the state (\\ref{sgs3}) becomes\n\\begin{equation}\n\\label{pf3}\n|pseudoFOCK\\rangle = \\frac{1}{\\sqrt{2}}(|1,2\\rangle +|2,1\\rangle)\n\\;,\\end{equation}\nthat is the state (\\ref{pseudo:fock}) with $N=3$. Finally, when $N=4$ and $\\xi\\rightarrow +\\infty$ , we retrieve the twin Fock state $|2,2\\rangle$.\n\n\nTo understand the role of the intra-well interaction-hopping interplay in determining the structure of the ground state of the two-site BH Hamiltonian, we have studied the changes experienced by the probabilities $|c_i|^2$ by varying the scaled on-site interaction $\\xi=U\/J$ in the presence of $N=2$, $N=3$, and $N=4$ bosons, see Fig. 1. From this figure, we can see that a crossover occurs when $\\xi$ ranges from $\\xi=-30$ (the two top panels of Fig. 1: the largest probabilities $|c_i|^2$ are located in correspondence to $|0,N\\rangle$ and $|N,0\\rangle$, this being representative of cat-like states (\\ref{cat}) with $N=2,4$ and $N=3$ bosons) to $\\xi=30$ (the two bottom panels of Fig. 1: $|c_i|^2$ reaches its largest value in correspondence to  $|1,1\\rangle$ when $N=2$ and $|2,2\\rangle$ when $N=4$  -  separable twin Fock state (\\ref{fock}) - and in correspondence to the states $|1,2\\rangle$ and $|2,1\\rangle$ when $N=3$, state (\\ref{pseudo:fock})) passing for $\\xi=0$ (the three panels at the fifth row from the top) describing an almost Gaussian distribution of the probabilities $|c_i|^2$, that is the atomic coherent state (\\ref{ACS}). Note that at fifth row of Fig. 1, the plot of $|c_i|^2$ with $N=4$ has been, nominally, labeled by $\\xi=0$; actually, this plot has been obtained by performing the limit $\\xi \\rightarrow 0$ in the equations for $|c_i|^2$ obtained by Eq. (\\ref{c4}).\n\n\\begin{figure}[htpb]\n\\centering\n{\\includegraphics[width=.24\\columnwidth]{c21.eps}} \\quad\n{\\includegraphics[width=.24\\columnwidth]{c31.eps}} \\quad\n{\\includegraphics[width=.24\\columnwidth]{c41new.eps}} \\\\\n{\\includegraphics[width=.24\\columnwidth]{c22.eps}} \\quad\n{\\includegraphics[width=.24\\columnwidth]{c32.eps}} \\quad\n{\\includegraphics[width=.24\\columnwidth]{c42new.eps}} \\\\\n{\\includegraphics[width=.24\\columnwidth]{c23.eps}} \\quad\n{\\includegraphics[width=.24\\columnwidth]{c33.eps}} \\quad\n{\\includegraphics[width=.24\\columnwidth]{c43new.eps}} \\\\\n{\\includegraphics[width=.24\\columnwidth]{c24.eps}} \\quad\n{\\includegraphics[width=.24\\columnwidth]{c34.eps}} \\quad\n{\\includegraphics[width=.24\\columnwidth]{c44new.eps}} \\\\\n{\\includegraphics[width=.24\\columnwidth]{c25.eps}} \\quad\n{\\includegraphics[width=.24\\columnwidth]{c35.eps}} \\quad\n{\\includegraphics[width=.24\\columnwidth]{c45new.eps}} \\\\\n{\\includegraphics[width=.24\\columnwidth]{c26.eps}} \\quad\n{\\includegraphics[width=.24\\columnwidth]{c36.eps}} \\quad\n{\\includegraphics[width=.24\\columnwidth]{c46new.eps}} \\\\\n{\\includegraphics[width=.24\\columnwidth]{c27.eps}} \\quad\n{\\includegraphics[width=.24\\columnwidth]{c37.eps}} \\quad\n{\\includegraphics[width=.24\\columnwidth]{c47new.eps}} \\\\\n{\\includegraphics[width=.24\\columnwidth]{c28.eps}} \\quad\n{\\includegraphics[width=.24\\columnwidth]{c38.eps}} \\quad\n{\\includegraphics[width=.24\\columnwidth]{c48new.eps}}\n\\caption{Vertical axis: probabilities $|c_i|^2$ for different values of $\\xi=U\/J$. Horizontal axis: kets $|i,N-i\\rangle$ ($i=0,...,N$). Left: $N=2$. Middle: $N=3$. Right: $N=4$. At the third row (from the top): plots for $\\xi$ signing the maximum of the entanglement entropies (\\ref{s2}) ($N=2$), (\\ref{s3}) ($N=3$), (\\ref{s4}) ($N=4$). All the quantities are dimensionless.}\n\\label{result:coeff}\n\\end{figure}\n\nIn conclusions, we can say that a very strong boson-boson attraction tends to establish in the system a ground state given by a symmetric superposition of two fully populated Fock states both with $N=2,4$ and $N=3$ bosons. On the other hand, a very large interatomic repulsion induces different ground states depending if $N=2$ (separable twin-Fock state) or $N=3$ (symmetric superposition of quasi-fully populated Fock states).\n\nOn the repulsive side, the above described crossover, for even $N$, is reminescent of the quantum phase transition with optical-lattice-confined bosons theoretically predicted in \\cite{jaksch} and experimentally observed by Greiner and co-workers \\cite{greiner}. This transition - induced by varying the depth of the optical potential - is a transition from the superfluid phase (the hopping dominates the Hamiltonian, $J \\gg U$. In this case each atom is spread out over the entire lattice) to the Mott insulator one (on-site interactions dominates the Hamiltonian, $U \\gg J$. In this case, exact numbers of atoms are localized at individual lattice sites). Note that the even-odd difference (separable twin Fock state versus symmetric superposition of non-fully populated Fock states) which tends to become less relevant for larger particle numbers, indeed, is a well known Mott insulators feature, as commented, for example, in Ref. \\cite{oguri}.\nAs commented before, we characterize the correlations of the ground state by calculating the Fisher information, the coherence visibility, and the entanglement entropy.\nIt is possible to achieve, in the case of $2$, $3$, and $4$ bosons, analytical formulas for these three parameters.\n\nLet us start by  evaluating the Fisher information. To this end we employ at the right-hand side of Eq. (\\ref {fi2}) the expressions for the coefficients $c_i$ given by Eq. (\\ref{c2}) when $N=2$, by Eq. (\\ref{c3}) when $N=3$, and by Eq. (\\ref{c4}) when $N=4$ (with the normalization factors $A_2$ ($N=2$), $A_3$ ($N=3$), and $A_4$ ($N=4$) given by Eq. (\\ref{a2}), Eq. (\\ref{a3}), and Eq. (\\ref{a4}) respectively; note that the $c_i$'s are real for any $N$, so that $c_{i}^*=c_{i}$). Then, for $N=2$ we obtain\n\\begin{equation}\n\\label{f2}\nF = \\frac{8}{16+\\xi^2+\\xi \\sqrt{16+\\xi^2}}\n\\;,\\end{equation}\nwhile for $N=3$ one gets\n\\begin{equation}\n\\label{f3}\nF = \\frac{27 + (1 + \\xi + \\sqrt{4 + \\xi (2 + \\xi)})^2}{9 (3 +\n[1 + \\xi + \\sqrt{4 + \\xi (2 + \\xi)}]^2)}\n\\;.\\end{equation}\nFor $N=4$, the Fisher information is given by\n\\begin{eqnarray}\n\\label{f4}\n&&F=\\frac{3\\,\\big(16+(\\tilde E-6\\xi)^2\\big)}{e_4}\\nonumber\\\\\n&&e_4=64-(\\tilde E-6 \\xi)(54\\xi^3-45\\tilde E \\xi^2+\\nonumber\\\\\n&&(\\tilde E^2+4) (12\\xi-\\tilde E))\n\\;\\end{eqnarray}\nwith $\\tilde E$ being given by Eq. (\\ref{egs4}).\n\nWe have studied the Fisher informations $F$ given by Eqs. (\\ref{f2}), (\\ref{f3}) and (\\ref{f4}) as functions of the dimensionless parameter $\\xi=U\/J$, see the top panel of Fig. 2. As it can be seen from this figure, when the boson-boson interaction is strongly attractive ($\\xi \\ll -1$, this being correspondent to states close to the cat state (\\ref{cat})) $F$ tends to $1$. In the deep repulsive regime, it can be observed that when $\\xi \\gg 1$ and $N=2,4$ (solid line, dot-dashed), when the ground state tends to a separable Fock state (\\ref{fock}), $F$ tends to zero. With $N=3$ (dashed line), when the ground state tends to a superposition of two separable Fock states given by Eq. (\\ref{pseudo:fock}), the Fisher information tends to a finite value (see also the Tabs. 1-3).\n\n\\begin{figure}[ht]\n\\epsfig{file=fas234v2.eps,width=1.00\\linewidth,clip=}\n\\caption{(Color online). Fisher information $F$ (top panel), coherence visibility $\\alpha$ (middle panel), entanglement entropy $S$ (bottom panel) vs scaled on-site interaction $\\xi=U\/J$. Solid line: $N=2$. Dashed line: $N=3$. Dot-dashed line: $N=4$. $F$, $\\alpha$, $S$, and $\\xi$ are dimensionless.}\n\\end{figure}\n\nTo obtain the coherence visibility $\\alpha$, we use at the right-hand side of Eq. (\\ref{visibility2}) the form of the $c_i$'s provided by Eq. (\\ref{c2}) when $N=2$, by Eq. (\\ref{c3}) when $N=3$, and by Eq. (\\ref{c4}) when $N=4$ (with the normalization factors $A_2$ ($N=2$), $A_3$ ($N=3$), $A_4$ ($N=4$) given by Eq. (\\ref{a2}), Eq. (\\ref{a3}), Eq. (\\ref{a4}), respectively). For $N=2$, we get\n\\begin{equation}\n\\label{alpha2}\n\\alpha = \\frac{4 \\left(\\xi+\\sqrt{16+\\xi^2}\\right)}{16+\\xi^2+\\xi \\sqrt{16+\\xi^2}}\n\\;,\\end{equation}\nfor $N=3$\n\\begin{equation}\n\\label{alpha3}\n\\alpha = \\frac{2(1 + \\xi + \\sqrt{4 + \\xi (2 + \\xi)}) (4 + \\xi + \\sqrt{4 + \\xi (2 + \\xi)})}{3 (3 +\n  [1 + \\xi + \\sqrt{4 + \\xi (2 + \\xi)}]^2)}\n\\;,\\end{equation}\nand for $N=4$\n\\begin{equation}\n\\label{alpha4}\n\\alpha=\\frac{6\\sqrt{(\\tilde E-6\\xi)^2}(\\sqrt{(18\\xi^2-9\\tilde E\\xi +\\tilde E^2-4)^2}+4)}{e_4}\n\\;\\end{equation}\nwith $\\tilde E$ given by Eq. (\\ref{egs4}) and $e_4$ by the second row of Eq. (\\ref{f4}).\n\nWe show the behavior of $\\alpha$ (Eq. (\\ref{alpha2}), Eq. (\\ref{alpha3})), Eq. (\\ref{alpha4})), when the scaled on-site interaction is varied, in the middle panel of Fig. 2. $\\alpha$ reaches its maximum value ($\\alpha=1$), both when $N=2,4$ (solid line, dot-dashed) and $N=3$ (dashed line), in the absence of boson-boson interaction that corresponds to the atomic coherent state (\\ref{ACS}). For strongly attractive bosons, $\\xi \\ll -1$, the ground state is a cat-like state, and the coherence visibility approches to zero (see the solid (dot-dashed) line, $N=2(4)$, and the dashed one, $N=3$). When the repulsion between the bosons is sufficiently strong, $\\xi \\gg 1$, we can see that for $N=2,4$ (solid line, dot-dashed line)  - when the ground state tends to a fully incoherent state - $\\alpha$ approaches to zero, while for $N=3$ (dashed line) - when the ground state is close to state (\\ref{pseudo:fock}) - $\\alpha$ is finite.\n\nFinally, by employing the expressions of the coefficients $c_i$ provided by Eq. (\\ref{c2}) in Eq. (\\ref{ee}), we calculate the entanglement entropy $S$ for $N=2$, and get\n\\begin{equation}\n\\label{s2}\nS=-A_{2}^{2}\\bigg(2\\log_2[2A_2^{2}] +\\frac{s_2}{4}\\log_2[\\frac{s_2A_2^{2}}{4}]\\bigg)\n\\;,\\end{equation}\nwhere $A_2$ is the normalization factor given by Eq. (\\ref{a2}) and\n\\begin{equation}\n\\label{s2constant}\ns_2=\\frac{(\\xi+\\sqrt{16+\\xi^2})^2}{2}\n\\;.\\end{equation}\n\nBy employing the expressions of the coefficients $c_i$ provided by Eq. (\\ref{c3}) in Eq. (\\ref{ee}), we calculate the entanglement entropy $S$ for $N=3$, which has the following expression\n\n\\begin{equation}\n\\label{s3}\nS=-\\frac{2A_{3}^2}{3}\\bigg(s_{3}^2\\,\\log_2[\\frac{s_3}{4\\sqrt{4+\\xi(\\xi+2)}}]+3\\log_2[A_{3}^2]\\bigg)\n\\;,\\end{equation}\nwhere $A_3$ is the normalization factor given by Eq. (\\ref{a3}) and\n\\begin{equation}\n\\label{s3constant}\ns_3=(1+\\xi+\\sqrt{4+\\xi(\\xi+2)})\n\\;.\\end{equation}\n\nBy following the same path for $N=4$ (i.e. by using formulas given by Eq. (\\ref{c4}) in Eq. (\\ref{ee})), one obtains the following entanglement entropy\n\\begin{eqnarray}\n\\label{s4}\n&&S=\\log_2 d_4-d_{4}^{-1}\\bigg(48 \\log_2 24+24(6\\xi-\\tilde E)^2\\log_2[6(6\\xi-\\tilde E)]\\nonumber\\\\\n&&+2(18\\xi^2-9\\tilde E\\xi+\\tilde E^2-4)^2\\log_2[(18\\xi^2-9\\tilde E\\xi+\\tilde E^2-4)]\\bigg)\\;,\\nonumber\\\\\n\\end{eqnarray}\nwhere $\\tilde E$ is given by Eq. (\\ref{egs4}) and $d_4$ by the second row of Eq. (\\ref{a4}).\n\n\nNote that the results stated for $N=2$ by Eqs. (\\ref{f2}) (Fisher information), (\\ref{alpha2}) (coherence visibility), and (\\ref{s2}) (entanglement entropy) coincide with those that we obtained in \\cite{main}.\n\nThe bottom panel of Fig. 2 shows the entanglement entropies (\\ref{s2}), (\\ref{s3}), and (\\ref{s4}) as functions of the scaled on-site interaction $\\xi=U\/J$. In the limit $\\xi \\rightarrow -\\infty$, corresponding to the emergence of the cat-like state (\\ref{cat}), $S$ tends to one both when $N=2,4$ (solid line, dot-dashed line) and $N=3$ (dashed line). By analyzing the plot of $S$ versus $\\xi$ it is possible to observe that - as discussed in \\cite{main} - the greater is the number of bosons the closest to zero is the negative $\\xi$ (which, in fact, is equal to $-\\sqrt{2}\\simeq -1.41421$ when $N=2$, $-1$ when $N=3$, and is $\\simeq -0.761472$ when $N=4$) for which $S$ attains its maximum value.\nFor $N=2$ and $N=3$, the latter value coincide with that predicted theoretically, i.e. $\\log_2(N+1)$ (see the comment at the end of the previous section): in the ground state of the two-site Bose-Hubbard Hamiltonian all the Fock states $|i,N-i\\rangle$ have the same probability $|c_i|^2=1\/(N+1)$ for any $i$, as one can see from the left and the middle panels of the third row (from the top) of Fig. 1. When $N=4$, instead, the maximum of $S$ does not coincide with $\\log_2(N+1)$, as it can be observed from the right plot of the third row (from the top) of Fig.1, where the $|c_i|^2$, corresponding to the interaction signing the maximum of $S$ (\\ref{s4}), are different from each other.\n\n\nAs conclusive remarks, we note that when $N=3$, $S$ approaches to $1$ both in the limit $\\xi \\rightarrow -\\infty$ and in the limit $\\xi \\rightarrow +\\infty$, as one can observe from the dashed line in the bottom panel of Fig. 2.\nMoreover, we observe that the plot reported in the bottom panel of Fig. 2 shows that the cat-like state (\\ref{cat}) ($\\xi \\ll -1$, deeply attractive bosons) is not the maximally entangled ground state achievable in our system.\n\n\nA remarkable point emerges from our analysis. The three ground-state characterizing parameters ($F$, $\\alpha$, $S$) in the deep repulsive regime exhibit very different behavior depending on if $N=2,4$ or $N=3$, as it can be seen from Fig. 2 and from Tabs. 1-3. In fact, when $\\xi=U\/J\\rightarrow +\\infty$, the Fisher information, the coherence visibility, and entanglement entropy are all equal to zero if the ground state is the state (\\ref{fock}) ($N=2,4$); they are finite, instead, if the ground state is the state (\\ref{pseudo:fock}) ($N=3$). Note that this circumstance is quite general. In fact, the states (\\ref{fock}) and (\\ref{pseudo:fock}) are the ground states of the two-site Bose-Hubbard Hamiltonian in the limit $\\xi \\rightarrow +\\infty$ for any even $N$ and any odd $N$, respectively. In particular, we want to stress that when the boson-boson interaction is strongly repulsive, the ground state of the two-site BH Hamiltonian is not quantum entangled when $N$ is even, while it is a quantum entangled state when $N$ is odd.\n\n\n\n\\subsection{Fisher information and coherence visibility via the Hellmann-Feynmann theorem}\n\nAt this point, it is interesting to observe that it is possible to achieve the above formulas for $F$ - Eqs. (\\ref{f2}),(\\ref{f3}),(\\ref{f4}) -  and $\\alpha$ - Eqs. (\\ref{alpha2}),(\\ref{alpha3}), (\\ref{alpha4})  - also by exploiting the Hellmann-Feynman theorem (HFT) \\cite{cohen}.\n\nBy using this theorem a relation can be established between the Fisher information $F$ and\nthe first partial derivative of the ground-state energy $E$ with respect to $U$, $\\displaystyle{\\frac{\\partial E}{\\partial U}}$. According to the HFT,\nwe have that \\cite{cohen}\n\\begin{equation}\n\\label{hft1}\n\\frac{\\partial E}{\\partial U}=\\langle E|\\frac{\\partial \\hat{H}}\n{\\partial U}|E\\rangle\n\\;.\\end{equation}\nIf the  properties $\\langle E |\\hat{n}_L|E\\rangle=\\langle  E|\\hat{n}_R|E\\rangle=N\/2$ (Eq. (\\ref{lrs})) and $\\langle E |\\hat{n}_L|E\\rangle+\\langle  E|\\hat{n}_R|E\\rangle=N$ are used in Eq. (\\ref{hft1}), we get\n\\begin{equation}\n\\label{ezerou1}\n\\frac{\\partial E}{\\partial U}=\\langle E| \\hat{n}^{2}_L |E\\rangle-\\frac{N}{2}\n\\;.\\end{equation}\nOn the other hand, again thanks to $\\langle E |\\hat{n}_L|E\\rangle=\\langle  E|\\hat{n}_R|E\\rangle=N\/2$ and  $\\langle E |\\hat{n}_L|E\\rangle+\\langle  E|\\hat{n}_R|E\\rangle=N$ ,  Eqs. (\\ref{qfi}) and (\\ref{fi}) give rise to\n\\begin{equation}\n\\label{ezerou2}\nF=\\frac{4}{N^2}\\langle E | \\hat{n}^{2}_L|E\\rangle-1\n\\;,\\end{equation}\nso that (as also commented in \\cite{main})\n\\begin{equation}\n\\label{fihf}\nF=\\frac{4}{N^{2}}\\frac{\\partial E}{\\partial U}+\\frac{2}{N}-1\n\\;.\\end{equation}\nWe have therefore to know the energy of the ground state. When $N=2$ this energy is given by Eq. (\\ref{egs2}), when $N=3$ by Eq. (\\ref{egs3}), and when $N=4$ by Eq. (\\ref{egs4}). By keeping in mind that in these three latter equations $\\xi=U\/J$ and the energies are measured in units of $J$, one can resort from $\\tilde E$ to $E$ and obtain Eq. (\\ref{f2}) for $N=2$, Eq. (\\ref{f3}) for $N=3$, and Eq. (\\ref{f4}) for $N=4$.\n\nThe Hellmann-Feynman theorem provides a relation between the coherence visibility $\\alpha$\nand the first partial derivative of the ground-state energy $E$ with respect to $J$ as well. Let us start from the fact that, according to the HFT,\none has that \\cite{cohen}\n\\begin{equation}\n\\frac{\\partial E}{\\partial J}=\\langle E|\n\\frac{\\partial \\hat{H}}{\\partial J}|E\\rangle\n\\;.\\end{equation}\nBy using the fact that the coefficients $c_i$ involved in the expansion of $|E\\rangle$ ($N=2,3,4$) are real (see Eqs. (\\ref{c2}), (\\ref{c3}), (\\ref{c4})  jointly to Eqs. (\\ref{a2}),(\\ref{a3}),(\\ref{a4})), we can write that\n$\\langle E| \\hat{a}^\\dagger_L\\hat{a}_R|E\\rangle=\\langle E| \\hat{a}^\\dagger_R\\hat{a}_L|E\\rangle$, and then\n\\begin{equation}\n\\label{ezeroj1}\n\\frac{\\partial E}{\\partial J}=-2\\langle E| \\hat{a}^\\dagger_L\\hat{a}_R|E\\rangle\n\\;.\\end{equation}\nOn the other hand, again in force to $\\langle E| \\hat{a}^\\dagger_L\\hat{a}_R|E\\rangle=\\langle E| \\hat{a}^\\dagger_R\\hat{a}_L|E\\rangle$, we have that the coherence visibility (\\ref{visibility}) is\n\\begin{equation}\n\\label{ezeroj2}\n\\alpha=\\frac{2 \\langle E| \\hat{a}^\\dagger_L\\hat{a}_R|E\\rangle}{N}\n\\end{equation}\nso that (see also \\cite{main})\n\\begin{equation}\n\\label{alphahf}\n\\alpha =-\\frac{1}{N}\\frac{\\partial E}{\\partial J}\\;.\n\\end{equation}\nAlso in this case, we use in Eq. (\\ref{alphahf}) the results given by Eq. (\\ref{egs2}), $N=2$, by Eq. (\\ref{egs3}), $N=3$, and by Eq. (\\ref{egs4}) (by keeping in mind that $\\xi=U\/J$ and the energies are measured in units of $J$) and get Eq. (\\ref{alpha2}) for $N=2$, Eq. (\\ref{alpha3}) for $N=3$, and Eq. (\\ref{alpha4}) for $N=4$.\\\\\n\n\n\n\\section{Conclusions}\n\nWe have investigated a finite number $N$ of (both attractively and repulsively) interacting bosonic atoms confined\nby a one-dimensional double-well shaped potential. Within the two-site Bose-Hubbard model framework, we have carried out the zero-temperature analysis for $N=2$, $N=3$, and $N=4$ bosons by finding analytical formulas for the eigenvectors and eigenvalues of the corresponding ground states. These have been characterized by analytically calculating the Fisher information, the coherence visibility, and the entanglement entropy. We have studied these parameters by varying the boson-boson interaction strength (ranging from strong attractions to strong repulsions) which is the key quantity in determining the kind of ground state sustained by the two-site Bose-Hubbard Hamiltonian. In particular, we have commented on the difference, existing in the deep repulsive regime, between the structure of the ground state in the presence of an even number of bosons and that with an odd number of particles. We have pointed out, in particular, that the ground state of the two-site Bose-Hubbard Hamiltonian is not quantum entangled when $N$ is even, while it is a quantum entangled state when $N$ is odd.\\\\\n\n\n\\begin{acknowledgement}\nThis work has been supported by MIUR (PRIN 2010LLKJBX). GM and LS acknowledge financial support from\nthe University of Padova (Progetto di Ateneo 2011) and Cariparo Foundation (Progetto di Eccellenza 2011). GM acknowledges financial support also from Progetto Giovani 2011\nof University of Padova.\n\\end{acknowledgement}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThroughout this paper, we work over the field $\\mathbb{C}$ of complex numbers. It has been well known  by people working in higher dimensional geometry that there is an analogue between the local object Kawamata log terminal (klt)  singularities $(X,o)$ and log Fano varieties as the global counterpart (cf. e.g. \\cite{Sho00, Xu14} etc.). From this comparison, since the stability theory of Fano varieties has been a central object in people's study in the last three decades, it is natural to expect there is a local stability theory on singularities.  The primary goal of this preprint is to develope such a theory.  In another word,  we want to investigate singularities using the tools from the theory of K-stability, a notion which was first defined in \\cite{Tia97} and  later algebraically formulated in \\cite{Don02}.  We note that this interaction between birational geometry and K-stability theory  has been proved to significantly fertilize  both subjects (cf. \\cite{Oda12, Oda13, LX14, WX14, LWX14, Fuj15} etc.).   \n\nFor the stability theory of log Fano varieties, a crucial ingredient is the CM weight. Philosophically, the stability of log Fano varieties is equivalent to minimizing the CM weight.  In the stability theory of singularities, we fix the singularity $(X,o)$ and look for `the most stable' valuations $v$. Thus the first step of establishing a local stability theory for $(X,o)$ would be to find the right counterpart of CM weight in the local setting.\nIn the work \\cite{Li15a}, the first named author defined the normalized volume function ${\\widehat{\\rm vol}}_{X,o}$ on the space of valuations centered at $o$, which we believe should be the right invariant. In fact, its derivative at the canonical divisorial valuation over the cone singularity along certain tangent direction gives the the CM weight. So in the local picture, the normalized volume function indeed carries more information than the CM weight!  \n\nBy the above discussion, inspired by the global theory, we focus on studying the valuation minimizing the normalized volume function, which is conjectured to uniquely exist and  ought to be thought as the `(semi-)stable' object.  This picture is understood well in the case of Sasakian geometry where one only considers the valuations coming from the Reeb vector fields on the torus (e.g. \\cite{MSY08, CoSz16}). Here we can naturally define the stability of the singularity using the one for the base. However, this requires the extra cone structure. By investigating the minimizer of the normalized volume function on all valuations, our plan, as we mentioned, built on the previous work (\\cite{Li15a, Li15b, LL16}), is to establish an intrinsic stability which only depends on the isomorphic class of the singularity.\nTo revisit the cone singularity case, in fact it was shown in \\cite{Li15b, LL16} that a Fano manifold $X$ is K-semistable, if and only if that among all valuations over the vertex $o$ of the cone $C(X)$ given by a multiple of $-K_X$,  the canonical valuation defined to be the one obtained by blowing up the vertex $o\\in C(X)$ minimizes the normalized volume function.  This gives evidence to justify that at least for such singularities, our study is on the right direction. \n\nFor an {\\it arbitrary} klt singularity, there is no direct way to associate a global object. Nevertheless, in differential geometry, when there is a `canonical' metric, the metric tangent cone around the singularity is the stable object in the category of metric space. With a similar philosophy,  we expect the minimizer of the normalized volume function always gives {\\it a degeneration} to a K-semistable Fano cone singularity in the Sasakian setting, and such degeneration should be provided by the minimizer of ${{\\widehat{\\rm vol}}}_{X,o}$.  In the current paper, we work out this picture in the case that the minimizer is divisorial, by  implementing the machinery of the minimal model program (based on the foundational results in \\cite{BCHM10}). So our treatment will be purely algebraic though it is strongly inspired by profound analytic results.  \n\n\n\n  One ingredient we introduce is to define the volume associated to a birational model and then connect it to the normalized volume of a valuation. For studying the divisorial valuations, the class of models which play a central role here are the ones obtained by the construction of {\\it Koll\\'ar} component (cf. \\cite{Xu14}): for an arbitrary $n$-dimensional klt singularity $(X,o)$, we can use minimal model program to construct a birational model whose exceptional locus is an $(n-1)$-dimensional $\\mathbb{Q}$-Fano variety. We will systematically develop the tools of using Koll\\'ar components to understand the normalized local volume and its minimizer.   In fact, Koll\\'ar components can be considered as the local analogue of special degenerations studied in \\cite{LX14}. \nIn the case of Sasakian geometry, one Reeb vector gives a Koll\\'ar component if and only if it is rational, i.e., it is quasi-regular. \n  \nTherefore, to summarize, the aim of this paper is of twofolds: on one hand, we aim at using the construction of Koll\\'ar components to get information of the valuations space, especially the minimizer of the normalized volume functions; On the other hand, in the reverse direction, we want to use the viewpoint of stability to study the birational geometry construction of Koll\\'ar components, and search out a more canonical object under suitable assumptions.   \n\nWe also expect for any klt singularity $(X,o)$, even when the minimizer is not necessarily divisorial, we can still use suitable birational models to degenerate $(X,o)$ to a K-semistable (possibly irregular) Sasakian singularity. However, it seems to involve a significant amount of new technical issues. \n\nIn the below, we will give more details. \n\n\\subsection{Koll\\'ar components}\n\\begin{defn}[Koll\\'ar component]\\label{d-kollar}\nLet $o\\in (X,D)$ be a klt singularity. We call a proper birational morphism $\\mu:Y\\to X$ provides a {\\bf Koll\\'ar component $E$}, if $\\mu$ is isomorphic over $X\\setminus \\{o\\}$, and $\\mu^{-1}(o)$ is an irreducible divisor $S$, such that $(Y,S+\\mu^{-1}_*D)$ is pure log terminal and $-S$ is $\\mathbb{Q}$-Cartier and ample over $X$.\n\\end{defn}\n\nWe easily see the birational model $Y$ is uniquely determined once the divisorial valuation $S$ is fixed, and if we denote \n$$(K_Y+S+\\mu_*^{-1}D)|_S=K_S+\\Delta_S,$$ \nthen $(S,\\Delta_S)$ is a klt log Fano pair. \n\n Given any klt singularity $(X,o)$, after the necessary minimal model program type result is established (see \\cite{BCHM10}), we know that there always exists a Koll\\'ar component  (see \\cite{Pro00} or \\cite[Lemma 1]{Xu14}), but  it is often not unique (nevertheless, see the discussion in \\ref{e-example}.4 for some known special cases for the uniqueness). From what we have discussed, instead of an arbitrary Koll\\'ar component, we want to study those which are `more stable', and show if it exists, it provides a more canonical object. Indeed, we prove that if there is a K-semistable Koll\\'ar component, then it gives the unique minimizer of ${\\widehat{\\rm vol}}(X,o)$ among all Koll\\'ar components.   \n \n \\vskip 1mm\n\n\\vskip 1mm \n \nCompared to the global theory of degeneration of Fano varieties, this fits into the philosophy that K-stability provides a more canonical degeneration(cf. \\cite{LWX14, SSY14}) and it should minimize the CM weight among all degenerations. But one surprising thing to us is that K-semistability is enough, instead of K-polystability which was required in the global setting.  \n \n \\bigskip\n \nThe following theorem is our main theorem. \n \\begin{thm}\\label{t-main1}Let $o\\in (X,D)$ be klt singularity.  \nA divisorial valuation \n${\\rm ord}_S$ is a minimizer of ${\\widehat{\\rm vol}}_{X,D, o}$ if and only if the following conditions are satisfied\n\\begin{enumerate}\n\\item\n$S$ is a Koll\\'{a}r component; \n\\item\n$(S, \\Delta_S)$ is K-semistable.\n\\end{enumerate}\nMoreover, such a minimizing divisorial valuation is unique.\n\\end{thm}\n \nMore precisely, we will prove Theorem \\ref{t-main1} by proving the following four theorems. For each of them, we need somewhat different techniques. \n\nFirst we prove\n\n\\begin{thmx}\\label{t-main}\nIf $o\\in (X,D)$ is an algebraic klt singularity. Let $S$ be a Koll\\'ar component over $X$.\nIf $(S,\\Delta_S)$ is (log-)$K$-semistable. Then ${\\widehat{\\rm vol}}_{X,o}$ is minimized at the valuation ${\\rm ord}_S$. \n\n\\end{thmx}\n \nThis extends the main theorem in \\cite{LL16} from cone singularities to the more general setting. For the proof, we indeed degenerate  a general singularity to a cone singularity induced by its Koll\\'ar components. However, instead of degenerating the valuation, we degenerate the associated ideals and then use the result in \\cite{Liu16}. An extra subtlety is to treat the case of the cone singularity, we can not directly use \\cite{LL16} as there they only proved the result for the cone singularity over an analytic K-semistable $\\mathbb{Q}$-Fano variety, which we still do not know to be equivalent to the algebraic definition. As a redemption, we first show that it suffices to concentrate on the equivariant data and then use \\cite{Li15b} to finish the argument. \nIn  Section \\ref{s-exam}, we use this criterion to find minimizers for various examples of singularities including: quotient singularities, $A_k$ and $E_k$ singularities etc.\n\n\\bigskip\n\nNext, we turn to the result on the uniqueness.\n\n\\begin{thmx}\\label{t-main2}\nIf $o\\in (X,D)$ is an algebraic klt singularity. Assume $S$ is a Koll\\'ar component over $X$ such that $(S,\\Delta_S)$ is $K$-semistable. Then \n$${\\widehat{\\rm vol}}_{X,o}({\\rm ord}_S)<{\\widehat{\\rm vol}}_{X,o}({\\rm ord}_T)$$ for any other divisorial valuation $T$. \n\\end{thmx}\n\nThis is done by a detailed study of the geometry when the equality holds. In the cone singularity case, we investigate the equality condition in the calculation in \\cite{Li15b}. It posts a strong assumption which enables us to compute the corresponding invariants including nef thresholds and pseudo-effective thresholds. The argument is partially inspired by the work in \\cite{Liu16}. Once this is clear, the rest follows from a simple application of Kawamata's base point free theorem. And the general case can be again reduced to the case of cone singularity using a degeneration process. \n\n\\bigskip\n\nNow we consider the converse direction. \nFor any klt singularity, a minimizer of the normalized volume function always exists by \\cite{Blu16b}. The following theorem says that if a minimizer is divisorial, it always yields a Koll\\'ar component. We indeed prove slightly more for a general rational rank 1 minimizer.\n\\begin{thmx}\\label{t-divisor}\nGiven an arbitrary algebraic klt singularity $o\\in (X,D)$ where $X=\\mathrm{Spec}(R)$. Let $v$ be a valuation that minimizes ${\\widehat{\\rm vol}}_{X,o}$. Assume the valuation group of $v$ is isomorphic to $\\mathbb{Z}$, i.e., $v$ has rational rank one,  and one of the following two assumptions holds\n\\begin{enumerate}\n\\item $v$ is a multiple of a divisorial valuation; or\n\\item the graded family of valuative ideals  $$\\mathfrak{a}_{\\bullet}=\\{\\mathfrak{a}_k\\} \\mbox{\\ where \\ } \\mathfrak{a}_k=\\{f\\in R\\ |\\ v(f)\\ge k\\} $$ is finitely generated, i.e., there exists $m\\in \\mathbb{N}$ such that $\\mathfrak{a}_{mk}=(\\mathfrak{a}_m)^k$ for any $k\\in \\mathbb{N}$.\n\\end{enumerate}\nThen up to a rescaling, $v$ is given by the divisorial valuation induced by a Koll\\'ar component $S$.\n\\end{thmx}\n\nThe fact that the assumption 1 above implies $v$ is given by a Koll\\'ar component is also independently proved in \\cite{Blu16b}. \nWe note that a minimizer is conjectured to be quasi-monomial ((cf. \\cite[Conjecture 6.1.3]{Li15a})) and  the graded family of valuative ideals for a minimizer of the normalized volume function is conjectured to be always finitely generated (cf. \\cite[Conjecture 6.1.5]{Li15a}). So granted any one of these two conjectures, this result should presumably characterize all the cases with minimizers of rational rank 1. \nThe proof uses the definition of the volumes of a model, and run the decreasing process of the volumes given by the minimal model program as in \\cite{LX14}.\n\n\\bigskip\n\nNext we turn to the stability of the minimizer. By using the techniques from the toric degeneration (see \\cite{Cal02, AB04, And13}) and the relation between CM weight and normalized volumes, we \nwill prove\n\\begin{thmx}\\label{t-mintok}\nWe use the same notation as in Theorem \\ref{t-divisor}. Let $\\mu\\colon Y\\to X$ be the morphism which extracts $S$, and write $(K_Y+S+\\mu_*^{-1}D)|_S=K_S+\\Delta_S$, then $(S,\\Delta_S)$ is a K-semistable log Fano pair. \n\\end{thmx}\n\n\n\\subsection{Approximation}\n\n\nIn a different direction,  we also obtain results which describe the minimizer of the normalized  volume function from the viewpoint of Koll\\'ar components. We show\nfor a general klt singularity, although the minimizer of its associated normalized volume function might not be given by a Koll\\'ar component, but we can always approximate it by a sequence of Koll\\'ar components. \n\\begin{thm}\\label{t-approx}\nGiven an arbitrary algebraic klt singularity $o\\in (X,D)$, and a minimizer $v^{\\rm m}$ of ${\\widehat{\\rm vol}}_{X,o}$, there always exists a sequence of Koll\\'ar components $\\{ S_i\\}$ and positive numbers $c_j$ such that \n$$\\lim_{j\\to \\infty}c_j\\cdot {\\rm ord}_{S_j}\\to v\\mbox{\\ in }{\\rm Val}_{X,o} \\mbox{\\ \\ \\ and \\ \\ \\ } \\lim_{i\\to \\infty} {\\widehat{\\rm vol}}({\\rm ord}_{S_i})={\\widehat{\\rm vol}}(v^{\\rm m}).$$\n\\end{thm}\nHere ${\\rm Val}_{X,o}$ consists of all valuations centered at $o$, and is endowed with the weakest topology as in \\cite[Section 4.1]{JM12}. \n\n\n\\subsection{Equivariant K-semistability}\nBy relating a Fano variety and the cone over it, we can compare the calculation in \\cite{Li15b} for a cone and \\cite{Fuj16} for its base.  Then an interesting by product of our method is the following theorem.\n\\begin{thm}\\label{t-equiK}\nLet $T\\cong (\\mathbb{C}^*)^r$ be a torus. Let $(V,\\Delta) $ be a log Fano variety with a $T$-action. Then $(V,\\Delta)$ is K-semistable if and only if any $T$-equivariant special test configuration $\\mathcal{S}\\to \\mathbb{A}^1$ of $(V,\\Delta)$ has its generalized Futaki invariant ${\\rm Fut}(\\mathcal{V})\\ge 0$. \n \\end{thm}\n \n When $S$ is smooth and $\\Delta=0$, this follows from the work of \\cite{DS15} with an analytic argument. Our proof is completely algebraic. It again uses the techniques of degenerating any ideal to an equivariant one and showing it has a smaller invariant. \n\n\\bigskip\n\nThe paper is organized in the following way: In Section \\ref{s-pre}, we give some necessary backgrounds.  In Section \\ref{s-vmodel}, we introduce one key new tool: the volume of a model. By combining the normalized volume function on valuations with the local volume defined in \\cite{Ful13}, and applying minimal model program, we prove Theorem \\ref{t-approx} and Theorem \\ref{t-divisor}. In Section \\ref{s-min}, we prove Theorem \\ref{t-main}, by connecting it to the infimum of ${\\rm lct}(X,D;\\mathfrak{a})^n \\cdot {\\rm mult}(\\mathfrak{a})$ for all $\\mathfrak{m}$-primary ideals $\\mathfrak{a}$ centered on $o$. We note that this latter invariant indeed has also been studied in other context (cf. \\cite{dFEM04}).  In Section \\ref{s-uni}, we prove Theorem \\ref{t-main2}. We first prove it for the cone singularity case, using heavily the ideas and calculations in \\cite{LL16}. Then we use a degeneration argument \nto reduce the general case to the case of cone singularities. In Section \\ref{s-Ksta}, we prove Theorem \\ref{t-mintok}, which verifies the K-semistability of a minimizing Koll\\'ar component.  In Section \\ref{s-exam}, we give some examples on how to apply our techniques to calculate the minimizer for various classes of klt singularities. \n \\vspace{5mm}\n \n \\noindent {\\it History:} Since \\cite{Li15a}, the study of the minimizer of the normalized volume function moves forward rapidly. Our preprint is inspired by the earlier work of \\cite{Li15a, Li15b} and  \\cite{LL16}. It also uses ideas from \\cite{Liu16}, which is written in the same period as the first version of our preprint.  After we posted our preprint, the existence of the minimizer is completely settled in \\cite{Blu16b}. Inspired by this work, in the revision, we improve our work by showing that for a $\\mathbb{C}^*$-equivariant singularity, we only need to consider the equivariant  valuations for minimizing the normalized local volume. We also include his result on the existence of the minimizer in the exposition. In particular, this allows us to stay with the algebraic definition of K-semistability.  Another major improvement we achieve in the revision is that we can indeed show that any Koll\\'ar component which minimizes the normalized local volume is always K-semistable.\n  \\vspace{5mm}\n \n\\noindent\n{\\bf Acknowledgement}: We thank Yuchen Liu, Dhruv Ranganathan and Xiaowei Wang for helpful discussions and many useful suggestions. We especially  want to thank Harold Blum and Mircea Musta\\c{t}\\v{a} for pointing out a gap in an earlier draft.  CL is partially supported by NSF DMS-1405936. CX is partially supported by `The National Science Fund for Distinguished Young Scholars (11425101)'. Part of the work was done when CX visited Imperial College London and Massachusetts Institute of Technology. He wants to thank Paolo Cascini and Davesh Maulik for the invitation and providing a wonderful environment.   \n\n \n\\section{Preliminary}\\label{s-pre}\n\n\\noindent{\\bf Notation and Conventiones:} We follow the standard notation in  \\cite{Laz, KM98, Kol13}. A {\\it log Fano} pair $(X,D)$ is a projective klt pair such that $-K_X-D$ is ample.  \n\nFor a local ring $(R,\\mathfrak{m})$ and $\\mathfrak{a}$ an $\\mathfrak{m}$-primary ideal, we denote by $l_R(R\/\\mathfrak{a})$ the length of $R\/\\mathfrak{a}$. \n\nFor the K-semistability of a log Fano pair, see \\cite{Tia97, Don02} (also see \\cite{Oda13,LX14}).\n\n\n\n\n\n\n\n\\subsection{Normalized volume}\n\nLet $(X,o)$ be a normal algebraic singularity and $v\\in {\\rm Val}_{X,o}$, which is the space of all valuations centered on $o$.  Let $D\\ge 0$ be a $\\mathbb{Q}$-divisor such that $K_X+D$ is $\\mathbb{Q}$-Cartier. We can define the volume ${\\rm vol}_{X,o}(v)$ and the log discrepancy $A_{X,D}(v)$ (if the context is clear, we will abbreviate it as ${\\rm vol}(v)$ and $A(v)$) following \\cite{ELS03} and \\cite{JM12} (see e.g. \\cite[Section 1.1]{Li15a}).  In particular, if $S$ is a divisor with center on $X$ to be $o$, we have \n$$A_{X,D}(S):=A_{X,D}({\\rm ord}_S)=a(S; X,D)+1$$ \nthe same as the standard log discrepancy. \n\n\\begin{defn}Notation as above. We denote the {\\bf normalized volume} by ${\\widehat{\\rm vol}}_{X,D,o}(v)$ (or ${\\widehat{\\rm vol}}_{X,o}(v)$ if $D$ is clear or ${\\widehat{\\rm vol}}_{X,D}(v)$ if $o$ is clear or simply ${\\widehat{\\rm vol}}(v)$ if there is no confusion) to be \n$${\\rm vol}_{X,o}(v)\\cdot A^n(v)$$ \nif $A^n(v)<+\\infty$; and  $+\\infty$  if $A^n(v)=+\\infty$. \n\\end{defn}\n\nRecall that it was conjectured that ${\\widehat{\\rm vol}}(v)$ achieves the minimum at a valuation $v\\in {\\rm Val}_{X,o}$.  In \\cite{Li15a}, it is showed that the space\n$$\\{v\\in {\\rm Val}_{X,o}|\\ v(\\mathfrak{m})=1, {\\widehat{\\rm vol}}(v) \\le C \\}$$\nfor any constant $C>0$ forms a compact set. However, in general the volume function ${\\rm vol}$ is only upper continuous on ${\\rm Val}_{X,o}$ although we expect it is continuous at a minimizer of ${\\widehat{\\rm vol}}$. \n\n\\begin{prop}\\label{p-upperconti}\nIf $\\{v_i\\}$ is a sequence of valuations, such that $v_i\\to v$ in the weakest topology, then \n$${\\rm vol}(v)\\ge \\limsup_i {\\rm vol}(v_i). $$ \n\\end{prop}\n\\begin{proof}The valuation $v$ determines a sequence of graded ideas \n$$\\mathfrak{a}_k=\\mathfrak{a}_k(v)=\\{f\\in R\\ |\\  v(f)\\ge k\\}.$$  \nBy \\cite{Mus02}, we know that for any $\\epsilon>0$, there exists a sufficiently large $k$ such that \n$$\\frac{1}{k^n}{\\rm mult}(\\mathfrak{a}_k)< {\\rm vol}(v)+\\epsilon.$$\nSince $R$ is Noetherian, we know that there exist  finitely many generators $f_p$ ($1\\le p\\le j$) of $\\mathfrak{a}_k=(f_1,...,f_j)$. As $v(f_p)\\ge k$, we know that for any $\\delta$, there exists sufficiently large $i_0$ such that for any $i\\ge i_0$, $v_i(f_p)\\ge k-\\delta$. Thus \n$$\\mathfrak{a}^{(i)}_{k-\\delta}=\\{f\\in R\\ |\\  v_i(f)\\ge k-\\epsilon \\}\\supset \\mathfrak{a}_k .$$\nTherefore,\n$${\\rm vol}(v_i)\\le \\frac{1}{(k-\\delta)^n}{\\rm mult}(\\mathfrak{a}^{(i)}_{k-\\epsilon})\\le  \\frac{1}{(k-\\delta)^n}{\\rm mult}(\\mathfrak{a}_{k})\\le  \\frac{k^n}{(k-\\delta)^n}({\\rm vol}(v)+\\epsilon).$$\n\\end{proof}\n\n\n\\begin{prop}\\label{p-valuation}\nLet $(X,o)=(\\mathrm{Spec}(R),\\mathfrak{m})$ be a singularity. Let $v$ and $v'$ be two real valuations in ${\\rm Val}_{X,o}$. Assume \n$${\\rm vol}(v)={\\rm vol}(v')>0\\qquad \\mbox{and}  \\qquad v(f)\\ge v'(f)$$ for any $f\\in R^*$, then $v=v'$.\n\\end{prop}\n\\begin{proof} We prove it by contradiction. Assume this is not true, we fix $g\\in R$ such that \n$$v(g)=l>v'(g)=s.$$\nDenote by $r= l-s>0.$ Fix $k\\in \\mathbb{R}_{>0}$. Consider \n$$\\mathfrak{a}_k:=\\{f\\in R | \\ v(f)\\ge k\\}\\qquad\\mbox{ and} \\qquad{\\mathfrak{b}}_k:=\\{f\\in R |\\  v'(f)\\ge k\\}. $$\nSo ${\\mathfrak{b}}_k\\subset \\mathfrak{a}_k$, and we want to estimate the dimension of \n$$\\dim(R\/{{\\mathfrak{b}}_k})-\\dim(R\/{\\mathfrak{a}_k})=\\dim (\\mathfrak{a}_k\/{\\mathfrak{b}}_k). $$\nFix a positive integer $m<\\frac{k}{l}$ and a set\n$$g^{(1)}_{m},...,g^{(k_m)}_{m} \\in {\\mathfrak{b}}_{k-ml}$$\nwhose images in ${\\mathfrak{b}}_{k-ml}\/{\\mathfrak{b}}_{k-ml+r}$ form a $\\mathbb{C}$-linear basis.\n\nWe claim that \n$$\\{f^m\\cdot  g^{(j)}_{m}\\}\\ \\ (1\\le m \\le \\frac{k}{l}, 1\\le j\\le k_m)$$ \nare $\\mathbb{C}$-linear independent in $\\mathfrak{a}_k\/{\\mathfrak{b}}_k$. Granted this for now, \nwe know since ${\\rm vol}(v)>0$, then \n$$\\limsup_{\\lim k\\to \\infty} \\frac{1}{k^n}\\sum_{1\\le m \\le \\frac{k}{l}}k_m=\\limsup_{\\lim k\\to \\infty} \\sum_{1\\le m\\le \\frac{k}{l}} \\frac{1}{k^n} \\dim ({\\mathfrak{b}}_{k-ml}\/{\\mathfrak{b}}_{k-ml+r})>0, $$\nwhich then implies ${\\rm vol}(v)>{\\rm vol}(v')$.\n\n\\bigskip\n\nNow we prove the claim.\n\n\\noindent{\\bf Step 1:} For any $1\\le m \\le \\frac{k}{l}, 1\\le j\\le k_m$,\n\\begin{eqnarray*}\nv(f^m\\cdot g^{(j)}_{m})&=&v(f^m)+v(g^{(j)}_{m})\\\\\n &\\ge& ml+v'(g^{(j)}_{m})\\\\\n &\\ge & ml+k-ml\\\\\n &\\ge &k.\n \\end{eqnarray*}\n Thus $f^m\\cdot g^{(j)}_{m}\\in \\mathfrak{a}_k$. \n \n \\vspace{3mm}\n \n \\noindent{\\bf Step 2:} If  $$\\{f^m\\cdot  g^{(j)}_{m}\\}\\ \\ (1\\le m \\le \\frac{k}{l}, 1\\le j\\le k_m)$$ are not $\\mathbb{C}$-linear independent in $\\mathfrak{a}_k\/{\\mathfrak{b}}_k$,\n then there is an equality\n $$\\sum_{m}h_m= b \\in  {\\mathfrak{b}}_k,$$\n where there exists $c_j\\in \\mathbb{C}$, such that \n $$h_m=f^m\\cdot \\sum_{1\\le j \\le k_m} c_jg^{(j)}_{m}$$ \n \n  and some $h_m\\neq 0$. Consider the maximal $m$, such that $h_m\\neq 0$.\n Since\n\\begin{eqnarray*}\n  v'(h_m)&=&v'(f^m\\cdot \\sum_{1\\le j \\le k_m} c_jg^{(j)}_{m}) \\\\\n  &=& v'(f^m)+v'(\\sum_{1\\le j \\le k_m} c_j g^{(j)}_{m})\\\\\n&< & ms+k-ml+r\\\\\n &= & k-(m-1)l+(m-1)s,\n \\end{eqnarray*}\n where the third  inequality follows from that \n $$ \\sum_{1\\le j \\le k_m} c_jg^{(j)}_{m} \\notin {\\mathfrak{b}}_{k-ml+r} .$$\n However, we have \n\\begin{eqnarray*}\n  v'(h_m)&=&v'(b-\\sum_{j<m}h_j) \\\\\n  &\\ge & \\min \\big\\{ v'(b), v'(h_1),....v'(h_{m-1}) \\big\\}\\\\\n&= & \\min_{1\\le j \\le m-1}\\{ k , js+k-jl\\} \\\\\n &=& k-(m-1)l+(m-1)s,\n \\end{eqnarray*}\nwhich is a contradiction. \n\\end{proof}\n\n\n\n\nSeveral results in our work depend on a relation between the normalized volume and some normalized multiplicity. The latter quantity was first considered in the smooth case in \\cite{dFEM04}, and since then it has been studied  in many other work, including positive characteristic versions (see e.g. \\cite{TW04}). Its relevance to the normalized volume appeared in \\cite[Example 3.7]{Li15a}. We have the following more precise observation in \\cite{Liu16}. \n\\begin{prop}[{\\cite[Section 4.1]{Liu16}, cf. \\cite[Example 3.7]{Li15a}}]\\label{p-inf}\nLet $(X,o)=({\\rm Spec}R, \\mathfrak{m})$ and $D\\ge 0$ a $\\mathbb{Q}$-divisor, such that $o\\in (X,D)$ is a klt singularity. Then we have \n\\begin{equation}\\label{eq-vol2mul}\n\\inf_{v} {\\widehat{\\rm vol}}_{X,o}(v)=\\inf_{\\mathfrak{a}}{\\rm lct}^n(X,D; \\mathfrak{a})\\cdot{\\rm mult}(\\mathfrak{a}),\n\\end{equation}\nwhere on the left hand side the infimum runs over all the valuations centered at $o$, and on the right hand side it runs over all the $\\mathfrak{m}$-primary ideals. Moreover, \nthe left hand side can be replaced by $\\inf_{v\\in {\\rm Div}_{X,o}} {\\widehat{\\rm vol}}_{X,o}(v)$ where ${\\rm Div}_{X,o}$ denotes the space of all divisorial valuations with center at $o$.\n\\end{prop}\nFor the reader's convenience we provide a sketch of the proof. \n\n\n\\begin{proof\nWe first use the same argument as in \\cite[Example 2.7]{Li15a}) to prove that the left hand side is greater or equal to the right hand side. For any real valuation $v$, consider \nthe graded family of valuative ideals \n$$\\mathfrak{a}_k=\\mathfrak{a}_k(v)=\\{f\\in R\\ |\\  v(f)\\ge k\\}.$$ \nThen $v(\\mathfrak{a}_k)\\ge k$ and we can estimate:\n\\begin{eqnarray*}\nA_{X,D}(v)^n \\cdot \\frac{{\\rm mult}(\\mathfrak{a}_k)}{k^n}&\\ge & \\left(\\frac{A_{X,D}(v)}{v(\\mathfrak{a}_k)}\\right)^n \\cdot {\\rm mult}(\\mathfrak{a}_k)\\ge {\\rm lct}^n(X,D; \\mathfrak{a}_k)\\cdot {\\rm mult}(\\mathfrak{a}_k).\n\\end{eqnarray*}\nSince for $\\mathfrak{a}_\\bullet=\\{\\mathfrak{a}_k\\}$ is a graded family of $\\mathfrak{m}$-primary ideals on $X$, \n\\[\n{\\rm mult}(\\mathfrak{a}_\\bullet)=\\lim_{k\\rightarrow+\\infty} \\frac{l_R(R\/\\mathfrak{a}_k)}{k^n}=\\lim_{k\\rightarrow+\\infty}\\frac{{\\rm mult}(\\mathfrak{a}_k)}{k^n}.\n\\]\n(see e.g. \\cite{ELS03, Mus02, LM09, Cut12}), as $k\\rightarrow +\\infty$, the left hand side converges to ${\\widehat{\\rm vol}}(v)$ and we get one direction.\n\nFor the other direction of the inequality, we follow the argument in \\cite{Liu16}. For any $\\mathfrak{m}$-primary ideal $\\mathfrak{a}$, we can choose a divisorial valuation $v$ calculating \n${\\rm lct}(\\mathfrak{a})$. Then $v$ is centered at $o$. Assume $v(\\mathfrak{a})=k$, or equivalently $\\mathfrak{a}\\subseteq \\mathfrak{a}_k(v)$. Then we have $\\mathfrak{a}^l \\subseteq \\mathfrak{a}_k(v)^l\\subseteq \\mathfrak{a}_{kl}(v)$ for any $l\\in \\mathbb{Z}_{>0}$. So we can estimate:\n\\begin{eqnarray*}\n{\\rm lct}^n(X,D; \\mathfrak{a})\\cdot {\\rm mult}(\\mathfrak{a})&=&\\frac{A_{X,D}(v)^n}{k^n}\\cdot {\\rm mult}(\\mathfrak{a})=A_{X,D}(v)^n\\cdot\\frac{{\\rm mult}(\\mathfrak{a}) l^n}{(kl)^n}\\\\\n&=&A_{X,D}(v)^n \\cdot \\frac{{\\rm mult}(\\mathfrak{a}^l)}{(kl)^n}\\ge A_{X,D}(v)^n\\cdot \\frac{{\\rm mult}(\\mathfrak{a}_{kl})}{(kl)^n}.\n\\end{eqnarray*}\nAs $l\\rightarrow +\\infty$, then again the right hand side converges to \n$$A_{X,D}(v)^n \\cdot {\\rm mult}(\\mathfrak{a}_\\bullet(v))={\\widehat{\\rm vol}}(v).$$ So we have proved the right-hand-side is bigger than the left-hand-side in \\eqref{eq-vol2mul}.\n\nThe last statement follows easily from the above proof.\n \\end{proof}\n \nIn a very recent preprint \\cite{Blu16b}, it is proved that a minimizer always exists.\n\n\\begin{thm}[Blum] For any klt singularity $o\\in (X,D)$, ${\\widehat{\\rm vol}}_{X,D}(v)$ always has a minimizer $v^{\\rm m}$ in ${\\rm Val}_{X,o}$.\n\\end{thm}\n\n\n\n\n\n\\subsection{Properties of Koll\\'ar component}\nThe concept of {\\it Koll\\'ar component} is defined in Definition \\ref{d-kollar}. It always exists (cf. see \\cite{Pro00} or \\cite[Lemma 1]{Xu14}). In this section, we  establish some of their properties using the machinery of the minimal model program.\n\nThe following statement is the local analogue of \\cite[Theorem 1.6]{LX14}, which can be also easily obtained by following the proof of the existence of Koll\\'ar component. (See e.g. the proof of \\cite{Xu14}.) \n\\begin{prop}\\label{p-special}Let $o\\in (X,D)$ be a klt singularlty. Let $\\mu\\colon Y\\to X$ be a model, such that $\\mu$ is an isomorphism over $X\\setminus\\{o\\}$ and $(Y,E+\\mu_*^{-1}D)$ is dlt where $E$ is the divisorial part of $\\mu^{-1}(o)$.\nThen we can choose a model $W\\to Y$ and  run MMP to obtain $W\\dasharrow Y'$, such that $Y'\\to X$ gives a Koll\\'ar component $S$ with $a(Y,E; S)=-1$. \n\\end{prop}\n\n\n\n\n\nWe also have the following straightforward lemma. \n\\begin{lem}\\label{l-inter}\nIf $S$ is a Koll\\'{a}r component, then ${\\rm vol}({\\rm ord}_S)=(-S|_S)^{n-1}$ and ${\\widehat{\\rm vol}}({\\rm ord}_S)=(-(K_Y+S+\\mu_*^{-1}D)|_S)^{n-1}\\cdot A_{X,D}(S)^n$.\n\\end{lem}\n\\begin{proof}\nFor any $k\\ge 0$, we have an exact sequence,\n$$0\\to \\mathcal{O}_{Y}(-(k+1)S)\\to \\mathcal{O}_{Y}(-kS)\\to \\mathcal{O}_{S}(-kS)\\to 0.$$\nBecause $-S$ is ample over $X$, we have the vanishing \n$$R^1f_*(\\mathcal{O}_{Y}(-(k+1)S))=0,$$\nfrom which we get \n\\[\nH^0(S, -kS|_S)\\cong \\frac{H^0(Y, -kS)}{H^0(Y, -(k+1)S)}=\\frac{\\mathfrak{a}_k({\\rm ord}_S)}{\\mathfrak{a}_{k+1}({\\rm ord}_S)}.\n\\]\nSo we get the identity:\n\\begin{eqnarray*}\n\\dim_{\\mathbb{C}}\\left(\\mathcal{O}_{X,o}\/\\mathfrak{a}_{m}({\\rm ord}_S)\\right)&=&\\sum_{k=0}^{m-1} \\dim H^0(S, -kS|_S).\n\\end{eqnarray*}\nThen the result follows easily from the Hirzebruch-Riemann-Roch formula.\n\nAs $K_Y+S+\\mu_*D\\sim_{\\mathbb{Q},X} A_{X,D}(S)\\cdot S$, the second identity is implied by the first statement. \n\\end{proof}\n\n\\begin{rem}Inspired by the above simple calculation, we indeed can extend the definition of normalized volumes to any model $f:Y\\to (X,o)$, such that $f$ is isomorphic over $X\\setminus \\{o\\}$. See Section \\ref{s-vmodel}. \n\\end{rem}\n\n\\begin{lem}\\label{l-finite}\nLet $f\\colon (X',o')\\to (X,o)$ be a finite morphism, such that $f^*(K_X+D)=K_{X'}+D'$ for some effective $\\mathbb{Q}$-divisors. We assume  $(X,D)$ and $(X',D')$ are klt. If $S$ is a Koll\\'ar component given by $Y\\to X$ over $o$, then $Y\\times_XX'\\to X'$ induces a Koll\\'ar component $S'$ over $o'\\in (X',D')$. \n\nConversely, if $X'\\to X$ is Galois with Galois group $G$, then any $G$-invariant Koll\\'ar component $S'$ over $o\\in (X',D')$ is the pull back from a Koll\\'ar component over $o\\in (X,D)$.\n\\end{lem}\n\\begin{proof} The first part is standard. In fact, denote by $S'=f^{-1}(S)$, then $(Y',\\mu'^{-1}_*D'+S')$ is log canonical, such that if we restrict to $T$ a component of $S'$, \n$$(K_{Y'}+\\mu'^{-1}_*D'+S')|_{T}=K_{T}+\\Delta_{T},$$\nthen $(T,\\Delta_{T})$ is klt, which by Koll\\'ar-Shokurov connectedness theorem implies that $T=S'$.\n\n\nFor the converse, let  $$L\\sim_{X'}-m(K_{Y'}+\\mu'^{-1}_*D'+E')$$  be a divisor of general position for sufficiently divisible $m$ and  $H:=\\frac{1}{m}L$, then $(Y', E'+\\mu'^{-1}_*D'+H)$ is plt. Replacing $H$ by $H_G:=\\frac{1}{|G|}(\\sum_{g\\in G} g^*H)$, we know that $(X', D'+ \\mu_*H_{G})$ is $G$-invariant, and there exists a $\\mathbb{Q}$-divisor $H_X\\ge 0$, such that\n$$f^*(K_X+D +H_X)=K_{X'}+D'+\\mu_*H_G.$$\nTherefore, $(X,D+H_X)$ is plt, and its unique log canonical place is a divisor $S$ which is a Koll\\'ar component over $o\\in (X,D)$ whose pull back gives the Koll\\'ar component $S'$ over $o'\\in (X',D')$.  \n\\end{proof}\n\nWe prove a change of volume formula for Koll\\'ar components under a finite map.\n\\begin{lem}\\label{l-finitevolume}With the same notation as in Lemma \\ref{l-finite}, then \n$$d\\cdot {\\widehat{\\rm vol}}_{X,D}({\\rm ord}_S)={\\widehat{\\rm vol}}_{X',D'}({\\rm ord}_{S'}),$$\nwhere $d$ is the degree of $X'\\to X$. \n\\end{lem}\n\\begin{proof}Since the pull back of $S$ is $S'$ which is irreducible by Lemma \\ref{l-finite}, let the degree of $S'\\to S$ be $a$ and the ramified degree be $r$, we have \n$$ar=d \\qquad \\mbox{and}\\qquad rA_{X,D}({\\rm ord}_S)=A_{X',D'}({\\rm ord}_{S'})$$ \n(see \\cite[5.20]{KM98}).\nBy Lemma \\ref{l-inter}, we know that \n\\begin{eqnarray*}\nd\\cdot {\\widehat{\\rm vol}}_{X,D}({\\rm ord}_S)& = & ar\\cdot A_{X,D}({\\rm ord}_S)\\cdot ((K_Y+S+\\mu_*^{-1}D)|_{S})^{n-1}\\\\\n&=&(rA_{X,D}({\\rm ord}_S))\\cdot \\big( (a(K_Y+S+\\mu_*^{-1}D)|_{S})^{n-1}\\big)\\\\\n&= &A_{X',D'}({\\rm ord}_{S'})\\cdot \\big( ((K_{Y'}+S'+\\mu_*'^{-1}D')|_{S'})^{n-1}\\big),\\\\\n&=&{\\widehat{\\rm vol}}_{X',D'}({\\rm ord}_{S'}),\n\\end{eqnarray*}\nwhere for the third equality we use the projection formula of intersection numbers. \n\\end{proof}\n\n\n\n\n\\subsection{Deformation to cone}\\label{ss-deformation}\nLet $(X, o)=({\\rm Spec}(R), \\mathfrak{m})$ be an algebraic singularity such that $(X,D)$ is klt for a $\\mathbb{Q}$-divisor $D\\ge 0$. \nLet $S$ be a Koll\\'{a}r component and $\\Delta=\\Delta_S$ be the different divisor defined by adjunction $(K_Y+S+\\mu_*^{-1}D)|_S=K_S+\\Delta_S$ where $Y\\rightarrow X$ is the extraction of $S$.\nDenote $v_0:={\\rm ord}_S$ and \n$$T=\\bigoplus_{k=0}^{+\\infty} \\mathfrak{a}_k(v_0)\/\\mathfrak{a}_{k+1}(v_0)=\\bigoplus_{k=0}^{+\\infty} T_k$$ and the $d$-th truncation\n$$T^{(d)}=\\bigoplus_{k=0}^{+\\infty} \\mathfrak{a}_{dk}(v_0)\/\\mathfrak{a}_{dk+1}(v_0)=\\bigoplus_{k=0}^{+\\infty} T_{dk}\\qquad \\mbox{for $d\\in \\mathbb{N}$}.$$\n\n\nNow we give a more geometric description of $\\mathrm{Spec}(T)$ and $\\mathrm{Spec}(T^{(d)})$ using the idea of degenerating  $o\\in (X,D)$ to an (orbifold) cone over the Koll\\'ar component $S$. Assume $\\mu\\colon Y\\rightarrow X$ is the extraction of the Koll\\'{a}r component $S$ of $(X, o)$. Then \n$\\mu_{\\mathbb{A}^1}\\colon Y\\times{\\mathbb{A}}^1\\rightarrow X\\times {\\mathbb{A}}^1$ has the exceptional divisor $S\\times {\\mathbb{A}}^1$. The divisor $S$ is not necessarily Cartier, but only $\\mathbb{Q}$-Cartier. Thus we can take the index 1 covering Deligne-Mumford stack $\\pi: \\mathcal{Y}\\to Y$ for $S$. So $\\pi$ is isomorphic over $Y\\setminus S$ and $\\pi^*(S)=\\mathcal{S}$ is Cartier on ${\\mathcal{Y}}$.\n\nWe consider the deformation to the normal cone construction for ${\\mathcal{S}}\\subset {\\mathcal{Y}}$. More precisely, we consider the blow up $\\pi_1: {\\mathcal{Z}} \\rightarrow {\\mathcal{Y}}\\times {\\mathbb{A}}^1$ along ${\\mathcal{S}}\\times\\{0\\}$. Denote by $E$ the exceptional divisor of $\\pi_1$ and by $\\mathcal{T}$ the strict\ntransform of ${\\mathcal{S}}\\times{\\mathbb{A}}^1$. We note that $E$ has a stacky structure along the 0 and $\\infty$ section, but a scheme structure at other places. \nThen $\\mathcal{T}\\subset {\\mathcal{Z}}$ is a Cartier divisor which is proper over ${\\mathbb{A}}^1$ and can be contracted to a normal Deligne-Mumford stack $\\Psi_1\\colon {\\mathcal{Z}}\\to {\\mathcal{W}}$ and in this way we get a flat family ${\\mathcal{W}}\\rightarrow {\\mathbb{A}}^1$ such that ${\\mathcal{W}}_t\\cong X$ and ${\\mathcal{W}}_0\\cong \\bar{{\\mathcal{C}}}\\cup \\mathcal{Y}_0$, where $\\mathcal{Y}_0$ is the birational transform of $Y\\times\\{0\\}$. To understand ${\\mathcal{C}}$ if we  denote by ${\\mathcal{Z}}^0:={\\mathcal{Z}}\\setminus {\\mathcal{Y}}_0$, then the fiber ${\\mathcal{Z}}^0$ over $0$ is isomorphic to ${\\mathcal{C}}$ which is an affine orbifold cone over ${\\mathcal{S}}$ with the polarization given by $\\mathcal{O}_{{\\mathcal{Y}}}(-{\\mathcal{S}})|_{{\\mathcal{S}}}$.\nMoreover, $\\bar{{\\mathcal{C}}}$ is the projective orbifold cone completing ${\\mathcal{C}}$.\n\n\nLet $d$ be a positive integer such that $d\\cdot S$ is Cartier in $Y$, then ${\\mathcal{C}}^{(d)}$ given by the cone over $\\mathcal{O}_{{\\mathcal{Y}}}(-d\\cdot{\\mathcal{S}})|_{{\\mathcal{S}}}$ is a degree $d$ cyclic quotient of ${\\mathcal{C}}$, which is a usual ($\\mathbb{A}^1$-)cone over ${\\mathcal{S}}$.\nWe denote by $C$ and $C^{(d)}$  the underlying coarse moduli space of ${\\mathcal{C}}$ and ${\\mathcal{C}}^{(d)}$. We also denote $S$ to be the coarse moduli space of ${\\mathcal{S}}$.\n\nApplying the exact sequence, \n$$0\\to \\mathcal{O}_Y(-(k+1)S)\\to \\mathcal{O}_Y(-kS)\\to \\mathcal{O}_{S}(-kS)\\to 0,$$\nsince $h^1(\\mathcal{O}_Y(-(k+1) S))=0$ by the Grauert-Riemenschneider vanishing theorem, we get:\n\\[\nH^0(S, \\mathcal{O}(-k S|_S))\\cong H^0(\\mathcal{O}_Y(-kS))\/H^0(\\mathcal{O}_Y(-(k+1)S)).\n\\]\nNotice that the right hand side is equal to:\n\\[\n\\frac{\\mu_*\\mathcal{O}_Y(-kS)}{\\mu_*\\mathcal{O}_Y(-(k+1)S)}=\\frac{\\mathfrak{a}_{k}(v_0)}{\\mathfrak{a}_{k+1}(v_0)}.\n\\]\nIn particular, $C={\\rm Spec} (T)$. Similarly, $C^{(d)}={\\rm Spec}(T^{(d)})$.\n\nThere is also a degree $d$ cyclic quotient morphism $h\\colon C\\to C^{(d)}$, and we know that\n$$h^*(K_{\\bar{C}^{(d)}}+C^{(d)}_\\Delta+C^{(d)}_{D})=K_{\\bar{C}}+C_{D},$$ \nwhere $C_D$ is the intersection of $\\bar{C}$ with the birational transform of $D\\times \\mathbb{A}^1$ and $C^{(d)}_{\\Delta}$ (resp. $C^{(d)}_{D}$) on $\\bar{C}^{(d)}$ is the induced cone over the branched $\\mathbb{Q}$-divisor  on $S$ of ${\\mathcal{S}}\\to S$ (resp. $\\mu^{-1}_*D|_S$). In particular, $C^{(d)}_{\\Delta}+C^{(d)}_{D}$ is the cone over $\\Delta_S$.\n\n\\subsection{Filtrations and valuations}\\label{sec-filtration}\n\n\nAssume ${\\mathcal{F}}$ is a $\\mathbb{Z}$-graded filtration on $R$. We have the Rees algebra and extended Rees algebra:\n\\begin{equation}\n{\\mathcal{R}}:={\\mathcal{R}}({\\mathcal{F}})=\\bigoplus_{k=0}^{+\\infty} ({\\mathcal{F}}^k R) t^{-k}, \\quad {\\mathcal{R}}':={\\mathcal{R}}'({\\mathcal{F}})=\\bigoplus_{k=-\\infty}^{+\\infty} ({\\mathcal{F}}^k R) t^{-k},\n\\end{equation}\nand the associated graded ring:\n\\begin{equation}\n{\\rm gr}_{\\mathcal{F}}(R)={\\mathcal{R}}'\/t {\\mathcal{R}}'=\\bigoplus_{k=0}^{+\\infty} ({\\mathcal{F}}^k R\/{\\mathcal{F}}^{k+1} R) t^{-k}=: \\bigoplus_{k=0}^{+\\infty} A_k.\n\\end{equation}\nAssuming ${\\mathcal{R}}'$ is finitely generated, ${\\mathcal{X}}:={\\rm Spec}_{\\mathbb{C}[t]} ({\\mathcal{R}}')$ can be seen as a $\\mathbb{C}^*$-equivariant flat degeneration of $X$ into \n${\\mathcal{X}}_0={\\rm Spec}_{\\mathbb{C}} ({\\mathcal{R}}'\/t{\\mathcal{R}}')={\\rm Spec}_{\\mathbb{C}}({\\rm gr}_{\\mathcal{F}} R)$. Denote $E={\\rm Proj} ({\\rm gr}_{\\mathcal{F}}(R))$, $\\tilde{X}={\\rm Proj}_{R} {\\mathcal{R}}$. Then the natural map $\\tilde{X}\\rightarrow X$ is the filtered blow up associated with the ${\\mathcal{F}}$ such that $E$ is the exceptional divisor. Moreover $\\tilde{X}$ can be seen as a flat deformation of a natural filtered blow up on ${\\mathcal{X}}_0$. Indeed following \\cite[5.15]{TW89}, we have a filtration ${\\mathcal{F}}$ on ${\\mathcal{R}}'$:\n\\[\n{\\mathcal{F}}^m {\\mathcal{R}}'=\\left\\{\\sum_{k=-\\infty}^{+\\infty} \\left({\\mathcal{F}}^{\\max(k,m)} R\\right) t^{-k}  \\right\\}.\n\\]\n\nThe objects associated to the corresponding Rees algebra and graded algebra over ${\\mathcal{R}}'$ are:\n\\[\n\\tilde{{\\mathcal{X}}}={\\rm Proj}_{{\\mathcal{R}}'} \\bigoplus_{r=0}^{+\\infty}({\\mathcal{F}}^r {\\mathcal{R}}' ) T^{-r}, \\quad \\mathcal{E}={\\rm Proj}_{\\mathbb{C}} \\bigoplus_{r=0}^{+\\infty} ({\\mathcal{F}}^r {\\mathcal{R}}' \/{\\mathcal{F}}^{r+1} {\\mathcal{R}}') T^{-r}.\n\\]\nThen we have the following commutative diagram (see \\cite[Proposition 5.17]{TW89}):\n\\[\n\\begin{CD}\n&& E @>>> {\\mathcal{E}} @<<< E \\\\\n&& @VVV @VVV @VVV \\\\\n\\tilde{\\mathbb{C}}^p@<<<  \\tilde{X} @>>> \\tilde{{\\mathcal{X}}} @<<< \\tilde{{\\mathcal{X}}}_0\\\\\n@VVV  @VVV @VVV @VVV\\\\\n\\mathbb{C}^p @<<< X@>>> {\\mathcal{X}}  @<<< {\\mathcal{X}}_0\n\\end{CD}\n\\]\n\n\nThere is a natural $\\mathbb{C}^*$-action on ${\\mathcal{X}}_0$ associated to the natural $\\mathbb{N}$-grading such that the quotient is isomorphic to $E$. Let $\\mathcal{J}=\\bigoplus_{k\\ge 0} {\\mathcal{F}}^{k+1}t^{-k}=t {\\mathcal{R}}'\\cap {\\mathcal{R}}$ so that ${\\mathcal{R}}\/\\mathcal{J}\\cong {\\rm gr}_{{\\mathcal{F}}}(R)\\cong {\\mathcal{R}}'\/t {\\mathcal{R}}'$.\nNow we assume furthermore that\n$E$ is a normal projective variety. This implies both ${\\mathcal{R}}$ and ${\\mathcal{R}}'$ are normal (see \\cite{TW89}).\nLet $\\mathfrak{P}$ be the unique minimal prime ideal of ${\\mathcal{R}}$ over $\\mathcal{J}$ that corresponds to $E$, and $w$ the valuation of $K(t)$ attached to $\\mathfrak{P}$.  Then the restriction of $w$ to $R$ is equal to $b \\cdot {\\rm ord}_E$. Assume $a=w(t)$. Thus the filtration ${\\mathcal{F}}$ is equivalent to the filtration that is given by:\n\\[\n(t^m {\\mathcal{R}}') \\cap R=\\{f\\in R; {\\rm ord}_E(f)\\ge m  a\/b\\}.\n\\]\n\\begin{rem}\nThere is a general Valuation Theorem about  the relation between finitely generated filtrations and valuations proved by Rees for which we refer the reader to \\cite{Ree88}. See also \\cite{BHJ15}.\n\\end{rem}\n\n\n\n\n\\section{Volume of models}\\label{s-vmodel}\n\nOne very useful tool for us to study the minimzer of the normalized local volume is the concept of  a local volume of a model. It is this concept which enables us to apply the machinery of the minimal model program to construct different models, especially those yielding Koll\\'ar components.  \n\n\\subsection{Local volume of models}\\label{s-lvmodel}\nIn this section, we extend the definition of volume to models in the `normalized' sense. We use the concept of local volumes as in \\cite{ELS03, Ful13}. Let us first recall the definition, which is from \\cite{Ful13}.\n\\begin{defn}[Local volume](cf. See \\cite{Ful13})\nLet $X$ be a normal algebraic variety of dimension at least two and let $o$ be a point on $X$. Fixing a proper birational map $\\mu\\colon Y \\to  X$, for an arbitrary Cartier divisor $D$ on $Y$,  we define the local volume of $D$ at $x$ to be\n$${\\rm vol}^F_o(D) = \\limsup_{m\\to \\infty} \\frac{h^1_o(mD)} {m^n \/n!},$$\nwhere $h^1_o(mD):=\\dim H^1_{\\{o\\}}(X, f_*\\mathcal{O}_Y(mD)).$\n\\end{defn}\n\nWe can define the volume of a $\\mathbb{Q}$-Cartier divisor $D$ to be\n$${\\rm vol}^F_{o}(D):=\\frac{{\\rm vol}^F_o(mD)}{m^n},$$\nfor sufficiently divisible $m$.\n\n\\begin{lem}\\label{l-pushforward}\nLet $\\mu\\colon Y\\to X$ be a birational morphism. If $D\\ge 0$ is an exceptional $\\mathbb{Q}$-divisor, such that ${\\rm Supp}(D)\\subset \\mu^{-1}(o)$.\nThen \n$${\\rm vol}^F_{o}(-D)=\\limsup_{k\\to \\infty} \\frac{l_R(\\mathcal{O}_X\/\\mathfrak{a}_k)}{k^n\/n!},$$\nwhere $k$ is sufficiently divisible and $\\mathfrak{a}_k=\\mu_*(\\mathcal{O}_Y(-kD))$.\n\\end{lem}\n\\begin{proof}This follows from \\cite[Remark 1.1(ii)]{Ful13} (see also \\cite[Remark 1.31 and 1.32]{Ful13}).\n\\end{proof}\nThe right hand side of the above display is also the volume ${\\rm vol}(\\mathfrak{a}_{\\bullet})$ defined in \\cite[Definition 3.1, Proposition 3.11]{ELS03}. In particular, given a prime divisor $D$ over $o$ with log discrepancy $a$, then we see that \n$${\\rm vol}^{F}_o(-aD)={\\widehat{\\rm vol}}_{X,o}({\\rm ord}_D).$$ \n\n\n\\begin{defn}\\label{d-modelv}Let $o\\in (X,D)$ be a klt singularity. \nLet $\\mu\\colon Y\\to (X,o)$ be a birational morphism, such that  $\\mu$ is an isomorphism over $X\\setminus \\{o\\}$. Let $E=\\sum_i G_i$ be the divisorial part of $\\mu^{-1}(o)$. Then we define the volume ${\\rm vol}_{(X,D),o}(Y)$ (abbreviated as ${\\rm vol}_{X,o}(Y)$ or ${\\rm vol}(Y)$ if $(X,D; o)$ is clear) of $Y$ to be \n$${\\rm vol}_{(X,D),o}(Y):={\\rm vol}^F_{o}(-K_Y-E-\\mu_*^{-1}(D))={\\rm vol}^F_{o}(\\sum_i -a_iG_i),$$\nwhere $a_i=A_{X,D}(G_i)$ is the log discrepancy.\n\\end{defn}\n\n\nWe mainly combine the above definition with the following construction.\n\\begin{defn}\\label{d-dlt}\nFor a klt pair $(X,D)$ with an idea $\\mathfrak{a}$, we denote by $c$ its log canonical threshold ${\\rm lct}(X,D;\\mathfrak{a})$.  We say $\\mu\\colon Y\\to X$ is a {\\it dlt modification} of $(X,D+c\\cdot \\mathfrak{a})$, if\n\\begin{enumerate}\n\\item denote the divisorial part of $\\mu^*(\\mathfrak{a})$ by $\\mathcal{O}(-\\sum m_iG_i)$ and denote by $\\mu^*(K_X+D)=K_Y+D_Y$, then \n$$D_Y+c\\cdot \\sum m_i G_i=\\mu^{-1}_*(D)+E$$ where $E$ is the reduced  divisor on ${\\rm Ex}(\\mu)$; \n\\item $(Y,D_Y+c\\cdot \\sum m_i G_i)$ is dlt; and\n\\item for any divisor $F$, \n$$a(F; Y,D_Y+c\\cdot \\sum m_i G_i)=-1\\mbox{ if and only if }a(F; X,D+c\\cdot\\mathfrak{a})=-1.$$ \n\\end{enumerate}\nBy the argument in \\cite{OX12}, we know that it follows the MMP result in \\cite{BCHM10} that the dlt modification of $(X,D+c\\cdot \\mathfrak{a})$ always exists. More precisely, we can choose general elements $f_j\\in\\mathfrak{a}$ $(1\\le j \\le l)$ which generate $\\mathfrak{a}$ such that $l> \\frac{1}{c}$. Let $D_j=(f_j)$, then $Y$ is the dlt modification of $(X,D+c\\cdot\\frac{1}{l}\\sum^l_{j=1}D_j)$. \n\\end{defn}\nWe do not need property 3 in our argument, but it seems to us it is natural to require this by comparing with \\cite[Theorem 1.34]{Kol13}.\n\\begin{lem}\\label{l-gooddlt}\nWe can indeed assume that  $-K_Y-{\\mu}_*^{-1}D-E$ is nef over $X$.\n\\end{lem}\n\\begin{proof}Since $(X,D)$ is klt, we know that \n$$K_Y+\\mu_*^{-1}D+E\\sim_{\\mathbb{Q},X} \\sum a_iG_i$$\nwith $a_i=A_{X,D}(G_i)>0$. Running an MMP with scaling by an ample divisor, we obtain a relative minimal model $Y\\dasharrow Y'$ of \n$$\nK_Y+\\mu^{-1}_*(D+c\\cdot \\frac{1}{l}\\sum_{j=1}^l D_j)+E \\sim_{{\\mathbb{Q}}, X} - c\\cdot \\sum m_i G_i+\\sum_i a_i G_i=0.\n$$\nSo we have\n$$\nK_Y+\\mu^{-1}_*(D+c\\cdot \\frac{1}{l}\\sum_{j=1}^l D_j)+E-\\epsilon \\sum a_i G_i=-\\epsilon (K_Y+\\mu_*^{-1}D+E),\n$$\nand hence $-K_{Y'}-{\\mu'}_*^{-1}D-E'$ is nef over $X$. Furthermore, since  \n$$K_Y+\\mu_*^{-1}D+E\\sim_{\\mathbb{Q},X}-c \\cdot\\frac{1}{l} \\mu_*^{-1} \\sum^l_{j=1}D_j,$$\n$Y'$ also gives a minimal model of the dlt pair $$\\big(Y, \\mu_*^{-1}(D+c(1+\\epsilon) \\cdot\\frac{1}{l}\\sum^l_{j=1}D_j)+E\\big),$$\nwhich implies $(Y', {\\mu'}_*^{-1}D+E')$ is a dlt modification of $(X,D+c\\cdot\\frac{1}{l}\\sum^l_{j=1}D_j)$. Therefore, we can replace $ Y$ by $Y'$.\n\\end{proof}\n\nWhen $E$ is irreducible, then ${\\rm vol}_{X,o}(Y)={\\widehat{\\rm vol}}_{X,o}({\\rm ord}_E)$.  We can also generalize Lemma \\ref{l-inter} to the dlt case.\n\\begin{lem}\\label{l-inter2}In the setting of Definition \\ref{d-modelv}, if we assume that $-K_Y-\\mu_*^{-1}D-E$ is nef over $X$. \nThen \n $${\\rm vol}_{X,o}(Y)=\\sum_i a_i \\big((-K_Y-\\mu_*^{-1}D-E)|_{E_i}\\big)^{n-1}.$$\n\\end{lem}\n\\begin{proof}Let $m$ be sufficiently divisible such that $L:=m(K_Y+\\mu_*^{-1}D+E)$ is Cartier. Denote by $F$ the effective Cartier divisor $F:=\\sum_i ma_iG_i$. \nThen \n$$0\\to \\mathcal{O}_Y(L^{\\otimes -(k+1)})\\to \\mathcal{O}_Y(L^{\\otimes -k})\\to \\mathcal{O}_F(L^{\\otimes -k})\\to  0.$$\nSince $L^{-1}$ is nef, we know that $R^1\\mu_*(L^{\\otimes -(k+1)})=0$.\nThus \n$${\\rm vol}_o^F(L)={\\rm vol}(L|_{F}),$$\nthen we conclude by dividing $m^n$ in both sides. \n\\end{proof}\n\n\\begin{lem}\\label{l-model1}\nLet $\\mathfrak{a}$ be an $\\mathfrak{m}$-primary ideal, we define $c={\\rm lct}(X,D;\\mathfrak{a})$ and $(Y,E)\\to X$ be the dlt modification of $ (X,D+c\\cdot \\mathfrak{a})$. Then\n$${\\rm vol}_{X,o }(Y)\\le {\\rm lct}^n(\\mathfrak{a} )\\cdot{\\rm mult}(\\mathfrak{a} ).$$\n\\end{lem}\n\\begin{proof}Write $K_Y+\\mu_*^{-1}D+E=\\mu^*(K_X+D)+\\sum_i a_iG_i,$ where $E$ is the reduced divisor on ${\\rm Ex}(\\mu)$.  If we denote the vanishing order of $\\mu^* \\mathfrak{a}$ along $G_i$ by $m_i$, then since  $c$ is the log canonical threshold and for every $i$, $G_i$ computes the log canonical threshold,  we know that $c\\cdot m_i=a_i$. Thus\n$$\\mathfrak{a}^k\\subset \\mu_*\\mathcal{O}_Y(-\\sum_i km_iG_i)=_{\\rm def} {\\mathfrak{b}}_k.$$\nIt suffices to show that \n$${\\rm mult}({\\mathfrak{b}}_{\\bullet})={\\rm vol}_o^F (-\\sum m_iG_i),$$\nbut this follows Lemma \\ref{l-pushforward}.\n\\end{proof}\n\\begin{lem}\\label{l-kollar}\nNotations as above. Then there exists a Koll\\'ar component $S$, such that \n$${\\widehat{\\rm vol}}({\\rm ord}_S)\\le {\\rm vol}(Y).$$ \n\\end{lem}\n\\begin{proof}It follows from Proposition \\ref{p-special} that we can choose a model $W\\to Y$ and  run MMP to obtain $W\\dasharrow Y'$, such that $Y'\\to X$ gives a Koll\\'ar component $S$ with $a(Y,E; S)=-1$. \nIf we fix a common resolution $p\\colon W'\\to Y $ and $q\\colon W'\\to Y'$, then since $-(K_Y+E)$ is nef and $A_{Y, E}(S)=0$, \nwe know $-p^*(K_Y+E)+q^*(K_{Y'}+S)$ is $q$-nef and $q$-exceptional. By the negativity lemma, we get\n$$p^*(K_Y+E)\\ge q^*(K_{Y'}+S).$$ \nThus \n$${\\widehat{\\rm vol}}({\\rm ord}_S)={\\rm vol}(-K_{Y'}-S)\\le {\\rm vol}(-K_Y-E)={\\rm vol}(Y).$$\n\\end{proof}\n\n\n\n\n\n\\subsection{Approximating by Koll\\'ar components}\nWith the above discussions, we can start to prove our theorems.\n\\begin{proof}[Proof of Theorem \\ref{t-approx}]\nBy Proposition \\ref{p-inf}, we know \n$$\\inf_{v} {\\widehat{\\rm vol}}_{X,o}(v)=\\inf_{\\mathfrak{a}}{\\rm lct}^n(\\mathfrak{a})\\cdot{\\rm mult}(\\mathfrak{a}).$$\nBy the above construction in Lemma \\ref{l-model1} and \\ref{l-kollar}, for any $\\mathfrak{m}$-primary ideal $\\mathfrak{a}$, we know that there exists a Koll\\'ar component $S$, \nsuch that \n$${\\widehat{\\rm vol}}({\\rm ord}_S)\\le {\\rm lct}^n(\\mathfrak{a})\\cdot{\\rm mult}(\\mathfrak{a}). $$  This finishes the proof of the first part of Theorem \\ref{t-approx}. \n\n\\bigskip\n\nWe continue to prove the second part of the theorem. Let $\\{\\mathfrak{a}_k\\}_{k\\in \\Phi}$ be the associated graded family of valuation ideals induced by $v$ where $\\Phi\\in \\mathbb{R}$ is the value semigroup. For each $\\mathfrak{a}_k$ ($k\\in \\Phi$), we denote by \n$$c_k:={\\rm lct}(X,D; \\mathfrak{a}_k).$$\nLet $\\mu_k\\colon Y_k\\to X$ be a dlt modification of $(X,D; c_k\\cdot \\mathfrak{a}_k)$ and $E_k$ the exceptional divisor of $Y_k$ over $X$. Assume the model we obtain from Lemma \\ref{l-kollar} is $Y'_k$ with the Koll\\'ar component $S_k$.\n \nWe consider the valuation \n$$v_k=\\frac{c_k\\cdot k}{A_{X,D}(S)}{\\rm ord}_{S_k}.$$ \nBy \\cite{JM12}, we know \n$$c_k\\cdot k={\\rm lct} (X,D; \\frac{1}{k}\\mathfrak{a}_k)=\\inf_{v'}\\frac{A_{X,D}(v')}{\\frac{1}{k}v'(\\mathfrak{a}^k)}\\le A_{X,D}(v) <\\infty.$$\nThen by the Izumi type estimate in \\cite[Proposition 1.2]{Li15a}, we know that \n$$v_k(\\mathfrak{m}){\\rm ord}_{o}\\le v_k \\le c \\cdot {\\rm ord}_{o},$$\nfor some positive constant $c$ and all $k$. And by \\cite[Theorem 2.3]{Li15a} and the fact that ${\\widehat{\\rm vol}}(v_k)$ is bounded from above, we know that $v_k(\\mathfrak{m})$ is bounded from below. In particular, by the compactness result \\cite[Proposition 5.9]{JM12} and Proposition \\ref{p-sequcom}, we know that there is an infinite sequence $\\{v_{k_j}\\}_{k_j\\in \\Phi}$ with $k_j\\to +\\infty$ which has a limit in ${\\rm Val}_{X,o}$.\n\nDenote by \n$$v'=\\lim_{i\\to \\infty} v_{k_i},$$ then we know that \n$$A_{X,D}(v')\\le \\liminf_{i\\to \\infty} A_{X,D}(v_{k_i})=\\lim_{i\\to \\infty} c_{k_i}\\cdot {k_i} \\le A_{X,D}(v)$$ as $A_{X,D}$ is lower semicontinuous (see \\cite[Lemma 5.7]{JM12}). We claim for any $f$, we have\n$$v'(f)\\ge v(f),$$\nand this clearly implies that ${\\rm vol}(v')\\le {\\rm vol}(v)$, which then implies ${\\widehat{\\rm vol}}(v')\\le {\\widehat{\\rm vol}}(v)$.  \n\nDenote by $v(f)= p$. For a fixed $k_j$, choose $l$ such that \n$$(l-1)p< k_j\\le lp .$$ Let $k=k_j$ in the previous construction, then \n\\begin{eqnarray*}\n v(f)= p&\\Longrightarrow & v(f^l)= pl,\\\\\n&\\Longrightarrow & f^l\\in \\mathfrak{a}_{pl},\\\\\n&\\Longrightarrow & f^l\\in \\mathfrak{a}_{k_j},\\\\\n&\\Longrightarrow &l\\cdot  {\\rm ord}_{E_i}(f)\\ge m_{k_j,i} \\mbox{\\ \\  for any $i$},\\\\\n&\\Longrightarrow& l\\cdot  {\\rm ord}_{S_{k_j}}(f)\\ge A_{X,D}(S_{k_j})\\cdot \\frac{1}{c_{k_j}},\\\\\n&\\Longrightarrow&v_{k_j}(f)\\ge \\frac{k_j}{l}>p-\\frac{p}{l}.\n\\end{eqnarray*}\nThe fourth arrow is because if $f^l\\in \\mathfrak{a}_{k_j}$, then $f^l$ vanishes along $m_{k_j,i}E_i$;  and\nthe  fifth arrow is because  that\n$$K_{Y_{k_j}}+Y_{k_j}+\\mu_{k_j*}^{-1}D+E_{k_j}\\sim_{\\mathbb{Q},X} c_{k_j}\\cdot \\sum m_{k_j,i}E_i,$$\nand the pull back of $K_{Y_{k_j}}+Y_{k_j}+\\mu_{k_j*}^{-1}D+E_{k_j}$ is larger than the one from \n$$K_{Y'_{k_j}}+\\mu_{k_j*}^{'-1}D+S_{k_j}\\sim_{\\mathbb{Q},X} A_{X,D}(S_{k_j})S_{k_j}.$$\nThus $v'(f)=\\lim v_{k_j}(f)\\ge p= v(f)$.\n\\end{proof}\n\n\\begin{prop}\\label{p-sequcom} Let $o\\in (X,D)$ be a klt singularity. Let $a$ and $b$ be two positive numbers. Then the subset $K_{a,b}$ of ${\\rm Val}_{x,X}$ which consists of all valuations with \n$$a\\le v(\\mathfrak{m}) \\qquad \\mbox{and} \\qquad A_{X,D}(v)\\le b$$\n is sequential compact. \n\\end{prop}\n\\begin{proof} Let $\\{v_i\\}$ be a sequence contained in $K$. Let $\\{\\mathfrak{a}_{i,k}\\}$ be its associated graded valuative ideals for $k\\in \\Phi_i$. We can find a countably generated field $F\\subset \\mathbb{C}$, such that $R={\\rm Spec}(R_F)\\times_F \\mathbb{C}$ for some finitely generated $F$-algebra $R_F$ and $D$, $x$ are defined over $F$. Furthermore, we can assume for each pair $(i,k)$, ${\\mathfrak{a}_{i,k}}=({\\mathfrak{a}_{i,k}})_F\\times_F {\\mathbb{C}}$, for some ideal $({\\mathfrak{a}_{i,k}})_F\\subset R_F$. Denote by $X_F:={\\rm Spec}(R_F)$ and $D_F$ the divisor of $D$ descending on $X_F$.\n\nNow let $(v_i)_F$ be the restriction of $v_i$ on $R_F$. By our definition, we know that\n$$\\mathfrak{a}_{i,k}=\\{f\\in R_F \\ | (v_i)_F(f)\\ge k\\},$$\nand  $(v_i)_F\\in (K_{a,b})_F$ where $(K_{a,b})_F$ is defined for all $v\\in {\\rm Val}_{x,X_F}$ with  $a\\le v(\\mathfrak{m}_F)$ and $A_{X_F,D_F}(v)\\le b$. By \\cite[Theorem 1.1]{HLP12}, ${\\rm Val}_{x,X_F}$ has the same topology as a set of some Euclidean space, thus $(K_{a,b})_F$ is sequential compact as it is compact by \\cite[Proposition 5.9]{JM12}. Therefore after passing through a subsequence, $(v_i)_F$ has a limit $(v_{\\infty})_F$, which can be extended to a valuation $v_{\\infty}:=(v_{\\infty})_F\\otimes \\mathbb{C}$. In fact, $v_{\\infty}$ is defined as follows: for any $f\\in R$ it can be written $f=\\sum^m_{j=1}f_j\\otimes_F h_j$ such that $0\\neq f_j\\in R$ and $h_1,...,h_m\\in \\mathbb{C}$ are linearly independent over $F$, then \n$$v_{\\infty}(f)=\\min^m_{j=1} (v_{\\infty})_F(f_j).$$ We claim $v_i=(v_i|_{R_F})\\otimes_F\\mathbb{C}$. In fact, for any $f$, if $v_i(f)=k$, then $f\\in \\mathfrak{a}_{i,k}=(\\mathfrak{a}_{i,k})_F\\otimes_F \\mathbb{C}$, thus $(v_i|_{R_F})\\otimes_F\\mathbb{C}(f)=k$. \n\nTo see that for any $f$, $v_{\\infty}(f)=\\lim v_i(f)$, we know for some $j$, \n$$v_{\\infty}(f)=(v_{\\infty})_F(f_j)=\\lim_{i}(v_i|_{R_F})(f_j)\\ge \\limsup_i v_i(f). $$\nFor another direction,  if we have a subsequence of $i$, such that $\\lim_i v_i(f)<v_{\\infty}(f)$, after passing to a subsequence again, we can find a $j$,  such that  $$\\lim_iv_i(f)=\\lim_iv_i(f_j)=\\lim_i (v_{\\infty})_F(f_j)\\ge v_{\\infty}(f), $$\na contradiction. \n\\end{proof}\n\nAs Example \\ref{ex-irreg} shows that for a general klt singularity $(X,o)$, the minimum is not always achieved by a Koll\\'ar component. Thus we have to take a limit process.  However, if the minimizer $v$ has rational rank 1, then it should always yield a Koll\\'ar component. First we have the following result which is inspired by the work in \\cite{Blu16a}. \nWe note that it is also independently obtained in \\cite{Blu16b}.\n\n\n\n\\begin{lem}\\label{l-lcplace}\nIf $E\\in {\\rm Val}_{X,o}$ such that ${\\rm ord}_E$ minimizes ${\\widehat{\\rm vol}}(X,D)$. Then the Rees algebra associated to ${\\rm ord}_E$ is finitely generated. \n\\end{lem}\n\\begin{proof}If we let $\\{\\mathfrak{a}_{\\bullet}\\}$ be the graded valuative ideas  associated to ${\\rm ord}_E$, then we know that \n\\begin{eqnarray*}\n{\\widehat{\\rm vol}}({\\rm ord}_{E})&= &\\lim_{k\\to \\infty}A_{X,D}({\\rm ord}_E)^n \\cdot \\frac{{\\rm mult}(\\mathfrak{a}_k)}{k^n}\\\\\n &\\ge& \\lim_{k\\to \\infty}{\\rm lct} (X,D;\\mathfrak{a}_k)^n \\cdot {\\rm mult}(\\mathfrak{a}_k)\\\\\n & \\ge&{\\widehat{\\rm vol}}({\\rm ord}_{E})\n\\end{eqnarray*}\nby Proposition \\ref{p-inf} and our assumption that ${\\rm ord}_E$ is a minimizer of ${\\widehat{\\rm vol}}(X,D)$. So we conclude that (see \\cite{Mus02})\n$${\\rm lct}(X,D;\\mathfrak{a}_{\\bullet}):=\\lim_{k\\to \\infty} k\\cdot {\\rm lct} (X,D;\\mathfrak{a}_k)=A_{X,D}({\\rm ord}_E),$$ \nwhich we denote by $c$. Therefore,  we can choose $\\epsilon$ sufficiently small,  such that the discrepancy $a(E;X,D+(1-\\epsilon)c\\cdot \\mathfrak{a}_{\\bullet})\\in (-1,0),$\n\n\nOn the other hand, we know \n$${\\rm lct}(X,D;\\mathfrak{a}_{\\bullet})=\\lim_{m\\to \\infty} m\\cdot {\\rm lct}(X,D;\\mathfrak{a}_{m}).$$ So for sufficiently large $m$, we know that for all $G$, the discrepancy\n$$a(G;X, D+\\frac{1}{m}(1-\\epsilon)c\\cdot\\mathfrak{a}_m)>-1.$$ \nWe also have $a(E;X, D+\\frac{1}{m}(1-\\epsilon)c\\cdot\\mathfrak{a}_m)<0$.\nThen similar to the discussion in \\ref{d-dlt}, we can find general $\\mathbb{Q}$-divisor $\\Delta$, such that\n$(X, D+\\frac{1}{m}(1-\\epsilon)c \\cdot \\Delta)$ is klt and $a(E;X, D+\\frac{1}{m}(1-\\epsilon)c\\cdot \\Delta)<0$. In particular, we can apply \\cite{BCHM10} to obtain a model $\\mu\\colon Y\\to X$ such that ${\\rm Ex}(\\mu)=E$ and $-E$ is $\\mu$-ample, which implies the finite generation. \n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{t-divisor}] Applying Lemma \\ref{l-lcplace},  the assumption in Case 1 which says $v$ is a divisorial valuation implies the assumption in Case 2, thus we only need to treat the Case 2.\n\n\\bigskip\n\nBy the proof of Proposition \\ref{p-inf}, we know that if we let $\\mathfrak{a}_k$ be the ideal of elements with values at least $k$,\nthen \n\\begin{eqnarray*}\nA_{X,D}(v)^n \\cdot \\frac{{\\rm mult}(\\mathfrak{a}_k)}{k^n}&\\ge & \\left(\\frac{A_{X,D}(v)}{v(\\mathfrak{a}_k)}\\right)^n \\cdot {\\rm mult}(\\mathfrak{a}_k)\\ge {\\rm lct}^n(X,D; \\mathfrak{a}_k)\\cdot {\\rm mult}(\\mathfrak{a}_k).\n\\end{eqnarray*}\nBy the finite generation assumption, we know that $\\mathfrak{a}_{kl}=\\mathfrak{a}^l_{k}$ for sufficiently divisible $k$ and any $l$.\nSo replace $k$ by $kl$ in the above display and let $l\\to +\\infty$, we know that\n$${\\widehat{\\rm vol}}_{X,o}(v)\\ge {\\rm lct}^n(X,D; \\mathfrak{a}_k)\\cdot {\\rm mult}(\\mathfrak{a}_k)\\ge  {\\widehat{\\rm vol}}_{X,o}(v).$$\nTake $\\mu\\colon Y\\to X$ to be the dlt modification of $(X,D+{\\rm lct}(X,D, \\mathfrak{a}_k)\\cdot \\mathfrak{a}_k)$ as given in Lemma \\ref{l-gooddlt}. The above discussion then implies that\n$$ {\\rm lct}^n(X,D; \\mathfrak{a}_k)\\cdot {\\rm mult}(\\mathfrak{a}_k)\\ge  {\\widehat{\\rm vol}}_{X,o}(v)={\\rm vol}_{X,o}(Y).$$\n\nIt follows from Proposition \\ref{p-special}, that we can choose a model $W\\to Y$ and  running MMP to obtain $W\\dasharrow Y'$, such that $\\mu'\\colon Y'\\to X$ gives a Koll\\'ar component $S$ with $a(S; Y,\\mu^{-1}D_*+E)=-1$. We only need to show that if $Y'$ and $Y$ are not isomorphic in codimension 1, then \n$${\\rm vol}_{X,o}(Y')<{\\rm vol}_{X,o}(Y).$$\n\nThis is the the local analog of the argument in \\cite[Proposition 5]{LX14}. We give the details for the reader's convenience. \n\nLet $\\pi\\colon Y\\to Y^{\\rm c}$ be the canonical model of $-K_Y-{\\mu}_*^{-1}D-E$ over $X$, which exists because \n$$-\\epsilon (K_Y+{\\mu}_*^{-1}D+E)\\sim_{\\mathbb{Q},X}K_Y+{\\mu}_*^{-1}(D+c\\cdot\\frac{1}{l}\\sum D_j)+E-\\epsilon \\sum_i A_{X,D}(G_i)G_i$$\nis a klt pair for $\\epsilon$ sufficiently small. The assumption that $Y'$ and $Y$ are not isomorphic in codimension 1 implies $Y^{\\rm c}\\neq Y$.\n\nTake $p\\colon \\hat{Y}\\to Y$ and $q\\colon \\hat{Y}\\to Y'$ a common log resolution, and write\n$$p^*(K_Y+{\\mu}_*^{-1}D+E)=q^*(K_{Y'}+{\\mu'}_*^{-1}D+S)+G.$$\n By negativity lemma (cf. \\cite[3.39]{KM98}), we conclude that $G\\ge 0$. Since \n $$K_Y+{\\mu}_*^{-1}D+E\\sim_{\\mathbb{Q},X}\\sum_i A_{X,D}(G_i)G_i $$\n and\n  $$K_{Y'}+{\\mu'}_*^{-1}D+S\\sim_{\\mathbb{Q},X} A_{X,D}(S),$$\n we know that \n $$p^*(\\sum_iA_{X,D}(G_i)G_i)=q^*(A_{X,D}(S))+G.$$\n \n  For $0\\le \\lambda \\le 1$, let \n  $$L_{\\lambda}=q^*(A_{X,D}(S))+\\lambda G=\\sum_{i} b_i(\\lambda) F_i,$$\n  where $F_i$ runs over all divisor supports on $\\hat{Y}_{o}:=\\hat{Y}\\times_X\\{o\\}$, and $-L_{\\lambda}|_{\\hat{Y}_o}$ is nef.\n  Define \n$$f(\\lambda) = \\sum_i b_i(\\lambda)(-L_{\\lambda}|_{F_i})^{n-1},$$\nthus $f(\\lambda)$ is non-decreasing as $G\\ge 0$. By Lemma \\ref{l-inter} and \\ref{l-inter2}, we know that\n$$f(1)={\\rm vol}_{X,o}(Y)\\qquad\\mbox{and}\\qquad f(0)={\\rm vol}_{X,o}(Y').$$\n\nSince $Y\\dasharrow Y'$ are not isomorphic incodimension 1, it must contract some component $G_1$ of $E$, and the coefficient of $G_1$ in $G$ is\n$$a:=A_{Y',{\\mu'}_*^{-1}D+S}(G_1)>0.\n$$\nThen\n\\begin{eqnarray*}\n\\frac{df (\\lambda)}{d \\lambda}|_{\\lambda=1}&=& n\\cdot G\\cdot \\big(-p^*(K_Y+{\\mu}_*^{-1}D+E)\\big)^{n-1}\\\\\n&\\ge&n \\cdot aG_1 \\cdot \\big(-\\pi_*(K_Y+{\\mu}_*^{-1}D+E)\\big)^{n-1}\\\\\n&>&0.\n\\end{eqnarray*}\nThus ${\\rm vol}_{X,o}(Y')=f(0)<f(1)={\\rm vol}_{X,o}(Y)$.\n\\end{proof}\n\nWith all this discussion, we also obtain the following result, which characterizes the equality condition in Proposition \\ref{p-inf} and is a corresponding generalization of \\cite[Theorem 1.4]{dFEM04} (see Remark \\ref{r-sharp}) for smooth point. See \\cite[9.6]{LazII} for more background. \n\n\\begin{thm}\\label{p-equality}\nLet $(X,o)=(\\mathrm{Spec} (R),\\mathfrak{m})$. Assume $(X,D)$ is a klt singularity for a $\\mathbb{Q}$-divisor $D\\ge 0$.\nIf there exists an $\\mathfrak{m}$-primary ideal ${\\mathfrak{b}}$, such that  ${\\rm lct}^n(X,D;{\\mathfrak{b}})\\cdot {\\rm mult} ({\\mathfrak{b}})$ is equal to $\\inf_{v\\in {\\rm Val}_{X,o}}{\\widehat{\\rm vol}}(v)$, then $(X,D)$ is minimized at a Koll\\'ar component.\n\nMoreover,\nfor an ideal $\\mathfrak{a}$, \n $${\\rm lct}^n(X,D;\\mathfrak{a})\\cdot {\\rm mult} (\\mathfrak{a})=\\inf_{v\\in {\\rm Val}_{X,o}}{\\widehat{\\rm vol}}(v),$$\n if and only if there exists a Koll\\'ar component $S$ and a  positive integer $k$ such that ${\\rm ord}_S$ computes the log canonical threshold of $\\mathfrak{a}$ on $o\\in (X,D)$ with\n $${\\widehat{\\rm vol}}({\\rm ord}_S)=\\inf_{v\\in {\\rm Val}_{X,o}}{\\widehat{\\rm vol}}(v), $$\nand the only associated Rees valuation of $\\mathfrak{a}^k$ is ${\\rm ord}_S$.  \n\\end{thm}\n\\begin{proof}By the argument in Theorem \\ref{t-divisor}, we see that \n$${\\rm lct}^n(X,D;{\\mathfrak{b}})\\cdot {\\rm mult} ({\\mathfrak{b}})$$\nreaches the minimum of ${\\widehat{\\rm vol}}_{X,o}$ if and only if the dlt modification of \n$$(X,D; c\\cdot {\\mathfrak{b}}) \\qquad \\mbox {where $c={\\rm lct}(X,D; {\\mathfrak{b}})$} $$\nis a model $\\mu:Y\\to X$ which only extracts a Koll\\'ar component $S$ of $(X,D)$ such that ${\\rm ord}_S$ is a minimizer of ${\\widehat{\\rm vol}}_{X,o}$.\n\nNow we fix an ideal $\\mathfrak{a}$ such that  $${\\rm lct}^n(X,D;\\mathfrak{a})\\cdot {\\rm mult} (\\mathfrak{a})=\\inf_{v\\in {\\rm Val}_{X,o}}{\\widehat{\\rm vol}}(v),$$ and $S$ is the Koll\\'ar component we obtain as above. \nLet $\\mu^*\\mathfrak{a}$ have the vanishing order $m$ along $S$. Since $S$ is $\\mathbb{Q}$-Cartier, then let $k$ be a positive integer such that $mk S$ is Cartier. We claim that\n$$ \\mu^*(\\mathfrak{a}^k)=\\mathcal{O}_Y(-mkS).$$\nGranted this for now, then we know that $Y$ coincides with the normalized blow up $X^{+}\\to X$ of $\\mathfrak{a}^k$, i.e., $S$ is the only associated Rees valuation for $\\mathfrak{a}^k$.\n\n To see the claim, since $-mkS$ is Cartier, we know that \n $$ \\mu^*(\\mathfrak{a}^k)={\\mathfrak{c}}\\cdot \\mathcal{O}_Y(-mkS)\\qquad \\mbox{for some ideal ${\\mathfrak{c}}\\subset \\mathcal{O}_Y$},$$\n and we aim to show that ${\\mathfrak{c}}$ is indeed trivial. If not, we take a normalized blow up $\\phi\\colon Y^+\\to Y$ of ${\\mathfrak{c}}$, so $\\phi^*{{\\mathfrak{c}}}=\\mathcal{O}_{Y^+}(-E)$ for some effective Cartier divisor $E$. Since $-S$ is ample over $X$, we can  choose $l$ sufficiently big, such that \n $$-D:=-\\phi^*(mklS)-E$$ on $Y^+$ is ample over $X$. \n \n Since \n $$(\\mu\\circ \\phi)^*\\mathfrak{a}^{kl}=\\mathcal{O}_{Y^+}(-\\phi^*(mklS)-lE)\\subset \\mathcal{O}_{Y^+}(-\\phi^*(mklS)-E),$$\n we know that \n \\begin{eqnarray*}\n {\\rm mult}(\\mathfrak{a}^{kl})&\\ge & {\\rm vol}^{F}_{o}(-\\phi^*(mklS)-E)\\\\\n  &= & {\\rm vol}^{F}_o(-D)\\\\\n  &=& mkl (-D|_{\\phi^*S})^{n-1}+ (-D|_{E})^{n-1}\\\\\n  &>&  mkl (mkl(-S)|_{S})^{n-1}\\\\\n  &=&(mkl)^n{\\rm vol}({\\rm ord}_S).\n\\end{eqnarray*}\nSince $ {\\rm lct}^n(X,D;\\mathfrak{a})=\\frac{1}{l}\\cdot A_{X,D}({\\rm ord}_S)$, we can easily see the above inequality is  contradictory to the assumption that \n$${\\rm lct}^n(X,D;\\mathfrak{a})\\cdot {\\rm mult} (\\mathfrak{a})={\\widehat{\\rm vol}}({\\rm ord}_S).$$\nHere the inequality in the fourth row comes from a similar but easier calculation as in the proof of of Theorem \\ref{t-divisor}. \n\\bigskip\n\n\n\nFor the converse direction, since $\\mathfrak{a}$ reaches the minimum if and only if $\\mathfrak{a}^k$ reaches the minimum, then we can replace $\\mathfrak{a}$ by $\\mathfrak{a}^k$ and assume its only associated Rees valuation is ${\\rm ord}_{S}$, i.e.,  we know that the normalized blow up $\\mu\\colon X^{+}\\to X$, has the property that $\\mu^*(\\mathfrak{a})=\\mathcal{O}_{X^+}(-mS)$. Then the valuative ideal \n$$\\mathfrak{a}_{mk}= \\overline{\\mathfrak{a}^k},$$\nwhere $\\overline{\\mathfrak{a}^k}$ means the integral closure of $\\mathfrak{a}^k$. And ${\\rm lct}(X,D;\\mathfrak{a})=\\frac{A_{X,D}(S)}{m}.$ \nWe claim that \n$${\\rm mult}(\\mathfrak{a})=\\lim_{k\\to +\\infty}\\frac{n!\\cdot l_R(R\/\\overline{\\mathfrak{a}^{k}})}{k^n} ,$$\nand this immediately implies that $${\\widehat{\\rm vol}}({\\rm ord}_S)={\\rm lct}(X,D;\\mathfrak{a})^n\\cdot {\\rm mult}(\\mathfrak{a}).$$\n\nTo see the claim,  if we denote by $\\mathcal{J}(\\mathfrak{a}^k)=\\mathcal{J}(X,D;\\mathfrak{a}^k)$ the multiplier ideal, then we know that \n$${\\rm mult}(\\mathfrak{a})=\\lim_{k\\to +\\infty}\\frac{n!\\cdot l_R(R\/\\mathcal{J}(\\mathfrak{a}^k) )}{k^n} ,$$\nby the local Skoda Theorem \\cite[9.6.39]{LazII}. On the other hand, since $(X,D)$ is klt, we have \n$$\\overline{\\mathfrak{a}^k} \\subset \\mathcal{J}(\\mathfrak{a}^k),  $$\nthus we are done. \n\\end{proof}\n\\begin{rem}\\label{r-sharp}\nIn the proof, we indeed show that  for any $k$ such that $mkS$ is Cartier on $Y$,  the integral closure $\\overline{\\mathfrak{a}^k}$ coincides with the valuative ideal $\\mathfrak{a}_{mk}$ of ${\\rm ord}_{S}$. \n\\end{rem}\n\n\n\n\n\\section{K-semistability implies minimum}\\label{s-min}\n\\subsection{Degeneration to initial ideals}\\label{s-degin}\nLet $(X, o)=({\\rm Spec}(R), \\mathfrak{m})$ be an algebraic singularity such that $(X,D)$ is klt for a $\\mathbb{Q}$-divisor $D\\ge 0$. Denote by $\\mathfrak{m}$ the maximal ideal of $o\\in X$. Suppose ${\\mathfrak{b}}$ is an $\\mathfrak{m}$-primary ideal on $X$. We consider its flat degeneration $W\/{\\mathbb{A}}^1$ where $W$ is the underlying coarse moduli space of ${\\mathcal{W}}$ defined in Section \\ref{ss-deformation}.  We keep the notation in Section \\ref{ss-deformation} and denote by $v_0$ the valuation ${\\rm ord}_S$. \n\nNext we will describe explicitly a way of obtaining an ideal $\\mathfrak{B}$ on ${\\mathcal{X}}$ such that $\\mathfrak{B}\\otimes \\mathcal{O}_{X\\times\\mathbb{C}^*}={\\mathfrak{b}}$ and $\\mathfrak{B}\\otimes \\mathcal{O}_C\\cong {\\bf in}({\\mathfrak{b}})$ by considering the closure of ${\\mathfrak{b}}\\times \\mathbb{C}^*$ on the pull back ${\\mathcal{Z}}$.\n\nFor this we consider the extended Rees algebra (see \\cite[6.5]{Eis94}):\n\\[\n\\mathcal{R}'=\\bigoplus_{k\\in \\mathbb{Z}}{\\mathcal{R}}'_k:=\\bigoplus_{k\\in \\mathbb{Z}} \\mathfrak{a}_k t^{-k}\\subset R[t, t^{-1}].\n\\]\nNotice that if $k\\le 0$, then $\\mathfrak{a}_k=R$. It is well known that:\n\\[\n\\mathcal{R}'\\otimes_{\\mathbb{C}[t]}\\mathbb{C}[t,t^{-1}]\\cong R[t, t^{-1}], \\quad \\mathcal{R}'\\otimes_{\\mathbb{C}[t]}\\mathbb{C}[t]\/(t)\\cong \\bigoplus_{k=0}^{+\\infty}(\\mathfrak{a}_k\/\\mathfrak{a}_{k+1})t^{-k}\\cong T.\n\\]\nGeometrically this exactly means $W=\\mathrm{Spec} (\\mathcal{R}')$ and\n\\[\nW\\times_{{\\mathbb{A}}^1}({\\mathbb{A}}^1\\setminus\\{0\\})=X\\times ({\\mathbb{A}}^1\\setminus\\{0\\}), \\quad W\\times_{\\mathbb{A}^1}\\{0\\}=C.\n\\]\nNotice that there is a natural ${\\mathbb{G}}_m$-action on ${\\mathcal{R}}'$ given by the $\\mathbb{Z}$-grading.\n\nFor any $f\\in R$, supposing $v_0(f)=k$ then we define \n$$\\tilde{f}=t^{-k}f\\in \\mathfrak{a}_k t^{-k}\\subset {\\mathcal{R}}',$$ and denote \n$${\\bf in}(f)=[f]=[f]_{\\mathfrak{a}_{k+1}}\\in \\mathfrak{a}_k\/\\mathfrak{a}_{k+1}=T_k,$$\nwhere we use $[f]_{\\mathfrak{a}}$ to denote the image of $f$ in $R\/\\mathfrak{a}$. \nThen we define the ideal ${\\mathfrak{B}}$ to be the ideal in ${\\mathcal{R}}'$ generated by $\\{\\tilde{f}; f\\in {\\mathfrak{b}}\\}$, and ${\\bf in}({\\mathfrak{b}})$ the ideal of $T$ generated by \n$\\{ {\\bf in} (f); f\\in {\\mathfrak{b}} \\}$.\nThe first two items of the following lemma is similar to (but not the same as) \\cite[Theorem 15.17]{Eis94} and should be well known to experts. Notice that here we degenerate both the ambient space and the ideal. A version of the equality \\eqref{eqdim} was proved in \\cite[Proposition 4.3]{Li15b}.\n\\begin{lem}[]\n\\begin{enumerate}\n\\item There are the identities:\n\\begin{equation*}\\label{eqflat}\n\\left({\\mathcal{R}}'\/{\\mathfrak{B}}\\right) \\otimes_{\\mathbb{C}[t]}\\mathbb{C}[t,t^{-1}]\\cong (R\/{\\mathfrak{b}})[t, t^{-1}], \\quad \\left({\\mathcal{R}}'\/{\\mathfrak{B}}\\right)\\otimes_{k[t]}k[t]\/(t)\\cong T\/{\\bf in}({\\mathfrak{b}}).\n\\end{equation*}\n\\item \nThe $\\mathbb{C}[t]$-algebra \n${\\mathcal{R}}'\/{\\mathfrak{B}}$ is free and thus flat as a $\\mathbb{C}[t]$-module. In particular, we have the identity of dimensions:\n\\begin{equation}\\label{eqdim}\n\\dim_{\\mathbb{C}} \\left(R\/{\\mathfrak{b}}\\right)=\\dim_{\\mathbb{C}}  \\left(T\/{\\bf in}({\\mathfrak{b}})\\right).\n\\end{equation}\n\\item\nIf ${\\mathfrak{b}}$ is $\\mathfrak{m}_R$-primary, then ${\\bf in}({\\mathfrak{b}})$ is an $\\mathfrak{m}_T$-primary homogeneous ideal.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\n\nThe statement (1) follows easily from the definition.\n\n\\bigskip\n\nNext we prove (2). Denote by ${\\mathfrak{c}}_k=T_k\\cap {\\bf in}({\\mathfrak{b}})$ the $k$-th homogeneous piece of ${\\bf in}({\\mathfrak{b}})$.\nWe fix a basis $\\left\\{{\\bf in}(f^{(k)}_i); 1\\le i\\le d_k \\right\\}$ of $T_k\/{\\mathfrak{c}}_k$. We want to show that \n\\[\n{\\mathcal{A}}':=\\left\\{\\left.\\left[\\widetilde{f^{(k)}_i}\\right]=\\left[f^{(k)}_i\\right]_{{\\mathfrak{B}}}\\; \\right|\\; 1\\le i\\le d_k \\right\\}\\subset {\\mathcal{R}}'\/{\\mathfrak{B}}\n\\]\nis a $\\mathbb{C}[t]$-basis of ${\\mathcal{R}}'\/{\\mathfrak{B}}$.\n\nWe first verify that ${\\mathcal{A}}'$ is a linearly independent set. To prove this, we just need to show that ${\\mathcal{A}}'$ is a $\\mathbb{C}[t, t^{-1}]$-linearly independent subset of $(R\/{\\mathfrak{b}})[t,t^{-1}]$. \nIt is then enough to show that \n\\begin{equation}\\label{eqcA}\n{\\mathcal{A}}:=\\left\\{[f^{(k)}_i]=[f^{(k)}_i]_{\\mathfrak{b}} \\; |\\; 1\\le i\\le d_k\\right\\}\\subset R\/{\\mathfrak{b}}.\n\\end{equation}\nis $\\mathbb{C}$-linearly independent, which can be verified directly as in \\cite[Proposition 4.3]{Li15b}. See also \\cite[Proposition 15.3]{Eis94}.\n\nSo we just need to show that ${\\mathcal{A}}'$ spans ${\\mathcal{R}}'\/{\\mathfrak{B}}$. Equivalently, we need to show that for any $f\\in R$, $[\\tilde{f}]=[\\tilde{f}]_{{\\mathfrak{B}}} \\in {\\mathcal{R}}'\/{\\mathfrak{B}}$ is in the $\\mathbb{C}[t]$-span of ${\\mathcal{A}}'$. \nThis can be shown again with the help of ${\\mathcal{A}}$ in \\eqref{eqcA}, that is, it is enough to prove that\n${\\mathcal{A}}$ $\\mathbb{C}$-spans $R\/{\\mathfrak{b}}$. Indeed, assuming the latter, for any $f\\in R$, there exists a linear combination $g=\\sum_{i,k}c_{ik} f^{(k)}_i$ such that \n$f-g=:h\\in {\\mathfrak{b}}$. If $m=v_0(f)$, then \n\\[\n\\tilde{f}=t^{-m}f=\\sum_{i,k}c_{ik}t^{-m}f^{(k)}_i+t^{-m}h\n\\]\nBecause $t^{-m}h\\in {\\mathfrak{B}}$, the above indeed implies $[\\tilde{f}]$ is in the $\\mathbb{C}[t]$-span of ${\\mathcal{A}}'$.\n\nTo prove that ${\\mathcal{A}}$ indeed $\\mathbb{C}$-spans $R\/{\\mathfrak{b}}$, we first claim that the following set is finite:  \n\\[\n\\{v_0(g)\\; |\\; g\\in R-{\\mathfrak{b}}\\}.\n\\]\nIndeed because ${\\mathfrak{b}}$ is $\\mathfrak{m}$-primary, there exists $N>0$ such that $\\mathfrak{m}^N\\subseteq {\\mathfrak{b}}\\subseteq \\mathfrak{m}$. So $R-{\\mathfrak{b}}\\subseteq R-\\mathfrak{m}^N$. Now the claim follows from the fact that for  any element $f\\in \\mathfrak{m}^N$,  \n$$v_0(f)\\le c \\cdot A(v_0)\\cdot N$$\nby Izumi's theorem,  where $c$ is a uniform constant not depending on $f$.\n\nIf there is $[f]\\neq 0\\in R\/{\\mathfrak{b}}$ that is not in the span of ${\\mathcal{A}}$, then we can choose a maximal $k=v_0(f)$ such that this happens. There are two cases:\n\\begin{enumerate}\n\\item If ${\\bf in}(f)\\in T_k\\setminus {\\mathfrak{c}}_k$, then because ${\\bf in}(f^{(k)}_i)$ is a basis of $T_k\/{\\mathfrak{c}}_k$, there exists $t_j\\in \\mathbb{R}$ such that ${\\bf in}(f)-\\sum_{j=1}^{d_k} t_j {\\bf in}(f^{(k)}_j)={\\bf in}(g) \\in {\\mathfrak{c}}_k$ for some $g\\in{\\mathfrak{b}}$. So we get:\n\\[\nv_0\\left(f-\\sum_{j=1}^{d_k} t_j f^{(k)}_j-g\\right)>k.\n\\]\n By maximality \nof $k$, $[f-\\sum_{j=1}^{d_k} t_j f^{(k)}_j-g]=[f]-\\sum_{j=1}^{d_k} t_j [f^{(k)}_j]$ and hence $[f]$ is in the span of ${\\mathcal{A}}$. Contradiction.\n\\item If ${\\bf in}(f)\\in {\\mathfrak{c}}_k={\\bf in}({\\mathfrak{b}})\\cap T_k$. Then ${\\bf in}(f)={\\bf in}(g)$ for some $g\\in {\\mathfrak{b}}$. So $v_0(f-g)>k$ and hence $[f-g]$ is in the span \nof ${\\mathcal{A}}$ by the maximal property of $k$. But then $[f]=[f-g]+[g]=[f-g]$ is in the span of ${\\mathcal{A}}$. Contradiction.\n\\end{enumerate}\n\n\\bigskip\nTo prove part 3 of the Lemma, we need to show that there exists $N\\in \\mathbb{Z}_{>0}$ such that $\\mathfrak{m}_T^{N}\\subseteq {\\bf in}({\\mathfrak{b}})\\subseteq \\mathfrak{m}_T$. Because\n${\\mathfrak{b}}$ is $\\mathfrak{m}_R$ primary, there exists $N_1\\in \\mathbb{Z}_{>0}$ such that $\\mathfrak{m}_R^{N_1}\\subseteq {\\mathfrak{b}} \\subseteq \\mathfrak{m}_R$. By Izumi's theorem, \nthere exists $l\\in\\mathbb{Z}_{>0}$ such that $\\mathfrak{a}_{l m}\\subseteq \\mathfrak{m}_R^m$ for any $m\\in \\mathbb{Z}_{>0}$. By letting $N=l N_1$, it's easy to see that $\\mathfrak{m}_{T}^{N}\\subseteq {\\bf in}({\\mathfrak{b}})\\subseteq\\mathfrak{m}_T$.\n\n\n\n\n\n\\end{proof}\n\n\n\\begin{lem}\nIf ${\\mathfrak{b}}_\\bullet=\\{{\\mathfrak{b}}_k\\}$ is a graded family of ideals of $R$, then ${\\bf in}({\\mathfrak{b}}_\\bullet):=\\{{\\bf in}({\\mathfrak{b}}_k)\\}$ is also a graded family\nof ideals of $T$. \n\\end{lem}\n\\begin{proof}\nWe need to show that:\n\\[\n{\\bf in}({\\mathfrak{b}}_k)\\cdot {\\bf in}({\\mathfrak{b}}_l)\\subseteq {\\bf in}({\\mathfrak{b}}_{k+l}).\n\\]\nIf $v_0(f)=k$ and $v_0(g)=l$, then $v_0(fg)=k+l$.\n\\[\n{\\bf in}(f)\\cdot {\\bf in}(g)=[f]_{\\mathfrak{a}_{k+1}}\\cdot [g]_{\\mathfrak{a}_{l+1}}=[f g]_{\\mathfrak{a}_{k+l+1}}={\\bf in}(f\\cdot g).\n\\]\n\\end{proof}\n\n\\begin{lem}\\label{l-deg}\nIf ${\\mathfrak{b}}_\\bullet$ is a graded family of ideals, then \n\\begin{equation}\\label{eqdeg}\n{\\rm lct}^n({\\mathfrak{b}}_\\bullet)\\cdot {\\rm mult}({\\mathfrak{b}}_\\bullet)\\ge {\\rm lct}^n({\\bf in}({\\mathfrak{b}}_\\bullet))\\cdot {\\rm mult}({\\bf in}({\\mathfrak{b}}_\\bullet)).\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nBy the flatness of ${\\mathfrak{B}}$ and the lower semicontinuity of log canonical thresholds, we have ${\\rm lct}({\\mathfrak{b}}_k)\\ge {\\rm lct}({\\bf in}({\\mathfrak{b}}_k))$. Therefore, by \\eqref{eqdim}\n\\begin{eqnarray*}\n{\\rm lct}^n({\\mathfrak{b}}_k)\\cdot l_R(R\/{\\mathfrak{b}}_k)&\\ge& {\\rm lct}^n({\\bf in}({\\mathfrak{b}}_k))\\cdot l_T(T\/{\\bf in}({\\mathfrak{b}}_k)).\n\\end{eqnarray*}\nTaking limits as $k\\rightarrow+\\infty$, we then get the inequality \\eqref{eqdeg}.\n\\end{proof}\n\n\\subsection{Equivariant K-semistability and minimizing}\\label{s-equiv}\n\nIn this section, we will take a detour to show the discussion in Section \\ref{s-degin} can be used to study the equivariant K-semistability. Here for a $\\mathbb{Q}$-Fano variety $(X,D)$ with an action by an algebraic group $G$, we call it {\\it $G$-equivariantly K-semistable (resp. Ding semistable)} if for any $G$-equivariant test configuration, its generalized Futaki (resp. Ding) invariant is non-negative.  Let $T=(\\mathbb{C}^*)^k$ be a torus. First we improve the two approximating results to the equivariant case.\n\\begin{lem}\\label{l-Tmini}\nLet $(X,o)=({\\rm Spec}R, \\mathfrak{m})$ and $D\\ge 0$ a $\\mathbb{Q}$-divisor, such that $o\\in (X,D)$ is a klt singularity. Assume $x\\in (X,D)$ admits a $T$-action.  Then we have \n\\begin{equation}\\label{eq-Tvol2mul}\n\\min_{v} {\\widehat{\\rm vol}}_{X,o}(v)=\\inf_{\\mathfrak{a}}{\\rm lct}^n(X,D; \\mathfrak{a})\\cdot{\\rm mult}(\\mathfrak{a})=\\inf_{S}{\\widehat{\\rm vol}}_{X,o}({\\rm ord}_{S}),\n\\end{equation}\nwhere on the left hand side the infimum runs over all the valuations centered at $o$, and on the middle it runs over all the $T$-equivariant $\\mathfrak{m}$-primary ideals; and at the end, it runs over all $T$-equivariant Koll\\'ar components. \n\\end{lem}\n\\begin{proof} Let $\\{\\mathfrak{a}_{\\bullet}\\}=\\{\\mathfrak{a}^k\\}$ be a graded sequence for an $\\mathfrak{m}$-primary ideal $\\mathfrak{a}$.\nThe discussion in Section \\ref{s-degin} implies that for $\\{{\\mathfrak{b}}_{\\bullet}\\}=_{\\rm defn} {\\bf in}(\\mathfrak{a}_{\\bullet})$\n$${\\rm lct}^n(X,D; {\\mathfrak{b}}_{\\bullet})\\cdot{\\rm mult}({\\mathfrak{b}}_{\\bullet})\\le {\\rm lct}^n(X,D; \\mathfrak{a}_{\\bullet})\\cdot{\\rm mult}(\\mathfrak{a}_{\\bullet}).$$ \nSince\n$${\\rm lct}^n(X,D; {\\mathfrak{b}}_{\\bullet})\\cdot{\\rm mult}({\\mathfrak{b}}_{\\bullet})=\\lim_{m} {\\rm lct}^n(X,D; {\\mathfrak{b}}_{m})\\cdot{\\rm mult}({\\mathfrak{b}}_{m}),$$\nwe conclude the first inequality as a corollary of Proposition \\ref{p-inf}. \n\n\\medskip\n\nFor the second equality, we just need to show that the construction in Section \\ref{s-lvmodel} can be established $T$-equivariantly. This is standard, which relies on two facts: first, we can always take an equivariant log resoltuion of $(X,D, \\mathfrak{a})$ (see \\cite{Kol07}); second,  as $T$ is a connected group, for any curve $C$ in a $T$-variety and any $t\\in T$, $t\\cdot C$ will always be numerically equivalent to $C$; as the minimal model program only depends on the numerical class $[C]$, we know that any MMP sequence is automatically $T$-equivariant. \nTherefore, for any $T$-equivariant $\\mathfrak{m}$-primary ideal $\\mathfrak{a}$, we can find a $T$-equivariant dlt modication $Y\\to X$ and then a $T$-equivariant Koll\\'ar component $S$, such that\n$${\\rm lct}^n(X,D; \\mathfrak{a})\\cdot{\\rm mult}(\\mathfrak{a})\\ge {\\rm vol}(Y)\\ge {\\widehat{\\rm vol}}( {\\rm ord}_{S}).$$\n\\end{proof}\n\nIn the below, we also need use the main idea from Fujita's work (see \\cite{Fuj15, Fuj16}). \n\n\\begin{prop}[\\cite{Fuj16}]\\label{p-fujita}\nLet $(V, \\Delta)$ be an $(n-1)$-dimensional log-Fano pair which is $T$-equivariantly log-Ding-semistable. Let $\\delta$ be a positive rational number such that $-\\delta^{-1}(K_V+\\Delta)$ is Cartier. \nLet $I_M\\subset \\cdots\\subset I_1\\subset \\mathcal{O}_V$ be a sequence of $T$-invariant coherent ideal sheaves and assume\n$${\\mathcal{I}}:=I_M+I_{M-1} t^1+\\cdots+I_1 t^{M-1}+(t^M)\\subset \\mathcal{O}_{V\\times \\mathbb{C}}$$\nform a flag ideal.  Let\n$\\Pi: {\\mathcal{V}}\\rightarrow V\\times\\mathbb{C}$ be the blowup along ${\\mathcal{I}}$. Denote ${\\mathcal{D}}=\\Pi^*(\\Delta\\times \\mathbb{C})$. Let $E\\subset {\\mathcal{X}}$ be the Cartier divisor defined by $\\mathcal{O}_{{\\mathcal{V}}}(-E)={\\mathcal{I}}\\cdot \\mathcal{O}_{{\\mathcal{V}}}$, and let\n\\[\n{\\mathcal{L}}:=\\Pi^*\\mathcal{O}_{V\\times \\mathbb{C}}\\left(-\\delta^{-1}(K_{V\\times\\mathbb{C}\/\\mathbb{C}}+\\Delta\\times \\mathbb{C})\\right)\\otimes \\mathcal{O}_{{\\mathcal{X}}}(-E).\n\\]\nAssume that ${\\mathcal{L}}$ is semi ample over $\\mathbb{C}$. Then $({\\mathcal{V}}, {\\mathcal{D}}; {\\mathcal{L}})$ is naturally seen as a (possibly non-normal) semi test configuration of $(S, \\Delta; -\\delta^{-1}(K_V+D))$. Under these conditions, \n$((V\\times\\mathbb{C}^1, \\Delta\\times \\mathbb{C}); {\\mathcal{I}}^\\delta\\cdot (t)^d)$ must be sub log canonical, where\n\\[\nd:=1+\\frac{\\delta^{n}\\bar{{\\mathcal{L}}}^n}{n (-K_V-\\Delta)^{n-1}}.\n\\]\n\\end{prop}\n\nThe following argument is essentially in \\cite{Li15b}.\n\\begin{prop}\\label{thm-Ksemi} \nIf $(V,\\Delta)$ is a projective log Fano variety and $o\\in (X,D)$ is the cone of $(V,\\Delta)$ induced by some ample Cartier divisor $L=-r^{-1}(K_V+\\Delta)$. Then the canonical valuation obtained by blowing up the vertex minimizes ${\\widehat{\\rm vol}}_{X,D}$ if and only if $(V,\\Delta)$ is K-semistable.\n\\end{prop}\n\\begin{proof}\n\nFirst we assume $(S, \\Delta)$ is $(n-1)$-dimensional log-K-semistable and prove the volume minimizing. By Lemma \\ref{l-Tmini}, we only need to prove that for any $\\mathbb{C}^*$-invariant divisorial valuation $v$ over $(X, o)$, \n$${\\widehat{\\rm vol}}({\\rm ord}_V)\\le {\\widehat{\\rm vol}}(v).$$ Let $c_1=v(V)>0$ and $R=\\bigoplus_{k=0}^{+\\infty}R_k=\\bigoplus_{k=0}^{+\\infty}H^0(V, k L)$ thus $X={\\rm Spec}(R)$. \n\nOn $R$, we define a filtration\n$${\\mathcal{F}} R^{(t)} = \\bigoplus_k {\\mathcal{F}}^{kt}R_k, $$ where \n$${\\mathcal{F}}^xR_k:=H^0(V, L^{\\otimes k}\\otimes \\mathfrak{a}_x), \\qquad \\mbox{and \\ \\   }  \\mathfrak{a}_x=\\{f\\in \\mathcal{O}_V\\ |\\  v(f)\\ge x\\}. $$\n  Then the volume is defined to be\n$${\\rm vol}({\\mathcal{F}} R^{(t)}) := \\limsup_{m\\to \\infty} \\frac{\\dim_{\\mathbb{C}} {\\mathcal{F}}^{mt}R_m}{m^n\/n!}.$$\n By \\cite[(18)]{Li15b}, we get a formula for\n${\\rm vol}(v)$:  \n\\begin{eqnarray*}\n{\\rm vol}(v)&=&\\lim_{m\\rightarrow+\\infty} \\frac{n!}{m^n}\\dim_{\\mathbb{C}} R\/\\mathfrak{a}_m(v)\\\\\n&=&\\frac{L^{n-1}}{c_1^n}-\\int^{+\\infty}_{c_1}{\\rm vol}\\left({\\mathcal{F}} R^{(t)}\\right)\\frac{dt}{t^{n+1}}\\\\\n&=&-\\int^{+\\infty}_{c_1}\\frac{d {\\rm vol}\\left({\\mathcal{F}} R^{(t)}\\right)}{t^n}.\n\\end{eqnarray*}\nThen we consider the following function \n\\begin{eqnarray*}\n\\Phi(\\lambda, s)&=&\\frac{L^{n-1}}{(\\lambda c_1 s+(1-s))^n}-n \\int^{+\\infty}_{c_1}{\\rm vol}\\left({\\mathcal{F}} R^{(t)}\\right)\\frac{\\lambda s dt}{(1-s+\\lambda st)^{n+1}}\\\\\n&=&\\int^{+\\infty}_{c_1}\\frac{-d \\; {\\rm vol}({\\mathcal{F}} R^{(t)})}{((1-s)+\\lambda st)^n}.\n\\end{eqnarray*}\n$\\Phi(\\lambda, s)$ satisfies the following properties:\n\\begin{enumerate}\n\\item For any $\\lambda \\in (0, +\\infty)$, we have:\n\\[\n\\Phi(\\lambda, 1)={\\rm vol}(\\lambda v)=\\lambda^{-n} {\\rm vol}(v), \\quad \\Phi(\\lambda, 0)={\\rm vol}(v_0)=L^{n-1}.\n\\]\n\\item For any $\\lambda \\in (0, +\\infty)$, $\\Phi(\\lambda, s)$ is continuous and convex with respect to $s\\in [0, 1]$.\n\\item The directional derivative of $\\Phi(\\lambda, s)$ at $s=0$ is equal to:\n\\[\n\\Phi_s(\\lambda, 0)=n \\lambda L^{n-1} \\left(\\lambda^{-1}-c_1-\\frac{1}{L^{n-1}}\\int^{+\\infty}_{c_1} {\\rm vol}\\left({\\mathcal{F}} R^{(t)}\\right)dt\\right).\n\\]\n\\end{enumerate}\nLet $\\lambda_*=\\frac{r}{A_{(X, C(D))}(v)}$. By Item 1 and 2, we just need to prove $\\Phi_s(\\lambda_*, 0)\\ge 0$. \nLet $\\bar{v}=v|_{\\mathbb{C}(S)}$ be the restriction of $v$ under the inclusion $\\mathbb{C}(S)\\hookrightarrow \\mathbb{C}(X)$. \nIt is known that $\\bar{v}=b\\cdot {\\rm ord}_E$ where $b>0$ and ${\\rm ord}_E$ is a divisorial valuation on $\\mathbb{C}(S)$.\n\nUsing Adjunction formula, it is easy to show that:\n\\[\n\\lambda^{-1}-c_1=\\frac{A_{(X, D)}(v)}{r}-c_1= A_{V,\\Delta}(\\bar{v})=b\\cdot A_{V,\\Delta}(E).\n\\]\nBy change of variables we get:\n\\[\n\\int^{+\\infty}_{c_1}{\\rm vol}\\left({\\mathcal{F}} R^{(t)}\\right)dt=\\int^{+\\infty}_0 {\\rm vol}\\left({\\mathcal{F}}_{\\bar{v}} R^{(t)}\\right) dt.\n\\]\nwhere\n\\[\n{\\mathcal{F}}_{\\bar{v}} R^{(t)} = \\bigoplus_k H^0(V, L^{\\otimes k}\\otimes \\mathfrak{a}_{kt}), \\qquad \\mbox{and \\ \\   }  \\mathfrak{a}_{kt}=\\{f\\in \\mathcal{O}_V\\ |\\  \\bar{v}(f)\\ge kt\\}.\n\\]\nSo we get:\n\\begin{eqnarray*}\n\\Phi_s(\\lambda_*, 0)&=&n \\lambda_* L^{n-1}\\left(A_{(V,\\Delta)}(\\bar{v})-\\frac{1}{L^{n-1}}\\int^{+\\infty}_{0} {\\rm vol}\\left({\\mathcal{F}}_{\\bar{v}}R^{(t)}\\right)dt\\right)\\\\\n&=&n \\lambda_* L^{n-1} b\\left(A_{(V,\\Delta)}(E)-\\frac{1}{L^{n-1}}\\int_0^{+\\infty} {\\rm vol}\\left({\\mathcal{F}}_{{\\rm ord}_E}R^{(t)}\\right)dt \\right).\n\\end{eqnarray*}\nBy the valuative criterion of (log-)K-semistability derived in \\cite{Li15b, Fuj16, LL16}, we get $\\Phi_s(\\lambda_*, 0)\\ge 0$ (see e.g. Proposition \\ref{p-fujita}). \n\nConversely the statement follows immediately by the valuative criterion for log-K-semistability in \\cite{Li15b, Fuj16, LL16} by choosing a family of valuation $v_s$ emanating from $v_0={\\rm ord}_S$ in the direction of ${\\rm ord}_E$ as in \\cite[Section 7]{Li15b}.  \n\n\\end{proof}\n\\begin{rem}\nThe fact that K-semistability implies the canonical valuation is a minimizer was proved in \\cite{LL16}, where $(V,\\Delta)$ is assumed to be analytic K-semistable, i.e., if there exists a special test configuration $({\\mathcal{X}}, \\mathcal{D})$ that degenerates $(X,D)$ to a log Fano $(X_0,D_0)$ with a conic K\\\"ahler-Einstein metric.  If $(V,\\Delta)$ is analytic K-semistable, then we know $(V,\\Delta)$ is K-semistable. The converse implication conjecturally true, and known for smooth $X$ when $D=0$ (see \\cite{CDS15, Tia15}).\n\\end{rem}\n\nWith all the techniques we have, we can prove Theorem \\ref{t-equiK}.\n\\begin{proof}[Proof of Theorem \\ref{t-equiK}]  \n Let $(X,D)$ be a cone of $-r(K_V+\\Delta)$ over $(V,\\Delta)$ for some sufficiently divisible positive integer $r$. And we consider the minimizing problem of the normalized local volume  at the $T$-singularity $o$ which is the vertex. We aim to show that if $(V, \\Delta)$ is $T$-equivariantly K-semistable then ${\\rm ord}_V$ \nminimizes ${\\widehat{\\rm vol}}_{X,D}$. \n\nFollowing the proof of Lemma \\ref{l-Tmini}, by degenerating the ideal  $r$ times, we can find a sequence of $T$-equivariant ideals $\\{\\mathfrak{a}\\}_{i\\in I}$ such that \n$$\\inf_{i\\in I}{\\rm lct}^n(X,D; \\mathfrak{a})\\cdot{\\rm mult}(\\mathfrak{a})=\\min_{v\\in {\\rm Val}_{X,o}} {\\widehat{\\rm vol}}(v).$$ Then using the equivariant resolution and running an MMP process, we can find a sequence of $T$-equivariant Koll\\'ar components  $S_i$ such that \n$$\\inf_{i} {\\widehat{\\rm vol}}({\\rm ord}_{S_i})= \\min_{v\\in {\\rm Val}_{X,o}} {\\widehat{\\rm vol}}(v). $$\n\nSince we assume that for any $T$-equivariant special degeneration of $ (S,\\Delta)$, the generalized Futaki invariant is non-negative, and for special test configuration, the generalized Futaki invariant is the same as Ding invariant, we know that the Ding invariant for any $T$-equivariant special test configuration is non-negative. Using the fact that MMP decreases Ding invariant (see \\cite[Corollary 3.4]{Fuj16} and its proof), we know this implies that the Ding invariant for any $T$-equvariant test configuration is nonnegative.\n\nThen for any $T$-equivariant Koll\\'ar component $S_i$, we consider $v={\\rm ord}_{S_i}\\in {\\rm Val}_{X,o}$. Denote its induced divisorial valuation $b\\cdot {\\rm ord}_E$ on $S$.\nRestricting the calculation in \\cite{Fuj16} to the equivariant setting, we conclude that \n$$A_{(V,\\Delta)}(E)-\\frac{1}{L^{n-1}}\\int_0^{+\\infty} {\\rm vol}\\left({\\mathcal{F}}_{{\\rm ord}_E}R^{(t)}\\right)dt\\ge 0.$$\nRunning the construction as in the proof of Proposition \\ref{thm-Ksemi},  then we know that $\\Phi_s(\\lambda_*, 0)\\ge 0$\nwhich gives that $${\\widehat{\\rm vol}}(v_*)\\le {\\widehat{\\rm vol}}({\\rm ord}_{S_i}).$$\nThis implies that  the canonical component is a minimizer which then implies that $(V,\\Delta)$ is K-semistable by Proposition \\ref{thm-Ksemi} .\n\\end{proof}\n\n\n\n\n\n\n\\subsection{Proof of Theorem \\ref{t-main}}\nLet $(X, o)=({\\rm Spec}(R), \\mathfrak{m})$ be an algebraic singularity such that $(X,D)$ is klt for a $\\mathbb{Q}$-divisor $D\\ge 0$. \nLet $S$ be a Koll\\'{a}r component and $\\Delta=\\Delta_S$ be the different divisor defined by adjunction $(K_Y+S+\\mu_*^{-1}D)|_S=K_S+\\Delta_S$ where $Y\\rightarrow X$ is the extraction of $S$. We follow the notation in Section \\ref{ss-deformation}. \n\\begin{lem}\\label{l-ksemi} \nLet ${\\mathfrak{b}}_\\bullet$ be a graded sequence of $\\mathfrak{m}_0$-primary ideal. \nIf $(S,\\Delta_S)$ is K-semistable, then we have\n$$ {\\rm lct}^n({\\mathfrak{b}}_\\bullet)\\cdot {\\rm mult}({\\mathfrak{b}}_\\bullet)\\ge {\\widehat{\\rm vol}}_{C,C_{\\Delta},o}({\\rm ord}_S).$$\n\\end{lem}\n\\begin{proof} \nUsing the result in \\cite{JM12}, we have \n\\begin{eqnarray*}\n{\\rm lct}^n({\\mathfrak{b}}_\\bullet)\\cdot {\\rm mult}({\\mathfrak{b}}_\\bullet)&=&\\lim_{k \\to +\\infty} \\big(k\\cdot {\\rm lct}({\\mathfrak{b}}_k) \\big)^n\\cdot \\frac {{\\rm mult}({\\mathfrak{b}}_k)}{k^n}\\\\\n&=&\\lim_{k\\to +\\infty} {\\rm lct}^n({\\mathfrak{b}}_k)\\cdot {\\rm mult}({\\mathfrak{b}}_k).\n\\end{eqnarray*}\nBy Proposition \\ref{p-inf},  it suffices to show that  ${\\widehat{\\rm vol}}_{C,C_{\\Delta},o}({\\rm ord}_S)$ is equal to\n$$\\min_{v} {\\widehat{\\rm vol}}_{C,C_{\\Delta},o}(v)$$\nfor $v$  runs over valuations centered on $o$. \n\nIt follows from  Proposition \\ref{thm-Ksemi} that if we choose $d$ sufficiently divisible, such that $C^{(d)}=C(S,H)$ is constructed as the cone over $S$ with some ample Cartier divisor $H$ proportional to $-(K_{S}+\\Delta_{S})$, then the canonical valuation ${\\rm ord}_S$ is a minimizer of ${\\widehat{\\rm vol}}_{C^{(d)}, C^{(d)}_{\\Delta}+C^{(d)}_D}$.   By Proposition \\ref{p-cover}, this implies the same for $C$.\n\\end{proof}\n\n\\begin{prop}\\label{p-cover}\nUnder the above notation,  ${\\widehat{\\rm vol}}_{({C}, C_D)}$ minimizes at ${\\rm ord}_S$ if and only if ${\\widehat{\\rm vol}}_{({C}^{(d)}, C^{(d)}_{\\Delta}+C^{(d)}_D)}$ minimizes at ${\\rm ord}_{S^{(d)}}$. \n\\end{prop}\n\\begin{proof}The degree $d$ cover $h\\colon C\\to C^{(d)}$ is a fiberwise map and the Galois group $G=_{\\rm defn}\\mathbb{Z}\/d$ is a natural subgroup $\\mathbb{C}^*$.\nLet $E$ be a Koll\\'ar component over $C^{(d)}$, by Lemma \\ref{l-finite} we know $h^*(E)$ is a Koll\\'ar component over $ C$, and it follows from Lemma \\ref{l-finitevolume} (or \\cite[Lemma 6.9]{Li15b}) that\n$$d\\cdot {\\widehat{\\rm vol}}({\\rm ord}_E)={\\widehat{\\rm vol}}(h^*E). $$\n So if ${\\rm ord}_S$ minimizes ${\\widehat{\\rm vol}}_{C,C_{\\Delta}}$, then the corresponding  canonical valuation also minimizes ${\\widehat{\\rm vol}}_{{C}^{(d)}, C^{(d)}_{\\Delta}+C^{(d)}_D}$. \n \n For the converse, let $E$ be a $T$-invariant Koll\\'ar component over $C$, since it is $G$-invariant, by Lemma \\ref{l-finite} we know it is a pull back of a Koll\\'ar component $F$ over $C^{(d)}$. If the canonical valuation minimizes ${\\widehat{\\rm vol}}_{{C}^{(d)}, C^{(d)}_{\\Delta}+C^{(d)}_D}$, then we see over $C$, ${\\widehat{\\rm vol}}({\\rm ord}_S)$ is less or equal to ${\\widehat{\\rm vol}}({\\rm ord}_E)$ for any $T$-equivariant Koll\\'ar component $E$, therefore ${\\rm ord}_S$ is a minimizer of ${\\widehat{\\rm vol}}_{C,C_{\\Delta}}$ by Lemma \\ref{l-Tmini}. \n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{t-main}]\nWith the notation as in Section \\ref{ss-deformation}, we denote by $ \\mathcal{D}$ the birational transform of $ D\\times \\mathbb{A}^1$ on $\\mathcal{Z}^0 $ and write $a\\mathcal{T}\\sim_{\\mathbb{Q},{\\mathcal{X}}}K_{{\\mathcal{Z}}^0}+\\mathcal{T}$. Restricting over a general fiber and taking the coarse moduli spaces, we obtain \n$$aS\\sim_{X,{\\mathbb{Q}}} K_Y+S+\\mu^{-1}_*(D),$$\nthen $a=A_{X,D}(S)$.  Similarly, over the central fiber over $0$, we get\n$$aS_0\\sim_{X,{\\mathbb{Q}}} K_{Y_0}+S_0+\\mu^{-1}_{0,*}C_D,$$\n thus $a=A_{C,C_D}(S_0)$, where $\\mu_0\\colon Y_0\\to C$  is the blow up of the vertex with the exceptional divisor $S_0$.\n\nWe also know that \n$${\\rm vol}_{X,o}({\\rm ord}_S)=(-S|_S)^{n-1}=(-S_0|_{S_0})^{n-1}={\\rm vol}_{C,p}({\\rm ord}_{S_0}).$$\n\nCombining all the above, we know that for any ideal ${\\mathfrak{b}}$ on $X$, if we  let ${\\mathfrak{b}}_{\\bullet}=\\{{\\mathfrak{b}}^k\\}$, then\n\\begin{eqnarray*} \n{\\widehat{\\rm vol}}_{X,o}({\\rm ord}_S) &= &{\\rm vol}_{X,o}({\\rm ord}_S)\\cdot A^n_{X,D}(S) \\\\\n & =  &{\\rm vol}_{C,p}({\\rm ord}_{S_0})\\cdot A^n_{C,{C_{D}}}(S_0)\\\\\n &= & \\inf_v {\\rm vol}_{C,p}({\\rm ord}_{v})\\cdot A^n_{C,{C_{D}}}(v)\\\\\n &\\le & {\\rm lct}^n({\\bf in}({\\mathfrak{b}}_\\bullet))\\cdot {\\rm mult}({\\bf in}({\\mathfrak{b}}_\\bullet))\\\\\n &\\le & {\\rm lct}^n({\\mathfrak{b}}) \\cdot  {\\rm mult} ({\\mathfrak{b}}),\n\\end{eqnarray*}\nwhere  the last two inequalities follow from Lemma \\ref{l-deg} and \\ref{l-ksemi}.  \nThus we conclude that \n$${\\widehat{\\rm vol}}_{X,o}({\\rm ord}_S)\\le \\inf_{{\\mathfrak{b}}} {\\rm lct}^n({\\mathfrak{b}}) \\cdot  {\\rm mult} ({\\mathfrak{b}})=\\inf_v {\\widehat{\\rm vol}}_{X,o}(v),$$\nwhere the second equality follows from Proposition \\ref{p-inf}.\n\\end{proof}\n\n\n\n\n\\section{Uniqueness}\\label{s-uni}\n\nIn this section, we will prove Theorem \\ref{t-main2} about the uniqueness of the minimizers among all Koll\\'ar components. There are two steps: first we prove this for cone singularities; then for a general singularity, we combine the deformation construction with some results from the minimal model program to essentially reduce it to the case of cone singularities.\n\n\\subsection{Case of cone singularity}\\label{ss-ucone}\n\nWe first settle the case of cone singularities. It can be proved using Proposition \\ref{p-Tequiv} and \\cite[Theorem 3.4]{Li15b}. Here we give a different proof, which analyzes the geometry in more details. A similar argument in the global case appears in the proof of \\cite[Theorem 3]{Liu16}, where a characterization of quotients of $\\mathbb{P}^n$ is given by achieving the maximal possible volumes among all K-semistable $\\mathbb{Q}$-Fano varieties with only quotient singularities.\n\n\\bigskip\n\nLet $(V,\\Delta)$ be an $(n-1)$-dimensional log Fano variety and $-(K_V+\\Delta)=r H$ for an ample Cartier divisor $H$ and $r\\le n$. Assume $X^0:=C(V, H)$ is the affine cone over the base $V$ with the vertex $o$ and let $X$ be the projective cone and $D$ be the cone divisor over $\\Delta$ on $X$. In the below, for a variety $\\bullet$, we denote the product $\\bullet \\times \\mathbb{A}^1$ by $\\bullet_{\\mathbb{A}^1}$.\n\nConsider a Koll\\'ar component $S$ over $o\\in (X,D)$ with the extraction morphism $\\mu\\colon Y\\to X$. Let \n$$\\mu_{\\mathbb{A}^1}(=\\mu \\times {\\rm id})\\colon Y_{\\mathbb{A}^1}\\rightarrow X_{\\mathbb{A}^1}$$ \nbe the extraction of $S_{\\mathbb{A}^1}$. We carry out the deformation process in Section \\ref{ss-deformation} with respect to $S$. Here $X$ is a projective variety instead of a local singularity, but the construction is exactly the same.  We denote by $Z$ (resp. $W$) the coarse moduli space of ${\\mathcal{Z}}$ (resp. ${\\mathcal{W}}$), so there are morphisms, $\\psi_1:Z\\to W$, $\\phi_1\\colon Z\\to Y_{\\mathbb{A}^1}$ and $\\pi\\colon W\\to X_{\\mathbb{A}^1}$. We denote by $\\phi=\\mu_{\\mathbb{A}^1}\\circ\\phi_1.$\n\n\\begin{center}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=10cm, height=6cm]{bir.jpg}\n\\end{figure}\n\\end{center}\n\nThen we have the following equalities:\n\\begin{enumerate}\n\\item\n$K_{Y_{\\mathbb{A}^1}}+(\\mu_{\\mathbb{A}^1})_*^{-1}(D_{\\mathbb{A}^1})=\\mu_{\\mathbb{A}^1}^* (K_{X_{\\mathbb{A}^1}}+D_{\\mathbb{A}^1})+a  S_{\\mathbb{A}^1} $ with $a=A_{X,D}(S)-1$;\n\\item\n$K_{Z}+\\phi_*^{-1}(D_{\\mathbb{A}^1})=\\phi_1^*\\big(K_{Y_{\\mathbb{A}^1}}+(\\mu_{\\mathbb{A}^1})_*^{-1}D_{\\mathbb{A}^1}\\big)+{\\mathcal{F}}$;\n\\item\n$K_{Z}+\\phi_*^{-1}D_{\\mathbb{A}^1}=\\psi_1^* \\big(K_{W}+(D_{\\mathbb{A}^1})_W\\big)+a  S_{\\mathbb{A}^1}$, where $(D_{\\mathbb{A}^1})_W:=\\psi_{1*}\\phi_*^{-1}(D_{\\mathbb{A}^1})$.\n\\end{enumerate}\nThe first two equalities imply:\n\\begin{eqnarray*}\nK_{Z}+\\phi_*^{-1}(D_{\\mathbb{A}^1})&=&\\phi_1^* \\big(K_{Y_{\\mathbb{A}^1}}+(\\mu_{\\mathbb{A}^1})_*^{-1}D_{\\mathbb{A}^1}\\big)+ {\\mathcal{F}}\\\\\n &=&\\phi_1^*\\mu_{\\mathbb{A}^1}^* (K_{X_{\\mathbb{A}^1}}+D_{\\mathbb{A}^1})+a \\phi_1^*S_{\\mathbb{A}^1}+{\\mathcal{F}}\\\\\n&=&\\phi^*(K_{X_{\\mathbb{A}^1}}+D_{\\mathbb{A}^1})+a S_{\\mathbb{A}^1}+(a+1){\\mathcal{F}}.\n\\end{eqnarray*}\nSo $A_{X_{\\mathbb{A}^1},D_{\\mathbb{A}^1}}({\\mathcal{F}})=a+2=A_{X,D}(S)+1$. This implies:\n\\[\nK_{W}+(D_{\\mathbb{A}^1})_W=\\pi^*K_{X_{\\mathbb{A}^1},D_{\\mathbb{A}^1}}+A_{X,D}(S) \\bar{C}.\n\\]\n\nRecall that $-K_X-D=(1+r) L$, where $L=\\mathcal{O}(1)$ for the cone construction, so we get:\n\\[\nK_{W}+(D_{\\mathbb{A}^1})_W=-(1+r)\\rho^*L+A_{X,D}(S) \\bar{C},\n\\]\nwhere $\\rho\\colon W\\to X$ the composite of $\\pi\\colon W\\to X_{\\mathbb{A}^1}$ with the first projection $X_{\\mathbb{A}^1}\\to X$. \n\\begin{rem}\n\\begin{enumerate}\n\\item As in \\cite{LL16}, we define the cone angle parameter $\\beta=\\frac{r}{n}$. Denote by $V_{\\infty}$ the section at infinite place of $X$ and ${\\mathbb{V}}_{\\infty}$ the birational transform of $(V_{\\infty})_{\\mathbb{A}^1}$. Then  \n$$-(K_X+D+(1-\\beta)V_\\infty)=\\delta L$$ with $\\delta=r\\frac{n+1}{n}$, we get:\n\\begin{eqnarray}\\label{lclY2}\n& &K_{W}+(D_{\\mathbb{A}^1})_W+(1-\\beta){\\mathbb{V}}_\\infty\\\\\n&=&\\pi^*\\big(K_{X_{\\mathbb{A}^1}}+D_{\\mathbb{A}^1}+(1-\\beta) {\\mathbb{V}}_\\infty\\big)+A_{X,D}(S)\\bar{C}\\nonumber\\\\\n&=&-\\delta\\rho^*L+A_{X,D}(S)\\bar{C}.\n\\end{eqnarray}\n\\item\nBecause $K_{Y}+\\mu_*^{-1}D=\\mu^*(K_X+D)+(A_{X,D}(S)-1)S$, we have:\n\\begin{eqnarray*}\nK_{Y}+\\mu^{-1}_*D+(1-\\beta)V_\\infty&=&\\mu^*\\big(K_X+D+(1-\\beta)V_\\infty\\big)+(A_{X,D}(S)-1)S\\\\\n& =&-\\delta \\mu^*L+(A_{X,D}(S)-1)S.\n\\end{eqnarray*}\n\\end{enumerate}\n\\end{rem}\n\n\nThe above construction works for any Koll\\'{a}r component. From now on we assume that $(V,\\Delta)$ is K-semistable and $S$ minimizes the normalized volume, i.e. it satisfies\n\\begin{equation}\n{\\widehat{\\rm vol}}({\\rm ord}_{S})={\\widehat{\\rm vol}}({\\rm ord}_{V_0})=r^n (H^{{n-1}}),\n\\end{equation}\nwhere $V_0$ denotes the canonical divisor obtained by blowing up the vertex of the cone and we aim to show $S=V$ is the canonical component. We note that by Proposition \\ref{thm-Ksemi} , we know that ${\\widehat{\\rm vol}}({\\rm ord}_{V_0})$ is the minimal normalized volume. \n\nThen we have:\n\\[\n{\\rm vol}({\\rm ord}_{S})=\\frac{{\\widehat{\\rm vol}}({\\rm ord}_{S})}{A_{X,D}(S)^n}=\\frac{r^n (H^{n-1})}{A_{X,D}(S)^n}.\n\\]\n\nIn Section \\ref{s-equiv}, we have used the filtration induced by a valuation (see also \\cite{BHJ15, Fuj15}).  Here we use the same construction but for sections on the projective cone instead of the base. \n\\begin{defn}[Filtration by valuation]\nFixed a valuation $v$. Let $S_m=H^0(X, L^{\\otimes m})$. Define ${\\mathcal{F}}^xS_m\\subset S_m$ to be a decreasing filtration (with respect to $x$) as follows:\n$${\\mathcal{F}}^xS_m:=H^0(X, L^{\\otimes m}\\otimes \\mathfrak{a}_x), \\qquad \\mbox{where \\ } \\mathfrak{a}_x=\\{f\\in \\mathcal{O}_X\\ |\\  v(f)\\ge x\\}. $$\n\nOn $\\bigoplus_{m=0} S_m$, we define ${\\mathcal{F}} S^{(t)} = \\bigoplus {\\mathcal{F}}^{kt}S_k$.  Then the volume is defined to be\n$${\\rm vol}({\\mathcal{F}} S^{(t)}) := \\limsup_{m\\to \\infty} \\frac{\\dim_{\\mathbb{C}} ({\\mathcal{F}}^{mt}S_m)}{m^n\/n!}.$$\n\n\\end{defn}\n\n\n\n\\begin{prop}\\label{p-fanocone}\nWith the above notation, if the case $(V,\\Delta)$ is log K-semistable, then $(X,D+(1-\\beta)V_{\\infty})$ is log K-semistable. As a consequence,\n\\[\nA_{X, D}(S)-\\frac{\\delta }{(L^{n})}\\int_0^{+\\infty}{\\rm vol}({\\mathcal{F}} S^{(x)})dx\\ge 0.\n\\]\n\\end{prop}\n\\begin{proof}By Theorem \\ref{t-equiK}, we only need to check this is true for $\\mathbb{C}^*$-equvariant special test configuration $(\\mathcal{X},\\mathcal{D}+(1-\\beta){\\mathcal{V}})\/\\mathbb{P}^1$ of $(X,D+(1-\\beta)V_{\\infty})$, where we consider the test configuration over $\\mathbb{P}^1$ by adding a trivial fiber over $\\{\\infty \\}$. \nConsider the closure of $\\mathcal{V}\\supset V_{\\infty}\\times( \\mathbb{P}^1\\setminus \\{0 \\})$. As $(1+r)V\\sim_{\\mathbb{Q}}  -(K_X+D) $, we know that if we denote by ${\\mathcal{L}}$ the polarization on the test configuration extending $L$, then \n$$(1+r)\\mathcal{V}\\sim_{\\mathbb{Q},\\mathbb{P}^1}-K_{\\mathcal{X}}-\\mathcal{D}=(1+r)\\mathcal{L}$$\nas the fiber over $0$ is irreducible. \nLet $\\Delta_{\\infty}(=\\Delta)=V_{\\infty}\\cap D$ and $\\mathcal{E}$ be the closure of $\\Delta_{\\infty}\\times ( \\mathbb{P}^1\\setminus \\{0\\})$, then\n$$(K_{\\mathcal{X}}+\\mathcal{D}+{\\mathcal{V}})|_{{\\mathcal{V}}}=K_{{\\mathcal{V}}}+\\mathcal{E}$$ as $\\mathcal{X}$  is smooth along the codimension 2 points over $0$ and so there is no different divisor. \n\n\nThen the generalized Futaki invariant of $(\\mathcal{X},\\mathcal{D}+(1-\\beta){\\mathcal{V}})\/\\mathbb{P}^1$ is \n$${\\rm Fut}(\\mathcal{X})= -\\frac{1}{(n+1)(\\delta L)^n}(-K_{\\mathcal{X}\/\\mathbb{P}^1}-\\mathcal{D}-(1-\\beta)\\mathcal{V})^{n+1}. $$\nSince $V_{\\infty}\\sim L$ and $\\delta=r\\frac{n+1}{n}$, it's easy to verify that the generalized Futaki invariant of the induced test configuration of $(V, \\Delta)$ is \n$${\\rm Fut}(\\mathcal{V})= -\\frac{1}{nr^{n-1} H^{n-1}}((-K_{\\mathcal{V}\/\\mathbb{P}^1}-\\mathcal{E})|_{\\mathcal{V}})^{n}={\\rm Fut}(\\mathcal{X}). $$\n\nRecall the log-K-semistability is equivalent to the log-Ding-semistablity (see e.g. \\cite{Fuj16}). Then the second part is a standard generalization of \\cite[Proposition 4.5]{LL16} to the log setting, and we skip it.  \n\\end{proof}\n\nThe key calculations are given by the following results proved in in \\cite{LL16}.\n\\begin{prop}\\label{p-equal}\nSuppose $(V, \\Delta)$ is log-K-semistable. If $S$ is a Koll\\'{a}r component obtaining the minimum of ${\\widehat{\\rm vol}}$ over $(X, o)$, then the graded filtration induced by $S$ satisfies the following two conditions:\n\\begin{enumerate}\n\\item\n\\[\nA_{X, D}(S)-\\frac{\\delta }{(L^{n})}\\int_0^{+\\infty}{\\rm vol}({\\mathcal{F}} S^{(x)})dx=0.\n\\]\n\\item Denote $\\tau:=\\sqrt[n]{\\frac{(L^{ n})}{{\\rm vol}({\\rm ord}_{S})}}$. We have:\n\\[\n{\\rm vol}\\left({\\mathcal{F}} S^{(x)}\\right)={\\rm vol}_Y(\\mu^*L-xS)=(L^{n})-{\\rm vol}({\\rm ord}_{S})x^n\n\\text{ for any } x\\in [0, \\tau].\n\\]\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}By Proposition \\ref{p-fanocone}, we know $(X,D+(1-\\beta)V_{\\infty})$ is log K-semistable. Then it follows from precisely the equality case of the Formula (25) and (29) in \\cite{LL16}. \n\\end{proof}\n\n\\begin{lem}We have $\\tau=\\frac{A_{X,D}(S)}{r}$.\n\\end{lem}\n\\begin{proof}\nCombining 1 and 2 in Proposition \\ref{p-equal}, we know that \n\\[\nA_{X,D}(S)-\\frac{r(1+n)}{n\\cdot L^n}\\int^{\\tau}_{0}\\big(L^n-{\\rm vol}({\\rm ord}_S)x^n\\big)dx= A_{X,D}(S)-r\\cdot \\tau= 0.\n\\]\n\\end{proof}\n\n\nBy arguing as in \\cite{Fuj15} (see also \\cite{Liu16}), we know that:\n\\begin{lem}\\label{lemamp}\nWe know $\\tau$ is the nef threshold, i.e. \n$$\\tau=\\sup\\left\\{x| \\ \\mu_{\\mathbb{A}^1}^*L-xS \\text{ is ample } \\right\\}.$$ \n\\end{lem}\n\\begin{proof} When the point is smooth, this follows from \\cite[Theorem 2.3(2)]{Fuj15}. Exactly the same argument can be used to treat the current case. \n\\end{proof}\n\\begin{thm}\\label{t-cone}\nIf $S$ is a Koll\\'{a}r component obtaining the minimum of the normalized volume, then $S$ is the canonical component $V_0$.\n\\end{thm}\n\nWe first show the following statements.\n\\begin{lem}\n\\begin{enumerate}\n\\item\n$\\rho^*L-\\tau \\bar{C}$ is base-point-free, and contracts $Y$ to $S_{\\infty}(\\cong S)\\subset \\bar{C}$ as the section at the infinite place. \n\\item\n$A_{X,D}(S)=r$ and there is a special test configuration degenerating $(V, \\Delta)$ into $(E, \\Delta_E)$.\nMoreover, there is a special test configuration of $(X, D+ (1-\\beta)V_\\infty; L)$\nto $(\\bar{C}, D_{\\bar{C}}+(1-\\beta) S_{\\infty}; {\\mathcal{L}}_0)$ which is indeed an isomorphism, where $D_{\\bar{C}}$ is the intersection of $ \\bar{C}$ with $(D\\times {\\mathbb{A}}^1)_W$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\n\nThe proof of this part is along the similar line in \\cite[Proof of Lemma 33]{Liu16}.\nFirst we observe the following restrictions of $\\rho^*L-x\\bar{C}$:\n\\begin{enumerate}\n\\item\n$\\left.\\rho^*L-x\\bar{C}\\right|_{X_t}=L$, $t\\neq 0$;\n\\item\n$\\left.\\rho^*L-x \\bar{C}\\right|_{Y_0}=\\mu^*L-x S$; \n\\item\n$\\left.\\rho^*L-x\\bar{C}\\right|_{\\bar{C}}=-x\\bar{C}|_{\\bar{C}}=x Y_0|_{\\bar{C}}=x S_{\\infty}=x\\mathcal{O}_{\\bar{C}}(1)$. \n\\end{enumerate}\nSo by Lemma \\ref{lemamp}, it is easy to see that $\\rho^*L-x\\bar{C}$ is ample when $x\\in (0, \\tau)$. \nTo show that $\\rho^*L-\\tau\\bar{C}$ is base-point-free, we calculate by using \\eqref{lclY2}:\n\\begin{eqnarray*}\nm(\\rho^*L-x \\bar{C})-K_{W}-(D_{{\\mathbb{A}}^1})_W&=&m(\\rho^*L-x\\bar{C})+(1+r) \\rho^*L-A_{X,D}(S)\\bar{C}\\\\\n&=&(m+1+r)\\left(\\rho^*L-\\frac{mx+A_{X,D}(S)}{m+1+r}\\bar{C}\\right).\n\\end{eqnarray*}\nNotice that:\n\\[\n\\frac{mx+A_{X,D}(S)}{m+1+r}<\\tau= \\frac{A_{X,D}(S)}{r}\n\\]\nif and only if \n\\[\nx< \\left(1+\\frac{1}{m}\\right)\\frac{A_{X,D}(S)}{r}.\n\\] \nBecause this is satisfied for \n$$x=\\tau=\\frac{A_{X,D}(S)}{r}\\qquad \\mbox{for any\\ } m>0,$$ the first statement holds by base-point-free theorem \\cite[Theorem 3.13]{KM98}. \nNext we claim that \n\\begin{equation}\\label{eqcontr1}\nH^0(Y, m(\\mu^*L-\\tau S))\\cong H^0(S, -m\\tau S)\n\\end{equation} \nfor any $m$ sufficiently divisible. To see this, we consider the exact sequence:\n\\begin{equation}\n0\\rightarrow \\mathcal{O}_{Y}(m(\\mu^*L-\\tau S)-S)\\rightarrow \\mathcal{O}_{Y}(m(\\mu^*L-\\tau S))\\rightarrow \\mathcal{O}_{Y}(m(\\mu^*L-\\tau S))\\otimes\\mathcal{O}_S\\rightarrow 0,\n\\end{equation}\nand its associated long exact sequence of cohomology groups.\nBy the above discussion, and\n\\[\nm(\\mu^*L-\\tau S)-S-K_Y=m(\\mu^*L-\\frac{A_{X,D}(S)}{r}S)+(1+r)\\mu^*L-A_{X,D}(S)S\n\\]\nis ample,  it follows from the Kawamata-Viehweg vanishing theorem that  \n\\[\nH^1\\big(Y, m(\\mu^*L-\\tau S)\\otimes \\mathcal{O}(-S)\\big)=0 \\text{ for any } m\\ge 0.\n\\]\nWe also have\n\\[\nH^0\\big(Y, m(\\mu^*L-\\tau S)\\otimes \\mathcal{O}(-S)\\big)=0 \\text{ for any } m\\ge 0.\n\\]\nas $\\tau$ is also the pseudo-effective threshold. \nThus we know $|m(\\rho^*L-\\tau \\bar{C})|$ contracts the fiber  $W\\times_{\\mathbb{A}^1} \\{0 \\}$ to $\\bar{C}$ for sufficiently divisible $m$. This finishes the proof of (1). We denote by $ \\theta\\colon W\\to \\mathcal{X}$ the induced morphism and there is an ample line bundle ${\\mathcal{L}}$ on ${\\mathcal{X}}$ such that $\\theta^*{\\mathcal{L}}=\\rho^*L-\\tau \\bar{C}$. \n\\bigskip\n\nNext we prove (2).  Let $(D_{\\mathbb{A}^1})_{{\\mathcal{X}}}$ be the push forward of $(D_{\\mathbb{A}^1})_{W}$ on ${\\mathcal{X}}$. Then $-K_{{\\mathcal{X}}}-(D_{\\mathbb{A}^1})_{{\\mathcal{X}}}$ and $(1+r) {\\mathcal{L}}$ coincide outside ${\\mathcal{X}}_0$, they must be linearly equivalent on the whole\n${\\mathcal{X}}$ because ${\\mathcal{X}}_0$ is irreducible. In particular, they are linearly equivalent when restricted to ${\\mathcal{X}}_0$.\n\nSince \n$$ (K_{Y}+\\mu^{-1}_*D+S)|_S=K_S+\\Delta_S\\sim_{\\mathbb{Q}}A_{X,D}(S) \\cdot S|_S,$$ \nwe know that\n$$-K_{{\\mathcal{X}}}-(D_{\\mathbb{A}^1})_{{\\mathcal{X}}}|_{{\\mathcal{X}}_0}=-K_{\\bar{C}}-D_{\\bar{C}}\\sim_{\\mathbb{Q}}(1+A_{X,D}(S))S_{\\infty}.$$\nSimilarly, we have\n${\\mathcal{L}}|_{{\\mathcal{X}}_0}\\sim_{{\\mathbb{Q}}}\\tau S$ with $\\tau=\\frac{A_{X,D}(S)}{r}$.\nTherefore,\n\\[\n1+A_{X,D}(S)=(1+r)\\frac{A_{X,D}(S)}{r},\n\\]\nwhich implies $A_{X,D}(S)=r$ and $\\tau=1$. \n\nThe degree of $V_{\\infty}$ under ${\\mathcal{L}}$ is\n\\begin{eqnarray*}\n{\\mathcal{L}}|_{{\\mathcal{X}}_0}^{n-1}\\cdot V_\\infty&=&L^{n-1}\\cdot V_\\infty\\\\\n&=&L^{ n},\n\\end{eqnarray*}\nwhile the degree of $S$ is\n\\begin{eqnarray*}\n{\\mathcal{L}}|_{{\\mathcal{X}}_0}^{ n-1}\\cdot S &=&\\tau^{-1}{\\mathcal{L}}_0^{ n}=L^n_t=L_0^n.\n\\end{eqnarray*}\n\nThe restriction $\\theta|_{V_{\\infty}} \\colon V_{\\infty}\\to S $ is finite since \n$$(\\rho^*L-\\tau \\bar{C})|_{V_{\\infty}}=L|_{V_{\\infty}}$$\nis ample. And the degree is 1 by the above calculation on degrees, which implies this is an isomorphism. We claim $Y$ is indeed the $\\mathbb{P}^1$-bundle over $V_{\\infty}$ induced by blowing up the vertex of $X$,  $S$ is a section and the morphism $\\theta$ is just contracting the $\\mathbb{P}^1$-bundle. \n Granted this for now, \nwe then indeed have an isomorphism from\n$(X, D+(1-\\beta)V_\\infty; L)$\nto $(\\bar{C}, D_{\\bar{C}}+(1-\\beta) S_{\\infty}; {\\mathcal{L}}_0)$. \n\nTo see the claim, let $l$ be a curve contracted by $\\theta$, we want to show that it is the birational transform of a ruling line of $X$. To see this, since $(\\rho^*(L)-\\bar{C})\\cdot l=0$, we know that $\\rho^*(L)\\cdot l=1$. So the image $\\rho_*l$ of $l$ in $X$ is a line, and it passes through the vertex. Therefore, it is a ruling of the cone. \n\n\n\\end{proof}\n\nBy the above proof, let ${\\mathcal{V}}$ be the birational transform of $(V_{\\infty})_{\\mathbb{A}^1}$ on ${\\mathcal{X}}$, and ${\\mathcal{H}}$ the extension of $H_{\\mathbb{A}^1}\\setminus\\{0\\}$ on ${\\mathcal{X}}$, we know that:\n\\[\n{\\mathcal{X}}={\\rm Proj}_{{\\mathcal{V}}}\\left( \\bigoplus_{k=0}^{+\\infty} {\\mathcal{S}}_k\\right),\n\\]\nwhere\n\\[\n{\\mathcal{S}}_k=\\bigoplus_{i=0}^{k} (H^0({\\mathcal{V}}, i {\\mathcal{H}})\\cdot u^{k-i}).\n\\]\nFrom this we easily see that \n$S$ and $V$ give the same component over the vertex. \n\n\n\n\\subsection{The general case}\n\nIn this section, we prove Theorem \\ref{t-main2} in the general case.\nWe first show that the cone case we prove in Section \\ref{ss-ucone} can be generalized to orbifold cone.  Let $T=\\mathbb{C}^*$.\n\n\n\\begin{prop}\\label{p-Tequiv}\nLet $x\\in (X,D)$ be a $T$-singularity. Assume a minimizer $v$ of ${\\widehat{\\rm vol}}_{X,x}$ is given by a rescaling of ${\\rm ord}_{F}$ for a Koll\\'ar component $F$, then $v$ is $T$-equivariant. \n\\end{prop}\n\\begin{proof}\n Let $\\mathfrak{a}$ be an ideal whose normalized blow up gives the model of extracting the Koll\\'ar component $F$ (see the proof of Proposition \\ref{p-equality}). Denote the degeneration of $\\{\\mathfrak{a}_{\\bullet}\\}:=\\{\\mathfrak{a}^p\\}$ by ${\\mathfrak{b}}:=\\{{\\bf in} (\\mathfrak{a}^p)\\}$ (which in general is not necessarily equal to but only contains $({\\bf in}(\\mathfrak{a}))^p$ ).  \n \n Since \n $${\\rm mult}(\\mathfrak{a})\\cdot {\\rm lct}^n(X,D; \\mathfrak{a})={\\widehat{\\rm vol}}({\\rm ord}_F)\\ge {\\rm mult}({\\mathfrak{b}}_{\\bullet})\\cdot {\\rm lct}^n(X,D; {\\mathfrak{b}}_{\\bullet}),$$\n  But ${\\rm mult}(\\mathfrak{a})={\\rm mult}({\\mathfrak{b}}_{\\bullet})$ and $ {\\rm lct}(X,D,\\mathfrak{a})\\ge  {\\rm lct}(X,D,{\\mathfrak{b}}_{\\bullet})$, we know that \n  $$ {\\rm lct}(X,D,\\mathfrak{a})= {\\rm lct}(X,D,{\\mathfrak{b}}_{\\bullet})=\\lim_{k\\to \\infty}{\\rm lct}(X,D, \\frac{1}{k}{\\mathfrak{b}}_k),$$\nwhich we denote by  $c$. \nIn particular, for we can choose $\\epsilon$ sufficiently small, and $k$ sufficiently large, such that the log discrepancy\n$$a_l(F, X,D+(c-\\epsilon)\\mathfrak{a})<1 $$\nand $({X},D+(c-\\epsilon)\\frac{1}{k}{\\mathfrak{b}}_k)$ is klt. \nThus  by \\cite{BCHM10} we can construct a model $Z\\to W=_{\\rm defn} X\\times \\mathbb{A}^1$ extracting only the irreducible divisor $ F_Z$ which gives $F$ over the generic fiber. Furthermore, we can assume $-F_W$ is ample over $W$ and we denote by $W_0=X\\times \\{0 \\}$. \n\n\n\nTherefore, $W$ and  $Y\\times \\mathbb{A}^1$ where $Y=W\\times_{\\mathbb{A}^1}\\{t\\}$ are isomorphic incodimension 1, with the exceptional divisors antiample over $X\\times \\mathbb{A}^1$. Thus we conclude that they are isomorphic.\n\\end{proof}\n\n\n\\begin{prop}\\label{p-cover2}\nUnder the notation  in Section \\ref{ss-deformation}, $S$ is the unique minimizer among all Koll\\'ar components for ${\\widehat{\\rm vol}}_{({C},C_{\\Delta})}$ if and only if the same holds for ${C}^{(d)}$ on\n${\\widehat{\\rm vol}}_{({C}^{(d)},C^{(d)}_{\\Delta}+C^{(d)}_D)}$.\n\\end{prop}\n\\begin{proof}By Proposition \\ref{p-Tequiv}, any minimizing Koll\\'ar component  $E$ of ${\\widehat{\\rm vol}}_{X,D}$ is $T$-equivalent, therefore it is $G=\\mathbb{Z}\/d$ equivalent. So $E$ is the pull back of a Koll\\'ar component on $C^{(d)}$ by Lemma \\ref{l-finite}, which can be only the canonical component obtained by blowing up the vertex by our assumption and Lemma \\ref{l-finitevolume}. \n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{t-main2}] We first notice that by Theorem \\ref{t-cone} and Proposition \\ref{p-cover2}, we know that for the coarse moduli space of an orbifold cone over a K-semistable log Fano pair, the only Koll\\'ar component which minimizes the normalized volume function is given by the canonical component. \n\nNow we consider the case of a general klt singularity $o\\in (X,D)$. Let us assume that there is another component $F$ such that \n$${\\widehat{\\rm vol}}_{X,o}({\\rm ord}_S)={\\widehat{\\rm vol}}_{X,o}({\\rm ord}_F),$$\nin particular, they take the minimal value of ${\\widehat{\\rm vol}}_{X,o}$ by Theorem \\ref{t-main}. \n\n Let $W\\to B$ be the family which degenerates $X$ to $X_0=Y_0\\cup \\bar{C}$, where $Y_0\\cong Y$ extracting $S$ over $X$ and $\\bar{C}$ is the the coarse moduli space of the orbifold cone  over $S=\\bar{C}\\cap Y_0$. Then we argue as in the proof Proposition \\ref{p-Tequiv}: Let $\\mathfrak{a}$ be an ideal whose normalized blow up gives the model of extracting the Koll\\'ar component $F$ (see the proof of Proposition \\ref{p-equality}). Denote the degeneration of $\\{\\mathfrak{a}_{\\bullet}\\}:=\\{\\mathfrak{a}^p\\}$ by ${\\mathfrak{b}}_{\\bullet}:=\\{{\\bf in} (\\mathfrak{a}^p)\\}$.  \n \n Since \n $${\\rm mult}(\\mathfrak{a})\\cdot {\\rm lct}^n(X,\\mathfrak{a})={\\widehat{\\rm vol}}_{X_0}({\\rm ord}_F)= {\\widehat{\\rm vol}}_{Y_0}({\\rm ord}_{S_0})\\le {\\rm mult}({\\mathfrak{b}}_{\\bullet})\\cdot {\\rm lct}^n(\\bar{C},{\\mathfrak{b}}_{\\bullet}),$$\n where the last inequality is from the assumption that $S_0$ is K-semistable and Theorem \\ref{t-main}. \n But ${\\rm mult}(\\mathfrak{a})={\\rm mult}({\\mathfrak{b}}_{\\bullet})$ and $ {\\rm lct}(X,\\mathfrak{a})\\ge  {\\rm lct}(\\bar{C},{\\mathfrak{b}}_{\\bullet})$, we know that \n  $$ {\\rm lct}(X,\\mathfrak{a})= {\\rm lct}(\\bar{C},{\\mathfrak{b}}_{\\bullet})=\\lim_{k\\to \\infty}{\\rm lct}(\\bar{C}, \\frac{1}{k}{\\mathfrak{b}}_k),$$\nwhich we denote by  $c$. \nIn particular, for we can choose $\\epsilon$ sufficiently small, and $k$ sufficiently large, such that the log discrepancy\n$$a_l(F, X,D+(c-\\epsilon)\\mathfrak{a})<1 $$\nand $({C},C_D+(c-\\epsilon)\\frac{1}{k}{\\mathfrak{b}}_k)$ is klt. \nThus  by \\cite{BCHM10} we can construct a model $Z\\to W$ extracting only the irreducible divisor $ F_Z$ which gives $F$ over the generic fiber. Furthermore, we can assume $-F_W$ is ample over $W$ and after a base change, we can assume $F_W|_{X_0}$ is reduced. \n \nWe claim that $Z_0\\to W_0$ also only extracts a Koll\\'ar component. In fact, locally over the vertex $v$ of $\\bar{C}$, since \n$$(-F_W|_{X})^n=(-F_W|_{X_0})^n,$$\nthen by Lemma \\ref{l-inter2},\n$${\\widehat{\\rm vol}}_X({\\rm ord}_F)={\\widehat{\\rm vol}}(Z_0\/W_0)= {\\widehat{\\rm vol}}_{Y_0}({\\rm ord}_{S_0}).$$\nThus we conclude that the volume of the model $Z_0$ is equal to ${\\widehat{\\rm vol}}_{X,o}(F)$, which is equal to the minimum of the normalized volume ${\\widehat{\\rm vol}}_{\\bar{C},v}$. It follows from the argument in the proof of Theorem \\ref{t-divisor} that over $v$, $Z_0\\to W_0$ yields a Koll\\'ar component $F_0$.  By the proof in the cone case Theorem \\ref{t-cone}, $F_0$ has to be the same as the canonical component $S_0$. In particular, this implies that the birational transform ${\\mathbb{P}}$ of $\\bar{C}$ in $Z$ is the extract of the canonical component. \n \n\nThus there is a morphism $\\mathbb{P}\\to S$. Let  $l$ be the fiber class of $\\mathbb{P}\\to S$. Consider $K_{Z}+\\mu_*^{-1}(D_{\\mathbb{A}^1})+F_Z$, which satisfies that \n$$\\big(K_{Z}+\\mu_*^{-1}(D_{\\mathbb{A}^1})+F_Z\\big)|_{F_0}=K_{F_0}+\\Delta_{F_0}$$\nis anti-ample, \n$$ \\big(K_{Z}+\\mu_*^{-1}(D_{\\mathbb{A}^1})+F_Z\\big)|_S=K_{S}+\\Delta_S$$\nis anti-ample, and  \n$$\\big(K_{Z}+\\mu_*^{-1}(D_{\\mathbb{A}^1})+F_Z\\big)\\cdot l=0.$$\nThus $l$ is an extremal ray in $N_1(Z\/X_{\\mathbb{A}^1})$. \n\nHence we know that there is a morphism $Z\\to W'$ which contracts $\\mathbb{P}$ and $W'$ admits a morphism $\\chi\\colon W'\\to X_{\\mathbb{A}^1}$. Restricting over $0$, the central fiber is the birational model $\\mu\\colon Y\\to X$ which extracts $S$. On the other hand, let $\\mu_F\\colon Y_F\\to X$ be the birational model which extracts the Koll\\'ar component $F$. \n\nAs $Y_F\\times \\mathbb{A}^1$ and $W'$ is isomorphic in codimension 1, if we denote by $F_{W'}$ the push forward of $F_Z$ on $W'$, we have\n\\begin{eqnarray*}\nW' &=&{\\rm Proj}\\bigoplus_m{\\mu_1}_* \\mathcal{O}_{W'}\\big(-m F_{W'}\\big)\\\\\n& =& {\\rm Proj}\\bigoplus_m{\\mu_2}_* \\mathcal{O}_{Y_F\\times {\\mathbb{A}}^1}\\big(-m(F\\times {\\mathbb{A}}^1)\\big)\\\\\n&=&Y_F\\times {\\mathbb{A}}^1.\n\\end{eqnarray*}\nConsider the central fiber over $0$, this implies that $Y_F=Y$. \n\\end{proof}\n\n\n\\section{Minimizing Koll\\'{a}r component is K-semistable}\\label{s-Ksta}\n\nIn this section, we aim to prove the a Koll\\'ar component is minimizing only if it is K-semistable. The method used in the proof of this result is motivated by a method used in the study of toric degenerations (see \\cite[Section 3.2]{Cal02}, \\cite[Proposition 2.2]{AB04} and \\cite[Proposition 3]{And13}). In particular this method allows us to reduce two-step degenerations to a one-step degeneration.\n\n\\begin{proof}[Proof of Theorem \\ref{t-mintok}]\nBy Proposition  \\ref{thm-Ksemi}\nwe know that the canonical valuation of $({C}^{(d)}, C^{(d)}_{\\Delta}+C^{(d)}_D)$ minimizes the normalized local volume if and only if $(S,\\Delta_S)$ is K-semistable. Thus by Proposition \\ref{p-cover}, to show that $(S,\\Delta_S)$ is K-semistable, it suffices to show that the canonical component is a minimizer of ${\\widehat{\\rm vol}}_{X_0,D_0}$ for $ (X_0,D_0)=_{\\rm defn} (C,C_D)$, which is the special degeneration  associated to $S$.\n\nLet $(X, D)$ be a $\\mathbb{Q}$-Gorenstein klt singularity with $X={\\rm Spec}_{\\mathbb{C}} (R)$. Assume that $S$ is a Koll\\'{a}r component that minimizes ${\\widehat{\\rm vol}}_{(X, D; o)}$ and appears as the exceptional divisor in a plt blow-up $X'\\rightarrow X$. Let $\\Delta_S$ be the divisor on $S$ satisfying $K_{X'}+S|_S=K_S+\\Delta_S$. \nBy Theorem \\ref{t-approx} (and Lemma \\ref{l-Tmini}), to show that $(S, \\Delta_S)$ is K-semistable, it suffices to show that  \n$${\\widehat{\\rm vol}}_{X_0,D_0}({\\rm ord}_S) \\le {\\widehat{\\rm vol}}_{X_0,D_0}({\\rm ord}_{F})$$\n for any $\\mathbb{C}^*$-invariant Koll\\'ar component $F$ over $o'\\in (X_0,D_0)$. \n\n Let $({\\mathcal{Y}}, {\\mathcal{E}})$ be the associated special degeneration which degenerates $(X_0, D_0)$ to a pair $(Y_0, E_0)$ where $Y_0$ is an orbifold cone over $(F, \\Delta_F)$.\n Then we have a $ \\mathbb{Z}_{\\ge 0}\\times \\mathbb{Z}_{\\ge 0}$-valued function on $R$:\n\\begin{eqnarray}\\label{eq-Z2val}\nw: R& \\longrightarrow &  \\mathbb{Z}_{\\ge 0} \\times \\mathbb{Z}_{\\ge 0}\\\\\nf & \\mapsto &\\left( {\\rm ord}_S(f), {\\rm ord}_F({\\bf in}(f)) \\right).\\nonumber\n\\end{eqnarray}\nWe give $  \\mathbb{Z}_{\\ge 0}\\times \\mathbb{Z}_{\\ge 0}$ the following lexicographic order: $(m_1, u_1)<(m_2, u_2)$ if and only if $m_1<m_2$, or $m_1=m_2$ and $u_1<u_2$. If we denote by $\\Gamma$ the valuative monoid and denote the associated graded ring by\n\\[\n{\\rm gr}_w R=\\bigoplus_{(m, u)\\in \\Gamma} R_{\\ge (m, u)}\/ R_{> (m, u)},\n\\]\nthen it's easy to see that ${\\mathcal{Y}}_0={\\rm Spec}_{\\mathbb{C}} \\left({\\rm gr}_w R\\right)$. \nWe will also denote:\n\\[\nA=\\bigoplus_{m\\in \\mathbb{N}} R_{\\ge m}\/R_{>m}=\\bigoplus_{m\\in \\mathbb{N}} A_m.\n\\]\nThen ${\\rm Spec}_{\\mathbb{C}} (A)={\\mathcal{X}}_0=Y$. Moreover if we define the extended Rees ring of $A$ with respect to the filtration associated to ${\\rm ord}_{F}$:\n\\[\n\\mathcal{A}':=\\bigoplus_{k\\in \\mathbb{Z}} {\\mathcal{A}}_k:=\\bigoplus_{k\\in \\mathbb{Z}} \\mathfrak{b}_k t^{-k} \\subset A[t, t^{-1}],\n\\]\nwhere $\\mathfrak{b}_k=\\{f\\in A; {\\rm ord}_{F}(f)\\ge k\\}$. Then the flat family ${\\mathcal{Y}}\\rightarrow \\mathbb{C}$ is given by the ${\\rm Spec}_{\\mathbb{C}[t]}\\left(\\mathcal{A}'\\right)$. In particular, we have\n\\[\n{\\mathcal{A}}' \\otimes_{\\mathbb{C}[t]}\\mathbb{C}[t, t^{-1}]\\cong A[t, t^{-1}], \\quad {\\mathcal{A}}' \\otimes_{\\mathbb{C}[t]}\\mathbb{C}[t]\/(t)\\cong {\\rm gr}_w R.\n\\]\n\n\\bigskip\n\nPick up a set of homogeneous generators $\\bar{f}_1, \\dots, \\bar{f}_p$ for ${\\rm gr}_w R$ with $\\deg(\\bar{f}_i)=(m_i, u_i)$. Lift them to generators $f_1, \\dots, f_p$ for $A$ such that $f_i\\in A_{m_i}$. \nSet $P=\\mathbb{C}[x_1, \\dots, x_p]$ and give $P$ the grading by $\\deg(x_i)=(m_i, u_i)$ so that the surjective map \n$$P\\rightarrow {\\rm gr}_w R \\qquad\\mbox{ given by }\\qquad x_i\\mapsto f_i$$ is a map of graded rings. \nLet $\\bar{g}_1, \\dots, \\bar{g}_q\\in P$ be a set of homogeneous generators of the kernel and assume $\\deg(\\bar{g}_j)=(n_j, v_j)$. \n\nSince $\\bar{g}_j(\\bar{f}_1, \\dots, \\bar{f}_p)=0 \\in {\\rm gr}_wR$, it follows \n$$\\bar{g}_j(f_1, \\dots, f_p) \\in (A_{n_j})_{>v_j} \\qquad \\mbox{ for each } j.$$ \nBy\nthe flatness of ${\\mathcal{A}}$ over $\\mathbb{C}[t]$, there exist liftings $g_j\\in \\bar{g}_j+(P_{n_j})_{> v_j}$ of the relation $\\bar{g}_j$ such that \n$$g_j(f_1, \\dots, f_p)=0  \\mbox{ for } 1\\le j\\le q .$$ \nSo $g_j{}'s$ form a Gr\\\"{o}bner basis of $J$ with respect to the order function ${\\rm ord}_V$, where $J$ is the kernel of the \nsurjection $P\\rightarrow A$. In other words, if we let $K=(\\bar{g}_1, \\dots, \\bar{g}_q)$ denote the kernel $P\\rightarrow {\\mathcal{A}}_0$, then $K$ is the initial ideal of $J$ with respect to the order determined by \n${\\rm ord}_F$. As a consequence, we have:\n\\[\n{\\mathcal{A}}=P[\\tau]\/(\\tilde{g}_1, \\dots, \\tilde{g}_q),\n\\]\nwhere $\\tilde{g}_j=\\tau^{v_j} g_j(\\tau^{-u_1} x_1, \\dots, \\tau^{-u_p} x_p)$.\n\n\\bigskip\n\nNow we lift $f_1, \\dots, f_p$ further to generators $F_1, \\dots, F_p$ of $R$. Then we have: \n$$g_j(F_1, \\dots, F_p)\\in R_{> m_j}.$$ By the flatness of ${\\mathcal{R}}$ over $\\mathbb{C}[t]$, there exist $G_j\\in g_j+P_{> n_j}$ such that \n$$G_j(F_1, \\dots, F_p)=0.$$ Let $I$ be the kernel of $P\\rightarrow R$. Then $G_j{}'s$ form a Gr\\\"{o}bner basis with respect to the order function $w$ in \\eqref{eq-Z2val} and the associated initial ideal is $I$. As a consequence, we have:\n\\[\n{\\mathcal{R}}=P[\\zeta]\/(\\tilde{G}_1, \\dots, \\tilde{G}_q)\n\\]\nwhere $\\tilde{G}_j=\\zeta^{n_j} G_j(\\zeta^{-m_1} x_1, \\dots, \\zeta^{-m_p} x_p)$. \n\nIn summary we have $(\\mathbb{C}^*)^2$ action on $\\mathbb{C}^p$ generated by two 1-parameter subgroups $\\lambda_0(t)=t^{\\bold m}$ and $\\lambda'(t)=t^{\\bold u}$. $\\lambda_0$ degenerates $(X, D)$ to $(X_0, D_0)$ and $\\lambda'$ degenerates $(X_0, D_0)$ further to $(Y_0, E_0)$. \n\n\\bigskip\n\nWe now claim that, for $0<\\epsilon\\ll 1$, there is a family of one parameter subgroups $\\lambda_\\epsilon(t)\\colon \\mathbb{C}^*\\to (\\mathbb{C}^*)^{2}$ such that the following conditions are satisfied:\n\\begin{enumerate}\n\\item $\\lambda_\\epsilon(t)$ degenerates $X$ to $Y_0$. For this to happen, we need to make sure that the initial term of $G_j$ with respect to the weight function $\\pi_\\epsilon$ defined by $\\lambda_\\epsilon (t)$ is exactly $\\bar{g}_j$. For the latter condition to hold it suffices: \n$$\\pi_\\epsilon (n'_j, v'_j)> \\pi_\\epsilon (n_j, v_j) \\mbox{\\ where \\ } (n'_j, v'_j)=\\deg(G_j-\\bar{g}_j).$$ \n\n\\item \nAs $\\epsilon\\rightarrow 0$, $\\lambda_\\epsilon\\rightarrow \\lambda_0$ in the sense that $\\lambda_\\epsilon(t)\\rightarrow \\lambda_0(t)$ for any $t\\in \\mathbb{C}^*$.\n\\item\nFor $0<\\epsilon\\ll 1\\in {\\mathbb{Q}}$, $S_\\epsilon=Y_0\/\\lambda_\\epsilon(t)$ is a Koll\\'{a}r component over $(X,o)$. Moreover as $\\epsilon\\rightarrow0$, ${\\rm ord}_{T_\\epsilon}\\rightarrow {\\rm ord}_{S}$.\n\\end{enumerate}\n\nDenote by $B\\subset \\mathbb{Z}\\times \\mathbb{Z}$ be the finite set consisting of the differences $(n'_j, v'_j)-(n_j, v_j)$, together with $0$ and the two generators of $\\mathbb{N}\\times\\mathbb{N}$. Let $M$ be a positive integer that is larger than all coordinates of $(m, u)-(n, v)$ for all pairs of elements $(m, u), (n, v)\\in B$ and let $\\epsilon$ be sufficiently small such that $1> M \\epsilon$. \nDefine\n$$\\pi_\\epsilon=e^*_0-\\epsilon e^*_{1}.$$ \nThen for $\\epsilon$ sufficiently small, $\\pi_\\epsilon$ is a linear projection $\\pi: \\mathbb{Z}\\times \\mathbb{Z}\\rightarrow \\mathbb{Z}$ such that $0<\\pi(n_j, v_j)<\\pi(n'_j, v'_j)$. \n\nSo by choosing $M\\gg 1$ so that $\\epsilon\\ll 1$ we see that indeed there is a family of linear projections $\\pi_\\epsilon$ approaches $\\pi_0$ which corresponds to ${\\rm ord}_S$. We can define $\\lambda_\\epsilon(t)$ to be one parameter subgroup corresponding to $\\pi_\\epsilon$.\n\nLet $\\lambda_\\epsilon=\\lambda_\\epsilon(t)$ be the $\\mathbb{C}^*$-action induced by a choice of rational $\\epsilon$.  Now we claim that $Y_0\/(\\lambda_\\epsilon(t))$ yields a Koll\\'{a}r component $S_\\epsilon$ over both $(X, o)$ and $(X_0, D_0)$. Moreover the associated special degeneration has the central fibre equal to $(Z, \\lambda_\\epsilon)$. \nBecause ${\\rm ord}_{S}$ is a minimizer on ${\\widehat{\\rm vol}}_{X,D}$, we have:\n\\[\n{\\widehat{\\rm vol}}_{X,D}({\\rm ord}_{S_\\epsilon})\\ge {\\widehat{\\rm vol}}_{X,D}({\\rm ord}_{S}).\n\\]\nOn the other hand, we have \n\\[\n{\\widehat{\\rm vol}}_{X,D}({\\rm ord}_{S_\\epsilon})={\\widehat{\\rm vol}}_{X_0,D_0}({\\rm ord}_{S_\\epsilon})={\\widehat{\\rm vol}}_{Y_0,E_0}({\\rm ord}_{S_\\epsilon}).\n\\]\nIt's known that ${\\widehat{\\rm vol}}_{Y_0, E_0}$ is a convex function with respect to $\\epsilon$. So we conclude\n${\\widehat{\\rm vol}}_{Y_0, E_0}({\\rm ord}_F)\\ge {\\widehat{\\rm vol}}_{Y_0, E_0}({\\rm ord}_{T_0})$. As a consequence we have:\n\\[\n{\\widehat{\\rm vol}}_{X_0, D_0}({\\rm ord}_F)={\\widehat{\\rm vol}}_{Y_0, E_0}({\\rm ord}_F)\\ge {\\widehat{\\rm vol}}_{Y_0, E_0}({\\rm ord}_{T_0})={\\widehat{\\rm vol}}_{X_0, D_0}({\\rm ord}_S).\n\\]\n\n\n\nTo see the construction of $S_\\epsilon$,\nwe define a filtration:\n\\begin{eqnarray*}\n{\\mathcal{F}}^N R&=&{\\rm Span}_{\\mathbb{C}}\\left\\{f_1^{a_1}\\dots f_p^{a_p}; \\pi_\\epsilon  \\left( \\sum_{i=1}^p a_i (m_i, u_i) \\right) \\ge N\\right\\}\\\\\n&=& \\{ f; \\text{ there exists } F\\in P \\text{ such that } F|_X=f \\text{ and } \\lambda_\\epsilon (F)\\ge N  \\}.\n\\end{eqnarray*}\nThen $\\{{\\mathcal{F}}^N R\\}$ is the weighted filtration induced by the weighted blow up $\\widehat{\\mathbb{C}}^p\\rightarrow \\mathbb{C}^p$ and the associated graded ring of the above filtration is isomorphic to ${\\rm gr}_w R$ with the grading given by the weight function $\\lambda_\\epsilon$. Denote the strict transform of $X$ by $\\hat{X}$. Then the exceptional divisor $\\hat{X}\\rightarrow X$ is isomorphic to $S_\\epsilon=Y_0\/\\lambda_\\epsilon$ by the discussion in Section \\ref{sec-filtration}.  By Proposition \\ref{prop-Kolseq}, $(S_\\epsilon, \\Delta_{\\epsilon})=(Y_0, E_0) \/ \\lambda_\\epsilon$ is a klt log-Fano-variety and a Koll\\'{a}r component over $(X, D; o)$.  \n\\end{proof}\n\n\\begin{prop}\\label{prop-Kolseq}\nFor any $0<\\epsilon\\ll 1$ with $\\epsilon\\in \\mathbb{Q}_{+}$, let $(S_\\epsilon, \\Delta_\\epsilon)=Y_0\/\\lambda_\\epsilon$. Then $S_\\epsilon$ is a Koll\\'{a}r component over  $o\\in (X,D)$.\n\\end{prop}\n\\begin{proof}\nFor $0<\\epsilon\\ll 1$ with $\\epsilon\\in \\mathbb{Q}_{+}$, $\\xi_\\epsilon$ generates a $\\mathbb{C}^*$-action. We have a log orbifold $\\mathbb{C}^*$-bundle\n$\\pi: (Y_0^{\\circ}, E_0^{\\circ}):=(Y_0\\setminus \\{v\\},  E_0\\setminus \\{v\\}) \\rightarrow (S_\\epsilon, \\Delta_\\epsilon)$ where $v$ is the vertex of $Y_0$. The Chern class of this orbifold $\\mathbb{C}^*$-bundle, denoted by $c_1(Y_0^{\\circ} \/S_\\epsilon)$, is contained in ${\\rm Pic}(S_\\epsilon)$ and is ample. $Y_0^\\circ$ can be compactified by adding the zero section $S_\\epsilon$ so that we get a birational morphism $\\mu: Y_\\epsilon\\rightarrow Y_0$ with the exceptional divisor isomorphic to  be $S_\\epsilon$. \n\nBecause $Y_0$ has klt singularity, by \\cite[40-42]{Kol04} (see also \\cite{BG08}) we know that $c_1(Y_0^{\\circ}\/S_\\epsilon)=-r^{-1} (K_{S_\\epsilon}+\\Delta_\\epsilon)$ and $(S_\\epsilon, \\Delta_\\epsilon)$ has klt singularities. \nSo $(S_\\epsilon, \\Delta_\\epsilon)$ is a Koll\\'{a}r component over $v\\in (Y_0,  E_0)$. \n\nTo transfer this to $(X, o)$, we notice that the graded ring of $w_\\epsilon$ is isomorphic to ${\\rm gr}_{{\\rm wt}_\\epsilon} \\mathbb{C}[Y_0]$. The exceptional divisor of the filtered blow-up associated to $w_\\epsilon$ is isomorphic to ${\\rm Proj}({\\rm gr}_{{\\rm wt}_\\epsilon} \\mathbb{C}[Y_0])$ which is isomorphic to $(S_\\epsilon, \\Delta_\\epsilon)$. Since $(S_\\epsilon, \\Delta_\\epsilon)$ is klt, by inversion of adjunction we know that the filtered blow up is indeed a plt blow up and hence $S_\\epsilon$ is a Koll\\'{a}r component over $(X, o)$. \n\\end{proof}\n\n\n\n\n\\section{Examples}\\label{s-exam}\n\nIn this section, we find out the minimizer for some examples of klt singularities $(X,o)=(\\mathrm{Spec} R,\\mathfrak{m})$. We note that by Proposition \\ref{p-inf} and \\ref{p-equality}, this also explicit calculates \n$$\\inf_{\\mathfrak{a}} {\\rm lct}(X,\\mathfrak{a})^n\\cdot {\\rm mult}(\\mathfrak{a})$$ for all $\\mathfrak{m}$-primary ideals $\\mathfrak{a}$ and gives the equality condition, which generalizes the results in \\cite{dFEM04} on a smooth point. \n\\begin{exmp}\\label{e-example}\nWe explicitly compute the minimizer for quotient, $A_k$, $E_k$ and weakly exceptional singularities in the below. \n\\end{exmp}\n\\begin{enumerate}\n\\item (cf. \\cite[Example 4.9]{LL16}) Let $(X,o)=(\\mathbb{C}^n,0)\/G$ be an $n$-dimensional quotient singularity. Let $E\\cong \\mathbb{P}^{n-1}$ be the exceptional divisor over $\\mathbb{C}^n$ obtained by blowing up $0$. Then denote by $S$ the valuation over $(X,o)$ which is the quotient of $E$ by $G$. Applying Lemma \\ref{l-finite} to the pull back of Koll\\'ar components on $X$, we know that \n$${\\widehat{\\rm vol}}_{X,o}({\\rm ord}_S)\\le {\\widehat{\\rm vol}}_{{\\rm ord}_E}$$ for any Koll\\'ar component $E$ over $(X,o)$. So ${\\widehat{\\rm vol}}_{X,o}$ minimizes at ${\\rm ord}_S$ with \n$${\\widehat{\\rm vol}}_{X,o}({\\rm ord}_S)=\\frac{n^n}{|G|}.$$ \n\n\\item\nConsider the $n$-dimensional $A_{k-1}$ singularity:\n$$\nX=A^{n}_{k-1}:=\\{z_1^2+\\cdots+z_n^2+z_{n+1}^k=0\\}.\n$$\nWe consider cases when $k>\\frac{2(n-1)}{n-2}$ (for other cases, see \\cite[Example 4.7]{LL16}). We want to show that the valuation corresponding to the weight\n$w_*=(n-1, \\cdots, n-1, n-2)$ is a minimizer among all valuations in ${\\widehat{\\rm vol}}_{X,o}$. In \\cite[Example 2.8]{Li15a}, these are computed out as the minimizer among all valuations obtained by weighted blow ups on the ambient space $\\mathbb{C}^{n+1}$. \n\nWe notice that under the weighted blow up corresponding to $w_*$, we have a \nbirational morphism $Y\\rightarrow X$ with exceptional divisor $S$ isomorphic to the weighted hypersurface\n$$\nS:=\\{Z_1^2+\\cdots+Z_n^2=0\\}\\subset {\\mathbb{P}}(n-1, \\cdots, n-1, n-2)=:{\\mathbb{P}}_{w_*}.\n$$\nBecause ${\\mathbb{P}}_{w_*}\\cong {\\mathbb{P}}(1,\\cdots, 1, n-2)$, it is easy to see that $S$ is isomorphic to $\\bar{C}(Q, -K_Q)$ where \n$Q=Q^{n-2}=\\{Z_1^2+\\cdots+Z_n^2=0\\}\\subset{\\mathbb{P}}^{n-1}$ (notice that $K_{Q}^{-1}=(n-2)H$). On the other hand, because ${\\mathbb{P}}_{w_*}$ is not well-formed, we have\ncodimensional 1 orbifold locus along the infinity divisor $Q_\\infty\\subset S$ with the isotropy group $\\mathbb{Z}\/(n-1)\\mathbb{Z}$. So the corresponding Koll\\'{a}r component\nis the log Fano pair $\\left(S, (1-\\frac{1}{n-1})Q_\\infty\\right)$. Because $Q_\\infty$ has KE, by \\cite{LL16} there is a conical KE on the pair $\\left(S, (1-\\frac{1}{n-1})Q_\\infty\\right)$.\nSo by Theorem \\ref{t-main} and Theorem \\ref{t-main2}, ${\\rm ord}_S$ is indeed a global minimizer of ${\\widehat{\\rm vol}}$ that is the unique minimizer among all Koll\\'{a}r components.\nNotice that for any higher order klt perturbation of these singularities, $w_*$ is also a minimizer.\n\n\\item \nWe can also use Theorem \\ref{t-main} to verify that the valuations in \\cite[Example 2.8]{Li15a} for $E_k$ (k=6,7,8) are indeed minimizers in ${\\widehat{\\rm vol}}_{X,o}$, which are unique among Koll\\'{a}r components. To avoid repetition, we will only do this for $E_7$ \nsingularities. The argument for other two cases are similar. So consider the $(n+1)$-dimensional $E_7$ singularity:\n\\[\nX^{n+1}=\\{z_1^2+z_2^2+\\cdots+z_n^2+z_{n+1}^3z_{n+2}+z^3_{n+2}=0\\}\\subset\\mathbb{C}^{n+2}.\n\\]\n\\begin{enumerate}\n\\item If $n+1=2$, then $X^2$ is a quotient singularity $\\mathbb{C}^2\/E_7$ and so we get the unique polystable component by \\cite[Example 4.9]{LL16} and example 1 above.\n\n\\item If $n+1=3$, then $X^3=\\{z_1^2+z_2^2+z_3^3z_4+z_4^3=0\\}\\subset \\mathbb{C}^4\\cong \\{w_1 w_2+w_3^3w_4+w_4^3=0\\}\\subset \\mathbb{C}^4$ by the change of variables. This singularity has a \n$(\\mathbb{C}^*)^2$-action and is an example of $T$-variety of complexity one. By the recent work in \\cite[Theorem 7.1 (II)]{CoSz16}, $X^3$ indeed has a Ricci flat cone K\\\"{a}hler metric \nassociated to the canonical $\\mathbb{C}^*$-action associated to $w_*$. So by \\cite[Theorem 1.7]{LL16}, the unique K-polystable Koll\\'{a}r component is given by the orbifold $X^3\/\\langle w_*\\rangle$.\n\n\\item $n+1=4$, then under the weighted blow up corresponding to $w_*=(9,9,9,5,6)$, we have a birational morphism \n$\\hat{X}\\rightarrow X$ with exceptional divisor $E$ isomorphic to the weighted hypersurface\n$$\nE=\\{z_1^2+z_2^2+z_3^2+z_5^3=0\\}\\subset {\\mathbb{P}}(9, 9, 9, 5, 6)={\\mathbb{P}}(w_*).\n$$\nSince ${\\mathbb{P}}(w_*)$ is not well-formed, we have:\n$$\nE\\cong \\{z_1^2+z_2^2+z_3^2+z_5^3=0\\}\\subset {\\mathbb{P}}(3, 3, 3, 5, 2)={\\mathbb{P}}'. \n$$\nwith orbifold locus of isotropy group $\\mathbb{Z}\/3\\mathbb{Z}$ along $$\nV=\\{z_1^2+z_2^2+z_3^2+z_5^3=0\\}\\subset {\\mathbb{P}}(3,3,3,2).\n$$\n\n\nAlternatively, $E$ is a weighted projective cone over the weighted hypersurface. It is easy to see that as an orbifold $(V, \\Delta) \\cong \\left({\\mathbb{P}}^2, (1-\\frac{1}{3})Q\\right)$ where $Q=\\{z_1^2+z_2^2+z_3^2=0\\}\\subset {\\mathbb{P}}^2$.\nBy \\cite{LS14}, there exists an orbifold K\\\"{a}hler-Einstein metric on $(V, \\Delta)$.\nNotice that $-(K_V+\\Delta)=3L-\\frac{4}{3}L=\\frac{5}{3}L$ where $L$ is the hyperplane bundle of ${\\mathbb{P}}^2$. Denoting by $H$ the hyperplane bundle of ${\\mathbb{P}}'$, then\n$H|_V=L\/3$. If $V$ is considered as a divisor of $E$, then \n$$V|_V=\\big(\\{z_4=0\\}\\cap E\\big)=5H|_V=\\frac{5}{3}L.$$ So $-(K_V+\\Delta)= V|_V$. \n\nThen by \\cite[Theorem 1.7]{LL16}, there exists an orbifold K\\\"{a}hler-Einstein metric on $E$ because the cone angle at infinity is $\\beta=1\/3$. Thus the unique log-K-semistable (actually log-K-polystable) Koll\\'{a}r component is given by the pair $\\left(E, \\left(1-\\frac{1}{3}\\right)V\\right)$.\n\n\\item\n$n+1=5$, under the weighted blow up corresponding to $w_*=(3, 3, 3, 3, 2, 2)$, we have a birational morphism $\\hat{X}\\rightarrow X$ with exceptional divisor $E$ isomorphic to the \nweighted hypersurface:\n\\[\nE=\\{z_1^2+z_2^2+z_3^2+z_4^2+z_6^3=0\\}\\subset {\\mathbb{P}}(3, 3, 3, 3, 2, 2)=:{\\mathbb{P}}(w_*).\n\\]\nThis is a weighted projective cone over the weighted hypersurface:\n\\[\nV=\\{z_1^2+z_2^2+z_3^2+z_4^2+z_6^3=0\\}\\subset {\\mathbb{P}}(3, 3, 3, 3, 2).\n\\]\nAs orbifold, we have $(V, \\Delta)=\\left({\\mathbb{P}}^3, (1-\\frac{1}{3})Q\\right)$. By \\cite{LS14, Li13}, $(V, \\Delta)$ is log-K-semistable and degenerates to a conical K\\\"{a}hler-Einstein pair. So by \\cite{LL16}, we know \nthat $(E, (1-\\beta)V_\\infty)$ is log-K-semistable. To determine $\\beta$, we notice that \n$$-(K_V+\\Delta)=4L-\\frac{4}{3}L=\\frac{8}{3}L=4\\cdot \\frac{2}{3}L=4\\cdot V_\\infty|_V.$$ \nSo $\\beta=1$ and we conclude that the unique (strictly) K-semistable Koll\\'{a}r component is indeed the ${\\mathbb{Q}}$-Fano variety $E$.\n\n\\item $n+1\\ge 6$. Under the weighted blow up corresponding to $w_*=(n-1, \\dots, n-1, n-2, n-2)$, we have a birational morphism $\\hat{X}\\rightarrow X$ with exceptional divisor $E$ isomorphic to\nthe weighted hypersurface:\n\\[\nE=\\{z_1^2+\\cdots+z_n^2=0\\}\\subset {\\mathbb{P}}(n-1, \\cdots, n-1, n-2, n-2)=:{\\mathbb{P}}(w_*).\n\\]\nThis is the weighted projective cone over \n\\[\nV=\\{z_1^2+\\cdots+z_n^2=0\\}\\subset {\\mathbb{P}}(n-1, \\cdots, n-1, n-2).\n\\]\nBy the discussion in the above $A^n_{k-1}$ singularity case, we know that as an orbifold,\n$(V, \\Delta)=\\left(\\bar{C}(Q, -K_Q), (1-\\frac{1}{n-1})Q_\\infty\\right)$, which has an orbifold K\\\"{a}hler-Einstein metric. Notice that\n\\[\n-(K_V+\\Delta)=(n(n-1)+n-2)H|_V-2(n-1)H|_V=n (n-2)H|_V.\n\\]\nBy \\cite[Theorem 1.7]{LL16}, the ${\\mathbb{Q}}$-Fano variety $E$ indeed has an orbifold K\\\"{a}hler-Einstein metric ($\\beta=n\/n=1$ at infinity) and hence by Theorem \\ref{t-main} is the unique K-semistable (actually K-polystable) Koll\\'{a}r component.\n\n\\end{enumerate}\n\nWe remark that, however, in the case of $D_{k+1}$ singularities, since the valuations computed out in \\cite[Example 2.8]{Li15a} could be irrational, our method can not directly tell whether it is a minimizer in ${\\rm Val}_{X,o}$. \n\n\\item A notion called weakly-exceptional singularity is introduced in \\cite{Pro00}. As the name suggested, this is a weaker notion than the exceptional singularity introduced in \\cite{Sho00}, which forms a special class of singularities in the theory of local complements. In our language, a singularity $(X,o)$ is {\\it weakly-exceptional} if and only if it has a unique Koll\\'ar component $S$. We know that if a singularity is weakly-exceptional, then the log $\\alpha$-invariant for the log Fano $(S,\\Delta_S)$ is at least 1 (see \\cite[Theorem 4.3]{Pro00} and \\cite{CS14}). In particular, we know that $(S,\\Delta_S)$ is K-semistable (see \\cite[Theorem 1.4]{OS14} or \\cite[Theorem 3.12]{Ber13}). And by Theorem \\ref{t-main} and \\ref{t-main2}, we know ${\\rm ord}_S$ is the unique minimizer of ${\\widehat{\\rm vol}}(S)$ among all Koll\\'ar components. See \\cite{CS14} for examples of weakly exceptional singularities. \n\\end{enumerate}\n\n\\bigskip\n\nThe following example is indeed a prototype of our study. \n\n\\begin{exmp}\\label{ex-irreg}\nIf a log terminal singularity $o\\in X$ has a quasi-regular Sasakian-Einstein metric, then the $\\mathbb{C}^*$-quotient provides a Koll\\'ar component which minimizes ${\\widehat{\\rm vol}}_{x,X}$.\n\\end{exmp}\nAssume $X={\\rm Spec} R$ is an affine variety with effective torus action and ${\\mathbb{Q}}$-Gorenstein klt isolated singularity at $o\\in X$. Denote the torus by ${\\mathbb{T}}:=({\\mathbb{G}}_m)^d$.\nThen we have a weight decomposition:\n\\[\nR=\\bigoplus_{\\alpha\\in \\Gamma} R_\\alpha,\n\\]\nwhere\n\\begin{itemize}\n\\item\nFor any $\\alpha\\in \\mathbb{Z}^d$, $R_\\alpha=\\{f\\in R\\; ; \\; t\\cdot f=t^{\\alpha}f \\text{ for any } t\\in ({\\mathbb{G}}_m)^d\\}$.\n\\item\n$\\Gamma=\\{\\alpha\\in \\mathbb{Z}^d; R_\\alpha\\neq 0\\}$.\n\\end{itemize}\nThe Reeb cone of $X$ is the following conic subset of the Lie algebra of ${\\mathbb{T}}=({\\mathbb{G}}_m)^d$:\n$$\n\\mathcal{RC}:=\\{\n\\xi\\in \\mathbb{R}^d; \\langle \\alpha, \\xi\\rangle>0 \\text{ for any } \\alpha\\in \\Gamma\n\\}.\n$$\nNotice that the elements in the Reeb cone $\\mathcal{RC}$ can be considered as holomorphic vector fields on $X$ via the map \n$\\xi\\mapsto \\sigma_\\xi:=\\sum_{i=1}^d \\xi_i \\sigma_i$\nwhere $\\sigma_i$ is the infinitesimal generator of the action of the $i$-th factor of $({\\mathbb{G}}_m)^d$. \nIf we denote by $\\{e_i\\}$ the standard basis of $\\mathbb{R}^d$, then\n\\[\n\\sigma_i(p)=\\left.\\frac{d}{ds}\\right|_{s=0}\\exp (s e_i)\\circ p - \\sqrt{-1} \\left.\\frac{d}{ds}\\right|_{s=0}\\exp(\\sqrt{-1} s e_i) \\circ p.\n\\]\nSuppose that there is a K\\\"{a}hler cone metric $g_\\xi=dr^2+r^2 g_{L}$ on $X$ where\n\\begin{enumerate}\n\\item\n$r: X-\\{o\\}\\rightarrow \\mathbb{R}_{>0}$ is a smooth radius function, $L=\\{r=1\\}\\subset X$ is the link of the isolated singularity $(X,o)$.\n\\item $r\\frac{\\partial}{\\partial r}- \\sqrt{-1} J(r\\frac{\\partial}{\\partial r})=\\sigma_\\xi$.\n\\end{enumerate}\nNow assume that $g_\\xi$ is a Ricci-flat K\\\"{a}hler metric, equivalently the link $L$ has a Sasaki-Einstein metric (see \\cite{BG08}). There are two different cases:\n\\begin{enumerate}\n\\item (quasi-regular case)\nIf the generic orbit of $\\partial_\\theta:=J(r\\frac{\\partial}{\\partial r})$ is closed, $g_L$ is called a quasi-regular Sasaki-Einstein metric.  $\\sigma_\\xi$ generates a ${\\mathbb{G}}_m$-subgroup of ${\\mathbb{T}}$. the quotient $X\/{\\mathbb{G}}_m$ is an Fano orbifold $(V, \\Delta)$ with orbifold K\\\"{a}hler-Einstein metric. By \\cite{LL16} \n${\\rm ord}_V$ is a minimizer of ${\\widehat{\\rm vol}}_{X,o}$. By Theorem \\ref{t-main2}, this is the unique minimizer among all Koll\\'{a}r components.\n\\item (irregular case)\nIf the generic orbit of $\\partial_\\theta$ is not closed, $g_L$ is called an irregular Sasaki-Einstein metric. Explicit examples can be found in \\cite{GMSW04a, GMSW04b, FOW09}. \nSince $\\xi\\in \\mathbb{R}^d$ is an irrational vector, we can approximate it by a sequence of rational vectors $\\{\\xi_k\\}\\subset \\mathcal{RC}\\cap {\\mathbb{Q}}^d$. Each $\\xi_k$ generates a ${\\mathbb{G}}_m$-subgroup $G_k$ of ${\\mathbb{T}}$, and we get a sequence of quotients $X\/G_k=(V_k, \\Delta_k)$. It is easy to see that $V_k$ are all Koll\\'{a}r components. It is possible to show that a suitable rescaling $c_k\\cdot {\\rm ord}_{V_k}$ converges to a minimizer of ${\\widehat{\\rm vol}}_{(X,o)}$, which corresponds to the vector field $\\sigma_\\xi$. The details will be discussed in a future work.\n\\end{enumerate}\n\n\\bigskip\n\nAs we mentioned, there is a related differential geometry study on the metric tangent cone. \n\n\\begin{exmp}\\label{ex-metric}\n In the notation of \\cite{DS15}, let $p\\in Z$ be a singularity appearing on the Gromov-Hausdorff limit of a sequence of KE Fano manifolds. Assume the Reeb vector of the weight tangent cone $W$ is quasi-regular, then the $\\mathbb{C}^*$-quotient of $W$ indeed gives a minimizer of $p\\in Z$.\n\\end{exmp}\nAssume $(X, o)$ is an algebraic germ on a normal affine variety that is embedded into $\\mathbb{C}^N$. Assume that we have a weight vector $w=(a_1, \\dots, a_N)\\in \\mathbb{Z}_{>0}^N$ satisfying that there is no common factor of $a_i$. Then it defines a filtration on $R=\\mathcal{O}_{X,o}$:\n\\begin{equation}\\label{eq-wf}\n{\\mathcal{F}}^k=\\{f\\in R; \\text{ there exists } g\\in \\mathcal{O}_{\\mathbb{C}^N,o} \\text{ with } g|_X=f \\text{ and } w(g)\\ge k\\}.\n\\end{equation}\nFrom this we can form the Rees algebra and extended Rees algebra:\n\\[\n{\\mathcal{R}}=\\bigoplus_{k=0}^{+\\infty} {\\mathcal{F}}^k t^{-k}, \\quad {\\mathcal{R}}'=\\bigoplus_{k\\in \\mathbb{Z}} {\\mathcal{F}}^k  t^{-k},\n\\]\nwhere ${\\mathcal{F}}^k=R$ if $k\\le 0$. We also have the associated graded ring of $(R, {\\mathcal{F}})$:\n\\[\nG=\\bigoplus_{k=0}^{+\\infty} \\left({\\mathcal{F}}^k\/{\\mathcal{F}}^{k+1}\\right) \\cdot t^{-k}\\cong {\\mathcal{R}}'\/t{\\mathcal{R}}'.\n\\]\nIn our discussion, we always assume that $G$ is a finitely generated $\\mathbb{C}$-algebra that is also a normal domain. Then we can get a normal affine variety $W={\\rm Spec}(G)$. As in \\cite{DS15}, we call $W$ to be a weighted tangent cone of $(X, o)$. Notice that there is a natural $\\mathbb{C}^*$-action on $G$ and hence it is an orbifold cone over the base $E={\\rm Proj}(G)$ (by Pinkham-Demazure's construction).\n\nOn the other hand, the weight $w$ defines a weighted blow up $\\pi_w: \\widehat{\\mathbb{C}^N} \\rightarrow \\mathbb{C}^N$ given by:\n\\[\n\\widehat{\\mathbb{C}^N}={\\rm Proj} \\left(\\bigoplus_{k=0}^{+\\infty} \\mathfrak{A}_k\\cdot t^{-k}\\right), \n\\]\nwhere \n\\[\n\\mathfrak{A}_k=\\{g\\in \\mathbb{C}[z_1, \\cdots, z_n]; w(g)\\ge k\\}.\n\\]\nNotice that the exceptional divisor of $\\pi_w$ is the weighted projective space with weight $w$. We have the following observation, which implies that the discussion can be put in the setting of\nfiltered blowing-ups as studied in \\cite{TW89}.\n\\begin{lem}\n\\begin{enumerate}\n\\item\nIf $Y$ denotes the strict transform of $X$ under $\\pi_w$, then\n$Y={\\rm Proj}({\\mathcal{R}})$. In particular the Rees algebra ${\\mathcal{R}}$ is finitely generated. Moreover, the exceptional divisor of the natural birational morphism\n$Y\\stackrel{\\psi}{\\longrightarrow} X$ is isomorphic to $E$.\n\\item\nThe filtration ${\\mathcal{F}}=\\{{\\mathcal{F}}^k\\}_{k\\ge 0}$ is the same as the filtration induced by the divisorial valuation ${\\rm ord}_E$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nDenoting by $\\iota_X: X\\rightarrow \\mathbb{C}^N$ the embedding morphism, then ${\\mathcal{F}}^k$ in \\eqref{eq-wf} coincides with the inverse image ideal $\\iota_X^{-1}\\mathfrak{A}_k\\cdot \\mathcal{O}_{X,o}$ (cf. \\cite[7.15]{Har77}). So $Y$ is indeed \nthe strict transform of $X$ under the weighted blow up $\\pi_w$. The second statement is standard (see \\cite[1.2.1]{TW89}). Item 2 follows from the discussion in Section \\ref{sec-filtration} (see \\cite[(2.2)]{TW89}).\n\\end{proof}\n\nIf $(X,o)$ is a klt singularity appearing in a Gromov-Hausdorff limit of a sequence of K\\\"{a}hler-Einstein Fano manifolds, Donaldson-Sun (\\cite{DS15})\nconstructed an affine variety $W$ using the metric structure on the Gromov-Hausdorff limit and showed that it specially degenerates to the metric tangent cone $C(Y)$. The degeneration can be realized under a common embedding of $W$ and $C(Y)$ into some ambient $\\mathbb{C}^N$. More precisely $W$ is a weighted tangent cone associated to some weight under the common embedding. The weight is determined by the Reeb vector field of the singular Ricci flat metric on $C(Y)$. If the Reeb vector field is quasi-regular, that is, if it generates a $\\mathbb{C}^*$-action, then the weight can be normalized to have integer components so that we are in the situation discussed above. Moreover it is shown in this case $E=W\\setminus \\{ 0 \\}\/\\mathbb{C}^*$ specially degenerates to $C(Y)\\setminus \\{ 0\\}\/\\mathbb{C}^*$ and $C(Y)\\setminus \\{ 0\\} \/\\mathbb{C}^*$ admits a weak K\\\"{a}hler-Einstein metric (see \\cite{DS15}). This implies $E$ is analytically K-semistable, which then implies it is (algebraically) K-semistable.  So we can apply our result Theorem \\ref{t-main} to see that ${\\rm ord}_E$ is indeed a global minimizer of ${\\widehat{\\rm vol}}$ over $(X, o)$, which is the unique minimizer among all Koll\\'{a}r components.\n \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThree decades ago, \\citet{Justin1986} introduced a set of 7 elementary single-fold operations which have become known as the ``axioms'' of origami, the Japanese art of paper folding \\citep{Alperin2000,Alperin2006,Ghourabi2013}. Each operation is defined by one or more alignments (incidences) between points and lines on a sheet of paper, that must be achieved with a single fold.  The number of solutions of each operation must be finite; however, depending on the relative position of the given points and lines, some of the operations may have none, one or multiple solutions. It has been shown that the set of operations constitutes a more powerful geometrical tool than the combination of straight edge and compass \\citep{Alperin2000}. For example, the operations allow for the trisection of arbitrary angles \\citep{Hull1996}, solving the problem of duplicating  the cube \\citep{Messer1986}, constructing heptagons \\citep{Geretschlaeger1997} and solving cubic and fourth order equations \\citep{Alperin2000,Geretschlager1995}, all of which may be not done by straight edge and compass alone. \n\nJustin's work \\citep{Justin1986} seems to have been overlooked at its time, and the same fold operations have been rediscovered later and expressed under various forms by \\citet{Huzita1989}, Hatori \\citep[2001, according to][]{Alperin2006} and other enthusiasts of origami mathematics \\citep[e.g.,]{Alperin2000,Auckly1995, Geretschlager1995,Kawasaki2011,Martin1998}. The operations are popularly known today as Huzita's axioms, Huzita-Justin's axioms or Huzita-Hatori's axioms.\\footnote{In his work, \\citet{Huzita1989} listed six of Justin's fold operations. The seventh was rediscovered by Hatori, in 2001 (according to \\citep{Alperin2006})} Let us note that the designation as ``axioms'' is not correct since some of the operations may be derived from others, and further, some of them may not be possible depending on the configuration of given points and lines. \nIt has been claimed that the set is complete, in the sense that it contemplates all possible alignments between points and lines with a finite number of solutions, and excluding redundant alignments \\citep{Alperin2006}. \n\nMore recently, \\citet{Kasem2011} showed several inconsistencies in the operations owing to the rather imprecise form in which they had been stated. The inconsistencies included impossibility of some folds, infinite solutions and superfluous conditions. Seeking a more rigorous treatment, \\citet{Ghourabi2013} expressed the operations in formal algebraic terms and analyzed their number of solutions and conditions of existence. Such a formalization is a necessary step for adapting folding techniques to industrial applications. In fact, recent years have seen a surge of applications of origami to science and technology, e.g., in aerospace and automotive technology \\citep{Cipra2001}, materials science \\citep{Zhou2016}, biology \\citep{Mahadevan2005}, civil engineering \\citep{Filipov2015}, robotics \\citep{Felton2014} and acoustics \\citep{Harne2016}. Further, a number of computational systems of origami simulation have been developed \\citep{Ida2006a}. \n   \nThe present article follows the call for normalization and analyzes the elementary operations by applying the geometry of reflections. Folding a sheet of paper along a straight line superposes the paper on one side of the fold line to the other side. \nAs a result, all points and lines on each side of fold line are reflected across it  onto the\nother side \\citep{Martin1998}. In fact, the geometry of paper folding may be reproduced by using a semi-reflective mirror called ``Mira'' \\citep{Demaine2007}. A full geometrical characterization of Mira constructions has been  given by \\citet{Emert1994} in terms of ``primitive actions'', which are equivalent to the fold operations studied here.\n\nFirst, the analysis will determine all possible incidences between given points and lines on a plane that may be achieved by a reflection. Next, it will derive all possible fold operations that may be defined so as to satisfy combinations those incidences. In this way, a total of eight elementary operations will be obtained, i.e., one more than the previous set, where the new operation is to fold along a given line. This operation has been commonly disregarded by previous studies on the argument that it does not create a new line; however, completeness of the set demands its inclusion. Further, it has applications in actual origami folding, as will be discussed later (Section \\ref{foldingalong}).  \n\n\n\\section{Reflections on a plane}\n\\label{reflections}\n\nThe medium on which all folds are performed is assumed to be an infinite Euclidean plane \\citep{Geretschlager1995}. Points are denoted by capital letters ($P$, $Q$ etc.), lines by small letters ($m$, $n$ etc.), except the fold line which is denoted by the special symbol $\\chi$, and $P\\in m$ means that point $P$ is on line $m$. \n\nA reflection is defined as follows \\citep{Martin1998}:\n\n\\begin{definition}\\label{defreflx} Given a line $\\chi$, the reflection $\\mathcal{F}$ in $\\chi$ is the mapping on the set of points in the plane such that for point $P$\n\\[\n\\mathcal{F}(P)=\n\\left\\{\n\\begin{array}{ll}\nP & \\text{if $P\\in \\chi$},\\\\\nP' & \\text{if $P\\notin \\chi$ and $\\chi$ is the perpendicular bisector of segment $\\overline{PP'}$}\n\\end{array}\n\\right.\n\\]\n(see Fig.~\\ref{reflx}).\n\\end{definition}\n\nIt is easy to see that $\\mathcal{F}(P)=P'$ iff $\\mathcal{F}(P')=P$.\n\n\\begin{figure}[!htb]\n\\centering\n\\begin{pspicture}(1,1.5)(5,5)\n\\psline[linecolor=foldline,linewidth=1pt](1.5,4.5)(4.5,1.5)\n\\psline[linestyle=dashed,linecolor=gray,linewidth=1pt](1.8,1.8)(4.2,4.2)\n\\qdisk(1.8,1.8){2pt}\n\\uput[110](1.8,1.8){$P'$}\n\\qdisk(4.2,4.2){2pt}\n\\uput[90](4.2,4.2){$P$}\n\\uput[45](2,4){$\\chi$}\n\\end{pspicture}\n\\caption{Reflection of point $P$ in line $\\chi$.} \n\\label{reflx}\n\\end{figure}\n\n\nReflection of a line $m$ in $\\chi$ is obtained by reflecting every point in $m$. Therefore, $\\mathcal{F}(m)=\\{\\mathcal{F}(P)|\\,P\\in m\\}$.\nLet $m'=\\mathcal{F}(m)$ and  consider the following cases:\n\n\\begin{enumerate}\n\n\\item If $m$ and $\\chi$ are parallel ($m\\parallel \\chi$), then $m\\parallel m'$ (Fig.~\\ref{linereflx}, left).\n\n\\item\\label{it2} If $m$ and $\\chi$ are not parallel ($m\\nparallel \\chi$)  then line $\\chi$ is a bisector of the angle between $m$ and $m'$ (Fig.~\\ref{linereflx}, center).\n\n\\item If $m=\\chi$, then every point in $m$ is its own reflection, and therefore $m'=m$.\n\n\\item \\label{R2.4} If $m$ and $\\chi$ are perpendicular ($m\\bot\\chi$), then the reflection of every point of $m$ is also on $m$; therefore, $m=m'$ (Fig.~\\ref{linereflx}, right). Note also that $\\chi$ divides $m$ into two halves, and each half is reflected onto the other. Thus, for every point $P$ on $m$ and not on the intersection with $\\chi$, $\\mathcal{F}(P)\\neq P$. \n\n\\end{enumerate}\n\n\n\\begin{figure}[!htb]\n\\begin{pspicture}(1,0.5)(6,5.5)\n\\psline[linecolor=foldline,linewidth=1pt](2,4)(4.5,1.5)\n\\psline[linewidth=1pt](5.3,2.3)(2.8,4.8)\n\\psline[linewidth=1pt](1.2,3.2)(3.7,0.7)\n\\uput[45](2.8,4.8){$m$}\n\\uput[45](1.2,3.2){$m'$}\n\\uput[45](2,4){$\\chi$}\n\\end{pspicture}\n\\begin{pspicture}(1,0.5)(6,5.5)\n\\psline[linewidth=1pt](3,5)(4,1)\n\\psline[linewidth=1pt](1,3)(5,2)\n\\psline[linecolor=foldline,linewidth=1pt](1.5,4.5)(4.5,1.5)\n\\uput[0](3,5){$m$}\n\\uput[45](1,3){$m'$}\n\\uput[45](1.5,4.5){$\\chi$}\n\\end{pspicture}\n\\begin{pspicture}(1,0.5)(5,5.5)\n\\psline[linewidth=1pt](1.5,1.5)(4.5,4.5)\n\\psline[linecolor=foldline,linewidth=1pt](1.5,4.5)(4.5,1.5)\n\\uput[90](4.5,4.5){$m=m'$}\n\\uput[45](1.75,4.25){$\\chi$}\n\\end{pspicture}\n\\caption{Reflection of line $m$ in line $\\chi$. Left: $m\\parallel \\chi$. Center: $m\\nparallel\\chi$. Right: $m\\bot\\chi$} \n\\label{linereflx}\n\\end{figure}\n\n\\section{Incidence constraints}\n\nElementary single-fold operations are defined in terms of incidence constraints  between pairs of objects (points or lines) that must be satisfied with a fold \\citep{Alperin2006, Ghourabi2013,Justin1986}. Each constraint involves an object $\\alpha$ and the image $\\mathcal{F}(\\beta)$ of an object $\\beta$ (including the case $\\alpha=\\beta$) by reflection in the fold line.  The symmetry of the reflection mapping implies that all incidence relations are also symmetric.\n\nA total of six different incidences are possible on a plane, an they are defined and analyzed in the next subsections (see also Table \\ref{table0}). In order to facilitate the posterior definitions of the fold operations, incidences involving distinct objects (i.e., $\\alpha\\neq \\beta$) are distinguished from those involving the same object (i.e., $\\alpha= \\beta$).\n\n\\subsection{Incidences involving distinct objects}\n\n\\begin{incidence}\n$\\mathcal{F}(P)=Q$, with $P\\neq Q$.\n\\label{I1}\n\\end{incidence}\n\n\\noindent In this incidence, the reflection of a given point $P$ coincides with another given point $Q$. According to Definition \\ref{defreflx}, its solution is the unique fold line $\\chi$ which is the perpendicular bisector of segment $\\overline{PQ}$. \n\n\\begin{incidence}\n$\\mathcal{F}(m)= n$, with $m\\neq n$.\n\\label{I2}\n\\end{incidence}\n\n\\noindent In this incidence, the reflection of a given line $m$ coincides with another given line $n$. Two cases are possible:\n\n\\begin{enumerate}\n\n\\item When $m\\nparallel n$, there are two possible fold lines that satisfy the incidence, which are the bisectors to the angles defined by $m$ and $n$ (Fig. \\ref{s2}).  \n\\item When $m\\parallel n$, there is only one solution, which is a fold line parallel and equidistant to both $m$ and $n$ (Fig. \\ref{linereflx}, left, with $m'=n$).\n\\end{enumerate}\n\n\\begin{figure}[!htb]\n\\centering\n\\begin{pspicture}(1,0.5)(6,5.5)\n\\psline[linewidth=1pt](3,5)(4,1)\n\\psline[linewidth=1pt](1,3)(5,2)\n\\psline[linecolor=foldline,linewidth=1pt](1.5,4.5)(4.5,1.5)\n\\psline[linecolor=foldline,linewidth=1pt](2.65,1.35)(5.65,4.35)\n\\uput[0](3,5){$m$}\n\\uput[45](1,3){$n$}\n\\uput[45](1.5,4.5){$\\chi_1$}\n\\uput[135](5.5,4.2){$\\chi_2$}\n\\end{pspicture}\n\\caption{Incidence \\ref{I2} in the case of $m\\nparallel n$.\n\\label{s2}\n\\end{figure}\n\n\\pagebreak\n\\begin{incidence}\n$\\mathcal{F}(P)\\in m$, with $P\\notin m$.\n\\label{I3}\n\\end{incidence}\n\n\\noindent In this incidence, the reflection of a given point $P$ is on a given line $m$, and the case in which $P$ is already on $m$ is excluded. It has been shown that the fold lines that satisfy the incidence are  tangents to a parabola with focus $P$ and directrix $m$ \\citep{Alperin2000,Martin1998}. \n\nIt is useful to derive equations of the fold lines and the associated parabola, for later application to the analysis of fold operations. Without loss of generality, choose a Cartesian system of coordinates $x, y$ so that $P$ is located at $(0, 1)$ and line $m$ is $y=-1$ (Fig.~\\ref{parab}). Also, let $P'=\\mathcal{F}(P)$ be located at $(t, -1)$, where $t$ is a free parameter. \n\n\\begin{figure}[!htb]\n\\centering\n\n\\begin{pspicture}(-3,-3)(4,4)\n\\psset{xunit=1.2cm,yunit=1.2cm}\n\\psaxes[linecolor=gray,labels=none,linewidth=1pt]{->}(0,0)(-1.5,-1.5)(3.5,3)[$x$,0][$y$,0]\n\\psplot[linewidth=1pt,linecolor=vector,plotpoints=100]{-1.5}{3.2}{x x mul 4 div}\n\\uput[135]{*0}(0,1){$P(0, 1)$}\n\\psline(-1.5,-1)(3,-1)\n\\uput[90]{*0}(-1.5,-1){$m$}\n\\uput[90]{*0}(-1.5,.6){$\\Psi$}\n\\uput[-90]{*0}(2.5,-1){$P'(t, -1)$}\n\\psline[linecolor=gray,linewidth=1pt, linestyle=dashed](2.5,-1)(2.5,1.56)\n\\psline[linecolor=gray,linewidth=1pt, linestyle=dashed](0,1)(2.5,1.56)\n\\psline[linecolor=gray,linewidth=1pt, linestyle=dashed](0,1)(2.5,-1)\n\\uput[90]{*0}(2.5,1.56){$T$}\n\\psline[linecolor=foldline,linewidth=1pt](-.2,-1.82)(3.2,2.44)\n\\uput[90]{*0}(.5,-.8){$\\chi$}\n\\qdisk(0,1){2pt}\n\\qdisk(2.5,-1){2pt}\n\\qdisk(2.5,1.56){2pt}\n\\qdisk(1.25,0){2pt}\n\\end{pspicture}\n\\caption{Incidence \\ref{I3}. The fold line $\\chi$ is tangent to a parabola with focus $P$ and directrix $m$.} \n\\label{parab}\n\\end{figure}\n\n\nThe slope of segment $\\overline{PP'}$ is $-2\/t$. The fold line $\\chi$ is perpendicular to $\\overline{PP'}$ and therefore has a slope of $t\/2$. Further, $\\chi$ passes through the midpoint of $\\overline{PP'}$, which is located at $(t\/2, 0)$. Thus, $\\chi$ has an equation \n\\begin{equation}\ny=\\frac{t}{2}\\left(x - \\frac{t}{2}\\right).\n\\label{chipar}\n\\end{equation}\n\nNext, consider point $T$ located at the intersection of $\\chi$ with a vertical line through $P'$. Its coordinates may be obtained by evaluating Eq.~(\\ref{chipar}) at $x=t$, which produces $(t, t^2\/4)$. Those coordinates describe parametrically a parabola with equation \n\\begin{equation}\ny=\\frac{x^2}{4},\n\\label{p1}\n\\end{equation}\nwhich is denoted by $\\Psi$. This is precisely the equation of a parabola with focus at $(0, 1)$ and directrix $y=-1$.\\footnote{The general equation of a parabola with vertical axis and vertex at $(0, 0)$ is $y=x^2\/(4a)$, where $a$ is the distance from the vertex to the directrix $y=-a$ or the focus $(0, a)$ \\citep{Weisstein2016a}.} Further, note that the slope of a tangent to $\\Psi$ at point $T$ is $y'(t)=t\/2$, which is the same slope of $\\chi$. Therefore, $\\chi$ is a line tangent to $\\Psi$ at point $T$. \n\n\n\n\nSince $t$ in Eq.~(\\ref{chipar}) is a free parameter, then the solution to this incidence is a family of fold lines with one parameter (Fig.~\\ref{s3}).\n\nThe case $P\\in m$ is excluded because under such condition any fold line passing through $P$ or perpendicular to $m$ satisfies the incidence. Those two possibilities are considered in incidences \\ref{I4} and \\ref{I5}, respectively.\n\n\\begin{figure}[!htb]\n\\centering\n\\begin{pspicture}(-2,-3)(4,4.5)\n\\psset{xunit=1.3cm,yunit=1.3cm}\n\\rput{-45}\n\\parabola[linecolor=vector,linewidth=1pt](2.2,2.4)(0,0)\n\\qdisk(0,.5){2pt}\n\\uput[90]{*0}(0,.5){$P$}\n\\psline[linewidth=1pt](-2,-.5)(2,-.5)\n\\uput[90]{*0}(-2,-.5){$m$}\n\\psline[linecolor=foldline,linewidth=1pt](-2.,0)(2,0)\n\\psline[linecolor=foldline,linewidth=1pt](-2.02,1.9)(-.12,-1)\n\\psline[linecolor=foldline,linewidth=1pt](-2,1.5)(.5,-1)\n\\psline[linecolor=foldline,linewidth=1pt](-2.05,.9)(1.75,-1)\n\\psline[linecolor=foldline,linewidth=1pt](-1.7,-1)(2.05,.9)\n\\psline[linecolor=foldline,linewidth=1pt](-.5,-1)(2,1.5)\n\\psline[linecolor=foldline,linewidth=1pt](.12,-1)(2.02,1.9)\n}\n\\end{pspicture}\n\\caption{Fold lines for incidence \\ref{I3}.}\n\\label{s3}\n\\end{figure}\n\n\\subsection{Incidences involving an object and its reflected image}\n\n\\begin{incidence}\n$\\mathcal{F}(P)=P$.\n\\label{I4}\n\\end{incidence}\n\n\\noindent In this incidence, the reflection of a given point $P$ coincides with itself, and it is satisfied by any fold line $\\chi$ passing through $P$ . An arbitrary direction for line $\\chi$ may be defined by its angle $\\theta$ with, e.g., the $x$-axis in a Cartesian coordinate system. Therefore, the solution to the incidence is a family of fold lines with one parameter (Fig.~\\ref{s4}).\n\n\\begin{figure}[!htb]\n\\centering\n\n\\begin{pspicture}(6,1)(10,5)\n\\psline[linecolor=foldline,linewidth=1pt](6.6,1.6)(9.4,4.4)\n\\psline[linecolor=foldline,linewidth=1pt](9.4,1.6)(6.6,4.4)\n\\psline[linecolor=foldline,linewidth=1pt](8,1.4)(8,4.6)\n\\psline[linecolor=foldline,linewidth=1pt](6.4,3)(9.6,3)\n\\qdisk(8,3){2pt}\n\\uput[80](8.1,3){$P$}\n\\end{pspicture}\n \n\\caption{Fold lines for incidence \\ref{I4}. } \n\\label{s4}\n\\end{figure}\n\n\n\\begin{incidence}\n$\\mathcal{F}(m)=m$, and $\\exists P\\in m$, $\\mathcal{F}(P)\\neq P$.\n\\label{I5}\n\\end{incidence}\n\n\\noindent Both this and the next incidence consider the reflection of a line $m$ to itself. As discussed in Section \\ref{reflections}, there are two ways in which such a reflection may be achieved. In the current case, one half of $m$, defined from an arbitrary point $R\\in m$, is reflected upon the opposite half. \n\nThe position of point $R$ may be specified by its distance $s$ from a particular point $P_0\\in m$. Therefore, the solution to the incidence is a family of fold lines with one free parameter (Fig.~\\ref{s5}). \n\n\\begin{figure}[!htb]\n\\centering\n\n\\begin{pspicture}(1,0.5)(5,5.5)\n\\psline[linewidth=1pt](1.25,1.25)(4.75,4.75)\n\\psline[linecolor=foldline,linewidth=1pt](2.5,5.5)(5.5,2.5)\n\\psline[linecolor=foldline,linewidth=1pt](2.,5.)(5,2)\n\\psline[linecolor=foldline,linewidth=1pt](1.5,4.5)(4.5,1.5)\n\\psline[linecolor=foldline,linewidth=1pt](1,4)(4,1)\n\\psline[linecolor=foldline,linewidth=1pt](.5,3.5)(3.5,.5)\n\\uput[135](1.5,1.5){$m$}\n\\end{pspicture}\n\\caption{Fold lines for incidence \\ref{I5}.} \n\\label{s5}\n\\end{figure}\n\n\\begin{incidence}\n$\\mathcal{F}(m)=m$, and $\\forall P\\in m$, $\\mathcal{F}(P)= P$.\n\\label{I6}\n\\end{incidence}\n\n\\noindent This is the second case of reflection of line $m$ to itself. In this case, each point $P\\in m$ is reflected to itself, and therefore the incidence is satisfied by the unique fold line $\\chi=m$.\n\n\\section{Elementary fold operations}\n\n\\subsection{Definition}\n \nA straight line on a plane is an object with two degrees of freedom.\\footnote{A fold line $\\chi$ may be defined by an equation of the form $ax+by+c=0$, where $a$, $b$ and $c$ are constants, and $(a, b)$ is a normal vector to $\\chi$. A vector in arbitrary direction may be defined by  letting $a=\\cos \\theta$, $b=\\sin\\theta$, with $0\\le \\theta < 2\\pi$. Therefore, two parameters must be set in order to define any fold line, namely, $\\theta$ and $c$.} \nWhen an incidence constraint is set for $\\chi$, satisfying the constraint consumes a number of degrees of freedom, and that number is called the codimension of the constraint. Incidences \\ref{I1}, \\ref{I2}  and \\ref{I6} have either a unique or a finite number of solutions; therefore, those incidences have codimension 2. On \nthe other hand, each of incidences  \\ref{I3}, \\ref{I4} and \\ref{I5} have a family of solutions with one free parameter and therefore they have codimension 1 (Table \\ref{table0}). \n \n\\begin{table}[!htb]\n\\centering\n\n\\begin{threeparttable}\n\\caption{Incidence constraints.}\n\\label{table0}\n\\centering\n\n\\begin{tabular}{clc}\n\\toprule\nIncidence & Definition\\tnote{a} & Codimension\\\\\n\\midrule\n\\ref{I1} & $\\mathcal{F}(P)=Q$, with $P\\neq Q$ & 2\\\\\n\\ref{I2} & $\\mathcal{F}(m)=n$, with $m\\neq n$ & 2 \\\\\n\\ref{I3} & $\\mathcal{F}(P)\\in m$, with $P\\notin m$ & 1 \\\\\n\\ref{I4} & $\\mathcal{F}(P)=P$ & 1\\\\\n\\ref{I5} & $\\mathcal{F}(m)=m$, and $\\exists P\\in m$, $\\mathcal{F}(P)\\neq P$ &1\\\\\n\\ref{I6} & $\\mathcal{F}(m)=m$, and $\\forall P\\in m$, $\\mathcal{F}(P)= P$ & 2\\\\\n\\bottomrule\n\\end{tabular}\n\n\\begin{tablenotes}\n \\small\n\\item [a]$P$ and $Q$ are points; $m$ and $n$ are lines.\n\\end{tablenotes}\n\n\\end{threeparttable}\n\n\\end{table}\n\nAn elementary single-fold operation is defined as a minimal set of alignments between points and lines that is satisfied with a single fold and has a finite number of solutions \\citep{Alperin2006}. Equivalently, it is the resolution of a set of incidence constraints which have a total codimension of 2.\n\nEach of the incidences \\ref{I1}, \\ref{I2} and \\ref{I6} already define an elementary operation. The other three incidences must be applied in pairs (including pairing incidences  of the same type), and there are a total of 6 possible pairs. However, incidence \\ref{I5} can be not be used twice. If it is, then for given lines $m$ and $n$ the fold line $\\chi$ has to satisfy both $\\mathcal{F}(m)=m$ and  $\\mathcal{F}(n)=n$. Therefore, $\\chi$  must be perpendicular to both $m$ and $n$, and two cases are possible:\n\\begin{enumerate}\n\\item If $m\\nparallel n$, then a perpendicular to both lines does not exist (in the Euclidean plane).  \n\\item If $m\\parallel n$, then any perpendicular to $m$ or $n$ is a valid fold line. \n\\end{enumerate}\n\nThus, the pair of constraints has none or infinite solutions, and so it does not define a valid elementary fold operation.\n\nA total of eight elementary fold operations may be then defined, and they are analyzed in the next subsections.\n\n\\subsection{Elementary operations defined by codimension 2 incidences}\n\nThe following three operations are defined by incidences \\ref{I1}, \\ref{I2} and \\ref{I6}, respectively. \n\n\\begin{operation}\nGiven points $P$ and $Q$, with $P\\neq Q$, construct a fold line  so that $\\mathcal{F}(P)=Q$.\n\\label{O1}\n\\end{operation}\n\n\\begin{operation}\nGiven lines $m$ and $n$, with $m\\neq n$, construct a fold line so that $\\mathcal{F}(m)=n$.\n\\label{O2}\n\\end{operation}\n\n\\begin{operation}\nGiven a line $m$, construct a fold line so that $\\mathcal{F}(m)=m$, and $\\forall P\\in m,$ $\\mathcal{F}(P)= P$.\n\\label{O3}\n\\end{operation}\n\n\\subsection{Elementary operations defined by pairs of codimension 1 incidences}\n\n\\begin{operation}\nGiven points $P$ and $Q$, with $P\\neq Q$, construct a fold line so that $\\mathcal{F}(P)=P$ and $\\mathcal{F}(Q)=Q$.\n\\label{O4}\n\\end{operation}\n\n\\noindent\nThis operation is defined by application of incidence \\ref{I4} to two distinct points. It has a unique solution, which is a fold line $\\chi$ passing through points $P$ and $Q$ (Fig.~\\ref{fO4}).\n\n\\begin{figure}[!htb]\n\\centering\n\n\\begin{pspicture}(1.5,1.5)(5,5.5)\n\\uput[135](2.3,2.3){$P$}\n\\uput[135](4.5,4.5){$Q$}\n\\psline[linecolor=foldline,linewidth=1pt](1.8,1.8)(5,5)\n\\qdisk(2.3,2.3){2pt}\n\\qdisk(4.5,4.5){2pt}\n\\uput[135](3.5,3.5){$\\chi$}\n\\end{pspicture}\n\\caption{Operation \\ref{O4}.}\n\\label{fO4}\n\\end{figure}\n\n\\begin{operation}\nGiven a point $P$ and a line $m$, construct a fold line such that $\\mathcal{F}(P)=P$ and $\\mathcal{F}(m)=m$, and $\\exists Q\\in m$, $\\mathcal{F}(Q)\\neq Q$.\n\\label{O5}\n\\end{operation}\n\n\\noindent\nThis operation is defined by application of incidences \\ref{I4} and \\ref{I5}. It has a unique solution, which is a fold line $\\chi$ perpendicular to $m$ and passing through $P$ (Fig.~\\ref{fO5}). Note that the case $P\\in m$ is allowed, which has the same unique solution.\n\n\\begin{figure}[!htb]\n\\centering\n\n\\begin{pspicture}(-2,-2)(2,2.5)\n\\psline(-1.2,1.2)(1.2,-1.2)\n\\uput[90](-1.,1.2){$m$}\n\\psline[linecolor=foldline,linewidth=1pt](-1.5,-1.5)(1.5,1.5)\n\\qdisk(1,1){2pt}\n\\uput[90]{*0}(1,1){$P$}\n\\uput[135]{*0}(-1,-1){$\\chi$}\n\\end{pspicture}\n\\caption{Operation \\ref{O5}.}\n\\label{fO5} \n\\end{figure}\n\n\\begin{operation}\nGiven points $P$, $Q$, and a line $m$, with $P\\notin m$, construct a fold line such that $\\mathcal{F}(P)\\in m$ and $\\mathcal{F}(Q)=Q$.\n\\label{O6}\n\\end{operation}\n\n\\noindent \nThis operation is defined by application of incidences \\ref{I3} and \\ref{I4}. Its solution is a fold line that is tangent to a parabola with focus $P$ and directrix $m$, and passes through point $Q$. \n\nAssume the same parabola of Fig.~\\ref{parab}, given by Eq.~(\\ref{p1}), and a point $Q$ at the position $(x_q, y_q)$. Replacing the coordinates of $Q$ in Eq.~(\\ref{chipar}) produces the quadratic equation\n\\begin{equation}\nt^2-2x_qt+4y_q=0.\n\\label{tyt}\n\\end{equation}\nThe discriminant of Eq.~(\\ref{tyt}) is $\\Delta=4x_q^2-16y_q$, and $\\Delta=0$ yields $y_q=x_q^2\/4$, which implies $Q\\in \\Psi$. Since $\\Psi$ is the location of points that are equidistant from $P$ and  $m$, we may conclude that the fold operation has a unique solution when $Q$  is equidistant to $P$ and $m$, two solutions when $Q$ is closer to $m$ (i.e., $y_q<x_q^2\/4$), and no solution when $Q$ is closer to $P$ (i.e., $y_q>x_q^2\/4$). \n\n\n\nFig.~\\ref{fO6} shows an example for the case of two solutions.\n\n\\begin{figure}[!htb]\n\\centering\n\n\\begin{pspicture}(-3,-2.5)(3,4.5)\n\\rput{-45}\n\\psplot[linecolor=vector,plotpoints=100,linewidth=1pt]{-2.5}{2.5}{x x mul 2 div}\n\\qdisk(0,.5){2pt}\n\\uput[90]{*0}(0,.5){$P$}\n\\psline(-2.5,-.5)(2.5,-.5)\n\\uput[-135]{*0}(-2,-.5){$m$}\n\\psline[linecolor=foldline,linewidth=1pt](-2.5,.53)(2.5,-.6)\n\\psline[linecolor=foldline,linewidth=1pt](-.4,-1.5)(2.5,2.45)\n\\qdisk(0.57,-.17){2pt}\n\\uput[-30]{*0}(0.7,-.17){$Q$}\n\\uput[0]{*0}(-1.8,1.56){$\\Psi$}\n\\uput[-135]{*0}(-2,.6){$\\chi_1$}\n\\uput[-45]{*0}(-.2,-1.3){$\\chi_2$}\n}\n\\end{pspicture}\n\\caption{Example of operation \\ref{O6} with two solutions.}\n\\label{fO6} \n\\end{figure}\n\n\\begin{operation}Given points $P$, $Q$, and lines $m$, $n$, with $P\\notin m$, $Q\\notin n$, and $P\\neq Q$ or $m \\neq n$, construct a fold line so that $\\mathcal{F}(P)\\in m$ and $\\mathcal{F}(Q)\\in n$.\n\\label{O7}\n\\end{operation}\n\n\\noindent\nThis operation derives from the application of incidence \\ref{I3} to two distinct point-line pairs. Its solution is a fold line that is tangent to both a parabola $\\Psi$ with focus $P$ and directrix $m$, and a parabola $\\Theta$ with focus $Q$ and directriz $n$. \n\nAgain, assume the same parabola $\\Psi$ of Fig.~\\ref{parab}, given by Eq.~(\\ref{p1}). Assume also that  $Q$ is located at $(x_q, y_q)$, and its reflection $Q'=\\mathcal{F}(Q)$ is at $(x_q', y_q')$. Then, segment $\\overline{QQ'}$ has a slope $(y_q - y_q')\/(x_q-x_q')$. \nThe fold line $\\chi$, given by Eq.~(\\ref{chipar}), has a slope $t\/2$ and is perpendicular to $\\overline{QQ'}$ (because it reflects $Q$ onto $Q'$). Therefore, \n\\begin{equation}\n\\frac{t}{2}=-\\frac{x_q-x_q'}{y_q-y_q'}.\n\\label{o61}\n\\end{equation}\nFurther, $\\chi$ passes through the midpoint of $\\overline{QQ'}$, which is located at $((x_q+x_q')\/2, (y_q + y_q')\/2)$. Replacing those coordinates into Eq.~(\\ref{chipar}) produces  \n\\begin{equation}\n2(y_q+y_q') = t\\left(x_q+x_q'-t\\right).\n\\label{o62}\n\\end{equation}\nFinally, eliminating $t$ from Eqs.~(\\ref{o61}) and (\\ref{o62}) produces\n\\begin{equation}\n(y_q+y_q')(y_q-y_q')^2=-(x_q^2-x_q'^2)(y_q-y_q')-2(x_q-x_q')^2.\n\\label{cubic}\n\\end{equation}\n\nFor a given line $n$, the coordinates of $Q'$ satisfy an equation of the form\n\\begin{equation}\nax_q'+by_q'+c=0,\n\\label{linen}\n\\end{equation}\nwhere $a$, $b$ and $c$ are constants.  \n\nEqs.~(\\ref{cubic}) and (\\ref{linen}) may be solved for $x_q'$ and $y_q'$. Substituting this solution into Eq.~(\\ref{o61}) yields $t$, which defines the fold line $\\chi$ in Eq.~(\\ref{chipar}). Two cases may be considered: \n\\begin{enumerate}\n\\item If $m\\parallel n$, then $Q'$ is on a horizontal line and so $y_q' = -c\/b$. In that case, Eq.~(\\ref{cubic}) is quadratic in $x_q'$ and may have zero to two solutions.\n\n\\item If $m\\nparallel n$, solving Eq.~(\\ref{linen}) for $x_q'$ or $y_q'$ and replacing in Eq.~(\\ref{cubic}) produces a cubic equation with one to three solutions. An example for the latter case is shown in Fig.~\\ref{fO7}. \n\\end{enumerate}\n\nLet us investigate further the conditions to ensure the existence of solutions. As noted above, the operation may not have a solution only in the case of $m\\parallel n$. Re-arranging Eq.~(\\ref{cubic}) produces\n\\begin{equation}\n(2-y_q+y_q')(x_q-x_q')^2+2x_q(y_q-y_q')(x_q-x_q')+(y_q+y_q')(y_q-y_q')^2=0,\n\\label{cubic2}\n\\end{equation}\nwhich is a quadratic equation in $(x_q-x_q')$. The discriminant is \n\\begin{equation}\n\\Delta=4x_q^2(y_q-y_q')^2-4(2-y_q+y_q')(y_q+y_q')(y_q-y_q')^2,\n\\end{equation}\nand letting $\\Delta\\ge 0$ produces\n\\begin{equation}\nx_q^2+(y_q-1)^2\\ge (y_q'+1)^2.\n\\label{cubic3}\n\\end{equation}\n\nThe left side of Eq.~(\\ref{cubic3}) is the squared distance between $P$ and $Q$, and the right side is the squared distance between $m$ and $n$.  This result does not seem reported in the literature, and may be stated as a theorem:\n\n\\begin{theorem}\nGiven points $P$, $Q$, and lines $m$, $n$, with $P\\notin m$, $Q\\notin n$, and $P\\neq Q$ or $m \\neq n$, a fold line that places $P$ on $m$ and $Q$ on $n$ exists iff the distance between $P$ and $Q$ is larger than or equal to the distance between $m$ and $n$.\n\\end{theorem}\n\n\\begin{figure}[!htb]\n\\centering\n\n\\psset{xunit=1.2cm,yunit=1.2cm}\n\\begin{pspicture}(-2,-4)(4,2.5)\n\\rput{-45}\n\\psplot[linecolor=vector,plotpoints=100,linewidth=1pt]{-1.5}{2.5}{x x mul 2 div}\n\\qdisk(0,.5){2pt}\n\\uput[90]{*0}(0,.5){$P$}\n\\psline(-1.5,-.5)(3.5,-.5)\n\\uput[90]{*0}(-1.5,-.5){$m$}\n\\qdisk(2,0){2pt}\n\\uput[0]{*0}(2,0){$Q$}\n\\psline(1,-2.5)(1,2.5)\n\\uput[-90]{*0}(1.05,-2){$n$}\n\\psplot[linecolor=vector,plotpoints=100,linewidth=1pt]{1.5}{4}{x 2 mul 3 sub sqrt}\n\\psplot[linecolor=vector,plotpoints=100,linewidth=1pt]{1.5}{3.5}{x 2 mul 3 sub sqrt -1 mul}\n\\psplot[linecolor=foldline,plotpoints=100,linewidth=1pt]{-1.5}{3.5}{x -1.5 add -.53 mul -.94 add}\n\\psplot[linecolor=foldline,plotpoints=100,linewidth=1pt]{-1.5}{4}{x -1.5 add .65 mul 0.76 add}\n\\psplot[linecolor=foldline,plotpoints=100,linewidth=1pt]{.7}{2.5}{x -1.5 add 2.85 mul 0.18 add}\n\\uput[-90]{*0}(-1.5,-1.2){$\\chi_2$}\n\\uput[135]{*0}(-1.4,0.5){$\\chi_3$}\n\\uput[180]{*0}(0.6,-1.8){$\\chi_1$}\n\\uput[0]{*0}(-1.5,1.1){$\\Psi$}\n\\uput[-90]{*0}(3.8,2.1){$\\Theta$}\n}\n\\end{pspicture}\n\\caption{Example of operation \\ref{O7} with three solutions.} \n\\label{fO7}\n\\end{figure}\n\n\\begin{operation}\nGiven point $P$ and lines $m$ and $n$, with $P\\notin m$, construct a fold line so that $\\mathcal{F}(P)\\in m$ and $\\mathcal{F}(n)=n$ and $\\exists Q\\in n, \\mathcal{F}(Q)\\neq Q$.\n\\label{O8}\n\\end{operation}\n\n\\noindent \nThis operation is defined by application of incidences \\ref{I3} and \\ref{I5}. Its solution is a fold line that is tangent to a parabola with focus $P$ and directrix $m$, and is perpendicular to line $n$. \n\nAs in the previous operation, assume the same parabola of Fig.~\\ref{parab} given by Eq.~(\\ref{p1}), and a line $n$ with equation $ax+by+c=0$. The fold line $\\chi$ has a slope $t\/2$ and is perpendicular to $n$. Two cases may be considered:\n\\begin{enumerate}\n\\item If $m\\nparallel n$, then $a\\neq 0$. Therefore,\n\\begin{equation}\n\\frac{t}{2}=-\\frac{b}{a}\n\\end{equation}\nwhich has a unique solution for $t$ (Fig.~\\ref{fO8}). Knowing $t$, Eq.~(\\ref{chipar}) defines the fold line $\\chi$. \n\n\\item If $m\\parallel n$ then $n$ is a horizontal line and cannot be perpendicular to any tangent to parabola $\\Psi$. In this case, the operation does not have a solution.\n\\end{enumerate}\n\n\\begin{figure}[!htb]\n\\centering\n\n\\begin{pspicture}(-3,-2.5)(3,3)\n\\psset{xunit=1.1cm,yunit=1.1cm}\n\\rput{-30}\n\\psplot[linecolor=vector,linewidth=1pt,plotpoints=100]{-2}{2.1}{x x mul 2 div}\n\\qdisk(0,1){2pt}\n\\uput[90]{*0}(0,1){$P$}\n\\psline(-2,-1)(2,-1)\n\\uput[90]{*0}(-2,-1){$m$}\n\\psplot[plotpoints=10]{0}{3}{x -2 add}\n\\psplot[linecolor=foldline,linewidth=1pt,plotpoints=10]{-2}{1.25}{x -1 mul -.5 add}\n\\uput[90]{*0}(2.5,.5){$n$}\n\\uput[0]{*0}(-1.8,1.56){$\\Psi$}\n\\uput[-30]{*0}(1.2,-1.6){$\\chi$}\n}\n\\end{pspicture}\n\\caption{Example of operation \\ref{O8} with one solution.}\n\\label{fO8} \n\\end{figure}\n\n\\subsection{Summary}\n\nTable \\ref{table2} lists the incidence constraints that define each operation and their number of solutions, Table \\ref{table2b} lists the conditions for the existence of solutions, and Table \\ref{table3} restates the operations as folding actions of a medium $\\mathcal{O}$.\n\n\\begin{table}[!htb]\n\\centering\n\\begin{threeparttable}\n\\caption{Incidence constraints and number of solutions of the elementary single-fold operations.}\n\\label{table2}\n\\centering\n\\begin{tabularx}{\\textwidth}{lXl}\n\\toprule\nOperation & Incidence constraints & Solutions\\\\\n\\midrule\n\\ref{O1} & $\\mathcal{F}(P)=Q$, with $P\\neq Q$ & 1\\\\\n\\ref{O2} & $\\mathcal{F}(m)=n$, with $m\\neq n$ & 1, 2\\\\\n\\ref{O3} & $\\mathcal{F}(m)=m$,and $\\forall P\\in m, \\mathcal{F}(P)=P$  &1\\\\\n\\ref{O4} & $\\mathcal{F}(P)=P$ and $\\mathcal{F}(Q)=Q$, with $P\\neq Q$ & 1\\\\\n\\ref{O5} & $\\mathcal{F}(P)=P$ and $\\mathcal{F}(m)=m$, and $\\exists Q\\in m, \\mathcal{F}(Q)\\neq Q$ & 1\\\\\n\\ref{O6} & $\\mathcal{F}(P)\\in m$, with $P\\notin m$, and $\\mathcal{F}(Q)=Q$ & 0 -- 2\\\\\n\\ref{O7} & $\\mathcal{F}(P)\\in m$, with $P\\notin m$, $\\mathcal{F}(Q)\\in n$, with $Q\\notin n$, and $P\\neq Q$ or $m\\neq n$ & 0 -- 3\\\\\n\\ref{O8} &  $\\mathcal{F}(P)\\in m$, with $P\\notin m$, and $\\mathcal{F}(n)=n$, and $\\exists Q\\in n$, $\\mathcal{F}(Q)\\neq~Q$ & 0, 1\\\\\n\\bottomrule\n\\end{tabularx}\n\\end{threeparttable}\n\\end{table}\n\n\\begin{table}[!htb]\n\\centering\n\\begin{threeparttable}\n\\caption{Conditions for the existence of solutions of the elementary single-fold operations.}\n\\label{table2b}\n\\centering\n\\begin{tabularx}{\\textwidth}{lX}\n\\toprule\nOperation & Conditions\\\\\n\\midrule\n\\ref{O1} to \\ref{O5} & none\\\\\n\\ref{O6} & distance between $P$ and $Q$ larger than or equal to the distance between $Q$ and $m$\\\\\n\\ref{O7} & distance between $P$ and $Q$ larger than or equal to the distance between $m$ and $n$\\\\\n\\ref{O8} &  $m\\nparallel n$\\\\\n\\bottomrule\n\\end{tabularx}\n\\end{threeparttable}\n\\end{table}\n\n\n\\begin{table}[!htb]\n\\centering\n\\begin{threeparttable}\n\\caption{Elementary single-fold operations restated as folding actions.}\n\\label{table3}\n\\centering\n\\begin{tabularx}{\\textwidth}{cX}\n\\toprule\nNo. &\\multicolumn{1}{c}{Action\\tnote{a}}\\\\\n\\midrule\n\\ref{O1} & Given two distinct points $P$ and $Q$, fold $\\mathcal{O}$ to place $P$ onto $Q$.\\\\\n\\ref{O2} &  Given two distinct lines $m$ and $n$, fold $\\mathcal{O}$ to align $m$ and $n$.\\\\\n\\ref{O3} & Fold along a given a line $m$.\\\\\n\\ref{O4} & Given two distinct points $P$ and $Q$, fold $\\mathcal{O}$ along a line passing through $P$ and $Q$.\\\\\n\\ref{O5} & Given a line $m$ and a point $P$, fold $\\mathcal{O}$ along a line passing through $P$ to reflect half of $m$ onto its other half.\\\\\n\\ref{O6} & Given a line $m$, a point $P$ not on $m$ and a point $Q$, fold $\\mathcal{O}$ along a line passing through $Q$ to place $P$ onto $m$.\\\\\n\\ref{O7} & Given two lines $m$ and $n$, a point $P$ not on $m$ and a point $Q$ not on $n$, where $m$ and $n$ are distinct or $P$ and $Q$ are distinct, fold $\\mathcal{O}$ to place $P$ onto $m$, and $Q$ onto $n$.\\\\\n\\ref{O8} & Given two lines $m$ and $n$, and a point $P$ not on $m$, fold $\\mathcal{O}$ to place $P$ onto $m$, and to reflect half of $n$ onto its other half.\\\\\n\\bottomrule\n\\end{tabularx}\n\\begin{tablenotes}\n \\small\n\\item [a]$\\mathcal{O}$ denotes the medium in which folds are performed; e.g., a sheet of paper, fabric, plastic, metal or any other foldable material.\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{table}\n\n\n\\section{Discussion}\n\\label{foldingalong}\n\nThe complete set of elementary single-fold operations contains eight operations, listed in Table \\ref{table3}. Operations \\ref{O1}, \\ref{O2} and \\ref{O4} to \\ref{O8} constitute Justin's  original set \\citep{Justin1986}, and operation \\ref{O3} is the new addition proposed here. \n\\ref{O3} does not create a new line and has been ignored in previous studies on origami constructions \\citep{Alperin2006, Ghourabi2013}.\\footnote{In his formulation, \\citet{Justin1986} allowed for solutions where the fold line itself coincides with an existent line. However, such action was not considered as a fold operation on its own.} However, it is a valid  elementary single-fold operation and completeness of the set demands its inclusion.\n\nThere is also a more practical reason for not ignoring \\ref{O3}. Folding a sheet of paper along a line superposes the paper on both sides of the fold line, in two layers. In origami mathematics, it is assumed that all lines and points marked on one layer are also defined on the layers above and below, as if the paper were ``transparent'' \\citep{Geretschlager1995, Martin1998}. However, it is not so in actual paper folding: a given origami work may require to fold, e.g., the top layer along a line marked on the layer below it. Such is the case when folding parallel lines for building tessellation grids \\citep[see instruction 5 for a triangle grid in p. 8]{Gjerde2008}. \n\nFig.~\\ref{parf} shows a simple example. Assume a sheet of paper (not necessarily square) in which two parallel lines $m$ and $n$ are marked, and assume that we want to create a third equidistant parallel line $o$. Both steps (1) and (2) require to fold along given lines ($n$ and $m$, respectively). Naturally, it would be also possible to use the points of intersections of $m$ and $n$ with the borders of the paper as references, instead of the lines themselves. Thus, the instruction for step (2) could be: fold the top layer along a line passing through points $P$ and $Q$ on the bottom layer, where $P$ and $Q$ are the intersections of $m$ with the upper and lower edges of the paper. However,  in actual practice it is much simpler and convenient to perform the fold by aligning it with line $m$. Further, it might be the case that the borders of the paper are not well defined (or have not been defined) or that it is considered as a theoretical infinite plane. \n\n\\begin{figure}[!htb]\n\\centering\n\\begin{pspicture}(0,-1)(15,4)\n\\psset{xunit=.8cm,yunit=.8cm}\n\\psframe[linewidth=0.5pt](0,0)(4,4) \n\\uput[45](.5,0){$m$}\n\\uput[45](1.5,0){$n$}\n\\psline[linewidth=1pt](.5,0.1)(1.5,3.9)\n\\psline[linewidth=1pt](1.5,0.1)(2.5,3.9)\n\\psarc[linecolor=darkgray]{->}(1.5,0){2}{40}{110}\n\n\\pspolygon[linewidth=0.5pt](6,0)(7.5,0)(8.5,4)(6,4) \n\\psline[linewidth=1pt](6.5,0.1)(7.5,3.9)\n\\pspolygon[linewidth=0.5pt,fillstyle=solid,fillcolor=backside](5.3,1.25)(7.5,0)(8.5,4)(7.25,4.7) \n\\psline[linestyle=dashed,linecolor=gray](6.62,0.5)(7.5,3.9)\n\\psline[linestyle=dashed,linecolor=gray](6.9,4)(8.5,4)\n\\uput[135](6.6,0){$m$}\n\\uput[45](7.5,0){$n$}\n\\psarcn[linecolor=darkgray]{->}(6.7,0.5){1.75}{95}{60}\n\n\\pspolygon[linewidth=0.5pt](10,0)(11.5,0)(12.5,4)(10,4) \n\\psline[linewidth=1pt](10.53,0.1)(10.62,.45)\n\\pspolygon[linewidth=0.5pt,fillstyle=solid,fillcolor=backside](10.62,.45)(11.5,0)(12.5,4)(11.62,4.5) \n\\pspolygon[linewidth=0.5pt,fillstyle=solid,fillcolor=white](10.61,.45)(12.1,0.45)(12.1,4.45)(11.61,4.5) \n\n\\uput[135](10.6,0){$m$}\n\\uput[45](11.5,0){$n$}\n\\psarcn[linecolor=darkgray]{->}(11.7,0.){1.75}{95}{60}\n\n\\psframe[linewidth=0.5pt](14,0)(18,4) \n\\uput[45](14.5,0){$m$}\n\\uput[45](15.5,0){$n$}\n\\uput[45](16.5,0){$o$}\n\\psline[linewidth=1pt](14.5,0.1)(15.5,3.9)\n\\psline[linewidth=1pt](15.5,0.1)(16.5,3.9)\n\\psline[linewidth=1pt](16.5,0.1)(17.5,3.9)\n\n\\uput[-90](2,-.3){(1)}\n\\uput[-90](7.25,-.3){(2)}\n\\uput[-90](11.25,-.3){(3)}\n\\uput[-90](16,-.3){(4)}\n\n\\end{pspicture}\n \n\\caption{Given parallel lines $m$ and $n$, create a third equidistant parallel line at the left of $n$. (1) Fold along line $n$. (2) Fold the top layer along line $m$ in the bottom layer. (3) Unfold. (4) Final result.} \n\\label{parf}\n\n\\end{figure}\n\n\\ref{O3} is also a common instruction also for building figurative models \\citep[see steps 8 and 16 of ``Baby'' in page 88, and step 6 of ``Songbird 2'' in page 340]{Lang2012}.  Thus, the operation should be in the repertoire of, e.g., computational systems for origami simulation and design \\citep{Ida2009,Tsuruta2009}.   \n\n\\section{Conclusion}\n\nAnalysis of reflections of points and lines on a plane subject to incidence constraints has determined a complete set of eight elementary single-fold operations. Precise definitions and conditions of existence of solutions of all operations are given in Tables \\ref{table2} to \\ref{table3}, which may be useful to scientific and technological applications of origami. \n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAll spaces in this paper are Hausdorff topological spaces. We will use\n$\\Phi:X\\smap Y$ to designate that $\\Phi$ is a map from $X$ to the\nnonempty subsets of $Y$, i.e.\\ a \\emph{set-valued mapping}. Such a\nmapping is \\emph{lower semi-continuous}, or l.s.c., if the set \\[\n\\Phi^{-1}[U]=\\{x\\in X:\\Phi(x)\\cap U\\neq \\emptyset\\}\n\\]\nis open in $X$, for every open $U\\subset Y$. Also, let us recall that a\nmap $f:X\\to Y$ is a \\emph{selection} for $\\Phi:X\\smap Y$ if $f(x)\\in\n\\Phi(x)$, for all $x\\in X$.\\medskip\n\nLet $n\\geq-1$. A family $\\mathscr{S}$ of subsets of a space $Y$ is\n\\emph{equi-$LC^n$} \\cite{michael:56b} if every neighbourhood\n$U$ of a point $y\\in \\bigcup\\mathscr{S}$ contains a neighbourhood $V$\nof $y$ such that for every $S\\in \\mathscr{S}$, every continuous map\n$g:\\s^k\\to V\\cap S$ of the $k$-sphere $\\s^k$, $k\\leq n$, can be\nextended to a continuous map $h:\\B^{k+1}\\to U\\cap S$ of the\n$(k+1)$-ball $\\B^{k+1}$.  A space $S$ is called $C^n$ if for every\n$k\\leq n$, every continuous map $g:\\s^k\\to S$ can be extended to a\ncontinuous map $h:\\B^{k+1}\\to S$. In these terms, a family\n$\\mathscr{S}$ of subsets of $Y$ is equi-$LC^{-1}$ if it consists of\nnonempty subsets; similarly, each nonempty subset $S\\subset Y$ is\n$C^{-1}$.\\medskip\n\nLet $\\mathscr{F}(Y)$ be the collection of all nonempty closed subsets\nof a space $Y$. The following theorem was proved by Ernest Michael,\nsee \\cite[Theorem 1.2]{michael:56b}, and is commonly called the\n\\emph{finite-dimensional selection theorem}.\n\n\\begin{theorem}\n  \\label{theorem-st-app-v10:1}\n  Let $X$ be a paracompact space with $\\dim(X)\\leq n+1$, $Y$ be a\n  com\\-pletely metrizable space, and $\\mathscr{S}\\subset \\mathscr{F}(Y)$\n  be an equi-$LC^n$ family such that each $S\\in \\mathscr{S}$ is\n  $C^n$. Then each l.s.c.\\ mapping $\\Phi:X\\to \\mathscr{S}$ has a\n  continuous selection.\n\\end{theorem}\n\nThe original proof of Theorem \\ref{theorem-st-app-v10:1} in\n\\cite{michael:56b} takes up most of that paper, and is accomplished in\n6 steps. Other proofs of this theorem can be found in the monograph\n\\cite{repovs-semenov:98}, and the book\n\\cite{zbMATH00193669}. Actually, in \\cite{repovs-semenov:98} are given\ntwo different approaches to obtain the theorem --- the one which\nfollows the original Michael's proof, and another one based on\nfiltrations \\cite{schepin-brodsky:96}. Other proofs were given by\nother authors, see e.g. \\cite{MR2192951} and \\cite{gutev:05}.\nHowever, what all these proofs have in common is that they may somehow\ndiscourage the casual reader and make Theorem\n\\ref{theorem-st-app-v10:1} not so accessible to wider audience. The\nmain purpose of this paper is to fill in this gap, and present a\nsimplified and self-contained proof of this theorem. \\medskip\n\nThe paper is organised as follows. The next section contains a brief\nreview of canonical maps and partitions of unity, which is essential\nfor the proper understanding of any of the available proofs of Theorem\n\\ref{theorem-st-app-v10:1}. In this regard, let us explicitly remark\nthat these considerations were not made readily available in previous\nproofs, so they are now included to make the exposition\nself-contained. The essential preparation for the proof of Theorem\n\\ref{theorem-st-app-v10:1} starts in Section\n\\ref{sec:asph-sequ-select}, which contains a selection theorem for\nfinite aspherical sequences of lower locally constant mappings\n(Theorem \\ref{theorem-nerves-v2:5}). This theorem is similar to a\ntheorem of Uspenskij, see \\cite[Theorem 1.3]{uspenskij:98}, and\nrepresents a relaxed version of another theorem proved by the author,\nsee \\cite[Theorem 3.1]{gutev:05}. Section \\ref{sec:gener-asph-sequ}\ncontains several simple constructions of finite aspherical sequences\nof sets providing the main interface between such sequences of sets\nand the property of equi-$LC^n$. Finally, the proof of Theorem\n\\ref{theorem-st-app-v10:1} is accomplished in Section\n\\ref{sec:select-lcn-valu}. It is based on two constructions which are\nalso present in Michael's proof. The one, Proposition\n\\ref{proposition-st-app-v15:2}, relates l.s.c.\\ mappings to lower\nlocally constant mappings; the other --- Proposition\n\\ref{proposition-st-app-v15:3}, relates selections for lower locally\nconstant mappings to approximate selections for l.s.c.\\\nmappings. These constructions are applied together with Theorem\n\\ref{theorem-nerves-v2:5} to deal with two selection properties of\nl.s.c.\\ equi-$LC^n$-valued mappings, see Theorems\n\\ref{theorem-st-app-v9:1} and \\ref{theorem-st-app-v17:1}. The\nproof of Theorem \\ref{theorem-st-app-v10:1} is then obtained as an\nimmediate consequence of these properties.\n\n\\section{Canonical maps and partitions of unity}\n\\label{sec:canon-maps-part}\n\nThe \\emph{cozero set}, or the \\emph{set-theoretic support}, of a\nfunction $\\xi:X\\to \\R$ is the set $\\coz(\\xi)=\\{x\\in X:\\xi(x)\\neq 0\\}$.\nA collection $\\xi_a:X\\to [0,1]$, $a\\in \\mathscr{A}$, of continuous\nfunctions on a space $X$ is a \\emph{partition of unity} if\n$\\sum_{a\\in \\mathscr{A}}\\xi_a(x)=1$, for each $x\\in X$.  Here,\n``$\\sum_{a\\in \\mathscr{A}}\\xi_a(x)=1$'' means that only countably many\nfunctions $\\xi_a$'s do not vanish at $x$, and the series composed by\nthem is convergent to 1.  For a cover $\\mathscr{U}$ of a space $X$, a\npartition of unity $\\{\\xi_U:U\\in \\mathscr{U}\\}$ on $X$ is\n\\emph{index-subordinated} to $\\mathscr{U}$ if $\\coz(\\xi_U)\\subset U$,\nfor each $U\\in \\mathscr{U}$, see Remark\n\\ref{remark-st-app-vgg-rev:1}. The following theorem is well known, it\nis a consequence of Urysohn's characterisation of normality\n\\cite{MR1512258} and the Lefschetz lemma \\cite{MR0007093}.\n\n\\begin{theorem}\n  \\label{theorem-st-app-v3:1}\n  Every locally finite open cover of a normal space has an\n  index-subordinated partition of unity.\n\\end{theorem}\n\nA partition of unity $\\{\\xi_a:a\\in \\mathscr{A}\\}$ on a space $X$ is\ncalled \\emph{locally finite} if $\\{\\coz(\\xi_a):a\\in \\mathscr{A}\\}$ is\na locally finite cover of $X$. Complementary to Theorem\n\\ref{theorem-st-app-v3:1} is the following important property of\npartitions of unity; it follows from a construction of M. Mather, see\n\\cite[Lemma]{MR0281155} and \\cite[Lemma 5.1.8]{engelking:89}.\n\n\\begin{theorem}\n  \\label{theorem-st-app-vgg-rev:1}\n  If a cover $\\mathscr{U}$ of a space $X$ has an index-subordinated\n  partition of unity, then $\\mathscr{U}$ also has an\n  index-subordinated locally finite partition of unity. \n\\end{theorem}\n\nBy a \\emph{simplicial complex} we mean a collection $\\Sigma$ of\nnonempty finite subsets of a set $S$ such that $\\tau\\in \\Sigma$,\nwhenever $\\emptyset\\neq \\tau\\subset \\sigma\\in \\Sigma$. The set\n$\\bigcup\\Sigma $ is the \\emph{vertex set} of $\\Sigma$, while each\nelement of $\\Sigma$ is called a \\emph{simplex}. The\n\\emph{$k$-skeleton} $\\Sigma^k$ of $\\Sigma$ ($k\\geq 0$) is the\nsimplicial complex\n$\\Sigma^{k}=\\{\\sigma\\in \\Sigma:\\card(\\sigma)\\leq k+1\\}$, where\n$\\card(\\sigma)$ is the cardinality of $\\sigma$. In the sequel, for\nsimplicity, we will identify the vertex set of $\\Sigma$ with its\n$0$-skeleton $\\Sigma^0$.  In these terms, a \\emph{simplicial map}\n$g:\\Sigma_1\\to \\Sigma_2$ is a map $g:\\Sigma_1^0\\to \\Sigma_2^0$ between\nthe vertices of simplicial complexes $\\Sigma_1$ and $\\Sigma_2$ such\nthat $g(\\sigma)\\in \\Sigma_2$, for each $\\sigma\\in \\Sigma_1$. If\n$g:\\Sigma_1\\to \\Sigma_2$ is a simplicial map and\n$g:\\Sigma_1^0\\to \\Sigma_2^0$ is bijective, then the inverse $g^{-1}$\nis also a simplicial map, and we say that $g$ is a \\emph{simplicial\n  isomorphism}.\\medskip\n\nThe set $\\Sigma_S$ of all nonempty finite subsets of a set $S$ is a\nsimplicial complex. Another natural example is the \\emph{nerve}\n$\\mathscr{N}(\\mathscr{U})$ of a cover $\\mathscr{U}$ of a set $X$,\nwhich is the subcomplex of $\\Sigma_\\mathscr{U}$ defined by\n\\begin{equation}\n  \\label{eq:st-app-vgg-rev:8}\n\\mathscr{N}(\\mathscr{U})= \\left\\{\\sigma\\in\n  \\Sigma_\\mathscr{U}:\\bigcap\\sigma\\neq\\emptyset\\right\\}.  \n\\end{equation}\nThe $k$-skeleton of $\\mathscr{N}(\\mathscr{U})$ is denoted by\n$\\mathscr{N}^k(\\mathscr{U})$, and the vertex set\n$\\mathscr{N}^0(\\mathscr{U})$ of $\\mathscr{N}(\\mathscr{U})$ is actually\n$\\mathscr{U}$ because we can always assume that\n$\\emptyset\\notin \\mathscr{U}$.\\medskip\n\nFor a set $\\mathscr{A}$, let $\\ell_1 (\\mathscr{A})$ be the linear\nspace of all functions $y:\\mathscr{A}\\to \\R$ with\n${\\sum _{a \\in \\mathscr{A}} |y (a)| < \\infty}$. In fact,\n$\\ell_1(\\mathscr{A})$ is a Banach space when equipped with the norm\n$\\|y\\|_1 = \\sum _{a\\in \\mathscr{A}} |y (a)| $, but this will play no\nrole in the paper.  The vertex set $\\Sigma^0$ of a simplicial complex\n$\\Sigma$ is a linearly independent subset of $\\ell_1(\\Sigma^0)$, where\neach $v\\in \\Sigma^0$ is identified with its characteristic function\n$v:\\Sigma^0\\to \\{0,1\\}$, namely with the function $v(u)=0$ for\n$u\\neq v$, and $v(v)=1$.  Then to each $\\sigma\\in \\Sigma$ one can\nassociate the \\emph{geometric simplex} $|\\sigma|=\\conv(\\sigma)$, which\nis the convex hull of $\\sigma$. Thus, $|\\sigma|$ is a\n\\emph{$k$-dimensional simplex} if and only if\n$\\hbox{Card}(\\sigma)=k+1$. The set\n$|\\Sigma|=\\bigcup_{\\sigma\\in \\Sigma}|\\sigma|\\subset\\ell_1(\\Sigma^0)$\nis called the \\emph{geometric realisation} of $\\Sigma$. As a\ntopological space, we will consider $|\\Sigma|$ endowed with the\n\\emph{Whitehead topology} \\cite{MR1576810,MR0030759}. In this\ntopology, a subset $U\\subset |\\Sigma|$ is open if and only if\n$U\\cap |\\sigma|$ is open in $|\\sigma|$, for every $\\sigma\\in \\Sigma$.\nLet us explicitly remark that the Whitehead topology on $|\\Sigma|$ is\nnot necessarily the subspace topology on $|\\Sigma|$ as a subset of the\nBanach space $\\ell_1\\left(\\Sigma^0\\right)$. However, both topologies\ncoincide on each geometric simplex $|\\sigma|$, for $\\sigma\\in\n\\Sigma$. \\medskip\n\nIf $p\\in |\\sigma|$ for some $\\sigma\\in \\Sigma$, then $p$ is both an\nelement $p\\in \\ell_1(\\Sigma^0)$ and a unique convex\ncombination of the elements of\n$\\sigma\\subset \\Sigma^0\\subset\n\\ell_1(\\Sigma^0)$. Hence,\nthe geometric realisation $|\\Sigma|$ is the set of all\n$p\\in\\ell_1(\\Sigma^0)$ such that\n\\begin{equation}\n  \\label{eq:st-app-vgg-rev:1}\n  p(v)\\geq 0,\\ v\\in \\Sigma^0,\\quad\\text{and}\\quad\n  \\coz(p)=\\left\\{v\\in \\Sigma^0: p(v)>\n    0\\right\\}\\in \\Sigma.\n\\end{equation}\nHere, $p(v)$ is called the $v$-th \\emph{barycentric} (or\n\\emph{affine}) \\emph{coordinate of} ${p\\in |\\Sigma|}$, while the\nsimplex $\\coz(p)\\in \\Sigma$ is called the \\emph{carrier} of $p$, and\ndenoted by $\\car(p)=\\coz(p)$. Since the representation\n$p= \\sum_{v\\in \\car(p)}p(v)\\cdot v$ is unique, the carrier $\\car(p)$\nis the minimal simplex of $\\Sigma$ with the property that\n$p\\in|\\car(p)|$.\\medskip\n\nTo each vertex ${v}\\in \\Sigma^0$, we can now associate the function\n$\\alpha_v:|\\Sigma|\\to [0,1]$, defined by\n\\begin{equation}\n  \\label{eq:st-app-vgg-rev:3}\n  \\alpha_v(p)=p(v),\\quad \\text{for every $p\\in\n    |\\Sigma|$.} \n\\end{equation}\n  It is\ncalled the $v$-th \\emph{barycentric coordinate function} and\nis continuous being affine on each simplex $|\\sigma|$, for\n$\\sigma\\in \\Sigma$. The cozero\nset $\\coz(\\alpha_v)$ of $\\alpha_v$ is called the\n\\emph{open star} of the vertex $v\\in \\Sigma^0$, and\ndenoted by\n\\begin{equation}\n  \\label{eq:st-app-v4:3}\n  \\st\\langle v\\rangle\n  =\\big\\{p\\in |\\Sigma|: \\alpha_v(p)>0\\big\\}.\n\\end{equation}\nClearly, the open star $\\st\\langle v\\rangle$ is open in\n$|\\Sigma|$ because $\\alpha_v$ is continuous.  The following\nproposition is an immediate consequence of \\eqref{eq:st-app-vgg-rev:1},\n\\eqref{eq:st-app-vgg-rev:3} and \\eqref{eq:st-app-v4:3}.\n\n\\begin{proposition}\n  \\label{proposition-st-app-v2:1}\n  If $\\Sigma$ is a simplicial complex, then the collection\n  $\\left\\{\\alpha_v:v\\in \\Sigma^0\\right\\}$ is a\n  partition of unity on $|\\Sigma|$ with\n  $\\coz(\\alpha_v)=\\st\\langle v\\rangle$, for each\n  $v\\in \\Sigma^0$.\n\\end{proposition}\n\nWe now turn to the other essential concept in this section. For a\ncover $\\mathscr{U}$ of a space $X$, a continuous map\n$f:X\\to |\\mathscr{N}(\\mathscr{U})|$ is called \\emph{canonical for\n  $\\mathscr{U}$} if\n\\begin{equation}\n  \\label{eq:st-app-v1:3}\n  f^{-1}(\\st\\langle U\\rangle)\\subset U,\\quad \\text{for every $U\\in\n    \\mathscr{U}$.} \n\\end{equation}\nCanonical maps are essentially partitions of unity, which are\nindex-subordinated to the corresponding cover of the space.\n\n\\begin{theorem}\n  \\label{theorem-st-app-v12:1}\n    A cover $\\mathscr{U}$ of a space $X$ has an index-subordinated\n  partition of unity if and only if $\\mathscr{U}$ has a canonical\n  map. \n\\end{theorem}\n\n\\begin{proof}\n  Let $\\mathscr{U}$ be a cover of $X$ and $\\alpha_U$, $U\\in \\mathscr{U}$,\n  be the barycentric coordinate functions of\n  $|\\mathscr{N}(\\mathscr{U})|$.\\smallskip\n\n  Suppose that $f:X\\to |\\mathscr{N}(\\mathscr{U})|$ is a canonical map\n  for $\\mathscr{U}$. Since $f$ is continuous, by Proposition\n  \\ref{proposition-st-app-v2:1},\n  $\\{\\alpha_U\\circ f:U\\in \\mathscr{U}\\}$ is a partition of unity on\n  $X$. By the same proposition and \\eqref{eq:st-app-v1:3}, we also\n  have that\n  \\[\n  \\coz(\\alpha_U\\circ f)=f^{-1}(\\coz(\\alpha_U))=f^{-1}(\\st\\langle\n  U\\rangle)\\subset U,\\quad U\\in \\mathscr{U}.\n  \\]\n  \n  Conversely, suppose that $\\mathscr{U}$ has an index-subordinated\n  partition of unity. Then by Theorem \\ref{theorem-st-app-vgg-rev:1},\n  $\\mathscr{U}$ also has an index-subordinated locally finite\n  partition of unity $\\{\\xi_U:U\\in \\mathscr{U}\\}$. For each $x\\in X$,\n  let $\\sigma_\\xi(x)\\in \\mathscr{N}(\\mathscr{U})$ be the simplex\n  determined by the point $x$ and the functions $\\xi_U$,\n  $U\\in \\mathscr{U}$, namely\n  $\\sigma_\\xi(x)=\\{U\\in \\mathscr{U}:\\xi_U(x)>0\\}$.  Next, define a map\n  $f:X\\to |\\mathscr{N}(\\mathscr{U})|$ by\n  \\begin{equation}\n    \\label{eq:st-app-v3:1}\n  f(x)=\\sum_{U\\in\n    \\sigma_\\xi(x)}\\xi_U(x)\\cdot U,\\quad x\\in X.\n\\end{equation}\n\nSince $\\{\\xi_U:U\\in \\mathscr{U}\\}$ is a locally finite partition of\nunity, each point $p\\in X$ has a neighbourhood $V_p\\subset X$ such\nthat\n$\\mathscr{U}_p=\\{U\\in \\mathscr{U}: V_p\\cap \\coz(\\xi_U)\\neq\n\\emptyset\\}$ is a finite set. According to \\eqref{eq:st-app-v3:1},\nthis implies that\n$f(V_p)\\subset |\\mathscr{N}(\\mathscr{U}_p)|\\subset\n\\ell_1(\\mathscr{U}_p)$. However, $\\ell_1(\\mathscr{U}_p)$ is now the\nusual Euclidean space $\\R^{\\mathscr{U}_p}$ because $\\mathscr{U}_p$ is\na finite set.  For the same reason, $\\mathscr{N}(\\mathscr{U}_p)$ has\nfinitely many simplices. Therefore, the Whitehead topology on\n$|\\mathscr{N}(\\mathscr{U}_p)|$ is the subspace topology on\n$|\\mathscr{N}(\\mathscr{U}_p)|$ as a subset of $\\R^{\\mathscr{U}_p}$.\n Since each\nfunction $\\xi_U=\\alpha_U\\circ f$, $U\\in \\mathscr{U}_p$, is continuous,\nso is the restriction $f\\uhr V_p$. This shows that $f$ is continuous\nas well. Finally, let $U\\in \\mathscr{U}$ and\n$x\\in f^{-1}\\left(\\st\\langle U\\rangle\\right)$. Then\n$f(x)\\in \\st\\langle U\\rangle$ and by \\eqref{eq:st-app-v4:3} and\n\\eqref{eq:st-app-v3:1}, we get that\n$\\xi_U(x)=\\alpha_U(f(x))>0$. Accordingly, $U\\in \\sigma_\\xi(x)$ which\nimplies that $x\\in U$ because $\\coz(\\xi_U)\\subset U$. Thus, $f$ is\ncanonical for $\\mathscr{U}$, see \\eqref{eq:st-app-v1:3}.\n\\end{proof}\n\nCanonical maps will be involved in the proof of Theorem\n\\ref{theorem-st-app-v10:1} with two properties, which are briefly\ndiscussed below.\\medskip\n\nFor a simplicial complex $\\Sigma$, as mentioned before, the carrier\n$\\car(p)$ of a point $p\\in |\\Sigma|$ is the minimal simplex of\n$\\Sigma$ with $p\\in |\\car(p)|$, see \\eqref{eq:st-app-vgg-rev:1}.\nAccording to \\eqref{eq:st-app-vgg-rev:3} and \\eqref{eq:st-app-v4:3},\nit has the following natural representation\n\\begin{equation}\n  \\label{eq:st-app-v4:2}\n  \\car(p)=\\left\\{v\\in \\Sigma^0:\n    p\\in \\st\\langle v\\rangle\\right\\}.\n\\end{equation}\n\nFor a cover $\\mathscr{U}$ of $X$ and $x\\in X$, we will associate the\nsimplicial complex\n\\begin{equation}\n  \\label{eq:st-app-vgg-rev:7}\n  \\Sigma_\\mathscr{U}(x)=\\left\\{\\sigma\\in \\Sigma_\\mathscr{U}: x\\in\\bigcap\n  \\sigma\\right\\}. \n\\end{equation}\nAccording to \\eqref{eq:st-app-vgg-rev:8}, we have\nthat $\\Sigma_\\mathscr{U}(x)\\subset \\mathscr{N}(\\mathscr{U})$, for\nevery $x\\in X$. Thus, \\eqref{eq:st-app-vgg-rev:7} defines a\nnatural set-valued mapping $\\Sigma_\\mathscr{U}:X\\smap\n\\mathscr{N}(\\mathscr{U})$.  To this mapping, we will associate the\nmapping $|\\Sigma_\\mathscr{U}|:X\\smap |\\mathscr{N}(\\mathscr{U})|$ which\nassigns to each $x\\in X$ the geometric realisation\n$|\\Sigma_\\mathscr{U}|(x)= |\\Sigma_\\mathscr{U}(x)|$. In terms of this\nmapping, we have the following selection interpretation of canonical\nmaps which extends an observation of Dowker \\cite{dowker:47}, see\nRemark \\ref{remark-st-app-vgg-rev:6}.\n\n\\begin{proposition}\n  \\label{proposition-st-app-v11:1}\n  Let $\\mathscr{U}$ be a cover of a space $X$. Then a continuous map\n  $f:X\\to |\\mathscr{N}(\\mathscr{U})|$ is canonical for $\\mathscr{U}$\n  if and only if $f$ is a selection for the mapping\n  $|\\Sigma_\\mathscr{U}|:X\\smap |\\mathscr{N}(\\mathscr{U})|$.\n\\end{proposition}\n\n\\begin{proof}\n  Let $f$ be a canonical map for $\\mathscr{U}$, and $x\\in X$. Whenever\n  $U\\in \\car(f(x))$, it follows from \\eqref{eq:st-app-v4:2} that\n  $f(x)\\in \\st\\langle U\\rangle$ and therefore, by\n  (\\ref{eq:st-app-v1:3}), $x\\in U$. Thus, by\n  \\eqref{eq:st-app-vgg-rev:7}, $\\car(f(x))\\in \\Sigma_\\mathscr{U}(x)$\n  and we have that\n  $f(x)\\in|\\car(f(x))|\\subset |\\Sigma_\\mathscr{U}|(x)$. Conversely,\n  suppose that $f$ is as selection for $|\\Sigma_\\mathscr{U}|$, and\n  $x\\in f^{-1}\\left( \\st\\langle U\\rangle\\right)$ for some\n  $U\\in \\mathscr{U}$. Then by (\\ref{eq:st-app-v4:2}), $U\\in\\car(f(x))$\n  because $f(x)\\in \\st\\langle U\\rangle$. Moreover, $f(x)\\in |\\sigma|$\n  for some $\\sigma\\in \\Sigma_\\mathscr{U}(x)$ because\n  $f(x)\\in |\\Sigma_\\mathscr{U}|(x)$. Since $\\car(f(x))$ is the minimal\n  simplex with this property, we get that\n  $U\\in \\car(f(x))\\subset \\sigma$ and, therefore, $x\\in U$.  That is,\n  $f^{-1}(\\st\\langle U\\rangle)\\subset U$.\n\\end{proof}\n\n\n\nEach {simplicial map} $g:\\Sigma_1\\to \\Sigma_2$, between simplicial\ncomplexes $\\Sigma_1$ and $\\Sigma_2$, can be extended to a continuous\nmap $|g|:|\\Sigma_1|\\to |\\Sigma_2|$ which is affine on each geometric\nsimplex $|\\sigma|$, for $\\sigma\\in \\Sigma_1$. This map is simply\ndefined by\n\\[\n  |g|(p)=\\sum_{v\\in \\car(p)}\n  \\alpha_v(p)\\cdot g(v),\\quad p\\in |\\Sigma_1|.\n\\]\n\nIf a cover $\\mathscr{V}$ of $X$ refines another cover $\\mathscr{U}$,\nthen there exists a natural simplicial map\n$r:\\mathscr{N}(\\mathscr{V})\\to \\mathscr{N}(\\mathscr{U})$ with \n$V\\subset r(V)$, for each $V\\in \\mathscr{V}$.  Such a map is commonly\ncalled a \\emph{canonical projection}, or a \\emph{refining simplicial\n  map}, or simply a \\emph{refining map}. Canonical maps are preserved\nby refinements in the following sense.\n\n\\begin{corollary}\n  \\label{corollary-st-app-vgg-rev:1}\n  Let $\\mathscr{U}$ and $\\mathscr{V}$ be covers of a space $X$ such\n  that $\\mathscr{V}$ refines $\\mathscr{U}$. If\n  $r:\\mathscr{N}(\\mathscr{V})\\to \\mathscr{N}(\\mathscr{U})$ is a\n  refining map and $g:X\\to |\\mathscr{N}(\\mathscr{V})|$ is canonical\n  for $\\mathscr{V}$, then the composite map\n  $|r|\\circ g:X\\to |\\mathscr{N}(\\mathscr{U})|$ is canonical for\n  $\\mathscr{U}$.\n\\end{corollary}\n\n\\begin{proof}\n  This follows from Proposition \\ref{proposition-st-app-v11:1} and the\n  fact that $r\\left(\\Sigma_\\mathscr{V}(x)\\right)\\subset\n  \\Sigma_\\mathscr{U}(x)$, $x\\in X$, because $V\\subset r(V)$ for every\n  $V\\in \\mathscr{V}$, see \\eqref{eq:st-app-vgg-rev:7}. \n\\end{proof}\n \nWe conclude this section with several remarks. \n\n\\begin{remark}\n  \\label{remark-st-app-vgg-rev:1}\n  For a space $X$, the \\emph{support} of a function $\\xi:X\\to \\R$,\n  called also the \\emph{topological support}, is the set\n  $\\supp(\\xi)=\\overline{\\coz(\\xi)}$. In several sources, a partition\n  of unity $\\{\\xi_U:U\\in \\mathscr{U}\\}$ on a space $X$ is called\n  \\emph{index-subordinated} to a cover $\\mathscr{U}$ of $X$ if\n  $\\supp(\\xi_U)\\subset U$, for every $U\\in \\mathscr{U}$; and\n  $\\{\\xi_U:U\\in \\mathscr{U}\\}$ is called \\emph{weakly\n    index-subordinated} to $\\mathscr{U}$ if $\\coz(\\xi_U)\\subset U$,\n  for every $U\\in \\mathscr{U}$, see e.g.\\ \\cite{MR3099433}. However,\n  these variations in the terminology do not affect the results of\n  this section. Namely, if $\\{\\eta_U:U\\in \\mathscr{U}\\}$ is a\n  partition of unity on $X$, then $X$ also has a (locally finite)\n  partition of unity $\\{\\xi_U:U\\in \\mathscr{U}\\}$ with\n  $\\supp(\\xi_U)\\subset \\coz(\\eta_U)$, for all $U\\in \\mathscr{U}$,\n  \\cite[Proposition 2.7.4]{MR3099433}. This property is essentially\n  the construction of M. Mather for proving Theorem\n  \\ref{theorem-st-app-vgg-rev:1}.\n\\end{remark}\n\n\\begin{remark}\n  \\label{remark-st-app-vgg-rev:2}\n  Canonical maps provide an isomorphism between simplicial complexes\n  and nerves of covers. Namely, if\n  $\\mathscr{O}_\\Sigma=\\big\\{\\st\\langle {v}\\rangle:\n  {v}\\in\\Sigma^0\\big\\}$ is the cover of $|\\Sigma|$ by the open stars\n  of the vertices of a simplicial complex $\\Sigma$ and\n  $\\sigma\\subset \\Sigma^0$, then $\\sigma\\in \\Sigma$ if and\n  only if\n  $\\bigcap_{{v}\\in \\sigma}\\st\\langle {v}\\rangle\\neq\n  \\emptyset$. That is, $\\sigma\\in \\Sigma$ precisely when\n  $\\st\\langle\\sigma\\rangle=\\{\\st\\langle {v}\\rangle:\n  {v}\\in \\sigma\\}\\in \\mathscr{N}(\\mathscr{O}_\\Sigma)$. Hence,\n  $\\st\\langle \\cdot\\rangle: \\Sigma\\to \\mathscr{N}(\\mathscr{O}_\\Sigma)$\n  is a simplicial isomorphism and the associated map\n  $|\\st\\langle\\cdot\\rangle|: |\\Sigma|\\to\n  |\\mathscr{N}(\\mathscr{O}_\\Sigma)|$ is both a homeomorphism and a\n  canonical map for $\\mathscr{O}_\\Sigma$.\n\\end{remark}\n\n\\begin{remark}\n  \\label{remark-st-app-vgg-rev:6}\n  In the case of a point-finite cover $\\mathscr{U}$ of $X$, Proposition\n  \\ref{proposition-st-app-v11:1} is reduced to the following selection\n  interpretation of canonical maps given by Dowker\n  \\cite{dowker:47}. Whenever $x\\in X$, let\n  $\\sigma(x)=\\{U\\in \\mathscr{U}: x\\in U\\}\\in \\mathscr{N}(\\mathscr{U})$\n  be the simplex determined by $x$. Then a continuous map\n  $f:X\\to |\\mathscr{N}(\\mathscr{U})|$ is canonical for $\\mathscr{U}$\n  if and only if $f(x)\\in |\\sigma(x)|$, for every $x\\in X$. While\n  $\\sigma(x)$ is only an element of $\\Sigma_\\mathscr{U}(x)$, we have\n  that $|\\sigma(x)|=|\\Sigma_\\mathscr{U}|(x)$ because\n  $\\sigma\\subset \\sigma(x)$, for each\n  $\\sigma\\in \\Sigma_\\mathscr{U}(x)$.\n\\end{remark}\n\n\\section{Aspherical sequences of mappings and selections}\n\\label{sec:asph-sequ-select}\n\nA  mapping $\\varphi:X\\smap Y$ is  \\emph{lower locally\n  constant} \\cite{gutev:05} if the set $\\{x\\in X:K\\subset\n\\varphi(x)\\}$ is open in $X$, for every compact subset $K\\subset Y$.\nThis property appeared in a paper of Uspenskij \\cite{uspenskij:98};\nlater on, it was used by some authors (see, for instance,\n\\cite{chigogidze-valov:00a,valov:00}) under the name ``strongly\nl.s.c.'', while in papers of other authors strongly l.s.c.\\ was\nalready used for a different property of set-valued mappings (see, for\ninstance, \\cite{gutev:95e}). Every lower locally constant\nmapping is l.s.c.\\ but the converse fails in general and\ncounterexamples abound. In fact, if we consider a single-valued map\n$f:X\\to Y$ as a set-valued one, then $f$ is l.s.c.\\ if and only if it\nis continuous, while $f$ will be lower locally constant if and only if\nit is locally constant. Thus, our terminology provides some natural\nanalogy with the single-valued case.\\medskip\n\nLet $k\\ge 0$. For subsets $S, B\\subset Y$, we will write that\n$S\\embed{k} B$ if every continuous map of the $k$-sphere in $S$ can be\nextended to a continuous map of the $(k+1)$-ball in $B$. Evidently,\nthe relation $S\\embed{k} B$ implies that $S\\subset B$. Similarly,\nfor mappings $\\varphi, \\psi :X\\smap Y$, we will write\n$\\varphi \\embed{k}\\psi$ to express that $\\varphi(x)\\embed{k}\\psi(x)$,\nfor every $x\\in X$.  In these terms, we shall say that a sequence of\nmappings $\\varphi_k: X\\smap Y$, $0\\leq k\\leq n$, is\n\\emph{aspherical} if $\\varphi_k\\embed{k}\\varphi_{k+1}$, for every\n$k<n$.  The\nfollowing theorem will be proved in this section.\n\n\\begin{theorem}\n\\label{theorem-nerves-v2:5}\nLet $X$ be a paracompact space with $\\dim(X)\\le n$, $Y$ be a space,\nand $\\varphi_{k}:X\\smap Y$, $0\\leq k\\leq n$, be an aspherical sequence\nof lower locally constant mappings. Then $\\varphi_{n}$ has a\ncontinuous selection.\n\\end{theorem}\n\nThe proof of Theorem \\ref{theorem-nerves-v2:5} is based on special\nskeletal selections motivated by the characterisation of canonical\nmaps in Proposition \\ref{proposition-st-app-v11:1}.  Namely, we shall say that a mapping\n$\\varphi:X\\smap Y$ has a \\emph{$k$-skeletal selection}, $k\\geq 0$, if\nthere exists and open cover $\\mathscr{U}$ of $X$ and a continuous map\n$u:|\\mathscr{N}^k(\\mathscr{U})|\\to Y$ such that\n\\begin{equation}\n  \\label{eq:st-app-vgg-rev:9}\n  u\\left(|\\Sigma_\\mathscr{U}^k(x)|\\right)\\subset \\varphi(x),\\quad\n  \\text{for every $x\\in X$.}\n\\end{equation}\nHere, $\\Sigma_\\mathscr{U}^k(x)$ is the $k$-skeleton of the\nsimplicial complex $\\Sigma_\\mathscr{U}(x)$, see\n\\eqref{eq:st-app-vgg-rev:7}. In fact, just like before, one can\nconsider $\\Sigma_\\mathscr{U}^k:X\\smap \\mathscr{N}^k(\\mathscr{U})$ as a\nset-valued mapping; similarly for\n$|\\Sigma_\\mathscr{U}^k|:X\\smap |\\mathscr{N}^k(\\mathscr{U})|$. Then a\ncontinuous map $u:|\\mathscr{N}^k(\\mathscr{U})|\\to Y$ is a $k$-skeletal\nselection for $\\varphi$ if and only if the composite mapping\n$u\\circ |\\Sigma_\\mathscr{U}^k|:X\\smap Y$ is a set-valued selection for\n$\\varphi:X\\smap Y$, see Remark \\ref{remark-st-app-vgg-rev:5}. \\medskip\n\nWe proceed with the following constructions of $k$-skeletal selections\nwhich furnish the essential part of the proof of Theorem\n\\ref{theorem-nerves-v2:5}.\n\n\\begin{proposition}\n  \\label{proposition-nerves-v2:2}\n  Each lower locally constant mapping $\\varphi:X\\smap Y$ has a \n  $0$-skeletal selection.\n\\end{proposition}\n\n\\begin{proof}\n  For each $x\\in X$, take a point $y(x)\\in \\varphi(x)$, and set\n  \\begin{equation}\n    \\label{eq:nerves-v2:3}\n    U(x)=\\big\\{z\\in X: y(x)\\in \\varphi(z)\\big\\}.\n  \\end{equation}\n  Then $\\mathscr{U}=\\{U(x):x\\in X\\}$ is an open cover of\n  $X$. Moreover, for each $U\\in \\mathscr{U}$ there is a point $x_U\\in\n  X$ with $U=U(x_U)$. Since\n  $|\\mathscr{N}^0(\\mathscr{U})|=\\mathscr{U}$, we may define a map\n  $u:|\\mathscr{N}^0(\\mathscr{U})|\\to Y$ by $u(U)=y(x_U)$, for each\n  $U\\in \\mathscr{U}$. If $x\\in U\\in \\mathscr{U}$, then $x\\in U(x_U)$\n  and by (\\ref{eq:nerves-v2:3}), we get that $u(U)=y(x_U)\\in\n  \\varphi(x)$.\n\\end{proof}\n\n\\begin{proposition}\n  \\label{proposition-st-app-vgg-rev:1}\n  Let $X$ be a paracompact space, and $\\psi:X\\smap Y$ be a mapping\n  which has a $k$-skeletal selection, for some $k\\geq 0$. Then $\\psi$\n  has a $k$-skeletal selection $u:|\\mathscr{N}^k(\\mathscr{U})|\\to Y$\n  for some open locally finite cover $\\mathscr{U}$ of $X$. \n\\end{proposition}\n\n\\begin{proof}\n  Let $v:|\\mathscr{N}^k(\\mathscr{V})|\\to Y$ be a $k$-skeletal\n  selection for $\\psi$, for some open cover $\\mathscr{V}$ of\n  $X$. Since $X$ is paracompact, the cover $\\mathscr{V}$ has an open\n  locally finite refinement $\\mathscr{U}$. Let\n  $r:\\mathscr{N}(\\mathscr{U})\\to \\mathscr{N}(\\mathscr{V})$ be a refining\n  map. Then by \\eqref{eq:st-app-vgg-rev:9}, $u=v\\circ |r|\\uhr\n  |\\mathscr{N}^k(\\mathscr{U})|: |\\mathscr{N}^k(\\mathscr{U})|\\to Y$ is\n  a $k$-skeletal selection for $\\psi$ because\n  $r(\\Sigma_\\mathscr{U}^k(x))\\subset \\Sigma_\\mathscr{V}^k(x)$, for\n  every $x\\in X$. \n\\end{proof}\n\nA cover $\\mathscr{V}$ of $X$ is a \\emph{star-refinement} of a cover\n$\\mathscr{U}$ if the cover\n$\\mathscr{V}^*=\\left\\{V^*:V\\in \\mathscr{V}\\right\\}$ refines\n$\\mathscr{U}$, where\n$V^*=\\bigcup\\{W\\in \\mathscr{V}:W\\cap V\\neq \\emptyset\\}$.  To reflect\nthis property, we shall say that a simplicial map\n$\\ell:\\mathscr{N}(\\mathscr{V})\\to \\mathscr{N}(\\mathscr{U})$ is a\n\\emph{star-refining map} if $V^*\\subset \\ell(V)$, for each\n$V\\in \\mathscr{V}$. Each star-refining map\n$\\ell:\\mathscr{N}(\\mathscr{V})\\to \\mathscr{N}(\\mathscr{U})$ has the\nproperty that\n  \\begin{equation}\n    \\label{eq:nerves-v2:2}\n    \\bigcup\\sigma \\subset  \\bigcap \\ell(\\sigma),\\quad\n    \\text{for each \n      $\\sigma\\in \\mathscr{N}(\\mathscr{V})$.}\n  \\end{equation}\n\n\\begin{proposition}\n\\label{proposition-nerves-v4:1}\nLet $X$ be a paracompact space, $Y$ be a space, and\n$\\psi,\\varphi:X\\smap Y$ be such that $\\varphi$ is lower locally\nconstant and $\\psi \\embed{k}\\varphi$ for some $k\\geq 0$. If $\\psi$ has\na $k$-skeletal selection, then $\\varphi$ has a $(k+1)$-skeletal\nselection.\n\\end{proposition}\n\n\n\\begin{proof}\n  By Proposition \\ref{proposition-st-app-vgg-rev:1}, $\\psi$ has a\n  $k$-skeletal selection $u:|\\mathscr{N}^k(\\mathscr{U})|\\to Y$ for\n  some open locally finite cover $\\mathscr{U}$ of $X$.  For each\n  $\\sigma\\in \\mathscr{N}(\\mathscr{U})$, let\n  $u_\\sigma=u\\uhr |\\mathscr{N}^k(\\sigma)|$ be the restriction of $u$\n  over the subcomplex\n  $|\\mathscr{N}^k(\\sigma)|=|\\sigma|\\cap |\\mathscr{N}^k(\\mathscr{U})|$.\n  Then, whenever $\\sigma\\in \\Sigma_\\mathscr{U}^{k+1}(x)$ for some\n  $x\\in X$, the map $u_\\sigma$ can be extended to a continuous map\n  $u_{(x,\\sigma)}:|\\sigma|\\to Y$ such that\n  \\begin{equation}\n    \\label{eq:nerves-v4:1}\n    u_{(x,\\sigma)}(|\\sigma|)\\subset \\varphi(x). \n  \\end{equation}\n  Indeed, if $\\sigma\\in \\Sigma_\\mathscr{U}^k(x)$, then by\n  \\eqref{eq:st-app-vgg-rev:9},\n  $u(|\\sigma|)\\subset \\psi(x)\\subset \\varphi(x)$ and we can take\n  $u_{(x,\\sigma)}=u_\\sigma$. If\n  $\\sigma\\notin \\Sigma_\\mathscr{U}^k(x)$, then\n  $| \\mathscr{N}^k(\\sigma)|=\\bigcup\\{|\\tau|: \\emptyset\\neq\n  \\tau\\subsetneq \\sigma\\}$ is homeomorphic to the $k$-sphere being the\n  boundary of $|\\sigma|$.  Hence, $u_\\sigma$ has a continuous\n  extension $u_{(x,\\sigma)}:|\\sigma|\\to \\varphi(x)$ because\n  $u(| \\mathscr{N}^k(\\sigma)|)\\subset u(\n  |\\Sigma_\\mathscr{U}^k(x)|)\\subset \\psi(x)\\embed{k} \\varphi(x)$, see\n  \\eqref{eq:st-app-vgg-rev:9}.\\medskip\n\n\n  Now, whenever $x\\in X$, set\n  \\begin{equation}\n    \\label{eq:nerves-v3:1}\n    K(x)=\\bigcup\\left\\{u_{(x,\\sigma)}(|\\sigma|):\n      \\sigma\\in \\Sigma_\\mathscr{U}^{k+1}(x)\\right\\}.\n  \\end{equation}\n  Then by (\\ref{eq:nerves-v4:1}), $K(x)\\subset \\varphi(x)$; moreover,\n  $K(x)$ is compact because $\\mathscr{U}$ is locally finite and,\n  therefore, $\\Sigma_\\mathscr{U}^{k+1}(x)$ contains finitely many\n  simplices.  Since $\\varphi$ is lower locally constant, for each\n  $U\\in \\mathscr{U}$, each point $x\\in U$ is contained in the open set\n  \\begin{equation}\n    \\label{eq:nerves-v3:3}\n    W_{(x,U)}=\\big\\{z\\in U: K(x)\\subset \\varphi(z)\\big\\}. \n  \\end{equation}\n  Since $X$ is paracompact, the cover\n  $\\{W_{(x,U)}: x\\in U\\in \\mathscr{U}\\}$ has an open star-refinement\n  $\\mathscr{V}$. So, there are maps $p:\\mathscr{V}\\to X$ and\n  ${\\ell:\\mathscr{V}\\to \\mathscr{U}}$ such that\n  \\begin{equation}\n    \\label{eq:st-app-vgg-rev:4}\n    V^*\\subset W_{(p(V),\\ell(V))},\\quad \\text{for every $V\\in\n      \\mathscr{V}$.} \n  \\end{equation}\n  Accordingly, $\\ell$ is a star-refining map because by\n  \\eqref{eq:nerves-v3:3},\n  $V^*\\subset W_{(p(V),\\ell(V))}\\subset \\ell(V)$. Finally, take a map\n  $q:\\mathscr{N}(\\mathscr{V})\\to \\mathscr{V}$ which selects from any\n  simplex $\\sigma\\in \\mathscr{N}(\\mathscr{V})$ a vertex\n  $q(\\sigma)\\in\\sigma$, and next set\n  $\\pi=p\\circ q:\\mathscr{N}(\\mathscr{V})\\to X$. Then \n  \\begin{equation}\n    \\label{eq:st-app-vgg-rev:2}\n    \\ell(\\sigma)\\in \\Sigma_\\mathscr{U}(\\pi(\\sigma)),\\quad \\sigma\\in\n    \\mathscr{N}(\\mathscr{V}), \n  \\end{equation}\n  because\n  $\\pi(\\sigma)=p(q(\\sigma))\\in q(\\sigma)\\subset\\bigcup \\sigma\\subset\n  \\bigcap \\ell(\\sigma)$, see (\\ref{eq:nerves-v2:2}).\\medskip\n\n  We complete the proof as follows. Using \\eqref{eq:nerves-v4:1} and\n  \\eqref{eq:st-app-vgg-rev:2}, one can define a continuous extension\n  $v:|\\mathscr{N}^{k+1}(\\mathscr{V})|\\to Y$ of the map\n  $u\\circ |\\ell|\\uhr\n  |\\mathscr{N}^k(\\mathscr{V})|:|\\mathscr{N}^k(\\mathscr{V})|\\to Y$ by\n  $v\\uhr |\\sigma|=u_{(\\pi(\\sigma),\\ell(\\sigma))}$, for every\n  $\\sigma\\in \\mathscr{N}^{k+1}(\\mathscr{V})$.  This $v$ is a\n  $(k+1)$-skeletal selection for $\\varphi$. Indeed, let\n  $\\sigma\\in \\Sigma_\\mathscr{V}^{k+1}(x)$ for some $x\\in X$. Then\n  $x\\in q(\\sigma)$ because $q(\\sigma)\\in \\sigma$, see\n  \\eqref{eq:st-app-vgg-rev:7}. Moreover, by\n  \\eqref{eq:st-app-vgg-rev:4},\n  $q(\\sigma)\\subset [q(\\sigma)]^*\\subset\n  W_{(\\pi(\\sigma),\\ell(q(\\sigma)))}$. Hence, by \\eqref{eq:nerves-v3:1},\n \\eqref{eq:nerves-v3:3} and \\eqref{eq:st-app-vgg-rev:2},\n  $ v(|\\sigma|)=u_{(\\pi(\\sigma),\\ell(\\sigma))}(|\\ell(\\sigma)|)\\subset\n  K(\\pi(\\sigma))\\subset \\varphi(x)$.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{theorem-nerves-v2:5}]\n  According to Propositions \\ref{proposition-nerves-v2:2} and\n  \\ref{proposition-nerves-v4:1}, the mapping $\\varphi_{n}$ has an\n  $n$-skeletal selection $u:|\\mathscr{N}^{n}(\\mathscr{U})|\\to Y$,\n  for some open cover $\\mathscr{U}$ of $X$.  Since $X$ is paracompact\n  and $\\dim(X)\\le n$, the cover $\\mathscr{U}$ has an open refinement\n  $\\mathscr{V}$ with\n  $\\mathscr{N}(\\mathscr{V})=\\mathscr{N}^{n}(\\mathscr{V})$, see\n  Remark \\ref{remark-st-app-vgg-rev:4}. Let\n  $r:\\mathscr{N}(\\mathscr{V})\\to \\mathscr{N}^{n}(\\mathscr{U})$ be a\n  refining map, and $g:X\\to |\\mathscr{N}(\\mathscr{V})|$ be a canonical\n  map for $\\mathscr{V}$ which exists because $X$ is paracompact, see\n  Theorems \\ref{theorem-st-app-v3:1} and\n  \\ref{theorem-st-app-v12:1}. Then by Corollary\n  \\ref{corollary-st-app-vgg-rev:1}, the composite map\n  $h=|r|\\circ g:X\\to |\\mathscr{N}^{n}(\\mathscr{U})|$ is a canonical\n  map for $\\mathscr{U}$. Finally, by \\eqref{eq:st-app-vgg-rev:9} and\n  Proposition \\ref{proposition-st-app-v11:1}, the composite map\n  $f=u\\circ h:X\\to Y$\n\\begin{center}\n  \\begin{tikzcd}\n    &\\lvert\\mathscr{N}^{n}(\\mathscr{U})\\rvert \\arrow[d, \"u\"]\\\\\n    {X} \\arrow[ur,  \"h\"] \\arrow[r, rightsquigarrow, \"\\varphi\"]\n    & Y\n  \\end{tikzcd}\n\\end{center}\nis a continuous selection for $\\varphi$.\n\\end{proof}\n\n\\begin{remark}\n  \\label{remark-st-app-vgg-rev:4}\n  Let $n\\geq -1$ be an integer and $X$ be a normal space.  The\n  \\emph{order} of a cover $\\mathscr{V}$ of $X$ doesn't exceed $n$ if\n  $\\bigcap\\sigma=\\emptyset$, for every $\\sigma\\subset \\mathscr{V}$\n  with $\\card(\\sigma)\\geq n+2$; equivalently, if\n  $\\mathscr{N}(\\mathscr{V})=\\mathscr{N}^n(\\mathscr{V})$. In these\n  terms, the covering dimension of $X$ is at most $n$, written\n  $\\dim(X)\\leq n$, if every finite open cover of $X$ has an open\n  refinement $\\mathscr{V}$ with\n  $\\mathscr{N}(\\mathscr{V})=\\mathscr{N}^n(\\mathscr{V})$. According to\n  a result of Dowker \\cite[Theorem 3.5]{dowker:47}, $\\dim(X)\\leq n$ if\n  and only if every locally finite open cover of $X$ has an open\n  refinement $\\mathscr{V}$ with\n  $\\mathscr{N}(\\mathscr{V})=\\mathscr{N}^n(\\mathscr{V})$. In\n  particular, for a paracompact space $X$, we have that\n  $\\dim(X)\\leq n$ if and only if every open cover of $X$ has an open\n  refinement $\\mathscr{V}$ with\n  $\\mathscr{N}(\\mathscr{V})=\\mathscr{N}^n(\\mathscr{V})$.\n\\end{remark}\n\n\\begin{remark}\n  \\label{remark-st-app-vgg-rev:5}\n  A mapping $\\psi:X\\smap Y$ is a \\emph{set-valued selection} (or\n  \\emph{set-selection}, or \\emph{multi-selection}) for\n  $\\varphi:X\\smap Y$ if $\\psi(x)\\subset \\varphi(x)$, for all $x\\in\n  X$. In terms of set-valued selections, a mapping $\\varphi:X\\smap Y$\n  has a $k$-skeletal selection, $k\\geq 0$, if there exists an open\n  cover $\\mathscr{U}$ of $X$ and a continuous map\n  $u:|\\mathscr{N}^k(\\mathscr{U})|\\to Y$ such that the composite\n  mapping $u\\circ |\\Sigma_\\mathscr{U}^k|:X\\smap Y$ \n\\begin{center}\n  \\begin{tikzcd}\n    &\\lvert\\mathscr{N}^{k}(\\mathscr{U})\\rvert \\arrow[d,\n    \"u\"]\\\\\n    {X} \\arrow[ur, rightsquigarrow, \"\\lvert\\Sigma_\\mathscr{U}^k\\rvert\"] \\arrow[r,\n    rightsquigarrow, \"\\varphi\"] & Y\n  \\end{tikzcd}\n\\end{center}\nis a set-valued selection for $\\varphi:X\\smap Y$.\n\\end{remark}\n\n\\section{Generating aspherical sequences of sets}\n\\label{sec:gener-asph-sequ}\n\nFor a point $y\\in Y$ of a metric space $(Y,d)$ and $\\varepsilon>0$,\nlet\n\\[\n\\mathbf{O}_\\varepsilon(y)=\\{z\\in Y: d(z,y)<\\varepsilon\\}\n\\]\nbe the open $\\varepsilon$-ball centred at $y$; and\n$\\mathbf{O}_\\varepsilon(S)=\\bigcup_{y\\in S}\\mathbf{O}_\\varepsilon(y)$\nbe the $\\varepsilon$-neighbourhood of a subset $S\\subset Y$. Also,\nrecall that a map $f:X\\to Y$ is an \\emph{$\\varepsilon$-selection} for\na mapping $\\varphi:X\\smap Y$ if $f(x)\\in\n\\mathbf{O}_{\\varepsilon}(\\varphi(x))$ for every $x\\in X$.\\medskip\n\nThroughout this section, $\\delta:(0,+\\infty)\\to (0,+\\infty)$ is a\nfixed function. To this function, we associate the sequence of\niterated functions $\\delta_n:(0,+\\infty)\\to (0,+\\infty)$, $ n\\geq 0$,\ndefined by\n\\begin{equation}\n  \\label{eq:st-app-v7:2}\n  \\delta_{0}(\\varepsilon)=\\varepsilon\\quad\\text{and}\\quad\n  \\delta_{n+1}(\\varepsilon)= \\delta(\\delta_{n}(\\varepsilon)). \n\\end{equation}\n\n\\begin{proposition}\n  \\label{proposition-st-app-v7:1}\n  Let $(Y,d)$ be a metric space and\n  ${S_0\\subset S_1\\subset\\dots\\subset S_{n}\\subset Y}$\n  be such that\n  $\\mathbf{O}_{\\delta(\\varepsilon)}(y)\\cap S_k\\embed{k}\n  \\mathbf{O}_\\varepsilon(y)\\cap S_{k+1}$, for every $y\\in Y$ and\n  $k< n$. Then\n    \\begin{equation}\n      \\label{eq:st-app-v7:3}\n      \\mathbf{O}_{\\delta_{n-k}(\\varepsilon)}(y)\\cap S_k\\embed{k}\n      \\mathbf{O}_{\\delta_{n-k-1}(\\varepsilon)}(y)\\cap S_{k+1},\\quad  k< n.\n  \\end{equation}\n\\end{proposition}\n\n\\begin{proof}\n  Follows from the fact that\n  $\\delta_{n-k}(\\varepsilon)=\\delta\\big(\\delta_{n-k-1}(\\varepsilon)\\big)$,\n  see (\\ref{eq:st-app-v7:2}).\n\\end{proof}\n\nWe now have the following ``local'' version of Theorem\n\\ref{theorem-nerves-v2:5}.\n\n\\begin{theorem}\n  \\label{theorem-st-app-vgg-rev:2}\n  Let $(Y,d)$ be a metric space, $X$ be a paracompact space with\n  $\\dim(X)\\leq n$, and $\\psi_k:X\\smap Y$, $0\\leq k\\leq n$, be\n  lower locally constant mappings such that\n  $\\mathbf{O}_{\\delta(\\varepsilon)}(y)\\cap\\psi_k(x)\\embed{k}\n  \\mathbf{O}_\\varepsilon(y)\\cap \\psi_{k+1}(x)$ for every $x\\in X$,\n  $y\\in Y$ and $k< n$. Then for each continuous\n  $\\delta_{n}(\\varepsilon)$-selection $g:X\\to Y$ for $\\psi_0$, there\n  is a continuous selection $f:X\\to Y$ for $\\psi_{n}$ with\n  $d(f(x),g(x))<\\varepsilon$, for all $x\\in X$.\n\\end{theorem}\n\n\\begin{proof}\n  Let $g:X\\to Y$ be a continuous $\\delta_{n}(\\varepsilon)$-selection\n  for $\\psi_0$. Next, for each $k\\leq n$, define a set-valued mapping\n  $\\varphi_k$ by\n  $\\varphi_k(x)=\\mathbf{O}_{\\delta_{n-k}(\\varepsilon)}(g(x))\\cap\n  \\psi_k(x)$, $x\\in X$. Since $g$ is a\n  $\\delta_{n}(\\varepsilon)$-selection for $\\psi_0$, the mapping\n  $\\varphi_0$ is nonempty-valued and, according to\n  (\\ref{eq:st-app-v7:3}), so is each $\\varphi_k$, $k\\leq n$. In fact,\n  by (\\ref{eq:st-app-v7:3}), the resulting sequence of mappings\n  $\\varphi_k:X\\smap Y$, $0\\leq k\\leq n$, is aspherical.  Moreover,\n  each $\\varphi_k$ is lower locally constant because so are $\\psi_k$\n  and the mapping $x\\to \\mathbf{O}_{\\delta_{n-k}(\\varepsilon)}(g(x))$,\n  $x\\in X$ (see Proposition \\ref{proposition-st-app-v15:2}).  Hence,\n  by Theorem \\ref{theorem-nerves-v2:5}, $\\varphi_{n}$ has a continuous\n  selection $f:X\\to Y$ because $X$ is a paracompact space with\n  $\\dim(X)\\leq n$. Evidently, $f$ is a selection for\n  $\\psi_{n}$ and, by (\\ref{eq:st-app-v7:2}),\n  $f(x)\\in \\mathbf{O}_{\\delta_{n-n}(\\varepsilon)}(g(x))=\n  \\mathbf{O}_{\\delta_{0}(\\varepsilon)}(g(x))=\\mathbf{O}_\\varepsilon(g(x))$,\n  $x\\in X$.\n\\end{proof}\n\nWe conclude this section with the following two applications of Theorem\n\\ref{theorem-st-app-vgg-rev:2} which will provide the main interface\nbetween selections for l.s.c.\\ mappings and Theorem\n\\ref{theorem-nerves-v2:5}, see Theorems \\ref{theorem-st-app-v9:1} and\n\\ref{theorem-st-app-v17:1}.\n\n\\begin{corollary}\n  \\label{corollary-st-app-v7:2}\n  Let $E$ be a normed space, and $\\emptyset\\neq S\\subset T\\subset E$\n  be such that $S\\embed{k} T$ and\n  $\\mathbf{O}_{\\delta(\\varepsilon)}(y)\\cap S\\embed{i}\n  \\mathbf{O}_\\varepsilon(y)\\cap S$, for every $y\\in E$ and\n  $0\\leq i< k$. Then\n  $\\mathbf{O}_{\\delta_k(\\varepsilon)}(S)\\embed{k}\n  \\mathbf{O}_\\varepsilon(T)$.\n\\end{corollary}\n\n\\begin{proof}\n  Let $\\ell:\\s^{k}\\to \\mathbf{O}_{\\delta_k(\\varepsilon)}(S)$ be a\n  continuous map from the $k$-sphere $\\s^k$. Consider the constant\n  mappings $\\psi_i(x)=S$, $x\\in\\s^k$ and $i\\leq k$. Then $\\ell$ is a\n  continuous $\\delta_k(\\varepsilon)$-selection for $\\psi_0$, and\n  $\\mathbf{O}_{\\delta(\\varepsilon)}(y)\\cap \\psi_i(x)\\embed{i}\n  \\mathbf{O}_\\varepsilon(y)\\cap \\psi_{i+1}(x)$ for every $x\\in \\s^k$,\n  $y\\in E$ and $i< k$. Hence, by Theorem\n  \\ref{theorem-st-app-vgg-rev:2}, there exists a continuous $q:\\s^{k}\\to\n  S$ with $\\|q(x)-\\ell(x)\\|<\\varepsilon$, for every $x\\in\\s^k$. Let\n  $h_{1}$ be the linear homotopy between $\\ell$ and $q$, i.e.\\\n  $h_{1}(x,t)=t q(x)+ (1-t)\\ell(x)$, whenever $(x,t)\\in \\s^{k}\\times\n  [0,1]$.  Then, $h_{1}(\\s^{k}\\times [0,1])\\subset\n  \\mathbf{O}_{\\varepsilon}(S)\\subset\n  \\mathbf{O}_{\\varepsilon}(T)$. Also, let $h_2:\\s^k\\times[0,1]\\to T$\n  be a homotopy between $q$ and a constant map, which exists because\n  $S\\embed{k}T$.  Finally, take $h$ to be the homotopy obtained by\n  combining $h_{1}$ and $h_{2}$. Then $h$ is a homotopy of $\\ell$ with\n  a constant map over a subset of $\\mathbf{O}_{\\varepsilon}(T)$.\n\\end{proof}\n\n\\begin{corollary}\n  \\label{corollary-st-app-v8:1}\n  Let $E$ be a normed space, and $\\emptyset\\neq S\\subset T\\subset E$\n  be such that\n  $\\mathbf{O}_{\\delta(\\varepsilon)}(y)\\cap T\\embed{k}\n  \\mathbf{O}_\\varepsilon(y)\\cap T$ and\n  $\\mathbf{O}_{\\delta(\\varepsilon)}(y)\\cap S\\embed{i}\n  \\mathbf{O}_\\varepsilon(y)\\cap S$, for every $y\\in E$ and\n  $0\\leq i< k$.  Define functions\n  \\begin{equation}\n    \\label{eq:st-app-v8:1}\n    \\eta(\\varepsilon)=\\delta(\\varepsilon)\/2\\quad\\text{and} \\quad\n    \\lambda(\\varepsilon,\\mu)=\\delta_k\\big(\\min\\left\\{\\eta(\\varepsilon),\n      \\mu\\right\\}\\big),\\\n    \\varepsilon,\\mu>0.      \n  \\end{equation}\n  Then $\\mathbf{O}_{\\eta(\\varepsilon)}(y)\\cap\n  \\mathbf{O}_{\\lambda(\\varepsilon,\\mu)}(S)\\embed{k}\n  \\mathbf{O}_\\varepsilon(y)\\cap \\mathbf{O}_\\mu(T)$, for every $y\\in\n  E$.\n\\end{corollary}\n\n\\begin{proof}\n  Let\n  $\\ell:\\s^{k}\\to \\mathbf{O}_{\\eta(\\varepsilon)}(y)\\cap\n  \\mathbf{O}_{\\lambda(\\varepsilon,\\mu)}(S)$ be a continuous map for\n  some $y\\in E$.  Then, precisely as in the previous proof, there\n  exists a continuous map $q:\\s^k\\to S$ such that\n  $\\|q(x)-\\ell(x)\\|<\\min\\{\\eta(\\varepsilon),\\mu\\}$, for every\n  $x\\in \\s^k$. Since $\\eta(\\varepsilon)=\\delta(\\varepsilon)\/2$, see\n  (\\ref{eq:st-app-v8:1}), just like before, using a linear homotopy,\n  we get that $\\ell$ and $q$ are homotopic in\n  $\\mathbf{O}_{\\delta(\\varepsilon)}(y)\\cap \\mathbf{O}_\\mu(S)$.\n  Moreover $q$ is homotopic to a constant map in\n  $\\mathbf{O}_\\varepsilon(y)\\cap T$ because\n  $q:\\s^k\\to \\mathbf{O}_{\\delta(\\varepsilon)}(y)\\cap S\\subset\n  \\mathbf{O}_{\\delta(\\varepsilon)}(y)\\cap T\\embed{k}\n  \\mathbf{O}_\\varepsilon(y)\\cap T$.  Accordingly, $\\ell$ is homotopic\n  to a constant map in\n  $\\mathbf{O}_\\varepsilon(y)\\cap \\mathbf{O}_\\mu(T)$.\n\\end{proof}\n\n\\section{Selections for equi-$LC^{n}$-valued mappings}\n\\label{sec:select-lcn-valu}\n\nIn this section, to each $\\Phi:X\\smap Y$ we associate the mapping\n$\\overline{\\Phi}:X\\to \\mathscr{F}(Y)$ defined by\n$\\overline{\\Phi}(x)=\\overline{\\Phi(x)}$, $x\\in X$. Moreover, for a\npair of mappings $\\Phi,\\Psi: X\\smap Y$, we will use $\\Phi\\wedge\\Psi$\nto denote their intersection, i.e.\\ the mapping which assigns to each\n$x\\in X$ the set $[\\Phi\\wedge\\Psi](x)=\\Phi(x)\\cap \\Psi(x)$. Finally,\nto each $\\varepsilon>0$ and a mapping $\\Phi:X\\smap Y$ in a metric\nspace $(Y,d)$, we will associate the mapping\n$\\mathbf{O}[\\Phi,\\varepsilon]:X\\smap Y$ defined by\n\\begin{equation}\n  \\label{eq:st-app-v18:1}\n \\mathbf{O}[\\Phi,\\varepsilon](x)=\n \\mathbf{O}_\\varepsilon(\\Phi(x)),\\quad \\text{$x\\in X$.}\n\\end{equation}\nThis convention will be also used in an obvious manner for usual maps\n$f:X\\to Y$ considering $f$ as the singleton-valued mapping\n$x\\to\\{f(x)\\}$, $x\\in X$.  In these terms, for maps\n$f,g:X\\to Y$ and $\\varepsilon,\\mu>0$, we have that $f$ is a\n$\\mu$-selection for $\\Phi:X\\smap Y$ with $d(f(x),g(x))<\\varepsilon$\nfor every $x\\in X$, if and only if $f$ is a selection for the mapping\n$\\mathbf{O}[\\Phi,\\mu]\\wedge\\mathbf{O}[g,\\varepsilon]$. \\medskip\n\nThe following two constructions are due to Michael, see \\cite[Lemma\n11.3]{michael:56b} and \\cite[Proof that Lemma 5.1 implies Theorem 4.1,\npage 569]{michael:56b}. They reduce the selection problem for l.s.c.\\\nmappings to that of lower locally constant mappings. For completeness,\nwe sketch their proofs following the original arguments in\n\\cite{michael:56b}.\n\n\\begin{proposition}\n  \\label{proposition-st-app-v15:2}\n  Let $(Y,d)$ be a metric space, $\\Phi:X\\smap Y$ be l.s.c.\\ and\n  $\\varepsilon>0$. Then the mapping\n  $\\mathbf{O}[\\Phi,\\varepsilon]:X\\smap Y$ is lower locally constant.\n\\end{proposition}\n\n\\begin{proof}\n  Take $x_0\\in X$ and a compact set\n  $K\\subset\n  \\mathbf{O}[\\Phi,\\varepsilon](x_0)=\\mathbf{O}_\\varepsilon(\\Phi(x_0))$. Then\n  $K\\subset \\mathbf{O}_\\delta(S)$ for some finite subset\n  $S\\subset \\Phi(x_0)$ and some $\\delta>0$ with\n  $\\delta<\\varepsilon$. Since $\\Phi$ is l.s.c.,\n  $U=\\bigcap_{y\\in S}\\Phi^{-1}[\\mathbf{O}_{\\varepsilon-\\delta}(y)]$ is\n  an open set containing $x_0$. Moreover, $x\\in U$ implies\n  $S\\subset \\mathbf{O}_{\\varepsilon-\\delta}(\\Phi(x))$ and, therefore,\n  $K\\subset \\mathbf{O}_\\delta(S)\\subset\n  \\mathbf{O}_\\varepsilon(\\Phi(x))=\\mathbf{O}[\\Phi,\\varepsilon](x)$.\n\\end{proof}\n\n\\begin{proposition}\n  \\label{proposition-st-app-v15:3}\n  Let $(Y,d)$ be a complete metric space,\n  ${\\xi:(0,+\\infty)\\to (0,+\\infty)}$ be a function with\n  $\\xi(\\varepsilon)\\leq \\varepsilon$, and $\\Phi:X\\smap Y$ be a mapping\n  such that for each continuous $\\xi(\\varepsilon)$-selection\n  $g:X\\to Y$ for $\\Phi$ and $\\mu>0$, then mapping\n  ${\\mathbf{O}[\\Phi,\\mu]\\wedge \\mathbf{O}[g,\\varepsilon]}$ has a\n  continuous selection. Then for every continuous\n  $\\xi(\\varepsilon\/2)$-selection $g:X\\to Y$ for $\\Phi$, the mapping\n  $\\overline{\\Phi}\\wedge\\mathbf{O}[g,\\varepsilon]$ also has a\n  continuous selection.\n\\end{proposition}\n\n\\begin{proof}\n  Let $f_1=g:X\\to Y$ be a continuous\n  $\\xi\\left(2^{-1}\\varepsilon\\right)$-selection for $\\Phi$. By\n  condition with $\\mu=\\xi\\left(2^{-2}\\varepsilon\\right)$, the mapping\n  $\\mathbf{O}\\left[\\Phi,\\xi\\left(2^{-2}\\varepsilon\\right)\\right]\\wedge\n  \\mathbf{O}[f_1,2^{-1}\\varepsilon]$ has a continuous selection\n  $f_2:X\\to Y$. Thus, by induction, there exists a sequence of\n  continuous maps $f_n:X\\to E$ such that $f_{n+1}$ is a selection for\n  $\\mathbf{O}\\left[\\Phi,\\xi\\left(2^{-(n+1)}\\varepsilon\\right)\\right]\\wedge\n  \\mathbf{O}\\left[f_n,2^{-n}\\varepsilon\\right]$, for every $n\\in\\N$.\n  Then the sequence $\\{f_n:n\\in \\N\\}$ is uniformly Cauchy because\n  $d(f_{n+1}(x),f_n(x))<2^{-n}\\varepsilon$, $x\\in X$. Hence, it\n  converges uniformly to some continuous map $f:X\\to Y$ because\n  $(Y,d)$ is complete. Since\n  $\\xi\\left(2^{-n}\\varepsilon\\right)\\leq 2^{-n}\\varepsilon$, each\n  $f_{n}$ is a $2^{-n}\\varepsilon$-selection for $\\Phi$ being a\n  selection for\n  $\\mathbf{O}\\left[\\Phi,\\xi\\left(2^{-n}\\varepsilon\\right)\\right]$, see\n  (\\ref{eq:st-app-v18:1}). Hence, $d(f(x),\\Phi(x))=0$, for each\n  $x\\in X$. Finally, we also have that\n  \\[\n  d(f(x),g(x))\\leq\\sum_{n=1}^\\infty\n  d(f_{n+1}(x),f_{n}(x))<\\sum_{n=1}^\\infty2^{-n}\\varepsilon=\\varepsilon,\\quad\n  x\\in X.  \\qedhere\n  \\]\n\\end{proof}\n\nLet $n\\geq -1$. A family $\\mathscr{S}$ of subsets of a metric space\n$(Y,d)$ is called \\emph{uniformly equi-$LC^{n}$} \\cite{michael:56b} if\nfor every $\\varepsilon>0$ there exists $\\delta(\\varepsilon)>0$ such\nthat, for every $S\\in\\mathscr{S}$, every continuous map of the\n$k$-sphere ($k\\leq n$) in $S$ of diameter$\\ < \\delta(\\varepsilon)$ can\nbe extended to continuous map of the $(k+1)$-ball into a subset of $S$\nof diameter$\\ <\\varepsilon$.  Just as in the case of equi-$LC^n$\nfamilies, a family $\\mathscr{S}$ is uniformly equi-$LC^{-1}$ iff it\nconsists of nonempty sets.  For such a family $\\mathscr{S}$, by\nreplacing $\\delta(\\varepsilon)$ with $\\frac{\\delta(\\varepsilon)}2$, we\nget that $\\mathscr{S}$ is uniformly equi-$LC^{n}$ if there exists a\nfunction $\\delta:(0,+\\infty)\\to (0,+\\infty)$ such that\n\\begin{equation}\n  \\label{eq:st-app-v7:4}\n  \\mathbf{O}_{\\delta(\\varepsilon)}(y)\\cap S\\embed{k} \\mathbf{O}_\\varepsilon(y)\\cap\n  S,\\quad \\text{for every $S\\in \\mathscr{S}$, $y\\in Y$ and $0\\leq k\\leq n$.}\n\\end{equation}\nEvidently, we may further assume that\n$\\delta(\\varepsilon)\\leq \\varepsilon$, for every $\\varepsilon>0$.\nBased on this and the results of the previous section, we now have the\nfollowing two applications of Theorem \\ref{theorem-nerves-v2:5}. The\nfirst one gives a simplified proof of \\cite[Theorem 4.1]{michael:56b}.\n\n\\begin{theorem}\n  \\label{theorem-st-app-v9:1}\n  Let $E$ be a Banach space and $\\mathscr{S}$ be a uniformly\n  equi-$LC^{n}$ family of subsets of $E$. Then there exists\n  a function $\\gamma:(0,+\\infty)\\to (0,+\\infty)$ with the following\n  property\\textup{:} If $X$ is a paracompact space with $\\dim(X)\\le\n  n+1$, $\\Phi:X\\to \\mathscr{S}$ is l.s.c.\\ and $g:X\\to E$ is a\n  continuous $\\gamma(\\varepsilon)$-selection for $\\Phi$, then\n  $\\overline{\\Phi}\\wedge \\mathbf{O}[g,\\varepsilon]$ has a continuous\n  selection.\n\\end{theorem}\n\n\\begin{proof}\n  Let $\\delta(\\varepsilon)\\leq \\varepsilon$ be as in\n  (\\ref{eq:st-app-v7:4}) with respect to the family\n  $\\mathscr{S}$. Also, let $\\lambda(\\varepsilon,\\mu)$ and\n  $\\eta(\\varepsilon)$ be as in (\\ref{eq:st-app-v8:1}) applied to this\n  particular function $\\delta(\\varepsilon)$. Next, define functions\n  $\\eta_k(\\varepsilon)$ and $\\lambda_k(\\varepsilon,\\mu)$,\n  $0\\leq k\\leq n+1$, by\n  \\begin{equation}\n    \\label{eq:st-app-v9:1}\n    \\begin{cases}\n      \\eta_{n+1}(\\varepsilon)=\\varepsilon &\\text{and}\\quad\n      \\eta_{k}(\\varepsilon)=\\eta\\big(\\eta_{k+1}(\\varepsilon)\\big)\\\\\n      \\lambda_{n+1}(\\varepsilon,\\mu)=\\mu &\\text{and}\\quad\n      \\lambda_{k}(\\varepsilon,\\mu)=\n      \\lambda\\big(\\eta_{k+1}(\\varepsilon),\\lambda_{k+1}(\\varepsilon,\\mu)\\big).\n    \\end{cases}\n  \\end{equation}\n  Then $\\gamma(\\varepsilon)=\\eta_{0}(\\varepsilon\/2)$ is as\n  required. Indeed, let $X$ and $\\Phi$ be as in the theorem.  Applying\n  Proposition \\ref{proposition-st-app-v15:3} with\n  $\\xi(\\varepsilon)=\\eta_{0}(\\varepsilon)$, it will be now\n  sufficient to show that for every $\\mu>0$ and a continuous\n  $\\eta_{0}(\\varepsilon)$-selection $g:X\\to E$ for $\\Phi$, the\n  mapping $\\mathbf{O}[\\Phi,\\mu]\\wedge \\mathbf{O}[g,\\varepsilon]$ has a\n  continuous selection. To this end, for every $0\\le k\\le n+1$, let\n  $\\varphi_k=\\mathbf{O}[\\Phi,\\lambda_{k}(\\varepsilon,\\mu)]\\wedge\n  \\mathbf{O}[g,\\eta_{k}(\\varepsilon)]$. According to Proposition\n  \\ref{proposition-st-app-v15:2}, each $\\varphi_{k}$ is lower locally\n  constant. Moreover, the resulting sequence of mappings\n  $\\varphi_k:X\\smap E$, $0\\leq k\\leq n+1$, is aspherical because by\n  (\\ref{eq:st-app-v7:4}) and Corollary \\ref{corollary-st-app-v8:1},\n  \\begin{align*}\n    \\varphi_k(x) =&\\ \n                    \\mathbf{O}_{\\lambda_{k}(\\varepsilon,\\mu)}(\\Phi(x))\\cap\n                    \\mathbf{O}_{\\eta_{k}(\\varepsilon)}(g(x))\\\\ \n    =&\\\n       \\mathbf{O}_{\\lambda(\\eta_{k+1}(\\varepsilon),\\lambda_{k+1}(\\varepsilon,\\mu))}(\\Phi(x))\n    \\cap \\mathbf{O}_{\\eta(\\eta_{k+1}(\\varepsilon))}(g(x))\\\\\n                  &\\embed{k} \n                    \\mathbf{O}_{\\lambda_{k+1}(\\varepsilon,\\mu)}(\\Phi(x))\\cap\n                    \\mathbf{O}_{\\eta_{k+1}(\\varepsilon)}(g(x)) \n                    =\\varphi_{k+1}(x),\\quad k\\leq n.\n  \\end{align*}\n  Hence by Theorem \\ref{theorem-nerves-v2:5},\n  \\[\n    \\varphi_{n+1}=\\mathbf{O}[\\Phi,\\lambda_{n+1}(\\varepsilon,\\mu)]\\wedge\n  \\mathbf{O}[g,\\eta_{n+1}(\\varepsilon)]=\\mathbf{O}[\\Phi,\\mu]\\wedge\n  \\mathbf{O}[g,\\varepsilon]\n\\]\nhas a continuous selection. The proof is complete.\n\\end{proof}\n\n\\begin{theorem}\n\\label{theorem-st-app-v17:1}\nLet $X$ be a paracompact space with $\\dim(X)\\le n+1$, $E$ be a Banach\nspace, and $\\Phi_{k}:X\\smap E$, $0\\leq k\\leq n+1$, be a sequence of\nl.s.c.\\ mappings such that $\\{\\Phi_{k}(x): x\\in X\\}$ is uniformly\nequi-$LC^{k}$ and $\\Phi_{k} \\embed{k}\\Phi_{k+1}$ for every $k\\leq\nn$. Then $\\Phi_{n+1}$ has a continuous $\\varepsilon$-selection, for\nevery $\\varepsilon>0$.\n\\end{theorem}\n\n\\begin{proof}\n  According to (\\ref{eq:st-app-v7:4}) and Corollary\n  \\ref{corollary-st-app-v7:2}, for each $0\\leq k\\leq n$ there exists a\n  function $\\delta_k:(0,+\\infty)\\to (0,+\\infty)$ such that\n  \\begin{equation}\n    \\label{eq:st-app-v17:1}\n    \\mathbf{O}_{\\delta_k(\\varepsilon)}(\\Phi_k(x))\\embed{k}\n    \\mathbf{O}_\\varepsilon(\\Phi_{k+1}(x)),\\quad x\\in X.\n  \\end{equation}\n  Next, define functions $\\gamma_k:(0,+\\infty)\\to (0,+\\infty)$,\n  $0\\leq k\\leq n+1$, by\n  \\begin{equation}\n    \\label{eq:st-app-v17:2}\n    \\gamma_{n+1}(\\varepsilon)=\\varepsilon\\quad \\text{and}\\quad \\gamma_{k}(\\varepsilon)=\n    \\delta_{k}(\\gamma_{k+1}(\\varepsilon)),\\quad k\\leq n.\n  \\end{equation}\n  Finally, define a sequence of mappings $\\varphi_k:X\\smap E$ by\n  $\\varphi_k=\\mathbf{O}[\\Phi_k,\\gamma_{k}(\\varepsilon)]$. It now\n  follows from (\\ref{eq:st-app-v17:1}) and (\\ref{eq:st-app-v17:2})\n  that\n  \\begin{align*}\n    \\varphi_k(x)=\\mathbf{O}_{\\gamma_{k}(\\varepsilon)}(\\Phi_k(x))\n    &\n      = \\mathbf{O}_{\\delta_k(\\gamma_{k+1}(\\varepsilon))}(\\Phi_k(x))\\\\\n    &\\embed{k}\n      \\mathbf{O}_{\\gamma_{k+1}(\\varepsilon)}(\\Phi_{k+1}(x))=\\varphi_{k+1}(x),\\quad\n      k\\leq n. \n  \\end{align*}\n  Hence, the mappings $\\varphi_k$, $0\\leq k\\leq n+1$, form an\n  aspherical sequence. Moreover, by Proposition\n  \\ref{proposition-st-app-v15:2}, each $\\varphi_k$ is lower locally\n  constant. Since $\\dim(X)\\leq n+1$, by Theorem\n  \\ref{theorem-nerves-v2:5},\n  $\\varphi_{n+1}=\\mathbf{O}[\\Phi_{n+1},\n  \\gamma_{n+1}(\\varepsilon)]=\\mathbf{O}[\\Phi_{n+1},\\varepsilon]$ has a\n  continuous selection, i.e. $\\Phi_{n+1}$ has a continuous\n  $\\varepsilon$-selection.\n\\end{proof}\n\nWe are also ready for the proof of Theorem \\ref{theorem-st-app-v10:1}.\n\n\\begin{proof}[Proof of Theorem \\ref{theorem-st-app-v10:1}]\n  Let $X$, $Y$, $\\mathscr{S}\\subset \\mathscr{F}(Y)$ and $\\Phi:X\\to\n  \\mathscr{S}$ be as in that theorem. Since $\\mathscr{S}$ is\n  equi-$LC^n$, by \\cite[Theorem\n  1]{dugundji-michael:56} (see, also, \\cite[Proposition\n  2.1]{michael:56b}), $\\bigcup\\mathscr{S}$ can be embedded into a\n  Banach space $E$  such that $\\mathscr{S}\\subset \\mathscr{F}(E)$ is\n  uniformly equi-$LC^n$.  Then by Theorem \\ref{theorem-st-app-v17:1},\n  applied with $\\Phi_k=\\Phi$, $0\\leq k\\leq n+1$, the mapping $\\Phi$\n  has a continuous $\\varepsilon$-selection, for every\n  $\\varepsilon>0$. Hence, by Theorem~\\ref{theorem-st-app-v9:1},\n  $\\overline{\\Phi}=\\Phi$ has a continuous selection as well. \n\\end{proof}\n\nAnother application of Theorems \\ref{theorem-st-app-v9:1} and\n\\ref{theorem-st-app-v17:1}  is the following generalisation of\nTheorem \\ref{theorem-st-app-v10:1}, see \\cite{schepin-brodsky:96},\n\\cite[Theorem 7.2]{repovs-semenov:98} and \\cite[Corollary\n7.10]{gutev:05}.\n\n\\begin{corollary}\n  \\label{corollary-st-app-v18:1}\n  Let $X$ be a paracompact space with $\\dim(X)\\le n+1$, $Y$ be a\n  completely metrizable space, and $\\Phi_{k}:X\\to \\mathscr{F}(Y)$,\n  $0\\leq k\\leq n+1$, be a sequence of l.s.c.\\ mappings such that\n  $\\Phi_{k} \\embed{k}\\Phi_{k+1}$ for $k\\leq n$, while each family\n  ${\\{\\Phi_{k}(x): x\\in X\\}}$, for $k\\leq n+1$, is\n  equi-$LC^{k}$. Then $\\Phi_{n+1}$ has a continuous selection.\n\\end{corollary}\n\n\\begin{proof}\n  As before, the proof is reduced to the case when $Y=E$ is a Banach\n  space, and each family\n  $\\{\\Phi_{k}(x):x\\in X\\}\\subset \\mathscr{F}(E)$, $0\\le k\\le n+1$, is\n  uniformly equi-$LC^{k}$. Then by Theorem \\ref{theorem-st-app-v17:1},\n  $\\Phi_{n+1}$ has a continuous $\\varepsilon$-selection, for every\n  $\\varepsilon>0$.  Finally, by Theorem \\ref{theorem-st-app-v9:1},\n  $\\Phi_{n+1}$ also has a continuous selection.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}