text:
Best Used Cars Under £100 Per Month
Top 8: Best Used Cars Under £100 Per Month
Modern, stylish and practical used cars for an affordable monthly sum.
What cars can I get for under £100 per month?
Owning a car doesn't have to cost the earth. It's quite simple to find a car that come with low road tax, insurance and fuel costs. The same goes for monthly finance payments too - there are lots of great used cars you can purchase for under £100 per month.
It's a common misconception that the only cars in this price bracket are high mileage, really old cars. The reality is that Personal Contract Purchase (PCP) finance means that newer cars can be financed for a much lower monthly payment than you might imagine.
With cars in this price bracket from popular manufacturers such as Ford, Vauxhall, Peugeot and Citroen all available at this price point, stylish and practical motoring can be achieved on a budget of just £100 per month.
Retro styling and fun to drive
One of the best-selling city cars of all time not just in the UK, but in Europe as well, the Fiat 500 is super cheap and super cheerful.
People love its cute styling, retro interior and the large amount of colours and personalisation options that were available when the car was new, which means there are lots of different colour schemes to choose from on the used market.
The driving experience is fun too, with peppy little engines and handling that suits urban life down to the ground. Because Fiat sold so many from new and their low purchase price, you can get a pretty modern Fiat 500 for under £100 per month.
Search Used Fiat 500
Affordable when new, even more affordable when used
Dacia are known for one thing and one thing only - being affordable. Affordable doesn't mean rubbish though, and used Dacias are built on time-proven parts from their parent company Renault, meaning they are robust and reliable for everyday use and abuse.
Sanderos are a fair size too, so you get a lot of metal for your £100 per month budget. The boot is bigger than the all-conquering Ford Fiesta and the rear seats have a decent amount of room too.
Access models are the least expensive Sanderos available, and you can pick them up for a very reasonable monthly cost.
Search Used Dacia Sandero
Vauxhall Viva
Surprisingly spacious city car
Vauxhall were onto a winner when they created the Viva, which is easy to find as a used car. Popular as new cars, there are plenty of great value used examples on the market.
The interior roominess of the Viva really has to be seen to be believed. Despite its extremely compact exterior dimensions, space for front and rear passengers is generous, and the boot is a decent size too.
Coupled with the incredibly economical 1.0-litre 3-cylinder that from which you'll struggle to get less than 60mpg, the Viva is an appealing prospect, especially at this price point.
Search Used Vauxhall Viva
Super economical Japanese hatch
It's fairly easy to pick up a used Nissan Micra for less than £100 per month. The version that was sold in the UK between 2010 and 2017 was inexpensive as a new car, so it's no surprise to see it feature in this list.
What's it good at? Well, like with most small cars, it's really easy to park and the controls are all really simple to use, too. It's very, very economical and the robust build quality means that they're generally considered to be very reliable.
This generation looked a lot better than Micras of old (though not quite as good as the new one) and the interior is comfy enough for two adults and two kids.
Search Used Nissan Micra
Practical and economical
Fiat really are the kings of inexpensive, small cars and that's why they've got not one but two cars in this list. Buying a used Fiat Panda for under £100 a month makes a lot of sense for those who want practical daily motoring on a limited budget.
Its boxy, sorry, "squircle", shape means that it has more head room than other cars in this sector, and with the rear seats down it can almost act like a miniature van - very useful for when you need to nip to the tip.
Motoring doesn't get much more economical than this, with engines that don't use much fuel, cheap road tax, low servicing costs and inexpensive replacement parts helping to keep the pounds in your pocket.
Search Used Fiat Panda
Peugeot 108/Citroën C1
A dynamic duo
The cars that can do it all. In the context of urban motoring, anyway. Cheap monthly finance deals made both the 108 and C1 popular as new car purchases but they're perhaps even more attractive as used vehicles because the cost simply keeps getting less and less.
You can even get the most recent versions that are only a few years old for under a £100 per month budget, thanks to their ability to be financed through PCP, which keeps costs down.
Given their size, they are cheap to run too with low insurance and road tax costs, plus they have super-economical 1.0-litre and 1.2-litre engines that are really good on fuel.
Search Used Peugeot 108 Search Used Citroen C1
Popular, practical supermini
You'll be able to find an older, but still relatively new, used Vauxhall Corsa for less than £100 per month on finance. Its seemingly tireless popularity means there are thousands of used examples available at any one time.
They are popular with young people as first cars because they look decent, drive well and are pretty practical too. Both modern '14-19 and older '06-14 versions can be found at this price point, depending on the type of car finance you choose.
You can get some quite well-equipped versions for your budget, with air con, alloy wheels and even Bluetooth connectivity available for the money.
Search Used Vauxhall Corsa
A great all-rounder
A used Fiesta is at the top of nearly every other list of everyday cars that you can think of, because it does so much so well. However, you won't find loads under £100 per month as they hold their value quite well.
The sub £100 a month Ford Fiestas are out there however, and if you manage to track one down then you'll be getting a very good car for not a lot of money.
They're Britain's best-selling new and used car and with good reason, its blend of great looks, brilliant handling, reasonable practicality and everyday comfort make it easy to see why they're so popular with UK buyers.
Search Used Ford Fiesta
Find your perfect used car for less than £100 per month
Above you've seen eight different ways you can get into a modern, practical and reliable used car for less than £100 per month on HP or PCP finance.
At Evans Halshaw, we stock all of the above cars from Fiat, Ford, Vauxhall and the rest. The best part is, you can use our smart search to filter by monthly payment when you're looking for your next used car.
Search all used cars under £100 per month.
Best Used Cars Under £10,000
Top 10: Used Cars with Zero Road Tax
Best Used 7-Seater Cars Under £10,000


text:
package com.amazonaws.services.codeartifact;

import javax.annotation.Generated;

import com.amazonaws.services.codeartifact.model.*;

/**
 * Abstract implementation of {@code AWSCodeArtifactAsync}. Convenient method forms pass through to the corresponding
 * overload that takes a request object and an {@code AsyncHandler}, which throws an
 * {@code UnsupportedOperationException}.
 */
@Generated("com.amazonaws:aws-java-sdk-code-generator")
public class AbstractAWSCodeArtifactAsync extends AbstractAWSCodeArtifact implements AWSCodeArtifactAsync {

    protected AbstractAWSCodeArtifactAsync() {
    }

    @Override
    public java.util.concurrent.Future<AssociateExternalConnectionResult> associateExternalConnectionAsync(AssociateExternalConnectionRequest request) {

        return associateExternalConnectionAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<AssociateExternalConnectionResult> associateExternalConnectionAsync(AssociateExternalConnectionRequest request,
            com.amazonaws.handlers.AsyncHandler<AssociateExternalConnectionRequest, AssociateExternalConnectionResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<CopyPackageVersionsResult> copyPackageVersionsAsync(CopyPackageVersionsRequest request) {

        return copyPackageVersionsAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<CopyPackageVersionsResult> copyPackageVersionsAsync(CopyPackageVersionsRequest request,
            com.amazonaws.handlers.AsyncHandler<CopyPackageVersionsRequest, CopyPackageVersionsResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<CreateDomainResult> createDomainAsync(CreateDomainRequest request) {

        return createDomainAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<CreateDomainResult> createDomainAsync(CreateDomainRequest request,
            com.amazonaws.handlers.AsyncHandler<CreateDomainRequest, CreateDomainResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<CreateRepositoryResult> createRepositoryAsync(CreateRepositoryRequest request) {

        return createRepositoryAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<CreateRepositoryResult> createRepositoryAsync(CreateRepositoryRequest request,
            com.amazonaws.handlers.AsyncHandler<CreateRepositoryRequest, CreateRepositoryResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<DeleteDomainResult> deleteDomainAsync(DeleteDomainRequest request) {

        return deleteDomainAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<DeleteDomainResult> deleteDomainAsync(DeleteDomainRequest request,
            com.amazonaws.handlers.AsyncHandler<DeleteDomainRequest, DeleteDomainResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<DeleteDomainPermissionsPolicyResult> deleteDomainPermissionsPolicyAsync(DeleteDomainPermissionsPolicyRequest request) {

        return deleteDomainPermissionsPolicyAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<DeleteDomainPermissionsPolicyResult> deleteDomainPermissionsPolicyAsync(DeleteDomainPermissionsPolicyRequest request,
            com.amazonaws.handlers.AsyncHandler<DeleteDomainPermissionsPolicyRequest, DeleteDomainPermissionsPolicyResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<DeletePackageVersionsResult> deletePackageVersionsAsync(DeletePackageVersionsRequest request) {

        return deletePackageVersionsAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<DeletePackageVersionsResult> deletePackageVersionsAsync(DeletePackageVersionsRequest request,
            com.amazonaws.handlers.AsyncHandler<DeletePackageVersionsRequest, DeletePackageVersionsResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<DeleteRepositoryResult> deleteRepositoryAsync(DeleteRepositoryRequest request) {

        return deleteRepositoryAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<DeleteRepositoryResult> deleteRepositoryAsync(DeleteRepositoryRequest request,
            com.amazonaws.handlers.AsyncHandler<DeleteRepositoryRequest, DeleteRepositoryResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<DeleteRepositoryPermissionsPolicyResult> deleteRepositoryPermissionsPolicyAsync(
            DeleteRepositoryPermissionsPolicyRequest request) {

        return deleteRepositoryPermissionsPolicyAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<DeleteRepositoryPermissionsPolicyResult> deleteRepositoryPermissionsPolicyAsync(
            DeleteRepositoryPermissionsPolicyRequest request,
            com.amazonaws.handlers.AsyncHandler<DeleteRepositoryPermissionsPolicyRequest, DeleteRepositoryPermissionsPolicyResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<DescribeDomainResult> describeDomainAsync(DescribeDomainRequest request) {

        return describeDomainAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<DescribeDomainResult> describeDomainAsync(DescribeDomainRequest request,
            com.amazonaws.handlers.AsyncHandler<DescribeDomainRequest, DescribeDomainResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<DescribePackageResult> describePackageAsync(DescribePackageRequest request) {

        return describePackageAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<DescribePackageResult> describePackageAsync(DescribePackageRequest request,
            com.amazonaws.handlers.AsyncHandler<DescribePackageRequest, DescribePackageResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<DescribePackageVersionResult> describePackageVersionAsync(DescribePackageVersionRequest request) {

        return describePackageVersionAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<DescribePackageVersionResult> describePackageVersionAsync(DescribePackageVersionRequest request,
            com.amazonaws.handlers.AsyncHandler<DescribePackageVersionRequest, DescribePackageVersionResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<DescribeRepositoryResult> describeRepositoryAsync(DescribeRepositoryRequest request) {

        return describeRepositoryAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<DescribeRepositoryResult> describeRepositoryAsync(DescribeRepositoryRequest request,
            com.amazonaws.handlers.AsyncHandler<DescribeRepositoryRequest, DescribeRepositoryResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<DisassociateExternalConnectionResult> disassociateExternalConnectionAsync(DisassociateExternalConnectionRequest request) {

        return disassociateExternalConnectionAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<DisassociateExternalConnectionResult> disassociateExternalConnectionAsync(DisassociateExternalConnectionRequest request,
            com.amazonaws.handlers.AsyncHandler<DisassociateExternalConnectionRequest, DisassociateExternalConnectionResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<DisposePackageVersionsResult> disposePackageVersionsAsync(DisposePackageVersionsRequest request) {

        return disposePackageVersionsAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<DisposePackageVersionsResult> disposePackageVersionsAsync(DisposePackageVersionsRequest request,
            com.amazonaws.handlers.AsyncHandler<DisposePackageVersionsRequest, DisposePackageVersionsResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<GetAuthorizationTokenResult> getAuthorizationTokenAsync(GetAuthorizationTokenRequest request) {

        return getAuthorizationTokenAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<GetAuthorizationTokenResult> getAuthorizationTokenAsync(GetAuthorizationTokenRequest request,
            com.amazonaws.handlers.AsyncHandler<GetAuthorizationTokenRequest, GetAuthorizationTokenResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<GetDomainPermissionsPolicyResult> getDomainPermissionsPolicyAsync(GetDomainPermissionsPolicyRequest request) {

        return getDomainPermissionsPolicyAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<GetDomainPermissionsPolicyResult> getDomainPermissionsPolicyAsync(GetDomainPermissionsPolicyRequest request,
            com.amazonaws.handlers.AsyncHandler<GetDomainPermissionsPolicyRequest, GetDomainPermissionsPolicyResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<GetPackageVersionAssetResult> getPackageVersionAssetAsync(GetPackageVersionAssetRequest request) {

        return getPackageVersionAssetAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<GetPackageVersionAssetResult> getPackageVersionAssetAsync(GetPackageVersionAssetRequest request,
            com.amazonaws.handlers.AsyncHandler<GetPackageVersionAssetRequest, GetPackageVersionAssetResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<GetPackageVersionReadmeResult> getPackageVersionReadmeAsync(GetPackageVersionReadmeRequest request) {

        return getPackageVersionReadmeAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<GetPackageVersionReadmeResult> getPackageVersionReadmeAsync(GetPackageVersionReadmeRequest request,
            com.amazonaws.handlers.AsyncHandler<GetPackageVersionReadmeRequest, GetPackageVersionReadmeResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<GetRepositoryEndpointResult> getRepositoryEndpointAsync(GetRepositoryEndpointRequest request) {

        return getRepositoryEndpointAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<GetRepositoryEndpointResult> getRepositoryEndpointAsync(GetRepositoryEndpointRequest request,
            com.amazonaws.handlers.AsyncHandler<GetRepositoryEndpointRequest, GetRepositoryEndpointResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<GetRepositoryPermissionsPolicyResult> getRepositoryPermissionsPolicyAsync(GetRepositoryPermissionsPolicyRequest request) {

        return getRepositoryPermissionsPolicyAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<GetRepositoryPermissionsPolicyResult> getRepositoryPermissionsPolicyAsync(GetRepositoryPermissionsPolicyRequest request,
            com.amazonaws.handlers.AsyncHandler<GetRepositoryPermissionsPolicyRequest, GetRepositoryPermissionsPolicyResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<ListDomainsResult> listDomainsAsync(ListDomainsRequest request) {

        return listDomainsAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<ListDomainsResult> listDomainsAsync(ListDomainsRequest request,
            com.amazonaws.handlers.AsyncHandler<ListDomainsRequest, ListDomainsResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<ListPackageVersionAssetsResult> listPackageVersionAssetsAsync(ListPackageVersionAssetsRequest request) {

        return listPackageVersionAssetsAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<ListPackageVersionAssetsResult> listPackageVersionAssetsAsync(ListPackageVersionAssetsRequest request,
            com.amazonaws.handlers.AsyncHandler<ListPackageVersionAssetsRequest, ListPackageVersionAssetsResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<ListPackageVersionDependenciesResult> listPackageVersionDependenciesAsync(ListPackageVersionDependenciesRequest request) {

        return listPackageVersionDependenciesAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<ListPackageVersionDependenciesResult> listPackageVersionDependenciesAsync(ListPackageVersionDependenciesRequest request,
            com.amazonaws.handlers.AsyncHandler<ListPackageVersionDependenciesRequest, ListPackageVersionDependenciesResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<ListPackageVersionsResult> listPackageVersionsAsync(ListPackageVersionsRequest request) {

        return listPackageVersionsAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<ListPackageVersionsResult> listPackageVersionsAsync(ListPackageVersionsRequest request,
            com.amazonaws.handlers.AsyncHandler<ListPackageVersionsRequest, ListPackageVersionsResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<ListPackagesResult> listPackagesAsync(ListPackagesRequest request) {

        return listPackagesAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<ListPackagesResult> listPackagesAsync(ListPackagesRequest request,
            com.amazonaws.handlers.AsyncHandler<ListPackagesRequest, ListPackagesResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<ListRepositoriesResult> listRepositoriesAsync(ListRepositoriesRequest request) {

        return listRepositoriesAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<ListRepositoriesResult> listRepositoriesAsync(ListRepositoriesRequest request,
            com.amazonaws.handlers.AsyncHandler<ListRepositoriesRequest, ListRepositoriesResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<ListRepositoriesInDomainResult> listRepositoriesInDomainAsync(ListRepositoriesInDomainRequest request) {

        return listRepositoriesInDomainAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<ListRepositoriesInDomainResult> listRepositoriesInDomainAsync(ListRepositoriesInDomainRequest request,
            com.amazonaws.handlers.AsyncHandler<ListRepositoriesInDomainRequest, ListRepositoriesInDomainResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<ListTagsForResourceResult> listTagsForResourceAsync(ListTagsForResourceRequest request) {

        return listTagsForResourceAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<ListTagsForResourceResult> listTagsForResourceAsync(ListTagsForResourceRequest request,
            com.amazonaws.handlers.AsyncHandler<ListTagsForResourceRequest, ListTagsForResourceResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<PutDomainPermissionsPolicyResult> putDomainPermissionsPolicyAsync(PutDomainPermissionsPolicyRequest request) {

        return putDomainPermissionsPolicyAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<PutDomainPermissionsPolicyResult> putDomainPermissionsPolicyAsync(PutDomainPermissionsPolicyRequest request,
            com.amazonaws.handlers.AsyncHandler<PutDomainPermissionsPolicyRequest, PutDomainPermissionsPolicyResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<PutPackageOriginConfigurationResult> putPackageOriginConfigurationAsync(PutPackageOriginConfigurationRequest request) {

        return putPackageOriginConfigurationAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<PutPackageOriginConfigurationResult> putPackageOriginConfigurationAsync(PutPackageOriginConfigurationRequest request,
            com.amazonaws.handlers.AsyncHandler<PutPackageOriginConfigurationRequest, PutPackageOriginConfigurationResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<PutRepositoryPermissionsPolicyResult> putRepositoryPermissionsPolicyAsync(PutRepositoryPermissionsPolicyRequest request) {

        return putRepositoryPermissionsPolicyAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<PutRepositoryPermissionsPolicyResult> putRepositoryPermissionsPolicyAsync(PutRepositoryPermissionsPolicyRequest request,
            com.amazonaws.handlers.AsyncHandler<PutRepositoryPermissionsPolicyRequest, PutRepositoryPermissionsPolicyResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<TagResourceResult> tagResourceAsync(TagResourceRequest request) {

        return tagResourceAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<TagResourceResult> tagResourceAsync(TagResourceRequest request,
            com.amazonaws.handlers.AsyncHandler<TagResourceRequest, TagResourceResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<UntagResourceResult> untagResourceAsync(UntagResourceRequest request) {

        return untagResourceAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<UntagResourceResult> untagResourceAsync(UntagResourceRequest request,
            com.amazonaws.handlers.AsyncHandler<UntagResourceRequest, UntagResourceResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<UpdatePackageVersionsStatusResult> updatePackageVersionsStatusAsync(UpdatePackageVersionsStatusRequest request) {

        return updatePackageVersionsStatusAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<UpdatePackageVersionsStatusResult> updatePackageVersionsStatusAsync(UpdatePackageVersionsStatusRequest request,
            com.amazonaws.handlers.AsyncHandler<UpdatePackageVersionsStatusRequest, UpdatePackageVersionsStatusResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

    @Override
    public java.util.concurrent.Future<UpdateRepositoryResult> updateRepositoryAsync(UpdateRepositoryRequest request) {

        return updateRepositoryAsync(request, null);
    }

    @Override
    public java.util.concurrent.Future<UpdateRepositoryResult> updateRepositoryAsync(UpdateRepositoryRequest request,
            com.amazonaws.handlers.AsyncHandler<UpdateRepositoryRequest, UpdateRepositoryResult> asyncHandler) {

        throw new java.lang.UnsupportedOperationException();
    }

}

redpajama_set_name: RedPajamaGithub


Our Unrefined Dark Muscovado Sugar will transform your Chocolate brownies into rich, indulgent squidgy pieces of brownie heaven.
Preheat the oven to 190°C/gas mark 5, line a 20cm square tin with baking parchment.
Whisk the eggs until pale and fluffy. Add the Dark Muscovado Sugar and whisk until thick. Gently fold in the chocolate.
Sift in the flour and fold in until the mixture is smooth. Mix in the glacé cherries.


Venice MainStreet and Venice Blue's Festival donated $3,000 to the Venice High School Band.
Venice MainStreet President, Tracy Ivey, presents a check to Jonathan Case, Venice High School Band Teacher. The donation of $3,000 was for the participation of Venice High School students in the 2013 and 2014 Venice Blue's Festival.


text:
package com.certusnet.xproject.common.support;

import com.certusnet.xproject.common.consts.GlobalConstants;
import com.certusnet.xproject.common.exception.SystemException;
import com.certusnet.xproject.common.support.Messages;
import com.certusnet.xproject.common.util.ExceptionUtils;

/**
 * 模块内异常解析逻辑
 *
 * @author	  	pengpeng
 * @date	  	2015年11月1日 下午11:08:50
 * @version  	1.0
 */
public class ModuleExceptionResolver {

	public static ExceptionMetadata resolveException(Throwable ex) {
		return resolveException(ex, "message.request.failure");
	}

	public static ExceptionMetadata resolveException(Throwable ex, String messageCode) {
		Throwable target = null;
		String code = GlobalConstants.RESULT_CODE_FAILURE;
		String stackTrace = null;
		String message = null;
		boolean found = false;
		Throwable cause = ex;
		while(cause != null){
			if (cause instanceof SystemException) { // 已知的异常信息
				target = cause;
				message = cause.getMessage();
				stackTrace = ExceptionUtils.getStackTrace(cause);
				if(ExceptionUtils.isContainsChineseChar(message)){
					found = true;
					break;
				}
			}
			cause = cause.getCause();
		}
		if(!found){
			target = ExceptionUtils.getRootCause(ex);
			message = Messages.getMessage(messageCode, target.getMessage()); // 未知的异常消息,需要转换成统一的,以增强用户体验
			stackTrace = ExceptionUtils.getStackTrace(ex);
		}
		return new ExceptionMetadata(target, message, stackTrace, code);
	}

	public static class ExceptionMetadata {

		private Throwable target;

		private String message;

		private String stackTrace;

		private String code;

		public ExceptionMetadata() {
			super();
		}

		public ExceptionMetadata(Throwable target, String message,
				String stackTrace, String code) {
			super();
			this.target = target;
			this.message = message;
			this.stackTrace = stackTrace;
			this.code = code;
		}

		public Throwable getTarget() {
			return target;
		}

		public void setTarget(Throwable target) {
			this.target = target;
		}

		public String getMessage() {
			return message;
		}

		public void setMessage(String message) {
			this.message = message;
		}

		public String getCode() {
			return code;
		}

		public void setCode(String code) {
			this.code = code;
		}

		public String getStackTrace() {
			return stackTrace;
		}

		public void setStackTrace(String stackTrace) {
			this.stackTrace = stackTrace;
		}

	}

}


If you are working for a mobile application development company for a long time, then you have certainly got many unique opportunities to work with hundreds of thousands of business owners, small, medium and large enterprises need and entrepreneurs. The majority of them have shared their stories of investing a large amount of energy, effort and time, as well as money to make their application the very best it could be.
Everything those are required from strategically placing the relevant widgets, amazingly crafted layout as well as engaging informative content. Even though the countless hours as well as expert attention to detail they have given, a large number of businesses are still not getting the kind of adoption they were looking for. Instead of hundreds of thousands of people flooding the application store to download and upload the application, they get about a dozen. After listening this story, you may assume the content is not up to the mark, problems may be related crawling, uploading or even downloading the apps.
Creating a mobile application not only open up the new channel by which you can communicate with but also the business opportunities you need to get. A misconception generally exist here is, your audience simply come to your app store and download your app just in one simple click. Your application is a space completely customizable to your audience need, but the problem is most of the time it is not the application itself, even, it is the most important step so sometimes missed after the application is built, promotion.
Here, through this blog post we will try to give you some proven methods to help you to market your application. You have two things to invest in, one is cost and another is confident, nothing more you need here, not even a single penny to invest to market and promote your application. If you follow these fifteen creative ways you can easily improve your mobile app development marketing as well as promotion, and your followers will be increased and at the same time your downloads will improve. Let us get to it then.
Feature the application in an official good blog – Additionally, rather than combining your application on your own website, you can include it in your blog post on a regular basis, which is another promotional strategy you can do. This way is beyond doubt a great way to improve your mobile application development and that is creatively.
Feature the application with your email – Your latest application needs to engrave into all facets of your advertising and marketing, which combines all of your emails, which is important. As a business owner of small or medium enterprise, you will send out different types of emails in different times and your application in each of those is no other than missed opportunity. Emails are still having charm when provided they are utilized under the right time at the right place.
Create a Lucrative Demo Video – Videos are nothing but an easy way for the purpose of your marketing to showcase everything your beautiful as well as well-built application has to offer. For the sake of your demo video, you can create a simple 30-45 second maximum commercial with the things, like WHY, HOW and WHAT.
Connect to large number of audience – Marketing as well as promoting your application begin long before you ever bring it to the application market. Prior to start telling your app development team to begin creating your application with a million different things, you should reach out your audience or customers and get in touch with them. As a matter of fact, you wish to make it sure that whatsoever are creating, is going to be at the right time and at the right place for them.
Reach Out To Your Potential Influencers – Connecting to influencers are important. Promoting and marketing your application can help you extend beyond your regular area of influence, and sometimes, these connections are what your organization need. When you develop honest, reliable and authentic relationship with your influencer, you will go a long way without any hindrance and help promote and market your mobile application.
Get in touch as much as possible – Marketing your mobile application starts long before you ever bring it to your target market. Prior to start saying your development team to begin creating mobile application with a hundreds of thousands of things you should talk to them, so that you can create a list of your visitors and get in touch with them.
Consider alternative application store – It is said that the more you promote your app content, the easier as well as faster the process becomes. You must agree that all small and medium sized business of all types do not have a large budget to dedicate to their marketing as well as promotion events. Even though those companies that do have their amount to invest my find themselves only having a few hands-on channels to promote their app content.
Apply for rewards – Participating in different types of application rewards are some of the most recommended channels and we would highly suggest you to take advantage of. These types of rewards can give you tons of press, reviews, maximum downloading option and reviews.
Search Engine Optimization Matters – Yes, we are talking about search engine optimization. When your SEO comes with many things and you not only apply to promote it, but also implement in your mobile app URL that shows up in search results in the time your uses key-in related to their queries.
Increase Visibility On Social Media – You should increase your app visibility on your social network, groups as well as communities, like Twitter, LinkedIn, Google+, Facebook and others.
Ask for Your Users Feedback – You should ask for your users' feedback on your Mobile Application and give them free promotional codes to try it out, so that the more they use mobile applications they use yours and give uses feedback.
Take a Screen Shot – Take a beautiful screen shot for your mobile app and upload it the app store with other sites for promotion. This will help your users understand your app and help them use it, directly download it and gauge an extra age for your app.
Application Store Optimization – App store optimization which is shortly known as ASO. At its core, the SEO or search engine optimization for the purpose of Google Play as well as Application store. Just like SEO, application store optimization not only focuses on keyword relevance, but also searches the relevance including the keyword density in your mobile app description.
Build a Strong and Beautiful Landing Page – A landing page will help you optimize your store, which is important for all. A great landing page is a must-have thing for all types of business, even whether you do not have a mobile application. Therefore, having a great landing page is like having a beautiful business card.
Include App Download Links on Your Website – There are many ways you already come to know to promote or market your mobile app. This is no other than a creative strong way to improve your mobile application development. When you do it, your users directly download the app by way of the links on your website. Here, your website will work as a tool or site to promote your mobile application.
Last, but certainly not the least, Mobile app development is one thing and showing creativity with mobile application development, advertising and marketing the app is another thing. You should see how your application is performing, and after a certain time of research, you should dive deep into a detail analysis of how creative ways can improve your mobile app development and make your app unparalleled in the market Visit Here. Do not be a copycat, better you follow the steps or ways we have given and implement each of them step-by-step for better possible outcome.
Sunny Chawla is a Marketing Manager at AIS Technolabs – a Web design and Development Company. Helping global businesses with unique and engaging tools for their business. He would love to share thoughts on Ecm Services, Web Designing and Andriod Web Development.


text:
Vallauri – famiglia italiana di origine piemontese
 Giancarlo Vallauri – scienziato e ammiraglio italiano
 Luigi Lombardi Vallauri – filosofo e professore universitario italiano
 Mario Vallauri – filologo e orientalista italiano
 Tommaso Vallauri – filologo classico, latinista e politico italiano

redpajama_set_name: RedPajamaWikipedia


Organic Chemistry 51C. Lecture 09. Reactions of Carboxylic Acids, Esters, Amides, & Nitriles.
Here we talk about the basics of Organic Compounds, Homologous Series and Functional Groups.
Jan 4, 2016 ... This video tutorial provides an introduction or basic overview on what you will learn in your first college semester of organic chemistry.
The Organic Chemistry Portal offers an overview of recent topics, interesting reactions and information on important chemicals for organic chemists.
An index of first-semester organic chemistry topics covered on Master Organic Chemistry, from structure and bonding to substitution and elimination.
Organic chemistry is the scientific study of the structure, properties, composition, reactions, and synthesis of organic compounds that by definition contain carbon.


So, yellow.....that's not a fun colour to paint! I know I wanted to challenge myself with this but I'm not sure I thought it through all the way. Why did I not go for a simple scheme that didn't involve yellow? Either way, I've made my bed and I'm damn well going to a smooth yellow finish on their helms.
Going back a couple of steps, in part one I'd just finished stippling the armour with the two blues and lightening the Helms and face masks ready for the yellows.
I next applied a Blue wash to the armour to try to make it stand out more and painted the helms and face masks with Ulthuan grey to bring them closer to grey before I applied the Casandora Yellow wash. the results of this wash are below. I also painted the weapons with Ulthuan grey to get them ready for the Ushabti Bone. This is the first time I've really bothered doing transition colours and I have to says its been very beneficial, especially working from a black undercoat as I always do.
Ushabti Bone was applied to the weapons next to start making the rifles look like wraith-bone. I also used the Alaitoc Blue for the visors and went over the yellow helms and masks with several thin layers of Flash Gitz yellow. This yellow I found the painstaking process but eventually was happy with the results after about 4-5 layers of watered down colour.
Then the cloaks got a layer of Baneblade and the weapons were given a heavy Agrax Earthshade wash.
Next steps are to put agrax earthshade on the cloaks then work on the pattern on the cloaks, I'm going to have to work out what is best for this so any suggestions would be good. I think a fair amount of practice on a piece of cardboard will be needed. Also the hair, weapons, backpacks and the remaining details will have to be done.
as always any thoughts, comments and criticism is welcomed below the line.
Looking good so far. If you aren't comfortable doing a cano then my suggestion would be- dont bother. With a wash on the cloaks and then highlighted back up I think they will look good, a nice contrast with he yellow and blue.


Perfect for DJs with a small budget. You can start off with 1 drop to tag your mixtapes or use it live as intro.
ATTENTION: Enter your drop script and details about the drop in the "Order Notes" form at checkout! Please include the right pronunciation, all necessary details and be as specific as possible. Thank you!


text:
Ammoconia reisseri är en fjärilsart som beskrevs av Ronkay och Zoltan Varga 1984. Ammoconia reisseri ingår i släktet Ammoconia och familjen nattflyn. Inga underarter finns listade i Catalogue of Life.

Källor

Nattflyn
reisseri


Pure Hawaiian beeswax ready for your projects! Our double rendered beeswax is ready for making cosmetics such as lip balms, salves, and lotions. Or make your own healthy air cleansing candles. There are over 101 uses for beeswax, no home should be without some. Our double rendered wax comes in a variety of shapes and sizes and can be purchased by the ounce or by the pound for more savings. A limited quantity of single rendered beeswax is available for a reduced price, which has impurities that can be filtered out, or let settle to the bottom of your melting pot (photo available upon request).


This gorgeous dude is still looking for his forever home!
Likes : Sleeping, spending time with people.
Dislikes : He doesn't like sharing or being alone.
Please note – location isn't an issue, transport is provided across the country.
Shadow loves company and just being with people, he's not really much of a player he likes to just sit and watch people although he is known to play from time to time, he can be a bit of a wind up with other dogs at times and he is dominant in some circumstances, although he hasn't shown this at all in his foster home where the other dogs are also dominant.
He doesn't like being on his own at all, he's always been used to contact in his life when he needs / wants it so it's hard him to be on his own. He can be destructive if left alone without a human or another dog. He doesn't like being locked away in a crate or locked in a room, he likes to have free roam of the home. Whilst being in foster Shadow has settled amazingly and does find comfort with having other dogs around when his humans can't be.
He does suffer with food aggression and doesn't like to share so he needs somebody who understands to give him his own area to eat and that bowls should be taken away when finished, treats similarly should be given only in this way. Shadow will take treats perfectly but his problems are around his food bowl, he WILL snap if you try and place your hand in or around his bowl so you must leave him alone to eat and remove the bowl after he has left the room , his bowl should never be left down after eating. This is a situation that is easily managed with the right steps in place!
apply for a home check!


text:
\section{Introduction}\label{sec:intro}

Interferometry, which began with Young's double slit experiment, has blossomed over the years into a technique now widely used in physics, biology and astronomy. Examples include the use of astronomical interferometry in the first determination of the double-lobed morphology of Cygnus~A \cite{Jennison+1953}, the first imaging of the event horizon of a black hole in the center of the M87 galaxy \cite{eht19-1,eht19-2,eht19-3,eht19-4,eht19-5,eht19-6,eht21-7} and inner jet structures around the black hole in Centaurus A \cite{Janssen+2021},
determination of crystal structures \cite[][and references therein]{Hauptman1986,Hauptman-Nobel,Hauptman1991,Giacovazzo+2002,Giacovazzo2014}, seismic imaging \cite[][and references therein]{Snieder+2013}, remote sensing using radar and sonar \cite[][and references therein]{Callow2003,Rosen+2000}.

In each of these applications, accurate measurement of both the amplitude and phase of the coherence (represented as complex values) is critical for success. Often, accurate measurements of the complex-valued correlations is made difficult by corruptions introduced by the propagation medium, and non-ideal behavior of the instrument as well as of the measurement process. For example, in radio interferometry, the electromagnetic wavefronts are corrupted by ionospheric and tropospheric turbulences at low and high frequencies, respectively, as well as by corrupting factors in the array element responses. In optical interferometry, the atmospheric turbulence and imperfections in the telescope's surface geometry tend to destroy the phase information.

These element-based effects can be calibrated if there is a standard object of known morphology available \cite{TMS2017,SIRA-II}. An alternative strategy is self-calibration \cite{Cornwell+1981,Ekers1984}, which is a self-consistent scheme which uses iteration and feedback to successively refine the image starting with an initial model. But this is not always possible at the desired level of accuracy when the signal-to-noise ratio or the \textit{a priori} knowledge of the calibrator object is inadequate. Therefore, there is considerable interest in calibration-independent quantities that are unaffected by these element-specific factors, and are hence true observable properties of the object's structure.

Such interferometric invariants are typically constructed using the product of pairwise correlations measured on a closed loop of array elements, and we call them \textit{closure invariants}. In co-polar interferometry (correlations between measurements of the same polarization at all the array elements) at radio and optical wavelengths, they are widely known as closure phases and amplitudes \cite[][and references therein]{TMS2017,SIRA-II,mon07b}.
The analog of closure invariants include the triplet and quartet invariants in X-ray crystallography \cite[][and references therein]{Hauptman1986,Hauptman-Nobel,Hauptman1991,Giacovazzo+2002,Giacovazzo2014},
and Bargmann invariants in quantum mechanics \cite{Bargmann1964}. A detailed geometric insight into closure phase and its connections to various disciplines is provided in \cite{Thyagarajan+2020c}.

In astronomical interferometry, closure quantities can be traced to \cite{jen58,Twiss+1960}. Since then, they have been invaluable tools for interferometry at optical \cite{mon03a,mon03b,mon06,mon07a,mon07b} and radio frequencies \cite[][and references therein]{Schwab1980,Cornwell+1981,Pearson+1984,bhatnagar+2001}. This is particularly true of high spatial resolution applications using very long baseline interferometry (VLBI) at radio frequencies. A famous example is the recent imaging of the event horizon of the supermassive black hole at the center of the M87 galaxy \cite{eht19-1,eht19-2,eht19-3,eht19-4,eht19-5,eht19-6,eht21-7}. Recently, new approaches of probing the structure formation in the intergalactic medium through application of closure phases on faint spectral line emission from the early universe during the cosmic reionization epoch ($z\gtrsim 6$) are also being explored \cite{thy18,thy20a,thy20b}.

In the context of co-polar correlations in astronomical interferometry, a detailed mathematical approach to determining the number of generic closure invariants (including closure phases and closure amplitudes), and implications for the resulting signal-to-noise ratios was presented by \cite{Lannes1991}. Recently, first steps in the extension of closure invariants (called ``closure traces'') to correlations of full polarimetric antenna measurements were taken in \cite{Broderick+2020}, which relied on including antenna auto-correlations to derive an independent set of closure traces involving four correlations.

In this and an accompanying paper \cite{polarimetric-invariants} (hereafter papers~I and II, respectively), we establish a formalism using a combination of group-theoretic and linear algebraic approaches that advances the previous work \cite{Lannes1991,Broderick+2020}. In contrast to \cite{Broderick+2020}, we do not rely on the use of auto-correlations (which in radio astronomy tend to be susceptible to significant noise biases, besides instabilities caused by radio frequency interference and instrumental systematics), but can incorporate them if reliably measured, and derive a complete and independent set of closure invariants using gauge theory. In this paper, we employ the $\mathrm{GL}(1,\mathbb{C})$ gauge group\footnote{In general, $\textrm{GL}(n,\mathbb{C})$ refers to the group of nonsingular $n\times n$ matrices under multiplication, so $\textrm{GL}(1,\mathbb{C})$ is the group of nonzero complex numbers under multiplication.}
and its associated gauge freedom to treat and derive the complete and independent set of closure invariants in the co-polar case. The closure phases and closure amplitudes familiar in radio interferometery emerge naturally from our formalism which treats triangular loops as fundamental units, and thus unifies the prescription for obtaining closure phases and closure amplitudes. In paper~II, we build on the foundations presented in this paper and provide a formalism using the $\textrm{GL}(2,\mathbb{C})$ gauge group and its associated gauge freedoms for deducing the complete and independent set of closure invariants in the general case of full polarimetric interferometry. In both papers, we also confirm our findings with a parallel and independent viewpoint using conventional linear algebra along with numerical simulations. We also generalize the analysis to provide a prescription for extracting all the independent closure invariants in an $N$-element interferometer array. These closure invariants represent true observables about the system under observation, which in the case of interferometric imaging in astronomy corresponds to the true physical properties of the target object's morphology.

This paper is structured as follows. Section~\ref{sec:context} lays out the co-polar interferometry context within which we seek a full set of independent invariants. In section~\ref{sec:counting}, we present the expected number of independent closure invariants through a dimension count analysis. Section~\ref{sec:closure-invariants} develops a formalism that provides multiple but equivalent methods to produce a complete and independent set of closure invariants, which can be readily identified with closure phases and closure amplitudes familiar in radio interferometry. These invariants are listed explicitly in the case of three, four, and $N$ array elements. Section~\ref{sec:summary} summarizes this work. In Appendix~\ref{sec:short-spacing}, we describe a scheme for reliably measuring auto-correlations as the coincidence limit of cross-correlations if an extra short-spaced pair of elements is available. In Appendix~\ref{sec:selfcal}, we place our work in relation to the widely used `self-calibration' technique, which implicitly uses closure quantities. Appendix~\ref{sec:numerics} describes the numerical scheme used to confirm our analytic results, and Appendix~\ref{sec:invariants-form} brings out how our approach clarifies some aspects of the choice of invarints as it relates to the effects of noise and to object model parameters. In Appendix~\ref{sec:Pancharatnam-closure}, we highlight the explicit connection between the closure phase and Pancharatnam phase \cite{Pancharatnam1956a,Pancharatnam1975} in optics. Motivated by the radio astronomy application, we extend the Pancharatnam  rule to include amplitudes as well.

\section{Co-polar interferometry}\label{sec:context}

Consider an interferometer array with $N$ elements labeled by the indices $a,b=0,\ldots,N-1$. Each element, $a$, in the array measures the amplitude and phase of the same polarization (co-polar, or scalar) state of electric field, $e_a$ (represented by a complex number), incident on it, which is stochastic. The true correlation of the stochastic fields between pairs of array elements is, by definition, obtained by cross-multiplication and averaging, $S_{ab}\coloneqq \left\langle e_a e_b^\dagger \right\rangle$, where, $\dagger$ denotes the conjugate transpose operator, which reduces to complex conjugation for $\textrm{GL}(1,\mathbb{C})$ matrices in the co-polar interferometric case dealt in this paper.

The complex electric fields, $e_a$, may be subject to arbitrary gains $G_a$ at each element, where, $G_a$ are non-zero complex numbers representing both amplitude and phase distortion of the measured signal. In ideal conditions, all the gains $G_a$ would be unity and the measured signal at each baseline would accurately reflect the true signal correlation, $S_{ab}$, of emission from the object. In fact, the measured correlation, $C_{ab}$, which is a corrupted form of the true correlation, can be written as
\begin{align}
    C_{ab} &= G_a \left\langle e_a e_b^\dagger \right\rangle G_b^\dagger = G_a \, S_{ab} \, G^\dagger_b \label{mtrelation}
\end{align}
because of local gain distortions at each element. Generally, the cross-correlations are complex-valued with non-zero amplitudes. From the definition, cross-correlations satisfy $C_{ab}=C_{ba}^\dagger$. The auto-correlations, $A_{aa}\coloneqq C_{aa}$, are real and positive. Our objective is to construct quantities which are immune to the gain distortions and actually reflect true properties of the object's morphology, rather than local conditions at the individual elements of the interferometer.

We relate this problem to the `gauge' theories of fundamental forces like electromagnetism. The local gains $G_a$ are regarded as gauge transformations and our objective is to isolate a maximal set of gauge invariant quantities which we will call closure invariants. This term encompasses both closure amplitudes and closure phases. Our treatment places them on an equal footing by working with a $\textrm{GL}(1,\mathbb{C})$ gauge group.

\section{Counting arguments}\label{sec:counting}

Let us first do a dimensional count to see how many closure invariants we would expect to find. The number of measured cross-correlations is $N(N-1)/2$, this being the number of baselines or element pairs, i.e., the number of non-repeating combinations among $N$ elements taken two at a time. Since each cross-correlation is a complex number we have $N(N-1)$ real numbers. If $n_\textrm{A}$ auto-correlations (described by $n_\textrm{A}$ real numbers) are also measured, we would have to add $n_\textrm{A}$ to the above count. Without loss of generality, we can assume that $n_\textrm{A}$ is either zero or one\footnote{In radio astronomy, measuring any more auto-correlations does not give us new information due to the implicit assumption of spatial stationarity.}. In radio astronomy, auto-correlation measurements are unreliable because they are dominated by non-astronomical systematics. Our formalism for closure invariants works with or without auto-correlations.

The unknown element-based gains are $N$ complex numbers, but note that there may be sets of gains $G_a$ which do not affect
the correlations. That is, they satisfy $G_a C_{ab}G_b^\dagger=C_{ab}$ for all $a,b$. This yields $G_a G_b^\dagger=1$ and so $G_aG_a^\dagger G_b G_b^\dagger=1$. If the correlations themselves are unchanged, then so will the triple products of correlations. Thus, using $G_a G_b^\dagger G_b G_c^\dagger G_c G_a^\dagger=1$ (assuming $N\ge 3$, and distinct $a,b,c$) gives us $G_c G_c^\dagger=1$ for all $c$. This means $G_a=e^{i\theta}$ for all $a$, which is simply an overall phase factor that cancels out in Eq.~(\ref{mtrelation}). This gives us a count of $2N-1$ real numbers in the unknown gains.

Assuming none of the measured correlations is redundant (they are all independent of each other) and all of them are used, the number of closure invariants (that are independent of any choice of the unknown gains) is the difference
\begin{align}
  N(N-1)+n_\textrm{A} - (2N-1) &= N^2-3N+1+n_\textrm{A} \, . \label{eqn:count}
\end{align}
Note that this equals the number of closure phases, $(N-1)(N-2)/2$, plus the number of closure amplitudes, $N(N-3)/2+n_\textrm{A}$ \cite{TMS2017,Lannes1991}.
For example, this equals 1 (one closure phase) or 5 (three closure phases and two closure amplitudes) when $n_\textrm{A}=0$ for $N=3$ or $N=4$, respectively, and 2 (one closure phase and one closure amplitude) or 6 (three closure phases and three closure amplitudes) when $n_\textrm{A}=1$ for $N=3$ and $N=4$, respectively. As $N$ becomes large, the structural information about the object that can be extracted using closure phases and closure amplitudes asymptotically approaches that which can be extracted using correlations.

\section{Formalism for closure invariants}\label{sec:closure-invariants}

From a gauge theory perspective, we can regard each of the $N$ elements $a=0,\ldots,N-1$ as vertices in a graph. The gains that represent element-based corruption factors are local variables and multiplication by the gain factor results in a gauge transformation at each vertex. Each baseline is an edge $ab$ or link carrying ``connection'' variables that are bilocal since they are defined on the link connecting two vertices. These are the cross-correlations. In the language of gauge theory, each triangle, $\Delta_{(abc)}\equiv (a,b,c)$ is called an ``elementary plaquette'' and defines a ``Wilson loop''. The Wilson loop of any closed circuit can be determined from those of the elementary triangular plaquettes. Our objective is to list a complete and independent set of elementary Wilson loops. These are the closure invariants of radio astronomy.

In a $U(1)$ gauge theory like electromagnetism, the gauge group would be unitary and complex conjugation in Eq. (\ref{mtrelation}) would be the same as inversion. One can then form independent closure invariants from each elementary triangular plaquette $abc$ by simply multiplying correlations around the triangle, as in $\mathcal{B}_{abc}=C_{ab}C_{bc}C_{ca}$. ($\mathcal{B}_{abc}$ is called the ``bispectrum'' in interferometry). However, our gauge group $\textrm{GL}(1,\mathbb{C})$ is {\it not} unitary.
Using $\mathcal{G}=\{G_a\}, a=0,\ldots,N-1$ to denote a general gauge transformation at every vertex, $a$, on the graph, the bispectrum and other higher order products (``polyspectra'') of a set of $C_{ab}$ around a closed loop, $\Gamma$, transform as
\begin{align}
  \mathcal{P}_\Gamma &\xmapsto{\mathcal{G}} |G_a|^2 \, |G_b|^2 \ldots \mathcal{P}_\Gamma \, .
\end{align}
The phase of $\mathcal{P}_\Gamma$ is invariant under gauge transformations $\mathcal{G}=\{G_a\},a=0,\ldots,N-1$ and this is familiarly known as the closure phase. However, because the gauge group is not unitary, the amplitude of $\mathcal{P}_\Gamma$ is not invariant as it still depends on the gain amplitudes, $|G_a|$.

Similarly, one can define closure amplitudes for every even-edged loop. For example, given the loop $\Gamma=abcda$, the quantity
\begin{align}
  \mathcal{Q}_{\Gamma} &= C_{ab} \, C^{-1}_{bc} \, C_{cd} \, C^{-1}_{da}
\end{align}
transforms as
\begin{align}
  \mathcal{Q}_\Gamma &\xmapsto{\mathcal{G}} G_a G^{\dagger -1}_a \, G_b G^{\dagger -1}_b \,G_c G^{\dagger -1}_c \, G_d G^{\dagger -1}_d \mathcal{Q}_\Gamma .
\end{align}
It follows, since $\left| G_a G^{\dagger -1}_a \right|=1$ that $\left|\mathcal{Q}_\Gamma\right|$, the modulus of $\mathcal{Q}_\Gamma$, is a closure invariant:
\begin{align}
  \left|\mathcal{Q}_\Gamma\right| &\xmapsto{\mathcal{G}} \left|\mathcal{Q}_\Gamma\right| .
\end{align}
However, the phase of $\mathcal{Q}$ is {\it not} an invariant because it still depends on the phases of $G_a$ at the loop vertices.

The virtues of $\mathcal{P}_\Gamma$ and $\mathcal{Q}_\Gamma$ can be combined in a single quantity $\mathcal{C}_\Gamma$ by defining a hat operator, $\widehat{Z}=(Z^\dagger)^{-1}$, on non-zero complex numbers. We introduce the term \textit{covariants} as the set of even number of products of correlations around a closed loop, $\Gamma$, with even numbered terms ``hatted'' starting with the second. For example, a covariant on a 4-vertex loop can be written as $\mathcal{C}_\Gamma = C_{ab}\widehat{C}_{bc}C_{cd}\widehat{C}_{da}$. Hence, covariants transform as
\begin{align}
  \mathcal{C}_\Gamma &\xmapsto{\mathcal{G}} G_0 \, \mathcal{C}_\Gamma \, G_0^{-1} \, .
\end{align}
Because the gauge group of $G_a$, $\textrm{GL}(1,\mathbb{C})$, is Abelian,
\begin{align}
  \mathcal{C}_\Gamma &\xmapsto{\mathcal{G}} \mathcal{C}_\Gamma \, .
\end{align}
Thus, the covariants are invariant\footnote{Note that a four-edged covariant essentially resembles a \textit{cross-ratio}, which is a well-known invariant in projective geometry \cite[][and references therein]{Klein1872,Penrose+1984}.} under the gauge transformation effected by the element-based corruption, and form the closure invariants we are seeking in co-polar interferometry. Some of the notation above is deliberately kept general so that it transfers easily to the more difficult non-Abelian case discussed in paper~II.

\subsection{Triangular Plaquettes}\label{sec:triangular-plaquettes}

The need to have an even number of correlations in the product around a closed loop appears to impose the requirement of elementary quadrilaterals rather than triangles. However, it is unclear beforehand which even-vertexed loops will provide us with a complete set of independent invariants. We show below that such a set can be obtained by considering variables on elementary triangular plaquettes that are independent, rather than quadrilateral plaquettes. The latter can be entirely derived from the former.

The independent triangles in an $N$-element interferometer can be obtained by fixing a base vertex (for example, 0) and choosing all the triangles that contain this vertex \cite{TMS2017}. There are $N_\Delta=(N-1)(N-2)/2$ such independent triangles. We construct elementary triangular plaquette variables pinned at vertex $0$ that includes just three elements ($0, a, b$): $\mathcal{A}_{ab}=C_{0a}\widehat{C}_{ab}C_{b0}$. This quantity is neither a bispectrum nor a covariant, but acts as a building block that can be used to construct all the closure invariants. These triangular variables are clearly independent and complete as all closed loops can be decomposed into triangles and each distinct triangle is independent. Around the closed loop pinned at vertex 0 ($\Gamma_0$), they undergo gauge transformations with $G_0$ as
\begin{align}
\mathcal{A}_{ab} &\xmapsto{\mathcal{G}} \left|G_0\right|^2\mathcal{A}_{ab} \, . \label{eqn:Atransform}
\end{align}
We term such quantities as {\it advariants} in this paper. A special advariant is $\mathcal{A}_0=A_{00}$, the auto-correlation. The notation clarifies that this auto-correlation advariant is a local variable based at $0$. Note that advariants have an advantage that their gauge freedom is restricted to just a scaling by a single $|G_0|^2$ in contrast to polyspectra (including bispectrum) whose gauge transformations depend on $|G_a|$ of all the vertices in the loop. Thus, $|G_0|^2$ is the only unknown quantity in the advariants that has to be eliminated to arrive at invariant quantities. However, the phase of a triangular advariant is equal to the phase of the corresponding bispectrum, and thus gives the closure phase \cite{jen58}, which is an invariant.

\subsection{Complete and Independent Set of Invariants}\label{sec:invariants}

\subsubsection{Method 1}\label{sec:method1}

Consider all the $N_\Delta$ elementary triangular plaquettes pinned at vertex $0$. Each of them gives us a complex number $\mathcal{A}_{ab}$. These complex numbers scale as in Eq.~(\ref{eqn:Atransform}). The real and imaginary parts of these complex numbers give us $M=2N_\Delta$ real numbers, $\{\widetilde{I}_m\}, m=1,\ldots,M$, all of which transform by the unknown scale, $|G_0|^2$, under gauge transformations. The {\it ratios} of these $\widetilde{I}_m$ with respect to any of them ($\widetilde{I}_0$, for example) or some symmetric combination ($[\sum_m \widetilde{I}_m^2]^{1/2}$, for example), will eliminate this unknown scale, and thus give us $M-1$ real closure invariants, $\{\widetilde{I}_k/\widetilde{I}_0\}, k=2,\ldots,M$. Their number is $N^2-3N+1$, which is in exact agreement with the expectation in Eq.~(\ref{eqn:count}). This gives us a complete and independent set of closure invariants. Also, there is no need for auto-correlations in this construction. If we were to add $A_{00}$ as a measured quantity, we would gain a single real invariant because $\mathcal{A}_0=A_{00}$ being real and positive, would only contribute one more real number to $\{\widetilde{I}_m\}$. Note that the practical problem of systematic effects affecting auto-correlations can be mitigated by the use of cross-correlations between a pair of closely spaced elements in cases where the angular size of the object is much smaller than the angular resolution corresponding to the short spacing, and is described in Appendix~\ref{sec:short-spacing}.

While this method of taking ratios completely solves the problem we sought to address, namely, the determination of a complete and independent set of co-polar interferometric invariants, we provide below an
alternative formalism that allows us to better visualize the closure phases and amplitudes, as they are familiarly known in radio interferometry, in terms of combinations of the invariants determined here.

\subsubsection{Method 2}\label{sec:method2}

The elementary covariant quantity defined earlier as the closed loop product of an even number of correlations with alternate terms hatted starting with the second (a minimum of 4 terms) can be equivalently defined as the multiplication of an even number (minimum of two) advariants with every alternate term hatted starting with the second. For example, consider two triangles, $\Delta_1\equiv \Delta_{(0,1,2)}$ (the base triangle) and $\Delta_\ell\equiv \Delta_{(0,a,b)}$ chosen from the set of $N_\Delta$ triangles, both pinned at vertex 0, and their reversed forms, $\nabla_1\equiv \Delta_{(0,2,1)}$ and $\nabla_\ell\equiv \Delta_{(0,b,a)}$.
The covariant, which is an invariant, can now be expressed by pairing the advariants on these triangles as
\begin{align}
  \mathcal{I}_{\Delta_1;\Delta_\ell} = \mathcal{C}_{\Delta_1;\Delta_\ell} &= \mathcal{A}_{\Delta_1}\widehat{\mathcal{A}}_{\Delta_\ell} \label{eqn:rect-covariant} \\
  \textrm{or,} \quad \mathcal{I}_{\Delta_1;0} = \mathcal{C}_{\Delta_1;0} &= \mathcal{A}_{\Delta_1}\widehat{\mathcal{A}}_0 \, ,\label{eqn:tri-covariant}
\end{align}
which are in general complex-valued. In the co-polar case studied in this paper, since covariants are indeed invariants, they will be used interchangeably. However, the same is not true in the full polarimetric case presented in paper~II.

The following properties of covariants are noted:
\begin{itemize}
  \item $\mathcal{A}_{\Delta_1}$ scales as $|G_0|^2$, whereas $\widehat{\mathcal{A}}_{\Delta_\ell}$, and $\widehat{\mathcal{A}}_0$ scale as $|G_0|^{-2}$, thereby cancelling the unknown scale factor in Eqs.~(\ref{eqn:rect-covariant}) and (\ref{eqn:tri-covariant}), and making the covariant an invariant.
  \item A covariant formed with $\Delta_1$ and $\Delta_\ell$ has the same amplitude as that formed with $\Delta_1$ and $\nabla_\ell$. That is, $|\mathcal{I}_{\Delta_1;\Delta_\ell}| = |\mathcal{I}_{\Delta_1;\nabla_\ell}|$, and thus do not provide independent amplitude information. Similarly, $|\mathcal{I}_{\Delta_\ell;\Delta_1}| = |\mathcal{I}_{\Delta_1;\Delta_\ell}|^{-1}$.
  \item If $\phi_{\Delta_1}$ and $\phi_{\Delta_\ell}$ are the phases of the advariants (closure phases), then the phases of $\mathcal{I}_{\Delta_1;\Delta_\ell}$ and $\mathcal{I}_{\Delta_1;\nabla_\ell}$ are $\phi_{\Delta_1}+\phi_{\Delta_\ell}$ and $\phi_{\Delta_1} - \phi_{\Delta_\ell}$, respectively.
  \item Each invariant, $\mathcal{I}_{\Delta_1;\Delta_\ell}$, is complex-valued and thus contains two real-valued invariants, except when $\Delta_\ell=\Delta_1$. In that case, $|\mathcal{I}_{\Delta_1;\Delta_1}|=1$, and therefore, contains only one real invariant, namely, its phase which is twice the closure phase, $2\phi_{\Delta_1}$.
\end{itemize}

Using the above properties, a set of covariants (invariants) can be constructed by pairing all the independent triangular advariants, $\mathcal{A}_{\Delta_\ell}$ with the advariant of the base triangle, $\mathcal{A}_{\Delta_1}$, as $\mathcal{I}_{\Delta_1\Delta_\ell}=\mathcal{A}_{\Delta_1}\widehat{\mathcal{A}}_{\Delta_\ell}, \ell=1,\ldots,N_\Delta$. Each of these complex invariants yields two real invariants for $\ell\ne 1$ and only one real invariant for $\ell=1$ as noted above. Therefore, the total number of real invariants is $2N_\Delta-1 = N^2-3N+1$. The presence of an auto-correlation measurement will increase this count to $N^2-3N+2$ due to one additional real-valued amplitude invariant in $\mathcal{A}_{\Delta_1}\widehat{\mathcal{A}}_0$. Therefore, this method provides $N^2-3N+1+n_\textrm{A}$ real invariants, which is a complete and independent set, and is consistent with the first method using ratios as well as with the dimension count analysis in section~\ref{sec:counting}. We also verified these results numerically as detailed in Appendix~\ref{sec:numerics}.

\subsection{Closure Phases and Amplitudes}\label{sec:closure-phase-amp}

Using the second method, we can directly identify the closure phases and amplitudes familiar in radio interferometry with the closure invariants obtained here.

The independent closure phases are directly given by the phases of the $N_\Delta$ independent triangular advariants. But they can be derived from the covariants, $\mathcal{I}_{\Delta_1\Delta_\ell}$, as well, where $\Delta_\ell$ is any triangle that shares vertex 0 with the reference triangle, $\Delta_1$. The phase of $\mathcal{I}_{\Delta_1\Delta_1}$ is $2\phi_{\Delta_1}$, which yields $\phi_{\Delta_1}$ to within an ambiguity of $\pi$. This ambiguity can be addressed by using the phase of the advariant, $\mathcal{A}_{\Delta_1}$, which gives $\phi_{\Delta_1}$ directly without the aforementioned ambiguity. The phase of the rest of the covariants $\mathcal{I}_{\Delta_1\Delta_\ell}$ is $\phi_{\Delta_1}+\phi_{\Delta_\ell}$, which will directly yield $\phi_{\Delta_\ell}$ via plain substitution of the previously determined $\phi_{\Delta_1}$. This will yield $N_\Delta$ independent closure phases. The same construct also yields closure amplitudes.

An inspection of $|\mathcal{I}_{\Delta_1\Delta_\ell}|$ shows that it is indeed a closure amplitude, $|C_{01}| |C_{21}|^{-1} |C_{20}| |C_{a0}|^{-1} |C_{ab}| |C_{0b}|^{-1}$. In general, closure amplitudes formed from a pair of triangular advariants involves six such terms. However, when $\Delta_\ell$ shares an edge with $\Delta_1$ ($a=2$, for example), then the contributions, $|C_{20}|$ and $|C_{a0}|^{-1}$, associated with that edge cancel each other, and the closure amplitude reduces to the familiar form containing four terms, $|C_{01}| |C_{21}|^{-1} |C_{2b}| |C_{0b}|^{-1}$. Noting that $|\mathcal{I}_{\Delta_1\Delta_1}|$ does not contribute an amplitude (it always has unit amplitude), there are $N_\Delta-1=N(N-3)/2$ independent closure amplitudes (from the $N_\Delta-1$ triangle pairs sharing vertex 0 with the base triangle $\Delta_1$). The auto-correlation measurement, if present, adds one more independent closure amplitude through $|\mathcal{A}_{\Delta_1}\widehat{\mathcal{A}}_0|=|C_{01}| |C_{21}|^{-1} |C_{20}| A_{00}^{-1}$. Therefore, there are $N_\Delta-1+n_\textrm{A}$ independent closure amplitudes in general. It may also be noted that if an auto-correlation can be reliably measured, we can obtain all the invariants just from triangles with the auto-correlation advariant as the base advariant using $\mathcal{I}_{0;\Delta_l} = \mathcal{A}_0\widehat{\mathcal{A}}_{\Delta_1}$, without requiring quadrilaterals or 6-edged loops.

The set of closure amplitudes and phases contains a total of $2N_\Delta-1+n_\textrm{A}=N^2-3N+1+n_\textrm{A}$ real invariants as given in Eq.~(\ref{eqn:count}), which is complete and independent, and is in agreement with all the earlier analyses. Our approach using advariants and covariants (invariants) constructed only from these triangular plaquettes provides a unifying approach that directly identifies the traditional invariants known as closure phases and amplitudes in radio interferometry.

Then the question arises: which form of closure invariants is preferable? In Appendix~\ref{sec:invariants-form}, we consider this question from the viewpoint of correlations within the chosen set of invariants and their overall contribution to the determination of the likelihood of parameterized models. In brief, the choice of invariants is immaterial in the determination of the likelihood function of the model parameters as long as the set of invariants employed is complete. However, certain choices of coordinates in which the invariants are represented could lead to coordinate-induced correlations among the invariants, and also to situations when one or more of the invariants become ill-defined in those coordinates. A well-known example occurs when representing real and imaginary parts of random variable in polar form, where the amplitude and phase angle can become correlated in general and the phase angle becomes ill-defined in low $S/N$ regimes even if the real and imaginary parts are perfectly uncorrelated and well-defined. For these reasons, even if the real and imaginary parts of the invariants in our approach may be correlated, we still prefer this representation over the polar form consisting of phases and amplitudes.

\subsection{Illustration with Examples}\label{sec:examples}

For illustration, we consider the familiar examples of 3- and 4-elements arrays below.

\subsubsection{3-element Array}

Consider an array with $N=3$ and one auto-correlation measured ($n_A=1$). We have 1 and 6 real values in the auto- and cross-correlations, respectively, totaling 7 real measurements. We have $2N-1=5$ unknown real-valued parameters in the three complex-valued element gains. From dimension counting used in Eq.~(\ref{eqn:count}), we expect to find two real-valued closure invariants. From these measurements, we can form a triangular advariant, $\mathcal{A}_{\Delta_1}$, corresponding to the triangle $\Delta_1\equiv \Delta_{(0,1,2)}$, and an auto-correlation advariant, $\mathcal{A}_0$. The complex invariant
\begin{align}
  \mathcal{I}_{\Delta_1;0} &= \mathcal{A}_{\Delta_1}\widehat{\mathcal{A}}_0 = \frac{|C_{01}| |C_{20}| }{|C_{12}| A_{00}} \, e^{i\phi_{\Delta_1}} \label{eqn:auto-corr-covariant}
\end{align}
gives both these real-valued invariants. The magnitude of this complex quantity is like a closure amplitude (with one auto-correlation) and the phase is the standard closure phase of three elements. These two quantities are evidently independent. $\mathcal{I}_{\Delta_1;\Delta_1}=e^{2i\phi_{\Delta_1}}$ will give no new invariants.

If auto-correlation is not measured ($n_A=0$), the number of real measurements reduces by 1 to 6. Hence, we now expect only one invariant from Eq~(\ref{eqn:count}). Using the only advariant available, $\mathcal{A}_{\Delta_1}$, we find
\begin{align}\label{three}
  \mathcal{I}_{\Delta_1;\Delta_1} &= \mathcal{A}_{\Delta_1}\widehat{\mathcal{A}}_{\Delta_1} = e^{2i\phi_{\Delta_1}} \,
\end{align}
contains that expected invariant, namely, its phase, which is twice the standard closure phase associated with $\Delta_1$, and carries an ambiguity of $\pi$. However, the closure phase, $\phi_{\Delta_1}$, can be unambiguously obtained from the phase of $\mathcal{A}_{\Delta_1}$ directly.

\subsubsection{4-element Array}

For an $N=4$ array, we have $N_\Delta=3$ independent triangles, $\Delta_1\equiv \Delta_{(0,1,2)}, \Delta_2\equiv \Delta_{(0,2,3)}, \Delta_3\equiv \Delta_{(0,1,3)}$, and hence, three complex advariants, $\mathcal{A}_{\Delta_1},\mathcal{A}_{\Delta_2}$, and $\mathcal{A}_{\Delta_3}$. If the auto-correlation $\mathcal{A}_0$ is measured we have an additional real advariant. The dimension count analysis predicts six invariants when the auto-correlation is measured and five when it is not.

When auto-correlation is not measured ($n_\textrm{A}=0$), we combine the reference triangle $\Delta_1$, first with itself, and then with the two other independent ones, $\Delta_2$ and $\Delta_3$.
\begin{align}
  \mathcal{I}_{\Delta_1;\Delta_1} &= \mathcal{A}_{\Delta_1} \widehat{\mathcal{A}}_{\Delta_1} = e^{2i\phi_{\Delta_1}} \\
  \mathcal{I}_{\Delta_1;\Delta_2} &= \mathcal{A}_{\Delta_1} \widehat{\mathcal{A}}_{\Delta_2} = \frac{|C_{01}||C_{23}|}{|C_{21}||C_{03}|} e^{i(\phi_{\Delta_1}+\phi_{\Delta_2})} \\
  \mathcal{I}_{\Delta_1;\Delta_3} &= \mathcal{A}_{\Delta_1} \widehat{\mathcal{A}}_{\Delta_3} = \frac{|C_{20}||C_{31}|}{|C_{21}||C_{30}|} e^{i(\phi_{\Delta_1}+\phi_{\Delta_3})} \, .
\end{align}
We see that there are five real-valued invariants in the equations above -- three independent closure phases and two independent closure amplitudes familiar in radio interferometry for a 4-element co-polar interferometer array.

When $n_\textrm{A}=1$, we can write down the following three complex invariants -- $\mathcal{A}_{\Delta_1} \widehat{\mathcal{A}}_0$, $\mathcal{A}_{\Delta_2} \widehat{\mathcal{A}}_0$, and $\mathcal{A}_{\Delta_3} \widehat{\mathcal{A}}_0$ --
which gives us six real invariants comprising of three closure phases and three closure amplitudes (two as above, and one more due to auto-correlation). The invariant phase in $\mathcal{A}_{\Delta_1} \widehat{\mathcal{A}}_{\Delta_1}$ is redundant due to the presence of $\mathcal{A}_{\Delta_1} \widehat{\mathcal{A}}_0$, and can therefore be ignored.

Thus, the closure phases and closure amplitudes emerge naturally in our approach which treats independent triangles as fundamental. We break this up into a single invariant associated with the reference triangle, $\Delta_1$, and two for each of the rest. The auto-correlation measurement will provide one additional closure amplitude.

\subsection{Quadrilateral Closure Amplitudes}\label{sec:quads}

We have so far emphasized the role of triangles in constructing advariants. The complete set of closure invariants we generate include four-sided loops (quadrilaterals) as well as six-sided loops. Traditionally, closure amplitudes have been discussed by astronomers using quadrilaterals. In order to connect with this discussion we ask: Can we obtain a complete, independent set with only quadrilateral-based closure amplitudes? A clear methodology is provided in \cite{Blackburn+2020} which involves placing the $N$ elements on a ring and considering two non-overlapping pairs of elements where each pair is made of consecutive elements. Here, we provide an alternate method based on our formalism.

One way to ensure independence is to ensure that each loop contains a unique element pair not present in any other. To do this, we fix the element pair $(0,1)$ and consider the quadrilateral made with two more elements $a$ and $b$. We arrange for $1<a<b<N$ and choose the set (which we call $\mathbb{A}$) of closure amplitudes, $|C_{01}||C_{1a}|^{-1}|C_{ab}||C_{b0}|^{-1}$. So, each such choice of a pair $(a,b)$ with $a<b$ gives us an independent closure amplitude. The number of closure amplitudes in the set $\mathbb{A}$ is therefore the number of such $(a,b)$ pairs, which is $(N-2)(N-3)/2$. However, we know from basic counting \cite{TMS2017} and linear algebra \cite{Lannes1991} that the total number of independent closure amplitudes is $N(N-3)/2$. We therefore need $(N-3)[N-(N-2)]/2=N-3$ more independent closure amplitudes to complete the desired list. We supply these in a set $\mathbb{B}$, by observing that element $0$ is only paired with $b$ and hence never paired with element $2$ in the set $\mathbb{A}$. We therefore add precisely $N-3$ more closure amplitudes by including the pair, $(0,2)$, as $|C_{01}||C_{1b}|^{-1}|C_{b2}||C_{20}|^{-1}$, which contains $C_{02}$ in addition to our base choice of $C_{01}$. The set $\mathbb{B}$ is only indexed by $b$ which can take $N-3$ values, and its members are clearly independent of each other, since each contains the element pair $(1,a)$ which is unique to it within this set $\mathbb{A}$. Every member of $\mathbb{B}$ is also independent of all those of set $\mathbb{A}$, since none of those contains the pair $(0,2)$. This completes the explicit construction of $N(N-3)/2$ independent quadrilateral closure amplitudes. Although this construction has been described in a self-contained way, it was arrived at by considering quadrilateral closure amplitudes as the product of two advariants, sharing an edge.

The closure amplitudes from our approach and the ring-based approach in \cite{Blackburn+2020} are essentially equivalent. However, a consequence of the fundamental difference in the methodologies leads to one notable difference in the invariants constituting the complete and independent set when an $N+1$-th element (indexed by $a=N$) is added to the array. In the approach of \cite{Blackburn+2020}, the inclusion of a new member, $a=N$, in the ring will remove the earlier pairing of $(N-1,0)$ because they are no longer adjacent and the corresponding second set of consecutive pairs will also get removed. In their place, two new pairs of consecutive indices, $(N-1,N)$ and $(N,0)$, and the corresponding sets of second adjacent pairings will have to be included. In other words, some members of the original set of closure amplitudes will be removed and replaced with a larger but different set. A similar removal and replacement process also occurs when an array element has to be excluded. In our approach, the additional element yields $N-1$ triangles (and advariants) pinned at base vertex 0 containing this element that were not present before. When paired with the base triangle, we get $N-1$ new complex-valued invariants in addition to the existing set. That is, the members of the original set are undisturbed and $2N-2$ real-valued invariants are added, amounting to a total of $N^2-N-1$ real invariants for the $N+1$-element array. This difference may have practical consequences when certain array elements are flagged at specific times or frequencies due to poor quality of data. Our approach will be robust to such practical challenges.

Thus, we have provided multiple methods of explicitly listing the closure invariants in an $N$-element interferometer array. These methods do not yield the same set of invariants, but carry exactly equivalent information. Our approach unifies the treatment of all co-polar closure invariants, including the triangular closure phases and quadrilateral closure amplitudes familiar in radio interferometry as seen earlier.

\section{Summary}\label{sec:summary}

In this paper, we have explicitly enumerated from co-polar interferometric measurements a complete and independent set of closure invariants that are immune to corruptions local to the elements of an interferometer array. The solution relies on mathematical ideas borrowed from gauge theories of particle physics and the geometric phase of quantum mechanics and optics. Specifically, the co-polar interferometric invariants presented here are invariant under gauge transformations by the element-based corruption factors which belong to the Abelian gauge group, $\textrm{GL}(1,\mathbb{C})$.

Invariants such as closure phases and closure amplitudes have been in wide use in many disciplines for decades. However, they relied on using closed triangular and quadrilateral loops, respectively, and thus required different treatments.
The solution presented here relies on using only independent triangular plaquettes, which form the simplest non-trivial loops and are well determined, as our basis. The quantity defined on these triangles, called advariants, can be combined with an advariant chosen on a reference triangle to produce a complete set of independent invariants. The main result of this work is that using this formalism we have unified the treatment of closure invariants, which have historically required separate methods. Thus, we have provided an altered and unified perspective which improves our global understanding of interferometric closure invariants from a symmetry viewpoint. The familiar closure phases and closure amplitudes emerge naturally from this approach.

Radio astronomers are aware that auto-correlation measurements are unreliable because they could be dominated by non-astrophysical systematic effects. Our formalism works even without auto-correlations, but can naturally accommodate them if they are reliably measured. We have also outlined a methodology for using a cross-correlation on a short-spaced element pair in place of an auto-correlation to increase the number of invariants by 1, without introducing systematic errors associated with a direct single-element auto-correlation measurement.

Our approach clarifies certain aspects related to the choice of the form of closure invariants and their implications for their covariance properties and their impact on the likelihood of model parameters. While the methods presented here are specific to co-polar interferometry, the concepts serve as a stepping stone and reveal their full power when applied to a discussion of invariants in full polarimetric interferometry in paper~II. We expect that this solution will aid astronomers in processing data collected from VLBI measurements and other radio interferometry experiments.

There is also an unexpected spinoff from this application which is described in more detail in Appendix E. As explained there, there is a strong
connection between closure phases in astronomy and the Pancharatnam phase in optics and quantum mechanics. The present application suggests how
the Pancharatnam phase rule can be generalized to include amplitudes as well as phases. This rule goes beyond existing
ideas in the physics literature and shows how an astronomical application can enrich the physics which is being applied.

\begin{acknowledgments}
We thank Supurna Sinha, Ron Ekers, and the anonymous reviewers for a careful reading of the paper. J.~S. acknowledges support by a grant from the Simons Foundation (677895, R.G.).
\end{acknowledgments}

\input{ms1-accepted.bbl}

redpajama_set_name: RedPajamaArXiv


I spent a part of my childhood in Kleinburg, attending a school adjacent to the McMichael Canadian Art Collection. I'll never forget the strange pictures full of colour and wavy forms on the walls of that gallery and in the halls leading into it. I had no idea at that time how important they, and painting in general, would become for me.
In the early nineties, I accompanied my Mother to the opening of an exhibition of David Milne's paintings. It was at that opening that I met A.J. Casson. He came in a wheelchair and was near deaf, but, had a sprightly personality that was not dimmed by his 94 years. I was only vaguely aware of who he was at that time. Untutored in art and only beginning that journey into my passion for painting, I was simply in awe of such a celebrity in our midst. People who knew who he was crowded around to say hello to the legendary painter. I was one of them. I said hello and made a comment I cannot remember and he responded with some cheerful joke. It was then I realized he probably couldn't hear me over the hum of the crowd. His eyes were piercing and he had a big smile. It was a landmark moment for me.
A.J. Casson has always been one of my favourite of the Group of Seven. His paintings of places I knew added to my curiosity. I worked in the building at the left and center of the painting "Kleinburg" as a teenager. The house in "The Client", I passed twice a day on the school bus. It was, and I believe still is, a Chiropractic Office. My fascination with old buildings was well underway and to see buildings I knew and loved through his eyes and the eyes of his times changed my view of art. My own voice has come through the merging of my love for old houses and the passion I have for Casson's and the paintings of the Group of Seven.
In the years since, I have learned about him, his generation and the impact his work and the work of his contemporaries had on the world of art. I have stood in awe of beautiful paintings by Casson, Thomson, Harris, Varley, Jackson, Carmichael and others in the Art Gallery of Ontario and the National Gallery in Ottawa. Canada's Group of Seven took aspects of European impressionists, expressionists and the Fauves and made it their own, creating canvases that brought the remote Canadian wilderness to the eyes of urbanites and were instrumental in helping create a national identity for a country that has long struggled to maintain one.
Casson, passed away only months after that meeting, in early 1992. I consider it a privilege to have had the opportunity, even briefly to meet such a great painter. At the time, I had no idea that I, too, would be a painter and how his work would influence mine.
I so much see the influence. Interesting how a life can, in hindsight fit together like pieces of an intricate puzzle.
Fascinating that you worked in one of those houses!
It is funny, isn't it? The place I worked in was originally the town post office and was a jeweler when I was there and as far as I know, it still is. You can actually find it on Google Street view – Look for Jeremy Jeweler.
What a sweet painting–thanks for sharing–I'd never seen it. I once met someone who claimed to be one of the Group of Seven, but I doubted the veracity of that for several reasons.


text:
Q: Web API 405 Method Not Allowed In Remote Following is one my action:
public IHttpActionResult PutUnit(int id, Unit unit)
{
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        return BadRequest(ModelState);

    if (id != unit.UnitId)
        return BadRequest();

    db.Entry(unit).State = EntityState.Modified;

    try
    {
        db.SaveChanges();
    }
    catch (DbUpdateConcurrencyException)
    {
        if (!UnitExists(id))
        {
            return NotFound();
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    return StatusCode(HttpStatusCode.NoContent);
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Following is the response I get:
Date: Tue, 19 Aug 2014 10:30:31 GMT
Server: Microsoft-IIS/8.5
X-Powered-By: ASP.NET
Allow: GET, HEAD, OPTIONS, TRACE
Access-Control-Allow-Methods: GET, POST, PUT, DELETE, OPTIONS
Content-Type: text/html
Access-Control-Allow-Origin: *
Access-Control-Allow-Headers: Content-Type
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All actions of my controller works fine locally but once I deployed the same project on a remote server. I could not access only PUT and DELETE actions of my controller instead got the 405/Method Not Allowed error. web.config and webapiconfig.cs are same and using the default structure. Any suggestion please?

A: Take a look at Enabling The PUT Verb with Handlers and IIS 7.0

redpajama_set_name: RedPajamaStackExchange


As a User, I would like to have RTM integrated with Foursquare, so that the #section.locations can be more easily populated.
Using this idea it would be easy to also have some lightweight integration between foursquare's TODO list and RTM's.
+1 for 4sq integration. It would be nice to be able to filter tasks, for that location, after a 4sq check-in.


text:
Fresno County Supervisors Recognize Artsakh
in Armenia, Artsakh, Community, Latest, News, Top Stories
Fresno County Board of Supervisors
FRESNO—The Fresno County Board of Supervisors on Tuesday unanimously adopted a resolution condemning the acts of violence against Artsakh and Armenia and calling upon the United States Congress to support an immediate ceasefire, and reaffirming its support for a free and independent Nagorno-Karabakh Republic (Artsakh).
The supervisors said that Turkey and Azerbaijan have attacked Artsakh and Armenia, targeting non-combatant civilians, threatening the security and sovereignty of the Nagorno-Karabakh Republic (Artsakh) through hostile acts.
The document recalled that in April 2013, the County of Fresno became the first County in the State of California to officially recognize the independence of the Nagorno-Karabakh Republic (Artsakh). In that resolution, the Supervisors of the Fresno County noted that Artsakh is a historic Armenian province and has since maintained all the characteristics of a free country, despite continued attacks by Azerbaijan.
On October 12, the Fresno City Council adopted a resolution recognizing Artsakh as a free
and independent country. The document also stated that Turkey and Azerbaijan have begun attacking Artsakh and Armenia, targeting civilians, using suicide drones and ISIS mercenaries, with the mission of continuing genocide.
Armenia's Consul General to Los Angeles Ambassador Armen Baibourtian commended the efforts of Berj Apkarian, the Honorary Consul of Armenia in Fresno.
Armenian Trade & Labor Association Urges Community to Take Action on Artsakh
Hagop hagopian says:
So was Prussia!!! Who cares we just lost all the territories now we will wait for the slaughter
Leave a Reply to Hagop hagopian Cancel reply

redpajama_set_name: RedPajamaCommonCrawl


text:
Ciencias Sociales / Ciencia Política / Relaciones internacionales / Conflictos internacionales. Violencia y terrorismo político /
The new age of terrorism and the international political system
Guelke ,A.
Editorial: I.B. Tauris & Co. Ltd.
Fecha de la edición: 2009
Lugar de la edición: London. Reino Unido
Nº Pág.: 238
Idiomas: Inglés
Papel: Rústica
Sin Stock. Disponible en 5/6 semanas.
Did the 9/11 attacks on the US usher in a new age of mass-casualty terrorism and war on "terrorist groups of a global reach?" Adrian Guelke questions whether terrorism has evolved into a new form of political violence or whether it remains essentially unchanged, with violent acts by small groups pursuing their struggles against #imperialism# and the West. He explores the varying perceptions of, and reactions to, violent acts in the developed and developing worlds both within and across national boundaries and examines how #terrorist# groups join the peace process and take their part in state, society and #legitimate# government. This stimulating and strongly-argued interpretation of terrorism uncovers its theoretical and practical mainsprings and shows that 9/11, the subsequent attacks in London, Madrid, Bali, and even the concept of a global war on terrorism, may have changed perceptions and increased the fear of mass-casualties, but have not altered its basic nature. The study is now even more vital for students, policy-makers, those involved in politics and conflict resolution and, especially, general readers.
Otros libros de Guelke ,A.
Politics in deeply divided societies


text:
Uno entre mil es el nombre del tercer álbum de estudio grabado por el intérprete mexicano Manuel Mijares. Fue lanzado al mercado por la empresa discográfica EMI Capitol México el 9 de agosto de 1988 ganando 3 discos de oro y 1 discos de platino en México. El álbum fue dedicado al padre del artista.

Historia
Este álbum fue el último en la relación artística entre el equipo de producción creado para EMI Capitol y conformada por Miguel Blasco, José Ramón Flórez, Jesús Glück, Gian Pietro Felisatti y Manuel Mijares.

El álbum fue editado en lenguaje Italiano titulado "Uno su mille".

Lista de canciones

Uno entre mil (Edición LP)

Sencillos
 Soldado del amor
 Uno entre mil
 Tan solo
 El breve espacio

Posicionamientos

Posicionamientos del álbum
El álbum logró la 19.ª posición en Cartelera Álbumes de Pop latino.

Equipo técnico
 Realizado y dirigido por: Jesús Glück
 Director de arte y productor ejecutivo: Miguel Blasco
Colabora en la producción artística: José Antonio Álvarez Alija
 Arreglos: Jesús Glück y Loris Ceroni
Ingeniero de Grabación y Mezclas: José Antonio Álvarez Alija

Referencias

Álbumes de 1988
Álbumes de México de 1988
Álbumes de Mijares
Álbumes en español
Álbumes de balada romántica
Álbumes de EMI Latin