text:
PA: Resident takes action
"With no phone available, a Harrah resident shot at two home invaders Sunday afternoon, according to the Harrah Police Department."
"Police Chief Eddie Holland told The Sun that the homeowner ... hadn't been able to contact them until at least 30 minutes after the incident because he lacked a phone. ..."
"According to the resident ... two black men ... pulled up into the driveway at approximately 10 a.m. and knocked on the door and rang the doorbell. The homeowner didn't answer after seeing the two look in through the blinds of the front window. The two men left and later returned ..." ...
"The homeowner said that if one of the men had actually entered the house, he would have shot them with the pistol he owns. ..." ...
VA: Personal safety should be entrusted to more than just the police
"On Monday, at about 9:20 AM my graduate level class was evacuated from the Squires Student Center. We were interrupted in class and not informed of anything other than the following words: 'You need to get out of the building.'"
"Upon exiting my classroom, we were met at the doors leading outside by two armor-clad policemen with fully automatic weapons, plus their side arms. ..."
"It was at this time that I realized that I had no viable means of protecting myself. Please realize that I am licensed to carry a concealed handgun in the Commonwealth of Virginia, and do so on a regular basis. However, because I am a Virginia Tech student, I am prohibited from carrying my gun at school because of Virginia Tech's student policy ..." ...
VA: Arming citizens could help security
"The recent escape and pursuit of William Morva caused us to see things we never expected to on campus: Squires evacuated, police with assault rifles and snipers on rooftops. In the aftermath we must be sure to fairly and accurately assess the actions of all parties to ascertain what could have been done better or worse." ...
"We must give due criticism to the University for its decisions that put students in greater danger than was necessary. By that I am not so much referring to the decision to allow morning classes to take place, but rather the decades-old policy that prohibits students, faculty and staff from legally carrying firearms on campus. This ban even goes so far as to include those who have valid Concealed Handgun Permits." ...
Ohio Gun Owners Critical to National GOP Hopes (BFA)
"As we roll into the campaign season for the upcoming fall elections, Ohio is taking on added importance as the national political parties seek to make the Buckeye State a scrimmage for the upcoming Presidential campaign season. ..." ...
"One hint I would like to pass on to GOP party at the outset: Ohio has passed far more gun control in the past 16 years than gun rights. So you can save all of the 'you are hurting the party/you�ll be worse off with Democrats' whining. The Ohio gun owner is interested in gun rights as their identifying issue ..." ...
"16 years of controlling the state government with only 1 gun rights bill to show for it, and that one law is the worst shall-issue law in the country. ..." ...
NY: Brownsville Residents Join Yassky for Congress to Demand Action Following Rash of Shootings
Submitted by: New York State Rifle & Pistol Association
Website: http://www.nysrpa.org
"Outraged residents and District Leader Lisa Kennner joined Congressional candidate David Yassky on Tuesday to call for stricter federal regulation of the firearms industry following a rash of shootings in Brownsville."
"This summer, 12 separate shootings have left four people dead in Brownsville. Members of the largest public housing development in Brownsville will join Yassky to call for a repeal of the federal law which immunizes gun companies from victims' lawsuits."
"'A dozen shootings in one neighborhood in just a few months should be a wake up call to Washington and our communities here in the City that gun violence is still epidemic in New York,' said Council Member Yassky ..." ...
Military recruiters cited for sex abuse
"More than 100 young women who expressed interest in joining the military in the past year were preyed upon sexually by their recruiters. Women were raped on recruiting office couches, assaulted in government cars and groped en route to entrance exams, a six-month investigation by the Associated Press found. The investigation found that more than 80 military recruiters were disciplined last year for sexual misconduct with potential enlistees. The cases occurred across all branches of the military and in all regions of the country."
"'This should never be allowed to happen,' one 18-year-old victim said. 'The recruiter had all the power. He had the uniform. He had my future. I trusted him.'" ...
VA: Judge Calls Mistrial In Former Deputy's Trial
"After a Loudoun jury returned from deliberations 'hopelessly deadlocked,' Circuit Court Judge Thomas D. Horne Tuesday called a mistrial in the case of a former sheriff's office investigator who is charged with stealing guns from a property room."
"Mike Grau, 48, who was a 15-year veteran of the sheriff's office ... is accused of stealing a Ruger rifle and scope and a Remington shotgun that were being held in the sheriff�s office's property room." ...
"Grau was charged only with larceny of two guns, but [Prosecutor] Olmstead said there were a total of 14 guns and an amount of drugs taken from the property room, and that Grau could be charged later for the remaining guns. The drugs were never located and Grau said he flushed them down a toilet." ...
NY: Mayor's son wounded in accidental shooting
"The youngest son of Mayor J. Michael Cindrich is recovering from minor gunshot wounds after he accidentally shot himself with one of his father's antique handguns, the mayor said yesterday."
"John Cindrich, 25, was treated for hand and leg wounds ... following the shooting Sunday morning in his room at the family's home."
"Village detectives and state police were investigating the shooting. Mount Kisco police Chief Louis Terlizzi said 'it appeared to be an accidental discharge.'" ...
"The mayor is a lieutenant in the Mamaroneck town Police Department and his older son, Michael, is a Westchester County police sergeant. John is not in law enforcement and his father would not say what type of work he does." ...
WA: Johnny, where's my gun?
"An off-duty Seattle police officer made a gaffe reminiscent of Chief Wiggum, the incompetent police chief on 'The Simpsons.'"
"Oak Harbor Police Capt. Rick Wallace said an officer accidentally left behind his gun, badge, ID and wallet when he checked out of the Auld Holland Inn Sunday, Aug. 13." ...
"Wallace said losing a badge and ID, and especially a service weapon, is not just embarrassing, but can be a serious matter."
"'There would be a discipline review in our office,' he said." ...
"Luckily for the Seattle officer, an honest housekeeper found his gun and badge. And Wallace said nobody reported the embarrassing oversight to the Seattle officer's boss." [emphasis added]
MA: Hearing focuses on police chief
"Town officials want to end the employment of Police Chief Francis R. Fox Jr., believing he showed professional misconduct in the administration of a self-defense course, a lawyer representing Belchertown said yesterday during a hearing at Town Hall." ...
"Presenting his case, Terry said that Fox improperly received compensation in connection with the program ..."
"Fox benefited personally from what was sold as a department-sponsored program, funded with community policing funds, Terry said." ...
"Fox's lawyer ... did not dispute the facts of the case, but argued that Fox did nothing purposefully to defraud the town ..."
"'The intentions behind the conduct were always good,' he said."
PA: Former Philly Police Officer Charged With Shoplifting
Submitted by: jac
"A former Philadelphia police officer is accused of shoplifting -- by telling her two teenaged nieces to steal for her."
"Prosecutors say 35-year-old Renell Cohen purchased $86 worth of merchandise this month at the Franklin Mills Mall in Philadelphia."
"But prosecutor Janet Turnbull says Cohen also directed her teenage nieces to take other merchandise worth nearly $260."
"Turnbull says security employees and cameras followed their moves right up to the arrest."
"According to Turnbull, when Cohen was confronted, she gave a false name and did not disclose the fact she was a police officer."
"Her attorney argues that Cohen didn't know the girls took the merchandise."
"Cohen, a four-year veteran of the force, was fired by the department after she was charged." ...
UK: Police sorry after father shot dead
"A senior Metropolitan Police officer has apologised to the family of a young father shot dead after standing up to a gang of youths in the same road where he was stabbed just months earlier."
"Peter Woodhams, a 22-year-old father-of-one, was shot moments after a confrontation with the gang ..." ...
"His fiancee, Jane Bowden, claims she phoned police every day for five weeks following the knife attack, but said officers never even took a statement from her."
"Commander Rod Jarman told a press conference ... 'We will robustly deal with any failings in our investigation of the previous incident and make sure that those matters are dealt with. ..." ...
South Africa: Heist kingpin used cop guns
"Mpumalanga police have launched a high-level internal investigation to find out how an alleged cash-heist mastermind managed to get hold of police weapons."
"Alleged cash-heist kingpin Robert Green used a police R-5 assault rifle to execute his wife and father-in-law on Tuesday, before committing suicide with a police-issue Z-88 9mm pistol."
"'We have no idea how Green, who was on trial on various armed robbery charges, got the weapons.'"
"'We are conducting ballistic and other tests to trace the weapons back to their original, registered police handlers or armouries,' said captain Calvin Mashile of Lowveld police on Wednesday." ...
Israel: Police crash into suspect under surveillance
Israeli police tailing a suspected senior underworld figure accidentally crashed into his vehicle, the Yedioth Ahronoth newspaper reported Wednesday.
Police put the suspected criminal under 24-hour surveillance a month ago and the man knew he was being followed, it said.
But Tuesday, a police officer fell asleep at the wheel of his car and crashed into the man's vehicle in the southern city of Ashkelon, slightly injuring him, Yedioth reported.
Ashkelon police could not be reached by Reuters but told the newspaper an accident involving a police car and a jeep occurred in the city Tuesday. They gave no further details.
FL: Car dealer: An armed driver is a happy one
"A creative marketing campaign conjured up by Prestige Nissan General Sales Manager Dan Crumpler offers buyers of used or new Nissan SUVs or trucks in August a choice of a Remington 1187 shotgun or a Remington 700 rifle."
"'We've given away five or six guns already,' Crumpler said Monday at the Thomasville-based dealership. 'The guns retail for about $699 each, so it's a good deal, especially with dove season starting Labor Day weekend and with deer season starting in early October.'"
"Crumpler's phone rings. A potential buyer wants to come up and work a deal on the truck and free gun offer." ...
NY: Lawmakers vote to keep trap-and-skeet open
"Just after the clock struck midnight, Suffolk legislators voted to keep open the trap-and-skeet shooting range in Southaven County Park in Yaphank, drawing enthusiastic applause from the small crowd that had awaited the outcome."
"The 15-3 tally was in opposition to a bill sponsored by Legis. Kate Browning (WFP-Shirley), who sought to close down the decades-old range, which reopened in July after having been closed in 2001."
"Enthusiasts of the sport strongly urged Suffolk lawmakers to allow the shooting range to remain open, saying it provides a needed form of recreation as well as a venue that generates revenue for the county."
Writer: Ohioans do well in rifle, shotgun matches (BFA)
"The Toledo Blade outdoor writer Steve Pollick is covering the just-ended National Rifle and Pistol Matches at Camp Perry near Port Clinton and at the Clay Target Championships in Sparta, IL (the Grand American World's Trapshooting Championships, held in Vandalia, OH for 81 straight years, were forced out last year by Dayton airport expansion)."
"From the story:"
"Ohio shooters have made their marks in the target shooting world this summer..."
"Ken Potter, a Toledo police patrolman for nearly 16 years, was crowned national high-power rifle police champion after firing a score of 2333-73X out of a possible 2400. The X-score refers to tie-breaking center-shot bull's-eyes." ...
OR: War relics featured in gun club event
"Big Timber Rifle & Pistol Club of Toledo hosts the John C. Garand Match to be held Saturday, Aug. 26. Registration starts at 9 a.m. and the match begins at 10 a.m."
"The match is open to everyone using a World War II U.S. issued service rifle such as the M1 Garand, M1917 Enfield, M1903 Springfield, M1941 Johnson and the U.S. carbine, caliber .30 M1. Club members will have loaner rifles available for anyone who wishes to participate in this match. The entry fee of $25 includes 60 rounds of M2 Ball ammo to be used in this match by all competitors. 30 cal carbine ammo also available." ...
MI: World Hunting Association to Switch Format
"The World Hunting Association today announced that the organization will shift to a traditional harvest format in time for the organization's inaugural tournament this fall. ..." ...
"The hunting tour will feature 12-day tournaments during which the pro hunters can take up to six deer--four does, one management buck and one trophy buck. Certain bucks that are featured and studied on the internet portal ... will not be eligible for harvest during competition, but the hunters can gain bonus points for bringing the deer into effective range and capturing the footage on camera. A bow, rifle and a muzzle loader will be used over the course of the tournament: At least one deer must be taken with each of the three throughout a competition." ...
Australia: Employer's Duty to Provide Security
"An interesting decision was handed down by District Court Judge Tutt on 2 June 2006 in the matter of White v Calstores Pty Ltd [2006] QDC 161 regarding the obligation placed on an employer to provide a safe system of work. It is uncertain at this time if an appeal will be lodged by the Defendant." ...
"Tutt DCJ ... found that:"
"'the employer breached its duty of care to provide a safe place of work ... in that when it required the plaintiff to work alone in the shop premises at night it should have provided her with the assistance of a security guard to police and/or secure the premises ...'" ...
Submitter's Note: So when self-defense is outlawed, businesses become liable to protect their employees. I wonder if US businesses are aware of this case?
UK: Teen questioned over gun killing
"A 14-year-old boy is being questioned after a man was shot dead outside his home in east London."
"Peter Woodhams, 22, who was engaged and had a young son, collapsed in Leslie Road, Canning Town, on Monday. He died from a gunshot wound to the chest."
Police think he may have been involved in an argument with a gang of youths, near shops in Freemasons Road, shortly before the shooting." ...
UK: Alarm scares off post office gunman
"A GUN-WIELDING robber fled empty-handed from a raid on Cuddington Post Office after staff sounded the alarm."
"The incident occurred shortly before noon on Thursday at the post office on Trickett Lane when the man threatened staff and customers with a weapon."
"The raider pulled up in a dark-coloured vehicle out-side the premises and left the engine running."
"Once inside, the robber threatened two members of staff and two customers with a weapon described as a 'rifle-type' gun, demanding money." ...
UK: The crime rate was far from low 25 years ago
"Householders living in London or any other big city will rub their eyes in disbelief at Sir Ian Blair's suggestion that they feel safe enough to leave their doors unlocked."
"In addition, his declaration that levels of security are on a par with those in 1981 makes a pretty big assumption that 25 years ago crime was low."
"The Metropolitan Police Commissioner, who would then have been a young CID officer in London, must have a short memory."
"It was in the 1960s and 1970s that crime really began to rise dramatically, with acquisitive offences such as burglary and car thefts peaking in the early 1990s." ...
Americans need not fear the federal government because they enjoy the advantage of being armed, which you possess over the people of almost every other nation. � James Madison.


text:
/* Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements.  See the NOTICE file distributed with
 * this work for additional information regarding copyright ownership.
 * The ASF licenses this file to You under the Apache License, Version 2.0
 * (the "License"); you may not use this file except in compliance with
 * the License.  You may obtain a copy of the License at
 *
 *     http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */

package org.teavm.classlib.java.util.regex;

import static org.junit.Assert.assertEquals;
import java.util.regex.Matcher;
import java.util.regex.Pattern;
import java.util.regex.PatternSyntaxException;
import org.junit.Test;
import org.junit.runner.RunWith;
import org.teavm.junit.TeaVMTestRunner;

@RunWith(TeaVMTestRunner.class)
public class ReplaceTest {

    @Test
    public void testSimpleReplace() throws PatternSyntaxException {
        String target;
        String pattern;
        String repl;

        target = "foobarfobarfoofo1";
        pattern = "fo[^o]";
        repl = "xxx";

        Pattern p = Pattern.compile(pattern);
        Matcher m = p.matcher(target);

        assertEquals("foobarxxxarfoofo1", m.replaceFirst(repl));
        assertEquals("foobarxxxarfooxxx", m.replaceAll(repl));
    }

    @Test
    public void testCaptureReplace() {
        String target;
        String pattern;
        String repl;
        String s;
        Pattern p = null;
        Matcher m;

        target = "[31]foo;bar[42];[99]xyz";
        pattern = "\\[([0-9]+)\\]([a-z]+)";
        repl = "$2[$1]";

        p = Pattern.compile(pattern);
        m = p.matcher(target);
        s = m.replaceFirst(repl);
        assertEquals("foo[31];bar[42];[99]xyz", s);
        s = m.replaceAll(repl);
        assertEquals("foo[31];bar[42];xyz[99]", s);

        target = "[31]foo(42)bar{63}zoo;[12]abc(34)def{56}ghi;{99}xyz[88]xyz(77)xyz;";
        pattern = "\\[([0-9]+)\\]([a-z]+)\\(([0-9]+)\\)([a-z]+)\\{([0-9]+)\\}([a-z]+)";
        repl = "[$5]$6($3)$4{$1}$2";
        p = Pattern.compile(pattern);
        m = p.matcher(target);
        s = m.replaceFirst(repl);
        // System.out.println(s);
        assertEquals(
                "[63]zoo(42)bar{31}foo;[12]abc(34)def{56}ghi;{99}xyz[88]xyz(77)xyz;",
                s);
        s = m.replaceAll(repl);
        // System.out.println(s);
        assertEquals(
                "[63]zoo(42)bar{31}foo;[56]ghi(34)def{12}abc;{99}xyz[88]xyz(77)xyz;",
                s);
    }

    @Test
    public void testEscapeReplace() {
        String target;
        String pattern;
        String repl;
        String s;

        target = "foo'bar''foo";
        pattern = "'";
        repl = "\\'";
        s = target.replaceAll(pattern, repl);
        assertEquals("foo'bar''foo", s);
        repl = "\\\\'";
        s = target.replaceAll(pattern, repl);
        assertEquals("foo\\'bar\\'\\'foo", s);
        repl = "\\$3";
        s = target.replaceAll(pattern, repl);
        assertEquals("foo$3bar$3$3foo", s);
    }
}

redpajama_set_name: RedPajamaGithub


According to The Wall Street Journal, Apple is aggressively forcing iPhone users to use Apple Pay by constantly reminding users to provide their credit card details to activate the mobile payment service. If you are an Apple iPhone user, you might notice a red badge on the settings icon on the home screen.
Tapping on the settings icon reveals a message from Apple reminding you to add the credit card details.
Apple benefits twice by recieving a 0.15% of the amount of a completed credit card transaction using Apple Pay. The company also recieves a flat amount for each debit card transaction. With this Apple generated $29 billion in revenue from Apple Pay last year.


Get In A Sales Program!
Anyone who has been in the business of selling and entrepreneurship understands how important it is to start in the right direction. An effective sales training program can do wonders for your team, but you should also remember that there are several companies out there that will leave you with empty promises and a hole … Read moreGet In A Sales Program!


text:
Ho Kham may refer to two administrative units, covering the same area

Ho Kham subdistrict
Ho Kham municipality

redpajama_set_name: RedPajamaWikipedia


But speaking to Neeson at the Ritz Carlton Hotel in downtown New York, where he was promoting his latest action thriller, Taken 2, I got the impression that he was deeply affected by the sounds of Islamic call for prayer that he heard 5 times a day while shooting the movie in Istanbul.
So moving, that he bought CD's of Islamic chants, which he puts on before bedtime to help him sleep.
Born in North Ireland, Neeson, whose father was a custodian at a catholic school, was brought up a very strict Catholic. In fact, he was the altar boy in the local church. But he became disenchanted by his Catholic faith, when it failed to soothe his soul and fill the spiritual void that was left by the tragic death of his wife Natasha Richardson in 2009, following a skiing accident in Canada. "I guess I'm somewhat lapsed as a practicing Catholic," he reflects in his soft voice.
Frankly, I have not seen the Oscar-nominated actor in such high spirit since 2009; he had invariably looked stricken by profound grief. Whether it's his new-found spiritually or the passing of time that healed the wounds of his agonizing loss, he wouldn't say. "It's personal," he insists. He wouldn't also elaborate on his own religious convictions other than speaking abstractly about spirituality.
"I am a big believer in acts of kindness, no matter how small that is. Thank you notes and giving to charity are very, very important," he says.
Although he began performing in the mid-70's as part of an Irish acting company, the 60-year-old gained international recognition when Spielberg cast him to portray the German industrialist in Schindler's List (1993), which earned him his first Academy Award nomination. He followed it with compelling performances in Michael Collins (1996) and Kinsey (2004), but it was his action roles in films such as Star Wars: Episode I- The Phantom Menace" (1999) The A-Team (2010) and Taken (2008) that endowed him with the widest appeal to moviegoers.
In addition to the sequel Taken 2, which is due to be released next week, earlier this year, he starred in the thriller The Grey. Currently, he is getting into shape -not spiritually- to play a grieved writer in a Paul Haggis movie.
"I've got a couple of bed scenes with Olivia Wilde, so the push ups continue," he quips.


After lunch, we will drive you to the city of Pisa, that according to the legend was founded by mythical Trojan refugees from the homonymous Greek city of Pisa.
To skip the long lines at the ticketing offices and to ensure the success of the trip, please let us arrange this for you. We will book the entrance ahead of time to get the best shift, according to the schedule of the day. Please in your booking confirm us the number of tickets and you will pay the tickets cost cash to your Driverguide (from € 12,00 to € 16,50 p.p. including the official booking fee and depending on the temporary exhibitions).


BrandlinkDC recently hosted a b*tchin' bash at Bethesda's Ginger Boutique to celebrate the release of Jill Kargman's new book, The Rockstar in Seat 3A. I wore my Rockstar jeans from Old Navy, my Ann Taylor strappy sandals (so comfortable) and a sequin Chico's top. Bet you didn't know Chico's could be so edgy!
The Rockstar in Seat 3A looks like an amazing beach read!
Author Jill Kargman signing my book!
The Brandlink DC interns fully committed to the Rockstar theme!
I love both your top and Liz's outfit!!! Both so cute!


Barstools Are Meant To Be In Multiples!
Finding the right stools for your home is a challenge, because there are so many factors and ideas to consider, including height, style and comfort.
Recently, we have found that demand for the bar stools is rising exponentially. The increasing number of spacious dining rooms and kitchens with counter height or bar tables require taller stools. Also, bar stools are becoming more popular because their varied styles not only are more appealing than the common wooden stool, but actually accentuate the theme of a home. They allow for a higher view when eating, drinking or socializing and can add uniqueness to the atmosphere. There are many different constructions. Bar stools are often made of wood or metal. With and without armrests, back, and padding on the seat surface. And can range from basic wooden designs to extremely detailed ones with a custom height for the perfect fit. Plus, there many stools with upholstered seats, allowing you to hang on comfortably.
Needless to say, barstools should be arranged in multiples for the complete look.
Pinnadel Swivel and Tall Swivel Stools have metal frames, wood planked seats with adjustable height swivel and metal footrests. The frames are made from tubular, stamped sheet and solid bar metal with a gun metal finish.
Pinnadel Counter Swivel and Tall Swivel Barstools have metal frames, wood cap rails and wood base with swivel, as well as cushion seats in a vintage brown faux leather with nailhead trim and metal footrests. The frame are made from tubular, stamped sheet and solid bar metal with a gun metal finish. It also features chair backs for your comfort.
Pinnadel Metal Counter and Metal Bar Stools are made from stamped sheet metal in a retro design with tubular X and stretcher bracing and a convenient handhold cut-out in the seats. The frames are made from tubular, stamped sheet and solid bar metal with a gun metal finish. The stools have nesting design for compact storage. They create a special design in your room.
Check our Top 10 Home Bar Section or browse FurniturePick catalog for other interesting ideas. Don't hesitate calling call us to obtain information.


text:
\section{Introduction}\label{section:intro}
The Discrete Fourier Transform (DFT) represents a signal as a combination of complex exponentials (or frequencies). Analysis of signals using the DFT has been a popular tool in many areas of engineering and science. Given an $N-$ length signal $\vec{x}$, its DFT is another $N-$ length signal given by $\mathcal{F}_N \vec{x}$, where \[\mathcal{F}_N = \left(e^{-2\pi i mn/N}\right)_{m,n=0}^{N-1}\] is the $N\times N$ DFT matrix. While a naive multiplication of the DFT matrix $\mathcal{F}_N$ with the signal $\vec{x}$ would incur a computational complexity of $O(N^2)$, the Fast Fourier Transform (FFT) is a suite of algorithms that compute this multiplication (by exploiting the structure of the matrix $\mathcal{F}_N$) with $O(N\log N)$ computational complexity\footnote{By computational complexity we mean the number of (complex) additions and multiplications used by the algorithm. Also recall that we say the complexity is $O(g(N))$ to mean that for sufficiently large $N$ the number of operations required is bounded above by $c g(N)$ for some constant $c$.}. Perhaps the most well known among these algorithms is the radix-$2$ FFT \cite{1162257}, which assumes $N$ to be a power of $2$ and exploits the block structure of the matrix $\mathcal{F}_N$. Many other algorithms, including Cooley-Tukey FFT \cite{cooley1965algorithm}, Good Thomas FFT \cite{good1958interaction} and Rader's FFT \cite{rader1968discrete} (to name a few) developed in the literature can be used to compute the DFT in $O(N\log N)$, even when $N$ is non-prime power composite, or itself a prime. The importance of FFT makes it one of the most important algorithms developed in the last century\cite{10.2307/29775194,dongarra2000guest}.

Many applications of DFT rely on the crucial assumption that most of the frequency components (i.e., the entries of $\mathcal{F}_N\vec{x}$) are zero (or close to zero). This naturally leads to the question: if the frequency components are known to be zero at certain locations, can we do better than $O(N\log N)$? In this work, we attempt to address the following problem:

\begin{problem}
Given a space of $N-$length signals $\mathbb{B}^\mathcal{J}$, such that for any signal $\vec{x} \in \mathbb{B}^\mathcal{J}$ in the space, its DFT $\mathcal{F}_N\vec{x}$ is known to be nonzero only at locations $\mathcal{J}$:
\[
\mathcal{F}_N\vec{x}(j)=0 \quad \text{ for any }j\notin \mathcal{J};
\]
What is the best achievable computational complexity for finding the DFT of signals in this space?
\end{problem}
Here we assume that $\mathcal{J}$ is known beforehand. We make some preliminary observations on this problem in Section \ref{section:pre_observations}: a naive computation of the $|\mathcal{J}|=k$ frequency coefficients incurs a complexity of $O(Nk)$, and with a simple argument, we can easily show that the DFT of such signals can be computed with $O(k^3)$ complexity, irrespective of the frequency support $\mathcal{J}$ (more on this in Section \ref{section:pre_observations}). However, it is also apparent with simple examples, that more structured $\mathcal{J}$ is, the lesser the complexity required to compute the DFT. We can take for instance, assuming $k$ divides $N$, the frequency support $\mathcal{J}$ to be periodic ( $\mathcal{J}=\{0,N/k, 2N/k, \ldots(k-1)N/k\}$), or a set of consecutive elements $\mathcal{J} =\{0,1,2,\ldots, k-1\}$: in both these cases we can argue (using basic properties of the DFT, again, in Section \ref{section:pre_observations}) that the DFT of signals with such frequency supports can be computed in $O(k \log k)$. We can then ask if there is any general structural property of the support set $\mathcal{J}$ that enables an $O(k \log k)$ computation of the DFT.

Towards this end, we assume the set $\mathcal{J}$ is \emph{spectral}:
\begin{defn}
We say that a set $\mathcal{J} \subseteq \mathbb{Z}_N$ is spectral if there exists a set $\mathcal{I} \subseteq \mathbb{Z}_N$ of the same size as $\mathcal{J}$, such that the (square) Fourier submatrix of $\mathcal{F}_N$ with rows indexed by $\mathcal{I}$ and columns indexed by $\mathcal{J}$ is unitary\footnote{Here and in the rest of the document, whenever we say unitary, we mean unitary up to scaling.} (up to scaling). If $M$ is such a submatrix, it satisfies $M^*M = k I$ where $k=|\mathcal{J}|$ and $I$ is the $k \times k$ identity, and  $\mathbb{Z}_N$ is the set of integers modulo $N$ \footnote{ For a matrix $M$, $M^*$ denotes the conjugate transpose of $M$.}.
\end{defn}
 Indeed, the periodic and consecutive element sets mentioned earlier are examples of spectral sets. These sets are relevant in the context of Fuglede's conjecture \cite{fuglede1974commuting} in Fourier analysis, and this conjecture is as yet open for the discrete case for arbitrary $N$ \cite{dutkay2014some, siripuram2018lp}.

When $\mathcal{J}$ is spectral and $N$ is a power of $2$, we provide a (deterministic) algorithm to compute the DFT of signals in $\mathbb{B}^\mathcal{J}$ that has complexity $O(k \log k)$. Our algorithm uses recent results on the structure of spectral sets \cite{fan2016compact,siripuram2019discrete,siripuram2020convolution} in terms of the binary expansion of the indices in $\mathcal{J}$. Our algorithm reads $k$ entries of the vector $\vec{x}$ at specific locations chosen according to the structure of the spectral set $\mathcal{J}$. The algorithm operates very similar to the radix-2 FFT algorithm (see Fig \ref{fig:block-diagram}): the crucial difference is that as opposed to the digit reversing permutation used by the radix-2 FFT, our algorithm reverses only a subset of digits. Since $\mathcal{J} = \mathbb{Z}_N$ is trivially a spectral set (Problem 1, in this case, reduces to finding the standard DFT), our algorithm can be seen as a generalization of the standard radix-2 FFT.

In Section \ref{section:motivation_SFFT}, we explain our motivations and try to place this result among other results in sparse FFT algorithms. In Section \ref{section:pre_observations}, we make some preliminary observations on Problem 1 and elaborate on some of the comments mentioned in the introduction. In Section \ref{section:main_result}, we present the main result (Theorem \ref{thm:block-structure}) that enables our algorithm to work, and finally in Section \ref{section:proofs}, we provide the proof for Theorem \ref{thm:block-structure}. Though we give our proof for the case when $N$ is a power of $2$ for ease of exposition; these techniques can be easily generalized to the case when $N$ is a power of any prime $p$.
\section{Motivations and connections to Sparse FFT literature}\label{section:motivation_SFFT}
The problem of efficiently finding the DFT is even more important considering the ever-increasing data sizes that emerging technologies generate and analyze. As such, much of the work on FFT in this century has focused on the sparse-FFT algorithms. Most of these algorithms assume the DFT $\mathcal{F}_N \vec{x}$ has only $k$ (typically $k << N$) non zero entries (but the locations of these $k$ non zero entries are unknown). In addition to computational complexity, algorithms in sparse DFT computation are also interested in minimizing the sample complexity, which is the number of the entries of $\vec{x}$ that the algorithm needs access to compute $\mathcal{F}_N \vec{x}$. What differentiates this area research from the allied areas of compressed sensing and sparse signal recovery \cite{2051150} is the emphasis on computational complexity, in addition to sample complexity.

The best known sample complexity is $O(k \log N)$ \cite{14819549}, and the best known time complexity is $O(k \log N \log(N/k))$ \cite{nearlyoptimalsparse}; but it is not yet known if the same algorithm can achieve both of these. Some of these algorithms are probabilistic, which assume that the frequency support is more or less uniform and provide algorithms that work with constant or high probability for large $N$ \cite{gilbert2014recent, pawar-ramachandran, nearlyoptimalsparse}.

More recently, there has been an increasing interest in developing algorithms for the case when the sparsity pattern is not completely arbitrary. Among these include results for block sparse signals \cite{3055462} that achieve a complexity of $O(k \text{ poly} (\log n))$.

Along these lines, the problem that motivated us was the DFT computation for signals with partially known support, similar to such models in compressed sensing \cite{2051150}. An optimal algorithm in such a setting has to make use of the (partially known) support structure in some way. At the extreme, one can ask about the optimal complexity when the support is fully known: this leads us to Problem 1. To the best of our knowledge, Problem 1 has not been tackled from the computation perspective.

In this context, our algorithm is deterministic, assumes the frequency support is known and spectral and has a sample complexity $k$ and computational complexity $O(k \log k)$.

\section{Preliminary observations} \label{section:pre_observations}
Let us start with some simple observations on Problem 1. Given an $\vec{x}$, we will often refer to $\vec{x}$ as being in the time domain and $\mathcal{F} \vec{x}$ as being in the frequency domain. We will denote the DFT matrix with $\mathcal{F}_N$ or $\mathcal{F}$ when $N$ is apparent from the context. We also denote by $\vec{x}_\mathcal{I}$ the vector obtained by taking only the entries indexed $\mathcal{I}$ from $\vec{x}$, and by $A(\mathcal{I},\mathcal{J})$ the submatrix $A$ with rows indexed $\mathcal{I}$ and columns indexed $\mathcal{J}$.

Now for $\vec{x}\in \mathbb{B}^\mathcal{J}$, we have
\[
\vec{x} = \underbrace{\mathcal{F}^{-1}}_{\substack{\text{inverse }\\ \text{DFT matrix }}}\underbrace{\mathcal{F} \vec{x}}_{ \text{DFT of }\vec{x}},
\]
and since $\mathcal{F} \vec{x}$ (the signal in the frequency domain) is non-zero on locations in $\mathcal{J}$, only the columns of $\mathcal{F}^{-1}$ indexed by $\mathcal{J}$ (in other words, only the complex exponentials with frequencies from $\mathcal{J}$) play a role in the reconstruction of $\vec{x}$. Suppose we read only $k$ entries of the vector $\vec{x}$ corresponding to the locations $\mathcal{I}$ in the time domain, we get
\[
\vec{x}_\mathcal{I} = \mathcal{F}^{-1}(\mathcal{I},\mathcal{J})(\mathcal{F}\vec{x})_\mathcal{J}.
\]
To find $(\mathcal{F}\vec{x})_\mathcal{J}$ (and hence $\mathcal{F}\vec{x}$), we could solve the above system of equations to get
\begin{equation}
\label{eq:recovery-from-time-samples}
(\mathcal{F}\vec{x})_\mathcal{J} = \left(\mathcal{F}^{-1}(\mathcal{I},\mathcal{J})\right)^{-1}\vec{x}_\mathcal{I},
\end{equation}
provided we pick $\mathcal{I}$ in such a way that the resulting submatrix $\mathcal{F}^{-1}(\mathcal{I},\mathcal{J})$ is invertible. One easy way to obtain an invertible submatrix is to pick $\mathcal{I}$ to be $k$ consecutive elements of $\mathbb{Z}_N$, this ensures the resulting submatrix is Vandermonde and hence invertible \cite{donoho-stark:uncertainty, DS-1}. The solution to \eqref{eq:recovery-from-time-samples} involves inverting a $k \times k$ matrix, and hence has a complexity of $O(k^3)$ \cite{boyd2018introduction}. Note that this works irrespective of $\mathcal{J}$, and scales only with the size of $\mathcal{J}$ and not the dimension $N$. However, the $O(k^3)$ complexity is applicable only in the noiseless case, as the resulting Vandermonde matrix, though invertible, maybe poorly conditioned \cite{DS-1}.

As discussed in the introduction, we can consider specific frequency support sets $\mathcal{J}$: for e.g., suppose we assume $k$ divides $N$ and set $\mathcal{J} = \{0,1,2,\ldots, k-1\}$. We can then downsample the signal $\vec{x}$ in the time domain to take samples at $\mathcal{I}=\{0,k',2k',\ldots,(k-1)k'\}$ where $k'=N/k$. We know from elementary Fourier analysis that the DFT undergoes aliasing \cite{osgood2018lectures}:
\[
\mathcal{F}_k \vec{x}_\mathcal{I} (m) = \sum_{i}\mathcal{F}_N \vec{x}(m-ik), \quad m=0,1,2\ldots,k-1.
\]
but since the signal in the frequency domain is limited to $\{0,1,\ldots,k-1\}$ this aliasing does not lead to any overlaps. This reduces the complexity of finding $\vec{x}$ to the complexity of finding a $k-$ point DFT, resulting in $O(k \log k)$ complexity\footnote{A similar argument can be applied when $\mathcal{J} = \{0,k',2k',\ldots, (k-1)k'\}$ is a periodic set.}. Note that this can be seen directly from \eqref{eq:recovery-from-time-samples}: for the given $\mathcal{I}$ and $\mathcal{J}$, the resulting submatrix $\mathcal{F}^{-1}(\mathcal{I}, \mathcal{J})$ has entries $\exp( 2\pi i mk' n/ N) = \exp(2\pi i mn/k)$ for $m,n = 0,1,\ldots, k-1$. Thus the submatrix $\left(\mathcal{F}^{-1}(\mathcal{I},\mathcal{J})\right)^{-1} = k'\mathcal{F}_k$ is the $k \times k$ DFT matrix and the $O(k \log k)$ complexity follows.

We can then ask, what is the structure on $\mathcal{J}$ which enables $O(k \log k)$ computation of the DFT of signals in $\mathbb{B}^\mathcal{J}$? To start with, we can assume that it is possible to pick an $\mathcal{I}$ such that the submatrix $F^{-1}(\mathcal{I}, \mathcal{J})$ is unitary, and this leads us directly to the definition of spectral sets. In this case, \eqref{eq:recovery-from-time-samples} reduces to
\begin{equation}
    \label{eq:orthogonal-recovery-from-time-samples}
(\mathcal{F}\vec{x})_\mathcal{J} = k'\left(\mathcal{F}(\mathcal{J},\mathcal{I})\right)\vec{x}_\mathcal{I}.
\end{equation}
This assumption directly reduces the complexity to $O(k^2)$, however, unlike in the example above, the submatrix $\mathcal{F}(\mathcal{J},\mathcal{I})$ may not be a $k \times k$ DFT matrix.Take for example, $N=2^{10}$, $\mathcal{J} = \{1, 292, 641, 932\}$, and $\mathcal{I} = \{316, 384, 828, 896\}$, the resulting submatrix is\\

\resizebox{0.95\hsize}{!}{%
$\begin{pmatrix}
-0.36 - 0.93i &  -0.71 - 0.71i &   0.36 + 0.93i &  0.71 + 0.71i \\
0.77 - 0.63i &  -1.00 + 0.00i  &  0.77 - 0.63i &  -1.00 + 0.00i \\
0.36 + 0.93i &  -0.71 - 0.71i &  -0.36 - 0.93i  &  0.71 + 0.71i \\
-0.77 + 0.63i &  -1.00 + 0.00i &  -0.77 + 0.63i &  -1.00 + 0.00i
\end{pmatrix}$}\\

\begin{comment}

Take for example, $N=2^{10}$, $\mathcal{J} = \{161, 545, 636, 1020\}$, and $\mathcal{I} = \{0, 4, 512, 516\}$, the resulting submatrix is

\[ \begin{pmatrix}
1.00  & -0.6895 + 0.7242i & -1.00 &  0.6895 - 0.7242i\\
1.00  &  0.6895 - 0.7242i & -1.00 & -0.6895 + 0.7242i\\
1.00  & -0.9952 - 0.0980i &  1.00 & -0.9952 - 0.0980i\\
1.00  &  0.9952 + 0.0980i &  1.00 &  0.9952 + 0.0980i
\end{pmatrix}\]
\end{comment}

Which can be checked to be unitary. However, this is not a $k\times k$ DFT matrix (nor can it be written as $D_1\mathcal{F}_kD_2$ for some diagonal matrices $D_1, D_2$). Thus, the $O(k \log k)$ complexity does not follow from the already known FFT algorithms.

Also, note that given a spectral set $\mathcal{J}$, there could be many possible time domain samples $\mathcal{I}$ that result in a unitary submatrix \cite{siripuram2019discrete}. For one specific choice of $\mathcal{I}$, we prove in Theorem \ref{thm:block-structure} that the resulting submatrix $\mathcal{F}(\mathcal{J}, \mathcal{I})$ has a block structure (similar to $\mathcal{F}_k$) that enables $O(k \log k)$ computation (down from $O(k^2)$) in \eqref{eq:orthogonal-recovery-from-time-samples}. It is not yet clear to us if this property extends to arbitrary unitary submatrices of the DFT matrix.

\section{Main result}\label{section:main_result}
In this section, we present our main result. We will discuss about the structure of spectral sets, but first, we note that for $N$ is a power of 2, any spectral set has a size $k=2^r$ (see Lemma \ref{lem:spectral-structure}). The following theorem is the main result of this work:
\begin{theorem}
\label{thm:block-structure}
Suppose that $N$ is a power of $2$ and that $\mathcal{J}\subseteq \mathbb{Z}_N$ is a spectral set. Then under a suitable choice of indices $\mathcal{I} \subseteq \mathbb{Z}_N$, a suitable permutation of the rows and columns, the submatrix $\mathcal{F}(\mathcal{J}, \mathcal{I})$ has the form
\[
\begin{pmatrix}
I & D \\ I & -D
\end{pmatrix}\begin{pmatrix}
M & 0 \\ 0 & M
\end{pmatrix}
\]
where $D$ is a diagonal matrix; and $M$ is a unitary submatrix of $\mathcal{F}$ of size $k/2$.
\end{theorem}
 Recall that to solve \eqref{eq:orthogonal-recovery-from-time-samples}; we need to do the multiplication $\mathcal{F}(\mathcal{J}, \mathcal{I})\vec{v}$. If $T(k)$ is the complexity to compute $\mathcal{F}(\mathcal{J}, \mathcal{I})\vec{v}$, then from the structure of $\mathcal{F}(\mathcal{J}, \mathcal{I})$ in Theorem \ref{thm:block-structure}, we have $T(k) = 2T(k/2) + 2k$. Since $k=2^r$ is a power of $2$, Theorem \ref{thm:block-structure} can be recursively applied and results in a complexity of $O(k \log k)$. Note that this is very similar to the block structure of the DFT matrix \cite{osgood2018lectures}: we may call the diagonal elements of $D$ as the twiddle factors (see Fig \ref{fig:block-diagram}).

To elaborate on the statement of Theorem \ref{thm:block-structure}, we need to explain the structure of spectral sets first. Consider writing, for each index in $\mathcal{J}$, the corresponding binary digits, arrayed in rows. The columns of such an arrangement represent the bits: starting with the least significant bit on the left, and each row of such an arrangement represents an index of $\mathcal{J}$ (see Fig \ref{fig:digit-table-example}).

Define the \emph{pivots} of such an arrangement to be the positions which contain the first (starting from the left) difference for some pair of rows. In Fig \ref{fig:digit-table-example}, for instance, indices $636$ and $545$ differ in the digit corresponding to $2^0$, whereas the indices $636$ and $1020$ first differ in the digit corresponding to $2^7$ (and are identical before that). Similarly, considering all the other differences, we see that $\{2^0, 2^7\}$ form the pivots. Finally, if we denote the pivots by $\mathcal{L}$, we say a digit table is \emph{conforming} if $|\mathcal{J}| = 2^{|\mathcal{L}|}$.

\begin{figure}[htb]
\begin{center}
\begin{tabular}{c|c|cccccc|c|cc}
\cline{2-2} \cline{9-9}
 & $2^0$ & $2^1$ & $2^2$ & $2^3$ & $2^4$ & $2^5$ & $2^6$ & $2^7$ & $2^8$ & $2^9$  \\
\hline
 636 & 0 & 0 & 1 & 1& 1 & 1 & 1 & 0 & 0& 1 \\
 545 & 1 & 0 & 0 & 0 & 0& 1 & 0 & 0 & 0& 1 \\
 1020 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1& 1 \\
 161 & 1 & 0 & 0 & 0 & 0& 1 & 0 & 1 & 0& 0 \\
 \cline{2-2} \cline{9-9}
 \end{tabular}
 \end{center}
 \caption{An example digit table for $\mathcal{J}=\{161,545,636,1020\}$, $N=1024$. The rows of the digit-table represent the digits of an element of $\mathcal{J}$. Here $\mathcal{L} = \{2^0,2^7\}$ (highlighted) represent the pivots. This is a conforming digit table, since $4=|\mathcal{J}|=2^{|\mathcal{L}|}$. Note that the pivoted digits take all $4$ possible values $00, 10, 01$ and $11$.}
 \label{fig:digit-table-example}
\end{figure}

Suppose we consider the $|\mathcal{L}|-$tuple representing the pivoted digits for each index. By definition of pivots, all these tuples must be distinct. Conformity enforces that these tuples must take all the possible $2^{|\mathcal{L}|}$ values. For the example $\mathcal{J}$ in Fig \ref{fig:digit-table-example}, the pivot digits for $636,545,1020, 161$ are $00,10,01,11$ respectively. With these definitions in place, we have the following:
 \begin{lemma}
 \label{lem:spectral-structure}
 when $N$ is a power of $2$, the submatrix $\mathcal{F}^{-1}(\mathcal{I},\mathcal{J})$ is unitary iff for some set of pivots $\mathcal{L}$
 \begin{enumerate}
     \item $\mathcal{J}$ corresponds to a conforming digit table with pivots $\mathcal{L}$,
     \item $\mathcal{I}$ corresponds to a conforming digit table with pivots $N/2\mathcal{L}$.
 \end{enumerate}
 \end{lemma}
Lemma \ref{lem:spectral-structure} is a direct consequence of earlier works \cite{siripuram2019discrete, siripuram2020convolution}; we discuss more about this in Section \ref{section:proofs}.

Suppose we start with a $\mathcal{J}$ that is spectral, construct its digit table, and read off the pivot columns $\mathcal{L}$. By Lemma \ref{lem:spectral-structure}, this must be a conforming digit table, and we must have $|\mathcal{J}| = 2^{|\mathcal{L}|}$. Now we take $\mathcal{L}' = N/2\mathcal{L}$ and construct a digit table by setting the pivoted digits to take all possible $|\mathcal{L}'|-$tuples and all non pivoted digits to zero. For e.g., consider the set $\mathcal{J}$ in Fig \ref{fig:digit-table-example}: the pivots are $\mathcal{L}=\{2^0,2^7\}$; so we set $\mathcal{L}' = N/2\mathcal{L} = \{2^2, 2^9\}$, and construct the digit-table, leading to Fig \ref{fig:I-digit-table-example}. This construction ensures the resulting digit-table is conforming. We then take the indices corresponding to this digit table as the time domain sampling locations $\mathcal{I}$. This is the choice of indices $\mathcal{I}$ referred to in the statement of Theorem \ref{thm:block-structure}.
\begin{figure}[htb]
\begin{center}
\begin{tabular}{ccc|c|cccccc|c|}
\cline{4-4} \cline{11-11}
 & $2^0$ & $2^1$ & $2^2$ & $2^3$ & $2^4$ & $2^5$ & $2^6$ & $2^7$ & $2^8$ & $2^9$  \\
\hline
 0 & 0 & 0 & 0 & 0& 0 & 0 & 0 & 0 & 0& 0 \\
 512 & 0 & 0 & 0 & 0 & 0& 0 & 0 & 0 & 0& 1 \\
 4 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0& 0 \\
 516 & 0 & 0 & 1 & 0 & 0& 0 & 0 & 0 & 0& 1 \\
 \cline{4-4} \cline{11-11}
 \end{tabular}
 \end{center}
 \caption{Construction of the time domain samples $\mathcal{I}$ for the example in Fig \ref{fig:digit-table-example}. The non pivot digits are set to zero, and the pivot digits take all possible values.}
 \label{fig:I-digit-table-example}
\end{figure}

For the structure in Theorem \ref{thm:block-structure} to be realized, we also need an appropriate sorting of the indices in $\mathcal{I}$ and $\mathcal{J}$. The sorting on $\mathcal{I}$ and $\mathcal{J}$ is related to the pivot digits:
\begin{enumerate}
    \item The indices of $\mathcal{I}$ (i.e., the columns of the submatrix $\mathcal{F}(\mathcal{J}, \mathcal{I})$) are sorted lexicographically based on the pivot digits, starting from the left to the right. For example, with two pivots (as in Fig \ref{fig:I-digit-table-example}), the index with pivot digits $00$ comes first, followed by the indices with pivot digits $01$, $10$, and $11$ respectively.
    \item The indices of $\mathcal{J}$ (i.e., the rows of the submatrix $\mathcal{F}(\mathcal{J}, \mathcal{I})$) are sorted lexicographically based on the pivot digits, starting from the right to the left. For example, with two pivots (as in Fig \ref{fig:digit-table-example}), the index with pivot digits $00$ comes first, followed by the indices with pivot digits $10$, $01$ and $11$ respectively.  The non pivot digits are ignored for the purpose of sorting.
\end{enumerate}
For e.g., if $\mathcal{J} =\mathcal{I} = \mathbb{Z}_N$, then all the digits are pivots. In this case, the rows $\mathcal{J}$ are sorted in the natural order, whereas the columns $\mathcal{I}$ are sorted in the bit-reversed order, as in the standard radix-2 FFT (\cite{oppenheim1999discrete, osgood2018lectures}). Our algorithm here can be seen as a generalization of the same.

With the specific choice of $\mathcal{I}$, and the sorting on the rows and columns of the submatrix $\mathcal{F}(\mathcal{J}, \mathcal{I})$ mentioned above, the structure in Theorem \ref{thm:block-structure} applies. We defer the proof to Section \ref{section:proofs}.

\begin{figure*}[ht]
\centering

\includegraphics[width=0.75\textwidth]{butterfly.JPG}

\caption{DFT computation of $1024-$ length signal $x$ with spectral support set $\mathcal{J} = \{252, 296, 472, 508, 552, 684, 728, 940\}$ and sparsity $k = 8$}
\label{fig:block-diagram}
\end{figure*}
\section{Proofs}\label{section:proofs}
In this section, we will prove the results presented earlier. For this, we will find it convenient to make some definitions first. Suppose $1_\mathcal{I}, 1_\mathcal{J} \in \mathbb{C}^N$ represents the indicators of the sets $\mathcal{I}$ and $\mathcal{J}$. For $i \in \mathbb{Z}_N,$ we denote by $(i,N)$ the greatest common divisor (gcd) of $i$ and $N$. Define
\begin{equation}
\label{eq:def-idempotents}
    h_\mathcal{I} = \mathcal{F}^{-1}1_\mathcal{I}, \quad h_\mathcal{J}=\mathcal{F}^{-1}1_\mathcal{J}.
\end{equation}
These are convolution idempotents (i.e., they satisfy $h*h=h$ where $*$ is the discrete circular convolution). These idempotents arise naturally when taking inner products of any two columns of $\mathcal{F}(\mathcal{J}, \mathcal{I})$, as in the lemma below.

\begin{lemma}
\label{lem:unitary-idempotent}
The submatrix $\mathcal{F}(\mathcal{J},\mathcal{I})$ is unitary iff $h_\mathcal{J}(i_1-i_2) = 0$ for any $i_1, i_2\in \mathcal{I}$ ($i_1\neq i_2$),
\end{lemma}
\begin{IEEEproof}
The inner product of any two columns indexed $n_1$ and $n_2$ is given by \(\sum_{m \in \mathcal{J}}\exp(2 \pi i m(n_1-n_2)/N = Nh_\mathcal{J}(n_1-n_2).\)
\end{IEEEproof}
So we want to construct $\mathcal{I}$ in such a way that the differences of any two elements in $\mathcal{I}$ fit in the zero set of $h_\mathcal{J}$. However, the zero set of $h_\mathcal{J}$ correspond to the roots of a polynomial with integer coefficients; and as such, they have a lot of structure. In particular, we have $h_\mathcal{J}(n)=0$ iff $h_\mathcal{J}(ns)=0$ for any $s$ coprime to $N$. This allows us to write the zero set of $h_\mathcal{J}$ as \(\{i \in \mathbb{Z}_N  \colon (i,N) \in \mathcal{D}_\mathcal{J}\}\), where $\mathcal{D}_\mathcal{J}$ is some set of divisors of $N$. Thus, the zero set contains all the indices in $\mathbb{Z}_N$ whose gcd with $N$ is in $\mathcal{D}_\mathcal{J}$. The proof is elementary: we direct the interested reader to the  references (\cite[Theorem~2.1]{malikiosis2016fuglede},\cite{siripuram2018lp}, \cite{siripuram2020convolution, reddy2020some}) for the proof and details. We refer to the set $\mathcal{D}_\mathcal{J}$ often as zero-set divisors of $h_\mathcal{J}$.

The involvement of gcd is why digit tables are very convenient to represent spectral sets. For any two indices $i_1, i_2$, we see that the gcd $(i_1-i_2, N)$ relates to the first non zero difference in the digits of $i_1$ and $i_2$. For any set $\mathcal{I}$, the set $\{(i_1-i_2,N):i_1, i_2 \in \mathcal{I}, i_1 \neq i_2\}$ is simply the set of pivots in the digit table for $\mathcal{I}$. This is the crucial observation we use next.

\subsection{Proof of Lemma \ref{lem:spectral-structure}}
Starting with $\mathcal{J}$, as in the preceding discussion, let $D_\mathcal{J}$ be the zero-set divisors of $h_\mathcal{J}$. Then $\mathcal{F}(\mathcal{J},\mathcal{I})$ is unitary, iff $(i_1-i_2,N) \in \mathcal{D}_\mathcal{J}$ for any $i_1, i_2\in \mathcal{I}$ (from Lemma \ref{lem:unitary-idempotent}). From the observations made previously, this means the digit table for $\mathcal{I}$ must have some subset of $\mathcal{D}_\mathcal{J}$ as pivots. The largest possible size of $\mathcal{I}$ (by the definition of pivots) is $2^{|\mathcal{D}_\mathcal{J}|}$: so we have $|\mathcal{I}| \leq 2^{|\mathcal{D}_\mathcal{J}|}$.

The rest of the proof of Lemma \ref{lem:spectral-structure} relies on the results on idempotents from \cite[Theorem 1]{siripuram2020convolution}: an idempotent $h_\mathcal{J}$ has zero set divisors $D_\mathcal{J}$ iff it is a concatenation of conforming digit tables with pivots $\mathcal{L}^*=N/2\mathcal{D}_\mathcal{J}$. Since there must be at least one table in this concatenation, the size of $\mathcal{J}$ is at least $2^{|\mathcal{L}^*|}$: so we have $|\mathcal{J}| \geq 2^{|\mathcal{L}^*|} = 2^{|\mathcal{D}_\mathcal{J}|}$. However, $|\mathcal{I}| = |\mathcal{J}|$ and so combining the inequalities obtained till now gives $|\mathcal{I}| = |\mathcal{J}| = 2^{|\mathcal{D}_\mathcal{J}|}$. Consequently, the digit tables for both $\mathcal{I}$ and $\mathcal{J}$ are conforming: with pivots $\mathcal{D}_\mathcal{J}$ for $\mathcal{I}$ and pivots $N/2\mathcal{D}_\mathcal{J}$ for $\mathcal{J}$.
\hspace*{\fill}~{\rule{2mm}{2mm}}\par\endtrivlist\unskip
\subsection{Proof of Theorem \ref{thm:block-structure}}
This proof relies heavily on the ordering of $\mathcal{I}$ and $\mathcal{J}$ introduced in Section \ref{section:main_result}.
Some notation before we proceed: suppose $k=2^r$, and that the pivots for $\mathcal{I}$, from left to right, are $d_0, d_1, \ldots, d_{r-1}$. Then the pivots for $\mathcal{J}$, from left to right, are $d'_{r-1}, d'_{r-2}, \ldots, d'_0 $, where $d'=N/2d$ (this follows from Lemma \ref{lem:spectral-structure}). The ordering of $\mathcal{I}$ and $\mathcal{J}$ splits them naturally into smaller sets as in Figure \ref{fig:digit-table-ordering}: we have $\mathcal{I} = \mathcal{I}_0 \cup \mathcal{I}_1$ where $\mathcal{I}_0$ contains all indices with the leftmost pivot digit zero, and $\mathcal{I}_1$ contains all indices with the leftmost pivot digit one. A similar split $\mathcal{J} = \mathcal{J}_0 \cup \mathcal{J}_1$ occurs for the row indices $\mathcal{J}$, with the split based on the rightmost pivot digit.

\begin{figure*}[ht]
\centering
\begin{tabular}{cc}
			\begin{tabular}{ccccc|c|ccc}
			\cline{6-6}
			$2^0$  & $\cdots$ & $d'_{r-1}$  & $\cdots$ & $d'_{r-2}$  $\cdots$& $d'_{0}$& $\cdots$\\
			\cline{1-9}
			 & &  & &  &  0& & &  \hspace{-.2in}\rdelim\}{3.5}{*}[$\mathcal{J}_0$]\\
			 & &   & & &  \vdots&  &  &\\
			&  &  & &  &  0& &  &\\
			\cdashline{1-9}
			&  & &  & &  1& & & \hspace{-.2in}\rdelim\}{3.5}{*}[$\mathcal{J}_1$]\\
		    &  &   & &  & \vdots& &  &\\
			&  &  &  &   & 1& &  &\\
			\cline{6-6}
		\end{tabular}
	\begin{tabular}{cc|c|cccccccc}
			\cline{3-3}
			$2^0$  & $\cdots$ & $d_0$  & $\cdots$ \, & $d_1$ & $\cdots$ & $d_2$ & $\cdots$& $d_{r-1}$& $\cdots$\\
			\cline{1-10}
			 & & 0 & &  & & & & & &  \hspace{-.2in}\rdelim\}{3.5}{*}[$\mathcal{I}_0$]\\
			 & & \vdots  & & & & & & & &\\
			&  & 0 & &  & & & & & &\\
			\cdashline{1-10}
			&  & $1$ &  & & & & & & & \hspace{-.2in}\rdelim\}{3.5}{*}[$\mathcal{I}_1$]\\
		    &  & \vdots  & & &  & & & &  &\\
			&  & $1$ &  & & &  & & & &\\
			\cline{3-3}
		\end{tabular

\end{tabular}
\caption{On the left: A general representation of confirming digit table (sorted) of frequency support set $\mathcal{J}$, here $d'_{0}$ is the largest pivot. On the right: Corresponding confirming digit table of time domain support set $\mathcal{I}$ constructed and sorted as explained in the Section \ref{section:main_result}. Here the smallest pivot is $d_{0} =N/2d_0'$.}
		\label{fig:digit-table-ordering}

\end{figure*}

This results in
\begin{equation}
\label{eq:block-split}
\mathcal{F}(\mathcal{J}, \mathcal{I}) = \begin{pmatrix}
\mathcal{F}(\mathcal{J}_0, \mathcal{I}_0) & \mathcal{F}(\mathcal{J}_0, \mathcal{I}_1) \\ \mathcal{F}(\mathcal{J}_1, \mathcal{I}_0) & \mathcal{F}(\mathcal{J}_1, \mathcal{I}_1)
\end{pmatrix}
\end{equation}

First, we note that both $\mathcal{I}_0$ and $\mathcal{I}_1$ have pivots $d_1, d_2, \ldots, d_{r-1}$; and both $\mathcal{J}_0$ and $\mathcal{J}_1$ have pivots $d'_{r-1}, d'_{r-2}, \ldots, d'_1$.  Since all these four sets have sizes $k/2 = 2^{|\mathcal{L} \setminus \{d_0\}|}$, it follows that all these four sets correspond to conforming digit tables. From Lemma \ref{lem:spectral-structure}, it follows that all the four submatrices in \eqref{eq:block-split} are unitary.

We also make the following observations:
\begin{enumerate}[label=(\alph*)]
    \item All the entries of $\mathcal{I}$ are multiples of $2^{d_0}$. The entries of $\mathcal{I}_0$ are even multiples of $2^{d_0}$, whereas the entries of $\mathcal{I}_1$ are odd multiples of $2^{d_0}$.
    \item The digit tables for $\mathcal{I}_0$ and $\mathcal{I}_1$ are identical except for the leftmost pivot ($d_0$): thus $\mathcal{I}_1 = 2^{d_0} + \mathcal{I}_0$.
    \item The digit tables for $\mathcal{J}_0$ and $\mathcal{J}_1$ are identical up to the last pivot digit: the entries in $\mathcal{J}_0$ has the last pivot digit as $0$, whereas the entries in $\mathcal{J}_1$ have the last pivot digit as $1$.

    To see this, note that due to the proposed sorting, the $i^{\textsf{th}}$ in $\mathcal{J}_0$ and the $i^{\textsf{th}}$ index in $\mathcal{J}_1$ will have the same digits in all pivots except the last ($d'_0$). From the definition of pivots, this forces all the digits (including the non pivot digits) before $d'_0$ to be identical.

\end{enumerate}
Now to prove that $\mathcal{F}(\mathcal{J},\mathcal{I})$ has the structure of Theorem \ref{thm:block-structure}, we show the following
\begin{enumerate}
    \item $\mathcal{F}(\mathcal{J}_0, \mathcal{I}_1) = D \mathcal{F}(\mathcal{J}_0,\mathcal{I}_0)$\\
    From observation (b) above, we can write
    \begin{align*}
    \mathcal{F}(\mathcal{J}_0, \mathcal{I}_1) &= \left(e^{2\pi i mn/N} \right) \quad m \in \mathcal{J}_0, n \in \mathcal{I}_1 \\&= \left(e^{2\pi i m(n+2^{d_0})/N} \right) \quad {m \in \mathcal{J}_0, n \in \mathcal{I}_0} \\ &=D \mathcal{F}(\mathcal{J}_0,\mathcal{I}_0),
    \end{align*}
    where $D$ is a diagonal matrix with entries $\exp(2 \pi i m2^{d_0}/N)$, for $m \in \mathcal{J}_0$.
     \item $F(\mathcal{J}_1, \mathcal{I}_0) =  \mathcal{F}(\mathcal{J}_0,\mathcal{I}_0)$ \\
     We simply take the ratio of corresponding entries of these matrices and show that the ratio is $1$. This ratio is of the form
     \[
     e^{2\pi i m_{0j}n/N} / e^{2\pi i m_{1j}n/N}=     e^{2\pi i n(m_{0j} -m_{1j})/ N},
     \]
     where $n \in \mathcal{I}_0$, and $m_{00},m_{01} ,m_{02}, \ldots,$ are (in order) the entries of $\mathcal{J}_0$; similarly $m_{10},m_{11} ,m_{12}, \ldots,$ are (in order) the entries of $\mathcal{J}_1$. We note from observation (c) above that $m_{0j}-m_{1j}=\alpha 2^{d'_0}$, where $\alpha$ is odd. Further, from observation (a) above, the index $n$ is an even multiple of $2^{d_0}$, resulting in the ratio being
     \[
     e^{2\pi i \text{(even)}2^{d_0}\alpha 2^{d'_0}/ N} = 1.
     \]

      \item $\mathcal{F}(\mathcal{J}_1, \mathcal{I}_1) = - \mathcal{F}(\mathcal{J}_0, \mathcal{I}_1)$ \\
      This is similar to 2) above. We take the ratio of corresponding entries we get
      \[
     e^{2\pi i m_{0j}n/N} / e^{2\pi i m_{1j}n/N}=     e^{2\pi i n(m_{0j} -m_{1j})/ N},
     \]
     where $n \in \mathcal{I}_1$, and $m_{ij}$ are as before. From observation (a) above, $n$ is an odd multiple of $2^{d_0}$, so the ratio becomes
     \[
     e^{2\pi i \text{(odd)}2^{d_0}\alpha 2^{d'_0}/ N} = -1.
     \]
    \hspace*{\fill}~{\rule{2mm}{2mm}}\par\endtrivlist\unskip
\end{enumerate}

redpajama_set_name: RedPajamaArXiv


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The Institute of Banking Personnel Selection (IBPS) will release the scores of Probationary Officers or PO preliminary examination 2018 by late evening. IBPS PO Prelims Examination was conducted on 13th and 14th of October 2018. All the candidates who had appeared in this exam can check their score on the official website of IBPS when the results are out.
Educational Qualification: The candidate must possess a Degree of Graduation from a recognized university.
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text:
Q: How do you the splitting field of this polynomial over GF(3) What is the splitting field of $x^4-x^2-2$ over GF(3), where GF(3) is the finite field with 3 elements.
I know how to get the splitting field of polynomials over $Q$ but I'm not sure how to do this since I'm not even sure what the elements of GF(3) are...
I would greatly appreciate if anyone can help. Thanks!

A: Note $$(x^2+1)^2=x^4+2x^2+1$$ In $GF(3)=\mathbb{Z}/3\mathbb{Z}$ we have $2=-1$ and $1=-2$. So  $$(x^2+1)^2=x^4-x^2-2$$
Now you just have to find the spliting field of $x^2+1$ over $GF(3)$. This is easier since it can only have linear factors. A quick check that $0^2=0$, $1^2=1$ and $2^2=1$ verifies that $x^2+1$ is irreducible. So the extension $GF(3)[x]/(x^2+1)=GF(3)[w]$ where $w$ is a solution to $x^2+1$ should be the splitting field (note the other root is $-w$ which will be in there for free). This field is in-fact isomorphic to $GF(9)$

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text:
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Posted in Iowa September 09, 2020 by Kim Magaraci
Seven Million Matchsticks Make Up The Remarkable Creations At Iowa's Matchstick Marvels
If you're looking for a fun, small town Iowa museum that's sure to impress any visitor, head to Gladbrook and check out Matchstick Marvels! It's sure to be unlike any place you've ever been.
Matchstick Marvels is a small town Iowa museum that shows off the work of artist Pat Acton. Pat creates enormous scale models with million of matchsticks!
Facebook / Matchstick Marvels
In 2014 Pat honored the International Space Station with 8 gallons of wood glue, 286,000 matchsticks, 8 gallons of wood glue and nearly 2,000 hours of labor.
You could say that this model was out of this world.
Star Wars fans will immediately recognize a matchstick Millenium Falcon, which took nearly a million little pieces to put together.
The Falcon was commissioned in 2014 and remains one of the most impressive creations ever made with matchsticks.
Back when the renewed World Trade Center in New York City was just a concept, Pat helped bring it to life with an enormous scale model.
After 2,100 hours of work, his version of the WTC 468,000 matchsticks and 17 gallons of wood glue into an impressive masterpiece.
In 2019, Apollo 11 came to life! The Saturn rocket and Lunar landing module were a tribute to the 50th anniversary of the 1969 moon landing.
Matchstick Marvels
One of the more recent creations was a replica 1970s Dodge Charger, the car that's been tied to the Fast and Furious Franchise.
As you can see, no detail was spared in the creation of this gorgeous matchstick muscle car.
Pat's largest creation, the Steampunk "Plane Loco", was one of his own creations. It was inspired by an early 1900s train and Leonardo da Vinci's designs.
"Plane Loco" was created over 3,000 hours, with one million matchsticks and 35 gallons of wood glue. The wingspan is 13 feet, and it's more than 20 feet long and 9 feet high.
This one is on display in the Ripley's Believe It Or Not Museum in New York City!
You can visit Matchstick Marvels and see the recent works of Pat Acton in the town of Gladbrook, Iowa. The building is handicap accessible and open April 1 through Nov. 30, 7 days a week, from 1-5 pm. Admission is just $5 for adults $3 for children ages 5-12, and free for kids under 5.
Visit the Matchstick Marvels website here for more information.
Address: 319 2nd St, Gladbrook, IA 50635, USA
Iowa In Your Inbox
Only In Iowa
The Vegan Burgers From Dirt Burger In Iowa Are Remarkable Creations You'll Want To Devour
In 2021, Take These 12 Incredible Iowa Hikes, One For Each Month Of The Year
8 Iconic Foods You'll Be Sure To Find On Any Iowa Road Trip
Try The Most Unique Pizza In The Midwest At Fong's Pizza in Iowa
Haunted towns in Iowa
Dubuque Is Allegedly One Of Iowa's Most Haunted Small Towns
Morning Star Chapel Is One Of The Smallest Churches In The Country And You Can Find It In Iowa
Artisan Pizza, Ice Cream, Homemade Sweets, And Smoked Meats Are All Waiting For You At Iowa's NewBo City Market
The Unique Day Trip To Czech Village In Iowa Is A Must-Do

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text:
Jose Antonio Rangel. (Las Vegas Metropolitan Police Department)
Man in custody in August death of Lesly Palacio; son still at large as murder indictment handed down
by: Greg Haas
Posted: Feb 12, 2021 / 02:21 PM PST
Updated: Feb 12, 2021 / 03:38 PM PST
LAS VEGAS (KLAS) — One of two men accused in the August death of 22-year-old Lesly Palacio is in custody, according to court records.
Lesly Palacio. (Photo provided by Las Vegas Metropolitan Police Department)
Jose Antonio Rangel, 46, is behind bars at the Clark County Detention Center, and authorities believe his son Erick Rangel, aka Erick Michel Rangel-Ibarra, remains in Mexico.
Officials had no further information on the son, who was 25 at the time the crime was committed.
The revelation comes as an indictment named both men in the murder of Palacio. The indictment was filed today in District Court.
Records show Jose Rangel was arrested on Jan. 19, five months after Palacio was killed. Her body was found Sept. 9 near Valley of Fire State Park.
Jose Rangel was detained by U.S. Customs and Border Patrol in San Diego for active warrants relating to this case. He was booked into Clark County Detention Center on Jan. 31 on warrants for harboring/concealing/aiding a felony offender and destroying/concealing evidence.
Surveillance video, statements built murder case …
Jose Rangel is jailed on $100,000 bond, and is scheduled to be arraigned before District Court Judge Tierra Jones at 8:30 a.m. on Feb. 19. The indictment will supersede previous charges in the case.
If Rangel is released on bail, he will be subject to high level electronic monitoring. He will also have to surrender his passport.
In addition to the murder charge, both men face charges of destroying evidence and accessory to murder.
Authorities believe the son killed Palacio, and the two men moved her body from Erick Rangel's residence. He was seen helping her into the home just after 6 a.m., and the two men took her body out to the back of Erick Rangel's Ford F-150 pickup less than two hours later.
Video surveillance from a neighbor's house was part of the evidence in the case, according to an arrest report.
The video, which police have not released, shows an attempt to clean the driveway at the home with a hose by the man fitting Jose Rangel's description.
Detectives executing a search warrant at Erick Rangel's residence found "a small amount of blood spatter in Rangel's bedroom," which analysis showed to be human blood.
Anyone with any information in the case is urged to contact the LVMPD Homicide Section by phone at 702-828-3521, or by email at homicide@lvmpd.com. To remain anonymous, contact Crime Stoppers by phone at 702-385-5555, or on the internet at www.crimestoppersofnv.com.
Erick Michel Rangel-Ibarra, left, and Jose Antonio Rangel. (Las Vegas Metropolitan Police Department)


March 4, 2019 March 4, 2019 - 196 Comments.
In yet another attack on the laws of the land, cities and towns in America are being granted federal funding from the government, despite their refusal to abide by the nation's immigration laws.
The Justice Department has officially confirmed that the majority of places which qualify as sanctuary cities have been given the green light to receive funding from the United States government. This comes in light of the fact that these cities are literally breaking the law by aiding and abetting known criminals and refusing to cooperate with the men and women who are responsible for enforcing the laws of this country.
This is truly unfortunate because it sends a horrible message. Cities which are allowed to flagrantly break the law and receive funding from the government are being told that what they're doing is OK. They're being told that it's OK to disregard the laws of the land and spit in the face of law enforcement. If cities can get away with not following the law, why should citizens have to follow the law? This is a very dangerous and slippery slope which we've found ourselves on.
At the end of the day, immigration laws exist for the purpose of keeping this country safe from harm. The reality is that we cannot allow just anyone to pour in. We have to screen people; we have to be sure that people who want to come to America are able to support themselves and positively contribute to society.
Sanctuary cities may believe that they are somehow on the right side of things, but history will not remember them in that manner.
What do you think about sanctuary cities still receiving federal funding? Let us know in the comments section down below!


Conestoga Insurance provides choice and experience - Conestoga Insurance Brokers Ltd.
Some would say don't do business with your friends but our family has not heeded this advice. We have always felt special with the service and personal attention we have received at Conestoga Insurance Broker's over the years, and it has always been a pleasure to catch up with the boss once business has been taken care of.
We recently partnered with Metro Land Media for the "In Your Neighbourhood" business section. This new section focusses on local businesses in the community and with 40 years of serving our insurance community, we were happy to tell our story.
There will be more articles to follow. At Conestoga Insurance, we recognize that your insurance needs are unique and we tailor our insurance solutions to meet your needs.


text:
Latest News / Headline
Pirelli announce tyre choices for 2016 season opener
Australia Pirelli
Pirelli will make their red-marked supersoft tyre available in Australia for the first time next year, after revealing the three compounds that will be used in the 2016 season-opener under the new tyre regulations.
Teams will also be able to use the white-marked medium and yellow-marked soft compounds over the Melbourne weekend.
As part of the new 2016 rules, every team must save at least one set of the supersoft tyre for Q3. Those who qualify inside the top ten will have to give the tyre back to Pirelli at the end of qualifying, while those who do not make the cut can carry the extra set into the race.
Teams must also have at least one set of mediums and one set of softs available for the race - although they are only obligated to race with one of those two compounds (should they also choose to use the supersoft compound during the race).
With 13 sets of tyres in total made available over the weekend, teams have free choice over their individual allocations for the remaining 10 sets.
Pirelli clarify new 'free choice' tyre regulations


FBLA OPERATION CHRISTMAS CHILD IS A SUCCESS!
Thanks to Mrs. Carelock's first period class who brought the most boxes and made the collection a success! Her first period will enjoy a spudnut party on Friday. Special thanks to Ian Girard and his mom for their extra special work. We collected around 60 boxes! Thanks for all who gave!


Come and join Sherry at 864 Hewetson this Saturday from 1-3pm as she opens this lovely home to the public to view!
Stunning lake views from this Upper Mission home. Over 3800 square feet of contemporary living with 4 bedroom plus office and 4 baths. Abundance of windows create a bright enviroment and frame the picturesque views. Quality materials and finishing through out. Brand new pool with paver deck and poolside firepit. Ready to move in and start the homes next chapter!!!


text:
Rondò Veneziano es una orquesta de cámara italiana, especializada en música barroca, tocando instrumentos originales, con la incorporación de una sección rítmica estilo rock de sintetizadores, guitarra bajo y batería, dirigido por el Maestro Gian Piero Reverberi, quien también es el compositor principal de toda la piezas originales de Rondo Veneziano. La adición inusual de instrumentos modernos, más adecuados para el jazz, combinados con arreglos de Reverberi y composiciones originales, han dado lugar a nuevas versiones pródigosas de obras clásicas en los últimos años. Por regla general, en sus giras de conciertos, los músicos, mayormente mujeres, se suman al efecto barroco en general con el uso de atuendos y peinados de la época barroca.

Historia

Rondò Veneziano nació en la primavera de 1979 en una idea conjunta de Freddy Naggiar (Baby Records) y el compositor Gian Piero Reverberi. este último deja RCA italiano por Baby Records, donde trabaja con Al Bano, Romina Power, Pupo y Ricchi e Poveri.

El proyecto inicial era crear una formación instrumental italiana, en línea con lo que se había propuesto Angelo Branduardi o Stephen Schlaks. Reverberi optó por una solución entre el barroco y el pop.

La restauración del Carnaval de Venecia a finales de 1970, contribuyó a que el carácter tan festivo y mundialmente famoso Carnaval de Venecia sea un tema esencial de inspiración. Para materializar el ambiente barroco, los solistas vienen vestidos al estilo del siglo XVIII, con pelucas. Estos elementos estaban destinados a fortalecer el impacto en el público y hacer que el grupo sea inmediatamente reconocible.

Las primeras cuatro canciones (incluyendo Rondò Veneziano, San Marco y Andante Veneziano) fueron registrados bajo la dirección de Reverberi en Varirecording Studios (Mílan), con una orquesta sinfónica de Génova así como un baterista y un bajista. Estas primeras piezas agradaron a Baby Records, y junto a otras cinco composiciones agregadas formaron un primer álbum: Rondò Veneziano.

Si bien, el disco estaba listo ya en el invierno 1979, este no se publicó hasta el año siguiente para preparar una estrategia de negocio más eficaz. El 22 de febrero de 1980, la canción Rondò Veneziano fue elegida por Silvio Berlusconi, por consejo de Freddy Naggiar para abrir y cerrar los programas de reciente Canal 5, provocando la curiosidad de los espectadores durante casi un año, ya que la pieza se publicó seis veces al día, sin que ni su nombre ni su origen fueran mencionados.

Evolución
La primera década de álbumes de la orquesta solo incluyó composiciones totalmente originales en el estilo barroco rondò, "una composición musical basada en la alternancia de un tema recurrente principal y episodios contrastantes".

En años posteriores, además de muchos discos nuevos y originales continuando el propio estilo y la tradición única de Gian Piero Reverberi, Rondò Veneziano también trajo la fusión de instrumentos clásicos y contemporáneos para un pequeño número de álbumes dedicados a algunos de los grandes compositores tradiciónales de música clásica y barroca.

En una entrevista, El Maestro Reverberi dijo sobre el sonido de Rondò Veneziano: "La música de Rondò Veneziano es ante todo positiva también cuando parece estar triste, nunca es triste Siempre es positivo en el sentido de que al final siempre hay un buen futuro. Así que creo que también la razón del éxito es que se trata de la música en la que no hay que pensar negativo o sentirse negativo o si se siente uno negativo debe ser algo que te lleva a pensar en positivo ".

Una versión de "La Serenissima" (Tema de Venice in Peril) fue lanzado en el Reino Unido como un sencillo y alcanzó el número 58 en la lista de singles del Reino Unido en octubre de 1983 La pista fue también ampliamente utilizado en ese momento por BBC TV, como la melodía del tema de Hospital Watch. La pista fue más tarde usada en el éxito mundial "Venice in Peril" álbum que fue lanzado como parte de una campaña internacional para salvar Venecia se hunda en la laguna.

En 1985, proporcionaron la banda sonora de la película Not Quite Jerusalem (conocida como Not Quite Paradise en los EE.UU.). Esto marco una reelaboración de muchas de las piezas originales que figuran en los álbumes Venice in Peril y The Genius of Venice.

La orquesta ha producido 70 álbumes en los 29 años desde su fundación en 1979.

Discografía Parcial
Todos compuestos por Gian Piero Reverberi, Ivano Pavesi y Laura Giordano:

 Rondò Veneziano (1980)
 La Serenissima  (1981)
 Notturno in gondola (1981)
 Scaramucce  (1982)
 Venezia 2000 (1983)
 Concerto Futurissimo (1984)
 Odissea Veneziana (1984)
 Casanova  (1985)
 Not Quite Jerusalem (1985)
 Odissea Veneziana (1985)
 Fantasia Veneziana (1986)
 Lagune (1986)
 Rapsodia Veneziana (1986)
 Arabesque (1987)
 Misteriosa Venezia (1987)
 Concerto  (1988)
 Poesia di Venezia (1988)
 Armonie (1989)
 Masquerade (1989)
 Visioni di Venezia (1989)
 Barocco  (1990)
 Musica...Fantasia (1990)
 The Genius of Mozart (1990)
 The Genius of Vivaldi (1990)
 Vivaldi, Mozart, Beethoven (1990)

 Magica Melodia (1991)
 Prestige (1991)
 Stagioni di Venezia (1992)
 Rondò 2000 (1992)
 Venezia Romantica (1992)
 Concerto per Beethoven (1993)
 Concerto per Mozart (1993)
 Concerto per Vivaldi (1993)
 Il mago di Venezia (1994)
 Back 2 back (1995)
 Eine nacht in Venedig (1995)
 I grandi successi vol.1 (1995)
 Sinfonia di Natale (1995)
 I grandi successi vol.2 (1996)
 Preludio all'amore (1996)
 Seduzione (1996)
 The best of - vol.1 (1996)
 Die Grossen Erfolge (1997)
 Fantasia Classica (1997)
 I grandi successi (1997)
 In concerto (1997)
 Marco Polo  (1997)
 Via dell'Amore (1997)

 Fantasia d'Autunno (1998)
 Fantasia d'Inverno (1998)
 Zodiaco (1998)
 Attimi Di Magia (1999)
 Fantasia d'Estate (1999)
 Fantasia di Primavera (1999)
 Honeymoon - Luna di miele (1999)
 Protagonisti vol.1 (1999)
 Splendore Di Venezia (1999)
 I grandi successi (2000)
 La storia del classico (2000)
 Nur Das Beste (2000)
 Protagonisti vol.2 (2000)
 The very best of (2000)
 Capriccio veneziano (2001)
 Papagena (2001)
 3 originals (2001)
 The Magic of Christmas (2001)
 Concertissimo (2002)
 La Piazza (2002)
 Vénitienne (2002)
 Nur Das Beste 2 (2004)
 25 Live in Concert (2005)
 Rondò Veneziano: Chamber Orchestra (2009)
 Once upon a time (2010)

Referencias

Música de Italia
Música de Italia del siglo XX
Música clásica del siglo XX
Orquestas de cámara
Artistas de Koch Records


text:
Q: Problem with hitTestObject I am writing HitTestObject for my character:
(everycode is in ENTERFRAME section)
-for the Character:
if(bullet.bulletList.length>0)
    for (var i:int = 0; i < bullet.bulletList.length; i++)
    {   if(this.hitTestObject(bullet.bulletList[i]))
           {
               updateHP();
                        }
             }

-for the enermy attack code:
bullet= new Bullet();
bulletList.push(bullet);

At the begining when there is not bullet in the stage, no error produced, but when the first bullet addchild to stage, an error like this appeared:
TypeError: Error #2007: 参数 hitTestObject 不能为空。(something's like hisTestOject cannot be empty)
at flash.display::DisplayObject/_hitTest()
at flash.display::DisplayObject/hitTestObject()
at com.miraclei::char/loop()

I think it's because when the first bullet removeFromStage, the hitTestObject function found nothing in the bulletList, then error happened. I tried many way but still not ok!
Someone has solution for this?

A: If I'm reading this correctly, the trick is to iterate through your array of bullets in descending numeric order:
for (var i:int = bullet.bulletList.length-1; i > -1; i-- )
{   etc..

you won't run into empty array positions this way.
hope this help!


text:
962 total votes
Guadalcanal Campaign
The Guadalcanal Campaign, also known as the Battle of Guadalcanal and codenamed Operation Watchtower by Allied forces, was a military campaign fought between 7 August 1942 and 9 February 1943 on and around the island of Guadalcanal in the Pacific theatre of World War II. It was the first major offensive by Allied forces against the Empire of Japan. On 7 August 1942, Allied forces, predominantly American, landed on the islands of Guadalcanal, Tulagi, and Florida in the southern Solomon Islands with the objective of denying their use by the Japanese to threaten the supply and communication routes between the US, Australia, and New Zealand. The Allies also intended to use Guadalcanal and ... more on Wikipedia
Guadalcanal Campaign is ranked on...
#13 of 75 The Most Important Battles in US History 35.9k VOTES The most significant major battles in American history, ranked by everyone. Anyone can vote on this list, making these ranki...
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