![]() All inquires regarding this position must be addressed to the email provided. Note : Do not submit application materials to Fresno County Human Resources and do not apply at. Please email a Comprehensive Resume (Include State Bar License Number) to. HOW TO APPLY : Applications for this recruitment must be submitted via e-mail only. In addition to the above, the following is also required : Note : Incumbents must obtain and maintain Federal Bureau of Investigation (FBI) and Department of Justice (DOJ) clearance as a condition of employment. The experience must be acceptable to the Director of Child Support Services. Experience : Two (2) years as a practicing attorney-at-law. Child Support Attorney III: License : Active membership, in good standing, with the State Bar of California. Experience : One (1) year as a practicing attorney-at-law. ![]() Child Support Attorney II: License : Active membership, in good standing, with the State Bar of California. Samples of Duties Please click the links below to review the Samples of Duties: Child Support Attorney I Child Support Attorney II Child Support Attorney III Knowledge, Skills and Abilities Please click the links below to review the Knowledge, Skills and Abilities: Child Support Attorney I Child Support Attorney II Child Support Attorney III Minimum Qualifications Child Support Attorney I: License : Active membership, in good standing, with the State Bar of California. Eligible candidates must be able to perform the essential functions of the position with or without reasonable accommodation. Considerable walking or standing may be involved. This group includes positions which require only light physical effort and may include lifting of small, light objects and some bending, stooping, squatting, twisting, and reaching. Positions in this classification are in Medical Group I - Light Physical Activity (Personnel Rule 8). A live scan fingerprint check will also be required. As a condition of employment, candidate's child support case, education, and employment histories, along with any criminal records will be reviewed. Incumbents assigned to the Child Support Attorney II/III levels are expected to handle a more complex caseload and exercise greater independence. Child Support Attorneys perform professional legal services in the preparation and litigation of a variety of child support matters and perform related work as required. ![]() The Fresno County Department of Child Support Services is offering an exceptional career opportunity for individuals interested in the position of Child Support Attorney I/II/III. Some exceptions to remote work apply and may include court appearances and necessary in-office work. Position Description Scheduled Salary Increase : 3% effective 3/6/23 3% effective 3/4/24 Child Support Attorney Hourly Salaries Child Support Attorney I: $33.43 - $40.63 Hourly Child Support Attorney II: $39.02 - $47.43 Hourly Child Support Attorney III: $46.82 - $56.92 Hourly Remote Work Availability This position allows for partial remote work opportunities. Handle travel arrangements and prepare expense forms for OCERS Board Members, Senior Executives, and team members of the Legal DivisionĪssist the Executive Secretary II and other Administrative team members with their duties as necessaryĪssist the Legal Analyst as Clerk of the Hearing Officers as necessaryĪssist the Legal Analyst with intake, logging, and processing of requests under the Public Records Act Oversee incoming mail for the Legal Division Organize and maintain electronic and hard files for the Legal Division Maintain the calendar for the Legal Division ![]() Type, compose, and edit correspondence, memos, legal briefs, reports, PowerPoint presentations, charts, policies, and procedures Interpret and apply administrative and departmental policies, laws, and rules with particular reference to the activities of the office Organize and prepare meeting space and order food and beverages for Board and Committee meetingsįinalize and assemble agenda information for Board and Committee meetings and upload it to electronic serversĪttend Board and Committee meetings to record and transcribe minutes ![]() Provide secretarial assistance for the General Counsel, other OCERS Senior Executives, and the team members of the Legal Division ESSENTIAL DUTIES & RESPONSIBILITIES: The successful candidate will possess the following attributes and qualities. ![]()
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![]() It enables you to integrate your team members onto the platform, such as copy editors and writers, and enjoy a smooth collaboration workflow. Happy Scribe converts audio to text in over 60 various languages, making it ideal for narration and subtitles. They found a solution a few days later, and this tool was born. They discovered how difficult and time-consuming it was to transcribe a number of interviews for academic studies after being requested to do so. Two Irish entrepreneurs came up with the idea for Happy Scribe. It's as simple as typing to correct your recordings. You can make and edit podcasts using the text editor. One of the features that sets this solution apart from the competition is that it allows you to edit your podcast episodes by just changing the text. With precise transcription, you can make your podcast more accessible. Podcast transcription that is simple to use. ![]() This service is dedicated to providing the most precise, accurate, and cost-effective AI transcription on the market. It includes free audio to text converter, as well as podcast editing tools, free podcast creation solutions, text to audio software, and more, giving modern podcasters everything they need to create high-quality content fast and efficiently. Podcatle is yet another industry-leading solution for podcast enthusiasts. If you have a podcast, educational video, journalistic audio, or any other type of audio, you can convert it to text and release it in a variety of formats, including eBooks, emails, web articles, white papers and more. It helps to distribute your content in other formats When a text version of your podcast is available, it can be extremely beneficial to people who are deaf or have hearing impairments. It improves the accessibility of your content People have a wide range of interests and preferences, and there is a segment of a target audience who would rather read than watching a video. Diversification assists in reaching a larger audience If you're a vlogger who doesn't provide subtitles for your videos, you're wasting out on a great opportunity to reach out to a wider audience. Search engines can easily find your content Whether you want to convert your audio to text for vlogging, podcasting, teaching, journalism, or any other reason, the tools listed in this article will be a great arsenal for you.īefore we get started, let review the top benefits of transcribing audios to text: It would be wonderful if there was a way to avoid that extra piece of workload, and that's where transcription software comes in. In addition, we are very sure that transcribing audios would be the most time-consuming activity on your to-do list, which you definitely don't want. It would be extremely tough, time-consuming, and would complicate your working process. Supports professional USB foot pedals to control playback.Imagine spending for hours attempting to manually transcribe a large audio recording. ![]() Load CD audio directly and start working as the audio loads.Automatically have finished transcriptions sent to your clients from the program.Automatically receives and loads files by internet (FTP), email or over a local computer network.Dock both analog and digital portable voice recorders to load recordings.Use ‘hotkeys’ for mouse-free playback when transcribing directly into Word or other software.Plays most formats including encrypted dictation files.Variable speed playback (constant pitch).More Transcription Software screenshots > If you are looking to install WATCHED in PC then read the rest of the article where you will find 2 ways to install WATCHED in PC using BlueStacks and Nox app player. WATCHED is developed by WATCHED AG and listed under Entertainment. Download and install WATCHED in PC and you can install WATCHED 115.0.0.9.100 in your Windows PC and Mac OS. Note: If your organization supports SSO, you’ll be signed in automatically after launching the app. You need an Microsoft 365 account to sign in. Sign in to Yammer and set your notification preferences. ![]() Mac: Drag the Yammer app to your applications folder. Follow the onscreen instructions to install Boot Camp and Windows support software (drivers). Supported FormatsSupports audio and video playbackĪfter Windows installation completes, your Mac starts up in Windows and opens a ”Welcome to the Boot Camp installer” window. ![]() ![]() ![]() Our team consists of a very talented mature guys, we want you to be sure that we are not going to let you down! We are not giving up on the server, we believe we can make a great game play scene for everyone.Also a simple reminder it's the “time”, to evolve and develop a game you need “time”, things could take less or more depending the difficulty of the feature/features we want to implement or fix. We came up with alot of thinking in 2012, our team investigated the private servers scene on that time and we came up with an popular MMORPG like Archlord, episode 1, then upgraded to episode 2, a mixture on that time because no source code was released. Was hard plan to decide which episode to open and what cap level could everyone suits, we thought of making an old school game play for some reasons, first the old school players are the real archlord players, which you will be entertained playing with them, that was our aim.We wanted to get those players and bring back them together, secondly, old school games was the real fun, everyone is equal and the only difference is your skills, experience and your playing style.Ĭalidus server game play is the reflection of simplicity, we dont want to build a maze inside the servers and broke things.We take care of peoples opinions wich it's the most important for us and for the entire community, we want to offer great old and new content for all hosted games,and high quality. Archlord 2 is a place that’s equally suited for novice gamers as well as those with plenty of experience behind them. INTRODUCTION WE ARE CALIDUS In the beginning, i have to thank you for giving us a chance because you lose your time reading this. ![]() We came up with alot of thinking in 2012, our team investigated the private servers scene on that time and we came up with an popular MMORPG like Archlord, episode 1, then upgraded to episode 2, a mixture on that time because no source code was released.Later in the time the project had different stages,wipes,problems and we was forced to close the development.Calidus has come back now on 2022 back to life. WE ARE CALIDUS!In the beginning, i have to thank you for giving us a chance because you lose your time reading this. ![]() ![]() ![]() ![]() Now you can play this game and torture the bear to relieve your stress.Now open the file and click on the bottom right on the install button.Open the android folder and then search for the Kick the Buddy Mod APK file.Wait till the download is completed and then go to the file manager.Now go to the browser and click on the downloading link.Enable the unknown sources and do not worry about the warning message as this game is totally safe.Now open the security option and further go to the unknown sources.In order to download this game, first of all go to the settings pf your device.You will get unlimited stuff to torture the toy bear including the weapons.The design of this game is also very excellent and exciting.The fire dragon and Thor's hammer are the powers of God which you can use additionally to torture the teddy bear.This game has a very nice sound system too which sounds really fun.Once you have earned enough money, you can. Punch and kick the crap out of a happy dummy standing on a colorful bedroom and earn coins for every time you hit it. You can play this game online and for free on. You will get watermelons, guns, stones and a lot of dangerous weapons to shoot, kick, smash and strangle the buddy. Super Buddy Kick is a cool and fun-addicting clicker game for those who enjoy a little bit of violence once in a while.There is a whole range of things which you can do to relieve your stress.Kick the Buddy (MOD, Unlimited Money/Gold) - a great game in which you can relieve the accumulated stress of the day by being given an immortal mannequin to deal a lot of damage to. Despite of breaking things at your home, you can simply smash the buddy. Play Kick the Buddy Voodoodoll test system Detonate obliterate discharge shoot freeze apply Divine beings superpower and let it go You presently have. Download (111.6Mb) Updated to version 1.5.10 Playgendary Limited. This game is best for relieving the stress.You can invite your friends as well to smash things together on the teddy bear.This game is suitable for both Android and iOS users. Poki Games have new games similar: Cristiano Ronaldo Kick N Run, 3D Free Kick World Cup 18, Hero 2 Super Kick.This game comes up with a guarantee of complete security so that the users can install it in their phones without worrying about the hacking, viruses and malware.The user interface of this game is quite simple and easy for the users to play.It attracts the players with such beautiful background. It has nice graphics which look cute as well as colorful.This game s 100% free without any kind of charges, so that you can download it easily and get a help in cooling down your anger.This game could be highly offensive for many people because it does not seem right to torture a cute faced toy.This game is great for changing your bad mood. Kick the Buddy: Explode, destroy, fire, shoot, smash, freeze, send the power of the Gods and dont even think about.There are a lot of weapons in this game provided. ![]() Install all open weapons in auto attack mode and start collecting even more coins to buy all kinds of weapons! This game will be a real killer of free time and a cool anti-stress for everyone !. Kick him all you want, the main thing is to make him as painful as possible. Now you’ll have a stylish version of Buddy in front of you and you can resume your attacks again. They can be spent on accessories for a Buddy doll: a cap, glasses in the style of steampunk or cool headphones. That way your kicks and attacks will be even stronger, and you can make a lot of money in a short period of time. Earn a certain amount and be able to open not only a new rifle or flamethrower, but even a tank. First, in your arsenal there will be only a knife that you can throw at the doll. The main hero of the game will be a rag doll named Buddy. Ahead of you is waiting for a huge portion of adrenaline! Get slates and money to collect the whole arsenal: rockets, rifles, cold weapons, and even nuclear bombs. Kick The Badi is a cool new game that will help you relieve stress and have a good time making fun of the poor hero of the game. Shoot, explode, destroy your Buddy will help you forget the stress and blow off steam Wild monsters, powerful weapons, and destructive elements are at your command. ![]() ![]() ![]() App Data : Documents, Attachments, etc.Media Data : Photos, Videos, Thumbnails, Music, Playlists, Ringtones, etc.Personal Data : Contacts, Call History, Messages, Message Attachment, Voicemail, Calendar, Reminders, etc.The PhoneRescue iPhone and iPad versions let you pick up only what you want. You don’t even have to run a full blind restore. This software can recover files stored in your iTunes or AnyTrans backup in a few seconds. PhoneRescue will analyze your device and recover all the data without overwriting anything. Īll you have to do is connect the phone to your computer with a USB cable. The software will find what you need for you and retrieve it in a timely fashion. The PhoneRescue app uses NO-DATA-LOSS technology. Unfortunately, not all phones have Recycle bins. We’ve all accidentally deleted a file, contact, or message we wanted to keep. Let’s have a quick look at some of them: Salvage Deleted Data PhoneRescue has some features that make it exciting and easy to use. PhoneRescue reviews by users are mostly positive.īut is it really the best service you can choose? You just have to connect it to your computer with a USB cable. You don’t even have to take it to the repair shop! The app lets you do everything yourself. You will be able to get everything from your phone back, no matter what happened to it. Whether your data has been deleted, got lost, or damaged, this software can help you. If you want everything to be taken care of automatically, there are programs created specifically for data recovery. Sure, most people these days use cloud storage. Although it has initially been made only for iOS, PhoneRescue has an Android version as well. It has been created to help you recover anything from your phone. Meet PhoneRescue by iMobie - your knight in shining armor! Although there are a few minor problems with the free version (mainly missing icons and other items), the paid version of PhoneRescue has a large number of great features including auto-backup and auto-complete as well as numerous other options to customize the program.It’s safe to say data loss is one of all phone owners’ worst nightmares. The developers of this program claim that this program is easy to use and effective. In addition to the free version of this excellent Mac program, there is also an option to buy the full version of PhoneRescue through the Mac app store for a fee of $9.99. ![]() If you do not have PhoneRescue, make sure you search for and install the version of the app which is compatible with your Mac OS before trying to download phonerescue. When downloading PhoneRescue to your Mac, make sure you have the latest version installed. If you travel frequently, you should consider getting iWork and Numbers for your Mac as these three apps offer turn-by-turn navigation. iBooks is also useful if you have multiple copies of documents stored on your Macs. iLocalis is available in both free and paid versions and can be used to access the Mapbox mapping service on your Mac, iPod Touch or iPhone. To use iBooks for Mac, you will need to purchase the software. PhoneRescue offers a bundle of five popular iPad applications: iLocalis, iBooks, iWork, Numbers and VoiceTime. The free download version has been found to work with most of the cellular phones which are manufactured by Apple, such as iPhone, iPod Touch, iPhone 3G and so on. iMobie Inc., the company behind PhoneRescue, claims that this application is designed specifically for use with the iPhone and not other mobile phones. This Mac-friendly application allows users to use their Macs with cellular phones. PhoneRescue is a great tool for any one who uses computers, especially Mac users. ![]() ![]() ![]() ![]() In other words, for zebras, the animals' default state is to produce black hair, making them black with white stripes, according to Brittanica (opens in new tab). The production of melanin from melanocytes is "prevented during the development of a white hair, but not of a black hair," Caro told Live Science in an email. Zebras' black fur is chock-full of melanin, but melanin is absent from white fur, in essence, because the follicles that make up the stripes of white hair have "turned off" melanocytes, meaning they don't churn out pigment. This pigment is known as melanin a lot of melanin leads to darker colors, like dark brown or black, while less melanin leads to lighter colors, such as hazel or blond, Live Science previously reported. ![]() Title: The White Stripes - Jolene (Official Music video) Duration: 03:51: File Format: mp3: Mime Type: audio/mpeg. These cells produce a pigment that determines the color of hair and skin. Details lagu The White Stripes - Jolene (Official Music video) bisa kamu lihat di tabel, untuk link download The White Stripes - Jolene (Official Music video) berada dibawah. Here's why: Every piece of hair - both light and dark - grows from a follicle filled with melanocyte cells, according to a 2005 review in the Journal of Investigative Dermatology (opens in new tab). In fact, zebras actually have more light-colored hair than dark - their bellies are usually light - so it may seem that zebras are white with black stripes.īut that's not the case. Although zebras have black skin, different developmental processes determine their fur color, just like a light-skinned person can have dark hair, Caro said. ![]() ![]() ![]() Whatever the reason, those looking for the best space flight simulation games may be better served looking backward rather than up into the sky. While it's certainly true that it can be difficult to be innovative in a crowded marketplace, there's perhaps an argument to be made that the simplistic graphics of yesteryear were just far better suited to capturing the desolate nature of the great unknown. Updated March 27, 2022, by Tom Bowen: Despite huge advancements in technology allowing for state-of-the-art graphics and endless sandbox environments, many modern space-based games have struggled to make the same impact as some of the games that came before them. There are many different varieties and all game players have their preferences, but which ones are the best of the best? This list will serve to highlight the best space flight simulation games, according to Metacritic. space man Reviews FTL: Faster Than Light. Find out why the latest indie hit, FTL Faster Than Light, is one of the most unlikely contenders for game of the year in our full review Halo gives you the space gunfights. It's risky, considering how easy it is to botch a simulation game. If you dont have this game and you like space/strategy/ship/RPG games. A top-down spaceship resource management strategy sim roguelike, with retro visuals and a punishing difficulty level can’t possibly be fun, right Wrong. Over the years, many companies have created their own takes on space flight simulations. RELATED: Space Case: Things Sci-Fi Games Get Wrong About Space (& Things They Actually Get Right) ![]() Its aim is to recreate the atmosphere of running a spaceship exploring the galaxy (like Firefly/Star Trek/BSG etc.) What is the gameplay like The game is split into two major parts: exploration and combat. ![]() This is especially true in sci-fi games some games lean into the fantastical, like Mass Effect, while others like Elite: Dangerous capitalize on the wonders of space travel itself. FTL is a spaceship simulation roguelike-like. In many ways, space and space flight are equal parts fantastical and realistic in a way. Properly positioning your crew, tactically firing your weapons at enemy weak spots, hull breaches, oxygen, specific and visual system damage that. FTL is so amazing to me because it is one of the few games where the things that other games skip over is super important. Space often finds itself as the landscape of various media properties. Wait, before you start suggesting a bunch of random space exploration strategy games, that isn't really why I love FTL so much. ![]() ![]() Throughout the course of his 11 years of service, Ice is said to have helped with a number of investigations, including discovering suspects who were hiding and discovering almost a kilo of heroin hidden in a wall in a home. Plus the overall public just doesn’t believe he is guilty.” The agency stated in a subsequent post: “We have been threatened with massive protests if we move forward with our investigation and/or charges. “100% innocent! How dare you slander his good name!,” wrote one lady.Īnother wrote, “Best news story of the century!” “Justice for Ice! I’ll defend pro bono!” commented one lawyer. The Facebook post was flooded with comments from people defending and showing their support for officer Ice For more information, visit the police asked the department’s social media followers for their opinions, thousands of users defended the dog. BrillDog believes simplicity and transparency are essential for every business shipper. The BrillDog technology builds on years of logistics expertise, a deep understanding of supply chain needs, and powerful new technologies. Customers benefit from the affordable, real-time, cloud-based supply chain functionality, analysis, as well as tactical and strategic supply chain advice. īrillDog is an advanced, integrated solution that makes the supply chain easy for small and medium-sized companies. Sam and his wife, Denise, reside in Maryland. In 2022, Sam founded BrillDog, a supply chain technology platform focusing on the needs of small and mid-sized companies. Nexterus, based in New Freedom, Pennsylvania, is America's oldest privately held third-party logistics firm. He is the third-generation owner of Nexterus Inc., a family business that began in 1946. Sam Polakoff was raised in Baltimore and Columbia, Maryland. Sam Polakoff is the award-winning author of medical and political thrillers with a hint of science fiction. ![]() “I've always had a vivid imagination, and writing provides me an outlet with fun and engaging plots and characters that fill my books.” “I wrote my first novel in my 50s on a challenge from an executive coach who asked me to complete a personal goal I have always wanted to achieve but never have,” adds Polakoff. His third novel, Escaping Mercy, won the 2022 American Fiction Award for medical thrillers and was an Amazon bestseller in the Crime/Mystery – Science Fiction category. Sam is an award-winning author of sci-fi, crime, and mystery fiction. In researching the mysterious growth, Dave and Ivy discover a conspiracy guided by criminals' intent to control a programmable public.” Ivy McDermott discovers a strange microdot in Ostrinsky's brain. “I think overstressed people, especially those that work in the supply chain industry, can take a break and read a book that will take their mind off all the issues they need to address.”Ī preview of this medical thriller, crime/mystery, and cybercrime novel: “ After being suspended from his job as a police detective, Dave Ostrinsky, needing a respite from his stress-filled life, rides his bike along a mountain trail and accidentally goes over a dangerous cliff. “This book will appeal to anyone who likes reading to escape life’s stressful reality, especially people in small-to-medium sized businesses that are challenged by their supply chains BrillDog was founded to solve those challenges, and I hope this book will give them a respite from the stress,” says Sam Polakoff, Founder and CEO, BrillDog. ![]() ![]() ![]() ![]() Spanish, based on Latin, is one of these. But sticking with the original meaning of “gender,” some languages have two genders for all or almost all words. In recent years, “gender,” originally only a linguistic word, has come to mean what we used to call “sex,” as in male and female in mammalian anatomy and physiology. Since the term Latinx gained popularity in 2016, it and its variations (for me, Latine offers more phonetic fluidity)-have become a source of fierce disagreements among Latine people of all races, ages, genders, and sexual identities. Patience was necessary, given that the debate over the use of Latinx (and more recently, Latine) to refer to people with origins in Latin America has gone in dizzying circles. I would have several conversations with family and friends about what this meant for me, what pronouns I would now use, and how we would have patience with each other in learning and moving forward. The Zocalo article, by Sebastian Ferrada, explains: Like the communist language changes in Russia after 1918 and China after 1949, this strictly is a top-down, dictatorial movement. But when I talk to working-class Latinos, such as those who work around my apartment complex, they’re not using Latinx. But Inclusivity Doesn’t Have to Be.”Īctually, it isn’t “complicated.” For one thing, I know a lot of Latinos, and none uses Latinx lingo. I bring this up because Zocalo Public Square ran an article on the Latinx controversy, “Why Is the Latinx Debate So Fierce? Gender, Language, and Identity Are Complicated. That article came out just before the Great Leap Forward (Second Five Year Plan) socialist industrialization scheme, 1958-62, failed-and starved to death as many as 55 million people. The present fundamental language reform … epitomizes the Communists’ … ability to alter the Chinese cultural heritage so that more efficient indoctrination of the people, tighter control of mass organizations, and a higher level of industrialization can be assured. In Far Eastern Survey in 1956, Tao-Tai Hsia, a Yale University professor, described what happened in “The Language Revolution in Communist China”:Ī veteran Far Eastern news analyst in the United States concluded that the main reason for Communist China’s doing away with the elegant and traditional ideographs was that the physical structure of Chinese characters suggests some very un-Marxian notions. What was shocking was how a dictatorship could quickly change the language at its own whim.Īlthough I don’t know Chinese, similar changes were pushed by the Chinese Communist Party after Mao’s 1949 takeover. Some reforms long had been proposed, and probably would have been adopted gradually, and organically. White émigrés who left the country after the Revolution refused to accept the new spelling, and accused the Bolsheviks of mutilating the Russian language. The pre-revolutionary spelling was virtually outlawed. In 1918, a decree on new spelling rules was issued and all printed publications were obliged to follow them. intention was to ditch everything ‘old’-the tsarist regime, religion, the economy, and the language. The Russia Beyond website features a good article explaining what happened: It went far beyond just insisting on tovarishch (comrade) instead of grazhdanin (citizen). One of the first things they taught us was the Marxist-Leninists, when they took over after the Russian Revolution in 1917, changed the Russian language. Our teachers in Monterey all were native Russian speakers, exiles from Soviet tyranny. And the Soviets invaded Afghanistan to prop up a puppet regime. In 1979, the communist Sandinista regime seized power in Nicaragua. In 1977, the communist Mengistu regime, backed by Moscow, took over Ethiopia, and soon perpetrated the worst famine since Mao’s Great Leap Forward of the late 1950s. That year, Castro’s troops installed a communist regime in Angola. South Vietnam fell to communist North Vietnam in 1975. It was a time of increasing Cold War tensions. Leonard Wood, Mo., in late April 1978 I arrived at the Defense Language Institute in Monterey, Calif., to learn Russian. Forty-five years ago, after two months of U.S. ![]() ![]() ![]() It is only recently that a renewed special attention has been drawn to the study of contagion dynamics over higher-order interactions and over more general graph structures, like simplexes. Classically, the multi-group SIS model has assumed pairwise interactions of contagion across groups and thus has been vastly studied in the literature. We focus on simplicial complexes and refer to the model as to the simplicial SIS model. This paper analyzes a Susceptible-Infected-Susceptible (SIS) model of epidemic propagation over hypergraphs. 2) makes another appearance, this time to prove a far-reaching generalization of the Schröder–Bernstein Theorem. For this result the fixed-point theorem of Knaster and Tarski (Theorem 1. We observed that existence of such a “mean” is equivalent to existence of a finitely additive probability “measure” on \(\mathcal\) with nonvoid interior, each can be partitioned into a finite collection of subsets that can be rigidly reassembled into the other. 10 we used the fixed-point theorem of Markov and Kakutani to show that every abelian group G is “amenable” in the sense that there is a G-invariant mean on the vector space B(G) of bounded, real-valued functions on G. This result, which far surpassed anything that seemed attainable at the time, is only part of what Lomonosov proved in an astonishing two-page paper that introduced nonlinear methods-in particular the Schauder Fixed-Point Theorem-into this supposedly hard-core-linear area of mathematics. If an operator T on a Banach space commutes with a non-zero compact operator, then T has a nontrivial invariant subspace. ![]() In this chapter we’ll see why invariant subspaces are of interest and then will prove one of the subject’s landmark theorems: Victor Lomonosov’s 1973 result, a special case of which states: ![]() For Hilbert space, however, the Invariant Subspace Problem remains open, and is the subject of much research. Here “operator” means “continuous linear transformation,” and “invariant subspace” means “closed (linear) subspace that the operator takes into itself.” To say that a subspace is “nontrivial” means that it is neither the zero subspace nor the whole space.Įxamples constructed toward the end of the last century show that in the generality of Banach spaces there do exist operators with only trivial invariant subspaces. Does every operator on Hilbert space have a nontrivial invariant subspace? This chapter is about the most vexing problem in the theory of linear operators on Hilbert space: The Schauder Theorem will also be important in the next chapter where it will provide a key step in the proof of Lomonosov’s famous theorem on invariant subspaces for linear operators on Banach spaces. After proving the theorem we’ll use it to prove an important generalization of the Picard–Lindelöf Theorem of Chap. This is the famous Schauder Fixed-Point Theorem (circa 1930) which will occupy us throughout this chapter. However all is not lost: the “convex” version does survive: compact, convex subsets of normed linear space do have the fixed-point property. It turns out that the “ball” version of Brouwer’s theorem does not survive the transition to infinitely many dimensions. The closed unit ball ofĪnd the seemingly more general, but in fact equivalent 4 we proved two versions of the Brouwer Fixed-Point Theorem: The “Ball” version (Theorem 4. ![]() Recall that to say a metric space has the fixed-point property means that every continuous mapping taking the space into itself must have a fixed point. The material is split into four parts: the first introduces the Banach Contraction-Mapping Principle and the Brouwer Fixed-Point Theorem, along with a selection of interesting applications the second focuses on Brouwer’s theorem and its application to John Nash’s work the third applies Brouwer’s theorem to spaces of infinite dimension and the fourth rests on the work of Markov, Kakutani, and Ryll–Nardzewski surrounding fixed points for families of affine maps. Readers will find the presentation especially useful for independent study or as a supplement to a graduate course in fixed-point theory. Appendices provide an introduction to (or refresher on) some of the prerequisite material and exercises are integrated into the text, contributing to the volume’s ability to be used as a self-contained text. The level of exposition increases gradually throughout the book, building from a basic requirement of undergraduate proficiency to graduate-level sophistication. This text provides an introduction to some of the best-known fixed-point theorems, with an emphasis on their interactions with topics in analysis. ![]() |
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