![]() The 13-inch MacBook Pro with M2 starts at $1,299 (US) and $1,199 (US) for education. It also adds a next-generation media engine and a powerful ProRes video engine for hardware-accelerated encode and decode, this makes it possible to play back more streams of 4K and 8K video than before. M2 delivers 100GB/s of unified memory bandwidth and supports up to 24GB of fast unified memory. M2 features a next-generation 8-core CPU with advancements in both performance and efficiency cores, along with Apple’s next-generation GPU, which now has up to 10 cores - two more than M1. The M2 and M1 chip stand side by side at WWDC What’s with the M2 processor announced at WWDC?ĭescribed in more detail here, the M2 chip brings industry-leading power efficiency, a unified memory architecture, and custom technologies to make Macs running the processor more powerful while not requiring more electrical power. Playing graphics-intensive games like Baldur’s Gate 3 is also nearly 40 percent faster than the previous 13-inch MacBook Pro, and up to 3.3x faster for customers upgrading from a model without Apple silicon.Working with RAW images in apps like Affinity Photo is nearly 40 percent faster than the previous generation, and up to 3.4x faster for users who are upgrading from a model without Apple silicon.Those aren’t the only performance boosts. ![]() Editors can convert their video projects to ProRes nearly 3x faster than before. It also supports ProRes encode and decode in the media engine of M2, so users can play back up to 11 streams of 4K and up to two streams of 8K ProRes video. The new processor means the Mac can support up to 24GB of unified memory and 50 percent more memory bandwidth. That makes sense given how underpowered the smallest pro Mac had become in comparison with its more powerful siblings. Most features remain unchanged – what happens in this upgrade is a good improvement in performance. The world’s second best-selling laptop the 13-inch MacBook Pro brings up to 24GB of unified memory, ProRes acceleration, and up to 20 hours of battery life, Apple said. The MacBook Pro shows its multitasking skills So, what’s new in the 13” MacBook Pro with an M2 chip? That is up to 5x faster for customers that haven’t yet upgraded to Apple silicon.Applying filters and effects in apps like Adobe Photoshop is up to 20 percent faster than before.If you are using an Intel system you can achieve up to 15x the performance by upgrading to Apple Silicon, compared to the 1.6GHz dual-core Intel Core i5-based MacBook Air.For intensive workloads like editing complex timelines in Final Cut Pro, performance is nearly 40 percent faster than the M1 MacBook Air.What does Apple claim about M2 MacBook Air performance?Īpple made a smattering of performance claims concerning the new M2 MacBook Air, teling us that: Additional technical specifications are available at /macbook-air-m2. ![]() MacBook Air with M2 starts at $1,199 (US) and $1,099 (US) for education. ![]() Finally, you get up to eighteen hours battery life. The Mac supports fast charge for charging up to 50 percent in just 30 minutes with an optional 67W USB-C power adapter. MacBook Air also supports immersive Spatial Audio for music and movies with Dolby Atmos. A three-mic array captures clean audio using advanced beamforming algorithms, while the speakers produce improved stereo separation and vocal clarity. MacBook Air includes a new 1080p FaceTime HD camera with a larger image sensor for twice the resolution and low-light performance of the previous generation. It’s bright – 500 nits, Apple said, and also supports 1 billion colors. You get a 13.6-inch Liquid Retina screen with much thinner borders, so you’ve got more screen than chassis. ![]()
0 Comments
![]() Though Holland was introduced in Captain America: Civil War, he’s not exactly a household name and he certainly hasn’t reached the one-name level of fame that Zendaya has hit at only 20 years old. Still, you can understand why it’s Zendaya’s star power that Marvel is leveraging to sell Spider-Man: Homecoming. There’s a scene where Zendaya’s Michelle watches from the sidelines as Spider-Man rushes off to save Liz, and that moment kind of sums up how important both characters are to the plot. We go to Liz’s house, we meet her parents, and several times, she presents Peter with the sort of plot-advancing quandary that’s at the core of his super-identity: How can he juggle his obligation to fight crime with the needs that come from simply being a normal teenager? Tall and willowy, she’s got the heavy-lidded eyes and a sleepy sensuality of a literal dream girl, and Peter moons over her for most of the movie, trying to muster up the courage to ask her to the titular dance. ![]() Though she’s not as prominent as past onscreen flings like Kirsten Dunst’s Mary Jane and Emma Stone’s Gwen Stacy, Harrier’s Liz fulfills a similar narrative function. (You’ll know it when you see it.) But insofar as Spider-Man: Homecoming has an actual female lead, it’s probably Laura Harrier, who plays Peter’s crush Liz Allan. Her punch lines manage the rare feat of seeming tossed-off but comically precise, her sarcasm is a great foil for Holland’s zippy pep, and her character participates in one scene that’s so woke it will have Fox News anchors frothing at the mouth like J. Now, is Zendaya an utter delight in the limited time she spends onscreen? Yes, absolutely. Spider-Man never saves her, and Peter Parker never kisses her: In fact, he barely seems to know she exists. She is not present for most of the movie’s major scenes or action set pieces, has no bearing on the plot, and disappears for wide swaths of the film. Zendaya’s character, Michelle, is one of several high-schoolers in Peter Parker’s loose-knit friend group, popping up every so often at the end of a scene to say something snarky. (The middle syllable is drawn out like “day,” not pinched like “Princess Di.” Now you know, Dad!)Ī word of advice, though: Enjoy all these moments you’re spending with Zendaya right now, because it all adds up to markedly more screen time than she’s got in Spider-Man: Homecoming.Īre you surprised? Would you think, given how often Marvel has pushed Holland and Zendaya out as a promotional pair, that she must be playing his love interest? She isn’t. ![]() The Disney-minted actress-singer seems to be everywhere promoting her role in Spider-Man: Homecoming: She’s on the cover of Vogue! She’s going toe-to-toe with Tom Holland, the new Peter Parker, on Lip-Sync Battle! She’s turning heads with her fashion choices on a global press tour! Truly, it is now Zendaya’s world, and all we can do is live in it … as well as teach our curious dads how to pronounce her name. ![]() ![]() Social link items are typically required to fuse the highest level persona of each arcana. This is important to note as some personae cannot be fused unless their respective item is obtained by the player. These are items granted to the player upon completion of a social link in the game. The final factor that has an influence on persona fusion is the possession of social link items. This guide will use the base level of personae when listing fusion recipes, so registering personae and overwriting their base level is strongly advised against. The difference between the base level and current level of a persona is important to establish as using a persona that is the wrong level in fusions may cause disparity between this guide and the results that you actually achieve in-game.The player can register the new level of a persona in the persona compendium, but this overwrites its base level. Using Anubis as an example, if the player levelled Anubis up to level 65 then the base level of Anubis would be 59, yet its current level would be 65. The Fusion Suit is the result of drastic alterations made to Samus Arans Power Suit during the removal of parts infected by the. The current level of a persona is the level of a persona after it has been used by the player and gained experience.So many aspects of Persona 4 Golden are random and unique to each individual playthrough. All personae can be summoned from the persona compendium at their base level unless they have been overwritten by the player. There are a few reasons for this, namely the very essence of the game itself. This tells us that the base level of Anubis is level 59. As you can see, the player is unable to fuse Anubis as they are not yet level 59.Here is an example of the base level of a persona: The base level of a persona is the level that the player must be to either fuse it in the Velvet Room or encounter it during shuffle time. This is not a major factor, but it is worth noting as a significant number of fusions will be unavailable to the player when at a low level. This is a factor that can impact persona fusion as the player cannot fuse a persona that is a higher level than themselves. Here is how you can do so: Knowledge Required to ace exams, Lv 3 is required to start Margarets (Empress) S-Link, Lv 5 (max) required to. This trophy requires you to achieve the Max level (5) in all of the five parameters/qualities. As the arcana of a persona cannot be changed, this is not a factor that will affect the accuracy of this guide. In Persona 4: Golden, just like in Persona 4, there are 5 different status parameters, each with 5 levels. ![]() ![]() The most common method of obtaining a new persona via fusion is by fusing two or more personae of different arcana. The possession of social link items that unlock powerful personae.The arcana of the personae involved in the fusion.In Persona 4 Golden, the result of your fusion is dictated by five factors: Just bought the game from steam and totally hooked with it.The only prob is that there are too many fusionto choose from and i hope theres some guide for fusion. Before this guide begins there are some foundations regarding persona fusion that must be made clear to avoid later confusion, mainly involving variables that can alter the results of the fusion recipes outlined in this guide. ![]() ![]() On his stuffed body he wears worn-out and faded hand-me down Munchkin clothing of blue, with an old wrinkled and pointed hat that stands a foot above his head. He is described as having an overstuffed sack filled with straw for a head with a mouth, nose, ears, and eyes painted on with blue paint, (one of his eyes is said to be larger than the other) and has two big blue buttons for pupils. Somewhere in the middle of the Munchkin Country of Oz, he was introduced as being placed high up on a wooden pole in the cornfields of the Munchkin farming community. The Scarecrow is being made by two Munchkins. In Baum's seventh Oz book titled the The Patchwork Girl of Oz, published in 1913, the Patchwork Girl named Scraps, who later served as the Scarecrow's love interest is brought to life with this same powder by a young Munchkin boy named Ojo.ĭorothy meets the Scarecrow by W.W. In this story, the Powder of Life is a special potion invented by a mysterious Magician and brings three Oz characters to life: Jack Pumpkinhead, Sawhorse, and Gump. ![]() ![]() The Powder of Life is an element that Baum introduces in his second Oz book which serves as a sequel to the first story titled The Marvelous Land of Oz, published in 1904. We may speculate that a Munchkin farmer, who made him added the powder hoping to more effectively scare the crows away. One possibility could be that the magic substance called " Powder of Life" (mentioned in a later book) changed the scarecrow to a living being. Scarecrow of Oz " There was a great cornfield beyond the fence, and not far away Dorothy saw a Scarecrow, placed high on a pole to keep the birds away from the ripe corn." ― The Wonderful Wizard of Oz (1900)īaum never wrote how the Scarecrow was given life, (in contrast to his Oz companion the Tin Woodman, who was a real man with a body of flesh and blood whose soul transferred to its current form). ![]() ![]() ![]() I've suggested these features to their support. There are times I want to look at a list of tasks while I'm in another app, but Be Focused Pro hides itself any time you click out of its dialog box. Be Focused Pro tucks away nicely in the menu bar-sometimes a little too nicely. A persistent dialog (similar to Focus List). Being able to arrow up and down between tasks and, perhaps, Enter to edit, Shift-Enter to start working, would be bonus. A shortcut key to add a new task would be enough. It would be nicer if I could collect ideas quickly from the keyboard without having to bother my mouse. ToDo view offers a great place to sketch out a day. Better keyboard control in the ToDo view. Nevertheless, I find myself wishing for a few tweaks to make it more useful in my workflow.ġ. ![]() It's one of this well-scoped apps that doesn't try to do too much and would suffer bloat if it added every user suggestion. It even has modest time-tracking capabilities, a report of intervals worked and CSV export if you want to geek out your analysis. Type in what you're working on, press the "Play" icon to start, and get to it. As a timer with reporting, Be Focused is great. I find the Windows version more convenient, since most of my work is at the computer, but I'm glad for the phone version syncing with it when I'm on the run or planning. ![]() I use Be Focused Pro daily, both on my iPhone and Windows. Overall, great product, but for my needs I'd love some adjustments! (I sometimes track my words written per hour in a spreadsheet, so the little differences matter there.) The only options are to let it run to 20 or "skip" entirely, so I can't get an accurate accounting of my time later. Also, if I work for 18 out of 20 minutes and have to stop, it won't record that interval correctly. It would be nice to have a default schedule in settings, but be able to modify the length of the current interval on the fly. If I sit down with 18 minutes to work, or 35, I can't edit the interval for a one-time session without going into settings and modifying the entire schedule, which I then have to change again next time. I have an ideal schedule set up in the app - e.g., 8 intervals of 20 minutes each - but more than half the time that's not how my day looks. Sometimes I'm setting out to work on something (writing specifically) based on the amount of time I have available, rather than an ideal schedule of X intervals of X minutes each. I'm a busy mom with several types of work on my plate every day. I do wish I could edit intervals on the fly, though. I honestly feel like Apple overlooked people being really heavy sleepers and sleeping through their first alarm when they designed this lol.I like the way this app works (Windows OS), and how it tucks away neatly while you're working. Haven't missed any of my alarms since and still get to use the sleep focus feature! I made this change a few days ago when I left my original comment and it looks like the Health app still tracks my sleep just fine, but the Sleep focus gets turned off by a time parameter, rather the Wake Up Alarm being dismissed. From there, you can toggle Wake Up Alarm to Off, tap done, and then go back to your Clocks app to set an alarm(s) for your desired wake up time. If you tap that, you can edit each schedule. The 2nd card listed should be Full Schedule & Options. Or, if you're in Health, go to the Sleep section, and scroll to Your Schedule. I figured out there's several ways to get to this, but from your Sleep Focus, it's toward the bottom, as a button directly below your Weekdays and Weekends bedtime and wake up alarm time. /r/AppleMusic for discussion of Apple's music streaming service./r/iOSSetups for iOS wallpapers, setups, and apps./r/iPhoneXSMax for XS Max specific issues./r/AppleSwap if you want to trade devices!.Keep an eye on temporarily free or discounted apps on /r/AppHookup!.Your iPhone broke? Head on over to /r/iphonehelp!.Looking for new Wallpaper? /r/iWallpaper!./r/applemaps for everything Apple Maps!./r/iOSgaming for all your iOS gaming needs!./r/Siri: That wonderful robot that lives in our phones./r/iOSBeta for the latest on iOS developer and open beta.r/AppleWatch is the place to be for all discussions on the Apple Watch.r/iPod for the classic device that started the revolution of mobile Apple devices. ![]() For Kensington, the focus is on security and authentication. r/iPhone for all about the iPhone and our sister sub. But no one is a fan of false alarms: They diminish brand satisfaction and can prove. ![]() ![]() Suppress explicit images with a prompt “dress” or a negative prompt “nude”. It is good at generating females with correct anatomy.Ĭaution: F222 is prone to generating explicit images. F222 F222į222 is good at generating photo-realistic images. v1.5 inpainting modelĪ special model trained for inpainting. You can treat the v1.5 model as the default v1 base model. Images from this model is very similar to v1.4. V1.4 model is the first publicly released Stable Diffusion base model. You will find a brief description of them in this section. Models availableįor your convenience, the notebook has options to load some popular models. Otherwise, you will continue to consume compute credits. When you finish using the notebook, don’t forget to click “Disconnect and delete runtime” in the top right drop-down menu. Since you are the one who created this link, you can ignore the safety warning and proceed. It is likely because someone used the same ngrok link to put up something malicious. Occasionally, you will see a warning message that the site is unsafe to visit. If you are a first-time user, you can select the v1.5 model.Ĭlick on Visit Site to Start AUOTMATIC1111 GUI. Currently we offer v1.4, v1.5, v1.5 inpainting, F222, anything v3, inkpunk diffusion, Mo Di diffusion, v2.1-512, v2.1-768 and v2 depth model. You must grant permission to access Google Drive if you choose the first or the second options. All data and images will be deleted after you disconnect. Embeddings (from output_path/embeddings folder)ģ.Lora models (from output_path/Lora folder).Upscalers (from output_path/ESRGAN folder).Models (from output_path/models folder).It will load the followings from your Google Drive. Small models, images and settings: Saves Lora models, Embeddings, GUI Settings and all image in your Google Drive.Everything: Save the whole AUTOMATIC1111 Stable Diffusion webui in your Google Drive. ![]() You will need the credential after you start AUTOMATIC11111. You should see the notebook with the second cell like below. Open the Colab notebook in Quick Start Guide. Step-by-step instructions to run the Colab notebook It will consume compute units when the notebook is kept open. When you use Colab for AUTOMATIC1111, be sure to disconnect and shut down the notebook when you are done. That comes in handy when you need to train Dreambooth models fast. With a paid plan, you have the option to use Premium GPU. (It’s a steal) You can also get high-RAM machines, which are useful for using v2 models and some extensions. It gives you 100 compute units per month which are about 50 hours on a standard GPU. ![]() They have three paid plans – Pay As You Go, Colab Pro, and Colab Pro+. Google Colab is an interactive computing service offered by Google. Many of the tutorials on this site are demonstrated with this GUI. Although it associates with AUTOMATIC1111’s GitHub account, it has been a community effort to develop this software.ĪUTOMATIC1111 is feature-rich: You can use text-to-image, image-to-image, upscaling, depth-to-image, and run and train custom models all within this GUI. Instead of writing codes, we write prompts in a text box and click buttons to generate images.ĪUTOMATIC1111 was one of the first GUIs developed for Stable Diffusion. Most users use a GUI (Graphical User Interface) to use Stable Diffusion. Stable Diffusion is a machine-learning model. But you need to know what it can do because it is the gold standard in features, though not necessarily stability… You should know what AUTOMATIC1111 is if you want to be a serious user of Stable Diffusion. ![]() ![]() New product development – Companies and individuals can bring different levels of expertise and skills to a joint venture that can aid the development of products and services that otherwise would be difficult for a company to create on its own.There are lots of reasons why your business may consider entering into a joint venture with other business partners, including: They should ideally benefit all parties, growing businesses or providing additional revenue streams that would be impossible without partnering with another business. Joint ventures usually have a defined timeframe or outcome, such as a one-off project – although they can also encompass long-term partnerships. Joint ventures don’t need to be equally split between partners, either, with different partners able to hold differing stakes. Joint ventures aren’t restricted to limited companies opens in new window, either.Īny number of companies or individuals may collaborate on a joint venture, and these agreements can involve all types of business structures including sole traders opens in new window and self-employed individuals opens in new window, limited companies and limited partnerships. It’s effectively a commercial agreement between two or more participants, usually entered into in order to achieve specific business goals such as launching a new type of business or selling products into a new market.Įach company maintains their separate business structure and legal status opens in new window, with joint ventures sometimes creating a new, jointly-owned child entity that is effectively at arms reach from the parent companies. It’s full of tips on energy efficiency, dealing with late payments, and accessing finance guidance to support your business, whatever the future holds.Īt its most basic, a joint venture is when two or more businesses agree to work together. To help make things a little easier we’ve created the Start Up Loans Guide to Business Resilience. We know it’s a tough time to be running a business or starting a new one. ![]() It pays to understand what joint ventures are, as well as their advantages and disadvantages. However, they aren’t without their pitfalls and poorly conceived partnerships can harm both parties. Joint ventures can dramatically increase the reach and scale of both businesses while reducing the risk. Joint ventures are usually formed by two businesses with complementary strengths.įor example, a technology company may create a partnership opens in new window with a marketing company opens in new window to bring an innovative product to market.Īn overseas business could join forces with a local distribution company in order to sell its products in that local market opens in new window. ![]() Today’s business often collaborate with other companies on joint ventures – pooling their resources and expertise to develop new products, expand into different markets opens in new window or increase operational capabilities.Ī joint venture allows businesses to grow and gain access to markets or expertise beyond their existing capability.īy teaming up with another company, many small businesses use joint venture agreements to share specialised expertise, such as technical skills or intellectual property opens in new window, as well as spread the risks and costs of developing a new market or product. ![]() ![]() ![]() ![]() Safety Recall Information - For important safety recall information, click here.įor any difficulty using this site with a screen reader or because of a disability, please contact us at 1-80 or For California consumers: For more information about our privacy practices, click here. LUMIMAN 48ft RGBW Smart Outdoor String Lights with 15 Dimmable LED Bulbs IP65 Waterproof Color Changing String Lights Compatible with Alexa, Google Assistant for Balcony, Backyard Garden Party. Mark the on side position of the switch with red marker. Taking pictures of the circuit board and wires will really make it easier if you are a newbie like me. If your lights have an on/off switch on the circuit board, set it to on. ![]() For more information, go to or see a Harbor Freight store associate. Open the battery/board case for the solar lights. No other meaning of "Compare to" should be implied. Prices advertised by others may vary by location. ![]() 24 ft., 12-Bulb Shatterproof Outdoor LED String Lights, Black. 12-Bulb Shatterproof Outdoor LED String Lights, Black. Compare our price of 24.99 to PORTFOLIO at 34.98 (model number: SLC12BK). Get up to 5500 lumens of brilliant light in all directions with this outdoor lighting. To receive the coupon prices online, the coupon code(s) must be entered into your shopping cart.Īt Harbor Freight Tools, the "Compare to" price means that the specified comparison, which is an item with the same or similar function, was advertised for sale at or above the "Compare to" price by another additional retailer in the U.S. 12 Bulb Outdoor Incandescent String Lights Black Item 63483 / 64486. PRICES - Although we make every effort to assure that our prices, products and coupons are advertised as accurately as possible, we are only human, and in the event an error is made, we reserve the right to correct it. No use on prior purchases after 30 days from original purchase or without original receipt. Indoor/Outdoor String Light with S14 Single Filament LED Bulbs The Hampton Bay LED String Lights combine The Hampton Bay LED String Lights combine the beauty of vintage, Edison-style bulbs with the latest in cutting-edge LED technology to bring you beautiful lighting where you want it. Add up to 50 strings total for an impressive 1200 ft.Original coupon only.Outdoor rated, heavy duty gauge construction withstands the elements. Leopcito 4 Pack Camping Lanterns Battery Powered, COB LED Camping Lights for Power Outages. Each strand features low-power LED lights with an attractive Edison style filament bulb that consumes 85 less electricity than typical outdoor light strands. Vigdur Outdoor String Lights - Plug-in G Type Bulb Incandescent String Light with 10-Smoky Glass Shades,Waterproof Connectable Lantern String Lights for Backyard Garden Holiday Party Decor 37 33.99 33. This string light is versatile enough for almost any installation, such as outdoor accent lighting on patios, porches, walkways, and restaurants. Get enchanting warm, incandescent lighting for your outdoor events and parties. The LUMINAR OUTDOOR 12 bulb LED filament string light is the perfect accessory to complement any backyard or outdoor setting. It's possible to connect up to 50 strands together for an impressive 1200 ft. Each strand features low-power LED lights with an attractive Edison style filament bulb that consumes 85% less electricity than typical outdoor light strands. 12-bulb LED filament string light is the perfect accessory to complement any backyard or outdoor setting. Need replacement bulbs Want to build your own string light and pick your cord and bulbs separately Choose from either LED or incandescent bulbs in three socket sizes medium, C9 and C7. ![]() ![]() ![]() It can be said that the best alternative for Adobe Acrobat, and any open-source PDF editors.Ĭompared with those open-source PDF editors, PDFgear offers all-around PDF editing solutions. PDFgear Desktop is a completely-free PDF management toolkit available on Windows and macOS. Markup PDF files online Convert PDF to/from others Password protect PDF filesīest Alternative to Open-source PDF Editors – PDFgear Desktop Manage PDF pagesĪnnotate PDF files Fill out PDF forms Password protect PDFs View and annotate PDFs Copy and paste the content into a PDFĪnnotate a PDF Insert images, links, etc. OCR PDF files Annotate PDF easily Cloud storage of PDFs Merge and split PDF files Extract pages from PDF files Rotate PDF pagesĪnnotate a PDF Insert the image into a PDF Spell-check PDF documentsĮdit PDF documents Fill PDF forms Password protect PDFsĬonvert PDF formats Edit PDF with annotating tools Create PDF formsĬonvert file formats Edit PDF with annotating tools Password protect PDF documentsĬonvert to PDF Annotate PDF documents Edit text in PDFĬonvert PDF formats Edit PDF with annotating tools e-sign PDF documents Comparison of the Best Open Source PDF Editors Open-source PDF EditorsĬonvert PDF file format Reduce PDF file size Add signature to a PDF Watermark a PDFĮdit existing words in a PDF Manage PDF pages Add a digital signatureĪnnotate a PDF Change words in a PDF Remove PDF texts, images, links, etc.Įdit existing content in a PDF Convert PDF to Word Annotate and comment on a PDF ![]() Check out the best online and desktop PDF editing software and pick one for long-term use. This article will list the 17 best open-source PDF editors for you to select. To make some edits to a PDF document, you have to resort to professional PDF editing tools. But this non-editable feature also brings several handicaps. It is a non-editable format, which maintains the same file formatting on different devices and systems. PDF is a widely used file format in the office. ![]() ![]() Quite the contrary, they have been discovered and rediscovered many times in the past, in very different areas of research. These facts have important consequences, as we will see in the next section.Īlthough Ellerman provided a solid and fruitful probabilistic interpretation of this definition of entropy, the formulae (1) and (2) are not new. Īgain we emphasize that, in contrast to the Shannon–Von Neumann entropy, the logical entropy S L represents both a probability (of obtaining different results in two consecutive draws, as mentioned above) and a norm in the Euclidean space. For a system known with certainty, both entropies yield, S VN = S L = 0, but for maximal uncertainty S VN = log n, whereas. In particular S VN is additive, while S L is not, at least not in the standard fashion, see. Of course, the logical entropy definition (1) implies very different properties from the standard Shannon–Von Neumann entropy For instance, one can define the scalar product: between two probability distributions, and their Euclidean distance d( p, q) as: This connection to Euclidean geometry allows one to use standard geometrical concepts when making use of the logical entropy. 1 The information I has the nice property of being the square of a norm in, actually the Euclidean norm. This quantity reflects the knowledge we have of the state of a physical system, being maximum when we know its state with certainty, and minimum when all states are equally probable. For equal probabilities (, ∀ i), one gets:, when n → ∞.įollowing, one can also define the information I as the complement of the entropy to unity: The lower bound is reached when one element has probability p i = 1, while for all others p j≠ i= 0. ![]() ![]() In this work, we will mainly consider this case, unless otherwise stated. The subsets B i can possibly contain one single element, in which case S L represents the probability that two consecutive draws yield different elements of U. ![]() Summing over all n subsets, we obtain the total number of distinctions, which is the definition of the logical entropy S L: This is precisely the concept of distinction, i.e., the ability establish that two independent draws are different from one another. If a certain set U is partitioned into a number n of subsets B i (such that ), each endowed with a probability p i of finding an element of U in that subset, then the probability that in two independent draws one will obtain elements in distinct subsets B i and B j≠ i is: p i(1 − p i). Succinctly, logical entropy is based on the concept of distinctions. “Logical entropy” is a concept introduced by David Ellerman in a series of works spanning the last decade see also Ellerman’s paper in this Special Issue. Here, we will introduce each topic separately, before bringing them together in the following sections. 1 IntroductionĪs its title suggests, this work sits at the crossroad of three different topics: (a) an alternative definition of entropy, (b) the extension of standard probabilities to negative values, and (c) the relevance of the first two items to our understanding of quantum mechanics. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This result suggest that the logical entropy plays a profound role in establishing the peculiar rules of quantum physics. For infinite-dimensional spaces (continuum), we show that, under the sole hypothesis that the logical entropy and the total probability are preserved in time, one obtains an evolution equation for the probability density that is basically identical to the quantum evolution of the Wigner function in phase space, at least when one considers only the momentum variable. Several intriguing quantum-like properties of the logical entropy are derived and discussed in finite dimensional spaces. Here, we analyze and reinterpret negative probabilities in the light of the concept of logical entropy. ![]() The quadratic form of the logical entropy lends itself to a generalization of the probabilities that include negative values, an idea that goes back to Feynman and Wigner. The same formula comes across as a useful definition of entropy in quantum mechanics, where it is linked to the notion of purity of a quantum state. Although the mathematical formula itself is not new, Ellerman provided a sound probabilistic interpretation of S L as a measure of the distinctions of a partition on a given set. * Corresponding author: concept of Logical Entropy, S L = 1-Σ n i=1 p i 2, where the p i are normalized probabilities, was introduced by David Ellerman in a series of recent papers. Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504, 67000 Strasbourg, France ![]() |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |