The kinematical phase space of classical gravitational field is flat (affine) and unbounded. Because of this, field variables may tend to infinity leading to appearance of singularities, which plague Einstein's theory of gravity. At the slides an idea of generalizing theory of gravity by compactification of the phase space is discussed. The procedure of compactification of the phase space of de Sitter cosmological model with two dimensional phase space is introduced. The model is analyzed at both classical and quantum level.
Towards the loop quantum gravity with compact phase spaceJakub Mielczarek
The kinematical phase space of the classical gravitational field
is flat (affine) and unbounded. Consequently, field variables may
tend to infinity, leading to the appearance of singularities, which
plague Einstein's theory of gravity. In loop quantum gravity (LQG),
the holonomies partially solve the problem by compacting the
connection variables. However, the conjugated fluxes remain
unbounded. The slides summarize some of our attempts toward
formulating LQG in a fully compact phase space manner, which
implies the finite-dimensional Hilbert space of the theory. The
compactification procedure is discussed first on the example
of a minisuperspace cosmological model (loop quantum cosmology)
with a single degree of freedom. The slides contain an overview of our recent results on applying the compact phase space approach to solving the Hamiltonian constraint using variational quantum methods.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms for those who already suffer from conditions like anxiety and depression.
This document discusses quantum space technologies and photons. It mentions horizontal and vertical polarization of photons, as well as the quantum state of a photon that is a superposition of horizontal and vertical polarization states. The document is licensed under a Creative Commons license.
This talk aims at emphasizing the potential of a synergy between quantum gravity and the quantum computing technologies. Such a combination would be beneficial for both understanding the Planck scale physics and the stimulation of development of the quantum technologies. This is especially important in the present early days of commercial quantum computers, when challenges originating from the basic research may catalyze the technological progress. Our attention will be focused on simulations of Planck scale physics with the use of existing universal and adiabatic quantum computers. Current possibilities, technological challenges and prospects for the future will be outlined.
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
Towards the loop quantum gravity with compact phase spaceJakub Mielczarek
The kinematical phase space of the classical gravitational field
is flat (affine) and unbounded. Consequently, field variables may
tend to infinity, leading to the appearance of singularities, which
plague Einstein's theory of gravity. In loop quantum gravity (LQG),
the holonomies partially solve the problem by compacting the
connection variables. However, the conjugated fluxes remain
unbounded. The slides summarize some of our attempts toward
formulating LQG in a fully compact phase space manner, which
implies the finite-dimensional Hilbert space of the theory. The
compactification procedure is discussed first on the example
of a minisuperspace cosmological model (loop quantum cosmology)
with a single degree of freedom. The slides contain an overview of our recent results on applying the compact phase space approach to solving the Hamiltonian constraint using variational quantum methods.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms for those who already suffer from conditions like anxiety and depression.
This document discusses quantum space technologies and photons. It mentions horizontal and vertical polarization of photons, as well as the quantum state of a photon that is a superposition of horizontal and vertical polarization states. The document is licensed under a Creative Commons license.
This talk aims at emphasizing the potential of a synergy between quantum gravity and the quantum computing technologies. Such a combination would be beneficial for both understanding the Planck scale physics and the stimulation of development of the quantum technologies. This is especially important in the present early days of commercial quantum computers, when challenges originating from the basic research may catalyze the technological progress. Our attention will be focused on simulations of Planck scale physics with the use of existing universal and adiabatic quantum computers. Current possibilities, technological challenges and prospects for the future will be outlined.
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
The cost of acquiring information by natural selectionCarl Bergstrom
This is a short talk that I gave at the Banff International Research Station workshop on Modeling and Theory in Population Biology. The idea is to try to understand how the burden of natural selection relates to the amount of information that selection puts into the genome.
It's based on the first part of this research paper:
The cost of information acquisition by natural selection
Ryan Seamus McGee, Olivia Kosterlitz, Artem Kaznatcheev, Benjamin Kerr, Carl T. Bergstrom
bioRxiv 2022.07.02.498577; doi: https://doi.org/10.1101/2022.07.02.498577
Authoring a personal GPT for your research and practice: How we created the Q...Leonel Morgado
Thematic analysis in qualitative research is a time-consuming and systematic task, typically done using teams. Team members must ground their activities on common understandings of the major concepts underlying the thematic analysis, and define criteria for its development. However, conceptual misunderstandings, equivocations, and lack of adherence to criteria are challenges to the quality and speed of this process. Given the distributed and uncertain nature of this process, we wondered if the tasks in thematic analysis could be supported by readily available artificial intelligence chatbots. Our early efforts point to potential benefits: not just saving time in the coding process but better adherence to criteria and grounding, by increasing triangulation between humans and artificial intelligence. This tutorial will provide a description and demonstration of the process we followed, as two academic researchers, to develop a custom ChatGPT to assist with qualitative coding in the thematic data analysis process of immersive learning accounts in a survey of the academic literature: QUAL-E Immersive Learning Thematic Analysis Helper. In the hands-on time, participants will try out QUAL-E and develop their ideas for their own qualitative coding ChatGPT. Participants that have the paid ChatGPT Plus subscription can create a draft of their assistants. The organizers will provide course materials and slide deck that participants will be able to utilize to continue development of their custom GPT. The paid subscription to ChatGPT Plus is not required to participate in this workshop, just for trying out personal GPTs during it.
SDSS1335+0728: The awakening of a ∼ 106M⊙ black hole⋆Sérgio Sacani
Context. The early-type galaxy SDSS J133519.91+072807.4 (hereafter SDSS1335+0728), which had exhibited no prior optical variations during the preceding two decades, began showing significant nuclear variability in the Zwicky Transient Facility (ZTF) alert stream from December 2019 (as ZTF19acnskyy). This variability behaviour, coupled with the host-galaxy properties, suggests that SDSS1335+0728 hosts a ∼ 106M⊙ black hole (BH) that is currently in the process of ‘turning on’. Aims. We present a multi-wavelength photometric analysis and spectroscopic follow-up performed with the aim of better understanding the origin of the nuclear variations detected in SDSS1335+0728. Methods. We used archival photometry (from WISE, 2MASS, SDSS, GALEX, eROSITA) and spectroscopic data (from SDSS and LAMOST) to study the state of SDSS1335+0728 prior to December 2019, and new observations from Swift, SOAR/Goodman, VLT/X-shooter, and Keck/LRIS taken after its turn-on to characterise its current state. We analysed the variability of SDSS1335+0728 in the X-ray/UV/optical/mid-infrared range, modelled its spectral energy distribution prior to and after December 2019, and studied the evolution of its UV/optical spectra. Results. From our multi-wavelength photometric analysis, we find that: (a) since 2021, the UV flux (from Swift/UVOT observations) is four times brighter than the flux reported by GALEX in 2004; (b) since June 2022, the mid-infrared flux has risen more than two times, and the W1−W2 WISE colour has become redder; and (c) since February 2024, the source has begun showing X-ray emission. From our spectroscopic follow-up, we see that (i) the narrow emission line ratios are now consistent with a more energetic ionising continuum; (ii) broad emission lines are not detected; and (iii) the [OIII] line increased its flux ∼ 3.6 years after the first ZTF alert, which implies a relatively compact narrow-line-emitting region. Conclusions. We conclude that the variations observed in SDSS1335+0728 could be either explained by a ∼ 106M⊙ AGN that is just turning on or by an exotic tidal disruption event (TDE). If the former is true, SDSS1335+0728 is one of the strongest cases of an AGNobserved in the process of activating. If the latter were found to be the case, it would correspond to the longest and faintest TDE ever observed (or another class of still unknown nuclear transient). Future observations of SDSS1335+0728 are crucial to further understand its behaviour. Key words. galaxies: active– accretion, accretion discs– galaxies: individual: SDSS J133519.91+072807.4
PPT on Direct Seeded Rice presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
PPT on Sustainable Land Management presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
Mending Clothing to Support Sustainable Fashion_CIMaR 2024.pdfSelcen Ozturkcan
Ozturkcan, S., Berndt, A., & Angelakis, A. (2024). Mending clothing to support sustainable fashion. Presented at the 31st Annual Conference by the Consortium for International Marketing Research (CIMaR), 10-13 Jun 2024, University of Gävle, Sweden.
Evidence of Jet Activity from the Secondary Black Hole in the OJ 287 Binary S...Sérgio Sacani
Wereport the study of a huge optical intraday flare on 2021 November 12 at 2 a.m. UT in the blazar OJ287. In the binary black hole model, it is associated with an impact of the secondary black hole on the accretion disk of the primary. Our multifrequency observing campaign was set up to search for such a signature of the impact based on a prediction made 8 yr earlier. The first I-band results of the flare have already been reported by Kishore et al. (2024). Here we combine these data with our monitoring in the R-band. There is a big change in the R–I spectral index by 1.0 ±0.1 between the normal background and the flare, suggesting a new component of radiation. The polarization variation during the rise of the flare suggests the same. The limits on the source size place it most reasonably in the jet of the secondary BH. We then ask why we have not seen this phenomenon before. We show that OJ287 was never before observed with sufficient sensitivity on the night when the flare should have happened according to the binary model. We also study the probability that this flare is just an oversized example of intraday variability using the Krakow data set of intense monitoring between 2015 and 2023. We find that the occurrence of a flare of this size and rapidity is unlikely. In machine-readable Tables 1 and 2, we give the full orbit-linked historical light curve of OJ287 as well as the dense monitoring sample of Krakow.
Discovery of An Apparent Red, High-Velocity Type Ia Supernova at 𝐳 = 2.9 wi...Sérgio Sacani
We present the JWST discovery of SN 2023adsy, a transient object located in a host galaxy JADES-GS
+
53.13485
−
27.82088
with a host spectroscopic redshift of
2.903
±
0.007
. The transient was identified in deep James Webb Space Telescope (JWST)/NIRCam imaging from the JWST Advanced Deep Extragalactic Survey (JADES) program. Photometric and spectroscopic followup with NIRCam and NIRSpec, respectively, confirm the redshift and yield UV-NIR light-curve, NIR color, and spectroscopic information all consistent with a Type Ia classification. Despite its classification as a likely SN Ia, SN 2023adsy is both fairly red (
�
(
�
−
�
)
∼
0.9
) despite a host galaxy with low-extinction and has a high Ca II velocity (
19
,
000
±
2
,
000
km/s) compared to the general population of SNe Ia. While these characteristics are consistent with some Ca-rich SNe Ia, particularly SN 2016hnk, SN 2023adsy is intrinsically brighter than the low-
�
Ca-rich population. Although such an object is too red for any low-
�
cosmological sample, we apply a fiducial standardization approach to SN 2023adsy and find that the SN 2023adsy luminosity distance measurement is in excellent agreement (
≲
1
�
) with
Λ
CDM. Therefore unlike low-
�
Ca-rich SNe Ia, SN 2023adsy is standardizable and gives no indication that SN Ia standardized luminosities change significantly with redshift. A larger sample of distant SNe Ia is required to determine if SN Ia population characteristics at high-
�
truly diverge from their low-
�
counterparts, and to confirm that standardized luminosities nevertheless remain constant with redshift.
CLASS 12th CHEMISTRY SOLID STATE ppt (Animated)eitps1506
Description:
Dive into the fascinating realm of solid-state physics with our meticulously crafted online PowerPoint presentation. This immersive educational resource offers a comprehensive exploration of the fundamental concepts, theories, and applications within the realm of solid-state physics.
From crystalline structures to semiconductor devices, this presentation delves into the intricate principles governing the behavior of solids, providing clear explanations and illustrative examples to enhance understanding. Whether you're a student delving into the subject for the first time or a seasoned researcher seeking to deepen your knowledge, our presentation offers valuable insights and in-depth analyses to cater to various levels of expertise.
Key topics covered include:
Crystal Structures: Unravel the mysteries of crystalline arrangements and their significance in determining material properties.
Band Theory: Explore the electronic band structure of solids and understand how it influences their conductive properties.
Semiconductor Physics: Delve into the behavior of semiconductors, including doping, carrier transport, and device applications.
Magnetic Properties: Investigate the magnetic behavior of solids, including ferromagnetism, antiferromagnetism, and ferrimagnetism.
Optical Properties: Examine the interaction of light with solids, including absorption, reflection, and transmission phenomena.
With visually engaging slides, informative content, and interactive elements, our online PowerPoint presentation serves as a valuable resource for students, educators, and enthusiasts alike, facilitating a deeper understanding of the captivating world of solid-state physics. Explore the intricacies of solid-state materials and unlock the secrets behind their remarkable properties with our comprehensive presentation.
Mechanisms and Applications of Antiviral Neutralizing Antibodies - Creative B...Creative-Biolabs
Neutralizing antibodies, pivotal in immune defense, specifically bind and inhibit viral pathogens, thereby playing a crucial role in protecting against and mitigating infectious diseases. In this slide, we will introduce what antibodies and neutralizing antibodies are, the production and regulation of neutralizing antibodies, their mechanisms of action, classification and applications, as well as the challenges they face.
BIRDS DIVERSITY OF SOOTEA BISWANATH ASSAM.ppt.pptxgoluk9330
Ahota Beel, nestled in Sootea Biswanath Assam , is celebrated for its extraordinary diversity of bird species. This wetland sanctuary supports a myriad of avian residents and migrants alike. Visitors can admire the elegant flights of migratory species such as the Northern Pintail and Eurasian Wigeon, alongside resident birds including the Asian Openbill and Pheasant-tailed Jacana. With its tranquil scenery and varied habitats, Ahota Beel offers a perfect haven for birdwatchers to appreciate and study the vibrant birdlife that thrives in this natural refuge.
ESA/ACT Science Coffee: Diego Blas - Gravitational wave detection with orbita...Advanced-Concepts-Team
Presentation in the Science Coffee of the Advanced Concepts Team of the European Space Agency on the 07.06.2024.
Speaker: Diego Blas (IFAE/ICREA)
Title: Gravitational wave detection with orbital motion of Moon and artificial
Abstract:
In this talk I will describe some recent ideas to find gravitational waves from supermassive black holes or of primordial origin by studying their secular effect on the orbital motion of the Moon or satellites that are laser ranged.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
Creative operations teams expect increased AI use in 2024. Currently, over half of tasks are not AI-enabled, but this is expected to decrease in the coming year. ChatGPT is the most popular AI tool currently. Business leaders are more actively exploring AI benefits than individual contributors. Most respondents do not believe AI will impact workforce size in 2024. However, some inhibitions still exist around AI accuracy and lack of understanding. Creatives primarily want to use AI to save time on mundane tasks and boost productivity.
The cost of acquiring information by natural selectionCarl Bergstrom
This is a short talk that I gave at the Banff International Research Station workshop on Modeling and Theory in Population Biology. The idea is to try to understand how the burden of natural selection relates to the amount of information that selection puts into the genome.
It's based on the first part of this research paper:
The cost of information acquisition by natural selection
Ryan Seamus McGee, Olivia Kosterlitz, Artem Kaznatcheev, Benjamin Kerr, Carl T. Bergstrom
bioRxiv 2022.07.02.498577; doi: https://doi.org/10.1101/2022.07.02.498577
Authoring a personal GPT for your research and practice: How we created the Q...Leonel Morgado
Thematic analysis in qualitative research is a time-consuming and systematic task, typically done using teams. Team members must ground their activities on common understandings of the major concepts underlying the thematic analysis, and define criteria for its development. However, conceptual misunderstandings, equivocations, and lack of adherence to criteria are challenges to the quality and speed of this process. Given the distributed and uncertain nature of this process, we wondered if the tasks in thematic analysis could be supported by readily available artificial intelligence chatbots. Our early efforts point to potential benefits: not just saving time in the coding process but better adherence to criteria and grounding, by increasing triangulation between humans and artificial intelligence. This tutorial will provide a description and demonstration of the process we followed, as two academic researchers, to develop a custom ChatGPT to assist with qualitative coding in the thematic data analysis process of immersive learning accounts in a survey of the academic literature: QUAL-E Immersive Learning Thematic Analysis Helper. In the hands-on time, participants will try out QUAL-E and develop their ideas for their own qualitative coding ChatGPT. Participants that have the paid ChatGPT Plus subscription can create a draft of their assistants. The organizers will provide course materials and slide deck that participants will be able to utilize to continue development of their custom GPT. The paid subscription to ChatGPT Plus is not required to participate in this workshop, just for trying out personal GPTs during it.
SDSS1335+0728: The awakening of a ∼ 106M⊙ black hole⋆Sérgio Sacani
Context. The early-type galaxy SDSS J133519.91+072807.4 (hereafter SDSS1335+0728), which had exhibited no prior optical variations during the preceding two decades, began showing significant nuclear variability in the Zwicky Transient Facility (ZTF) alert stream from December 2019 (as ZTF19acnskyy). This variability behaviour, coupled with the host-galaxy properties, suggests that SDSS1335+0728 hosts a ∼ 106M⊙ black hole (BH) that is currently in the process of ‘turning on’. Aims. We present a multi-wavelength photometric analysis and spectroscopic follow-up performed with the aim of better understanding the origin of the nuclear variations detected in SDSS1335+0728. Methods. We used archival photometry (from WISE, 2MASS, SDSS, GALEX, eROSITA) and spectroscopic data (from SDSS and LAMOST) to study the state of SDSS1335+0728 prior to December 2019, and new observations from Swift, SOAR/Goodman, VLT/X-shooter, and Keck/LRIS taken after its turn-on to characterise its current state. We analysed the variability of SDSS1335+0728 in the X-ray/UV/optical/mid-infrared range, modelled its spectral energy distribution prior to and after December 2019, and studied the evolution of its UV/optical spectra. Results. From our multi-wavelength photometric analysis, we find that: (a) since 2021, the UV flux (from Swift/UVOT observations) is four times brighter than the flux reported by GALEX in 2004; (b) since June 2022, the mid-infrared flux has risen more than two times, and the W1−W2 WISE colour has become redder; and (c) since February 2024, the source has begun showing X-ray emission. From our spectroscopic follow-up, we see that (i) the narrow emission line ratios are now consistent with a more energetic ionising continuum; (ii) broad emission lines are not detected; and (iii) the [OIII] line increased its flux ∼ 3.6 years after the first ZTF alert, which implies a relatively compact narrow-line-emitting region. Conclusions. We conclude that the variations observed in SDSS1335+0728 could be either explained by a ∼ 106M⊙ AGN that is just turning on or by an exotic tidal disruption event (TDE). If the former is true, SDSS1335+0728 is one of the strongest cases of an AGNobserved in the process of activating. If the latter were found to be the case, it would correspond to the longest and faintest TDE ever observed (or another class of still unknown nuclear transient). Future observations of SDSS1335+0728 are crucial to further understand its behaviour. Key words. galaxies: active– accretion, accretion discs– galaxies: individual: SDSS J133519.91+072807.4
PPT on Direct Seeded Rice presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
PPT on Sustainable Land Management presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
Mending Clothing to Support Sustainable Fashion_CIMaR 2024.pdfSelcen Ozturkcan
Ozturkcan, S., Berndt, A., & Angelakis, A. (2024). Mending clothing to support sustainable fashion. Presented at the 31st Annual Conference by the Consortium for International Marketing Research (CIMaR), 10-13 Jun 2024, University of Gävle, Sweden.
Evidence of Jet Activity from the Secondary Black Hole in the OJ 287 Binary S...Sérgio Sacani
Wereport the study of a huge optical intraday flare on 2021 November 12 at 2 a.m. UT in the blazar OJ287. In the binary black hole model, it is associated with an impact of the secondary black hole on the accretion disk of the primary. Our multifrequency observing campaign was set up to search for such a signature of the impact based on a prediction made 8 yr earlier. The first I-band results of the flare have already been reported by Kishore et al. (2024). Here we combine these data with our monitoring in the R-band. There is a big change in the R–I spectral index by 1.0 ±0.1 between the normal background and the flare, suggesting a new component of radiation. The polarization variation during the rise of the flare suggests the same. The limits on the source size place it most reasonably in the jet of the secondary BH. We then ask why we have not seen this phenomenon before. We show that OJ287 was never before observed with sufficient sensitivity on the night when the flare should have happened according to the binary model. We also study the probability that this flare is just an oversized example of intraday variability using the Krakow data set of intense monitoring between 2015 and 2023. We find that the occurrence of a flare of this size and rapidity is unlikely. In machine-readable Tables 1 and 2, we give the full orbit-linked historical light curve of OJ287 as well as the dense monitoring sample of Krakow.
Discovery of An Apparent Red, High-Velocity Type Ia Supernova at 𝐳 = 2.9 wi...Sérgio Sacani
We present the JWST discovery of SN 2023adsy, a transient object located in a host galaxy JADES-GS
+
53.13485
−
27.82088
with a host spectroscopic redshift of
2.903
±
0.007
. The transient was identified in deep James Webb Space Telescope (JWST)/NIRCam imaging from the JWST Advanced Deep Extragalactic Survey (JADES) program. Photometric and spectroscopic followup with NIRCam and NIRSpec, respectively, confirm the redshift and yield UV-NIR light-curve, NIR color, and spectroscopic information all consistent with a Type Ia classification. Despite its classification as a likely SN Ia, SN 2023adsy is both fairly red (
�
(
�
−
�
)
∼
0.9
) despite a host galaxy with low-extinction and has a high Ca II velocity (
19
,
000
±
2
,
000
km/s) compared to the general population of SNe Ia. While these characteristics are consistent with some Ca-rich SNe Ia, particularly SN 2016hnk, SN 2023adsy is intrinsically brighter than the low-
�
Ca-rich population. Although such an object is too red for any low-
�
cosmological sample, we apply a fiducial standardization approach to SN 2023adsy and find that the SN 2023adsy luminosity distance measurement is in excellent agreement (
≲
1
�
) with
Λ
CDM. Therefore unlike low-
�
Ca-rich SNe Ia, SN 2023adsy is standardizable and gives no indication that SN Ia standardized luminosities change significantly with redshift. A larger sample of distant SNe Ia is required to determine if SN Ia population characteristics at high-
�
truly diverge from their low-
�
counterparts, and to confirm that standardized luminosities nevertheless remain constant with redshift.
CLASS 12th CHEMISTRY SOLID STATE ppt (Animated)eitps1506
Description:
Dive into the fascinating realm of solid-state physics with our meticulously crafted online PowerPoint presentation. This immersive educational resource offers a comprehensive exploration of the fundamental concepts, theories, and applications within the realm of solid-state physics.
From crystalline structures to semiconductor devices, this presentation delves into the intricate principles governing the behavior of solids, providing clear explanations and illustrative examples to enhance understanding. Whether you're a student delving into the subject for the first time or a seasoned researcher seeking to deepen your knowledge, our presentation offers valuable insights and in-depth analyses to cater to various levels of expertise.
Key topics covered include:
Crystal Structures: Unravel the mysteries of crystalline arrangements and their significance in determining material properties.
Band Theory: Explore the electronic band structure of solids and understand how it influences their conductive properties.
Semiconductor Physics: Delve into the behavior of semiconductors, including doping, carrier transport, and device applications.
Magnetic Properties: Investigate the magnetic behavior of solids, including ferromagnetism, antiferromagnetism, and ferrimagnetism.
Optical Properties: Examine the interaction of light with solids, including absorption, reflection, and transmission phenomena.
With visually engaging slides, informative content, and interactive elements, our online PowerPoint presentation serves as a valuable resource for students, educators, and enthusiasts alike, facilitating a deeper understanding of the captivating world of solid-state physics. Explore the intricacies of solid-state materials and unlock the secrets behind their remarkable properties with our comprehensive presentation.
Mechanisms and Applications of Antiviral Neutralizing Antibodies - Creative B...Creative-Biolabs
Neutralizing antibodies, pivotal in immune defense, specifically bind and inhibit viral pathogens, thereby playing a crucial role in protecting against and mitigating infectious diseases. In this slide, we will introduce what antibodies and neutralizing antibodies are, the production and regulation of neutralizing antibodies, their mechanisms of action, classification and applications, as well as the challenges they face.
BIRDS DIVERSITY OF SOOTEA BISWANATH ASSAM.ppt.pptxgoluk9330
Ahota Beel, nestled in Sootea Biswanath Assam , is celebrated for its extraordinary diversity of bird species. This wetland sanctuary supports a myriad of avian residents and migrants alike. Visitors can admire the elegant flights of migratory species such as the Northern Pintail and Eurasian Wigeon, alongside resident birds including the Asian Openbill and Pheasant-tailed Jacana. With its tranquil scenery and varied habitats, Ahota Beel offers a perfect haven for birdwatchers to appreciate and study the vibrant birdlife that thrives in this natural refuge.
ESA/ACT Science Coffee: Diego Blas - Gravitational wave detection with orbita...Advanced-Concepts-Team
Presentation in the Science Coffee of the Advanced Concepts Team of the European Space Agency on the 07.06.2024.
Speaker: Diego Blas (IFAE/ICREA)
Title: Gravitational wave detection with orbital motion of Moon and artificial
Abstract:
In this talk I will describe some recent ideas to find gravitational waves from supermassive black holes or of primordial origin by studying their secular effect on the orbital motion of the Moon or satellites that are laser ranged.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
Creative operations teams expect increased AI use in 2024. Currently, over half of tasks are not AI-enabled, but this is expected to decrease in the coming year. ChatGPT is the most popular AI tool currently. Business leaders are more actively exploring AI benefits than individual contributors. Most respondents do not believe AI will impact workforce size in 2024. However, some inhibitions still exist around AI accuracy and lack of understanding. Creatives primarily want to use AI to save time on mundane tasks and boost productivity.
Organizational culture includes values, norms, systems, symbols, language, assumptions, beliefs, and habits that influence employee behaviors and how people interpret those behaviors. It is important because culture can help or hinder a company's success. Some key aspects of Netflix's culture that help it achieve results include hiring smartly so every position has stars, focusing on attitude over just aptitude, and having a strict policy against peacocks, whiners, and jerks.
PEPSICO Presentation to CAGNY Conference Feb 2024Neil Kimberley
PepsiCo provided a safe harbor statement noting that any forward-looking statements are based on currently available information and are subject to risks and uncertainties. It also provided information on non-GAAP measures and directing readers to its website for disclosure and reconciliation. The document then discussed PepsiCo's business overview, including that it is a global beverage and convenient food company with iconic brands, $91 billion in net revenue in 2023, and nearly $14 billion in core operating profit. It operates through a divisional structure with a focus on local consumers.
Content Methodology: A Best Practices Report (Webinar)contently
This document provides an overview of content methodology best practices. It defines content methodology as establishing objectives, KPIs, and a culture of continuous learning and iteration. An effective methodology focuses on connecting with audiences, creating optimal content, and optimizing processes. It also discusses why a methodology is needed due to the competitive landscape, proliferation of channels, and opportunities for improvement. Components of an effective methodology include defining objectives and KPIs, audience analysis, identifying opportunities, and evaluating resources. The document concludes with recommendations around creating a content plan, testing and optimizing content over 90 days.
How to Prepare For a Successful Job Search for 2024Albert Qian
The document provides guidance on preparing a job search for 2024. It discusses the state of the job market, focusing on growth in AI and healthcare but also continued layoffs. It recommends figuring out what you want to do by researching interests and skills, then conducting informational interviews. The job search should involve building a personal brand on LinkedIn, actively applying to jobs, tailoring resumes and interviews, maintaining job hunting as a habit, and continuing self-improvement. Once hired, the document advises setting new goals and keeping skills and networking active in case of future opportunities.
A report by thenetworkone and Kurio.
The contributing experts and agencies are (in an alphabetical order): Sylwia Rytel, Social Media Supervisor, 180heartbeats + JUNG v MATT (PL), Sharlene Jenner, Vice President - Director of Engagement Strategy, Abelson Taylor (USA), Alex Casanovas, Digital Director, Atrevia (ES), Dora Beilin, Senior Social Strategist, Barrett Hoffher (USA), Min Seo, Campaign Director, Brand New Agency (KR), Deshé M. Gully, Associate Strategist, Day One Agency (USA), Francesca Trevisan, Strategist, Different (IT), Trevor Crossman, CX and Digital Transformation Director; Olivia Hussey, Strategic Planner; Simi Srinarula, Social Media Manager, The Hallway (AUS), James Hebbert, Managing Director, Hylink (CN / UK), Mundy Álvarez, Planning Director; Pedro Rojas, Social Media Manager; Pancho González, CCO, Inbrax (CH), Oana Oprea, Head of Digital Planning, Jam Session Agency (RO), Amy Bottrill, Social Account Director, Launch (UK), Gaby Arriaga, Founder, Leonardo1452 (MX), Shantesh S Row, Creative Director, Liwa (UAE), Rajesh Mehta, Chief Strategy Officer; Dhruv Gaur, Digital Planning Lead; Leonie Mergulhao, Account Supervisor - Social Media & PR, Medulla (IN), Aurelija Plioplytė, Head of Digital & Social, Not Perfect (LI), Daiana Khaidargaliyeva, Account Manager, Osaka Labs (UK / USA), Stefanie Söhnchen, Vice President Digital, PIABO Communications (DE), Elisabeth Winiartati, Managing Consultant, Head of Global Integrated Communications; Lydia Aprina, Account Manager, Integrated Marketing and Communications; Nita Prabowo, Account Manager, Integrated Marketing and Communications; Okhi, Web Developer, PNTR Group (ID), Kei Obusan, Insights Director; Daffi Ranandi, Insights Manager, Radarr (SG), Gautam Reghunath, Co-founder & CEO, Talented (IN), Donagh Humphreys, Head of Social and Digital Innovation, THINKHOUSE (IRE), Sarah Yim, Strategy Director, Zulu Alpha Kilo (CA).
Trends In Paid Search: Navigating The Digital Landscape In 2024Search Engine Journal
The search marketing landscape is evolving rapidly with new technologies, and professionals, like you, rely on innovative paid search strategies to meet changing demands.
It’s important that you’re ready to implement new strategies in 2024.
Check this out and learn the top trends in paid search advertising that are expected to gain traction, so you can drive higher ROI more efficiently in 2024.
You’ll learn:
- The latest trends in AI and automation, and what this means for an evolving paid search ecosystem.
- New developments in privacy and data regulation.
- Emerging ad formats that are expected to make an impact next year.
Watch Sreekant Lanka from iQuanti and Irina Klein from OneMain Financial as they dive into the future of paid search and explore the trends, strategies, and technologies that will shape the search marketing landscape.
If you’re looking to assess your paid search strategy and design an industry-aligned plan for 2024, then this webinar is for you.
5 Public speaking tips from TED - Visualized summarySpeakerHub
From their humble beginnings in 1984, TED has grown into the world’s most powerful amplifier for speakers and thought-leaders to share their ideas. They have over 2,400 filmed talks (not including the 30,000+ TEDx videos) freely available online, and have hosted over 17,500 events around the world.
With over one billion views in a year, it’s no wonder that so many speakers are looking to TED for ideas on how to share their message more effectively.
The article “5 Public-Speaking Tips TED Gives Its Speakers”, by Carmine Gallo for Forbes, gives speakers five practical ways to connect with their audience, and effectively share their ideas on stage.
Whether you are gearing up to get on a TED stage yourself, or just want to master the skills that so many of their speakers possess, these tips and quotes from Chris Anderson, the TED Talks Curator, will encourage you to make the most impactful impression on your audience.
See the full article and more summaries like this on SpeakerHub here: https://speakerhub.com/blog/5-presentation-tips-ted-gives-its-speakers
See the original article on Forbes here:
http://www.forbes.com/forbes/welcome/?toURL=http://www.forbes.com/sites/carminegallo/2016/05/06/5-public-speaking-tips-ted-gives-its-speakers/&refURL=&referrer=#5c07a8221d9b
ChatGPT and the Future of Work - Clark Boyd Clark Boyd
Everyone is in agreement that ChatGPT (and other generative AI tools) will shape the future of work. Yet there is little consensus on exactly how, when, and to what extent this technology will change our world.
Businesses that extract maximum value from ChatGPT will use it as a collaborative tool for everything from brainstorming to technical maintenance.
For individuals, now is the time to pinpoint the skills the future professional will need to thrive in the AI age.
Check out this presentation to understand what ChatGPT is, how it will shape the future of work, and how you can prepare to take advantage.
The document provides career advice for getting into the tech field, including:
- Doing projects and internships in college to build a portfolio.
- Learning about different roles and technologies through industry research.
- Contributing to open source projects to build experience and network.
- Developing a personal brand through a website and social media presence.
- Networking through events, communities, and finding a mentor.
- Practicing interviews through mock interviews and whiteboarding coding questions.
Google's Just Not That Into You: Understanding Core Updates & Search IntentLily Ray
1. Core updates from Google periodically change how its algorithms assess and rank websites and pages. This can impact rankings through shifts in user intent, site quality issues being caught up to, world events influencing queries, and overhauls to search like the E-A-T framework.
2. There are many possible user intents beyond just transactional, navigational and informational. Identifying intent shifts is important during core updates. Sites may need to optimize for new intents through different content types and sections.
3. Responding effectively to core updates requires analyzing "before and after" data to understand changes, identifying new intents or page types, and ensuring content matches appropriate intents across video, images, knowledge graphs and more.
A brief introduction to DataScience with explaining of the concepts, algorithms, machine learning, supervised and unsupervised learning, clustering, statistics, data preprocessing, real-world applications etc.
It's part of a Data Science Corner Campaign where I will be discussing the fundamentals of DataScience, AIML, Statistics etc.
Time Management & Productivity - Best PracticesVit Horky
Here's my presentation on by proven best practices how to manage your work time effectively and how to improve your productivity. It includes practical tips and how to use tools such as Slack, Google Apps, Hubspot, Google Calendar, Gmail and others.
The six step guide to practical project managementMindGenius
The six step guide to practical project management
If you think managing projects is too difficult, think again.
We’ve stripped back project management processes to the
basics – to make it quicker and easier, without sacrificing
the vital ingredients for success.
“If you’re looking for some real-world guidance, then The Six Step Guide to Practical Project Management will help.”
Dr Andrew Makar, Tactical Project Management
Unlocking the Power of ChatGPT and AI in Testing - A Real-World Look, present...Applitools
During this webinar, Anand Bagmar demonstrates how AI tools such as ChatGPT can be applied to various stages of the software development life cycle (SDLC) using an eCommerce application case study. Find the on-demand recording and more info at https://applitools.info/b59
Key takeaways:
• Learn how to use ChatGPT to add AI power to your testing and test automation
• Understand the limitations of the technology and where human expertise is crucial
• Gain insight into different AI-based tools
• Adopt AI-based tools to stay relevant and optimize work for developers and testers
* ChatGPT and OpenAI belong to OpenAI, L.L.C.
The document discusses various AI tools from OpenAI like GPT-3 and DALL-E 2, as well as ChatGPT. It explores how search engines are using AI and things to consider around AI-generated content. Potential SEO uses of ChatGPT are also presented, such as generating content at scale, conducting topic research, and automating basic coding tasks. The document encourages further reading on using ChatGPT for SEO purposes.
1. Quantum cosmology
with compact phase space
Jakub Mielczarek
Jagiellonian University, Cracow
Based on:
D. Artigas, J. M. and C. Rovelli,
Phys. Rev. D 100, 043533 (2019)
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PoToR 2019
Szczecin, 25.09.19
2. Nonlinear Field Space Theory (NFST)
J. M. and T. Trześniewski, The Nonlinear Field Space Theory,
Phys. Lett. B 759, 424 (2016)
J. M., Spin-Field Correspondence,
Universe 3, 29 (2017)
T. Trześniewski, Nonlinear Field Space Theory and Quantum Gravity,
Acta Phys. Pol. B Proc. Suppl. 10, 329 (2017)
J. M. and T. Trześniewski, Nonlinear Field Space Cosmology,
Phys. Rev. D 96, 043522 (2017)
J. Bilski, S. Brahma, A. Marcianò and J. M.,
Klein–Gordon field from the XXZ Heisenberg model,
Int. J. Mod. Phys. D 28, 1950020 (2018)
D. Artigas, J. M. and C. Rovelli,
A minisuperspace model of compact phase space gravity,
Phys. Rev. D 100, 043533 (2019)
Grant Sonata Bis DEC-2017/26/E/ST2/00763 of the
National Science Centre, Compact phase space field theories,
2018-2021
3. Scalar field
Mechanical
systems
Gauge fields
Fermionic field (in progress)
Minisuperspace
models
Cosmology
Midisuperspace
models
(in progress)
General
Relativity
Scalar field on cosmological background
Where we are?
Scalar multiplet (in progress)
4. Compact phase spaces arise in the semiclassical description of
quantum systems with finite dimensional Hilbert spaces.
dimH ⇠
V
(2⇡})n
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V =
Z
!
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Compact phase space Finite dim Hilbert space
We are focused on the compact case rather on the phase space with boundaries.
dim = 2n<latexit sha1_base64="di84GEhON1ThgBVYa5K+Prggwhk=">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</latexit><latexit sha1_base64="di84GEhON1ThgBVYa5K+Prggwhk=">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</latexit><latexit sha1_base64="di84GEhON1ThgBVYa5K+Prggwhk=">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</latexit><latexit sha1_base64="OltJ7dcckV/ivblgW+2UZZZLVxI=">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</latexit>
5. Why compact phase space for gravity?
Locally finite dimensional Hilbert space of quantum gravitational
field is supported at the theoretical ground.
Finite horizon entropy
Spikes (IR divergences)Singularities (UV divergences)
(But the result may depend on how microstates are counted, e.g. LQG)
S ⇠ A ln dimH<latexit sha1_base64="OWJR1xY4tR/g7Rmx2fJJkYLSafs=">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</latexit><latexit sha1_base64="OWJR1xY4tR/g7Rmx2fJJkYLSafs=">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</latexit><latexit sha1_base64="OWJR1xY4tR/g7Rmx2fJJkYLSafs=">AAAC6XicjVHLTtwwFD2kDyjQkpZlN1YHJFajpEKC5UA3sxxUBkYiCDkeD1jjPJQ4CDSaH+iuu6pbfoBt+yOIPyh/wbUxUgtCxVGS43PvOfa9Ny21qk0UXc8EL16+ej0792Z+YfHtu6Xw/Ye9umgqIfui0EU1SHkttcpl3yij5aCsJM9SLffT8Rcb3z+VVa2KfNecl/Iw48e5GinBDVFH4cpXltQqY1ss0TlLjDwzk6HKpizJuDkRXE+6U3YUtqJ25BZ7DGIPWvCrV4RXSDBEAYEGGSRyGMIaHDU9B4gRoSTuEBPiKkLKxSWmmCdtQ1mSMjixY/oe0+7AszntrWft1IJO0fRWpGRYJU1BeRVhexpz8cY5W/Yp74nztHc7p3/qvTJiDU6I/Z/uPvO5OluLwQibrgZFNZWOsdUJ79K4rtibs7+qMuRQEmfxkOIVYeGU931mTlO72m1vuYv/cZmWtXvhcxvc2FvSgOOH43wM9j6346gd76y3Ott+1HP4iE9Yo3luoIMueuiT9zdc4hd+B+Pge/Aj+HmXGsx4zTL+WcHFLbnBnYw=</latexit><latexit sha1_base64="ATa3zLCqMDarwQmSOjvO3CJ0w4w=">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</latexit>
Spikes in Ponzano-Regge amplitudes
are cured by q-deformations, which
introduce finite dimension of a Hilbert
space -> Turaev-Viro amplitude.
Compact phase spaces introduce
bounds on the the phase space
variables, which regularize curvature,
energy density, etc.
Model-independent argument: N. Bao, S. M. Carroll and A. Singh, The Hilbert
Space of Quantum Gravity Is Locally Finite-Dimensional,
Int. J. Mod. Phys. D 26 (2017) no.12, 1743013
SBH =
1
4
A
l2
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SdS ⇠
1
l2
Pl⇤<latexit sha1_base64="o8zLmax7SoV9k7RXd9Tl93kvyFA=">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</latexit><latexit sha1_base64="o8zLmax7SoV9k7RXd9Tl93kvyFA=">AAAC+nicjVHLThRBFD20qCMojrJ0U3FC4mrSTSC6JLBx4WLIOEBCw6S6pgYqVD9SXU0gbX+JO3fGLT/AVj+A8Af6F94qexKUEKxOd586955Tde9NCq1KG4bXc8GD+YePHneeLCw+fbb0vPvi5U6ZV0bIkch1bvYSXkqtMjmyymq5VxjJ00TL3eRky8V3T6UpVZ59tOeFPEj5UaamSnBL1Li7PhzXsZVntp4Mm4bFpUoZi6eGizpqan24OgsPdNPEH8h4whs27vbCfugXuw2iFvTQrkHevUKMCXIIVEghkcES1uAo6dlHhBAFcQeoiTOElI9LNFggbUVZkjI4sSf0PaLdfstmtHeepVcLOkXTa0jJsEKanPIMYXca8/HKOzv2Lu/ae7q7ndM/ab1SYi2Oib1PN8v8X52rxWKKd74GRTUVnnHVidal8l1xN2c3qrLkUBDn8ITihrDwylmfmdeUvnbXW+7jP32mY91etLkVfrlb0oCjf8d5G+ys9qOwH22v9TY221F38Aqv8Ybm+RYbeI8BRuT9GZf4jh/Bp+BL8DX49ic1mGs1y/hrBRe/AVzipbs=</latexit><latexit sha1_base64="o8zLmax7SoV9k7RXd9Tl93kvyFA=">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</latexit><latexit sha1_base64="y+20wniumFcasvp6MEg/CagtrNE=">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</latexit>
Simple counting of microstates:
6. De Sitter model
related to scale factorrelated to Hubble factor
Symplectic form:
Hamiltonian:
Poisson bracket:
Hubble factor:
Friedmann equation:
(|q| := V0a3
)<latexit sha1_base64="rnk4rieXR+iFU10oBdKFygfnWu4=">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</latexit><latexit sha1_base64="rnk4rieXR+iFU10oBdKFygfnWu4=">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</latexit><latexit sha1_base64="rnk4rieXR+iFU10oBdKFygfnWu4=">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</latexit><latexit sha1_base64="OZow8DheX0vGhJLueanSFsU/vd4=">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</latexit>
Equation of motion: ˙q = {q, HGR} =
3
2
Nqp
<latexit sha1_base64="dQGZ6rQVprP6fnWSrS8ey2qQovc=">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</latexit><latexit sha1_base64="dQGZ6rQVprP6fnWSrS8ey2qQovc=">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</latexit><latexit sha1_base64="dQGZ6rQVprP6fnWSrS8ey2qQovc=">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</latexit><latexit sha1_base64="7E0r767JOJVZ8w5yFk9y7KjNN+8=">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</latexit>
H :=
1
3
˙q
q
=
1
2
p
<latexit sha1_base64="R3S10L1lSoNu/NXPP2tCsBep7TU=">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</latexit><latexit sha1_base64="R3S10L1lSoNu/NXPP2tCsBep7TU=">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</latexit><latexit sha1_base64="R3S10L1lSoNu/NXPP2tCsBep7TU=">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</latexit><latexit sha1_base64="zBJRKpQ03XuZBmfnNyPv3JIi0Ao=">AAAC/3icjVHNTtwwGBxCS5efwrYcuVhdIXFhlSwHEBLSqr1wpBILSCxCjtdLo80mxnEqoSiHvklv3BDXvkCvcEN9A/oW/eyaih+h4ijJeL6ZsT87VmlSmDD8NRFMvno99aYxPTM793Z+ofnu/V6Rl1rInsjTXB/EvJBpksmeSUwqD5SWfByncj8efbL1/a9SF0me7ZozJY/G/CRLhonghqjj5uY229xi/aHmoorqaq32uD/ITXVa1/SyLbb6T9Cp+yOuFGeKHTdbYTt0gz0FkQct+LGTN2/QxwA5BEqMIZHBEE7BUdBziAghFHFHqIjThBJXl6gxQ96SVJIUnNgRfU9odujZjOY2s3BuQauk9GpyMiyTJyedJmxXY65eumTLPpdduUy7tzP6xz5rTKzBF2L/57tTvtRnezEYYsP1kFBPyjG2O+FTSncqdufsXleGEhRxFg+orgkL57w7Z+Y8hevdni139VuntKydC68t8dvuki44enydT8Fepx2F7ehzp9X96K+6gSV8wArd5zq62MYOepT9HT9xhevgW3AeXASXf6XBhPcs4sEIfvwBNEmmTA==</latexit>
N = 1<latexit sha1_base64="89gxBv5EJwAAmy2v6N5F1YpMsUk=">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</latexit><latexit sha1_base64="89gxBv5EJwAAmy2v6N5F1YpMsUk=">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</latexit><latexit sha1_base64="89gxBv5EJwAAmy2v6N5F1YpMsUk=">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</latexit><latexit sha1_base64="MRDBXiwF92ZrKOXPN4+xcBsvANw=">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</latexit>
for
0 =
@HGR
@N
= q
✓
3
4
p2
+
⇤
◆
<latexit sha1_base64="DdmPK75dnWwXln/T+j8aPIYFxuE=">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</latexit><latexit sha1_base64="DdmPK75dnWwXln/T+j8aPIYFxuE=">AAADMXicjVFNT9wwEJ2kLR/bQpf2yMXqqhIIsUooElyQUHsoB4QoYgGJwMrxehdrvYlxHFQU5VfxT7hU3BAnUP9Ajx0bo36gqjhK8vxm3vOMJ1VSFCaKroLw2fMXY+MTk42Xr6amXzdn3uwVeakZ77Bc5vogpQWXIuMdI4zkB0pzOkol30+Hn2x8/4zrQuTZrjlX/GhEB5noC0YNUt2mjNaSvqasShTVRlBJNrpVYvhXU33eqev6F79VkzVymkjeN3OL5F70oa6W62RIlaJEHS8teDrZxAJ6FMUuVCdaDE7MfKPbbEXtyC3yGMQetMCv7bz5DRLoQQ4MShgBhwwMYgkUCnwOIYYIFHJHUCGnEQkX51BDA7UlZnHMoMgO8TvA3aFnM9xbz8KpGZ4i8dWoJPAeNTnmacT2NOLipXO27L+8K+dpazvHf+q9RsgaOEH2f7qHzKfqbC8G+rDqehDYk3KM7Y55l9Ldiq2c/NaVQQeFnMU9jGvEzCkf7pk4TeF6t3dLXfzWZVrW7pnPLeHOVokDjv8e52Owt9SOo3b8Zbm1/tGPegJm4R3M4TxXYB02YBs66H0JP4IgCMOL8Cq8Dm/uU8PAa97CHyv8/hPParlU</latexit><latexit sha1_base64="DdmPK75dnWwXln/T+j8aPIYFxuE=">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</latexit><latexit sha1_base64="2KwQlw+G9cQV1+ysZgu2TekWafQ=">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</latexit>
Scalar constraint:
7. Spherical phase space
The spherical space is consider mostly because of
its relation to spin. It is also a non-trivial case
since the sphere is not a cotangent bundle, but
equipped with symplectic form is a well defined
symplectic manifold.
In this case, kinematics and dynamics can be
associated with a certain spin system. This is
important due to both practical reason (convenient
to perform calculations) and fundamental reason
(spin degrees of freedom for QG, relation to
condensed-matter systems - analog models).
change of variables
R1R2 = S<latexit sha1_base64="bIZhG1QqI09pAWDU9o2gQaghXbU=">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</latexit><latexit sha1_base64="bIZhG1QqI09pAWDU9o2gQaghXbU=">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</latexit><latexit sha1_base64="bIZhG1QqI09pAWDU9o2gQaghXbU=">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</latexit><latexit sha1_base64="EQQGZxdPbWiCIMWfS4rffAw+7yo=">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</latexit>
Dimensional parameters
Symplectic form on a sphere:
Volume of phase space:
V =
Z
⌦
! = 4⇡S
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8. Employing the above, the su(2) algebra is obtained:
with the following components:
Poisson bracket:
~S = (Sx, Sy, Sz)<latexit sha1_base64="m/xY74MHlz+mAfq/RBRpw/5Ip2Q=">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</latexit><latexit sha1_base64="m/xY74MHlz+mAfq/RBRpw/5Ip2Q=">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</latexit><latexit sha1_base64="m/xY74MHlz+mAfq/RBRpw/5Ip2Q=">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</latexit><latexit sha1_base64="TJZ5tBzh19LPZSWaEzYMfbqYv0I=">AAAC23icjVHLSsQwFD1T3+OrKrhxUxwEBRlaN7oRBt24VMYZBR1KG6MGO21JU3EcZ+VO3PoDbvV/xD/Qv/AmVvCBaErSk3PvOcnNDdNIZMp1n0tWX//A4NDwSHl0bHxi0p6abmZJLhlvsCRK5H4YZDwSMW8ooSK+n0oetMOI74Vnmzq+d85lJpJ4V3VS3moHJ7E4FixQRPn27OE5Z916b32x7l8s1/0Ozculsm9X3KprhvMTeAWooBjbif2EQxwhAUOONjhiKMIRAmT0HcCDi5S4FrrESULCxDl6KJM2pyxOGQGxZ7Se0O6gYGPaa8/MqBmdEtGUpHSwQJqE8iRhfZpj4rlx1uxv3l3jqe/WoX9YeLWJVTgl9i/dR+Z/dboWhWOsmRoE1ZQaRlfHCpfcvIq+ufOpKkUOKXEaH1FcEmZG+fHOjtFkpnb9toGJv5hMzeo9K3JzvOpbUoO97+38CZorVc+tejsrldpG0ephzGEei9TPVdSwhW00yPsK93jAo9Wyrq0b6/Y91SoVmhl8GdbdG7xhl1w=</latexit>
~S<latexit sha1_base64="llYAybiqQdrYUSZLV8CxWNAxoKM=">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</latexit><latexit sha1_base64="llYAybiqQdrYUSZLV8CxWNAxoKM=">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</latexit><latexit sha1_base64="llYAybiqQdrYUSZLV8CxWNAxoKM=">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</latexit><latexit sha1_base64="TBxPMM/yPmf0T6MMvEYGyXsdj0c=">AAACy3icjVHLTsJAFD3UF+ILdemmkZi4Ii0bXRLduDHBKI8EiWmHASeUtmmnJIgs/QG3+l/GP9C/8M44JCoxOk3bM+eec2fuvX4ciFQ6zmvOWlhcWl7JrxbW1jc2t4rbO400yhLG6ywKoqTleykPRMjrUsiAt+KEe0M/4E1/cKrizRFPUhGFV3Ic887Q64eiJ5gniWpdjzibXE4LN8WSU3b0sueBa0AJZtWi4guu0UUEhgxDcISQhAN4SOlpw4WDmLgOJsQlhISOc0xRIG9GKk4Kj9gBffu0axs2pL3KmWo3o1MCehNy2jggT0S6hLA6zdbxTGdW7G+5JzqnutuY/r7JNSRW4pbYv3wz5X99qhaJHo51DYJqijWjqmMmS6a7om5uf6lKUoaYOIW7FE8IM+2c9dnWnlTXrnrr6fibVipW7ZnRZnhXt6QBuz/HOQ8albLrlN2LSql6Ykadxx72cUjzPEIVZ6ihruf4iCc8W+dWat1Z959SK2c8u/i2rIcPWVCSOA==</latexit>
One can introduce a vector
{Si, Sj} = ✏ijkSk<latexit sha1_base64="YX9p7dM1VtrYwrYhlp09dB4VSdM=">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</latexit><latexit sha1_base64="YX9p7dM1VtrYwrYhlp09dB4VSdM=">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</latexit><latexit sha1_base64="YX9p7dM1VtrYwrYhlp09dB4VSdM=">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</latexit><latexit sha1_base64="I7UOLjFAAj//XAWx8T1brYw7Vag=">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</latexit>
The is therefore a vector of angular momentum (spin) with the norm S.
9. Compact phase space FRW
Kinematics (symplectic form)
Dynamics (Hamiltonian)
Friedmann
equation:
where
The procedure is
ambiguous! We make
the simplest choice.
Affine 2-sphere
:=
4
3
⇤
R2
12
2 [0, 1]
<latexit sha1_base64="y7CalWv79SyRMqSyy1dqgvmsa0k=">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</latexit><latexit sha1_base64="y7CalWv79SyRMqSyy1dqgvmsa0k=">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</latexit><latexit sha1_base64="y7CalWv79SyRMqSyy1dqgvmsa0k=">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</latexit><latexit sha1_base64="89Xa6GSnIb2JPqG3sm0wmExRhx0=">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</latexit>
The affine case is recovered in the limit.R1, R2 ! 1<latexit sha1_base64="LNmqQtKkrDUFLwrPZMyKh+db234=">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</latexit><latexit sha1_base64="LNmqQtKkrDUFLwrPZMyKh+db234=">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</latexit><latexit sha1_base64="LNmqQtKkrDUFLwrPZMyKh+db234=">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</latexit><latexit sha1_base64="MaR+F5j7siTjGesrWgPJZwhvMBE=">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</latexit>
10. Polymer limit
This is the case of loop quantum cosmology.
R2 ! 1<latexit sha1_base64="HiI/N4+bdyKk6MJWpdoP8nv6DhU=">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</latexit><latexit sha1_base64="HiI/N4+bdyKk6MJWpdoP8nv6DhU=">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</latexit><latexit sha1_base64="HiI/N4+bdyKk6MJWpdoP8nv6DhU=">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</latexit><latexit sha1_base64="i0BtGXKYQ1woH+zsYAye3Punzfc=">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</latexit>
In the limit the so-called
polymerization of momentum is obtained.
The su(2) algebra reduces to the cylindrical case:
The Hamiltonian reduces to
where the polymerization
scale is:
:=
1
R1<latexit sha1_base64="Djrz7m1f9f39QkJvA10DkbB/15g=">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</latexit><latexit sha1_base64="Djrz7m1f9f39QkJvA10DkbB/15g=">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</latexit><latexit sha1_base64="Djrz7m1f9f39QkJvA10DkbB/15g=">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</latexit><latexit sha1_base64="IpcgmU5scOhVCdLM71hbptMDtxU=">AAAC3XicjVHLSsNAFD2Nr1pfVTeCm2ARXJXEjSIIRTcuq9hWaEuZTKcaTJMwmQil1J07cesPuNXfEf9A/8I7Ywo+EJ2Q5My555yZO+PFgZ8ox3nJWROTU9Mz+dnC3PzC4lJxeaWeRKnkosajIJJnHktE4IeipnwViLNYCtb3AtHwLg91vXElZOJH4akaxKLdZ+eh3/M5U0R1imutgMRdZu/t262eZHzojoYnHXfUKZacsmOG/RO4GSghG9Wo+IwWuojAkaIPgRCKcACGhJ4mXDiIiWtjSJwk5Ju6wAgF8qakEqRgxF7S95xmzYwNaa4zE+PmtEpArySnjU3yRKSThPVqtqmnJlmzv2UPTabe24D+XpbVJ1bhgti/fGPlf326F4Uedk0PPvUUG0Z3x7OU1JyK3rn9qStFCTFxGnepLglz4xyfs208ieldny0z9Vej1Kye80yb4k3vki7Y/X6dP0F9u+w6Zfd4u1Q5yK46j3VsYIvucwcVHKGKGmVf4wGPeLI61o11a919SK1c5lnFl2HdvwOvq5iJ</latexit>
The Friedmann equation is where
11. Cosmological evolution
The leading correction behaves as a
phantom matter with the equation of state:
~ de Sitter expansion ~ de Sitter contraction
Cosmological recollapse (IR effect)
(UV effect)
(UV effect)
⇤ ! ⇤e↵ = ⇤(1 )<latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">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</latexit><latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">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</latexit><latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">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</latexit><latexit sha1_base64="tMpofGH0WR6riQZeZTjXuvBAuG0=">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</latexit>
⇤ ! ⇤e↵ = ⇤(1 )<latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">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</latexit><latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">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</latexit><latexit sha1_base64="KFctk0LTyOzTF6eUfqpJ7HoQ6Us=">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</latexit><latexit sha1_base64="tMpofGH0WR6riQZeZTjXuvBAuG0=">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</latexit>
de Sitter expansion may terminate…
=
⇢⇤
⇢c
2 [0, 1]
<latexit sha1_base64="8nwwv8e0XHClhmhCuOxDNKky930=">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</latexit><latexit sha1_base64="8nwwv8e0XHClhmhCuOxDNKky930=">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</latexit><latexit sha1_base64="8nwwv8e0XHClhmhCuOxDNKky930=">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</latexit><latexit sha1_base64="j4/HWccULijMCRJsJrBIHjusaik=">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</latexit>
and the effective phantom can be a sign of it.
12. Evolution on the phase space
Exact solutions:
= 0.1<latexit sha1_base64="peqAXdMVhDI9wGrxixOaG6Z+A3I=">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</latexit><latexit sha1_base64="peqAXdMVhDI9wGrxixOaG6Z+A3I=">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</latexit><latexit sha1_base64="peqAXdMVhDI9wGrxixOaG6Z+A3I=">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</latexit><latexit sha1_base64="zs0Q4bWdZsUJj9bry7G2E35WgXA=">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</latexit>
= 0.5<latexit sha1_base64="UMP/i6Z8/z4Y70jemoPLarttSlQ=">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</latexit><latexit sha1_base64="UMP/i6Z8/z4Y70jemoPLarttSlQ=">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</latexit><latexit sha1_base64="UMP/i6Z8/z4Y70jemoPLarttSlQ=">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</latexit><latexit sha1_base64="n7rCGcVm8TztJDnGePDDtHfx+AI=">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</latexit>
= 0.9<latexit sha1_base64="oLcKaX0pusbV5i4A4hGBUmrQSKs=">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</latexit><latexit sha1_base64="oLcKaX0pusbV5i4A4hGBUmrQSKs=">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</latexit><latexit sha1_base64="oLcKaX0pusbV5i4A4hGBUmrQSKs=">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</latexit><latexit sha1_base64="/8b2BwzT4ViCy1omV12vrBaQYhg=">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</latexit>
There are four symmetric solutions
for the four quadrants of a sphere.